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interacting fermionic particles play a central role in the structure of matter and exist over a very broad range of energies , from extremely low temperature trapped atomic fermi gases , where @xmath10 k @xcite , to very high temperature primordial matter , like quark - gluon plasmas , where @xmath11 k @xcite . for all of these systems , the most intriguing physics is related to very strong interactions between fermionic particles , such as the strong coupling between electrons in high-@xmath8 superconductors and the strong interactions between neutrons in neutron matter . current many - body quantum theories face great challenges in solving problems for strongly interacting fermi systems , due to the lack of a small coupling parameter . for example , the critical temperature of a superfluid - normal fluid transition in a strongly interacting fermi gas has been controversial for many years . the critical temperature @xmath12 has been predicted to have values in the range between 0.15 and 0.35 by different theoretical methods @xcite . a complete understanding of the physics of strongly interacting systems can not yet be obtained from a theoretical point of view . there is a pressing need to investigate strongly interacting fermions experimentally . in recent years , based on progress in optical cooling and trapping of fermionic atoms , a clean and controllable strongly interacting fermi system , comprising a degenerate , strongly interacting fermi gas @xcite , is now of interest to the whole physics community . strongly interacting fermi gases are produced near a broad feshbach resonance @xcite , where the zero energy s - wave scattering length @xmath13 is large compared to the interparticle spacing , while the interparticle spacing is large compared to the range of the two - body interaction . in this regime , the system is known as a unitary fermi gas , where the properties are universal and independent of the details of the two - body scattering interaction @xcite . in contrast to other strongly interacting fermi systems , in atomic gases , the interactions , energy , and spin population can be precisely adjusted , enabling a variety of experiments for exploring this model system . intense studies of strongly interacting fermi gases have been implemented over the past several years from a variety of perspectives . some of the first experiments observed the expansion hydrodynamics of the strongly interacting cloud @xcite . evidence for superfluid hydrodynamics was first observed in collective modes @xcite . collective modes were later used to study the @xmath14 equation of state throughout the crossover regime @xcite . recently , measurements of sound velocity have also been used to explore the @xmath14 equation of state @xcite . below a feshbach resonance , fermionic atoms join to form stable molecules and molecular bose - einstein condensates @xcite . fermionic pair condensation has been observed by projection experiments using fast magnetic field sweeps @xcite . above resonance , strongly bound pairs have been probed by radio frequency and optical spectroscopy @xcite . phase separation has been observed in spin polarized samples @xcite . rotating fermi gases have revealed vortex lattices in the superfluid regime @xcite as well as irrotational flow in both the superfluid and normal fluid regimes @xcite . measurement of the thermodynamic properties of a strongly interacting fermi gas was first accomplished by adding a known energy to the gas , and then determining an empirical temperature that was calibrated using a pseudogap theory @xcite . recent model - independent measurements of the energy and entropy @xcite provide a very important piece of the puzzle , because they enable direct and precision tests that distinguish predictions from recent many - body theories , without invoking any specific theoretical model @xcite . one of the major challenges for the experiments in strongly interacting fermi gases is the lack of a precise model - independent thermometry . two widely - used thermometry methods are model - dependent , in that they rely on theoretical models for calibration . the first relies on adiabatic magnetic field sweeps between the molecular bec regime and the strongly interacting regime @xcite . subsequently , the temperature of the strongly interacting gas is estimated from the measured temperature in the bec regime using a theoretical model of the entropy @xcite . the second method , used by our group @xcite , is based on determining an empirical temperature from the cloud profiles that is calibrated by comparing the measured density distribution with a theoretical model for the density profiles . currently two model - independent thermometry methods have been reported for strongly - interacting gases . one is the technique employed by the mit group @xcite , which is only applicable to imbalanced mixtures of spin - up and spin - down atoms . that method is based on fitting the noninteracting edge for the majority spin after phase separation . another model - independent method is demonstrated in ref . @xcite , which is applicable to both balanced and imbalanced mixtures of spin - up and spin - down fermions . the energy @xmath1 and entropy @xmath2 are measured and then parameterized to determine a smooth curve @xmath4 . then the temperature in both the superfluid and normal fluid regime is obtained from the fundamental thermodynamic relation @xmath5 . in this paper , we will describe our model - independent thermodynamic experiments on a strongly interacting fermi gas of @xmath0li , which we have conducted at duke university . first , we will describe our measurements of both the total energy @xmath1 and the total entropy @xmath2 of a trapped strongly - interacting fermi gas tuned near a feshbach resonance . then , we determine the temperature @xmath5 after showing that the @xmath4 data are very well parameterized by using two different power laws that are joined with continuous @xmath1 and @xmath6 at a certain entropy @xmath7 that gives the best fit . to examine the sensitivity of the temperature to the form of the fit function , we employ two different fit functions that allow for a heat capacity jump or for a continuous heat capacity at @xmath7 . we find that the @xmath6 values closely agree for both cases . we find a significant change in the scaling of @xmath1 with @xmath2 above and below @xmath7 , in contrast to the behavior for an ideal fermi gas , where a single power - law well parameterizes @xmath4 over the same energy range . by interpreting @xmath7 as the critical entropy for a superfluid - normal fluid transition in the strongly interacting fermi gas , we estimate the critical energy @xmath15 and critical temperature @xmath8 . both the model - independent @xmath4 data and the estimated critical parameters are compared with several recent many - body theories based on both analytic and quantum monte carlo methods . we also show how parameterizing the @xmath4 data provides experimental temperature calibrations , which helps to unify , in a model - independent way , the results obtained by several groups @xcite . first we relate the endpoint temperatures for adiabatic sweeps of the bias magnetic field between the strongly interacting and ideal noninteracting regimes , as used in the jila experiments to characterize the condensed pair fraction @xcite . this enables the ideal gas temperature observed for the onset of pair condensation @xcite to be related to the critical temperature of the strongly interacting fermi gas . the temperature obtained by parameterizing the strongly interacting gas data also calibrates the empirical temperature based on the cloud profiles , as used in our previous studies of the heat capacity @xcite . these temperature calibrations yield values of @xmath8 close to that estimated from our @xmath4 data . next , we discuss three different methods for determining the universal many - body parameter , @xmath16 @xcite , where @xmath17 is the energy per particle in a uniform strongly interacting fermi gas at @xmath14 in units of the energy per particle of an ideal fermi gas at the same density . first , we describe the measurement of the sound velocity at resonance and its relationship to @xmath16 . then , we determine @xmath16 from the ground state energy @xmath18 of the trapped gas . here , @xmath18 is obtained by extrapolating the @xmath4 data to @xmath9 , as suggested by hu et al . this avoids a systematic error in the sound velocity experiments arising from the unknown finite temperature . finally , to explore the systematic error arising from the measurement of the number of atoms , @xmath16 is determined in a number - independent manner from the ratio of the cloud sizes in the strongly and weakly interacting regimes . all three results are found to be in very good agreement with each other and with recent predictions . finally , we obtain three universal thermodynamic functions from the parameterized @xmath4 data , the energy @xmath19 , heat capacity @xmath20 , and global chemical potential @xmath21 . our experiments begin with an optically - trapped highly degenerate , strongly interacting fermi gas of @xmath0li @xcite . a 50:50 mixture of the two lowest hyperfine states of @xmath0li atoms is confined in an ultrastable co@xmath22 laser trap with a bias magnetic field of 840 g , just above a broad feshbach resonance at @xmath23 g @xcite . at 840 g , the gas is cooled close to the ground state by lowering the trap depth @xmath24 @xcite . then @xmath24 is recompressed to a final trap depth of @xmath25k , which is much larger than the energy per particle of the gas , for the highest energies employed in the experiments . this suppresses evaporation during the time scale of the measurements . the shallow trap yields a low density that suppresses three body loss and heating . the low density also yields a weakly interacting sample when the bias magnetic field is swept to 1200 g , although the scattering length is @xmath26 bohr , as discussed in detail in [ sec : entropy ] . the shape of the trapping potential is that of a gaussian laser beam , with a transverse gaussian profile determined by the spot size and an axial lorentzian profile determined by the rayleigh length . to simplify the calculations of the ideal gas properties in subsequent sections , as well as the theoretical modelling , we take the trap potential to be approximated by a three dimensional gaussian profile , @xmath27 where @xmath28 is the @xmath29 width of trap for each direction . here , we take the zero of energy to be at @xmath30 . when the cold atoms stay in the deepest portion of the optical trap , where @xmath31 , the gaussian potential can be well approximated as a harmonic trap with transverse frequencies @xmath32 , @xmath33 and axial frequency @xmath34 , where @xmath35 here @xmath36 is the @xmath0li mass . at our final trap depth @xmath37 , the measured transverse frequencies are @xmath38 hz and @xmath39 hz . the axial frequency is weakly magnetic field dependent since the total axial frequency has both an optical potential contribution @xmath40 determined by eq . [ eq : trapfreqwelldepth ] and a magnetic potential contribution arising from magnetic field curvature , @xmath41 . the net axial frequency is then @xmath42 . we find @xmath43 hz at 840 g and @xmath44 hz at 1200 g. the total number of atoms is @xmath45 . the corresponding fermi energy @xmath46 and fermi temperature @xmath47 at the trap center for an ideal noninteracting harmonically trapped gas are @xmath48 , where @xmath49 . for our trap conditions , we obtain @xmath50k . using @xmath51 , we can rewrite eq . [ eq : udipole ] as a symmetric effective potential , @xmath52 where @xmath53 is the scaled position vector . here , @xmath54 with @xmath55 . to obtain the anharmonic corrections for the gaussian trap , we expand eq . [ eq : reducedtruepotential ] in a taylor series up to second order in @xmath56 , @xmath57 model - independent energy measurement is based on a virial theorem , which for an ideal gas in a harmonic confining potential @xmath58 yields @xmath59 . since the harmonic potential energy is proportional to the mean square cloud size , measurement of the cloud profile determines the total energy . remarkably , a trapped unitary fermi gas at a broad feshbach resonance obeys the same virial theorem as an ideal gas , although it contains superfluid pairs , noncondensed pairs , and unpaired atoms , all strongly interacting . this has been demonstrated both theoretically and experimentally @xcite . the virial theorem shows that the total energy of the gas at all temperatures can be measured from the cloud profile using @xmath60 where @xmath24 is the trapping potential and @xmath61 is the position vector . . [ eq : energymeas ] can be shown to be valid for any trapping potential @xmath24 and for any spin mixture , without assuming either the local density approximation or harmonic confinement @xcite . using eq . [ eq : truepotentialquartic ] in eq . [ eq : energymeas ] and keeping the lowest order anharmonic corrections , we obtain the energy per atom in terms of the axial mean square size , @xmath62 . \label{eq : energygaussian}\ ] ] here , we have used the local density approximation with a scalar pressure , which ensures that @xmath63 . for the ground state , where the spatial profile is a zero temperature thomas - fermi profile , we have @xmath64 . for energies @xmath65 , where the spatial profile is approximately gaussian , we have @xmath66 . since the anharmonic correction is small at low temperatures where the cloud size is small , we use the gaussian approximation over the whole range of energies explored in our experiments . for the conditions of our experiments , there is no evidence that the local density approximation breaks down for a 50:50 spin mixture . in this case , measurement of the mean square size in any one direction determines the total energy . from eq . [ eq : energygaussian ] , we see that by simply measuring the axial mean square size @xmath67 at 840 g and measuring the axial trap frequency by parametric resonance , we actually measure @xmath68 , the total energy per particle of the strongly interacting fermi gas at 840 g. this determines the total energy per particle in a model - independent way @xcite . the entropy @xmath2 of the strongly interacting gas at 840 g is determined by adiabatically sweeping the bias magnetic field from 840 g to 1200 g , where the gas is weakly interacting @xcite . the entropy @xmath69 of the weakly interacting gas is essentially the entropy of an ideal fermi gas in a harmonic trap , which can be calculated in terms of the mean square axial cloud size @xmath70 measured after the sweep . since the sweep is adiabatic , we have @xmath71 the adiabaticity of the magnetic field sweep is verified by employing a round - trip - sweep : the mean square size of the cloud at 840 g after a round - trip - sweep lasting 2s is found to be within 3% of mean square size of a cloud that remains at 840 g for a hold time of 2s . the nearly unchanged atom number and mean square size proves the sweep does not cause any significant atom loss or heating , which ensures entropy conservation for the sweep . the background heating rate is the same with and without the sweep and increases the mean square size by about 2% over 2s . the mean square size data are corrected by subtracting the increase arising from background heating over the 1 s sweep time @xcite . at 1200 g in our shallow trap , we have @xmath72 , where the fermi wavevector @xmath73 and the s - wave scattering length @xmath74 bohr @xcite . we find that the gas is weakly interacting : for the lowest temperatures attained in our experiments , the gas at 1200 g is a normal fluid that we observe to expand ballistically . we have calculated the ground state mean square size at 1200 g in our gaussian trap , based on a mean - field theory , @xmath75 @xcite , which is close to that of an ideal harmonically trapped gas , @xmath76 . here , @xmath77 is the mean square size corresponding to the fermi energy of an ideal noninteracting fermi gas at magnetic field @xmath78 , which includes the magnetic field dependence of the axial trapping frequency : @xmath79 . we expect that the entropy of the gas at 1200 g is close to that of an ideal gas , except for a mean field shift of the energy . we therefore assume that a reasonable approximation to the entropy is that of an ideal fermi gas , @xmath80 , where @xmath81 is the ground state mean square size of an ideal fermi gas in the gaussian trapping potential of eq . [ eq : reducedtruepotential ] . here , we apply an elementary calculation based on integrating the density of states for the gaussian trap with the entropy per orbital @xmath82 $ ] , where @xmath83 is the ideal fermi gas occupation number at temperature @xmath6 for an orbital of energy @xmath84 . by calculating @xmath85 as a function of the _ difference _ between the finite temperature and ground state mean square cloud sizes , we reduce the error arising from the mean field shift at 1200 g , and ensure that @xmath86 for the ground state . the exact entropy of a weakly interacting gas @xmath69 at 1200 g , @xmath87 , has been calculated using many - body theories @xcite for the gaussian potential of eq . [ eq : reducedtruepotential ] . in the experiments , we determine the value of @xmath88 , where we take @xmath89 , the value measured at our lowest energy at 1200 g by extrapolation to @xmath14 using the sommerfeld expansion for the spatial profile of an ideal gas . this result is close to the theoretical value , @xmath90 . the entropy versus cloud size curve for an ideal noninteracting fermi gas and the exact value for a weakly interacting gas @xmath69 at 1200 g are plotted in fig . [ fig : entropysizeoverlap ] . we find that the entropies @xmath91 and @xmath92 ) , agree within a few percent over most of the energy range we studied , except at the point of lowest measured energy , where they differ by 10% . the results show clearly that the shape of the entropy curve of a weakly interacting fermi gas is nearly identical to that of an ideal gas when the mean field shift of the ground state size is included by referring the mean square cloud size to that of the ground state . so we have to a good approximation , @xmath93 since the corrections to ideal gas behavior are small , the determination of @xmath94 by measuring the axial mean square size @xmath70 relative to the ground state provides an essentially model - independent estimate of the entropy of the strongly interacting gas . sound velocity measurements have been implemented for fermi gases that are nearly in the ground state , from the molecular bec regime to the weakly interacting fermi gas regime @xcite . a sound wave is excited in the sample by using a thin slice of green light that bisects the cigar - shaped cloud . the green light at 532 nm is blue detuned from the 671 nm transition in lithium , creating a knife that locally repels the atoms . the laser knife is pulsed on for 280 @xmath95s , much shorter than typical sound propagation times @xmath96 ms and excites a ripple in the density consisting of low density valleys and high density peaks . after excitation , the density ripple propagates outward along the axial direction @xmath97 . after a variable amount of propagation time , we release the cloud , let it expand , and image destructively . in the strongly interacting regime , we use zero - temperature thomas - fermi profiles for a non - interacting fermi gas to fit the density profiles , and locate the positions of the density valley and peak . by recording the position of the density ripple versus the propagation time , the sound velocity is determined . a detailed discussion of potential sources of systematic error is given by joseph et al . @xcite . for a strongly interacting fermi gas in the unitary limit , the sound velocity @xmath98 at the trap center for the ground state is determined by the fermi velocity of an ideal gas at the trap center , @xmath99 and the universal constant @xmath16 , @xmath100 a precision measurement of the sound speed therefore enables a determination of @xmath16 @xcite . as discussed below , the values of @xmath16 determined from the @xmath4 data and the sound velocity data are in very good agreement . in the experiments , the raw data consists of the measured mean square cloud sizes at 840 g and after an adiabatic sweep of the magnetic field to 1200 g. using this data , we determine both the energy and the entropy of the strongly interacting gas . the data is then compared to several recent predictions . we begin by determining the axial mean square cloud sizes at 840 g and after the adiabatic sweep to 1200 g. since the atom number can vary between different runs by up to 20% , it is important to make the comparison independent of the atom number and trap parameters . for this purpose , the mean square sizes are given in units of @xmath77 , as defined above . the measured mean square sizes are listed in table [ tbl : energyentropy ] . in the experiments , evaporative cooling is used to produce an atom cloud near the ground state . energy is controllably added by releasing the cloud and then recapturing it after a short time @xmath101 as described previously @xcite . for a series of different values of @xmath101 , the energy at 840 g is directly measured from the axial cloud size according to eq . [ eq : energygaussian ] . then the same sequence is repeated , but the cloud size is measured after an adiabatic sweep to 1200 g. in each case , the systematic increase in mean square size arising from background heating rate is determined and subtracted . the total data comprise about 900 individual measurements of the cloud size at 840 g and 900 similar measurements of the cloud size after a sweep to 1200 g. to estimate the measurement error , we split the energy scale at 840 g into bins with a width of @xmath102 . measured data points within the width of the energy bin are used to calculate the average measured values of the cloud sizes and the corresponding standard deviation at both 840 g and 1200 g. ' '' '' & @xmath103&@xmath104&@xmath105&@xmath106&@xmath107&@xmath108 + ' '' '' 1&0.568(4)&0.743(6)&0.548(4)&0.63(8)&0.91(23)&0.97(5 ) + ' '' '' 2&0.612(5)&0.776(13)&0.589(5)&0.99(11)&1.18(22)&1.24(9 ) + ' '' '' 3&0.661(5)&0.803(11)&0.634(5)&1.22(8)&1.36(20)&1.42(7 ) + ' '' '' 4&0.697(9)&0.814(15)&0.667(8)&1.30(10)&1.43(18)&1.49(8 ) + ' '' '' 5&0.74(1)&0.87(4)&0.71(1)&1.6(2)&1.72(18)&1.8(2 ) + ' '' '' 6&0.79(1)&0.89(2)&0.75(1)&1.7(1)&1.79(15)&1.9(1 ) + ' '' '' 7&0.83(1)&0.94(2)&0.79(1)&2.0(1)&2.03(16)&2.1(1 ) + ' '' '' 8&0.89(2)&1.02(2)&0.84(2)&2.3(1)&2.32(18)&2.4(1 ) + ' '' '' 9&0.91(1)&1.02(3)&0.86(1)&2.3(1)&2.31(16)&2.4(1 ) + ' '' '' 10&0.97(1)&1.10(1)&0.91(1)&2.55(4)&2.57(17)&2.64(4 ) + ' '' '' 11&1.01(1)&1.17(2)&0.94(1)&2.74(7)&2.75(19)&2.82(7 ) + ' '' '' 12&1.05(1)&1.18(1)&0.98(1)&2.78(4)&2.80(17)&2.87(4 ) + ' '' '' 13&1.10(1)&1.22(1)&1.03(1)&2.89(2)&2.90(15)&2.97(2 ) + ' '' '' 14&1.25(2)&1.35(5)&1.15(1)&3.21(12)&3.20(14)&3.27(12 ) + ' '' '' 15&1.28(1)&1.39(3)&1.18(1)&3.28(6)&3.28(14)&3.35(6 ) + ' '' '' 16&1.44(2)&1.49(2)&1.31(2)&3.49(4)&3.48(9)&3.55(4 ) + ' '' '' 17&1.53(2)&1.62(6)&1.39(2)&3.74(10)&3.73(11)&3.80(10 ) + ' '' '' 18&1.58(1)&1.63(1)&1.42(1)&3.76(2)&3.74(8)&3.81(2 ) + ' '' '' 19&1.70(2)&1.73(6)&1.52(1)&3.94(9)&3.92(7)&3.99(9 ) + ' '' '' 20&1.83(5)&1.79(2)&1.62(4)&4.03(3)&4.01(1)&4.08(3 ) + ' '' '' 21&1.93(3)&1.96(3)&1.70(2)&4.28(4)&4.26(6)&4.32(4 ) + ' '' '' 22&2.11(5)&2.17(3)&1.83(4)&4.55(3)&4.53(7)&4.59(3 ) + the ratio of the mean square axial cloud size at 1200 g ( measured after the sweep ) to that at 840 g ( measured prior to the sweep ) is plotted in fig . [ fig : sizeratio ] as a function of the energy of a strongly interacting gas at 840 g. the ratio is @xmath109 , since for an adiabatic sweep of the magnetic field from the strongly interacting regime to the weakly interacting regime , the total entropy in the system is conserved but the energy increases : the strongly interacting gas is more attractive than the weakly interacting gas . a similar method was used to measure the potential energy change in a fermi gas of @xmath110k , where the bias magnetic field was adiabatically swept between the strongly interacting regime at the feshbach resonance and a noninteracting regime above resonance @xcite . the resulting potential energy ratios are given as a function of the temperature of the noninteracting gas @xcite . in contrast , by exploiting the virial theorem which holds for the unitary gas , we determine both the energy and entropy of the strongly interacting gas , as described below . [ fig : sizeconvertentropy ] shows the entropy which is obtained from the mean square size at 1200 g @xmath111 as listed in table [ tbl : energyentropy ] . first , we find the mean square size relative to that of the ground state , @xmath112 . we use the measured value @xmath113 for the lowest energy state that we obtained at 1200 g , as determined by extrapolation to @xmath14 using a sommerfeld expansion for the spatial profile of an ideal fermi gas . then we determine the entropy in the noninteracting ideal fermi gas approximation : @xmath114 $ ] , where we have replaced @xmath81 by the ground state value at 1200 g. as discussed above , this method automatically ensures that @xmath9 corresponds to the measured ground state @xmath115 at 1200 g , and compensates for the mean field shift between the measured @xmath116 for a weakly interacting fermi gas and that calculated @xmath117 for an ideal fermi gas in our gaussian trapping potential . as shown in fig . [ fig : entropysizeoverlap ] , the entropy obtained from a more precise many - body calculations is in close agreement with the ideal gas entropy calculated in the ideal gas approximation . the energy is determined from the cloud profiles at 840 g using eq . [ eq : energygaussian ] . finally , we generate the energy - entropy curve for a strongly interacting fermi gas , as shown in fig . [ fig : energyentropy ] . here , the energy @xmath1 measured from the mean square axial cloud size at 840 g is plotted as a function of the entropy @xmath2 measured at 1200 g after an adiabatic sweep of the magnetic field . we note that above @xmath118 ( @xmath119 ) the @xmath4 data ( blue dots ) for the strongly interacting gas appear to merge smoothly to the ideal gas curve ( dashed green ) . for an ideal fermi gas . for this figure , the ideal gas approximation to the entropy is used , @xmath106 of table [ tbl : energyentropy ] . [ fig : energyentropy],width=384 ] in addition to the entropy calculated in the ideal gas approximation , table [ tbl : energyentropy ] also provides a more precise entropy @xmath120 versus the axial mean square cloud size . these results are obtained by hu et al . @xcite using a many - body calculation for @xmath121 at 1200 g in the gaussian trap of eq . [ eq : reducedtruepotential ] . perhaps the most important application of the energy - entropy measurements is to test strong coupling many - body theories and simulations . since the energy and entropy are obtained in absolute units without invoking any specific theoretical model , the data can be used to distinguish recent predictions for a trapped strongly interacting fermi gas . fig . [ fig : comparetheory ] shows how four different predictions compare to the measured energy and entropy data . these include a pseudogap theory @xcite , a combined luttinger - ward - de dominicis - martin ( lw - ddm ) variational formalism @xcite , a t - matrix calculation using a modified nozires and schmitt - rink ( nsr ) approximation @xcite , and a quantum monte carlo simulation @xcite . the most significant deviations appear to occur near the ground state , where the precise determination of the energy seems most difficult . the pseudogap theory predicts a ground state energy that is above the measured value while the prediction of ref . @xcite is somewhat low compared to the measurement . all of the different theories appear to converge at the higher energies . the temperature @xmath6 is determined from the measured @xmath4 data using the fundamental relation , @xmath5 . to implement this method , we need to parameterize the data to obtain a smooth differentiable curve . at low temperatures , one expects the energy to increase from the ground state according to a power law in @xmath6 and a corresponding power law in @xmath2 , i.e. , @xmath122 . for a harmonically trapped ideal fermi gas , we have in the sommerfeld approximation an energy per particle in units of @xmath46 given by @xmath123 . the corresponding entropy per particle in units of @xmath124 is @xmath125 , so that @xmath126 . we attempt to use a single power law to fit the @xmath127 curve for a noninteracting fermi gas in a gaussian trapping potential , with @xmath128 , as in our experiments . the energy and entropy are calculated in the energy range @xmath129 and displayed as dots in fig . [ fig : idealgasfit ] . we find that a single power law @xmath130 fits the curve very well over this energy range . note that the power law exponent is @xmath131 , close to the low temperature value . using the fit function , we can extract the reduced temperature @xmath132 as a function of @xmath85 and compare it to the theoretical reduced temperature @xmath133 at the same @xmath85 . the results are shown as the green dashed line in fig . [ fig : idealgasfit ] . we see that the agreement is quite good except below @xmath134 and above @xmath135 , where the deviation is @xmath136% . to improve the fit and to make a more precise determination of the temperature , we employ a fit function comprising two power laws that are joined at a certain entropy @xmath7 , which gives the best fit . when used to fit the data for the strongly interacting fermi gas , we consider two types of fits that incorporate either a jump in heat capacity or a continuous heat capacity at @xmath7 . in this way , we are able determine the sensitivity of the temperature and critical parameters to the form of the fit function . the two types of fits yield nearly identical temperatures , but different values of @xmath7 and hence of the critical parameters , as discussed below . we take the energy per particle @xmath1 in units of @xmath46 to be given in terms of the entropy per particle in units of @xmath124 in the form @xmath137 we constrain the values of @xmath138 and @xmath3 by demanding that energy and temperature be continuous at the joining point @xmath7 : @xmath139 by construction , the value of @xmath140 does not affect these constraints and is chosen in one of two ways . fixing @xmath141 , the fit incorporates a heat capacity jump at @xmath7 , which arises from the change in the power law exponents at @xmath7 . alternatively , we choose @xmath140 so that the second derivative @xmath142 is continuous at @xmath7 , making the heat capacity continuous . the final fit function has 5 independent parameters @xmath143 , and takes the form @xmath144+e(s - s_c)^2 ; \,\,s\geq s_c . \label{eq : evsslowengtwopower}\end{aligned}\ ] ] here , when @xmath140 is not constrained to be zero , it is given by @xmath145 fig . [ fig : idealgasfit ] shows the improved fit to the calculated energy versus entropy of a noninteracting fermi gas in a gaussian trap for @xmath128 , using eq . [ eq : evsslowengtwopower ] with @xmath146 , since the ideal gas has no heat capacity jump . in this case , both power law exponents @xmath147 and @xmath148 are close to @xmath149 as for the single power law fit . the temperature determined from the fit agrees very closely with the exact temperature , as shown in fig . [ fig : idealgasfit ] ( red solid line ) . in contrast to the noninteracting case , we have found that the energy - entropy data of a strongly interacting fermi gas is not well fit by a single power law function @xcite . however , the two power - law function fits quite well , with a factor of two smaller value of @xmath150 than for the single power - law fit . here , we use @xmath151 , where @xmath152 ( @xmath153 ) is the fitted ( data ) value for the @xmath154 point , and @xmath155 is the corresponding the standard error . motivated by the good fits of the two power - law function to the ideal gas energy versus entropy curve and the good agreement between the fitted and exact temperature , we apply the two power - law fit function to the data for the strongly interacting fermi gas . ) . for comparison , the dot - dashed green curve shows @xmath4 for an ideal fermi gas . for this figure , the ideal gas approximation to the entropy is used , @xmath106 of table [ tbl : energyentropy ] . [ fig : energyentropyfit],width=384 ] fig . [ fig : energyentropyfit ] shows the fit ( red solid curve ) obtained with a heat capacity jump using eq . [ eq : evsslowengtwopower ] with @xmath141 and @xmath156 , @xmath157 , @xmath158 , the ground state energy @xmath159 , and the critical entropy @xmath160 . also shown is the fit ( blue dashed curve ) with continuous heat capacity ( @xmath146 ) and @xmath161 , @xmath162 , @xmath163 , the ground state energy @xmath164 , and the critical entropy @xmath165 . the fit functions for the @xmath4 data for the strongly interacting fermi gas exhibit a significant change in the scaling of @xmath1 with @xmath2 below and above @xmath7 . the dramatic change in the power law exponents for the strongly interacting gas suggests a transition in the thermodynamic properties . the power law exponent is @xmath166 above @xmath7 , comparable to that obtained for the ideal gas , where @xmath167 . the power law exponent below @xmath7 is @xmath168 , which corresponds to the low temperature dependence @xmath169 , close to that obtained in measurements of the heat capacity , where the observed power law was @xmath170 after the model - dependent calibration of the empirical temperature @xcite , see [ sec : ttwiddle ] . if we interpret @xmath7 as the critical entropy for a superfluid - normal fluid transition in the strongly interacting fermi gas , then we can estimate the critical energy @xmath15 and the critical temperature @xmath171 . for the fits of eq . [ eq : evsslowengtwopower ] with a heat capacity jump @xmath172 ) or with continuous heat capacity ( @xmath146 ) , we obtain @xmath173 using the fit parameters in eq . [ eq : criticalparam ] yields critical parameters of the strongly interacting fermi gas , which are summarized in table [ tbl : criticalparam ] . the statistical error estimates are from the fit , and do not include systematic errors arising from the form of the fit function . ' '' '' & @xmath174 & @xmath175 & @xmath176 + ' '' '' expt @xmath4 fit@xmath177 & 2.2(1 ) & 0.83(2 ) & 0.21(1 ) + ' '' '' expt @xmath4 fit@xmath178 & 1.6(3 ) & 0.70(5 ) & 0.185(15 ) + ' '' '' heat capacity experiment@xmath179 & & 0.85 & 0.20 + ' '' '' theory ref . @xcite & & & 0.30 + ' '' '' theory ref . @xcite & & & 0.31 + ' '' '' theory ref . @xcite & & & 0.27 + ' '' '' theory ref . @xcite & & & 0.29 + ' '' '' theory ref . @xcite & 2.15 & 0.82 & 0.27 + ' '' '' theory ref . @xcite & 1.61(5 ) & 0.667(10 ) & 0.214(7 ) + we note that the fit function for @xmath180 previously used in ref . @xcite to determine the temperature was continuous in @xmath2 and @xmath1 , but intentionally ignored the continuous temperature constraint in order to determine the entropy as a power of @xmath181 both above and below the joining energy @xmath15 . as the continuous temperature constraint is a physical requirement , we consider the present estimate of the temperature @xmath6 to be more useful for temperature calibrations and for characterizing the physical properties of the gas than the estimate of ref . @xcite . in contrast to the temperature @xmath6 , the estimate of @xmath8 depends on the value of the joining entropy @xmath7 that optimizes the fit and is more sensitive to the form of fit function than the temperature that is determined from the @xmath1 and @xmath2 data . for the fit function @xmath180 used in ref . @xcite , the temperatures determined by the fit function just above @xmath15 , @xmath182 , and below @xmath15 , @xmath183 , were different . an average of the slopes @xmath184 and @xmath185 was used to estimate the critical temperature . from those fits , the critical energy was found to be @xmath186 , the critical entropy per particle was @xmath187 . the estimated critical temperature obtained from the average was @xmath188 , significantly higher than than the value @xmath189 obtained using eq . [ eq : evsslowengtwopower ] , which incorporates continuous temperature . we are able to substantiate the critical temperature @xmath189 by using our data to experimentally calibrate the temperature scales in two other experiments . in [ sec : onsetpaircond ] , we find that this value is in very good agreement with the estimate we obtain by calibrating the ideal gas temperature observed for the onset of pair condensation . nearly the same transition temperature is obtained in [ sec : ttwiddle ] by using the @xmath4 data to calibrate the empirical transition temperature measured in heat capacity experiments @xcite . table [ tbl : criticalparam ] compares the critical parameters estimated from the power - law fits to the @xmath4 data with the predictions for a trapped unitary fermi gas from several theoretical groups . we note that calculations for a uniform strongly interacting fermi gas at unitarity @xcite yield a lower critical temperature , @xmath190 , than that of the trapped gas , where @xmath191 is the fermi temperature corresponding to the uniform density @xmath192 . extrapolation of the uniform gas critical temperature to that of the trapped gas shows that the results are consistent @xcite . using the parameters from the fits and eq . [ eq : evsslowengtwopower ] , the temperature of the strongly interacting fermi gas , in units of @xmath47 can be determined as a function of the entropy per particle , in units of @xmath124 , @xmath193 here @xmath7 is given in table [ tbl : criticalparam ] from the fits to the @xmath4 data for the strongly interacting gas , eq . [ eq : criticalparam ] gives @xmath8 . [ fig : tempvsentropy ] shows the temperature as a function of entropy according to eq . [ eq : temperaturestrongint ] for fits with a heat capacity jump and for continuous heat capacity . the estimates of the temperature of the strongly interacting fermi gas as a function of the entropy can be used to experimentally calibrate the temperatures measured in other experiments , without invoking any specific theoretical models . the jila group measures the pair condensate fraction in a strongly interacting fermi gas of @xmath110k as a function of the initial temperature @xmath194 in the noninteracting regime above the feshbach resonance @xcite . in these experiments , a downward adiabatic sweep of the bias magnetic field to resonance produces a strongly interacting sample . using our @xmath4 data , we relate the endpoint temperatures for adiabatic sweeps of the bias magnetic field between the ideal and strongly interacting fermi gas regimes . we therefore obtain the critical temperature for the onset of pair condensation in the strongly interacting fermi gas , and find very good agreement with our estimates based on entropy - energy measurement . in addition , we calibrate the empirical temperature based on the cloud profiles , which was employed in our previous measurements of the heat capacity @xcite . we relate the endpoint temperatures for an adiabatic sweep between the strongly interacting and ideal fermi gas regimes . [ eq : temperaturestrongint ] gives the temperature of the strongly interacting gas as a function of entropy , i.e. , @xmath195 . next , we calculate the entropy per particle @xmath196 for an ideal fermi gas in our gaussian trap , in units of @xmath124 , with @xmath197 in units of @xmath47 , as used in [ sec : idealgas ] to determine @xmath198 . for an adiabatic sweep between the strongly interacting and ideal fermi gas regimes , where @xmath199 , the temperature of the strongly interacting gas is related to that for the ideal fermi gas by @xmath200 , \label{eq : adiabactictempcal}\ ] ] which is shown in fig . [ fig : tempsivni ] . + for an adiabatic sweep from the ideal fermi gas regime to the strongly interacting fermi gas regime at low temperature @xmath201 , the reduced temperature of the strongly interacting gas is greater than or equal to that of the ideal gas . this arises because the entropy of the strongly interacting gas scales as a higher power of the temperature than that of the ideal gas . in our present experiments , we could not take data at high enough temperatures to properly characterize the approach of the temperature to the ideal gas regime . above @xmath8 , our @xmath4 data are obtained over a limited range of energies @xmath202 to avoid evaporation in our shallow trap . in this energy range , our data are reasonably well fit by a single power law . however , such a power law fit can not completely describe the higher temperature regime . we expect that the temperatures of the strongly interacting gas and ideal gas must start to merge in the region @xmath203 , where the @xmath4 data for the strongly interacting gas nearly overlaps with the @xmath4 curve for an ideal gas , as shown in fig . [ fig : energyentropy ] . from fig . [ fig : tempvsentropy ] , @xmath204 corresponds to @xmath205 , approximately the place where the calibrations from the two different power law fits ( for @xmath141 and @xmath146 ) begin to differ in fig . [ fig : tempsivni ] . we therefore expect that the single power law fit overestimates the temperature @xmath6 of the strongly interacting gas for @xmath206 , yielding a trend away from ideal gas temperature , in contrast to the expected merging at high temperature . in ref . @xcite , projection experiments measure the ideal fermi gas temperature @xmath194 where pair condensation first appears . in those experiments , @xmath194 is estimated to be @xmath207 @xcite . from the calibration , [ fig : tempsivni ] , we see that for @xmath208 , the corresponding temperature of the strongly interacting gas is @xmath209 for both the red solid and blue dashed curves , which is almost the same as the ideal gas value . the critical temperature of the strongly interacting gas for the onset of pair condensation is then @xmath210 , in very good agreement with the values @xmath211 and @xmath212 that we obtain from the two fits to the @xmath4 measurements . this substantiates the conjecture that the change in the power law behavior observed at @xmath8 in our experiments corresponds to the superfluid transition . in our previous study of the heat capacity , we determined an empirical temperature @xmath213 as a function of the total energy of the gas @xcite . the gas was initially cooled close to the ground state and a known energy was added by a release and recapture method . then a thomas - fermi profile for an ideal fermi gas was fit to the low temperature cloud profiles to determine the fermi radius . holding the fermi radius constant , the best fit to the cloud profiles at higher temperatures determined the effective reduced temperature , which is denoted @xmath214 . the @xmath215 data @xcite was observed to scale as @xmath216 for @xmath217 , while below @xmath218 , the energy was found to scale as @xmath219 . the transition point occurs at an energy @xmath220 , which is close to the value @xmath221 obtained from power - law fit to the @xmath4 data for the fit with a heat capacity jump . assuming that @xmath222 corresponds to the superfluid - normal fluid transition , we can determine the corresponding value of @xmath12 for the strongly interacting gas . to calibrate the empirical temperature we start with @xmath215 . then , as discussed in [ sec : energyvstemp ] , eq . [ eq : evsslowengtwopower ] determines @xmath19 and hence @xmath223from the fits to our @xmath4 data . hence @xmath224 $ ] , where @xmath225 is the reduced temperature of the strongly interacting gas and @xmath226 is the reduced energy . for simplicity , we give the analytic results obtained using the @xmath141 fit to the @xmath4 data , @xmath227 fig . [ fig : tempsivni ] shows the full calibration ( green dashed curve ) . for comparison , the calibration obtained from the pseudogap theory of the cloud profiles gave @xmath228 for @xmath229 , and @xmath230 above @xmath218 . for @xmath222 , we obtain from eq . [ eq : tildetcalib ] @xmath231 ( see fig . [ fig : tempsivni ] ) , in good agreement with the value obtained for the onset of pair condensation and with the values @xmath232 and @xmath233 determined from the fits to the @xmath4 data . measurement of the ground state energy of a unitary fermi gas provides a stringent test of competing many - body theoretical predictions and is therefore of great interest . for a unitary fermi gas of uniform density in a 50 - 50 mixture of two spin states , the ground state energy per particle can be written as @xmath234 where @xmath235 is the local fermi energy corresponding to the density @xmath192 . the ground state energy of the unitary fermi gas differs by a universal factor @xmath236 from that of an ideal fermi gas at the same density . the precise value of @xmath237 has been of particular interest in the context of neutron matter @xcite , and can be measured in unitary fermi gas experiments @xcite . the sound speed at temperatures near the ground state determines @xmath16 according to eq . [ eq : soundres ] . we have made precision measurements of the sound speed in a trapped fermi gas at the feshbach resonance @xcite . at 834 g , we vary the density by a factor of 30 to demonstrate universal scaling and obtain the value @xmath238 . using eq . [ eq : soundres ] then yields @xmath239 . note that the reference fermi velocity @xmath240 depends on the fermi energy of an ideal gas at the trap center and hence on both the trap frequencies and atom number ( as @xmath241 ) , which are carefully measured to minimize systematic errors @xcite . while the energy of the gas as measured from the mean square cloud size was close to the ground state value , the precise temperature of the gas was not determined . the universal parameter @xmath16 also can be determined by measuring the ground state energy @xmath18 of a harmonically trapped unitary fermi gas , which is given by @xmath242 our @xmath4 data enables a new determination of @xmath18 by extrapolating the measured energy @xmath4 to @xmath9 . as pointed out by hui et al . @xcite , this method avoids a systematic error arising when the finite temperature is not determined in the measurements . from both of our fit functions below @xmath7 , we obtain @xmath243 . [ eq : hobeta ] yields @xmath244 . this result is slightly more negative than that obtained in the sound speed experiments , which is reasonable since the sound speed measurements are done at finite temperature . both results are in very good agreement . one possible systematic error in these measurements arises from the determination of the atom number . the measurements of @xmath16 from the sound speed and from the energy - entropy measurements were done in different laboratories . the close agreement is gratifying , considering that the imaging systems that determine the atom number employed @xmath245-polarized light for the sound speed experiments , while the entropy - energy measurements used @xmath246-polarized light , for which the resonant optical cross section is a factor of two smaller than for @xmath245 polarization . to examine the systematic error arising from the atom number determination , we employ a third method to measure @xmath16 based on the measured ratio of the cloud size at 840 g and at 1200 g , which is number independent . the ratio of the ground state mean square sizes for the weakly and strongly interacting gases is predicted to be @xmath247 note that we obtain @xmath248 from a mean field calculation @xcite , in agreement with that obtained using a many - body calculation @xcite . our measurements for the ground state mean square size at 1200 g are accomplished by fitting a sommerfeld expansion of the axial density for an ideal fermi gas to the cloud profile @xcite . the fit determines the fermi radius @xmath249 and reduced temperature @xmath133 , yielding @xmath250 for @xmath14 , close to the predicted value of @xmath251 . the ground state energy @xmath252 at 840 g from the entropy - energy experiments determines the ground state mean square size as @xmath253 . hence , @xmath254 . the corresponding @xmath255 from eq . [ eq : groundstateratio ] . since the mean square sizes are determined from the images and the ratio @xmath256 is number independent , this result shows that the systematic error arising from the number measurement is within the quoted error estimate . we also can determine @xmath16 by directly extrapolating to zero entropy the ratio of the axial mean square size of the weakly interacting fermi gas at 1200 g to that of strongly interacting gas at 840 g. when this is done , we obtain @xmath257 , in very good agreement with the estimates based on the sound speed and ground state energy . finally , we can estimate the correction to the ground state energy arising from the finite scattering length at 840 g , @xmath258 . for the trap conditions in the @xmath4 measurements , @xmath259 , where @xmath260 is the wavevector for an ideal fermi gas at the trap center . to estimate the true unitary ground state energy at @xmath261 , we first determine the leading order @xmath262 correction to the trapped atom density , where @xmath263 is the local fermi wavevector corresponding to the density @xmath192 . the local chemical potential is estimated from ref . @xcite . using the notation of eq . [ eq : reducedtruepotential ] and a harmonic approximation , the corrected density yields @xmath264 $ ] , where @xmath265 is the fermi radius for the unitary gas . according to the virial theorem ( see eq . [ eq : energymeas ] ) , the mean square size and energy of the unitary gas are corrected by the same factor . the unitary ground state energy is then @xmath266 for @xmath267 , we obtain @xmath268 and the value of @xmath269 obtained directly from @xmath270 is shifted to @xmath271 . we also obtain the corrected value of @xmath272 in eq . [ eq : groundstateratio ] and @xmath273 . table [ tbl : beta ] compares the values of @xmath16 obtained in our experiments to several recent predictions . note that the table does not include the finite @xmath274 correction for the @xmath4 measurement at 840 g described above . ' '' '' & @xmath16 + ' '' '' @xmath4 experiment & -0.59(2 ) + ' '' '' sound velocity experiment & -0.565(15 ) + ' '' '' cloud size ratio experiment & -0.61(2 ) + ' '' '' ref . @xcite & -0.58(1 ) + ' '' '' ref . @xcite & @xmath275 + ' '' '' ref . @xcite & -0.599 + ' '' '' ref . @xcite & -0.60(1 ) + ' '' '' ref . @xcite & -0.646(4 ) + using the @xmath4 data for the strongly interacting fermi gas and the temperature determined from the two power - law fits , we estimate several universal functions . first , we determine the dependence of the energy on temperature @xmath19 and the corresponding heat capacity , @xmath20 . then we find the global chemical potential of the trapped gas as a function of the energy @xmath21 . the energy is readily determined as a function of temperature using eq . [ eq : evsslowengtwopower ] for the case where there is a heat capacity jump and @xmath141 , @xmath276;\,\,\,0\leq t\leq t_c\nonumber\\ e_>(t)&=&e_c+{\frac{s_ct_c}{d}}\left[\left(\frac{t}{t_c}\right)^{\frac{d}{d-1}}-1\right ] ; \,\,\,t\geq t_c , \label{eq : energyvstemp}\end{aligned}\ ] ] where the energy ( temperature ) is given in units of @xmath46 ( @xmath47 ) and the critical energy @xmath15 is @xmath277 with @xmath18 the ground state energy . for the case with @xmath146 , where the heat capacity is continuous , we determine the ordered pairs @xmath278 $ ] as a function of @xmath2 and plot @xmath19 . fig . [ fig : energyvstemp ] shows the results using the best fits for both cases . of particular interest is the low temperature power law . for @xmath141 , we obtain @xmath279 and @xmath280 . since @xmath147 is near @xmath281 , the energy relative to the ground state scales approximately as @xmath282 . this is consistent with sound modes dominating the low energy excitations . however , one would expect instead that the free fermions on the edges of the trapped cloud would make an important contribution to the low energy excitations @xcite . over an extended range of @xmath201 , the net entropy arising from the bose and fermi excitations has been predicted to scale as @xmath283 , yielding an energy scaling @xcite as @xmath284 . in this case , one would expect that @xmath285 , i.e. , @xmath286 in eq . [ eq : energyvstemp ] , so that @xmath287 . hence , the low energy power law exponents for the entropy should be between @xmath281 and @xmath288 , which is barely distinguishable for our data . the heat capacity at constant trap depth @xmath289 is readily obtained from eq . [ eq : energyvstemp ] ( where there is a heat capacity jump , since we have constrained @xmath141 in eq . [ eq : evsslowengtwopower ] ) . for this parameterization , @xmath290 where @xmath6 and @xmath8 are given in units of @xmath47 , and @xmath7 is given in units of @xmath124 . for the fit with a continuous heat capacity , we use @xmath195 to find @xmath291 , and plot the ordered pairs @xmath292 $ ] . the heat capacity curves for both cases are shown in fig . [ fig : cvst ] . according to eq . [ eq : heatcapacity ] , a jump in heat capacity occurs at @xmath7 : @xmath293 and @xmath294 differ when the power law exponents @xmath147 and @xmath148 are different . this is a consequence of the simple two power - law structure assumed for the fit function @xmath4 given by eq . [ eq : evsslowengtwopower ] for @xmath141 , and can not be taken as proof of a true heat capacity jump . at present , the precise nature of the behavior near the critical temperature can not be determined from our data , and it remains an open question whether the data exhibits a heat capacity jump or a continuous heat capacity . the global chemical potential @xmath295 is readily determined from the fits to the @xmath4 data for a strongly interacting fermi gas , which obeys universal thermodynamics . the local energy density generally takes the form @xmath296 , where @xmath297 is the local internal energy , which includes the kinetic energy and the interaction energy . here , @xmath192 is the local density , @xmath95 is the local chemical potential , @xmath298 is the pressure and @xmath299 is the total entropy per unit volume . the local chemical potential can be written as @xmath300 , where @xmath24 is the trap potential . in the universal regime , where the local pressure depends only on the local density and temperature , we have @xmath301 , as noted by ho @xcite . hence , @xmath302 . integrating both sides over the trap volume and using @xmath303 , where @xmath1 and @xmath304 ) are the total energy and average potential energy per particle , respectively , we obtain @xmath305 where @xmath2 is the entropy per particle . for simplicity , we assume harmonic confinement and use the virial theorem result , @xmath306 , from eq . [ eq : energymeas ] , which holds in the universal regime . then , eq . [ eq : globalchempot1 ] yields the global chemical potential of a harmonically trapped fermi gas in the universal regime , @xmath307 by using the fit to the measured entropy - energy data to obtain the temperature @xmath5 from eq . [ eq : evsslowengtwopower ] , the global chemical potential of a trapped unitary fermi gas can be calculated from eq . [ eq : globalchempot ] . for @xmath141 , where the heat capacity has a jump , the simple power law fits above and below @xmath15 each yield a different linear dependence of @xmath295 on @xmath1 , @xmath308 where @xmath309 . we plot the chemical potential in fig . [ fig : chemenergy ] . the data points are obtained using eq . [ eq : globalchempot ] with the measured energy @xmath1 and entropy @xmath2 and the temperature determined from the fit to the @xmath4 data , using @xmath141 in eq . [ eq : evsslowengtwopower ] , i.e. , with a heat capacity jump . the solid red curve is given by eq . [ eq : globalchempotvse ] . we note that the low temperature data points in @xmath4 are best fit with the power law @xmath279 , which is close to @xmath281 . according to eq . [ eq : globalchempotvse ] , this produces a slope near zero for @xmath310 . since the power - law fit above @xmath15 gives @xmath311 , the slope according to eq . [ eq : globalchempotvse ] changes from nearly zero for @xmath310 to negative for @xmath312 . note that from eq . [ eq : globalchempot ] , we obtain the slope @xmath313 since the entropy @xmath2 is continuous , we see that a jump in the heat capacity produces a corresponding jump in the slope of @xmath295 versus @xmath1 . for comparison , fig . [ fig : chemenergy ] also shows the chemical potential obtained for @xmath146 in eq . [ eq : evsslowengtwopower ] , where the heat capacity is continuous ( blue dashed curve ) . we have studied the thermodynamic properties of a strongly interacting fermi gas by measuring both the energy and the entropy . the model - independent data obtained in both the superfluid and the normal fluid regimes do not employ any specific theoretical calibrations , and therefore can be used as a benchmark to test the predictions from many - body theories and simulations . parameterizing the energy - entropy data determines the temperature of the strongly interacting fermi gas and also yields estimates of the critical parameters . we use the measured data to calibrate two different temperature scales that were employed in observations of the onset of pair condensation and in heat capacity studies . these calibrations yield critical temperatures in good agreement with the results estimated from our energy - entropy data . our data does not determine whether the heat capacity exhibits a jump or is continuous at the critical temperature . however , for a finite system with nonuniform density , the latter is most likely . considering that there is huge interest in determining the detailed behavior of the superfluid transition in a strongly interacting fermi gas @xcite , more precise determinations of the critical temperature , the heat capacity , and the chemical potential near the critical point , as well as the high temperature behavior and the approach to the ideal gas limit , will be important topics for future research . this research has been supported by the physics divisions of the army research office and the national science foundation , the physics for exploration program of the national aeronautics and space administration , and the chemical sciences , geosciences and biosciences division of the office of basic energy sciences , office of science , u. s. department of energy . we thank willie ong for a careful reading of the manuscript .
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strongly interacting fermi gases provide a clean and controllable laboratory system for modeling strong interparticle interactions between fermions in nature , from high temperature superconductors to neutron matter and quark - gluon plasmas .
model - independent thermodynamic measurements , which do not require theoretical models for calibrations , are very important for exploring this important system experimentally , as they enable direct tests of predictions based on the best current non - perturbative many - body theories . at duke university , we use all - optical methods to produce a strongly interacting fermi gas of spin-1/2-up and spin-1/2-down @xmath0li atoms that is magnetically tuned near a collisional ( feshbach ) resonance .
we conduct a series of measurements on the thermodynamic properties of this unique quantum gas , including the energy @xmath1 , entropy @xmath2 , and sound velocity @xmath3 .
our model - independent measurements of @xmath1 and @xmath2 enable a precision study of the finite temperature thermodynamics .
the @xmath4 data are directly compared to several recent predictions .
the temperature in both the superfluid and normal fluid regime is obtained from the fundamental thermodynamic relation @xmath5 by parameterizing the @xmath4 data using two different power laws that are joined with continuous @xmath1 and @xmath6 at a certain entropy @xmath7 , where the fit is optimized .
we observe a significant change in the scaling of @xmath1 with @xmath2 above and below @xmath7 . taking the fitted value of @xmath7 as an estimate of the critical entropy for a superfluid - normal fluid phase transition in the strongly interacting fermi gas
, we estimate the critical parameters .
our @xmath4 data are also used to experimentally calibrate the endpoint temperatures obtained for adiabatic sweeps of the magnetic field between the ideal and strongly interacting regimes .
this enables the first experimental calibration of the temperature scale used in experiments on fermionic pair condensation , where the ideal fermi gas temperature is measured before sweeping the magnetic field to the strongly interacting regime .
our calibration shows that the ideal gas temperature measured for the onset of pair condensation corresponds closely to the critical temperature @xmath8 estimated in the strongly interacting regime from the fits to our @xmath4 data .
we also calibrate the empirical temperature employed in studies of the heat capacity and obtain nearly the same @xmath8 .
we determine the ground state energy by three different methods , using sound velocity measurements , by extrapolating @xmath4 to @xmath9 and by measuring the ratio of the cloud sizes in the strongly and weakly interacting regimes .
the results are in very good agreement with recent predictions . finally , using universal thermodynamic relations , we estimate the chemical potential and heat capacity of the trapped gas from the @xmath4 data .
pacs numbers : 03.75.ss
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multi - wavelength light curves of blazar flares show complex and diversified features . while in some cases there is a time lag between gamma - ray and x - ray / optical flares ( e.g. * ? ? ? * ; * ? ? ? * ; * ? ? ? * ; * ? ? ? * ) , in other cases an orphan flare in a certain wavelength was detected ( e.g. * ? ? ? * ; * ? ? ? even if a time - dependent model is adopted , such a variety of behaviors may be difficult to reproduce with by a one - zone model @xcite . while spatial gradients of the physical parameters in the emission regions @xcite may explain some fraction of the lags , some flares have spectral evolutions thatare too complex to be modeled , even with time - dependent multi - zone radiative transfer simulations @xcite . this may imply that inhomogeneous emission regions evolve with a longer timescale than a dynamical one . such nontrivial properties in a blazar flare make it difficult to probe physical processes such as electron acceleration or cooling . in 2013 december , the _ fermi_-large area telescope ( lat ) detected one of the most intense flares in the gamma - ray band from flat spectrum radio quasar ( fsrq ) 3c 279 , reaching @xmath0 for the integrated flux above 100mev ( * ? ? ? * hereafter h15 ) . the flux level is comparable to the historical maximum of this source observed at the gamma - ray band @xcite . the gamma - ray flare showed a very rapid variability with an asymmetric time profile with a shorter rising time of @xmath1hr and a longer falling time of @xmath2hr . we can expect that this extraordinary flare was emitted from a sufficiently compact region that can be regarded as homogeneous , which is different from other usual flares . in this case , the decaying timescale may directly correspond to the cooling timescale , and the flare is an ideal target for discussing the physical processes . another important property of the flare event of 3c 279 , a very hard photon index of @xmath3 , was observed in the @xmath4mev band by _ such a hard photon index has been rarely observed in fsrqs , whose luminosity peak from inverse - compton ( ic ) scattering is usually located below 100mev . while the mean of the @xmath5 distribution in fsrqs corresponds to about 2.4 @xcite , hard photon indices @xmath6 only have been occasionally observed in some bright fsrqs during rapid flaring events @xcite . in order to reproduce the hard photon index by ic scattering in the fast cooling regime , the index of parent electrons should be much harder than two , which can hardly be generated in a normal shock acceleration process . in addition , the flare event of 3c 279 indicates a high compton dominance parameter @xmath7 , leading to extremely low jet magnetization with @xmath8 ( h15 ) . to explain the flare event of 3c 279 , rather than assuming prompt electron injection by the shock acceleration , we propose the stochastic acceleration ( sa ) model , which is phenomenologically equivalent to the second - order fermi acceleration ( fermi - ii ; e.g. * ? ? ? * ; * ? ? ? * ; * ? ? ? * and references therein ) . the sa may be driven by magnetic reconnection @xcite . otherwise , hydrodynamical turbulences that drive the acceleration are possibly induced via the kelvin helmholtz instability as an axial mode ( e.g. * ? ? ? * ) , or the rayleigh taylor and richtmyer meshkov instabilities as radial modes @xcite . broadband spectra of blazars in the steady state have been successfully fitted with recent sa models @xcite . the flare state should be also tested with such models to show the wide - range applicability of the sa . this is the first attempt to apply a fermi - ii model to explain both broadband spectra and light curves of fsrqs simultaneously . in this letter , we perform time - dependent simulations of the emissions from 3c 279 with the sa . starting from modeling a steady emission , the gamma - ray flare is reproduced by decreasing the magnetic field for the steady model . we demonstrate that the sa model agrees with the observed spectrum and light curve . we adopt the numerical simulation code used in @xcite . in this model , a conical outflow with an opening angle @xmath9 , where @xmath10 is its bulk lorentz factor , is ejected at radius @xmath11 from the central engine . the evolutions of electron and photon energy distributions are calculated in the comoving frame taking into account the sa , synchrotron emission , ic scattering with the klein nishina effect , adiabatic cooling , @xmath12 pair production , synchrotron self - absorption , and photon escape . the sa is characterized by the energy diffusion coefficient , @xmath13 . @xcite conservatively assumed the kolmogorov - like diffusion as @xmath14 . here , we adopt , however , @xmath15 , which corresponds to the hard - sphere scattering . this choice leads to reasonable spectra of 3c 279 without complicated assumptions such as nontrivial temporal evolution of the diffusion coefficient or electron injection rate . if the cascade of the turbulence stops at a certain length scale larger than the gyro - radius of the highest - energy electrons , the mean free path of electrons becomes comparable to this scale independently of electron energies . in this case , the energy diffusion can be approximated as the hard - sphere scattering ( e.g. * ? ? ? * ) . the recent magnetohydrodynamical simulations accompanying the inverse cascade shows the @xmath16 spectrum in turbulences @xcite , which also support the hard - sphere approximation . the volume we consider is a conical shell with a constant width of @xmath17 , then the isotropically equivalent volume @xmath18 ( the actual volume is @xmath19 ) . hereafter , the values in the shell frame are denoted with prime characters . during the dynamical timescale @xmath20 in the plasma frame , electrons are injected at a constant rate @xmath21 in the volume @xmath22 monoenergetically ( @xmath23 ) and accelerated with a constant coefficient @xmath24 . as done in @xcite , we can consider the temporal evolutions of the injection rate and the diffusion coefficient . however , this simple model with constant @xmath21 and @xmath24 is sufficient to reproduce the photon spectrum of 3c 279 . the average magnetic field in the comoving frame is assumed to behave as @xmath25 . the evolution of the photon spectrum for observers is computed taking into account the relativistic motion and curvature of the jet surface . as a reference case , we consider one of the most active periods in the gamma - ray band during the first two years of the _ fermi_-lat observations . in @xcite , this period is denoted as period `` d '' in 2009 . although this period corresponds to an event with a prominent flare , the broadband spectrum in the paper is averaged over five days . if the emission zone is inside the broad emission region as suggested by the short variability timescale reported in h15 , the flare state is significantly longer than the variability timescale . therefore , we adopt a steady emission model for period d in 2009 . by assuming continuous steady ejection of the shells from @xmath11 , we model the steady photon spectrum , though the plasma and its emission evolves with @xmath26 in the shell frame . the model parameters are @xmath27 pc , @xmath28 , @xmath29 ( @xmath30 ) , @xmath31 ( @xmath32 ) , and @xmath33 g. we adopt the same model as that of @xcite for the external radiation of the broad emission lines with the photon temperature @xmath34 ev and the energy density @xmath35 in the shell frame . the cooling timescale in this external radiation field is written as @xmath36 as shown in figure [ fig : ele1 ] ( a ) , electrons are continuously accelerated between @xmath11 and @xmath37 , then they are rapidly cooled via ic scattering after the shutdown of the acceleration . the electron spectrum at @xmath37 in the low - energy region is consistent with the assumed power - law index of @xmath38 in the broken power - law model of @xcite . in the highest - energy region , though the klein nishina effect suppresses the ic cooling effect , the synchrotron cooling ( @xmath39 ) prevent the acceleration above @xmath40 mev . the resultant photon spectrum well reproduces the observed spectrum from far - infrared to gamma - ray bands ( see figure [ fig : ele1 ] ( b ) ) . the low - energy cutoff at @xmath41 ev is due to the synchrotron self - absorption . the x - ray flux is originated from the synchrotron self - compton ( ssc ) emission . those x - ray data strongly constrain the emission radius @xmath42 . thus , our sa model can naturally produce a hard electron spectrum , and the steady photon spectrum agrees with the observed one in 2009 . based on this result , we will probe the intensive flare in 2013 in the next section . the most intensive flare denoted as period `` b '' ( on mjd 56646 ) in h15 shows a very hard spectrum with @xmath43 and a short variability with an hourly scale in the gamma - ray band observed with _ during a short time interval of the gamma - ray flare period ( 0.2 days ) , there were simultaneous optical observations , whose results did not show any correlated variability with the gamma - ray flare as presented in h15 . the x - ray observations in the period are available from _ swift_-bat transient monitor results by the _ swift_-bat team @xcite . the data provided an upper limit in the 15 - 50 kev band . we adopt the same values for @xmath42 , @xmath10 , @xmath44 , and @xmath45 as those in the previous section . by changing @xmath24 , @xmath21 , and @xmath46 , we attempt to fit the spectrum of the flaring period b in 2013 . we consider one shell that contributes to the flare emission . the energy diffusion coefficient and injection rate are slightly increased from the values in the steady model to @xmath47 ( @xmath48 ) , and @xmath49 ( @xmath50 ) , respectively . hereafter , we call this the `` fiducial '' flare model . no significant concurrent flare in the optical bands implies that the optical photons are emitted from other steady components . as discussed in section 4.3 in h15 , the lack of overall correlation between the optical and gamma - ray bands in 2013 - 2014 also suggests the different origin of the optical component . the synchrotron flux of the flare should be below the observed flux level so that an upper limit for the magnetic field in the flare zone will be given . here , we adopt a very low value of @xmath51 g. figure [ fig : ele2 ] ( a ) shows the evolution of the electron energy distribution in this flare model . electrons are accelerated to higher energies compared to the case in the steady model . the power - law distributions above @xmath52 ev are due to not only the larger @xmath24 but also the inefficiency of the synchrotron cooling owing to the low magnetic field . the secondary bumps at @xmath53 ev for @xmath54-@xmath55 are attributed to the generation of secondary electron positron pairs via internal @xmath12 absorption . the resultant gamma - ray spectra shown in figure [ fig : ele2 ] ( b ) agree well with the observed gamma - ray data . here , we add an underlying component ( that overlaps the solid gray line in the figure ) to explain the gamma - ray flux before the flare and the optical data . the flare spectrum has a higher synchrotron peak energy than the model in h15 , since our flare model shows a drastic growth of the maximum energy of electrons compared to the steady model . as remarked above , the weak magnetic field strikingly increases the maximum energy of electrons . even for this low magnetic field , the optical flux is slightly enhanced during the flare . the steady behavior of the optical light curve may prefer a weaker magnetic field , but we regard this as a conservative upper limit . the sharp cutoff at @xmath56 ev in the photon spectrum is due to the @xmath12 absorption inside the emission region . some fraction of photons above @xmath57 ev escape from the shell . the model flux at the @xmath40 gev band is still higher than the detection limit for erenkov telescopes . however , it should be noted that we have neglected the @xmath12 absorption after the escape from the shell . the absorption by the broad emission lines during propagation may greatly suppress the flux around @xmath40 gev . even for the same values of @xmath42 and @xmath10 as in the steady model , the light curve is well reproduced as shown in figure [ fig : lc ] . thus , the emission zones of the intense flare in 2013 and the active period in 2009 may be located at similar distances from the central engine . the observed asymmetric profile in the light curve is favorable for our simple one - shell emission - zone model . the strong cooling due to the external ic yields the rapid decay of the light curve . the evolutions of energy densities in figure [ fig : ened ] clearly show the energy input by the sa and rapid cooling just after the end of the acceleration . at @xmath58 , the energy density ratio of the magnetic field to electrons is quite low as @xmath59 . the observational constraints , of course , do not determine the model parameter uniquely . however , the essential parameter for determining the gamma - ray spectral shape is only @xmath24 in our model ( the role of @xmath21 is just normalizing the flux level , and the value of @xmath46 does not affect the gamma - ray spectral shape ) . in figure [ fig : comp ] , we compare several photon spectral models derived with different parameter sets . when we reduce @xmath24 by a factor of two from the fiducial model ( `` low-@xmath60 '' model : @xmath61 , @xmath62 , the others are the same ) , the peak photon energy does not reach 10 gev . conversely , we increase the diffusion coefficient as shown in the `` high-@xmath60 '' model ( @xmath63 , @xmath64 , @xmath65 , the others are the same ) . in this case , we need an even weaker magnetic field . the peak time of the light curve is delayed due to the lower @xmath21 compared to the fiducial model . in figure [ fig : lc ] , we shift the light curve by 1.5 hr earlier . other physical parameters ( @xmath10 , etc . ) of the jet were derived from the steady model . however , as h15 supposed , we also try to increase @xmath10 in our model . the initial radius should be increased as @xmath66 to keep the variability timescale . such an example ( `` high-@xmath10 '' model : @xmath67 , @xmath68 , @xmath69 , @xmath70 , @xmath24 is the same ) is shown in figure [ fig : comp ] . due to the relatively short @xmath71 ( @xmath72 ) , the maximum electron energy grows as high as @xmath73 ev . a very low magnetic field ( @xmath74 ) is necessary again to suppress the synchrotron flux in the x - ray band . the strong cooling due to the higher @xmath75 makes the gev spectrum too soft compared to the observed gamma - ray spectrum . the simple sa model can reasonably explain the very hard spectrum and short variability in the intensive flare in 2013 . the turbulence driving the particle acceleration may be generated by the hydrodynamical instability or the magnetic reconnection . compared to the steady model for the active period in 2009 , the drastic alteration we need is the decrease of the magnetic field . the other parameters have almost similar values . the absence of the optical flare implies the weak magnetic field ( @xmath76 g ) . the requirement of the magnetic field decrease at gamma - ray flare stages was suggested by @xcite as well . the required low magnetic field seems irrelevant to the energy source for the particle acceleration . therefore , a hydrodynamical instability is responsible for driving the sa . when @xmath28 , the variability timescale is consistent with @xmath77 pc as shown in figure [ fig : lc ] . this distance from the engine also agrees with the constraint by the x - ray ssc component in the active period in 2009 . the size of the central engine may be @xmath78 pc for the black hole mass of @xmath79 . if we adopt the simplest model for the jet acceleration due to the magnetic energy dissipation @xcite , the bulk lorentz factor at @xmath11 should be @xmath80 , which is inconsistent with the postulated value of @xmath10 . given the variability timescale @xmath81 , the initial radius can be scaled as @xmath82 when we change @xmath10 . however , even in this case , the maximum lorentz factor at @xmath42 inferred from the magnetic dissipation model increases by a factor of only @xmath83 . for the jet acceleration model by the poynting flux dissipation , not only the low magnetic field but also the short variability timescale are problematic . this problem is also raised for the very short gamma - ray flare ( a few hundreds seconds ) of bl lac objects like pks 2155304 @xcite . for such ssc - dominant objects , however , the distance from the engine is not well constrained compared to fsrqs . the tiny change of the diffusion coefficient @xmath24 , in spite of the drastic decrease of the magnetic field , seems enigmatic . the assumed value of @xmath15 may be favorable for this invariant behavior of @xmath24 . in this low magnetic field case , the average energy gain per scattering may be proportional to @xmath84 , where @xmath85 is the average turbulence velocity , rather than the alfvn velocity . the pitch angle diffusion approximation @xcite and power - law magnetic turbulence of @xmath86 , where @xmath87 is the wavenumber , leads to @xmath88 . the last factor of @xmath89 implies that electrons interact with higher ( lower ) amplitude turbulence at longer ( shorter ) wavelengths for a weaker ( stronger ) magnetic field . if @xmath90 , @xmath15 results in @xmath91 , which is independent of @xmath92 . alternatively , magnetic bottles as `` hard spheres '' @xcite may be formed in turbulence independently of the strength of the magnetic field . such requirements for the turbulence property motivate us to probe the hydrodynamical instabilities in blazar jets . the authors thank the anonymous referee for the useful comments . we also thank f. takahara , m. kusunose , k. toma , j. kakuwa , k. nalewajko and g. m. madejski for useful discussion . this study is partially supported by grants - in - aid for scientific research no.80399279 from the ministry of education , culture , sports , science and technology ( mext ) of japan . abdo , a. a. , ackermann , m. , ajello , m. , et al . 2010 , , 710 , 810 abdo , a. a. , ackermann , m. , ajello , m. , et al . 2010 , , 463,919 ackermann , m. , ajello , m. , atwood , w. , et al . 2015 , arxiv:1501.06054 aharonian , f. , akhperjanian , a. g. , bazer - bachi , a. r. , et al . 2007 , , 664 , l71 asano , k. , takahara , f. , kusunose , m. , toma , k. , & kakuwa , j. 2014 , , 780 , 64 beresnyak , a. , yan , h. , & lazarian , a. 2011 , , 728 , 60 blandford , r. , & eichler , d. 1987 , phr , 154 , 1 baejowski , m. , blaylock , g. , bond , i. h. , et al . 2005 , , 630 , 130 brandenburg , a. , kahniashvili , t. , & tevzadze , a. g. 2015 , , 114 , 075001 chen , x. , fossati , g. , liang , e. p. , & bttcher , m. 2011 , , 416 , 2368 diltz , c. , & bttcher , m. 2014 , j. high ene . astrop . , 1 , 63 drenkhahn , g. 2002 , , 387 , 714 fossati , g. , buckley , j. h. , bond , i. h. , et al . 2008 , , 677 , 906 hayashida , m. , madejski , g. m. , nalewajko , k. , et al . 2012 , , 754 , 114 hayashida , m. , nalewajko , k. , madejski , g. m. , et al . 2015 , , 807 , 79 janiak , m. , sikora , m. , nalewajko , k. , et al . 2012 , , 760 , 129 kakuwa , j. , toma , k. , asano , k. , kusunose , m. , & takahara , f. 2015 , , 449 , 551 katarzyski , k. , ghisellini , g. , mastichiadis , a. , tavecchio , f. , & maraschi , l. 2006 , , 453 , 47 krawczynski , h. , coppi , p. s. , & aharonian , f. 2002 , , 336 , 721 krawczynski , h. , hughes , s. b. , horan , d. , et al . 2004 , , 601 , 151 krimm , h. a. , holland , s. t. , corbet , r. h. d. , et al . 2013 , , 209 , 14 kusunose , m. , takahara , f. , & li , h. 2000 , , 536 , 299 lazarian , a. , vlahos , l. , kowal , g. , yan , h. , beresnyak , a. , de gouveia dal pino , e. m. 2012 , space sci . , 173 , 557 lefa , e. , rieger , f. m. , & aharonian , f. 2011 , , 740 , 64 matsumoto , j. , & masada , y. 2013 , , 772 , l1 mizuno , y. , hardee , p. e. , & nishikawa , k. 2007 , , 662 , 835 pacciani , l. , tavecchio , f. , donnarumma , i. , et al . 2014 , , 790 , 45 stawarz , . , & petrosian , v. 2008 , , 681 , 1725 wehrle , a. e. , et al . 1998 , , 497 , 178 zrake , j. 2014 , , 794 , 26
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the very short and bright flare of 3c 279 detected with _ fermi_-lat in 2013 december is tested by a model with stochastic electron acceleration by turbulences .
our time - dependent simulation shows that the very hard spectrum and asymmetric light curve are successfully reproduced by changing only the magnetic field from the value in the steady period .
the maximum energy of electrons drastically grows with the decrease of the magnetic field , which yields a hard photon spectrum as observed .
rapid cooling due to the inverse - compton scattering with the external photons reproduces the decaying feature of the light curve .
the inferred energy density of the magnetic field is much less than the electron and photon energy densities .
the low magnetic field and short variability timescale are unfavorable for the jet acceleration model from the gradual poynting flux dissipation .
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the problem of obtaining additional physical effects only due to multiple repetitions of the same measurement or interaction has been discussed both analytically @xcite and experimentally @xcite . these phenomena , in which one may preserves in time an initially prepared state or even `` guide '' its time evolution to another final predetermined state @xcite in contrast to the known rules of quantum mechanics by which the result of measurement can not be known beforehand , are collectively termed quantum zeno effect @xcite . they were discussed exclusively at the level of either the schroedinger equation @xcite , or by using the density matrix @xcite . we show in this work , using specific examples , that these phenomena may be found also in the context of quantum field theory . moreover , it has been shown @xcite , using the spin example , that these repetitions not only preserves or guide to some predetermined state but also may result in entirely new effects as will be explained . we show , using quantum field theory , that this is indeed the case and not only in the spin case . we show this for the two most discussed cases in relation to the many body problem in quantum field theory @xcite : 1 ) the many body system in which the constituent particles are not interacting with one another , but are submitted to an external potential @xmath1 , and 2 ) the many body system in which the constituent particles are interacting with one another . in both cases the single particle propagator can be represented by an infinite series from which we can get the energies and the lifetime of the relevant system @xcite . in the expression `` single particle propagator '' we mean especially the specific green function @xmath2 which is the probability amplitude that if at the time @xmath3 we add a particle in state @xmath4 to the system in its ground state , then at the time @xmath5 the system will be found in its ground state with an added particle in the state @xmath6 @xcite . the propagator @xmath2 is termed the `` dressed '' or `` clothed '' propagator to differentiate it from the free ( bare ) propagator @xmath7 which has the same meaning of a probability amplitude as that of @xmath2 , but with no perturbing interaction resulting from either an external potential or from some interaction among the particles composing the system . we remark that the `` clothed '' propagator is conventionally estimated @xcite by summing to an infinite order over some selective series which is always characterized by the same basic diagram ( from a very large number of possible diagrams ) repeated to all orders . from the sum over this series one derives physical results like the ground and excited energy states of the system @xcite . that is , the physical phenomena appear after summing to infinite order over this set of series of repetitions of the same diagram . there exists a large number of examples corroborating this . the known hartree @xcite and hartree - fock @xcite quantum field realizations of physical phenomena are the results of summing to an infinite order over only the same repeated diagram . that is , over only the bubble terms @xcite in the first case , and over only the bubble and open oyster terms in the second case @xcite . likewise , the random phase approximation method ( rpa ) is based upon summing over only the terms called the ring terms @xcite . the basic phonon relations are derived @xcite from summing to an infinite order over only the same repeated ( to all orders ) process which represents the einstein constant frequency phonon . the plasmon characteristics have been derived by summing over only the `` pair bubbles '' terms @xcite . even the two particle propagator is handled by summing over only what is termed the ladder terms @xcite . for all the above and many other cases this summing over the same repeated process results in a new particle , the quasi particle @xcite , with a characteristic energy , an effective mass , and a finite lifetime . these infinite repetitions over the same process dress the initial `` bare '' particle and transform it to another one with different energy , mass , and lifetime . we will show in section 3 that if we have no repetitions then we have also no quasiparticles and no excited energy states . thus , according to the previous discussion , the starting point will not be the general series which is not summable @xcite , but a selective series which is generally a series of only one process ( from actually a very large number of possible processes ) and all its different orders . here , in order to emphasize this element of repetition and its essential role in the formation of the zeno effect @xcite we discuss a special version of the last series in which the terms of these series are not all the orders of the once performed relevant interaction , but _ all the orders of the @xmath0 times repetitions of it _ , as will be explained in the following sections . also , using the bubble and open - oyster examples we illustrate the aharonov - vardi conclusion @xcite , with respect to spin rotations , that even when the physical mechanisms ( potentials and interactions ) , that cause the time evolutions of the physical systems , are absent , nevertheless , the large number of repetitions of the `` measurement '' of the corresponding observables induces this type of time evolution . in our case we obtain , by these repetitions , an induced continuous spectrum of excited state energies in a finite interval . in section 2 use is made of the vacuum amplitude @xmath8 @xcite and the unique nature of the zeno effect @xcite to show this effect for the bubble process @xcite , and for the general unlinked diagram with @xmath0 identical links @xcite . in section 3 the zeno effect is shown also for the case in which the initial and final states of the system are different . this is demonstrated for the specific open - oyster process @xcite , and for the general case of different initial and final states of the system in which the amplitude has a value greater than unity . the vacuum amplitude , as defined in the literature ( see , for example , @xcite ) , takes into account all the various processes that lead from the ground state , back to the same state . here , in order to discuss the zeno effect @xcite which is characterized by a large number of repetitions of the same process , we adopt a restricted vacuum amplitude formalism that involves repetitions of only one particular process . as we have pointed out , the hartree and hartree - fock procedures , for example , belong to this category . as mentioned , our basic diagram is the @xmath0 times repetitions of the process that begins and ends at the same state , where in the limit of dense measurement @xmath0 tends to be a very large number . that is , this basic diagram is , actually , composed of @xmath0 identical parts . thus , the terms of the infinite series representing the vacuum amplitude must signify the different orders of this basic @xmath0-times - repeated interaction . the first term of this infinite series is the free term when no interaction occurs in the time interval @xmath9 ( we specify the initial time by @xmath10 ) . the value of this first term of the vacuum amplitude is unity @xcite , since it expresses the fact that in the unperturbed case the probability amplitude for the quantum system to stay in its ground state is unity . the second term denotes the basic diagram , just described . the third term denotes the probablity when this @xmath0-times - repeated interaction is performed twice in the time interval @xmath9 etc . as an example for this process we take the bubble interaction @xcite in which an external potential lifts the system at the time @xmath11 out of its initial state @xmath12 creating a hole , and instantaneously puts it back in , destroying the hole . in the energy - time representation the probability amplitude for the occurence of the bubble process is given by @xcite @xmath13 where @xmath14 is the external potential that transmits the system from the state @xmath12 back again to the same state @xmath12 . @xmath14 does not depends on @xmath11 so it can be moved out of the integral sign in eq ( [ e1 ] ) . the point correlation function @xmath15 is the probability amplitude that at the time @xmath3 a hole in state @xmath12 has been added and instantaneously removed ( destroyed ) from the system in its ground state @xcite . the value of @xmath15 is -1 ( see @xcite ) . the minus sign in eq ( [ e1 ] ) is for the fermion loop @xcite of the bubble process . the integration time from @xmath10 to @xmath11 is the time it takes this process to occur . if this bubble interaction is repeated @xmath0 times over the same total finite time @xmath9 , we obtain for the probability amplitude to find the system at time @xmath11 to have the same state it has at time @xmath10 @xcite @xmath16v^n_{ll } \nonumber \end{aligned}\ ] ] where @xmath17 is the dyson time ordered product operator @xcite . the division by @xmath18 is because we take into account all the possible orders of the times @xmath19 . here each @xmath20 have the same constant value ( of @xmath21 as we have remarked ) , so we obtain from the equation ( [ e2 ] ) @xmath22 the last equation is the probability amplitude to find the system at the time @xmath11 , after it has been interacted upon @xmath0 times by the same bubble interaction , to have the same state it has at the time @xmath10 . now , as we have mentioned we must take into account all the possible orders of this @xmath0 times repeated interaction . if , for example , this @xmath0-th order interaction is repeated two , three , and four times over the same finite total time @xmath9 , we obtain for the relevant probability amplitudes @xmath23 , @xmath24 , and @xmath25 respectively . the divisions by @xmath26 , @xmath27 , and @xmath28 take into account the possible time orders among these @xmath0-th order interactions ( repeated two , three , and four times ) besides the extra @xmath18 times permutations for each such @xmath0 times repeated interaction . we note that since , as we have remarked , each such @xmath0-th order interaction is treated as _ the basic interaction _ its @xmath0 parts are not time permuted with the @xmath0 parts of any other identical basic interaction . repeating this @xmath0th order bubble process @xmath0 times , and taking the former equations into account we obtain for the probability amplitude ( denoted by @xmath29 ) to find the system in the time @xmath11 to be in the same state it was in the initial time @xmath10 . @xmath30 we are interested in showing the existence of the zeno effect in the limit of dense measurement , that is , of a very large @xmath0 . we obtain @xmath31 that is , the probability to remain with the initial state after all these interactions is unity which is the zeno effect @xcite . we can generalize from the specific bubble interaction to a general one . the only condition this general interaction has to fulfil is to start and end at the same state , so that when it is repeated @xmath0 times , the resulting @xmath0-th order diagram is composed of @xmath0 unlinked identical links . now , it is known @xcite that the value of an unlinked diagram with @xmath0 unlinked links @xmath32 is @xmath33 , no matter what is the character of @xmath32 . thus , denoting our fundamental generalized interaction by @xmath32 , and repeating the same process , as we have done for the bubble interaction , we obtain the following vacuum probability amplitude @xmath34 ( to start and end at the same state ) in the zeno limit @xmath35 that is , the quantum zeno effect may occur in the framework of quantum field theory . this derivation is general in that we do not have to specify the fundamental repeated interaction @xmath32 . the same conclusion can also be obtained by considering the ground state energy of the perturbed system which is obtained by using the vacuum amplitude from eq ( [ e6 ] ) . this ground state energy is obtained from the following relation , known as the linked cluster theorem @xcite @xmath36 where @xmath37 is the ground state energy of the unperturbed hamiltonian corresponding to the unperturbed ground state @xmath38 which is assumed to be the initial state of the system , and @xmath39 is a positive infinitesimal such that @xmath40 , and @xmath41 for any finite @xmath42 . @xmath8 , in our case , is the @xmath43 from eq ( [ e6 ] ) . one sees from the general linked cluster expansion given , for example , by mattuck ( in @xcite p. 110 ) that the expansion ( [ e6 ] ) results from including only the bubble contribution . thus , substituting in eq ( [ e7 ] ) for @xmath8 ( @xmath43 from eq ( [ e6 ] ) ) we obtain @xcite @xmath44 thus , we see that in the zeno limit the initial energy ( the initial state ) is preserved . this is true for any general process @xmath32 , such that when repeated @xmath0 times the value of its @xmath0 unlinked parts diagram ( we are restricted here to the vacuum amplitude case ) is @xmath33 . all we have to do is to use the general @xmath43 from eq ( [ e6 ] ) , and eq ( [ e7 ] ) . the result we obtain is identical to eq ( [ e8 ] ) . all our discussion thus far of the bubble zeno effect uses the vacuum amplitude , and so is restricted to the case where the initial and final states of the system were the ground state . we generalize now to any other state and take into account explicitly the unperturbed propagators which connect neighbouring interactions . here also our basic unit is , because of the zeno effect , the @xmath0-times - repeated bubble interaction . this general bubble process is now more natural than the former , since each bubble interaction is naturally related to the former and to the following identical ones by connecting paths which are the free propagators @xmath45 defined as the free propagation of the system from the time @xmath3 to @xmath5 without any disturbance whatever . thus , in order to accomodate to this situation we have to multiply each fundamental bubble process given by eq ( [ e1 ] ) by the free propagators @xmath46 and @xmath47 , the first leads from the initial time @xmath10 to the time of the interaction @xmath3 and the second from @xmath3 to the time after the interaction @xmath5 , so that eq ( [ e1 ] ) would be written as @xmath48 where @xmath49 is the initial and final state of each such fundamental bubble process . the interaction is denoted now by @xmath50 that signifies that our system begins and ends at the same state @xmath49 , creating and destroying a hole in state @xmath12 ( if the system interacts only with an external potential then this interaction is denoted by @xmath51 as is done for the vacuum amplitude case ) . @xmath50 is a probability amplitude that does not depend on time and is given by @xcite @xmath52 and @xmath53 has the same meaning as in the former case . the free propagator @xmath47 has the following value @xcite @xmath54 with @xmath55 substituting from eq ( [ e10 ] ) into eq ( [ e9 ] ) we obtain @xmath56 now , since we deal with identical repetitions of the same interaction all the @xmath50 s are the same . also all the @xmath57 s are , for the same reason , identical to each other moreover , we can take also the time differences @xmath58 , especially for large @xmath0 , to be the same . thus , taking these considerations into account , we write the relevant modified form of eq ( [ e2 ] ) as follows @xmath59\int_{t_0}^t\int_{t_0}^{t_1 } \ldots \int_{t_0}^{t_{n-1 } } \cdot \nonumber \\ & & \cdot e^{-i\epsilon_k(t_n - t_0 ) } dt_1dt_2 \ldots dt_n= ( v_{klkl})^n\int_{t_0}^t\int_{t_0}^{t_1 } \ldots \int_{t_0}^{t_{n-2}}(\frac{e^{-i\epsilon_k(t_{n-1}-t_0)}}{-i\epsilon_k}- \frac{1}{(-i\epsilon_k)})dt_1dt_2\ldots dt_{n-1}= \nonumber \\ & & = ( v_{klkl})^n ( \frac{e^{-i\epsilon_k(t - t_0)}}{(-i\epsilon_k)^{n-1}}-\frac{1}{(-i\epsilon_k)^{n-1 } } -\frac{(t - t_0)}{(-i\epsilon_k)^{n-2}}- \frac{(t - t_0)^2}{(-i\epsilon_k)^{n-3}}-\ldots)= \label{e12 } \\ & & = ( v_{klkl})^n(\frac{e^{-i\epsilon_k(t - t_0)}}{(-i\epsilon_k)^{n-1}}- \sum_{m=0}^{n-1}\frac{(t - t_0)^m}{m!(-i\epsilon_k)^{n-1-m } } ) \nonumber \end{aligned}\ ] ] here , we have taken @xcite @xmath60 . expanding the exponent @xmath61 in a taylor series we obtain from the last equation @xmath62 the left hand side of figure 1 shows the @xmath0 times repetitions of the bubble process which is represented as a circle . these unconnected repetitions conform to eq ( [ e2 ] ) . the right hand side of the figure shows these @xmath0 times repetitions connected by leading paths , and so they conform to eq ( [ e13 ] ) . we note that since what interests us in this work is the limit of very large @xmath0 of these @xmath0-times repeated interactions , represented by equations ( [ e12])-([e13 ] ) in this section and eq ( [ e28 ] ) in the following one , these @xmath0 multiple interactions are to be regarded as one connected unseparated process ( see the discussion before eq ( [ e4 ] ) ) and not as repetitions over improper self energy parts @xcite , so we can use the following dyson s equation @xcite as we have done in equations ( [ e18 ] ) , ( [ e29 ] ) and ( [ e34 ] ) . @xmath63,\ ] ] where @xmath17 is the dyson s time ordered product . the right hand side of eq ( [ e14 ] ) is generally used because the @xmath64 s do not commute . here the @xmath64 s take numerical values ( see equations ( [ e12 ] ) , ( [ e13 ] ) , and ( [ e28 ] ) ) , and so we do not have here any commutation problems . thus , the @xmath65 from eq ( [ e11 ] ) , for example , could have been written and substituted in eq ( [ e12 ] ) as @xmath66 now , we have to take into account all the possible orders of the @xmath0 times repeated interaction process given by eq ( [ e12 ] ) . for example , the second order process , is @xmath67 and the @xmath0th order process @xmath68 where the expression @xmath69 contains @xmath0 terms . we want to demonstrate the zeno effect in the dense measurement limit , that is , for very large @xmath0 . so , repeating this @xmath0th order bubble interaction to all orders , taking the former equations into account , adding and subtracting 1 , and using the dyson s equation we obtain for the probability amplitude to find the system at time @xmath11 in the same state it was at the initial time @xmath10 ( compare with eq ( [ e5 ] ) ) @xmath70 the last outcome is obtained by using the last results of equations ( [ e12 ] ) and ( [ e13 ] ) from which we obtain @xmath71 . @xmath72 is the probability amplitude to begin and end at the same state without any interaction . this no - interaction process , like the basic bubble interaction discussed here , is an @xmath0-times - repeated process . that is , @xmath72 is the @xmath0 times repetitions of the free propagator given by eq ( [ e10 ] ) , so that the time allocated for each is @xmath73 . thus , @xmath72 , with the help of eq ( [ e10 ] ) and in the zeno limit where @xmath74 , is @xmath75 from equations ( [ e18])-([e19 ] ) we obtain for the zeno limit of the probability of the bubble process @xmath76 that is , in the limit of the zeno effect we obtain for the bubble process , when it is represented by either eq ( [ e1 ] ) ( in the vacuum amplitude case ) or by the more general eq ( [ e9 ] ) , a probability of unity to begin and end in the same state . we must again note that taking into account only the bubble process , from the large number of possible different processes , is the earlier hartree method @xcite of dealing with the interacting many body system . but unlike this hartree point of view in which the bubble interaction is taken once to all orders , here in order to emphasize the important role of these identical repetitions to the zeno effect this bubble interaction is taken @xmath0 times to all orders where @xmath77 . now , we discuss the other ( excited ) states of the system . the conventional procedure that yields the excited state energies is to find the poles of the propagator @xmath78 @xcite which is the fourier transform of the propagator @xmath79 . the last propagator is the probability amplitude to find the system at the time @xmath11 , _ after interaction _ , in the same state it has started from at the time @xmath10 , and it is , for the zeno process , no other than the @xmath80 we found in eq ( [ e18 ] ) . thus , we must transform this equation from the @xmath81 representation to the @xmath82 one . we do this by finding the @xmath82 representation of @xmath72 from eq ( [ e19 ] ) using the fourier transform method @xmath83 the @xmath84 in the exponent comes from multiplying by @xmath85 , where @xmath84 is an infinitesimal satisfying @xmath86 , and @xmath87 , ( @xmath88 is a constant ) @xcite . we do this in order to remain with a finite result for this exponent when @xmath89 . the @xmath90 is the @xmath0 times repetitions of the free propagator @xmath91 which is the @xmath82 representation of @xmath47 from eq ( [ e10 ] ) . we are interested in the limit of very large @xmath0 , and as seen from eq ( [ e21 ] ) when @xmath92 we , actually , have a pole for each value of @xmath93 that satisfies @xmath94 , that is , @xmath95 . there are no excited energies outside this range . we note that in the many body interaction picture the excited energy @xmath57 is equal to @xcite the difference between the excited state energy of the interacting @xmath96-particle system and the ground state of the interacting @xmath97-particle system . thus , if the bubble process is performed once and the selective series of this once performed process is summed to all orders , as in the hartree method , one obtains excited state energies at the value given by eq ( [ e23 ] ) . but when this bubble process is repeated @xmath0 times and the selective series of this @xmath0-times repeated process is summed to all orders , as we have just done in equation ( [ e12])-([e18 ] ) , we obtain from eq ( [ e21 ] ) excited state energies for all values of @xmath93 that satisfy @xmath94 . that is , we obtain a large number ( continuum ) of extra excited energies that has been added _ only because of these identical repetitions of the same bubble process_. this mechanism of obtaining physical results as a consequence of just repeating the same process which by itself , without these repetitions , does not yield these results has already been noted in @xcite in connection with rotations that occur only because of a large number of repetitions of the same measurement . speaking in terms of quasi - particles @xcite we can write the @xmath82 representation of @xmath98 from eq ( [ e18 ] ) , using eq ( [ e21 ] ) , as @xmath99 @xmath100 is the lifetime of the quasi - particle , and since @xmath84 is small @xmath100 is very large , so these quasi - particles with the extra excited energies just mentioned have a very large lifetime . we must note that the relevant excited state energies @xmath101 obtained when the bubble process is performed once and the selective series of this once performed process is summed to all orders is just the hartree @xmath101 @xcite . @xmath102 when the bubble process is repeated @xmath0 times , then as can be seen from equations ( [ e13]),([e18])-([e19 ] ) , and ( [ e21])-([e22 ] ) the @xmath101 s obtained do not depend on any potential @xmath1 . this , as we have remarked , is in accord with the aharonov - vardi conclusion @xcite that the physical mechanisms that trigger the time evolutions of the system does not play an essential role , since the mere large number of repetitions of the same measurement is the cause of this time evolution . we note that aharonov and vardi show this for the spin @xmath103 particle example , but it is obvious from their representation that this conclusion is a general one . we have shown this for the bubble process for which a large number of repetitions results in excited energies that do not depend upon any potential . we show in the next section that if we have no repetitions then we do not have any excited energies . in summary , we find that when the interaction involved does not end at the same state it has began from and if it is not repeated then no excited state results from such an interaction ( see eqs ( [ e31])-([e32 ] ) in the next section and the discussion that follows ) . if this interaction begins and ends at the same state as in the hartree model then a single pole @xmath104 is found ( see eq ( [ e23 ] ) ) . and when this interaction is repeated @xmath97 times then in the limit of @xmath105 one find a continuum of poles ( cut ) for all values of @xmath106 that satisfy @xmath107 , where @xmath108 is the energy by which the involved system propagates during the interaction . that is , as has been remarked in @xcite the large number of repetitions produces new stable physical effects ( see also eq ( [ e22 ] ) and the discussion that follows it ) that do not appear in the absence of them . and the more larger the number of these repetitions on the same time interval , as in the discussion here in which the repeated interaction is not taken by itself but by its @xmath97 time repetitions where @xmath105 , the more larger is the new stable physical effect as the cut found here ( see eqs ( [ e21])-([e22 ] ) and the relevant discussion there ) instead of the single pole of the hartree model . we note that the quasi - particles related to these poles have a very long lifetime so that once they are formed they do not decay fastly . we , now , show that we can apply the zeno effect @xcite also for the general case , where the system ends at the time @xmath11 in some specific state which is not identical to the initial one from which it has started at the time @xmath10 . in this context we do not use the standard zeno effect at a state ( where the system returns to the same state it has started from ) , as discussed in the previous section , but apply a zeno effect along some definite feynman path of possible states in the sense of aharonov and vardi @xcite . that is , if we do dense measurement along any definite feynman path of states then we make it actual in the sense that its probability amplitude is unity . here we begin at some predetermined initial state and end at another predetermined final one . this aspect of the quantum zeno effect in which the evolution of the relevant quantum system is _ guided _ , by means of dense measurement , to the corresponding prefixed final state is termed in @xcite the dynamical quantum zeno effect , in contrast to the usual quantum zeno effect ( in which the system starts and ends at the same state ) which is termed in @xcite the static quantum zeno effect . the propagator in this general case is the probability amplitude that if the system begins at the initial time @xmath10 in a specific state , then it will be found at another specific one at the later time @xmath11 . as in the former section , in order to emphasize the important role of repetitions for the zeno effect , the basic diagram is the @xmath0 times repetitions of this interaction , where in the limit of dense measurement @xmath0 becomes very large number . thus , the terms of the infinite series representing the propagator signify the different orders of this @xmath0-repeated - interaction . in this case the repetitions is along some definite path connecting the initial and final states , and not local repetition as in the bubble example . we choose , as in the bubble case , some example that may be described from two points of view . one is the situation when the interaction is triggered by an external potential that acts @xmath0 , @xmath109 , @xmath110 times etc . the other , more natural , interaction is that caused by the correlations between different particles that comprise the system . unlike the bubble case , in both points of view there must be a connecting path between any two neighbouring interactions since they are not identical to each other , as will be explained in detail later . here the initial state of each such interaction is not identical to the initial state of the former one , _ but to its final state_. the only difference between the external potential situation and the correlation - between - particles one is in the character of the interaction which in the former case is denoted by @xmath111 , that is , a particle that begins at state @xmath49 is interacted upon by an external potential that changes its state to that of @xmath12 ( compare with the external potential situation of the bubble case in which a particle begins and ends at the same state , and therefore the external potential is denoted by @xmath51 ) . in the correlation - between - particles situation this interaction is denoted by @xmath112 @xcite ( compare with the @xmath50 of the correlation - between - particles situation of the bubble case @xcite ) a fundamental interaction in which the system ends at the time @xmath11 in a state different from the one with which it has started from at the initial time @xmath10 is , for example , what is termed the open - oyster diagram @xcite . we must remark that this interaction is calculated to be @xcite as one in which the particle that left the interaction site at the later time @xmath11 is in the same state @xmath49 with which another particle enters the interaction site at the initial time @xmath10 . nevertheless , we discuss here another version of this interaction in which the particle that leaves the interaction site at the time @xmath11 is in the state @xmath113 , and not in the initial one @xmath49 . we also call this interaction open - oyster . in the external potential version of this interaction an incoming particle at state @xmath49 enters the potential region at the time @xmath10 . then at time @xmath11 the potential knocks another particle out of the state @xmath114 into state @xmath12 , thus creating a particle in state @xmath12 , and a hole in state @xmath114 . at the same time @xmath11 the particle in @xmath49 is knocked into the hole in @xmath114 , and thus annihilated with it . the particle in @xmath12 continues propagating out of the potential region . this process is referred to as an exchange scattering @xcite , compared to the forward scattering of the bubble process in which the particle emerges in the same direction ( i.e , momentum state ) as it has entered . on the right hand side of figure 2 we see this open - oyster interaction , and on the left hand side of it we see @xmath0 times repetitions of this process over the same time interval @xmath9 . in the energy - time representation the probability amplitude for the occurence of the open - oyster process is given by @xcite : @xmath115 the difference between the bubble process that may represent the static zeno effect @xcite ( when repeated a large number of times ) , and the open - oyster process , that may be regarded as an example of the dynamic zeno effect @xcite ( when performed many times ) , can be understood in the following way @xcite : suppose we have a family of states denoted as @xmath116 , where @xmath117 , such that @xmath118 , where @xmath119 is the initial state of the quantum system . we assume that successive states differ infinitesimally from one another , so that we have @xmath120 . denoting , as before , the total finite time of the @xmath0 repeated interactions by @xmath9 , and the time it takes to perform each such interaction by @xmath121 we have @xmath122 . now , the open - oyster interaction may be regarded as , actually , projecting the evolving wave function at the time @xmath123 on the state @xmath116 . so when @xmath0 becomes very large in the limit of the zeno effect we obtain actually @xmath124 . this is the dynamic zeno effect of @xcite . the static zeno effect is the special case when @xmath125 for all @xmath49 . if we describe this process in terms of the correlation between the different particles of the system then in this interaction an incoming particle in state @xmath49 performs in a simultaneous manner several tasks ; 1 ) it strikes another particle from state @xmath114 to state @xmath12 , 2 ) creates a hole in @xmath114 , 3 ) is annihilated with the hole in @xmath114 , and the particle in @xmath12 leaves the system . the open - oyster interaction is written now as @xmath126 now , since the last two equations ( [ e24 ] ) and ( [ e25 ] ) are identical to each other , except for the subscripts of the potential @xmath1 , we concentrate our attention on eq ( [ e25 ] ) with the understanding that what we say about it holds also for eq ( [ e24 ] ) . @xmath112 denotes the interaction just described , and the @xmath127 s are the free propagators given by eq ( [ e10 ] ) . we must note again that the successive repetitions of the open - oyster interaction , required for the discussion of the dynamic zeno effect , are not characterized as being identical to each other , as in the bubble process , but that each such fundamental interaction begins from the point ( state ) in which the former interaction ends . thus , we have to take into account the path that connects each two such neighbouring interactions . this connecting path is , of course , the free propagator @xmath128 . substituting now from eq ( [ e10 ] ) into eq ( [ e25 ] ) , and assuming that @xmath112 does not depend on @xmath11 we obtain @xmath129 where we have used the value of 1 for @xmath130 . using eq ( [ e25 ] ) we write for the @xmath0-th order open - oyster process @xmath131 where we have assumed that for large @xmath0 all the potentials that transfer the system between two neighbouring states are equal to each other , that is , @xmath132 . carrying out the @xmath0 integrals of the last equation we obtain an expression with @xmath133 terms , each of which is a fraction with a numerator that is a difference of exponentials in the energies @xmath134 s multiplied by the times @xmath135 , and the denominator is a multiplication of @xmath0 different factors . this @xmath133 terms expression can be grouped into @xmath0 different groups in which the number of terms are arranged as @xmath136 . all the terms of the same group have the same numerator up to a sign , but a different denominator , so we can reduce the number of all the terms of each group to 1 by taking the common denominator of all the terms that belong to the same group . in such a way the total number of terms of the original expression is reduced from @xmath133 to @xmath0 . thus , we obtain @xmath137 it can be seen that all the @xmath0 numerators of the last equation are differences of sines and cosines , whereas each one of the corresponding @xmath0 denominators is a product of @xmath0 factors that are differences of energies . when @xmath0 is very large , which we always assume in this work , we have @xmath138 ( since neighbouring states differ infinitesimally ) , so in this limit we have at least two factors in each denominator that tend to zero . thus , although all the @xmath0 terms of equation ( [ e28 ] ) are multiplied by the factor @xmath139 ( @xmath1 is a probability amplitude that satisfies @xmath140 ) we obviously have @xmath141 . we are interested , as in the bubble case , in the repetitions to all orders of @xmath142 from eq ( [ e28 ] ) . beginning from this equation it is not hard to obtain the various orders of @xmath142 . so , if we take the infinite series ( that denotes the various orders of the @xmath0 repetitions process @xmath142 ) , adding and subtracting 1 , and taking the relation @xmath141 into account we obtain , using the dyson s equation , for the general probability amplitude in the zeno limit @xmath143 @xmath144 is the amplitude for our system to begin in some specific initial state @xmath116 at the time @xmath10 , and end in another different state @xmath145 at the time @xmath11 without any interaction whatever on our system . this no - interaction process is obviously zero if the final state is different from the initial one ( see , for example , @xcite ) , so we obtain for the _ probability _ of the open - oyster process in the zeno limit @xmath146 thus , we see that in this limit we obtain for the open - oyster process a probability of unity to end at a specific prescribed state different from another specific initial one . we now show that we have no excited state energies for the open - oyster process in the zeno limit . for this purpose we must find , in this limit , the poles of the propagator @xmath147 which is the fourier transform of the propagator @xmath148 given by eq ( [ e29 ] ) . thus , using the fourier transform procedure , multiplying by @xmath149 @xcite , and using @xmath150 we obtain @xmath151 where the @xmath84 is , as in eq ( [ e21 ] ) ( see the discussion after eq ( [ e21 ] ) ) , an infinitesimal quantity that satisfies @xmath152 , and @xmath87 , where @xmath88 is some finite number . this @xmath84 has been introduced in order to have a finite result for the exponent of eq ( [ e31 ] ) in the limit @xmath89 ( see appendix i in @xcite ) . from the last equation we obtain that the poles of @xmath153 , which are the excited energy states of the physical system are @xmath154 that is , there exists no excited energy states in the zeno limit of the open - oyster process . the reason , as we have remarked , is the absence of local repetitions in the version we have adopted here for the open - oyster process . that is , we discuss here a process in which the state of the particle that leaves the system is different from the state of the one that enters . and when this process is repeated the initial state of the entering particle in the repeated process is the final state of the leaving particle in the former one . thus , this process is not locally repeated , and this absence of repetitions entails the absence of excited states for the system . that is , all the energies of the @xmath96-particle system are equal , in the zeno limit , to each other and to the ground state energy of the @xmath97-particle system ( see @xcite , p. 41 ) . in contrast to this situation , when we have local repetitions of some process , then we have excited states of the physical system . that is , if the selective series of this process is composed of repeated to all order terms like the hartree selective series of the bubble process , then excited states are obtained ( see eq ( [ e23 ] ) ) . many more additional excited states are obtained when this summation to all orders is over the @xmath0-times repetitions of this process as we have obtained for the bubble process in the former section ( see eq ( [ e21 ] ) ) . now , if we discuss this open - oyster process from the conventional point of view @xcite where the energy of the leaving particle is the same as that of the entering one , and the summation to all orders is over the once - performed open - oyster process and not over the @xmath0th times repetitions of it , then we obtain for the @xmath101 @xcite @xmath155 where @xmath112 is the physical interaction that generates this open - oyster interaction . that is , the excited state energies of the system are determined by these repetitions , as has been remarked in @xcite ( see the discussion after eq ( [ e21 ] ) ) we must note that the result of eq ( [ e30 ] ) is obtained not only for the open - oyster case , but also for any other arbitrary interaction for which the amplitude @xmath156 to ends in a specific state different from the initial one satisfies @xmath157 . if we denote the propagator ( the full propagator , not the free one ) of such interaction by @xmath158 , its free propagator by @xmath159 , and adding and subtracting 1 the propagator takes the following form @xmath160 in obtaining the result of eq ( [ e34 ] ) we made use of the facts that @xmath161 , and @xmath157 so that @xmath162 . we see , therefore , that also for the general case , where the system reaches at the time @xmath11 a different state from that in which it started , we get a probability of 1 in the dense measurement limit . thus , we see that the zeno effect @xcite may be effective in the framework of quantum field theory . we show that the zeno effect may be discussed also in the context of quantum field theory . we have used in section 2 the dyson s equation and the bubble example to demonstrate the static zeno effect , in which the initial and final states of the system are the same . in section 3 we have used the open - oyster example and the dyson s equation to demonstrate the dynamic zeno effect , in which the initial and final states of the system are different . in this work the dyson s equation has been used to infinitely sum to all orders over the @xmath0 times repetitions of these two processes . it has been shown in sections 2 and 3 that the probability amplitudes to find the final state of the system identical to the initial one in the bubble case , or different in the open oyster case tend both to unity as the number of repetitions @xmath0 becomes large . we have found in section 2 that repeating the bubble process a large number of times in a finite total time results in obtaining a large number ( cut ) of additional excited energy states that emerge only because of these repetitions . by this we have corroborated the same conclusion arrived to by aharonov and vardi with respect to spin rotation . we have found , accordingly , in section 3 for the open - oyster process that the absence of any repetition results in the absence of excited state energies . i wish to thank l.p.horwitz for discussions on this subject , and for his review of the manuscript 99 1 in b. misra and e. c. sudarshan , j. math . phys , * 18 * , 756 ( 1977 ) ; `` decoherence and the appearance of a classical world in quantum theory '' , d. giulini , e. joos , c. kiefer , j. kusch , i. o. stamatescu and h. d. zeh , springer - verlag , ( 1996 ) ; saverio pascazio and mikio namiki , phys . rev a * 50 * , 6 , 4582 , ( 1994 ) ; a. peres , phys . rev d * 39 * , 10 , 2943 , ( 1989 ) , a. peres and amiram ron , phys . rev a * 42 * , 9 , 5720 ( 1990 ) r. a. harris and l. stodolsky , j. chem . phys , * 74 * , 4 , 2145 ( 1981 ) ; mordechai bixon , chem . phys , * 70 * , 199 - 206 , ( 1982 ) ; marcus simonius , phys . rev . lett , * 40 * , 15 , 980 ( 1978 ) r. j. cook , physica scripta t * 21 * , 49 - 51 ( 1988 ) ; w. m. itano , d. j. heinzen , j. j. bollinger , and d. j. wineland , phys . rev a , * 41 * , 2295 - 2300 , ( 1990 ) ; a.g.kofman and g.kurizki , _ phys.rev a _ * 54 * , 3750 - 3753 ( 1996 ) ; g.kurizki , a.g.kofman and v.yudson , _ phys.rev a _ * 53 * r35 ( 1995 ) ; s.r.wilkinson , c.f.bharucha , m.c.madison , p.r.morrow , q.niu , b.sundaram , and m.g.raizen , _ nature _ * 387 * , 575 - 577 ( 1997 ) y. aharonov and m. vardi , phys.rev d * 21 * , 2235 , ( 1980 ) p. facchi , a. g. klein , s. pascazio and l. schulman , phys.lett a * 257 * , 232 - 240 , ( 1999 ) , richard . d. mattuck , @xmath163 edition , mcgraw - hill international book company , ( 1976 ) , @xmath163 edition , g. mahan , plenum press new york ( 1993 ) ; `` a course on many body theory applied to solid state physics '' , c. enz , world scientific , ( 1992 ) , by richard . d. mattuck , annals of physics _ 27 _ , 216 - 226 , ( 1964 ) richard . p. feynman , rev . phys,*20 * , 2 , 367 ( 1948 ) ; `` quantum mechanics and path integrals '' , richard . p. feynman and a. r. hibbs , mcgraw - hill book company ( 1965 ) , h. haken , north - holland publishing company , ( 1981 ) the diagrams that represent repetitions over improper self energy parts can be separated into unconnected self energy parts by removing the connecting particles lines . generally , it is known @xcite that such diagrams can not be summed over by using dyson s equation as we do below ( see , for example , equations ( [ e18 ] ) and ( [ e34 ] ) ) , since these improper diagrams would have to be counted more than once @xcite . , gert roepstorff , springer - verlag , ( 1994 ) , @xmath163 edition , e. merzbacher , john wiley and sons , ( 1961 ) i. m. gelfand and a. m. yaglom , j. math . phys * 1*. 1 , 48 - 69 , ( 1960 ) , volume 4 , i. m. gelfand and n. ya . vilenkin , academic press , 1964
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_ we show , using quantum field theory , that performing a large number of identical repetitions of the same measurement does not only preserve the initial state of the wave function ( the zeno effect ) , but also produces additional physical effects .
we first demonstrate that a zeno type effect can emerges also in the framework of quantum field theory , that is , as a quantum field phenomenon .
we also derive a zeno type effect from quantum field theory for the general case in which the initial and final states are different .
the basic physical entities dealt with in this work are not the conventional once - perfomed physical processes , but their @xmath0 times repetition where @xmath0 tends to infinity .
we show that the presence of these repetitions entails the presence of additional excited state energies , and the absence of them entails the absence of these excited energies . _
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the recent advance in radioactive nuclear beam experiments provides us with novel information on unstable nuclei far away from the stability line ( tanihata et al . 1985 ) . the symmetry energy becomes more essential to understand the nuclear structure as we go further away from stability . the symmetry energy is also important to provide the equation of state ( eos ) for neutron stars and supernovae . the properties of dense matter under neutron rich environment determine the structure and chemical composition of stars and may change their evolution drastically . therefore , it is very interesting to study the influence of the symmetry energy on astrophysical problems . the effects of the symmetry energy on the evolution of neutron stars and supernovae have been studied by several authors . the possibility of the rapid cooling of neutron stars due to the direct urca process has been discussed in the case of the large proton fraction of neutron star matter , which is sensitive to the density dependence of the symmetry energy ( boguta 1981 ; lattimer et al . 1991 ) . bruenn ( 1989 ) studied the effects on supernova explosions systematically by doing numerical simulations of gravitational core collapse with the parameterized eos . swesty et al . ( 1994 ) also studied the role in the prompt phase of the supernova explosion . however , there has been no systematic study on the effect of the symmetry energy on the birth of neutron stars and supernova neutrinos as far as we know . moreover , in most of previous studies , the parameterized formula have been used to provide the eos of dense matter beyond the normal nuclear matter density . recently , there has been a great progress in the study of nuclei and dense matter within the relativistic many body framework ( serot & walecka 1986 ) . it was demonstrated that the relativistic brueckner hartree fock ( rbhf ) theory is capable of reproducing the saturation property of nuclear matter starting from the nucleon - nucleon interaction determined by the scattering experiments ( brockmann & machleidt 1990 ) . the relativistic mean field ( rmf ) theory has been shown to be very successful as an effective theory to describe the ground state properties of nuclei in the wide mass range of the periodic table ( gambhir et al . 1990 ) and has been applied to the eos for neutron stars ( serot & walecka 1986 ) . it is amazing that the rmf theory describes also the properties of unstable nuclei away from stability extremely well ( hirata et al . 1991 ; sugahara & toki 1994 ) . having the framework constrained by unstable nuclei , sumiyoshi and toki applied the same rmf theory to provide the data table of the eos for neutron stars and supernovae ( sumiyoshi & toki 1994 ; sumiyoshi et al . 1995 ) , which enables us to do numerical simulations of the thermal evolution of neutron stars and supernovae quantitatively while taking care of the experimental data of unstable nuclei . the properties of unstable nuclei have been shown to be very sensitive to the symmetry energy in the rmf theory ( sumiyoshi et al . hence , it would be nice if we could see how sensitive are the properties of neutron stars to the symmetry energy and what is the influence of the symmetry energy on the birth of neutron stars and supernova neutrinos . the numerical simulations of the cooling of protoneutron stars and neutrino burst have been done by several groups ( burrows & lattimer 1986 ; burrows 1988 ; suzuki 1993 ) . the influence on the numerical simulation of the birth of neutron stars due to the difference of eos was studied by adopting the eos tables in the two different many body frameworks ( sumiyoshi et al . the effects of various thermodynamical properties were pointed out there besides the stiffness of eos , which has been mainly studied so far ( burrows 1988 ) . here , we focus on the influence of the symmetry energy by changing solely the strength of the isovector interaction in the rmf theory in order to see its effect clearly . the purpose of this paper is to explore the influence of the symmetry energy on the cooling of protoneutron stars just born in supernova explosions . we construct the tables of eos for supernova simulations in the rmf theory and make comparisons of eos s when we change the symmetry energy drastically . then , we perform numerical simulations of the birth of neutron stars and supernova neutrinos adopting the two eos s with different symmetry energy . we investigate the influence of the symmetry energy on the thermal evolution of protoneutron stars and the properties of neutrino burst emitted during the cooling stage . this paper is arranged as follows . in section [ s : eos ] , we briefly describe the relativistic eos for neutron stars and supernovae . in section [ s : results ] , after a short introduction on the birth of neutron stars , we present the results of our numerical simulations . we discuss the effects of the symmetry energy on the deleptonization , the thermal evolution and the supernova neutrinos in the subsections . we summarize this paper in section [ s : summary ] . we start with a brief explanation of the tables of eos for numerical simulations . we calculate all the physical quantities of dense matter within the relativistic mean field theory . we refer the review article by serot & walecka ( 1986 ) as for the relativistic many body framework for nuclei and dense matter . all the details on the relativistic eos for neutron stars and supernovae in the rmf theory has been reported in the recent papers . ( sumiyoshi & toki 1994 ; sumiyoshi et al . 1995 ) . we adopt the phenomenological lagrangian with the non - linear @xmath0 and @xmath1 terms , which is motivated by the recent success of the rbhf theory ( brockmann & machleidt 1990 ) and has been shown to be very successful both for nuclear properties and dense matter ( sugahara & toki 1994 ) . the best parameter set for the lagrangian , named tm1 , was determined by the least square fitting to a set of nuclei including unstable ones , which is important to constrain the isovector interaction in the theory . it is remarkable that the rmf theory with the parameter set tm1 has been demonstrated to reproduce successfully the properties of unstable nuclei other than the ones used in the fitting . the properties of nuclear matter in the rmf theory with tm1 thus constrained has been shown to be quite similar to the properties of nuclear matter derived in the rbhf theory ( sugahara & toki 1994 ) . extending the rmf theory to the case at finite temperature , the table of the numerical data of physical quantities under various conditions of chemical composition , temperature and density , which are required for the numerical simulations , was constructed for the parameter set tm1 ( sumiyoshi et al . we use this table of eos as a standard one . in order to explore the influence of the symmetry energy , we newly construct the table of eos with a reduced value of the symmetry energy in the rmf theory . we reduce the value of the coupling constant @xmath2 between isovector - vector @xmath3 meson and nucleon in the lagrangian , which is essential to determine the symmetry energy , while keeping other parameters of tm1 unchanged . hereafter , we call the modified parameter set as tms . the symmetry energy is @xmath4mev for tm1 and @xmath5mev for tms : the corresponding coupling constant is @xmath6 for tm1 and @xmath7 for tms . we note that the symmetry energy in tm1 has been checked by unstable nuclei and this modification in tms is to explore the effect of the symmetry energy on astrophysical applications . we display in fig . [ f : epb ] the energy per baryon of symmetric nuclear matter and pure neutron matter for the cases of tm1 and tms . the incompressibility k at the normal nuclear matter density is 281mev for both cases , since the isovector meson does not contribute to symmetric nuclear matter . we comment here on the density dependence of the symmetry energy . the symmetry energy for both cases of tm1 and tms has a monotonically increasing feature with density , which is common for all the relativistic many body calculations due to the contribution of the isovector meson . this feature is demonstrated by the microscopic calculation with the use of the rbhf theory ( li et al . the reduction of the symmetry energy at the saturation density by changing the strength of the isovector interaction clearly corresponds to the reduction of the symmetry energy at high density . in contrast , the symmetry energy in the non - relativistic many body calculations such as the one by wiringa et al . ( 1988 ) has generally a weak density dependence and has sometimes a decreasing feature at high density . a variety of the density dependence depends on the choice of the density dependent potential , which is introduced to reproduce the saturation of nuclear matter . therefore , the relation between the symmetry energy at the saturation density and its behavior at high density is ambiguous in non - relativistic many body calculations . when we apply the eos s in the two cases to neutron stars , we found that the chemical compositions are very different while the hydrostatic structures are quite similar . in fig . [ f : eos]-a , we display the proton fraction of neutron star matter , which is the ratio between the proton density and the baryon density , as a function of the baryon mass density . the proton fraction in the case of tms is smaller than the case of tm1 because of the reduced symmetry energy . we show in fig . [ f : eos]-b the neutron star mass as a function of the central baryon mass density . in contrast to the difference in the chemical composition , the neutron star masses in the two cases are found very similar . the maximum mass turns out to be almost the same value of 2.2m_. the difference in the central baryon mass density is only about 1% for the case of neutron stars with the gravitational mass of 1.4m_. first of all , we describe briefly the supernova explosions and the birth of neutron stars ( suzuki 1994 ) . gravitational collapse of the core of a massive star leads to explosion of the envelope ( supernova explosion ) and formation of the neutron star . we shall divide the series of stages starting from the onset of the core collapse to the birth of neutron star into two phases . the first phase is the dynamical phase . the core of a massive star becomes unstable when it grows to the point near the chandrasekhar mass ( @xmath8 1.4 m _ ) , and it begins to collapse . the collapse never ceases until the central density exceeds the nuclear density . sudden stiffening of the eos above the nuclear density stops the inner core ( 0.5 0.8 m _ ) and the bounce of the inner core launches a shock wave into the falling outer core . the matter of the falling outer core is swept and decelerated by the shock wave and , then , accretes onto the unshocked inner core which has been in hydrostatic equilibrium in its dynamical time scale of milliseconds . the shock wave expels the envelope and we identify it as supernova explosion . the remnant at the center which consists of the unshocked inner core and the shocked outer core is called as a protoneutron star . it contains so many leptons and protons ( @xmath8 30% ) that we can not call it a neutron star at this stage . it takes only about 1 second for this dynamical process to take place . the second phase , which is the quasistatic phase of protoneutron star cooling , follows this dynamical phase . after the shock wave breaks out of the core surface and the accretion onto the inner core ceases , the protoneutron star evolves quasistatically keeping hydrostatic configuration . neutrinos which are trapped in the core diffuse out of the protoneutron star in a time scale of neutrino diffusion , which is of order of 10 seconds . they drive the evolution of the protoneutron star into the cold neutron star by carrying out thermal energy and lepton number from the protoneutron star . concerning supernova neutrinos , about a half of the total energy is emitted during the dynamical phase and the rest is emitted during the quasistatic phase . in this paper , we study the influence of the nuclear symmetry energy on the second stage : the evolution of the hot protoneutron star into the normal neutron star . the quasistatic cooling of the protoneutron star is simulated numerically by solving the general relativistic equations for hydrostatic structure ( oppenheimer - volkoff equation ) and for neutrino transport with the deleptonization and the entropy change of the matter simultaneously using the henyey - type method . as for the neutrino transfer , we adopt the multigroup flux limited diffusion scheme ( bruenn 1985 ) . we take into account the energy dependence of neutrino transport coefficients in the multigroup scheme . the flux limiter should be introduced in order to express the neutrino flux in the transparent regime in terms of the diffusion flux . we adopt mayle and wilson s flux limiter ( mayle et al . 1987 ) in this work . we include the general relativistic effects such as the time dilation and the red shift of the neutrino energy . we treat explicitly , _ e , |_e and @xmath9 , where @xmath9 represents the average of _ , | _ , _ and |_. this is a good approximation in the case where we can neglect the existence of @xmath10 and @xmath11 leptons because of the low temperature ( @xmath12100mev ) . the following neutrino interactions are included as opacity sources or collision terms in the neutrino transfer equations . [ cols= " < , < , < , < , < , < " , ] where @xmath13 represents all species of neutrinos , a is a representative heavy nucleus , and n is either a proton or a neutron . most of the interaction rates are taken from bruenn ( 1985 ) with some modifications . at present , many body effects on neutrino opacity are not included except for the multiple scattering suppression effects on nucleon bremsstrahlung process ( raffelt & seckel 1991 ) . the evolution of the protoneutron star is driven by the exchange of energy and lepton number between the matter and neutrinos due to the above interactions and the neutrino transport . descriptions of our numerical code are also given in suzuki ( 1993 ) . we perform numerical simulations of the protoneutron star evolution using the tables of eos in the rmf theory with tm1 and tms . as for the eos of the low density matter ( @xmath14 ) , wolff s eos ( hillebrandt & wolff 1985 ) is used for both cases . we construct the initial models for our numerical simulation by referring to the numerical results of mayle and wilson at 0.4sec after the core bounce in the hydrodynamical simulations of the supernova explosion ( mayle & wilson 1989 ; wilson 1990 ) . initial models for the simulations using the two eos s are constructed using the same entropy profile and the same electron fraction profile in order to study the effects of the difference in the symmetry energy . of course , in principle , these profiles should differ from each other because the difference of eos should also affect on the dynamical phase . to extract the direct influence of the nuclear symmetry energy on the protoneutron star evolution , we neglect its influence on the dynamical phase . the calculated initial hydrostatic structures of the two protoneutron stars are found similar ; the differences in the initial densities and the temperature profiles are smaller than 3% . on the contrary , there is a large difference in @xmath15 which is the chemical potential of _ e in the @xmath16-equilibrium with the matter because @xmath17 and @xmath18 are directly affected by the symmetry energy . in figs . [ f : entropy][f : mu ] , we present the profiles of the two initial models . @xmath19 at the center of the initial models are 142mev ( @xmath20 ) for tm1 and 170mev for tms ( @xmath21 ) , respectively . the resultant lepton fractions , @xmath22 are 0.330 for tms and 0.315 for tm1 . starting from these initial models we simulate the evolution of the protoneutron stars for the following 15 seconds with the same numerical code . figs . [ f : entropy][f : mu ] also show the profiles of the protoneutron stars at the end of the calculation ( @xmath23 ) for the two models . while the density profiles are still nearly identical , the distributions of the temperature and the electron fraction differ largely from each other . especially , the lepton fraction at the center for tm1 ( 0.179 ) is larger than that for tms ( 0.155 ) , while the former is smaller than the latter at the initial stage . this means that the deleptonization proceeds faster for tms than for tm1 . we find that this is caused by the difference of @xmath19 . in the central region of protoneutron stars , electron type neutrinos are degenerate and their diffusion fluxes are roughly proportional to @xmath24 , where @xmath25 is the mean free path of _ e and @xmath26 is the number density of _ e. @xmath25 is roughly inversely proportional to the neutrino energy squared , @xmath27 , and @xmath28 in the degenerate limit . consequently , the diffusion flux in the degenerate limit is proportional to @xmath29 ; the smaller symmetry energy results in the larger @xmath19 , the larger _ e flux , and therefore the faster deleptonization . the influence of the symmetry energy on the evolution of the temperature is complicated in our models . the central temperature at @xmath30 are 23.2mev for tm1 and 26.3mev for tms . higher temperature for tms is due to the higher total ( matter + neutrinos ) entropy per baryon ( 1.29 for tms and 1.20 for tm1 ) and the lower lepton fraction ( 0.155 for tms and 0.179 for tm1 ) . the lower lepton fraction which is the consequence of fast deleptonization means the smaller lepton number density and the temperature corresponding to a given total entropy becomes higher . as for the difference in the central entropy , we analyze the results of numerical simulations in detail and find the following reason for the present case . in the first 10 seconds of the protoneutron star cooling , the matter entropy of the central region increases . this entropy change has three main origins ; inward flux of @xmath9 , the downscattering of _ e and the emission of _ e due to electron capture . since , during the first 10 seconds , the temperature profile has its peak in the middle region of the protoneutron star , not at the center , there is the negative gradient in the number density of @xmath9 at the central region . the flux of @xmath9 is proportional to the gradient . therefore , at the early stage , @xmath9 flow inwards in the central region and transport the heat into the central region from the middle region . the pair annihilations of @xmath9 ( @xmath31 ) increase the matter entropy . in addition , in the central region where the electrons are strongly degenerate , _ e - electron scattering ( @xmath32 ) leads to the increase of the matter entropy . neutrinos lose their energy at the scattering ( downscattering ) because the scattered electrons which were within the fermi sea should have energy greater than the fermi energy . on the other hand , emission of _ e ( @xmath33 ) decreases the matter entropy . the two heating processes and the one cooling process result in the net heating of the central region . detailed analysis of the numerical simulations reveals that , among the above three processes which alter the central entropy , the largest difference of heating / cooling rate ( @xmath34 ) due to each process between the two models is the difference of the cooling rate due to the _ e emission . furthermore it is found that the difference in @xmath34 due to the _ e emission is caused mainly by the difference of @xmath19 as the case of the deleptonization rate . @xmath35 due to @xmath36 can be expressed as @xmath37 where @xmath38 is the energy density of _ e. since , in the central region at the early stage where the electron capture proceeds , @xmath39 is positive , it can be seen that the larger @xmath19 for the smaller symmetry energy results in the larger @xmath35 due to _ eemission , that is , the less entropy loss of the matter because @xmath40 is negative there . this small cooling rate due to the _ e emission leads to the higher entropy and higher temperature for the case of smaller symmetry energy . we note , however , that the temperature profile is affected by many factors . the difference in the evolutions of the protoneutron star due to the symmetry energy is reflected with the properties of supernova neutrinos emitted during the cooling stage of the star . we show in fig . [ f : lepton : flux ] the net flux of the electron type lepton number from the protoneutron star ; the flux of _ e minus |_e . the net flux is larger for tms than for tm1 during 15 seconds because of the difference in the diffusion fluxes as we have discussed in section [ s : deleptonization ] . this fact corresponds to the higher deleptonization rate in tms having the smaller symmetry energy . since the final electron fraction of the neutron star at the end of the deleptonization is smaller for tms , the total net neutrino number emitted during the cooling is larger for tms than for tm1 . it may be possible to extract the information of the final electron fraction of the neutron star from this quantity we show in fig . [ f : mean : energy ] the calculated time profile of the mean energy of |_e . the mean energy for tms becomes higher than that for tm1 at a later stage . this is due to the difference of the temperature profile of the protoneutron stars , as shown in fig . [ f : temperature ] . the mean energy of neutrinos depends mainly on the temperature at the neutrinosphere . the difference of the mean energies is larger for @xmath9 and smaller for _ e depending on the position of its neutrinosphere . the neutrinosphere for _ ewhich interact most strongly with the matter locates in the outermost region , where both the temperature and the density are low and the difference between the temperature in the two cases is small . we show in fig . [ f : luminosity ] the luminosity of |_e , which is higher for tms than for tm1 . the general feature is quite similar for the other types of neutrinos . these results correspond to the larger flux and the higher mean energy of neutrinos . the total energy carried out by neutrinos is larger for tms than for tm1 , since the gravitational mass of the neutron star at zero temperature is smaller for tms . we note that the total baryon mass of the protoneutron stars is fixed to be 1.62 m _ ( the gravitational mass of the initial protoneutron star is 1.5479 m _ for tms and 1.5482 m _ for tm1 ) in the present study and the gravitational mass of the cold neutron star turns out to be 1.475 m _ for tms and 1.485 m _ for tm1 . therefore , the total energy released from the protoneutron star amounts to 0.073 m _ ( @xmath41 ) for tms and 0.063m_(@xmath42 ) for tm1 , which means the larger luminosity for tms , provided that the time duration of neutrino emission is similar . the main influence of the symmetry energy of the eos on the birth of neutron stars comes from the change of the chemical composition rather than the stiffness . a change in the chemical potential for neutrinos affects the rates of the interaction with the matter even if the matter density is unchanged . it affects the deleptonization rate through the change of the diffusion flux , and the temperature inside stars through the change of the heating rate . those changes influence the properties of the neutrino burst such as the time profile of the mean energy while the density profile remains unchanged . the difference in the properties of the neutrino burst is emphasized moreover by the difference in the proton fraction and the binding energy of the cold neutron stars at the end of the birth stage . the above results come from our study on the evolution of protoneutron stars by assuming the same initial conditions following the result of mayle & wilson . here we comment on expected difference in the initial protoneutron star configuration when we simulate also the dynamical phase , the birth of the protoneutron stars , using the two eos s . since tms has a smaller coupling constant @xmath2 , it is expected that the symmetry energy and hence the difference between the chemical potential of neutrons and that of protons are smaller for tms than for tm1 even at the density of the collapsing core . this means a larger fraction of free protons for tms , which will result in a larger electron - capture rate during the collapse and a smaller trapped lepton fraction at the core bounce ( bruenn 1989 ) . throughout the birth stage of neutron stars including both the dynamical phase and the quasistatic phase , deleptonization will proceed faster for tms than for tm1 and electron - type lepton number flux will be larger for tms . the work in this direction to simulate also the dynamical phase is in progress . further studies are required to predict precisely the profile of the neutrino burst and to extract the information on the symmetry energy from the observational signals of the next supernova explosion in our galaxy or close - by . the influence of the thermodynamical properties of eos such as the symmetry energy should be carefully examined besides the one by the stiffness , which has been mainly focused . the symmetry energy is one of the most interesting keys to clarify the current problems in nuclear physics and astrophysics . it would be interesting to apply the many body framework such as the rmf theory , in which the symmetry energy is being checked by the properties of unstable nuclei , to solve consistently the current issues on the cooling of the neutron stars and the supernova explosion together with the ones in nuclear physics . we would like to thank k. oyamatsu , s. nishizaki , t. takatsuka and k. sato for comments and discussions , and r. mayle , j. r. wilson and r. g. wolff for providing us with their data . we are grateful to y. sugahara , d. hirata and i. tanihata for providing us with valuable information and for fruitful discussions on the aspect of unstable nuclei . k. s. acknowledges the support by the special researchers basic science program . boguta j. , 1981 , phys . b106 , 255 brockmann r. , machleidt r. , 1990 , phys . c42 , 1965 bruenn s. w. , 1985 , apjs 58 , 771 bruenn s. w. , 1989 , apj 340 , 955 burrows a. , lattimer j. m. , 1986 , apj 307 , 178 burrows a. , 1988 , apj 334 , 891 gambhir y. k. , ring p. , thimet a. , 1990 , ann . of phys . 198 , 132 hillebrandt w. , wolff r. g. , 1985 , in : arnett w. d. , truran j. w. , ( eds . ) nucleosynthesis : challenges and new developments , the university of chicago press , chicago , p.131 hirata d. , toki h. , watabe t. , tanihata i. , carlson b. v. , 1991 , phys . c44 , 1467 lattimer j. m. , pethick c. j. , prakash m. , haensel p. , 1991 66 , 2701 li g. q. , machleidt r. , brockmann r. , 1992 , phys . c45 , 2782 mayle r. , wilson j. r. , 1989 , private communication mayle r. , wilson j. r. , schramm d. n. , 1987 , apj 318 , 288 raffelt g. , seckel d. , 1991 , phys . 67 , 2605 serot b. d. , walecka j. d. , 1986 , adv . 16 , 1 sugahara y. , toki h. , 1994 , nucl . a579 , 557 sumiyoshi k. , hirata d. , toki h. , sagawa h. , 1993a , nucl . a552 , 437 sumiyoshi k. , suzuki h. , toki h. , 1993b , in : morrissey d. j. ( ed . ) proceedings of the third international conference on radioactive nuclear beams , michigan , usa , 1993 , editions frontieres , gif - sur - yvette , p.501 sumiyoshi k. , toki h. , 1994 , apj 422 , 700 sumiyoshi k. , kuwabara h. , toki h. , 1995 , nucl . a581 , 725 suzuki h. , 1993 , in : suzuki y. , nakamura k. ( eds . ) proceedings of the international symposium on frontiers of neutrino astrophysics , takayama , japan , 1992 , universal academy press inc . , tokyo , p.219 suzuki h. , 1994 , in : fukugita m. , suzuki a. ( eds . ) physics and astrophysics of neutrinos , springer - verlag , tokyo , p.763 , and references therein swesty f. d. , lattimer j. m. , myra e. s. , 1994 , apj 425 , 195 tanihata i. , hamagaki h. , hashimoto o. , et al . , 1985 , 55 , 2676 wilson j. , 1990 , nucl . b.(proc . suppl . ) 13 , 380 wiringa r. b. , fiks v. , fabrocini a. , 1988 , phys .
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we study the influence of the symmetry energy of the equation of state on the thermal evolution of protoneutron stars and the properties of supernova neutrinos by the numerical simulations after the protoneutron stars are formed .
as for the equation of state ( eos ) of nuclear matter , we take two eos s with different symmetry energies obtained by the relativistic mean field theory .
we find the symmetry energy plays the essential role on the evolution of lepton profiles and the neutrino fluxes .
@referee = 1.0 in
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You are an expert at summarizing long articles. Proceed to summarize the following text:
low - frequency radio telescopes have relatively large fields - of - view , because the telescope primary beam size scales with wavelength . current high - resolution ( sub - arcminute ) radio interferometers that operate at sub - ghz frequencies , like the giant metrewave radio telescope ( gmrt ) , the low - frequency array ( lofar ) , or the very large array ( vla ) , have field diameters measured in ( sometimes tens of ) degrees . typical challenges of handling high - resolution , low - frequency continuum observations are ( i ) imaging and deconvolving many sources inside ( and outside ! ) the large primary beam area , ( ii ) direction - dependent ( dd ) visibility amplitude and phase variations across the field - of - view due to antenna beam patterns , pointing errors , ionosphere , etc . , and the abundance of radio frequency interference ( rfi ) . processing data coming from these telescopes can be daunting , since all the challenges mentioned above ( and more ) needs to be dealt with at the same time . in data reduction , the _ art _ is to determine at any given time which effect is dominant in limiting the image quality . generally , a few _ big effects _ stand out and are generally easy to identify and fix . but subsequenty , many _ smaller effect _ will present themelves , which may be more difficult to recognize , and harder to identify and disentangle . a effective way to mitigate the smaller effects is to perform iterations of ( dd ) calibration , imaging and flagging , in each step refining the quality of both the visibility data and the reconstructed sky model . developments towards a semi - automated data reduction package started in 2006 , driven by the complicated reduction of gmrt 150 mhz observations on galaxy cluster abell 2256 ( and other similar data sets ) . several challenges presented themselves at the same time : ( i ) the target contains faint , diffuse emission in the presence of bright interfering sources , ( ii ) the data was strongly affected by rfi on many baselines , ( iii ) the data was strongly affected by ionospheric phase distortions , ( iv ) the data was affected by instrumental instabilities . it quickly became clear that this data required an algorithm to correct for ionospheric phase errors , and possibly other dd effects . simultaneously followed the desire to automate several trivial but very time - consuming data reduction steps . for the basis of these new developments a stable data reduction package was sought , with the possibility to easily expand the functionality . this was found in the combination of the data reduction package aips @xcite and the powerful high - level programming language python ( and its standard scientific libraries like scipy , pylab , matplotlib , and numpy ) , with the parseltongue interface @xcite providing access to aips tasks , data files ( images and visibilities ) and tables from python . a new python module was created named _ spam _ , an acronym for _ source peeling and atmospheric modeling _ , encapsulating high - level data reduction functions and new algorithm development . spam - based data reductions , i.e. for gmrt observations , have been captured in python scripts that execute aips tasks directly ( mostly during initial reduction steps ) , call high - level functions that encompass multiple aips /parseltongue calls , and require few manual operations . these scripted data reductions automatically keep a history of the data reduction steps , and provide a standard that is well - tested and reproducible . at the same time , the spam user is required to perform some manual steps , and can easily decide to change the order of data reduction steps depending on the target field geometry and the intended science , therefore the term data reduction _ recipe _ is more appropriate than _ pipeline_. note that spam assumes an unpolarized sky , observed with limited ( @xmath0 percent ) fractional bandwidth . spam recipes are available for all sub - ghz gmrt frequencies , developed and refined over the last several years , with minimal differences between frequencies . the general structure of this data reduction is described in , and includes _ measuring _ dd ionospheric phase errors through peeling of bright sources within the science field ( e.g. , * ? ? ? * ) , _ modeling _ these errors with a single- or multi - layer ionospheric phase model , and _ applying _ ionospheric phase corrections during wide - field ( re-)imaging of the science field ( see section [ sec : ioncal ] ) . a combination of rfi- and bad data mitigation routines are used inbetween rounds of calibration and imaging . _ classical outlier removal _ excises visibilities mostly based on excessive visibility amplitudes and statistical outlier rejection along the time and frequency axes . subtraction _ models and subtracts low - level , quasi - continuous , ground - based rfi based on its fringe - rotation signature ( e.g. , * ? ? ? * ) . and _ ripple killing _ excises bad visibility data based on a combination of high visibility amplitudes , high visibility weights and high imaging weights ( i.e. , though uniform weighting ) . this is done by fourier - transforming residual images back onto the uv - plane , identifying high - amplitude uv - cells , and rejecting all visibilities that fall in these cells . a more elaborate description will be given in an upcoming paper ( intema et al . , in preparation ) . following @xcite there are four ionospheric calibration regimes , depending on the size of the radio interferometer array and the size of the field - of - view , both relative to the scale of phase structure in the ionosphere . this makes most sense when imagining a single horizontal density wave ( or a bubble ) in a thin layer at a fixed height . regime 4 is the most complex case , when both the horizontal size of the array and the projected field - of - view at ionospheric height are similar or larger than the horizontal scale of the ionospheric phase structure . in this regime , when looking towards a single source in the field - of - view , the higher - order phase structure over the array will cause both an apparent position shift and a source deformation . when looking at a second source in the same field - of - view at sufficient distance from the first , both the apparent position shift and the source distortion will be different . the spam ionospheric calibration strategy is designed to operate in regime 4 , meaning that it will also work in the less complex regimes 13 . in an automated peeling routine , measurements of the ionospheric phase structure are obtained by phase calibrating on ( typically 1020 ) bright sources within the field - of - view . per time interval ( typically 1020 seconds ) , the measured phases of all source antenna pairs are mapped onto a common domain along their line - of - sight , i.e. , at the pierce points through a virtual phase screen at fixed height . the phases are fit with an optimized set of base functions , reproducing the measured phases , and predicting dd phases corrections in arbitrary viewing directions during imaging . the automated wide - field imager applies any relevant dd phase corrections on the fly in a modified version of polyhedron ( facet - based ) imaging and cotton - schwab clean deconvolution ( e.g. , see * ? ? ? the intrinsic book - keeping issues of faceted imaging are largely hidden from the user . the imager also includes an iterative , robust clean boxing scheme , and simultaneous imaging of the primary beam area and bright outlier fields . in principle , the generic imager can handle other dd effects as well , like asymmetric primary beams or pointing errors , as long as a model is available to generate the dd correction tables . spam is an aips - based python package that provides semi - automated data reduction scripts for all sub - ghz frequencies at gmrt . apart from well - tested standard data reduction steps , spam includes direction - dependent ( ionospheric ) calibration and imaging , and alternative rfi- and bad data mitigation methods . with spam , gmrt data reductions are highly efficient , highly reproducible , and give good quality results . spam is available for everyone to use . spam for casa , currently under development , will also provide wide - bandwidth ionospheric calibration , direction - dependent ionospheric faraday rotation calibration , and will naturally benefit from the advanced modes ( and continuous improvements ) of the casa imager ( e.g. , see * ? ? ? * and references therein ) .
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high - resolution astronomical imaging at sub - ghz radio frequencies has been available for more than 15 years , with the vla at 74 and 330 mhz , and the gmrt at 150 , 240 , 330 and 610 mhz .
recent developments include wide - bandwidth upgrades for vla and gmrt , and commissioning of the aperture - array - based , multi - beam telescope lofar .
a common feature of these telescopes is the necessity to deconvolve the very many detectable sources within their wide fields - of - view and beyond .
this is complicated by gain variations in the radio signal path that depend on viewing direction .
one such example is phase errors due to the ionosphere . here
i discuss the inner workings of spam , a set of aips - based data reduction scripts in python that includes direction - dependent calibration and imaging .
since its first version in 2008 , spam has been applied to many gmrt data sets at various frequencies .
many valuable lessons were learned , and translated into various spam software modifications .
nowadays , semi - automated spam data reduction recipes can be applied to almost any gmrt data set , yielding good quality continuum images comparable with ( or often better than ) hand - reduced results . spam is currently being migrated from aips to casa with an extension to handle wide bandwidths .
this is aimed at providing users of the vla low - band system and the upcoming wide - bandwidth gmrt with the necessary data reduction tools .
[ firstpage ] atmospheric effects methods : data analysis instrumentation : interferometers
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You are an expert at summarizing long articles. Proceed to summarize the following text:
search for the quark - gluon plasma ( qgp ) at relativistic heavy - ion collision experiments ( rhice ) has reached a very exciting stage with the ongoing experiments at rhic and with the upcoming heavy - ion experiments at the large hadron collider . no one doubts that the qgp phase has already been created at rhic but conclusive evidence for the same is still lacking . many signatures have been proposed for the detection of the qgp phase @xcite and these have been thoroughly investigated both theoretically and experimentally . along with continued investigation of these important signatures of qgp , there is a need for investigating novel signals exploring qualitatively non - trivial features of the qgp phase and/or the quark - hadron phase transition . with this view we focus on the non - trivial vacuum structure of the qgp phase which arises when one uses the expectation value of the polyakov loop @xmath1 as the order parameter for the confinement - deconfinement phase transition @xcite . this order parameter transforms non - trivially under the center @xmath0 of the color su(3 ) group and is non - zero above the critical temperature @xmath2 . this breaks the global @xmath0 symmetry spontaneously above @xmath2 , while the symmetry is restored below @xmath2 in the confining phase where this order parameter vanishes . in the qgp phase , due to spontaneous breaking of the discrete @xmath0 symmetry , one gets domain walls ( interfaces ) which interpolate between different @xmath0 vacua . the properties and physical consequences of these @xmath0 interfaces have been discussed in the literature@xcite . it has been suggested that these interfaces should not be taken as physical objects in the minkowski space @xcite . similarly , it has also been subject of discussion whether it makes sense to talk about this @xmath0 symmetry in the presence of quarks @xcite . however , we will follow the approach where the presence of quarks is interpreted as leading to explicit breaking of @xmath0 symmetry , lifting the degeneracy of different @xmath0 vacua @xcite . thus , with quarks , even planar @xmath0 interfaces do not remain static and move away from the region with the unique true vacuum . our main discussion will be for the pure gauge theory which we discuss first . later we will briefly comment on the situation with quarks . a detailed study with inclusion of quark effects is postponed for a future work . in earlier works some of us had shown that there are novel topological string defects which form at the intersection of the three @xmath0 interfaces . these strings are embedded in the qgp phase and their cores consist of the confining phase . structure of these strings and interfaces were discussed in these earlier works @xcite . it was also shown that reflection of quarks from collapsing @xmath0 interfaces can lead to large scale baryon inhomogeneities in the early universe @xcite . this effect is also utilized to argue for @xmath3 enhancement of quarks of heavy flavor ( consecutively corresponding hadrons ) in relativistic heavy - ion collisions @xcite . in this paper we carry out numerical simulation of formation of these @xmath0 interfaces and associated strings at the initial confinement - deconfinement transition which is believed to occur during the pre - equilibrium stage in relativistic heavy - ion collision experiments . for the purpose of numerical simulation we will model this stage as a quasi - equilibrium stage with an effective temperature which first rises ( with rapid particle production ) to a maximum temperature @xmath4 , where @xmath2 is the critical temperature for the confinement - deconfinement phase transition , and then decreases due to continued plasma expansion . we use the effective potential for the polyakov loop expectation value @xmath1 as proposed by pisarski @xcite to study the confinement - deconfinement ( c - d ) phase transition . within this model , the c - d transition is weakly first order . even though lattice results show that quark - hadron transition is most likely a smooth cross - over at zero chemical potential , in the present work we will use this first order transition model to discuss the dynamical details of quark - hadron transition . one reason for this is that our study is in the context of relativistic heavy - ion collision experiments ( rhice ) where the baryon chemical potential is not zero . for not too small values of the chemical potential , the quark - hadron phase transition is expected to be of first order , so this may be the case relevant for us any way ( especially when collision energy is not too high ) . further , our main interest is in determining the structure of the network of z(3 ) domain walls and strings resulting during the phase transition . these objects will form irrespective of the nature of the transition , resulting entirely from the finite correlation lengths in a fast evolving system , as shown by kibble @xcite . the kibble mechanism was first proposed for the formation of topological defects in the context of the early universe @xcite , but is now utilized extensively for discussing topological defects production in a wide variety of systems from condensed matter physics to cosmology @xcite . essential ingredient of the kibble mechanism is the existence of uncorrelated domains of the order parameter which result after every phase transition occurring in finite time due to finite correlation length . a first order transition allows easy implementation of the resulting domain structure especially when the transition proceeds via bubble nucleation . with this view , we use the polyakov loop model as in @xcite to model the phase transition and confine ourselves with temperature / time ranges so that the first order quark - hadron transition proceeds via bubble nucleation . the @xmath0 wall network and associated strings ( as mentioned above ) formed during this early confinement - deconfinement phase transition evolve in an expanding plasma with decreasing temperature . eventually when the temperature drops below the deconfinement - confinement phase transition temperature @xmath2 , these @xmath0 walls and associated strings will melt away . however they may leave their signatures in the form of extended regions of energy density fluctuations ( as well as @xmath3 enhancement of heavy - flavor hadrons @xcite ) . we make estimates of these energy density fluctuations which can be compared with the experimental data . especially interesting will be to look for extended regions of large energy densities in space - time reconstruction of hadron density ( using hydrodynamic models ) . in our model , we expect energy density fluctuations in event averages ( representing high energy density regions of domain walls / strings ) , as well as event - by - event fluctuations as the number / geometry of domain walls / strings and even the number of qgp bubbles , varies from one event to the other . a detailed analysis of energy fluctuations , especially the event - by - event fluctuations , is postponed for a future work . we also determine the distribution / shape of @xmath0 wall network and its evolution . in particular , our results provide an estimate of domain wall velocities ( for the situations studied ) to range from 0.5 to 0.8 . these results provide crucial ingredients for a detailed study of the effects of collapsing @xmath0 walls on the @xmath3 enhancement of heavy flavor hadrons @xcite in rhice . we emphasize that the presence of @xmath0 walls and string may not only provide a qualitatively new signature for the qgp phase in these experiments , it may provide the first ( and may be the only possible ) laboratory study of such topological objects in a relativistic quantum field theory system . the paper is organized in the following manner . in section ii , we discuss the polyakov loop model of confinement - deconfinement phase transition . we describe the effective potential proposed by pisarski @xcite and discuss the structure of the @xmath0 walls and associated strings . in section iii , we present the physical picture of the formation of these @xmath0 walls and associated strings in the confinement - deconfinement phase transition via the kibble mechanism which provides a general framework for the production of topological defects in symmetry breaking phase transitions . we confine ourselves with temperature / time ranges so that the first order transition ( in the pisarski model ) proceeds via bubble nucleation . the other possibility of spinodal decomposition is of completely different nature and we will present it in a future work . ( we mention here that a simulation of spinodal decomposition in polyakov loop model has been carried out in ref . @xcite , where fluctuations in the polyakov loop are investigated in detail . in comparison , the main focus of our work is on the topological objects , z(3 ) walls , and strings , and energy density fluctuations resulting therefrom . ) in section iv , we discuss the calculation of the profile of the critical bubble using bounce technique @xcite and also the estimates of nucleation rates for bubbles for the temperature / time range relevant for rhice . section v presents the numerical technique of simulating the phase transition via random nucleation of bubbles . in section vi we discuss the issue of the effects of quarks in our model . section vii presents the results of the numerical simulations . here we discuss distribution of @xmath0 wall / string network formed due to coalescence of qgp bubbles ( in a confining background ) . we also calculate the energy density fluctuations associated with @xmath0 wall network and strings . we discuss evolution of these quantities with changing temperature via bjorken s hydrodynamical model . in section viii we discuss possible experimental signatures resulting from the presence of @xmath0 wall network and associate strings . section ix presents conclusions . we first discuss the case of pure su(n ) gauge theory . we will later discuss briefly the case with quarks . the order parameter for the confinement - deconfinement ( c - d ) phase transition is the expectation value of the polyakov loop @xmath5 which is defined as @xmath6 here p denotes the path ordering , g is the gauge coupling , and @xmath7 , where t is the temperature . @xmath8 , @xmath9 being the generators of su(n ) in the fundamental representation , is the time component of the vector potential at spatial position @xmath10 and euclidean time @xmath11 . under a global z(n ) symmetry transformation , @xmath5 transforms as @xmath12 the expectation value of @xmath1 is related to @xmath13 where @xmath14 is the free energy of an infinitely heavy test quark . for temperatures below @xmath2 , in the confined phase , the expectation value of polyakov loop is zero corresponding to the infinite free energy of an isolated test quark . ( hereafter , we will use the same notation @xmath1 to denote the expectation value of the polyakov loop . ) hence the z(n ) symmetry is restored below @xmath2 . z(n ) symmetry is broken spontaneously above @xmath2 where @xmath1 is non - zero corresponding to the finite free energy of the test quark . for qcd , @xmath15 , and we take the effective theory for the polyakov loop as proposed by pisarski @xcite . the effective lagrangian density is given by @xmath16 @xmath17 is the effective potential for the polyakov loop @xmath18 the values of various parameters are fixed to reproduce the lattice results @xcite for pressure and energy density of pure su(3 ) gauge theory . we make the same choice and give those values below . the coefficients @xmath19 and @xmath20 have been taken as , @xmath21 and @xmath22 . we will take the same value of @xmath23 for real qcd ( with three massless quark flavors ) , while the value of @xmath20 will be rescaled by a factor of 47.5/16 to account for the extra degrees of freedom relative to the degrees of freedom of pure su(3 ) gauge theory @xcite . the coefficient @xmath23 is temperature dependent @xcite . @xmath23 is taken as , @xmath24 , where r is taken as t/@xmath2 . with the coefficients chosen as above , the expectation value of order parameter approaches to @xmath25 for temperature @xmath26 . as in @xcite , we use the normalization such that the expectation value of order parameter @xmath27 goes to unity for temperature @xmath26 . hence the fields and the coefficients in @xmath28 are rescaled as @xmath29 , @xmath30 , @xmath31 and @xmath32 to get proper normalization of @xmath27 . for the parameters chosen as above , the value of @xmath2 is taken to be 182 mev . we see that the @xmath19 term in eq.(4 ) gives a cos(3@xmath33 ) term , leading to z(3 ) degenerate vacua structure . here , the shape of the potential is such that there exists a metastable vacuum upto a temperature @xmath34 250 mev . hence first order transition via bubble nucleation is possible only upto @xmath35 mev . we show the plot of v@xmath36 in @xmath37 direction in fig.(1a ) for a value of temperature t = 185 mev . this shows the metastable vacuum at @xmath38 . fig.(1b ) shows the structure of vacuum by plotting v(@xmath39 ) as a function of @xmath33 for fixed @xmath27 , where @xmath40 is the vacuum expectation value of @xmath17 at @xmath41 mev . + \(a ) plot of @xmath28 in @xmath42 direction for @xmath43 showing the metastable vacuum at @xmath44 . in ( a ) and ( b ) , plots of @xmath45 are given in units of @xmath46 . the value of critical temperature is taken to be @xmath47 . the @xmath0 structure of the vacuum can be seen in ( b ) in the plot of the potential @xmath28 as a function of @xmath33 for fixed @xmath48 . here , @xmath40 corresponds to the absolute minimum of @xmath28 . [ fig.1 ] in fig.(1b ) , the three degenerate vacua are separated by large barrier in between them . while going from one vacuum to another vacuum , the field configuration is determined from the field equations . for the case of degenerate vacua , there are time independent solutions which have planar symmetry . these solutions are called domain wall . for the non - degenerate case , as will be appropriate for the case when quarks are included as dynamical degrees of freedom in discussing the quark hadron transition , the solutions of the interfaces separating these vacua will be similar to the bounce solutions @xcite , though the standard bounce techniques need to be extended for the case of complex scalar field . the resulting planar domain wall solutions will not be static . as mentioned above , we will be neglecting such effects of quarks , and hence will discuss the case of degenerate vacua only . later , in section vi , we will examine the justification of using the approximation of neglecting quark effects . in physical space , after the phase transition , regions with different z(3 ) vacua are separated by domain walls . inside a domain wall , @xmath49 becomes very small as @xmath50 @xcite . in ref.@xcite the intersection of the three z(3 ) domain walls was considered and , using topological arguments , it was then shown that at the line - like intersection of these interfaces , the order parameter @xmath1 should vanish . this leads to a topological string configuration with core of the string being in the confining phase . properties of these new string configurations were determined in @xcite using the model of eq.(3 ) . this string configuration is interesting , especially when we note that such a string has exactly reverse physical behavior compared to the standard qcd string . the qcd string exists in the confining phase , connecting quarks and antiquarks , or forming baryons , glueballs etc . inside the qcd string , the core region is expected to behave as a deconfined region . in contrast the string discussed in @xcite , arising at the intersection of z(3 ) walls , exists in the high temperature deconfined phase . its core is characterized by restored @xmath0 symmetry , implying that it is in the confined phase . to differentiate it with the standard qcd string , this new string structure was called as _ the qgp string _ in ref.@xcite . it is also important to note that although the standard qcd string breaks by creating quark - antiquark pairs , the qgp string can not break as it originates from topological arguments . this qgp string , thus should either form closed loops , or it should end at the boundary separating the deconfined phase from the confined phase . its structure is very similar to certain axionic strings discussed in the context of early universe @xcite . note that as these strings contain the confining phase ( with @xmath44 ) in the core , while they are embedded in the qgp phase , a transition from deconfining phase to the confining phase , in the presence of such strings may begin from regions near the strings . similarly , the presence of domain walls may lead to heterogeneous bubble nucleation in a first order quark - hadron transition . we will be studying the formation of z(3 ) domain walls and these qgp strings in the initial confinement - deconfinement phase transition , and their subsequent evolution . we now briefly describe the physical picture of the formation of topological defects via the kibble mechanism . kibble first gave a detailed theory of the formation of topological defects in symmetry breaking phase transitions in the context of the early universe @xcite . subsequently it was realized that the basic physics of kibble mechanism is applicable to every symmetry breaking transition , from low energy physics of condensed matter systems to high energy physics relevant for the early universe @xcite . basic physics of the kibble mechanism can be described as follows . after a spontaneous symmetry breaking phase transition , the physical space consists of regions , called domains . in each domain the configuration of the order parameter field can be taken as nearly uniform while it varies randomly from one domain to another . in a numerical simulation where the phase transition is modeled to implement the kibble mechanism , typically the physical region is divided in terms of elementary domains of definite geometrical shape . the order parameter is taken to be uniform within the domain and random variations of order parameter field within the vacuum manifold are allowed from one domain to the other . the order parameter field configuration in between domains is assumed to be such that the variation of the order parameter field is minimum on the vacuum manifold ( the so - called geodesic rule ) . with this simple construction , topological defects arise at the junctions of several domains if the variation of the order parameter in those domains traces a topologically non - trivial configuration in the vacuum manifold . see , ref . @xcite for a detailed discussion of this approach . we , however , will follow a more detailed simulation as in ref.@xcite where the kibble mechanism was implemented in the context of a first order transition . bubbles of true vacuum ( determined from the bounce solution ) were randomly nucleated in the background of false vacuum . each bubble was taken to have the uniform orientation of the order parameter in the vacuum manifold , while the order parameter orientation varied randomly from one bubble to another . this provided the initial seed domains , as needed for the kibble mechanism . evolution of this initial configuration via the field equation led to expansion of bubbles which eventually coalesce and lead to the formation of topological defects at the junctions of bubbles when the order parameter develops appropriate variation ( winding ) in that region . important thing is that in this case one does not need to assume anything like the geodesic rule . as different bubbles come into contact during their expansion , the value of the order parameter in the intermediate region is automatically determined by the field equations . an important aspect of the kibble mechanism is that it does not crucially depend on the dynamical details of the phase transition . although the domain size depends on the dynamics of phase transitions , the defect number density ( per domain ) and type of topological defects produced via the kibble mechanism depends only on the topology of order parameter space and spatial dimensions . if the vacuum manifold m has disconnected components , then domain walls form . if it is multiply connected ( i.e. , if m contains unshrinkable loops ) , strings will form . when m contains closed two surfaces which can not be shrunk to a point , then monopoles will form in three dimensional physical space . in our case domain walls and string network will be produced in the qgp phase . as discussed above , domain walls arise due to interpolation of field between different z(3 ) vacua . at the intersection of these interfaces , string is produced . as we explained in the introduction , we will study the z(3 ) domain wall and string formation with the first order transition model given by eq.(3 ) , such that the transition occurs via bubble nucleation . the semiclassical theory of decay of false vacuum at zero temperature has been given in ref . @xcite , and the finite temperature extension of this theory was given in ref . the process of barrier tunneling leads to the appearance of bubbles of the new phase . resulting bubble profile is determined using the bounce solution for the false vacuum decay which we will discuss below . first we note general features of the dynamics of a standard first order phase transition at finite temperature via bubble nucleation . a region of true vacuum , in the form of a spherical bubble , appears in the background of false vacuum . the creation of bubble leads to the change in the free energy of the system as , @xmath51 where r is the radius of bubble , @xmath52 is the surface energy contribution and @xmath53 is the volume energy contribution . @xmath54 is the difference of the free energy between the false vacuum and the true vacuum and @xmath55 is the surface tension which can be determined from the bounce solution . a bubble of size @xmath56 will expand or shrink depending on which process leads to lowering of the free energy given above . the bubbles of very small sizes will shrink to nothing since surface energy dominates . if the radius of bubble exceeds the critical size @xmath57 , it will expand and lead to the transformation of the metastable phase into the stable phase . eq.(5 ) is useful for the so - called _ thin wall _ bubbles where there is a clear distinction between the surface contribution to the free energy and the volume contribution . for the temperature / time relevant for our case in relativistic heavy - ion collisions this will not be the case . instead we will be dealing with the _ thick wall _ bubbles where surface and volume contributions do not have clear separations . we will determine the profiles of these thick wall bubbles numerically following the bounce technique @xcite . first we note that for the effective potential in eq.(4 ) , the barrier between true vacuum and false vacuum vanishes at temperatures above about 250 mev . so first order transition via bubble nucleation is possible only within the temperature range of @xmath58 mev - 250 mev . above @xmath59 250 mev , spinodal decomposition will take place due to the roll down of field . implementation of the dynamics of phase transition via spinodal decomposition is of completely different nature , and we hope to discuss this in a future work . as we are discussing the initial confinement - deconfinement transition in the context of rhice , clearly the discussion has to be within the context of longitudinal expansion only , with negligible effects of the transverse expansion . however , bjorken s longitudinal scaling model @xcite can not be applied during this pre - equilibrium phase , even with the assumption of quasi - equilibrium ( as discussed above ) , unless one includes a heat source which could account for the increase of effective temperature during this phase to the maximum equilibrium temperature @xmath60 . as indicated above , this heat source can be thought of as representing the rapid particle production ( with subsequent thermalization ) during this early phase . we will not attempt to model such a source here . instead , we will simply use the field equations resulting from bjorken s longitudinal scaling model for the evolution of the field configuration for the entire simulation , including the initial pre - equilibrium phase from @xmath61 to @xmath62 . the heating of the system until @xmath63 will be represented by the increase of the temperature upto @xmath64 . thus , during this period , the energy density and temperature evolution will not obey the bjorken scaling equations @xcite . after @xmath65 , with complete equilibrium of the system , the temperature will decrease according to the equations in the bjorken s longitudinal scaling model . we will take the longitudinal expansion to only represent the fact that whatever bubbles will be nucleated , they get stretched into ellipsoidal , and eventually cylindrical , shapes during the longitudinal expansion ( ignoring the boundary effects in the longitudinal direction ) . the transverse expansion of the bubble should then proceed according to relative pressure difference between the false vacuum and the true vacuum as in the usual theory of first order phase transition . we will neglect transverse expansion for the system , and focus on the mid rapidity region . with this picture in mind , we will work with effective 2 + 1 dimensional evolution of the field configuration , ( neglecting the transverse expansion of qgp ) . however , for determining the bubble profile and the nucleation probability of bubbles , one must consider full 3 + 1 dimensional case as bubbles are nucleated with full 3-dimensional profiles in the physical space . it will turn out that the bubbles will have sizes of about 1 - 1.5 fm radius . taking the initial collision region during the pre - equilibrium phase also to be of the order of 1 - 2 fm in the longitudinal direction , it looks plausible that the nucleation of 3-dimensional bubble profile as discussed above may provide a good approximation . of course the correct thing will be to consider the bounce solutions for rapidly longitudinally expanding plasma , and we hope to return to this issue in some future work . we neglect transverse expansion in the present work , which is a good approximation for the early stages when wall / string network forms . however , this will not be a valid approximation for later stages , especially when temperature drops below @xmath2 and wall / string network melts . the way to account for the transverse expansion in the context of our simulation will be to take a lattice with much larger physical size than the initial qgp system size , and allow free boundary conditions for the field evolution at the qgp system boundary ( which will still be deep inside the whole lattice ) . this will allow the freedom for the system to expand in the transverse direction automatically . with a suitable prescription of determining temperature from local energy density ( with appropriate account of field contributions and expected contribution from a plasma of quarks and gluons ) in a self consistent manner , the transverse expansion can be accounted for in this simulation . we hope to come back to this in a future work . let us consider the effective potential in eq.(4 ) , at a temperature such that there is a barrier between the true vacuum and the three z(3 ) vacua . an example of this situation is shown in fig.1 for the case with @xmath66 185 mev . the initial system ( of nucleons ) was at zero temperature with the order parameter l(x ) = 0 , and will be superheated as the temperature rises above the critical temperature . it can then tunnel through the barrier to the true vacuum , representing the deconfined qgp phase . at zero temperature , the tunneling probability can be calculated by finding the bounce solution which is a solution of the 4-dimensional euclidean equations of motion however , at finite temperature , this 4-dimensional theory will reduce to an effectively 3 euclidean dimensional theory if the temperature is sufficiently high , which we will take to be the case . for this finite temperature case , the tunneling probability per unit volume per unit time in the high temperature approximation is given by @xcite ( in natural units ) @xmath67 where @xmath68 is the 3-dimensional euclidean action for the polyakov loop field configuration that satisfies the classical euclidean equations of motion . the condition for the high temperature approximation to be valid is that @xmath69 , where @xmath70 is the radius of the critical bubble in 3 dimensional euclidean space . the values of temperature for our case ( relevant for bubble nucleation ) will be above @xmath71 mev . as we will see , the bubble radius will be larger than 1.5 fm ( @xmath34 ( 130 mev)@xmath72 ) which justifies our use of high temperature approximation to some extent . the determination of the pre - exponential factor is a non - trivial issue and we will discuss it below . the dominant contribution to the exponential term in @xmath73 comes from the least action @xmath74 symmetric configuration which is a solution of the following equation ( for the lagrangian in eq.(3 ) ) . @xmath75 where @xmath76 , subscript e denoting the coordinates in the euclidean space . the boundary conditions imposed on @xmath77 are @xmath78 bounce solution of eq.(7 ) can be analytically obtained in the _ thin wall _ limit where the difference in the false vacuum and the true vacuum energy is much smaller than the barrier height . this situation will occur for very short time duration near @xmath79 for the effective potential in eq.(4 ) . however , as the temperature is rapidly evolving in the case of rhice , there will not be enough time for nucleating such large bubbles ( which also have very low nucleation rates due to having large action ) . thus the case relevant for us is that of thick wall bubbles whose profile has to be obtained by numerically solving eq.(7 ) . as we have mentioned earlier , in the high temperature approximation the theory effectively becomes 3 ( euclidean ) dimensional . for a theory with one real scalar field in three euclidean dimensions the pre - exponential factor arising in the nucleation rate of critical bubbles has been estimated , see ref . the pre - exponential factor obtained from @xcite for our case becomes @xmath80 it is important to note here that the results of @xcite were for a single real scalar field and one of the crucial ingredients used in @xcite for calculating the pre - exponential factor was the fact that for a bounce solution the only light modes contributing to the determinant of fluctuations were the deformations of the bubble perimeter . even though we are discussing the case of a complex scalar field @xmath1 , this assumption may still hold as we are calculating the tunneling from the false vacuum to one of the z(3 ) vacua ( which are taken to be degenerate here as we discussed above ) . this assumption may need to be revised when light modes e.g. goldstone bosons are present which then also have to be accounted for in the calculation of the determinant . a somewhat different approach for the pre - exponential factor in eq.(6 ) is obtained from the nucleation rate of bubbles per unit volume for a liquid - gas phase transition as given in ref . @xcite , @xmath81 here @xmath82 is the dynamical prefactor which determines the exponential growth rate of critical droplets . @xmath83 is a statistical prefactor which measures the available phase space volume . the exponential term is the same as in eq.(6 ) with @xmath84 being the change in the free energy of the system due to the formation of critical droplet . this is the same as @xmath85 in eq.(6 ) . the bubble grows beyond the critical size when the latent heat is conducted away from the surface into the surrounding medium which is governed by thermal dissipation and viscous damping . for our case , in the general framework of transition from a hadronic system to the qgp phase , we will use the expression for the dynamical prefactor from ref.@xcite @xmath86 here @xmath55 is the surface tension of the bubble wall , @xmath87 is the difference in the enthalpy densities of the qgp and the hadronic phases , @xmath88 is thermal conductivity , @xmath89 is the critical bubble radius , and @xmath54 and @xmath90 are shear and bulk viscosities . @xmath90 will be neglected as it is much smaller than @xmath54 . for @xmath88 and @xmath54 , the following parametrizations are used @xcite . @xmath91 @xmath92 here @xmath93 is the ratio of the baryon density of the system to the normal nuclear baryon density , @xmath94 is in mev , @xmath54 is in mev/@xmath95c , and @xmath88 is in c/@xmath95 . with this , the rate in eq.(10 ) is in @xmath96 . for the range of temperatures of our interest ( @xmath97 ) , and for the low baryon density central rapidity region under consideration , it is the last @xmath98 independent term for both @xmath54 and @xmath82 which dominates , and we will use these terms only for calculating @xmath54 and @xmath82 for our case . for the statistical prefactor , we use the following expression @xcite @xmath99 the correlation length in the hadronic phase , @xmath100 , is expected to be of order of 1 fm and we will take it to be 0.7 fm @xcite ) . we will present estimates of the nucleation rates from eq.(6 ) as well as eq.(10 ) . one needs to determine the critical bubble profile and its 3-dimensional euclidean action @xmath85 ( equivalently , @xmath84 in eq.(10 ) ) . we solve eq.(7 ) using the fourth order runge - kutta method with appropriate boundary conditions ( eq.(8 ) ) , to get the profile of critical bubble @xcite . the critical bubble profiles ( for the 3 + 1 dimensional case ) are shown in fig.(2a ) for different temperatures . the bubble size decreases as temperature increases , since the energy difference between true vacuum and false vacuum increases ( relative to the barrier height ) as temperature increases . we choose a definite temperature @xmath101 mev for the nucleation of bubbles , which is suitably away from @xmath2 to give acceptable bubble size and nucleation probabilities for the relevant time scale . making @xmath94 larger ( up to @xmath102 mev when the barrier disappears ) leads to similar bubble profile , and the nucleation probability of same order . + \(a ) critical bubble profiles for different values of the temperature . ( b ) solid curve shows the critical bubble for the 3 + 1 dimensional case ( which for finite temperature case becomes 3 euclidean dimensional ) for @xmath94 = 200 mev and the dotted curve shows the same for 2 + 1 dimensional case ( i.e. 2 euclidean dimensions for finite temperature case ) . [ fig.2 ] recall that we are calculating bubble profiles using eq.(7 ) relevant for the 3 + 1 dimensional case , however , the field evolution is done using 2 + 1 dimensional equations as appropriate for the mid rapidity region of rapidly longitudinally expanding plasma . in fig.(2b ) solid curve shows the critical bubble for the 3 + 1 dimensional case ( which for finite temperature case becomes 3 euclidean dimensional ) for @xmath94 = 200 mev and the dashed curve shows the same for 2 + 1 dimensional case ( i.e. 2 euclidean dimensions for finite temperature case ) . it is clear that the 3 dimensional bubble is of supercritical size and will expand when evolved with 2 + 1 dimensional equations . this avoids the artificial construction of suitable supercritical bubbles which can expand and coalesce as was done in ref.@xcite . ( recall that for the finite temperature case , a bubble of exact critical size will remain static when evolved by the field equations . in a phase transition , bubbles with somewhat larger size than the critical size expand , while those with smaller size contract . ) for the bubble profile given by the solid curve in fig.(2b ) , the value of the action @xmath68 is about 240 mev . using eq.(6 ) for the nucleation rate , we find that the nucleation rate of qgp bubbles per unit time per unit volume is of the order of @xmath103 . the thermalization time for qgp phase is of the order of 1 fm at rhic ( say , for au - au collision at 200 gev energy ) . hence the time available for the nucleation of qgp bubbles is at most about 1 fm . we take the region of bubble nucleation to be of thick disk shape with the radius of the disk ( in the transverse direction ) of about 8 fm and the thickness of the disk ( in the longitudinal direction ) of about 1 fm . total space - time volume available for bubble nucleation is then about 200 @xmath104 ( in practice , less than this ) . for the case of eq.(6 ) , net number of bubbles is then equal to 5 . for the case of the nucleation rate given by eq.(10 ) , one needs an estimate of the critical bubble size @xmath89 as well as bubble surface energy @xmath55 , ( along with other quantities like @xmath88 etc . as given by eqs.(11)-(14 ) ) . determination of @xmath89 is somewhat ambiguous here as the relevant bubbles are thick wall bubbles as seen in fig.(2 ) . here there is no clear demarcation between the core region and the surface region which could give an estimate of @xmath89 . essentially , there is no core at all and the whole bubble is characterized by the overlap of bubble wall region . we can take , as an estimate for the bubble radius @xmath89 any value from 1 fm - 1.5 fm . it is important to note here that this estimate of @xmath89 is only for the calculation of nucleation rate @xmath73 , and not for using the bubble profile for actual simulation . when bubbles are nucleated in the background of false vacuum with @xmath105 , a reasonably larger size of the bubble is used so that cutting off the profile at that radius does not lead to computational errors and field evolution remains smooth . once we have an estimate of @xmath89 , we can then estimate the surface tension @xmath55 ( which also is not unambiguous here ) as follows . with the realization that essentially there is no core region for the bubbles in fig(2 ) , we say that the entire energy of the bubble ( i.e. the value of @xmath85 ) comes from the surface energy . then we write @xmath106 for @xmath85 = 240 mev , we get @xmath107 mev/@xmath95 if we take @xmath89 = 1.5 fm . with @xmath89 = 1.0 fm , we get @xmath55 = 20 mev/@xmath95 . number of bubbles expected can now be calculated for the case when nucleation rate is given by eq.(10 ) . we find number of bubbles to be about 10@xmath108 with @xmath89 taken as 1.5 fm . this is in accordance with the results discussed in ref . the bubble number increases by about a factor 5 if @xmath89 is taken to be about 1 fm . thus , with the estimates based on eq.(10 ) , bubble nucleation is a rare event for the time available for rhice . as we have mentioned above , for us the bubble nucleation on one hand represents the possibility of actual dynamics of a first order transition , while on the other hand , it represents the generic properties of the domain structure arising from a c - d transition , which may very well be a cross - over , occurring in a finite time . with this view , and with various uncertainties in the determination of pre - exponential factors in the nucleation rate , we will consider a larger number of bubbles also and study domain wall and string production . first we will consider nucleation of 5 bubbles and then we will consider nucleation of 9 bubbles to get a better network of domain walls and strings . in our simulation critical bubbles are nucleated at a time when the temperature @xmath94 crosses the value @xmath101 mev during the initial stage between @xmath61 to @xmath109 fm ( during which we have modeled the system temperature to increase linearly from 0 to @xmath60 ) . we take @xmath110 mev so bubble nucleation stage is taken to be at @xmath111 fm when @xmath94 reaches the value 200 mev . again , this is an approximation since in realistic case bubbles will nucleate over a span of time given by the ( time dependent ) nucleation rate , which could lead to a spectrum of sizes of expanding bubbles at a given time . however , due to very short time available to complete the nucleation of qgp bubbles in the background of confined phase , bubbles will have very little time to expand during the nucleation period of all bubbles ( especially as initial bubble expansion velocity is zero ) . thus it is reasonable to assume that all the bubbles nucleate at the same time . after nucleation , bubbles are evolved by the time dependent equations of motion in the minkowski space @xcite as appropriate for bjorken s longitudinal scaling model . @xmath112 with @xmath113 at @xmath11 = 0 . here @xmath114 , and dot indicates derivative with respect to the proper time @xmath11 . the bubble evolution was numerically implemented by a stabilized leapfrog algorithm of second order accuracy both in space and in time with the second order derivatives of @xmath115 approximated by a diamond shaped grid . here we follow the approach described in @xcite to simulate the first order transition . we need to nucleate several bubbles randomly choosing the corresponding z(3 ) vacua for each bubble . this is done by randomly choosing the location of the center of each bubble with some specified probability per unit time per unit volume . before nucleating a bubble , it is checked if the relevant region is in the false vacuum ( i.e. it does not overlap with some other bubble already nucleated ) . in case there is an overlap , the nucleation of the new bubble is skipped . the orientation of @xmath77 inside each bubble is taken to randomly vary between the three z(3 ) vacua . for representing the situation of relativistic heavy ion collision experiments , the simulation of the phase transition is carried out by nucleating bubbles on a square lattice with physical size of 16 fm within a circular boundary ( roughly the gold nucleus size ) . we use fixed boundary condition , free boundary condition , as well as periodic boundary condition for the square lattice . to minimize the effects of boundary ( reflections for fixed boundary , mirror reflections for periodic boundary conditions ) , we present results for free boundary conditions ( for other cases , the qualitative aspects of our results remain unchanged ) . even for free boundary conditions , spurious partial reflections occur , and to minimize these effects we use a thin strip ( of 10 lattice points ) near each boundary where extra dissipation is introduced . we use 2000 @xmath116 2000 lattice . for the physical size of 16 fm , we have @xmath117 = 0.008 fm . to satisfy the courant stability criteria , we use @xmath118 , as well as @xmath119 , ( which we use for the results presented in the paper ) . for au - au collision at 200 gev , the thermalization is expected to happen within 1 fm time . as mentioned above , in this pre - equilibrium stage , we model the system as being in a quasi - equilibrium stage with a temperature which increases linearly with time ( for simplicity ) . the temperature of the system is taken to reach upto 400 mev in 1 fm time , starting from @xmath94 = 0 . after @xmath109 fm , the temperature decreases due to continued longitudinal expansion , i.e. @xmath120 the stability of the simulation is checked by checking the variation of total energy of the system during the evolution . the energy fluctuation remains within few percent , with no net increase or decrease in the energy ( for fixed and periodic boundary conditions , and without the dissipative @xmath121 term in eq.(16 ) ) showing the stability of the simulation . the bubbles grow and eventually start coalescing , leading to a domain like structure . domain walls are formed between regions corresponding to different z(3 ) vacua , and strings form at junctions of z(3 ) domain walls . recall that the domain wall network is formed here in the transverse plane , appearing as curves . these are the cross - sections of the walls which are formed by elongation ( stretching ) of these curves in the longitudinal direction into sheets . at the intersection of these walls , strings form . in the transverse plane , these strings looks like vortices , which will be elongated into strings in the longitudinal direction . we now discuss the effects of quarks . as we mentioned above , we will follow the approach where the presence of quarks is interpreted as leading to explicit breaking of the z(3 ) symmetry , lifting the degeneracy of different z(3 ) vacua @xcite . this has important effects in the context of our model . first of all , different vacua having different energies implies different nucleation rates for the qgp bubbles with different z(3 ) vacua . further , for non - degenerate vacua , even planar @xmath0 interfaces do not remain static , and move away from the region with the unique true vacuum . thus , while for the degenerate vacua case every closed domain wall collapses , for the non - degenerate case this is not true any more . a closed wall enclosing the true vacuum may expand if it is large enough so that the surface energy contribution does not dominate ( this is essentially the same argument as given for the bubble expansion , see eq.(5 ) and the discussion following it ) . to see the importance of these effects we need an estimate of the explicit symmetry breaking term arising from inclusion of quarks . for this we use the estimates given in @xcite . even though the estimates in ref.@xcite are given in the high temperature limit , we will use these for temperatures relevant for our case , i.e. @xmath122 mev , to get some idea of the effects of the explicit symmetry breaking . the difference in the potential energy between the true vacuum with @xmath123 and the other two vacua ( @xmath124 , and @xmath125 , which are degenerate with each other ) is estimated in ref . @xcite to be , @xmath126 where @xmath127 is the number of massless quarks . if we take @xmath128 then @xmath129 . at the bubble nucleation temperature ( which we have taken to be about @xmath101 mev , the difference between the false vacuum and the true vacuum is about 150 mev/@xmath130 while @xmath131 at @xmath101 mev is about four times larger , equal to 600 mev/@xmath130 . as @xmath94 approaches @xmath2 , this difference will become larger as the metastable vacuum and the stable vacuum become degenerate at @xmath2 , while @xmath131 remains non - zero . for @xmath94 near 250 mev ( where the barrier between the metastable vacuum and the stable vacuum disappears ) , @xmath131 becomes almost comparable to the difference between the potential energy of the false vacuum ( the confining vacuum ) and the true vacuum ( deconfined vacuum ) . it does not seem reasonable that at temperatures of order 200 mev a qgp phase ( with quarks ) has higher free energy than the hadronic phase . this situation can be avoided if the estimates of eq.(18 ) are lowered by about a factor of 5 so that these phases have lower free energy than the confining phase . ( a more desirable situation will be when @xmath131 approaches zero as the confining vacuum and the deconfining ( true ) vacuum become degenerate at @xmath2 . ) it is in the spirit of the expectation that explicit breaking of z(3 ) is small near @xmath2 for finite pion mass @xcite . even with such lower estimates , the effects of quarks may give different nucleation probabilities for different z(3 ) vacua . however , in this paper we will ignore this possibility . this may not be very unreasonable as for thick wall bubbles thermal fluctuations may be dominant in determining the small number of bubbles nucleated during the short span of time available . for very few bubbles nucleated , there may be a good fraction of events where different z(3 ) vacua may occur in good fraction . also note that the pre - exponential factor for the bubble nucleation rate of eq.(6 ) , as given in eq.(9 ) increases with the value of @xmath68 . thus , for the range of values of @xmath68 for which the exponential factor in eq.(6 ) is of order 1 , which is likely in our case , the nucleation rate may not decrease with larger values of @xmath68 , i.e. for the z(3 ) vacua with higher potential energies than the true vacuum . ( of course for very large values of @xmath68 the exponential term will suppress the nucleation rate . ) thus our assumption of neglecting quark effects for the bubble nucleation rate may not be unreasonable . we now consider the effect of non - zero @xmath131 , as in eq.(18 ) , on the evolution of closed domain walls . the temperature range relevant for our case is @xmath132 mev . in an earlier work we had numerically estimated the surface tension of z(3 ) walls to be about 0.34 and 7.0 gev/@xmath95 for @xmath94 = 200 and 400 mev respectively . the effects of quarks will be significant if a closed spherical wall ( with true vacuum inside ) starts expanding instead of collapsing . again , using the bubble free energy eq.(5 ) , with @xmath133 and @xmath55 as the surface energy of the interface , we see that the critical radius @xmath134 of the spherical wall is @xmath135 for @xmath94 = 200 mev and 400 mev we get @xmath136 and 1.5 fm respectively . though these values are not large , these are not too small either when considering the fact that relevant sizes and times for rhice are of order few fm anyway . the values of @xmath134 we estimated here are very crude as for these sizes wall thickness is comparable to @xmath134 hence application of eq.(5 ) , separating volume and surface energy contributions , is not appropriate . further , as we discussed above , the estimate in eq.(18 ) which is applicable for high temperature limit , seems an over estimate by about an order of magnitude at these temperatures . thus uncertainties of factors of order 1 may not be unreasonable to expect . in that case the dynamics of closed domain walls of even several fm diameter will not be affected by the effects of quarks via eq.(18 ) . we will see in the next section of simulation results that the domain walls and strings typically have large velocities ( e.g. about 0.5 - 0.8 ) at the time of formation . these result from momentum of colliding bubble walls and from curvature in the shape of these walls ( as well as asymmetries in the profiles of strings ) at the time of formation . with such large velocities present , the effects of pressure differences between different z(3 ) vacua due to quarks may become subdominant in studying the evolution of these structures for the short time duration available for rhice . with this , we will assume that for small closed walls , of order few fm diameter , as is expected in rhice , the quark effects in the evolution of wall network may be neglected . we plan to remove these assumptions and include the effects of non - zero @xmath131 due to quarks on bubble nucleation and wall evolution in a future work . as we mentioned above , the number of bubbles expected to form in rhice is small . we first present and discuss the case of 5 bubbles which is more realistic from the point of view of nucleation estimates given by eq.(6 ) ( though a gross over - estimate for eq.(10 ) ) . note that a domain wall will form even if only two bubbles nucleate ( with different z(3 ) vacua ) . however , to see qgp string formation , we need nucleation of at least three bubbles . next we will discuss the case of 9 bubbles which is a much more optimistic estimate of the nucleation rate ( even for eq.(6 ) ) . alternatively , this case can be taken as better representation of the case when the transition is a cross over and bubbles only represent a means for developing a domain structure expected after the cross - over is completed . ( in this case only relevant energy density fluctuations , as discussed below , will be those arising from z(3 ) walls and strings , and not the ones resulting from bubble wall coalescence . ) fig.3 - 7 show the results of simulation when five bubbles are nucleated with random choices of different z(3 ) vacua inside each bubble . fig.3 shows a time sequence of surface plots of the order parameter @xmath1 in the two dimensional lattice . fig.3a shows the initial profiles of the bubbles of the qgp phase embedded in the confining vacuum with @xmath44 at @xmath11 = 0.5 fm with the temperature @xmath66 200 mev . ( recall that for initial 1 fm time the temperature is taken to linearly increase from zero to @xmath60 = 400 mev . ) the radial profile of each bubble is truncated with appropriate care of smoothness on the lattice for proper time evolution . fig.3b shows the profile of each bubble at @xmath137 1.5 fm showing the expansion of the bubbles . near the outer region of a bubble the field grows more quickly towards the true vacuum . if bubbles expand for long time then bubble walls become ultra - relativistic and undergo large lorentz contraction . this causes problem in simulation ( see , e.g. @xcite ) . in our case this situation arises at outer boundaries ( for the inner regions bubbles collide quickly ) . for outer regions also it does not cause serious problem because of the use of dissipative boundary strip ( as explained above in sect.v ) . \(a ) and ( b ) show plots of profiles of @xmath77 at @xmath138 0.5 fm and 1.5 fm respectively . ( c)-(i ) show plots of @xmath139 at @xmath137 1.5 , 2.5 , 4.0 , 6.0 , 9.0 , 11.0 , and 13.7 fm . @xmath94 drops to below @xmath2 around at @xmath137 10.5 fm and @xmath66 167 mev at @xmath137 13.7 fm . formation of domain walls and string and antistring ( at junctions of three walls ) can be seen in the plots in ( e ) - ( h ) . [ fig.3 ] figs . 3c-3i show plots of @xmath139 clearly showing formation of domain walls and strings ( junctions of three walls ) . here @xmath40 is a reference vacuum expectation value of @xmath77 calculated at the maximum temperature @xmath140 400 mev . formation of domain walls , extending through the entire qgp region , is directly visible from fig.3e ( at @xmath141 fm ) onwards . the temperature drops to below @xmath142 mev at @xmath143 fm . the last plot in fig.3i is at @xmath144 fm when the temperature @xmath66 167 mev , clearly showing that the domain walls have decayed away in the confined phase and the field is fluctuating about @xmath44 . plots of the phase @xmath33 of the order parameter @xmath77 . ( a ) shows the initial distribution of @xmath33 in the bubbles at @xmath138 0.5 fm . ( b ) - ( f ) show plots of @xmath33 at @xmath137 2.0 , 4.6 , 11.0 , 12.2 , and 13.7 fm respectively . location of domain walls and the string ( with positive winding ) and antistring ( with negative winding ) are clearly seen in the plots in ( b)-(e ) . the motion of the antistring and associated walls can be directly seen from these plots and an estimate of the velocity can be obtained . [ fig.4 ] in fig.3f-3h we see two junctions of three domain walls where the qgp strings form . this is seen more clearly in fig.4 where the phase @xmath33 of @xmath77 is plotted ( with the convention that @xmath33 is the angle of the arrow from the positive @xmath145 axis ) . the domain walls are identified as the boundaries where two different values of @xmath33 meet , and strings correspond to the non - trivial winding of @xmath33 at the junctions of three walls . from fig.4b , c we clearly see that at one of the junctions we have a string ( at @xmath146 5 fm , @xmath147 8 fm ) with positive winding and we have an anti - string at @xmath146 9 fm , @xmath148 8 fm with negative winding . note the rapid motion of the walls forming the antistring towards positive @xmath149 axis from fig.4c ( at @xmath138 4.6 fm ) to fig.4e ( at @xmath138 12.2 fm ) . the average speed of the antistring ( and wall associated with that ) can be directly estimated from these figures to be about 0.5 ( in natural units with @xmath150 ) . this result is important in view of the discussion in the preceding section showing that effects of pressure differences between different z(3 ) vacua , arising from quarks , may be dominated by such random velocities present for the walls and strings at the time of formation . the motion of the walls here is a direct result of the straightening of the @xmath151 shaped wall structure due to its surface tension . for the same reason the wall in the left part of fig.4b also straightens from the initial wedge shape . fig.4f shows almost random variations of @xmath33 at @xmath144 fm when the temperature is 167 mev , well below the critical temperature . though it is interesting to note that a large region of roughly uniform values of @xmath33 still survives at this stage . surface plots of the local energy density @xmath152 in gev/@xmath130 . ( a ) - ( f ) show plots at @xmath137 3.0 , 3.6 , 5.0 , 6.0 , 8.0 , and 13.2 fm respectively . extended domain walls can be seen from these plots of @xmath153 in ( b ) - ( e ) . small peaks in @xmath153 exist at the locations of string and antistring ( larger peaks arise from oscillations of field where bubbles coalesce ) . plot in ( f ) is at the stage when @xmath66 169 mev and domain walls have decayed away . [ fig.5 ] fig.5 shows surface plots of the local energy density @xmath153 . @xmath153 is plotted in units of gev/@xmath130 . although the simulation is 2 + 1 dimensional representing the transverse plane of the qgp system , we calculate energy in 3 + 1 dimensions by taking a thickness of 1 fm in the central rapidity region . fig.5a shows plot at @xmath138 3 fm when bubbles have coalesced . in fig.5b at @xmath154 3.6 fm we see that bubble walls have almost decayed ( in ripples of @xmath77 waves ) between the bubbles with same @xmath155 ( i.e. same z(3 ) vacua ) as can be checked from @xmath33 plots in fig.4 . energy density remains well localized in the regions where domain walls exist . also , one can see the small peaks in the energy density where strings and antistrings exist . large peaks arise from oscillations of @xmath77 when bubble walls coalesce , as discussed in @xcite . large values of @xmath153 near the boundary of the lattice are due to relativistically expanding bubble walls . motion of walls and generation of increased fluctuations in energy density are seen in fig.5c-5e . fig.5f at @xmath138 13.2 fm ( with @xmath94 = 169 mev ) shows that walls have decayed . however , some extended regions of high energy density can be seen at this stage also . contour plots of the local energy density @xmath153 at different stages . plots in ( a ) - ( f ) correspond to @xmath137 3.0 , 3.6 , 5.0 , 7.6 , 10.2 , and 13.2 fm respectively . structure of domains walls formed near the coalescence region of bubbles with different @xmath33 is clear in ( b ) , whereas the bubble walls at lower half of @xmath149 region , and near @xmath156 10 fm , are seen to simply decay away due to same vacuum in the colliding bubbles . motion of the antistring and associated domain walls is clear from plots in ( b ) - ( e ) . the last plot in ( f ) is at @xmath157 fm when @xmath94 = 169 mev . [ fig.6 ] fig.6 shows contour plots of energy density @xmath153 . fig.6a shows coalescence of bubbles at @xmath137 3 fm . fig.6b at @xmath154 3.6 fm clearly shows the difference in the wall coalescence depending on the vacua in the colliding bubbles . where domain walls exist , we see extended regions of high energy density contours whereas where the two vacua in colliding bubbles are same , there are essentially no high energy density contours . motion of domain walls ( and strings at wall junctions ) is clearly seen in these contour plots in fig.6b-6d . fig.6e is at @xmath138 10.2 fm when @xmath94 drops to @xmath2 . wall structures are still present . fig.6f is at @xmath137 13.2 fm ( @xmath94 = 169 mev ) when walls have decayed away , though some extended structures in contours still survive . we have also calculated the variance of energy density @xmath158 at each time stage to study how energy fluctuations change during the evolution . in fig.7 we show the plot of @xmath159 as a function of proper time . here @xmath153 is the average value of energy density at that time stage . the energy density @xmath153 decreases due to longitudinal expansion , hence we plot this ratio to get an idea of relative importance of energy density fluctuations . fig.7 shows initial rapid drop in @xmath159 due to large increase in @xmath153 during the heating stage upto @xmath11 = 1 fm , followed by a rise due to increased energy density fluctuations during the stage when bubbles coalesce and bubble walls decay , as expected . interesting thing to note is a slight peak in the plot near @xmath138 10.5 fm when @xmath94 drops below @xmath2 . this should correspond to the decay of domain walls and may provide a signal for the formation and subsequent decay of such objects in rhice . plot of the ratio of variance of energy density @xmath160 and the average energy density @xmath153 as a function of proper time . energy fluctuations increase during the initial stages when bubbles coalesce and bubble walls decay . after that there is a slow decrease in energy fluctuation until the stage when the temperature drops below @xmath2 and @xmath143 fm . energy fluctuations increase after this stage . note small peak near the transition stage . [ fig.7 ] fig.8 - 12 show the results of simulation where nine bubbles are nucleated . fig.8 shows a time sequence of surface plots of @xmath1 ( similar to fig.3 ) . fig.8a shows the initial profiles of the qgp bubbles at @xmath11 = 0.5 fm with the nucleation temperature of @xmath66 200 mev . fig.8b shows the profile of @xmath77 for the bubbles at @xmath137 1.5 fm showing the expansion of the bubbles . fig.8c-8i show plots of @xmath139 at different stages . noteworthy here is the formation of a closed domain wall near the central region which is clearly first seen in fig.8e at @xmath137 5 fm . the collapse of this closed domain wall is seen in the subsequent plots with the closed wall completely collapsing away in fig.8h at @xmath137 9.6 fm . only surviving structure is an extended domain wall along the @xmath145 axis . fig.8i is at @xmath157 fm when @xmath66 169 mev . the domain walls have decayed away and @xmath77 fluctuates about the value zero as appropriate for the confined phase . \(a ) and ( b ) show plots of profiles of @xmath77 at @xmath138 0.5 fm and 1.5 fm respectively for the case when 9 bubbles are nucleated . ( c)-(i ) show plots of @xmath139 at @xmath137 2.0 , 3.0 , 5.0 , 7.0 , 8.6 , 9.6 , and 13.2 fm respectively . formation of a closed domain wall is first clearly seen in the plot in ( e ) . this closed domain wall collapses as seen in plots in ( e ) through ( h ) . only surviving domain wall is an extended wall along @xmath145 axis in ( h ) . plot in ( i ) is when the temperature @xmath66 169 mev . [ fig.8 ] fig.9 shows plots of the phase @xmath33 of @xmath77 at different stages . initial phase distribution in different bubbles is shown in fig.9a at @xmath161 fm . fig.9b shows the formation of closed , elliptical shaped domain wall at @xmath162 fm . strings and antistrings can also be identified by checking the windings of @xmath33 . the closed domain wall collapses , and in the process becomes more circular , as shown in the plots in fig.9b-9h . fig.9h shows the plot at @xmath137 9.6 fm when the closed domain wall completely collapses away , leaving only an extended domain wall running along @xmath145 axis between @xmath163 fm . the final fig.9i at @xmath164 fm is when the temperature @xmath165 167 mev showing random fluctuations of @xmath33 when domain walls have decayed away in the confining phase . plots of the phase @xmath33 of @xmath77 . ( a ) shows initial distribution of @xmath33 in the bubbles at @xmath138 0.5 fm . ( b ) - ( i ) show plots of @xmath33 at @xmath137 2.6 , 5.0 , 6.0 , 7.0 , 8.0 , 8.6 , 9.6 , and 13.8 fm respectively . ( b ) shows formation of elliptical shaped closed domain wall which subsequently becomes more circular as it collapses away by @xmath137 9.6 fm as shown by the plot in ( h ) . the plot in ( i ) is at @xmath166 mev showing random fluctuations of @xmath33 . [ fig.9 ] fig.10 shows the surface plot of energy density @xmath153 at different stages . extended thin regions of large values of @xmath153 are clearly seen in the plots corresponding to domain walls . collapse of the closed domain wall is also clearly seen in fig.10c-10 g . important thing to note here is the surviving peak in the energy density plot at the location of domain wall collapse . this peak survives even at the stage shown in fig.10i at @xmath157 fm when @xmath167 mev , well below the transition temperature . such _ hot spots _ may be the clearest signals of formation and collapse of z(3 ) walls . surface plots of the local energy density @xmath153 in gev/@xmath130 . ( a ) - ( i ) show plots at @xmath137 2.6 , 3.5 , 4.6 , 5.6 , 7.0 , 8.6 , 9.6 , 11.2 , and 13.2 fm respectively . formation and subsequent collapse of closed domain wall is clearly seen in plots in ( c ) through ( g ) . note that the strong peak in @xmath153 resulting from domain wall collapse ( the _ hot spot _ ) survives in ( i ) when @xmath94 = 169 mev . [ fig.10 ] contour plots of @xmath153 are shown in fig.11 . though closed domain wall can be seen already in fig.11b ( at @xmath137 3.5 fm ) , the domain wall is still attached to outward expanding bubble walls near @xmath156 3 fm , @xmath168 12 fm which affects the evolution / motion of that portion of the domain wall . formation of distinct closed wall structure is first visible in fig.11c at @xmath1384.6 fm . subsequent plots clearly show how the domain wall becomes circular and finally collapses away by fig.11k at @xmath169 fm . note the survival of the _ hot spot _ even at the stage shown in fig.11_l _ at @xmath157 fm when @xmath66 169 mev . one can make a rough estimate of the velocity of the closed wall during its collapse from these plots . in fig.11c , at @xmath137 4.6 fm , the @xmath145 extent of the closed wall is about 8 fm and the @xmath149 extent is about 5 fm . the wall collapses away by the stage in fig.11k at @xmath137 9.6 fm . this gives rough velocity of collapse in @xmath145 direction to be about 0.8 while the velocity in @xmath149 direction is about 0.5 . note that here , as well as in fig.4 for the five bubble case , the estimate of the wall velocity is not affected by the extra dissipation which is introduced only in a very thin strip ( consisting of ten lattice points ) near the lattice boundary . formation and collapse of such closed domain walls is important as the resulting hot spot can lead to important experimental signatures . further , such closed domain wall structures are crucial in the studies of @xmath3 enhancement , especially for heavy flavor hadrons as discussed in @xcite contour plots of the local energy density @xmath153 at different stages . plots in ( a ) - ( l ) correspond to @xmath137 2.6 , 3.5 , 4.6 , 6.0 , 6.6 , 7.0 , 7.6 , 8.0 , 8.6 , 9.0 , 9.6 , and 13.2 fm respectively . formation of distinct closed wall structure is first visible in ( c ) at @xmath1384.6 fm . subsequent plots show the collapse of this domain wall as it becomes more circular . the wall finally collapses away in ( k ) at @xmath169 fm . note that concentration of energy density at the location of domain wall collapse ( the _ hot spot _ ) survives even at the stage shown in ( l ) at @xmath157 fm when @xmath66 169 mev . [ fig.11 ] fig.12 shows the evolution of the ratio of the variance of energy density and the average energy density . as for the five bubble case , initial drop and rise are due to heating stage upto @xmath137 1 fm and subsequent bubble coalescence and decay of bubble walls . in this case the ratio remain roughly constant upto @xmath170 10.5 fm which is the transition stage to the confining phase . this is the stage when the surviving extended domain wall starts decaying . this is also the stage soon after the closed domain wall collapses away . the prominent peak at this stage should be a combined result of both of these effects . the large increase in the variance of energy density at this stage should be detectable from the analysis of particle distributions and should be a clear signal of hot spots resulting from collapse of closed walls and the decay of any surviving domain walls . plot of the ratio of variance of energy density @xmath158 and the average energy density @xmath153 as a function of proper time . energy fluctuations increase during the initial stages when bubbles coalesce and bubble walls decay . after that @xmath159 remains roughly constant until the stage when the temperature drops below @xmath2 at @xmath143 fm . this is also the stage just after the collapse of the closed domain wall . energy fluctuations sharply increase around this stage . note the prominent peak at this stage . the @xmath0 wall network and associated strings form during the early confinement - deconfinement phase transition . they undergo evolution in an expanding plasma with decreasing temperature , and eventually melt away when the temperature drops below the deconfinement - confinement phase transition temperature . they may leave their signatures in the distribution of final particles due to large concentration of energy density in extended regions as well as due to non - trivial scatterings of quarks and antiquarks with these objects . first , we focus on the extended regions of high energy density resulting from the domain walls and strings . this is clearly seen in our simulations and some extended structures / hot spots also survive after the temperature drops below the transition temperature @xmath2 . note that even the hot spot resulting from the collapse of closed domain wall in fig.9,10 will be stretched in the longitudinal direction into an extended linear structure ( resulting from the collapse of a cylindrical wall ) . we know that at rhic energies , the final freezeout temperature is not too far below the transition temperature @xmath2 . this means that the energy density concentrated in any extended ( sheet like for domain walls and line like for strings / hot spots ) regions may not be able to defuse away effectively . assuming local energy density to directly result in multiplicity of particles coming from that region , an analysis of particle distribution in @xmath3 and in rapidity should be able to reflect any such extended regions . in this context , it will be interesting to investigate if the ridge phenomenon seen at rhic @xcite could be a manifestation of an underlying z(3 ) domain wall / string structure . correlation of particle production over large range of rapidity will naturally result from longitudinally extended regions of high energy density ( _ hot spots _ in the transverse plane ) . combined with flow effects it may lead to ridge like structures @xcite . if extended domain wall structure survives in the transverse plane also , this will then extend to sheet like regions in the longitudinal direction . decay of such a region of high energy density may directly lead to a ridge like structure , without requiring flow effects . we expect non - trivial signatures resulting from the consideration of interactions of quarks and antiquarks with domain walls . it was shown in an earlier work @xcite using generic arguments that quarks and antiquarks should have non - zero reflection coefficients when traversing across these domain walls . a collapsing domain wall will then concentrate any excess baryon number enclosed , leading to formation of baryon rich regions . this is just like witten s scenario for the early universe @xcite ( which was applied for the case of rhice in ref . however , for these works it was crucial that the quark - hadron transition be of ( strong ) first order . as we have emphasized above , in our case formation of z(3 ) walls and strings will be a generic feature of any c - d phase transition . even though we have implemented it in the context of a first order transition via bubble nucleation , these objects will form even if the transition is a cross - over . thus , concentration of baryons in small regions should be expected to occur in rhice which should manifest in baryon concentration in small regions of rapidity and @xmath3 . another important aspect of quark / antiquark reflection is that inside a collapsing wall , each reflection increases the momentum of the enclosed particle . when closed domain walls collapse then enclosed quarks / antiquarks may undergo multiple reflections before finally getting out . this leads to a specific pattern of @xmath3 enhancement of quarks with heavy flavors showing more prominent effects @xcite . the modification of @xmath3 spectrum of resulting hadrons can be calculated , and the enhancement of heavy flavor hadrons at high @xmath3 can be analyzed for the signal for the formation of z(3 ) domain walls in these experiments @xcite . in our simulations extended domain walls also form which show bulk motion with velocities of order 0.5 . quarks / antiquarks reflected from such moving extended walls will lead to anisotropic momentum distribution of emitted particles which may also provide signature of such walls . for collapsing closed domain walls , spherical domain walls were used for estimates in ref . @xcite and in ref . our simulation in the present work provides a more realistic distribution of shapes and sizes for the resulting domain wall network . we have estimated the velocity of moving domain walls to range from 0.5 to 0.8 for the situations studied . these velocities are large enough to have important effect on the momentum of quarks / antiquarks undergoing reflection from these walls . one needs to combine the analysis of @xcite with the present simulation to get a concrete signature for baryon concentration and heavy flavor hadron @xmath3 spectrum modification . we plan to carry this out in a future work . we also plan to study effects of spontaneous violation of cp due to formation of these z(3 ) walls in rhice . our results show interesting pattern of the evolution of the fluctuations in the energy density . as seen in fig.7 and fig.12 , energy density fluctuations show rapid changes during stages of bubble wall coalescence and during collapse / decay of domain walls . even string - antistring annihilations should be contributing to these fluctuations . fluctuations near the transition stage may leave direct imprints on particle distributions . it is intriguing to think whether dileptons or direct photons may be sensitive to these fluctuations , which could then give a time history of evolution of such energy density fluctuations during the early stages as well . even the presence of domain walls and strings during early stages may affect quark - antiquark distributions in those regions which may leave imprints on dileptons / direct photons . an important point to note is that in our model , we expect energy density fluctuations in event averages ( representing high energy density regions of domain walls / strings as discussed above ) , as well as event - by - event fluctuations . these will result due to fluctuation in the number / geometry of domain walls / strings from one event to the other resulting from different distribution of ( randomly occurring ) z(3 ) vacua in the qgp bubbles . even the number of qgp bubbles , governed by the nucleation probability , will vary from one event to the other contributing to these event - by - event fluctuations . we have carried out numerical simulation of formation of @xmath0 interfaces and associated strings at the initial confinement - deconfinement phase transition during the pre - equilibrium stage in relativistic heavy - ion collision experiments . a simple model of quasi - equilibrium system was assumed for this stage with an effective temperature which first rises ( with rapid particle production ) to a maximum temperature @xmath4 , and then decreases due to continued plasma expansion . using the effective potential for the polyakov loop expectation value @xmath1 from ref . @xcite we study the dynamics of the ( c - d ) phase transition in the temperature / time range when the first order transition of this model proceeds via bubble nucleation . as we have emphasized above , though our study is in the context of a first order transition , its results are expected to be valid even when the transition is a cross - over . ( though for non - zero chemical potential the transition may indeed be of first order ) . the generic nature of our results arises due to the fact that the formation of z(3 ) domain walls and associated strings happens due to the a general domain structure resulting after any transition ( occurring in a finite time ) . this is the essential physics of the kibble mechanism underlying the formation of topological defects in symmetry breaking transitions . the @xmath0 wall network and associated strings formed during this early c - d transition are evolved using field equations in a plasma which is longitudinally expanding , with decreasing temperature . we have neglected here the transverse expansion which is a good approximation for the early stages near the formation stage of these objects , but may not be a good approximation for the later parts of simulations when temperature drops below @xmath2 and z(3 ) domain walls and strings melt away . we have studied size / shape of resulting closed domain wall as well as extended domain walls and have estimated the velocities of walls to range from 0.5 to 0.8 . we also calculate the energy density fluctuations expected due to formation of these objects . various experimental signals which can indicate the formation of these topologically non - trivial objects in rhice have been discussed . for example , existence of these objects will result in specific patterns of energy density fluctuations which may leave direct imprints on particle distributions . in our model , we expect energy density fluctuations in event averages ( representing high energy density regions of domain walls / strings ) , as well as event - by - event fluctuations as the number / geometry of domain walls / strings and even the number of qgp bubbles , varies from one event to the other . extended regions of large energy densities arising from z(3 ) walls and associated strings may be manifested in space - time reconstruction of hadron density ( using hydrodynamic model ) . the correlation of particle production over large range of rapidity will be expected from such extended regions . this , combined with the flow effects ( for string like regions ) , or possibly directly ( for sheet like extended region ) may provide an explanation for the ridge phenomena observed at rhic @xcite . also , from the reflection of quarks and antiquarks from collapsing domain walls , baryon number enhancement in localized regions ( due to concentration of net baryon number ) as well as enhancement of heavy flavor hadrons at high @xmath3 is expected . we emphasize again that the presence of @xmath0 walls and string may not only provide qualitatively new signatures for the qgp phase in these experiments , it may provide the first ( and may be the only possible ) laboratory study of such topological objects in a relativistic quantum field theory system . we are very grateful to sanatan digal , anjishnu sarkar , ananta p. mishra , p.s . saumia , and abhishek atreya for very useful comments and suggestions . usg , ams , and vkt acknowledge the support of the department of atomic energy- board of research in nuclear sciences ( dae - brns ) , india , under the research grant no 2008/37/13/brns . usg and vkt acknowledge support of the computing facility developed by the nuclear - particle physics group of physics department , allahabad university under the center of advanced studies ( cas ) funding of ugc , india . u. w. heinz , lectures given at 2nd latin american school of high - energy physics , san miguel regla , mexico , 1 - 14 jun 2003 , published in _ san miguel regla 2003 , high - energy physics _ 165 . e - print : hep - ph/0407360 ; j. adams et al . [ star collaboration ] , j. phys . * g 32 * , l37,(2006 ) [ arxiv : nucl - ex/0509030 ] ; j.putschke , talk at quark matter 2006 for star collaboration ) , shanghai , nov.2006 ; s. a. voloshin , phys . lett . * b 632 * , 490 ( 2006)[arxiv : nucl - th/0312065 ] .
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we study the dynamics of confinement - deconfinement ( c - d ) phase transition in the context of relativistic heavy - ion collisions within the framework of effective models for the polyakov loop order parameter .
we study the formation of @xmath0 walls and associated strings in the initial transition from the confining ( hadronic ) phase to the deconfining ( qgp ) phase via the so called kibble mechanism .
essential physics of the kibble mechanism is contained in a sort of domain structure arising after any phase transition which represents random variation of the order parameter at distances beyond the typical correlation length .
we implement this domain structure by using the polyakov loop effective model with a first order phase transition and confine ourselves with temperature / time ranges so that the first order c - d transition proceeds via bubble nucleation , leading to a well defined domain structure . the formation of @xmath0 walls and associated strings results from the coalescence of qgp bubbles expanding in the confining background .
we investigate the evolution of the @xmath0 wall and string network .
we also calculate the energy density fluctuations associated with @xmath0 wall network and strings which decay away after the temperature drops below the quark - hadron transition temperature during the expansion of qgp . we discuss evolution of these quantities with changing temperature via bjorken s hydrodynamical model and discuss possible experimental signatures resulting from the presence of @xmath0 wall network and associate strings .
key words : quark - hadron transition , relativistic heavy - ion collisions , z(3 ) domain walls
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the discovery of hot jupiters that transit in front of their parent stars has advanced our knowledge of extrasolar planets adding a fundamental datum : the planetary radius . there has been considerable activity revising the measured radii , owing to uncertainties in the differential image analysis ( see pont et al . 2006 ) . it is important to obtain accurate radii from photometry , in order to compare these exoplanets with the giant planets of the solar system , and with the models . in addition , if accurate photometry of transits is available , one can use timing for future studies of multiplicity in these systems ( e.g. sartoretti & schneider 1999 , miralda - escude 2002 , holman & murray 2005 , agol et al . 2005 ) . new samples of transiting hot jupiters should become available soon ( see for example fischer et al . 2005 , sahu et al . 2006 ) , but up to now the ogle search has provided the largest number of transiting candidates . in particular , udalski et al . ( 2002 ) discovered very low amplitude transits in the @xmath5 , @xmath6 magnitude star ogle - tr-111 , located in the carina region of the milky way disk , at @xmath7 , @xmath8 . they monitored 9 individual transits , measuring an amplitude @xmath9 mag , and a period @xmath10 days . the period is a near - multiple of a day , therefore , the window for transit observations is restricted to a couple of months per year . the planet ogle - tr-111-b was discovered by pont et al . ( 2004 ) with precise velocity measurements . they measured @xmath11 , @xmath12 , and @xmath13 . they call this planet the `` missing link '' because of the relatively long period , which overlaps with the planets discovered by radial velocity searches . ogle - tr-111-b is one of the least irradiated known transiting extrasolar planets , ( baraffe et al . 2005 , laughlin et al . 2005 ) , and therefore it is also an interesting case to study because it may probe the transition region between strongly irradiated and isolated planets . we have previously carried out a selection of the most promising ogle planetary candidates using low dispersion spectroscopy in combination with optical and near - infrared photometry ( gallardo et al . 2005 ) . this work identified ogle - tr-111 as one of the most likely candidates to host exoplanets . gallardo et al . ( 2005 ) classify ogle - tr-111 as a k - type main sequence star with @xmath14 k , located at a distance @xmath15 pc , with magnitudes @xmath5 , @xmath6 , and @xmath16 , and reddening @xmath17 . their low dispersion spectrum shows strong mgb band characteristic of a metal - rich dwarf . they find that this star is intrinsically fainter ( @xmath18 ) , and smaller ( @xmath19 ) than the sun . based on the high dispersion spectroscopy , pont et al . ( 2004 ) derive similar stellar parameters for ogle - tr-111 : temperature @xmath20 k , gravity @xmath21 , mass @xmath22 , radius @xmath23 , and metallicity @xmath24=0.12 $ ] dex . the stellar parameters were further improved by santos et al . ( 2006 ) , based on high s / n spectra , deriving @xmath25 , @xmath26 , and @xmath24=+0.19 \pm 0.07 $ ] , and assume @xmath27 . the values from these independent studies agree within the uncertainties . the known planetary parameters are in part based on the ogle photometry . there has been recent revisions of the radii of other confirmed ogle planets using high cadence , high s / n photometry with large telescopes ( see pont et al . recently , winn et al . ( 2006 ) presented accurate photometry of two transits for ogle - tr-111 in the @xmath28-band , revising the ephemeris , obtaining a period @xmath29 , and measuring the system parameters , including an accurate stellar radius @xmath30 , and planet radius @xmath31 . this planet radius is @xmath32 larger than the recent value of santos et al . ( 2006 ) . in this paper we present new high cadence @xmath0-band photometry covering a transit of ogle - tr-111 , giving an independent determination of the planetary radius , and deriving an accurate period for the system . the observations and photometry are described by fernndez et al . ( 2006 ) and daz et al . the photometric observations were taken with vimos at the unit telescope 4 ( ut4 ) of the european southern observatory very large telescope ( eso vlt ) at paranal observatory during the nights of april 9 to 12 , 2005 . the vimos field of view consists of four ccds , each covering 7@xmath338 arcmin , with a separation gap of 2 arcmin , and a pixel scale of 0.205 arcsec / pixel . the large field of view of this instrument allows to monitor simultaneously a number of ogle transit candidates , in comparison with fors at the vlt , which has a smaller field of view ( fernndez et al . however , for high precision photometry of an individual candidate fors should be preferred because its finer pixel scale allows better sampling ( e.g. pont et al . 2006 ) . here we report on the observations of ogle - tr-111 , which was located in one of the four monitored fields , and it happened to have a transit during the first night of our run . we used the bessell @xmath0 filter of vimos , with @xmath34 , @xmath35 . the @xmath0-band was chosen in order to complement the ogle light curves which are made with the @xmath28-band filter . in addition , the @xmath0-band is more sensitive to the effects of limb darkening during the transit , and is adequate for the modeling of the transit parameters . we have monitored two fields on april 9 , 2005 , one of which included the star ogle - tr-111 . the fields were observed alternatively with three exposures of 15s before presetting to the next field . for this program we managed to reduce the observation overheads for telescope presets , instrument setups , and the telescope active optics configuration to an absolute minimum . this ensured adequate sampling of the transit : we obtained 224 points during the first night in the field of ogle - tr-111 . the observations lasted for about 9.5 hours , until the field went below 3 airmasses . in order to reduce the analysis time of the vast dataset acquired with vimos , the images of ogle - tr-111 analyzed here are 400@xmath33400 pix , or 80 arcsec on a side . each of these small images contains about 500 stars with @xmath36 that can be used in the difference images , and light curve analysis . the 7 best seeing images ( @xmath37 ) taken near the zenith were selected , and a master image was made in order to serve as reference for the difference image analysis ( see alard 2000 , alard & lupton 1998 ) . the candidate star is not contaminated by faint neighbours , judging from our deep @xmath0 and @xmath38-band images . there is a @xmath39 mag star 2 arcsec south of our target , which does not affect our photometry . the difference image photometry yields a noisier light curve with a low amplitude transit , and we decided to apply the ogle pipeline dia ( udalski et al . this new reduction showed significantly reduced photometric scatter . for ogle - tr-111 , with a mean visual magnitude @xmath5 , we achieved a photometric accuracy of @xmath40 @xmath41 magnitudes . the errors mainly depend on the image quality , which was worse at the beginning and end of the time series , when the target had large airmass . figure 1 shows the light curve for the first night of observations , when the ogle - tr-111 transit was monitored . every single datapoint is shown , no mesurement is discarded . for comparison , figure 1 also shows the phased light curve of the ogle @xmath28-band photometry ( in a similar scale ) . the transit is well sampled in the @xmath0-band , and the scatter is smaller . there are @xmath42 points in our single transit shown in figure 1 , and the minimum is well sampled , allowing us to measure an accurate amplitude . in the case of ogle , the significance of the transits is in part judged by the number of transits detected . in the case of the present study , we compute the signal - to - noise of the single , well sampled transit . for ogle - tr-111 we find the s / n of this transit to be @xmath43 following gaudi ( 2005 ) . however , this does not include systematic effects ( red noise ) , which we consider below . after the ogle 2002 transit campaign , the field of ogle - tr-111 was observed by the ogle survey regularly but less frequently with the main aim of improving the ephemeris . altogether more than @xmath44 new epochs were collected in the observing seasons 20032005 . unfortunately , there were no eclipses observed between 2002 and april 2005 , when ogle started recovering eclipses . this is because this system has a near multiple of a day period . the full ogle photometric dataset covering almost 350 cycles made it possible to significantly refine the ephemeris of ogle - tr-111 : @xmath45 ( udalski et al . 2005 , private communication ) . the vimos transit reported here also agrees with this ephemeris , as discussed in section 5 . in light of the high dispersion follow up of this target by pont et al . ( 2004 ) that confirmed the low mass of ogle - tr-111-b , we can consider that the transit measured here is representative of all transits of ogle - tr-111-b . unfortunately , just a single transit was measured here , and only in the @xmath0-band filter , but it is of value because we used a different passband that previous works and can therefore independently check the parameters for the system . for example , the stellar limb darkening is different from the @xmath28-band , with transits that are shallower at the edges but about @xmath46 deeper in the central parts ( e.g. claret & hauschildt 2003 ) . there are a few published measurements of the radius of the ogle - tr-111 companion , based on the ogle photometric data ( table 1 ) . udalski et al . ( 2002 ) measured @xmath9 mag , and estimated @xmath47 for ogle - tr-111 , and a lower limit of @xmath48 for its companion , arguing this was one of the most promising extrasolar planetary transit candidates . based on a fit to the same ogle data , silva & cruz ( 2006 ) estimated @xmath49 for the companion . pont et al . ( 2004 ) give @xmath50 for ogle - tr-111 , and @xmath51 for its companion , also using on the ogle data . gallardo et al . ( 2005 ) measure @xmath52 for ogle - tr-111 , and @xmath53 for its companion , also based on the ogle amplitude . figure 2 shows our best fit to the transit curve , following mandel & agol ( 2002 ) , using appropriate limb - darkenning coefficients for the @xmath0-band . this fit yields @xmath54 mag , @xmath55 , and @xmath56 . using @xmath57 gives @xmath58 . the uncertainties of the fit parameters were estimated from the @xmath59 surface . as shown by pont ( 2006 ) , the existence of covariance between the observations produces a low - frequency noise which must be considered to obtain a realistic estimation of the uncertainties . to model the covariance we followed gillon et al . ( 2006 ) and obtained an estimate of the systematic errors in our observations from the residuals of the lightcurve . the amplitude of the white ( @xmath60 ) and red ( @xmath61 ) noise can be obtained by solving the equation system presented in their equations ( 5 ) and ( 6 ) . repeating our procedure for similar vlt data on ogle - tr-113 ( daz et al . 2006 ) , we estimated the white noise amplitude , and the low - frequency red noise amplitude , and then , the surface which determines the uncertainty interval . the projections of the @xmath59 surface to estimate the uncertainties of the fit parameters are shown in figure 3 , as done by daz et al . ( 2006 ) for ogle - tr-113 . the light curve was also fitted to obtain the transit time , by fixing the other parameters ( a , r@xmath62 and i ) . for example , the regions for @xmath63 ( only white noise , without systematics ) , and @xmath64 ( with white + red noise , including systematics ) for the fit parameters as function of the transit time are marked in figure 4 . at this point , the major uncertainty on the ogle - tr-111-b planetary radius arises from the uncertainties in the stellar properties . note that we do not fit the star radius simultaneously using the photometry as done by winn et al . ( 2006 ) . adopting @xmath19 from gallardo et al . ( 2005 ) , we obtain @xmath65 . adopting @xmath57 from santos et al . ( 2006 ) , we obtain @xmath66 . the unweighted mean of these two independent determinations is @xmath67 . however , we will adopt the spectroscopic determination of santos et al . ( 2006 ) of @xmath57 , instead of the one from gallardo et al . ( 2005 ) for two main reasons . first , the santos spectroscopic data are more recent and of higher quality , resulting on a complete analysis of the composition and stellar parameters based on high dispersion spectroscopy , while gallardo give an indirect determination based on the surface brightness . second , in order to allow a direct comparison with the results of winn et al . ( 2006 ) , who also adopt the mass from santos et al . ( 2006 ) . table 1 lists the previous estimates of the size for the ogle - tr-111 transiting planet @xmath68 from the literature , and this work , along with the stellar parameters . the agreement of the most recent values to within about @xmath32 implies that the radius of this planet is known . the unweighted mean of the radii measured by santos et al . ( 2006 ) , winn et al . ( 2006 ) , and this work is : @xmath69 with this radius , ogle - tr-111-b does not seem to be oversized for its mass , its gross properties ( mass , radius , mean density ) being similar to the jovian planets of the solar system , as listed for example by guillot ( 2005 ) . ogle - tr-111-b is the least irradiated of the known transiting extrasolar planets , with equilibrium temperature @xmath70 ( baraffe et al . 2005 , laughlin et al . 2005 , lecavelier des etangs 2006 ) . it lies at the cool end of the distribution of the other transiting hot jupiters ( @xmath71 ) , but it is still warmer than the solar system giants ( @xmath72 ) . thus , ogle - tr-111-b is not only a missing link regarding its orbital properties , as suggested by pont et al . ( 2004 ) , but also may probe the transition between strongly irradiated and more isolated planets . it is interesting to compare ogle - tr-111-b with hd209458b , which has a similar mass ( @xmath73 ) , and orbital semimajor axis ( @xmath74 ) . yet the radius of hd209458b is about 40% larger than the radius of ogle - tr-111-b measured here ( laughlin et al . 2005 , baraffe et al . 2005 ) . two main effects could produce this large difference . first , hd209458b is inflated by stellar irradiation , which is smaller for ogle - tr-111-b . the difference in incident flux is a factor of 4 according to baraffe et al . this irradiation difference is mostly because the primary star hd209458 ( @xmath75 ) , is hotter than ogle - tr-111 ( @xmath76 ) . second , the presence of a massive solid core in ogle - tr-111-b might make its radius smaller in comparison with hd209458b . models predict that the presence of a massive solid core should reduce the radius of a giant planet significantly ( saumon et al . 1996 , burrows et al . 2003 , bodenheimer et al . 2003 , sato et al . 2006 , guillot et al . for example , the reduction in radius is @xmath77% for a @xmath78 planet with a @xmath79 core with respect to a planet with a small core . it has been recently realized that it is very important to measure the transit times accurately , because of the exciting possibility of using these times for future studies of multiplicity in these systems ( holman & murray 2005 , agol et al . 2005 ) . winn et al . ( 2006 ) measure two transits accurately , giving an improved period of : @xmath80 the mean transit times measured by winn et al . ( 2006 ) are : @xmath81= 2453787.70854 \pm 0.00035,$ ] and @xmath81= 2453799.75138 \pm 0.00030.$ ] these transits are separated by 3 cycles , and we use their mean along with our own observations to compute a more accurate period . the vlt transit occurred 80.5 cycles before the mean of the transits observed by winn et al . the mean transit time measured at the vlt is : @xmath81= 2453470.56397 \pm 0.00076,$ ] and the final period measured here is : @xmath82 this period is independent of the ogle photometry , but it is consistent with the ogle data . the errors include the systematic errors ( figure 4 ) . therefore , our improved ephemeris for the mean transit times of ogle - tr-111-b is : @xmath83 where the numbers in parenthesis indicate the errors in the last two digits of their respective quantities . adopting the ephemeris of winn et al . ( 2006 ) , the mean predicted time is off from the center of our transit , which occurs about 5 minutes earlier . in fact , the period determined here is more than @xmath84 away from the period measured by winn et al . ( 2006 ) , or @xmath85 away using our larger errorbars ( that include systematics ) . nonetheless , we believe that it is accurate , because it relies on their two well sampled transits as well as our single transit . interestingly , there is a difference with the previous measured periods.such difference can arise in the presence of another massive planet in the system , which is the main motivation for measuring accurate timing of this planet . with the present data , however , it can not be claimed that this is the case : we can not yet rule out a constant period . more transits should be accurately measured in the following seasons to follow up this interesting system and confirm or rule out variations in the mean transit times . due to the period very close to @xmath86 , about 20 consecutive full transits can be observed during a season for a period of three months , and then they are not observable again for a period of about six months . with the new ephemeris we are able to predict that the next observable series of transits of ogle - tr-111 occur around december 2006 , and then again in mid-2008 , as shown in figure 5 . it is evident that the window of opportunity for accurate photometric measurements of the transits for this target is small . an alternative way to track another massive planet in the system is to perform radial velocity follow - up with no critical time of observational windows , even though this method is time consuming and difficult for such a faint candidate . udalski et al . ( 2002 ) discovered low amplitude transits in the main sequence star ogle - tr-111 , which we observed with vimos at the eso vlt . the planet ogle - tr-111-b is a massive planet with mass @xmath2 ( pont et al . we were able to accurately sample one low amplitude transit on this star with high cadence observations , complementing the recent measurement of two transits by winn et al . we improve upon the parameters of the transit , in particular measuring the transit time , @xmath87 hours , the orbital inclination @xmath88 , and the orbital semimajor axis @xmath89 . these data are in agreement with the orbital parameters measured by radial velocities and with the stellar parameters . \(1 ) we measure an accurate radius based on vlt transit photometry and revised stellar parameters . we obtain @xmath66 , which for a mass of @xmath11 , gives a density of @xmath3 . thus , the planet of ogle - tr-111 has a radius similar to jupiter , and a mean density that resembles that of saturn . the ogle - tr-111-b planet does not appear to be significantly inflated by stellar radiation like hd209458-b . \(2 ) using newly available transits from winn et al . ( 2006 ) , we are able to measure accurately the orbital period of this interesting system , @xmath90 as well as to update the ephemeris . the timing is different from previously published values , but there is not yet sufficient data to claim time variations caused by another massive planet in the system . follow up of the ogle - tr-111-b transits is warranted in the next observable transit windows around december 2006 and mid-2008 . llllll udalski + 2002 & @xmath91 & @xmath92 & @xmath93 & lower limit , 9 transits , @xmath28-band phot.pont + 2004 & @xmath94 & @xmath95 & @xmath96 & using ogle phot . , new stellar par.gallardo + 2005& & @xmath97 & @xmath98 & using ogle amp . , new stellar par.silva + 2006 & @xmath99 & & @xmath100 & re - analysis of ogle phot.santos + 2006 & @xmath101 & @xmath102 & @xmath103 & using ogle phot . , new stellar par.winn + 2006 & @xmath101 & @xmath104 & @xmath105 & two transits , @xmath28-band phot.this work & @xmath101 & @xmath104 & @xmath106 & single transit in @xmath0-band
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we present accurate @xmath0-band photometry for a planetary transit of ogle - tr-111 acquired with vimos at the eso very large telescope . the measurement of this transit allows to refine the planetary radius , obtaining @xmath1 .
given the mass of @xmath2 previously measured from radial velocities , we confirm that the density is @xmath3 .
we also revise the ephemeris for ogle - tr-111-b , obtaining an accurate orbital period @xmath4 days , and predicting that the next observable transits would occur around december 2006 , and after that only in mid-2008 .
even though this period is different from previously published values , we can not yet rule out a constant period .
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main - belt comets ( mbcs ; * ? ? ? * ) exhibit cometary activity indicative of sublimating ice , yet orbit entirely within the main asteroid belt ( figure [ fig_aeimbcs ] ) . seven mbcs 133p / elst - pizarro , 176p / linear , 238p / read , 259p / garradd , p/2010 r2 ( la sagra ) , p/2006 vw@xmath6 , and p/2012 t1 ( panstarrs ) are currently known . in addition , three other objects p/2010 a2 ( linear ) , ( 596 ) scheila , and p/2012 f5 ( gibbs ) have been observed to exhibit comet - like dust emission , though their active episodes have been attributed to impact events and are not believed to be sublimation - driven @xcite . as such , we do not consider these objects to be ice - bearing main - belt objects , and refer to them as disrupted asteroids ( figure [ fig_aeimbcs ] ) . p/2012 t1 was discovered on 2012 october 6 by the 1.8 m pan - starrs1 ( ps1 ) survey telescope on haleakala @xcite . ps1 employs a @xmath7 1.4 gigapixel camera , consisting of 60 orthogonal transfer arrays , each comprising 64 @xmath8 pixel ccds . our discovery observations were made using sloan digital sky survey ( sdss ) @xmath9- and @xmath10-like filters designated @xmath11 and @xmath12 @xcite . comet candidate identification in ps1 data is accomplished using automated point - spread function ( psf ) analysis procedures @xcite implemented as part of ps1 s moving object processing system ( mops ; * ? ? ? follow - up observations were obtained in photometric conditions between october 2012 and february 2013 using the university of hawaii ( uh ) 2.2 m and the 10 m keck i telescopes , both on mauna kea , the 6.5 m baade and clay magellan telescopes at las campanas , the 2.0 m faulkes telescope south ( fts ) at siding spring , the 1.8 m perkins telescope ( pt ) at lowell observatory , and the southern astrophysical research ( soar ) telescope on cerro pachon ( table [ table_obslog ] ; figure [ fig_observations]a , b ) . we employed a 2048@xmath132048 pixel textronix ccd for uh observations , the low resolution imaging spectrometer ( lris ; * ? ? ? * ) for keck observations , the inamori magellan areal camera and spectrograph ( imacs ) for baade observations , the megacam mosaic camera ( consisting of 36 2048@xmath134608 pixel ccds ) for clay observations , a 4096@xmath134096 pixel fairchild ccd for fts observations , the perkins reimaging system for lowell observations , and the soar optical imager ( soi ; * ? ? ? * ) for soar observations . we used sdss - like filters for clay observations , bessell filters for fts observations , and kron - cousins filters for all other observations . uh 2.2 m , keck , lowell , and soar observations were conducted using non - sidereal tracking at the apparent rate and direction of motion of p/2012 t1 on the sky , while other observations were conducted using sidereal tracking . ps1 data were reduced using the system s image processing pipeline ( ipp ; * ? ? ? * ) . we performed bias subtraction and flat - field reduction for follow - up data using image reduction and analysis facility ( iraf ; * ? ? ? * ) software and using flat fields constructed either from images of the illuminated interior of the telescope dome or dithered images of the twilight sky . some photometric calibration was performed using field star magnitudes provided by the sloan digital sky survey ( sdss ; * ? ? ? * ) converted to kron - cousins @xmath14-band equivalents using the transformation equations derived by r. lupton ( available online at http://www.sdss.org/ ) . photometry of @xcite standard stars and field stars was performed for all data using iraf and obtained by measuring net fluxes within circular apertures , with background sampled from surrounding circular annuli . conversion of @xmath9-band magnitudes measured from ps1 and clay data to their @xmath14-band equivalents was performed assuming approximately solar colors for the object . comet photometry was performed using circular apertures , where to avoid dust contamination from the comet itself , background sky statistics are measured manually in regions of blank sky near , but not adjacent , to the object . photometry aperture sizes were chosen to encompass @xmath1595% of the total flux from the comet ( coma and tail ) while minimizing interference from nearby field stars or galaxies , and varied from @xmath16 to @xmath17 in radius depending on seeing conditions . field stars in comet images were also measured to correct for any extinction variations during each night . in addition to imaging , we also obtained optical spectra of p/2012 t1 on 2012 october 19 with lris in spectroscopic mode on keck . two g2v solar analog stars , hd28099 and hd19061 , were also observed to allow removal of atmospheric absorption features and calculation of p/2012 t1 s relative reflectance spectrum . we utilized a @xmath18-wide long - slit mask and lris s d500 dichroic , with a 400/3400 grism on the blue side ( dispersion of 1.09 pixel@xmath3 and spectral resolution of @xmath07 ) , and 150/7500 grating on the red side ( dispersion of 3.0 pixel@xmath3 and spectral resolution of @xmath018 ) . exposure times totaled 1320 s and 1200 s on the blue and red sides , respectively , where the comet was at an airmass of @xmath01.2 during our observations . data reduction was performed using iraf . photometry results from follow - up observations are listed in table [ table_obslog ] . for reference , we also compute @xmath19 @xcite for each of our observations , though we note that it is not always a reliable measurement of the dust contribution to comet photometry in cases of non - spherically symmetric comae ( e.g. , * ? ? ? while much of our photometry are based on snapshot observations ( meaning that unknown brightness variations due to nucleus rotation could be present ) , we find that the object s intrinsic brightness roughly doubles from the time of its discovery in early october until mid - november ( @xmath040 days ; over a true anomaly range of @xmath20 ) , and then decreases by @xmath060% between late december and early february ( @xmath050 days ; @xmath21 ) ( figure [ fig_observations]c ) . similar photometric behavior is observed for several other mbcs @xcite . for comparison , the brightness of disrupted asteroid ( 596 ) scheila declined by 30% in just 8 days @xcite . mbcs 133p and 238p both exhibited long - lived brightening and did so during multiple apparitions , making us extremely confident in their cometary natures @xcite . while long - lived activity is no guarantee of cometary activity @xcite , the photometric behavior of p/2012 t1 is certainly inconsistent with dust particles ejected impulsively in an impact . its steady brightening implies the action of a prolonged dust ejection mechanism like sublimation . furthermore , while apparently long - lived activity could be due to the long dissipation times of large particles ejected by an impact , p/2012 t1 s eventual fading after several weeks suggests that this is not the case here , since such large particles would be expected to persist much longer ( cf . * ; * ? ? ? multi - filter observations using lris on keck i ( which permits simultaneous @xmath22- and @xmath14-band imaging , eliminating the effects of rotational brightness variations ) and the baade telescope indicates that coma of p/2012 t1 had approximately solar colors of @xmath23 ( measured on keck ) , and @xmath24 and @xmath25 ( measured on baade ) . our lris red - side spectrum ( figure [ fig_spectra ] ) of p/2012 t1 is approximately linear with a slightly blue slope of @xmath26 % /1000 , similar to the spectrum of 133p when it was active during its 2007 perihelion passage @xcite . this result differs significantly , however , from the red slopes measured for mbc p/2006 vw@xmath6 when it was active ( 7.2%/1000 ; * ? ? ? * ) as well as for other cometary dust comae @xcite . to derive the cn production rate ( cf * ; * ? ? ? * ) , we employ a simple @xcite model , using a resonance fluorescence efficiency of @xmath27=3.63\times10^{-13}$ ] erg s@xmath3 molecule@xmath3 @xcite . we find an upper limit to the cn production rate of @xmath2 mol s@xmath3 . the cn to water production rate in mbcs is unknown , but we adopt the average ratio in other observed comets ( @xmath28\sim-2.5 $ ] ; @xmath29% ) @xcite , and infer a water production rate of @xmath30 mol s@xmath3 . we also search for 0.7 @xmath1 m absorption due to a charge transfer transition in oxidized iron in phyllosilicates , indicative of the presence of hydrated minerals @xcite . thermal evolution models suggest that aqueous alteration occurred within asteroid parent body interiors @xcite . if these models are correct , mbcs could be icy fragments from the outer shells of asteroid parent bodies where temperatures were never high enough to melt ice . if an mbc happened to be a fragment from near an ice and hydrated rock boundary in such a parent body , it could contain hydrated minerals . to date , no mbcs have shown evidence of having hydrated minerals on their surfaces . our keck spectrum of p/2012 t1 likewise shows no signs of absorption at 0.7 @xmath1 m , and thus , no detectable evidence of hydrated minerals . to determine whether p/2012 t1 is likely to be native to the main belt , or if it could be a recently implanted interloper from the outer solar system , we analyze its long - term dynamical stability in a manner similar to that performed for other mbcs ( cf . * ; * ? ? ? * ; * ? ? ? * ; * ? ? ? we generate nine sets of 100 dynamical clones of p/2012 t1 that are gaussian - distributed in orbital element space and centered on the object s osculating orbital elements as of 2012 december 1 . three of these sets are characterized by @xmath31 values equivalent to the uncertainties on those orbital elements ( @xmath32 au , @xmath33 , @xmath34 ) , three sets are characterized by @xmath31 values 10 times larger than those uncertainties , and three sets are characterized by @xmath31 values 100 times larger . we then perform forward integrations for 100 myr using the n - body integration package , mercury @xcite . we include the gravitational effects of all eight major planets and treat all dynamical clones as massless test particles . non - gravitational forces are not considered in this analysis . in these simulations , less than 5% of the test particles reach heliocentric distances of @xmath35 au ( and are therefore considered to have been ejected from the asteroid belt ) over the 100 myr test period . the remaining test particles diverge to occupy regions of orbital element space that are larger than their initial distributions but that are also essentially independent of those initial distributions , i.e. , the 1-@xmath31 sets of test particles diverge to occupy similar regions as the 100-@xmath31 sets ( figure [ fig_stability ] ) . this divergence occurs quickly ( within @xmath36 years ) and remains approximately constant over the 100 myr test period . we therefore find that p/2012 t1 is largely dynamically stable and is unlikely to be a recently implanted interloper , though we do note that the ejection of a small number of test particles indicates that the region is not perfectly stable over the considered time period . mbcs 133p and p/2006 vw@xmath6 have recently been found to be dynamical members of very young ( @xmath3710 myr ) asteroid families @xcite . these findings are interesting because the surfaces of ice - bearing main - belt objects may become significantly collisionally devolatilized on timescales of @xmath38 gyr @xcite . however , if mbcs only originated in the recent fragmentation events that created the aforementioned young families , their surfaces should have experienced far less collisional devolatilization , and thus remain susceptible to activation by small impactors ( cf . * ; * ? ? ? while we currently lack a sufficient sample size to ascertain whether there is a significant overabundance of mbcs in young families , the fact that two of seven mbcs ( @xmath030% ) are found to belong to such families is suggestive of a physical correlation . to test whether p/2012 t1 originated in a recent fragmentation event , we search for an associated dynamical family utilizing a hierarchical clustering analysis @xcite . using computed proper elements of @xmath39 au , @xmath40 , and @xmath41 , we find that p/2012 t1 belongs to the @xmath0155 myr - old lixiaohua asteroid family @xcite . while the lixiaohua family is of intermediate age , p/2012 t1 could belong to an even younger sub - family , much as the young beagle and p/2006 vw@xmath6 families are both sub - groups of the much older themis family @xcite . unfortunately , the density of asteroids in the region of orbital element space occupied by the lixiaohua family is extremely high . in fact , most lixiaohua family members , including p/2012 t1 , are linked to the family at cut - off velocities as low as 20 m s@xmath3 . as such , identification of a younger sub - family for p/2012 t1 will be extremely difficult . the slight instability of the p/2012 t1 s orbit ( section [ stability ] ) also interferes with our ability to establish young family linkages using techniques such as those applied in the case of p/2006 vw@xmath6 @xcite . currently , the key question that must be answered when a main - belt object exhibits comet - like activity is whether that activity is sublimation - driven , implying the presence of ice , or is produced by another means . definitive evidence of sublimation would be the direct detection of a gaseous sublimation product such as cn or h@xmath42o in the coma of such an object . unfortunately , unsuccessful attempts to detect cn have now been made for four of the most recently discovered mbcs ( * ? ? ? * ; * ? ? ? * ; * ? ? ? * ; * ? ? ? * this work ) , where each work has found similar upper limit cn production rates of @xmath43 mol s@xmath3 , corresponding to water production rates of @xmath44 mol s@xmath3 . searches for line emission from the ( @xmath45 ) rotational transition of ortho - water at 557 ghz with the _ herschel _ space observatory for 176p and p/2012 t1 were also unsuccessful , finding @xmath46 mol s@xmath3 and @xmath47 mol s@xmath3 , respectively ( * ? ? ? * orourke et al . , private communication ) . while these results do not definitively rule out sublimation , they do indicate that the production rates of sublimation products by mbcs are too low to detect from current earth - bound facilities . as such , we must rely on indirect evidence to determine the likely source of comet - like activity in main - belt objects . @xcite examined various mechanisms by which an asteroid - like body could undergo comet - like mass loss , including ice sublimation , impact ejection , rotational instability , and electrostatic levitation , finding that in many individual cases of comet - like objects , the cause of observed mass loss could not be definitively identified due to insufficient observational data . nonetheless , certain mechanisms could sometimes be ruled out based on physical and observational constraints . for example , electrostatic levitation was ruled out as a cause of 133p s observed activity because it would have depleted the supply of mobile surface dust during a single active episode , leaving no obvious source of mobile dust for subsequent active episodes @xcite . the rapid rotation of 133p also minimizes the amount of electrostatic charging that can occur given the short time that any portion of the object s surface spends in sunlight @xcite . this mechanism s efficacy furthermore depends on unknown cohesive properties of asteroid regolith dust grains @xcite . finally , given the many asteroids similar to 133p with more favorable rotational properties , it is unclear why 133p would exhibit observable dust levitation while other asteroids do not . dust ejection via rotational spin - up , perhaps via the yarkovsky - okeefe - radzievsky - paddack ( yorp ) effect @xcite , was also ruled out due to the lack of a plausible mechanism for producing repeated activity or explaining the rarity of similar activity on other asteroids . @xcite did note that for an object exhibiting repeated activity , sublimation appears to be the only reasonable explanation . repeated activity has only been established for 133p and 238p , however , with others either failing to exhibit repeated activity upon completion of a full orbit period since its previously observed active episode ( 176p ) , or not yet having completed one full orbit since their first observed active episodes ( 259p , p/2010 r2 , p/2006 vw@xmath6 , and p/2012 t1 ) . as discussed in section [ photresults ] , p/2012 t1 s observed photometric behavior indicates ongoing and even increasing dust production over several weeks ( figure [ fig_observations]c ) , consistent with continuous sublimation - driven dust ejection and inconsistent with impulsive impact - driven dust ejection . the comet also exhibits a diffuse coma and a featureless fan - like tail that remains aligned with the antisolar direction , distinctly different from the crossed filamentary structure of p/2010 a2 s tail , the three - plumed morphology of ( 596 ) scheila s dust tail , and the orbit - plane - aligned dust trail of p/2012 f5 @xcite . the post - perihelion peaking of p/2012 t1 s activity is also consistent with the post - perihelion peaking of activity for other mbcs @xcite . while we can not yet definitively conclude that p/2012 t1 s activity is sublimation - driven , we note that all evidence examined thus far is consistent with sublimation . we therefore find that p/2012 t1 is most likely a true mbc , and not a disrupted asteroid , though additional observations ( e.g. , to search for repeated activity during its next perihelion passage in mid-2018 ) and more detailed dust modeling will be required to definitively rule out other dust ejection mechanisms . a primary ultimate objective of mbc studies is to connect observations of the distribution and composition of volatiles in small primitive bodies to the distribution of volatiles in the protoplanetary disk , and to link this through disk observations to other forming planetary systems @xcite . presently , we have insufficient information to make these connections , in part because we have few direct constraints on the volatile content of small bodies and because our solar system s precise dynamical history remains poorly understood . further work on both fronts would help this situation , though significantly advancing our understanding of volatile material in the asteroid belt may require in - situ investigation , e.g. , by a visiting spacecraft . is supported by nasa through hubble fellowship grant hf-51274.01 awarded by the space telescope science institute , which is operated by the association of universities for research in astronomy , inc . , for nasa , under contract nas 5 - 26555 . , n.h . , and k.j.m . acknowledge support through the nasa astrobiology institute under cooperative agreement nna09da77a . is supported by the ministry of education and science of serbia under project 176011 . , t.r . , and l.u . acknowledge support through nsf grant ast-1010059 . we thank larry wasserman and brian taylor at lowell for assistance in obtaining observations . some data presented was acquired using the ps1 system operated by the ps1 science consortium ( ps1sc ) and its member institutions . the ps1 survey was made possible by contributions from ps1sc member institutions and nasa through grant nnx08ar22 g issued through the planetary science division of the nasa science mission directorate . sdss - 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borro , m. , rezac , l. , hartogh , p. , et al . 2012 , , 546 , l4 vilas , f. 1994 , icarus , 111 , 456 wainscoat , r. , hsieh , h. , denneau , l. , et al . 2012 , central bureau electronic telegrams , 3252 , 1 wilson , l. , keil , k. , browning , l. b. , krot , a. n. , & bourcier , w. 1999 , meteoritics and planetary science , 34 , 541 york , d. g. , et al . 2000 , , 120 , 1579 zappal , v. , cellino , a. , farinella , p. , & kne@xmath48evi , z. 1990 , , 100 , 2030 lcrrcrrrrrrrccc 2012 sep 11 & & 0.0 & 2.411 & 1.753 & 21.4 & 262.1 & 243.5 & 6.3 & ... & ... & ... + 2012 oct 06 & ps1 & 1 & 40 & @xmath49 & 7.4 & 2.414 & 1.540 & 14.4 & 273.5 & 244.1 & 6.8 & [email protected] & [email protected] & [email protected] + 2012 oct 08 & ps1 & 1 & 40 & @xmath49 & 8.0 & 2.415 & 1.527 & 13.7 & 274.9 & 244.1 & 6.8 & [email protected] & [email protected] & [email protected] + 2012 oct 12 & clay & 38 & 2280 & @xmath9 & 9.1 & 2.416 & 1.507 & 12.4 & 278.1 & 244.2 & 6.7 & [email protected] & [email protected] & [email protected] + 2012 oct 14 & uh2.2 & 1 & 300 & @xmath14 & 9.8 & 2.418 & 1.496 & 11.5 & 280.3 & 244.3 & 6.6 & [email protected] & [email protected] & [email protected] + 2012 oct 15 & uh2.2 & 2 & 600 & @xmath14 & 10.0 & 2.418 & 1.491 & 11.2 & 281.4 & 244.3 & 6.6 & [email protected] & [email protected] & [email protected] + 2012 oct 15 & fts & 8 & 480 & @xmath14 & 10.0 & 2.418 & 1.491 & 11.1 & 281.6 & 244.3 & 6.6 & [email protected] & [email protected] & [email protected] + 2012 oct 18 & keck & 4 & 1440 & @xmath22 & 10.9 & 2.419 & 1.479 & 10.0 & 285.3 & 244.4 & 6.4 & [email protected] & ... & ... + & & 4 & 1200 & @xmath14 & ... & ... & ... & ... & ... & ... & ... & [email protected] & [email protected] & [email protected] + 2012 oct 19 & keck & 4 & 1440 & @xmath22 & 11.2 & 2.419 & 1.475 & 9.7 & 286.8 & 244.4 & 6.4 & [email protected] & ... & ... + & & 4 & 1200 & @xmath14 & ... & ... & ... & ... & ... & ... & ... & [email protected] & [email protected] & [email protected] + 2012 oct 22 & uh2.2 & 14 & 4200 & @xmath14 & 12.0 & 2.421 & 1.466 & 8.5 & 292.0 & 244.5 & 6.2 & [email protected] & [email protected] & [email protected] + 2012 oct 25 & baade & 3 & 180 & @xmath22 & 12.9 & 2.422 & 1.459 & 7.6 & 298.2 & 244.5 & 6.0 & [email protected] & ... & ... + & & 2 & 120 & @xmath51 & ... & ... & ... & ... & ... & ... & ... & [email protected] & ... & ... + & & 5 & 300 & @xmath14 & ... & ... & ... & ... & ... & ... & ... & [email protected] & [email protected] & [email protected] + 2012 nov 8 & uh2.2 & 2 & 600 & @xmath14 & 17.0 & 2.431 & 1.455 & 5.3 & 0.5 & 244.8 & 4.8 & [email protected] & [email protected] & [email protected] + 2012 nov 9 & uh2.2 & 2 & 1200 & @xmath14 & 17.2 & 2.432 & 1.457 & 5.4 & 5.5 & 244.8 & 4.7 & [email protected] & [email protected] & [email protected] + 2012 nov 13 & uh2.2 & 26 & 7800 & @xmath14 & 18.4 & 2.434 & 1.467 & 6.3 & 22.6 & 244.8 & 4.2 & [email protected] & [email protected] & [email protected] + 2012 nov 14 & uh2.2 & 14 & 4200 & @xmath14 & 18.7 & 2.435 & 1.470 & 6.5 & 26.1 & 244.8 & 4.1 & [email protected] & [email protected] & [email protected] + 2012 nov 22 & lowell & 4 & 2400 & @xmath14 & 21.0 & 2.441 & 1.502 & 9.1 & 44.9 & 244.7 & 3.1 & [email protected] & [email protected] & [email protected] + 2012 nov 23 & lowell & 2 & 1400 & @xmath14 & 21.3 & 2.442 & 1.507 & 9.5 & 43.9 & 244.7 & 3.0 & [email protected] & [email protected] & [email protected] + 2012 dec 18 & uh2.2 & 30 & 9000 & @xmath14 & 28.3 & 2.467 & 1.712 & 17.7 & 64.5 & 244.1 & 0.1 & [email protected] & [email protected] & [email protected] + 2012 dec 19 & uh2.2 & 49 & 15000 & @xmath14 & 28.6 & 2.468 & 1.723 & 17.9 & 64.8 & 244.1 & 0.2 & [email protected] & [email protected] & [email protected] + 2012 dec 20 & lowell & 4 & 4800 & @xmath14 & 28.9 & 2.470 & 1.734 & 18.2 & 65.1 & 244.1 & 0.3 & [email protected] & [email protected] & [email protected] + 2013 jan 08 & uh2.2 & 10 & 8400 & @xmath14 & 34.2 & 2.493 & 1.962 & 21.6 & 69.5 & 243.7 & 2.1 & [email protected] & [email protected] & [email protected] + 2013 feb 04 & soar & 1 & 600 & @xmath14 & 41.4 & 2.531 & 2.334 & 22.9 & 73.3 & 243.9 & 3.5 & [email protected] & [email protected] & [email protected] + 2015 jun 27 & & 180.0 & 3.896 & 3.656 & 15.0 & 114.2 & 296.1 & 0.5 & ... & ... & ... + 2018 apr 12 & & 0.0 & 2.402 & 3.392 & 3.1 & 134.3 & 243.4 & 2.9 & ... & ... & ...
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we present initial results from observations and numerical analyses aimed at characterizing main - belt comet p/2012 t1 ( panstarrs ) .
optical monitoring observations were made between october 2012 and february 2013 using the university of hawaii 2.2 m telescope , the keck i telescope , the baade and clay magellan telescopes , faulkes telescope south , the perkins telescope at lowell observatory , and the southern astrophysical research ( soar ) telescope .
the object s intrinsic brightness approximately doubles from the time of its discovery in early october until mid - november and then decreases by @xmath060% between late december and early february , similar to photometric behavior exhibited by several other main - belt comets and unlike that exhibited by disrupted asteroid ( 596 ) scheila .
we also used keck to conduct spectroscopic searches for cn emission as well as absorption at 0.7 @xmath1 m that could indicate the presence of hydrated minerals , finding an upper limit cn production rate of @xmath2 mol s@xmath3 , from which we infer a water production rate of @xmath4 mol s@xmath3 , and no evidence of the presence of hydrated minerals .
numerical simulations indicate that p/2012 t1 is largely dynamically stable for @xmath5 myr and is unlikely to be a recently implanted interloper from the outer solar system , while a search for potential asteroid family associations reveal that it is dynamically linked to the @xmath0155 myr - old lixiaohua asteroid family .
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dna microarrays allow the comparison of the expression levels of all genes in an organism in a single experiment , which often involve different conditions ( _ i.e. _ health - illness , normal - stress ) , or different discrete time points ( _ i.e. _ cell cycle ) @xcite . among other applications , they provide clues about how genes interact with each other , which genes are part of the same metabolic pathway or which could be the possible role for those genes without a previously assigned function . dna microarrays also have been used to obtain accurate disease classifications at the molecular level @xcite . however , transforming the huge amount of data produced by microarrays into useful knowledge has proven to be a difficult key step @xcite . on the other hand , clustering techniques have several applications , ranging from bioinformatics to economy @xcite . particularly , data clustering is probably the most popular unsupervised technique for analyzing microarray data sets as a first approach . many algorithms have been proposed , hierarchical clustering , k - means and self - organizing maps being the most known @xcite . clustering consists of grouping items together based on a similarity measure in such a way that elements in a group must be more similar between them than between elements belonging to different groups . the similarity measure definition , which quantifies the affinity between pairs of elements , introduces _ a priori _ information that determines the clustering solution . therefore , this similarity measure could be optimized taking into account additional data acquired , for example , from real experiments . some works with _ a priori _ inclusion of bioinformation in clustering models can be found in @xcite . in the case of gene expression clustering , the behavior of the genes reported by microarray experiments is represented as @xmath0 points in a @xmath1-dimensional space , being @xmath0 the total number of genes , and @xmath1 the number of conditions . each gene behavior ( or point ) is then described by its coordinates ( its expression value for each condition ) . genes whose expression pattern is similar will appear closer in the @xmath1-space , a characteristic that is used to classify data in groups . in our case , we have used the superparamagnetic clustering algorithm ( spc ) @xcite , which was proposed in 1996 by domany and collaborators as a new approach for grouping data sets . however , this methodology has difficulties dealing with different density clusters , and in order to ameliorate this , we report here some modifications of the original algorithm that improve cluster detection . our main contribution consists on increasing the similarity measure between genes by taking advantage of transcription factors , special proteins involved in the regulation of gene expression . the present paper is organized as follows : in section 2 , the spc algorithm is introduced , as well as our proposal to include further biological information and our considerations for the selection of the most natural clusters . results for a real data set , as well as performance comparisons , are presented in section 3 . finally , section 4 is dedicated to a summary of our results and conclusions . a potts model can be used to simulate the collective behavior of a set of interacting sites using a statistical mechanics formalism . in the more general inhomogeneous potts model , the sites are placed on an irregular lattice . next , in the spc idea of domany _ et al . _ @xcite , each gene s expression pattern is represented as a site in an inhomogeneus potts model , whose coordinates are given by the microarray expression values . in this way , a particular lattice arrangement is spanned for the entire data set being analyzed . a spin value @xmath2 , arbitrarily chosen from @xmath3 possibilities , is assigned to each site , where @xmath4 corresponds to the site of the lattice @xmath5 . the main idea is to characterize the resulting spin configuration by the ferromagnetic hamiltonian : @xmath6 where the sum goes over all neighboring pairs , @xmath2 and @xmath7 are spin values of site @xmath4 and site @xmath8 respectively , and @xmath9 is their ferromagnetic interaction strength . each site interacts only with its neighbors , however since the lattice is irregular , it is necessary to assign the set of nearest - neighbors of each site using the so - called @xmath10-mutual - nearest - neighbor criterion @xcite . the original interaction strength is as follows : @xmath11 with @xmath12 the average number of neighbors per site and @xmath13 the average distance between neighbors . the interaction strength between two neighboring sites decreases in a gaussian way with distance @xmath14 and therefore , sites that are separated by a small distance have more probability of sharing the same spin value during the simulation than the distant sites . on the other hand , said probability , @xmath15 , also depends on the temperature @xmath16 , which acts as a control parameter . at low temperatures , the sites tend to have the same spin values , forming a ferromagnetic system . this configuration is preferred over others because it minimizes the total energy . however , the probability of encountering aligned spins diminishes as temperature increases , and the system could experience either a single transition to a totally disordered state ( paramagnetic phase ) , or pass through an intermediate phase in which the system is partially ordered , which is known as the superparamagnetic phase . in the latter case , varios regions of sites sharing the same spin value emerge . sites within these regions interact among them with a stronger force , exhibiting at the same time weak interactions with sites outside the region . these regions could fragment into smaller grains , leading to a chain of transitions within the superparamagnetic phase until the temperature is so high that the system enters the paramagnetic phase , where each spin behaves independently . this hierarchical subdivision in magnetic grains reflects the organization of data into categories and subcategories . regions of aligned spins emerging during simulation correspond to groups of points with similar coordinates , _ i.e. _ , similar gene expression patterns @xcite . this subdivision can be simulated , for example , by using the monte carlo approach , by which one can compute and follow the evolution of system properties such as energy , magnetization and susceptibility , while the temperature is modified . in addition , the temperature ranges in which each phase transition takes place can be localized . rather than thresholding the distances between pairs of sites to decide their assignment to clusters , the pair correlation @xmath17 , indicating a collective aspect of the data distribution , is preferred . it can be calculated as follows @xcite @xmath18 in this way , @xmath17 is the normalized probability for finding two potts spins @xmath2 and @xmath7 sharing the same value for a given temperature step . if both spins belong to the same ordered region , their correlation value would be close to one , otherwise their correlation would be close to zero @xcite . thus , for each temperature step , two sites are assigned to the same cluster if their correlation exceeds a threshold value of @xmath19 . if a site does not have a single correlation value greater than @xmath20 , it is joined with its neighbor showing the highest value . + for our spctf algorithm , we also accept sites whose @xmath17 are larger than @xmath20 in order to build a cluster . however , differently from the traditional spc algorithm @xcite , if two sites do not reach the @xmath17 value greater than @xmath20 they are not connected . this is because with our data we have found that the original condition led to unnatural growth of some clusters when the temperature is increased . as already mentioned , the data are fragmented in various clusters for each temperature value , and for higher temperatures , the number of clusters increases due to finer and finer segmentation . in order to select the more representative clusters through all temperature steps , we assign a stability value to each obtained cluster , based on its evolution . we define @xmath21 as the number of temperature steps until the system reaches the paramagnetic phase and @xmath22 as the number of temperature steps a cluster @xmath23 survives , while @xmath24 and @xmath25 are defined as the total number of sites and the number of elements in a given cluster , respectively . we assign a stability parameter @xmath26 to each cluster , as follows : @xmath27 where @xmath28 is the fraction of temperature steps a cluster @xmath23 survives , while @xmath29 is the fraction of total elements belonging to @xmath23 . the advantage of using the stability parameter @xmath26 is that it gives preference to clusters that survive several temperatures , but also have an acceptable number of elements . we added a small positive real number @xmath30 to the denominator in the expression of @xmath26 for the special case when @xmath31 , where @xmath32 belongs to the range @xmath33 $ ] , leading to @xmath34 instead of the infinity . it has been reported that the main drawback of the spc algorithm consists of dealing with data showing regions of different density @xcite . in this case , either depending on temperature or the number of neighbors selected , some clusters will easily get prominent whereas the detection of others will be hindered . to overcome this problem , at least two techniques have been proposed _ e.g. _ , sequential superparamagnetic clustering @xcite and a modularity approach @xcite . our idea is to take advantage of already available biological information to improve lattice connectivity in such a way that biologically significant clusters have more probability of being detected by the algorithm . indeed , at the transcriptional level , the expression of a gene could be promoted / suppressed by the binding of the proteins named transcription factors to specific sequences on the gene promoter region . then , if a group of genes shows the same expression behavior in a microarray experiment , it is quite possible that they are being regulated by a specific transcription factor , forming a group of coregulated genes @xcite . thus , available information about which genes are targeted by the same transcription factors may be useful in the detection of groups of genes with similar expression profiles . to make effective this idea , we downloaded from _ www.yeastract.com _ a list of yeast transcription factors that are well documented , and whenever two neighboring genes are controlled by the same transcription factor , we increased their interaction strength . it is important to note that the list provided by _ www.yeastract.com _ includes transcription factors associated with several processes and are not only cell cycle related . the formula that takes this into account replaces eq . ( [ eq : js ] ) of the original algorithm , and has the following form : @xmath35 here , @xmath36 is the number of common transcription factors shared by @xmath4 and @xmath8 ( @xmath32 , which varies for each pair of neighboring genes ) , multiplied by a factor @xmath37 which was chosen to be 2.0 after comparing the results obtained with several other values . the selected value has the characteristic of preserving well - defined susceptibility peaks as well as obtaining larger clusters . the objective is to strengthen some connections without preventing the natural fragmentation of clusters caused by the temperature parameter . if two elements do not share a transcription factor , then @xmath38 , recovering the original spc formula therefore , the modified interaction strength between each site and its neighbors is governed by two aspects : the distance between them , which comes from gene expression values generated through microarray experiments , and the number of transcription factors regulating both genes , obtained from documented biological data . any time two genes share a transcription factor , their interaction strength becomes larger , and this favors that the clusters including these sites remain stable for longer temperature ranges , with the corresponding increase of their stability values . we analyzed spellman _ et al . _ @xcite microarray data in which gene expression values from synchronized yeast cultures were obtained at various time moments , aiming to identify cell cycle genes . yeast cultures were synchronized by three methods : adding alpha pheromone , which arrests cells in the g1 phase ; using centrifugal elutration for separating small g1 cells ; and using a mutation that arrests cells late in mitosis at a given temperature . combining the three experiments and using fourier and correlation algorithms , spellman _ @xcite reported @xmath39 cell cycle regulated genes . the goal was to compare the performance of spc and spc with transcription factors ( spctf ) , which are algorithms that do not make assumptions about periodicity . nonetheless , the overall analysis is time consuming and we only selected the data set treated with the alpha pheromone , available at _ http://cellcycle - www.stanford.edu_. genes with missing values were discarded , leaving an input matrix of @xmath40 genes and @xmath41 time courses that included only @xmath42 of the genes reported by spellman _ _ @xcite . furthermore , as we do not include the other two synchronization experiments , we expect to loose some of their cell cycle genes . it is worth mentioning that getz _ _ @xcite also analyzed the spellman alpha synchronized set with the spc algorithm . they took @xmath43 genes which have characterized functions and introduced a fourier transform to take into account the oscillatory nature of the cell cycle . in our case , however , we decided not to introduce any considerations about the periodicity of the data , mainly because the time series cover only two cell cycle periods @xcite . we obtain compact gene clusters implementing spc original algorithm and spctf , both with parameter values @xmath44 and @xmath45 . the cluster with the highest stability value contains an extremely large number of elements without a clear biological linkage between them . it is mainly composed of genes whose expression do not change significantly over time , thus it is possible that they are included here for this very reason . we discard this cluster from our analysis , although it could always be taken apart and analyzed again with spctf by choosing the appropiate number of neighbors to obtain more information . to compare in more detail both approaches , it is necessary to correlate each cluster in the spc method with its equivalent in spctf . in order to do this , we calculate the euclidian distance between the mean position vector of every cluster in each approach , and choose the pairs with the shortest distance between them . ( we recall that the mean position vector of a cluster is obtained by averaging each coordinate between all its elements ) . although different measures could have been used , this one performed adequately , as can be seen in the supplementary information file , where we provide a more detailed comparison between spctf and spc clusters . in table [ tabla1 ] , we present the differences in cluster size as well as the hits , the number of genes reported by spellman _ @xcite , which have been included in the clusters . when going through the spctf approach , one can see that the first largest cluster looses some genes , while the number of the rest of the clusters augments . besides , hits or coincidences with spellman _ et al . _ @xcite cell cycle genes in clusters of six or more elements increase by @xmath46 , from @xmath47 to @xmath48 . therefore , we were able to incorporate several genes to these clusters , mainly from outliers . * comparison between spc and spctf * in the following analysis , we focus on clusters of six or more elements , because we are interested in finding groups of several genes sharing the same expression pattern ( coregulated genes ) . results of the comparison for the first @xmath49 most stable clusters , discarding the first one , are shown in fig . [ fig : todos_cc ] . generally , these clusters incorporate more elements with spctf , including more cell cycle genes as those reported by spellman _ _ @xcite and thus improving the matching . clusters , discarding the first one . gray bars correspond to the clusters obtained with the spc algorithm and black bars to the equivalent clusters in spctf . groups tend to increase in size and also in hits with cell cycle genes reported by spellman _ @xcite , with the exception of cluster @xmath50.,height=211 ] depending on the available information about the genes , we classify the clusters in three groups . the first cluster type , cell cycle genes , cc , corresponds to groups formed in their majority ( @xmath51 ) by already reported cell cycle genes ( fig . [ fig : cc_sp ] ) . the second type , mixed genes , m , contains clusters with non - reported genes as well as already known cell cycle genes ( fig . [ fig : m_n ] ) , and in the third type , no hits , n , we include the clusters that contain only one hit or are entirely composed of non - previously identified cell cycle genes ( fig . [ fig : m_n ] ) . it is worth mentioning that more cell cycle experiments have been done since spellman _ _ @xcite and new genes have been classified meanwhile as cell cycle regulated . some of these newly reported cell cycle genes were obtained by cho _ @xcite , pramila _ et al . _ @xcite , rowicka _ et al . _ @xcite and lichtenberg _ et al . we analize our @xmath49 clusters taking now as hits , genes reported either by spellman _ @xcite or by one of the above mentioned studies . in this way , we gained thirty additional hits in the spc clusters , while in spctf clusters we have fifty - two extra genes . the results including all the aforementioned cell cycle studies are presented in figs . [ fig : all_studies1][fig : m_n_all ] @xcite . most stable clusters . hits are now taken as cell cycle genes reported by all studies . gray bars correspond to the clusters obtained with the spc algorithm and black bars to the equivalent clusters in spctf.,height=211 ] in addition , we analyze the expression profiles of the genes conforming each cluster using the sceptrans tool @xcite , and we notice that all the genes grouped in the same cluster had the same expression pattern . this gives us further confidence that our algorithm is grouping data correctly . the expression profiles for a representative member of each cluster type are shown in fig . [ fig : exp_pro ] . we also find two clusters ( @xmath52 and @xmath49 ) that present an oscillating behaviour that is due to an artifact in the manner the microarray experiment was performed , see @xcite . in the supplementary information file , we include the list of oscillating genes identified in @xcite and the number of these genes inside each of our first @xmath49 clusters . we also include the expression profiles of these clusters as well as those of size @xmath53 and @xmath54 which contain hits with cell cycle genes identified by spellman _ these clusters have also similar expression profiles but were not further analyzed because of their low number of elements . in the case of gene annotation , it is important to have clusters of many elements to effectively assure that an unknown gene shares the biological function already assigned to the other genes in the same cluster . + + + the cc clusters are almost entirely composed of cell cycle regulated genes reported either by spellman _ @xcite or by other authors , besides , their expression patterns are similar , which leaves no doubt on their validity . for the m and n clusters , we know that they are well grouped because their elements share the same expression patterns , but in order to select those of worth for further analysis ( for example in a laboratory experiment ) we analyze them through musa , motif finding using an unsupervised approach algorithm , that can be found at _ www.yeastract.com_. this program searches for the most common sequences ( motifs ) in the regulatory region of a set of genes , and compare them to the transcription factor binding sites already described in yeastract database @xcite . results of this analysis are shown in table [ tabla2 ] , which includes the quorum or percentage of genes containing a motif in each cluster , and the alignment score , which quantifies the level of similarity between the encountered motif and the known transcription factor associated with it . the clusters that probably would give us the best results would be those associated with cell cycle transcription factors with high percentages and scores . we select in this way , the clusters @xmath55 , @xmath53 , @xmath56 , @xmath57 , @xmath58 and @xmath59 because they have percentages higher than @xmath60 and scores higher than @xmath61 . in order to validate the musa analysis , we also constructed various clusters with sizes ranging from six to thirty - seven genes that were composed by genes selected at random from the original data . when analyzing these random clusters in the same way in musa , we obtain at most two cell cycle transcription factor coincidences . * musa analysis * large amounts of biological information are constantly obtained by throughput techniques and clustering algorithms have taken an important place in the unraveling of this information . however , the clustering analyses offer a difficult challenge because any data set can be grouped in numerous ways , depending on the level of resolution asked for and the applied similarity measure . in this work , we propose the use of available biological information in order to strengthen the interaction between genes which share a transcription factor involved in any metabolic process , improving the similarity measure . this information is introduced in the natural evolution of the spc algorithm , and in this way , we are able to enhance the creation and endurance of groups of possible coregulated genes . as the network spanned by the transcription factors information connects all genes , clustering directly _ a posteriori _ using only this information in the present case results into a single massive cluster ( see section iv of the supplementary information ) . however , by having the distance play an important weight in the interaction formula , the far - located clusters will not join , despite sharing transcription factors between their genes . with this in mind , we have modified the spc algorithm , and applied both the original and modified spctf algorithm to one of the three spellman _ @xcite data sets of the yeast cell cycle . the expression profiles of the genes in all resulting clusters show a similar behavior , but we obtain larger clusters with spctf . we classified them in three types , cc , m , and n , depending on the amount of cell cycle reported elements inside each cluster . with spctf , the cc type clusters increase in size including more cell cycle genes , and for the m and n type clusters , we also looked for common sequences in its regulatory regions and selected various groups worth of further research in order to report possible new cell cycle genes . as expected , some of these clusters include already known cell cycle genes sharing a transcription factor , _ but more importantly , at the predictive level , they promote the inclusion of new genes with similar expression patterns_. it is also important to note that the modified algorithm can be applied to any data set , and the followed methodology leads to the selection of the potential gene subsets feasible to be experimentally investigated . our work can serve as an example of how the inclusion of available biological information , such as transcription factors , and bioinformatic tools , such as musa , can lead to better and more confident results , aiding in the analysis of data coming from microarray experiments . the authors thank drs . s. ahnert and g. sherlock for useful discussions and comments . we also thank conacyt for providing support for two of the authors ( m.p.m.a . and j.c.n.m . ) and also the referees for helpful remarks and information . this work was partly supported through the project sep - conacyt-2005 - 49039 . p. t. monteiro , n. d. mendes , m. c. teixeira , s. dorey , s. tenreiro , n. p. mira , h. pais , a. p. francisco , a. m. carvalho , a. b. lourenco , i. sa - correia , a. l. oliveira , and a. t. freitas , nucleic acids res . * 36 * , d132 ( 2008 ) .
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in this work , we modify the superparamagnetic clustering algorithm ( spc ) by adding an extra weight to the interaction formula that considers which genes are regulated by the same transcription factor . with this modified algorithm that we call spctf , we analyze spellman _ et al .
_ microarray data for cell cycle genes in yeast , and find clusters with a higher number of elements compared with those obtained with the spc algorithm .
some of the incorporated genes by using spcft were not detected at first by spellman _
et al .
_ but were later identified by other studies , whereas several genes still remain unclassified .
the clusters composed by unidentified genes were analyzed with musa , the motif finding using an unsupervised approach algorithm , and this allow us to select the clusters whose elements contain cell cycle transcription factor binding sites as clusters worth of further experimental studies because they would probably lead to new cell cycle genes .
finally , our idea of introducing available information about transcription factors to optimize the gene classification could be implemented for other distance - based clustering algorithms .
superparamagnetic clustering , similarity measure , microarrays , cell cycle genes , transcription factors .
+ paper - physa-3 20100901.tex physica a 389(24 ) , 5689 - 5697 ( 2010 ) + doi : 10.1016/j.physa.2010.09.006
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You are an expert at summarizing long articles. Proceed to summarize the following text:
thz emission from intrinsic josephson junctions of the bscco type has received much attention recently . several experiments have been reported @xcite , in which thz radiation emitted from bscco single crystals were observed . however in most cases the detected power is rather small , or the frequency is rather low , or the emitted radiation is detected indirectly on an on - chip detector . it has also been demonstrated that bscco can be considered a josephson junction with ac josephson effect even at frequencies as high as 2 thz @xcite . recently a very convincing experiment was reported @xcite and it has attracted much focus and renewed experimental efforts . parallel to the experimental work there has been theoretical / numerical work on fluxon dynamics in in layered superconductors of the bscco type @xcite . the calculations demonstrate that the best way to obtain thz radiation is by having in - phase motion of the fluxons in the different layers . this poses an interesting problem since the in - phase state of traveling fluxons is an energetically unfavorable state , and two fluxons of equal polarity will consequently repel each other . however , due to the disparity of wave speeds for in - phase and out - of - phase solutions , it has been shown @xcite that the energetically unfavorable in - phase state of traveling fluxons are be stable above the asymptotic speed of the out - of - phase mode . it has been assumed that the best way to obtain that is by having flux flow generated by a magnetic field applied parallel to the a - b plane , as studied theoretically in ref . . however the successful experiment in ref . was done without a magnetic field , and it was suggested that an internal cavity ( based on the so - called fiske steps ) played a major role in the thz generation . in this paper we study the fluxon modes in a stack of josephson junctions interacting with an external cavity . we derive the conditions under which a large amount of current is induced in the cavity , and under which the external cavity may induce bunching of the josephson junction fluxons . .,width=312 ] assuming that all the junctions in the stack are identical , the equations for a stack of long josephson junctions with @xmath0 superconducting layers and @xmath1 insulating layers can be written as @xcite @xmath2 where the @xmath3th element of @xmath4 , @xmath5 , is the gauge invariant phase difference across insulating layer @xmath3 . the @xmath6 coupling matrix , @xmath7 , is given by ( only non - zero elements are shown ) @xmath8 with @xmath9 being the coupling parameter between the layers@xcite . the vector @xmath10 has the components @xmath11 where dampng parameter @xmath12 represents dissipation , and @xmath13 is the bias - current in the z - direction . each component of @xmath14 is a current in the @xmath15-direction . equations ( [ stackeq])-([current ] ) have been written in normalized units . space @xmath16 is normalized to the josephson penetration depth , @xmath17 , and time @xmath18 is normalized to the inverse plasma frequency @xmath19 , where @xmath20 is the vacuum permeability , @xmath21 is the critical current of the individual josephson junctions , @xmath22 is the effective thickness of the insulating layer , and @xmath23 is the capacitance of the individual junctions , see refs . and for details . the model of the josephson stack coupled to a series cavity is shown in fig . [ model ] , where @xmath24 is the cavity inductance , @xmath25 is the cavity resistance , and @xmath26 is the total cavity capacitance . the boundary conditions for the phases can be written as@xcite @xmath27 where @xmath28 is the normalized capacitance and @xmath29 is the normalized charge in the cavity . defining @xmath30 to be the normalized cavity frequency and and @xmath31 as the quality factor , the linear cavity equation becomes @xmath32 for more details on these equations see ref . . two terms are present in eq . ( [ bc2l ] ) . the first term couples the junctions to the cavity , by equally dividing the cavity current between the @xmath1 junctions . the second term represents a direct coupling between the junction through the capacitors @xmath33 . in a real world situation , the junctions would be embedded in a resonator and couple through electro - magnetic radiation from the edges . the second term then models the radiation leaving one junction and ending up in another junction without being reflected by the cavity . this is clearly not very efficient due to the geometry of the stack . with an efficient cavity , the second term is therefore expected to be much smaller than the first term and may safely be neglected . this may be justified by numerical calculations . we thus choose to consider @xmath34 as the boundary conditions for eqs . ( [ stackeq])-([current ] ) and ( [ cavity ] ) . we analyze the system in eqs . ( [ stackeq])-([current ] ) and ( [ cavity])-([bc ] ) in the case of weak inductive coupling where @xmath35 valid to first order in @xmath36 . the solution to the linear cavity equation with initial conditions @xmath37 and @xmath38 is @xmath39 with @xmath40 . the junction voltage at @xmath41 , @xmath42 , thus generates the cavity charge . we look at the case where there is one fluxon in each junction and @xmath42 then becomes a voltage pulse . to simplify the integrations , these pulses are approximated by delta functions , i.e. , @xmath43 approximating voltage pulses at @xmath44 , @xmath45 , @xmath46 is the fluxon shuttling frequency in junction @xmath3 and @xmath47 is the phase shift of junction @xmath3 . note , that since the present analysis is performed in the case of small @xmath36 , all the @xmath48-functions will have approximate the same amplitude , @xmath49 . with this ansatz , the cavity current becomes @xmath50e^{-\frac{\omega}{2q}\tilde{t}_n^i}\ , \label{gencur}\end{aligned}\ ] ] with @xmath51 and where @xmath52 is the heaviside step function . limiting the analysis to the case of a high @xmath53 resonator , the steady state cavity current becomes @xmath54 for @xmath55 . the phases , @xmath56 , are determined by [ eqlabel ] @xmath57 and @xmath58 only fluxons shuttling with the same frequency , @xmath59 for @xmath60 will be considered . in this case , eq . ( [ curgen ] ) can be reduced to @xmath61 with @xmath62 and @xmath63 for all @xmath3 . thus , the cavity current is very simple when the cavity has reached a steady state . note that the amplitude of the cavity current for an in - phase mode ( @xmath64 ) is @xmath65 . for an anti - phase mode ( @xmath66 ) the amplitude is @xmath67 for @xmath1 even and @xmath68 for @xmath1 odd . using eq . ( [ sinv ] ) , the hamiltonian of the stack of weakly coupled josephson junctions is @xmath69 with @xmath70 being the kronecker delta function . using eqs . ( [ stackeq])-([current ] ) , ( [ cavity ] ) and ( [ sinv ] ) the rate of change in energy is @xmath71_0^l\nonumber \ .\end{aligned}\ ] ] to determine the amplitude of the @xmath48-functions , we require that in the phase - locked state the energy - exchange of a `` collision '' with the boundary is the same for both a fluxon solution and the @xmath48-function approximation . this energy - exchange is given by the time - integral of the last term in eq . ( [ dhdt ] ) , @xmath72_0^l dt\ , \nonumber \end{aligned}\ ] ] where @xmath73 and @xmath74 are taken such that they cover one collision with the boundary . to model a fluxon collision with the boundary , the following profiles are used@xcite @xmath75 with @xmath76 determining the fluxon polarity , @xmath77 being the lorentz factor , and the lowest characteristic velocity @xmath78 to first order in @xmath36 . we take the same fluxon polarity in all junctions , thus @xmath79 for all @xmath3 . following refs . , using eqs . ( [ bc ] ) , ( [ iphaselock ] ) , and ( [ profile ] ) in eq . ( [ dhb ] ) yields @xmath80 \label{deltahb}\end{aligned}\ ] ] with @xmath81 and where the integration was carried out from @xmath82 to @xmath83 for mathematical convenience . calculation of @xmath84 for the @xmath48-function approximation in eq . ( [ deltaapprox ] ) gives @xmath85\ .\end{aligned}\ ] ] requiring @xmath86 determines the amplitude of the @xmath48-functions to @xmath87 @xmath88 being given by eq . ( [ zeta ] ) and @xmath89 . the asymptotic velocity , @xmath90 , present in eq . ( [ profile ] ) may be determined similarly to what is outlined in ref . @xmath91 when the fluxons are shuttling with frequency @xmath92 . the conditions for a steady state require that the energy averaged over one period is zero , thus @xmath93 using eq . ( [ dhdt ] ) , @xmath94 gives the condition @xmath95 where @xmath96 is determined from@xcite @xmath97 the current - voltage characteristics in eq . ( [ phaselock ] ) include the phase @xmath98 , such that at a given bias current the system can adjust this phase together with the collision times , @xmath47 , to satisfy condition ( [ phaselock ] ) ( if possible ) . the phase , @xmath98 , is related to the fluxon shuttling frequency , @xmath92 , through eqs . ( [ cphi ] ) and ( [ sphi ] ) and one may thus change the fluxon shuttling frequency by changing the bias current . ( [ phaselock ] ) @xmath13 is thus obtained as a function of @xmath92 . in the numerical simulations in section iv we shall , inversely , obtain @xmath92 as function of @xmath13 . to calculate the conditions for the cavity to induce bunching ( in - phase motion ) , we consider a triangular fluxon configuration with one fluxon in each junction , modeled by @xmath99 . the interaction energy between the fluxons is first calculated by considering an infinite line with a lattice spacing of @xmath100 , thus @xmath101 the interaction energy of this configuration is @xmath102 resulting in the well - known fluxon - fluxon force @xmath103 the force on the fluxons from the boundary can be calculated from @xmath104 using @xmath105 , valid for @xmath106 , resulting in @xmath107\ . \nonumber\end{aligned}\ ] ] the separation between the fluxons , @xmath100 , may now be determined from the condition @xmath108 solving eq . ( [ balance ] ) for @xmath100 enables one to calculate the current voltage characteristics , eq . ( [ phaselock ] ) , and the cavity current , eq . ( [ iphaselock ] ) , in the steady state for a triangular fluxon lattice . + for the case of only two junctions , eq . ( [ fb ] ) reads @xmath109 in fig . [ figfb ] we plot @xmath98 from eqs . ( [ cphi ] ) and ( [ sphi ] ) as well as @xmath110 from eq . ( [ fi ] ) and @xmath111 from eq . ( [ fb2 ] ) as a function of @xmath92 at constant @xmath100 . it is seen that when the system is above the resonance frequency , the force from the boundary is negative while the @xmath110 is positive , thus they may balance each other . below the resonance frequency the two forces has the same sign and the only steady state solution must be the one where the fluxons move in anti - phase . the perturbation to the current - voltage characteristic by the cavity is contained in the @xmath84-term in eq . ( [ phaselock ] ) , given by eq . ( [ deltahb ] ) . in the case of anti - phase motion for two coupled junctions , this term will be zero . for three junctions , however , it will be non - zero . ( [ fi ] ) and ( [ fb ] ) will have the same direction for @xmath112 resulting in anti - phase motion below the resonance frequency . this suggests that only for an odd number of junctions , we may observe an area of negative differential resistance in the current - voltage characteristics . + + + + + + + + numerical simulations of the full non - linear eqs . ( [ stackeq])-([current ] ) with boundary conditions ( [ cavity ] ) and ( [ bc ] ) has been done using second order finite differences for the spatial derivatives and a 5th order runge - kutta method with adaptive step size for the temporal integration@xcite . the spatial resolution was kept at @xmath113 for all considered systems . the initial fluxon configuration had one fluxon in each junction , each moving in anti - phase with the one in the neighboring junction(s ) . the system was integrated until a stabilized cavity current was obtained or 20000 time units had passed . the stable cavity current , @xmath114 , and the individual voltages at @xmath41 , @xmath115 , was analyzed using interpolation and fft@xcite to determine the most significant frequency in the power spectrum which is used as the cavity current frequency and the fluxon shuttling frequency , @xmath92 . the difference in collision times , @xmath116 , can be calculated directly from the simulation using @xmath42 . it may also be calculated analytically using eqs . ( [ uasymptotic ] ) and ( [ balance ] ) , where the latter eq . was solved numerically for @xmath100 . sometimes there were multiple solutions , @xmath117 , and we have chosen @xmath118 . when no solution was found in the interval , we used @xmath119 corresponding to anti - phase motion . the amplitude of the stabilized cavity current can be determined using a simple line search in the @xmath114 data and compared to the amplitude of eq . ( [ iphaselock ] ) using the value of @xmath100 obtained from eq . ( [ balance ] ) . the frequency versus applied bias current can be compared to eq . ( [ phaselock ] ) , again using @xmath100 obtained from eq . ( [ balance ] ) . when a steady state is found , we observe that @xmath120 with @xmath121 or @xmath122 , with @xmath123 giving a very low cavity current and therefore no significant difference from the unperturbed system . for high bias currents , the system was observed to be in the @xmath123 state and a switching to @xmath121 occurred when the system came close to the resonance in the current voltage characteristic . below resonance frequency , the system again switched to the @xmath123 state . to compare the analytical and numerical results , we only show the numerical simulation points where a steady state was reached for the @xmath121 case . in figs . [ l8 ] and [ l4 ] we have used eqs . ( [ iphaselock ] ) , ( [ phaselock ] ) , ( [ fi ] ) , and ( [ fb])/([fb2 ] ) for the analytical results ( shown with dashed lines ) . fig . [ l8 ] shows the results on the system with @xmath124 and @xmath125 . well above the resonance frequency we get very little current in the cavity . as the shuttling frequency approaches the cavity frequency , the cavity current increases and reaches a maximum at near cavity frequency but suddently drops to near zero slightly above the cavity frequency . note that all numerical results are only shown for frequency values larger than the resonance frequency this in general agreement with the findings in ref . where only oscillators with frequencies higher than the resonance frequency can be syncronised . in addition , the forces shown in fig . [ figfb ] are seen to be directed in opposite directions for frequencies larger than the resonance frequency and in the same direction for frequencies smaller than the resonance frequency . the fluxon - fluxon distance will thus decrease only if the shuttling frequency is larger than the resonance frequency . the corresponding current - voltage characteristic thus shows a deviation from the case without a cavity only above the resonance frequency . the fluxon separation shows a similar behavior . exactly at the resonance frequency , where the cavity current is at its maximum , the system exhibit anti - phase motion due to the boundary force being zero and we find the system to be in the @xmath123 state in the numerical simulations . slightly above resonance frequency , the separation has a minimum and then it increases until it reaches at maximum at some point and then it decreases again . the cavity current is thus largest slightly above resonance frequency , since the boundary force is zero exactly at resonance frequency , resulting in anti - phase fluxon motion . [ l4 ] shows the corresponding results on the system with @xmath126 and @xmath125 . the general behavior is the same as the one in fig . [ l8 ] , except for the fluxon - fluxon separation . below and at resonance frequency , we again see anti - phase behavior . slightly above the resonance frequency , we see that the fluxon - fluxon separation has decreased to zero , i.e. the system has switched to a bunched state . at higher fluxon shuttling frequency , the fluxon are separated at some distance @xmath100 and at some point this distance become so great that we again see anti - phase motion . the minor discrepancies observed in figs . [ l8 ] and [ l4 ] between theory and numerical experiment are primarily caused by the two core assumptions in the perturbation analysis ; namely the rigid collective coordinate approximation for the fluxon , and the idealized treatment of the fluxon reflection at the boundaries of the junction . among the approximations inherent to these assumptions are omission of phonons and the change in fluxon dynamics during reflections . we notice , however , that the agreement between theory and simulations is very good , as can be seen in the figures . the rather weak force induced by the cavity on the fluxons can only be used to obtain bunching in the weakly coupled case . it is , however , essential that the fluxons do not move in perfect anti - phase in order to induce current into the cavity . the top plot of fig . [ l8highs ] the amplitude of the cavity current is shown for a simulation with similar parameters as fig . [ l8 ] but with a much higher inductive coupling , @xmath127 , approaching the case of intrinsic junctions . the simulation was started with a high bias current , resulting in a high fluxon - shuttling frequency and the bias current was gradually lowered resulting in lower fluxon shuttling frequencies . ( [ iphaselock ] ) gives zero cavity current in the case of a perfect anti - phase mode . in the numerical simulations , however , we do not find zero cavity current , but rather that the system is in the @xmath123 state , i.e. the cavity is oscillating with twice the fluxon shuttling frequency . as the fluxon shuttling frequency is lowered , the cavity current increases enough to slightly break the anti - phase motion and the cavity starts to oscillate at the fluxon shuttling frequency , seen in fig . [ l8highs ] by following the line marked with a from the high frequency part to the low frequency part . as the fluxon shuttling frequency is lowered still , the amplitude of the cavity current increases and the fluxon separation , @xmath128 , decreases . near the resonance the cavity current gets smaller and the fluxon separation start to increase again . at some point , the fluxons start to behave erratic , meaning that we can not find a definitive value of @xmath100 in the simulations and thus we do not obtain steady state motion . the part of the curve near @xmath129 where the value of @xmath128 oscillates heavily shows this . at some point the system again switches to the anti - phase motion with the cavity oscillating at twice the fluxon frequency , seen by following the b-line from the low frequency part to the high frequency part of the figure . we have not been able to determine if the erratic behaviour near @xmath129 is due to a too short simulation time before we give up finding a steady state of if this is the true behaviour of the system . we have analytically calculated the cavity current and the current - voltage relation for @xmath1 weakly inductively coupled stacked josephson junctions coupled to a resonance cavity . we have shown that the cavity introduces a force between the natively repulsive fluxons which may be used to obtain bunching in the weakly coupled case . the effect is strongest for short junction , where the boundaries have larger influence . our simple analysis show overall good agreement with numerical simulations . in the case of high inductive coupling our perturbation results deviate from the simulations but the overall picture is still consistent with the theory . sm and nfp would like to acknowledge the stvf framework program `` new superconductors : mechanisms , processes and products '' for financial support . sm would also like to acknowledge financial support from the lundbeck foundation . h. b. wang , p. h. wu , and t. yamashita , appl . phys . lett . * 78 * , 4010 ( 2001 ) ; g. hetchfischer , w. walkenhorst , g. kunkel , k. schlenga , r. kleriner , p. mller , ieee trans . . supercond . * 7 * , 2723 ( 1997 ) ; k. kadowaki , i. kakeya , t. yamamoto , t. yamazaki , m. kohri , and y. kubo , physica c * 111 * , 437 ( 2006 ) .
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we consider fluxon dynamics in a stack of inductively coupled long josephson junctions connected capacitively to a common resonant cavity at one of the boundaries .
we study , through theoretical and numerical analysis , the possibility for the cavity to induce a transition from the energetically favored state of spatially separated shuttling fluxons in the different junctions to a high velocity , high energy state of identical fluxon modes .
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You are an expert at summarizing long articles. Proceed to summarize the following text:
we assume that the classical limit of the massive gluons is represented by the equation of motion with a mass : - d _ f _ - m^2 a _ = 0 , [ eqm ] where f_^a = _ a_^a - _ a_^a + g f_abc a_^b a_^c , and @xmath0 is the covariant derivative . the landau - gauge condition @xmath1 follows from this equation by applying @xmath2 on the equation . at the zeroth order in @xmath3 , we find the following solution & & a_0 = 0 , + & & a = ^-1(_2 t + _ 3 t ) h_0(z , r ) , [ h0 ] where h_0 = ( _ x , _ y , _ z - ( m^2 -^2 ) dz ) ( z , r ) , with @xmath4 and @xmath5 . the electric and magnetic field is e^a & = & -a_0^a - i g f_abca^b a_0^c - ^a , + b^a & = & a^a . this solution may be interpreted as having two point charges placed at @xmath6 and @xmath7 with the opposite signs . this electric field has no divergence @xmath8 except for the points on the charges with the magnitude @xmath9 . the electric flux looks like that of magnetic field in superconducting material . the color of the magnetic field and the electric field are 90 degrees different in the space of color , and the magnetic flux keeps inducing the electric flux . the shape of electric flux and magnetic field are shown in figure [ fig : em4 ] . the electric flux looks like a tube elongated in @xmath10-direction . when we pull the electric charges apart , the tube will get longer but it wo nt get thicker and be about @xmath11 . therefore , a constant tension will occur between the charges , which is quite consistent with stringy picture@xcite of confinement . . the mass @xmath12 . electric flux on the vertical plane and magnetic flux on a tube for the zero - th order classical solution are shown.,width=340 ] the zeroth - order solution above may be improved to all the orders perturbatively . the equation of motion ( [ eqm ] ) in a expanded form is a^a - _ _ a^a - m^2 a^a = g f_abc ^ a_^b a_^c + g f_abc a_^b f_^c . [ eqm2 ] we can perturbatively obtain the solution by repeatedly applying this equation . in addition , we have @xmath13 not only for the zeroth - order solution ( [ h0 ] ) but also to all the orders in @xmath3 . the right hand side of eq.([eqm2 ] ) gives zero if we apply @xmath14 on it , since the first term gives @xmath15 , and the second term @xmath16 will become ^ j_^a = ( ^ a^a ) ( _ a_^a - _ a_^a ) + a^a^ ( _ a_^a - _ a_^a ) + g ^ a^a f_abc a_^b a_^c . using symmetry , the equation of motion and jacobi s identity , we have ^ j_^a & = & g a^a f_abc ( _ a_^ba_^c + a_^b ( _ a_^c - _ a_^c ) + g a^b f_cde a_^d a_^e + m^2a^a ) & & + g ^ a^a f_abc a_^b a_^c = 0 . then @xmath17 follows at the @xmath18-th order , if @xmath17 holds at the @xmath19-th order in @xmath3 . this classical solution gives a picture that @xmath20 and @xmath21 are rotating within a color plane that includes @xmath22 and @xmath23 direction , and quark charges are rotating too . quantum mechanically , a quark must change its color after emitting a gluon . this should have been the reason why it was difficult to understand the static force between confined quarks in the analogy of electric force . further this picture clarifies why non - abelian nature is essential for confinement . in the solution of massive yang - mills theory such as weak theory , the solution does not form conserved flux tubes and its electric field vanishes at longer distance . the reason why we do nt have such a decaying solution is because we do nt have current conservation @xmath24 for the theory with broken global symmetry . this confinement picture will be valid for all the theories that have the same equation of motion classically , including real qcd and lattice qcd . here we additionally present a toy but quantum model , in which the mechanism presented in the previous section holds and easier to analyze . we consider a theory with its lagrangian : ( f_^a)^2 + |_- ig a_^a ^a |^2 - ( ^)^2 + ^ , and with a spontaneous symmetry break of _ ij , [ vev ] where @xmath25 is a complex valued @xmath26 matrix field , and its left index couples to the gauge field but its right index does not . the gauge symmetry is broken , but the global color - rotation symmetry would not be broken under the vacuum expectation value ( [ vev ] ) since the non - gauged ( right ) index of @xmath25 may be rotated together . using the faddeev - popov method , the lagrangian is l & - & ( _ a^a - i g m ( - ^ ) ) ^2 & + & i |c^a ( ( d)_ab + g m ( ^a ^b+ ^b^a ^ ) ) c^b . it is invariant under the brst transformation : _ ba^a _ & = & _ c^a + g f_abc a^b _ c^c , _ b _ ij & = & i g c^a^a_ik _ kj , _ b c^a & = & - g f_abc c^b c^c/2 , + _ b|c^a & = & i b^a , _ b b^a & = & 0 . this model gives the equation of motion ( [ eqm ] ) as the classical counterpart . under the spontaneous symmetry breakdown of eq.([vev ] ) , the gauge bosons and ghosts acquire mass , and massive scalar particles appear due to higgs mechanism . we may take the mass of the higgs particle large enough to make it physically irrelevant . it is easy to show existence of mass gap in this model . here @xmath27 is the full hamiltonian with @xmath28 shifted variables , and @xmath29 . for any eigenstate @xmath30 with its energy @xmath31 , we have e = e|h + h_m|e > e|h_m|e , and the right - hand side should be a positive value unless the number of the gluon is zero . in general , |e= |n_g = 0 + |n_g = 1 + |n_g = 2 + , where @xmath32 is a state with its gluon number @xmath33 . any energy eigenstate with quarks should include @xmath34 states . if it does not , ( h_0 + h_int + h_a)|e= e|e , where @xmath35 includes @xmath36 states and @xmath37 should only include @xmath38 and @xmath39 states but the right - hand side can not include @xmath34 states . due to nonbreaking of the brst symmetry , the quarks are confined in this model because no colored state may appear as follows . the color current is j_^a = f_abc a^b f_0^c(x ) + j_0^a + ( a_b)^a - i ( |cd_c ) + i ( _ |cc ) , where @xmath40 is the color current from the quarks and the higgs field @xmath41 , where @xmath42 . the maxwell equation d^f_^a + g j_^a = _ b^a - i g f_abc ( _ |c^b ) c^c , may be written as @xmath43 . under brst formalism , physical states must be annihilated by the generator @xmath44 @xcite . therefore @xmath45 for any physical states @xmath46 and @xmath47 , which means that ( ^f_^a + g j_^a ) |f= 0 inside the physical space . the operator on the left - hand side is the generator of color gauge transformation , and then this equation means that only color - neutral state can appear as a physical state . in this letter , we presented a new classical solution of qcd and discussed possible relation to confinement . further , we found that confinement and mass gap may occur in a model with a explicit mass introduction with higgs mechanism . dual meissner effect , i.e. massiveness of magnetic field , has mainly been considered to be the mechanism that confines color charged particles , i.e. quarks . however , the picture presented here shows that mass acquisition of electric field is rather appropriate to show the confinement string if the vector particle could acquire mass without breaking the conservation of color current . this mechanism will be valid in the real qcd because the lattice qcd , which is equivalent to the real qcd , dynamically acquires mass from analytical and lattice studies@xcite . further , we have another possibility that some of higgsless massive vector field theories @xcite works . in that case , the analyses in the latter sections would be useful . bogolubsky , e .- ilgenfritz , m. mller - preussker , a. sternbeck , lattice gluodynamics computation of landau - gauge green s functions in the deep infrared , phys.lett.b676:69-73,2009 [ arxiv:0901.0736 ]
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recent researches on the solution of schwinger - dyson equations , as well as lattice simulations of pure qcd , suggest that the gluon propagator is massive . in this letter
, we assume that the classical counterpart of this massive gluon field may be represented with the equation of motion for yang - mills theory with a mass term added .
a new classical solution is given for this equation .
it is discussed that this solution may have some role in confinement .
these days , evidences are accumulating that the lattice qcd , which is equivalent to the real qcd , dynamically acquires mass from analytical and lattice studies@xcite ( and see @xcite for a list of references ) .
the analytical studies with the schwinger - dyson equation ( sde ) nicely agree @xcite with the lattice data .
those sde analyses are based on landau gauge . in this letter , the classical counterpart of massive gluons in the landau gauge is considered . in the next section , a new classical solution is given for the equation of motion , and its relation to confinement is discussed .
the mechanism here will equally be valid as far as the theory has a mass but the color symmetry is unbroken .
though this theory is intended to give insight into real qcd , a non - qcd toy model is additionally analyzed in section 2 .
this model is not for the real world but for facilitating analysis in a toy world .
this model shows interesting behavior of the mass gap and absence of colored states .
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You are an expert at summarizing long articles. Proceed to summarize the following text:
we develop new entropy based statistics @xmath2 called _ slide statistics _ which can be computed from any sample data in a metric space . as an application , we use these statistics to test whether financial returns are independent observations from a particular distribution . for example , figure [ fig : fig1 ] shows plots of @xmath3 against @xmath14 for data @xmath15 consisting of n - tuples of consecutive returns regarded as a subset of the metric space @xmath16 with the usual metric . as can be seen , the lower curve corresponding to the s@xmath0p @xmath1 is very different from the ones obtained for either the normal or laplace distributions . any potential model for the returns of the s@xmath0p @xmath1 must be able simulate the @xmath3 curve in figure [ fig : fig1 ] which is a new requirement that is apparently difficult to meet . the statistics @xmath17 contain information about the interplay between the dimension of a space and distributions on that space and provide a new way of describing continuous random variables and stochastic processes in general . the @xmath17 will take some time to define but we will provide explicit formulas for slide statistics @xmath3 and @xmath4 that are easily computed for any sample in a metric space . against @xmath14 for the s@xmath0p @xmath1 for the @xmath18 trading days ending december 31 , 2014 . the middle cluster of five graphs are the corresponding plots obtained from five different simulated sequences of returns with the laplace distribution . the upper cluster of five graphs are the corresponding plots obtained from five different simulated sequences of normally distributed returns . , width=384 ] our applications will focus on financial returns but many other important processes in science and mathematics also yield data in the form of a set of points in a metric space that must be quantified and interpreted . fields like fractal analysis have developed in order to obtain dimensional information from a wide range of real world data using quantities like the hausdorff , information and correlation dimensions @xcite . one of the problems with these traditional measures of dimension is that they tend to concentrate on the geometric properties of a set while ignoring the statistical origins of the data . we will give examples where @xmath19 appears to converge to the reciprocal of the hausdorff dimension when @xmath20 is taken to be a a larger and larger random sample from a fractal . it is not clear in general how to assign a dimension to a point process but our theory and simulations suggest a purely statistical way to identify a class of them for which it does make sense to assign a dimension . specifically , we think the dimension of a point process @xmath21 should be defined to be @xmath22 provided @xmath23 and @xmath24 for @xmath25 in which case we say the process is tangible with dimension @xmath22 . in the case of a tangible process , the values of @xmath26 are entirely determined by the number @xmath22 but this relationship does not hold in general . a possible example of a tangible process is the random generation of points @xmath27 in @xmath8^d$ ] , where the @xmath28 are chosen independently and uniformly at random from @xmath8 $ ] , for which we provide evidence that @xmath29 and @xmath30 . in particular , @xmath3 and @xmath4 are new spatial statistics for testing hypotheses concerning the distribution of points in @xmath8^d$ ] . although we will sometimes use fractals as examples , these statistics can be applied to any random variable with values in a metric space . in particular , the slide statistics appear to describe continuous random variables @xmath7 in an entirely new way that has nothing to do with mean and standard deviation since @xmath31 for @xmath32 . for example , the first two slide statistics for any normal variable @xmath11 appear to converge to @xmath12 and @xmath13 which provide the basis for a new goodness of fit test for normality . for any exponential random variable @xmath33 , simulation gives that @xmath34 and @xmath35 and we conjecture from these and other simulations that @xmath36 for any continuous real - valued random variable @xmath7 . if we think of @xmath37 as the _ dimension _ of the random variable @xmath7 , then this last conjecture says the uniform distribution on @xmath8 $ ] has the maximum dimension of any continuous real - valued random variable . the second slide statistic @xmath4 is negative for the examples given so far but @xmath4 is positive for the cauchy distribution and is often found to be positive for the log daily returns of the standard and poor 500 index which is yet another demonstration of the non - normality of these returns . the construction of @xmath38 requires several ideas working in concert and can be summarized as follows . in section [ s:2 ] , we introduce a variant of the differential entropy called the genial entropy which is the starting point for defining the slide statistics . unlike the differential entropy , the genial entropy is scale invariant and in section [ s : inequality ] we prove it is never negative which will give a new lower bound for the differential entropy . given a finite set of distinct points @xmath20 in a metric space , we find the distance from each point to its nearest neighbour and arrange these distances in descending order so @xmath39 . let @xmath40 be the function on @xmath41 whose value on @xmath42 is @xmath43 . let @xmath44 be the area under @xmath45 and let @xmath46 denote the genial entropy of the density @xmath47 which turns out to be @xmath48 at @xmath49 . we then define @xmath50 to be the @xmath14th derivative from the right at @xmath48 of the function @xmath46 which is developed in section [ s : calculus ] . in section [ s : calculus ] , we derive an explicit formula for @xmath19 and state a conjectured formula for @xmath51 . the fact that @xmath3 and @xmath4 are easily evaluated using a computer makes them particularly suitable for practical applications . the results obtained from the simulation of @xmath3 and @xmath4 in a variety of contexts are summarized in section [ s : slide ] and demonstrate the interplay between dimension and distribution that is captured by these statistics . as applications in section [ s : returns ] , we consider a variety of possible distributions as models for financial returns and use the slide statistics to illustrate how poorly these models fit the empirical data . our starting point for the development of the slide statistics is the differential entropy @xmath52 of a density @xmath53 which is well known in statistics and information theory @xcite . our focus will be on a variant of the differential entropy called the genial entropy or g - entropy which will be described in definition [ d:2.02 ] . in the simplest case of a probability density @xmath53 that has an inverse function such as @xmath54 on @xmath55 $ ] , the genial entropy is just the sum of the differential entropies of @xmath53 and @xmath56 . the next proposition shows how this sum can be written in a form that makes sense for densities that may not have inverses . [ p:2.01 ] suppose @xmath53 is a continuous function on @xmath57 $ ] with the properties that the derivative of @xmath53 exists and is negative on @xmath58 , @xmath59 and @xmath60 . also assume the differential entropies of @xmath53 and @xmath56 both exist . then the sum of their differential entropies is given by @xmath61 . substitute @xmath62 into the integral @xmath63 to get @xmath64 which equals @xmath65 . after integrating by parts this last integral becomes @xmath66 and the result follows . the genial entropy will only be defined for densities of the following special form . [ d:2.01 ] a corner density is a function @xmath67 where @xmath68 is a connected interval contained in @xmath69 , @xmath53 is monotone decreasing and @xmath70 . here is our definition of genial entropy which is motivated by the conclusion of proposition [ p:2.01 ] . [ d:2.02 ] let @xmath67 be a corner density . the genial entropy or g - entropy of @xmath53 is defined by @xmath71 when this integral exists , with the usual convention that @xmath72 . if @xmath53 satisfies the conditions of proposition [ p:2.01 ] , then @xmath73 so in particular @xmath74 and @xmath75 must have the same genial entropy which happens to be euler s constant as shown in table [ t:2.01 ] . in section [ s : inequality ] , we will prove the genial entropy is always nonnegative . the genial entropy of some corner densities . . [ cols="^,^,^",options="header " , ] the stable distributions are often considered @xcite as possible models for financial returns . we now consider the family of stable distributions @xmath76 described in @xcite where @xmath77 $ ] is the stable parameter , @xmath78 $ ] is the skewness parameter , @xmath79 is the scale parameter and @xmath80 is the location parameter . by proposition [ p:5.01 ] , the values of @xmath17 are not affected by changes in @xmath81 and @xmath82 so we will simply work with the family @xmath83 which we write as @xmath84 . figure [ fig : fig5 ] shows the points @xmath85 from table [ t:5.03 ] plotted together with @xmath86 samples from stable distributions with parameters @xmath87 and @xmath88 chosen uniformly at random from the intervals @xmath89 and @xmath90 respectively . we see that the values in table [ t:5.03 ] lie outside the region corresponding to the stable distributions . figure [ fig : fig6 ] show an expanded view of figure [ fig : fig5 ] showing that the slide statistics for the financial returns in table [ t:5.03 ] are well outside the region corresponding to samples from stable distributions . this failure of the stable distributions to fit financial returns is consistent with the findings of @xcite . obtained for @xmath86 samples from stable distributions with parameters @xmath87 and @xmath88 chosen uniformly at random from the intervals @xmath89 and @xmath90 respectively . the values in table [ t:5.03 ] are shown plotted with squares . , width=384 ] , width=384 ] as we have seen , the statistics @xmath3 and @xmath4 can be used as the basis for goodness of fit test for financial returns . much more research needs to be done to better understand the role of the slide statistics in characterizing financial data . in particular , we would like to understand the relationship between the sign of @xmath4 and the behaviour of financial markets . the simulations we have described suggest that @xmath3 and @xmath4 can be used to distinguish between probability distributions and are capapble of detecting dimensional information . the values of @xmath91 we obtained through simulation point to a larger theory of these statistics which is currently being developed . at present however , the @xmath91 are quite mysterious and much work will need to be done to understand all they are telling us about sets of points in metric spaces , random variables or point processes in general . we gave examples of point processes on the cantor set and the sierpinski triangle for which @xmath5 converged to the dimension of the fractal . we would like to understand when this occurs in general and also the relationship between @xmath5 and the usual definitions of dimension . more generally , we would like to know when a point process is tangible in the sense of definition [ d:5.04 ] . when a process is tangible it is possible to use the expression @xmath92{\frac{(-1)^{n+1}(n-1)!(n-1)\zeta(n)}{\rho_n(u)}}$ ] as a statistic for estimating the dimension . we gave examples in which it converged to the dimension faster for @xmath93 than for @xmath94 and we would like to know what happens for larger values of @xmath14 . in terms of calculations , we need to prove conjecture [ cj:4.01 ] concerning the calculation of @xmath51 and we need formulas for @xmath50 for @xmath95 . given the complexity of our conjecture for @xmath51 , the formulas for larger values of @xmath14 are likely to be very complicated . the convergence of the @xmath50 for real - valued random variables can sometimes be improved by using the distances between consecutive points rather than the distances to nearest neighbours . we would like to have a better understanding of this situation and also to know if there are any higher dimensional analogues . in defining the slide statistics , we used the functions @xmath45 which can be thought of as a continuous deformation of @xmath40 at @xmath96 into the constant function @xmath97 at @xmath49 . we can achieve the same effect using the functions @xmath98 and calculate the derivatives corresponding to the slide derivatives in section [ s : calculus ] . it turns out that only the first of these statistics is interesting and while it is easier to calculate than @xmath3 it often does nt converge . bonetti , m. , forsberg , l. , ozonoff , a. and pagano , m. , _ the distribution of interpoint distances . mathematical modeling applications in homeland security _ , ht banks and c castillo chavez , eds . , 2003,87 - 106 .
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we present a new approach to financial returns based on an infinite family of statistics called _ slide statistics _ that we introduce .
the evidence these statistics provide suggests that certain distributions such as the stable distributions are not good models for the financial returns from various securities or indexes like the s@xmath0p @xmath1 and the dow jones .
formally , we associate with any finite subset of a metric space an infinite sequence of scale invariant numbers @xmath2 derived from a variant of differential entropy called the genial entropy .
we give explicit formulas for @xmath3 and @xmath4 that are easily evaluated by a computer and make this theory particularly suitable for applications . as statistics for point processes , these numbers often appear to converge in simulations and we give examples where @xmath5 converges to the hausdorff dimension and we prove that @xmath6 . for a uniform random variable @xmath7 on @xmath8^n$ ] , the evidence from simulations suggests that @xmath9 and @xmath10 which yields new tests for spatial randomness .
the slide statistics describe continuous random variables in an entirely new way . for example , if @xmath11 is any normal variable then simulations suggest that @xmath12 and @xmath13 which provides new goodness of fit tests for normality .
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You are an expert at summarizing long articles. Proceed to summarize the following text:
in recent years higher dimensional gravity is attracting much interest . one reason is the possibility that these higher dimensions could be detectable at cern . the possibility that space time may have more than four dimensions is initiated by high energy physics and inspired by d - brane ideology in string theory . our 4-dimensional space time ( brane ) is embedded in the 5-dimensional bulk . it is assumed that all the standard model degrees of freedom reside on the brane , where as gravity can propagate into the bulk @xcite . the effect of string theory on classical gravitational physics is investigated by the low - energy effective action . if our 5-dimensional space time is obtained as an effective theory , the matter fields , for example the u(1 ) field , can exists in the bulk . in general relativity(gr ) , gravitating non - abelian gauge field , i.e. , the yang - mills(ym ) field , can be regarded as the most natural generalization of einstein - maxwell(em ) theory . in particular , particle - like , soliton - like and black hole solutions in the combined einstein - yang - mills(eym ) models , shed new light on the complex features of compact object in these models . see @xcite for an overview . the reason for adding a cosmological constant to these models , was inspired by the study of the so - called ads / cft correspondence @xcite , since the 5-dimensional einstein gravity with cosmological constant gives a description of 4-dimensional conformal field theory in large n limit . brane world scenarios predict a negative cosmological constant . there is a relationship between the frw equations controlling the cosmological expansion and the formulas that relate the energy and entropy of the cft @xcite , indicating that both sets of equations may have a common origin . in the ads - brane cosmological models , the ads / cft model describes a cft dominated universe as a co - dimension one brane , with fixed tension , in the background of an ads black hole @xcite . the brane starts out inside the black hole , passes through the horizon and keeps expanding until it reaches a maximal radius , after which it contracts and falls back into the black hole . at these moments of horizon crossing , it turns out that the frw equation turns into an equation that expresses the entropy density in terms of the energy density and coincides with the entropy of the cft . however , in these models , one adds on an artificial way tension into the equations . more general , one could solve the equations of einstein together with the matter field equations , for example , the ym field and try to obtain the same correspondence . string theory also predicts quantum corrections to classical gravity theory and the gauss - bonnet(gb ) term is the only one leading to second order differential equations in the metric . in the 4-dimensional eym - gb model with a dilaton field ( eymd - gb ) @xcite , it was found that the gb contribution can lead to possible new types of dilatonic black holes . further , for a critical gb coupling @xmath0 the solutions cease to exist . the ads / cft correspondence can also be investigated in the einstein - gb gravity . for a recent overview , see @xcite . from the viewpoint of ads / cft correspondence , it is argued that the gb term in the bulk corresponds to the next leading order corrections in the @xmath1 expansion of a cft . further , it is argued that the entropy of an einstein - gb black hole and the cft entropy induced on the brane are equal in the high temperature limit . in this paper we investigate the possibility of regular and singular solutions in the 5-dimensional eym - gb model and the effect of a cosmological constant on the behaviour of the solutions . the action of the model under consideration is @xcite @xmath2,\ ] ] with @xmath3 the gravitational constant , @xmath4 the cosmological constant , @xmath5 the gauss - bonnet coupling and @xmath6 the gauge coupling . the coupled set of equations of the eym - gb system will then become @xmath7 @xmath8 with the einstein tensor @xmath9 and gauss - bonnet tensor @xmath10 further , with @xmath11 the ricci tensor and @xmath12 the energy - momentum tensor @xmath13 and with @xmath14 , and @xmath15 where @xmath16 represents the ym potential . consider now the spherically symmetric 5-dimensional space time @xmath17 with the ym parameterization @xmath18 where @xmath19 and @xmath20 are functions of and @xmath21 and @xmath22 . it turns out that no time evolution of the metric component @xmath23 can be found from the equations , so @xmath23 depends only on r. the equations become ( we take @xmath24 ) @xmath25 @xmath26 @xmath27 and @xmath28 the independent field equations then read @xmath29 @xmath30 while the equation for @xmath23 decouples and can be integrated : @xmath31 the equations are easily solved with an ode solver and checked with maple . we will take for the initial value of @xmath20 the usual form @xmath32 . then the other variables can be expanded around @xmath33 : @xmath34 so we have 3 initial parameters and 4 fundamental constants @xmath35 , @xmath6 and @xmath5 . we solved the equations with a two point boundary value solver . epsf fig.1 solution for @xmath36 for @xmath37 and @xmath38 respectively . + fig . 2 solution for @xmath39 for @xmath37 and @xmath38 respectively . 3 solution for @xmath40 for @xmath37 and @xmath38 respectively . + from our numerical solutions , we see that the gb term increases the number of nodes of the yang - mills field . further , we see that for positive @xmath4 the solution develops a singularity , while for negative @xmath4 it remains singular free . a matter field term in the action will lead to an extra term inside the square root of eq.(16 ) , for example in the case of the 5-dimensional einstein - maxwell - gb model @xcite and the 5-dimensional einstein - yang - mills - gb model with the wu - yang ansatz @xcite . in these models , however , there are no additional equations for the maxwell field and ym field respectively . so an analytic solution for @xmath41 is obtained . when one simultaneously tries to solve the einstein equation and matter field equations , then it is not easy to obtain an analytic expression for @xmath41 , as is the case of our eym - gb model . however we can analyse the equation for @xmath41 when @xmath20 becomes a constant : @xmath42 the solution is @xmath43 with @xmath44 an integration constant . since horizons occur where @xmath45 , we can expect cosmological- and event horizons . one can easily check that the zero s of @xmath41 are @xmath46 so the horizon radius depends only on suitable combinations of @xmath4 and @xmath44 . the expression inside the square root becomes negative ( and hence @xmath41 is singular ) for @xmath47 depending on the parameters , this singular surface can be shielded by the event horizon ( otherwise , it will be naked ) . this is well known behaviour in the models where the equation for @xmath41 decouples from the matter field equation . one should like to prove that for negative @xmath4 that f(r ) has no zero s and is regular everywhere in our model . this is currently under study . a 5-dimensional spherically symmetric particle - like solution is found in the einstein - yang - mills gauss - bonnet model . as in other studies in higher dimensional cosmological models , a negative cosmological constant seems to favor for stability and results in most cases in asymptotically anti de sitter space time . in our 5-dimensional eym - gb model , we also find a profound influence of a negative cosmological constant on the behaviour of horizons . the appearance of horizons in e - gb models is not surprising . these gb black holes are found by many authors . however , the lacking of horizons in the eym - gb model for suitable negative cosmological constant is quite new . the explanation for this behaviour must come from the ym term on the right hand side of eq.(9 ) . the zero s of @xmath41 will depend on the behaviour @xmath48 . there could be a connection of the solution presented here with the ads / cft correspondence . as mentioned before , no analytic expression for @xmath41 available . moreover , to obtain the ( n-1)-dimensional entropy on the brane , one needs the junction conditions at the brane ( @xcite ) , which becomes very complicated in the eym - gb model . the junction condition also introduces a brane tension . this tension must cancel the cosmological constant , in order to obtain the desired cft correspondence . the contribution of ym field on the junction could have profound impact on the tension of the brane and the role of a cosmological constant could be different . so the strong influence of a small cosmological constant on the eventually formed gb black hole in our model is quite clear from the consideration mentioned above . 10 randall l and sundrum r 1999 _ phys . rev . lett . * 83 * 3370 , 4690 _ volkov m and galtsov d v 1999 _ phys . rep . * 319 * 1 _ maldcena j 1999 _ adv . phys * 2 * 3370 , 231 _ bjoraker j and hosotani y 2000 _ phys . lett . * 84 * 1853 _ savonije i and verlinde e 2001 _ phys . lett . b*507 * 305 _ gubser s s 2001 _ phys . d_63 _ 084017 _ donets e e and galtsov d v 1995 _ phys b * 352 * 261 _ torii t , yajima h and maeda k 1997 _ phys d * 55 * 739 _ ogushi s and sasaki m 2005 _ prog . . phys . * 113 * 979 _ okuyama n and maeda k 2003 _ phys . d * 67 * 104012 _ boulware d g and deser s 1985 _ phys . . lett . * 55 * 2656 _ cai r and guo q 2004 _ phys . rev . d*69 * 104025 _ clunan t , ross s f and smith d j 2004 _ class . * 21 * 3447 _ thibeault m simeone c and eiroa e f 2005 _ preprint gr - qc/0512029 _ mazharimousavi s and halilsoy m 2008 _ preprint gr - qc/08011562 _ slagter r j 2006 _ in the proceedings of the conference on general relativity and gravitation , paris , 2005 _
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we present a numerical solution on a 5-dimensional spherically symmetric space time , in einstein - yang - mills - gauss - bonnet theory using a two point boundary value routine .
it turns out that the gauss - bonnet contribution has a profound influence on the behaviour of the particle - like solution : it increases the number of nodes of the ym field . when a negative cosmological constant in incorporated in the model , it turns out that there is no horizon and no singular behaviour of the model . for positive cosmological constant
the model has singular behaviour .
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You are an expert at summarizing long articles. Proceed to summarize the following text:
since its very origins , the exact renormalization group ( erg ) @xcite has proved to be a powerful tool for studies of non - perturbative effects in quantum field theory ( see recent reviews in @xcite ) . a particularly interesting case is that of an effective scalar field theory in two dimensions . as it was first conjectured by zamolodchikov @xcite , for a @xmath0 symmetric theory there should exist an infinite set of non - perturbative fixed points corresponding to the unitary minimal series of @xmath3 conformal field theories , where @xmath4 @xcite . morris @xcite showed numerically that such points do exist . the calculation was performed with a reparametrization invariant version of the legendre erg equation @xcite expanded in powers of derivatives . it was also pointed out there that to the level of the local potential approximation only the continuum limits described by periodic solutions and corresponding to critical sine - gordon models could be obtained . to find the expected set of fixed points the calculations had to be taken to the next order in the derivative expansion . this constituted a manifestation of the non - perturbative nature of the phenomena , and remarkably the legendre erg equation was powerful enough to locate and describe with good accuracy the expected set of 2d field theories . in this work we study the same @xmath0 symmetric scalar field theories in two dimensions but now with the polchinski erg equation @xcite . we present preliminary results which complement the results obtained with the legendre erg equation . in sect . 2 we follow the article by ball et al . @xcite to present the basic equations of the formalism this will allow us to set up notation for sect . 3 where we analyse the equations to second order in the derivative expansion . in sect . 4 we discuss the results and present our conclusions . the polchinski equation @xcite for a scalar theory can be written as follows @xcite & & = _ k(^2)+ds+ + & & _ - _ _ p [ cv1 ] . here @xmath5 is a general wilsonian action which can be written in terms of dimensionless variables as follows [ ; t ] & = & _ p _ p p^2 ( k(^2 ) ) ^-1 _ -p + _ int[;t ] , [ action1 ] + _ int[;t ] & = & dy . [ action2 ] in eq . ( [ cv1 ] ) the partial derivative on @xmath5 means it only acts on the explicit @xmath6 dependence of the couplings and the prime in the momentum derivative means it does not act on the delta function of the energy - momentum conservation , and @xmath7 . @xmath8 is a ( smooth ) regulating function which damps the high energy modes satisfying the normalization condition @xmath9 . the renormalized field @xmath10 changes with scale according to @xmath11 \hat{\varphi}_{p},\ ] ] where @xmath12 is the anomalous scaling dimension . to the second order in the derivative expansion we consider the two terms which are written explicitly in eq . ( [ action2 ] ) . within this approximation the polchinski erg equation reduces to the following system @xcite & = & f+2az-2ff+^+f+^-xf , [ eq1 ] + & = & z+bf^2 - 4zf-2zf+^-xz- z-/2 , [ eq2 ] where @xmath13 , @xmath14 , @xmath15 and the potentials @xmath16 and @xmath17 are defined in eq . ( [ action2 ] ) . the dots and primes denote the partial derivatives with respect to @xmath18 and @xmath19 respectively . the parameters @xmath20 and @xmath21 reflect the scheme dependence of the equations and are equal to @xmath22 , @xmath23 . here @xmath24 and @xmath25 , @xmath26 parametrize the regulating function in eq . ( [ action1 ] ) and are defined by & = & ( -1)^n+1k^(n+1)(0 ) , + i_n & = & -_(^2)^nk ( ^2)=-_d_0^dz z^d/2 - 1-nk(z ) , where @xmath27 stands for the @xmath28-th derivative of @xmath29 and @xmath30 . in the next section we search for fixed - point solutions , i.e. for functions @xmath31 and @xmath32 which are independent of @xmath18 and satisfy the system & & f+2az-2ff+^+f+^-xf=0 , [ eqfp1 ] + & & z+bf^2 - 4zf-2zf+^-xz- z-/2=0 . [ eqfp2 ] we will choose the initial conditions ( according to the terminology adopted in the literature on the erg equations ) set by the @xmath0 symmetry : @xmath33 and @xmath34 and by the normalization condition : @xmath35 . for the value of the first derivative of @xmath31 at the origin we will take the condition @xmath36 , where @xmath37 is a free parameter . the anomalous dimension @xmath12 at a fixed point becomes the critical exponent @xmath38 . to solve eqs . ( [ eqfp1 ] ) , ( [ eqfp2 ] ) for @xmath39 we consider the recursive numerical method already tested for @xmath40 @xcite . the physical fixed point solutions @xmath41 , @xmath42 at the fixed point value @xmath43 are regular for @xmath44 and have a certain asymptotic behavior as @xmath45 . thus the natural method for finding the correct numerical solution is to select those which can be integrated as far as possible in @xmath19 . a generic solution will end at a sharp singularity for a finite value of @xmath19 . the difficulty lies in the nonlinear and stiff nature of the equations and the need to fine tune @xmath1 and @xmath37 . this makes the direct integration of the system too hard . one way out is to solve it recursively . unlike the case @xmath40 studied in a number of articles @xcite , one faces an additional difficulty in two dimensions . it is not possible to start the iterative procedure by setting in eq . ( [ eqfp1 ] ) @xmath46 and @xmath47 as it is prescribed by the consistency of the leading approximation . for @xmath39 the polchinski equation in the leading order has only periodic or singular solutions for all values of @xmath37 . to overcome this difficulty one has to consider @xmath48 as the initial value to start the iterations consequently , an analysis of the leading order polchinski equation @xmath49 with the initial conditions @xmath33 , @xmath36 for non - zero @xmath1 is required . we studied eq . ( [ eqfp01 ] ) for @xmath39 numerically for a wide range of values of @xmath1 and @xmath37 . our results show that for each @xmath50 we can fine tune @xmath37 in such a way as to obtain a non - trivial regular fixed point solution . the set of such values @xmath51 form a discrete series of continuous lines @xmath52 ( see fig . 1 ) . in fact simple arguments can be presented which explain the appearance of the lines in the parameter space corresponding to regular fixed - point solutions @xmath41 . let @xmath53 denote position of the pole of a generic solution of eq . ( [ eqfp01 ] ) . suppose that for some values @xmath54 the solution is regular , i.e. @xmath55 . let us take another value @xmath56 sufficiently close to @xmath57 . assuming that the function @xmath53 is continuous , it is clear that there should exist the value @xmath58 such that again @xmath59 . hence there is a line of the `` constant value '' @xmath60 in the parameter space . when we move along a fixed line the solutions @xmath41 do not change their shape significantly . moreover , their shape follows a regular pattern when passing from one curve to the other similar to solutions obtained by morris @xcite . this can be considered as a sign for the existence of the infinite discrete set of fixed points corresponding to the minimal unitary series of conformal models . we also would like to note that for @xmath61 and @xmath62 there are no other non - trivial fixed - point solutions besides the ones corresponding to the lines discussed here . = 1.0 we would like to note that a similar picture takes place in other dimensions . for @xmath40 we found the same discrete set of lines in the @xmath63-plane corresponding to regular solutions of eq . ( [ eqfp01 ] ) , but in this case they are situated in the interval @xmath64 ( see fig . 1 ) . as one can see there is a line ( upper line in fig . 1 ) which crosses the @xmath37-axis at @xmath65 . this is the value of the parameter @xmath37 for which a non - trivial fixed point solution of the polchinski equation was found in the leading order ( local potential approximation ) @xcite . the important observation is that there is only one line in the @xmath63-plane with positive values of @xmath1 . since according to general arguments at physical fixed points @xmath66 , this suggests that for @xmath40 there is only one non - trivial fixed point . this is the wilson - fischer fixed point found in numerous previous studies @xcite . recall that for @xmath39 all the lines are situated in the @xmath67 half - plane , hence one can expect an infinite number of non - trivial fixed points . one more remark is relevant here . by a simple scaling analysis of eq . ( [ eqfp01 ] ) it can be shown that there is a certain mapping between the lines @xmath52 in different dimensions . when we pass from one dimension to another the line experiences a vertical shift and scaling transformation . more details about this mapping will be presented elsewhere . we now pass to the study of the system ( [ eqfp1 ] ) , ( [ eqfp2 ] ) . we have seen that there are families of solutions of eq . ( [ eqfp01 ] ) corresponding to a given fixed point . it turns out that when the second equation of the system is taken into account , this degeneracy disappears . we solved the system using the following iteration procedure developed by ball et al . first we set @xmath68 , choose some initial value @xmath69 and fine tune @xmath37 to the value @xmath70 corresponding to the regular solution @xmath71 of the first equation ( [ eqfp1 ] ) ( or ( [ eqfp01 ] ) ) . of course , the point @xmath72 lies on one of the lines described above ( see fig . 1 ) ) . as the next step we insert the function @xmath71 into the second equation ( [ eqfp2 ] ) for a fixed @xmath21 and fine tune @xmath1 to the value @xmath73 for which a regular solution @xmath74 exists . then we substitute @xmath74 and @xmath75 into the first equation of the system and find a regular solution for a fixed value of @xmath20 thus obtaining a new value @xmath76 and a new function @xmath77 . we repeat this process keeping @xmath20 and @xmath21 fixed . as a result a sequence of functions @xmath78 , @xmath79 , @xmath80 , @xmath81 , and a sequence of numbers @xmath82 , @xmath83 , @xmath84 , @xmath85 , are obtained , and we test them for convergence . for @xmath40 and for the relevant line , associated with the wilson - fischer fixed point , we confirmed the results by ball et al . @xcite . the new feature in our calculations is that we took @xmath86 as the initial value of the iterating procedure , whereas in @xcite only @xmath87 was considered . we conclude that the numerical method converges and that the rate of convergence is controlled by @xmath20 for fixed @xmath88 . the best @xmath20 was shown to correspond to the inflexion point where @xmath89 changes sign . the important observation is that the final values @xmath38 and @xmath90 to which the iterations converge ( i.e. the fixed - point values ) do not depend on the initial value @xmath91 . when @xmath88 is closer to the fixed - point value the rate of convergence is of course faster . the final value @xmath38 depends on @xmath21 linearly . for @xmath92 the two equations decouple and there is no need for iterations to find a solution of the system . we just have to adjust @xmath21 such that @xmath93 . for the wilson - fischer fixed point @xmath94 we found @xmath95 and @xmath96 . = 1.0 for @xmath39 the situation is totally different . for any line @xmath97 we start with and the initial value @xmath98 the iterative procedure turns out to be divergent if @xmath99 . only for @xmath92 we have been able to find a solution to the now decoupled system ( [ eqfp1 ] ) , ( [ eqfp2 ] ) by adjusting @xmath21 . similar to the case @xmath40 we have not found any natural criteria to select the value of @xmath21 since for each line @xmath21 depends monotonically on @xmath1 decreasing as @xmath100 . to determine the fixed - point solutions , corresponding to the minimal unitary series of conformal field theories , we have fixed the value of @xmath21 by a fit to the series of exact values for the anomalous scaling dimension @xmath38 . in this way we found @xmath101 . the fixed - point solutions for @xmath41 display regular behaviour and are reminiscent of those obtained by morris @xcite . in particular they have @xmath102 extrema , @xmath103 ( see fig . the fixed point solutions for @xmath42 have the same pattern of extrema , though their profiles are different from those of morris . next for @xmath104 and and corresponding values @xmath43 we have calculated the critical exponent @xmath2 . for this we considered perturbations of the functions @xmath31 and @xmath32 around the fixed - point solutions , @xmath105 and substituted them into eqs . ( [ eq1 ] ) , ( [ eq2 ] ) . after linearization we obtained the system -2g_nf_*-2f_*g_n+_*^+g_n+ _ * ^-xg_n&= & _ ng_n , + h_n-2f_*h_n-2g_nz_*-4f_*h_n-4g_nz _ * & & + + _ * ^-xh_n-_*h_n+ 2bf_*g_n&=&_nh_n , where @xmath106 and @xmath107 are calculated for @xmath43 . for @xmath0 perturbations the initial conditions are @xmath108 , @xmath109 . we also imposed the normalization condition @xmath110 . away from the fixed point ( but sufficiently close to it ) we relaxed @xmath111 to be different from zero , @xmath112 . then @xmath113 and @xmath114 were fine tuned so that polynomially growing eigenfunctions were obtained . for @xmath39 , @xmath92 and @xmath104 we have calculated the critical exponent @xmath115 . the results for @xmath38 and @xmath2 are given in table 1 . we conclude that the polchinski erg equation gives the values for the critical exponent which match quite well the exact conformal field theory values @xcite and the results by morris @xcite , also included in the table . we note that the best fit value for @xmath21 that we have found actually corresponds to a well defined subset of the regulating functions . an explicit example is @xmath116 , where @xmath117 to ensure @xmath118 and @xmath119 , and @xmath120 to give @xmath104 . values obtained by morris @xcite ( @xmath57 , @xmath121 ) and the exact results of conformal field theory ( @xmath122 , @xmath123 ) . in this work we have studied the solutions of the polchinski erg equation for an effective @xmath0-symmetric scalar field theory in the two - dimensional space @xmath124 . we have seen that this equation provides a reliable non - perturbative evidence for the existence of the fixed - point solutions corresponding to the minimal unitary series of conformal field theories and allows to calculate the anomalous dimension and the critical exponents with good accuracy . this constitutes another positive test of the power of the erg approach . at the same time our studies are complementary to similar calculations within the erg approach based on the equation for the legendre action @xcite . as mentioned above , in the leading order of the derivative expansion ( local potential approximation ) the consistent value for @xmath1 is 0 and only periodic sine - gordon type fixed - point solutions can be obtained . however , we have found that there are continuous families of fixed - point solutions corresponding to a series of lines @xmath52 in the @xmath63 plane which have a part with positive values @xmath125 . it was also argued that these lines correspond to the multicritical fixed points of the theory . it is by taking into account the second order in the derivative expansion that we find isolated fixed - point solutions . we have found the first 10 points out of the infinite series and calculated the critical exponent @xmath2 for them . the results depend on the choice of the regulating function . the value of @xmath21 , for which a regular solution exists , depends linearly on @xmath1 , so the criterium of minimal sensitivity can not be applied to fix @xmath21 . the best fit to the conformal field theory values of @xmath1 gives us @xmath92 , @xmath101 . no other regulating functions have been found to work . for @xmath39 whenever @xmath126 , the iterative procedure is not seen to converge . in fact , for @xmath92 there is no need for iterations since the two equations of the polchinski approach decouple . this is in sharp contrast to the case of @xmath40 where convergence was checked for @xmath126 @xcite . for the values @xmath92 , @xmath104 , giving the best fit , our results are comparable in accuracy and sometimes better ( the accuracy also increases with multicriticality ) than those of morris @xcite . we would like to note that fixing @xmath21 by the best fit to exact results for the anomalous dimensions @xmath1 is reminiscent of fixing the renormalization scheme dependence in the perturbative renormalization group . it is also similar to fixing the regulator by the condition of the reparametrization invariance for the erg equation for the legendre action ( that corresponds to the limiting case of @xmath92 , @xmath127 ) @xcite . another important point we would like to mention is that in our analysis we have not found non - trivial fixed points other than those corresponding to the minimal models . this is what one expects from zamolodchikov s @xmath128-theorem @xcite . the conclusion is already clear from the analysis of the leading order polchinski equation for @xmath48 when the lines in the @xmath129-plane corresponding to non - trivial fixed points are plotted . as final comments we would like to mention that to obtain an estimate of the error of our numerical results in table 1 one needs to carry out the calculations to the next order of the derivative expansion . it would be also interesting to expand the analysis for higher dimension operators . we would like to thank tim morris and jos latorre for some fruitful discussions during the workshop . we also acknowledge financial support by fundao para a cincia e a tecnologia under grant number cern / s / fae/ + 1177/97 . yu.k . acknowledges financial support from fellowship praxis xxi + /bcc/4802/95 . r.n . acknowledges financial support from fellowship praxis xxi / bpd/14137/97 . morris , int . j. mod . b12 ( 1998 ) 1343 ; in : `` zakopane 1997 . new developments in quantum field theory '' ( nato workshop on theoretical physics , zakopane , 1997 ) , p. 147 ( hep - th/9709100 ) ; preprint , hep - th/9802039 . a. hasenfratz and p. hasenfratz , nucl . b270 ( 1986 ) 687 ; t.r . morris , phys . b329 ( 1994 ) 241 ; p.e . haagensen , yu . kubyshin , j.i . latorre and e. moreno , phys . b323 ( 1994 ) 330 ; n. tetradis and c. wetterich , nucl . b422 ( 1994 ) 541 ; j. comellas , nucl . phys . b509 ( 1998 ) 662 .
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we investigate a @xmath0-symmetric scalar field theory in two dimensions using the polchinski exact renormalization group equation expanded to second order in the derivative expansion .
we find preliminary evidence that the polchinski equation is able to describe the non - perturbative infinite set of fixed points in the theory space , corresponding to the minimal unitary series of 2d conformal field theories .
we compute the anomalous scaling dimension @xmath1 and the correlation length critical exponent @xmath2 showing that an accurate fit to conformal field theory selects particular regulating functions .
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for a family @xmath8 of @xmath9-graphs , let @xmath10 denote the maximum number of edges in an @xmath3-vertex @xmath9-graph which contains no member of @xmath8 . determining @xmath10 is a fundamental question in graph theory which becomes extremely difficult when @xmath11 . let @xmath12 and call this value the _ turn density _ of @xmath8 ( as has been pointed out many times , it is easy to show that this limit exists ) . when @xmath8 consists of a single graph @xmath1 , we write @xmath13 for @xmath14 . let @xmath15 denote the complete @xmath0-graph on four vertices . over 70 years ago , turn famously conjectured that @xmath16 , but this conjecture is still unproved @xcite . in fact , when @xmath11 there are very few @xmath9-graphs for which the turn density is known ( see @xcite for a detailed account ) . despite this general difficulty , there is a special @xmath0-graph called the _ fano plane _ for which much is known . the fano plane , denoted @xmath17 , is the projective geometry of dimension @xmath18 over the field with @xmath18 elements ; alternatively , @xmath17 is the @xmath0-graph on seven vertices @xmath19 with the seven edges @xmath5 . let @xmath20 denote the balanced complete bipartite @xmath0-graph , which is obtained by partitioning a set of @xmath3 vertices into parts of size @xmath21 and @xmath22 and taking as edges all the triples intersecting both parts . since @xmath20 is @xmath18-colorable and it is easy to see that @xmath4 is not , @xmath20 contains no copy of @xmath4 . therefore , @xmath23 . ss @xcite conjectured that this lower bound is asymptotically best possible and hence @xmath24 . a few decades later , de caen and fredi @xcite proved ss conjecture via a clever use of so - called link graphs . a few years later , keevash and sudakov @xcite and independently fredi and simonovits @xcite proved the exact counterpart of this result ; that is , @xmath25 for sufficiently large @xmath3 . let @xmath26 be a @xmath9-graph with vertex set @xmath27 . given any subset @xmath28 , @xmath29 , the _ degree _ of @xmath30 , denoted by @xmath31 , is the number of edges of @xmath26 that contain @xmath30 . for simplicity , when @xmath30 consists of one vertex @xmath32 or two vertices @xmath32 and @xmath33 , we write @xmath34 , and @xmath35 instead of @xmath36 and @xmath37 , respectively . when @xmath38 , we call @xmath35 the _ co - degree _ of @xmath32 and @xmath33 , while the set of vertices @xmath39 such that @xmath40 is called the _ co - neighborhood _ of @xmath41 and will be denoted by @xmath42 . for each integer @xmath43 , let @xmath44 . we call @xmath45 the minimum _ @xmath46-degree _ of @xmath26 . for a family @xmath8 of @xmath9-graphs , let @xmath47 denote the maximum value of @xmath48 in an @xmath3 vertex @xmath9-graph @xmath26 which contains no member of @xmath8 and let @xmath49 . mubayi and zhao @xcite prove that this limit exists in the case @xmath50 and lo and markstrm @xcite prove that this limit exists for all @xmath51 ( a fact previously sketched by keevash @xcite ) . note that the case @xmath52 just reduces to @xmath14 . when @xmath38 , we call @xmath53 the _ co - degree density _ of @xmath8 . for general @xmath9-graphs , a simple averaging argument shows that @xmath54 when @xmath55 ( see @xcite section 13.2 ) . it is also pointed out in @xcite section 13.2 that for any graph @xmath1 , @xmath56 . the same argument applies to any finite family @xmath57 as well . [ min - degree ] for a finite family @xmath8 of @xmath9-graphs , @xmath58 . ( sketch ) . let @xmath59 . let @xmath60 be any small positive real . let @xmath3 be sufficiently large as a function of @xmath61 . let @xmath26 be a @xmath9-graph with @xmath62 . by @xcite proposition 4.2 , @xmath26 contains a subgraph @xmath63 on @xmath64 vertices with @xmath65 . since @xmath66 and @xmath67 as @xmath68 , when @xmath3 is large enough , we have @xmath69 . so @xmath63 contains a member of @xmath57 and therefore @xmath26 contains a member of @xmath57 . so the minimum degree problem is essentially the same as the turn problem . the minimum co - degree problem however is drastically different . for instance , there are @xmath0-graphs @xmath1 with @xmath13 arbitrarily close to @xmath70 and yet @xmath71 ( see @xcite ) . in general , there has not been a very good understanding of the relationship between @xmath13 and @xmath72 ( see @xcite and @xcite for detailed discussions ) . similar to the situation with the turn density , not much is known about @xmath72 even for small graphs @xmath1 such as @xmath15 ( in this case czygrinow and nagle @xcite conjectured that @xmath73 ) . mubayi @xcite initiated the study of @xmath74 , where @xmath4 is the fano plane . as pointed out earlier , @xmath20 contains no copy of @xmath4 . so , @xmath75 . mubayi @xcite proved an asymptotically matching upper bound thus establishing @xmath76 . he further conjectured that @xmath77 , for sufficiently large @xmath3 . this was later proved by keevash @xcite using a very sophisticated argument involving hypergraph regularity , quasi - randomness , and stability ( we should mention that keevash proves the stronger statement that the extremal example is stable " . also , the scope of keevash s paper is not limited to the problem of determining the co - degree threshold for the fano plane . ) . in this paper , we give a simple proof of mubayi s conjecture which is in the same spirit as mubayi s original proof of @xmath76 . our main result is [ main ] there exists @xmath78 such that if @xmath79 , then @xmath80 . since we are giving a new proof of an old result , it is worth mentioning that we only need @xmath78 to be large enough so that supersaturation " holds ( see section [ lemmas ] ) . while we do not make an attempt to compute the value of @xmath78 , it is considerably smaller than the value of @xmath78 needed for the use of regularity in @xcite . the paper is organized as follows . in section [ lemmas ] we give some lemmas and introduce a family of @xmath0-graphs called _ rings_. in section [ fano ] we prove theorem [ main ] by making use of the family of rings . in section [ turan ] we determine the turn density of the family of rings . finally , in section [ remarks ] , we conclude with some remarks and open problems . for any @xmath9-graph @xmath26 , the _ @xmath81-blowup _ of @xmath26 , denoted @xmath82 , is the graph obtained from @xmath26 by cloning each vertex @xmath81 times . for a family of @xmath9-graphs @xmath8 , let @xmath83 . erds @xcite used supersaturation to show [ erdos - blowup]@xcite for any finite family of @xmath9-graphs @xmath57 and any positive integer @xmath81 , @xmath84 . keevash and zhao @xcite proved an analogous result for the co - degree density . [ codegreeblowup][kz ] for any finite family of @xmath9-graphs @xmath57 and any positive integer @xmath81 , @xmath85 . the same supersaturation argument in fact gives [ blowup ] for any finite family of @xmath9-graphs @xmath57 and any positive integer @xmath81 , and any @xmath86 , @xmath87 , @xmath88 . we also make the following trivial observation based on the definitions . [ subgraph ] let @xmath57 and @xmath89 be two families of @xmath9-graphs . let @xmath90 . suppose that for every member @xmath91 , some subgraph of @xmath26 belongs to @xmath57 . then @xmath92 . so , in particular , @xmath93 . we now define a family of @xmath0-graphs , called _ rings _ , which will play a central role in our proof of theorem [ main ] . let @xmath94 and let @xmath27 be a set of at most @xmath95 vertices surjectively labeled with @xmath96 . let @xmath97 be the family of @xmath0-graphs on @xmath27 with edge set @xmath98 , where addition is defined modulo @xmath99 . let @xmath100 be the ( unique ) member of @xmath97 which has exactly @xmath95 vertices and call @xmath100 a ring on @xmath95 vertices . let @xmath101 and @xmath102 . [ familyblowup ] for all positive integers @xmath94 and @xmath103 , we have @xmath104 and @xmath105 . since @xmath106 , we have @xmath107 . on the other hand , for every @xmath108 , @xmath109 clearly contains a copy of @xmath110 , since in any member of @xmath109 there are @xmath99 distinct copies of @xmath111 . by proposition [ subgraph ] and lemma [ codegreeblowup ] , @xmath112 . thus , @xmath113 . by a similar argument , we have @xmath114 . [ lmproperty ] a hypergraph @xmath1 on @xmath115 vertices is said to have the _ @xmath116-property _ if every subset of @xmath117 vertices contains at least one edge of @xmath1 . mubayi and rdl @xcite recursively constructed for every @xmath94 a family @xmath118 of @xmath0-graphs with the @xmath119-property . they showed that @xmath120 for each fixed @xmath94 and used this to establish an upper bound on the turn density of @xmath121 ( sometimes referred to as the @xmath0-book with @xmath0 pages ) . this family @xmath118 also played a key role in mubayi s proof of @xmath122 . here , we observe that for every @xmath99 the graph @xmath100 has the @xmath123-property and we will also show that @xmath124 is small . then , by using @xmath125 instead of @xmath126 we are able to establish @xmath127 . [ edge ] @xmath100 has the @xmath123-property . clearly @xmath100 has @xmath95 vertices . let @xmath128 be any set of vertices in @xmath100 that contains no edge . we show that @xmath129 . for each @xmath130 , if @xmath131 then @xmath132 ( addition modulo @xmath99 ) otherwise we would have an edge . this implies @xmath129 . next , we show that @xmath124 is small by using an auxiliary directed graph . first we recall some old results concerning short directed cycles in directed graphs . as usual , for a directed graph @xmath133 , let @xmath134 and @xmath135 denote the minimum out - degree and in - degree of @xmath133 respectively . caccetta and hggkvist @xcite conjectured that if @xmath133 is a directed graph on @xmath3 vertices with @xmath136 , then @xmath133 contains a cycle of length at most @xmath137 . while their conjecture remains open , chvtal and szemerdi @xcite gave a simple proof of a slightly weaker statement . [ chsz ] let @xmath133 be a directed graph on @xmath3 vertices . if @xmath136 ( or @xmath138 ) , then @xmath133 contains a directed cycle of length at most @xmath139 . there have been improvements on this result . however , theorem [ chsz ] suffices for our purposes . [ ring - codegree ] for all @xmath94 we have @xmath140 . by lemma [ familyblowup ] , it suffices to prove that @xmath141 . let @xmath142 . let @xmath143 be a small positive real and let @xmath144 . let @xmath3 be sufficiently large as a function of @xmath143 . let @xmath26 be a @xmath0-graph on @xmath3 vertices with @xmath145 . let @xmath133 be an auxiliary digraph with vertex set @xmath146 such that @xmath147 is an edge of @xmath133 if and only if @xmath148 and @xmath149 are edges of @xmath26 ( in other words , if and only if @xmath150 ) . let @xmath151 . then @xmath133 has @xmath152 vertices . for any @xmath153 , its out - neighbors in @xmath133 are precisely all the @xmath18-subsets of @xmath154 and thus ( using @xmath3 being sufficiently large ) @xmath155 by theorem [ chsz ] , @xmath133 contains a directed cycle @xmath156 of length at most @xmath157 . the subgraph of @xmath26 corresponding to @xmath156 is a member of @xmath158 . let @xmath159 be the @xmath0-graph obtained from the complete @xmath0-partite @xmath0-graph with vertex set @xmath160 by adding the vertex @xmath161 and the three edges @xmath162 . notice that @xmath163 . we obtain theorem [ main ] as a corollary of the following more general theorem . [ f * ] for sufficiently large @xmath3 , @xmath164 for all @xmath165 . in the introduction we pointed out that @xmath20 gives the lower bound @xmath166 , thus it suffices to prove @xmath167 . by theorem [ ring - codegree ] and lemma [ codegreeblowup ] , @xmath168 . let @xmath3 be large enough such that @xmath169 . let @xmath26 be a graph on @xmath3 vertices with @xmath170 . then @xmath26 contains a copy @xmath1 of @xmath171 for some @xmath172 . for each vertex @xmath173 in @xmath100 , let @xmath174 denote the clone of @xmath173 in @xmath171 . for all @xmath175 , let @xmath176 . summing over all @xmath175 and using the exact condition @xmath170 ( the only place where the exact condition is needed ) , gives @xmath177 this implies that there exists some @xmath178 which is contained in more than @xmath99 different sets @xmath179 . therefore , by lemma [ edge ] , there are vertices @xmath180 such that @xmath181 is an edge in @xmath100 with @xmath182 , @xmath183 and @xmath184 . so in @xmath171 , @xmath185 induces a complete @xmath0-partite @xmath0-graph and thus @xmath186 ( see figure [ fplus ] ) . let @xmath187 denote the family @xmath188 . the fact that @xmath100 has the @xmath189-property and the family @xmath125 has small co - degree density was key to our short proof of mubayi s conjecture . conceivably , the family @xmath125 can be useful elsewhere in the study of the turn problem for @xmath0-graphs . for instance , if @xmath125 also has relatively small turn density , then it could potentially be used in bounding the turn densities of other @xmath0-graphs , just like how @xmath126 was used by mubayi and rdl @xcite . in this section , we show that similar to @xmath126 the family @xmath187 also has turn density at most @xmath190 . in fact , we will show that the turn density of @xmath191 is exactly @xmath190 . the family @xmath187 does , however , have some advantages over @xmath126 . one , it has the @xmath123-property versus @xmath126 having the @xmath192-property . two , the structure of @xmath100 is simple and explicit , while in forcing a member of @xmath126 , we do not quite know which particular structure that member has . next , we show that @xmath187 has turn density at least @xmath190 via a construction inspired by the `` half - graph '' constructions from bandwidth problems . [ ring - lower - construction ] let @xmath193 and @xmath194 . let @xmath195 be a @xmath0-graph on @xmath196 whose edges are all the triples of the form @xmath197 and @xmath198 where @xmath199 . it is easy to check that @xmath200 . [ ring - lower ] for all @xmath3 the graph @xmath195 given in example [ ring - lower - construction ] contains no member of @xmath191 and hence @xmath201 . observe first that , based on the definition of @xmath195 , for any @xmath202 with @xmath203 , the pair @xmath204 has no co - neighbor in @xmath205 and the pair @xmath206 has no co - neighbor in @xmath207 . suppose for a contradiction that @xmath26 contains a copy @xmath1 of @xmath100 , for some @xmath99 . suppose @xmath208 @xmath209 and @xmath210 . for each @xmath173 in @xmath100 , let @xmath174 denote its image in @xmath195 under a fixed isomorphism from @xmath100 to @xmath1 . for any @xmath211 in @xmath212 ( or @xmath213 ) , let @xmath214 denote its subscript in @xmath212 ( or @xmath213 ) . in other words , if @xmath215 , then @xmath216 . there are two cases to consider . _ for some @xmath217 , @xmath218 and @xmath219 are in the same set . without loss of generality , we may assume that @xmath220 and that @xmath221 are both in @xmath212 . then @xmath222 must both be in @xmath213 . furthermore , by the observation we made at the beginning of this proof , @xmath223 . by repeating this argument , we get @xmath224 , which is a contradiction . _ for all @xmath217 , @xmath218 and @xmath219 are in different sets . by the symmetry of @xmath100 , we may assume that all the @xmath218 s are in @xmath212 and all the @xmath219 s are in @xmath213 . based on the observation we made at the beginning of the proof , we now must have @xmath225 for all @xmath226 ( with addition defined modulo @xmath99 ) . this leads to a contradiction like in case 1 . we now prove the main result of this section . this follows immediately from the following lemma . given a @xmath0-graph @xmath26 and a vertex @xmath32 , the _ link graph _ @xmath227 of @xmath32 is a @xmath18-graph whose edges are all the pairs @xmath228 such that @xmath229 . [ ring - upper ] @xmath230 . by proposition [ min - degree ] and lemma [ familyblowup ] , it suffices to prove @xmath231 . let @xmath3 be sufficiently large as a function of @xmath99 . let @xmath26 be a @xmath0-graph on @xmath3 vertices with @xmath232 . we prove that @xmath26 contains a member of @xmath158 . create an auxiliary digraph with vertex set @xmath146 ( all @xmath18-subsets of @xmath233 ) where @xmath147 is an edge of @xmath133 if and only if @xmath148 and @xmath149 are edges of @xmath26 ( in other words , if and only if @xmath234 is in the link graph of both @xmath235 and @xmath174 ) . let @xmath151 . let @xmath236 be a vertex in @xmath133 . since @xmath237 , the link graph of @xmath161 has at least @xmath238 edges and the link graph of @xmath173 has at least @xmath238 edges . therefore there are at least @xmath239 edges in the intersection of their link graphs , which implies @xmath240 . so we can apply theorem [ chsz ] to the directed graph @xmath133 to obtain a directed cycle @xmath156 of length at most @xmath241 . notice that the directed cycle @xmath156 corresponds to a subgraph of @xmath26 which is a member of @xmath158 . proposition [ ring - lower ] and lemma [ ring - upper ] now yield @xmath242 . we have now determined the turn density of the entire family of rings . however , computing its value for any single member @xmath100 appears to be difficult . after all , @xmath243 is just @xmath15 and determining @xmath244 has been notoriously difficult . a quick observation that one can make is [ evenupper ] for any positive integers @xmath245 we have @xmath246 . thus , for all even @xmath99 , we have @xmath247 . since @xmath248 is contained in the @xmath249-blowup of @xmath250 , we have @xmath251 . now suppose @xmath99 is even . since @xmath252 , we have @xmath253 . recall that the conjectured value for @xmath244 is @xmath254 . for the lower bound , turn s construction @xmath255 is obtained by partitioning @xmath3 vertices as equally as possible into three sets @xmath256 and including as edges all triples of the form , @xmath257 , @xmath258 , @xmath259 , @xmath260 for all @xmath261 ( see figure [ turanexample ] ) . it is straightforward to check that if @xmath255 contains @xmath100 for some @xmath99 , then @xmath99 must be divisible by @xmath0 . hence , @xmath255 contains no @xmath100 when @xmath262 . so we have the following . [ evenlower ] for @xmath263 , @xmath264 . so by propositions [ evenupper ] and [ evenlower ] , if turn s conjecture is true , then we would have @xmath265 for every even @xmath99 with @xmath262 . [ bothexamples ] finally , for odd @xmath99 , the following construction shows that @xmath266 is larger than @xmath267 . let @xmath268 be a @xmath0-graph on @xmath3 vertices where the vertices are partitioned into three sets @xmath256 with sizes @xmath269 , @xmath270 whose edges are all triples of the form , @xmath271 , @xmath257 for all @xmath261 and @xmath272 ( see figure [ oddexample ] ) . it is easy to check that @xmath273 and that if @xmath268 contains @xmath100 then @xmath99 must be even . furthermore , we can iterate this construction inside @xmath274 and @xmath275 to push the density above @xmath267 while maintaining the fact that there are no odd rings . suppose @xmath276 , then the density of @xmath268 before iterating is @xmath277 ; this gives an optimal value of @xmath278 when @xmath279 . after iterating , the density becomes @xmath280 ; numerical methods give an approximate optimal value of @xmath281 when @xmath282 . thus we have the following . [ oddlower ] for odd @xmath99 , @xmath283 . let @xmath284 be obtained by adding the edges @xmath285 to @xmath286 . note the following simple observation . [ q3 ] for odd @xmath99 at least @xmath287 , @xmath100 is contained in the blow - up of @xmath284 . the final results in this section are obtained by using razborov s flag algebra calculus . since the upper bounds are not tight ( and we do nt intend to formally publish the bounds obtained from these calculations ) , we refer the reader to @xcite , @xcite , @xcite for an explanation of the method and its applications . [ uppers ] 1 . for even @xmath99 , @xmath288 . 2 . for odd @xmath289 , @xmath290 3 . for @xmath99 an odd multiple of @xmath0 , @xmath291 . 4 . for @xmath99 an even multiple of @xmath0 , @xmath292 . in each case we use lemma [ codegreeblowup ] to transfer a statement about the blow - up of a graph to a statement about @xmath100 . 1 . proposition [ evenupper ] shows @xmath100 is contained in the blow - up of @xmath15 and flag algebra calculations give @xmath293 ( see @xcite ) . 2 . proposition [ q3 ] shows @xmath100 is contained in the blow - up of @xmath284 and flag algebra calculations give @xmath294 . 3 . @xmath100 is contained in the blow - up of @xmath286 and flag algebra calculations give @xmath295 4 . @xmath100 is contained in the blow - up of @xmath243 and @xmath286 and flag algebra calculations give @xmath296 the results of this section are summarized below , with the lower bounds coming from propositions [ ring - lower ] , [ evenlower ] , [ oddlower ] and the upper bounds coming from proposition [ uppers ] . perhaps the most interesting thing about the upper bounds for rings is that it is possible to get nearly tight results for every value of @xmath99 using only flag algebra calculations for small @xmath0-graphs and lemma [ codegreeblowup ] . given the results of this section and theorem [ ring - codegree ] , it would be interesting to solve the following problem . determine @xmath266 or @xmath298 for each fixed value of @xmath99 . let @xmath249 be a prime power and let @xmath299 be @xmath300-graph with vertex set equal to the one dimensional subspaces of @xmath301 and edges corresponding to the two - dimensional subspaces of @xmath301 . we call @xmath299 the _ projective geometry _ of dimension @xmath18 over @xmath302 ; note that @xmath303 is the fano plane . in @xcite , keevash also proved the following more general theorem about projective geometries furthermore , there is a nearly matching lower bound when @xmath249 is an odd prime power ( see @xcite and @xcite ) . the proof we present in section [ fano ] relies on the fact that there is a family of @xmath0-graphs @xmath305 , such that each member @xmath306 has the @xmath307-property and @xmath308 . in fact , our same proof could be used to give a simple proof of theorem [ keevashpg ] if there was an affirmative answer to the following question . [ lm ] let @xmath309 . does there exist a finite family @xmath310 of @xmath9-graphs such that for each @xmath311 there exists a positive integer @xmath99 such that @xmath312 has the @xmath189-property and @xmath313 . it seems conceivable that obtaining a @xmath9-graph with the @xmath116 property for some other values of @xmath115 and @xmath117 might give us the same benefit and be easier to obtain ; however , this is not the case . on one hand , we must have @xmath314 so that equation holds . on the other hand , when @xmath315 the complete balanced bipartite @xmath9-graph @xmath316 , which has @xmath317 , does not contain any subgraph with the @xmath116-property ( any subgraph of @xmath316 with @xmath115 vertices must contain an independent set of size @xmath318 ) . so we must have @xmath319 , which implies that @xmath115 is even and @xmath320 . a different problem is the following . instead of determining the co - degree threshold for a single copy of @xmath1 in @xmath26 , one can ask about the co - degree threshold for @xmath321 vertex disjoint copies of @xmath1 in @xmath26 ( assuming @xmath322 divides @xmath323 ) . this has been referred to as the _ tiling _ or _ factoring _ problem and received much attention lately . interestingly , the co - degree threshold for tiling with @xmath15 and @xmath324 have been determined ( see @xcite , @xcite ) , but the co - degree threshold for a single copy of @xmath15 or @xmath324 is still unknown and appears to be difficult . since the co - degree threshold for a single copy of @xmath4 is known and seems to be much easier than @xmath15 or @xmath324 , it would be interesting to determine the co - degree threshold for tiling with @xmath4 . let @xmath3 be divisible by @xmath325 and let @xmath26 be a @xmath0-graph on @xmath3 vertices . determine the minimum value @xmath326 such that @xmath327 implies that @xmath26 contains @xmath328 vertex disjoint copies of @xmath4 . the relationship between the edge density of a hypergraph and its subgraphs with large co - degree is also very intriguing . even the following simple questions do not seem to have an easy answer . a @xmath9-graph @xmath1 is said to _ cover pairs _ if @xmath1 has at least @xmath329 vertices and every pair of vertices lies in some edge , i.e. @xmath330 . l. caccetta and r. hggkvist , on minimal digraphs with given girth , proceedings of the ninth southeastern conference on combinatorics , graph theory , and computing ( florida atlantic univ . , boca raton , fla . , 1978 ) , pp . 181187 , congress . xxi , utilitas math . , winnipeg , man . , 1978 . v. ss , remarks on the connection of graph theory , finite geometry and block designs , colloquio internazionale sulle teorie combinatorie ( roma , 1973 ) , tomo ii , pp . 223233 . atti dei convegni lincei , no . 17 , accad . lincei , rome , 1976 .
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given a @xmath0-graph @xmath1 , let @xmath2 denote the maximum value of the minimum co - degree of a @xmath0-graph on @xmath3 vertices which does not contain a copy of @xmath1 .
let @xmath4 denote the fano plane , which is the @xmath0-graph @xmath5 .
mubayi @xcite proved that @xmath6 and conjectured that @xmath7 for sufficiently large @xmath3 . using a very sophisticated quasi - randomness argument , keevash @xcite proved mubayi s conjecture .
here we give a simple proof of mubayi s conjecture by using a class of @xmath0-graphs that we call rings .
we also determine the turn density of the family of rings .
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You are an expert at summarizing long articles. Proceed to summarize the following text:
recently we proposed a new algorithm for computation of cohomology of a wide class of lie superalgebras . this algorithm reduces the computation for the whole cochain complex to a number of smaller tasks within smaller subcomplexes . one can demonstrate that if @xmath7 is the computation time for the whole complex , then partition of the complex into @xmath8 subcomplexes reduces the computation time roughly to the value @xmath9 . thus , the approach appeared to be efficient enough to cope with several difficult tasks in computing cohomology for particular lie ( super)algebras @xcite . more detailed experiments with the _ * c * _ implementation of the algorithm , including profiling , reveal that arithmetic operations over @xmath2 take the main part of computation time ( usually more than 90% for large tasks ) . the same is true if @xmath2 is replaced by @xmath10 ( though computation becomes somewhat faster ) . a standard way to reduce negative influence of this bottleneck is to compute several modular images of the problem with subsequent restoring the result over @xmath10 or @xmath2 by the chinese remaindering or an algorithm for recovering a rational number from its modular residue @xcite . though , as is clear , the sum of sizes of modules used for constructing images can not be less than the size of maximum integer in the final result , the modular approach allows the intermediate swelling of coefficients to be avoided . moreover the use of modular images is much more advantageous in the case of ( co)homology computation than in the traditional problems of linear algebra . as we demonstrate further , the overwhelming part of computation can be accomplished using only one modular image . recall that the gauss elimination , the basic constituent of algorithms for computing ( co)homology , over @xmath11 has only _ cubic _ computational complexity in contrast to the _ exponential _ one over @xmath2 or @xmath12 note that the approach presented here can be applied not only for the lie superalgebras , but in more general case of computation of homology or cohomology , especially when there is a practical method of splitting ( co)chain complex into smaller subcomplexes . to demonstrate the power of the new algorithm and program , we present the results of computation of cohomology in the trivial module for the algebra @xmath13 this is an example ( for @xmath14 ) of the lie superalgebra of special ( divergence free ) vector fields on the @xmath15-dimensional supermanifold preserving the odd version of symplectic ( periplectic , as a. weil called it ) structure @xcite . these superalgebras , being super counterparts of the lie algebras of hamiltonian vector fields , are vital in the batalin vilkovisky formalism , see @xcite . the @xmath16th cohomology is defined as the quotient group @xmath17 for the _ cochain complex _ @xmath18 here , the @xmath19 are abelian groups of cochains , graded by the integer @xmath16 ( called _ dimension _ or _ degree _ ) ; the @xmath20 are differentials ( @xmath21 ) ; the @xmath22 and @xmath23 are the subgroups of cocycles and coboundaries , respectively ( see @xcite for details ) . in order to apply without restrictions the linear algebra algorithms , we assume that the groups of cochains are additive groups of certain linear spaces or modules and we shall use the corresponding terms in the subsequent text . to compute the @xmath16th cohomology , it suffices to consider the following part of ( [ whole ] ) : @xmath24 first of all we split ( [ part ] ) using the @xmath10-grading in the cochain spaces induced by the gradings in the lie ( super)algebra ( and the module over this algebra ) involved in the construction of the cochain spaces : @xmath25 here , @xmath26 is a grading subset . it appears that , as a rule , any subcomplex in a given degree @xmath27 can be split , in turn , into smaller subcomplexes : @xmath28 here @xmath29 is a finite or infinite set of subcomplexes . equation ( [ spart ] ) means that the spaces @xmath30 split into the direct sum of subspaces @xmath31 and the matrices of the linear mappings @xmath32 can be represented in the block - diagonal form @xmath33 the construction of these subcomplexes is the _ central part _ of the splitting algorithm . thus , the whole task reduces to a collection of easier tasks of computing @xmath34 as a basis of the cochain space @xmath35 we choose the set of super skew - symmetric monomials of the form @xmath36 here , @xmath37 and @xmath38 are basis elements of the algebra and module , respectively , and @xmath39 is the dual element to @xmath40 that is , @xmath41 the degrees of factors in ( [ monom ] ) satisfy the relation @xmath42 notice that @xmath43 and this is a serious obstacle to extraction of finite - dimensional subcomplexes for infinite - dimensional lie ( super)algebras when computing cohomology in the adjoint module ( important in the deformation theory ) . we also assume that @xmath44 to construct a subcomplex @xmath45 from the sum in right hand side of ( [ spart ] ) , we begin with choosing somehow an arbitrary _ starting _ monomial @xmath46 of the form ( [ monom ] ) . there are various choices of the starting monomial and the time and space efficiency of computation depends on the choice . having no better idea , we use at present the following three strategies : choice of a lexicographically minimal , or lexicographically maximal , or random monomial . we call these strategies _ bottom _ , _ top _ and _ random _ , respectively . among these , the top strategy seems to be most efficient ( see experimental data in tables [ timesh ] and [ timessle ] ) and it is used by default . nevertheless , other strategies help sometimes to push through difficult tasks when the top strategy fails . then we construct the three sets @xmath47 @xmath48 and @xmath49 of basis monomials for @xmath50 @xmath51 and @xmath52 respectively , by the procedure _ * constructsubcomplex * _ presented on page . constructsubcomplex[constructsubcomplex ] * input : * @xmath46 , starting @xmath53-monomial + * output : * @xmath54 , @xmath55 , @xmath49 , monomial bases of cochain spaces in the current + subcomplex @xmath56 : @xmath57 such that @xmath58 + * local : * @xmath59 @xmath60 , @xmath61 , working subsets + of currently `` new '' ( not yet processed ) monomials ; + @xmath62 , @xmath63 , @xmath64 , working sets of monomials ; + @xmath65 , @xmath66 , @xmath67 , working monomials + * initial setting : * + 1 : @xmath68 + 2 : @xmath69 + 3 : @xmath70 + * loop over @xmath16-monomials : * + 4 : _ * while * _ @xmath71 _ * do * _ + 5 : @xmath72 + * supplement the set @xmath54 * : + 6 : @xmath73 + 7 : @xmath74 + 8 : @xmath75 + * supplement the set @xmath49 * : + 9 : @xmath76 + 10 : @xmath77 + 11 : @xmath78 + * exclude processed monomial @xmath66 * : + 12 : @xmath79 + * loop over @xmath80-monomials : * + 13 : _ * while * _ @xmath81 _ * do * _ + 14 : @xmath82 + * supplement the set @xmath83 * + 15 : @xmath84 + 16 : @xmath85 + 17 : @xmath86 + * exclude processed monomial @xmath87 * + 18 : @xmath88 + 19 : _ * od * _ + * loop over @xmath89-monomials : * + 20 : _ * while * _ @xmath90 _ * do * _ + 21 : @xmath91 + * supplement the set @xmath55 : * + 22 : @xmath92 + 23 : @xmath85 + 24 : @xmath93 + * exclude processed monomial @xmath65 : * + 25 : @xmath94 + 26 : _ * od * _ + 27 : _ * od * _ + 28 : _ * return * _ @xmath54 , @xmath55 , @xmath49 + the function _ * takemonomialfromset * _ called within _ * constructsubcomplex * _ takes the current monomial from a set of monomials . the function _ * inverseimagemonomials * _ generates the set of @xmath95-monomials whose images with respect to the mapping @xmath96 contain a given @xmath97-monomial . the function _ * imagemonomials * _ generates the set of @xmath98-monomials whose inverse images with respect to the @xmath99 contain a given @xmath97-monomial . in the finite - dimensional case , the loops in the procedure _ * constructsubcomplex * _ are finite and we obtain in the end a minimal subcomplex of the form ( [ subcomplex ] ) . this is the unique minimal subcomplex involving the starting monomial @xmath46 . let us consider in more detail the procedure of computation of cohomology within the subcomplex in accordance with formula ( [ insubcomplex ] ) . from now on we assume that @xmath100 and @xmath101 in ( [ subcomplex ] ) are finite - dimensional spaces over @xmath2 or @xmath11 or modules over @xmath12 since important in mathematics and physics fields @xmath102 and @xmath103 are , in principle , non - algorithmic objects , our main interest will be focused on the cohomology over the field @xmath2 ( or its algebraic extentions ) . in accordance with a general theorem in the homological algebra , called the _ universal coefficient theorem _ @xcite , ( co)homology with coefficients from an arbitrary abelian group @xmath104 can be expressed in terms of ( co)homology with coefficients in @xmath10 . thus , we can carry out the computation over @xmath10 and then go to the coefficient group we are interested in . let us consider now the connection between @xmath105 and @xmath106 in the finite - dimensional case , the group @xmath107 is a finitely generated abelian group having the following canonical representation @xmath108 here , @xmath109 , the number of copies of the integer group @xmath10 , is called the _ rank _ of the abelian group @xmath107 or the _ betti number_. the cyclic groups @xmath110 are called the _ torsion subgroups _ and their orders @xmath111 , having the property @xmath112 @xmath113 @xmath114 and so on , are called the _ torsion coefficients_. in the case of cohomology , the universal coefficient theorem is expressed by the following split short exact sequence @xmath115 where the operation @xmath116 is the _ periodic product _ of abelian groups . in our context @xmath117 the term `` split '' , in application to sequence ( [ exact ] ) , means the possibility to construct the isomorphism @xmath118 replacing @xmath104 by @xmath2 in ( [ isomorphism ] ) and taking into account that @xmath119 for any abelian group @xmath120 , we have @xmath121 . since @xmath122 for arbitrary @xmath123 and @xmath124 the dimension of @xmath125 interpreted as vector space over @xmath2 , coincides with the rank ( betti number ) @xmath109 of the group @xmath106 our modular approach is based on the following [ theorem ] @xmath126 * remarks : * 1 . inequality ( [ inequality ] ) means that non - trivial cohomology classes computed over the field of rational numbers @xmath2 can exist only in the subcomplexes with non - trivial cohomology classes computed over the finite field @xmath11 with arbitrary prime @xmath127 2 . h. khudaverdian turned author s attention to the fact that inequality ( [ inequality ] ) can be deduced immediately from the universal coefficient theorem : considering the product @xmath128 and taking into account representation ( [ abelgroup ] ) and isomorphism @xmath129 , we see that the dimension of @xmath130 , as a vector space over @xmath11 , can not be less than @xmath109 ( only additional dimensions may appear , if the torsions in @xmath107 or @xmath131 contain cyclic groups of the form @xmath132 ) . + nevertheless , we give here a direct constructive proof in order to demonstrate in parallel the main ideas of ( co)homology computation . _ proof . _ to prove inequality ( [ inequality ] ) , we have to compute ( [ insubcomplex ] ) in such a way as to avoid cancellations of integers and apply the modular homomorphism @xmath133 at the end of computation . thus , it is convenient to consider ( [ insubcomplex ] ) over @xmath10 instead of @xmath134 we assume that @xmath135 is odd and use a symmetric representation of @xmath11 , i. e. , @xmath136 we will also apply @xmath137 component - wise to multicomponent objects over @xmath138 like vectors and matrices . + we begin with the following setup : * @xmath139 and @xmath140 are represented as finite - dimensional modules @xmath141 respectively , i. e. , @xmath142 + * the differentials @xmath143 and @xmath144 are represented ( in the monomial bases of the form ( [ monom ] ) , in our case ) as integer @xmath145 and @xmath146 matrices @xmath147 d = d = , @xmath147 respectively . we write @xmath148 \end{tabular } } \hspace*{-7pt}\stackrel{\mbox{\scriptsize{$j$}}}{a } } $ ] to indicate that matrix @xmath120 has @xmath149 rows and @xmath150 columns . * the matrices @xmath151 and @xmath152 satisfy the relation @xmath153 the computation of cohomology , i.e. , construction of quotient module , can be reduced to the construction of so - called _ ( co)homology decomposition _ @xcite based on the computation of the smith normal forms @xcite of the matrices representing differentials . now we should consider the coboundary submodule @xmath168 combining the relation @xmath169 with the structure of the matrix @xmath29 ( see formula ( [ s0 ] ) ) , we can reduce the matrix @xmath152 determining coboundaries to the matrix @xmath170 acting in the submodule of cocycles : @xmath147 v^-1d = . @xmath147 computing the smith normal form @xmath171 for the reduced coboundary matrix we get @xmath147 s = = , @xmath147 where @xmath172 and @xmath173 \end{tabular } } \hspace*{-8pt}\stackrel{\mbox{\scriptsize{$~r'$}}}{\widetilde{s ' } } } = \mathrm{diag}\left(s'_1,\ \ldots\ , \ s'_{r'}\right).$ ] we can extend the transformation matrix @xmath174 \end{tabular } } \hspace*{-10pt}\stackrel{\mbox{\scriptsize{$m\!-\!r$}}}{\widetilde{u ' } } } $ ] acting in the submodule of cocycles to the transformation matrix acting in the whole module @xmath175 @xmath147 u = = . @xmath147 here , @xmath159 \end{tabular } } \hspace*{-8pt}\stackrel{\mbox{\scriptsize{$\,r$}}}{i } } $ ] is the @xmath176 identity matrix . using the transformation matrices @xmath177 and @xmath178 we can transform the initial ( monomial in our case ) basis @xmath179 in the module @xmath180 into the basis @xmath181 making the cohomology decomposition explicit @xmath182 in this decomposition we have @xmath183 and @xmath184 the formula for the dimension of cohomology ( betti number ) follows from decomposition ( [ decomposition ] ) @xmath185 now let us consider how ( [ dimension ] ) changes under @xmath186 the image of ( [ dimension ] ) takes the form @xmath187 since @xmath137 is a ring homomorphism , we have for arbitrary unimodular matrix @xmath120 with integer entries @xmath188 that is @xmath137 maps the above transformation matrices into invertible matrices . hence the number of elements in the decomposition basis @xmath189 remains unchanged , @xmath190 on the other hand , the invariant factors @xmath191 and @xmath192 of the matrices @xmath193 and @xmath29 divisible by @xmath135 vanish , hence @xmath194 and inequality ( [ inequality ] ) is proved by comparing ( [ dimension ] ) and ( [ dimensionp ] ) . @xmath195 an algorithm based on the above ideas was implemented in the _ * c * _ language . the program called _ * liecohomologymodular * _ has the following structure : 1 . input lie ( super)algebra @xmath120 , module @xmath196 over @xmath120 , cohomology degree ( dimension ) @xmath16 and grade @xmath27 . @xmath120 and @xmath196 should be defined over ( some algebraic extension of ) @xmath10 or @xmath134 2 . construct the full set @xmath197 of @xmath16-cochain monomials in grade @xmath27 . 3 . choose a prime @xmath135 for searching subcomplexes with non - trivial cocycles by computing over @xmath11 . [ choosemonom ] choose an element @xmath198 ( the starting monomial ) . 5 . construct a minimal subcomplex @xmath199 such that @xmath200 6 . compute @xmath201 . [ computeh ] if @xmath202 then compute @xmath203 over @xmath10 or @xmath2 ( or their extensions ) . we can use here the chinese remaindering or the rational recovery algorithm as more efficient procedures than direct computation over @xmath10 or @xmath2 . delete all basis monomials of @xmath51 from @xmath204 9 . if @xmath197 is empty , then stop computation , otherwise go to step [ choosemonom ] . in the current implementation we obtain the relations determining cocycles and coboundaries ( in fact , the rows of matrices of differentials ) within the procedure _ * constructsubcomplex*_. these relations are generated one by one as by - product of the functions _ * inverseimagemonomials * _ and _ * imagemonomials*_. to prevent unnecessary memory consumption , every newly arising relation is reduced modulo the system of relations existing to the moment and , if the result is not zero , the new relation is added to the system . thus , we automatically have the matrices of differentials in the normal form just after completion of the procedure _ * constructsubcomplex*_. this process is obviously equivalent to the gauss elimination method , the most standard method for the computation of the smith normal form of a matrix . in recent years , a number of new fast algorithms for the determination of smith normal form have been elaborated @xcite . these algorithms appear to be well suited to the ( co)homology computation . it is worth to study the possibility to incorporate these algorithms in our implementation . we could , for example , remove the generation of relations from the functions _ * inverseimagemonomials * _ and _ * imagemonomials * _ making them as fast as possible . then , after construction of subcomplex with the help of these modified functions , we should generate the matrices of differentials separately and apply the fast algorithms to these matrices . of course , this modification should be done if the total computation time decreases without substantial increase in the memory consumption . the works @xcite contain a detailed analysis of the properties of the sets of primes most appropriate for application of modular algorithms to a given matrix . here we give only a few comments concerning the choice of prime @xmath135 in our algorithm . these comments are based mainly on experiments with the program . from the practical point of view , we should use only primes @xmath135 for which all operations in @xmath11 can be done within one machine word . thus , for 32bit architecture we should choose @xmath135 from the set of 8951 primes @xmath205 a good choice should not produce excessive cocycles . of course , such cocycles will be removed at step [ computeh ] anyway , but at the expense of additional work . in our context , an `` unlucky '' prime is the one which divides the invariant factors of matrices of differentials ( or , in other words , the torsion coefficients of cohomology over @xmath10 ) and , as is clear , the probability for a given prime to be unlucky diminishes as the prime grows . on the other hand , there is an increase of time ( in the examples we have computed , up to factor 2 or 3 ) and space expenditures with increase of @xmath135 within the set @xmath206 so it makes sense not to use too large primes for searching subcomplexes with potentially non - trivial cohomology . in our practice , we use , as a rule , a compromise : a prime near the half of 32bit word , namely , @xmath207 i. e. , the 4th fermat number . however , quite satisfactory results can be obtained even with much smaller primes , as is illustrated in table [ tab : dimh ] . the symbols @xmath208 in the boxes of this table mean that ( non - zero ) @xmath209 , whereas @xmath210 means the cohomology with coefficients in the trivial module for the lie algebra @xmath211 of hamiltonian vector fields on the @xmath212-dimensional symplectic manifold . we performed computation over all modular fields @xmath213 through @xmath214 as is seen in table [ tab : dimh ] , the results for @xmath215 fully coincide with those for @xmath2 for all computed grades @xmath216 $ ] ( and for all cohomology degrees @xmath16 ) . the table also illustrates theorem [ theorem ] , i.e. , all boxes containing non - zero dimensions for the field @xmath2 contain also non - zero ( @xmath217 same for @xmath2 ) dimensions for all fields @xmath11 considered . in tables [ timesh ] and [ timessle ] we present ( considering both algebra and superalgebra cases ) the running times for computation over @xmath215 of cohomology @xmath218 ( for @xmath219 ) and @xmath220 ( for @xmath221 ) . the columns presented in these tables are : the dimensions of cochain spaces , i. e. , the sizes of matrices of differentials ; the running times in seconds for the _ top _ strategy of the choice of the starting monomial and comparison of the _ bottom _ and _ top _ strategies . the times in both tables were obtained on a 1133mhz pentium iii pc with 512 mb . note that the maximum memory consumption is near 46 mb and near 14 mb for the tasks in table [ timesh ] and in table [ timessle ] , respectively . .timing for @xmath222 , @xmath223 [ cols="^,^,^,^,^,^",options="header " , ] & @xmath224 + when computing cohomology we start with the construction of the full set of @xmath53-monomials . at the moment we do not see how to avoid this in deterministic algorithms . to represent the set of monomials , we need to allocate @xmath225 elements of memory representing basis elements of algebra and module ( @xmath226 for the trivial and @xmath227 for any non - trivial module ) . in our implementation we represent basis elements by two - byte integers . for the last box in column 9 ( @xmath228 ) of table [ tab : h1 ] we have @xmath229 and the set of monomials occupies near 12 mb . the dimensions grow very rapidly , so , in fact , we are working on the brink of abilities of 32bit architecture and , even theoretically , we can add only a few rows to table [ tab : h1 ] . but , as to arithmetical difficulties , we have some progress . i would like to thank h. khudaverdian , d. leites and i. shchepochkina for helpful comments on drafts of this text . 99 kornyak , v.v . : a new algorithm for computing cohomologies of lie superalgebras . in : _ computer algebra in scientific computing / casc01 _ , v.g.ganzha , e.w.mayr and e.v.vorozhtsov ( eds . ) , springer - verlag berlin heidelberg ( 2001 ) 391398 kornyak , v.v . : a method of splitting cochain complexes to compute cohomologies of lie ( super)algebras . _ russian journal for computer science ( `` programmirovanie '' ) . _ * 2 * ( 2002 ) 7680 ( in russian ) kornyak , v.v . : extraction of `` minimal '' cochain subcomplexes for computing cohomologies of lie algebras and superalgebras . in : _ computer algebra and its application to physics / caap-2001 _ , v.p . gerdt ( ed . ) , jinr dubna ( 2002 ) 186195 kornyak , v.v . : computation of cohomology of lie algebra of hamiltonian vector fields by splitting cochain complex into minimal subcomplexes . in : _ computer algebra in scientific computing / casc02 _ , v.g.ganzha , e.w.mayr and e.v.vorozhtsov ( eds . ) , springer - verlag berlin heidelberg ( 2002 ) 201206 kornyak , v.v . : a method of splitting cochain complexes for computing cohomology : lie algebra of hamiltonian vector fields @xmath230 _ russian journal for computer science ( `` programmirovanie '' ) . _ * 2 * ( 2003 ) 9499 ( in russian ) wang , p.s . , guy , m.j.t . , davenport , j.h . : @xmath135-adic reconstruction of rational numbers . _ sigsam bulletin , _ * 16 * ( 1982 ) 23 collins , g.e . , encarnacin , m.j . : efficient rational number reconstruction . _ j. symb . comp . , _ * 20 * ( 1995 ) 287297 leites , d. : lie superalgebras . in : _ modern problems of mathematics . recent developments _ , * 25 * , viniti , moscow ( 1984 ) p. 3 ( in russian ; english translation in josmar * 30(6 ) * ( 1985 ) p. 2481 ) leites , d. , shchepochkina , i. : _ classification of simple lie superalgebras of vector fields _ , preprint mpim-2003 - 28 ( www.mpim-bonn.mpg.de ) gomis j. , pars j. , samuel s. : antibracket , antifields and gauge - theory quantization . * 259 * ( 1995 ) , no . 12 , 145 pp fuks , d.b . : _ cohomology of infinite dimensional lie algebras . _ consultants bureau , new york ( 1987 ) fomenko , a.t . , fuks , d.b . : _ a course in homotopic topology . nauka ( 1989 ) . ( in russian ; an english translation of an earlier version was published by akadmiai kiad , budapest ( 1986 ) ) hilton , p. j. , wylie , s. : _ homology theory : an introduction to algebraic topology . _ cambridge university press , new york ( 1960 ) mac lane , s. , birkhoff , g. _ algebra . _ chelsea publishing co. , new york , third edition ( 1988 ) dumas , j - g . , saunders , b.d . , villard , g. : on efficient sparse integer matrix smith normal form computations . _ j. symb . comp . _ * 32*(12 ) ( 2001 ) 7199 dumas , j - g . , heckenbach , f. , saunders , b.d . , welker , v. : computing simplicial homology based on efficient smith normal form algorithms . in _ algebra , geometry and software systems , _ m. joswig , n. takayama ( eds . ) springer ( 2003 ) 177206 batalin , i.a . , vilkovisky , g.a . : gauge algebra and quantization . lett . _ * 102b * ( 1981 ) 2731 batalin , i.a . , vilkovisky , g.a . : closure of the gauge algebra , generalized lie equations and feynman rules . phys . _ * b234 * ( 1984 ) 106124 shander , v.n . : analogues of the frobenius and darboux theorems for supermanifolds . _ comptes rendus de l academie bulgare des sciences , _ * 36 * , no . 3 ( 1983 ) 309311 khudaverdian , h. m. : laplacians in odd symplectic geometry . in _ quantization , poisson brackets and beyond _ , th . voronov , ( ed . ) , contemp . , v.15 , amer . math . soc . , providence , ri ( 2002 ) 199212
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we describe an essential improvement of our recent algorithm for computing cohomology of lie ( super)algebra based on partition of the whole cochain complex into minimal subcomplexes .
we replace the arithmetic of rational numbers or integers by a much cheaper arithmetic of a modular field and use the inequality between the dimensions of cohomology @xmath0 over any modular field @xmath1 and over @xmath2 : @xmath3 . with this inequality we can , by computing over arbitrary @xmath4 quickly find the ( usually , rare ) subcomplexes for which @xmath5 and then carry out the full computation over @xmath2 within these subcomplexes .
we also present the results of application of the corresponding _ * c * _ program to the lie superalgebra of special vector fields preserving an `` odd - symplectic '' structure on the @xmath6-dimensional supermanifold .
for this algebra , we found some new basis elements of the cohomology in the trivial module .
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compact diode - pumped sources of the femtosecond pulses tunable within the wavelength range between 2 and 3 @xmath0 m are of interest for various applications , such as laser surgery , remote sensing and monitoring , spectroscopy of semiconductors etc . to date only cryogenically operated pb - salt diode lasers , optical parametrical oscillators and difference - frequency convertors were available for the operation in this spectral range . therefore the possibility of the direct mid - ir lasing from the new class of the transition - metal doped chalcogenides @xcite has attracted much attention . the impressive advantages of these media are room - temperature operation between 2 and 3 @xmath0 m , possibility of direct diode pumping , high emission and absorption cross - sections , negligibly low excited - state absorption and , as consequence , low thermal load ( the basic laser material characteristics will be described in the next section ) . the most remarkable examples of such lasers are cr@xmath1-doped znse , zns and znte . to date the following achievements for these media have been demonstrated : 1 ) for cr : znse cw operation with over 1.7 w power @xcite , over 1100 nm tunability @xcite , over 350 nm tunable diode - pumped cw operation @xcite , active mode locking @xcite and active modulator assisted passive mode locking @xcite were achieved ; 2 ) for cr : zns pulsed @xcite and tunable cw operation @xcite were obtained . cr : znte , which is a member of the considered media class , remains unexplored . in spite of the numerous advantages , there exist some obstacles for femtosecond pulse generation from these lasers . as they are the semiconductors , i.e. possess a comparatively narrow band - gap , the nonlinear refraction in the active crystal is large ( see below ) . hence the self - focusing has a comparatively low threshold which in the combination with the self - phase modulation produce a tendency to the multiple pulse operation in the kerr - lens mode locking ( klm ) regime @xcite . thus , there is a need in the study of the klm stability limits and methods of the stability enhancement in the considered lasers . moreover , the large absorption and emission cross sections tend to the stabilization of the cw operation , which prevents the klm self - start . in this paper we present an analysis of the nonlinear refraction in the zinc - chalcogenides ( by the example of znse ) . in the combination with the lasing properties this defines the main requirements to the klm optimization . then , the results of the numerical optimization of the klm aimed at the multipulsing suppression and taking into account a strong saturation of the kerr - lens induced fast absorber are presented . we demonstrate the possibility of the few optical cycle pulse generation from the cr - doped zinc - chalcogenodes . the problem of the self - starting klm is analyzed on the basis of the generalized momentum method taking into account real - world laser configurations , gain guiding , soft - aperture , thermo - lensing and other properties of the considered laser materials . the presented models are quite general and can be applied to the overall optimization of the different klm lasers . simulation of the klm can be based on the two quite different approaches . first one supposes the full - dimensional modelling taking into account the details of the field propagation in the laser cavity @xcite . the minimal dimension of such models is 2 + 1 and they allow the description of the spatio - temporal dynamics of the ultrashort pulses and their mode pattern . its main disadvantages are a large number of the parameters resulting in ambiguity of the optimization procedure and complexity of the interpretation of the obtained results . second approach is based on 1 + 1 dimensional model in the framework of the so - called nonlinear ginzburg - landau equation @xcite , which describes the klm as an action of the fast saturable absorber governed by the few physically meaningful parameters , viz . , its modulation depth @xmath2 and the inverse saturation intensity @xmath3 . this method allows the analytical realization in the week - nonlinear limit @xcite , however in the general case the numerical simulations are necessary . we shall base our analysis on the both approaches . in the beginning let us consider the master equation describing the ultrashort pulse generation in the klm solid - state laser : @xmath4a(z , t),\ ] ] where @xmath5 is the field amplitude ( so that @xmath6 has a dimension of the intensity ) , @xmath7 is the longitudinal coordinate normalized to the cavity length ( thus , as a matter of fact , this is the cavity round - trip number ) , @xmath8 is the local time , @xmath9 is the saturated gain coefficient , @xmath10 is the linear net - loss coefficient taking into account the intracavity and output losses , @xmath11 is the group delay caused by the spectral filtering within the cavity , @xmath12 are the @xmath13-order group - delay dispersion ( gdd ) coefficients , @xmath14 = @xmath15 = @xmath16 is the self - phase modulation ( spm ) coefficient , @xmath17 and @xmath18 are the frequency and wavelength corresponding to the gain band maximum , @xmath19 and @xmath20 are the linear and nonlinear refraction indexes , respectively , @xmath21 is the double length of the gain medium ( we suppose that the gain medium gives a main contribution to the spm ) . the last term in eq . describes the self - steepening effect and for the simplification will be not taken into account in the simulations . as an additional simplification we neglect the stimulated raman scattering in the active medium @xcite . these two factors will be considered hereafter . the gain coefficient obeys the following equation : @xmath22 here @xmath23 and @xmath24 are the absorption and emission cross - sections of the active medium , respectively , @xmath25 is the gain relaxation time , @xmath26 is the absorbed pump intensity , @xmath27 is the pump frequency , @xmath28 = @xmath29 is the maximum gain coefficient , @xmath30 is the concentration of the active centers . the assumption @xmath31 ( @xmath32 is the pulse duration , @xmath33 is the cavity period ) allows the integration of eq . . then for the steady - state gain coefficient we have : @xmath34 where @xmath35 = @xmath36 is the gain saturation energy flux , @xmath37 = @xmath38 is the pulse energy . for the numerical simulations in the framework of the distributed model it is convenient to normalize the time and the intensity to @xmath11 = @xmath39 and @xmath40 , respectively ( @xmath41 is the gain bandwidth ) . the simulation were performed on the @xmath42 mesh . only steady - state pulses were considered . as the criterion of the steady - state operation we chose the peak intensity change less than 1% over last 1000 cavity transits . the klm in the considered model ( [ lg ] , [ alpha ] ) is governed by the only four basic parameters : @xmath43 , @xmath44 , @xmath2 , and @xmath3 . this allows unambiguous multiparametric optimization . in the presence of the higher - order dispersions , the additional @xmath12 parameters appear . this complicates the optimization procedure , but keeps its physical clarity . now let us consider the basic active media parameters . .[params - exp ] material parameters of the cr - doped zinc - chalcogenides . [ cols="^,^,^,^,^,^,^,^,^",options="header " , ] in the previous section we considered the ultrashort pulse stability on the basis of the distributed 1 + 1-dimensional model . however , this model can not provide answers to the basic questions : what is the self - start ability of the kerr - lens mode locking and what has the real - world laser configuration to be ? these questions require an analysis on the basis of the time - spatial model . the simplest way is the assumption of the gaussian spatial distribution for the laser beam reducing problem to the 1 + 2-dimensions . the free - space propagation of the gaussian beam can be considered on the basis of the usual abcd - matrix formalism , while the propagation inside the nonlinear active medium is described by the following equation : @xmath45a\left ( { z , r , t } \right ) + i\chi \left| { a\left ( { z , r , t } \right ) } \right|^2 a\left ( { z , r , t } \right ) = \\ \alpha \exp \left ( { - \frac{{2r^2 } } { { w_p^2 } } } \right)a\left ( { z , r , t } \right ) + t_f ' ^2 \frac{{\partial ^2 a\left ( { z , r , t } \right ) } } { { \partial t^2 } } . \nonumber\end{gathered}\ ] ] here @xmath46 and @xmath47 are the group - velocity dispersion and the inverse group - velocity delay coefficients ( for znse we used @xmath48=2054 fs@xmath49/cm and @xmath47=13 fs / cm ) . the left - hand side of eq . ( [ self ] ) describes the non - dissipative factors : thermo - lensing @xcite ( @xmath50=@xmath51/ @xmath52 , @xmath53 is the wave number , @xmath54 is the coefficient of the refractive index thermo - dependence ( 5.35@xmath55 for znse ) , @xmath56 is the loss coefficient at the pump wavelength , @xmath57 is the pump power , @xmath58 is the thermo - conductivity coefficient ( 0.172 @xmath59 for znse ) ) ; diffraction ( in the cylindrically symmetrical case ) ; group - velocity dispersion and self - phase modulation ( providing self - focusing for radially varying beam , @xmath60=@xmath61 ) . the right - hand side of eq . ( [ self ] ) describes the dissipative factors inside the gain medium : radially varying gain ( providing gain guiding and soft aperture action , @xmath9 and @xmath62 are the saturated gain coefficient and the pump beam size , respectively ) ; spectral filtering caused by the gain band profile . it is convenient to rewrite eq . ( [ gain ] ) in the following way : @xmath63 where @xmath64=@xmath65 ( @xmath66 is the generation power ) , @xmath67 is the generation mode beam size . @xmath64=@xmath68 for the pulse with the gaussian time - profile , 2 for the @xmath69-shaped pulse and 1 for the cw ( in the latter case @xmath32=@xmath33 ) . the crucial simplification in the analysis of eq . ( [ self ] ) is based on the so - called aberrationless approximation : the propagating field has the invariable spatial - time profile , which is described by the set of the @xmath7-dependent parameters . in the non - dissipative case this approximation allows the variational approach providing rigorous description of the gaussian beam propagation outside the parabolical approximation @xcite . in the dissipative case we have to use the momentum method @xcite however , in contrast to refs . @xcite we shall consider the momentums resulting from the symmetries of eq . ( [ self ] ) . the @xmath70 invariance , the transverse and time translating invariance suggest the following momentums @xcite : @xmath71 like the variational approach we can substitute to eqs . ( [ momentum ] ) the trial solution describing the ultrashort pulse . if we take the gaussian time - spatial profile @xmath72 ( @xmath73 is the complex amplitude , @xmath74=@xmath75 , @xmath76 is the pulse amplification parameter excepting the geometrical magnification for the gaussian beam ) , the equations describing the evolution of the pulse and beam parameters are @xcite : @xmath77 t_f^2 + 2 \beta_2 ' ^2 \tau \left ( z \right)\psi \left ( z \right),\\ \label{iter } \frac{{dg\left ( z \right)}}{{dz } } = \frac{\alpha } { { 1 + \frac{{2w'\left ( z \right)^2 } } { { w_p^2 } } } } - \frac{{2t_f^2 } } { { \tau \left ( z \right)^2 } } - \beta_2 ' ^2 \psi \left ( z \right),\\ \nonumber \frac{{db\left ( z \right)}}{{dz } } = \frac{{2\vartheta } } { { w_p^2 } } + \frac{{2b\left ( z \right)^2 } } { k } + \frac{{\sqrt 2 p_0 e^{2g\left ( z \right ) } } } { { \pi p_{cr } k w\left ( z \right ) ' ^4 } } - \frac{1}{{2kw'\left ( z \right)^4 } } , \\ \nonumber \frac{{d\psi \left ( z \right)}}{{dz } } = 2 \beta_2 ' ^2 \left ( { \frac{1}{{\tau \left ( z \right)^4 } } - \psi \left ( z \right)^2 } \right ) - \frac{{8t_f^2 \psi \left ( z \right)}}{{\tau \left ( z \right)^2 } } + \frac{{2 p_0 e^{2g\left ( z \right ) } } } { { \pi p_{cr } w\left ( z \right ) ' ^2 \tau \left ( z \right)^2 } } , \\ \nonumber p_g = p_0 e^{2g\left ( z \right)},\end{gathered}\ ] ] where @xmath78 and @xmath79 are the power and the beam size before the active medium , respectively . eq . ( [ iter ] ) in the combination with the abcd - formalism allows defining the stability regions for cw , single- and multiple pulsing ( the latter requires the trivial generalization of the trial function ) . such regions can predict the real - world laser configurations providing the self - starting kerr - lens mode locking . as it is seen from table 2 the main features of the considered active media are the comparatively low @xmath80 ( e.g. @xmath80=2.6 mw for ti : sapphire ) and the extremely low @xmath81 ( 300 kw for ti : sapphire ) . this causes the stabilization of the cw against the mode locking due to the strong gain saturation . however , as it is seen from eqs . ( [ gain2 ] , [ iter ] ) ( note that the gain saturation is intensity - dependent , but the self - focusing is power - dependent ) , we can enhance the tendency to the mode locking due to the suppression of the gain saturation . this can be achieved by the growth of the generation mode within the active medium owing to , for example , the cavity shortening or the decrease of the folding mirrors curvature ( fig . [ zones ] ) . at the same time we have to keep the moderate level of the absorbed pump power and to use the asymmetrical cavity in order to separate the stability zones . the preferable configurations correspond to the pulse operation without the cw . this occurs between the cw stability zones ( see fig . [ zones ] ) . in conclusion , we presented the models , which can be usable for the numerical optimization of the klm lasers . on the basis of these models , the klm abilities of the cr - doped zinc - chalcogenides were estimated . it was shown , that the strong spm inherent to these media and destabilizing the single pulse operation can be overcomed by the choice of the appropriate gdd , pump , modulation depth and saturation parameter of the kerr - lensing induced fast saturable absorber . as a result , the cr : znte possesses the lowest klm threshold , however strong spm constrains the achievable pulse power for this laser . the best stability for the highest energy flux and the shortest pulse duration ( 19 fs ) are achievable in cr : zns . cr : znse . the presence of the third - order dispersion increases the minimum achievable pulse durations up to 34 - 40 fs and causes the strong ( up to 140 nm ) stokes shift of the generation wavelength . however , the latter effect does not reduce noticeably the pulse energy . the kerr - lens mode locking self - start ability of the considered active media is reduced by the strong gain saturation so that the cavity tuning providing the ultrashort pulsing differs essentially from that for the ti : sapphire . we have to avoid over - pumping and over - shortening of the generation mode . as a result , the short and asymmetrical cavities with the comparatively large curvature of the folding mirrors are preferable . on the whole , the cr - doped zinc - chalcogenides have the prospects for the sub-50 fs generation that amounts to only few optical cycles around 2.5 @xmath0 m . this work was supported by austrian national science fund project m688 . l.d.deloach , r.h.page , g.d.wilke , s.a.payne , and w.f.krupke . `` transition metal - doped zinc chalcogenides : spectroscopy and laser demonstration of a new class of gain media , '' _ ieee j. quantum electron . _ * 32 * , pp . 885895 , 1996 . r.h.page , k.i.schaffers , l.d.deloach , g.d.wilke , f.d.patel , j.b.tassano , s.a.payne , w.f.krupke , k .- t.chen , a.burger . `` cr@xmath1:doped zinc chalcogenides as efficient widely tunable mid - infrared lasers , '' _ ieee j. quantum electron . _ * 33 * , pp . 609619 , 1997 . u.hmmerich , x.wu , v.d.davis , s.b.trivedi , k.grasze , r.j.chen , s.w.kutcher . `` demonstration of room - temperature laser action at 2.5 @xmath0 m from cr@xmath1:cd@xmath82mn@xmath83te , '' _ optics lett . _ * 22 * , pp . 11801182 , 1997 . g.j.wagner , t.j.carrig . `` power scaling of cr@xmath84:znse lasers , '' in _ osa trends in optics and photonics _ , vol . 50 , _ advanced solid - state lasers _ marshall , ed . ( osa , washington dc 2001 ) , pp . 506510 . i.t.sorokina , e.sorokin , a.dilieto , m.tonelli , r.h.page . `` tunable diode - pumped continuous - wave operation and passive mode - locking of cr@xmath1:znse laser , '' in _ conference on lasers and electro - optics / europe 2001 _ , conference digest , mnich , p. 151 , 2001 . e.sorokin , i.t.sorokina , r.h.page . room - temperature cw diode - pumped cr@xmath85:znse laser , in _ osa trends in optics and photonics _ , vol . 50 , _ advanced solid - state lasers _ , cr . marshall , ed . ( osa , washington dc 2001 ) , pp . 101105 . k.graham , s.mirov , v.fedorov , m.e.zvanut , a.avanesov , v.badikov , b.ignatev , v.panutin , g.shevirdyaeva . `` spectroscopic characteristics and laser performance of diffusion doped cr@xmath1:zns , '' in _ osa trends in optics and photonics _ , vol . 50 , _ advanced solid - state lasers _ , cr . marshall , ed . ( osa , washington dc 2001 ) , pp . 561567 . v.l.kalashnikov , e.sorokin , i.t.sorokina . `` multipulse operation and limits of the kerr - lens mode locking stability for cr@xmath1:znse laser , '' in _ osa trends in optics and photonics _ , vol . 65 , _ advanced solid - state lasers _ , m.e . fermann , l.r . marshall , eds . ( osa , washington dc 2002 ) , pp . 374378 . detailed investigations show some interesting modification of the single pulse stability and its spectral characteristics , see v.l . kalashnikov , s. naumov , e. sorokin , i. sorokina , `` spectral broadening and shift of the few optical cycle pulses in cr@xmath86:yag laser , '' in program of assp2003 , san antonio , usa , february 2 - 5 , tub1 , 2003 .
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the kerr - lens mode locking ability and the ultrashort pulse characteristics are analyzed numerically for the cr - doped znte , znse , zns active media .
the advantages of these materials for the femtosecond lasing within 2 - 3 @xmath0 m spectral range are demonstrated .
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non - abelian anyons have a topological charge that provides a nonlocal encoding of quantum information @xcite . in superconducting implementations @xcite the topological charge equals the electrical charge modulo @xmath0 , shared nonlocally by a pair of midgap states called majorana fermions @xcite . this mundane identification of topological and electrical charge by no means diminishes the relevance for quantum computation . to the contrary , it provides a powerful way to manipulate the topological charge through the well - established sub-@xmath1 charge sensitivity of superconducting electronics @xcite . following this line of thought , three of us recently proposed a hybrid device called a _ top - transmon _ , which combines the adjustable charge sensitivity of a superconducting charge qubit ( the _ transmon _ @xcite ) to read out and rotate a topological ( _ top _ ) qubit @xcite . a universal quantum computer with highly favorable error threshold can be constructed @xcite if these operations are supplemented by the pairwise exchange ( braiding ) of majorana fermions , which is a non - abelian operation on the degenerate ground state @xcite . here we show how majorana fermions can be braided by means of charge - sensitive superconducting electronics . ( braiding was not implemented in ref . @xcite nor in other studies of hybrid topological / nontopological superconducting qubits @xcite . ) we exploit the fact that the charge - sensitivity can be switched on and off _ with exponential accuracy _ by varying the magnetic flux through a split josephson junction @xcite . this provides a macroscopic handle on the coulomb interaction of pairs of majorana fermions , which makes it possible to transport and exchange them in a josephson junction array . we compare and contrast our approach with that of sau , clarke , and tewari , who showed ( building on the work of alicea et al . @xcite ) how non - abelian braiding statistics could be generated by switching on and off the tunnel coupling of adjacent pairs of majorana fermions @xcite . the tunnel coupling is controlled by a gate voltage , while we rely on coulomb interaction controlled by a magnetic flux . this becomes an essential difference when electric fields are screened too strongly by the superconductor to be effective . ( for an alternative non - electrical approach to braiding , see ref . @xcite . ) the basic procedure can be explained quite simply , see sec . [ braiding ] , after the mechanism of the coulomb coupling is presented in sec . we make use of two more involved pieces of theoretical analysis , one is the derivation of the low - energy hamiltonian of the coulomb coupled majorana fermions ( using results from refs . @xcite ) , and the other is the calculation of the non - abelian berry phase @xcite of the exchange operation . to streamline the paper the details of these two calculations are given in appendices . the basic building block of the josephson junction array is the cooper pair box @xcite , see fig . [ fig_box ] , consisting of a superconducting island ( capacitance @xmath2 ) connected to a bulk ( grounded ) superconductor by a split josephson junction enclosing a magnetic flux @xmath3 . the josephson energy @xmath4 is a periodic function of @xmath3 with period @xmath5 . if the two arms of the split junction are balanced , each with the same coupling energy @xmath6 , the josephson energy @xmath7 varies between @xmath8 and @xmath9 as a function of @xmath10 . cooper pair box , consisting of a superconducting island ( brown ) connected to a bulk superconductor by a split josephson junction ( black , with the gauge - variant phase differences indicated ) . the island contains majorana fermions ( yellow ) at the end points of a nanowire ( grey ) . these are coupled by the coulomb charging energy , tunable via the flux @xmath3 through the josephson junction . ] when the island contains no majorana fermions , its hamiltonian has the usual form @xcite @xmath11 in terms of the canonically conjugate phase @xmath12 and charge @xmath13 of the island . the offset @xmath14 accounts for charges on nearby gate electrodes . we have chosen a gauge such that the phase of the pair potential is zero on the bulk superconductor . a segment of a semiconductor nanowire ( typically inas ) on the superconducting island can have majorana midgap states bound to the end points @xcite . for @xmath15 segments there can be @xmath16 majorana fermions on the island . they have identical creation and annihilation operators @xmath17 satisfying @xmath18 the topological charge of the island equals the fermion parity @xmath19 the eigenvalues of @xmath20 are @xmath21 , depending on whether there is an even or an odd number of electrons on the island . the majorana operators do not enter explicitly in @xmath22 , but affect the spectrum through a constraint on the eigenstates @xcite , @xmath23 this ensures that the eigenvalues of @xmath24 are even multiples of @xmath1 for @xmath25 and odd multiples for @xmath26 . since @xmath20 contains the product of all the majorana operators on the island , the constraint effectively couples distant majorana fermions without requiring any overlap of wave functions . we operate the cooper pair box in the regime that the josephson energy @xmath4 is large compared to the single - electron charging energy @xmath27 . the phase @xmath12 ( modulo @xmath28 ) then has small zero - point fluctuations around the value @xmath29 which minimizes the energy of the josephson junction , with occasional @xmath28 quantum phase slips . in appendix [ cinteraction ] we derive the effective low - energy hamiltonian for @xmath30 , @xmath31 the energy minimum @xmath32 at @xmath33 is increased by @xmath34 due to zero - point fluctuations of the phase . this offset does not contain the majorana operators , so it can be ignored . the term @xmath35 due to quantum phase slips depends on the majorana operators through the fermion parity . this term acquires a dynamics for multiple coupled islands , because then the fermion parity of each individual island is no longer conserved . we generalize the description to multiple superconducting islands , labeled @xmath36 , each connected to a bulk superconductor by a split josephson junction enclosing a flux @xmath37 . ( see fig . [ fig_islands ] . ) the josephson junctions contribute an energy @xmath38 we assume that the charging energy is dominated by the self - capacitance @xmath2 of each island , so that it has the additive form @xmath39 while both @xmath6 and @xmath2 may be different for different islands , we omit a possible @xmath40-dependence for ease of notation . there may be additional fluxes enclosed by the regions between the islands , but we do not include them to simplify the expressions . none of these simplifications is essential for the operation of the device . two cooper pair boxes , each containing a pair of majorana fermions . single electrons can tunnel between the superconducting islands via the overlapping majorana s @xmath41 and @xmath42 . this tunnel coupling has a slow ( cosine ) dependence on the enclosed fluxes , while the coulomb coupling between the majorana s on the same island varies rapidly ( exponentially ) . ] the set of majorana s on the @xmath40-th island is indicated by @xmath43 with @xmath44 . the fermion parities @xmath45 of neighboring islands @xmath40 and @xmath46 are coupled with strength @xmath47 by the overlapping majorana s @xmath43 and @xmath48 . we denote the gauge - invariant phase difference @xcite by @xmath49 . the corresponding tunnel hamiltonian @xcite @xmath50 is @xmath51-periodic in the gauge - invariant phase difference , as an expression of the fact that single electrons ( rather than cooper pairs ) tunnel through the midgap state . for example , in the two - island geometry of fig . [ fig_islands ] one has [ h12def ] @xmath52 in appendix [ cinteraction ] we derive the effective low - energy hamiltonian in the regime @xmath53 , @xmath54 the single sum couples majorana s within an island , through an effective coulomb energy @xmath55 . the double sum couples majorana s in neighboring islands by tunneling . both the coulomb and tunnel couplings depend on the fluxes through the josephson junctions , but in an entirely different way : the tunnel coupling varies slowly @xmath56 with the flux , while the coulomb coupling varies rapidly @xmath57 $ ] . three cooper pair boxes connected at a tri - junction via three overlapping majorana fermions ( which effectively produce a single zero - mode @xmath58 at the center ) . this is the minimal setup required for the braiding of a pair of majorana s , controlled by the fluxes through the three josephson junctions to a bulk superconductor . ] since @xmath59 and @xmath60 in the majorana - coulomb hamiltonian do not commute , the evolution of the eigenstates upon variation of the fluxes is nontrivial . as we will demonstrate , it can provide the non - abelian braiding statistic that we are seeking . similarly to earlier braiding proposals @xcite , the minimal setup consists of three superconductors in a tri - junction . ( see fig . [ fig_trijunction ] . ) each superconductor contains a pair of majorana fermions @xmath61 , with a tunnel coupling between @xmath62 , and @xmath63 . the majorana - coulomb hamiltonian takes the form @xmath64 with gauge - invariant phase differences [ alphatrijunction ] @xmath65 as we vary @xmath66 between @xmath8 and @xmath67 , the coulomb coupling @xmath55 varies between two ( possibly @xmath40-dependent ) values @xmath68 and @xmath69 . we require @xmath70 , which is readily achievable because of the exponential flux sensitivity of the coulomb coupling expressed by eqs . and . we call the coulomb couplings @xmath69 and @xmath68 _ on _ and _ off _ , respectively . we also take @xmath71 , meaning that the coulomb coupling is weaker than the tunnel coupling . this is not an essential assumption , but it allows us to reduce the 6majorana problem to a 4majorana problem , as we will now show . consider first the case that @xmath72 for all @xmath40 . then the hamiltonian has four eigenvalues equal to zero : three of these represent the majorana s @xmath73 far away from the junction , while the fourth majorana , @xmath74 is situated at the tri - junction . the tri - junction contributes also two nonzero eigenvalues @xmath75 , separated by the gap @xmath76 for @xmath77 well below @xmath78 and @xmath71 these two gapped modes can be ignored , and only the four majorana s @xmath79 need to be retained . the hamiltonian @xmath80 that describes the coulomb interaction of these four majorana s for nonzero @xmath55 is given , to first order in @xmath81 , by @xmath82 the hamiltonian describes four flux - tunable coulomb - coupled majorana fermions . although the coupling studied by sau , clarke , and tewari @xcite has an entirely different origin ( gate - tunable tunnel coupling ) , their hamiltonian has the same form . we can therefore directly adapt their braiding protocol to our control parameters . schematic of the three steps of the braiding operation . the four majorana s of the tri - junction in fig . [ fig_trijunction ] ( the three outer majorana s @xmath83 and the effective central majorana @xmath58 ) are represented by circles and the coulomb coupling is represented by lines ( solid in the _ on _ state , dashed in the _ off _ state ) . white circles indicate majorana s with a large coulomb splitting , colored circles those with a vanishingly small coulomb splitting . the small diagram above each arrow shows an intermediate stage , with one majorana delocalized over three coupled sites . the three steps together exchange the majorana s 1 and 2 , which is a non - abelian braiding operation . ] .[table1 ] variation of the flux through the three josephson junctions during the braiding operation , at time steps corresponding to the diagrams in fig . [ fig_braiding ] . the flux @xmath84 is varied in the opposite direction as @xmath85 , to ensure that the coupling parameters @xmath86 do not change sign during the operation . [ cols="<,^,^,^",options="header " , ] we have three fluxes @xmath87 to control the couplings . the braiding operation consists of three steps , see table [ table1 ] and fig . [ fig_braiding ] . @xcite had more steps , involving 6 rather than 4 majorana s . ) at the beginning and at the end of each step two of the couplings are _ off _ ( @xmath88 ) and one coupling is _ on _ ( @xmath89 ) . we denote by @xmath90 the step of the operation that switches the coupling that is _ on _ from @xmath40 to @xmath46 . this is done by first increasing @xmath91 from @xmath8 to @xmath92 and then decreasing @xmath66 from @xmath92 to @xmath8 , keeping the third flux fixed at @xmath8 . during this entire process the degeneracy of the ground state remains unchanged ( twofold degenerate ) , which is a necessary condition for an adiabatic operation . if , instead , we would first have first decreased @xmath66 and then increased @xmath91 , the ground state degeneracy would have switched from two to four at some point during the process , precluding adiabaticity . we start from coupling 3 _ on _ and couplings 1,2 _ off_. the braiding operation then consists , in sequence , of the three steps @xmath93 , @xmath94 , and @xmath95 . note that each coupling @xmath96 appears twice in the _ on _ state during the entire operation , both times with the same sign @xmath97 . the step @xmath90 transfers the uncoupled majorana at site @xmath46 to site @xmath40 in a time @xmath98 . the transfer is described in the heisenberg representation by @xmath99 . we calculate the unitary evolution operator @xmath100 in the adiabatic @xmath101 limit in appendix [ berry ] , by integrating over the berry connection . in the limit @xmath102 we recover the result of ref . @xcite , @xmath103 the result after the three steps is that the majorana s at sites 1 and 2 are switched , with a difference in sign , @xmath104 the corresponding unitary time evolution operator , @xmath105 has the usual form of an adiabatic braiding operation @xcite . for a nonzero @xmath68 the coefficient @xmath106 in the exponent acquires corrections of order @xmath107 , see appendix [ berry ] . if one repeats the entire braiding operation , the majorana s 1 and 2 have returned to their original positions but the final state differs from the initial state by a unitary operator @xmath108 and not just by a phase factor . that is the hallmark of non - abelian statistics @xcite . in summary , we have proposed a way to perform non - abelian braiding operations on majorana fermions , by controlling their coulomb coupling via the magnetic flux through a josephson junction . majorana fermions are themselves charge - neutral particles ( because they are their own antiparticle ) , so one may ask how there can be any coulomb coupling at all . the answer is that the state of a pair of majorana fermions in a superconducting island depends on the parity of the number of electrons on that island , and it is this dependence on the electrical charge modulo @xmath0 which provides an electromagnetic handle on the majorana s . the coulomb coupling can be made exponentially small by passing cooper pairs through a josephson junction between the island and a bulk ( grounded ) superconductor . the control parameter is the flux @xmath3 through the junction , so it is purely magnetic . this is a key difference with braiding by electrostatically controlled tunnel couplings of majorana fermions @xcite . gate voltages tend to be screened quite efficiently by the superconductor , so magnetic control is advantageous . another advantage is that the dependence of the coulomb coupling on the flux is governed by macroscopic electrical properties ( capacitance of the island , resistance of the josephson junction ) . tunnel couplings , in contrast , require microscopic input ( separation of the majorana fermions on the scale of the fermi wave length ) , so they tend to be more difficult to control . both ref . @xcite and the present proposal share the feature that the gap of the topological superconductor is not closed during the braiding operation . ( the measurement - based approach to braiding also falls in this category @xcite . ) two other proposals @xcite braid the majorana s by inducing a topological phase transition ( either by electrical or by magnetic means ) in parts of the system . since the excitation gap closes at the phase transition , this may be problematic for the required adiabaticity of the operation . the braiding operation is called topologically protected , because it depends on the _ off / on _ sequence of the coulomb couplings , and not on details of the timing of the sequence . as in any physical realization of a mathematical concept , there are sources of error . non - adiabaticity of the operation is one source of error , studied in ref . low - lying sub - gap excitations in the superconducting island break the adiabatic evolution by transitions which change the fermion parity of the majorana s . another source of error , studied in appendix [ berry ] , is governed by the _ off / on _ ratio @xmath107 of the coulomb coupling . this ratio depends exponentially on the ratio of the charging energy @xmath109 and the josephson energy @xmath110 of the junction to the bulk superconductor . a value @xmath111 is not unrealistic @xcite , corresponding to @xmath112 . the sign of the coulomb coupling in the _ on _ state can be arbitrary , as long as it does not change during the braiding operation . since @xmath113 , any change in the induced charge by @xmath114 will spoil the operation . the time scale for this quasiparticle poisoning can be milliseconds @xcite , so this does not seem to present a serious obstacle . a universal quantum computation using majorana fermions requires , in addition to braiding , the capabilities for single - qubit rotation and read - out of up to four majorana s @xcite . the combination of ref . @xcite with the present proposal provides a scheme for all three operations , based on the interface of a topological qubit and a superconducting charge qubit . this is not a topological quantum computer , since single - qubit rotations of majorana fermions lack topological protection . but by including the topologically protected braiding operations one can improve the tolerance for errors of the entire computation by orders of magnitude ( error rates as large as 10% are permitted @xcite ) . a sketch of a complete device is shown in fig . [ fig_device ] . josephson junction array containing majorana fermions . the magnetic flux through a split josephson junction controls the coulomb coupling on each superconducting island . this device allows one to perform the three types of operations on topological qubits needed for a universal quantum computer : read - out , rotation , and braiding . all operations are controlled magnetically , no gate voltages are needed . ] this research was supported by the dutch science foundation nwo / fom and by an erc advanced investigator grant . considering first a single island , we start from the cooper pair box hamiltonian with the parity constraint on the eigenstates . following ref . @xcite , it is convenient to remove the constraint by the unitary transformation @xmath115.\label{gaugetr}\ ] ] the transformed wave function @xmath116 is then @xmath28-periodic , without any constraint . the parity operator @xmath20 appears in the transformed hamiltonian , @xmath117 for a single junction the parity is conserved , so eigenstates of @xmath22 are also eigenstates of @xmath20 and we may treat the operator @xmath20 as a number . is therefore the hamiltonian of a cooper pair box with effective induced charge @xmath118 . the expression for the ground state energy in the josephson regime @xmath30 is in the literature @xcite , @xmath119 the first term @xmath120 is the minimal josephson energy at @xmath29 . zero - point motion , with josephson plasma frequency @xmath121 , adds the second term @xmath122 . the third term is due to quantum phase slips with transition amplitudes @xmath123 by which @xmath12 increments by @xmath124 . using @xmath125 , the ground state energy may be written in the form @xmath126 with @xmath127 defined in eq . . higher levels are separated by an energy @xmath128 , which is large compared to @xmath127 for @xmath30 . we may therefore identify @xmath129 with the effective low - energy hamiltonian of a single island in the large-@xmath4 limit . we now turn to the case of multiple islands with tunnel coupling . to be definite we take the geometry of two islands shown in fig . [ fig_islands ] . the full hamiltonian is @xmath130 , where @xmath131 and @xmath132 are two copies of the cooper box hamiltonian and @xmath133 is the tunnel coupling from eq . . to obtain @xmath28-periodicity in both phases @xmath134 and @xmath135 , we make the unitary transformation @xmath136 with @xmath137 the cooper pair box hamiltonians are transformed into @xmath138 with @xmath139 . the tunnel coupling transforms into @xmath140 where @xmath141 and h.c . stands for hermitian conjugate . since @xmath142 , the transformed tunnel coupling @xmath143 is @xmath28-periodic in @xmath134 and @xmath135 . for @xmath30 the phases remain close to the value which minimizes the sum of the josephson energies to the bulk superconductor and between the islands . to leading order in @xmath144 this minimal energy is given by @xmath145\nonumber\\ & + { \cal o}(e_{m}^{2}/e_{j}).\label{emindef}\end{aligned}\ ] ] the josephson coupling of the islands changes the plasma frequency @xmath146 for phase @xmath147 by a factor @xmath148 , so the zero - point motion energy is @xmath149 the transition amplitudes @xmath150 for quantum phase slips of phase @xmath147 are similarly affected , @xmath151 these are the contributions to the effective hamiltonian @xmath152 for @xmath53 , @xmath153\biggr)\nonumber\\ & \times[1+{\cal o}(e_{m}/e_{j})]+{\rm const}.\label{heffmultiapp}\end{aligned}\ ] ] eq . in the main text generalizes this expression for two islands to an arbitrary number of coupled islands . we evaluate the unitary evolution operator @xmath154 of the braiding operation in the adiabatic limit . this amounts to a calculation of the non - abelian berry phase ( integral of berry connection ) of the cyclic variation of the interaction hamiltonian @xmath155 . the eigenvalues are doubly degenerate at energy @xmath160 ( up to a flux - dependent offset , which only contributes an overall phase factor to the evolution operator ) . the two degenerate ground states at @xmath161 are distinguished by an even ( @xmath1 ) or odd ( @xmath162 ) quasiparticle number , the braiding path in three - dimensional parameter space along which the berry phase is evaluated . this path corresponds to the flux values in table [ table1 ] , with couplings @xmath165 for @xmath88 and @xmath166 for @xmath89 . the ratio @xmath167 in the figure is exaggerated for clarity . ] if we avoid this line the berry connection can be readily evaluated . it consists of three anti - hermitian @xmath168 matrices @xmath169 , @xmath170 off - diagonal terms in @xmath169 are zero because of global parity conservation . explicitly , we have @xmath171 a closed path @xmath172 in parameter space has berry phase @xcite @xmath173 the path @xmath172 corresponding to the braiding operation in fig . [ fig_braiding ] and table [ table1 ] is shown in fig . [ fig_path ] . we take all couplings @xmath96 positive , varying between a minimal value @xmath174 and maximal value @xmath175 . the parametrization is well - defined along the entire contour . the contour integral evaluates to @xmath176,\;\;\sigma_{z}=\begin{pmatrix } 1&0\\ 0&-1 \end{pmatrix},\\ & \epsilon=\frac{3}{\sqrt{2}}\frac{\delta_{\rm min}}{\delta_{\rm max}}+{\cal o}\left(\frac{\delta_{\rm min}}{\delta_{\rm max}}\right)^{2}. \label{braiding_error}\end{aligned}\ ] ] the limit @xmath177 corresponds to the braiding operator in the main text ( with @xmath178 and @xmath179 ) . 99 c. nayak , s. simon , a. stern , m. freedman , and s. das sarma , rev . mod . phys . * 80 * , 1083 ( 2008 ) . r. m. lutchyn , j. d. sau , and s. das sarma , phys . lett . * 105 * , 077001 ( 2010 ) . y. oreg , g. refael , and f. von oppen , phys . lett . * 105 * , 177002 ( 2010 ) . kitaev , phys . * 44 * ( suppl . ) , 131 ( 2001 ) . d. v. averin and yu . v. nazarov , phys . lett . * 69 * , 1993 ( 1992 ) . m. h. devoret , a. wallraff , and j. m. martinis , arxiv : cond - mat/0411174 . j. a. schreier , a. a. houck , j. koch , d. i. schuster , b. r. johnson , j. m. chow , j. m. gambetta , j. majer , l. frunzio , m. h. devoret , s. m. girvin , and r. j. schoelkop , phys . b * 77 * , 180502(r ) ( 2008 ) . f. hassler , a. r. akhmerov , and c. w. j. beenakker , new j. phys . * 13 * , 095004 ( 2011 ) . s. bravyi and a. yu . kitaev , phys . a * 71 * , 022316 ( 2005 ) ; s. bravyi , phys . a * 73*. 042313 ( 2006 ) . n. read and d. green , phys . b * 61 * , 10267 ( 2000 ) . d. ivanov , phys . lett . * 86 * , 268 ( 2001 ) . f. hassler , a. r. akhmerov , c .- y . hou , and c. w. j. beenakker , new j. phys . * 12 * , 125002 ( 2010 ) . j. d. sau , s. tewari , and s. das sarma , phys . a * 82 * , 052322 ( 2010 ) . k. flensberg , phys . * 106 * , 090503 ( 2011 ) . l. jiang , c. l. kane , and j. preskill , phys . * 106 * , 130504 ( 2011 ) . p. bonderson and r. m. lutchyn , phys . . lett . * 106 * , 130505 ( 2011 ) . j. alicea , y. oreg , g. refael , f. von oppen , and m. p. a. fisher , nature phys . * 7 * , 412 ( 2011 ) . j. d. sau , d. j. clarke , and s. tewari , phys . b * 84 * , 094505 ( 2011 ) . a. romito , j. alicea , g. refael , and f. von oppen , arxiv:1110.6193 . l. fu , phys . 104 * , 056402 ( 2010 ) ; c. xu and l. fu , phys . rev . b * 81 * , 134435 ( 2010 ) . b. van heck , f. hassler , a. r. akhmerov , and c. w. j. beenakker , phys . b * 4 * , 180502(r ) ( 2011 ) . f. wilczek and a. zee , phys . rev 52 * , 2111 ( 1984 ) . makhlin , g. schn , and a. shnirman , rev . phys . * 73 * , 357 ( 2001 ) . m. tinkham , _ introduction to superconductivity _ ( mcgraw - hill , new york , 1996 ) . p. bonderson , m. freedman , and c. nayak , phys . * 101 * , 010501 ( 2008 ) . m. cheng , v. galitski , and s. das sarma , phys . b * 84 * , 104529 ( 2011 ) . p. j. de visser , j. j. a. baselmans , p. diener , s. j. c. yates , a. endo , and t. m. klapwijk , phys . lett . * 106 * , 167004 ( 2011 ) . j. koch , t. m. yu , j. gambetta , a. a. houck , d. i. schuster , j. majer , a. blais , m. h. devoret , s. m. girvin , and r. j. schoelkopf , phys . rev . a * 76 * , 042319 ( 2007 ) . v. nazarov and ya . m. blanter , _ quantum transport : introduction to nanoscience _ ( cambridge , 2009 ) .
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we show how to exchange ( braid ) majorana fermions in a network of superconducting nanowires by control over coulomb interactions rather than tunneling . even though majorana fermions are charge - neutral quasiparticles ( equal to their own antiparticle ) , they have an effective long - range interaction through the even - odd electron number dependence of the superconducting ground state .
the flux through a split josephson junction controls this interaction via the ratio of josephson and charging energies , with exponential sensitivity . by switching the interaction on and off in neighboring segments of a josephson junction array
, the non - abelian braiding statistics can be realized without the need to control tunnel couplings by gate electrodes .
this is a solution to the problem how to operate on topological qubits when gate voltages are screened by the superconductor .
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optimization and ( feedback ) control of dynamical systems is often computationally infeasible for high dimensional plant models . therefore , one tries to reduce the order of the system , so that the input - output mapping is still computable with sufficient accuracy , but at considerably smaller cost than for the original system , @xcite . to guarantee the desired accuracy , computable error bounds are required . moreover , system properties which are relevant in the context of control system design like asymptotic stability need to be preserved . it has long been known that for linear time - invariant ( lti ) systems the method of balanced truncation preserves asymptotic stability and provides an error bound for the @xmath1-induced input - output norm , that is the @xmath0-norm of the associated transfer function , see @xcite . when considering model order reduction of more general system classes , it is natural to try to extend this approach . this has been worked out for descriptor systems in @xcite , for time - varying systems in @xcite , for bilinear systems in @xcite and general nonlinear systems e.g. in @xcite . yet another generaliztion of lti systems is obtained considering dynamics driven by noise processes . this leads to the class of stochastic systems , which have been considered in a system theoretic context e.g. in @xcite . quite recently , balanced truncation has also been described for linear stochastic systems of it type in @xcite . already the formulation of the method leads to two different variants that are equivalent in the deterministic case , but not so for stochastic systems . it is natural to ask which of the above mentioned properties of balanced truncation also hold for these variants . the aim of this paper is to answer this question . let us first recapitulate balanced truncation for linear deterministic control systems of the form @xmath2 here @xmath3 , @xmath4 , @xmath5 , and @xmath6 , @xmath7 and @xmath8 are the state , output , and input of the system , respectively . moreover @xmath9 denotes the spectrum of @xmath10 and @xmath11 the open left half complex plane . let @xmath12 denote the lyapunov operator and @xmath13 its adjoint with respect to the frobenius inner product . then @xmath14 if and only if there exists a positive definite solution @xmath15 of the lyapunov inequality @xmath16 , by lyapunov s classical stability theorem , see e.g. @xcite . balanced truncation means truncating a balanced realization . this realization is obtained by a state space transformation computed from the gramians @xmath17 and @xmath18 , which solve the dual pair of _ lyapunov equations _ [ eq : lyap ] @xmath19 or more generally the _ inequalities _ @xmath20 these ( in)equalities are essential in the characterization of stability , controllability and observability of system . if @xmath21 , the inequalities can be written as [ eq : lyapequiv ] @xmath22 in the present paper we discuss extensions of and for stochastic linear systems . as indicated above , the equivalent formulations and lead to different generalizations , if we consider it - type stochastic systems of the form @xmath23 where @xmath24 are as in ( [ eq : lti ] ) and @xmath25 . system is asymptotically mean - square stable ( e.g. @xcite ) , if and only if there exists a positive definite solution @xmath15 of the generalized lyapunov inequality @xmath26 here @xmath27 and @xmath28 . this stability criterion indicates that in the stochastic context the generalized lyapunov operator @xmath29 takes over the role of @xmath30 . substituting @xmath30 by @xmath29 in and , we obtain two different dual pairs of generalized lyapunov inequalities . we call them _ type i _ : [ eq : gram1 ] @xmath31 and _ type ii _ : [ eq : gram2 ] @xmath32 note that corresponds to in the sense that @xmath33 has been replaced by @xmath34 , while corresponds to , where @xmath35 has been replaced by @xmath36 . in general ( if @xmath37 and @xmath17 do not commute ) , the inequalities and are not equivalent . at first glance it is not clear which generalization is more appropriate . if the system is asymptotically mean - square stable and certain observability and reachability conditions are fulfilled , then for both types there are solutions @xmath38 . by a suitable state space - transformation , it is possible to balance the system such that @xmath39 is diagonal . consequently , the usual procedure of balanced truncation can be applied to reduce the order of . for simplicity , let us refer to this as _ type i _ or _ type ii balanced truncation_. under natural assumptions , this reduction preserves mean - square asymptotic stability . for type i , this nontrivial fact has been proven in @xcite . moreover , in @xcite , an @xmath40-error bound has been provided . however , different from the deterministic case , there is no @xmath0-type error bound in terms of the truncated entries in @xmath41 . this will be shown in example [ ex : noerrbound ] . in contrast , for type ii , an @xmath0-type error bound has been obtained in @xcite . in the present paper , as one of our main contributions , we show in theorem [ satz : trunc_stab2 ] that type ii balanced truncation also preserves mean - square asymptotic stability . the proof differs significantly from the one given for type i. using this result , we are able to give a more compact proof of the error bound , theorem [ thm : newgramianbound ] , which exploits the stochastic bounded real lemma @xcite . we illustrate our results by analytical and numerical examples in section iv . consider a stochastic linear control system of it - type @xmath42 where @xmath43 are uncorrelated zero mean real wiener processes on a probability space @xmath44 with respect to an increasing family @xmath45 of @xmath46-algebras @xmath47 ( e.g. @xcite ) . + to simplify the notation , we only consider the case @xmath48 and set @xmath49 , @xmath50 . but all results can immediately be generalized for @xmath51 . + let @xmath52 denote the corresponding space of non - anticipating stochastic processes @xmath53 with values in @xmath54 and norm @xmath55 where @xmath56 denotes expectation . let the homogeneous equation @xmath57 be asymptotically mean - square - stable , i.e. @xmath58 , for all solutions @xmath59 . then , by theorem [ thm : hans ] the equations @xmath60 have unique solutions @xmath61 and @xmath62 . under suitable observability and controllability conditions , @xmath18 and @xmath17 are nonsingular . a similarity transformation @xmath63 of the system implies the contragredient transformation as @xmath64 choosing e.g. @xmath65 , with cholesky factorizations @xmath66 , @xmath67 and a singular value decomposition @xmath68 , we obtain @xmath69 and @xmath70 after suitable partitioning @xmath71,\ ; s=\left [ \begin{array}{cc } s_1&s_2 \end{array } \right],\ ; s^{-1}=\left [ \begin{array}{c } t_1\\t_2 \end{array } \right]\end{aligned}\ ] ] a truncated system is given in the form @xmath72 the following result has been proven in @xcite . [ satz : trunc_stab1 ] let @xmath73 satisfy @xmath74 for a block - diagonal matrix @xmath75 with @xmath76 , assume that @xmath77 then , with the usual partitioning of @xmath10 and @xmath37 , we have @xmath78 its implication for mean - square stability of the truncated system is immediate . consider an asymptotically mean square stable stochastic linear system @xmath79 assume that a matrix @xmath80 is given as in theorem [ satz : trunc_stab1 ] and @xmath10 and @xmath37 are partitioned accordingly . + then the truncated system @xmath81 is also asymptotically mean square stable . if the diagonal entries of @xmath82 are small , it is expected that the truncation error is small . in fact this is supported by an @xmath40-error bound obtained in @xcite . additionally , however , from the deterministic situation ( see @xcite ) , one would also hope for an @xmath0-type error bound of the form @xmath83 with some number @xmath84 . the following example shows that no such general @xmath85 exists . [ ex : noerrbound ] let @xmath86 $ ] with @xmath87 , @xmath88 $ ] , @xmath89 $ ] , @xmath90 $ ] . + solving with equality , we get @xmath91 $ ] , @xmath92 $ ] with @xmath93 so that @xmath94 , where @xmath95 and @xmath96 . the system is balanced by the transformation @xmath97^{1/4}$ ] . + then @xmath98 $ ] so that @xmath99 for the truncated system of order @xmath100 . thus the output of the reduced system is @xmath101 , and the truncation error @xmath102 is equal to the stochastic @xmath0-norm ( see @xcite ) of the original system , @xmath103 we show now that this norm is equal to @xmath104 . thus , depending on @xmath105 , the ratio of the truncation error and @xmath106 can be arbitrarily large . according to the stochastic bounded real lemma , theorem [ thm : sbrl ] , @xmath107 is the infimum over all @xmath108 so that the riccati inequality @xmath109\nonumber\end{aligned}\ ] ] possesses a solution @xmath110<0 $ ] . if a given matrix @xmath15 satisfies this condition , then so does the same matrix with @xmath111 replaced by @xmath112 . hence we can assume that @xmath113 , and end up with the two conditions @xmath114 and ( after multiplying the upper left entry with @xmath115 ) @xmath116 thus necessarily @xmath117 , i.e. @xmath118 . this already proves that @xmath119 , which suffices to disprove the existence of a general bound @xmath85 in . taking infima , it is easy to show that indeed @xmath120 . we now consider the inequalities . [ lemma : gramexists ] assume that @xmath57 is asymptotically mean - square - stable . then inequality is solvable with @xmath121 . * proof : * by theorem [ thm : hans ] , for a given @xmath122 , there exists a @xmath123 , so that @xmath124 . then @xmath125 , for sufficiently small @xmath126 , satisfies @xmath127 so that holds even in the strict form . @xmath128 + it is easy to see that like in the previous section a state space transformation @xmath129 leads to a contragredient transformation @xmath130 , @xmath131 of the solutions . that is , @xmath18 and @xmath17 satisfy and , if and only if @xmath132 and @xmath133 do so for the transformed data . as before , we can assume the system to be balanced with @xmath134\;,\label{eq : defsigma}\end{aligned}\ ] ] where @xmath135 and @xmath136 , @xmath137 . hence , we will now assume ( after balancing ) that a diagonal matrix @xmath41 as in is given which satisfies @xmath138 [ eq : pq_ineq ] partitioning @xmath10 , @xmath37 , @xmath139 , @xmath140 like @xmath41 , we write the system as @xmath141 the reduced system obtained by truncation is @xmath142 the index @xmath143 is the number of different singular values @xmath144 that have been kept in the reduced system . in the following subsections , we consider matrices @xmath145,\quad n=\left [ \begin{array}{cc } n_{11}&n_{12}\\n_{21}&n_{22 } \end{array } \right]\ ; , \end{aligned}\ ] ] @xmath80 as in , and equations of the form [ eq : cbtilde ] @xmath146 with arbitrary right - hand sides @xmath147 and @xmath148 . for convenience , we write out the blocks of these equations explicitly : @xmath149\nonumber a_{12}^t\sigma_1+\sigma_2a_{21}+n_{12}^t&\sigma_1n_{11}\\&=-n_{22}^t\sigma_2 n_{21}-\tilde c_2^t\tilde c_1\label{eq : allcomps2}\\[2 mm ] \nonumber a_{22}^t\sigma_2+\sigma_2a_{22}+n_{22}^t&\sigma_2n_{22}\\&=-n_{12}^t\sigma_1 n_{12}-\tilde c_2^t\tilde c_2\label{eq : allcomps3}\\[2 mm ] \nonumber a_{11}^t\sigma^{-1}_1+\sigma^{-1}_1a_{11}+n_{11}^t&\sigma^{-1}_1n_{11}\\&=-n_{21}^t\sigma^{-1}_2 n_{21}-\tilde b_1\tilde b_1^t\label{eq : allcomps4 } \\[2mm]\nonumber a_{12}^t\sigma^{-1}_1+\sigma^{-1}_2a_{21}+n_{12}^t&\sigma^{-1}_1n_{11}\\&=-n_{22}^t\sigma^{-1}_2 n_{21}-\tilde b_2\tilde b_1^t\label{eq : allcomps5}\\[2 mm ] \nonumber a_{22}^t\sigma^{-1}_2+\sigma^{-1}_2a_{22}+n_{22}^t&\sigma^{-1}_2n_{22}\\&=-n_{12}^t\sigma^{-1}_1 n_{12}-\tilde b_2\tilde b_2^t\label{eq : allcomps6}\end{aligned}\ ] ] the following theorem is the main new result of this paper . [ satz : trunc_stab2 ] let @xmath10 and @xmath37 be given such that @xmath150 assume further that for a block - diagonal matrix @xmath75 with @xmath76 , we have [ eq : sigma_ineq ] @xmath151 then , with the usual partitioning of @xmath10 and @xmath37 , we have @xmath152 again we have an immediate interpretation in terms of mean - square stability of the truncated system . consider an asymptotically mean square stable stochastic linear system @xmath79 assume that a matrix @xmath80 is given as in theorem [ satz : trunc_stab2 ] and @xmath10 and @xmath37 are partitioned accordingly . + then the truncated system @xmath81 is also asymptotically mean square stable . * proof of theorem [ satz : trunc_stab2 ] : * note that the inequalities are equivalent to the equations with appropriate right - hand sides @xmath153 and @xmath154 . + by way of contradiction , we assume that does not hold . then by theorem [ thm : kreinrutman ] , there exist @xmath155 , @xmath156 , @xmath157 such that @xmath158 taking the scalar product of the equation with @xmath159 , we obtain @xmath160 whence @xmath161 and @xmath162 , @xmath163 by corollary [ cor : semidefy ] . hence @xmath164 analogously , we have @xmath165 by . + in particular , from @xmath163 , we get @xmath166\right)\\ & = \left [ \begin{array}{cc } a_{11}v+va_{11}^t+n_{11}vn_{11}^t & va_{21}^t+n_{11}vn_{21}^t \\ a_{21}v+n_{21}vn_{11}^t&n_{21}vn_{21}^t \end{array } \right]\\&=\left [ \begin{array}{cc } 0 & va_{21}^t \\ \end{array } \right]\;.\end{aligned}\ ] ] we will show that @xmath167 , which implies @xmath168 in contradiction to , and thus finishes the proof . we first show that @xmath169 is invariant under @xmath170 and @xmath171 . to this end let @xmath172 . then by , @xmath173 whence also @xmath174 , i.e. @xmath175 . from this , we have @xmath176 implying @xmath177 . thus @xmath178 and @xmath179 . since @xmath180 , it follows further that @xmath169 is invariant under @xmath170 and @xmath171 . let @xmath181 , where @xmath182 has full column rank , i.e. @xmath183 . then by the invariance , there exist square matrices @xmath15 and @xmath184 , such that @xmath185 it follows that @xmath186 whence @xmath187 . moreover , from , we get @xmath188 using this substitution in the following computation , we obtain @xmath189 we will show that the right hand side has nonnegative trace . this then implies that the whole term vanishes . note that @xmath190 taking the trace in , we have @xmath191^t m \left [ \begin{array}{c } v_1y\\v_1 \end{array } \right ] \;.\end{aligned}\ ] ] where @xmath192\;.\end{aligned}\ ] ] the matrix @xmath193 is positive semidefinite , because the upper left block is positive definite , and the corresponding schur complement @xmath194 vanishes . hence @xmath195\left [ \begin{array}{c } v_1y\\v_1 \end{array } \right]&=0\end{aligned}\ ] ] implying via the first block row that @xmath196 . from , using also again , we thus have @xmath197 i.e. @xmath198 . it follows that for arbitrary @xmath199 , the eigenvector @xmath159 in can be replaced by @xmath200 because @xmath201 induction leads to @xmath202 as above , we conclude that @xmath203 , @xmath204 , and @xmath205 . multiplying with @xmath206 and with @xmath207 , we get @xmath208 hence ( after multiplication with @xmath82 ) , for all @xmath209 , we have @xmath210 applying this identity repeatedly , we get @xmath211 if @xmath212 is the minimal polynomial of @xmath213 , then @xmath76 implies @xmath214 and @xmath215 whence @xmath216 and also @xmath167 . hence we obtain the contradiction . @xmath128 + the following theorem has been proven in @xcite using lmi - techniques . exploiting the stability result in the previous subsection , we can give a slightly more compact proof based on the stochastic bounded real lemma , theorem [ thm : sbrl0 ] . [ thm : newgramianbound ] let @xmath10 and @xmath37 satisfy @xmath217 assume furthermore that for @xmath75 with @xmath218 and @xmath76 , the following lyapunov inequalities hold , @xmath219 if @xmath220 and @xmath221 , then for all @xmath222 , it holds that @xmath223)}\le2 ( \sigma_{r+1}+\ldots+\sigma_\nu)\|u\|_{l^2_w([0,t])}\;.\ ] ] * proof : * we adapt a proof for deterministic systems e.g. ( * ? ? ? * theorem 7.9 ) . in the central argument we treat the case where @xmath224 and show that @xmath225}\le 2\sigma_\nu \|u\|_{l^2_w[0,t]}\;.\end{aligned}\ ] ] from and , we can see that also @xmath226 hence we can repeat the above argument to remove @xmath227 successively . by the triangle inequality we find that @xmath228}&\le \sum_{j = r}^{\nu-1}\|y_{j+1}-y_j\|_{l^2_w[0,t]}\\ & \le 2(\sigma_\nu+\ldots+\sigma_{r+1 } ) \|u\|_{l^2_w[0,t]}\;.\end{aligned}\ ] ] which then concludes the proof . + to prove , we make use of the stochastic bounded real lemma . in the following let @xmath229 and consider the error system defined by @xmath230 where @xmath231,\quad a_e=\left [ \begin{array}{ccc } a_{11}&a_{12}&0\\a_{21}&a_{22}&0\\0&0&a_{11 } \end{array } \right],\\ n_e&=\left [ \begin{array}{ccc } n_{11}&n_{12}&0\\n_{21}&n_{22}&0\\0&0&n_{11 } \end{array } \right],\quad b_e=\left [ \begin{array}{c } b_1\\b_2\\b_1 \end{array } \right],\\ c_e&=\left [ \begin{array}{ccc } c_1&c_2&-c_1 \end{array } \right]\;.\end{aligned}\ ] ] applying the state space transformation @xmath232&=\left [ \begin{array}{c } x_1-x_r\\x_2\\x_1+x_r \end{array } \right]=\underbrace{\left [ \begin{array}{ccc } i_r&0&-i_r\\0&i_{n - r}&0\\i_r&0&i_r \end{array } \right]}_{=s^{-1}}\left [ \begin{array}{c } x_1\\x_2\\x_r \end{array } \right ] , \end{aligned}\ ] ] we obtain the transformed system @xmath233\;,\\\tilde n_e&=s^{-1}n_es=\left [ \begin{array}{ccc } n_{11}&n_{12}&0\\\tfrac12n_{21}&n_{22}&\tfrac12n_{21}\\0&n_{12}&n_{11 } \end{array } \right]\;,\\ \tilde b_e&=s^{-1}b\left [ \begin{array}{c } 0\\b_2\\2b_1 \end{array } \right]\;,\quad\tilde c_e = c_es=\left [ \begin{array}{ccc } c_1&c_2&0 \end{array } \right]\;.\ ] ] by theorem [ thm : sbrl0 ] , we have @xmath234 , if the riccati inequality @xmath235 possesses a solution @xmath236 . we will show now that the block - diagonal matrix @xmath237 satisfies . partitioning @xmath238 $ ] , we have @xmath239 with the permutation matrix @xmath240 $ ] we define @xmath241 where @xmath242 by . using , we have @xmath243\\ & -\frac{\sigma_\nu}2\left [ \begin{array}{c } n_{21}^t\\0\\-n_{21}^t \end{array } \right]\left [ \begin{array}{c } n_{21}^t\\0\\-n_{21}^t \end{array } \right]^t+\sigma_\nu^2\left [ \begin{array}{c|c } 0&0\\\hline0 & m \end{array } \right]\le 0\;,\end{aligned}\ ] ] which is inequality . @xmath128 + [ ex : noerror_cont ] let the system @xmath244 and @xmath18 be as in example [ ex : noerrbound ] . the matrix @xmath245^{-1}>0\;,\text { where } 0<p\le 1\;,\end{aligned}\ ] ] satisfies inequality . as in example [ ex : noerrbound ] , we have @xmath246 for the corresponding reduced system of order @xmath100 , so that the truncation error again is @xmath247 , independently of @xmath2480,1]$ ] . on the other hand we have @xmath249 with equality for @xmath250 . theorem [ thm : newgramianbound ] thus gives the sharp error bound @xmath251 . note , that there is no @xmath121 satisfying the _ equation _ . the previous example illustrates the problem of optimizing over all solutions of inequality . to compare the reduction methods we need to compute @xmath252 from or . instead of the inequalities , , we can consider the corresponding equations , for which quite efficient algorithms have been developed recently , e.g. @xcite . these also allow for a low - rank approximation of the solutions . in contrast we can not replace by the corresponding equation , because this may not be solvable ( see example [ ex : noerror_cont ] ) . even worse , we do not have any solvability or uniqueness criteria nor reliable algorithms . therefore , in general , we have to work with the inequality , which is solvable according to lemma [ lemma : gramexists ] , but of course not uniquely solvable . in view of our application , we aim at a solution @xmath17 of , so that ( some of ) the eigenvalues of @xmath253 are particularly small , since they provide the error bound . choosing a matrix @xmath122 and a very small @xmath254 along the lines of the proof of lemma [ lemma : gramexists ] can be contrary to this aim . hence some optimization over all solutions of is required . note also that a matrix @xmath121 satisfies , if and only if it satisfies the linear matrix inequality ( lmi ) @xmath255&\le 0\;. \end{aligned}\ ] ] thus , lmi optimal solution techniques are applicable . however , their complexity will be prohibitive for large - scale problems . therefore further research for alternative methods to solve adequately is required . by @xmath256 and @xmath257 , we always denote the original and the @xmath143-th order approximated system . the stochastic @xmath0-type norm @xmath102 is computed by a binary search of the infimum of all @xmath108 such that the riccati inequality is solvable . the latter is solved via a newton iteration as in @xcite . finally , the lyapunov equations are solved by preconditioned krylov subspace methods described in @xcite . + unfortunately , for small @xmath108 , i.e. for small approximation errors , this method of computing the error runs into numerical problems , because contains the term @xmath258 . this apparently leads to cancellation phenomena in the newton iteration , if e.g. @xmath259 . therefore we mainly concentrate on cases where the error is larger , that is we make @xmath143 sufficiently small . in many examples we observe that type ii reduction gives a valid error bound , but the approximation error still is better with type i. this , however , is not always true , as the example @xmath260,\left [ \begin{array}{rr } 0&0\\1&0 \end{array } \right ] , \left [ \begin{array}{r } 0\\3 \end{array } \right ] , \left [ \begin{array}{r } 3\\0 \end{array } \right]\right)\end{aligned}\ ] ] shows . it can easily be verified that the type i lyapunov equations are solved by @xmath261\text { and } p=\left [ \begin{array}{rr } 3&3\\3&6 \end{array } \right]\;.\end{aligned}\ ] ] the type ii inequalities are e.g. solved by @xmath261\text { and } p=\left [ \begin{array}{rr } 8&0\\0&12 \end{array } \right]\;.\end{aligned}\ ] ] reduction to order @xmath262 gives the following error bounds and approximation errors for both types : [ cols="^,^,^",options="header " , ] the main contributions of this paper are the preservation of asymptotic stability for type ii balanced truncation proved in theorem [ satz : trunc_stab2 ] and the new proof of the @xmath0 error bound in theorem [ thm : newgramianbound ] . the efficient solution of is an open issue and requires further research . the same is true for the computation of the stochastic @xmath0-norm . consider the stochastic linear system of it - type @xmath263 where @xmath264 is a zero mean real wiener process on a probability space @xmath44 with respect to an increasing family @xmath45 of @xmath46-algebras @xmath47 ( e.g. @xcite ) . + let @xmath52 denote the corresponding space of non - anticipating stochastic processes @xmath53 with values in @xmath54 and norm @xmath55 where @xmath56 denotes expectation . by definition , system is asymptotically mean - square - stable , if @xmath58 , for all initial conditions @xmath265 . we have the following version of lyapunov s matrix theorem , see @xcite . here @xmath266 denotes the kronecker product . [ thm : hans ] the following are equivalent . ( i ) : : system is asymptotically mean - square stable . ( ii ) : : @xmath267 ( iii ) : : @xmath268 : @xmath269 ( iv ) : : @xmath270 : @xmath269 ( v ) : : @xmath271 : @xmath269 the theorem ( like all other results in this paper ) carries over to systems @xmath272 with more than one noise term , and many more equivalent criteria can be provided , see e.g. @xcite or ( * ? ? ? * theorem 3.6.1 ) . + the following theorem does not require any stability assumptions ( see ( * ? ? ? * theorem 3.2.3 ) ) . it is central in the analysis of mean - square stability . [ thm : kreinrutman ] let @xmath273 then there exists a nonnegative definite matrix @xmath156 , such that @xmath274 we also note a simple consequence of this theorem ( * ? ? ? * corollary 3.2 ) . here @xmath275 is the frobenius inner product for symmetric matrices . [ cor : semidefy ] let @xmath276 as in the theorem . for given @xmath277 assume that @xmath278 then @xmath279 . moreover , if @xmath161 then @xmath280 . now let us consider system with input @xmath281 and output @xmath282 . if system is asymptotically mean - square stable , then defines an input output operator @xmath283 from @xmath284 to @xmath285 , see @xcite . by @xmath107 we denote the induced operator norm , which is an analogue of the deterministic @xmath0-norm . it can be characterized by the stochastic bounded real lemma . * system is asymptotically mean - square stable and @xmath287 . * there exists a negative definite solution @xmath288 to the riccati inequality @xmath289 * there exists a positive definite solution @xmath290 to the riccati inequality @xmath291 we have stated the obviously equivalent formulations ( ii ) and ( iii ) to avoid confusion arising from different formulations in the literature . under additional assumptions also non - strict versions can be formulated . the following sufficient criterion is given in ( * ? ? ? * corollary 2.2.3 ) ( where also the signs are changed ) . unlike in the previous theorem , here asymptotic mean - square stability is assumed at the outset . w. s. gray , j. mesko , energy functions and algebraic gramians for bilinear systems , in : preprints of the 4th ifac nonlinear control systems design symposium , enschede , the netherlands , 1998 , pp . 103108 . p. benner , t. damm , m. redmann , y. rocio rodriguez cruz , positive operators and stable truncation , linear algebra appl . doi:10.1016/j.laa.2014.12.005 , in press . published electronically , dec . 30 , 2014 .
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we consider two approaches to balanced truncation of stochastic linear systems , which follow from different generalizations of the reachability gramian of deterministic systems . both preserve mean - square asymptotic stability , but
only the second leads to a stochastic @xmath0-type bound for the approximation error of the truncated system .
generalized lyapunov equation , model order reduction , balanced truncation , stochastic linear system , asymptotic mean square stability 15a24 , 93a15 , 93b36 , 93b40 , 93d05 , 93e15 ,
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You are an expert at summarizing long articles. Proceed to summarize the following text:
carbon chains play a central role in the chemistry and spectroscopy of interstellar space . the detection of cyanopolyacetylenes in dense interstellar clouds @xcite led to the suggestion by @xcite that carbon chain species be considered as candidates for the diffuse interstellar bands ( dibs ) , which are found in the 4000 - 8500 spectral region of stars reddened by interstellar dust @xcite . since then , many other molecules with a carbon chain backbone have been identified at radio frequencies in dense clouds @xcite . meanwhile , advances in laboratory measurements have provided an understanding of the types and sizes of carbon chains which have strong electronic transitions in the dib range @xcite . it is thus somewhat surprising that as yet among the bare carbon species only diatomic c@xmath11 has been identified in interstellar clouds where dib are detected . the 4052 electronic band system of c@xmath0 was first detected in comets @xcite and then in circumstellar shells by infrared spectroscopy @xcite . most recently c@xmath0 was identified in a dense cloud using sub - mm measurements of its low frequency bending mode and n@xmath12(c@xmath0)@xmath13 @xmath7 was estimated @xcite . @xcite established an upper limit of 5@xmath1410@xmath15 @xmath7 for the column density of c@xmath0 in the direction of @xmath2 oph , some two orders of magnitude lower than that set by @xcite . @xcite made a tentative detection of c@xmath0 towards an eighth magnitude star in the same part of the sky , hd 147889 , at a column density of 4@xmath1410@xmath16 @xmath7 . unfortunately , the star turned out to be a ( previously unknown ) double - lined spectroscopic binary which limited their sensitivity . this letter presents the detection of c@xmath0 towards three stars and infers the column densities in the diffuse clouds . although c@xmath17 , c@xmath18 and c@xmath19 were not detected , upper limits for their column densities are estimated . diatomic species , such as ch , cn , c@xmath11 , and ch@xmath20 , have been detected towards two of the stars chosen and their column densities are considered standards with which to compare models for the physical and chemical processes in diffuse regions @xcite . the four bare carbon chains , c@xmath0 , c@xmath17 , c@xmath18 and c@xmath19 , were selected for the present search because gas phase electronic transitions for these species have been identified in the laboratory in the 4000 - 5500 region and their oscillator strengths are known ( see table [ results ] ) . observations of the reddened stars @xmath2 oph ( hd 149757 ) , @xmath2 per ( hd 24398 ) and 20 aql ( hd 179406 ) were made with the gecko echellette spectrograph on 2000 july 16 and 19 , fiber fed from the cassegrain focus of the canada - france - hawaii 3.6-m telescope ( cfht ) @xcite . all three stars have a visual extinction , a@xmath21 , near 1 and were chosen because they are bright with sharp interstellar k i lines indicating either single clouds or little doppler distortion ( in the case of @xmath2 oph , @xcite resolved the c@xmath11 at 8756 into two close velocity components separated by 1.1 km s@xmath22 ) . the detector was a rear illuminated eev1 ccd ( 13.5 @xmath23m@xmath24 pixels ) and the spectral regions were centered at 4047 in the 14th order , and at 5060 and 5400 in the 11th and 10th orders , respectively . the ultraviolet gecko prism was used to isolate the 14th order , the blue grism for the 11th order , while the stock cfht filter # 1515 was used for the 10th order observations . individual spectra had exposure times ranging from 5 to 20 minutes and were obtained with continuous fiber agitation to overcome modal noise . the resulting combined spectra for the individual stars at each wavelength had unusually high signal - to - noise ratios ( s / n@xmath5800 - 4000 ) for ccd observations . the th / ar comparison arc spectra , taken before and after each spectrograph wavelength reconfiguration , had a typical fwhm of 2.8 pixels , which corresponds to resolutions of @xmath25 = 121000 , 113000 and 101000 at 4047 , 5060 and 5400 , respectively . processing of the spectra was conventional . groups of biases were taken several times throughout each night and at each grating setting a series of flat - field spectra of a quartz - iodide lamp were recorded . the biases and appropriate flats were averaged and used to remove the zero - level offset and pixel - to - pixel sensitivity variations of the ccd . one - dimensional spectra were extracted using standard iraf routines . spectra of vega and other unreddened stars were used to search for contaminating telluric water vapor lines and stellar photospheric features . heliocentric corrections were applied to each spectrum . the observations are summarised in table [ observations ] which lists exposure times and s / n per pixel for each spectral region . the final column gives the radial velocities measured from the interstellar k i 4044.1 and 4047.2 lines . these velocities have been applied to each spectrum to put the interstellar features on a laboratory scale . the @xmath3 origin band of c@xmath0 is quite clearly detected towards all three stars . figure [ figure1 ] compares the observed spectra with a simulated c@xmath0 absorption spectrum based on the spectrograph resolution and assuming a boltzmann distribution in the ground state rotational levels with a temperature of 80 k. the continuum noise level in the observations is @xmath50.1 % . low order polynomials have been applied to the stellar data to give a level continuum ( base line ) and , in the case of @xmath2 per , a weak , broad stellar feature at 4053.2 has been removed . residual broad features in the final spectra are only a few tenths of a percent deep , much less than in the original , and they in no way mask the sharp c@xmath0 lines . in the simulation the rotational line intensities were calculated using the hnl - london factors , while the line positions were taken from the laboratory measurements ( this avoids the problem of a perturbation affecting the low @xmath4 ground state levels which is not accounted for by the fitted spectroscopic constants ) . the individual rotational p , q and r lines are clearly resolved in the spectra of all three stars . table [ lines ] lists the observed positions and equivalent widths of each rotational line assigned in the spectra of the three stars . the table also gives the corresponding positions measured for these transitions in the laboratory ( gausset et al . the positions of some 30 lines in the spectrum of @xmath2 oph agree with the laboratory data to within 0.1 cm@xmath22 providing an unambiguous identification of c@xmath0 in these diffuse clouds . figures [ figurec2p ] , [ figurec2 m ] , and [ figurec3 m ] show the results of equivalent searches for @xmath27 origin band of c@xmath17 at 5078.1 ( @xmath2 oph only ) , the @xmath28 origin band of c@xmath18 at 5415.9 and the @xmath29 origin band of c@xmath19 at 4040.4 , together with simulated spectra for these transitions at 80 k based upon the published spectroscopic constants ( @xcite , @xcite , @xcite ) . in the cases of c@xmath30 and c@xmath31 the linewidth was assumed to be determined by the spectrograph resolution . for c@xmath32 the excited state has been identified as a short - lived feshbach resonance and the measured natural linewidth of @xmath51 cm@xmath22 is employed in the simulation . weak telluric lines were been removed from the c@xmath17 and c@xmath18 observations using standard procedures . unlike the 4050 region for c@xmath0 , each of these spectral regions is contaminated by weak stellar features . nonetheless , for the c@xmath17 and c@xmath18 ions , there are no sharp features corresponding to the rotational lines in the simulations . on the other hand , for c@xmath32 there are features in the spectrum of @xmath2 per and ( much less convincing ) in the magnified plot for @xmath2 oph which appear to coincide in position and shape with the simulated band heads . it is unlikely that these are due to c@xmath32 because the stellar lines in @xmath2 per have exactly the same shape as the coincident features . for @xmath2 oph the whole c@xmath32 spectrum sits within a weak stellar feature ( the lines are broadened by rapid rotation @xmath5400 km s@xmath22 ) . the photospheric lines in @xmath2 oph show nonradial pulsation ` ripples ' which will be washed out to some extent by the long exposure time employed . the spectrum of 20 aql , which normally has the strongest interstellar lines of the three , is free of stellar features but has no features coincident with the c@xmath32 simulation . it is concluded that , while interstellar c@xmath32 might be absent for 20 aql and present for the other two stars , it is more likely that in the latter cases the features are instead stellar . table [ results ] gives the measured equivalent widths for the most intense c@xmath0 line ( q(8 ) ) for each star together with an 1@xmath33 error estimate . for c@xmath17 , c@xmath18 and c@xmath19 , 3@xmath33 detection limits are given . the 1@xmath33 level errors and detection limits are derived from : where the 1@xmath33 limiting equivalent width , @xmath35 , and the fwhm of the feature , @xmath36 , are both measured in , the spectrograph dispersion , @xmath37 , in pixel@xmath22 , and s / n is the signal to noise per pixel . from the simulations , @xmath36 = 0.045 , 1.0 , 0.13 and 0.045 for c@xmath0 , c@xmath19 , c@xmath17 and c@xmath18 , respectively . in the case of c@xmath0 , equivalent widths , @xmath35 , were determined for each rotational line ( varying between 0.3 - 2.7@xmath38 ) and , in combination with the transition oscillator strength , f@xmath39 , and hnl - london factors , the column densities , @xmath40(c@xmath0 ) , of each rotational level ( @xmath4 ) in the ground electronic state were calculated @xcite . in cases where several rotational lines originating from the same level were assigned ( e.g. p(8 ) , q(8 ) , r(8 ) ) the mean of the determined column densities was taken . figure [ figure5 ] shows a boltzmann plot of ln(@xmath41 ) vs. the rotational energy @xcite where the slope is inversely proportional to the rotational temperature . among the lowest rotational levels ( @xmath4214 ) the populations are reasonably approximated by a distribution at 50 - 70 k , whereas the higher rotational levels correspond to a temperature of 200 - 300 k. the simulation in figure 1 uses 80 k as this represents an average temperature for the entire rotational population and allows both the high and low @xmath4 lines to be identified . the high temperature component of the distribution is apparent in the astronomical spectra where the r band head and the higher q lines are more intense than in the simulation ( figure 1 ) . @xcite also found a bimodal population distribution for c@xmath11 in diffuse clouds , with similar characteristic temperatures for the low and high @xmath4 values . the lower temperature is interpreted as the kinetic energy of the cloud and for both c@xmath11 and c@xmath0 the values obtained are comparable to those used in models of diffuse clouds @xcite . the higher temperature component is attributed to repopulation of the levels in the ground electronic state by radiative pumping from excited states . in the case of c@xmath0 it is expected that both the @xmath43 state and the higher lying @xmath44 state will contribute to the radiative pumping . the sensitivity of these measurements is such that @xmath40(c@xmath0 ) in the range 0.2 - 2@xmath45 @xmath7 is determined for rotational levels up to @xmath4=30 . the @xmath40 values were summed to give the estimated lower limits in the range 1 - 2@xmath6 @xmath7 for the total column density , @xmath46(c@xmath0 ) , in table [ results ] . a previous search for c@xmath0 in the direction of @xmath2 oph did not identify the molecule @xcite . it is unclear why this was the case as , in the light of the present observations , the signal - to - noise quoted for these measurements was adequate and the upper limit given was some thirty times lower than the column density reported here . the present measurements for the column densities of c@xmath0 are of the same order of magnitude as the tentative estimate for a translucent cloud @xcite and can be compared to those of other polyatomic molecules observed in diffuse interstellar clouds . column densities ( also towards @xmath2 oph ) in the 10@xmath47 @xmath7 range have been inferred for hco@xmath20 , c@xmath11h and c@xmath0h@xmath11 from observations in the mm region by @xcite , while h@xmath48 has been identified in diffuse regions and n@xmath12(h@xmath48 ) estimated @xmath49 @xmath7 by @xcite . column densities of c@xmath11 towards @xmath2 oph and @xmath2 per have been determined in the 2 - 3@xmath50 @xmath7 range by @xcite . a current model of the diffuse clouds by @xcite predicts n@xmath51(c@xmath11)/n@xmath12(c@xmath0 ) @xmath5 20 ( on a 10@xmath52 year time scale ) , implying n@xmath12(c@xmath0)@xmath53 @xmath7 , in agreement with the values deduced from the table [ results ] . the main production route to c@xmath0 is presumed to be the dissociative recombination process : c@xmath0h@xmath20 + @xmath54 @xmath55 c@xmath0 + h , where c@xmath0h@xmath20 is produced from smaller species by c@xmath20 ion insertion . under conditions where ultra - violet radiation penetrates , photodissociation of c@xmath0 takes place at a threshold of 1653 . as the strong @xmath44@xmath56@xmath57 electronic transition of c@xmath0 is predicted to occur around 1700 @xcite , the dissociation process , c@xmath0 @xmath55 c@xmath11 + c , may be an important destruction pathway in diffuse clouds . although only upper limits for the column densities of c@xmath19 , c@xmath17 and c@xmath18 could be presently established , these species are of interest as small ionic carbon fragments play a crucial role in the ion molecule schemes for diffuse cloud chemistry @xcite . the c@xmath17 ion is the only bare carbon cation for which the gas phase electronic spectrum is known @xcite . in diffuse clouds it is supposed to be the product of the fundamental step : c@xmath20 + ch @xmath55 c@xmath17 + h. its main destruction mechanism is hydrogenation : c@xmath17 + h@xmath11 @xmath55 c@xmath11h@xmath20 + h , which dominates over recombination with electrons in diffuse regions . the diffuse cloud model @xcite predicts a c@xmath17 abundance a factor of 10@xmath58 lower than c@xmath11 , implying a column density @xmath59 @xmath7 , in accord with the upper limit in table [ results ] . published models for diffuse regions do not include the smallest pure carbon anions , c@xmath18 and c@xmath19 in their reaction libraries . unlike c@xmath17 , c@xmath18 does not react with h@xmath11 so its main destruction mechanism is expected to be photodetachment . the similar rotational line widths and oscillator strengths of the c@xmath18 and c@xmath17 transitions lead to similar upper limits for their total column densities . the width of the unresolved bands for c@xmath19 and the presence of weak stellar features in this spectral region means that a higher column density of this ion could have escaped detection . the detection of c@xmath0 provides a powerful incentive for the laboratory study of the electronic transitions of longer carbon chains in the gas phase with the aim of comparison with dib data . the question as to what types and sizes of carbon chains will have strong transitions in the 4000 - 9000 range has already been answered : for example , @xmath44@xmath56@xmath57 transitions of the odd - number bare chains , c@xmath60 , @xmath61=8 - 30 @xcite . the existence of linear carbon chains up to c@xmath62 has been confirmed by the observation of their electronic spectra in neon matrices @xcite . as the oscillator strength scales almost linearly with the length of the molecule , one can expect f@xmath6310 - 20 for these carbon chains . with such an oscillator strength , a species with a column density @xmath64 @xmath7 would be enough to give rise to a strong dib , with an equivalent width of 1 . in view of the column density @xmath510@xmath16 @xmath7 for c@xmath0 determined in this work for three diffuse clouds , this appears to be a reasonable expectation . the support of the swiss national science foundation ( project no . 20 - 055285.98 ) , the canadian natural sciences and engineering research council and the national research council of canada is gratefully acknowledged . the authors thank the staff of the cfht for their care in setting up the fiber feed and agitator , thereby making such high signal - to - noise spectra possible . clccccccccl @xmath2 per & b1 ib & 2.85 & 0.28 & 4800 & 1200 & & & 2700 & 1900 & + 13.91 @xmath650.24 + @xmath2 oph & o9.5 v & 2.56 & 0.30 & 5400 & 2400 & 5400 & 4000 & 3000 & 2200 & @xmath5614.53 @xmath650.18 + 20 aql & b3 v & 5.36 & 0.27 & 10800 & 800 & & & 8400 & 900 & @xmath5612.53 @xmath650.08 + cc|lc|lc|lc 4049.784 & r(22 ) & & & 4049.770@xmath66&1.016 & 4049.795@xmath66&1.708 + 49.770 & r(24 ) & & & 49.770@xmath66&1.016 & 49.795@xmath66&1.708 + 49.810 & r(20 ) & 4049.782@xmath66 & 1.658 & 49.807@xmath66&1.162 & & + 49.784 & r(26 ) & 49.782@xmath66 & 1.658 & 49.807@xmath66&1.162 & & + 49.861 & r(18 ) & 49.865 & 0.309 & 49.877 & 0.511 & 49.865 & 0.821 + 49.963 & r(16 ) & 49.959 & 0.726 & 49.961 & 0.773 & & + 50.081 & r(14 ) & 50.079 & 0.792 & 50.091 & 0.759 & & + 50.206 & r(12 ) & 50.198 & 0.920 & 50.198 & 0.680 & 50.211 & 1.165 + 50.337 & r(10 ) & 50.329 & 1.034 & 50.342 & 0.679 & 50.339 & 1.450 + 50.495 & r(8 ) & 50.483 & 1.525 & 50.497 & 0.383 & 50.489 & 2.204 + 50.670 & r(6 ) & 50.669 & 1.562 & 50.662 & 0.943 & 50.667 & 2.330 + 50.865 & r(4 ) & 50.863 & 1.018 & 50.864 & 1.068 & 50.853 & 2.225 + 51.069 & r(2 ) & 51.073 & 0.896 & & & 51.074 & 1.657 + 51.309 & r(0 ) & 51.267@xmath67 & 0.371 & & & 51.386@xmath67&0.730 + 51.461 & q(2 ) & 51.457 & 1.045 & 51.457 & 0.923 & 51.455 & 1.180 + 51.521 & q(4 ) & 51.515 & 2.187 & 51.518 & 0.752 & 51.506 & 1.876 + 51.590 & q(6 ) & 51.586 & 2.719 & 51.588 & 2.183 & 51.583 & 1.965 + 51.682 & q(8 ) & 51.679 & 2.336 & 51.680 & 2.016 & 51.669 & 2.229 + 51.793 & q(10 ) & 51.788 & 2.138 & 51.795 & 2.294 & 51.787 & 2.508 + 51.929 & q(12 ) & 51.930 & 1.060 & 51.922 & 2.020 & 51.929 & 2.186 + 52.062 & p(4 ) & 52.074@xmath66&1.266 & 52.085@xmath66&1.043 & 52.065@xmath66 & 3.130 + 52.089 & q(14 ) & 52.074@xmath66&1.266 & 52.085@xmath66&1.043 & 52.065@xmath66 & 3.130 + 52.271 & q(16 ) & 52.262 & 1.217 & & & 52.271 & 1.821 + 52.424 & p(6 ) & & & & & 52.456 & 2.499 + 52.473 & q(18 ) & 52.466 & 1.233 & & & & + 52.698 & q(20 ) & 52.701 & 0.923 & & & & + 52.792 & p(8 ) & 52.784 & 0.605 & & & 52.772 & 1.592 + 52.900 & q(22 ) & 52.929 & 0.985 & & & 52.939 & 1.122 + 53.180 & p(10 ) & 53.197@xmath66&1.883 & & & + 53.207 & q(24 ) & 53.197@xmath66&1.883 & & & + 53.590 & p(12 ) & 53.593 & 0.678 & & & 53.588 & 1.237 + 53.795 & q(28 ) & 53.786 & 0.469 & & & 53.781 & 1.098 + 54.112 & q(30 ) & 54.113 & 0.523 & & & & + 54.459 & p(16 ) & 54.445 & 0.870 & & & & + 54.908 & p(18 ) & 54.904 & 1.122 & & & & + c@xmath0&@xmath3&4051.5&0.016&@xmath2 oph&[email protected]&20&1.6 + & & & & 20 aql&[email protected]&19&2.0 + & & & & @xmath2 per & [email protected]&17&1.0 + c@xmath19&@xmath29&4040.4&0.04&@xmath2 oph&6.0&&@xmath680.3 + & & & & 20 aql&20&&@xmath681.2 + & & & & @xmath2 per&12&&@xmath680.7 + c@xmath17&@xmath27&5066.9&0.025&@xmath2 oph&0.35&@xmath681.1&@xmath680.04 + c@xmath18&@xmath28&5408.6&0.044&@xmath2 oph & 0.30&@xmath680.5&@xmath680.02 + & & & & 20 aql&0.55&@xmath680.9&@xmath680.03 + & & & & @xmath2 per&0.35&@xmath680.6&@xmath680.02 +
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the smallest polyatomic carbon chain , c@xmath0 , has been identified in interstellar clouds ( a@xmath11 mag ) towards @xmath2 ophiuchi , 20 aquilae , and @xmath2 persei by detection of the origin band in its @xmath3 electronic transition , near 4052 .
individual rotational lines were resolved up to @xmath4=30 enabling the rotational level column densities and temperature distributions to be determined .
the inferred limits for the total column densities ( @xmath51 to 2@xmath6 @xmath7 ) offer a strong incentive to laboratory and astrophysical searches for the longer carbon chains .
concurrent searches for c@xmath8 , c@xmath9 and c@xmath10 were negative but provide sensitive estimates for their maximum column densities .
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You are an expert at summarizing long articles. Proceed to summarize the following text:
wireless sensor networks ( wsns ) can be utilized as target tracking systems that detect a moving target , localize it and report its location to the sink . so far , the wsn - based tracking systems have found various applications , such as battlefield monitoring , wildlife monitoring , intruder detection , and traffic control @xcite . this paper deals with the problem of target tracking by a mobile sink which uses information collected from sensor nodes to catch the target . main objective of the considered system is to minimize time to catch , i.e. , the number of time steps in which the sink reaches the moving target . moreover , due to the limited energy resources of wsn , also the minimization of data communication cost ( hop count ) is taken into consideration . it is assumed in this study that the communication between sensor nodes and the sink involves multi - hop data transfers . most of the state - of - the - art data collection methods assume that the current location of the target has to be reported to sink continuously with a predetermined precision . these continuous data collection approaches are not suitable for developing the wsn - based target tracking applications because the periodical transmissions of target location to the sink would consume energy of the sensor nodes in a short time . therefore , the target tracking task requires dedicated algorithms to ensure the amount of data transmitted in wsn is as low as possible . intuitively , there is a trade - off between the time to catch minimization and the minimization of data communication cost . in this study two algorithms are proposed that enable substantial reduction of the data collection cost without significant increase in time to catch . the introduced communication - aware algorithms optimize utilization of the sensor node energy by selecting necessary data readings ( target locations ) that have to be transmitted to the mobile sink . simulation experiments were conducted to evaluate the proposed algorithms against state - of - the - art methods . the experimental results show that the presented algorithms outperform the existing solutions . the paper is organized as follows . related works are discussed in section 2 . section 3 contains a detailed description of the proposed target tracking methods . the experimental setting , compared algorithms and simulation results are presented in section 4 . finally , conclusion is given in section 5 . in the literature , there is a variety of approaches available that address the problem of target tracking in wsns . however , only few publications report the use of wsn for chasing the target by a mobile sink . most of the previous works have focused on delivering the real - time information about trajectory of a tracked target to a stationary sink . this section gives references to the wsn - based tracking methods reported in the literature that deal explicitly with the problem of target chasing by a mobile sink . a thorough survey of the literature on wsn - based object tracking methods can be found in references @xcite . kosut et al . @xcite have formulated the target chasing problem , which assumes that the target performs a simple random walk in a two - dimensional lattice , moving to one of the four neighbouring lattice points with equal probability at each time step . the target chasing method presented in @xcite was intended for a system composed of static sensors that can detect the target , with no data transmission between them . each static sensor is able to deliver the information about the time of the last target detection to the mobile sink only when the sink arrives at the lattice point where the sensor is located . a more complex model of the wsn - based target tracking system was introduced by tsai et al . this model was used to develop the dynamical object tracking protocol ( dot ) which allows the wsn to detect the target and collect the information on target track . the target position data are transferred from sensor nodes to a beacon node , which guides the mobile sink towards the target . a similar method was proposed in @xcite , where the target tracking wsn with monitor and backup sensors additionally takes into account variable velocity and direction of the target . in this paper two target tracking methods are proposed that contribute to performance improvement of the above - mentioned target tracking approaches by reducing both the time to catch ( i.e. , the time in which mobile sink can reach the target ) and the data communication costs in wsn . in this study , the total hop count is analysed to evaluate the overall cost of communications , however it should be noted that different metrics can also be also used , e.g. , number of data transfers to sink , number of queries , number of transmitted packets , and energy consumption in sensor nodes . the introduced algorithms provide decision rules to optimize the amount of data transfers from sensor nodes to sink during target chasing . the research reported in this paper is a continuation of previous works on target tracking in wsn , where the data collection was optimized by using heuristic rules @xcite and the uncertainty - based approach @xcite . the algorithms proposed in that works have to be executed by the mobile sink . in the present study the data collection operations are managed by distributed sensor nodes . to reduce the number of active sensor nodes the proposed algorithms adopt the prediction - based tracking method @xcite . according to this method a prediction model is applied , which forecasts the possible future positions of the target . on this basis only the sensor nodes expected to detect the target are activated at each time step . in this section two methods are proposed that enable reduction of data transfers in wsn during target tracking . the wsn - based target tracking procedure is executed in discrete time steps . at each time step both the target and the sink move in one of the four directions : north , west , south or east . their maximum velocities ( in segments per time step ) are assumed to be known . movement direction of the target is random . for sink the direction is decided on the basis of information delivered from wsn . during one time step the sink can reach the nearest segments @xmath0 that satisfy the maximum velocity constraint : @xmath1 , where coordinates @xmath2 describe previous position of the sink . sink moves into segment @xmath0 for which the euclidean distance @xmath3 $ ] takes minimal value . note that @xmath4 are the coordinates of target that were lately reported to the sink . let @xmath5 denote coordinates of the segment where the target is currently detected . the sensor node that detects the target will be referred to as the target node . according to the proposed methods the information about target position is transmitted from the target node to the sink only at selected time steps . if this information is transmitted then the destination coordinates at sink @xmath4 are updated , i.e , @xmath6 . it means that the current position of the target is available for sink only at selected time steps . in remaining time periods the sink moves toward the last reported target position , which is determined by coordinates @xmath4 . hereinafter , symbol @xmath7 will be used to denote the direction chosen by sink when moving toward segment @xmath8 . at each time step , the coordinates @xmath4 and @xmath5 are known for the target node . therefore , the target node can determine the direction which will be chosen by the sink in both cases : if the current target position is transmitted to the sink and if the data transfer is skipped . according to the first proposed method , the coordinates @xmath5 are transmitted to the sink only if @xmath9 , i.e. , if the direction chosen on the basis of coordinates @xmath4 is different than the one selected by taking into account the current position @xmath5 . in the second proposed method , the target node evaluates probability @xmath10 $ ] that the move of sink in direction @xmath11 will minimize its distance to the segment in which the target will be caught . the target coordinates @xmath5 are transferred to the sink only if the difference @xmath12-p[dir(x_d , y_d)]$ ] is above a predetermined threshold . to evaluate probabilities @xmath10 $ ] , the target node determines an area where the target can be caught . this area is defined as a set of segments : @xmath13 where @xmath14 and @xmath15 are the minimum times required for target and sink to reach segment @xmath8 . let @xmath16 and @xmath17 denote the segments into which the sink will enter at the next time step if it will move in directions @xmath18 and @xmath19 respectively . in area @xmath20 two subsets of segments are distinguished : subset @xmath21 that consists of segments that are closer to @xmath16 than to @xmath17 and subset @xmath22 of segments that are closer to @xmath17 than to @xmath16 : @xmath23 < d[(x , y),(x_s , y_s)_d]\},\ ] ] @xmath24 < d[(x , y),(x_s , y_s)_c]\},\ ] ] on this basis the probabilities @xmath10 $ ] are calculated as follows : @xmath25= \frac{|a_c|}{|a| } , \ ; p[dir(x_d , y_d)]= \frac{|a_d|}{|a|},\ ] ] where @xmath26 denotes cardinality of the set . $ ] calculations , width=226 ] the operations discussed above are illustrated by the example in fig . 1 , where the positions of target and sink are indicated by symbols @xmath27 and @xmath28 respectively . velocity of the target is 1 segment per time step . for sink the velocity equals 2 segments per time step . gray color indicates the area @xmath20 in which the sink will be able to catch the target . the direction @xmath18 is shown by the arrow with number 1 and @xmath19 is indicated by the arrow with number 2 , thus @xmath29 and @xmath30 . subset @xmath21 includes gray segments that are denoted by 1 . the segments with label 2 belong to @xmath22 . in the analyzed example @xmath31 , @xmath32 , and @xmath33 . according to eq . ( 4 ) @xmath12 = 0.54 $ ] , @xmath34 = 0.38 $ ] and the difference of these probabilities equals to 0.16 . if the first proposed method is applied for the analysed example then the data transfer to sink will be executed , since @xmath9 , as shown by the arrows in fig , 1 . in case of the second method , the target node will send the coordinates @xmath5 to the sink provided that the difference of probability ( 0.16 ) is higher than a predetermined threshold . the threshold value should be interpreted as a minimum required increase in the probability of selecting the optimal movement direction , which is expected to be obtained after transferring the target position data . experiments were performed in a simulation environment to compare performance of the proposed methods against state - of - the - art approaches . the comparison was made by taking into account two criteria : time to catch and hop count . the time to catch is defined as the number of time steps in which the sink reaches the moving target . hop count is used to evaluate the cost of data communication in wsn . in the experiments , it was assumed that the monitored area is a square of 200 x 200 segments . each segment is equipped with a sensor node that detects presence of the target . thus , the number of sensor nodes in the analysed wsn equals 40 000 . communication range of each node covers the eight nearest segments . maximum velocity equals 1 segment per time step for the target , and 2 segments per time step for the sink . experiments were performed using simulation software that was developed for this study . the results presented in sect . 4.3 were registered for 10 random tracks of the target ( fig . each simulation run starts with the same location of both the sink ( 5 , 5 ) and the target ( 100 , 100 ) . during simulation the hop counts are calculated assuming that the shortest path is used for each data transfer to sink , the time to catch is measured in time steps of the control procedure . the simulation stops when target is caught by the sink . in the present study , the performance is analysed of four wsn - based target tracking algorithms . algorithms 1 and 2 are based on the approaches that are available in literature , i.e. the prediction - based tracking and the dynamical object tracking . these algorithms were selected as representative for the state - of - the - art solutions in the wsn - based systems that control the movement of a mobile sink which has to reach a moving target . the new proposed methods are implemented in algorithms 3 and 4 . the pseudocode in tab . 1 shows the operations that are common for all the examined algorithms . each algorithm uses different condition to decide if current position of the target will be transmitted to the sink ( line 6 in the pseudocode ) . these conditions are specified in tab . 2 . for all considered algorithms , the prediction - based approach is used to select the sensor nodes that have to be activated at a given time step @xmath35 . prediction of the possible target locations is based on a simple movement model , which takes into account the assumptions on target movement directions and its maximum velocity . if for previous time step @xmath36 the target was detected in segment @xmath37 , then at time step @xmath38 the set of possible target locations @xmath39 can be determined as follows : @xmath40 where @xmath41 is the maximum velocity of target in segments per time step . .pseudocode for wsn - based target tracking algorithms [ cols= " < , < " , ] algorithm 2 is based on the tracking method which was proposed for the dynamical object tracking protocol . according to this approach sink moves toward location of so - called beacon node @xmath4 . a new beacon node is set if the sink enters segment @xmath4 . in such case , the sensor node which currently detects the target in segment @xmath5 , becomes new beacon node and its location is communicated to the sink . when using this approach , the cost of data communication in wsn can be reduced because the data transfers to sink are executed less frequently than for the prediction - based tracking method . the proposed communication - aware tracking methods are applied in algorithm 3 and algorithm 4 ( see tab . details of these methods were discussed in sect . simulation experiments were carried out in order to determine time to catch values and hop counts for the compared algorithms . as it was mentioned in sect . 3 , the simulations were performed by taking into account ten different tracks of the target . average results of these simulations are shown in fig . it is evident that the best results were obtained for algorithm 4 , since the objective is to minimise both the time to catch and the hop count . it should be noted that fig . 3 . presents the results of algorithm 4 for different threshold values . the relevant threshold values between 0.0 and 0.9 are indicated in the chart by the decimal numbers . according to these results , the average time to catch increases when the threshold is above 0.2 . for the threshold equal to or lower than 0.2 the time to catch takes a constant minimal value . the same minimal time to catch is obtained when using algorithm 3 , however in that case the hop count is higher than for algorithm 4 . in comparison with algorithm 1 both proposed methods enables a considerable reduction of the data communication cost . the average hop count is reduced by 47% for algorithm 3 and by 87% for algorithm 4 with threshold 0.2 . algorithm 2 also reduces the hop count by about 87% but it requires much longer time to catch the target . the average time to catch for algorithm 2 is increased by 52% . detailed simulation results are presented in fig . these results demonstrate the performance of the four examined algorithms when applied to ten different tracks of the target . the threshold value in algorithm 4 was set to 0.2 . the shortest time to catch was obtained by algorithms 1 , 3 and 4 for all tracks except the 5th one . in case of track 5 , when using algorithm 4 slightly longer time was needed to catch the target . for the remaining tracks the three above - mentioned algorithms have resulted in equal values of the time to catch . in comparison with algorithm 1 , the proposed algorithms ( algorithm 3 and algorithm 4 ) significantly reduce the data communication cost ( hop count ) for all analysed cases . for each considered track algorithm 2 needs significantly longer time to reach the moving target than the other algorithms . the hop counts for algorithm 2 are close to those observed in case of algorithm 4 . according to the presented results , it could be concluded that algorithm 4 , which is based on the proposed method , outperforms the compared algorithms . it enables a significant reduction of the data communication cost . this reduction is similar to that obtained for algorithm 2 . moreover , the time to catch for algorithm 4 is as short as in case of algorithm 1 , wherein the target position is communicated to the sink at each time step . the cost of data communication in wsns has to be taken into account when designing algorithms for wsn - based systems due to the finite energy resources and the bandwidth - limited communication medium . in order to reduce the utilization of wsn resources , only necessary data shall be transmitted to the sink . this paper is devoted to the problem of transferring target coordinates from sensor nodes to a mobile sink which has to track and catch a moving target . the presented algorithms allow the sensor nodes to decide when data transfers to the sink are necessary for achieving the tracking objective . according to the proposed algorithms , only selected data are transmitted that can be potentially useful for reducing the time in which the target will be reached by the sink . performance of the proposed algorithms was compared against state - of - the - art approaches , i.e. , the prediction - based tracking and the dynamical object tracking . the simulation results show that the introduced algorithms outperforms the existing solutions and enable substantial reduction in the data collection cost ( hop count ) without significant decrease in the tracking performance , which was measured as the time to catch . the present study considers an idealistic wsn model , where the information about current position of target @xmath5 is always successfully delivered through multi - hop paths to the sink and the transmission time is negligible . in order to take into account uncertainty of the delivered information , the precise target coordinates @xmath5 should be replaced by a ( fuzzy ) set . relevant modifications of the presented algorithms will be considered in future experiments . although the proposed methods consider a simple case with a single sink and a single target , they can be also useful for the compound tracking tasks with multiple targets and multiple sinks @xcite . such tasks need an additional higher - level procedure for coordination of the sinks , which has to be implemented at a designated control node , e.g. , a base station or one of the sinks . the extension of the presented approach to tracking of multiple targets in complex environments is an interesting direction for future works . zheng , j. , yu , h. , zheng , m. , liang , w. , zeng , p. : coordination of multiple mobile robots with limited communication range in pursuit of single mobile target in cluttered environment . journal of control theory and applications , vol . 441 - 446 ( 2010 )
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this paper introduces algorithms for target tracking in wireless sensor networks ( wsns ) that enable reduction of data communication cost .
the objective of the considered problem is to control movement of a mobile sink which has to reach a moving target in the shortest possible time .
consumption of the wsn energy resources is reduced by transferring only necessary data readings ( target positions ) to the mobile sink .
simulations were performed to evaluate the proposed algorithms against existing methods .
the experimental results confirm that the introduced tracking algorithms allow the data communication cost to be considerably reduced without significant increase in the amount of time that the sink needs to catch the target .
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You are an expert at summarizing long articles. Proceed to summarize the following text:
the discovery of thousands of extrasolar planets and planet candidates in recent years ( see , e.g. , * ? ? ? * ; * ? ? ? * ; * ? ? ? * ; * ? ? ? * ; * ? ? ? * and references therein and see exoplanet.org for a complete list ) , coupled with the rapidly increasing interest in the potential existence of extrasolar life , raise again in a big way the question of whether or not our solar system is special in any sense . more specifically , we are interested in understanding whether the planetary and orbital properties in our solar system are typical or extremely unusual compared to those of extrasolar planets . the solar system contains eight planets and two main belts ( the asteroid belt and the kuiper belt ) . while tens of debris disks and warm dust belts ( similar perhaps to the solar system s asteroid belt ) have been observed and resolved , belts with dust masses as low as those in the solar system would currently be undetectable in extrasolar systems ( e.g. * ? ? ? * ; * ? ? ? consequently , we can quantitatively assess in detail how special the solar system is , only on the basis of its planetary components and properties such as its age and metallicity . however , there are now hundreds of unresolved debris disk candidates ( e.g. * ? ? ? * ) . of these , about two thirds of the systems are better modelled by a two component dust disk rather than a single dust disk . the two temperature components likely arise from two separate belts @xcite . thus , the two belt configuration of our own solar system is plausibly fairly typical . @xcite made an initial attempt to explore to what extent jupiter s periastron could be considered atypical compared to those of the giant planets known at the time . their analysis , however , included only fewer than 100 exoplanets , most of which had been detected via radial velocity measurements . consequently , selection effects dominated their conclusions a possibility fully acknowledged by the authors . in the present work we re - examine the question of how special the solar system is . in section [ special ] we use the much larger currently available database to consider the planetary orbital parameters . we identify the semi major axis of the innermost planet as the most discrepant characteristic of the solar system and the low mean eccentricity as being somewhat special . in section [ prop ] we compare the masses and densities of the planets in our solar system with those in exosolar systems . while the lack of a super earth in the solar system is somewhat unusual , we argue that none of the characteristics identified make the solar system very special . we discuss potential implications of our results in section 4 . some of the apparent differences between the solar system and exoplanetary systems continue to be driven by strong selection effects that affect the sample . we draw our conclusions in section 5 . we first consider the statistical distribution of orbital separation and eccentricity of the observed planetary orbits . to allow for a more meaningful quantitative analysis , we perform a @xcite transformation on the data . this transformation makes the data closer to a normal distribution so that we can more accurately evaluate properties such as the mean and the standard deviation . the transformation takes a skewed data set to approximate normality . it is based on the geometric mean of the measurements and is independent of measurement units . it is possible to do a multivariate box - cox transformation ( e.g. * ? ? ? however , because of the selection effects associated with different parameters we choose to consider each parameter separately . we transform the data with the function @xmath0 where @xmath1 is the parameter we are examining , such as the semi - major axis or eccentricity , and @xmath2 is a constant that depends upon the original distribution , that we discuss below . the maximum likelihood estimator of the mean of the transformed data is @xmath3 where @xmath4 and @xmath5 is the @xmath6-th measurement of a total of @xmath7 . similarly , the maximum likelihood estimator of the variance of the transformed data is @xmath8 we choose @xmath2 such that we maximise the log likelihood function @xmath9 this new distribution , @xmath10 , will be an exact normal distribution if @xmath11 or @xmath12 is an even integer . we can measure how well the transformed distribution compares to a normal distribution with two parameters . the skewness is @xmath13 where @xmath14 is the standard deviation of the distribution and thus we take @xmath15 . the skewness measures the asymmetry of the distribution , a positive number implying the right hand tail of the distribution is longer , and a negative number that the left hand tail is longer . furthermore , we can compare the kurtosis , @xmath16 this is a measure of the `` peakedness '' of the distribution and the heaviness of the tails . a normal distribution has a kurtosis value of zero . a positive value means that the distribution is tightly peaked but the tails are broad , and vice versa for a negative value . in the following subsections we take the samples of the eccentricity and semi - major axis of the observed exoplanets and compare them to the planets in our own solar system . there are a total of 539 exoplanets with a measured eccentricity . in the left hand panel of fig . [ ecc ] we show the eccentricity distribution for this sample . we find the maximum of the log likelihood function to be when @xmath17 . the right hand panel of fig . [ ecc ] shows the histogram of the box - cox transformed data . for our data , we find the skewness to be @xmath18 . for a normal distribution we expect the magnitude of the skewness to be up to @xmath19 . thus the data are not heavily skewed . the kurtosis is @xmath20 whereas for a normal distribution we would expect the magnitude to have values up to @xmath21 . thus , the data have a slightly large kurtosis , or the distribution is not so tightly peaked as a normal one . for the transformed data , the mean is @xmath22 and the standard deviation is @xmath23 . jupiter lies at @xmath24 from the mean . similarly the earth lies at @xmath25 . thus , the eccentricities of the planets in our solar system ( that range from venus at @xmath26 to mercury at @xmath27 ) are all relatively small compared to those of exoplanets , but not altogether significantly different . recently @xcite considered how the mean eccentricity of planets in a system is correlated with the number of planets in the system . they found a strong anti - correlation of eccentricity with multiplicity in systems observed by radial velocity . an extrapolation of their relation up to 8 planets fits well with the mean eccentricity observed in the solar system . furthermore , @xcite used the kepler exoplanets with asteroseismically determined stellar mean densities to derive a rather low eccentricity distribution of the multi planet kepler systems . thus , while the eccentricities in our solar system are low , those may be expected in a system with so many planets . there is some bias in the exoplanet eccentricity data . for the rv planets , the best fit eccentricity is biased upwards from the true value leading to a reduced number of systems with a low eccentricity ( e.g. * ? ? ? * ; * ? ? ? * ; * ? ? ? * ; * ? ? ? however , the detection efficiency decreases only mildly with increasing eccentricity because despite being more difficult to detect , they have a larger rv amplitude for a fixed planet mass and semi - major axis @xcite . while planets found with the transit method require follow up observations ( for example , with rv ) to determine the eccentricity , the distribution of eccentricities is consistent with those of the rv planets @xcite . without the bias , the eccentricities of the planets in our solar system would appear to be less special . to date , there are a total of 1580 planets with a determined semi - major axis . this increases up to a total of 5289 if we include planet candidates from kepler . the false positive rate of the kepler exoplanets is low , especially for multi planet systems and the non giant planets ( e.g. * ? ? ? * ) and thus we consider this much larger dataset also . we find the maximum of the log likelihood function to be when @xmath28 ( @xmath29 , including planet candidates ) . [ data ] shows the histogram of the box - cox transformed data . the left hand panel includes only confirmed exoplanets . for this data , we find the skewness to be @xmath30 . this is only slightly higher than the upper value expected for a normal distribution of @xmath31 . we find the kurtosis to be @xmath32 whereas for a normal distribution we would expect magnitudes smaller than @xmath33 . thus , the data have a large kurtosis . for the transformed data , the mean is @xmath34 and the standard deviation is @xmath35 . thus , jupiter lies at @xmath36 . the right hand panel of fig . [ data ] repeats this analysis but includes unconfirmed kepler planets . the skewness for this is small at @xmath37 ( expected magnitude less than @xmath38 ) and the kurtosis is much smaller also , @xmath39 ( expected magnitude less than @xmath40 ) . jupiter lies at @xmath41 , suggesting on the face of it that jupiter is rather special . however , as we explain below , this is most likely a result of selection effects . the majority of the planets in this distribution have been found by transit methods . the planet with the largest semi - major axis found by this method is only at @xmath42 ( the planet is kic 11442793 h , * ? ? ? kepler , is thought to be complete only for planets at least as large as the earth and for orbital periods up to a year ( e.g * ? ? ? microlensing surveys preferentially find planets at radial distances of a few au from their host star , that is often an m dwarf ( e.g * ? ? ? * ; * ? ? ? this scale is dictated by the size of the einstein ring radius around the lensing star . the radial velocity method has found planets in the range @xmath43 to @xmath44 ( e.g. * ? ? ? * ) . direct imaging can detect planets at much larger distances ( e.g. * ? ? ? * ; * ? ? ? * ) , but so far only 8 planets have been detected by the method . in order to test the possibility that jupiter s outlier status is largely due to selection effects , we repeated the analysis but removed planets found by the transit method . there remain @xmath45 planets in the sample . we find the maximum of the log likelihood function to be when @xmath46 . [ datano ] shows the histogram of the transformed data . for our data , we find the skewness to be @xmath47 . this is less than the value expected for a normal distribution of @xmath19 . for our data we find a high kurtosis of @xmath48 whereas for a normal distribution we would expect values with magnitude less than @xmath49 . for the transformed data , the mean is @xmath50 and the standard deviation is @xmath51 . thus , jupiter lies at @xmath52 and continues to be somewhat of an outlier , but the trend suggests that this is most likely still due to selection effects . the fact that direct imaging repeatedly reveals planets at separations much larger than jupiter s also may indicate that the current relative dearth of planets at large separations could be due to selection biases but more complete observations are required to test this possibility . we should also note that @xcite used only the planet with the largest velocity semi - amplitude in each observed system in their plots . however , they also performed the analysis with the most massive planet in each system and again with all the planets . they reported no difference in the significance of jupiter as an outlier . in 2004 , they found that jupiter was at @xmath53 and half a sigma from its nearest neighbour . we find that jupiter is not such an outlier as it was with the much smaller data set in 2004 , and selection effects continue to affect the distribution . this analysis should again be repeated once we have more reliable observations around the orbital radius of jupiter . currently , our inner solar system appears to be rather special compared to observations of exoplanet systems . the inner edge of our solar system is at the orbit of mercury at @xmath54 , while exoplanetary systems are observed to habor planets much closer to their star . we find that mercury lies at @xmath55 above the mean , while the earth is at @xmath56 above the mean of the distribution of confirmed exoplanet semi - major axes ( as shown in the left hand panel of fig . [ data ] ) . when we include all of the kepler candidates ( right hand panel of fig . [ data ] ) , these increase to @xmath57 for mercury and @xmath58 for the earth . thus , all of the planets in our solar system have orbital semi - major axes that are larger than the mean observed in exoplanetary systems . however , it is possible that this could be the result of selection effects as it is easier to find planets in this region , if they are there . in terms of the radial location of observed exoplanets , the lack of close in planets in our solar system is the parameter that makes our solar system most special . we should note though that if we use only the planets found by methods other than the transit , then mercury is at @xmath59 and the earth is at @xmath60 . consequently it is difficult to say how significant this discrepancy is . @xcite suggested that the migration of jupiter and saturn into the terrestrial planet forming region of our solar system ( down to @xmath61 ) led to the depletion of mass in @xmath62 . the planets are then thought to migrate outwards to their current location ( see also * ? ? ? however , whatever the formation mechanism for the giant planets in our solar system might have been , it is not thought to have been specific to our solar system , and thus neither is a depleted inner solar system . we discuss this point further in section [ discussion ] . finally in this section , we note that migration of planets through the protoplanetary disk or planet - planet interactions or secular interactions of a binary star could affect these distributions , especially that of the semi - major axes . for example , it may be theoretically impossible for jupiter mass planets to form at the radial location of hot jupiters ( e.g. * ? ? ? * ) . instead , they are supposed to form outside of the snow line and migrate inwards through the protoplanetary disk before the latter disperses ( e.g. * ? ? ? migration could also occur by the kozai lidov mechanism increasing the eccentricity of the planet followed by tidal circularization @xcite . evidence obtained by the rossiter mclaughlin effect suggests that some fraction of the hot jupiters may have been produced through dynamical interactions ( see e.g. * ? ? ? * ; * ? ? ? * ; * ? ? ? * ; * ? ? ? * ) . hot jupiter planets dominated the initial planet discoveries because they are large and close to their host star . however , we now know that they are quite rare and orbit only about one percent of solar type stars ( e.g. * ? ? ? opinions vary on whether in the solar system jupiter has significantly migrated . on one hand @xcite suggested that jupiter did not migrate much from its formation location , and on the other , @xcite proposed that the low mass of mars could be explained by gas - driven early migration of jupiter . @xcite further suggested that this formation process could explain the lack of objects in our inner solar system . @xcite assumed that planets form constantly at a radius of @xmath63 and found theoretically that about @xmath64 to @xmath65 of systems will have a jupiter mass planet that does not migrate significantly . however , this conclusion will be affected by the presence of a dead zone ( a region of the disk with no turbulence , e.g. * ? ? ? * ; * ? ? ? * ; * ? ? ? * ) that may slow or halt migration altogether . furthermore , the inner and outer edges of a dead zone may act as planet traps that stop migration ( e.g. * ? ? ? * ; * ? ? ? thus , the distribution of planet semi - major axes definitely does not represent the initial distribution at the time of planet formation . given that jupiter mass planets are thought to form in the vicinity of jupiter s current radial location , theory suggests that the radial location of jupiter is not particularly special . the current observational bias towards planets that are close to their host star means that it is easier to find planets that have migrated inwards , rather than those that have not , or even those that may have migrated outwards . we expect in the future that with more complete observations of jupiter mass planets at jupiter s radial location we will be able to constrain the migration mechanisms and uncover how special jupiter really is for its small ( net ) distance of migration . in this section we consider how the masses and densities of the planets in our solar system compare to those in exosolar systems . in this context , we discuss also the potential significance of the lack of a super - earth in our solar system . [ mass ] shows the distribution of the approximate masses of the exoplanets that have been observed to date , where @xmath6 is the orbital inclination . for directly imaged planets , the mass is predicted by theoretical models of the planets evolution . for the planets that have been observed by microlensing , it s the ratio of the planet to star mass that is measured with accuracy . ] . the masses of the planets within our solar system are shown with arrows at the top ( but are not included in the data ) . the masses of the gas giants fit well with those of exosolar planets , but the terrestrial planets are all on the low side . this is most likely due ( at least partially ) to the difficulty in finding low - mass planets . the masses of the exoplanets are strongly biased towards high mass and short period planets . kepler has shown us that small planets are very common but the mass measurement of small mass planets is difficult and thus currently they appear to be rare . there are two peaks in the data , the first of which is at a mass between that of the earth and that of uranus , at around @xmath66 , where @xmath67 is the mass of jupiter . planets with a mass in the range of @xmath68 to @xmath69 ( where @xmath70 is the mass of the earth ) are known as super - earths ( e.g. * ? ? ? our solar system does not contain any super - earths thus in that sense it is somewhat unusual . we discuss this further in subsection [ superearth ] . the second peak is around the mass of jupiter . we fit the exoplanet mass data with a binormal probability density function ( pdf ) @xmath71 where @xmath72 . with a kolmogorov - smirnov ( ks ) test we find the best fitting parameters to be @xmath73 , @xmath74 , @xmath75 , @xmath76 , @xmath77 and we show the pdf as the solid line in fig . [ mass ] . with this distribution , we find that jupiter is very typical at only @xmath78 from the higher - mass peak , while saturn is at @xmath79 . the terrestrial planets are hard to compare because there are so few data points for those small masses . however , neptune lies at @xmath80 and uranus at @xmath58 from the lower - mass peak . gas giants are thought to form outside of the snow line radius in the protoplanetary disk where there is more solid material available to form massive planets ( e.g. * ? ? ? * ; * ? ? ? * ; * ? ? ? the surface density of the protoplanetary disk decreases with increasing distance from the star and the timescale to form a planet increases with radius . thus , lower mass gas giants could preferentially form farther away from the star . low mass and large orbital radius planets are certainly harder to detect than those with higher mass and lower orbital radius . this can explain the lack of observed small mass planets , but also perhaps the dip in the observed distribution . for example , if uranus and neptune were around another star , at their large orbital radii , they would also be difficult to detect . the only method that could currently detect them is direct imaging . however , the smallest mass planet that has been found with this method is formaulhaut b that has an approximate mass of @xmath81 @xcite . it therefore remains a possibility that the double peaked mass distribution is solely the result of selection effects . . the total number of exoplanets is 1516 . , width=317 ] . , width=283 ] fig . [ density ] shows the approximate densities of the observed planets as a function of their mass . the exoplanets are shown in blue and the planets in our solar system in red . while the lower mass exoplanets have a large range in their density for a given mass ( see also * ; * ? ? ? * ; * ? ? ? * ; * ? ? ? * ) , the giant planets show a clear correlation of increasing density with mass . thus , the super - earths may have a wide range of compositions @xcite . despite the large spread in the data for the low mass planets , there appears to be a trend of decreasing density with increasing mass . this could be attributed qualitatively to the peak in the theoretical radius against mass of a planet ( see e.g. * ? ? ? * ; * ? ? ? * ; * ? ? ? * ; * ? ? ? more recently , it has been suggested that the super - earths ( with radii in the range @xmath82 ) show two trends separated by a critical planet radius . the smallest planets increase in density with radius while those that are larger decrease suggesting that the larger planets have a large amount of volatiles on a rocky core . there is some uncertainty over the value of the critical radius , as estimates range from about @xmath83 to @xmath84 @xcite , if it exists at all @xcite . the data suggest that the densities of the giant planets within our solar system are very typical of those of observed exoplanets . the masses of the terrestrial planets in our solar system are on the edge of our current sensitivity and thus it is hard to draw any conclusions about their densities . however , recently , @xcite found that the earth ( and venus ) can be modelled with the same ratio of iron to magnesium silicate as the low mass exoplanets observed and thus the earth may not be special in this respect . it is interesting to examine whether our solar system s lack of a super - earth is truly unusual . there have been several attempts to calculate an occurrence rate for super - earths taking into account the selection biases . the results for rv observations predict an occurrence rate in the range @xmath85 in the period range @xmath86 @xcite . the transiting planet observations imply a range of occurrence rates that is at most as high as @xmath87 in the period range @xmath88 @xcite . more recently @xcite examined the kepler sample for planets with radii in the range @xmath89 with orbital periods in the range 50 to @xmath90 and found an occurrence rate of @xmath91 . although these results include earth - size planets and some of the periods are longer than that of mercury , the occurrence rate increases with short orbital periods , making the existence of close - in planets more likely . the high occurrence rate of these types of planets offers perhaps the strongest argument against the solar system being very common , but even that does not necessarily make it extremely rare . typically , systems that have an observed super - earth , have more than one and this is theoretically expected if the planets form by mergers of inwardly migrating cores ( e.g. * ? ? ? * ; * ? ? ? it is possible that the presence of a super - earth can affect terrestrial planet formation . many of the super - earths observed are at small radial locations , where theoretically they could not have formed ( e.g. * ? ? ? * ; * ? ? ? * ) . @xcite found that if a super - earth migrates sufficiently slowly through the habitable zone ( defined as the radial range of distances from the star at which a rocky planet can maintain liquid water on its surface ) then any terrestrial planet that later forms there would be volatile - rich and not very earth - like . in conclusion , the masses and densities of the planets of our solar system appear to be very typical of those of exoplanets . however , the lack of a planet with a mass in the range @xmath92 , a super - earth , makes the solar system appear somewhat special . and mercury , being the inner most planet , at @xmath93 . , width=317 ] generally , the physical properties of the planets in our solar system are quite typical when compared to those of the observed exoplanets , although the lack of a super earth is unusual . the orbital properties , however , may be somewhat special and perhaps more conducive to life . low eccentricity planets have a more stable temperature throughout the orbit and therefore may be more likely to host life ( e.g. * ? ? ? * ; * ? ? ? . furthermore , planetary systems with a low average eccentricity are more likely to have long term dynamical stability . for example , the terrestrial planets in our solar system are expected to be dynamically stable at least until the sun becomes a red giant and engulfs the inner planets ( e.g. * ? ? ? * ) . there are a few other factors that could , in principle at least , make our solar system special with respect to the emergence of life . first we can consider the age . the current age of our sun is about half the age of the disk of our galaxy , and also half of the sun s total lifetime . thus , we expect that roughly half of the stars in our galaxy s disk are older and half are younger than our sun . this implies that the age of our solar system is definitely not special . furthermore , @xcite considered the planet formation history of the milky way and determined that our solar system formed close to the median epoch for giant planet formation . they also found that about @xmath94 of the currently - existing earth - like planets were already formed at the time of the earth s formation . we also note that the fact that the solar system contains a single star does not make it particularly special , since the binary fraction in the kepler sample , for example , is about 50% @xcite . the presence of terrestrial planets in the habitable zone around their host star appears to be quite common . for example , @xcite examined the kepler data for m dwarfs and found that for orbital periods shorter than @xmath95 , the occurrence rate of earth - size planets in the habitable zone is around @xmath96 . this conservative estimate could be as high as @xmath97 depending on how the habitable zone is defined . this is consistent with radial velocity surveys that find @xmath98 potentially habitable planets per m dwarf @xcite . thus , an earth - size planet in a habitable zone is not uncommon . a habitable planet may require a large moon which in turn may require an asteroid collision @xcite . thus , systems which contain an asteroid belt may be more conducive to initiating life . however , habitability may be sensitive to the size of the asteroid belt @xcite . as we have noted in the introduction , asteroid belts could be a common feature of planetary systems @xcite . the metallicity of a protoplanetary disk ( and hence the host star ) determines the structure of a planetary system that forms ( e.g. * ? ? ? * ; * ? ? ? the higher the metallicity of a star , the more giant planets that are observed ( e.g. * ? ? ? * ; * ? ? ? * ; * ? ? ? * ; * ? ? ? however , the correlation for lower mass planets is unclear @xcite . planets with radii less than four times that of the earth are observed around stars with a wide range of metallicities . however , the average metallicity of stars hosting small planets ( @xmath99 ) in @xcite is very close to solar . although such planets can form at a wide range of metallicities , the fact that the average metallicity of the small planets is solar may not be a coincidence . thus , while the metallicity of our solar system may not be especially promotive to the formation of a habitable planet , it s unclear whether the solar system is special or not in this respect . the variability of our sun has been compared to the activity of stars in the kepler sample with conflicting conclusions . @xcite and @xcite found that the sun is rather typical with only a quarter to a third of stars in the kepler sample being more active than the sun . on the other hand , @xcite found the sun to be relatively quiet with @xmath100 of stars being more active . the difference in the conclusions stems from choices in defining the activity level of our sun and the inclusion of stars with fainter magnitudes in @xcite . however , the studies agree that the active fraction of stars becomes larger for cooler stars . m dwarfs have a fraction of @xmath101 that are more active than the sun . thus , compared to other sun - like stars , our sun could be typical , but compared to cooler stars , our sun is certainly quiet . in general , there are three aspects in which the solar system differs most from other observed multi - planet systems . first , the low mean eccentricity of the planets in the solar system maybe somewhat special , although this may be accounted for by selection effects . secondly , there is in the total lack of planets inside mercury s orbit . massive planets migrating through the habitable zone can change the course of planet formation in that region . overall , however , processes that could act to clear the inner part of the solar systems ( such as giant planet migration ) , are believed to be operating within a non - negligible fraction of the exoplanet systems ( e.g. * ? ? ? third , the lack of super - earths in our solar system is somewhat special and could have allowed the earth to become habitable . a close - in super - earth could also affect the dynamical stability of a terrestrial planet in the habitable zone and this should be investigated in future work . we consider the first two of these special parameters in more detail in fig . [ average ] . for planets with measured eccentricity , we plot the mean eccentricity of the planetary system and the semi major axis of the innermost planet observed within the system . in this parameter space , the solar system appears to be somewhat special , but far from being rare . although , most of the systems with three or more planets , do have a planet with an orbital semi - major axis smaller than that of mercury , this could be at least partly due to selection effects . the observed semi major axis of the innermost planet may be close to complete but the number of planets in each system is certainly not . if the innermost planet is very close in , then it is easier to detect the planets outside of its orbit . if on the other hand the innermost planet is farther out ( e.g. some of the small blue and red points in fig . 6 ) then additional planets will be difficult to find ( see also discussion in section [ a ] ) . there are many factors that may be required in order to form a habitable planet . when we multiply the probabilities for each together , we may end up with a small probability for such an event . however , since we currently do not know which factors are truly important for life to emerge , such an exercise does not make much sense . if we consider too many details , clearly the solar system is special because all systems are different . however , at the moment we have not identified any parameter that makes the solar system so significantly different that it would make it rare . we find that the properties of the planets in our solar system are not so significantly special compared to those in exosolar systems to make the solar system extremely rare . the masses and densities are typical , although the lack of a super earth sized planet appears to be somewhat unusual . the orbital locations of our planets seem to be somewhat special but this is most likely due to selection effects and the difficulty in finding planets with a small mass or large orbital period . the mean semi - major axis of observed exoplanets is smaller than the distance of mercury to the sun . the relative depletion in mass of the solar system s terrestrial region may be important . the eccentricities are relatively low compared to observed exoplanets , although the observations are biased towards finding high eccentricity planets . the low eccentricity , however , may be expected for multi - planet systems . thus , the two characteristics of the solar system that we find to be most special are the lack of super - earths with orbital periods of days to months and the general lack of planets inside of the orbital radius of mercury . from the perspective of habitability the solar system does not appear to be particularly special . if exosolar life happens to be rare it would probably not be because of simple basic physical parameters , but because of more subtle processes that are related to the emergence and evolution of life . since at the moment we do nt know what those might be , we can allow ourselves to be optimistic about the prospects of detecting exosolar life . we should make every possible effort to detect and characterise the atmospheres of a few dozen earth - size planets in the habitable zone , in the coming two decades . we thank the anonymous referee for useful comments . this research has made use of the exoplanet orbit database and the exoplanet data explorer at exoplanets.org .
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with the availability of considerably more data , we revisit the question of how special our solar system is , compared to observed exoplanetary systems . to this goal , we employ a mathematical transformation that allows for a meaningful , statistical comparison .
we find that the masses and densities of the giant planets in our solar system are very typical , as is the age of the solar system . while the orbital location of jupiter is somewhat of an outlier , this is most likely due to strong selection effects towards short - period planets .
the eccentricities of the planets in our solar system are relatively small compared to those in observed exosolar systems , but still consistent with the expectations for an 8-planet system ( and could , in addition , reflect a selection bias towards high - eccentricity planets ) .
the two characteristics of the solar system that we find to be most special are the lack of super - earths with orbital periods of days to months and the general lack of planets inside of the orbital radius of mercury .
overall , we conclude that in terms of its broad characteristics our solar system is not expected to be extremely rare , allowing for a level of optimism in the search for extrasolar life .
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recently , within the framework of a coarse - grained nonlinear network model ( nnm ) , we have shown that dbs in proteins feature strongly site - modulated properties @xcite . more precisely , we have shown that spatially localized band - edge normal modes ( nm ) can be continued from low energies to db solutions centered at the same sites as the corresponding nms ( the nm sites ) . note that the latters lie , as a rule , within the stiffest regions of a protein @xcite . more generally , however , dbs display a gap in their excitation spectrum . as a consequence , they can `` jump '' to another site as their energy is varied , following spatial selection rules matching the pattern of dbs localized elsewhere @xcite . as a matter of fact , such jumps realize efficient _ energy transfers_. hereafter , we show that events of this kind , connecting with high yields even widely separated locations , can be triggered by a localized excitation , so long as its energy @xmath4 lies above a given threshold . energy transfer : all - site analysis . percentage of sites in subtilisin that transmit most of the kick energy to the band - edge nm site , val 177 ( black diamonds ) , or to the nm site of the second edge mode , met 199 ( stars ) . for a given kick energy , each site is kicked once , the most energetic nonlinear mode obtained is analyzed , and the site the most involved in this mode is recorded . when initial excitations are not imparted along the local stiffest direction , but are oriented at random , energy transfer towards val 177 is less likely ( open diamonds ) . ] -5 mm fig . [ ekept ] summarizes the outcome of one such experiment , where energy is initially either localized in nm ( m ) or in real ( r ) space . typically , the initial excitation is found to spark the formation of a discrete breather , pinning a variable amount of energy @xmath5 at a specific location . when less than 10 kcal / mole of kinetic energy is injected into the edge nm , nearly all this energy is kept by the db , whose overlap with the edge nm is large at low energies . increasing @xmath4 further , the frequency of the excited mode detaches from the linear band , while the excitation efficiency @xmath6 is eroded . in fact , as db localization builds up with energy ( see lower left panel ) , the spatial overlap with the edge nm diminishes , thus reducing excitation efficiency @xcite . the same db is also excited when the edge nm site is `` kicked '' along an _ appropriate _ direction , namely the maximum stiffness ( ms ) one @xcite ( see data marked ( r ) in fig . [ ekept ] ) . in this case , however , the excitation becomes more efficient as @xmath4 is increased , since the db asymptotically approaches a single - site vibration . for @xmath7 kcal / mole , the db looses its energy , which flows rapidly into the system . we find that the maximum strain direction invariably allows for the most efficient excitation of a nonlinear mode at a given site . [ eangle ] illustrates the efficiency of kicks given along the ms direction , with respect to kicks imparted along random directions . the correlation with the squared cosine of the angle between the kick and the ms unit vectors indicates that it is the amount of energy injected along the ms vector which is the dominant factor allowing for efficient excitation of a discrete breather . + interestingly , kicking away from the ms direction can promote energy transfer to another site . for instance , while a kick along the ms unit vector at the nm site of the band - edge mode invariably results in a db sitting at the same site , when the direction of the kick is picked at random discrete breathers localized elsewhere are also observed ( see again fig . [ eangle ] ) . in the following , we take advantage of the fact that ms directions can be easily calculated at any site in any structure @xcite in order to investigate energy transfer in a systematic manner . energy transfer as a function of distance from excitation site . the figure illustrates the outcome of an all - site kick experiment in myosin , a large 746 amino - acids enzyme involved in muscle contraction ( pdb code 1vom ) . the fraction of excitation energy found in the db is plotted versus the distance ( in units of links in the connectivity graph ) between the kicked site and the site where the nonlinear mode self - excites . the maximum amount of energy found in the db decreases with the number of links separating the feed and the target sites . for instance , when gln 246 is kicked , more than 40% of the energy ends up in a db localized at ala 125 ( the band - edge nm site ) . this amounts to four links , corresponding to a span of about 25 in real space . otherwise , when a kick is given to ile 351 , gln 246 or tyr 34 , 25 - 65% of the excitation energy flows either to ala 125 or leu 296 , the nm site of the third edge normal mode . in cases where more than 30% of the kick energy is transferred away , three sites turn out to be targeted half of the times , namely ala 125 ( 27% ) , leu 296 ( 13% ) and gly 451 ( 7% ) . when only long - range energy transfers are considered ( covering three or more links ) , the shares raise to 71 % and 18 % for ala 125 and leu 296 , respectively . in the remaining cases , the db is found either at leu 516 ( 7% , 14@xmath8 mode ) or at arg 80 ( 4% , 10@xmath8 mode ) . ] -5 mm when a given residue is kicked along the ms direction , a transfer event can occur when @xmath9 kcal / mol ( see an example in fig . [ etrans ] ) . at peak transfer , more than 75 % of such kicks excite a db localized at the band - edge nm site , while otherwise energy flows towards the nm site of another edge mode . conversely , when the kick is imparted along a random direction , energy transfer is found to be less efficient . + quite generally , a transfer event can be observed when almost any site is kicked , and in the majority of cases only a handful of well - defined sites are targeted . this means that energy transfer can occur between widely separated locations . indeed , as illustrated in fig . [ elost ] for myosin , only about 5 % of 55 kcal / mole kicks result in a db localized at the same location . for all other kicked sites , a transfer occurs to a db pinning a decreasing fraction of the excitation energy , one to eleven links away . note that all high - yield and long - range energy transfers aim at the nm sites of one of the edge nms , the nm site of the bande - edge mode being the most likely target . thus , energy systematically flows toward the stiffest regions of the structure . interestingly , this is where functionally relevant residues tend to be located @xcite . + in one occurrence , more than 20% of the kick energy ends up in a nonlinear mode localized more than five links away : following a kick at tyr 34 a remarkable nine - link stretch is covered up to leu 296 , making a jump of more than 60 . however , cases of ultra long - range energy transfer like this are more rare and , at the same time , less efficient . in fact , as a consequence of the rather small amount of energy transferred ( nearly 14 kcal / mole ) , the db that self - excites at the target site is poorly localized ( like in fig . [ ekept ] ) . site to site energy transfer in myosin . the local energies at sites ile 351 ( dotted line ) and leu 296 ( solid line ) are plotted as functions of time , after a 55 kcal / mole kick at ile 351 . the fluctuations occurring well before and after the transfer reflect the fact that the corresponding nonlinear modes are not perfectly localized on both sites . as a consequence , they exchange significant amounts of energy with their _ environs_. ] -5 mm a more efficient transfer event , covering two links ( about 11 ) , is analyzed in fig . [ edet ] . at first , a db is excited at the kicked site . however , due to interactions with the background , its energy slowly but steadily flows into the system . after approximately 1 ns , about 65 % of the excitation energy is still there . at @xmath10 ns , this amount of energy is rapidly and almost entirely transferred to leu 296 , marking the self - localization of another db . although the transfer itself is a quite complex process , involving several intermediate sites , it may well prove to be an example of _ targeted energy transfer _ @xcite . indeed , as the energy of the db at the the initial site drops , its frequency diminishes as well . this may allow for a transfer to occur if a resonance condition with the frequency of another db is met . the transmission should be irreversible , as a consequence of both dbs frequency drifts during energy exchange @xcite . note that , as the energy of the first db is eroded , the mode becomes also less and less localized @xcite . this , in turn , is likely to increase the overlap between the two db displacement patterns , thus allowing for more efficient energy channelling @xcite . to gain further understanding on the transfer mechanism , we investigated energy circulation in a dimeric form of rhodopsin . very few high - yield and long - range energy transfers were recorded between sites belonging to different monomers , the vast majority of transfer events being confined within the same domain . indeed , in less than 1% of the instances more than 30% of the kick energy ( 55 kcal / mole ) injected at one monomer is transmitted to the other . here , at variance with most protein dimers , the stiffest regions are located in monomer bulks , so that the edge nms are localized far away from the interface . this strongly suggests that energy transfers not only target stiff regions , but can couple any two sites efficiently only through rather stiff channeling pathways . on the other hand , when kicking one of the two ( almost ) equivalent sites of rhodopsin that are covalently linked to the retinal chromophore , up to about 50 % of the excitation energy ends up in a db localized at one of three specific sites , the targeted location depending upon where ( which monomer ) the kick is imparted and on the magnitude of the latter . interestingly , fig . [ k296 ] reveals that transfer efficiency is optimum in the narrow range 50 - 55 kcal / mole , _ i.e. _ exactly the energy of photons that can be absorbed by the retinal chromophore when it is embedded within rhodopsin ( @xmath11 nm ) . interestingly , the preferentially targeted residue in this energy range ( glu 113 ) is known to be involved in the early stages of the signaling cascade following rhodopsin activation @xcite . energy transfer in rhodopsin ( pdb code 3cap ) . the fraction of energy @xmath6 found in the discrete breather when kicking the site attached to the retinal chromophore ( lys 296 ) of monomer b is plotted versus the excitation energy . symbols indicate at which site the db self - localizes : glu 113 ( black diamonds ) , cys 185 ( open diamonds ) , met 86 ( open circle ) or another one ( stars ) . ] -5 mm in summary , despite its coarse - grained nature , the nnm framework is able to provide biologically sensible clues about energy circulation in proteins . high - yield and long - range energy transfers systematically pin energy at the sites the most involved in a small subset of band - edge linear modes , that is , within the stiffest parts of protein structures . these , in turn , are the regions preferentially hosting residues involved in catalytic mechanisms @xcite . thus , what our study suggests is that protein structures may have been designed , during the course of evolution , so as to allow energy to flow where it is needed , _ e.g. _ to , or close to catalytic sites , with the aim of lowering the energy barriers that have to be overcome during catalytic processes . interestingly , in view of the coarse - grained nature of the nnm scheme , the same site - specific , high - yield and long - range energy transfers observed in proteins are also likely to occur in other physical systems , possibly simpler to engineer and to handle , so long as they share with proteins both spatial and stiffness heterogeneity . proteins are modelled as networks of nodes of mass @xmath12 ( the @xmath13-carbons of their amino - acid residues ) linked by springs . specifically , in the nonlinear network model ( nnm ) @xcite , the potential energy of a protein , @xmath14 , has the following form : @xmath15 \notag\ ] ] where @xmath16 is the distance between particles @xmath17 and @xmath18 , @xmath19 their distance in the equilibrium structure ( as _ e.g. _ solved through x - ray crystallography ) and @xmath20 is a distance cutoff that specifies which pairs of nodes are interacting . note that @xmath21 corresponds to the widely used elastic network model ( enm ) @xcite , which has proven useful for quantitatively describing amino - acid fluctuations at room temperature @xcite , as well as for predicting and characterizing large - amplitude functional motions of proteins @xcite , in agreement with all - atom models @xcite , paving the way for numerous applications in structural biology @xcite , such as fitting atomic structures into low - resolution electron density maps @xcite , or providing templates for molecular replacement techniques @xcite . as in previous nnm studies @xcite , we take @xmath2210 , @xmath23 kcal / mol / @xmath24 and fix @xmath25 so that the low - frequency part of the linear spectrum matches actual protein frequencies , as calculated using realistic force fields @xcite . when @xmath26 a.m.u . ( the average amino - acid residue mass ) , this gives @xmath27 kcal / mol / @xmath28 . for each site in a given structure , the maximum - stiffness ( ms ) direction is computed through the sequential maximum strain algorithm @xcite . following an initial kinetic - energy impulse ( kick ) at a specific site along the local ms unit vector , a 2-ns microcanonical simulation is performed . after a 1-ns transient period during which a part of the excitation energy flows into the system , the velocity - covariance matrix is computed . its first eigenvector provides the pattern of correlated site velocities involved in the dominant ( most energetic ) nonlinear mode ( the db ) . accordingly , a transfer is recorded to the site at which the first principal mode ( pm1 ) is found localized . projecting the system trajectory on pm1 yields fair estimates of the db frequency and average energy @xcite . the localization index @xmath29 of a db centered at site @xmath30 is obtained from the weight of the latter in the normalized displacement pattern of the db , namely @xmath31 ^ 2 $ ] , where @xmath32 are the components at site @xmath30 of pm1 . ishikura , t , yamato , t ( 2006 ) energy transfer pathways relevant for long - range intramolecular signaling of photosensory protein revealed by microscopic energy conductivity analysis . _ chemical physics letters _ 432:533537 . dauxois , t , litvak - hinenzon , a , mackay , r , spanoudaki , a , eds ( 2004 ) _ energy localisation and transfer in crystals , biomolecules and josephson arrays . advanced series in nonlinear dynamics , vol.22 _ ( world scientific , singapore ) . perahia , d , mouawad , l ( 1995 ) computation of low - frequency normal modes in macromolecules : improvements to the method of diagonalization in a mixed basis and application to hemoglobin . _ 19:241246 . tama , f , miyashita , o , brooks iii , cl ( 2004 ) flexible multi - scale fitting of atomic structures into low - resolution electron density maps with elastic network normal mode analysis . _ 337:985999 .
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proteins are large and complex molecular machines . in order to perform their function ,
most of them need energy , _
e.g. _ either in the form of a photon , like in the case of the visual pigment rhodopsin , or through the breaking of a chemical bond , as in the presence of adenosine triphosphate ( atp ) .
such energy , in turn , has to be transmitted to specific locations , often several tens of away from where it is initially released .
here we show , within the framework of a coarse - grained nonlinear network model , that energy in a protein can jump from site to site with high yields , covering in many instances remarkably large distances . following single - site excitations ,
few specific sites are targeted , systematically within the stiffest regions .
such energy transfers mark the spontaneous formation of a localized mode of nonlinear origin at the destination site , which acts as an efficient energy - accumulating centre .
interestingly , yields are found to be optimum for excitation energies in the range of biologically relevant ones . ,
0 mm protein dynamics is encoded in their structures and is often critical for their function @xcite . since the early eighties
, it is well known that vibrational non - harmonicity has to be accounted for to understand intra - structure energy redistribution @xcite . among nonlinear effects ,
localized modes were suggested to play a key role @xcite , including topological excitations , such as solitons @xcite as well as discrete breathers ( db ) @xcite .
the latter , also known as intrinsic localized modes ( ilms ) , are spatially localized , time - periodic vibrations found generically in many systems as a combined effect of nonlinearity and spatial discreteness @xcite .
notably , dbs are able to _ harvest _ from the background and pin down for long times amounts of energy much larger than @xmath0 . indeed , their ability to pump energy from neighboring sites is a distinctive signature of db self - excitation @xcite , _
e.g. _ observed as a consequence of surface cooling @xcite or due to modulational instability of band - edge waves in nonlinear lattices @xcite .
therefore , provided such phenomena are compatible with cellular constraints , it is tempting to speculate that evolution has found a way to put such long - lived modes at work for lowering energy barriers associated with chemical reactions , _
e.g. _ for boosting enzyme efficiency during catalytic processes @xcite . optimum kick direction for exciting discrete breathers in dimeric citrate synthase ( pdb code 1ixe ) .
percentage of the system energy found in a nonlinear mode as a function of the direction of the initial kick given to ser 213a , the nm site of the band - edge mode .
the latter is measured by the angle @xmath1 between the kick direction and the ms unit vector . in all simulations ,
the ( kinetic ) energy of the kick is 55 kcal / mole and its direction is chosen at random , except when the maximum strain ( ms ) direction is picked instead ( black diamond at @xmath2 ) .
filled circles : ser 213a is found to be the most energetic site during the analysis timespan .
stars : it is another one . in one instance , while the kick was given in a direction close to the ms direction ( @xmath3 ) , the db jumped on a neighboring site ( namely , thr 208a ) . ]
-5 mm
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You are an expert at summarizing long articles. Proceed to summarize the following text:
for the past twenty years , much work has been done to understand the spin structure of the nucleons . there has been progress in determining the contribution of the lightest quarks to the spin , but there is still uncertain knowledge about the gluon contribution . transversity studies have contributed additional insight about quark dynamics , but little is known about the the orbital angular momentum of the constituents.@xcite this paper will summarize a project that provides a method of gaining insight into the nature of the orbital angular momentum of the nucleon constituents . recent experiments @xcite have significantly lowered the measurement errors of the quark longitudinal spin contribution ( @xmath2 ) to the proton . the compass collaboration analysis quotes a result @xmath3 while the hermes collaboration analysis quotes a result @xmath4 these groups and others @xcite have been working on providing a significant measure of the proton s spin weighted gluon density , @xmath5 where @xmath6 is the bjorken scaling variable and @xmath7 is the @xmath8 evolution variable . the combination of these measurements is summarized in terms of the @xmath9 sum rule : @xmath10 here @xmath11 and @xmath12 are the projections of the spin carried by all quarks and the gluons on the @xmath13-axis , respectively . also @xmath1 is the net @xmath13-component of the orbital angular momentum of the constituents . we do not attempt to separate the flavor components of @xmath1 within the sum rule . experimental groups at the compass , hermes and rhic collaborations are measuring both the gluon polarization and the asymmetry , @xmath14 to determine the gluon polarization @xcite . since there is no suitable theoretical model for @xmath15 , we have devised a way to model the asymmetry , @xmath16 to gain insight into the structure of @xmath15 . this , coupled with the @xmath9 sum rule can then shed light on the nature of the orbital angular momentum of the constituents , @xmath1 . to model @xmath16 , we write the polarized gluon asymmetry using the decomposition @xmath17 where @xmath18 \label{a0def}\ ] ] is a scale invariant calculable reference form @xcite . here @xmath19 represents the difference between the calculated and gauge - invariant asymmetry . since @xmath15 is unknown , a useful form is to write equation ( [ adef ] ) as @xmath20 although the quantity @xmath21 is not a physical parameter , it allows the theoretical development of the calculable quantity , @xmath22 . once an asymmetry is generated from equations ( [ a0def ] ) and ( [ dg ] ) , the gauge - invariant quantity @xmath16 can be compared to data . thus , each ansatz for @xmath21 gives a corresponding form for @xmath15 and a parametrization for @xmath1 . these can be compared to existing data to provide a range of suitable models for these contributions . with the definition for the asymmetry in equation ( [ a0def ] ) , the dglap equations can then be used to evaluate the evolution terms on the right side . @xmath23 . \label{a00}\ ] ] the polarized gluon distribution in the numerator of equation ( [ a00 ] ) is replaced by @xmath24 . for certain unpolarized distributions , there are points at which the denominator vanishes . to avoid this , we write equation ( [ a00 ] ) as : @xmath25 \\ \nonumber & = & a_0\cdot{{\partial{g}}\over { \partial{t } } } \\ & = & ( 2/\beta_0 ) a_0 \bigl[p_{gq}^{lo}\otimes q+p_{gg}^{lo}\otimes g\bigr ] . \nonumber \label{adge}\end{aligned}\ ] ] the nlo form is essentially the same as equation ( 9 ) with the splitting functions @xmath26 replaced with their nlo counterparts . the quark and gluon unpolarized distributions are cteq5 and the polarized quark distributions are a modified ggr set . @xcite there are constraints on @xmath27 that must be imposed to satisfy the physical behavior of the gluon asymmetry , @xmath28 . these are : * positivity : @xmath29 for all x , and * endpoint values : @xmath30 and @xmath31 note that the constraint of @xmath32 is built in to satisfy the assumption that the large @xmath6 parton distributions are dominated by the valence up quarks in the proton . the convolutions are dominated by the quark terms , which force the asymmetry to unity as @xmath33 . to investigate the possible asymmetry models , we use a parameterization for @xmath22 in the form @xmath34 which automatically satisfies the constraints that @xmath30 and @xmath31 . once a parametrization for @xmath21 is chosen , equation ( 9 ) is used to determine the parameters in equation ( [ a0form ] ) . the models for @xmath21 that led to asymmetries that satisfied these constraints were all in the range @xmath35 , with positive and negative values included . larger values of @xmath36 violate one or both of the constraints . a representative sample of models that satisfy the constraints are listed in table 1 . note that the integrals for @xmath15 are all positive , ranging from about 0.01 to 0.42 . the models that gave negative values for these integrals did not agree with the existing asymmetry data , reported at this workshop to be : * @xmath37 at @xmath38 from compass , @xmath39 gev@xmath40 * @xmath41 at @xmath42 from compass , @xmath43 gev@xmath40 * @xmath44 at @xmath45 from hermes , factorization method * @xmath46 at @xmath47 from hermes , approximate method . the models in table 1 that are within one @xmath48 of the preliminary data stated above are in the third , fourth and sixth rows , respectively . plots of the full asymmetry are shown in figure 1 . none of the models in table 1 are ruled out by the data since they fall within two @xmath48 of the data for our values of @xmath39 gev@xmath40 . all of these models except for the fourth row in table 1 ( impulses in figure 1 ) generate total asymmetries @xmath49 that are close to @xmath50 . ironically , early assumptions of the polarized gluon assumed this functional form as a naive estimate to the asymmetry . next - to - leading order corrections to these asymmetries tend to bring them less positive , but with the same general shape . a full set of viable asymmetries will be presented in an upcoming paper . @xcite using the data on @xmath2 in section 1 , the relation between @xmath51 and @xmath52 can be written as : @xmath53 the three models of the asymmetry that agree most closely with existing data give values of @xmath15 in the approximate range of @xmath54 . thus , the existing data with equation [ dglz ] imply the approximate relation @xmath55 . thus , the contribution of the orbital motion of the constituents to the proton spin may be comparable to the total quark contribution . a recent lattice calculation of the contribution of the quark orbital motion to the proton spin ( @xmath56 ) is consistent with zero . @xcite thus , the gluonic orbital motion appears to provide the majority contribution to @xmath1 in the @xmath9 sum rule . it is clear that future measurements of @xmath15 and @xmath57 must be made in a wider kinematic range of @xmath6 and @xmath8 with improved precision to better specify the appropriate model of the asymmetry and to extract the @xmath6 and @xmath8 dependence of the orbital angular momentum of the constituents . 99 x. ji , aip conf . proc . * 915 * : 16 , 2007 , x. ji , phys . rev . lett . * 78 * , 610 ( 1997 ) . see talk by y. bedfer , this proceedings , i. savin , aip conf . * 915 * : 399 , 2007 , and s. platchkov , nucl . phys . * a790 * : 58 , 2007 . see talk by s. belostotski , this proceedings and l. de nardo , aip conf . * 915 * : 404 , 2007 , see talk by g. bunce , this proceedings and r. fatemi , arxiv 0710.3207 ( hep - ex ) . g. ramsey , proceedings of the 16th international spin physics symposium , trieste , italy , c2005 , world scientific press , p. 310 . gordon , m. goshtaspbour and g.p . ramsey , phys . rev . * d58 * , 094017 ( 1997 ) . y. binder , g. ramsey and d. sivers , in preparation . j. w. negele , hep - lat/0509100 .
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determination of the orbital angular momentum of the proton is a difficult but important part of understanding fundamental structure .
insight can be gained from suitable models of the gluon asymmetry applied to the @xmath0 sum rule .
we have constrained the models of the asymmetry to gain possible scenarios for the angular momentum of the protons constituents .
results and phenomenology for determining @xmath1 are presented .
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You are an expert at summarizing long articles. Proceed to summarize the following text:
the environments of quasars provide important clues to the physical processes of their formation and also yield important information about the relations between the distribution of quasars and the large - scale structure of the universe . for more than three decades , we have known that quasars are associated with enhancements in the spatial distributions of galaxies ( @xcite ) . studies of the environments of quasars in the nearby universe ( @xmath2 ) have shown that quasars reside in environments ranging from small to moderate groups of galaxies rather than in rich clusters ( e.g. @xcite ; @xcite ; @xcite ) . in order to interpret the observational results of the environments of quasars at low redshifts and predict the environments of quasars at high redshifts , a physical model of quasar formation based on cosmological context is required . it has become widely accepted that quasars are fueled by accretion of gas onto supermassive black holes ( smbhs ) in the nuclei of host galaxies since @xcite proposed this idea on quasars . recent observations of galactic centers suggest that a lot of nearby galaxies have central black holes and their estimated masses correlate with the luminosities of spheroids of their host galaxies ( e.g. @xcite ; @xcite ; @xcite ) . the connection between smbhs and their host spheroids suggests that the formation of smbhs physically links the formation of the spheroids which harbor the smbhs . thus , this implies that the formation of quasars is closely related to the formation of galaxies , especially of spheroids . therefore , in order to study the formation and evolution of quasars , it is necessary to construct a unified model which includes both galaxy formation and quasar formation . recently , some authors have tried to construct galaxy formation models on the basis of the theory of hierarchical structure formation in cold dark matter ( cdm ) universe . these efforts are referred to as semi - analytic models ( sams ) of galaxy formation . in the cdm universe , dark matter halos cluster gravitationally and merge together in a manner that depends on the adopted power spectrum of initial density fluctuations . in each of the merged dark halos , radiative gas cooling , star formation , and supernova feedback occur . the cooled dense gas and stars constitute _ galaxies_. these galaxies sometimes merge together in a common dark halo and more massive galaxies form . in sams , the merger trees of dark matter halos are constructed using a monte - carlo algorithm and simple models are adopted to describe the above gas processes . stellar population synthesis models are used to calculate the luminosities and colors of model galaxies . it is therefore straightforward to understand how galaxies form and evolve within the context of this model . sams successfully have reproduced a variety of observed features of local galaxies such as their luminosity functions , color distribution , and so on ( e.g. @xcite ; @xcite , ; @xcite ; @xcite , ) . in these models , it is assumed that disk stars are formed by cooling of gas in the halo . if two galaxies of comparable mass merge , it is assumed that starbursts occur and form the spheroidal component in the center of the galaxy . @xmath3-body simulations have shown that a merger hypothesis for the origin of spheroids can explain their detailed internal structure ( e.g. @xcite ; @xcite , ; @xcite ) . kauffmann and charlot ( ) have demonstrated that the merger scenario for the formation of elliptical galaxies is consistent with the color - magnitude relation and its redshift evolution ( see also @xcite ) . on the other hand , hydrodynamical simulations have shown that a merger of galaxies drives gas to fall rapidly to the center of a merged system and to fuel nuclear starburst ( @xcite ; @xcite , ; @xcite ) . moreover , observed images of quasar hosts show that many quasars reside in interacting systems or elliptical galaxies ( @xcite ) . therefore , it has often been thought that the major merger of galaxies would be a possible mechanism for quasar and spheroid formation . so far , a lot of studies on quasar evolution based on the hierarchical clustering scenario have been carried out with the assumption that the formation of quasars is linked to the first collapse of dark matter halos with galactic mass and that these models can explain the decline of quasar number density at @xmath4 ( e.g. @xcite ; @xcite ) and properties of luminosity functions of quasars ( e.g. @xcite ; @xcite ; @xcite ) . however , if quasars are directly linked to spheroids of host galaxies rather than to dark matter halos , the approximation of a one - to - one relation between quasar hosts and dark matter halos would be very crude , especially at low redshift . therefore , it is necessary to construct a model related to spheroid formation and smbh formation directly . kauffmann and haehnelt ( ) introduced a unified model of the evolution of galaxies and quasars within the framework of sam ( see also @xcite ) . they assumed that smbhs are formed and fueled during major galaxy mergers and their model reproduces quantitatively the observed relation between spheroid luminosity and black hole mass in nearby galaxies , the strong evolution of the quasar population with redshift , and the relation between the luminosities of nearby quasars and those of their host galaxies . in this paper , we investigate properties of quasar environments , using a sam incorporated simple quasar evolution model . we assume that smbhs are formed and fueled during major galaxy mergers and the fueling process leads quasar activity . while this assumption is similar to the model of kauffmann and haehnelt ( ) , our galaxy formation model and the adopted model of fueling process are different from their model . here we focus on optical properties of quasars and attempt to consider the number of quasars per halo , effective bias parameter of quasars and the number of galaxies around quasars as characterizations of environments of quasars , because a ) these quantities provide a direct measure of bias in their distribution with respect to galaxies and b ) comparing results of the model with observations will enable us to constrain our quasar formation model . the paper is organized as follows : in [ model ] we briefly review our sam for galaxy formation ; in [ qsomodel ] we introduce the quasar formation model ; in [ env ] we calculate the galaxy number distribution function around quasars ; in [ disc ] we provide a summary and discussion . in this study , we use a low - density , spatially flat cold dark matter ( @xmath5cdm ) universe with the present density parameter @xmath6 , the cosmological constant @xmath7 , the hubble constant in units of @xmath8 @xmath9 and the present rms density fluctuation in spheres of @xmath10 radius @xmath11 . in this section we briefly describe our sam for the galaxy formation model , details of which are shown in @xcite . our present sam analysis obtains essentially the same results as the previous sam analyses , with minor differences in a number of details . first , we construct monte carlo realizations of merging histories of dark matter halos using the method of @xcite , which is based on the extended press - schechter formalism ( @xcite ; @xcite ; @xcite ; @xcite ) . we adopt the power spectrum for the specific cosmological model from @xcite . halos with circular velocity @xmath1240 km s@xmath13 are treated as diffuse accretion matter . the evolution of the baryonic component is followed until the output redshift coincides with the redshift interval of @xmath14 , corresponding to the dynamical time scale of halos which collapse at the redshift @xmath15 . note that @xcite recently pointed out that a much shorter timestep is required to correctly reproduce the mass function given by the press - schechter formalism . however , a serious problem exists only at small mass scales ( @xmath16 ) . thus we use the above prescription of timestep . if a dark matter halo has no progenitor halos , the mass fraction of the gas in the halo is given by @xmath17 , where @xmath18 is the baryonic density parameter constrained by primordial nucleosynthesis calculations ( e.g. @xcite ) . note that a recent measurement of the anisotropy of the cosmic microwave background by the boomerang project suggests a slight higher value , @xmath19 ( @xcite ) . @xcite have already investigated the effect of changing @xmath20 and showed that this mainly affects the value of the invisible stellar mass fraction such as brown dwarfs parameterized by @xmath21 ( see below ) . when a dark matter halo collapses , the gas in the halo is shock - heated to the virial temperature of the halo . we refer to this heated gas as the _ hot gas_. at the same time , the gas in dense regions of the halo cools due to efficient radiative cooling . we call this cooled gas the _ cold gas_. assuming a singular isothermal density distribution of the hot gas and using the metallicity - dependent cooling function by @xcite , we calculate the amount of cold gas which eventually falls onto a central galaxy in the halo . in order to avoid the formation of unphysically large galaxies , the above cooling process is applied only to halos with @xmath12400 km s@xmath13 . this handling would be needed because the simple isothermal distribution forms so - called `` monster galaxies '' due to the too efficient cooling at the center of halos . while this problem will probably solved by adopting another isothermal distribution with central core ( @xcite ) , we take the above approach for simplicity . stars are formed from the cold gas at a rate of @xmath22 , where @xmath23 is the mass of cold gas and @xmath24 is the time scale of star formation . we assume that @xmath24 is independent of @xmath15 , but dependent on @xmath25 as follows : @xmath26 the free parameters of @xmath27 and @xmath28 are fixed by matching the observed mass fraction of cold gas in neutral form in the disks of spiral galaxies . in our sam , stars with masses larger than @xmath29 explode as type ii supernovae ( sne ) and heat up the surrounding cold gas . this sn feedback reheats the cold gas to hot gas at a rate of @xmath30 , where @xmath31 is the efficiency of reheating . we assume that @xmath31 depends on @xmath25 as follows : @xmath32 the free parameters of @xmath33 and @xmath34 are determined by matching the local luminosity function of galaxies . with these @xmath35 and @xmath36 thus determined , we obtain the masses of hot gas , cold gas , and disk stars as a function of time during the evolution of galaxies . given the star formation rate as a function of time , the absolute luminosity and colors of individual galaxies are calculated using a population synthesis code by @xcite . the initial stellar mass function ( imf ) that we adopt is the power - law imf of salpeter form with lower and upper mass limits of @xmath37m@xmath38 and @xmath39m@xmath38 , respectively . since our knowledge of the lower mass limit is incomplete , there is the possibility that many brown dwarf - like objects are formed . therefore , following @xcite , we introduce a parameter defined as @xmath40 , where @xmath41 is the total mass of luminous stars with @xmath42 and @xmath43 is that of invisible brown dwarfs . to account for extinction by internal dust we adopt a simple model by @xcite in which the optical depth in @xmath44-band is related to the luminosity as @xmath45 . optical depths in other bands are calculated by using the galactic extinction curve , and the dust distribution in disks is assumed to be the slab model considered by @xcite . when several progenitor halos have merged , the newly formed larger halo should contain at least two or more galaxies which had originally resided in the individual progenitor halos . we identify the central galaxy in the new common halo with the central galaxy contained in the most massive of the progenitor halos . other galaxies are regarded as satellite galaxies . these satellites merge by either dynamical friction or random collision . the time scale of merging by dynamical friction is given by @xmath46 where @xmath47 and @xmath25 are the radius and the circular velocity of the new common halo , respectively , @xmath48 is the coulomb logarithm , and @xmath49 is the mass of the satellite galaxy including its dark matter halo @xcite . when the time passed after a galaxy becomes a satellite exceeds @xmath50 , a satellite galaxy infalls onto the central galaxy . on the other hand , the mean free time scale of random collision is given by @xmath51 where @xmath3 is the number of satellite galaxies , @xmath52 is their radius , and @xmath53 and @xmath54 are the 1d velocity dispersions of the common halo and satellite galaxies , respectively @xcite . with a probability of @xmath55 , where @xmath56 is the timestep corresponding to the redshift interval @xmath57 , a satellite galaxy merges with another randomly picked satellite . consider the case that two galaxies of masses @xmath58 and @xmath59 merge together . if the mass ratio @xmath60 is larger than a certain critical value of @xmath61 , we assume that a starburst occurs and all the cold gas turns into stars and hot gas , which fills the halo , and all of the stars populate the bulge of a new galaxy . on the other hand , if @xmath62 , no starburst occurs and a smaller galaxy is simply absorbed into the disk of a larger galaxy . these processes are repeated until the output redshift . we classify galaxies into different morphological types according to the @xmath44-band bulge - to - disk luminosity ratio @xmath63 . in this paper , galaxies with @xmath64 , and @xmath65 are classified as ellipticals / s0s and spirals , respectively . this method of type classification well reproduces the observed type mix . the above procedure is a standard one in the sam for galaxy formation . model parameters are determined by comparison with observations of the local universe . in this study , we use the astrophysical parameters determined by @xcite from local observations such as luminosity functions , and galaxy number counts in the hubble deep field . the adopted parameters of this model are tabulated in table [ tab : astro ] . in figure [ fig : gal - lum ] we plot the results of local luminosity functions of galaxies represented by solid lines . note that the resultant luminosity functions hardly change if the smbh formation model is included ( dashed lines ; see the next section ) . symbols with errorbars indicate observational results from the @xmath44-band redshift surveys ( apm , @xcite ; 2df , @xcite ) and from the @xmath66-band redshift surveys ( @xcite ; 2mass , @xcite ) . as can be seen , the results of our model using these parameters are generally consistent with the observations , at least with the apm result . [ tab : astro ] .model parameters [ cols="^,^,^,^,^,^,^,^,^,^,^,^,^ " , ] ( 120mm,80mm)fig1.eps in this section , we introduce a quasar formation and evolution model into our sam . as mentioned earlier , the masses of smbhs have tight correlation with the spheroid masses of their host galaxies ( e.g. @xcite ; @xcite ; @xcite ) and the hosts of quasars found in the local universe are giant elliptical galaxies or galaxies displaying evidence of major mergers of galaxies ( @xcite ) . moreover , in sams for galaxy formation , it is assumed that a galaxy - galaxy major merger leads to the formation of a spheroid . therefore , we assume that smbhs grow by merging and are fueled by accreted cold gas during major mergers of galaxies . when host galaxies merge , pre - existing smbhs sink to the center of the new merged galaxy owing to dynamical friction and finally coalesce . the timescale for this process is unknown , but for the sake of simplicity we assume that smbhs merge instantaneously . gas - dynamical simulations have demonstrated that the major merger of galaxies can drive substantial gaseous inflows and trigger starburst activity ( @xcite ; @xcite , ; @xcite ) . thus , we assume that during major merger , some fraction of the cold gas that is proportional to the total mass of stars newly formed at starburst is accreted onto the newly formed smbh . under this assumption , the mass of cold gas accreted on a smbh is given by @xmath67 where @xmath68 is a constant and @xmath69 is the total mass of stars formed at starburst . @xmath69 is derived in the appendix . the free parameter of @xmath68 is fixed by matching the observed relation between a spheroid luminosity and a black hole mass found by @xcite and we find that the favorable value of @xmath68 is nearly @xmath70 . in figure [ fig : bulge - bh ] we show scatterplots ( open circles ) of the absolute @xmath71-band magnitudes of spheroids versus masses of smbhs of model for @xmath72 . the thick solid line is the observational relation and the dashed lines are the @xmath73 scatter in the observations obtained by @xcite . for @xmath74 , changing @xmath68 shifts the black hole mass almost linearly . the obtained gas fraction ( @xmath75 ) is so small that the inclusion of smbh formation does not change the properties of galaxies in the local universe . in figure [ fig : gal - lum ] , the dashed lines show the results of the model with the smbh formation . this result differs negligibly from the result of the model without smbh formation . therefore , we use the same astrophysical parameters tabulated in table [ tab : astro ] regardless of inclusion of the smbh formation model . figure [ fig : bh - mass ] ( a ) shows black hole mass functions in our model at a series of redshifts . this indicates that the number density of the most massive black holes increases monotonically with time in the scenario where smbhs grow by accretion of gas and by merging . in figure [ fig : bh - mass ] ( a ) , we superpose the present black hole mass function obtained by @xcite . they derived this black hole mass function from the observed radio luminosity function of nearby radio - quiet galaxies and the empirical correlation between radio luminosities from the nuclei of radio - quiet galaxies and the mass of their black holes . our model result is consistent with their mass function . for comparison , we also plot the mass functions of bulge and disk for all galaxies in figure [ fig : bh - mass ] ( b ) and ( c ) , respectively . the steep slopes at low masses of mass functions of black hole and bulge are mainly due to random collisions between satellite galaxies in this model . to obtain the observed linear relation between a spheroid luminosity and a black hole mass , kauffmann and haehnelt ( ) adopted model of fueling process in which the ratio of accreted mass to total available cold gas mass scales with halo circular velocity in the same way as the mass of stars formed per unit mass of cooling gas . while their approach is similar to ours , their star formation and feedback models are different from ours and they do not consider random collisions . therefore , their resultant model description is slightly differ from ours in equation ( [ eq : bhaccret ] ) . ( 70mm,70mm)fig2.eps ( 120mm,80mm)fig3.eps next , we consider the light curve of quasars . we assume that a fixed fraction of the rest mass energy of the accreted gas is radiated in the @xmath44-band and the quasar life timescale @xmath76 scales with the dynamical time scale @xmath77 of the host galaxy where @xmath78 . here we adopt the @xmath44-band luminosity of a quasar at time @xmath79 after the major merger as follows ; @xmath80 the peak luminosity @xmath81 is given by @xmath82 where @xmath83 is the radiative efficiency in @xmath44-band , @xmath84 is the quasar life timescale and @xmath85 is the speed of light . in order to determine the parameter @xmath83 and the present quasar life timescale @xmath86 , we have chosen them to match our model luminosity function with the observed abundance of bright quasars at @xmath87 . we obtain @xmath88 and @xmath89 . the resulting luminosity functions at four different redshifts are shown in figure [ fig : qso - lum ] . we superpose the luminosity functions derived from the 2df 10k catalogue ( @xcite ) for a cosmology with @xmath90 and @xmath9 , which is analyzed and kindly provided by t. t. takeuchi . he used the method of @xcite for the estimation of the luminosity functions . in order to reanalyze the error with greater accuracy , they applied bootstrap resampling according to the method of @xcite . absolute @xmath44-band magnitudes were derived for the quasars using the @xmath91-corrections derived by @xcite . our model reproduces reasonably well the evolution of observed luminosity functions . thus , in the next section , we use these model parameters in order to investigate the environments of quasars . for comparison , we also plot the result of model with @xmath88 and @xmath92 in figure [ fig : qso - lum ] ( dot - dashed lines ) . in this case , the abundance of luminous quasars decreases . to prolong a quasar life timescale affects the quasar luminosity function due to the following two factors : a decrease in the peak luminosity @xmath93 ( eq.[[eq : qso - peak ] ] ) and an increase in the exponential factor @xmath94 in equation ( [ eq : qso - lc ] ) . for the majority of bright quasars , the elapsed time @xmath79 since the major merger is much smaller than the quasar life timescale @xmath95 , @xmath96 . therefore , the former factor dominates the latter and the number of luminous quasars decreases . thus , a long quasar life timescale results in a very steep quasar luminosity function . note that if we change the radiative efficiency @xmath83 , the quasar luminosities simply scale by a constant factor in our model . thus , changing @xmath83 shifts the luminosity function horizontally . in this section , we investigate the environments of quasars using our model . we consider the halo mass dependence of the mean number of quasars per halo and the probability distribution of the number of galaxies around quasars as characterizations of the environments of quasars . this is because the former is one of measures of the relation between quasars and dark matter distributions and the latter reflects the relationship between galaxies and quasars . in figure [ fig : number - gal ] , we plot @xmath97 and @xmath98 that denote the mean number of galaxies and quasars per halo with mass @xmath99 , respectively , at ( a ) @xmath100 and ( b ) @xmath101 . we select galaxies with @xmath102 and quasars with @xmath103 , where @xmath104 is absolute @xmath44-band magnitude . it should be noted that changing the magnitude of selection criteria for galaxies and quasars would alter these results , but qualitative features are not altered . as is seen in figure [ fig : number - gal ] , there are more galaxies and quasars at high @xmath15 . at higher redshift , halos have more cold gas available to form stars and to fuel smbhs because there has been relatively little time for star formation to deplete the cold gas at these redshifts . thus , the number of luminous galaxies grows . furthermore , at higher redshift , both timescales of the dynamical friction and the random collisions are shorter because the mass density of a halo is higher . therefore , the galaxy merging rate increases . consequently , the number of quasars also grows . moreover , the decrease in the quasar life timescale @xmath105 with redshift also contributes to the increase in the number of quasars because quasars become brighter as a result of decrease in @xmath105 ( eq . [ [ eq : qso - peak ] ] ) . from figure [ fig : number - gal ] , we find that the dependence of @xmath106 on halo mass @xmath99 is different from the dependence of @xmath97 . furthermore , figure [ fig : qg - ratio ] shows that the ratio of @xmath107 to @xmath108 varies with redshift and halo mass . @xcite used a combination of cosmological @xmath3-body simulation and semi - analytic modeling of galaxy formation and showed that the galaxy spatial distribution is sensitive to the efficiency with which galaxies form in halos with different mass . @xcite also obtained the same conclusion using an analytic model of galaxy clustering . these results are applicable to the quasar spatial distribution . therefore , our result indicates that the clustering properties of galaxies are not the same as those of quasars and that the bias in the spatial distribution of galaxies relative to that of dark matter is not the same as the bias in the spatial distribution of quasars . assumed that biases are independent of scale , we can calculate effective biases using the method of @xcite as follows ; @xmath109 where @xmath110 is the bias parameter for dark matter halos of mass @xmath99 at @xmath15 , @xmath111 denotes the mean number of objects ( galaxies or quasars ) in a halo of mass @xmath99 at @xmath15 that satisfy the selection criteria and @xmath112 is the dark halo mass function at @xmath15 . our sam adopts the press - schechter mass function which is given by @xmath113 dm , \label{eq : psmass}\ ] ] where @xmath114 is the present mean density of the universe , @xmath115 is the rms linear density fluctuation on the scale @xmath99 at @xmath116 and @xmath117 . @xmath118 is the linear growth factor , normalized to unity at the present day and @xmath119 is the linear critical density contrast at the collapse epoch . here , we use an approximate formula of @xmath120 for spatially flat cosmological model ( @xcite ) . the bias parameter for dark matter halos is given by @xcite ; @xmath121 \right\ } \left [ \frac{\sigma^{4}(m)}{2 \delta^{4}_{c}(z ) } + 1 \right]^{(0.06 - 0.02n_{\rm eff } ) } , \label{eq : bias}\ ] ] where @xmath122 is the effective spectral index of the power spectrum , @xmath123 , at the wavenumber defined by the lagrangian radius of the dark matter halo , @xmath124 and @xmath125 . figure [ fig : bias ] shows the evolution of effective bias for galaxies with @xmath102 and quasars with @xmath103 . as is seen in figure [ fig : bias ] , quasars are higher biased tracer than galaxies . furthermore , the evolution of quasar bias is different from that of galaxy bias . this reflects the difference in th dependence on halo mass @xmath99 and redshift of @xmath126 and @xmath127 . note that these effective biases are valid for large scale where objects ( galaxies or quasars ) which contribute two - point correlation function populate different halos . ( 120mm,80mm)fig5.eps ( 70mm,70mm)fig6.eps ( 70mm,70mm)fig7.eps next , we formulate the conditional probability that a halo with @xmath128 quasars has @xmath129 galaxies . the number density of the halos which contains @xmath129 galaxies and @xmath130 quasars at @xmath15 is obtained from the following expression : @xmath131 where @xmath132 denotes the number of the halos with mass @xmath99 which contains @xmath133 galaxies and @xmath134 quasars at @xmath15 and @xmath112 is the dark halo mass function at @xmath15 . the number density of the halos which contain @xmath130 quasars at @xmath15 is obtained from the following expression : @xmath135 where @xmath136 denotes the number of the halos with mass @xmath99 which contain @xmath137 quasars at @xmath15 . from equation ( [ eq : ng - q ] ) and ( [ eq : nq ] ) , the conditional probability that the halo with @xmath130 quasars has @xmath138 galaxies at @xmath15 is given by @xmath139 as is seen in the above formulation , given @xmath140 and @xmath141 from the quasar formation model , one can calculate the probability distribution for the number of galaxies around quasars . figure [ fig : gnd ] shows these galaxy number distribution functions around quasars estimated by our model . the results are shown for quasars brighter than @xmath142 and for galaxies brighter than @xmath143 . note that at @xmath144 and @xmath100 @xmath145 and @xmath146 for all @xmath129 ( fig . [ fig : gnd](a ) and ( b ) ) and that at @xmath147 @xmath146 for all @xmath129 ( fig . [ fig : gnd](c ) ) . at lower redshift , a halo has at most one quasar . fig [ fig : gnd](a ) and ( b ) show that the halo which has one quasar contains several galaxies by high probability . these results indicate that most quasars tend to reside in groups of galaxies at @xmath0 and is consistent with the observation at @xmath148 ( e.g. @xcite ; @xcite ; @xcite ) . on the other hand , at higher redshift , the numbers of galaxies in the halo with one or two quasars is from several to dozens ( fig [ fig : gnd](c ) and ( d ) ) . these results indicate that quasars locate in ranging from small groups of galaxies to clusters of galaxies . thus at @xmath1 quasars seem to reside in more varied environments than at lower redshift . kauffmann and haehnelt ( ) used a combination of cosmological @xmath3-body simulation and semi - analytic modeling of galaxy and quasar formation , and showed that the ratio of the amplitude of the quasar - galaxy cross correlation function to that of the galaxy autocorrelation function decrease with redshift . this indicates that the difference between galaxy and quasar distribution becomes smaller at higher redshift . thus , our results obtained by @xmath149 is not in conflict with their results . we have constructed a unified semi - analytic model for galaxy and quasar formation and have predicted the mean number of quasars per halo with mass @xmath99 , @xmath107 , the effective bias parameter of quasars @xmath150 and probability distribution of the number of galaxies around quasars , @xmath151 , as characterizations of the environments of quasars . these quantities reflect the processes of quasar formation such as the amount of cold gas available for fueling , the galaxy merger rate and the quasar life timescale . therefore , by comparing these predictions with observations , one will be able to constrain quasar formation models . our model can reproduce not only general form of the galaxy luminosity functions in the local universe but also the observed relation of the smbh mass to spheroid luminosity , and the quasar luminosity functions at different redshifts ( fig.[fig : bulge - bh ] and fig.[fig : qso - lum ] ) . using this model , we have shown @xmath107 and @xmath151 . the ratio of @xmath107 to @xmath97 varies with halo mass in our model ( fig[fig : number - gal ] ) . these results of our model suggest that the clustering of galaxies is not the same as the clustering of quasars and the effective bias parameter of quasars and its evolution are different from these of galaxies ( fig.[fig : bias ] ) . furthermore , we predict the galaxy number distribution function around quasars , @xmath151 ( fig[fig : gnd ] ) . at lower redshifts ( @xmath0 ) , most halos which have quasars have at most several galaxies . this indicates that most quasars reside in groups of galaxies . on the other hand , at higher redshift ( @xmath152 ) , the number of galaxies in the halo with quasars is from several to dozens ; quasars reside in ranging from small groups of galaxies to clusters of galaxies . these results show that most quasars at higher redshift reside in more varied environments than at lower redshift . this model prediction is checkable by statistics of galaxies around quasars which will be obtained in future . it is still controversial whether the environments of quasars depend on their optical and radio luminosities . some authors have claimed that radio - loud quasars were located in richer environments than radio - quiet quasars at at @xmath153 ( e.g. @xcite ; @xcite ; @xcite ; @xcite ) . however , other people obtained a different result . for example , @xcite observed the galaxy environment of radio - loud quasars and radio - quiet quasars and concluded that there is no significant difference in the richness . recent studies support this conclusion ( e.g. @xcite ) . the discrepancies between different studies may be caused partly by too small quasar samples and by differences in sample selection of quasars . this situation will soon improve with the availability of a new generation of very large quasar surveys such as the 2df quasar redshift survey ( @xcite ) and the sloan digital sky survey ( @xcite ) . although we do not deal with radio properties of quasars in this paper , our investigation of quasar environments will also provide a clue for understanding the radio character of quasar environments . the mean number of quasars per halo , @xmath107 , and probability distribution of the number of galaxies around quasars , @xmath151 , used in this study can provide some useful features of the quasar environments . furthermore , the spatial galaxy - quasar correlation function is used in order to quantify the galaxy environments around a quasar . therefore , for the further investigation of environments and clustering of quasars and in order to constrain the quasar formation model , it is also necessary to predict spatial distribution of galaxies and quasars . we will show the results using the combination of cosmological @xmath3-body simulation and sam for formation of galaxy and quasar in the near future . we would like to thank t. t. takeuchi for providing us with the reanalyzed data of the quasar luminosity functions derived from the 2df 10k catalogue . we are also grateful to k. okoshi , h. yahagi and s. yoshioka for useful comments and discussions . we also thank to the anonymous referee for a thorough reading of the manuscript and for his valuable suggestions and comments , which improved our paper very much . numerical computations in this work were partly carried out at the astronomical data analysis center of the national astronomical observatory , japan . this work has been supported in part by the grant - in - aid for the scientific research funds ( 13640249 ) of the ministry of education , culture , sports , science and technology of japan . in this appendix , we summarize our model of star formation and gas evolution . we use a simple instantaneous recycling approximation of model star formation , feedback and chemical enrichment . the following difference equations describe the evolution of the mass of cold gas @xmath23 , hot gas @xmath154 , and long lived stars @xmath155 at each time step . @xmath156 where @xmath157 is star formation rate , @xmath158 is the gas fraction returned by evolved stars , and @xmath31 is the efficiency of reheating . in this paper , @xmath159 . the solutions of these equations are the following : @xmath160 , \label{eq : coldsol } \\ m_{\rm hot } & = & m_{\rm hot}^{0 } + \beta \delta m _ { * } , \label{eq : hotsol } \\ m_{\rm star } & = & m_{\rm star}^{0 } + ( 1-r)\delta m _ { * } , \label{eq : starsol } \end{aligned}\ ] ] where @xmath161 and @xmath162 are the masses of cold gas , hot gas and long - lived stars from the previous time step , @xmath79 is the time sine the start of the time step , and @xmath163 is the mass of total formed stars . when a starburst occurs , stars are formed in a very short timescale . thus , the starburst corresponds to @xmath164 in the above solutions . in this case , the changes of masses are given by @xmath165 and the total star mass formed at starburst becomes @xmath166 from equation ( [ eq : totstar ] ) , we can obtain the mass of accreted cold gas onto a black hole ( eq.[[eq : bhaccret ] ] ) .
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we investigate the environments of quasars such as number distribution of galaxies using a semi - analytic model which includes both galaxy and quasar formations based on the hierarchical clustering scenario .
we assume that a supermassive black hole is fueled by accretion of cold gas and that it is a source of quasar activity during a major merger of the quasar host galaxy with another galaxy .
this major merger causes spheroid formation of the host galaxy .
our model can reproduce not only general form of the galaxy luminosity functions in the local universe but also the observed relation between a supermassive black hole mass and a spheroid luminosity , the present black hole mass function and the quasar luminosity functions at different redshifts . using this model ,
we predict the mean number of quasars per halo , bias parameter of quasars and the probability distribution of the number of galaxies around quasars . in our model ,
analysis of the mean number of quasars per halo shows that the spatial distribution of galaxies is different from that of quasars .
furthermore , we found from calculation of the probability distribution of galaxy numbers that at @xmath0 , most quasars are likely to reside in galaxy groups . on the other hand , at @xmath1 most quasars seem to reside in more varied environments than at a lower redshift ; quasars reside in environments ranging from small groups of galaxies to clusters of galaxies .
comparing these predictions with observations in future will enable us to constrain our quasar formation model .
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gamma - ray bursts ( grbs ) are instantaneously the most luminous objects in the universe , produced by the deceleration of ultra - relativistic outflow ( lorentz factors @xmath14 ) . the core - collapse of massive stars are the progenitor of long grbs , and the merger of binary compact stellar objects such as neutron stars ( ns ) and black holes ( bh ) are the possible progenitor of short grbs ( woosley & bloom 2006 , nakar 2007 , berger 2014 ) . in both cases accretion onto a compact object is likely to power the relativistic outflow and the same physical processes are involved . the outflow energy is first dissipated by internal shocks ( or another form of internal dissipation ) which produces the prompt @xmath15-rays . later the interaction of the outflow with the ambient medium produces an external shock which expands and produces the subsequent afterglow ( e.g. piran 2004 ; zhang & mszros 2004 ) . relativistic motion is an essential ingredient in the grb model although the exact outflow formation process is not known . understanding the nature of the outflow , especially the acceleration , collimation , and energy content is a major focus of international research efforts in the context of grb and other astrophysical jets . grb outflows are conventionally assumed to be a baryonic jet ( paczynski 1986 ; shemi & piran 1990 ) , although polarization measurements imply that magnetic fields play a role in the jet acceleration ( e.g. steele et al . 2009 ; mundell et al . 2013 ; gtz et al . 2009 ; yonetoku et al . relativistic outflows and possibly magnetic acceleration are features that grbs , active galactic nuclei ( agn ) , and microquasars have in common . stellar tidal disruption by a massive bh is also likely to produce a relativistic jet ( bloom et al . 2011 ; burrows et al . 2011 , zauderer et al . 2011 , levan et al . 2011 , cenko et al . 2012 ) . by studying and comparing the properties of these objects , we could gain an insight into the processes that govern the formation of relativistic jets ( e.g. marscher 2006a , nemmen et al . 2012 ) . in the case of blazars , we can measure apparent superluminal motion ( i.e. lower limits of @xmath1 ) , where reported apparent velocities are as high as @xmath16 for @xmath15-ray bright blazars ( jorstad et al . 2005 ; lister et al 2009 ; piner et al . 2012 , liodakis & pavlidou 2015 ) . the lorentz factor for agn is typically @xmath17 ( e.g. marscher 2006a ; saikia et al . 2016 , etc . ) or @xmath18 ( lister et al . 2009 ) . blazars with a high@xmath19 overpopulate centimeter - wave surveys of bright flat - spectrum sources because of beaming bias . alternatively , a volume - limited sample of radio - loud agn would be dominated by objects with more mundane jets . a power - law distribution of lorentz factors for agn can be assumed , @xmath20 , where population synthesis studies show that a value of @xmath21 between 1.5 and 1.75 provides a good match between a synthetic and observed distribution of apparent velocities ( lister & marscher 1997 ; marscher 2006b ) . recent work indicates a value of @xmath22 for blazars ( saikia et al . 2016 ) . many observations indicate that grbs are produced by ultra - relativistic outflows with @xmath23 . however , grb progenitors might not always eject such a high@xmath19 flow . for example , if the outflow is baryonic , the baryon loading might not always be optimal , resulting in lower lorentz factors . for an outflow with low @xmath1 , the internal dissipation processes ( i.e. @xmath15-ray production ) happen when the outflow is still optically thick . since we are currently discovering grb events through wide field monitoring of the @xmath15-ray sky ( e.g. swift , fermi , ipn ) , a population of low@xmath19 outflows might be undiscovered . compact stellar mergers are the most promising targets for ground - based gravitational wave ( gw ) detectors such as advanced ligo , virgo and kagra . the merger of a binary bh system produced the advanced ligo detection gw150914 , the first direct observation of gw ( abbott et al . em counterparts to bh - bh mergers are not expected , fermi gamma - ray burst monitor ( gbm ) however , claimed a 2.9@xmath24 detection of a weak @xmath15-ray burst 0.4 seconds after the gw detection ( connaughton et al . 2016 ) , if this burst is associated with gw150914 then an electromagnetic ( em ) afterglow would also be present ( yamazaki et al . 2016 ) . to maximize the science returns from further gw detections , the identification of an em counterpart will be crucial . the @xmath15-ray emission from short grbs are an ideal em counterpart to ns - ns / ns - bh mergers , and potentially bh - bh mergers . however , they occur relatively rarely within the range of gw detectors ( 300 mpc for face - on ns - ns mergers ) , this is possibly because @xmath15-ray emission is highly collimated , or the mis - match between short grb peak energies and the swift detection band can make detection more difficult . additionally the intrinsic rate of compact object mergers within this volume is relatively low . more isotropic em components such as macronovae are often discussed to localize a large sample of gw events ( e.g. metzger & berger 2012 ; nakar & piran 2011 ; gao et al . 2013 ; kisaka , ioka & takami 2015 ) . in this paper , we discuss the possibility that a significant fraction of compact stellar mergers result in the production of low@xmath19 jets ( @xmath25 ) . if such jets are common , x - ray , optical , and radio transients , i.e. on - axis orphan afterglows ( dermer et al . 2000 , nakar & piran 2002a , huang et al . 2002 , rhoads 2003 , cenko et al . 2013 , cenko et al . 2015 ) , would be more frequent than short grbs . such low frequency transients would accompany a good fraction of gw events and they allow for the accurate determination of the sky positions of the gw sources . the time lag between gw signals , where we can assume that the jet launch time @xmath26 is coincident with the merging time when the gw amplitude becomes maximal , and em jet emission will enable us to determine the @xmath1 distribution of jets from compact stellar mergers and it will provide another constraint on the acceleration process of relativistic jets . in @xmath27 [ motion ] we discuss the background of relativistic motion in the standard grb fireball model and the implications for the prompt @xmath15-ray emission . @xmath27 [ oaoag ] the case for a population of low lorentz factor jets is made . @xmath27 [ model ] details the assumptions and conditions made by the monte carlo model plus the numerical results . @xmath27 [ gwimp ] highlights the implications for gw rates within the ligo / virgo detection volume . in @xmath27 [ conc ] conclusions are given . observed grbs contain a large fraction of high energy @xmath15-ray photons , which can produce electron - positron pairs if they interact with lower energy photons . if the optical depth for this process is large , pairs will form rapidly and compton scatter other photons , resulting in an increased optical depth . the optical depth for the pair creation is very sensitive to the lorentz factor of the source @xmath28 ( e.g. piran 1999 ; lithwick & sari 2001 for the typical high energy spectral index @xmath29 ) . the source becomes optically thin if it is expanding with a lorentz factor @xmath23 . if there are baryons in grb outflows , another limit on @xmath1 can be obtained by considering the scattering of photons by electrons associated with these baryons ( e.g. lithwick & sari 2001 ) . note that high polarization results still suggest magnetized baryonic jets , rather than poynting - flux dominated jets ( steele et al . 2009 ; mundell et al . the optical depth due to these electrons at radius @xmath30 is @xmath31 where @xmath32 is the thomson cross - section , @xmath33 is the total isotropic explosion energy and @xmath34 proton mass . outflows become optically thin at the photospheric radius , @xmath35 where @xmath36ergs and @xmath37 . on the other hand , the variability timescale @xmath38 in grbs constrains the radius from which the radiation is emitted , @xmath39 where @xmath40 seconds . requiring @xmath41 , we obtain @xmath42 . for outflows with a small lorentz factor @xmath43 , the internal dissipation happens when the outflow is still optically thick . the photons will remain trapped and the thermal energy will be converted back to the kinetic form ( kobayashi & sari 2001 ; kobayashi et al . 2002 ) , and the prompt @xmath15-ray emission would be suppressed ( i.e. failed grbs ) . usually outflows are assumed to have a sub - relativistic temperature after the internal dissipation , and the internal energy density is comparable to the mass energy density @xmath44 . if a significant fraction of the internal energy is converted to electron - positron pairs , the number density of the electrons and positrons @xmath45 could be larger by a factor of @xmath46 than that of electrons that accompany baryons , where @xmath47 is the electron mass . a more detailed discussion ( lithwick & sari 2001 ) also shows that the scattering of photons by pair - created electrons and positrons is nearly always more important than that by electrons that accompany baryons . since the lepton pairs create an effective photosphere further out than the baryonic one , the approximation in equation ( [ eq : rph ] ) will provide conservative estimates when we discuss failed grb rates in @xmath27 4 and @xmath27 5 . even if a jet does not have a velocity high enough to emit @xmath15-rays , it eventually collides with the ambient medium to emit at lower frequencies . such synchrotron shock radiation has been well studied in the context of grb afterglows ( e.g. mszros & rees 1992 , 1997 ; sari & piran 1999 ; kobayashi et al . 1999 ) . because of relativistic beaming , the radiation from a jet can be described by a spherical model when @xmath48 where @xmath49 is the jet half - opening angle . we here consider a relativistic shell with an energy @xmath33 and an initial lorentz factor @xmath1 expanding into ism with particle density @xmath50 . the deceleration of the shell happens at , @xmath51 where @xmath52 protons @xmath53 , and @xmath54 is measured in the grb rest frame . the typical frequency and the spectral peak flux of the forward shock emission at the deceleration time @xmath54 are , @xmath55 ( sari et al . 1998 ; granot & sari 2002 ) where @xmath56 and @xmath57 are the microscopic parameters , @xmath58 , @xmath59 , and @xmath60 cm ( i.e. the ligo range for face - on ns - ns mergers ) . the optical emission , assumed to be between the peak frequency @xmath61 and the cooling frequency @xmath62 , is expected to rise as @xmath63 and decay as @xmath64 after the peak @xmath65 . self - absorption can significantly reduce synchrotron shock emission at low frequencies . the upper limit can be approximated as black body flux for the forward shock temperature ( e.g. sari & piran 1999 ) , the limit at @xmath54 is @xmath66 where @xmath67ghz and the observable blast - wave size @xmath68 . equalizing the synchrotron emission and the black body limit , we obtain the self - absorption frequency @xmath69 ghz at the deceleration time @xmath54 . the self - absorption limit initially increases as @xmath70 , and then steepens as @xmath71 after @xmath61 crosses the observational frequency @xmath72 . considering that the synchrotron flux at @xmath73 also increases as @xmath70 , if @xmath74 at @xmath54 , the synchrotron emission would be reduced by the self - absorption at least until the passage of @xmath61 through the observational band at @xmath75 days . if the jet break happens while the flux is still self - absorbed , the light curve becomes flat @xmath76 const ( sari , piran & halpern 1999 ) . however , this estimate is obtained by assuming the rapid lateral expansion ( i.e. @xmath77 ) . recently studies show that the sideways expansion is rather slow especially for mildly - relativistic jets ( granot & piran 2012 ; van eerten & macfadyen 2012 ) . we will assume that the blast - wave emission starts to decay at the jet break , @xmath78 even if it is in the self - absorption phase . at low frequencies @xmath79 ghz and early times , forward shock emission would be affected by synchrotron self - absorption . however , currently most radio afterglow observations are carried out at higher frequencies ( e.g. vla 8.5 ghz ) at which self - absorption is more important for the reverse shock emission . just before the deceleration time @xmath54 , a reverse shock propagates through the jet and heats the original ejecta from the central engine . the reverse shock region contains energy comparable to that in the forward shock region . however , it has a lower temperature due to a higher mass ( i.e. lower energy per particle ) . the shock temperature and the typical frequency are lower by a factor of @xmath80 and @xmath81 compared to those of the forward shock ( e.g. kobayashi & zhang 2003 ) . although reverse shocks in low-@xmath1 jets could emit photons in the radio band , the self - absorption limit is tighter due to the lower shock temperature ; we find that the forward shock emission always dominates . note that we rarely catch the reverse shock emission even for regular grbs with detectable @xmath15-ray emission . we will discuss only the forward shock ( i.e. blast wave ) emission in this paper . by using the estimates of lorentz factors based on long grb afterglow peak times , hascot et al . ( 2014 ) demonstrated that an apparent correlation between isotropic @xmath15-ray luminosity @xmath82 and lorentz factor @xmath1 can be explained by a lack of bright bursts with low lorentz factors . they have also predicted the existence of on - axis orphan afterglows of long grb events . we here extend their argument to short grbs , and we apply their formalism to cosmological ( i.e. @xmath15-ray satellite range ) and local ( i.e. gw detector range ) events to study the on - axis orphan afterglows of failed short grbs ( i.e. low-@xmath1 events ) . the following assumptions are made in our simple monte carlo simulation of a synthetic population of merger events : 1 . the redshift for each event is randomly determined using a distribution with a constant time delay with respect to the star formation rate , where the peak rate is at @xmath83 . the redshift limits of @xmath84 are used for the cosmological sample , and @xmath85 for local sample , i.e advanced ligo / virgo detectable range @xmath86 mpc = 300 mpc for ns - ns mergers where the factor of 1.5 accounts for the stronger gw signal from face - on mergers ( kochanek & piran 1993 ) . we use the event rate per unit comoving volume for short grbs obtained by wanderman & piran ( 2015 ) , which is a function of @xmath87 as @xmath88 numerical results for the cosmological cases are insensitive to the value of the maximum @xmath87 as long as it is much larger than unity . 2 . a power - law distribution of lorentz factors @xmath89 is assumed with reasonable limits @xmath90 . motivated by agn studies ( e.g. lister & marscher 1997 ; marscher 2006b ) , we choose @xmath91 as our fiducial value and the cases of @xmath92 and @xmath93 will be briefly discussed . the isotropic @xmath15-ray luminosity @xmath82 is randomly generated in the limit @xmath94erg / s @xmath95erg / s where the limits come from observational constraints and the luminosity distribution follows the form obtained by wanderman & piran ( 2015 ) , @xmath96 where this luminosity function is logarithmic in the interval d@xmath97 . for each event , the dissipation radius @xmath98 is evaluated by using a random @xmath1 and the typical pulse width in short grb light curves @xmath99 sec ( nakar & piran 2002b ) . @xmath15-ray photons are assumed to be emitted at @xmath100 with a random @xmath15-ray luminosity @xmath101 or equivalently a random isotropic @xmath15-ray energy @xmath102 where @xmath103 is the duration of short grbs . we assume @xmath104 sec for all bursts as this is the median value for a log normal distribution of durations for short gamma - ray bursts ( zhang et al . the spectral peak energy in the @xmath105 spectrum is known to be correlated with @xmath101 ( yonetoku et al . 2004 , ghirlanda et al . the correlation is consistent for both long and short grbs ( zhang et al . 2012 ) , and given by @xmath106 the @xmath105 spectrum is assumed to follow a broken power - law with low - energy index ( below @xmath107 ) of @xmath108 ( @xmath109 ) , and a high - energy index of @xmath110 ( @xmath111 ) , where @xmath112 and @xmath113 are the photon number spectral indices . the mean index values for all grbs are @xmath114 and @xmath115 ( gruber et al . 2014 ) but as short grbs are typically harder than average we use the values @xmath116 and @xmath117 . the spectral peak is normalized as the value integrated between 1 kev and 10 mev giving @xmath101 . if the outflow is optically thin , all the photons released at @xmath118 are radiated away . the event is considered to be detectable if the photon flux at the detector in the swift band ( 15 - 150 kev ) is @xmath119 photons s@xmath120 @xmath121 ( band 2006 ) . we take into account the redshift of the spectrum when the photon flux is evaluated . if the optical depth at the dissipation radius @xmath100 is more than unity , or equivalently the photospheric radius @xmath122 is larger than the dissipation radius , the @xmath15-ray emission would be suppressed where @xmath123 is the explosion energy and @xmath124 is the conversion efficiency from the explosion energy to @xmath15-rays . we use @xmath125 , this is consistent with theoretical predictions ( kobayashi et al . 1997 ) and the fiducial value used in other works ( liang et al . 2010 ; ghirlanda et al . the @xmath15-ray energy injected at @xmath100 is adiabatically cooled , and the photons decouple from the plasma at @xmath126 . assuming a sharp transition from optically thick to thin regime ( see beloborodov 2011 for the discussion of fuzzy photosphere ) , we use hydrodynamic scalings to estimate the cooling factor . the internal energy density ( photon energy density ) decays as @xmath127 and the lorentz factor is constant for the outflow with a sub - relativistic temperature ( piran et al . considering that the internal energy in the outflow shell with width @xmath128 is @xmath129 , the luminosity of photons released at @xmath118 is @xmath130 where we have assumed no shell spreading @xmath131 const . the spectral peak energy is similarly shifted as @xmath132 . the photons in the coupled plasma undergo pair production and compton down - scattering that progressively thermalises the distribution ( hascot et al . the electron temperature at @xmath118 can be approximated by a black - body temperature @xmath133 where @xmath21 is the radiation constant . the optical depth at @xmath118 is given by @xmath134 . the condition for efficient thermalisation is @xmath135 ( peer et al . 2005 , thomson et al . 2007 ) where @xmath47 is the mass of an electron and @xmath136 the boltzmann constant . the peak energy @xmath107 for such a case is given by @xmath137 , above which the distribution is exponentially suppressed . for simplicity we assume @xmath138 . if @xmath139 , the photons are not efficiently thermalised . the distribution is then limited by the efficiency of pair production where the maximum energy is @xmath140 kev . the distribution is cut - off above this energy . we generate a sample of @xmath141 events and evaluate the @xmath15-ray flux for each in the swift band . to allow for clarity without losing the general trend , the results for a population of 2000 events are shown in figure [ fig : f1 ] ; the blue circles and red crosses show the events detectable and undetectable by swift , respectively . the isotropic kinetic energy @xmath142 is the energy in the blast wave after deceleration time , @xmath143 , where @xmath33 is the total isotropic explosion energy , and @xmath144 is the isotropic @xmath15-ray energy at the photospheric radius @xmath126 . the lorentz factor @xmath1 of an outflow at @xmath145 is shown against this . the top panel shows the results with @xmath84 , where we find a small fraction @xmath146 of the total population and @xmath147 of the events with @xmath148 are detectable by swift . for the local population @xmath149 , these fractions are higher , at @xmath150 and @xmath151 respectively , due to the proximity ( see the bottom panel ) . the dashed line indicates the lower limit for a successful grb , events below this line have the prompt @xmath15-ray emission fully suppressed ; the cut - off , with the parameters used , is given by @xmath152 . in figure [ fig : f1 ] , the low - energy limit of @xmath153 is basically set by the monte - carlo luminosity distribution ( i.e. @xmath154 erg / s . note that the explosion energy @xmath33 is higher than the @xmath15-ray energy @xmath155 at the dissipation radius @xmath100 by a factor of @xmath156 ) . if we consider the local population ( the bottom panel ) , for the events above the dashed line ( i.e. the blue circles ) all of the @xmath15-ray energy is successfully radiated away , whereas for the events below the dashed line ( i.e. the red crosses ) , almost all of the @xmath15-ray energy is reabsorbed into the outflow . thus the distribution of @xmath153 for the blue circles has a slightly lower limit . if we consider the cosmological population ( the top panel ) , a fraction of events are distant and intrinsically dim . they are undetectable by swift even if all gamma - ray energy is successfully radiated away at @xmath100 . this is why there are red crosses above the dashed line for the cosmological population . the fraction of the events detectable by swift weakly depends on @xmath157 . if we assume @xmath158 erg / s , swift would be able to detect @xmath159 of the total cosmological population , and @xmath160 of the total local population . liang et al . ( 2010 ) , ghirlanda et al . ( 2012 ) and tang et al . ( 2015 ) report correlations between lorentz factor @xmath1 and the isotropic luminosity @xmath101 ( or the isotropic energy @xmath161 ) for long grbs : @xmath162 ; @xmath163 ; and @xmath164 , respectively . however , such power - law relations could indicate a lower limit on @xmath1 for observable long grbs with a given burst energy ( hascot et al . 2014 ) . in our simulation , we find that the detectable short bursts are always located above a line @xmath165 giving a lower limit relation @xmath166 . as discussed in section 3 , the kinetic energy @xmath153 of the failed grbs will be released as on - axis orphan afterglows at late times . figure [ fig : f2 ] shows the distributions of the peak flux ( the top panel ) and peak time ( the bottom panel ) of such x - ray , optical , and radio transients . to estimate these distributions , we have used the monte carlo results for the local sample ( @xmath167 mpc ) with model parameters : @xmath168 protons @xmath53 ( berger 2014 ; metzger & berger 2012 ) , @xmath169 , @xmath170 ( panaitescu & kumar 2002 ; yost et al . 2003 ; berger 2014 ) , the index of the power - law distribution of random electrons accelerated at shock @xmath171 ( sari , narayan & piran 1996 ; daigne et al 2011 ; metzger & berger 2012 ) , and the jet half - opening angle @xmath172 ensuring @xmath173 for our sample and is within the limits @xmath174 found by fong et al . ( 2015 ) for short grb . the jet opening angle plays a role only when we estimate the jet break time . the dotted green lines ( figure [ fig : f2 ] ) indicate the distribution for x - ray transients . the typical frequency of the blast wave emission @xmath61 is sensitive to the lorentz factor @xmath175 . since for the local population the on - axis orphan afterglows are produced by low-@xmath1 jets ( @xmath176 ) , the typical frequency @xmath61 is expected to already be below the x - ray and optical band at the deceleration time @xmath54 . the x - ray and optical light curves should peak at @xmath54 and they have the same peak time distribution . considering that the deceleration time @xmath177 is mainly determined by @xmath1 , we can roughly estimate the peak - time distribution @xmath178 . for @xmath179 , the distribution is wide and a large fraction of the events have the peak - time @xmath54 around several days after the merger event . if the minimum lorentz factor @xmath180 is assumed , the peak - time distribution would achieve the peak around a few weeks after the merger event . the distribution of the peak flux for x - ray , where the frequency is above the cooling frequency @xmath181 , is @xmath182 is shown in the top panel . given good localisation , all of the x - ray peak afterglow flux is above the minimum senstivity of the swift xrt @xmath183 erg @xmath121 s@xmath120 for @xmath184 seconds ( the vertical green thick solid line ) . the x - ray afterglows are below the trigger sensitivities of swift bat and maxi ; and too faint to be detectable by the swift bat survey . the solid red line in the top panel and the dotted green line in the bottom panel indicates the distribution for optical ( _ g_-band ) transients . the ab magnitude m@xmath185 axis is added in the top panel to indicate the optical flux . for optical transients , peak flux is @xmath186 , and @xmath187 of the optical orphan afterglows are brighter than @xmath188 ( the vertical solid red line indicates this typical limit for mid - sized ( @xmath189 m ) telescopes ) . the peak - time distribution for the bright events ( @xmath190 ) is shown as the the dashed magenta line in the bottom panel . the difference between the dotted green ( representing both x - ray and optical in peak time ) and dashed magenta line corresponds to the dim event population ( @xmath191 ) . since these events tend to have low-@xmath1 , their typical frequencies are much lower than the optical band , and they peak at late times . the solid blue lines give the distribution for radio ( 10 ghz ) transients . the typical frequency @xmath61 is expected to be above 10 ghz at the deceleration time @xmath54 . the light curve peaks when the typical frequency @xmath192 crosses the observational band : @xmath193 . since the dynamics of the blast wave at @xmath194 depends only on the sedov length @xmath195 and not on the initial lorentz factor @xmath1 , the peak - time distribution should be narrowly clustered , compared to the distribution of the optical transients . the monte carlo results actually give a narrow peak around @xmath196 10 days . the peak flux @xmath197 is bright : typically @xmath198 mjy . vla ( the vertical solid blue line ) can easily detect the transients . the dashed - dotted black lines indicate the distribution for radio ( 150 mhz ) transients . as we have discussed , this low frequency emission is suppressed by the self - absorption , and jet break is likely to happen before it becomes optically thin . the peak - time of the light curve is determined by the jet break time @xmath199 . for the fixed @xmath172 , we find that the peak - time distribution is similar to that for 10 ghz transients and it peaks around @xmath200 days . however , since the emission is still suppressed by the self - absorption at the peak time , the peak flux is much lower : @xmath201 mjy . approximately @xmath202 of the 150 mhz transients are brighter than the sensitivity limit of 48 lofar stations ( the vertical dashed black line ) , and all are brighter than the sensitivity limit for ska1-low ( the vertical dashed - dotted black line ) . typical afterglow light curves for a selection of on - axis orphan afterglows are shown in figure [ fig : f4 ] . an average luminosity distance for ns - ns gw detectable mergers from our sample is used of @xmath203 220 mpc . x - ray , optical , and radio ( 10 ghz ) are shown for 4 combinations of @xmath1 and @xmath153 . the vertical dashed line in each panel represents the deceleration time @xmath54 , as @xmath54 is most sensitive to @xmath1 ( see equation [ tdec ] ) the lower lorentz factor cases ( top two panels ) have a significantly later deceleration time . the vertical dotted line in each panel represents the jet - break time @xmath204 , a jet half - opening angle @xmath172 is used throughout , for narrower(wider ) jet half - opening angles the break time will be at earlier(later ) times . the jet - break time is only weakly dependent on the kinetic energy ( see equation [ eq : tb ] ) . in all cases the x - ray ( green dash - dotted line ) and the optical ( thin red line ) peak at the deceleration time , the 10 ghz ( thick blue line ) is shown to peak at a later time @xmath205 when the typical frequency @xmath206 crosses the radio frequency . in all cases at times earlier than @xmath54 the flux is @xmath207 , for the x - ray and optical the flux at @xmath208 is @xmath209 . at 10 ghz the flux is @xmath210 at @xmath211 , and @xmath212 after @xmath205 and before @xmath204 . in all cases at @xmath213 the flux is @xmath214 . the swift satellite has been detecting short grbs at a rate of @xmath215 yr@xmath120 since the launch in 2004 , and @xmath216 of the detected events have measured redshifts ( swift grb catalogue ) . unfortunately no swift short grb with known redshift has been detected within the advanced ligo / virgo range for face - on ns - ns mergers @xmath217 mpc , and only three ( 061201 , 080905a , and 150101b ) have occurred within the face - on ns - bh range @xmath218 mpc ( abadie et al . metzger & berger ( 2012 ) estimate that @xmath219 short grbs per year , with redshift measurements , are currently being localized by swift within @xmath220mpc ( @xmath221mpc ) . considering that the field of view of the swift bat is @xmath189sr , the all - sky rate of detectable short grbs with or without redshift information is higher by a factor of @xmath222 . if the distribution of @xmath1 is described by the power - law @xmath89 , when we consider the rate of jets from mergers regardless of inclination or detectability , the rate for failed grbs would be higher than the short grb rate . for local population @xmath223 mpc , we find that the fraction of failed events is about @xmath224 for @xmath225 , @xmath226 for @xmath227 , and @xmath228 for @xmath93 ( the same rates are obtained for a population of @xmath229mpc ) . if @xmath230(@xmath93 ) , the failed grb rate is higher by a factor of @xmath231(6.7 ) than the short grb rate ( i.e. the ratio of failed to successful grbs ) . the all - sky rate of the failed grbs with or without redshift information is about 2.6(5.1 ) per year for the ns - ns range and 26(51 ) per year for the ns - bh range . here we assumed the jet opening angle distribution does not depend on the lorentz factor of the jets ( i.e. grb and failed grb jets have the same opening angle ) . the jet half - opening angle is not well constrained for short grb jets ( the median value for 248 long grbs is @xmath232 ; fong et al . 2015 ) . using four short grbs which have temporal steepenings on timescale of @xmath233 days , the median value is estimated as @xmath234 ( fong et al . however , the majority of short grbs do not have detected jet break , the inclusion of these bursts is essential in understanding the true opening angle distribution . based on a probability argument , fong et al . ( 2015 ) obtain the median value @xmath235 16@xmath236 and 33@xmath236 if the maximum possible angle is 30@xmath236 and 90@xmath236 , respectively . if the typical jet half - opening angle of short grbs is @xmath237 , the beaming factor is @xmath238 where @xmath239 , only a small fraction of short grb jets point toward us ( see the black dashed line in figure [ fig : f3 ] ) . however , since the gw polarization components @xmath240 and @xmath241 depend on the inclination angle @xmath242 of the binary , mergers emit gws much more strongly along the polar axis than in the orbital plane . considering that the jets from the mergers are also likely to be directed along the polar axis , kochanek & piran ( 1993 ) show that when a grb is associated , the gw amplitude @xmath243 is stronger by a factor of @xmath244 than the amplitude averaged over the sky ( as seen from the source ) . the distances out to which gw detectors could detect the binary increases by a factor of @xmath245 if the jet points toward us ( we define an on - axis event as any jet where the inclination is within the half - opening angle , @xmath246 ) . when we consider a sample of merger gw events detected by a gw detector with sensitivity @xmath247 , their jets would tend to be directed to us . this is because on - axis events are detectable at a larger distance . the on - axis probability could be higher by roughly the volume factor of @xmath248 ( the blue dashed - dotted line , figure [ fig : f3 ] ) than the simple geometric estimate @xmath249 ( i.e. our line - of - sight falls within the opening angle of the jet with a higher probability ) . we also conduct a monte carlo simulation to estimate the on - axis probability . in the simulation , mergers are uniformly distributed in space , with a random inclination angle , and they emit gws with amplitude @xmath250 . after selecting the events detectable by a gw detector : @xmath251 , we evaluate the fraction of the events which have an inclination angle smaller than a given jet half - opening angle @xmath49 ; we assume uniform jets with a top hat distribution throughout from the jet symmetry axis ( e.g. @xmath252 outside of some core angle ) , only the central part could have lorentz factors high enough to produce @xmath15-rays . although the detailed study is beyond the scope of this paper , the failed grb rate could be even higher for structured jets . the result ( the red solid line ) does not depend on the detector sensitivity as long as the merger distribution is homogeneous . if we consider gw trigger events , the on - axis probability ( the red solid line ; @xmath253 and @xmath254 for @xmath255 and 33@xmath236 , respectively ) is much higher than the beaming factor ( the black dashed line ) . although isotropic em counterparts such as macronovae could be ideal to localize a large sample of gw events , @xmath256 of gw events would still be associated with the on - axis orphan afterglow of failed grbs especially when they have wider jet opening angles compared to short grb jets . for long grb jets , observational results indicate such a correlation @xmath257 with @xmath258 ( panaitescu & kumar 2002 ; salmonson & galama 2002 ; kobayashi et al . 2002 ; ghirlanda et al . the failed grb rates could be higher than those discussed at the beginning of this section . we have shown that failed grbs are much more frequent than short grbs when the lorentz factors of jets from compact stellar mergers follow a similar power - law distribution as those observed for agn . for most events the internal dissipation process happens when the jet is still optically thick , and the photons produced by the dissipation process will be converted back to the kinetic energy of the jet . by using a simple monte carlo model , we have shown that even for the local merger population within the ligo / virgo range , the @xmath15-ray emission from jets with @xmath176 will not be detected by @xmath15-ray satellites ( e.g. swift ) . for a power - law distribution of the jet lorentz factors @xmath259 , 78% of compact object mergers that have jets result in a failed grb . the failed grb events will produce on - axis orphan afterglow at late times . using the local short grb rate as normalization , the all - sky rate of the on - axis orphan afterglow is about 2.6 and 26 per year for the ns - ns range ( 300 mpc ) and ns - bh range ( 600 mpc ) , respectively . the opening angle of jets for long grbs was found to be a function of @xmath1 ( e.g. ghirlanda et al . 2013 ) , if low-@xmath1 jets from compact - binary mergers have wider half - opening angles @xmath49 than those of short grbs then the real rate would be higher than these . we have evaluated the peak time and peak luminosity of the on - axis orphan afterglows in x - ray , optical , and radio bands . although it is usually difficult to model observational data for orphan afterglow candidates when the explosion time is unknown ( i.e. the @xmath26 issue ) . for gw trigger events , gw signals will provide the explosion time @xmath26 . the peak time distribution in the x - 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ray afterglow are shown as dashed green lines , optical are shown as red thin solid lines , and radio ( 10 ghz ) are shown as blue thick solid lines . the vertical black dotted lines represent the deceleration time @xmath54 and the jet - break time @xmath204 ( assuming a @xmath172 ) ]
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short gamma - ray bursts ( grbs ) are believed to be produced by relativistic jets from mergers of neutron - stars ( ns ) or neutron - stars and black - holes ( bh ) .
if the lorentz - factors@xmath0of jets from compact - stellar - mergers follow a similar power - law distribution to those observed for other high - energy astrophysical phenomena ( e.g. blazars , agn ) , the population of jets would be dominated by low-@xmath1 outflows .
these jets will not produce the prompt gamma - rays , but jet energy will be released as x - ray / optical / radio transients when they collide with the ambient medium . using monte carlo simulations , we study the properties of such transients .
approximately@xmath2of merger - jets@xmath3mpc result in failed - grbs if the jet@xmath0follows a power - law distribution of index @xmath4 .
x - ray / optical transients from failed - grbs will have broad distributions of their characteristics : light - curves peak @xmath5days after a merger ; flux peaks for x - ray @xmath6mjy ; and optical flux peaks at @xmath7 .
x - ray transients are detectable by swift xrt , and @xmath8 of optical transients will be detectable by telescopes with limiting magnitude @xmath9 , for well localized sources on the sky .
x - ray / optical transients are followed by radio transients with peak times narrowly clustered around @xmath10days , and peak flux of @xmath11mjy at 10@xmath12ghz and @xmath13mjy at 150@xmath12mhz . by considering the all - sky rate of short grbs within the ligo / virgo range ,
the rate of on - axis orphan afterglows from failed - grb would be 2.6(26 ) per year for ns - ns(ns - bh ) mergers , respectively .
since merger jets from gravitational - wave ( gw ) trigger events tend to be directed to us , a significant fraction of gw events could be associated with the on - axis orphan afterglow .
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[ cols="^,^,^ " , ] next we study polarization properties of 3d models . motivated by 3d simulations of neutrino - driven explosions , where various sizes of complex structure appear , we set up 3d models by randomly placing different numbers of spherical clumps with different sizes . here we introduce two parameters to depict the model : the size parameter of the clumps @xmath3 , i.e. , the radius of the clump normalized by the photospheric radius ( @xmath4 ) and the photospheric covering factor ( @xmath5 ) . since the optical depth near the photosphere is the most important for line formation , the covering factor is evaluated by taking into account the clumps only in a shell between @xmath6 and @xmath7 km s@xmath8 . note that , as in the 2d cases , the line optical depth in our models has a spherical component and it is enhanced by a factor of @xmath9 within the clumps . the top panels of figure [ fig:3d ] show the polarization properties of the 3d model with the clump size of @xmath10 and the covering factor of @xmath11 . in the polarization spectrum , both stokes @xmath12 and @xmath13 parameters vary across the lines ( middle panel ) , and polarization shows a loop in the @xmath0 diagram ( right panel ) , as also found by @xcite . the @xmath0 loop in the 3d clumpy models can be understood as follows . in the 3d clumpy models , depending on the doppler velocities , different parts of the photospheric disk are hidden by the clumps . since the distribution of the clumps does not have a common symmetric axis , the position angle of the polarization can change depending on the doppler velocities . in general , the change in the position angle across the line can be arbitrary large , i.e. , the polarization in the @xmath0 diagram can be scattered around . but for the relatively large size of the clumps as in the case of @xmath10 , the same clump keeps contributing to the absorption even for different doppler velocities , and thus the change in the position angle tends to be smooth as a function of doppler velocities . therefore , the polarization tends to show a loop in the @xmath0 diagram in the 3d clumpy distribution with relatively large clumps . note that the @xmath0 loop can also be produced by other geometries , e.g. , a combination of the ellipsoidal photosphere and ellipsoidal line scattering shell whose symmetric axes are misaligned with each other @xcite , but even in such a case , it is required that the axisymmetry of the system is broken . ] ( top , red ) , @xmath14 ( middle , blue ) , and @xmath15 ( bottom , green ) . for each panel , four different colors ( lighter to darker colors from left to right ) represent models with four different line strengths at the photosphere ( @xmath16 = 3.0 , 10.0 , 30.0 , and 100.0 , respectively ) . for each model , results of 100 lines of sight are shown . the black points with error bars are observational data of the ( filled ) and ( open ) lines for 6 type ib and ic sne analyzed in @xcite : type ib sne 2005bf @xcite , 2008d @xcite , 2009jf @xcite , type ic sne 2002ap @xcite , 2007gr @xcite , and 2009mi @xcite . the solid line shows @xmath17 $ ] ( see section [ sec : methods ] ) . [ fig : pobs_fd],title="fig : " ] + ( top , red ) , @xmath14 ( middle , blue ) , and @xmath15 ( bottom , green ) . for each panel , four different colors ( lighter to darker colors from left to right ) represent models with four different line strengths at the photosphere ( @xmath16 = 3.0 , 10.0 , 30.0 , and 100.0 , respectively ) . for each model , results of 100 lines of sight are shown . the black points with error bars are observational data of the ( filled ) and ( open ) lines for 6 type ib and ic sne analyzed in @xcite : type ib sne 2005bf @xcite , 2008d @xcite , 2009jf @xcite , type ic sne 2002ap @xcite , 2007gr @xcite , and 2009mi @xcite . the solid line shows @xmath17 $ ] ( see section [ sec : methods ] ) . [ fig : pobs_fd],title="fig : " ] + ( top , red ) , @xmath14 ( middle , blue ) , and @xmath15 ( bottom , green ) . for each panel , four different colors ( lighter to darker colors from left to right ) represent models with four different line strengths at the photosphere ( @xmath16 = 3.0 , 10.0 , 30.0 , and 100.0 , respectively ) . for each model , results of 100 lines of sight are shown . the black points with error bars are observational data of the ( filled ) and ( open ) lines for 6 type ib and ic sne analyzed in @xcite : type ib sne 2005bf @xcite , 2008d @xcite , 2009jf @xcite , type ic sne 2002ap @xcite , 2007gr @xcite , and 2009mi @xcite . the solid line shows @xmath17 $ ] ( see section [ sec : methods ] ) . [ fig : pobs_fd],title="fig : " ] . one characteristic polarization degree is assigned for each object by taking the average of the corrected polarization ( @xmath18 ) of the and lines . color lines show the cumulative distribution of polarization degree for 100 lines of sight . three models with @xmath10 ( red ) , @xmath14 ( blue ) , and @xmath15 ( green ) are shown . in this plot , we use the models with @xmath19 since these models approximately give @xmath20 ( figure [ fig : pobs_fd ] ) , where the corrected polarization is defined . [ fig : pdist_size ] ] the size of the clumps is of interest to study the origin of the 3d structure in the sne . we show the first attempt to quantify the size of the clumps by comparing the results of the modelling and the observed polarization degrees , i.e. , the maximum polarization level at the absorption line . we calculate the polarization spectra with different sizes of the clumps by keeping the covering factor of @xmath21 and other parameters to be the same . the middle and bottom panels in figure [ fig:3d ] show the results for the 3d models with @xmath14 and @xmath22 , respectively . as shown in the figures , for a given covering factor , models with smaller clumps show a lower polarization . in such models , the photospheric disk is hidden by many small clumps , and polarization vectors tend to be cancelled out ( figure [ fig : schematic ] ) . this behavior was also pointed out by @xcite in the context of type ia sne . since stripped - envelope sne generally show non - zero line polarization , the typical size of the clumps should not be too small . since the polarization degree depends not only on the geometry but also on the absorption strength , it is important to compare the models and observations for similar absorption strengths . therefore , in figure [ fig : pobs_fd ] , we compare models and observations in the plane of the polarization degree and the fractional absorption depth . the black points with error bars are observational data of the ( filled ) and ( open ) lines for 6 type ib and ic sne analyzed in @xcite . the small dots show the polarization degree of the models for 100 lines of sight . in each panel , we show four sets of the models with the same size and distribution of the clumps but with the different line optical depth at the photosphere ( @xmath16 = 3.0 , 10.0 , 30.0 , and 100.0 from left to right ) . when the clump is as small as @xmath15 ( bottom panels of figures [ fig:3d ] and [ fig : pobs_fd ] ) , the polarization degree can not be @xmath23 for any line of sight . for the larger sizes of the clumps , a higher polarization can be obtained . when the size of the clumps is @xmath24 ( middle panels ) , the polarization degrees of these models are still short of some of the observed polarization . when the size of the clumps is relatively large , @xmath10 ( top panels ) , the polarization degree can be as high as @xmath25 for the fractional depth of 0.5 . ideally the polarization properties of the models should be compared with the statistical distribution of the observed polarization . although the number of objects with good data is still small , figure [ fig : pdist_size ] shows a cumulative distribution of polarization properties of 6 type ib and ic sne . to define one characteristic polarization for each object , we take the average of the corrected polarization ( @xmath18 ) for the and lines . color lines show the cumulative distribution of the modelled polarization for 100 lines of sight . we choose models with @xmath19 , which approximately give @xmath20 ( figure [ fig : pobs_fd ] ) . the comparison in the cumulative distribution clearly shows that the model with too small clumps ( @xmath15 ) is not consistent with the observations . the @xmath26 value for a kolmogorov - smirnov ( ks ) test is @xmath27 . since the number of objects is so small , we can not distinguish the model with the clump size of @xmath14 ( @xmath28 ) and @xmath10 ( @xmath29 ) . nevertheless , the model with @xmath14 is already short to explain the polarization level of @xmath30 , and seems to close to the lower limit for the clump size . here it is noted that our models adopt an enhancement factor of @xmath31 . for a higher enhancement factor , the polarization degree is not largely affected because models with @xmath31 already give an optically thick absorption in the clumps near the photosphere . on the other hand , for a smaller enhancement factor , the polarization degree decreases for a given fd . in such cases , even larger clumps is required to reproduce a high polarization degree . therefore , we conclude that a typical size of the 3d clumps should be @xmath32 of the photospheric radius to reproduce observed polarization degrees . to obtain possible constraints on the number or the covering factor of the clumps in the ejecta , we vary the covering factors of clumps keeping their size to be @xmath10 . figure [ fig : pdist_fc ] shows the model input ( left ) and cumulative distributions of the resultant polarization ( right ) . for the models , we choose the line strength at the photosphere to have @xmath20 , i.e. , @xmath16 = 30 for the models with @xmath33 , 0.2 , and 0.3 . @xmath16 = 10 for the models with @xmath34 and 0.7 . the observed distribution is the same as in figure [ fig : pdist_size ] . for the model with a smaller covering factor ( @xmath33 ) , the probability to have a high polarization is also low . then , by increasing the covering factor of the clumps , a higher polarization can be more frequently observed ( @xmath35 ) . however , if the covering factor of the clumps is too large ( @xmath36 ) , the distribution of resultant polarization shifts toward a lower value again since the system restores the symmetry again . since the observational samples are small , it is difficult to draw a firm conclusion on the covering factor of the clumps . however , the models with @xmath33 and @xmath36 are already at the edge of the distribution . by taking into account the fact that models with @xmath31 tend to give an upper limit of the polarization level ( see section [ sec:3d_size ] ) , it seems that current data do not support models with too small covering factors ( @xmath37 ) and too large covering factors ( @xmath38 ) . we have modelled line polarization of stripped - envelope core - collapse sne . the results of modelling are summarized as follows . ( 1 ) the observed @xmath0 loop can not be explained by the 2d axisymmetric models , but can be explained by the 3d clumpy models . ( 2 ) by comparing the results of the 3d clumpy models with the observed degrees of line polarization , it is found that a typical size of the clumps is relatively large , i.e. , @xmath32 of the radius , and a covering factor of the clumps in the ejecta is not too small and not too large ( @xmath39 ) . it is intriguing that such a large - scale clumpy structure is also seen in the element distribution of cassiopeia a ( e.g. , * ? ? ? * ; * ? ? ? * ; * ? ? ? * ; * ? ? ? * ; * ? ? ? * ) , which is a supernova remnant produced by a type iib sn @xcite . the similarity suggests that the element distribution as seen in cassiopeia a may also be able to reproduce the polarization properties observed in early phase of sne . here we discuss possible origins for the clumpy structure suggested by observations and modelling . one scenario is the rayleigh - taylor ( rt ) instability , causing matter mixing in the sn ejecta . by the rt instability , many clumps are produced and metal - rich ejecta inside are delivered toward the outer layers ( e.g. , * ? ? ? * ; * ? ? ? * ; * ? ? ? * ; * ? ? ? * ; * ? ? ? * ; * ? ? ? * ; * ? ? ? however , the rt instability alone usually produces small fingers in many directions . this is similar to the case of @xmath15 in figure [ fig : schematic ] and not consistent with the observations . the clumpy structure suggested by observations is more in favor of large - scale convection or sasi developed in the initial stage of the explosion . when the large - scale convection or sasi takes place , the subsequent evolution of the shock becomes asymmetric , which produces the large - scale asymmetry in the element distribution ( e.g. , * ? ? ? * ; * ? ? ? * ; * ? ? ? * ; * ? ? ? also , results of long - term simulations show that the ejecta structure near the shock breakout still keeps an imprint of the large - scale asymmetry generated by neutrino - driven convection and sasi , with the small - scale structures by the rt instability added on top of it @xcite . note that such long - term simulations for neutrino - driven explosion also nicely reproduce the geometry of cassiopeia a @xcite . it is worth noting that , although the loop in the @xmath0 diagram does not support a purely axisymmetric element distribution ( figure [ fig:2d ] ) , spectropolarimetric data do not rule out the presence of an overall bipolar structure or a dominant axis in the sn ejecta . as long as some large - scale , non - axisymmetric components exist , they can produce a large enough polarization level and the loop in the @xmath0 diagram . in fact , analysis of the [ ] line profiles in the late - phase spectra suggest a torus - like distribution of oxygen , which is consistent with a bipolar explosion ( e.g. , * ? ? ? * ; * ? ? ? * ; * ? ? ? * ) . since polarization at the early phase and nebular line profile are sensitive to the outer and inner ejecta , respectively , the combination of early and late phase observations may indicate that global 2d structure exists more in the inner ejecta and 3d clumpy structure is added in the outer ejecta . it is noted that , even by the late phase observations , presence of clumpy structure has also been suggested by the studies of line profiles ( e.g. , * ? ? ? * ; * ? ? ? * ; * ? ? ? * ) , ionization states ( e.g. , * ? ? ? * ; * ? ? ? * ) , and dust ( e.g. , * ? ? ? * ; * ? ? ? * ; * ? ? ? * ; * ? ? ? * ; * ? ? ? * ; * ? ? ? thus , the transition from the inner 2d to the outer 3d structures may be somewhat gradual . interestingly , our studies suggest that the shape of the loop in the @xmath0 diagram can be used as a probe of such a combined ( 2d @xmath40 3d ) geometry . as expected from the results of 2d ( figure [ fig:2d ] ) and 3d models ( figure [ fig:3d ] ) , if the ejecta has an overall 2d geometry + 3d clumpy structure , it tends to produce an elongated loop in the @xmath0 diagram . although current observational data do not allow us to extract such information , detailed studies will be possible in future with more observational samples with high signal - to - noise ratio . it is emphasized that our modelling includes many simplifications . for example , we parameterize the line optical depth and enhancement factor , but they must be determined by the combination of element abundance , temperature , and ionization states . thus , our models shown in the left panels of figures [ fig:2d ] and [ fig:3d ] are not readily connected with the element distribution . full radiation transfer modelling using 3d hydrodynamic models is required to obtain a closer link between the explosion models and observations . also , the comparison with observed polarization degree is done by averaging the polarization degrees of different lines . since polarization at different absorption lines reflects the distribution of each element and ion , the direct comparison for each element is necessary when larger observed samples and full transfer calculations are available . we have performed 3d radiation transfer simulations for the analysis of line polarization in stripped - envelope sne . we demonstrate that a purely axisymmetric , 2d structure always produces a straight line in the stokes @xmath0 diagram , and can not explain the commonly observed loop in the @xmath0 diagram . on the contrary , 3d clumpy structures naturally reproduce the loop . comparison of the results of the modelling and the observed polarization degrees enables to constrain a typical size of the clumps from polarization data for the first time . to reproduce the distribution of the observed polarization degrees ( 0.5 - 2.0 % ) , a typical size of the clump should be relatively large , i.e. , @xmath41 of the photospheric radius ( or the radius where the clump is located ) . the covering factor of the clump in the ejecta is only weakly constrained i.e. , to @xmath42 % . such a large - scale clumpy structure inferred by polarization is similar to that seen in the sn remnant cassiopeia a. the large - scale clumpy structure is unlikely to be produced only by the rt instability as it tends to produce small fingers in many directions . instead , the presence of the large - scale clumpy structure in the ejecta suggests that large - scale convection or sasi takes place at the onset of the explosion . polarization properties do not necessarily exclude the presence of a dominant axis in the sn ejecta since non - axisymmetric structure on top of the 2d axisymmetric structure can also reproduce the loop in the @xmath0 diagram . in fact , the analysis of the nebular spectra supports a bipolar geometry in the innermost layer . these observational constraints suggest that sn ejecta may have an overall 2d bipolar structure inside and 3d clumpy structure outside . we speculate that such a hybrid structure could be produced by sasi . in order to obtain further constraints on the explosion mechanism , polarization modelling using realistic sn models will be worthwhile as more and more long - term realistic simulations from core collapse to the shock breakout are becoming available . we thank takashi hattori , kentaro aoki , masanori iye , elena pian , toshiyuki sasaki , and masayuki yamanaka for their contribution to the spectropolarimetric observations with the subaru telescope , and the referee for valuable comments . mt thanks thomas janka , takashi moriya , and takaya nozawa for fruitful discussion . numerical simulations presented in this paper were carried out with cray xc30 at center for computational astrophysics , national astronomical observatory of japan . this research has been partly supported by the grant - in - aid for scientific research from jsps ( 24740117 , 26800100 , 15h02075 ) and mext ( 25103515 , 15h00788 ) , and by world premier international research center initiative ( wpi initiative ) , mext , japan . 135 natexlab#1#1 , a. , & barlow , m. j. 2016 , , 456 , 1269 , j. m. , mezzacappa , a. , & demarino , c. 2003 , , 584 , 971 , j. c. , & mclean , 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, b. 2015 , , 446 , 2089 , b. a. 2011 , bulletin of the astronomical society of india , 39 , 101 , b. a. , & hartmann , l. 1992 , , 395 , 529 , a. , janka , h .- mueller , e. , pllumbi , e. , & wanajo , s. 2016 , arxiv:1610.05643 , a. , janka , h .- t . , & mller , e. 2013 , , 552 , a126 , a. , mller , e. , & janka , h .- 2015 , , 577 , a48 , s. , & sawai , h. 2004 , , 608 , 907 we have developed a new 3d radiation transfer code to compute polarization spectrum of one line from arbitrary 3d distribution of the line optical depth . the code uses the monte carlo method , which is a common method to compute polarization by scattering processes ( e.g. , * ? ? ? * ; * ? ? ? * ; * ? ? ? * ; * ? ? ? * ; * ? ? ? for the application to sne , see @xcite . we set up the 3d cartesian spatial mesh with the @xmath43 meshes . the velocity is used as a spatial coordinate because the sn ejecta are homologously expands ( @xmath44 ) , the outer velocity of the grid is @xmath45 25000 km s@xmath8 , and thus , the resolution is @xmath46 500 km s@xmath8 . this spatial resolution gives the wavelength resolution of @xmath47 , which is sufficient to make comparison with observed data . since the code computes only one ( arbitrary ) line , the wavelength range used in the computation is very small . if the rest wavelength of the line is @xmath48 , we compute the spectrum only at the wavelength range between @xmath49 and @xmath50 . within this wavelength range , the energy spectrum is assumed to be constant ( @xmath51 = const ) . our code assumes a sharply defined inner boundary , and solves radiation transfer above the boundary by tracking the photon packets in the expanding ejecta . every photon packet has assigned energy , wavelength , and stokes parameters . especially each photon packet in the simulation has a constant energy , irrespective of the wavelength of the packet . because of this treatment , any photon is not lost during the simulation , which results in the accurate energy conservation @xcite . the position of the inner boundary is determined so that the electron scattering optical depth from the inner boundary to infinity is @xmath52 . in the simulations used in the main text of the paper , we always adopt @xmath53 ( table [ tab : param ] ) as in @xcite and @xcite . the radiation from the inner boundary is assumed to be thermalized , and thus , to be unpolarized ; @xmath54 the direction of the photon is determined by @xmath55 @xcite ( hereafter we use @xmath56 to denote a random number , @xmath57 ) , where @xmath58 is cosine of the angle between the radial and photon direction . the azimuthal angle around the radial direction @xmath59 is uniformly distributed , i.e. , @xmath60 . the emitted photon packets experience the electron scattering and the line scattering , which are treated in a similar way to that by @xcite . for the electron scattering , we assume a power - law density structure with the power - law index @xmath61 . we also have the photospheric velocity ( @xmath62 ) and the epoch from the explosion ( @xmath63 ) as input parameters . the photospheric radius ( @xmath64 ) is defined to be the radius where the optical depth for the electron scattering is unity . by setting @xmath62 and @xmath63 , the normalization of the electron density is determined . for the line scattering , we use the sobolev approximation @xcite , and assume a power - law optical depth profile with the same index @xmath61 . the parameter for the line scattering is @xmath16 , the optical depth at the photosphere . in addition , we assume enhancement of the optical depth by a factor of @xmath65 in some region . the parameters used in the simulations are summarized in table [ tab : param ] . a photon packet propagating in one computational grid can have 3 possible events ; ( 1 ) escaping from the grid , ( 2 ) the electron scattering , and ( 3 ) the line scattering . the event that actually occurs is judged by calculating the length to the 3 events . it is simple to compute the length to the next grid @xmath66 for the given position and the direction vector of the photon packet . the direction to the electron scattering event is computed by the randomly selected event optical depth @xmath67 . when the optical depth reaches this value , a scattering event occurs . thus , the distance to the electron scattering @xmath68 can be computed by @xmath69 . when @xmath68 is shorter than @xmath66 , the electron scattering occurs if there is no contribution of line scattering . since the line scattering is treated as a resonance , the distance to the line scattering event is @xmath70 , where @xmath71 is the comoving wavelength of the photon packet . if @xmath72 is shortest among 3 lengths , the line scattering is taken into account . the line scattering event actually occurs when the sum of the line scattering optical depth ( @xmath73 ) and the electron scattering optical depth in @xmath74 ( @xmath75 ) exceeds @xmath76 . if this sum does not reach @xmath76 , then the electron scattering opacity is evaluated and added again , and the fate of the packet is the electron scattering or the escape from the grid . for illustration of this process , see figure 1 of @xcite . when the scattering event occurs , the next direction vector of the photon packets is determined . for the electron scattering , this scattering angle depends on the polarization , which is discussed in the next section . for the line scattering , the direction is determined by the isotropic probability function in the comoving frame . by the scattering event , the energy and the wavelength of the packet are changed . for the energy , by the energy conservation in the rest frame , @xmath77 where @xmath78 and @xmath79 are the rest - frame energy of the incoming and outgoing packets , respectively . and @xmath80 and @xmath81 are the cosines of the angles between the radial direction and incoming / outgoing propagating directions , respectively . similarly , the change in the wavelength is given by @xmath82 where @xmath83 and @xmath84 are the rest - frame wavelength of the incoming and outgoing packet , respectively . scattering events change the polarization properties of the photon packets . for the electron scattering , the phase matrix can be written as follows @xcite ; @xmath85 where @xmath86 is the scattering angle on the plane of the scattering . this matrix should be operated in the scattering frame . in general , the rotation matrix for the stokes parameters is written as follows @xcite ; @xmath87 by using these matrices , the effect on the stokes parameters is given by @xmath88 here @xmath89 and @xmath90 is stokes parameter in the rest frame before and after the scattering , respectively . the angles @xmath91 and @xmath92 are the angles on the spherical triangle defined as in ( @xcite , see figure 1 of @xcite ) . equation [ eq : pol_escat ] means that the the angle - dependence of the intensity of the scattered light depends the polarization properties of the incident radiation . from equation [ eq : pol_escat ] , the probability distribution function ( @xmath93 ) of the total intensity is @xmath94 by using this function with the rejection method as outlined in @xcite , we determine the scattering angle of the electron scattering . for the computation of polarization for the electron scattering , the code was tested with the analytic formulae by @xcite for optically thin cases , and with numerical results by @xcite for optically thick cases . for both cases , we got an excellent agreement . for the application to a sn , expanding ejecta with the steep density slope , we checked our results with those by @xcite . we confirmed that our code gives the consistent results on the radial profile of polarization for several power - law indexes ( @xmath61 ) and the inner boundaries ( @xmath52 ) .
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we present modelling of line polarization to study multi - dimensional geometry of stripped - envelope core - collapse supernovae ( sne ) .
we demonstrate that a purely axisymmetric , two - dimensional geometry can not reproduce a loop in the stokes @xmath0 diagram , i.e. , a variation of the polarization angles along the velocities associated with the absorption lines . on the contrary
, three - dimensional ( 3d ) clumpy structures naturally reproduce the loop .
the fact that the loop is commonly observed in stripped - envelope sne suggests that sn ejecta generally have a 3d structure .
we study the degree of line polarization as a function of the absorption depth for various 3d clumpy models with different clump sizes and covering factors .
comparison between the calculated and observed degree of line polarization indicates that a typical size of the clump is relatively large , @xmath1 of the photospheric radius .
such large - scale clumps are similar to those observed in the sn remnant cassiopeia a. given the small size of the observed sample , the covering factor of the clumps is only weakly constrained ( @xmath2 ) .
the presence of large - scale clumpy structure suggests that the large - scale convection or standing accretion shock instability takes place at the onset of the explosion .
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the concept of _ compact galaxies _ was introduced by zwicky @xcite , who has described them as galaxies barely distinguishable from stars " on the palomar sky survey plates . the term blue compact dwarf ( bcd ) galaxies @xcite identify those objects that show low luminosity , small linear dimensions , strong emission lines superposed on a blue continuum , and spectral properties that indicate low chemical abundances . bcds form a subset of hii galaxies , a large number of which have been identified on objective prism surveys by @xcite , @xcite , @xcite and @xcite by the presence of strong emission lines , similar to giant hii regions ( ghiirs ) in our galaxy . here , we will refer to hii / bcd galaxies as objects with a metallicity 7.0@xmath112+log(o / h)@xmath18.4 ( e.g. , @xcite ) , low luminosity ( m@xmath2 @xmath3 -18 ) and gas - rich objects ( e.g. , @xcite ) undergoing vigorous starburst activity in a relatively small physical size ( @xmath41 kpc ) . the star - forming component , in these objects , typically contains multiple knots of star - formation with unresolved ensembles of young star clusters ( e.g. , @xcite , @xcite ) . the hypothesis of these systems being young , forming their first generation of stars has been discarded by the detection of an evolved underlying stellar host with an age @xmath51 gyr , in the majority of the nearby hii / bcd population ( e.g. , @xcite,@xcite ) . figure [ fig1 ] shows the optical spectrum of the galaxy tol 2146 - 391 obtained using integral field unit ( ifu ) observations with gemini / gmos . in this figure we label the most important emission lines used in our studies , in particular , the strong balmer hydrogen recombination lines and collisionally excited emission lines , such as [ o ii ] @xmath63726,3729 , [ o iii ] @xmath04363 , [ o iii ] @xmath05007 , [ s ii ] @xmath66717,6731 , [ n ii ] @xmath06584 , which have been used for the determination of physical conditions ( e.g. , electron temperature and density ) and chemical abundances ( e.g. , oxygen , nitrogen , etc ) . we also detect in some of our galaxies the high - ionization emission line he ii @xmath04686 . although progress has been made in this field , important unsolved questions remain with regard to the mode of star formation ( e.g. , quasi - continuous vs fluctuating ) , and the triggering mechanism of ongoing starburst activity in hii / bcds . it has been suggested @xcite that the cluster formation efficiency is lower in compact hii / bcd galaxies than the one found in more luminous galaxies . these luminous systems generally show an irregular outer shape and kinematical signatures of merging in their interstellar medium ( ism ) . in some cases , the formation of star cluster complexes occurs coevally @xcite , whereas in others star formation occurs in a propagating manner @xcite . in any case , the mechanism which may trigger the current star - formation in these galaxies is not well understood ; in particular the relative importance of intrinsic and environmental properties remains a subject of investigation . another important issue is the chemical and kinematical imprints of star cluster formation and evolution on the spatially resolved properties of the ism in hii / bcd galaxies . as a natural consequence of star formation driven feedback , the newly synthesized elements will be dispersed and mixed across the ism via hydrodynamic mechanisms ( e.g. , @xcite ) , leading to _ chemical homogeneity in the oxygen abundance _ @xcite of hii / bcd galaxies ( e.g. , @xcite and references therein ) . the nitrogen - to - oxygen ratio n / o has also been found to be rather constant ( log(n / o)@xmath7 - 1.6 ; @xcite ) at low metallicity ( 12+log(o / h)@xmath87.6 ) , suggesting primary production by massive stars @xcite as the main contribution to nitrogen enrichment at those very low metallicities . a small fraction of hii / bcd galaxies fall in this very - low metallicity regime @xcite and are commonly referred to as extremely metal poor ( xmp ) galaxies or xmp bcds . these galaxies are the best nearby candidates for cosmologically young objects , as various arguments imply that they have formed most of their stellar mass in the past 13 gyr @xcite . at intermediate metallicity ( 7.6@xmath112+log(o / h)@xmath18.3 ) the large observed spread in n / o has been attributed mainly to the loss of heavy elements via galactic winds @xcite , and/or to the delayed release of nitrogen by intermediate and/or massive stars and oxygen by massive stars @xcite . however , the delayed - release scenario can not explain the presence of some hii / bcd galaxies with a high n / o ratio at low metallicities . the most plausible explanation for the high n / o ratio observed in these objects is the chemical pollution of the ism by nitrogen released by massive wolf - rayet ( wr ) stars as is , apparently , the case of the well - studied bcd ngc 5253 @xcite . finally , at higher metallicities ( 12+log(o / h)@xmath98.3 ) the n / o ratio clearly increases with increasing oxygen abundance and the nitrogen content is mainly due to secondary production by intermediate - mass stars . so far , an increasing number of hii / bcd galaxies has been studied with ifu spectroscopy ( see table [ tab1 ] where we provide an overview of the literature ) with main focus on the spatial properties of the ism . recently , we have started a program investigating with ifu spectroscopy the physical conditions in the ism of the most compact hii / bcd galaxies , laying special emphasis on the extinction patterns , emission line ratios , oxygen and nitrogen abundances , kinematics and the relation on the intrinsic properties of star formation as well as possible evolutionary effects @xcite . to this end , we observed a sample of hii / bcd galaxies using the gmos - ifu on gemini south and north and , more recently , with vlt / vimos . the gmos - ifu observations were performed using the gratings b600 and r600 in one slit mode , covering a total spectral range from @xmath103000 to @xmath107230 @xmath11 . this observational setup provides a pattern of 500 hexagonal elements with a projected diameter of 0``.2 , covering a total 3''.5 @xmath125 `` field of view ( fov ) . the vimos - ifu observations were obtained using the gratings hr@xmath13blue and hr@xmath13orange covering a spectral range from @xmath103710 to @xmath107700 @xmath11 . our data yield a scale on the sky of 0.''33 per fiber , and cover a fov of 13 `` @xmath12 13 '' . in figure [ fig2 ] we show the g - band acquisition image of the xmp bcd galaxy hs2236 + 1344 , in which we indicated the total field of view of 4``@xmath128 '' and the h@xmath14 map of the galaxy obtained from the composition of two different pointings with gmos - ifu . in figure [ fig3 ] we show the h@xmath14 emission line map of the galaxies um 461 and tol 65 obtained using vimos - ifu . table 2 lists the general parameters of our sample of galaxies . [ cols="<,^,^,^,^,^,^ , > , < , > " , ] [ tab2 ] using the reddening corrected emission line intensities of the spectra of each one of the spaxels we can derive the physical conditions ( electron temperature and density ) and the chemical abundances ( o and n ) across the ism of the galaxies . we calculate oxygen abundances in regions where the [ oiii]@xmath04363 emission line has been detected assuming o / h = o@xmath15/h@xmath15 + o@xmath16/h@xmath15 , while nitrogen abundances are obtained assuming n / h = icf(n ) @xmath12 n@xmath15/h@xmath15 , with icf(n ) denoting the ionization correction factor ( o@xmath15+o@xmath17)/o@xmath15 . for the sake of illustration , in figure [ fig4 ] we show the spatial distribution of 12+log(o / h ) in the gmos - ifu fov of the galaxy tol 2146 - 391 . we can see in this figure that , despite a slight depression in the inner part of the galaxy , the oxygen abundance appears to be uniform across the galaxy ( see figure 18 in @xcite ) . in @xcite we compare the spatial distribution of 12+log(o / h ) , found in @xcite , with the position of the star cluster / complexes detected in the galaxy um 408 by @xcite using high resolution near - ir k@xmath18-band images . we found that the variation of the observed data points ( see figure 9 in @xcite and figure 1 in @xcite ) may not be statistically significant , indicating that these regions have identical chemical properties within the errors . it is interesting to note that we observed a marginal gradient of decreasing abundance from the center outward in um 408 , indicating that the highest abundance values are found near the peak of h@xmath14 emission and extinction c(h@xmath19 ) , and coincident with the position of the brightest star cluster / complex @xcite . in any case , the absence of chemical overabundances in the ism of um 408 , tol 2146 - 391 , tol 0104 - 388 and hs2236 + 1344 and in the dwarf galaxies studied in the literature ( e.g. , @xcite ) indicates that the population of young star clusters is not producing localized oxygen overabundances . the most likely explanation for this is that metals formed in the current star - formation episode reside in a hot gas phase ( t@xmath1010@xmath20 k ; @xcite ) ; thus , they are not observable in optical wavelengths . whereas metals from previous star - formation events are well mixed and homogeneously distributed through the whole galaxy . in tol 2146 - 391 , the 12+log(n / h ) radial distribution shows a slight decrease with radius . this would argue in favor of heavy elements being produced in a previous burst of star - formation and dispersed within the ism by starburst - driven super - shells @xcite , while the depressed central region could be attributed to radial inflow of relatively low metallicity gas from large radii to the center , thus diluting the abundance of the gas in the nuclear region . 5 '' covered by our gmos - ifu observation . right : 12 + log(o / h ) spatial distribution . the isocontours display the h@xmath14 emission . the maximum h@xmath14 emission is indicated in the maps by an x symbol . we only considered spaxels with signal to noise ratio ( s / n ) @xmath5 3 in the [ oiii ] @xmath04363 line . more details in @xcite . ] regarding the integrated properties of the galaxies , @xcite , @xcite and @xcite suggest that there is a dependence between n / o and the ew(h@xmath19 ) , in the sense of an increasing n / o ratio with decreasing ew(h@xmath19 ) . izotov et al . @xcite argue that this trend is naturally explained by nitrogen ejection from wr stars . in the following analysis , we mainly concentrate on the spatially resolved physical properties of the ism in individual galaxies and their possible relation to the star formation process ( e.g. , the star formation history , burst parameter , wr star content ) . in figure 5 ( see @xcite ) we show the log ( n / o ) versus ew(h@xmath19 ) and 12+log(o / h ) versus log ( n / o ) for all spaxels of the galaxies tol 0104 - 388 and tol 2146 - 391 . from that figure , it can be seen that the ew(h@xmath19 ) values are rather constant , with a very small variation of equivalent widths as the n / o ratio increases . a comparison of log ( n / o ) versuss 12+log(o / h ) in tol 2146 - 391 ( figure 3b ) reveals that the log ( n / o ) values increase with the 12+log(o / h ) . this data point distribution has similar patterns to those found in hii / bcd galaxies by @xcite of increasing n / o ratios with respect to the oxygen abundance . the inner region of tol 2146 - 391 ( near the peak of h@xmath14 ) presents n / o ratios which are larger than those expected by pure primary production of nitrogen . this might be a signature of time delay between the release of oxygen and nitrogen @xcite , or gas infall or outflow . in any case , for the metallicity of tol 2146 - 391 purely secondary nitrogen enrichment appears implausible . in the case of hs2236 + 1433 , we reported in @xcite evidence for a high n / o ratio in one of the three ghiirs of the galaxy . but again , the spatial distribution of these abundances , at large scales , lead us to consider that oxygen and hydrogen are well mixed and homogeneously distributed over the ism of the galaxy . ) . bottom panel : 12 + log ( o / h ) ratio versus log(n / o ) . triangles correspond to the data points of tol 0104 - 388 and circles corresponds with the data points of tol 2146 - 391 . more details in @xcite.,width=268 ] in summary , the results obtained in our studies suggest that the chemical properties ( o , n and n / o ) across hii / bcd galaxies are fairly uniform , although a slight gradient of o and n are observed in the ism of um 408 and tol 2146 - 391 , respectively . we suggest that global hydro - dynamical processes , such as starburst - driven super - shells or / and inflow of gas might be governing the transport and mixing of metals across these galaxies , keeping the n / o ratio constant through the ism at large scales @xcite . the origin of high - ionization emission lines , such as [ ne v ] @xmath03426 , [ fe v ] @xmath04227 and he ii @xmath04686 in starburst and hii / bcd galaxies has been a subject of study in the last years , given that photoionization models of hii regions generally fail to reproduce the observed intensities of these lines ( see , e.g. , @xcite ) . several mechanisms for producing hard ionizing radiation have been proposed in the literature , such as wr stars @xcite , primordial ( zero - metallicity ) stars , high - mass x - ray binaries ( hmxb ; @xcite ) , radiative shocks @xcite and o stars at low metallicity @xcite . in @xcite and @xcite we studied with gmos - ifu the spatial distribution of he ii @xmath04686 in the compact hii galaxies tol 0104 - 388 and tol 2146 - 391 , and in the xmp bcd galaxy hs 2236 + 1344 , respectively , in order to gain insights into the nature of their hard ionization radiation and its possible dependence on the properties of the ism @xcite . based on a spaxel - by - spaxel analysis , instead of the integrated properties of the galaxies ( see figure 15 in @xcite ) , our results indicate that the spatial distribution of he ii @xmath04686 relative to h@xmath19 does not depend on the ew(h@xmath19 ) , oxygen abundance or log(n / o ) . in particular , the oxygen abundance appears to be constant through the whole extent of our sample galaxies , as already is observed in other hii / bcd galaxies ( e.g. , @xcite ; and references therein ) . the opposite trend is found if we consider the integrated spectra of galaxies , in the sense that this emission line is stronger in galaxies at low metallicity @xcite . the lack of a relationship between the hardness of the ionizing radiation and the ew(h@xmath19 ) , or age , @xcite suggests that the presence of high - ionization lines , in particular he ii @xmath04686 , is not due to a single excitation mechanism . for instance , in @xcite , it was found for a sample of galaxies , with detected and non - detected wr features , the same dependence of i(he ii @xmath04686)/i(h@xmath19 ) on the ew(h@xmath19 ) . this indicates that wr stars are not the sole origin of he ii @xmath04686 in star - forming regions ( see also @xcite ) . in galaxies with detected wr stars , the he ii @xmath04686 commonly appears to not be coincident with the location of the wr bumps ( e.g. , mrk 178 ) and these stellar features are not always seen when nebular he ii is observed ( e.g. , tol 2146 - 391 , hs2236 + 1344 , tol 65 ) . the spatial offset between wr stars and he ii @xmath04686 , in mrk 178 , is interpreted by @xcite as an effect of the mechanical energy injected by wr star winds , so wr stars are not ruled out as the main source of the observed he ii @xmath04686 in that galaxy . an examination of individual spaxels in our data cubes , and also in the integrated spectra of our sample galaxies ( e.g. , figure [ fig1 ] in this contribution ) , does not reveal any clear stellar wr features . in the case of the xmp bcd galaxy hs 2236 + 1344 , we detected the heii @xmath04686 emission line in only one of the ghiirs of the galaxy ( the brightest one ) . in this galaxy , the he ii @xmath04686 line appears to be excited through point sources within a compact volume which , interestingly , does not coincide with the position where a high n / o abundance ratio has been observed . we discuss , in @xcite , the possibility that the heii @xmath04686 emission line , in hs2236 + 1344 , is associated with wr stars , high - mass x - ray binaries ( hmxbs ) , o stars at low metallicities , and/or a low - luminosity active galactic nucleus . however , since clear wr features have not been detected in that galaxy , wr stars are excluded as the primary excitation source of he ii @xmath04686 emission . as far as the spatial distribution of oxygen abundances is concerned , we did not detect localized overabundances in any of our sample galaxies . however , we find evidence for a marginal negative radial abundance gradient , with the highest abundances seen at the position of the brightest star cluster complexes ( peak of h@xmath14 emission ) , in the hii / bcd galaxy um 408 at least . if real , the slight trend for an increasing 12+log(n / h ) abundance , in the galaxy tol 2146 - 391 , suggests rapid self - enrichment by the freshly produced heavy elements in the present starburst on scales of hundreds of pc , or , alternatively , metal pollution by a previous star formation episode . in any case , the oxygen and nitrogen appear to be well mixed across the ism of hii / bcd galaxies , suggesting efficient transport by expanding starburst - driven supershells and/or gas infall from the halo . our spectroscopic ifu studies suggest a mixture of compact sources as the main excitation source for localized he ii @xmath04686 emission in hii / bcd galaxies , without clear wr signatures , with wr stars probably being of secondary importance . in the galaxy tol 2146 - 391 , we favor the idea of extended he ii @xmath04686 emission being primarily due to radiative shocks in the ism . p.l . is supported by a post - doctoral grant sfrh / bpd/72308/2010 , funded by fct ( portugal ) , and p.p . by ciencia 2008 contract , funded by fct / mctes ( portugal ) and poph / fse ( ec ) . we are very thankful to andrew humphrey for their very useful suggestions which have improved the paper . we would like thank the anonymous referee for his / her comments and suggestions which substantially improved the paper . we acknowledge support by the fundao para a cincia e a tecnologia ( fct ) under project fcomp-01 - 0124-feder-029170 ( reference fct ptdc / fis - 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we review the results from our studies , and previous published work , on the spatially resolved physical properties of a sample of hii / bcd galaxies , as obtained mainly from integral - field unit spectroscopy with gemini / gmos and vlt / vimos .
we confirm that , within observational uncertainties , our sample galaxies show nearly spatially constant chemical abundances , similar to other low - mass starburst galaxies . they also show he ii @xmath04686 emission with properties being suggestive of a mix of excitation sources , with wolf - rayet stars being excluded as the primary one .
finally , in this contribution we include a list of all hii / bcd galaxies studied thus far with integral - field unit spectroscopy .
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in this paper we continue the geometric construction of a sequence of flips associated to an embedded projective variety begun in @xcite . we give hypotheses under which this sequence of flips exists , and state some conjectures on how positive a line bundle on a curve must be to satisfy these hypotheses . these conjectures deal with the degrees of forms defining various secant varieties to curves and seem interesting outside of the context of the flip construction . as motivation , we have the work of a. bertram and m. thaddeus . in @xcite this sequence of flips is constructed in the case of smooth curves via git , in the context of the moduli space of rank two vector bundles on a smooth curve . an understanding of this as a sequence of log flips is given in @xcite , and further examples of sequences of flips of this type , again constructed via git , are given in @xcite,@xcite . our construction , however , does not use the tools of geometric invariant theory and is closer in spirit to @xcite,@xcite . in section [ review ] , we review the constructions in @xcite and @xcite and describe the relevant results from @xcite . in section [ genofsecants ] we discuss the generation of @xmath0 by cubics . in particular , we show ( theorem [ settheo ] ) that large embeddings of varieties have secant varieties that are at least set theoretically defined by cubics . we also offer some general conjectures and suggestions in this direction for the generation of higher secant varieties . the construction of the new flips is somewhat more involved than that of the first in @xcite . we give a general construction of a sequence of birational transformations in section [ genflipconst ] , and we describe in detail the second flip in section [ secondflip ] . we mention that some of the consequences of these constructions and this point of view are worked out in @xcite . * notation : * we will decorate a projective variety @xmath1 as follows : @xmath2 is the @xmath3 cartesian product of @xmath1 ; @xmath4 is @xmath5 , the @xmath3 symmetric product of @xmath1 ; and @xmath6 is @xmath7 , the hilbert scheme of zero dimensional subschemes of @xmath1 of length @xmath8 . recall ( cf . @xcite ) that if @xmath1 is a smooth projective variety then @xmath6 is also projective , and is smooth if and only if either @xmath9 or @xmath10 . write @xmath11 for the ( complete ) variety of @xmath12-secant @xmath13-planes to @xmath1 . as this notation can become cluttered , we simply write @xmath14 for @xmath15 and @xmath0 for @xmath16 . note also the convention @xmath17 . if @xmath18 is a @xmath12-vector space , we denote by @xmath19 the space of 1-dimensional quotients of @xmath18 . unless otherwise stated , we work throughout over the field @xmath20 of complex numbers . we use the terms locally free sheaf ( resp . invertible sheaf ) and vector bundle ( resp . line bundle ) interchangeably . recall that a line bundle @xmath21 on @xmath1 is _ nef _ if @xmath22 for every irreducible curve @xmath23 . a line bundle @xmath21 is _ big _ if @xmath24 induces a birational map for all @xmath25 . * acknowledgments : * i would like to thank aaron bertram , sheldon katz , zhenbo qin , and jonathan wahl for their helpful conversations and communications . fix a line bundle @xmath26 on a fixed smooth curve @xmath1 , and denote by @xmath27 the moduli space of semi - stable rank two vector bundles @xmath28 with @xmath29 . there is a natural rational map , the _ serre correspondence _ @xmath30 given by the duality @xmath31 , taking an extension class @xmath32 to @xmath28 . one has an embedding @xmath33 ( at least in the case @xmath34 ) and @xmath35 , defined only for semi - stable @xmath28 , is a morphism off @xmath36 where @xmath37 $ ] @xcite . this map is resolved in @xcite by first blowing up along @xmath1 , then along the proper transform of @xmath0 , then along the transform of @xmath38 and so on until we have a morphism to @xmath27 . a different approach is taken in @xcite . there , for a fixed smooth curve @xmath1 of genus at least @xmath39 and a fixed line bundle @xmath26 , the moduli problem of semi - stable pairs @xmath40 consisting of a rank two bundle @xmath28 with @xmath29 , and a section @xmath41 , is considered . this , in turn , is interpreted as a git problem , and by varying the linearization of the group action , a collection of ( smooth ) moduli spaces @xmath42 ( @xmath12 as above ) is constructed . as stability is an open condition , these spaces are birational . in fact , they are isomorphic in codimension one , and may be linked via a diagram & & & & & & + m_1 & & m_2 & & & & m_k where there is a morphism @xmath43 . the relevant observations are first that this is a diagram of flips ( in fact it is shown in @xcite that it is a sequence of log flips ) where the ample cone of each @xmath44 is known . second , @xmath45 is the blow up of @xmath46 along @xmath1 , @xmath47 is the blow up of @xmath45 along the proper transform of the secant variety , and all of the flips can be seen as blowing up and down various higher secant varieties . finally , the @xmath44 are isomorphic off loci which are projective bundles over appropriate symmetric products of @xmath1 . our approach is as follows : the sequence of flips in thaddeus construction can be realized as a sequence of geometric constructions depending only on the embedding of @xmath48 . an advantage of this approach is that the smooth curve @xmath1 can be replaced by any smooth variety . even in the curve case , our approach applies to situations where thaddeus construction does not hold ( e.g. for canonical curves with @xmath49 ) . in @xcite , we show how to construct the first flip using only information about the syzygies among the equations defining the variety @xmath48 . we summarize this construction here . [ kd ] let @xmath1 be a subscheme of @xmath50 . the pair @xmath51 * satisfies condition @xmath52 * if @xmath1 is scheme theoretically cut out by forms @xmath53 of degree @xmath8 such that the trivial ( or koszul ) relations among the @xmath54 are generated by linear syzygies . we say @xmath55 satisfies @xmath52 for @xmath56 if @xmath18 is spanned by forms @xmath54 satisfying the above condition . we say simply @xmath1 satisfies @xmath52 if there exists a set @xmath57 such that @xmath51 satisfies @xmath52 , and if the discussion depends only on the existence of such a set , not on the choice of a particular set . as @xmath58 is a weakening of green s property @xmath59@xcite , examples of varieties satisfying @xmath58 include smooth curves embedded by complete linear systems of degree at least @xmath60 , canonical curves with @xmath61 , and sufficiently large embeddings of arbitrary projective varieties . to any projective variety @xmath62 defined ( as a scheme ) by forms @xmath63 of degree @xmath8 , there is an associated rational map @xmath64 defined off the common zero locus of the @xmath54 , i.e. off @xmath1 . this map may be resolved to a morphism @xmath65 by blowing up @xmath50 along @xmath1 , or equivalently by projecting from the closure of the graph @xmath66 . we have the following results on the structure of @xmath67 : [ embed]@xcite let @xmath51 be a pair that satisfies @xmath52 . then : 1 . @xmath68 is an embedding off of @xmath69 , the variety of @xmath8-secant lines . 2 . the projection of a positive dimensional fiber of @xmath67 to @xmath70 is either contained in a linear subspace of @xmath1 or is a linear space intersecting @xmath1 in a @xmath8-tic hypersurface . if , furthermore , @xmath1 does not contain a line then @xmath67 is an embedding off the proper transform of @xmath69 . @xmath71 [ getabundle]@xcite let @xmath55 satisfy @xmath58 and assume @xmath62 is smooth , irreducible , contains no lines and contains no quadrics . then : 1 . the image of @xmath72 under @xmath67 is @xmath73 . 2 . @xmath74 is a rank two vector bundle on @xmath73 , where @xmath75 is the proper transform of the hyperplane section on @xmath70 . @xmath76 is the @xmath77-bundle @xmath78 . + @xmath71 this implies @xmath79 , and hence @xmath80 , are smooth . to complete the flip , we construct a base point free linear system on @xmath47 , and take @xmath81 to be the image of the associated morphism . denoting @xmath82 , the sheaf @xmath83 is locally free of rank @xmath84 on @xmath73 . write @xmath85 and rename @xmath67 as @xmath86 : [ flip]@xcite let @xmath55 satisfy @xmath58 and assume @xmath62 is smooth , irreducible , contains no lines and contains no plane quadrics . then there is a flip as pictured below with : 1 . @xmath87 , @xmath47 , and @xmath81 smooth 2 . @xmath88 , hence if @xmath89 then @xmath90 3 . @xmath91 is the blow up of @xmath81 along @xmath92 4 . @xmath93 is the blow up of @xmath87 along @xmath94 5 . @xmath95 , induced by @xmath96 , is an embedding off of @xmath92 , and the restriction of @xmath95 is the projection @xmath97 6 . @xmath86 , induced by @xmath98 , is an embedding off of @xmath94 , and the restriction of @xmath86 is the projection @xmath99 & e_2 _ ^h_1 & & & _ ^h_1 & + (e ) _ ^+ & & (f ) ^^- & ^^+ & & m_2 ^^- + & h^2x & & ^s_0 |>__1 & ^s_1 & to continue this process following thaddeus , we need to construct a birational morphism @xmath100 which contracts the transforms of @xmath101-secant @xmath39-planes to points , and is an embedding off their union . the natural candidate is the map induced by the linear system @xmath102 . we discuss two different reasons for this choice that will guide the construction of the entire sequence of flips . section [ genofsecants ] addresses the question of when this system is globally generated . note that we abuse notation throughout and identify line bundles via the isomorphism @xmath103 . the first reason is quite naive : just as quadrics collapse secant lines because their restriction to such a line is a quadric hypersurface , so too do cubics vanishing twice on a variety collapse every @xmath101-secant @xmath104 because they vanish on a cubic hypersurface in such a plane . similarly , to collapse the transform of each @xmath105-secant @xmath106 via a morphism @xmath107 , the natural system is @xmath108 . another reason is found by studying the ample cones of the @xmath44 . note that the ample cone on @xmath109 is bounded by the line bundles @xmath75 and @xmath98 . both of these bundles are globally generated , and by theorems [ embed ] and [ getabundle ] , they each give birational morphisms whose exceptional loci are projective bundles over hilbert schemes of points of @xmath1 ( @xmath110 and @xmath73 respectively ) . on @xmath81 , the ample cone is bounded on one side by @xmath96 . this gives the map @xmath111 mentioned in theorem [ flip ] ; in particular it is globally generated , the induced morphism is birational , and its exceptional locus is a projective bundle over @xmath73 . on the other side , the ample cone contains a line bundle of the form @xmath112 ( @xcite ) . in fact , if @xmath1 is a smooth curve embedded by a line bundle of degree at least @xmath113 , it is shown in @xcite that the case @xmath114 suffices , i.e. that the ample cone is bounded by @xmath96 and @xmath102 . therefore , it is natural to look for conditions under which @xmath102 is globally generated . thaddeus further shows that under similar positivity conditions , the ample cone of @xmath115 is bounded by @xmath116 and @xmath108 . noting the fact that @xmath117 , it is not difficult to see ( using zariski s main theorem ) that this system will be globally generated if @xmath118 is scheme theoretically defined by cubics , because a cubic vanishing twice on a variety must also vanish on its secant variety . unfortunately , there are no general theorems on the cubic generation of secant varieties analogous to quadric generation of varieties . we address this question in the next section . examplesomecubics some examples of varieties whose secant varieties are _ ideal _ theoretically defined by cubics include : 1 . @xmath1 is any veronese embedding of @xmath50 @xcite 2 . @xmath1 is the plcker embedding of the grassmannian @xmath119 for any @xmath120 @xcite . @xmath1 is the segre embedding of @xmath121 @xcite . + @xmath71 we prove a general result : [ settheo ] let @xmath62 satisfy condition @xmath58 . then @xmath122 is * set * theoretically defined by cubics for @xmath123 . we begin with the case @xmath124 , the higher embeddings being more elementary . let @xmath125 , @xmath126 , and @xmath127 the linear subspace of @xmath128 defined by the hyperplanes corresponding to all the quadrics in @xmath70 vanishing on @xmath1 . then @xmath129 as schemes and we show , noting that @xmath130 is ideal theoretically defined by cubics , that @xmath131 as sets . note that the map @xmath132 can be viewed as the composition of the embedding @xmath133 with the projection from @xmath127 , @xmath134 . let @xmath135 . if @xmath136 , then @xmath137 hence @xmath138 . otherwise , any secant line @xmath139 to @xmath18 through @xmath140 intersects @xmath18 in a length two subscheme @xmath141 . @xmath141 considered in @xmath70 determines a unique line in @xmath70 whose image in @xmath128 is a plane quadric @xmath142 spanning a plane @xmath143 . if @xmath144 is non - empty then @xmath145 , hence either @xmath127 intersects @xmath143 in a line @xmath146 through @xmath140 or @xmath147 . in the first case @xmath146 is a secant line to @xmath148 , in the second @xmath149 . in either situation @xmath138 . cubics.ps31.5 all that remains is the case @xmath150 and @xmath151 is empty . however in this case the line @xmath139 , and hence the scheme @xmath152 is collapsed to a point by the projection . as the rational map @xmath153 is an embedding off @xmath0 , this implies @xmath141 lies on the image of a secant line to @xmath62 . as a length two subscheme of @xmath70 determines a unique line , @xmath154 must be the image of a secant line to @xmath62 contradicting the assumption that @xmath151 is empty . for @xmath155 , note that the projection from @xmath127 is an embedding off @xmath156 ( this can be derived directly from theorem [ embed ] or see @xcite ) . therefore , if @xmath127 intersects a secant line , the line lies in @xmath127 , hence is a secant line to @xmath148 . examplecubics as green s @xmath59 implies @xmath58 , this shows that the secant varieties to the following varieties are set theoretically defined by cubics : 1 . @xmath1 a smooth curve embedded by a line bundle of degree @xmath157 , @xmath158 . @xmath1 a smooth curve with @xmath49 , embedded by @xmath159 , @xmath160 . 3 . @xmath1 a smooth variety embedded by @xmath161 , @xmath162 , @xmath139 very ample . @xmath1 a smooth variety embedded by @xmath163 for all @xmath164 , @xmath139 ample . + @xmath71 remarkalmostk3 notice that in the case @xmath124 of proposition [ settheo ] , the cubics that at least set theoretically define the secant variety satisfy @xmath165 . this is because : 1 . the ideal of the secant variety of @xmath166 is generated by cubics , and the module of syzygies is generated by linear relations @xcite . hence @xmath167 satisfies @xmath165 . it is clear from the definition that if @xmath168 satisfies @xmath52 , then any linear section does as well . + @xmath71 examplequadhyp if @xmath168 is a smooth quadric hypersurface , then @xmath169 is given by the intersection of @xmath170 with a hyperplane @xmath127 . furthermore , the intersection of @xmath171 with @xmath127 is a scheme @xmath172 with @xmath173 . therefore , a general smooth quadric hypersurface has @xmath174 as schemes , hence @xmath175 satisfies @xmath165 . @xmath71 we record here a related conjecture of eisenbud , koh , and stillman as well as a partial answer proven by m.s . ravi : @xcite let @xmath139 be a very ample line bundle that embeds a smooth curve @xmath1 . for each @xmath12 there is a bound on the degree of @xmath139 such that @xmath36 is ideal theoretically defined by the @xmath176 minors of a matrix of linear forms . @xcite if @xmath177 , then @xmath36 is * set * theoretically defined by the @xmath176 minors of a matrix of linear forms . these statements provide enough evidence to make the following basic : let @xmath139 be an ample line bundle on a smooth variety @xmath1 , @xmath178 fixed . then for all @xmath25 , @xmath179 embeds @xmath1 so that @xmath36 is ideal theoretically defined by forms of degree @xmath180 , and furthermore satisfies condition @xmath181 . remark5secant3plane if @xmath1 is a curve with a @xmath182-secant @xmath101-plane , then any cubic vanishing on @xmath0 must vanish on that @xmath101-plane . hence @xmath0 can not be set theoretically defined by cubics . this should be compared to the fact that if @xmath1 has a trisecant line , then @xmath1 can not be defined by quadrics . in particular , this shows that green s condition @xmath59 is not even sufficient to guarantee that their exists a cubic vanishing on @xmath0 . for example , if @xmath1 is an elliptic curve embedded in @xmath183 by a line bundle of degree @xmath182 , then @xmath0 is a quintic hypersurface . therefore , any uniform bound on the degree of a linear system that would guarantee @xmath0 is even set theoretically defined by cubics must be at least @xmath184 . @xmath71 we can use earlier work to give a more geometric necessary condition for @xmath0 to be defined as a scheme by cubics . specifically , in @xcite it is shown that the intersection of @xmath79 with the exceptional divisor @xmath28 of the blow up of @xmath70 along @xmath1 is isomorphic to @xmath185 . this implies that if @xmath186 is the blow up along @xmath1 , then @xmath187 . in fact , it is easy to verify that if @xmath1 is embedded by a line bundle @xmath139 , then @xmath188 where @xmath189 is identified with the fiber over @xmath140 of the projectivized conormal bundle of @xmath62 . now , if @xmath0 is defined as a scheme by cubics , then the base scheme of @xmath190 is precisely @xmath79 . the restriction of this series to @xmath189 is thus a system of quadrics whose base scheme is @xmath191 . in other words , if @xmath1 is a smooth variety embedded by a line bundle @xmath139 that satisfies @xmath58 and if @xmath0 is scheme theoretically defined by cubics , then for every @xmath192 the line bundle @xmath193 is very ample on @xmath191 and @xmath194 is scheme theoretically defined by quadrics . in the case @xmath1 is a curve , this implies that a uniform bound on @xmath195 that would imply @xmath0 is defined by cubics must be at least @xmath184 , the same bound encountered in remark [ 5secant3plane ] . the construction in @xcite shows similarly that any uniform bound that would imply @xmath36 is defined by @xmath196-tics must be at least @xmath197 . we combine these observations with the degree bounds encountered in the constructions of @xcite and @xcite to form the following : [ degreeboundsconj ] let @xmath1 be a smooth curve embedded by a line bundle @xmath139 . if @xmath198 then @xmath199 is defined as a scheme by forms of degree @xmath105 . if @xmath200 then @xmath199 satisfies condition @xmath201 . suppose that @xmath1 satisfies @xmath58 , is smooth , and contains no lines and no plane quadrics . suppose further that @xmath0 is scheme theoretically defined by cubics @xmath202 , and that @xmath0 satisfies @xmath165 . under these hypotheses , we construct a second flip as follows : we know that @xmath102 is globally generated by the discussion above ; hence this induces a morphism @xmath100 which agrees with the map given by the cubics @xmath203 on the locus where @xmath81 and @xmath70 are isomorphic . by theorem [ embed ] , @xmath204 is a birational morphism . we wish first to identify the exceptional locus of @xmath204 . it is clear that @xmath204 will collapse the image of a @xmath101-secant @xmath39-plane to a point , hence the exceptional locus must contain the transform of @xmath38 . however by theorem [ embed ] , we know that the rational map @xmath205 is an embedding off @xmath206 , the trisecant variety to the secant variety . this motivates the following [ secantsareequal ] let @xmath48 be an irreducible variety . assume either of the following : 1 . @xmath36 is defined as a scheme by forms of degree @xmath207 . 2 . @xmath1 is a smooth curve embedded by a line bundle of degree at least @xmath208 . then @xmath209 as schemes . first , choose a @xmath210-secant @xmath12-plane @xmath143 . @xmath143 then intersects @xmath199 in a hypersurface of degree @xmath105 , hence every line in @xmath143 lies in @xmath211 . as @xmath212 is reduced and irreducible , @xmath213 as schemes . for the converse , assume the first condition is satisfied choose a line @xmath139 that intersects @xmath199 in a scheme of length at least @xmath105 . it is easy to verify that @xmath36 is singular along @xmath199 , hence every form that vanishes on @xmath36 must vanish @xmath214 times on @xmath139 . by hypothesis , however , @xmath36 is scheme theoretically defined by forms of degree @xmath207 , hence each of these forms must vanish on @xmath139 . the sufficiency of the second condition follows from thaddeus construction and ( * ? ? ? * , ( i ) ) . this implies that if @xmath36 satisfies @xmath181 and if @xmath209 , then the map @xmath215 given by the forms defining @xmath36 is an embedding off of @xmath216 . we use theorem [ embed ] to understand the structure of these maps via the following two lemmas : [ nolines ] if the embedding of a projective variety @xmath168 is @xmath217-very ample , then the intersection of two @xmath196-secant @xmath210-planes , if nonempty , must lie in @xmath36 ( in fact , it must be an @xmath218 secant @xmath219 for some @xmath220 ) . in particular , @xmath216 has dimension @xmath221 . the first statement is elementary : assume two @xmath196-secant @xmath210-planes intersect at a single point . if the point is not on @xmath1 , then there are @xmath222 points of @xmath1 that span a @xmath223-plane , which is impossible by hypothesis . hence the intersection lies in @xmath17 . a simple repetition of this argument for larger dimensional intersections gives the desired result . the statement of the dimension follows immediately ; or see @xcite . [ possiblepreimages ] let @xmath62 be an irreducible variety whose embedding is @xmath217-very ample . assume that @xmath36 satisfies @xmath181 , and that @xmath224 as schemes . let @xmath225 be the closure of the graph of @xmath226 with projection @xmath227 . if @xmath228 is a point in the closure of the image of @xmath226 and @xmath229 is the fiber over @xmath228 then @xmath230 is one of the following : 1 . a reduced point in @xmath231 2 . a @xmath196-secant @xmath210-plane 3 . * contained * in a linear subspace of @xmath36 the first and third possibilities follow directly from theorem [ embed ] for the second , note that a priori @xmath230 could be any linear space intersecting @xmath36 in a hypersurface of degree @xmath180 . however , lemma [ nolines ] and the hypothesis that @xmath224 immediately imply that any such linear space must be @xmath105 dimensional ; hence a @xmath196-secant @xmath210-plane . with these results in hand we present the general construction . let @xmath232 be an irreducible projective variety and suppose @xmath233 , is a collection of dominant , birational maps . define the * dominating variety * of the collection , denoted @xmath234 , to be the closure of the graph of @xmath235 denote by @xmath236 the projection of @xmath234 to @xmath237 . note that @xmath236 is birationally isomorphic to @xmath238 for all @xmath239 . note further that if the @xmath240 are all morphisms then @xmath241 , in other words only rational maps contribute to the structure of the dominating variety . definition / notationk2l we say @xmath62 satisfies condition @xmath242 if @xmath243 satisfies condition @xmath244 for @xmath245 ; hence @xmath1 satisfies @xmath246 if and only if @xmath1 satisfies @xmath58 , @xmath1 satisfies @xmath247 if and only if @xmath1 satisfies @xmath58 and @xmath0 satisfies @xmath165 , etc . @xmath71 if @xmath62 satisfies condition @xmath242 , then each rational map @xmath248 is birational onto its image for @xmath249 , and assuming the conclusion of lemma [ secantsareequal ] each @xmath250 is an embedding off @xmath243 . therefore @xmath251 is the closure of the image of @xmath250 , @xmath252 is the closure of the graph of @xmath250 , and in the notation of theorem [ flip ] @xmath253 and @xmath254 . note @xmath255 . [ allblowups ] @xmath256 is the blow up of @xmath257 along the proper transform of @xmath258 , @xmath249 . this is immediate from the definition ( or see @xcite ) . remarkbert - thadspaces the spaces constructed in @xcite are of the type @xmath259 . the spaces @xmath260 and @xmath115 constructed in @xcite are @xmath261 and @xmath262 . @xmath71 our goal is to understand explicitly the geometry of this web of varieties generalizing theorem [ flip ] . in the next section we describe in detail the structure of the second flip . as each subsequent flip requires the understanding of @xmath263 for larger @xmath12 , it is not clear that the process will continue nicely beyond the second flip ( at least for varieties of arbitrary dimension ) . let @xmath62 be a smooth , irreducible variety that satisfies @xmath247 . the diagram of varieties we study in this section is : where @xmath264 is the dominating variety of the pair of birational maps @xmath265 and @xmath266 ; and where we have yet to construct the two rightmost varieties . we write @xmath267 and @xmath268 ( recall all three spaces are smooth by theorem [ flip ] ) . [ wtv2 ] let @xmath269 be a smooth , irreducible variety of dimension @xmath270 that satisfies @xmath247 . assume that @xmath1 is embedded by a complete linear system @xmath271 and that the following conditions are satisfied : 1 . @xmath139 is @xmath272-very ample 2 . if @xmath160 , then for every point @xmath192 , @xmath273 3 . @xmath274 as schemes 4 . the projection of @xmath1 into @xmath275 , from any embedded tangent space is such that the image is projectively normal and satisfies @xmath58 as the proof of theorem [ wtv2 ] is somewhat involved , we break it into several pieces . we begin with a lemma and a crucial observation , followed by the proof of the theorem . the observation invokes a technical lemma whose proof is postponed until the end . remark on the hypotheseseasilysatisfied note that if @xmath1 is a smooth curve embedded by a line bundle of degree at least @xmath113 , then conditions @xmath278 are automatically satisfied . conjecture [ degreeboundsconj ] would imply condition @xmath247 holds also . furthermore , if @xmath279 and @xmath280 then condition @xmath281 implies condition @xmath39 . if @xmath160 , then the image of the projection from the space tangent to @xmath1 at @xmath140 is @xmath282 furthermore , by the discussion after remark [ 5secant3plane ] any such projection of @xmath1 will be generated as a scheme by quadrics when @xmath0 is defined by cubics , hence condition @xmath283 is not unreasonable . [ projectionsarenotsobad ] with hypotheses as in theorem [ wtv2 ] , the image of the projection of @xmath1 into @xmath284 , @xmath285 , is @xmath191 , hence is smooth . furthermore , it contains no lines and it contains no plane quadrics except for the exceptional divisor , which is the quadratic veronese embedding of @xmath286 . if @xmath287 the statement is clear . otherwise , let @xmath288 denote the closure of the image of projection from the embedded tangent space to @xmath1 at @xmath140 . as mentioned above , @xmath289 , hence is smooth . let @xmath290 denote the exceptional divisor . the existence of a line or plane quadric not contained in @xmath291 is immediately seen to be impossible by the @xmath272-very ampleness hypothesis . observationcrucialobs let @xmath295 be the projection and let @xmath296 be the fiber over @xmath192 ; hence @xmath296 is the blow up of @xmath284 along a copy of @xmath191 . we again denote this variety by @xmath297 , and the embedding of @xmath298 into @xmath284 satisfies @xmath58 by hypothesis . the restriction of @xmath299 to @xmath296 can thus be identified with @xmath300 , and , noting lemma [ projectionsarenotsobad ] , it seems that theorem [ embed ] could be applied . unfortunately , it is not clear that this restriction should be surjective on global sections . however , by lemma [ nearsurjrest ] below , the image of the morphism on @xmath296 induced by the restriction of global sections is isomorphic to the image of the morphism given by the complete linear system @xmath301 . hence by the fourth hypothesis and lemma [ projectionsarenotsobad ] , the only collapsing that occurs in @xmath296 under the morphism @xmath302 is that of secant lines to @xmath288 . now , for some @xmath192 , suppose that a secant line @xmath172 in @xmath296 is collapsed to a point by the projection @xmath302 . then @xmath172 is the proper transform of a secant line to @xmath303 , but every such secant line is the intersection of @xmath296 with a @xmath101-secant @xmath104 through @xmath192 . for example , if @xmath304 is the secant line through @xmath305 , @xmath306 , then @xmath172 is the intersection of @xmath296 with the proper transform of the plane spanned by @xmath307 . it should be noted that the two dimensional fiber associated to the collapsing of a plane spanned by a quadric in the exceptional divisor ( lemma [ projectionsarenotsobad ] ) will take the place of a @xmath101-secant @xmath104 spanned by a non - curvilinear scheme contained in the tangent space at @xmath140 . ( of theorem [ wtv2 ] ) let @xmath308 be a point in the image of @xmath204 . the fiber over @xmath228 is mapped isomorphically into @xmath309 by the projection @xmath310 . we are therefore able to study @xmath311 by looking at the fiber of the projection @xmath302 , and projecting to @xmath312 and to @xmath309 . by applying lemma [ possiblepreimages ] to the map @xmath313 , the projection to @xmath312 is contained as a scheme in the _ total _ transform of one of the following ( note the more refined division of possibilities ) : 1 . a point in @xmath314 2 . a @xmath101-secant @xmath39-plane to @xmath1 not contained in @xmath0 3 . a linear subspace of @xmath0 not tangent to @xmath1 4 . a linear subspace of @xmath0 tangent to @xmath1 if the projection is a @xmath101-secant @xmath39-plane , then by observation [ crucialobs ] the projection to @xmath312 is a @xmath101-secant @xmath39-plane blown up at the three points of intersection , and so the image in @xmath309 is a @xmath104 that has undergone a cremona transformation . in the third case , observation [ crucialobs ] shows that either the projection to @xmath312 is the _ proper _ transform of a secant line to @xmath1 , or that the projection to @xmath315 is a linear subspace of @xmath0 that is not a secant line . in the first case , every such space is collapsed to a point by @xmath86 . the second implies @xmath204 has a fiber of dimension @xmath8 that is contained in @xmath316 . because @xmath317 is a @xmath77-bundle , this implies the projection of the fiber to @xmath315 is contained in a linear subspace @xmath143 of @xmath0 of dimension @xmath318 . furthermore , the proper transform of @xmath143 is collapsed to a @xmath8 dimensional subspace of @xmath309 , in particular the general point of @xmath143 lies on a secant line _ contained in @xmath143 _ by theorem [ getabundle ] . therefore @xmath319 has @xmath320 , hence @xmath321 but this is impossible by lemma [ nolines ] and the restriction that @xmath143 not be tangent to @xmath1 . in the final case , the proper transform in @xmath312 of a linear space @xmath322 tangent to @xmath1 at a point @xmath140 is @xmath323 . denote the exceptional @xmath324 by @xmath154 ; lemma [ projectionsarenotsobad ] implies @xmath325 is the quadratic veronese embedding of @xmath326 . a simple dimension count shows that the restriction to @xmath154 of the projective bundle @xmath327 arising from the blow up of @xmath312 along @xmath79 is precisely the restriction to @xmath154 of the projective bundle arising from the _ induced _ blow up of @xmath328 along @xmath191 ; denote this variety @xmath329 . furthermore , the transform of @xmath323 in @xmath264 is a @xmath77-bundle over @xmath330 . now by lemma [ nearsurjrest ] , every fiber of @xmath204 contained in @xmath330 is either a point or is isomorphic to a @xmath104 spanned by a plane quadric in @xmath154 . remarkaltprooffork=2 for curves , parts @xmath101 and @xmath283 of the proof can also be concluded by showing that any line contained in @xmath0 must be a secant or tangent line ( this is immediate from the @xmath331-very ample hypothesis ) . [ killideal ] let @xmath332 be a flat morphism of smooth projective varieties . let @xmath333 be a smooth fiber and let @xmath139 be a locally free sheaf on @xmath1 . if @xmath334 and @xmath335 for all @xmath336 , then @xmath337 for all @xmath336 . the hypotheses easily give the vanishing @xmath337 for all @xmath338 . for @xmath339 , take the exact sequence on @xmath148 @xmath340 because @xmath341 is supported at the point @xmath140 , it suffices to check that @xmath342 . @xmath93 flat implies @xmath343 , hence @xmath344 is trivial . now , @xmath345 implies @xmath346 . [ nearsurjrest ] under the hypotheses of theorem [ wtv2 ] , the image of @xmath296 under the projection @xmath302 is isomorphic to the image of @xmath296 under the morphism induced by the complete linear system associated to @xmath347 . * step 1 : * _ if @xmath348 are mapped to the same point under the projection to @xmath349 , then @xmath228 and @xmath350 map to the same point under the projection to @xmath351 . _ this is clear from the construction of the maps in question as the projections @xmath352 and @xmath353 respectively . * step 2 : * _ re - embed @xmath354 via the map associated to @xmath355 . _ this gives a map @xmath356 induced by a subspace of @xmath357 where @xmath358 . as @xmath354 is an embedding , the induced maps on @xmath296 have isomorphic images for all @xmath178 . we have , therefore , only to show @xmath359 surjects onto @xmath360 for some @xmath12 . * step 3 : * _ the map @xmath361 is surjective for all @xmath362 . _ this follows directly from the fact that @xmath0 is scheme theoretically defined by cubics and the construction of @xmath363 as @xmath364 . we show @xmath366 . let @xmath367 be the projection . by the projective normality assumption of theorem [ wtv2 ] , @xmath368 for all @xmath369 since @xmath370 is flat . ampleness of @xmath371 implies @xmath372 for all @xmath373 , where @xmath374 may depend on @xmath13 . from the exact sequence @xmath375 a finite induction shows that if @xmath376 for @xmath377 , some @xmath378 then @xmath366 for all @xmath362 . as @xmath379 , we have @xmath380 as soon as @xmath381 , the right side is @xmath382-nef and , because @xmath382 is birational , the restriction of the right side to the general fiber of @xmath382 is big . hence by @xcite , @xmath383 for @xmath384 . again by the ampleness of @xmath371 , we have @xmath376 for @xmath377 , @xmath13 as above . as in theorem [ getabundle ] , we show that the restriction of @xmath204 to the transform of @xmath38 is a projective bundle over @xmath386 . by a slight abuse of notation , write @xmath387 for the image of the proper transform of @xmath38 . note the following : a point @xmath395 determines a unique @xmath39-plane @xmath396 in @xmath397 by theorem [ wtv2 ] . for every such @xmath140 , the homomorphism @xmath398 has rank @xmath101 , hence gives a point in @xmath399 . the image of the associated morphism clearly coincides with the natural embedding of @xmath386 into @xmath399 described in @xcite . as in @xcite , there is a morphism @xmath400 so that the composition factors @xmath401 . this is constructed by associating to every @xmath389 the rank @xmath281 homomorphism : @xmath402 where @xmath396 is the @xmath104 in @xmath403 associated to @xmath141 . we wish to show further that blowing up @xmath38 along @xmath1 and then along @xmath0 resolves the singularities of @xmath38 . by theorem [ flip ] , @xmath407 is the blow up of @xmath403 along @xmath92 , hence it suffices to show @xmath408 is a smooth subvariety of @xmath409 . let @xmath414 denote the universal subscheme . we have morphisms @xmath415 and @xmath416 , and it is routine to check that @xmath417 . hence ( cf . @xcite ) @xmath418 maps @xmath419 to the nested hilbert scheme @xmath411 , where closed points of @xmath420 correspond to pairs of subschemes @xmath421 with @xmath422 . furthermore , via the description of the structure of the map @xmath204 , it is clear that the morphism of @xmath386-schemes @xmath423 is finite and birational . it is shown in @xcite that @xmath420 is smooth , hence this is an isomorphism . let @xmath424 be the blow up of @xmath403 along @xmath409 ; note @xmath424 is smooth . to construct @xmath425 , we first construct the exceptional locus as a projective bundle over @xmath386 . write @xmath426 . it is important to note that @xmath438 for all @xmath439 as the direct image on the right will differ from @xmath440 by a line bundle . hence for all @xmath439 the same morphism @xmath436 is induced by the surjection @xmath441 @xcite[free ] let @xmath21 be an invertible sheaf on a complete variety @xmath1 , and let @xmath442 be any locally free sheaf . assume that the map @xmath443 induced by @xmath444 is a birational morphism and that @xmath445 is an isomorphism in a neighborhood of @xmath192 . then for all @xmath120 sufficiently large , the map @xmath446 is surjective . taking @xmath447 and @xmath448 , the map induced by the linear system associated to @xmath449 is base point free off @xmath450 for @xmath451 . to show this gives a morphism , one shows the restriction of above linear system to the divisor @xmath450 induces a surjection on global sections , hence restricts to the map @xmath452 above . for this , define @xmath453 and write @xmath454 where @xmath455 and @xmath456 . by the above discussion , @xmath457 is nef for @xmath362 and it is routine to verify that @xmath458 is a big and nef @xmath459-divisor ; hence @xmath460 . [ topofflip2 ] with hypotheses as in theorem [ wtv2 ] and for @xmath12 sufficiently large , the morphism @xmath461 induced by the linear system @xmath462 is an embedding off of @xmath450 and the restriction of @xmath463 to @xmath450 is the morphism @xmath436 described above . because @xmath425 is the image of a smooth variety with reduced , connected fibers it is normal ( cf . let @xmath466 be a fiber of @xmath463 over a point @xmath467 . @xmath468 is a fiber of a @xmath433 bundle over @xmath386 , hence the normal bundle sequence becomes : @xmath469 this sequence splits , and allowing the elementary calculations @xmath470 and @xmath471 for all @xmath472 , @xmath425 is smooth by a natural extension of the smoothness portion of castelnuovo s contractibility criterion for surfaces given in @xcite . [ flip2 ] let @xmath269 be a smooth , irreducible variety of dimension @xmath270 that satisfies @xmath247 , with @xmath475 . assume that @xmath1 is embedded by a complete linear system @xmath271 and that the following conditions are satisfied : 1 . @xmath139 is @xmath272-very ample and @xmath274 as schemes 2 . the projection of @xmath1 into @xmath275 , from any embedded tangent space is such that the image is projectively normal and satisfies @xmath58 3 . if @xmath160 , then for every point @xmath192 , @xmath273 1 . @xmath476 and @xmath477 smooth 2 . @xmath478 ; as @xmath475 , @xmath479 3 . @xmath480 and @xmath481 4 . @xmath482 is the blow up of @xmath476 along @xmath483 5 . @xmath484 is the blow up along @xmath485 6 . @xmath486 , induced by @xmath487 , is an embedding off of @xmath483 , and the restriction of @xmath486 is the projection @xmath488 7 . @xmath489 , induced by @xmath490 , is an embedding off of @xmath491 , and the restriction of @xmath489 is the projection @xmath492 8 . @xmath493 is isomorphic to the nested hilbert scheme @xmath494 , hence is smooth . m andreatta and j a winiewski , a view on contractions of higher - dimensional varieties , in _ algebraic geometry - santa cruz 1995 _ , proc . pure math . , 62 , part 1 , amer soc . , providence , ri 1997 , pp . 153 - 183 . t jzefiak , p pragacz , and j weyman , resolutions of determinantal varieties and tensor complexes associated with symmetric and antisymmetric matrices , in _ tableaux de young et foncteurs de schur en algbre et gomtrie _ , astrisque 87 - 88 , socit mathmatique de france , 1981 , pp . 109 - 189 .
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we show how to use information about the equations defining secant varieties to smooth projective varieties in order to construct a natural collection of birational transformations .
these were first constructed as flips in the case of curves by m. thaddeus via geometric invariant theory , and the first flip in the sequence was constructed by the author for varieties of arbitrary dimension in an earlier paper .
we expose the finer structure of a second flip ; again for varieties of arbitrary dimension .
we also prove a result on the cubic generation of the secant variety and give some conjectures on the behavior of equations defining the higher secant varieties .
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the sense of hearing requires exquisite mechanical detection , with barely audible tones evoking displacements of the basilar membrane on the order of angstroms @xcite . the auditory system is also highly tuned , with frequency selectivity in various species reaching 0.1% , and the frequency range reaching as high as 100 khz . over the last 60 years , there have been significant advances in our understanding of the inner ear . however , the detailed mechanisms of the auditory system are still not understood , and thus , deficits are being mostly aided by technological solutions , such as cochlear implants . nonlinear effects have been shown to be important for the extreme sensitivity and robustness of the inner ear @xcite . compressive nonlinearity plays a role both in protecting the cells from damage , and for ensuring that the lowest levels of incoming sound receive the highest degree of amplification @xcite . nonlinear response has been demonstrated both at the organism level @xcite and in the motility of individual hair cells @xcite . here , we focus on the latter , since hair cells constitute the main functional elements in the detection process . on the apical surface of the hair cell , 20 - 300 stereocilia comprise the hair bundle ; tips of neighboring stereocilia are connected by tip links @xcite . during stimulus , deflections due to incoming sound induce shearing of the stereocilia comprising the hair bundle , increasing the tension on the tip links between them . the links are coupled to mechanically sensitive ion channels , which open and close in response to the stimulus forces @xcite . the resulting influx of ions depolarizes the cell , and thus leads to the release of neurotransmitters . when the bundles are deflected by sound waves , they move in a highly viscous medium . an active process has therefore been proposed to explain the high acuity of hearing @xcite . several models were developed to explain how the hair cell generates the forces needed to pump energy into the oscillation . these include amplification by active hair bundle motility @xcite and electromotility , a process of elongation and contraction of the hair cell soma in response to electrical stimulation @xcite . empirical evidence provided by otoacoustic emissions @xcite indicates an underlying active mechanism in the auditory response . synchronization of hair bundle oscillations by periodic perturbations , exhibited over a wide range of frequencies @xcite , suggests that the auditory system is analogous to general forced oscillatory media , similar to autocatalysis in chemical media or enzymatic dynamics in physiology @xcite . to explain the role of active amplification in hearing , theoretical models proposed that hair cell response follows temporal dynamics that arise through a _ hopf _ bifurcation @xcite , a generic mechanism that describes the birth of oscillatory behavior once the critical value of a control parameter is exceeded @xcite . the model equations predict high gain and sharp frequency selectivity at low - amplitude stimuli , and a reduction of both with increasing amplitudes . the amplification gain was shown to diverge as the control parameter approaches a critical value and to diminish away from the critical point @xcite . consequently , proximity to a hopf bifurcation is recognized to provide important advantages in explaining the phenomenology of hearing @xcite . experimental studies have shown that the dynamic response of a hair cell intertwines many degrees of freedom and complex processes , including biochemical feedback on the control parameter @xcite . yet , theoretical models have typically included only additive forcing terms @xcite . we develop a general theoretical framework that allows a systematic study of the impact of parametric versus additive forcing on the resonant response in the cochlea . the two types of forcing reflect the coupling between the driving force and the original ( unforced ) system , with the parametric term reflecting a situation in which the periodic forcing directly impacts one or more parameters . for example , the light - sensitive oscillatory belousov zhabotinsky chemical reaction under periodic illumination is a parametrically forced system , since the light affects a chemical reaction @xcite . the oscillatory nature of the belousov zhabotinsky chemical reaction is analogous to the spontaneous hair bundle oscillations , while the role of illumination @xcite is analogous to feedback by calcium ions . frequency locking is a generic feature of periodically driven oscillatory systems that exhibit resonant behavior , examples of which include faraday waves , nonlinear optical solitons , josephson junctions , and chemical reactions . frequency locked response has also been shown to be a crucial feature of auditory detection @xcite . we examine the properties of the 1:1 resonance domain ( arnold tongue ) in the cochlear response . we consider hair cells to be poised in the vicinity of the hopf bifurcation @xcite and derive a universal normal form equation that includes both additive and parametric driving forces . specifically , we examine the distinctions in the transition from unlocked to locked oscillations , under different types of forcing . this model of the cochlear response allows for the coexistence of multi - modal frequency locking ( i.e. , beyond 1:1 resonance ) and elucidates the presence of super- vs. sub - critical forms of the frequency locking transition @xcite . thus , this study provides a framework for incorporating both biochemical and mechanical feedback in the description of the auditory system . periodically forced oscillatory systems can be mathematically represented as follows : @xmath1 where @xmath2 is a set of observables ( with @xmath3 being an integer ) , @xmath4 and @xmath5 denote functions that describe interactions between observables , and @xmath6 is the frequency of the driving force . in what follows , we will consider , without a loss of generality , the two variable activator inhibitor fitzhugh - nagumo ( fhn ) model @xcite , in which @xmath7 . forcing acts on the activator variable @xmath8 ; up to linear order , it is given by @xmath9 , and @xmath10 . we note that higher order forcing terms do not contribute to the discussed results , which focus on 1:1 resonance . magnitudes of the additive and parametric forcing terms are represented by @xmath11 and @xmath12 , respectively . additional details on the fhn model are given in the appendix . near the hopf bifurcation and in the absence of periodic forcing , model equations of type can be reduced to a universal nonlinear equation , which is often referred to as the stuart - landau ( a.k.a . complex ginzburg - landau ) equation @xcite . under externally applied periodic forcing ( at frequency @xmath6 ) , the phase invariance along the limit cycle is destroyed , and the system exhibits either unlocked oscillations or entrained oscillations with discrete phase shifts . increasing the amplitude of the drive increases the range of detuning under which a resonant solution can arise . this detuning from an unforced characteristic hopf frequency @xmath13 is given by @xmath14 and thus corresponds to @xmath15 frequency locking ( where @xmath16 is an integer ) , for which the original system responds at translations @xmath17 @xcite . under small detuning values @xmath18 , the stuart - landau equation is modified as @xcite : @xmath19 where @xmath20 is a complex amplitude that describes weak temporal modulations of a primary limit cycle that is generated at the hopf onset , @xmath21 measures the distance from the hopf bifurcation , @xmath22 is the nonlinear frequency correction , @xmath23 is a ( real ) forcing magnitude , and @xmath24 is the complex conjugate of @xmath20 . notably , the @xmath15 resonant solutions are invariant under @xmath25 @xcite ; hence , the frequency locking condition implies that @xmath26 is constant . this resonance condition is fulfilled over a finite range of driving frequencies and amplitudes , which defines the domain of an arnold tongue . we focus this study on the 1:1 resonant response @xcite . the amplitude equation ( [ eq : fcgl ] ) with @xmath27 has been employed in several contexts , such as studies of fluctuations and the response to pitches @xcite . here , we examine the physical nature of the forcing and its implications for the resonant response . in what follows , we show that for 1:1 resonance , one can obtain a generalized amplitude equation @xmath28 and explore its frequency locking properties ; for details , we refer the reader to the appendix . to determine the regions of frequency locked solutions , we rewrite and separate the contributions of the additive and parametric components by the transformation @xmath29 next , we introduce the following notation to distinguish the additive and the parametric forcing terms : @xmath30 for which : @xmath31 using the polar representation @xmath32 and looking for stationary solutions , we impose the conditions : [ eq : amp_rho_phi ] @xmath33 solutions to ( [ eq : amp_rho_phi ] ) thus satisfy an equation for the amplitude @xcite : @xmath34 and for the phase [ eq : amp_phi ] @xmath35 for the additive case ( @xmath36 ) , the phase displays @xmath37 symmetry shifts , while for the parametric case ( @xmath38 ) , one obtains phase shifts of @xmath0 . the latter implies bistability of frequency locked solutions , viz . @xmath39 and @xmath40 , where @xmath41 and @xmath42 are solutions to ( [ eq : amp_rho_phi ] ) . this result is analogous to a situation where a spatially periodic system is driven by a space - dependent modulation @xcite . thus , the @xmath43 resonant solutions can exhibit different phase symmetries , with one of them displaying properties similar to the @xmath44 resonant solutions @xcite . the linear stability of these solutions @xmath45 is determined by the sign of the real part of the eigenvalues @xmath46},\ ] ] where the solutions are stable if @xmath47 and unstable otherwise . combining the results on existence and stability of frequency locked solutions , we obtain the resonance regimes , for parameter space that is spanned by the forcing amplitude ( @xmath48 or @xmath49 ) and detuning ( @xmath50 ) . in particular , we distinguish between two cases : the oscillatory regime ( @xmath51 ) and the quiescent regime ( @xmath52 ) , as shown in fig . [ fig : above_hopf ] and fig . [ fig : below_hopf ] , respectively . ( a ) ) , in a parameter space of detuning ( @xmath50 ) and forcing magnitudes for ( a ) parametric forcing , @xmath53 and ( b ) additive forcing , @xmath54 . the bottom panel describes the resonant region ( shaded area ) , while the top panels describe the amplitudes of resonant solutions and unlocked oscillations ( light dashed lines ) at two distinct @xmath55 values ( @xmath56 , @xmath57 ) as a function of @xmath50 ; solid lines in top / mid panels mark stable solutions , and the dashed line in the bottom panel of ( a ) marks the locus of points at which the nontrivial solutions bifurcate . equation [ eq : modfcgl ] was solved with parameters : @xmath58 and ( a ) @xmath59 , ( b ) @xmath60 . ] ( b ) ) , in a parameter space of detuning ( @xmath50 ) and forcing magnitudes for ( a ) parametric forcing , @xmath53 and ( b ) additive forcing , @xmath54 . the bottom panel describes the resonant region ( shaded area ) , while the top panels describe the amplitudes of resonant solutions and unlocked oscillations ( light dashed lines ) at two distinct @xmath55 values ( @xmath56 , @xmath57 ) as a function of @xmath50 ; solid lines in top / mid panels mark stable solutions , and the dashed line in the bottom panel of ( a ) marks the locus of points at which the nontrivial solutions bifurcate . equation [ eq : modfcgl ] was solved with parameters : @xmath58 and ( a ) @xmath59 , ( b ) @xmath60 . ] ( a ) ) , in a parameter space of detuning ( @xmath50 ) and forcing magnitudes for parametric forcing , @xmath53 . the bottom panel describes the resonant region ( shaded area ) , while the top panel describes a typical behavior at a specific @xmath55 ( @xmath61 ) value as a function of @xmath50 ; solid lines in the top panel mark stable solutions , and the dashed line in the bottom panel marks the locus of sub - critical bifurcation onsets for the nontrivial solution . ( b ) a typical amplitude dependence at a specific @xmath55 value ( @xmath62 ) as a function of @xmath50 , for the additive case ( dark line ) , with a superimposed parametric case ( light line ) that is taken from ( a ) at slice ( i ) . equation [ eq : modfcgl ] was solved with parameters : ( a ) @xmath63 , @xmath59 and ( b ) @xmath64 , @xmath60 . ] ( b ) ) , in a parameter space of detuning ( @xmath50 ) and forcing magnitudes for parametric forcing , @xmath53 . the bottom panel describes the resonant region ( shaded area ) , while the top panel describes a typical behavior at a specific @xmath55 ( @xmath61 ) value as a function of @xmath50 ; solid lines in the top panel mark stable solutions , and the dashed line in the bottom panel marks the locus of sub - critical bifurcation onsets for the nontrivial solution . ( b ) a typical amplitude dependence at a specific @xmath55 value ( @xmath62 ) as a function of @xmath50 , for the additive case ( dark line ) , with a superimposed parametric case ( light line ) that is taken from ( a ) at slice ( i ) . equation [ eq : modfcgl ] was solved with parameters : ( a ) @xmath63 , @xmath59 and ( b ) @xmath64 , @xmath60 . ] above the onset of the hopf bifurcation , both the parametric and the additive forcing terms lead to a similar arnold tongue response , as shown by the shaded domains in fig.[fig : above_hopf ] . at weak forcing magnitudes ( @xmath65 ) , the frequency locked solutions form isolas and coexist with an additional unstable solution ; in the parametric case , this unstable state is a trivial one . the shaded region is identified with solutions that are linearly stable with respect to temporal perturbations . outside of the resonance domain , unlocked oscillations prevail , with the maximal amplitude denoted by the light dashed line , as shown in the horizontal slices ( i ) at fixed @xmath55 . at larger values of @xmath12 , two new solutions bifurcate from the trivial state ( see the dark dashed lines in the resonance domain of fig . [ fig : above_hopf](a ) ) , where the right line denotes solutions of a super - critical nature , while the left is sub - critical ; both are characteristic of the 2:1 resonance @xcite . the left non - trivial solution continues towards negative detuning values , folds back toward positive values , and connects with the right super - critical branch . stability of this top branch defines the resonant solutions . in the additive forcing case ( @xmath11 ) , the isola merges with the bottom solution through a cusp bifurcation @xcite , which results in a single amplitude throughout the whole detuning region ( not shown here ) . both behaviors are described by slices ( ii ) at fixed @xmath55 . further details on the coexistence and stability of such solutions are of secondary significance and will be discussed elsewhere . properties of the frequency locked solutions in the quiescent regime are fundamentally different from those of the innately oscillatory regime . in the parametric case , the resonant solutions also take the form of an arnold tongue . however , outside of the resonance region , the trivial state is stable , and thus the system can be either quiescent ( outside of the resonance regime ) or frequency locked ( within the resonance regime ) . nevertheless , since the resonance boundaries preserve the super- and sub - critical properties , part of the resonance boundary exhibits _ hysteresis _ , as shown in fig . [ fig : below_hopf](a ) . these properties imply a smooth transition to frequency locked oscillations if one approaches the arnold tongue from large positive detuning , and an abrupt transition if the resonance is approached from negative detuning values . the former case is reversible upon detuning , while the latter is associated with two distinct transition onsets . the frequency locked solutions in the additive case persist throughout the whole parameter range , as shown in fig . [ fig : below_hopf](b ) by a typical slice along @xmath55 . nevertheless , even in the absence of a distinct transition to resonance , we can identify amplification of the response amplitude . notably , while the resonant behavior is smooth for the fhn model , it is possible to observe a hysteresis here as well , via a cusp bifurcation @xcite . however , the hysteresis designates a transition from one oscillatory state to another , rather than a transition from the quiescent state , as in the parametric case . hair cells were shown to be the sources of amplification in the inner ear @xcite , operating either via hair bundle motility @xcite or somatic electromotility @xcite . according to theoretical models , the amplification gain and frequency selectivity of the auditory response are dependent on the value of an internal control parameter . as empirical evidences support the existence of a feedback mechanism , which modulates this control parameter in response to external forcing @xcite , it is important to incorporate its dynamics into the theoretical models . a universal framework has therefore been formulated to describe the 1:1 frequency locking of a system poised near the hopf bifurcation and exposed to both additive and parametric forcing . while additive forcing ( @xmath11 term in ( [ eq : modfcgl ] ) ) has been employed in previous studies , we incorporate here a periodic forcing term that couples directly to biochemical processes ( @xmath66 term in ( [ eq : modfcgl ] ) ) modulating the internal control parameter . a related study of synchronization dynamics of coupled oscillators also supports the inclusion of biochemical feedback @xcite . we discuss below the empirical data indicating the influence of both types of forcing on the frequency locked response . for the 1:1 resonance , we explored the scaling of the response amplitude with the additive and parametric forcing magnitudes . as a result , we obtain responses that are characterized by distinct power laws . a number of power laws have been experimentally observed in the response of the auditory system and are discussed in @xcite . the experimental results show three distinct regimes in the response : ( i ) at low forcing magnitudes , the response scales linearly , ( ii ) as the magnitude is increased , there is a crossover to a nonlinear regime with @xmath67 exponent , and ( iii ) an additional transition is observed at high forcing magnitudes . the additive forcing can explain the first two cases , which result from the competition between nonlinear and linear terms . however , the crossover to linearity in the third regime @xcite is not captured by including only the additive forcing term . the response to parametric forcing exhibits linear scaling @xmath68 , and hence supports the emergence of the third regime . most of the model equations have employed additive forcing , where the 1:1 resonant solutions obey @xmath69 symmetry @xcite . the generalized equations developed here show that the inclusion of a parametric forcing term introduces bistability of solutions differing by phase shifts of @xmath0 . phase shifts of @xmath0 have apparently been observed in experiments @xcite , but not accounted for in a theoretical model . construction of resonance domains ( i.e. , arnold tongues ) provides insight into the transition from unlocked to frequency locked dynamics . indeed , experimental measurements have been performed and revealed complex dynamics in the transition from spontaneous oscillation into the resonance regime @xcite . experimental results have raised conjectures on the possible presence of both super- and sub - critical forms of the hopf instability @xcite . our results demonstrate that a hysteresis can be described solely by bifurcations of the phase - locked states , which leave the super - criticality of the hopf bifurcation intact . resolving the contributions from the additive versus parametric forcing is difficult in the regime above the hopf onset ( spontaneously oscillatory regime ) , as the magnitude of unlocked oscillations is equal to that of the locked states . however , the differences become clear in the regime at or slightly below the hopf onset , where the hysteresis is conjectured to occur in the transition from a quiescent to a frequency locked state . the results presented here are not limited to a specific model and should arise as general features of 1:1 frequency locking @xcite in other systems , such as elastically driven cardiomyocytes @xcite . we believe that this framework provides a more realistic description of the biological system , as it can incorporate both biochemical and mechanical feedback in descriptions of the auditory response . further , the theoretical framework that incorporates parametric behavior could be generalized to include the coexistence of multiple @xmath15 resonances , which have not been included in previous spatially extended models . hence , the methodology developed here could provide a framework for future spatiotemporal models of the cochlear response . finally , this study of frequency locking dynamics can be generalized to other systems , such as faraday waves , shaken granular media , forced oscillatory chemical reactions , and elastically forced cardiomyocytes . we thank robijn bruinsma , oreste piro , and ehud meron for helpful discussions on the subject . this work was partially supported by the adelis foundation ( a.y . ) , and by nih grant r01dc011380 , nsf grant ios-1257817 ( db ) . the fitzhugh - nagumo model is a general bonhoeffer - van der pol type equation , which has been employed as a prototypical model for many biological and chemical systems @xcite . the response of the system to periodic forcing is described by : where @xmath8 is an activator , @xmath71 is an inhibitor , and @xmath72 and @xmath73 are parameters . we note that the bonhoeffer - van der pol type equations have already been employed in modeling the dynamics of the auditory system @xcite . the trivial solution to the unforced equation ( [ eq : fhn ] ) crosses the hopf instability at @xmath74 and a critical frequency @xmath75 . near the instability onset , @xmath76 , and under 1:1 periodic forcing , eqs . [ eq : fhn ] obey the approximation : @xmath77 where @xmath78 employing standard multiple time - scale expansion , and letting @xmath79 and @xmath80 @xcite , we obtain , up to order @xmath81 , the generalized amplitude equation for 1:1 forcing that incorporates both additive and parametric components : @xmath82 where , @xmath83 68 ifxundefined [ 1 ] ifx#1 ifnum [ 1 ] # 1firstoftwo secondoftwo ifx [ 1 ] # 1firstoftwo secondoftwo `` `` # 1 '' '' @noop [ 0]secondoftwo sanitize@url [ 0 ] + 12$12 & 12#1212_12%12 @startlink[1 ] @endlink[0 ] @bib@innerbibempty @noop * * , ( ) @noop * * , ( ) @noop * * , ( ) @noop * * , ( ) @noop * * , ( ) @noop * * , ( ) @noop * * , ( ) @noop * * , ( ) @noop * * , ( ) @noop * * , ( ) @noop * * , ( ) @noop * * , ( ) @noop * * , ( ) @noop * * , ( ) @noop * * , ( ) @noop * * , ( ) @noop * * , ( ) @noop * * , ( ) @noop * * , ( ) @noop * * , ( ) @noop * * , ( ) @noop * * , ( ) @noop * * , ( ) @noop * * , ( ) @noop * * , ( ) @noop * * , ( ) @noop * * , ( ) @noop * * , ( ) @noop * * , ( ) @noop * * , ( ) @noop * * , ( ) @noop * * , ( ) @noop * * , ( ) @noop _ _ ( , , ) @noop _ _ ( , , ) @noop * * , ( ) @noop * * , ( ) @noop * * , ( ) @noop * * , ( ) @noop * * , ( ) @noop * * , ( ) @noop * * , ( ) @noop * * , ( ) @noop * * , ( ) @noop * * , ( ) @noop * * , ( ) @noop * * , ( ) @noop * * , ( ) @noop * * , ( ) @noop * * , ( ) @noop * * , ( ) @noop * * , ( ) @noop * * , ( ) @noop * * , ( ) @noop * * , ( ) @noop * * , ( ) @noop * * , ( ) @noop * * ( ) @noop * * , ( ) @noop * * , ( ) @noop * * , ( ) @noop * * , ( ) @noop * * , ( ) @noop * * , ( ) @noop * * , ( ) @noop * * , ( ) @noop * * , ( ) @noop * * , ( )
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the auditory system displays remarkable sensitivity and frequency discrimination , attributes shown to rely on an amplification process that involves a mechanical as well as a biochemical response . models that display proximity to an oscillatory onset ( a.k.a .
hopf bifurcation ) exhibit a resonant response to distinct frequencies of incoming sound , and can explain many features of the amplification phenomenology . to understand the dynamics of this resonance , frequency locking
is examined in a system near the hopf bifurcation and subject to two types of driving forces : additive and parametric .
derivation of a universal amplitude equation that contains both forcing terms enables a study of their relative impact on the hair cell response . in the parametric case , although the resonant solutions are 1:1 frequency locked , they show the coexistence of solutions obeying a phase shift of @xmath0 , a feature typical of the 2:1 resonance .
different characteristics are predicted for the transition from unlocked to locked solutions , leading to smooth or abrupt dynamics in response to different types of forcing .
the theoretical framework provides a more realistic model of the auditory system , which incorporates a direct modulation of the internal control parameter by an applied drive .
the results presented here can be generalized to many other media , including faraday waves , chemical reactions , and elastically driven cardiomyocytes , which are known to exhibit resonant behavior .
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one of the key problems to overcome when conducting multi - wavelength surveys is determining which sources are associated with one another in different wave - bands , and which are unrelated . when multiple observations have been conducted at similar wavelengths and with similar resolution and sensitivity , this problem can be reliably addressed by using a simple nearest neighbour match . however , in the situation where the two distinct sets of observations to be matched have considerably different resolution for example matching far - infrared or sub - millimetre survey data to an optical catalogue ( e.g. sutherland et al . , 1991 , clements et al . , 1996 , serjeant et al . , 2003 , clements et al . , 2004 , ivison et al . 2005 , 2007 , wang & rowan - robinson , 2009 , biggs et al . 2010 ) the large positional uncertainties in the longer - wavelength data can make it much more difficult to find reliable associations between sub millimetre sources and their optical / near infrared counterparts . one method which can be used to identify the most likely counterpart to a low - resolution source , is the likelihood ratio technique ( hereafter lr ) , first suggested by richter ( 1975 ) , and expanded by sutherland & saunders ( 1992 ) and ciliegi et al . the crucial advantage of the lr technique over other methods is that it not only uses the positional information contained within the two catalogues , but also includes brightness information ( both of the individual potential counterparts , and of the higher resolution catalogue as a whole ) to identify the most reliable counterpart to a low - resolution source . the _ herschel _ astrophysical terahertz large area survey ( _ herschel_atlas , eales et al . , 2010 ) is the largest open - time key project that will be carried out with the _ herschel space observatory _ ( pilbratt et al . , 2010 ) . the _ herschel_atlas will survey in excess of 550 deg@xmath7 in five channels centred on 100 , 160 , 250 , 350 and 500@xmath0 m , using the pacs ( poglitsch et al . , 2010 ) and spire instruments ( griffin et al . , 2010 ) . this makes _ herschel_atlas currently the largest area extragalactic _ herschel _ survey . herschel_atlas observations consist of two scans in parallel mode reaching 5@xmath8 point source sensitivities of 132 , 126 , 32 , 36 and 45 mjy in the 100@xmath0 m , 160@xmath0 m , 250@xmath0 m , 350@xmath0 m and 500@xmath0 m channels respectively , with beam sizes of approximately 9 , 13 , 18 , 25 and 35 arcsec in the same five bands . the spire and pacs map - making procedures are described in the papers by pascale et al . ( 2010 ) and ibar et al . ( 2010 ) , while the catalogues are described in rigby et al . one of the primary aims of the _ herschel_atlas was to obtain the first unbiased survey of the local universe at sub - mm wavelengths , and as a result the survey was designed to overlap with existing large optical and infrared surveys . in this paper , we present a discussion of our implementation of the lr technique to identify the most reliable counterparts to 250@xmath0m selected sources in the _ herschel_atlas science demonstration phase ( sdp ) data field ( eales et al . , 2010 ) . this field was chosen in order to take advantage of multi wavelength data from the sloan digital sky survey ( sdss york et al . , 2000 ) , and the uk infrared deep sky survey large area survey ( ukidss - las lawrence et al . , 2007 ) . this field also overlaps with the 9 hour field of the galaxy and mass assembly survey ( gama driver et al . , 2010 ) . the gama catalogue ( hill et al . , 2011 ) , comprises not only thousands of redshifts ( for galaxies selected as described in baldry et al . , 2010 , and observed with the maximum possible tiling efficiency robotham et al . , 2010 ) , but also @xmath9band defined aperture matched photometry in the @xmath10 bands . in addition , the gama fields are being systematically observed using the _ galaxy evolution explorer ( galex ) _ satellite ( martin et al . , 2005 ) at medium imaging survey depth to provide aperture matched fuv and nuv counterparts to the catalogued gama sources ( the _ galex_gama survey ; seibert et al . , in prep ) . these counterparts will potentially be of great scientific value once the most reliable optical counterpart can be established for each _ herschel_atlas source . in section [ lrcalc ] we present the specific lr method that we have used to identify counterparts to 250@xmath0m selected sources from the _ herschel_atlas sdp catalogue in an @xmath9band catalogue of model magnitudes derived from the sdss dr7 . in section [ sec : redshifts ] we present the redshift properties of our catalogue , which covers @xmath11deg@xmath7 over the gama 9 hour field . section [ results ] contains some basic results based on our reliable catalogue , and in section [ conclusions ] we present some concluding remarks about the likelihood ratio technique and the resulting catalogue . the likelihood ratio , i.e. the ratio between the probability that the source is the correct identification and the corresponding probability for an unrelated background source , is calculated as in sutherland & saunders ( 1992 ) : in which @xmath13 , and @xmath14 correspond to the sdss @xmath9band magnitude probability distributions of the full @xmath9band catalogue and of the true counterparts to the sub millimetre sources , respectively , while @xmath15 represents the radial probability distribution of offsets between the 250@xmath0 m positions and the sdss @xmath9band centroids . we will now describe how we calculate each component of this relationship in turn . for _ herschel_atlas sdp observations , it was necessary to determine the spire positional uncertainties . since this information was not available _ a priori _ , we empirically estimated @xmath17 using the sdss dr7 @xmath9band catalogue positions , assuming that the sdss positional errors were negligible in comparison to the spire errors . to determine @xmath17 , we derived histograms of the separations between the positions in the mad - x spire catalogue ( rigby et al . , 2010 ) of the @xmath18 250@xmath0 m sources , and all of those objects in the @xmath9band sdss dr7 catalogue within 50 arcsec , doing this for both the north south and east west directions ( figure [ fig : clust ] ) . these histograms can be well described as the sum of the gaussian positional errors plus the clustering signal for sdss sources convolved with gaussian errors , @xmath19 , with @xmath20 : where @xmath23 , with @xmath24 being measured in degrees for the purposes of comparison with the literature . we determined the values of @xmath25 and @xmath26 empirically based solely on galaxies in the sdss catalogue over @xmath27 deg@xmath7 centred on the _ herschel_atlas sdp field ( limited to @xmath28 ) , with the best fit parameters @xmath29 and @xmath30 , in reasonable agreement with the values of connolly et al . the effects of clustering ( i.e. @xmath31 ) are shown in the top panel of figure [ fig : clust ] . in order to determine the 1@xmath8 positional error of the 250@xmath0 m selected catalogue , we conducted a simple @xmath32 fit of our model ( equations [ xhist ] & [ yhist ] ) to the histograms . the results are shown in figure [ fig : clust ] for the summations in the east - west and north - south directions in the middle and bottom panels respectively . the clustering signal is shown in the bottom two panels by the dotted lines , with the histograms and their poisson error bars overlaid with the best fit model ( solid lines ) . the 1@xmath8 positional errors were found to be @xmath33 arcsec and @xmath34 arcsec in the two directions , consistent with one another within the errors . the advantages of this method are two fold ; firstly , it is not necessary to identify the counterparts to the 250@xmath0 m sources _ a priori _ , and secondly , the centroids of the best fit gaussians may be used to determine astrometric corrections in the spire maps ( e.g. pascale et al . , 2010 ) . the value for @xmath17 that we adopted was the weighted mean @xmath35 arcsec . positional errors of the spire 250@xmath0m selected sources , we produced histograms of the total number of sdss sources within a box 50 arcsec on a side around the spire 250@xmath0 m centres . after accounting for the clustering of sdss sources ( the top panel shows the signal expected for the clustering of sdss sources in the ra and dec directions convolved with gaussian positional errors the results are shown as solid and dashed lines for ra and dec , respectively , and these results appear as the dotted lines in the bottom two panels ) , we can add in an appropriate gaussian distribution of centres to account for the actual positions of the spire sources ( equations [ xhist ] & [ yhist ] ) . performing a @xmath32 minimisation allows us to then empirically determine the 1@xmath8 positional uncertainty for these sources , which are shown as @xmath36 and @xmath37 . ] theoretically , the positional uncertainty should depend on the signal - to - noise ratio ( snr ) of the detection and on the full - width at half maximum ( fwhm ) of the spire 250 @xmath0 m beam ( 18.1 arcsec , pascale et al . , 2010 ) , following the results derived in ivison et al . ( 2007 ; @xmath38 ) and assuming the case of uncorrelated noise . we use our empirical results in figure [ fig : clust ] to calibrate the theoretical relation presented in ivison et al . ( 2007 ) to our data , and assume that our results are symmetric in ra and dec . this leads us to introduce a factor of 1.09 to give equation [ ivison2007 ] : * whilst it is acceptable to neglect the sdss dr7 positional errors for the purposes of determining @xmath17 ( section [ sec : f_r ] ) , the astrometric precision for sources in the sdss dr7 catalogue is non - zero ( @xmath40 arcsec abazajian et al . , * large sources , especially those without gaussian surface brightness profiles ( e.g. bright spiral galaxies ) , have considerably larger positional uncertainties associated with them . * confusion provides a lower limit to the positional errors of the spire catalogue , although the snr in equation [ ivison2007 ] does include confusion noise as described in rigby et al . ( 2010 ) and pascale et al ( 2010 ) . to account for these effects , we do not allow the positional uncertainty to fall below 1 arcsec , and we also include a term which adds 5 percent of the sdss @xmath9band isophotal major axis in quadrature to the value determined by equation [ ivison2007 ] , for those sources with @xmath9band model magnitudes @xmath41 . finally , @xmath15 must be renormalised so that calculating the lr requires two further pieces of magnitude information , @xmath13 and @xmath14 . the quantity @xmath13 is simply the probability that a background source is observed with magnitude @xmath43 . to estimate this , we calculate the distribution of sdss dr7 @xmath9band model magnitudes for all of the primary photometry sources in the catalogue , normalised to the total area of the catalogue ( which is approximately 36.0 deg@xmath7 for the sdss catalogue that we use for this purpose ) . the non - triviality lies in the calculation of @xmath14 the probability that a true counterpart to a 250@xmath0 m source has a magnitude @xmath43 . to estimate this we calculate the @xmath9band magnitude distribution of the counterparts to the 250@xmath0 m sources using the method of ciliegi et al . this method involves counting all objects in the optical catalogue within some fixed maximum search radius ( @xmath44 ) of the spire positions . to avoid influencing the results of this analysis with erroneous deblends in the sdss dr7 catalogue ( which artificially alter the number counts ) , we eyeballed the sdss @xmath9band images of each of the 5@xmath8 250@xmath0 m sources , removing 370 sdss sources from the input catalogue . the magnitude distribution of the remaining objects is referred to as total@xmath45 . here we have adopted @xmath46arcsec , which encloses @xmath4799.996% of the real counterparts to the 250@xmath0 m sources based on our derived value for @xmath17 . the distribution total@xmath45 is then background subtracted to leave the magnitude distribution of excess sources around the 250@xmath0 m centres , real@xmath45 : where @xmath49 is the number of 250@xmath0 m sources in the catalogue . this enables us to empirically estimate @xmath14 from the sources in our optical catalogue rather than modelling the @xmath9band magnitude distribution of 250@xmath0m selected _ herschel_atlas sources . the distribution @xmath14 is given by equation [ eq : q_m ] : here @xmath53 represents the number of possible ids within 10.0 arcsec of the spire positions , and @xmath49 is defined as above . since the value of @xmath51 will be different for galaxies and unresolved sources in our catalogue , we must calculate @xmath14 , @xmath13 and @xmath51 separately for each population . we separate resolved and unresolved sources using a slightly modified version of the gama colour colour relation from baldry et al . ( 2010 , modified such that @xmath54 rather than 0.20 to avoid adding an unphysical sharp edge to the stellar locus in figure [ fig : stargalsep ] ) . having separated the two populations , we corrected the positions of the unresolved sources for known proper motions in the usno / sdss dr7 catalogue ( munn et al . , 2004 ) , precessing their co ordinates to the epoch of the _ herschel_atlas sdp observations . only those unresolved sources with proper motions detected at a snr @xmath55 were updated . sources in our catalogue classified as unresolved , we find that three satisfy the gama colour selection criteria for being stellar , and so are potentially evolved stars , dust - obscured qsos or debris disk candidates possibly indicative of a proto - planetary system ( e.g. thompson et al . , 2010 ) . the dashed line describes the first order star galaxy separation locus ( for more details see baldry et al . galaxy separation locus has been modified slightly from the baldry et al . value due to the fainter magnitudes considered in our survey . ] for our sdss dr7 @xmath9band catalogue , @xmath56 , i.e. 58.3 percent of the galaxy counterparts are brighter than our magnitude limit . for the unresolved sources the value is @xmath57 , indicating that only 1 percent of the unresolved sources in the catalogue are detected at @xmath58 in our 250@xmath0 m data ( although see section [ sec : randcat ] ) . thus we determine that overall @xmath59 . the distributions of @xmath14 , and @xmath13 ( as well as the magnitude dependence of the lr @xmath14/@xmath13 ) are shown in figure [ fig : nm ] , in which the left and right columns show the values for the resolved and unresolved sources , respectively . while the @xmath14 distribution for galaxies is well sampled at @xmath60 mag , we assume that @xmath61 is constant for all sources brighter than this , enabling us to use our well - defined @xmath13 to estimate @xmath14 for the brightest galaxies . since the fraction of _ sources associated with unresolved counterparts is low ( reflected in @xmath62 ) , the method used to determine @xmath14 for these sources differs . in order to ensure that the lr results for stars / qsos are not dominated by small number statistics , we assume a flat prior on @xmath14 , normalised to retain @xmath63 ( figure [ fig : nm ] ) . we can correct our value for @xmath51 for the clustering of sdss sources by simply dividing @xmath51 by @xmath64 ( remembering that @xmath24 is measured in degrees ) , giving a clustering corrected value of @xmath65 . this value is broadly consistent with the recent results of dunlop et al . ( 2010 ) , who recover optical counterparts to 8 out of 20 250@xmath0 m sources brighter than 36@xmath66 in data from the blast observations of the goods - south field to a comparable @xmath67band magnitude ( albeit with lower angular resolution at 250@xmath0 m and much more sensitive optical , infrared and radio data ) , while dye et al . ( 2009 ) found 80 counterparts to the 175 blast 250@xmath0 m sources brighter than 55@xmath66 down to similar magnitude limits in @xmath9 or @xmath68band data ( s. dye , private communication ) . to account for the fact that an _ herschel_atlas source may have more than one possible counterpart , we also define a reliability @xmath69 for each object @xmath70 being the correct counterpart out of all those counterparts within @xmath44 , again following sutherland & saunders ( 1992 ) : where the lr values have been determined for the resolved and unresolved counterparts separately ( see figure [ fig : lrhist ] ) . the reliability is a key statistic ; we recommend using only those counterparts with reliability @xmath72 for analysis , since this ensures not only that the contamination rate is low ( see below ) , but also that only one @xmath9band source dominates the far infrared emission ( as required for e.g. deriving spectral energy distributions for 250@xmath0m selected galaxies in the _ herschel_atlas catalogue , smith et al . _ in prep _ ) this is more conservative than other works in the literature ( e.g. chapin et al . , 2010 ) , where the chosen lr limit was defined based on a 10 percent sample contamination rate . sources in the spire sdp catalogue . the lr values for the resolved sources ( i.e. galaxies ) are shown as the solid histogram , with unresolved sources shown as the dotted histogram . middle : reliabilities for each counterpart . once more , the solid histogram represents the resolved sources , while the dotted histogram represents the unresolved sources . there are a total of 2423 sources which have a reliability @xmath73 , of which five are unresolved using the star / galaxy separation criteria of baldry et al . bottom : the variation of the reliability as a function of the likelihood ratio . this is not a linear relation since some sources have more than one counterpart with a high likelihood ratio . there are 263 sdss @xmath9band sources with reliability @xmath74 but @xmath75 ( the value above which @xmath4 for a single counterpart within the 10.0 arcsec maximum search radius ) . these may be interacting systems , as discussed in section [ mergers ] . these sources also demonstrate a possible limitation of the lr method , since the method implicitly assumes that there is only one true counterpart to a given 250@xmath0 m source . ] as a result we expect 103 false ids in our sample , which corresponds to a contamination rate of 4.2% . for those investigations in which it is desirable only to determine whether an optical source is associated with an _ herschel_atlas object ( with additional caveats about lensed sources and the de - blending efficiency in the optical catalogue ) , it is sufficient to use a likelihood ratio cut ( e.g. @xmath77 , i.e. the source is 5 times more likely to be associated with the sub millimetre object than it is to be a chance superposition of sources ) . this aspect of the likelihood ratio technique is discussed in more detail in section [ mergers ] . in table [ tab : nmatch ] , we present the number of possible optical counterparts within 10.0 arcsec of the 250@xmath0 m sample , including the relative fractions of reliable associations . only half of the 250@xmath0 m sources with a single optical counterpart within the search radius are deemed reliable . .the distribution of the number of sdss @xmath9band sources within 10.0 arcsec of the 250@xmath0 m positions , and the fraction of reliable counterparts . there are 2869 sources with only one possible match within 10.0 arcsec , and yet only 1389 of these are determined to be reliable ; the vast superiority of the lr technique over a simple nearest - neighbour algorithm is evident . [ cols="^,^,^,^",options="header " , ] we also compared the results of our likelihood ratio analysis to data from the faint images of the radio sky at twenty centimetres ( first ) survey ( becker , white & helfand , 1995 ) . the first survey covers 9,000 square degrees of sky with a resolution of 5 arcsec , with a source density of approximately 90 per square degree brighter than the detection threshold of 1@xmath66 . at these relatively bright flux limits , the source population is dominated by active galactic nuclei ( agn ) rather than star forming sources ( e.g. wilman et al . , 2008 ) ; as a result the overlapping population of sources between the _ herschel_atlas and first catalogues is not expected to dominate the number counts . to make the comparison between our lr analysis and first sources , we used the frequentist identification procedure of downes et al . ( 1986 ) , commonly used to quantify the formal significance of possible counterparts to sub - millimetre galaxies in radio survey data ( e.g. lilly et al . , 1999 , ivison et al . 2007 ) . in this procedure , the statistic used to assess the probability that a nearby radio source is _ not _ associated with the spire source is @xmath78 , where @xmath9 is the angular distance between the spire source and the radio source , @xmath79 is the flux density of the radio source , and @xmath80 is the surface density of radio sources with flux densities greater than this . for each spire source , we looked for radio sources in the first catalogue within 10.0 arcsec , and treated the radio source with the lowest value of @xmath81 as the one most likely to be associated with the spire source . we used a monte - carlo simulation ( e.g. eales et al . 2009 ) to determine the probability distribution of @xmath82 on the null hypothesis that there are no genuine associations between radio sources and spire sources . we then used this probability distribution to determine the probability that each measured value of @xmath82 would have occured by chance . we call this probability @xmath83 . of the 6621 @xmath18 250@xmath0 m spire sources , 105 have radio counterparts within 10.0 arcsec , all with values of @xmath84 . however , this does not take account of the fact that with such a large sample of spire sources one expects to find some low values of @xmath83 even if there were no genuine associations between the spire sources and first objects . we used a monte - carlo simulation to determine that 15 of the 105 associations are likely to be spurious . to correct for this , we calculated a new probability for each association , @xmath85 , where @xmath86 is a constant that we calculated using @xmath87 . we took the conservative decision to treat associations with @xmath88 as counterparts which are likely to be genuine , which rejected 29 of the original 105 associations . there were a total of 76 spire sources with @xmath89 counterparts , and each of these was scrutinised using the first and sdss images displayed side by side with the downes et al . and lr analysis overlaid . in this manner , we compared the results of the two independant identification methods . in forty - two cases , the @xmath89 radio counterpart is also identified as having @xmath90 in the @xmath9band data , and the two methods choose the same counterpart . there are thirty spire sources with high quality ( @xmath89 ) first counterparts which we do not recover in our lr analysis , including twenty three spire sources which do not have any @xmath9band counterparts in our sdss dr7 data ( presumably distant , optically faint radio sources ) . of the remaining seven sources with @xmath89 first counterparts : * four counterparts are detected in the optical data but have low reliabilities due to their faint magnitudes , or large separations in comparison to the value of @xmath17 derived based on the 250@xmath0 m source snr . * two sources have multiple , possibly interacting components with @xmath91 but @xmath92 , only one of which is a radio source ( these sources are discussed in section [ mergers ] ) . * in one further instance , the radio source has a double - lobed structure ( a so - called fr - ii , following fanaroff & riley , 1974 ) , not coincident with either the dust emission or the starlight in the plane of the sky . the lobes of this frii are extremely bright ; as a result , the @xmath93 statistic suggests that there is a low probability of a chance association , even though the separation between the spire position and the first centroid is large . the lr technique identifies the apparent host galaxy aligned at the centre , between the two luminous radio jets as having @xmath94 due to its large separation ( @xmath95 arcsec ) from the spire centroid ; this is an example of the limitations of the downes et al . method . however , these possibilities do not contaminate the 250@xmath0 m selected sample with incorrect associations . there are however , four instances where distinct counterparts have @xmath96 and @xmath97 ; here , the opposite is potentially true and the two methods conflict . these sources have derived reliabilities of 0.87 , 0.98 , 0.81 and 0.93 as compared with distinct downes et al . counterparts with @xmath93 statistics of 0.08 , 0.07 , 0.02 and 0.19 , respectively . these sources are shown in figure [ fig : downes_lr_dis ] , in which the 10.0 arcsec search radius centred on the 250@xmath0 m position is shown in red , any unreliable optical counterparts in black , the reliable optical i d in light blue , and the radio contours overlaid in royal blue . in two of the four cases , the additional sources implied by the radio data are visible in @xmath98band observations from viking ( sutherland , 2009 ) , indicating that these sources are not merely effects of the larger positional uncertainty in first as compared with sdss . futhermore , three of the four sources have spire colours @xmath99 , suggesting high redshifts ( @xmath6 ) or cold dust temperatures , with the former being at odds with the photometric redshifts of their most reliable counterparts ( @xmath100 ) . sources with similar spire colours and low - redshift counterparts are discussed in more detail in section [ submmcolours ] . finally , we note that probabilistic arguments such as those discussed here will inevitably present apparent disagreements for a small number of sources within large samples . in the remaining 101 out of 105 cases however , the results of our lr analysis are consistent with those using the first catalogue and the @xmath93 statistic , and crucially we recover an additonal 2,348 counterparts , compared with 31 extra counterparts to the 250@xmath0 m sources obtained by using only the radio data . an additional check on the identification process was conducted by searching for mid infrared data from the _ spitzer space telescope _ heritage archive , in order to compare the reliabilities from our @xmath9band catalogue with near- and mid infrared images between 3.6 and 160@xmath0 m . four sets of observations were found which overlapped with the _ herschel_atlas sdp observations these data can be used to examine the regions surrounding the spire ids for additional sources which may not be present in the @xmath9band catalogue used for the identification process , as a visual check on the effectiveness of the lr technique . there are a total of 49 sources that have _ spitzer _ data , and although these data vary in sensitivity , there is no evidence that would suggest a mis - identification from the @xmath9band catalogue . such indications of wrong ids would include reliable ( @xmath72 ) @xmath9band counterparts indicated for spire sources which have previously unrevealed bright _ spitzer _ sources nearer to the centre of the spire centroid . indeed , in one case in particular ( h atlas j090913.2 + 012111 ) , the sensitive irac data reveal the power of the lr technique . although there are three potential counterparts in the sdss dr7 @xmath9band catalogue all within 6 arcsec of the spire centroid , they have all been given low reliability ( @xmath101 , and also @xmath102 ) . the irac 3.6@xmath0 m data reveal a fourth candidate counterpart within 1 arcsec of the spire position , which is presumably the true counterpart . the @xmath9band and irac 3.6@xmath0 m data are presented in figure [ fig : rband_irac1 ] , with the various source positions overlaid to demonstrate the robustness of the lr method for this particular source , but also the need for longer - wavelength observations in order to be able to reliably identify the counterparts to higher redshift sources . the forthcoming data from the vista kilo - degree infrared galaxy ( viking ) survey and from the wide - field infrared survey explorer ( _ wise _ duval et al . , 2004 ) satellite will enable this . this example also highlights one crucial advantage of using the lr technique for _ herschel _ surveys rather than opting simply for the downes method ; the lr method takes into account the fact that not every source has a counterpart that is brighter than the detection limit in ancillary survey data .
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we present a technique to identify optical counterparts of 250@xmath0m - selected sources from the _ herschel_-atlas survey . of the 6621 250@xmath0 m @xmath1mjy sources in our science demonstration catalogue
we find that @xmath2 percent have counterparts brighter than @xmath3mag in the sloan digital sky survey . applying a likelihood ratio technique
we are able to identify 2423 of the counterparts with a reliability @xmath4 .
this is approximately 37 percent of the full 250@xmath0 m catalogue .
we have estimated photometric redshifts for each of these 2423 reliable counterparts , while 1099 also have spectroscopic redshifts collated from several different sources , including the gama survey .
we estimate the completeness of identifying counterparts as a function of redshift , and present evidence that 250@xmath0m - selected _ herschel_-atlas galaxies have a bimodal redshift distribution .
those with reliable optical identifications have a redshift distribution peaking at @xmath5 , while sub - mm colours suggest that a significant fraction with no counterpart above the r - band limit have @xmath6 .
we also suggest a method for selecting populations of strongly - lensed high redshift galaxies .
our identifications are matched to uv
nir photometry from the gama survey , and these data are available as part of the _ herschel_-atlas public data release .
[ firstpage ] galaxies : local , galaxies : infrared , galaxies : star - forming , methods : statistical , submillimetre : galaxies
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You are an expert at summarizing long articles. Proceed to summarize the following text:
we recently obtained a new hamiltonian for a lattice which includes interactions with internal nuclear degrees of freedom @xcite . we started with a dirac model for electrons and for nucleons on equal footing , then allowed the nuclei to be described by a finite basis expansion , and finally developed an approximation appropriate for low nuclear velocity . our original motivation for this was to obtain a model capable of describing the mass shift associated with excited nuclear states in a configuration interaction calculation . however , the new model unexpectedly contains a new coupling term which provides for a direct interaction between lattice vibrations and nuclear transitions . on the face of it , this new interaction term looks like it should allow for the strongly - coupled transitions that we have sought @xcite in connection with the generalized lossy spin - boson models @xcite that we proposed to account for some of the anomalies ( such as excess heat @xcite and collimated x - ray emission @xcite ) that have been reported in experiments in recent years . what is needed at this point is an explicit calculation for some nuclear system to see how it works , what states are coupled to , and how big the interaction is . in general nuclear structure models are much more complicated than atomic structure models due to the more complicated strong force interaction . we would like to work with empirical strong force models derived from scattering experiments and few - body bound state binding energies . in recent years these models have achieved impressive results @xcite ; however , some of these strong force models involve a fair amount of work to implement . if we go back a few decades we can find simpler versions of strong force models that are easier to work with , and are sufficiently accurate to clarify the issues of interest here . in the computations that follow we will focus on the old hamada - johnston potential model @xcite . without question the simplest compound nucleus which should show the effects of interest is the deuteron , and so we will focus on this system in what follows . in this formulation we have modeled the nucleons as elementary dirac particles . as nucleons are made up of strongly interacting quarks , we know that they are not elementary dirac particles . to do better in the case of coupling with the deuteron , we would require a description in terms of the six constituent quarks . we would expect from such a model a coupling matrix element likely somewhat different from what we calculated in this work . even so , it makes sense here to pursue this simpler deuteron model based on simple dirac nucleons as a step forward in the modeling process . in a recent paper we discussed the derivation of a finite basis approximation for a moving nucleus in the many - particle dirac model which leads to the new coupling that we are interested in . we begin with the ( relativistic ) finite basis model that we obtained . in @xcite we developed finite basis eigenvalue relations in the form @xmath2 where the off - diagonal matrix elements were written as @xmath3 here @xmath4 is the relative coupling matrix element @xmath5 we defined @xmath6 as @xmath7 the notation for the two - body version of the problem is a bit different than what we used for the many - particle problem . it is useful to recast the relative matrix element as @xmath8 the relative part of the off - diagonal matrix element corresponds to the rest frame interaction terms , which might come about from strong force interactions as in the development above for the nonrelativistic deuteron problem . what is new is the coupling with the center of mass momentum @xmath9 that appears in @xmath10 . we are interested in these new matrix elements . in @xcite we discussed the reduction of the new interaction matrix element to the nonrelativistic case . the results can be expressed as @xmath11 @xmath12 ( { \mbox{\boldmath$\sigma$}}_j \cdot c \hat{{\mbox{\boldmath$\pi$}}}_j ) \bigg | \phi_i \bigg \rangle\ ] ] @xmath13 ( { \mbox{\boldmath$\sigma$}}_j \cdot c \hat{\bf p } ) \bigg | \phi_i \bigg \rangle \bigg ] \ ] ] as above , this is written for the many - particle problem , and we wish to recast it in terms of the two - body problem ; we may write @xmath14 @xmath15 ( { \mbox{\boldmath$\sigma$}}_1 \cdot c \hat{\bf p } ) \bigg | \phi_i \bigg \rangle\ ] ] @xmath16 ( { \mbox{\boldmath$\sigma$}}_1 \cdot c \hat{\bf p } ) \bigg | \phi_i \bigg \rangle \bigg ] \ ] ] @xmath17 ( { \mbox{\boldmath$\sigma$}}_2 \cdot c \hat{\bf p } ) \bigg | \phi_i \bigg \rangle\ ] ] @xmath18 ( { \mbox{\boldmath$\sigma$}}_2 \cdot c \hat{\bf p } ) \bigg | \phi_i \bigg \rangle \bigg ] \ ] ] it is possible to split up this new interaction term into a contribution that takes the nucleon masses to be equal , and a small correction term that depends on the difference between the nucleon masses . in what follows our focus will be on the larger equal mass terms , which is equivalent to making an equal mass approximation . in this case we may write @xmath19 ( { \mbox{\boldmath$\sigma$}}_1 \cdot c \hat{\bf p } ) \bigg | \phi_i \bigg \rangle \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ ] ] @xmath20 ( { \mbox{\boldmath$\sigma$}}_1 \cdot c \hat{\bf p } ) \bigg | \phi_i \bigg \rangle \bigg ] \ ] ] @xmath21 ( { \mbox{\boldmath$\sigma$}}_2 \cdot c \hat{\bf p } ) \bigg | \phi_i \bigg \rangle\ ] ] @xmath22 ( { \mbox{\boldmath$\sigma$}}_2 \cdot c \hat{\bf p } ) \bigg | \phi_i \bigg \rangle \bigg ] \ ] ] where we have assumed that @xmath23 in this approximation there is no longer an explicit dependence on the state energy @xmath24 . it is possible to develop a nonrelativistic approximation using @xmath25 in this case , a finite basis model that includes center of mass dynamics to lowest order could be developed starting from a nonrelativistic hamiltonian of the form @xmath26 where @xmath27 is an operator that returns the rest mass energy of the nuclear state , and where @xmath28 in the equal mass approximation is @xmath29 @xmath30\ ] ] we know from the literature that the deuteron at rest can be modeled using a triplet s and triplet d state , since the tensor interaction mixes the two . since the kinetic energy and potential terms preserve @xmath31 and @xmath32 , each of the triplet s states mixes with a triplet d state that has the same @xmath31 and @xmath32 . the new interaction term causes these states to mix with singlet p states . in general , the new term does not preserve @xmath32 , so that we would require a finite basis approximation that distinguishes the different sublevels . however , it is possible to focus on a special case of the new interaction which does preserve @xmath32 . this occurs if we restrict our attention to @xmath34 we find in this case that mixing occurs for @xmath35 , but not for @xmath36 . in response , we might write @xmath37 with the understanding that @xmath38 nuclear state construction is usually carried out in the isospin scheme , with antisymmetry enforced through the application of the generalized pauli principle . the two - body problem is particularly simple in this regard , with spin , isospin and spatial components restricted to being either symmetric @xmath39 or antisymmetric @xmath40 ; we may write for the three states @xmath41 @xmath42 @xmath43 the antisymmetric spin and isospin terms @xmath44 and @xmath45 are singlets , and the symmetric spin and isospin terms @xmath46 and @xmath47 are triplets . the s state is a triplet spin state , so we may write it as @xmath48 @xmath49 the @xmath50 are spherical harmonics ; we choose @xmath51 and @xmath52 since we are working with an s state . the @xmath53 are spin functions for the neutron and proton spins ; the @xmath54 are isospin functions , and we have used an isospin singlet function here . we can develop a d state by applying the tensor @xmath55 operator on an s state . this approach was used early on as a convenient way of generating few - body wavefunctions for variational calculations in nuclear physics . we may write @xmath56\ ] ] this construction is convenient since @xmath57 - 2 \psi_{^3d}\ ] ] the singlet p state for a particular calculation can be specified using @xmath58 @xmath59 including an @xmath60 here leads to real coupling coefficients in what follows . we can evaluate the normalization integral for these states simply ; we write @xmath61 @xmath62 we are interested in developing coupled channel equations that include the new interaction . for the problem in the rest frame , this is most easily accomplished by developing an expression for the total energy and then using the variational principle . we can use the same basic approach here for the moving frame version of the problem . we begin with @xmath63 @xmath64 @xmath65 @xmath66 we can evaluate the diagonal matrix elements directly using mathematica to obtain @xmath67 u(r ) dr\ ] ] @xmath68 v(r)dr\ ] ] @xmath69 w(r ) dr\ ] ] in the case of the hamada - johnston potential , there occur off - diagonal matrix elements between the triplet s and singlet d states , which are given by @xmath70 the superscript @xmath71 in the associated potentials here is connected with the even triplet channel , since the hamada - johnston potentials are fit for the different channels separately . for the off - diagonal matrix elements of the new interaction , we have used mathematica to compute @xmath72 @xmath73 u(r ) dr\ ] ] @xmath74 dr\ ] ] @xmath75 u(r ) dr\ ] ] @xmath76 dr\ ] ] @xmath77 dr \bigg \rbrace\ ] ] @xmath78 @xmath79 v(r ) dr\ ] ] @xmath80 dr\ ] ] @xmath81 v(r ) dr\ ] ] @xmath82 dr\ ] ] @xmath83 v(r ) dr\ ] ] @xmath84 dr\ ] ] @xmath85 v(r ) dr \bigg \rbrace\ ] ] @xmath86 @xmath87 w(r ) dr\ ] ] @xmath88 dr\ ] ] @xmath89 w(r ) dr\ ] ] @xmath90 dr\ ] ] @xmath91 w(r ) dr\ ] ] @xmath92 dr\ ] ] @xmath93 w(r ) dr \bigg \rbrace\ ] ] @xmath94 @xmath95 2(r ) dr\ ] ] @xmath96 dr\ ] ] @xmath97 w(r ) dr\ ] ] @xmath98 dr\ ] ] @xmath99 v(r ) dr\ ] ] @xmath100 dr\ ] ] @xmath101 v(r ) dr \bigg \rbrace\ ] ] we have specified a finite basis problem with three channels , which would produce three complicated coupled - channeled equations if we decided to treat the different basis states on equal footing . however , since the momentum @xmath9 that we are interested in for applications of this model is small , the triplet s and d channels are then best considered to constitute the unperturbed deuteron problem , and the singlet p channel will contain the weak response of the deuteron to the @xmath102 perturbation . in this case , it seems appropriate to develop the coupled triplet s and d channels consistent with the rest frame deuteron problem . once the associated wavefunctions are known , then we can use them to approximate the occupation of the singlet p channel . given the approach outlined above , we can optimize the channel wavefunctions @xmath103 and @xmath104 by minimizing the rest frame energy @xmath105 @xmath106 u(r ) dr\ ] ] @xmath107 v(r)dr\ ] ] @xmath108 the minimization of this rest frame energy leads to the constraints @xmath109 u(r ) + \bigg [ - 3\sqrt{8 } v_{t}^{et}(r ) \bigg ] v(r)\ ] ] @xmath110 v(r)\ ] ] @xmath111 u(r)\ ] ] where @xmath112 is the relative energy . we recognize these as the rarita - schwinger equations based on the hamada - johnston potential model . we have solved the rarita - schwinger equations to obtain the channel wavefunctions plotted in figure [ deuteron1 ] . the triplet s channel wavefunction @xmath103 is larger and extends out to a relatively large radial separation , and the triplet d channel wavefunction @xmath104 is smaller and localized to much smaller radial separation . we can see the effect of the hard core potential in the zero boundary condition at the cut off radius . = 4.00 in = 3.200 in [ t ] in the perturbation theory approach outlined above , we can approximate the occupation of the singlet p channel in terms of known triple s and d channel wavefunctions . the associated constraint on the channel wavefunction can be written as @xmath113 w(r ) ~=~ m_j \left ( { 1 \over 2 mc^2 } \right ) \left ( { 1 \over 2 m_{av } c^2 } \right ) ( \hbar c ) ( c \hat{p}_z ) \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ ] ] @xmath114 u(r ) + 12 \sqrt{3 } v_t^{et}(r ) \left [ { d \over dr } u(r ) + { u(r ) \over r } \right ] + 8 \sqrt{3 } \left [ { d \over dr } v_t^{et}(r ) \right ] u(r)\ ] ] @xmath115 - 2 \sqrt{3 } v_{ll}^{et}(r ) \left [ { d \over dr } u(r ) - { u(r ) \over r } \right ] \bigg \rbrace\ ] ] @xmath116 v(r)\ ] ] @xmath117 + 4 \sqrt{6 } \left [ { d \over dr } v_t^{et}(r ) \right ] v(r ) - \sqrt{8 \over 3 } v_{ls}^{et}(r ) \left [ { d \over dr } v(r ) + 2{v(r ) \over r } \right ] \ ] ] @xmath118 v(r ) - 2 \sqrt{6 } v_{ll}^{et}(r ) \left [ { d \over dr } v(r ) + 2{v(r ) \over r } \right ] - \sqrt{6 } \left [ { d \over dr } v_{ll}^{et}(r ) \right ] v(r ) \bigg \rbrace\ ] ] we have solved this equation numerically assuming that @xmath103 and @xmath104 are fixed solutions of the rarita - schwinger equations , and the resulting normalized solution for @xmath119 is shown in figure [ singletp ] . = 4.00 in = 3.200 in [ t ] from the computation outlined above we can derive an equivalent two - level system model in the form @xmath120 we compute @xmath121 @xmath122 @xmath123 the off - diagonal coupling matrix elements are somewhat smaller than we were hoping for , and future work will be needed to understand if this coupling is sufficiently large to account for experimental results . in addition , we have found that these off - diagonal matrix element depend on the nuclear spin , which suggests that the system may respond to net spin alignment . we recently proposed a new fundamental hamiltonian for condensed matter lattice problems that includes coupling to nuclear internal degrees of freedom . from our perspective this new coupling seems to be what is needed to account for the excess heat effect in the fleischmann - pons experiment . what has been needed in order to evaluate the models that result is an estimate for the coupling matrix element . the development of an estimate for this matrix element is challenging for a variety of reasons . we have presumed in the derivation of the fundamental hamiltonian that it is sufficient to model the nucleons as elementary dirac particles . however , we know that nucleons are composite particles made up of quarks and gluons , and that it is unlikely that using a dirac model as we have done is going to give accurate results . to do better we probably need to go back and develop a better fundamental hamiltonian based on quarks and electrons . if it is possible to obtain reasonable nucleon models from empirical potentials , then we may be able to develop a better estimate for the deuteron coupling matrix element . working directly with bound state qcd at this stage does not seem to be an attractive option . once we have decided on the simpler model that adopts an elementary dirac particle model for nucleons , then it is an issue of whether to use a relativistic or nonrelativistic model , and further it is an issue of what potential to use . since these computations involve a fair amount of work , it seemed sensible to adopt a nonrelativistic model since it is simpler , and to work with an older relatively simple nuclear model . the hamada - johnston potential fits the bill in this regard , as it is sufficiently simple that we are able to complete a calculation in relatively short order . perhaps the most work in this computation was the evaluation of the spin , isospin , and angular momentum algebra ; for this we relied on brute force mathematica calculations . in the end , we have developed a model for the coupling between the different nuclear spin states of the ground state deuteron and lattice - induced coupling to a highly - excited singlet p virtual state . the energy of this virtual state is about 125 mev in this model , which is consistent with our expectations . the coupling matrix element fell short of what we had hoped for by about an order of magnitude . we will need to clarify in future calculations if this is sufficiently large to be relevant to experimental results . the coupling matrix element in this model is proportional to @xmath32 , which is interesting in connection with the reported dependence of excess heat on the strength of an applied magnetic field . since the matrix element is proportional to @xmath32 , there is the potential for a larger coupling if the deuteron spins can be aligned . we are interested in pursuing this possibility in future work . 0 a. b. karabut and s. a. kolomeychenko , `` experiments characterizing the x - ray emission from a solid - state cathode using a high - current glow discharge , '' _ condensed matter nuclear science , proc . iccf10 _ , edited by p. l. hagelstein and s. r. chubb , p. 585 ( 2003 ) . a. b. karabut , `` research into characteristics of x - ray emission laser beams from solid state cathode medium of high - current glow discharge , '' _ condensed matter nuclear science , proc . iccf11 _ , edited by j. p. biberian , p. 253 ( 2004 ) . a. b. karabut , `` study of energetic and temporal characteristics of x - ray emission from solid state cathode medium of high - current glow discharge , '' _ condensed matter nuclear science , proc . iccf12 _ , edited by a. takahashi , k .- ota , and y. iwamura , p. 344 ( 2005 ) . a. b. karabut , e. a. karabut , p. l. hagelstein , `` spectral and temporal characteristics of x - ray emission from metal electrodes in a high - current glow discharge , '' _ j. cond . mat . ( in press ) . h. kamada , a. nogga , w. glckle , e. hiyama , m. kamimura , k. varga , y. suzuki , m. viviani , a. kievsky , and s. rosati , `` benchmark test calculation of a four - nucleon bound state , '' _ phys . c _ * 64 * ( 2001 ) 044001 .
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we recently put forth a new fundamental lattice hamiltonian based on an underlying picture of electrons and deuterons as elementary dirac particles . within this model
there appears a term in which lattice vibrations are coupled to internal nuclear transitions .
this is interesting as it has the potential to provide a connection between experiment and models that describe coherent energy transfer between two - level systems and an oscillator . in this work
we describe a calculation of the coupling matrix element in the case of the deuteron based on the old empirical hamada - johnston model for the nucleon - nucleon interaction .
the triplet s and d states of the the deuteron in the rest frame couples to a singlet p state through this new interaction .
the singlet p state in this calculation is a virtual state with an energy of 125 mev , and a coupling matrix element for @xmath0-directed motion given by @xmath1 .
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You are an expert at summarizing long articles. Proceed to summarize the following text:
the first indication of the existence of dark matter has already been found in the 1930s @xcite . by now astrophysicists have strong evidence @xcite-@xcite to believe that a large fraction ( more than 80% ) of the matter in the universe is dark ( i.e. , interacts at most very weakly with electromagnetic radiation ) . the dominant component of this cosmological dark matter must be due to some yet to be discovered , non baryonic particles . weakly interacting massive particles ( wimps ) @xmath0 are one of the leading candidates for dark matter . wimps are stable particles which arise in several extensions of the standard model of electroweak interactions . typically they are presumed to have masses between 10 gev and a few tev and interact with ordinary matter only weakly ( for reviews , see @xcite ) . currently , the most promising method to detect many different wimp candidates is the direct detection of the recoil energy deposited in a low background laboratory detector by elastic scattering of ambient wimps on the nuclei in a detector @xcite-@xcite . the event rate of direct wimp detection depends strongly on the velocity distribution of the incident particles . usually and for simplicity , the local velocity distribution of wimps is assumed to be a shifted maxwell distribution , as would arise if the milky way halo is isothermal @xcite-@xcite . however , our halo is certainly not a precisely isothermal sphere . possibilities that have been considered in the literature include axisymmetric halo models @xcite , the so called secondary infall model of halo formation @xcite , and a possible bulk rotations of the halo of our galaxy @xcite . if the halo of our galaxy consists of wimps , about @xmath1 wimps should pass through every square centimeter of the earth s ( and our ! ) surface per second ( for @xmath2 ) . however , the cross section of wimps on ordinary materials is very low and makes these interactions quite rare @xcite . for example , in typical susy models with neutralino wimp , the event rate is about @xmath3 event / kg / day and the energy deposited in the detector by a single interaction is about @xmath4 kev . typical event rates due to cosmic rays and ambient radioactivity are much larger . the annual modulation of the event rate due to the orbital motion of the earth around the sun has been suggested as a way to discriminate signal from background @xcite . actually , the dama collaboration has claimed that they have observed this annual modulation of the event rate @xcite . note , however , that the annual modulation of the signal is expected to amount only to a few percent ; this method can therefore only be used once more than one hundred signal events have been accumulated . in the meantime , a more promising approach is to reduce the background by vetoing events that do not look like nuclear recoil . this method is , e.g. , being used by the cdms @xcite , cresst @xcite and edelweiss @xcite collaborations . the presently best null result , from cdms @xcite , contradicts the dama claim for standard halo models . gev , and fast wimps with large scattering cross section @xcite . another possible way out is to postulate that the detected events are actually inelastic , leading to the production of a second particle that is almost , but not exactly , degenerate with the wimp @xcite . ] so far most theoretical analyses of direct wimp detection have predicted the detection rate for a given ( class of ) wimp(s ) , based on a specific model of the galactic halo . the goal of this paper is to invert this process . that is , we wish to study , as model independently as possible , what future direct detection experiments can teach us about the wimp halo . in other words , we want to start the ( theoretical ) exploration of `` wimp astronomy '' . in this first study we use a time averaged recoil spectrum , and assume that no directional information exists . we can thus only hope to construct the ( time - averaged ) one dimensional velocity distribution @xmath5 , where @xmath6 is the absolute value of the wimp velocity in the earth rest frame . note that our ansatz is quite different from that of the recent paper @xcite , which assumes a wimp velocity distribution and then analyses with which precision the wimp mass can be determined from the direct detection experiment . the remainder of this article is organized as follows . in sec . 2 we show how to find the velocity distribution of wimps from the functional form of the recoil spectrum ; our assumption here is that this functional form has been determined by fitting the data of some ( future ) experiment(s ) . we then derive formulae for moments of the velocity distribution function , such as the mean velocity and the velocity dispersion of wimps , which can be compared with model predictions . we also discuss some simple halo models . in sec . 3 we will develop a method that allows to reconstruct the wimp velocity distribution function directly from recorded signal events . this allows statistically meaningful tests of predicted distribution functions . we will also show how to calculate the moments of the velocity distribution directly from these data . in sec . 4 we conclude our work and discuss some further projects . some technical details of our calculations are given in the appendices . in this section we first show how to reconstruct ( moments of ) the wimp velocity distribution , and then discuss some simple model distributions . the differential rate for elastic wimp nucleus scattering is given by @xcite : [ eqn201 ] = dv . here @xmath7 is the direct detection event rate , i.e. , the number of events per unit time and unit mass of detector material , @xmath8 is the energy deposited in the detector , @xmath9 is the elastic nuclear form factor , and @xmath5 is the one dimensional velocity distribution function of the wimps impinging on the detector . the constant coefficient @xmath10 is defined as [ eqn202 ] , where @xmath11 is the wimp density near the earth and @xmath12 is the total cross section ignoring the form fact suppression . the reduced mass @xmath13 is defined by [ eqn203 ] m_r , where @xmath14 is the wimp mass and @xmath15 that of the target nucleus . finally , @xmath16 is the minimal incoming velocity of incident wimps that can deposit the energy @xmath8 in the detector : [ eqn204 ] v_min = , where we define [ eqn205 ] . in eqs.([eqn201])([eqn205 ] ) we have assumed that the detector essentially only consists of nuclei of a single isotope . if the detector contains several different nuclei ( e.g. nai as in the dama detector ) , the right hand side ( rhs ) of eq.([eqn201 ] ) has to be replaced by a sum of terms , each term describing the contribution of one isotope . for simplicity , in the remainder of this article we will focus on mono isotopic detectors . in this section we wish to invert eq.([eqn201 ] ) , i.e. , we want to find an expression for the one dimensional velocity distribution function @xmath5 for given ( as yet only hypothetical ) measured recoil spectrum @xmath17 . to that end , we first define = , i.e. , @xmath18 is the primitive of @xmath19 . eq.([eqn201 ] ) can then be rewritten as [ eqn211 ] = dv = f_1(v ) - f_1(v_min ) . since wimps in today s universe are quite slow , @xmath5 must vanish as @xmath6 approaches infinity , f_1(v ) 0 . hence |_v this means that @xmath20 approaches a finite value . differentiating both sides of eq.([eqn211 ] ) with respect to @xmath16 and using eq.([eqn204 ] ) , we find & = & - \ { _ q = v_min^2/^2 } + + & = & \{-2 q } _ q = v_min^2/^2 . since this expression holds for arbitrary @xmath16 , we can write down the following result directly : [ en1 ] = = \{-2 q } . the rhs of eq.([en1 ] ) depends on the as yet unknown constant @xmath10 . recall , however , that @xmath21 is the _ normalized _ velocity distribution , i.e. , it is defined to satisfy [ eqn214 ] f_1(v ) ~ dv = 1 . therefore , the normalized one dimensional velocity distribution function can be written as [ eqn212 ] f_1(v ) = , with normalization constant @xmath22 ( see appendix [ appn ] ) [ eqn213 ] = ^-1 . note that the integral on the rhs of eq.([eqn213 ] ) starts at @xmath23 . in the next step we wish to compute _ moments _ of the distribution function @xmath21 : [ en2 ] v^n = _ v_min(q_thre)^v^n f_1(v ) ~ dv . in eq.([en2 ] ) we have introduced a threshold energy @xmath24 . this is needed experimentally , since at very low recoil energies , the signal is swamped by electronic noise . moreover , we will later meet expressions that ( formally ) diverge as @xmath25 . @xmath26 is calculated as in eq.([eqn204 ] ) . inserting eq.([eqn212 ] ) into eq.([en2 ] ) and integrating by parts , we find ( see appendix [ appn ] ) [ eqn215 ] v^n = , with [ en3 ] i_n(q_thre ) = _ q_thre^q^(n-1)/2 dq . physically , @xmath27 is the average wimp velocity , while @xmath28 gives the velocity dispersion . in the statistical sense is given by @xmath29 . ] we emphasize that eqs.([eqn215 ] ) and ( [ en3 ] ) can be evaluated directly once the recoil spectrum is known ; knowledge of the functional form of @xmath5 is not required . note that the first term in eq.([eqn215 ] ) vanishes for @xmath30 if @xmath31 . in the same limit , @xmath32 by virtue of eq.([eqn213 ] ) . on the other hand , as written in eqs.([eqn212 ] ) and ( [ eqn213 ] ) , the velocity distribution integrated over the experimentally accessible range of wimp velocities gives a value smaller than unity . using only quantities that can be measured in the presence of a nonvanishing energy threshold @xmath24 , we can replace eq.([eqn213 ] ) by [ en4 ] ( q_thre ) = ^-1 . using @xmath33 in eq.([eqn212 ] ) ensures that the velocity distribution integrated over @xmath34 gives unity . we emphasize again that our final results in eqs.([eqn212 ] ) and ( [ eqn215 ] ) are independent of the as yet unknown quantity @xmath10 defined in eq.([eqn202 ] ) . they do , however , depend on the wimp mass @xmath14 through the coefficient @xmath35 defined in eq.([eqn205 ] ) . this mass can be extracted from a single recoil spectrum only if one makes some assumptions about the velocity distribution @xmath5 . in the kind of model independent analysis pursued here , @xmath14 has to be determined by requiring that the recoil spectra using two ( or more ) different target nuclei lead to the same distribution @xmath5 through eq.([eqn212 ] ) . note that this can be done independent of the detailed particle physics model , which determine the value of @xmath12 for the two targets . however , one will need to know both form factors , which strongly depend on whether spin dependent or spin independent interactions dominate @xcite . within a given particle physics model , the best determination of @xmath14 should eventually come from experiments at high energy colliders . the simplest semi realistic model halo is a maxwellian halo . the normalized one dimensional velocity distribution function in the rest frame of our galaxy is then @xcite [ eqn221 ] f_1,(v ) = e^-v^2/v_0 ^ 2 , where @xmath36 is the orbital speed of the sun around the galactic center , which characterizes the velocity of all virialized objects in the solar vicinity . on the other hand , when we take into account the orbital motion of the solar system around the galaxy , as well as the orbit of the earth around the sun , the distribution function must be modified to @xcite [ eqn222 ] f_1,(v , v_e ) = . here [ eqn223 ] v_e(t ) = v_0 , where @xmath37 june 2nd is the date on which the velocity of the earth relative to the wimp halo is maximal @xcite . eq.([eqn223 ] ) includes the effect of the rotation of the earth around the sun ( second term ) , but does not allow for the possibility that the halo itself might rotate . substituting these two expressions into eq.([eqn201 ] ) , one easily obtains the corresponding recoil energy spectra that wimp direct detection would measure @xcite : [ eqn224a ] _ = e^-^2 q / v_0 ^ 2 , and [ eqn225a ] _ = . here @xmath38 is the error function , defined as ( x ) = _ 0^x e^-t^2 dt . hence [ eqn224b ] _ e^-^2 q / v_0 ^ 2 , and [ eqn225b ] _ - . for future reference we note that the ( unrealistic ) `` reduced '' spectrum ( i.e. , the recoil spectrum divided by the squared nuclear form factor ) in eq.([eqn224b ] ) is exactly exponential ; this remains approximately true for the potentially quasi realistic spectrum in eq.([eqn225b ] ) as well . in order to test our formulae , we calculated @xmath39 and @xmath40 , first from the normalized distribution functions in eqs.([eqn221 ] ) and ( [ eqn222 ] ) , [ eqn226a ] _ = v_0 , [ eqn226b ] _ = v_0 ^ 2 , and [ eqn227a ] _ = e^-v_e^2/v_0 ^ 2 + ( + v_e ) , [ eqn227b ] _ = v_0 ^ 2+v_e^2 . then we used the spectra of eqs.([eqn224b ] ) and ( [ eqn225b ] ) in eqs.([eqn212 ] ) , ( [ eqn213 ] ) and ( [ eqn215 ] ) , taking @xmath41 . they reproduced the normalized velocity distribution functions in eqs.([eqn221 ] ) and ( [ eqn222 ] ) , as well as the results in eqs.([eqn226a ] ) to ( [ eqn227b ] ) . in the previous section we have derived formulae for the normalized one dimensional velocity distribution function of wimps , @xmath5 , and for its moments @xmath42 , from the recoil energy spectrum , @xmath17 . in order to use these expressions , one would need a functional form for @xmath17 . in practice this might result from a fit to experimental data . note that our expression for @xmath5 in eq.([eqn212 ] ) requires knowledge not only of @xmath17 , but also of its derivative with respect to @xmath8 , i.e. , we need to know both the spectrum and its slope . this will complicate the error analysis considerably , if @xmath17 is the result of a fit . in this section we therefore go one step further , and derive expressions that allow to reconstruct @xmath5 and its moments _ directly from the data_. the assumption we have to make is that the sample to be analyzed only contains signal events , i.e. , is free of background . this should be possible in principle : the most copious backgrounds ( from radioactive @xmath43 and @xmath44 decays ) can be discriminated on an event by event basis in many modern wimp detectors . the remaining background is dominated by elastic scattering of fast neutrons . while these look just like signal events , this background can at least in principle be made arbitrarily small by careful shielding and the use of a muon veto system ( to veto cosmic ray induced neutrons ) . having a sample of pure signal events , we can proceed with a complete statistical analysis of the precision with which we can reconstruct @xmath5 and its moments . in the absence of a true experimental sample of this kind , we had to resort to monte carlo experiments . to that end we wrote an unweighted event generator . to do so , we had to specify the form factor @xmath9 appearing in eq.([eqn201 ] ) ; this is the topic of the first subsection . we then proceed to analyze the reconstruction of @xmath5 and its moments in two subsequent subsections . we start by presenting the two most commonly used parameterizations of the squared nuclear form factor @xmath45 appearing in eq.([eqn201 ] ) . we focus on spin independent scattering , which usually dominates the event rate ; moreover , spin dependent form factors are still only poorly understood . the simplest form factor is the exponential one , first introduced by ahlen _ _ @xcite and freese _ et al . _ @xcite : [ eqn301]f_ex^2(q ) = e^-q / q_0 , where @xmath8 is the recoil energy transferred from the incident wimp to the target nucleus , q_0 = is the nuclear coherence energy and r_0 = fm is the radius of the nucleus . the exponential form factor implies that the radial density profile of the nucleus has a gaussian form . this gaussian density profile is simple , but not very realistic . engel has therefore suggested a more accurate form factor @xcite , inspired by the woods saxon nuclear density profile , [ eqn302]f_ws^2(q ) = ^2 e^-(q s)^2 . here @xmath46 is a spherical bessel function , q = is the transferred 3momentum , and r_1 = with r_a 1.2 ~ a^1/3 fm , s 1 fm . in our simulation we used the more accurate woods saxon form factor in eq.([eqn302 ] ) . since we assume a detector without directional sensitivity , a single event is uniquely characterized by the measured recoil energy @xmath8 . existing experiments such as cresst @xcite and cdms @xcite can determine the recoil energy quite accurately . we will see shortly that the statistical errors on the reconstructed velocity distribution @xmath5 will be quite sizable even for next generation experiments , given existing bounds on the scattering rate . we can therefore to good approximation ignore the error of @xmath8 in our analysis . in the following we do not distinguish between the recoil spectrum @xmath17 and the actual differential counting rate @xmath47 . since @xmath17 is usually given per unit detector weight and unit time , the two quantities differ by a multiplicative constant . this constant cancels in eq.([eqn212 ] ) , since it will also appear in the normalization constant ( [ eqn213 ] ) . we divide the total energy range into @xmath48 bins with central points @xmath49 and widths @xmath50 . in each bin , @xmath51 events will be recorded . our simulated data set can therefore be described by [ e3_0 ] q_n- , i = 1 , 2 , , n_n , n = 1 , 2 , , b. the standard estimate for @xmath17 at @xmath52 is then and for the ideal recoil spectrum . ] [ e3_1 ] r_n _ q = q_n = , n = 1 , 2 , , b. the squared statistical error on @xmath17 is accordingly [ e3_2 ] ^2 ( r_n ) = . as noted earlier , we also need the _ slope _ of the recoil spectrum to reconstruct the velocity distribution ; see eq.([eqn212 ] ) . a rather crude estimator of this slope is [ e3_3 ] s_1,n _ q = q_n = , where @xmath53 and @xmath54 are the numbers of events in bin @xmath55 which have measured recoil energy @xmath8 larger and smaller than @xmath49 , respectively . this estimator is rather crude , since it only uses the information in which half of its bin a given event falls . it is clear intuitively that an estimator that makes use of the exact @xmath56value of each event should be better . this can e.g. be obtained from the average @xmath56value in a given bin , defined as [ e3_4 ] q_n = _ i=1^n_n q_n , i . expanding @xmath57 around @xmath58 , keeping terms up to linear order , gives [ e3_5 ] _ q q_n _ q = q_n + ( q - q_n ) = r_n + ( q - q_n ) s_n . eq.([e3_5 ] ) allows to calculate @xmath59 as expectation value of @xmath8 in the @xmath60th bin : [ e3_6 ] q_n - q_n = = . an improved estimator of the slope is thus [ e3_7 ] s_2,n = . a simple calculation shows that the estimator ( [ e3_7 ] ) indeed has a smaller statistical error than the one in eq.([e3_3 ] ) . the definition ( [ e3_3 ] ) immediately implies [ e3_8 ] ^2(s_1,n ) = = , where we have used eqs.([e3_1 ] ) and ( [ e3_2 ] ) . in order to calculate the statistical error of the estimator @xmath61 , we first have to compute [ e3_9 ] _ n = = . treating the number of events and the average @xmath56value in a given bin as independent variables and using . ] ^2 = , this yields [ e3_10 ] ^2(s_2,n ) = ( ) ^2 ^2(r_n ) + ( ) ^2 ^2 = . this is smaller than the error ( [ e3_8 ] ) , by a factor 3/4 . an important observation is that the statistical error of both estimators of the slope of the recoil spectrum scale like the bin width to the power @xmath62 . this can intuitively be understood from the argument that the variation of @xmath17 , which we are trying to determine , will be larger for larger bins . one would therefore naively conclude that the errors of the estimated slopes would be minimized by choosing very large bins . however , both eq.([e3_3 ] ) and eq.([e3_7 ] ) reproduce the actual slope at @xmath58 only if the taylor expansion ( [ e3_5 ] ) holds ; in the presence of higher powers of @xmath63 neither of these estimates exactly reproduces the true slope at @xmath58 . not surprisingly , the influence of these higher powers , which induce uncontrolled systematic errors , will _ increase _ quickly with increasing bin width @xmath64 . we had seen at the end of sec . 2 that the predicted recoil spectrum resembles a falling exponential . this is confirmed in fig . [ fig301 ] , which shows the predicted recoil spectrum of a 100 gev wimp on @xmath65ge , using the woods saxon form factor ( [ eqn302 ] ) and the `` shifted maxwellian '' velocity distribution of eq.([eqn222 ] ) ; as usual , we cut the velocity distribution off at a velocity @xmath66 , here taken to be 700 km / s , since wimps with @xmath67 can escape the gravitational well of our galaxy . this figure also shows the result of a simulated experiment , where the exposure time and cross section are chosen such that the expected number of events is 5,000 ; these have been collected in seven bins in recoil energy . while an approximately exponential function can be approximated by a linear ansatz , as in eq.([e3_5 ] ) , only over a narrow range of @xmath8 , i.e. , for small bin widths , the _ logarithm _ of this function can be approximated by a linear ansatz for much wider bins . this corresponds to the ansatz [ eqn311 ] _ n _ q q_n _ n e^(q - q_n ) e^(q - q_s , n ) . here @xmath68 is the recoil spectrum at the point @xmath52 , _ n_q = q_n , while @xmath69 is the _ logarithmic slope _ of the recoil spectrum in the @xmath60th bin . our next task is to find estimators for @xmath68 and @xmath69 using ( simulated ) data . note that for @xmath70 , @xmath68 can not be estimated from the number of events @xmath51 in the @xmath60th bin alone . instead , one has [ eqn312 ] n_n = _ n dq = b_n _ n , where we have introduced the dimensionless quantities [ e3_11 ] x_n . hence , _ n= [ eqn313 ] depends on @xmath69 . on the other hand , the second , equivalent expression in eq.([eqn311 ] ) still uses the quantities @xmath71 as normalization . evidently , they describe the spectrum @xmath17 at the shifted points @xmath72 . equivalence of the two expressions in eq.([eqn311 ] ) implies [ e3_12 ] q_s , n = q_n + ( ) . the second expression in eq.([eqn311 ] ) thus has the advantage that the prefactor @xmath73 can be computed more easily than @xmath68 ; on the other hand , while the @xmath49 are simply the midpoints of the @xmath60th bin , and can thus be chosen at will , the @xmath72 are derived quantities ; they depend on the logarithmic slopes @xmath69 , which we havent determined yet . to do so , we again use the average @xmath56value in the @xmath60th bin . from eq.([eqn311 ] ) we find : [ eqn314 ] q_n - q_n = = x_n - . unfortunately this expression can not be solved analytically for @xmath69 . it is , however , straightforward to find @xmath69 numerically once @xmath59 is known . alternatively , we can make use of the second moment of the recoil spectrum in the @xmath60th bin , defined as [ eqn315 ] _ n = = ^2 . multiplying both sides of eq.([eqn314 ] ) with @xmath74 and adding to eq.([eqn315 ] ) , we can calculate the logarithmic slopes as [ eqn316 ] = . note that @xmath75 determined either from eq.([eqn314 ] ) or from eq.([eqn316 ] ) is independent of the normalization @xmath73 or @xmath68 . in the following we will estimate the logarithmic slopes from eq.([eqn314 ] ) , since it simplifies the error analysis somewhat ; note that the statistical errors of @xmath59 and @xmath76 are correlated . using standard error propagation , we have [ e3_13 ] ^2(k_n ) = ^2 ^-2 . the error on the average energy transfer can be estimated directly from the data , using [ e3_14 ] ^2 = , where now @xmath76 is estimated from the data , analogously to the experimental definition of @xmath59 in eq.([e3_4 ] ) . the second factor in eq.([e3_13 ] ) can be calculated straightforwardly from eq.([eqn314 ] ) : [ e3_15 ] ^-1 = , where we have defined the auxiliary function [ e3_15a ] f(x ) = 1 - ( ) ^2 . for given input values @xmath77 and @xmath64 , eqs.([e3_13 ] ) and ( [ e3_15 ] ) also allow to calculate the expected statistical error of the estimated @xmath69 , using eq.([eqn315 ] ) to calculate the expected error of @xmath78 . the result is shown in fig . [ fig302 ] . by normalizing the bin width to the inverse slope , and the expected error to its value for a given bin width , the result becomes independent of @xmath73 , and can in fact be used for all combinations of @xmath69 and @xmath64 . we observe that for small bins , the expected error again scales like @xmath79 , just as the expected errors ( [ e3_8 ] ) and ( [ e3_10 ] ) of our two estimators for the linear slope . if the bin width is significantly larger than the absolute value of the inverse of the logarithmic slope , the error decreases even faster with increasing bin width . ) one might assume that the statistical error of the estimated values of the @xmath69 could be minimized by estimating them from the average values of @xmath80 $ ] , for some fixed value of @xmath81 . for sufficiently small @xmath82 this in fact amounts to using the average value of @xmath8 , as described in the text . increasing @xmath82 leads to slightly _ larger _ expected statistical errors . ] this again argues in favor of using large bins . however , we again have to consider systematic errors . after all , it is quite unlikely that the ( as yet unknown ) recoil spectrum @xmath57 exactly satisfies our ansatz ( [ eqn311 ] ) over an extended range of @xmath8 . rather , this ansatz should be considered as the first terms in a taylor expansion of the logarithm of @xmath57 . in this case the next order term , which contributes @xmath83 in the exponent , will already modify @xmath59 . since we estimate @xmath69 from the numerical value of @xmath59 using eq.([eqn314 ] ) , which is exact only for @xmath84 , any non zero @xmath85 will introduce some systematic error in our estimate of @xmath69 . fortunately much of this error can be absorbed by a simple trick . according to ( [ eqn311 ] ) the logarithmic slope is constant over the entire bin , i.e. , we could use @xmath69 extracted from eq.([eqn314 ] ) as estimate of the logarithmic slope at any point @xmath8 between @xmath86 and @xmath87 . once @xmath88 the true logarithmic slope will in fact vary with @xmath8 . however , one may hope that the expectation value of our estimator still reproduces the true logarithmic slope at _ some _ value of @xmath8 within the @xmath60th bin . this is illustrated by fig . [ fig303 ] , which shows various evaluations of the logarithmic slope within one bin as function of the quadratic coefficient @xmath89 . the true logarithmic slope at the center of the bin is , of course , still given by @xmath90 , independent of the correction @xmath89 . as argued above , the expectation value of our estimator , shown by the dashed ( red ) line , does depend on @xmath89 . note , however , that our estimator comes quite close to the true logarithmic slope at the shifted value @xmath91 defined in eq.([e3_12 ] ) , which is shown by the dot dashed ( blue ) line ; this is true for both signs of @xmath89 . we therefore conclude that we can minimize the leading systematic error by interpreting our estimator of @xmath69 as logarithmic slope of the recoil spectrum , not at the center of the bin @xmath49 , but at the shifted point @xmath72 . note that @xmath72 itself depends on @xmath69 ; this , however , does not introduce any additional error , if we simply interpret eq.([e3_12 ] ) as an admittedly somewhat complicated prescription for the determination of the @xmath56values where we wish to estimate the logarithmic slope of the recoil spectrum . using large bins has a second , obvious disadvantage : the number of bins scales inversely with their size , i.e. , by using large bins we d be able to estimate @xmath21 only at a small number of velocities . this can be alleviated by using overlapping bins , or equivalently by combining several relatively small bins into overlapping `` windows '' . this means that a given data point @xmath92 may well contribute to several different windows , and hence to the measurement of @xmath21 at several values of @xmath6 . this can increase the total amount of information about @xmath21 since the only information we use about the data points in a given window is encoded in the average recoil energy in this window . this averaging effectively destroys information . by letting a given data point contribute to several overlapping windows , this loss of information can be reduced . a final disadvantage of using large bins or windows is that it would lead to a quite large minimal value of @xmath6 where @xmath21 can be reconstructed , simply because the central value @xmath93 , and also the shifted point @xmath94 , of a large first bin would be quite large . this can be again be alleviated by first collecting our data in relatively small bins , and then combining varying numbers of bins into overlapping windows . in particular , the first window would be identical with the first bin . a final consideration concerns the size of the bins . choosing fixed bin sizes , and therefore also mostly fixed window sizes , would lead to errors on the estimated logarithmic slopes , and hence also on the estimates of @xmath21 , that increase quickly with increasing @xmath8 or @xmath6 . this is due to the essentially exponential form of the recoil spectrum , which would lead to a quickly falling number of events in equal sized bins . we found that we get roughly equal errors in all bins if we instead take linearly increasing bins . these considerations motivate the following set up for our mock experimental analysis . we start by binning the data , as in eq.([e3_0 ] ) , where the bin widths satisfy [ e3_16 ] b_n = b_1 + ( n-1 ) ; here the increment @xmath95 satisfies [ e3_17 ] = , @xmath48 being the total number of bins , and @xmath96 being the ( kinematical or instrumental ) extrema of the recoil energy . we then collect up to @xmath97 bins into a window , with smaller windows at the borders of the range of @xmath8 . in the following we use latin indices @xmath98 to label bins , and greek indices @xmath99 to label windows ; later on we will use latin indices @xmath100 to label all events in the sample . for @xmath101 the @xmath102th window simply consists of the first @xmath103 bins ; for @xmath104 , the @xmath102th window consists of bins @xmath105 ; and for @xmath106 , the @xmath102th window consists of last @xmath107 bins . this can also be described by introducing the indices @xmath108 and @xmath109 which label the first and last bin contributing to the @xmath102th window , with [ e3_18 ] i_- = \ { ll 1 , & n_w + - n_w + 1 , & n_w . , i_+ = \ { ll , & b + b , & b . , ( 1 b + n_w - 1 ) . the center of the @xmath110th bin is called @xmath111 , as before . the total number of windows defined through eq.([e3_18 ] ) is evidently @xmath112 . the basic observables needed for the reconstruction of @xmath21 are then the number of events @xmath113 in the @xmath110th bin , as well as the average @xmath114 defined as in eq.([e3_4 ] ) . from these one easily calculates the number of events per window , [ e3_19 ] n_= _ i = i_-^i_+ n_i as well as the averages [ e3_20 ] q_= _ i = i_-^i_+ n_i q_i . one drawback of the use of overlapping windows in the analysis is that the observables defined in eqs.([e3_19 ] ) and ( [ e3_20 ] ) are all correlated ( for @xmath115 ) . the slope in a given window will again be calculated as in eq.([eqn314 ] ) , with `` bin '' quantities replaced by `` window '' quantities . we thus need the covariance matrix for the @xmath116 , where @xmath117 is the midpoint of the @xmath102th window ; it follows directly from the definition ( [ e3_20 ] ) : [ e3_21 ] cov(q_- q _ , q_- q _ ) = _ i = i_-^i_+ , where @xmath118 is defined as in eq.([e3_14 ] ) . in eq.([e3_21 ] ) we have assumed @xmath119 ; the covariance matrix is , of course , symmetric . moreover , the sum is understood to vanish if the two windows @xmath120 do not overlap , i.e. , if @xmath121 . the ansatz ( [ eqn311 ] ) is now assumed to hold over an entire window . we again estimate the prefactor as [ e3_22 ] r_= , @xmath122 being the width of the @xmath102th window . this implies [ e3_23 ] cov(r _ , r _ ) = _ i = i_-^i_+ n_i , where we have again taken @xmath119 . finally , the mixed covariance matrix is given by [ e3_24 ] cov(r _ , q_- q _ ) = _ i = i_-^i_+ n_i ( q_i - q _ ) . matrix is not symmetric under the exchange of @xmath103 and @xmath123 . in the definition of @xmath124 and @xmath125 we therefore have to distinguish two cases : @xmath126 as before , the sum in eq.([e3_24 ] ) is understood to vanish if @xmath127 . the covariance matrices involving our estimators of the logarithmic slopes @xmath128 , derived from eq.([eqn314 ] ) with @xmath129 everywhere , can be calculated in terms of the covariance matrices in eqs.([e3_21 ] ) and ( [ e3_24 ] ) : [ e3_26 ] cov(k _ , k _ ) = ( q_- q _ , q_- q _ ) , where @xmath130 is as in eq.([e3_11 ] ) with @xmath129 , and the function @xmath131 has been defined in eq.([e3_15a ] ) ; and [ e3_26a ] cov(r _ , k _ ) = ( r _ , _ - q _ ) . we are now ready to put all pieces together to compute the reconstructed velocity distribution and its statistical error . inserting the ansatz ( [ eqn311 ] ) with the substitution @xmath129 into eq.([eqn212 ] ) , one finds the reconstructed velocity distribution [ e3_27 ] f_1,r(v _ ) = n . here , @xmath132 is given by eq.([e3_12 ] ) with @xmath129 , and [ e3_28 ] v_= , see eq.([eqn204 ] ) . finally , the normalization @xmath133 defined in eq.([eqn213 ] ) can be estimated directly from the data : [ e3_29 ] n^-1 = _ a where the sum runs over all events in the sample . since neighboring windows overlap , the estimates of @xmath21 at adjacent values of @xmath134 are correlated . this is described by the covariance matrix @xmath135 the covariance matrices involving the normalized counting rates @xmath136 and logarithmic slopes @xmath128 have been given in eqs.([e3_23 ] ) , ( [ e3_26 ] ) and ( [ e3_26a ] ) . in principle eq.([e3_30 ] ) should also include contributions involving the statistical error of our estimator ( [ e3_29 ] ) for @xmath133 . however , we find this error , and its correlations with the errors of the @xmath136 and @xmath128 , to be negligible compared to the errors included in eq.([e3_30 ] ) . we are finally in a position to present some numerical results . we first validate our results by presenting @xmath137 distributions , defined via [ e3_31 ] ^2_f _ , c _ . here @xmath138 is our estimate ( [ e3_27 ] ) of the velocity distribution , @xmath21 is the true ( input ) distribution , and @xmath139 is the inverse of the covariance matrix of eq.([e3_30 ] ) . we expect @xmath140 to be ( roughly ) distributed according to the standard @xmath141 distribution when the results of sufficiently many simulated experiments , with sufficiently many events per experiment , are analyzed . figs . [ fig304 ] show @xmath137 distributions for 5,000 simulated experiments , with on average 500 ( top ) and 5,000 ( bottom ) events per experiments . note that the actual number of events in a given simulated experiment varies according to the poisson distribution ; otherwise one would introduce an artificial correlation between the normalized counting rates @xmath142 in different bins . moreover , the number of bins has been fixed a priori in these analyses . the last bin is typically empty , and has therefore been ignored in the analysis . this also reduces the number of windows used in the analysis by one , i.e. , the upper ( lower ) frame shows results for @xmath143 ( @xmath144 ) . + the two histograms in each figure differ by the number of terms that have been included in the estimate of the covariance matrix for @xmath138 . the solid ( blue ) histograms have been obtained by only including the second term in eq.([e3_30 ] ) , while the dashed ( red ) histograms also include the other terms , which are due to the statistical errors on the rescaled event numbers @xmath136 . we note that including these terms on average leads to a slight overestimate of the true error of @xmath138 , i.e. , the average of @xmath140 is somewhat smaller than unity . this is partly due to the fact that we have ignored the error on the normalization @xmath133 , which is correlated quite strongly with the errors on the @xmath136 . the lower figure demonstrates that for an average of 5,000 events per experiment the distribution of @xmath140 values becomes quite similar to the well known @xmath141 distribution , shown by the smooth curve . at least two effects contribute to the difference . first , we heavily relied on gaussian error propagation in our estimate ( [ e3_30 ] ) of the covariance matrix of the reconstructed velocity distribution . this is essentially a taylor expansion , including only the first non trivial term . it therefore becomes exact only in the limit of small errors , i.e. , for large numbers of events in a given window . since the recoil spectrum is falling essentially exponentially , this condition is practically always violated at least in the last bin(s ) , see fig . [ fig301 ] . we discard windows containing less than 3 events , but it is clear that this at best alleviates the problem . in the case at hand , this evidently results in an overestimate of the true error . secondly , our estimator ( [ e3_27 ] ) for the velocity distribution relies on the estimate of the logarithmic slopes @xmath128 , which in turn is based on eq.([eqn314 ] ) . as illustrated in fig . [ fig303 ] this estimate of @xmath128 in general has some systematic error , which would tend to increase @xmath140 . however , this figure also led us to expect small systematic errors if @xmath128 is interpreted as estimator of the logarithmic slope at the shifted points @xmath132 . indeed , as stated above , the total expression ( [ e3_30 ] ) somewhat overestimates the true error even in the lower frame of figs . [ fig304 ] , which assumes a large number of events but uses a rather small number of bins , which thus have to be quite large . had we instead interpreted @xmath128 as estimator of the slope at @xmath117 , the average @xmath140 would be about 2.9 , indicating that the systematic error would have dominated . [ fig304 ] also show an excess of simulated experiments with rather large values of @xmath140 if the covariance matrix for @xmath138 is estimated based on the errors on the @xmath128 only . this is true also for the upper frame , even though in this case the average value of @xmath140 is only about 0.93 . to be conservative , from now on we therefore take the full eq.([e3_30 ] ) as our estimator of the covariance matrix of @xmath138 , leading to average @xmath145 for the upper ( lower ) frame of figs . [ fig304 ] . in figs . [ fig305 ] we show results for the reconstructed velocity distribution , for `` typical '' simulated experiments with 500 ( top ) and 5,000 ( bottom ) events . in the top frame we choose @xmath146 bins , the first bin having a width @xmath147 kev , and combine up to three bins into a window . since the last bin is in fact empty , this leaves us with @xmath143 windows , i.e. , we can determine @xmath21 for six discrete values of the wimp velocity @xmath6 ; recall that these `` measurements '' of @xmath21 are correlated , as indicated by the horizontal bars in the figure . in the lower frame we choose @xmath148 bins with @xmath149 kev , and combine up to four bins into one window . the bins are thus significantly smaller than in the upper frame . as a result , the last two bins are now ( almost ) empty , leaving us with @xmath150 windows . figs . [ fig305 ] indicate that one will need at least a few hundred events for a meaningful direct reconstruction of @xmath21 . recall that @xmath21 is normalized to unity . the overall magnitude of @xmath21 is therefore essentially fixed by the range of observed @xmath56values ; only the _ shape _ of this distribution then remains to be determined . one measure of the information content of the reconstructed @xmath138 is therefore the confidence level with which one can exclude a constant @xmath21 . + in fig . [ fig306 ] we show one minus this confidence level , i.e. , the probability that a reconstructed velocity distribution is compatible with a constant . this has been estimated by defining a @xmath140 variable as in eq.([e3_31 ] ) for the hypothesis @xmath151 const . , and integrating the theoretical @xmath141 distribution over the range @xmath152 . here the constant has been chosen as @xmath153 , where @xmath154 have been calculated as in eq.([eqn204 ] ) using the largest and smallest recoil energy , respectively , that has been measured in a given experiment . since this probability differs quite widely from one ( simulated ) experiment to the next , we show both the mean and the median probability . we see that we ll need at least 200 events if we want to reject the hypothesis of a constant @xmath21 at the 90% c. l. ( on average ) . the confidence level then increases very quickly as additional events are added ; by the time 1,000 events have been accumulated , we can be quite sure that a constant @xmath21 can be excluded with high confidence . .this table illustrates how the binning , and in particular the combination of bins into `` windows '' , affects the information that can be gleaned from the reconstructed wimp velocity distribution . the first four columns show the average number of events in a given experiment , the number of bins , the size of the first bin in kev , and the number of bins per window . the remaining four columns show the mean and median probability that the reconstructed @xmath21 is compatible with a constant , the mean of the quantity @xmath155 defined in eq.([e3_32 ] ) , and the average @xmath140 of eq.([e3_31 ] ) . [ cols="^,^,^,^,^,^,^,^",options="header " , ] this confidence level , as well as more general measures of the information that can be extracted from a given experiment , depend on the choices of @xmath156 and , in particular , @xmath97 . this is illustrated in table 1 , which shows results for different combinations of @xmath157 and @xmath97 for 500 ( first five rows ) and 5,000 ( last six rows ) expected events per experiment . here the mean and median probabilities are the same as in fig . [ fig306 ] . in addition we show the mean of the quantity @xmath155 , defined as [ e3_32 ] = _ , c _ f_1,r(v _ ) f_1,r(v _ ) . formally @xmath155 determines the significance with which the hypothesis @xmath158 can be rejected . since @xmath21 is normalized , this hypothesis is unphysical . nevertheless @xmath155 can be regarded as a measure of the information content of a set of reconstructed @xmath159 ; in the absence of correlations , it becomes the sum over the inverse squares of the _ relative _ errors . note that , in contrast to @xmath140 , @xmath155 does not have a @xmath160 factor in front ; after all , by adding more windows we also add more values @xmath134 at which @xmath21 is determined , which can increase the information content . the first four rows , as well as the last four rows , show the effect of varying @xmath97 , the maximal number of bins that are collected in a window . we see that there is an optimal choice for this quantity . reducing @xmath97 leads to loss of information , as indicated by greatly increased values for the probability that @xmath138 is compatible with @xmath21 being constant as well as reduced values of @xmath155 . on the other hand , making the windows too large introduces too large systematic uncertainties in the estimates of the logarithmic slopes @xmath128 , which in turn leads to too large average values of @xmath140 . this is illustrated by rows four and seven , which have large windows due to our choice of a large @xmath97 ( row 4 ) or a large @xmath161 ( row 7 ) . the table also shows that the choice of @xmath161 has some impact on the confidence level with which the hypothesis of a constant @xmath21 can be rejected . we saw in figs . [ fig305 ] that our input @xmath21 has a broad maximum at @xmath162 km / s . rejection of the hypothesis of a constant @xmath21 is therefore optimized by maximizing the information about the outer reaches of @xmath21 . getting accurate information about @xmath21 at large velocities is very difficult ; this would need a large number of events at large @xmath8 , where the counting rate is very small . this leaves the region of small wimp velocity . by choosing a large first bin , one greatly reduces the error on @xmath138 in this first bin , which is also the first window ; this was illustrated in fig . [ fig302 ] . in fact , for 500 events and @xmath163 one can formally maximize the confidence level with which a constant @xmath21 can be rejected by considering only two bins , and making the first bin very large ; this is shown in the fifth row . note that this leads to an average @xmath140 well below unity , indicating that in spite of the large bins , systematic errors are still insignificant . however , we note that in this case our assumption that the error on @xmath133 is negligible is clearly not justified , since @xmath133 receives almost its entire contribution from the large first bin . by including the error on @xmath164 but ignoring the ( strongly correlated ! ) error on @xmath133 we clearly over estimate the total statistical error in this case . recall also that a large first bin leads to a large value for the smallest velocity , @xmath165 , where @xmath21 is determined . our `` figure of merit '' @xmath155 is less dependent on the details of binning , although , as stated earlier , it does strongly benefit from combining several bins into windows . we also note that the optimal achievable @xmath155 is essentially proportional to the number of events in the sample . this is expected , since @xmath155 is something like an inverse squared relative error . we saw in the previous subsection that a direct reconstruction of the wimp velocity distribution @xmath21 will only be possible once several hundred elastic nuclear recoil events have been collected . this is a tall order , given that not a single such event has so far been detected ( barring the possible dama observation ) . the basic reason for the large required event sample is that , @xmath21 being a normalized distribution , only information on the _ shape _ of @xmath21 is meaningful . in order to obtain such shape information via direct reconstruction , we have to separate the events into several bins or windows . moreover , each window should contain sufficiently many events to allow an estimate of the _ slope _ of the recoil spectrum in this window . on the other hand , at the end of sec . 2 we also gave expressions for the _ moments _ of @xmath21 . with the exception of the moment with @xmath166 , these are entirely inclusive quantities , i.e. , each moment is sensitive to the entire data set ; no binning is required , nor do we need to determine any slope ( with one possible minor exception ; see below ) . it thus seems reasonable to expect that one can obtain meaningful information about these moments with fewer events . an independent motivation for the determination of these moments is that they are sensitive also to @xmath21 at large values of the wimp velocity @xmath6 . we saw above that direct reconstruction of @xmath21 at large @xmath6 is very difficult , due to the small number of events expected in this region . moreover , a delta function like contribution to @xmath21 at the highest velocity , @xmath167 is very difficult to detect using direct reconstruction ; such a contribution is expected in `` late infall '' models of galaxy formation @xcite . the experimental implementation of eq.([eqn215 ] ) is quite straightforward . for @xmath168 , the normalization @xmath133 has already been given in eq.([e3_29 ] ) . the case of non vanishing threshold energy @xmath24 can be treated straightforwardly , using eq.([en4 ] ) . to that end we need to estimate the recoil spectrum at the threshold energy . one possibility would be to choose an artificially high value of @xmath24 , and simply count the events in a bin centered on @xmath24 . however , in this case the events with @xmath169 would be left out of the determination of the moments . we therefore prefer to keep @xmath24 as small as experimentally possible , and to estimate the counting rate at threshold using the ansatz ( [ eqn311 ] ) . since we need the recoil spectrum only at this single point , we only have to determine the quantities @xmath164 and @xmath170 parameterizing @xmath17 in the first bin ; this can be done as described in the previous subsection , without the need to distinguish between bins and `` windows '' . introducing the shorthand notation [ e3_32a ] r_thre _ q = q_thre , the resulting error can be written as [ e3_33 ] ^2 ( r_thre ) = r_thre^2 . the squared errors for @xmath164 and @xmath170 are simply the corresponding diagonal entries of the respective covariance matrices given in eqs.([e3_23 ] ) and ( [ e3_26 ] ) . finally , the definition ( [ e3_12 ] ) of @xmath94 implies [ e3_34 ] q_s,1 + k_1 = q_1 - + x_1 , where @xmath171 as before and @xmath93 is the central @xmath56value in the first bin . it should be noted that the first term in eq.([eqn215 ] ) is negligible for all @xmath172 if @xmath173 kev ; however , even for this low threshold energy it contributes significantly to the normalization constant @xmath133 , as described by eq.([en4 ] ) . of course , the first term in eq.([eqn215 ] ) always dominates for @xmath174 . this is not surprising , since the very starting point of our analysis , eq.(1 ) , already shows that the counting rate at @xmath24 is proportional to the `` minus first '' moment of the velocity distribution . the integral appearing in eq.([eqn215 ] ) can be estimated through the sum [ e3_35 ] i_n = _ a , see eq.([e3_29 ] ) . since all @xmath175 are determined from the same data , they are correlated , with [ e3_36 ] cov(i_n , i_m ) = _ a . this can e.g. be seen by writing eq.([e3_35 ] ) as a sum over narrow bins , such that the recoil spectrum within each bin can be approximated by a constant . each term in the sum would then have to be multiplied with the number of events in this bin ; eq.([e3_36 ] ) then follows from standard error propagation . note that , when re converted into an integral , the expression for @xmath176 will diverge logarithmically for @xmath31 ; equivalently , the numerical estimate of this entry can become very large if the sample contains events with very small @xmath56values . our numerical results presented below have therefore been obtained with @xmath177 kev ; many existing experiments in fact require significantly larger energy transfers in their definition of a wimp signal . with this correction , the reconstructed @xmath187 indeed closely reproduce the input values after averaging over sufficiently many experiments , even if the number of events in a given experiment is small . however , the numerical analysis revealed a number of additional problems . these can be understood from the observation that the @xmath175 in eq.([e3_35 ] ) , and even more the entries in their covariance matrix ( [ e3_36 ] ) , receive significant contributions from large @xmath8 values , where the counting rate itself is already very small . this is illustrated in fig . [ fig307 ] . the @xmath188axis shows the quantity [ e3_45 ] r(q_min ) _ q_min^q_max dq , divided by the total counting rate @xmath189 . here , @xmath190 is the kinematic maximum of @xmath8 for given input parameters , and @xmath191 is varied freely between 0 and @xmath190 . the @xmath192axis shows analogously the contributions to some @xmath175 ( lower curves ) and to the corresponding diagonal elements of the covariance matrix , i.e. , the squared errors ( upper curves ) , that come from the region @xmath193 . in the latter case we have converted the sums in eq.([e3_36 ] ) back into integrals . the figure shows that the region of @xmath56values that contributes 99% of the counting rate only contributes about 92% to @xmath194 , 73% to @xmath195 and 51% to @xmath196 ; for the given input parameters , this corresponds to the region @xmath197 kev . even worse , this region only contributes about 35% to @xmath198 and 5% to @xmath199 ! this implies that an experiment collecting only a small number of events will typically underestimate @xmath187 and , especially , its error ; the problem will become worse with increasing @xmath55 . on the other hand , as mentioned above , when averaged over sufficiently many experiments , our estimates for the @xmath175 do reproduce the true ( input ) values . this implies that occasionally an experiment will greatly _ over_estimate the @xmath175 , the problem again getting worse for larger @xmath55 . our numerical analysis also shows that , after averaging over ( very ) many experiments , eq.([e3_36 ] ) reproduces the mean square deviation between our estimated @xmath175 and the true ( input ) value . nevertheless , we just saw that in most cases this error is being underestimated . in order to be conservative , we therefore added `` the error on the error '' to the diagonal entries of the covariance matrix ; the off diagonal entries are then scaled up such that the correlation matrix remains unaltered . the squared `` error on the error '' is defined as [ e3_46 ] ^2(cov(i_n , i_n ) ) = _ a . with this modification , the average @xmath141 , again averaged over many experiments , is in the vicinity of unity at least for the first few moments . , where @xmath200 are the values of our estimators based on eq.([e3_35 ] ) and @xmath175 are the true ( input ) values , in general does not imply that @xmath201 . adding the `` error of the error '' to the covariance matrix brings the average of this ratio closer to unity . ] another problem is that the errors of the @xmath175 are very highly correlated . this can also be understood from fig . [ fig307 ] : a single event at high @xmath8 will contribute greatly to all moments with sufficiently large @xmath55 . numerically we find correlations of more than 98% between @xmath202 and @xmath203 for all @xmath204 ; the correlation between @xmath27 and @xmath205 still amounts to more than 87% . this implies that the higher moments unfortunately add only little to the available information . worse , attempting to include high moments in a @xmath141 fit often leads to numerical instabilities ; recall that a covariance matrix containing 100% correlated entries become singular , i.e. , can no longer be inverted . in practice only the moments with @xmath206 therefore seem to be useful . we are now ready to present some representative numerical results . [ fig308 ] shows the first 10 moments reconstructed with 100 events , using our standard input parameters ( see fig . [ fig301 ] ) . the estimated values of the moments have been divided by the true values . we see that in this `` typical '' example the high moments are indeed underestimated . we also see that the estimated relative errors at first increase with increasing @xmath55 ; this reproduces correctly the trend of the actual deviation of the estimated moments from the exact values . however , even after adding `` the error on the error '' , we find that the relative errors start to decline again for @xmath207 . this effect is probably entirely spurious ; recall that the errors are likely to be even more underestimated than the moments themselves . nevertheless we find it encouraging that already with 100 events a couple of moments can be determined with errors of about 15% . [ fig309 ] show a example of the information that might be gleaned from analyses of reconstructed moments of the wimp velocity distribution . here we attempt to constrain a possible `` late infall '' component in @xmath21 @xcite , defined by the ansatz [ e3_47 ] f_1(v ; v_esc , n_l ) = n_s f_1,(v ) ( v_esc - v ) + n_l ( v - v_esc ) . here @xmath208 is the standard `` shifted gaussian '' distribution ( [ eqn222 ] ) . as before , we have multiplied it with a cut off at @xmath66 . in addition , we introduce a contribution of wimps with fixed velocity , which we set equal to @xmath66 ; these wimps are just falling into our galaxy . is significantly larger than @xmath209 , this smearing should not matter very much . see ref . @xcite for a discussion of the effect of late wimp infall on the recoil spectrum . ] @xmath210 and @xmath66 are our free parameters ; @xmath211 is then chosen such that the total @xmath21 is normalized to unity . we then plot contours of @xmath212 , defined as the deviation of @xmath141 from its minimal value , where @xmath141 is defined as [ e3_48 ] ^2 = _ m , n _ mn ; here @xmath213 are the reconstructed moments in our mock experiment , @xmath214 are the predictions for these quantities based on eq.([e3_47 ] ) , and @xmath215 is the inverse of the covariance matrix ( [ e3_41 ] ) . figs . [ fig309 ] show a strong degeneracy in the fit . if the galactic escape velocity @xmath66 is kept fixed at its input value of 700 km / s , a quite significant upper bound on the normalization @xmath210 of the late infall component could be derived already from our simulated experiment with 25 events . however , if @xmath66 is kept free , no significant upper bound can be derived even from the simulated experiment with 100 events . note that the two experiments have been simulated independently , i.e. , the 25 events used for the analysis in the upper frame are not part of the 100 events used in the lower frame . the simulated experiment with 100 events was a bit `` unlucky '' in that the input values lie just outside the @xmath216 contour . as a result , the upper bound on @xmath210 for fixed @xmath217 km / s actually comes out a little worse in this case than in the experiment with only 25 events . this is in spite of the fact that the larger data sample allowed us to include one more moment in the fit . note that , according to the definition ( [ e3_47 ] ) , all wimps in our galactic neighborhood have velocity @xmath218 . this implies that a lower bound on @xmath66 can be derived from the highest observed @xmath56value , @xmath219 , see eq.([eqn204 ] ) . unfortunately for our standard set of input parameters , this method on average only yields lower bounds of about 400 ( 460 ) km / s for experiments with 25 ( 100 ) events . even the experiment with 100 events would then still allow 60% of all wimps to originate from a late infall component ; this is to be compared with the theoretical expectation @xmath220 . finally , we note that for @xmath221 , it will be essentially impossible to derive a meaningful upper bound on @xmath66 from these experiments : because the original `` shifted gaussian '' velocity distribution is already very small at our input value @xmath217 km / s , increasing @xmath66 has essentially no effect on the measured recoil spectrum . in this paper , we have developed methods that allow to extract information on the wimp velocity distribution from the recoil energy spectrum @xmath17 measured in elastic wimp nucleus scattering experiments . in the long term this information can be used to test or constrain models of the dark halo of our galaxy ; this information would complement the information on the density distribution of wimps , which can be derived e.g. from measurements of the galactic rotation curve . to this end , in sec . 2 we derived expressions that allow to directly reconstruct the normalized one dimensional velocity distribution function of wimps , @xmath5 , given an expression ( e.g. , a fit to data ) for the recoil spectrum . we have also derived formulae for the moments of @xmath21 . all these expressions are independent of the as yet unknown wimp density near the earth as well as of the wimp nucleus cross section ; the only information about the nature of the wimp that is needed is its mass . furthermore , in sec . 3 we have developed methods that allow to apply our expressions directly to data , without the need to fit the recoil spectrum to a functional form . we found that a good variable that allows direct reconstruction of @xmath21 is the average recoil energy in a given bin ( or `` window '' ; see sec . this average energy is sensitive to the slope of the recoil spectrum , which is the quantity we need to reconstruct @xmath21 . we carefully analyzed the statistical errors . unfortunately we found that several hundred events will be needed for this method to be able to extract meaningful information on @xmath21 . this is partly due to the fact that @xmath5 is normalized , i.e. , only the _ shape _ of this distribution contains meaningful information , and partly because this shape depends on the _ slope _ of the recoil spectrum , which is intrinsically difficult to determine . we therefore turned to an analysis of the moments of @xmath21 . we found that reliably estimating higher moments , and in particular estimating their errors , is difficult . the main reason is that these higher moments get large contributions from very rare events with large recoil energies . nevertheless we found that , based only on the first two or three moments , some non trivial information can already be extracted from @xmath222 events . the main emphasis of this exploratory study was on the basic expressions as well as on their implementation in actual experiments . the models for @xmath21 we tried to constrain in our applications ( a constant in sec . 3.2 , a `` late infall '' component with fixed velocity in the earth rest frame in sec . 3.3 ) are not physical ; nevertheless they illustrate the difficulties one will have in extracting information from these experiments that are of interest for the modeling of the galactic wimp distribution . our analysis is based on several simplifying assumptions . first , we ignored all experimental systematic uncertainties , as well as the uncertainty on the determination of the recoil energy @xmath8 . this is probably quite a good approximation , given that we found that we ll have to live with quite large statistical uncertainties in the foreseeable future ; recall that not a single wimp event has as yet been unambiguously recorded . we also assumed that our detector consists of a single isotope . this is quite realistic for the current semiconductor ( si or ge ) detectors . the cresst detector @xcite contains three different nuclei . however , by simultaneously measuring heat and light , one might be able to tell on an event by event basis which kind of nucleus has been struck . in this case , our methods can be applied straightforwardly to the three separate sub spectra . our analyses treat each recorded event as signal , i.e. , we ignore backgrounds altogether . at least after introducing a lower cut @xmath24 on the recoil energy , this may in fact not be unrealistic for modern detectors , which contain cosmic ray veto and neutron shielding systems . background subtraction should be relatively straightforward when fitting some function to the data , which would allow to use the expressions of sec . 2 . it should also be feasible in the method described in sec . 3.2 , if its effect on the average @xmath56values in the bins can be determined ; in particular , an approximately flat ( @xmath56independent ) background would not change the slope of the recoil spectrum . subtracting the background in the determination of the moments as described in sec . 3.3 might be ( even ) more difficult . as noted earlier , we need to know the wimp mass @xmath223 . this is true for any reconstruction method based on data taken with a mono isotopic detector . in this case one can always `` reconstruct '' @xmath5 , for any ( assumed ) value of @xmath223 . fortunately in well motivated wimp models , @xmath223 can be determined with high accuracy from future collider data . even in this case one will want to check experimentally that the wimps seen in dark matter detection experiments are in fact the same ones produced at colliders . this can be done by using the methods developed here on two different data sets , obtained with different detector materials , and demanding consistent results for ( the moments of ) @xmath21 . the feasibility of such an analysis is currently under investigation . in our analysis we ignored the annual modulation of the wimp flux . again , given the large statistical errors expected in the foreseeable future , this is a reasonable first approximation . nevertheless , eventually one will have to extend the formulae and methods developed here to allow for an annual modulation . this is fairly straightforward if the background is again negligible . on the other hand , new methods may be needed to extract information on @xmath21 in a situation where the total counting rate is dominated by background events ; this is most likely the case for the dama data @xcite , even if they indeed contain a signal , which remains highly controversial . in summary , we have begun to explore what direct dark matter detection experiments can teach us about the velocity distribution of dark matter particles in our galactic neighborhood . our analyses show that this will require substantial data samples . we hope this will encourage our experimental colleagues to plan future experiments well beyond the stage of `` merely '' detecting dark matter . we thank holger drees for illuminating discussions on stochastics . this work was partially supported by the marie curie training research network `` universenet '' under contract no . mrtn - ct-2006 - 035863 , as well as by the european network of theoretical astroparticle physics entapp ilias / n6 under contract no . rii3-ct-2004 - 506222 . since v = , [ eqna01 ] we have dv = dq , [ eqna02 ] from eq.([eqn212 ] ) and according to the normalization condition in eq.([eqn214 ] ) , we have , f_1(v ) ~ dv & = & \{-2 q } dq + & = & dq + & = & + & = & dq + & = & 1 , [ eqna03 ] where we have used the conditions |_q 0 and |_q 0 . eq.([eqn213 ] ) immediately followed from eq.([eqna03 ] ) . using eqs.([eqna01 ] ) , ( [ eqna02 ] ) and integration by parts , we can also find the moments of @xmath21 , defined with a lower cut off @xmath24 on the energy transfer , as follows : & = & _ v_min(q_thre)^v^n f_1(v ) ~ dv + & = & _ q_thre^^n dq + & = & _ q_thre^ q^(n+1)/2 dq + & = & ^n+1 \ { _ q = q_thre + _ q_thre^q^(n-1)/2 dq } . this reproduces eq.([eqn215 ] ) in sec . 2 . starting point is the observation that we wish to compute the ratio of two integrals , [ eb1 ] = . in the second step the integrals have been discretized , i.e. , replaced by sums over bins @xmath224 with @xmath225 events per bin . we now write the @xmath225 as sum of average value @xmath226 and fluctuation @xmath227 : [ eb2 ] = . introducing the notation g_a = _ i second order in the @xmath227 , we have : @xmath228 \nonumber \\ \&\simeq\ & \frac { \bar g_1 } { \bar g_2 } + \frac { 1}{\bar g_2 } \abrac{\sum_i \delta n_i g_1(x_i ) } - \frac { \bar g_1 } { \bar g_2 ^ 2 } \abrac{\sum_i \delta n_i g_2(x_i ) } \nonumber \\ \&~\ & ~~~~~~~~ - \frac{1}{\bar g_2 ^ 2 } \abrac{\sum_i \delta n_i g_1(x_i ) } \abigg{\sum_j \delta n_j g_2(x_j ) } + \frac { \bar g_1 } { \bar g_2 ^ 3 } \left ( \sum_i \delta n_i g_2(x_i ) \right)^2 \ , .\end{aligned}\ ] ] we now consider the average over many experiments . of course , @xmath227 averages to zero , but the product @xmath229 averages to @xmath230 , i.e. , it is non zero for @xmath231 . hence : [ eb4 ] - + . the sums appearing in the two correction terms also appear in the definition of the covariance matrix between @xmath232 and @xmath233 . note that we wish to compute the first term on the right hand side , since in our case the estimators for @xmath232 and @xmath233 indeed average to the correct values . this then leads to the final result [ eb5 ] - = ( g_1,g_2 ) - ( g_2,g_2 ) . applying this result to eqs.([eqn215 ] ) and ( [ en4 ] ) then immediately leads to eq.([e3_44 ] ) . v. c. rubin and w. k. ford , _ astrophys . j. _ * 159 * , 379 ( 1970 ) ; s. m. faber and j. s. gallagher , _ annu . rev astrophys . _ * 17 * , 135 ( 1979 ) ; v. c. rubin , w. k. ford , and n. thonnard , _ astrophys . j. _ * 238 * , 471 ( 1980 ) ; k. g. begeman , a. h. broeils , and r. h. sanders , _ mon . not . r. astron . * 249 * , 523 ( 1991 ) ; r. p. olling and m. r. merrifield , _ mon . not . r. astron . soc . _ * 311 * , 361 ( 2000 ) . j. f. navarro , c. s. frenk , and s. d. m. white , _ astrophys . j. _ * 462 * , 563 ( 1996 ) ; a. v. kravtsov _ _ , _ astrophys . j. _ * 502 * , 48 ( 1998 ) ; b. moore _ _ , _ mon . not . r. astron . soc . _ * 310 * , 1147 ( 1999 ) .
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weakly interacting massive particles ( wimps ) are one of the leading candidates for dark matter . currently , the most promising method to detect many different wimp candidates is the direct detection of the recoil energy deposited in a low background laboratory detector due to elastic wimp nucleus scattering . so far
the usual procedure has been to predict the event rate of direct detection of wimps based on some model(s ) of the galactic halo .
the aim of our work is to invert this process . that is , we study what future direct detection experiment can teach us about the wimp halo . as the first step
we consider a time averaged recoil spectrum , assuming that no directional information exists .
we develop a method to construct the ( time averaged ) one dimensional velocity distribution function from this spectrum .
moments of this function , such as the mean velocity and velocity dispersion of wimps , can also be obtained directly from the recoil spectrum .
the only input needed in addition to a measured recoil spectrum is the mass of the wimp ; no information about the scattering cross section or wimp density is required .
march 2007 * reconstructing the velocity distribution of wimps + from direct dark matter detection data * + and chung - lin shan +
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You are an expert at summarizing long articles. Proceed to summarize the following text:
the two - level model of atoms interacting with light fields @xcite has often been used to explore optical cooling mechanisms @xcite . its inherent simplicity the atom has one ground state and one excited state makes the resulting models amenable to analysis , but also suppresses mechanisms @xcite that , in the appropriate parameter regimes , dominate the interaction . a notable example of such an initially overlooked mechanism in atomic physics is three - dimensional optical molasses @xcite . by means of the two - level model , one can predict the equilibrium temperature , the so - called `` doppler '' temperature @xmath0 , of atoms in molasses to be @xmath1 , where @xmath2 is the ( half - width at half - maximum ) linewidth of the transition from the excited to the ground level @xcite . data from early three - dimensional molasses experiments contradicted this @xcite , showing that the achievable equilibrium temperature was in fact much lower . this discrepancy was resolved independently by two groups @xcite , both explanations relying on the inclusion of the manifold of magnetic sublevels in each of the ground and excited states . in particular , the motion of the atoms in the optical field leads to a non - adiabatic following of the magnetic sublevel populations , which gives rise to a strong viscous force and efficient cooling to temperatures significantly lower than the doppler temperature . we recently @xcite explored a new scattering theory that deals with the interaction of light and matter in a unified form applicable from microscopic to macroscopic systems . in that work we only considered the two - level atom model and showed , in particular , how our model can explain such mechanisms as standard optical molasses and mirror - mediated cooling @xcite . in this paper we extend this model to deal with magnetic sublevels , in much the same spirit as ref . in due course , this extension will enable us to deal with multilevel atoms interacting with an arbitrarily complex system composed of immobile mirrors , cavities , mems devices , etc . , without resorting to a quantized model for such a system . + after we introduce the general extension in the next section , we then proceed to explore two prototypical systems the @xmath3 transition , leading to the `` sisyphus '' cooling mechanism , and the @xmath4 transition in section [ sec : linperplin ] and section [ sec : sigmasigma ] , respectively . , @xmath5 , @xmath6 , and @xmath7 . the scatterwe has velocity @xmath8 and is described by means of its polarizability tensor @xmath9 . the field mode amplitudes are , in general , functions of the wavenumber @xmath10 . ] we investigate the interaction of atoms with light of different polarizations . to this end , we denote the two polarization basis vectors by @xmath11 and @xmath12 , whereby the standard circular polarization basis is equivalent to setting @xmath13 and @xmath14 . starting from the transfer matrix model explored in ref . @xcite and using the definitions in fig . [ fig : system ] , we replace each of the field modes by a corresponding jones vector , similar to the model used in ref . @xcite . thus , for example , @xmath15 and similarly for @xmath16 , @xmath17 and @xmath18 . the transfer matrix @xmath19 , describing the effect of the scatterer on the four field modes by means of the relation @xmath20 is now transformed into an order @xmath21 tensor of the form @xmath22 where each of @xmath23 ( @xmath24 ) is a @xmath25 matrix relating the respective jones vector components . a general recipe for transforming the formulae for the field mode amplitudes , as given in ref . @xcite , can be summarized by means of the two replacements @xmath26 wherever necessary . in particular , then , @xmath27 we follow ref . @xcite , complement e@xmath28 3-b , in defining the polarizability tensor @xmath9 as the steady - state expectation value of the polarizability operator @xmath29 ; @xmath9 is therefore given by the trace @xmath30 where @xmath31 is the steady - state density matrix describing the system and the summation runs over all the internal sublevels of the atom , and where we construct the order 4 polarizability operator tensor @xmath29 similarly to ref . @xcite , eq . ( 14.9 - 24 ) . in the general @xmath11 , @xmath12 basis : @xmath32\,,\end{aligned}\ ] ] with @xmath33 being the characteristic polarizability of the atom . in eq . ( [ eq : chime ] ) , the dipole moment operator @xmath34 ( @xmath35 ) is related to the @xmath11 ( @xmath12 ) polarized light field and the sum runs over all the internal sublevels , @xmath36 , of the atom . the matrix elements of @xmath34 ( @xmath35 ) are given by the appropriate clebsch - gordan coefficients . importantly , this new transfer matrix still retains all its properties , allowing us to model the interaction of the multilevel atom with an arbitrary system of immobile optical elements such as mirrors , cavities , waveplates , etc . as in our previous work @xcite , this interaction is accounted for by the multiplication of the various transfer matrices of the elements making up the system ; this model is , in principle , applicable to systems of arbitrary complexity . finally , we recall that the diagonal elements , @xmath37 , of @xmath31 are the populations in each of the sublevels , whereas its off - diagonal elements , @xmath38 , are the respective coherences . the matrix elements of @xmath31 are obtained from the appropriate optical bloch equations ( see , for example , the procedure outlined in ref . we note here that , through its dependence on @xmath31 , @xmath39 depends on the fields that it helps to determine , and thus eq . ( [ eq : standardtmm ] ) will in general become a set of nonlinear equations . in cases , like the ones considered in the following sections , where only one multilevel atom is interacting with a linear optical system , this problem may be solved using a procedure similar to the one outlined below : the fields surrounding the atom are obtained from the input fields through linear operations and then used with the optical bloch equations to obtain the populations and coherences of the atom s various levels . knowledge of these quantities then determines the fields , and hence the forces acting on the atom , completely . + in the following sections we will restrict our discussion to the case where the input field is not modified by other transfer matrices . we will apply this mechanism to investigate the behaviour of atoms in two cases where the polarization of the light varies in space on scales of the order of the wavelength to verify the validity of the model given by eq . ( [ eq : newmatrix ] ) to eq . ( [ eq : chime ] ) . in the first instance , we illuminate our atom with two counterpropagating linearly polarized beams . we choose the planes of polarization of the two beams to be orthogonal to each other . the second configuration we will investigate involves illuminating the atom with two circularly polarized beams , choosing opposite handedness for the two beams . these two cases mirror those in ref . @xcite . transition . ] in this and the following sections , we will adopt the low - intensity hypothesis . this allows us to simplify the optical bloch equations and resulting system considerably by neglecting the populations and coherences of the excited state sublevels . we can thus replace @xmath31 by the ground state steady - state density matrix , @xmath40 . we denote the diagonal element @xmath41 of @xmath40 , the population in sublevel @xmath42 , by @xmath43 , and the off - diagonal element @xmath44 , the coherence between sublevels @xmath42 and @xmath45 , by @xmath46 . here we will discuss what is perhaps the simplest transition between two levels with multiple magnetic sublevels : the @xmath3 transition . in this case , we have two ground sublevels so that @xmath47 is a @xmath25 matrix . fig . [ fig:1 - 2_to_3 - 2 ] tabulates the clebsch - gordan coefficients required to evaluate @xmath9 . we thus have : @xmath48 and @xmath49 whereby @xmath50 suppose , now , that we illuminate the atom with two counterpropagating beams having orthogonal linear polarization and equal intensity . this can be represented by setting @xmath51 and @xmath52 where the shift in the @xmath53 coordinate is introduced to simplify our expressions . using the optical bloch equations , we can show that the steady state populations in the ground sublevels at zero atomic velocity are given by @xmath54 noting that the populations do not depend on the field amplitudes in the low intensity regime . we work to lowest order in @xmath33 and make use of the above relations to find the net force acting on the atom : @xmath55\cdot\bigl(\mathbf{b}-\mathbf{c}\bigr)^\star\right\}}\nonumber\\ & \phantom{=\ } + 4\tfrac{v}{c}\hbar k{\,\text{im}\!\left\{\bigl(\boldsymbol{\zeta}\mathbf{b}\bigr)\cdot\mathbf{c}^\star+\bigl(\boldsymbol{\zeta}\mathbf{c}\bigr)\cdot\mathbf{b}^\star\right\}}\nonumber\\ & \phantom{=\ } -2\tfrac{v}{c}\hbar k^2{\,\text{im}\!\left\{\biggl[\frac{\partial\boldsymbol{\zeta}}{\partial k}\bigl(\mathbf{b}+\mathbf{c}\bigr)\biggr]\cdot\bigl(\mathbf{b}+\mathbf{c}\bigr)^\star\right\}}\nonumber\\ & \approx2\hbar k\lvert b\rvert^2{\,\text{im}\!\left\{\bigl[\boldsymbol{\zeta}\bigl(\mathbf{b}+\mathbf{c}\bigr)\bigr]\cdot\bigl(\mathbf{b}-\mathbf{c}\bigr)^\star\right\}}\nonumber\\ & \phantom{\approx\ } + 2\tfrac{v}{c}\hbar k^2{\,\text{im}\!\left\{\biggl[\frac{\partial\boldsymbol{\zeta}}{\partial k}\bigl(\mathbf{b}+\mathbf{c}\bigr)\biggr]\cdot\bigl(\mathbf{b}+\mathbf{c}\bigr)^\star\right\}}\,,\end{aligned}\ ] ] where we have assumed that @xmath56 . the velocity - dependent force terms in the above expression arise through the doppler shifting of photons both between field modes in the same polarization and between field modes in different polarizations ; these mechanisms are accounted for by the diagonal and off - diagonal terms in @xmath9 , respectively . these terms emerge through the velocity - dependent terms in the generalised transfer matrix . + in the present case , eq . ( [ eq : generalforce ] ) simplifies approximately to @xmath57 assuming that @xmath33 is real for simplicity . + we now let @xmath58 be a characteristic residence time of the two ground state sublevels ; this will introduce a non - adiabatic following term , proportional to @xmath8 , in the populations of each of the sublevels and emerges from the optical bloch equations . thus , we obtain the expression @xmath59 which agrees precisely with the standard literature ( cf . ( 4.20 ) and ( 4.23 ) in ref . transition . ] if we illuminate an atom with two counterpropagating beams of light in a @xmath60@xmath61 configuration , rich dynamics are obtained not in the simplest ( @xmath3 ) case , but in the next simplest , where the ground state has three magnetic sublevels ( @xmath62 ) and the excited state five ( @xmath63 ) . in this case , then , we can express @xmath47 and @xmath29 as @xmath64 and @xmath65 using the clebsch - gordan coefficients in fig . [ fig:1_to_2 ] . together , these give @xmath66 with @xmath67 representing the nonzero coherence between the @xmath68 and the @xmath69 sublevels . note that we again apply the low intensity hypothesis , thereby replacing @xmath31 with @xmath40 . we now illuminate the atom with two counterpropagating beams of equal intensity , @xmath5 and @xmath6 , possessing @xmath60 and @xmath61 polarization , respectively : @xmath70 and @xmath71 we again use eq . ( [ eq : generalforce ] ) to derive the force acting on the atom , which is given by @xmath72 where the populations and coherences are again obtained from the optical bloch equations , and can be found in ref . @xcite . by observing the natural correspondence between @xmath33 and @xmath73 in this latter reference , we can see that our expression for the force acting on the atom again agrees with the standard literature to first order in @xmath74 ( cf . ( 5.9 ) in ref . @xcite ) . the resulting friction force is thus due to both the doppler shift , as evident in the terms shown explicitly in eq . ( [ eq : sigmasigmaforce ] ) , as well as through the non - adiabatic following of the atomic sublevel populations . by revisiting the transfer matrix formalism and expressing the polarizability of a scatterer as the expectation value of a quantum operator , we have endowed it with a strong quantum character that allows us to handle atoms with multiple ground and excited state sublevels . in principle , our extended formalism is only limited by its reliance on the optical bloch equations to give expressions for the ground state populations and coherences ; we have retained the character of our earlier formalism that allowed us to work to arbitrary order in the polarizability . we have applied this theory to two standard sub - doppler cooling configurations , the so - called `` lin@xmath75lin '' and `` @xmath60@xmath61 '' configurations , and thereby reproduced the known expressions for the force acting on the atom . this work was supported by the uk engineering and physical sciences research council ( epsrc ) grant ep / e058949/1 and by the _ cavity - mediated molecular cooling _ network within the euroquam programme of the european science foundation ( esf ) , as well as by the national scientific fund of hungary ( contract no . nf68736 ) .
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we present a generic transfer matrix approach for the description of the interaction of atoms possessing multiple ground state and excited state sublevels with light fields .
this model allows us to treat multi - level atoms as classical scatterers in light fields modified by , in principle , arbitrarily complex optical components such as mirrors , resonators , dispersive or dichroic elements , or filters .
we verify our formalism for two prototypical sub - doppler cooling mechanisms and show that it agrees with the standard literature .
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You are an expert at summarizing long articles. Proceed to summarize the following text:
ultra - luminous x - ray sources ( ulxs ) are very bright extragalactic x - ray point sources , with observed fluxes which would correspond to luminosities greater than @xmath0 ergs per second if they were radiating isotropically . since this is above the eddington limit for normal stellar - mass compact objects , it has been widely thought that they are associated with intermediate - mass black holes ( imbhs ) @xcite , although non - spherical accretion and beamed emission could give rise to inferred luminosities significantly above the eddington limit ( see , for example * ? ? ? * ) , allowing also for lower masses . @xcite have reported nustar observations of ulx m82 x-2 ( also known as nustar j095551 + 6940.8 ) which reveal periodic changes in the hard x - ray luminosity of this source , indicative of a rotating magnetized neutron star being involved rather than a black hole . the measured peak flux ( in the @xmath1 band ) would correspond to @xmath2 if the radiation were isotropic , and is challenging to explain with a neutron star . the period ( taken to be the neutron - star spin period ) was found to be @xmath3 , with a 2.53-day sinusoidal modulation , interpreted as being an orbital period @xmath4 corresponding to motion around an unseen companion which would be the mass donor in the accreting system . the time derivative of the spin period @xmath5 was also measured . values for this coming from different individual observations show considerable variations but a relevant underlying spin - up tendency was found , with @xmath6 . the mass donor is indicated as having a mass larger than @xmath7 , so that the system should be categorized as a high mass x - ray binary ( hmxb ) . taking canonical neutron star parameters as a rough guide ( @xmath8 and @xmath9 ) , the luminosity relation @xmath10 , gives the mass accretion rate corresponding to @xmath11 as being @xmath12 . there are three main mechanisms by which the mass transfer might occur : ( i ) via a spherical wind ( as for o - type hmxbs ) , ( ii ) via a disc - shaped wind ( as for be - type hmxbs ) , or ( iii ) by roche lobe overflow ( rlof ) . because of the large inferred @xmath13 , the third option was suggested as the mechanism by @xcite and subsequent studies @xcite . here , we investigate each of these scenarios in turn to see which may be appropriate for ulx m82 x-2 . in section 2 , we discuss the strength required for the neutron - star magnetic field , and show that it needs to be moderately strong but not at a magnetar level . in section 3 , we discuss the applicability of scenarios ( i)-(iii ) , finding that ( i ) is excluded but that ( ii ) and ( iii ) could be viable possibilities . in section 4 , we discuss the role of the propeller effect and transient behaviour , and section 5 contains conclusions . in the standard picture for hmxbs , the system has to be fairly young because the companion donor star is massive enough to have only a rather short main - sequence life - time . matter coming from the donor star falls towards its neutron star companion , becomes included in a keplerian accretion disc , and eventually becomes entrained by the neutron star s magnetic field , creating hot x - ray emitting accretion columns above the magnetic poles ( cf . sufficiently young neutron stars typically have magnetic - field strengths above @xmath14 ( see , for example , the data in the atnf pulsar catalogue , * ? ? ? * ) , with a tail of the distribution extending beyond @xmath15 and eventually joining with the magnetar regime at @xmath16 . this can be relevant for explaining how this source can be so luminous , because if the magnetic field is stronger than the quantum limit , @xmath17 , the scattering cross - section would be suppressed , reducing the opacity of matter in the accretion columns above the magnetic poles and allowing higher luminosities . with this in mind , @xcite suggested that this source might contain a magnetar . the computational results of @xcite and the evidence of propeller effect from @xcite support this idea . in other works , however , @xcite , @xcite and @xcite have explored different scenarios with standard pulsar fields @xmath18 , while other authors have advocated weaker fields , @xmath19 @xcite . as the present work was being completed , we have seen a new paper by @xcite , advocating a model with strong beaming and a magnetic field of @xmath20 . the work presented here represents a line of study parallel to theirs . we focus here on a scenario with a field at the top end of the range for standard pulsars . in the rest of this section , we apply some simple assumptions for testing the relevance of a solution of this type . we take the full entrainment of the accreting matter by the magnetic field to occur close to the magnetic radius , @xmath21 , where the magnetic pressure balances the ram pressure of the infalling matter . using the condition of mass continuity , we then obtain the following expression for the magnetic radius : @xmath22 here @xmath23 is the field strength at the surface of the neutron star , and @xmath24 is the ratio of the accretion velocity to the free - fall velocity @xcite . at @xmath25 , the accreting matter is taken to come into corotation with the neutron star , with the corotation speed being @xmath26 . since the system is probably close to spin equilibrium @xcite , it is reasonable to take @xmath27 as being approximately equal to the keplerian velocity at @xmath25 . we can then estimate the appropriate value of the field strength in order to be consistent with the observed spin period for ulx m82 x-2 . this gives @xmath28 interestingly , this value is just above the critical quantum limit @xmath29 at which electron scattering is suppressed , as mentioned earlier : @xmath30 @xcite . on the other hand , we can also consider the spin - up rate , focusing on the measured underlying tendency @xmath31 @xcite , rather on the variations seen in the individual measurements , as mentioned earlier . we use the classical ghosh & lamb model @xcite , in order to identify appropriate parameter values ; the spin - up rate is then given by eq . ( 15 ) in their paper . assuming that the moment of inertia of the neutron star has the canonical value , @xmath32 , and using the spin - up rate given by @xcite , we find that the magnetic field strength corresponding to @xmath33 would be @xmath34 , which is very similar to the value obtained above , in eq . ( [ eq : bns ] ) , from considerations of the spin period . based on these two independent estimates , we conclude that having a field of @xmath35 can be consistent with the interpretation that m82 x-2 could be an example ( albeit a rather extreme one ) of previously - known classes of hmxbs ( see , for example , @xcite ) . a field of this strength can be just large enough to permit the quantum suppression of electron scattering as invoked by @xcite . in this section , we examine the different possibilities mentioned in the introduction for the accretion mode of ulx m82 x-2 . the two mechanisms operating for standard hmxbs are : ( i ) accretion from a spherical wind emitted by the donor star , and ( ii ) accretion from a disc - shaped wind , thought to come from a `` decretion disc '' around the donor star ( in a decretion disc , angular momentum is continually added to the inner edge of the disc from the rapidly rotating central star , and the matter drifts outwards rather than inwards ) . type ( i ) occurs for ob - type hmxbs , while type ( ii ) fuels be - type hmxbs . additionally , although there is no observational evidence for this in standard hmxb studies , it is possible that roche lobe overflow ( rlof ) could be relevant here ; we are referring to this as type ( iii ) . @xcite and @xcite both suggested rlof as a likely accretion mode for m82 x-2 , but did not discuss it in detail . in a recent paper , however , @xcite presented an argument suggesting that a certain proportion of ulxs should indeed be accreting neutron stars with rlof mass - transfer . here , we will consider each of the mechanisms in turn to assess which of them may be appropriate for our case . in order to give a luminosity as high as that observed , the accretion rate needs to be at least as high as @xmath36 , as shown earlier . first , we consider whether a spherical stellar wind coming from the massive donor could successfully fuel ulx m82 x-2 ( type ( i ) accretion ) . in simple stellar wind models , the wind velocity is frequently described by an expression of the form @xmath37 @xcite , where @xmath38 is the terminal velocity of the wind , @xmath39 is the stellar radius , @xmath40 is the distance from the centre of the donor star , and @xmath41 is a parameter whose value is close to unity ( we will take @xmath42 exactly here ) . the accretion rate from a spherical wind is usually described by the hoyle - lyttleton formula @xcite : @xmath43 where @xmath44 is the accretion radius defined by @xmath45 and @xmath46 is the velocity of the neutron star relative to the accreting matter : @xmath47 , where @xmath48 is the orbital velocity of the neutron star . the wind density can be written as @xmath49 where @xmath50 is the rate of mass loss from the donor star . if one specifies a particular value for @xmath50 , the accretion rate @xmath13 can be calculated from eq . ( [ eq : dotm ] ) , and then the corresponding x - ray luminosity of the neutron star can be obtained from @xmath51 . @xcite have given analytic formulae from which @xmath50 can be calculated for high - mass stars in terms of their mass @xmath52 , luminosity @xmath53 , effective temperature @xmath54 and the ratio @xmath55 , where @xmath38 is the wind s terminal velocity and @xmath56 is the escape velocity from the stellar surface . a change in behaviour occurs when @xmath54 passes a value @xmath57 and separate formulae are given for temperatures above and below that . here we use the mass loss rates given by equations ( 24 ) and ( 25 ) in their paper . the ratio @xmath55 is known to be @xmath58 for @xmath59 and @xmath60 for @xmath61 @xcite while values for @xmath53 and @xmath54 can be obtained as functions of @xmath52 using a stellar evolution code ( we used the analytic fitting formulae from the paper by @xcite ) . putting this data into eqs . ( [ eq : dotm ] ) ( [ eq : rhow ] ) , we could then calculate predicted values of the x - ray luminosity for a sequence of masses of the donor star , and our results are shown in fig . [ fig : wind ] . the two curves in fig . [ fig : wind ] correspond to different evolutionary phases ; the zero - age main sequence phase ( solid curve ) and the terminal main sequence phase ( dashed curve ) . for stars more massive than 30 @xmath62 , the stellar radius becomes larger than that of the binary orbit , @xmath63 ^{1/3 } , \label{eq : a}\ ] ] before it reaches the terminal main sequence ( here @xmath64 is the mass of the donor star ) . it is clear , from this figure , that even if we consider the most efficient conditions , and an optimal viewing angle , an x - ray luminosity of ulx level ( @xmath65 ) can not be achieved . hence , we conclude that ulx m87 x-2 can not be fed by a spherical wind because the density of the wind would not be high enough . the corbet diagram ( where @xmath4 is plotted against @xmath66 ) is frequently used in studies of hmxbs , and it can be useful to consider it also in the present context . ob - type and be - type hmxbs show clearly different distributions in these diagrams , with the majority of the be - type ones being close to the diagonal from lower - left to upper - right ( they are the points marked with a @xmath67 in fig . [ fig : be ] ) . @xcite obtained an expression for the sequence followed by the be - type hmxbs by assuming the hoyle - lyttleton accretion rate given by eq . ( [ eq : dotm ] ) . the decretion disc wind parameters @xmath68 and @xmath69 could be taken from any appropriate wind model ; @xcite adopted a simple disc - shaped model : @xmath70 where @xmath71 and @xmath72 are the density and velocity at the stellar radius @xmath73 , and @xmath74 is a constant which needs to be fixed . in order to explain the positions of known standard be - hmxbs in the corbet diagram , @xcite used @xmath75 , and we will also use this value in the following this choice is roughly consistent with recent numerical computations of be - disc winds @xcite . with this wind model and any specified value of the magnetic field strength , one can obtain a corresponding relationship between @xmath66 and @xmath4 by substituting equation ( [ eq : dotm ] ) into equation ( [ eq : bns ] ) ( @xmath4 is related to @xmath76 via equation ( [ eq : a ] ) ) . this gives @xmath77 where @xmath78 in fig . [ fig : be ] we show @xmath4-@xmath66 curves , obtained from eq . ( [ eq : pp ] ) , for three values of both @xmath79 and @xmath71 . the values used for @xmath72 and the radius of the neutron star are the same as those used by @xcite : @xmath80 and @xmath81 . in the two frames , we show the results for two different donor masses : @xmath82 and @xmath83 , with the corresponding radii being those for the terminal main sequence phase on the stellar evolution tracks of @xcite : @xmath84 for the @xmath83 donor and @xmath85 for the @xmath82 donor . it can be seen that the results in the two frames are very similar . in each of them , ulx m82 x-2 ( marked with the circles ) takes a rather extreme position in comparison with the be - hmxbs but , for an appropriate set of parameters , it does fit with the be - hmxb sequences . as shown in the figure , assuming a magnetic field strength of @xmath86 , the model given by eq . ( [ eq : pp ] ) with the density @xmath87 does fit the position of ulx m82 x-2 very well in both cases . of course , the position of ulx m82 x-2 on the figure could also be explained by models with lower @xmath79 and extremely low @xmath71 , or higher @xmath79 and extremely high @xmath71 . from the limited available polarization data , @xcite suggested that the @xmath71 for discs around be - stars is between @xmath88 and @xmath89 , based on the nine systems which they studied . @xcite tested values of @xmath71 between @xmath90 and @xmath91 for fitting the ir radiation from be discs , and suggested that the best fit value is @xmath92 . in view of these results , our best fit density ( @xmath93 ) seems slightly high but still a reasonable value . additionally , if we take this value for @xmath71 , eq . ( [ eq : dotm ] ) gives @xmath94 for the mass accretion rate which is again quite reasonable for ulx m82 x-2 . we note that a field as low as @xmath20 would be difficult to accommodate within this picture . with this large mass loss rate , however , the donor be - star can not maintain a high density disc for very long and so , after the bright ( high accretion rate ) phase , it would run out of decretion disc matter . hence , in this scenario , the accreting system would inevitably become a transient . since , according to the archival survey , ulx m82 x-2 does show transient tendencies , this prediction seems to be consistent with the observations @xcite . we note , however , that transient behaviour could also be understood in the context of propeller switching @xcite , as we will discuss later . plotted against the spin period of the neutron star @xmath66 . the upper and lower panels show results for donor masses of @xmath82 and @xmath83 , respectively . the @xmath4-@xmath66 relation given by equation ( [ eq : pp ] ) is shown by the continuous curves with the three sets of curves in each panel showing results for different values of @xmath79 ( measured in gauss ) , and the different curves of each set being for different disc densities ( in @xmath95 ) . the stellar radii are set to the terminal main sequence values obtained from the stellar evolution tracks of @xcite . the positions on the diagram of known standard be - type hmxbs are shown with crosses , while that for ulx m82 x-2 is shown with a circle . the data for the be - hmxbs is taken from @xcite and @xcite . , title="fig:",width=302 ] plotted against the spin period of the neutron star @xmath66 . the upper and lower panels show results for donor masses of @xmath82 and @xmath83 , respectively . the @xmath4-@xmath66 relation given by equation ( [ eq : pp ] ) is shown by the continuous curves with the three sets of curves in each panel showing results for different values of @xmath79 ( measured in gauss ) , and the different curves of each set being for different disc densities ( in @xmath95 ) . the stellar radii are set to the terminal main sequence values obtained from the stellar evolution tracks of @xcite . the positions on the diagram of known standard be - type hmxbs are shown with crosses , while that for ulx m82 x-2 is shown with a circle . the data for the be - hmxbs is taken from @xcite and @xcite . , title="fig:",width=302 ] @xcite suggested that the accretion mode for ulx m82 x-2 may involve roche lobe overflow ( rlof ) and we need to consider this scenario as well . an approximate value for the roche radius @xmath96 is given by @xmath97 @xcite , where @xmath98 is the mass ratio @xmath99 ( with @xmath64 being the mass of the donor star ) and @xmath76 is an orbital separation given by eq . ( [ eq : a ] ) . for having rlof accretion , the donor radius needs to be larger than the roche radius but smaller than the orbital radius . [ fig : radii ] shows evolutionary tracks for the radii of stars with selected masses ( again using data from @xcite ) with the dotted lines linking the points where each track crosses its corresponding roche radius ( lower curve ) and orbital radius ( upper curve ) . the shaded region in fig . [ fig : rl ] shows where rlof can occur and is plotted as a function of the mass of the donor star . the upper bound of this shaded region corresponds to the orbital radius , and the lower bound shows the roche radius . stellar radii at several evolutionary stages ( zero - age main sequence , the end of the main sequence and the beginning of the giant branch ) are also shown . if @xmath64 is larger than @xmath100 , the donor star already fills its roche lobe during the main sequence stage , leading to the type of mass transfer known as case a @xcite . this occurs on the thermal timescale @xmath101 for a massive star , this timescale is shorter than @xmath102 and the mass transfer rate reaches @xmath103 , which is sufficient for feeding ulx m87 x-2 . however , for the most massive stars ( @xmath104 ) , the main - sequence stellar radius would be larger than the orbital radius as well , so that a common envelope would form around the binary in a short time scale . in that case , the system would be luminous at infra - red frequencies rather than in x - rays @xcite ; the donor must therefore be less massive than this in order to give the right sort of rlof accretion . furthermore , the mass of the donor ought to be restricted by the fact that the companion is a neutron star : the progenitor of the neutron star would have been the primary when this binary system was born and , according to @xcite , it should have had an original mass of less than @xmath105 in order to give rise to a neutron star rather than a black hole . the present primary should be less massive than the original one ( even if its mass may have increased slightly during the first mass transfer stage ) , and hence the donor mass is constrained to be less than roughly @xmath106 . another issue which needs to be taken into account is that if the mass ratio @xmath107 is above a certain limit ( @xmath108 for a red giant , and @xmath109 for a high - mass main sequence star ) , the binary would be subject to the darwin instability , meaning that it could not sustain a circular orbit as required in order for rlof to be a viable candidate mechanism here @xcite . for @xmath64 between @xmath110 , the star will be inside its roche lobe during its main sequence lifetime but may overflow it during its expansion at the end of the main sequence ( hertzsprung gap phase ) , giving an early case b type of mass transfer . this also proceeds on the thermal timescale , as given above , and could in principle be sufficient to feed ulx m87 x-2 . however , in general the timescale on which the star crosses the hertzsprung gap is quite short . for instance , an @xmath111 star evolves from the end of the main sequence to the giant branch in 0.35 myr @xcite and would have a radius between @xmath96 and @xmath76 for only some fraction of that ( determining which would require a detailed stellar evolution calculation ) . the relevant time is then very short compared with the overall life - time of the star ( @xmath112 for @xmath113 ) . however , the duration of the hertzsprung gap phase depends sensitively on the mass of the donor star and for the minimum donor mass envisaged , @xmath7 @xcite , it becomes @xmath114 . while it would still require a lucky chance to see any ulx fed by this case b type of rlof accretion , it would become less unlikely with the least massive possible donor stars . in connection with this : according to the population synthesis simulation of @xcite , the number of ulxs containing a neutron star should peak for systems having donor stars in a small mass range around @xmath115 , and so it is possible that the short ulx lifetime of these systems could be compensated by there being so many of them , and that they might actually predominate in the observations . to summarise : accretion via rlof might possibly be the mechanism for ulx m82 x-2 if the donor is an evolved star of @xmath116 passing through the hertzsprung gap . an rlof scenario with a main sequence donor of @xmath117 could also be a possibility although , for the higher masses , such a system would be prone to the orbit becoming eccentric due to the darwin instability . an rlof scenario with the donor being evolved up to the giant phase is ruled out completely . in the previous sections , we have discussed similarities between ulx m82 x-2 and known standard hmxbs , in terms of the strength of the neutron star s magnetic field and the accretion mode . in section 2 , we concluded that a fairly strong magnetic field , of around @xmath118 is favoured for this object . while our considerations have been quite simple ones , this conclusion is consistent with more detailed analyses @xcite . in section 3 , we considered the possible feeding mechanisms , concluding that accretion modes similar to those of standard be - hmxbs are possible here , and that some mechanisms involving roche lobe overflow would also be possible . in the present section , we discuss two further related issues . the 1.37 second spin period of the neutron star in ulx m82 x-2 is shorter than those of most hmxbs ( which are typically in the range of @xmath119 ) . in view of this and the rather high magnetic field which we are inferring , one needs to consider whether it would be able to continue accreting matter from its disc , and continue to spin up , or whether it would enter the propeller regime where the disc matter is instead forced away @xcite . the critical period for onset of the propeller regime is given by @xmath120 where @xmath121 is the magnetic moment measured in units of @xmath122 , as before , and @xmath123 is the x - ray luminosity in units of @xmath124 @xcite . [ fig : prop ] shows the spin period - luminosity plane , with the straight lines marking the relation given by eq . ( [ eq : ps ] ) for various values of @xmath79 . also shown are the locations on the plot of known hmxbs ( marked with crosses ) and of ulx m82 x-2 ( marked with the circle ) . for the latter , we are taking the values @xmath125 and @xmath126 . below the line for the relevant value of @xmath79 , the propeller regime is operative ; above it , accretion can proceed . according to this , ulx m82 x-2 would avoid the propeller regime if it has @xmath127 and so field strengths between the quantum limit of @xmath128 and the maximum of @xmath129 could give accretion and spin - up , but magnetar - strength fields of above @xmath130 could not do so . the allowed range for @xmath79 to exceed the quantum critical limit but avoid the propeller effect , is small but non - vanishing . ( dashed ) , @xmath128 ( thick solid ) and @xmath129 ( dotted ) are shown . to avoid the propeller effect operating , the location of ulx m82 x-2 needs to be above the line corresponding the surface magnetic field strength of its neutron star , i.e. @xmath79 needs to be smaller than @xmath129 for the parameter values given in the text , which is nevertheless significantly greater than the quantum limit @xmath131 . , width=302 ] it has been reported that archival data for m82 x-1/x-2 from xmm newton shows only tiny fluctuations at a level below 2.2 % @xcite . this could be understood in two different ways . firstly , the pulsations seen by @xcite could be a transient behaviour so that while xmm newton did not see the pulsations during the time - span of its observations , nustar did see them . the second possibility is that the pulsations might be observable only in the high energy range above 10 kev , so that nustar could detect them while xmm newton could not . @xcite , noting the variations seen by @xcite in different measurements of @xmath5 , have suggested that the system may be marginal for exhibiting the propeller effect , switching it on and off , and that the strong x - ray emissions and pulsations may only be observed when the propeller mechanism is not operating whereas during the propeller phase , only radiation from the hot disc is seen . in their latest and more extended analysis @xcite , they conclude that the magnetically threaded disc model with @xmath132 could explain the observed properties of this object very well . @xcite also argue that the observed bimodal distribution of x - ray luminosity can be interpreted as being due to the propeller effect switching on and off . in their picture , even when the system is in the propeller regime , a small fraction of matter still leaks through the magnetic field lines and continues to give rise to lower luminosity emission . another point is that hmxbs can often show different pulsation behaviour in different energy ranges , with the higher energy x - rays being emitted as a _ pencil beam _ , while lower energy ones are emitted as a _ fan beam _ @xcite . in this case , the emission directions would be different for the different energy ranges . in fact , more than 20% of hmxbs show different light - curve shapes in high and low energy bands @xcite . at present , the discussion on the transient behavior of m82 x-2 can not be concluded at all . more data is needed for deciding between these different possibilities . also one needs to consider the effect of irradiation of the accreting matter by the strong x - ray emission coming from the neutron star . the direct effect of the radiation pressure coming from this is , of course , to act as another obstacle for the accretion . in order to achieve a high accretion rate in luminous x - ray systems , some means is required for avoiding this obstacle , such as anisotropy of the x - ray radiation ( see , for example , * ? ? ? * ; * ? ? ? * ) . in the wind accretion case , however , there are effects of the irradiation which act in the opposite direction , enhancing the accretion rate onto the neutron star . in detached wind - fed binaries , intra - binary matter can not avoid some photo dissociation under these circumstances , and this reduces the efficiency of the acceleration mechanism driving the line - accelerated stellar wind away from the donor star @xcite . from equation ( [ eq : racc ] ) one can see that the resulting slower wind velocity would cause the accretion radius to be larger , and lead to _ increasing _ the accretion rate onto the neutron star , as given by equation ( [ eq : dotm ] ) . however , the wind velocity is limited by @xmath133 , where @xmath56 is the escape velocity at the donor surface , and so it can not be slow enough to significantly affect our discussion here . another potential positive effect of x - ray irradiation is that it could in principle cause the roche lobe to become swamped with heated outer - envelope matter which could then overflow the roche lobe even if the stellar radius was smaller than roche radius @xcite . this would require very strong irradiation though ; it could only work in tight systems with orbital periods of a few hours , and would not play any significant role for m82 x-2 . in this paper , we have discussed the magnetic field strength and accretion mode for the recently identified neutron star in the source ulx m82 x-2 . we have considered the conditions required for producing the observed values of the key parameters : @xmath134 , @xmath135 , @xmath136 and @xmath137 , and have argued that a consistent explanation can be given involving a moderately strong magnetic field at around the quantum limit @xmath138 . having a field strength above the quantum limit is favourable for explaining the high observed luminosity , because of the reduced opacity in this case . however , we note that there have been a number of other suggestions regarding the magnetic field strength @xcite ; further observations and discussions are required for resolving the issue . we then went on to examine whether the standard accretion modes for hmxbs can be appropriate for this object . we concluded that spherical wind accretion , which drives ob - type hmxbs , can not be the mechanism here but that an extension of the standard mechanism for be - type hmxbs can provide a natural explanation . we have shown that if the neutron star has a moderately strong magnetic field ( @xmath139 ) and there is wind accretion from a reasonably high - density ( @xmath140 ) decretion disc around the be - companion , then all of the main properties of ulx m82 x-2 , including its position on the corbet diagram , would be consistent with the relations followed by standard be - hmxbs . roche lobe overflow accretion is also a possibility . if the donor star has sufficiently high mass , it should still be on the main sequence and case a rlof accretion could then feed the ulx ; if it has lower mass , it would need to be undergoing case b rlof while passing through the hertzsprung gap during its post - main - sequence expansion . we gratefully acknowledge helpful discussions with odele straub and hitoshi yamaoka during the course of this work .
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periodic pulsations have been found in emission from the ultra - luminous x - ray source ( ulx ) m82 x-2 , strongly suggesting that the emitter is a rotating neutron star rather than a black hole .
however , the radiation mechanisms and accretion mode involved have not yet been clearly established . in this paper
, we examine the applicability to this object of standard accretion modes for high mass x - ray binaries ( hmxbs ) .
we find that spherical wind accretion , which drives ob - type hmxbs , can not apply here but that there is a natural explanation in terms of an extension of the picture for standard be - type hmxbs .
we show that a neutron star with a moderately strong magnetic field , accreting from a disc - shaped wind emitted by a be - companion , could be compatible with the observed relation between spin and orbital period .
a roche lobe overflow picture is also possible under certain conditions .
accretion , accretion disk stars : neutron x - rays : binaries x - rays : individuals : m82 x-2
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You are an expert at summarizing long articles. Proceed to summarize the following text:
double parton scattering ( dps ) means that two short - distance subprocesses occur in a given hadronic interaction , with two initial partons being active from each of the incident protons in a collision at the large hadron collider ( lhc ) . the concept is shown for illustrative purposes in fig . [ fig : feyn - diag ] , and it may be contrasted with conventional single parton scattering ( sps ) in which one short - distance subprocess occurs , with one parton active from each initial hadron . since the probability of single parton scattering is itself small , it is often expected that the chances are considerably suppressed for two or more short - distance interactions in a given collision . however , expectations such as these bear quantitative re - examination at the lhc where the high overall center - of - mass energy provides access to very small values of the fractional momentum @xmath2 carried by partons , a region in which parton densities grow rapidly . a large contribution from double parton scattering could result in a larger than otherwise anticipated rate for multi - jet production and produce significant backgrounds in searches for signals of new phenomena . the high energy of the lhc also provides an increased dynamic range of available phase space for detailed investigations of dps . and @xmath3 from one proton and @xmath4 and @xmath5 from the second proton . the two hard scattering subprocess are @xmath6 and @xmath7 . [ fig : feyn - diag ] ] investigations of double parton scattering have a long history theoretically @xcite , and there is evidence for their presence in collider data from the cern intersecting storage rings @xcite and fermilab tevatron @xcite . a significantly greater role for double - parton processes may be expected at the lhc where higher luminosities are anticipated along with the higher collision energies . of substantial importance is to know empirically how large the double parton contribution may be and its dependence on relevant kinematic variables . our aim is to calculate characteristic final states at lhc energies in which it may be straightforward to discern a double parton signal . we show in this paper that double parton scattering produces an enhancement of events in regions of phase space in which the `` background '' from single parton scattering is relatively small . if such enhancements are observed experimentally , with the kinematic dependence we predict , then we will have a direct empirical means to measure the size of the double parton contribution . in addition to its role in general lhc phenomenology , this measurement will have an impact on the development of partonic models of hadrons , since the effective cross section for double parton scattering measures the size in impact parameter space of the incident hadron s partonic hard core . from the perspective of sensible rates and experimental tagging , a good process to examine should be the 4 parton final state in which there are @xmath8 hadronic jets plus a @xmath9 quark and a @xmath10 antiquark , _ viz . _ @xmath11 . if the final state arises from double parton scattering , then it is plausible that one subprocess produces the @xmath12 system and another subprocess produces the two jets . there are , of course , many single parton scattering ( 2 to 4 parton ) subprocesses that can result in the @xmath11 final state , and we look for kinematic distributions that show notable separations of the two contributions . as we show , the correlations in the final state are predicted to be quite different between the double parton and the single parton subprocesses . for example , the plane in which the @xmath12 pair resides is uncorrelated with the @xmath13 plane in double parton scattering , but not in the single parton case . the state - of - the - art of calculations of single parton scattering is well developed whereas the phenomenology of double parton scattering is as yet much less advanced . in the remainder of this introduction , we first describe the approach we adopt for the calculation of double parton scattering , specializing to the proton - proton situation of the lhc . then we outline the paper and summarize our main results . our calculations are done at leading - order in perturbative qcd , adequate for the points we are trying to make . making the usual factorization assumption , we express the single - parton hard - scattering differential cross section for @xmath14 as @xmath15 indices @xmath16 and @xmath4 run over the different parton species in each of the incident protons . the parton - level subprocess cross sections @xmath17 are functions of the fractional partonic longitudinal momenta @xmath18 and @xmath19 from each of the incident hadrons and of the partonic factorization / renormalization scale @xmath20 . the parton distribution functions @xmath21 express the probability that parton @xmath16 is found with fractional longitudinal momentum @xmath18 at scale @xmath20 in the proton ; they are integrated over the intrinsic transverse momentum ( equivalently , impact parameter ) carried by the parton in the parent hadron . a formal theoretical treatment of double parton scattering would begin with a discussion of the hadronic matrix element of four field operators and an explicit operator definition of two - parton correlation functions . this procedure would lead to a decomposition of the hadronic matrix element into non - perturbative two - parton distribution functions and the corresponding hard partonic cross sections for @xmath22 . an operator definition of two - parton correlation functions may be found in ref . @xcite where the two - parton correlation function is reduced to a product of single parton distributions . an explicit operator definition of two - parton distributions with different values of the two fractional momenta @xmath18 and @xmath23 is presented in ref . @xcite , along with a model for the two - parton distributions in terms of normal parton distributions . in this paper , we follow a phenomenological approach along lines similar to refs . @xcite . in a double parton process , partons @xmath16 and @xmath3 are both active in a given incident proton . we require the joint probability that parton @xmath3 carries fractional momentum @xmath23 , given that parton @xmath16 carries fractional momentum @xmath18 . in general , this joint probability @xmath24 should also depend on the intrinsic transverse momenta @xmath25 and @xmath26 of the two partons ( or , equivalently , their impact parameters ) . the hard scales @xmath27 and @xmath28 are characteristic of the two hard subprocesses in which partons @xmath16 and @xmath3 participate . in the sections below , we discuss the choice we make of the hard - scale and do not explore in this paper theoretical uncertainties associated with higher - order perturbative contributions . in contrast to single parton distributions functions @xmath21 for which global analyses have produced detailed information , very little is known phenomenologically about the magnitude and functional dependences of joint probabilities @xmath24 . a common assumption made in estimates of double parton rates is to ignore possibly strong correlations in longitudinal momentum and to use the approximation @xmath29 for reasons of energy - momentum conservation , if not dynamics , the simple factorized form of eq . ( [ eq : approx1 ] ) can not be true for all values of the fractional momenta @xmath2 . the values of @xmath23 available to the second interaction are always limited by the values of @xmath18 in the initial interaction since @xmath30 . the approximation certainly fails even at the kinematic level if both partons carry a substantial fraction of the momentum of the parent hadron . however , it may be adequate for applications in which the values of @xmath18 and @xmath23 are small . we remark that the momentum integral @xmath31 as long as we can run the upper limits of the @xmath18 and @xmath23 integrations to @xmath32 , independently . the large phase space at the lhc may make it possible to explore dynamic correlations that break eq . ( [ eq : approx1 ] ) . in fig . [ fig : partonx ] , for the region of phase space of interest to us , we show the contributions to the @xmath33 cross section as a function of @xmath2 from both dps and sps , after minimal acceptance cuts are imposed ( sec . [ sect : calc ] ) . the center - of - mass energy is @xmath34 tev . it is evident that the majority of dps events are associated with low @xmath2 values , in essence never exceeding @xmath35 . the momentum carried off by the beam remnant is @xmath36 in dps and @xmath37 in sps . the results in fig . [ fig : partonx ] show that this remnant momentum is not too different in dps and sps . thus , the use of eq . ( [ eq : approx1 ] ) in calculations of event rates at the lhc appears adequate as a good first approximation . while available tevatron data on double parton scattering @xcite are insensitive to possible correlations in @xmath2 , the greater dynamic range at the lhc may make it possible to observe them . ) holds at one hard scale , evolution of the parton densities with @xmath20 will induce violations at larger scales . ] in the dps and sps events . most dps events have low @xmath2 values . the events used for this plot include the requirements @xmath38 , @xmath39 , and the threshold cuts discussed in sec . ii . ] assuming next that the two subprocesses @xmath6 and @xmath7 are dynamically uncorrelated , we express the double parton scattering differential cross section as : @xmath40 the symmetry factor @xmath41 is @xmath32 if the two hard - scattering subprocesses are identical and is @xmath8 otherwise . in the denominator , there is a factor @xmath42 with the dimensions of a cross section . given that one hard - scatter has taken place , @xmath43 measures the size of the partonic core in which the flux of accompanying short - distance partons is confined . it should be at most proportional to the transverse size of a proton . for the first process of interest in this paper , @xmath44 , eq . ( [ eq : doubpartcross ] ) reduces to @xmath45 tevatron collider data @xcite yield values in the range @xmath46 mb . we use this value for the estimates we make , but we emphasize that the goal should be to make an empirical determination of its value at lhc energies . in sec . [ sect : calc ] , we present our calculation of the double parton and the single parton contributions to @xmath47 . we identify variables that discriminate the two contributions quite well . in sec . [ sect:4jets ] , we treat the double parton and the single parton contributions to @xmath48 jet production , again finding that good separation is possible despite the combinatorial uncertainty in the pairing of jets . we show in both cases that the double parton contribution falls off significantly more rapidly with @xmath49 , the transverse momentum of the leading jet . for the value of @xmath46 mb and the cuts that we use , we find that , in the region in which it is most identifiable , double parton scattering is dominant for @xmath50 gev in @xmath11 at lhc energies , and @xmath51 gev in @xmath48 jet production . our conclusions are found in sec . [ sect : conclusions ] . in this section , we describe the calculation of the dps and sps event rates for @xmath33 production at the lhc . for our purposes , light jets ( denoted by @xmath4 ) are assumed to originate only from gluons or one of the four lighter quarks ( @xmath52 or @xmath53 ) and , as stated above , we perform all calculations for the lhc with a center - of - mass energy of @xmath34 tev . event rates are quoted for 10 pb@xmath54 of data . the prediction for the dps event rate is based on the assumption that the two partonic interactions which produce the @xmath55 and @xmath56 systems occur independently ( as expressed in eq . ( [ eq : doubpartcross ] ) ) . at leading order , the only contribution is : @xmath57 where the symbol @xmath58 denotes the combination of one event each from the @xmath55 and the @xmath56 final states . in an attempt to model some of the effects expected from initial- and final - state radiation , we also account for the possibility of an additional jet which is undetected because it is either too soft or outside of the accepted rapidity range . thus , we include several other contributions to the dps event : @xmath59 where the parentheses surrounding a jet indicate that it is undetected . we compute processes such as @xmath60 and @xmath61 at lo as 3 parton final - state processes . the 2 to 3 parton amplitudes for @xmath61 [ and @xmath60 ] diverge as the undetected jet @xmath63 becomes soft or collinear to one of the other final state partons or to an initial parton . the divergences are removed in a full next - to - leading order ( nlo ) treatment , in which real emission and virtual ( loop ) contributions are incorporated , and the finite @xmath64 , @xmath61 , and @xmath65 contributions are present with proper relative normalization . in the lo parton level simulations done in this paper , we employ a cut at the generator level to remove the divergences . all the final state objects in the processes listed above are required to have transverse momentum @xmath66 gev . in this fashion , we model some aspects of the expected momentum imbalance between the @xmath9 and @xmath10 arising from the 2 to 3 process @xmath67 , but we can not claim to include the relative normalization between the @xmath64 and @xmath65 contributions that would result from a full nlo treatment . we leave a complete nlo analysis for future work . the sps cross section is computed according to eq . ( [ eq : singlescat ] ) . it receives contributions at lowest order from the 2 parton to 4 jet final state process : @xmath68 and , in the case where a jet is undetected , from the 5-jet final states ( computed at lo ) : @xmath69 we also investigate the possibility of @xmath70 and @xmath71 final state contributions to the sps cross section where two of the jets `` fake '' @xmath9 jets . we find that the effects from these final states are subdominant compared to the processes listed in eqs . ( [ eq : sps - loprocess ] ) and ( [ eq : sps - nloprocesses ] ) . in our numerical analysis , we use the leading - order cteq6l1 parton distribution functions ( pdfs ) @xcite to compute both dps and sps cross sections , and we evaluate all cross sections using one - loop evolution of @xmath72 . for the renormalization and factorization scales , we choose the dynamic scale : @xmath73 where @xmath74 is the transverse momentum of the @xmath75 jet and @xmath76 ( @xmath77 gev ) for light ( bottom ) jets . in the case of roughly equal values of the transverse momenta @xmath74 , eq . ( [ eq : mu ] ) yields @xmath78 in sps and @xmath79 in dps . at lo there is no obviously `` right '' hard scale , and the choice in eq . ( [ eq : mu ] ) seems as good as any other . the dps events are generated as two separate sets of events with madgraph / madevent @xcite and then combined as described above . for example , at leading order , we generate events separately for @xmath80 and @xmath81 , and these events are then combined as indicated in eq . ( [ eq : dps - loprocess ] ) . to increase the speed of the simulations , the sps events are generated with alpgen @xcite since the sps processes of interest are hard - coded in alpgen , which contains more compact expressions for the squared - matrix - elements than madgraph . the events accepted after generation are required to have 4 jets @xmath38 with 2 of these tagged as @xmath9 s @xmath39 . at the generator level , all the final state objects in the processes listed in eq . ( [ eq : dps - loprocess ] ) through eq . ( [ eq : sps - nloprocesses ] ) must have transverse momentum @xmath66 gev , as mentioned above . furthermore , at the analysis level , all events ( dps and sps ) are required to pass the following acceptance cuts : @xmath82 where @xmath83 is the jet s pseudorapidity , and @xmath84 is the separation in the azimuthal angle ( @xmath85 ) - pseudorapidity plane between jets @xmath16 and @xmath4 : @xmath86 we model detector resolution effects by smearing the final state energy according to : @xmath87 where we take @xmath88 and @xmath89 for jets . to account for @xmath9 jet tagging efficiencies , we assume a @xmath9-tagging rate of 60% for @xmath9-quarks with @xmath90 and @xmath91 . we also apply a mistagging rate for charm - quarks as : @xmath92 while the mistagging rate for a light quark is : @xmath93 over the range @xmath94 , we linearly interpolate the fake rates given above @xcite . having detailed the calculation of the @xmath33 event rates from dps and sps , we now discuss some of the distinguishing characteristics of the two contributions . first , however , it is important to check that our simulations of dps events are not introducing an _ artificial _ correlation between the @xmath55 and @xmath56 final states . we do this by inspecting the angle @xmath96 between the plane defined by the @xmath55 system and the plane defined by the @xmath56 system . if the two scattering processes @xmath97 and @xmath98 which produce the dps final state are truly independent , one would expect to see a flat distribution in the angle @xmath96 . by contrast , many diagrams , including some with non - trivial spin correlations , contribute to the 2 parton to 4 parton final state in sps , and naively one would expect some correlation between the two planes . to avoid possible effects from boosting to the lab frame , we define the two planes in the partonic center - of - mass frame . we specify the planes by using the three - momenta of the outgoing jets . then , the angle between the two planes defined by the @xmath56 and @xmath55 systems is : @xmath99 where @xmath100 is the unit three - vector normal to the plane defined by the @xmath101 system . the normal is undefined when @xmath102 and @xmath103 are back - to - back or @xmath104 and @xmath105 are back - to - back , as occurs in a large fraction of the dps events . therefore , when @xmath106 , we use a different procedure . we use the three - momentum of one of the incoming partons along with the three - momentum of one of the outgoing @xmath9 quarks to define the @xmath55 plane . let @xmath107 be the three - momentum of an incoming parton , and @xmath108 be the three - momentum of the final - state @xmath9 ( or @xmath10 ) quark . we then define @xmath109 to be the azimuthal angle of the three - vector normal to the @xmath110 plane . note that we use @xmath85 here since the normal to any three - vector and the beam - line will be transverse to the beam - line ( not the case in the sps process ) . in this way , the jet which is not used to define the plane is guaranteed to lie in the plane . the plane for the @xmath56 system is defined in an analogous manner . finally , the angle between the planes is then : @xmath111 in fig . [ fig : phi - planes ] , we display the number of events as a function of the angle between the two planes . there is an evident correlation between the two planes in sps , while the distribution is flat in dps , indicative that the two planes are uncorrelated . and @xmath56 systems . in sps events , there is a correlation among the planes which is absent for dps events . , scaledwidth=59.0% ] another interesting difference between dps and sps is the behavior of event rates as a function of transverse momentum . as an example of this , in fig . [ fig : ptj_1 ] , we show the transverse momentum distribution for the leading jet ( either a @xmath9 or light @xmath4 ) for both dps and sps . several characteristics are evident . first , sps produces a relatively hard spectrum , and for the value of @xmath43 and the cuts that we use , we see that sps tends to dominate over the full range of transverse momentum considered . on the other hand , dps produces a much softer spectrum which ( up to issues of normalization in the form of @xmath43 ) can dominate at small values of transverse momentum . the cross - over between the two contributions to the total event rate is @xmath112 gev for the acceptance cuts considered here . a smaller ( larger ) value of @xmath43 would move the cross - over to a larger ( smaller ) value of the transverse momentum @xmath1 of the leading jet . distribution of the leading jet in @xmath113 after minimal cuts . , scaledwidth=59.0% ] we turn next to the search for variables that may allow for a clear separation of the dps and sps contributions . since the topology of the dps events includes two @xmath114 hard scattering events , the two pairs of jet objects are roughly back - to - back . we expect the azimuthal angle between the pairs of jets corresponding to each hard scattering event to be strongly peaked near @xmath115 . real radiation of an additional jet , where the extra jet is missed because it fails the threshold or acceptance cuts , allows smaller values of @xmath116 . the relevant distribution is shown for light jets ( non @xmath9-tagged ) in fig . [ fig : delphi]a . there is a clear peak near @xmath117 for dps events , while the events are more broadly distributed in sps events . the secondary peak near small @xmath118 arises from gluon splitting which typically produces nearly collinear jets . the suppression at still lower @xmath118 comes from the isolation cut @xmath119 . in the azimuthal angles of light jet pairs for dps and both sps+dps events the dijet pairs are back - to - back in dps events . ( b ) the variable @xmath120 for dps and sps+dps events provides a stronger separation of the underlying dps events from the total sample when compared to @xmath121 for any pair.,title="fig:",scaledwidth=49.0% ] in the azimuthal angles of light jet pairs for dps and both sps+dps events . the dijet pairs are back - to - back in dps events . ( b ) the variable @xmath120 for dps and sps+dps events provides a stronger separation of the underlying dps events from the total sample when compared to @xmath121 for any pair.,title="fig:",scaledwidth=49.0% ] the separation of dps events from sps events becomes more pronounced if information is used from both the @xmath55 and @xmath56 systems . as an example , we consider the distribution built from a combination of the azimuthal angle separations of both @xmath56 and @xmath122 pairs , using a variable adopted from ref . @xcite : @xmath123 in fig . [ fig : delphi]b , we present a distribution in @xmath124 for both dps and sps+dps events . again , as in the case of the @xmath125 distribution , we see that the sps events are broadly distributed across the allowed range of @xmath120 . however , the combined information from both the @xmath55 and @xmath56 systems shows that the dps events produce a sharp and substantial peak near @xmath126 which is well - separated from the total sample . the narrow peaks near @xmath117 in fig . [ fig : delphi]a and near @xmath127 in fig . [ fig : delphi]b will be smeared somewhat once soft qcd radiation and other higher - order terms are included in the calculation . another possibility for discerning dps is the use of the total transverse momentum of both the @xmath55 and @xmath56 systems . at lowest order for a @xmath128 process , the vector sum of the transverse momenta of the final state pair vanishes . in reality , radiation and momentum mismeasurement smear the expected peak near zero . nevertheless , we still expect dps events to show a distribution in the transverse momenta of the jet pairs that is reasonably well - balanced . to encapsulate this expectation for both light jet pairs and @xmath9-tagged pairs , we use the variable @xcite : @xmath129 here @xmath130 is the vector sum of the transverse momenta of the two final state @xmath9 jets , and @xmath131 is the vector sum of the transverse momenta of the two ( non @xmath9 ) jets . the distribution in @xmath132 is shown in fig . [ fig : sptprimecut ] . as expected , we observe that the dps events are peaked near @xmath133 and are well - separated from the total sample . the sps events , on the other hand , tend to be far from a back - to - back configuration and , in fact , are peaked near @xmath134 . this behavior of the sps events is presumably related to the fact that a large number of the @xmath55 or @xmath56 pairs arise from gluon splitting which yields a large @xmath135 imbalance and , thus , larger values of @xmath132 . for the dps and sps samples . due to the back - to - back nature of the @xmath114 events in dps scattering , the transverse momenta of the jet pair and of the @xmath9-tagged jet pair are small , resulting in a small value of @xmath132 . in ( a ) we show the @xmath132 distribution for our standard cuts , and in ( b ) we increase the cut on the transverse momentum of the leading jet , @xmath136 gev . the fraction of dps events in the whole sample decreases with increasing @xmath1.,title="fig:",scaledwidth=49.0% ] for the dps and sps samples . due to the back - to - back nature of the @xmath114 events in dps scattering , the transverse momenta of the jet pair and of the @xmath9-tagged jet pair are small , resulting in a small value of @xmath132 . in ( a ) we show the @xmath132 distribution for our standard cuts , and in ( b ) we increase the cut on the transverse momentum of the leading jet , @xmath136 gev . the fraction of dps events in the whole sample decreases with increasing @xmath1.,title="fig:",scaledwidth=49.0% ] in this subsection , we find that extraction of the dps `` signal '' for @xmath33 production from the sps `` background '' can be enhanced by combining information from both @xmath55 and @xmath56 systems . our simulations suggest that the variable @xmath132 may be a more effective discriminator than @xmath124 . however , given the leading order nature of our calculations and the absence of smearing associated with initial state soft radiation , this picture may change and a variable such as @xmath120 ( or some other variable ) may become a clearer signal of dps at the lhc . realistically , it would be valuable to study both distributions once lhc data are available in order to determine which is more instructive . in the following , we use the clear separation shown in fig . [ fig : sptprimecut ] in our exploration of the distinct properties of dps and sps events . the evidence in fig . [ fig : delphi ] and fig . [ fig : sptprimecut ] for distinct regions of dps dominance prompts the search for greater discrimination in a plane represented by a two dimensional distribution of one variable against another . we examined scatter plots involving the inter - plane angle @xmath96 , the jet - jet azimuthal angle difference @xmath116 , @xmath124 , and @xmath137 . strong kinematic correlations are evident in the plot of @xmath124 _ vs. _ @xmath137 at the level of our leading order calculation , and we observe no additional separation of dps and sps beyond that evident in figs . [ fig : delphi ] and [ fig : sptprimecut ] . likewise , there are strong correlations between @xmath116 and @xmath124 . one scatter plot with interesting features is displayed in fig . [ fig : scatter1 ] . the dps events are seen to be clustered near @xmath138 and are uniformly distributed in @xmath96 . the sps events peak toward @xmath139 and show a roughly @xmath140 character . while already evident in figs . [ fig : phi - planes ] and [ fig : sptprimecut ] , these two features are more apparent in the scatter plot fig . [ fig : scatter1 ] . moreover , the scatter plot shows a valley of relatively low density between @xmath141 and @xmath142 . in an experimental one - dimensional @xmath96 distribution such as fig . [ fig : phi - planes ] , one would see the sum of the dps and sps contributions . if structure is seen in data similar to that shown in the scatter plot fig . [ fig : scatter1 ] , one could make a cut at @xmath143 or @xmath35 and verify whether the experimental distribution in @xmath96 is flat as expected for dps events . and @xmath132 for the dps and sps samples.,scaledwidth=99.0% ] in fig . [ fig : ptj_1 ] , we show that dps produces a softer transverse momentum distribution for the leading jet ( either a @xmath9 or light @xmath4 ) . in data one would see only the sum of the dps and sps components in a plot like fig . [ fig : ptj_1 ] . a scatter plot of @xmath132 _ vs. _ the transverse momentum of the leading jet motivates an empirical separation of the two components . in figs . [ fig : sptprimecut](a ) and [ fig : sptprimecut](b ) we compare the @xmath132 distributions for two different selections on the transverse momentum @xmath1 of the leading jet in the @xmath144 sample . this comparison of the distributions confirms that events in the dps region , defined empirically by the region @xmath145 or @xmath35 , fall off more steeply with @xmath1 than the rest of the sample . it will be important and interesting to see whether the selection @xmath145 or @xmath35 in lhc data also produces events that show a more rapid decrease with @xmath1 . the leading - jet transverse momentum distributions are shown in figs . [ fig : ptjcrossover](a ) and [ fig : ptjcrossover](b ) for two different cuts on @xmath132 . in both cases , we see that the sps sample has a broader distribution in @xmath1 and that the dps sample dominates for small enough values of @xmath1 . for our chosen value of @xmath46 mb , and for cuts we employ , the crossover points are roughly @xmath146 gev for @xmath147 and @xmath148 gev for @xmath149 . for ( a ) @xmath147 and ( b ) @xmath150 . as the signal region becomes more dominated by sps events ( i.e. moving from ( a ) to ( b ) ) , the resulting distribution becomes harder and shifts the sps - dps cross - over from @xmath151 gev to @xmath152 gev . , title="fig:",scaledwidth=49.0% ] for ( a ) @xmath147 and ( b ) @xmath150 . as the signal region becomes more dominated by sps events ( i.e. moving from ( a ) to ( b ) ) , the resulting distribution becomes harder and shifts the sps - dps cross - over from @xmath151 gev to @xmath152 gev . , title="fig:",scaledwidth=49.0% ] in addition to @xmath153 , we can also ask how important dps can be for a generic @xmath154 final - state , where none of the jets are @xmath9-tagged . in this section , we describe our calculation of the double parton scattering and the single parton scattering contributions to the production of a @xmath154 final state , for which the cross section is larger . our exposition can be brief since we repeat the procedure described in some detail in sec . [ sect : calc ] . the dps process for @xmath154 production is topologically equivalent to @xmath155 . however , in the @xmath154 system , we lose the @xmath9-tagging ability that reduces the combinatorial background in @xmath153 , and the prospects for isolating and measuring dps over the sps background may appear less promising . fortunately , in going from the @xmath122 subprocess to the @xmath56 subprocess , a much larger dps rate is possible due to the much larger cross section for @xmath56 production . as we show below , we find that the dps signature can be extracted in this @xmath154 mode as well . the dps cross section for @xmath154 production receives contributions from the following sub - processes at the lowest order : @xmath156 where both @xmath9-quarks fail the @xmath9-tag . we do not include the @xmath157 process due to its relatively small rate ( @xmath158 nb ) . this rate is further reduced by requiring no @xmath9-tags , yielding roughly 40 events in the 10 pb@xmath54 of luminosity assumed here . following sec . [ sect : calc ] , we account for the possibility of an additional jet which is undetected because it is too soft or outside of the accepted rapidity range . thus , we include several other contributions to the dps cross section : @xmath159 where the parentheses surrounding a jet signify that it is not detected . the sps cross section receives contributions at lowest order from the final state : @xmath160 where both @xmath9-quarks fail the @xmath9-tag , and , in the case where a jet is not detected , from the final states : @xmath161 we refer to sec . [ sect : calc ] for the specification of acceptance cuts and detector resolution , and for our treatment of the potential divergences present in the amplitudes for the processes in eqs . ( [ eq : dps - loprocess-4j])-([eq : sps - nloprocesses-4j ] ) . similar to the @xmath153 process , the leading jet in the @xmath154 dps sample is typically softer than in the sps channels ( see fig . [ fig : ptj_1 - 4j ] ) . in this case , again using @xmath162 mb , we find that the cross - over between dps and sps dominance occurs near @xmath163 gev , higher than in the @xmath144 case shown in fig . [ fig : ptj_1 ] . , but for @xmath154 events . similar to the @xmath164 sample , the sps sample exhibits a harder @xmath135 spectrum.,scaledwidth=59.0% ] improvement in the separation between dps and sps in the @xmath154 case can be achieved with an analogous version of the @xmath132 variable introduced in eq . ( [ eq : sptprime ] ) : @xmath165 here @xmath166 is the vector sum of the transverse momenta of two final state jets , @xmath167 and @xmath9 , chosen among the four . the remaining @xmath53 and @xmath168 jets are then fixed . this choice is unique if a separation of the two hard interactions is possible . in the @xmath153 system , the separation into the @xmath122 and @xmath56 subsystems via @xmath9-tagging removed most of the degeneracy ( some degeneracy still remained via tagging efficiencies or light jet mistagging ) . in the @xmath154 system , the degeneracy can at first glance be problematic as there are 3 possible pairings of the four jets . distribution for @xmath154 events shows much more combinatorial background than in the @xmath155 events . even after accepting two mis - matched jet pairs , we see that the dps and sps samples can still be separated well.,scaledwidth=59.0% ] one might be tempted to take the pairing of jets which minimizes the value of @xmath132 . unfortunately , this choice places a bias on the distribution that makes it potentially problematic to trust the discrimination . instead , to construct @xmath132 we take all three combinations of pairings , which includes one `` correct '' pairing and two incorrect pairings in the dps process . this `` democratic '' @xmath132 distribution is shown in fig . [ fig : sptprimedem ] and is re - weighted by 1/3 for proper normalization . as in the @xmath153 case , we see that the dps distribution peaks near @xmath133 , indicative that two back - to - back hard interactions are present . in addition to this expected feature , we also see a continuum that extends above @xmath169 , associated with the wrong combination taken in the democratic approach . in fig . [ fig : sptprimedem ] we see that dps produces a secondary peak at @xmath134 , not present in the @xmath144 case in fig . [ fig : sptprimecut ] . it appears to arise from the wrong pairings of jets associated with the combinatorial background . in these instances , the wrong combination of two jets that are close together in @xmath170 , meaning that their momenta are aligned , can maximize the value of @xmath132 . overall , we see that the dps peak near @xmath171 provides a good means to separate dps events from sps events . , but for @xmath154 events with ( a ) democratic @xmath147 and ( b ) democratic @xmath150 . as in @xmath172 events , as one increases the cut on @xmath132 , the sps fraction increases and the total distribution is harder . , title="fig:",scaledwidth=49.0% ] , but for @xmath154 events with ( a ) democratic @xmath147 and ( b ) democratic @xmath150 . as in @xmath172 events , as one increases the cut on @xmath132 , the sps fraction increases and the total distribution is harder . , title="fig:",scaledwidth=49.0% ] as in the @xmath144 case , we inspect the distribution in the @xmath135 of the leading jet after cuts on the @xmath132 variable . since there are three jet pairings per event , we now require that at least one of the three pairings has @xmath132 in the given window . due to this softer constraint , the hardening of the @xmath135 spectrum of the leading jet is less dramatic than in the @xmath153 case ( e.g. compare figs . [ fig : ptjcrossover ] and [ fig : ptj_1 - 4j - crossover ] ) . the crossover of the sps and dps contributions occurs near @xmath146 gev for @xmath147 and near @xmath173 gev for @xmath149 our goal is to develop a method to search for a double parton scattering contribution in the @xmath95 and 4 jet final states at lhc energies and to measure the magnitude of its contribution relative to the single parton contribution to the same final states . based on our parton level simulations , we find that variables such as @xmath132 and @xmath124 that take into account information from the entire final state , thereby including both of the hard subprocesses in dps , are more effective at discrimination than variables such as @xmath116 that reflect only a subset of the final - state . the enhancement at low values of @xmath132 shown in figs . [ fig : sptprimecut ] , [ fig : scatter1 ] and [ fig : sptprimedem ] provides a good signature for the presence of double parton scattering . we urge experimenters to search for such a concentration of events in data at the lhc . having found this enhancement , we then suggest that the magnitude of this peak be examined as a function of the transverse momentum @xmath1 of the leading jet in the event sample . the double parton scattering contribution in the peak region should fall off more rapidly with @xmath1 than the rest of the sample . the distribution of events in the region of small values of @xmath132 should also be examined as a function of the inter - plane angle @xmath96 to see whether the flat behavior is seen , as expected for two independent production processes . once these characteristics of double parton scattering are established , the data can be used to determine the effective normalization @xmath43 , defined and discussed in the introduction . it will be interesting to see whether the values extracted for @xmath43 are about the same in the @xmath95 and 4 jet final states and how they compare with values measured at the fermilab tevatron . once double parton scattering is established in data , and @xmath43 is determined , in a relatively clean process such as @xmath144 , double parton contributions to a wide range of other processes can be computed with more certainty about their expected rates at lhc energies . to be sure , given the approximations described in the introduction , some variation in the values of @xmath43 might be expected and appropriate for different processes and in different kinematic regions . the connection of @xmath43 with the effective size of the hard - scattering core of the proton may mean that @xmath43 will have different values for @xmath174 , @xmath175 , and @xmath176 scattering . there are several avenues for future work . of great importance is the proper inclusion of next - to - leading order contributions @xcite . they are needed to make more robust predictions of the relative normalization of the dps and sps contributions , of the shape of the @xmath135 distribution of the leading jet , and for proper softening of the sharp peaks seen near @xmath177 in figs . [ fig : sptprimecut ] and [ fig : sptprimedem ] , and near @xmath178 in fig . [ fig : delphi]b . it will also be important to develop joint probabilities @xmath24 that are more sophisticated theoretically than the first approximation represented by eq . 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we evaluate the kinematic distributions in phase space of 4-parton final - state subprocesses produced by double parton scattering , and we contrast these with the final - state distributions that originate from conventional single parton scattering . our goal is to establish the distinct topologies of events that arise from these two sources and to provide a methodology for experimental determination of the relative magnitude of the double parton and single parton contributions at large hadron collider energies .
we examine two cases in detail , the @xmath0 and the 4 jet final states . after full parton - level simulations ,
we identify a few variables that separate the two contributions remarkably well , and we suggest their use experimentally for an empirical measurement of the relative cross section . we show that the double parton contribution falls off significantly more rapidly with the transverse momentum @xmath1 of the leading jet , but , up to issues of the relative normalization , may be dominant at modest values of @xmath1 .
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You are an expert at summarizing long articles. Proceed to summarize the following text:
two - dimensional ( 2d ) materials present some fascinating challenges to condensed - matter theory , with even the most simple 2d systems harboring surprises . one of the most famous problems involves the precise description of melting in 2d solids made up of hard , disk - like particles with short - range repulsive interactions.@xcite specifically , does the fluid undergo a weak first - order transition to the solid , or is there an intermediate hexatic phase linked by two continuous phase transitions ? related avenues of research concern the existence of exotic phases in systems made up of more complex particles , such as ( non)periodic solids of hard - disk dimers,@xcite pentamers and hexamers,@xcite tetratic phases of hard squares@xcite and hard rectangles,@xcite and orientationally ordered solids of hard pentagons and heptagons.@xcite the effects of additional interactions on the phase behavior and dynamics of 2d systems are also of interest , as evidenced by recent studies on dipolar potentials in the context of magnetic colloids.@xcite such models provide an ideal testing ground for condensed - matter theories , and in some cases challenge our most fundamental understanding of the properties of matter . despite their simplicity , 2d models can provide reliable descriptions of some real , and rather complex , experimental situations . for example , in a number of recent studies , 2d models have been employed to help interpret and understand the clustering and crystallization of proteins at interfaces . the conformations and interactions of proteins are central to biological activity , and ideally one would like to investigate these properties _ in vivo_. unfortunately , structural information is most commonly obtained from x - ray diffraction studies on crystals . there is a class of proteins , however , that can be studied under conditions resembling those _ in vivo_. membrane proteins constitute a large class of molecules found within the lipid bilayers that constitute cell walls . they fulfill a variety of roles , such as controlling the selective transport of ions and molecules across cell membranes , or providing binding sites for other molecules on to the membrane . the structures of membrane proteins can be studied by deposition on to a surface , alongside lipids , to form either low surface - coverages or 2d crystals ; the hydrophobic lipids help to mimic the interior of the membrane . electron microscopy or atomic - force microscopy can then be used to image directly the clustering and packing of proteins at the solid - air interface.@xcite in many cases , the ordering of proteins can be rationalized on the basis of their gross shapes ( the way in which those shapes would ` tile the plane ' ) and the presence of specific binding interactions between domains on different molecules . for example , the surface structure of bacteriorhodopsin ( a transmembrane protein ) is comprised of a close - packed array of trimers , each made up of monomers that resemble @xmath0 sectors of a circle . monte carlo ( mc ) simulations of hard sectors with an additional attractive square - well potential to mimic specific binding interactions yield insight on the self - assembly and subsequent crystallization processes.@xcite in another application , the ordering in 2d crystals of annexin v another ` triangular ' membrane protein was reproduced in simulations of a hard - disk model decorated with an appropriate orientation - dependent potential to mimic the locations of the specific binding sites on the protein . experimentally observed honeycomb and triangular structures were captured by the molecular model . these examples show that the basic physics of large - scale structural order in 2d protein crystals can be studied with simple models , and without resorting to atomistically detailed and hence very expensive computer simulations . there are a large number of proteins which are either inherently triangular in shape , or otherwise form trimeric structures . for example , rotavirus inner capsid protein v6 forms trimers resembling equililateral triangles , which pack in 2d crystals ( space group @xmath1).@xcite specific fragments of prion proteins found _ in vivo _ form trimeric units that crystallize in to a 2d structure ( space group @xmath2).@xcite finally , we highlight an example in which a membrane fusion protein ( from the semiliki forest virus ) is seen to form pentagons of trimers , with the center of the pentagon raised slightly out of the plane.@xcite some semblance of local five - fold coordination can also been seen in teta a roughly triangular transporter protein at moderate surface coverages.@xcite motivated by the diversity of 2d crystal structures exhibited by trimeric protein units , and also by the observation of five - fold coordination,@xcite we have investigated the structure and phase behavior of model trimeric molecules made up of hard disks . in order to mimic specific binding interactions , such as those that might give rise to local five - fold coordination , we focus on an equililateral triangle of three hard disks at contact , in which one disk can interact with the corresponding disks on other molecules _ via _ a short - range attractive square - well potential . as we will show below , this raises the possibility of generating orientational order within simple close - packed structures , and also offers the opportunity of forming clusters at low surface coverages . using mc simulations , we map out the phase diagram of the model system , and characterize the structures of the low - density clustered fluid and high - density solids which are formed at low temperature . the remainder of the article is organized as follows . in section [ sec : model ] we describe the molecular model , and summarize the simulation methods . the results are presented in section [ sec : results ] , and section [ sec : discussion ] concludes the paper . the molecular model consists of three hard disks , each of diameter @xmath3 , fused at mutual contact to form an equilateral triangle . two of the disks on each molecule are purely repulsive , and interact with all other disks in the system through the potential @xmath4 where @xmath5 is the separation between the centers of two disks . the third disk on each molecule carries a central attractive interaction site ; these ` attractive ' disks interact with each other _ via _ the potential @xmath6 where @xmath7 controls the range of the attraction . this potential crudely mimics an effective attraction between vertices of the molecular triangles , which might arise through specific interactions ( e.g. hydrogen bonding , disulfide bridges , effective solvophobic interactions ) . the parameter @xmath8 will clearly have a crucial role to play in the thermodynamics of the system . if @xmath9 then one should anticipate a conventional phase diagram containing a vapor - liquid transition , and a fluid - solid transition . the orientation of a trimer can be defined by a vector @xmath10 joining the geometrical center of the trimer with the center of the attractive disk . it is unlikely that there would be any periodic orientational ordering of @xmath10 in the solid phase ; if two trimers can interact favorably irrespective of the mutual orientation , then on entropy grounds the orientations will be disordered . in the opposite extreme , @xmath11 , the molecules will feel the orientation dependence of the net trimer - trimer potential , and ultimately we might expect the vapor - liquid transition to disappear from the equilibrium phase diagram . indeed , in a pure square - well hard - sphere fluid , condensation becomes metastable with respect to freezing when @xmath12.@xcite in the present case , an interaction range @xmath13 guarantees that attractive sites must face each other directly in order to interact ; when @xmath14 it is possible for an attractive disk to be within interaction range of a trimer even if it approaches from ` behind ' . with these comments in mind , we have chosen to study a system with @xmath15 . the ratio of @xmath16 to the ( angle - averaged ) diameter of the trimer is smaller than that in a pure square - well hard - sphere system with the same value of @xmath8 , and assuming some sort of correspondence between two and three - dimensional systems , we do not anticipate there being a vapor - liquid transition in the equilibrium phase diagram . on the other hand , because the trimers have to attain quite specific mutual orientations in order to interact favorably ( since @xmath13 ) , we should expect to see some sort of non - trivial structure in fluid and solid phases at low temperatures . systems of @xmath17 trimers were studied using mc simulations either in the isothermal - isobaric ( @xmath18 ) ensemble or the canonical ( @xmath19 ) ensemble.@xcite the simulation cell was rectangular with dimensions @xmath20 and @xmath21 , and area @xmath22 . each mc cycle consisted of one translational trial move and one orientational trial move for each of @xmath23 randomly selected molecules . to help equilibrate dense phases , every fifth mc cycle included @xmath23 trial moves in which a randomly selected trimer was rotated by @xmath24 . in @xmath18 simulations of solid phases , @xmath20 and @xmath21 were varied independently ; in @xmath18 simulations of fluid phases , the simulation cell was constrained to be square . for most thermodynamic state points typical equilibration runs consisted of @xmath25 mc cycles , but some points ( close to phase transitions ) required @xmath26 mc cycles . production runs were typically @xmath25 mc cycles . we define the following dimensionless units in terms of the square - well depth , @xmath27 , and the hard - disk diameter , @xmath3 : number density @xmath28 ; temperature @xmath29 ; pressure @xmath30 . the phase diagram of the model trimers in the density - temperature ( @xmath31-@xmath32 ) plane is sketched in fig . [ fig : phasediag ] . before detailing the determination of the phase boundaries , the characteristics of the different phases will be described . there are four distinct regions in the phase diagram . at low density and high temperature , a normal fluid phase is in evidence ( fluid i ) . a typical simulation configuration is shown in fig . [ fig : snapshot](a ) . there is neither translational nor long - range orientational order in the system . phase diagram of the model trimer system in the density - temperature ( @xmath31-@xmath32 ) plane : ( solid points and solid lines ) approximate fluid - solid phase boundaries , assumed to be first order ; ( open points and dashed lines ) boundaries between high - temperature unclustered states and low - temperature clustered states , as evidenced by maxima in the heat capacity along isobars ; ( dot - dashed line ) close - packed density , @xmath33 . ] at high density and high temperature , the stable solid phase ( solid i ) possesses an orientationally disordered structure ( in the sense that @xmath34 is disordered ) with the trimers close - packed to form alternating rows displaced by @xmath35 . figure [ fig : snapshot](b ) shows both the lack of orientational order , and the registry between alternating rows . notice the black bonds showing how the disks are connected within the trimers ; we call this an ` @xmath36 ' structure to denote the alternating alignment of the rows . the close - packed rows resemble those formed by vp6,@xcite although the registry between the rows is different . at the end of this section , we will briefly discuss the possibility of solids with other close - packed structures . at low temperature and low density we find a highly associated fluid ( fluid ii ) , in which the attractive disks aggregate to form distinct clusters . a typical configuration is shown in fig . [ fig : snapshot](c ) , which exhibits a broad distribution of cluster sizes . to identify clusters , we employ the obvious criterion that two trimers with attractive disks within interaction range belong to the same cluster . with this definition in mind , [ fig : snapshot](c ) shows that , in general , the attractive disks within the clusters form close - packed motifs , rather than loose arrangements of disks on the circumference of a ring . for clusters of three trimers there is no distinction , whereas for four or more trimers the close - packed arrangement is more favorable ; in a ring , each disk would have two nearest neighbors , whereas close - packed motifs can accommodate more than two direct contacts . in fig . [ fig : cluster ] we show the probability distribution function of clusters containing @xmath37 molecules , at different pressures along an isotherm with @xmath38 . as the pressure and density increase , the distributions show peaks at progressively higher values of @xmath37 . at the highest fluid - density shown @xmath39 , fig . [ fig : cluster](e ) the most probable cluster size is @xmath40 . we had hoped that these clusters would adopt a pentagonal structure , but instead the attractive disks form ` olympic rings ' motifs , such as those shown in fig . [ fig : snapshot](c ) . the maximum disk - disk separation in a perfect pentagon of disks is @xmath41 , which is longer than the range of the potential studied in this work . hence , to minimize the energy , the cluster will contract to form a close - packed structure . perhaps pentagonal clusters would be formed in a system with @xmath42 ? ( the upper limit means that there can be no other disks between two interacting attractive disks . ) we did some test runs in the fluid phase with @xmath43 , but no pentagonal clusters were observed . if anything , fewer distinct clusters were in evidence as compared to @xmath15 , presumably because it is less crucial that the trimers attain a specific mutual orientation in order to interact . cluster distributions for systems along the isotherm @xmath38 : ( a ) @xmath44 , @xmath45 ; ( b ) @xmath46 , @xmath47 ; ( c ) @xmath48 , @xmath49 ; ( d ) @xmath50 , @xmath51 ; ( e ) @xmath52 , @xmath39 ; ( f ) @xmath53 , @xmath54 . in ( a)-(e ) the system is fluid , whilst in ( f ) the system is solid ( ii ) . ] upon compression of the low - temperature fluid we often encountered metastable structures , such as that shown in fig . [ fig : snapshot](d ) . this clearly shows a predominance of @xmath55 clusters , with the attractive disks close packed to form a parallelogram motif , but the clusters are not yet fully packed in to a solid structure . this process is completed upon further compression , to form a @xmath56 periodic solid ( solid ii ) , a defective example of which is shown in fig . [ fig : snapshot](e ) . in simulations of the high - density solid ii phase , the initial configuration consisted of the appropriate @xmath36 structure , but with @xmath10 for each molecule chosen randomly from the three molecular arms ; the orientational structure shown in fig . [ fig : snapshot](e ) develops spontaneously . the cluster distribution for such a solid at temperature @xmath38 and density @xmath54 is shown in fig . [ fig : cluster](f ) . the primary peak is at @xmath55 , but the presence of defects such as those shown in fig . [ fig : snapshot](e ) gives rise to smaller ` clusters ' of attractive disks . the fluid - solid phase boundaries were located by monitoring the equation of state @xmath57 along selected isotherms in @xmath18 simulations . for each isotherm , two sets of simulations were performed : a compression branch , starting from a low - density fluid configuration ; and an expansion branch , starting from the perfect solid structure corresponding to that found in the compression branch at high pressure . portions of two representative examples ( @xmath38 and @xmath58 ) are shown in fig . [ fig : eos ] . of course , the fluid equations of state extend to much lower densities , but these exhibit entirely conventional behavior and hence are not shown ; in particular , there is no sign of a ` van der waals ' loop which would indicate a vapor - liquid phase transition . the main features of interest are the apparent discontinuities in the density at what are assumed to be first - order phase transitions ( we will not open up the can of worms associated with the precise nature of two - dimensional melting and freezing ) . in fig . [ fig : eos ] we indicate distinct fluid and solid branches in the equations of state , a number of putative metastable states ( as discussed above ) , and approximate tie - lines connecting the fluid and solid coexistence densities , obtained as follows . the fluid branch was fitted with a virial expansion containing terms up to @xmath59 , i.e. , @xmath60 , while the solid branch was found to be fitted rather well by a simple van der waals equation@xcite of the form @xmath61 , which contains a free - volume term arising from repulsive interactions , and a mean - field term arising from the attractions . the coexistence densities were then estimated by extrapolating the fitted branches of the equation of state to a pressure half way between those in the highest - density stable fluid and the lowest - density stable solid ; the metastable states were identified as those that did not fit on to either branch and/or for which the simulation configuration was clearly neither pure solid nor pure fluid , e.g. fig . [ fig : snapshot](d ) . obviously this approach provides only very rough locations for the phase boundaries shown in fig . [ fig : phasediag ] , but some general trends are nonetheless apparent . at very low temperatures , the coexistence densities decrease as the system is cooled , and the transition appears to be getting weaker . at high temperatures ( @xmath62 ) the fluid coexistence density ( @xmath63 ) is very similar to the density at which the pure hard - disk fluid undergoes its transition , either to a hexatic or a solid ( disk density @xmath64,@xcite ` trimer ' density @xmath65 ) . the apparent trimer solid coexistence density ( @xmath66 ) is significantly larger than the melting density of hard disks ( disk density @xmath67,@xcite ` trimer ' density @xmath68 ) . equations of state along isotherms with @xmath38 ( solid symbols , solid lines ) , and @xmath58 ( open symbols , dashed lines ) : ( circles ) state points from @xmath18 simulations , with @xmath36 solid phases ; ( squares ) state points from @xmath18 simulations , with @xmath69 solid phases ( @xmath58 only ) ; ( crosses ) putative metastable state points ; ( triangles ) approximate coexistence densities ; ( lines ) fits to the fluid and solid branches ( see text ) . the statistical errors in the @xmath18 simulation points are smaller than the symbols . ] the final piece of the equilibrium phase diagram concerns the crossover from high - temperature orientationally disordered states to low - temperature states that possess structural motifs arising from the clustering of the attractive disks . to delineate the boundary between these two regimes , we calculated the heat capacity appropriate to the statistical mechanical ensemble being sampled . in general we used @xmath18 simulations to measure @xmath70 where @xmath71 is the enthalpy ( minus the kinetic contribution ) as a function of temperature along an isobar . since clustering must be accompanied by a drop in the configurational energy , and enthalpy , a peak in @xmath72 would seem to be an obvious signal of a crossover from unclustered to clustered states . in simulations we evaluated the usual fluctuation formula , @xmath73/k_{b}t^{2}$ ] , and , as a check , differentiated an @xmath74 $ ] pad approximant fitted to the enthalpy as a function of @xmath75 ; @xmath76 these two approaches yielded consistent results , and the peak in @xmath72 was easy to locate accurately . in general the peak height is less pronounced at high densities , mainly due to the fact that even in the high - temperature phase there must be some attractive disks within interaction range due to the confinement . thus , the most difficult situation obtains at close packing of the trimers , @xmath77 . in this case we studied a perfect close - packed @xmath36 solid , and carried out @xmath19 mc simulations with @xmath78 rotations only . we show results for the configurational energy , @xmath79 , and the excess constant - area heat capacity , @xmath80 , in fig . [ fig : heatcap ] . a [ 5,5 ] pad fit provides a reliable description of the energy , and the corresponding results for @xmath81 are consistent with those obtained _ via _ the fluctuation formula . configurational energy @xmath79 ( left ) and excess heat capacity @xmath81 ( right ) as functions of reduced temperature @xmath32 at the close - packed density @xmath82 : ( circles ) simulation results ; ( lines ) results derived from a pad [ 5,5 ] fit ( see text ) . ] in fig . [ fig : phasediag ] we show the positions of the maxima in @xmath72 and @xmath81 at @xmath83 along with separate cubic fits to the points in the fluid and solid regions of the phase diagram . it appears that the two branches would meet up somewhere in the fluid - solid coexistence region . we stress that the boundaries indicated do not represent thermodynamic phase transitions ; rather , they separate different regimes of trimer association . finally , we briefly consider the possibility of the trimer system adopting other solid structures , such as the @xmath56 @xmath69 structure shown in fig . [ fig : snapshot](f ) , in which the close - packed ( horizontal ) rows are matched with the neighboring rows . in this case , the low - temperature , orientationally ordered solid exhibits rhombic cluster - motifs containing only four attractive disks . out of those four disks , two are interacting with two other disks , and two are interacting with three other disks . hence , the minimum configurational energy for an @xmath69 solid is @xmath84 per trimer . in the @xmath36 structure , there are six attractive disks per parallelogram motif , of which two have two neighbors , two have three neighbors , and two have four neighbors , giving a minimum energy of @xmath85 per trimer . hence , on energetic grounds , we should expect the @xmath36 structure to be thermodynamically favored . even at high temperature , the @xmath69 structure appears to be less stable with respect to the @xmath36 structure . as an example , in fig . [ fig : eos ] , we show an @xmath69-solid branch of the equation of state at @xmath58 , alongside the @xmath36-solid branch . for a given pressure , the @xmath36 solid has the higher density which makes this state at least mechanically stable with respect to @xmath69 . indeed , we only ever observed the fluid spontaneously freezing in to an @xmath36 structure . although we have not performed free - energy calculations , it would be very surprising if an entropic effect could compensate for the relative energetic and mechanical stability of the @xmath36 phase with respect to the @xmath69 phase . another possible close - packed structure is illustrated in fig . [ fig : hexagonal](a ) , without any indication of the attractive disks . this structure resembles that adopted by 2d crystals of teta,@xcite although we never saw this packing structure emerge from our simulations . as far as our model is concerned , the absence of this structure at low temperature is easy to understand . in figs . [ fig : hexagonal](b ) and [ fig : hexagonal](c ) we illustrate mirror images of the most obvious periodic arrangement of the attractive disks ( space group @xmath2 ) . the energy per trimer is only @xmath86 , and so this is not competitive with the @xmath36 structure that is seen to emerge spontaneously in our simulations . free - energy calculations would be of interest , particularly at high temperatures where entropy is everything ! ( color online ) illustrations of an alternative close - packed structure : ( a ) without an assignment of attractive disks ; ( b ) and ( c ) mirror images of a possible structural motif for a periodic arrangement of attractive disks . the attractive disks are colored dark gray ( red online ) , the repulsive disks are colored light gray , and all disks are drawn with diameter @xmath87 . ] in this article we have described the structure and phase behavior of a generic model of trimeric molecules , largely motivated by recent experimental 2d microscopy studies of clustering and crystallization in triangular proteins and protein trimers . the molecular model consists of a triangle of hard disks , with one of the disks participating in attractive square - well interactions with similar disks on other trimers . the range of the square - well potential , @xmath16 , was @xmath88 times the disk diameter . this system crudely mimics the general shape and specific interactions of a wide range of proteins . the model system exhibits fluid and solid phases which , at low temperatures , possess interesting structural motifs arising from the clustering of the ` attractive ' disks . in the fluid , a distribution of clusters is in evidence , including tetramers , pentamers , and hexamers ( of trimers ) . in the pentamers and hexamers , the attractive disks close - pack to form ` olympic rings ' and parallelogram shapes , respectively . we had hoped to find more open pentagonal clusters of trimers , such as those reported in ref . . to investigate the formation of such clusters further , it might be interesting to study a system of hard isosceles triangles with the unique angle equal to @xmath89 , and a short - range attraction operating between the corresponding vertices . in the low - temperature solid , the basic structural motif consists of clusters of six molecules , with the attractive disks close - packed to form a parallelogram . a metastable solid possessing a motif made up of four molecules was also identified . the fundamental difference between the two situations is the registry between neighboring close - packed rows of trimers ( @xmath36 versus @xmath69 ) . even at high temperatures , the orientationally disordered @xmath36 solid is at least mechanically stable with respect to the @xmath69 solid . we identified a third structure based on hexagonal close packing , but this structure is not competitive either , at least in terms of energy . it would be worth performing free - energy calculations to study these issues further . finally , it is worth commenting that a diverse range of 2d structures can be generated from very simple molecular models . fully atomistic calculations of 2d protein structures are expensive , and , it could be argued , yield little insight on the fundamental physics behind clustering and crystallization . as has been shown in a variety of cases , including the present study , the process of developing and studying simple models of complex systems can yield some surprising results .
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motivated by the diversity and complexity of two - dimensional crystals formed by triangular proteins and protein trimers , we have investigated the structures and phase behavior of hard - disk trimers . in order to mimic specific binding interactions ,
each trimer possesses on ` attractive ' disk which can interact with similar disks on other trimers _ via _ an attractive square - well potential . at low density and low temperature ,
the fluid phase mainly consists of tetramers , pentamers , or hexamers .
hexamers provide the structural motif for a high - density , low - temperature periodic solid phase , but we also identify a metastable periodic structure based on a tetramer motif . at high density there is a transition between orientationally ordered and disordered solid phases .
the connections between simulated structures and those of 2d protein crystals as seen in electron microscopy are briefly discussed .
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You are an expert at summarizing long articles. Proceed to summarize the following text:
l dwarfs are a class of objects with spectral types cooler than the latest m dwarfs @xcite . their population consists of both substellar ( i.e. brown dwarfs ) and stellar mass objects , with the brown dwarfs being identified by the presence of li absorption at 6708 in their spectra . kelu-1 was one of the first such brown dwarfs discovered . it was found at a trigonometric distance of 18.7 pc @xcite as part of a proper motion survey @xcite . since kelu-1 s discovery , more than 400 l dwarfs have been classified in large surveys such as the deep near - infrared survey ( denis ) , the 2-micron all - sky survey ( 2mass ) , and the sloan digital sky survey ( sdss ) . a spectral class sequence was established @xcite , and kelu-1 was subsequently assigned a spectral type of l2 in the optical @xcite and l3 in the near - ir @xcite . these other discoveries provided a baseline against which to compare kelu-1 s other characteristics . for example , it was quickly realized that kelu-1 was much brighter than other l2 dwarfs , causing @xcite to first suggest kelu-1 a binary . this super - luminosity has been seen in other studies @xcite where the main source of this over - brightness was attributed to either an unresolved binary or a young age . @xcite observed kelu-1 on 1998 august 14 with the nicmos camera on hst and no companions were found for separations greater than 300 mas and magnitude differences @xmath76.7 mag ( f165 m filter ) . @xcite observed kelu-1 in a non - ao , seeing - limited program on keck i using the nirc instrument on 1998 february 14 and 1999 february 9 . because kelu-1 had a fairly circular point spread function during these observations ( psf [email protected] ) , it was used as a psf star for the other companion search targets of their project . while these observations did not discount kelu-1 as being a very close , unresolved binary , they did warrant a look at the young age hypothesis . based on the strength of the li i absorption feature , @xcite constrained the age of kelu - i to be 0.3 - 1 gyr . with this age range and kelu-1 s brightness , it should have an effective temperature of 21002350 k , about 400 k higher than objects with similar spectral types @xcite . for it to have an effective temperature consistent with other l3 dwarfs , @xcite indicate kelu-1 must have an age of @xmath010 myr , more than an order of magnitude lower than the li -established lower age limit . these discrepancies mean that a young age is unlikely to be the cause of the over - luminosity . aside from the luminosity , there is other evidence that supports the binarity of kelu-1 . it is reported to be photometrically and spectroscopically variable with a period of 1.8 hrs @xcite . if this period corresponds to an orbital period , then the system would likely be circularized and the orbital radius of the system would be less than a solar radius @xcite . in addition , @xcite claim that the @xmath0120 km s@xmath9 radial velocity produced by a very close binary such as this could account for the large line broadening ( 60@xmath15 km s@xmath9 ) measured by @xcite . we are conducting an adaptive optics ( ao ) survey of l and t dwarfs at keck observatory in an effort to determine binarity in these objects ( an article describing the full details of this project is in preparation ) . because of the intrinsic faintness of l and t dwarfs ( magnitudes for kelu-1 are @xmath10=22.1 and @xmath11=19.2 ; * ? ? ? * ) , it is not possible to use them as natural guide stars ( ngs ) where the magnitude limit is about @xmath10=14 . instead , we utilize the newly commissioned laser guide star adaptive optics system ( lgs ao ) where the wavefront corrections are performed on a layer of sodium atoms 80 km high in the earth s mesosphere that are excited by a 10 - 14 watt laser tuned to 598 nm . the projected laser and returned light propagate along the same path , requiring the use of a natural star ( or other point - like astronomical source ) for tip - tilt measurements . the tip - tilt reference can be as faint as @xmath11=18.5 , but must be within 60 of the science target . thus , the area of sky reachable with ao is greatly increased by the less stringent requirements for a reference star provided by the used of the lgs , and many l and t dwarfs binaries can be observed with high resolution imaging from the ground . we included kelu-1 in our survey in an effort to resolve the controversy over its binarity and over - luminosity and to search for any low - mass companions . in this article we discuss the detection of a companion to kelu-1 , first reported in @xcite , and show the first evidence of orbital motion of the companion . observations of kelu-1 were obtained with nirc2 ( k. matthews et al . in preparation ) behind the lgs ao system on 2005 march 4 and 2005 april 30 during shared - risk time . a suitable tip - tilt reference star ( id=0643 - 0289866 , @xmath11=14.3 , separation=27 ) was identified in the usno - b 1.0 catalog @xcite . skies were photometric on both nights and the ao system provided very good corrections on our targets . kelu-1 was observed with the @xmath2 filter during both epochs , with additional exposures in the @xmath3 filter during follow - up observations in april . these filters were chosen to maximize the strehl ratio in our images . the data were reduced using custom and public idl scripts . for each night , two sets of reduced images were produced from the data : a photometric set and an astrometric set . the only difference in these sets is that an image distortion correction was performed on the astrometric data set . the distortion in the narrow camera is negligible within a region a few hundred pixels on a side around the center of the chip and gradually increases to as much as 4 - 5 pixels near the edge . while the correction fixes the distortion with residuals of 0.81 and 0.62 pixels in the x and y directions , respectively , it does not preserve flux information . consequently , its use in images for photometric analysis is not suitable . the plate scale for both nights was determined using astrometric images of the core of the globular cluster m5 . this field was observed with wfpc2 on hst , with the stars in common with our observations present in the planetary camera . we imaged this field with a 4 ( in april ) or 5 ( in march ) position dither pattern in the narrow camera ( @xmath810@xmath12 field - of - view ) . from the wcs information in the hst image headers , we determined the sky positions of several stars present in our nirc2 images . these positions were then used to calculate the plate scale and rotation angle for each position in the dither . the average of these values is the resultant scale and rotation for that night . the errors in these quantities are equal to the standard deviation of the measurements . table [ scale ] presents the plate scale for these nights . in our march 4 images of the kelu-1 field , we saw what appeared to be a binary object with a separation @xmath0300 mas and a differential @xmath2 magnitude of @xmath00.4 . in order to obtain proper motion confirmation of the companion , we re - observed this system on april 30 . on both epochs , we determined the separation and position angle of the candidate companion by using an average of measurements on the individual reduced astrometric images . the errors are the standard deviations of these measurements . table [ astrom ] shows these results . the moderately large proper motion of kelu-1 made it possible to confidently confirm physical companionship based on observations separated by only two months . the measured positions of the companion were about 15@xmath13 away from its predicted background position , clearly demonstrating that these two objects have a common proper motion and are physically associated . our binary conclusion firmly supports the recent announcement of kelu-1 s binarity from keck lgs ao observations obtained in may 2005 @xcite . for both nights , all of the photometric images were offset such that northeast object was in the same location in the overlap region . this image stack was then median averaged to obtain the final image ( figure [ nirc2 ] ) , on which the photometry was performed . the components were well separated and no psf subtraction routines were needed . no photometric standards were observed on either night , so only differential photometry is shown in table [ photom ] . the northeast object is the brighter component in both the @xmath3 and @xmath2 . it will henceforth be called kelu-1a and the fainter southwest component kelu-1b . our measured astrometry and differential photometry are consistent with those of @xcite . we re - examined e. martn s 1998 hst nicmos images using the psf modeling technique described in @xcite . in short , this technique involves the sub - pixel shifting of two model psfs and iteratively changing the relative brightness of the psfs to obtain the minimum residual for each shifted pair . the lowest minimum residual for all shifted pairs represents the best fit for the observed psf . our analysis discovered that the observed psfs in all three filters ( f110 m , f145 m , and f165 m ) were better fit by double psf solutions than they were by single psf solutions . the derived astrometry of the components is presented in table [ astrom ] . because the objects are unresolved , a 180@xmath5 ambiguity exists in the value of the position angle . it is also possible to estimate the magnitudes of the two components ( listed in table [ proptable ] ) using this technique . because the two components are unresolved , the determination of the individual magnitudes is subject to systematic uncertainties in the fitting process . therefore , it is difficult to quantify the uncertainty in these magnitudes . consequently , although these calculations may not determine the actual difference in magnitudes between the two components , they do indicate the reliability of a binary solution . furthermore , the computed magnitudes are also correlated , so if one object is actually fainter than computed , the other component would be brighter , thus conserving the total flux in the system . the spectral types of the components can be estimated using the photometry and absolute magnitude - spectral type relation in @xcite and the differential photometry presented here . for this exercise we assume that the @xmath14@xmath3 and @xmath14@xmath2 measurements in this study are equivalent to @xmath14@xmath3 and @xmath14@xmath15 in the cit photometric _ system _ as employed by @xcite while we know that this is not the case , our interest is only in _ approximately _ determining the spectral types of the components ; the _ only _ way to determine spectral types for these or any other objects is to obtain properly calibrated spectra . based on the multi - system , near - ir photometry study of @xcite , we conservatively impose an additional photometric error of 0.15 mag for using this assumption . this error is added in quadrature to our measured photometric errors . the @xmath16 and @xmath17 apparent magnitudes of kelu-1ab are [email protected] and [email protected] , respectively @xcite . coupled with our measured @xmath14@[email protected] and @xmath14@[email protected] , the apparent magnitudes of the resolved components become @xmath3[a]=12.99 , @xmath15[a]=12.37 , @xmath3[b]=13.49 , and @xmath15[b]=12.76 ( table [ proptable ] ) . converting these magnitudes from apparent to absolute and applying equations ( 2 ) and ( 3 ) from @xcite yield a spectral type of @xmath0l2@xmath11 for kelu-1a and @xmath0l3.5@xmath11 for kelu-1b . these estimates are consistent with the optical and near - ir spectral types of the composite system and with the spectral types derived by @xcite . with this photometry and spectral type estimates , we can place these objects on magnitude - spectral type diagram . figure [ magvsst ] presents a modified version of figure 4 from @xcite . in agreement with @xcite , it is quite clear that the over - luminosity of kelu-1ab was caused by the unresolved binary and not due to an unusually young effective temperature - based age @xcite . finally , we can estimate the effective temperature from the photometry presented here . @xcite calculate a @xmath15-band bolometric correction ( bc@xmath18 ) for kelu-1ab of [email protected] . while this calculation is independent of the spectral type of kelu-1ab , the spectral type used for kelu-1ab and all of the other objects in that study were based on near - ir spectra . the spectral types for kelu-1a and b presented here are based on a relation derived from optical spectral types . the near - ir spectral type of kelu-1ab is l3@xmath11 @xcite , whereas the optical spectral type is [email protected] @xcite . as discussed in @xcite , it is neither uncommon nor unexpected for l and t dwarfs to be classified differently in the optical and near - ir . consequently , it is unclear what value of bc@xmath18 should be used from @xcite . fortunately , for early l dwarfs such as kelu-1 a and b , the range of bc@xmath18 is quite small . the average value of bc@xmath18 for objects with near - ir spectral types l0l5 , inclusive , is 3.32 with a standard deviation of 0.07 . since this average value is consistent with the value of bc@xmath18 computed for the unresolved kelu-1ab , we choose the unresolved value as the bolometric correction for the individual components . we note , however , that the uncertainty in the effective temperature determination is largely dominated by the unknown age of kelu-1 and , given the constraints on the bolometric corrections of early l dwarfs , the exact value of bc@xmath18 is unimportant . based on our photometry , the absolute @xmath15 magnitudes of kelu-1a and b are 11.02 and 11.41 , respectively . assuming a @xmath15-band bolometric correction of 3.31 for each component yields bolometric magnitudes of 14.33 and 14.72 and luminosities ( log@xmath19 @xmath20 ) of -3.83 and -3.99 . finally , using an age range of 0.3 - 1 gyr @xcite and the @xcite dusty atmosphere models , we determine an effective temperature range for kelu-1a of 19002100 k and for kelu-1b 17001900 k ( table [ proptable ] ) . @xcite computed effective temperatures for kelu-1a and b ( 2020 k and 1840 k ) by scaling the effective temperature of kelu-1ab @xcite by their luminosity calculations . despite the differences between our method and theirs , our results are completely consistent . it is not known if the li i absorption at 6708 detected by @xcite and others is from one or both components . @xcite suggest that the presence of any lithium in the unresolved spectrum indicates both components are substellar and , at the very least , kelu-1b bears lithium in its atmosphere . consequently , the masses of both components must be @xmath210.065 m@xmath22 @xcite . to estimate component masses , we use the photometry derived here along with the age estimate of @xcite to place the system components on the dusty atmosphere model evolutionary tracks @xcite . from the combined mass estimates presented in figure [ dusty ] , we estimate the mass of kelu-1a to be [email protected] m@xmath22 and that of kelu-1b to be [email protected] m@xmath22 , in good agreement with the mass estimates derived by @xcite . even though the conservative errors on these masses do include stellar masses , as mentioned above the true masses are likely to not be greater than about 0.065 m@xmath22 . of course , by following the orbital motion of the companion over time , it will be possible to derive dynamical masses , which will provide much needed constraints on the models . the computation of these dynamical masses will be some time in coming . table [ astrom ] presents all of the known astrometry of the kelu-1ab system as of 2006 january . it is quite clear from this astrometry that kelu-1b has exhibited orbital motion from the time it was first resolved ( our 2005 march 4 observation ) to the time of its last observation ( a public archival hst observation on 2005 july 31 obtained by w. brandner ) . in fact , the displayed motion suggests that the companion is still moving toward apoapsis . during this nearly 5 month period the rate of change in the separation has not changed significantly and there is insufficient phase coverage to determine a reliable orbit . nonetheless , we can make some crude estimations of the orbit based on the available data . given so few points in the orbit , it is impossible to determine how much of the nearly linear motion is caused by a high inclination , very high eccentricity , or both . in order to simplify the discussion , we assume that the orbit is circular and thus , is being viewed nearly edge - on . the most recently measured 2005 positions and the 1998 psf - modeled position provide some tight constraints on the inclination of the orbit . although the position angle of the 4 resolved measurements is not significantly changing , it is not possible to extend a line connecting all four of these points directly back to kelu-1a . therefore , the inclination of the orbit must be @xmath790@xmath5 . we can get a rough estimate of the minimum inclination by assuming the maximum extent perpendicular to the 2005 july position is the separation of the modeled psfs in 1998 , i.e. 45 mas . this assumption results in an inclination @xmath481@xmath5 and is much more constrained than the minimum inclination suggested by @xcite based on the size of the nicmos psf . finally , since it is quite clear that kelu-1b is still moving away from kelu-1a , this minimum inclination estimate will only increase with time . the period of the binary can be estimated using the 2005 july 31 separation ( 298 mas or 5.4 au at the distance of kelu-1 ) and the masses derived above . with the caveat that the most recent separation is not the maximum orbital separation , the calculated period is @xmath23 years . this period is important for two reasons . first , it means that it is possible to obtain dynamical masses for the components in a reasonable time - frame , something that has been done for only a few other binary brown dwarf systems @xcite . second , this period means that the 1.8 hour period ( or 3.6 hours if the variations are ellipsoidal in nature ) reported by @xcite is _ not _ due to this binary . perhaps one of the components is itself a very close binary , or perhaps the shorter period is simply the rotation period for one of the components . in addition , the 60 km s@xmath9 rotation velocity measured by @xcite on 1997 june 2 is from a composite spectrum . the maximum orbital velocity for kelu-1ab is only @xmath03 km s@xmath9 , which translates to a displacement of @xmath10.09 and @xmath10.08 for the cs i and rb i atomics lines used by @xcite this wavelength shift can not account for the large equivalent widths of these lines ( 1.7 and 2.54 for cs i and rb i , respectively ) , so some other mechanism must be at work . in time it might be possible to separate the lines from each component using high resolution spectroscopy and resolve some of these issues . we have used the lgs ao system on the keck ii telescope to observe the binary brown dwarf kelu-1ab . we have also re - examined the 1998 hst observations of kelu-1 and show that the psfs in those images are best fit by a binary object solution . images from multiple epochs confirm the pair as having a common proper motion and demonstrate that kelu-1b is still heading towards apoapsis . we constrain the inclination of a circular orbit to @xmath481@xmath5 . while this companion detection does rectify the over - luminosity `` problem , '' it can not account for the 1.8 hour photometric period or the 60 km s@xmath9 rotation velocity . periodic monitoring of this system with high resolution imaging and spectroscopy is needed to ensure the maximum extent of the orbit is observed and measured , allowing for a more robust determination of the physical orbit . , c. c. , harris , h. c. , vrba , f. j. , guetter , h. h. , canzian , b. , henden , a. a. , levine , s. e. , luginbuhl , c. b. , monet , a. k. b. , monet , d. g. , pier , j. r. , stone , r. c. , walker , r. l. , burgasser , a. j. , gizis , j. e. , kirkpatrick , j. d. , liebert , j. , & reid , i. n. 2002 , , 124 , 1170 , d. a. , leggett , s. k. , marley , m. s. , fan , x. , geballe , t. r. , knapp , g. r. , vrba , f. j. , henden , a. a. , luginbuhl , c. b. , guetter , h. h. , munn , j. a. , canzian , b. , zheng , w. , tsvetanov , z. i. , chiu , k. , glazebrook , k. , hoversten , e. a. , schneider , d. p. , & brinkmann , j. 2004 , , 127 , 3516 , g. r. , leggett , s. k. , fan , x. , marley , m. s. , geballe , t. r. , golimowski , d. a. , finkbeiner , d. , gunn , j. e. , hennawi , j. , ivezi ' c , z. , lupton , r. h. , schlegel , d. j. , strauss , m. a. , tsvetanov , z. i. , chiu , k. , hoversten , e. a. , glazebrook , k. , zheng , w. , hendrickson , m. , williams , c. c. , uomoto , a. , vrba , f. j. , henden , a. a. , luginbuhl , c. b. , guetter , h. h. , munn , j. a. , canzian , b. , schneider , d. p. , & brinkmann , j. 2004 , , 127 , 3553 , d. g. , levine , s. e. , canzian , b. , ables , h. d. , bird , a. r. , dahn , c. c. , guetter , h. h. , harris , h. c. , henden , a. a. , leggett , s. k. , levison , h. f. , luginbuhl , c. b. , martini , j. , monet , a. k. b. , munn , j. a. , pier , j. r. , rhodes , a. r. , riepe , b. , sell , s. , stone , r. c. , vrba , f. j. , walker , r. l. , westerhout , g. , brucato , r. j. , reid , i. n. , schoening , w. , hartley , m. , read , m. a. , & tritton , s. b. 2003 , , 125 , 984 , f. j. , henden , a. a. , luginbuhl , c. b. , guetter , h. h. , munn , j. a. , canzian , b. , burgasser , a. j. , kirkpatrick , j. d. , fan , x. , geballe , t. r. , golimowski , d. a. , knapp , g. r. , leggett , s. k. , schneider , d. p. , & brinkmann , j. 2004 , , 127 , 2948 lccc 1998 aug 14 & 45@xmath118 & [ 38.0 or 218.0]@xmath111.9 & this work + 2005 mar 4 & [email protected] & [email protected] & this work + 2005 apr 30 & [email protected] & [email protected] & this work + 2005 may 1 & 291@xmath12 & [email protected] & @xcite + 2005 jul 31 & 298@xmath13 & [email protected] & this work lcc spectral type & l2@xmath11 & l3.5@xmath11 + @xmath3 magnitude & [email protected] & [email protected] + @xmath15 magnitude & [email protected] & [email protected] + f110 m magnitude & [email protected] & [email protected] + f145 m magnitude & [email protected] & [email protected] + f165 m magnitude & [email protected] & [email protected] + mass ( m@xmath22 ) & [email protected] & [email protected] + t@xmath24 ( k ) & 19002100 & 17001900 +
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we have resolved kelu-1 into a binary system with a separation of @xmath0290 mas using the laser guide star adaptive optics system on the keck ii telescope . we have also re - analyzed a 1998 hst observation of kelu-1 and find that the observed psf is best fit by a binary object separated by 45 mas .
observations on multiple epochs confirm the two objects share a common proper motion and clearly demonstrate the first evidence of orbital motion .
kelu-1b is fainter than kelu-1a by [email protected] magnitudes in the @xmath2 filter and [email protected] magnitudes in the @xmath3 filter .
we derive spectral types of l2@xmath11 and l3.5@xmath11 for kelu-1a and b , respectively .
the separation of flux into the two components rectifies kelu-1 s over - luminosity problem that has been known for quite some time . given the available data we are able to constrain the inclination of the system to @xmath481@xmath5 and the orbital period to @xmath640 years .
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one of the primary goals of the current generation of colliders is the discovery of the mechanism by which the mass scale of the weak interactions is generated and stabilized . whether that mechanism is supersymmetry ( susy ) , technicolor , extra dimensions or something not yet considered , it will generically require a number of new particle states . one or more of these particles are often stable , at least on experimental time scales , thanks to discrete symmetries in the model or suppressed couplings . the archetypes for such particles are the `` lsp '' ( lightest susy particle ) in the most susy models , and the nlsp ( next - to - lightest susy particle ) in gauge - mediated models . if a new particle produced at a collider lives long enough to escape the detector , there is no simple way to measure its lifetime . of utmost importance will be determining whether the lifetime is indeed finite or if the particle is stable on cosmological timescales . it has become standard lore that any stable remnant of new weak - scale physics must be , at most , weakly interacting . the possibility that the stable particle is electrically charged or strongly interacting has long been dismissed based on a combination of astrophysical and cosmological arguments . if such a particle , often called a champ @xcite , exists in abundance in our galaxy , then a steady flux of champs would have been captured by the earth over the course of our planet s lifetime . such stable , charged particles would form hydrogen - like atoms which could in turn form superheavy water molecules . these superheavy atoms and molecules would collect in our oceans in significant abundances . searches for superheavy isotopes of water in ocean water have all yielded negative results @xcite . these searches have effectively ruled out charged dark matter at the weak scale . this result is obviously dependent on the flux of champs onto the earth . there are two theoretical avenues for determining this flux . the first assumes that these relics represent the majority component of the galactic dark matter halo ; then their local abundance can be estimated directly from astrophysical data . from this we easily deduce the rate at which the champs would be captured and accumulate in our oceans . the second approach is to use standard cosmological assumptions and techniques to calculate the relic abundance of the champs from first principles . then using calculations of dark matter clumping , a local density can be estimated . these calculations also predict far greater abundances of superheavy water than are observed experimentally , even when the abundance is too small to account for the dark matter . there has been recent interest in whether charged stable relics are ruled out in more cosmologies . motivated in part by theories with large extra dimensions @xcite , studies have been done to calculate relic abundances for the case of extremely low reheat temperatures ( as low as an mev ) . the surprising result of these studies is that particles with tev masses can still be created during the reheat process even if the final temperature is around the mev scale @xcite . when applied to electrically charged relics , a bound of roughly 1 tev on the relic s mass can be deduced using experimental search bounds @xcite . but can we bound , exclude or search for stable , charged relics without recourse to cosmology ? in some models , physics near the tev scale is so non - canonical that it is unclear whether existing cosmological bounds apply at all ( see for example the model of ref . @xcite ) . in this paper , we will approach the same problem but from an original point of view . we will consider charged relics created by cosmic rays in the earth s upper atmosphere . in order to be specific , we will consider susy as our template model ; however extension of these bounds to any other model of weak - scale physics with a stable charged relic would be trivial . we will not place a bound on the mass of the susy relic itself . instead we will place a bound on the susy scale , or more specifically , the masses of the squarks and gluinos . direct production of the charged relic in cosmic ray collisions may be very small ( for example , the relic could be a stau with only small electromagnetic cross - sections ) . however , production of any susy state would , after a cascade of decays , result in one or more of the charged lsps . thus the production cross - section will generally be dominated by production of squarks and gluinos , not the lsp itself . none of these results depend in any way on new models which attempt to explain cosmic rays at the very highest energies . these results are generated using the usual spectrum of incident protons and heavy nuclei . our only assumption is that this spectrum has remained essentially unchanged over the last 3 billion years of the earth s history . while the energy frontier in particle accelerators is slowing moving higher and higher , collisions with center - of - mass energies in the tev range have been common throughout the history of the earth in the form of cosmic rays scattering off atmospheric nuclei . while the vast majority of these cosmic rays lose their energy through one or more hard qcd interactions , a small fraction can undergo new - physics interactions . with energies ranging up to @xmath1 ( in the earth s frame ) , the incident cosmic rays are capable of generating center - of - mass energies above @xmath2 when scattering off a proton . and with the exception of the very highest energies , the incident cosmic ray energy spectrum is well - measured . while these weak interactions in the upper atmosphere are useless for studying short - lived susy states , such states will be produced nonetheless . and if r - parity is conserved , those susy states will eventually decay down to the lsp . thus lsps are produced in our atmosphere in standard particle physics processes , independent of cosmological or astrophysical assumptions the all - particle spectrum for cosmic rays is well - measured up to energies of about @xmath1 . at energies up to about @xmath3 the spectrum follows a simple @xmath4 power law . around @xmath3 , the so - called `` knee '' is hit at which point the spectrum begins falling more dramatically , as @xmath5 . finally , at energies around @xmath6 ( the `` ankle '' ) the spectrum flattens out slightly to @xmath7 @xcite . the composition of the incident cosmic rays is not as well known . the primary spectrum is composed of a number of elements including protons , helium , iron , etc . for energies below the `` knee '' ( @xmath8 per nucleus ) , protons are the most abundant constituents , representing roughly 50% to 80% of the spectrum . at the knee , iron nuclei begin to dominate the spectrum ; however , only qualitative details of the composition can be inferred using , for example , the depth of shower maximum . the data seems to indicate that there is a relative rise in heavier elements and then a gradual decrease again as the `` ankle '' is approached @xcite . above @xmath9 per nucleus , lighter elements appear to be more abundant again , consistent with fragmentation of the heavier elements by the cmbr . extractions from the data of the average particle number as a function of primary energy vary from 5 to 15 for different experiments @xcite . given this level of ignorance , we assume a fixed fraction of protons ( 50% ) at energies below @xmath10 , exponentially decreasing to a constant 1% at energies above @xmath11 . we will also assume that the remainder of the primary composition ( the part that is not protons ) has on average @xmath12 nucleons / incident nucleus ; we will choose @xmath13 as a typical experimental value @xcite . we will model these nuclei as a collection of @xmath12 loosely bound nucleons , each carrying @xmath14 of the incident energy @xmath15 of the entire nucleus ; in the center - of - mass frame , the total energy of the collision , @xmath16 , is therefore reduced by a factor @xmath17 . to be conservative , we assume that any one nuclei can only participate in a single hard interaction and we do not consider the interactions of the daughter nuclei , if any are formed , at all . we find that nuclei heavier than hydrogen are responsible for only about 10% of the susy interactions and therefore our result is almost independent of how we model their interactions . we believe the above choices to be very conservative . looser constraints with more protons in the spectrum could allow our calculated rates to more than double . protons incident on our atmosphere will usually lose their energy either by qcd or qed processes . typical qed energy loss rates through brehmstrahlung and ionization are about 2 mev/(g/@xmath18 ) @xcite so that charged particles lose only a few @xmath19 traversing 15 km of atmosphere . this is negligible compared to losses in hard qcd interactions . energetic primary protons have a nuclear interaction length of @xmath20g/@xmath18 , corresponding to 12 qcd interactions over the depth of the atmosphere @xcite . for simplicity , we only consider the first such interaction and neglect secondaries produced in the resulting air showers . that is , we assume that once an incident proton has had a single hard qcd interaction , its energy is degraded beyond the point at which it is kinematically possible to create susy particles . in effect , we are discarding the possibility that a secondary could participate in a susy interaction , a choice which again results in a conservative bound on the number of susy states produced . a complete analysis , unneccesary here , involves solving a set of coupled cascade equations with appropriate boundary conditions , taking into account energy loss processes and energy - dependent cross sections . our method reduces to analyzing a fixed - target @xmath21 collision with one of the protons at very high energies . now we present the details of our calculation . after passing through a distance @xmath22 of the atmosphere with local density @xmath23 , the flux of protons with energies between @xmath15 and @xmath24 which have undergone hard qcd interactions and are therefore `` lost '' for susy interactions is simply given by @xmath25 where @xmath26 is the flux of protons measured in gev@xmath27@xmath28s@xmath27sr@xmath29 , @xmath30 is the number density of protons in the atmosphere at a depth @xmath31 , and @xmath15 is the energy of the incident proton in the earth s frame . at the energies under consideration here , the qcd @xmath21 cross - section , @xmath32 , is roughly 100 mb , independent of energy . over that same slice of atmosphere , the total number of susy interactions ( integrated over all energies ) , corresponding to the flux of lsps , is @xmath33 here @xmath34 represents the inclusive cross - section for @xmath35 , independent of whether @xmath36 is squarks , sleptons , etc . ( @xmath34 is a function of @xmath37 . ) this system of equations is simple to solve . for a spectrum of only protons , the total number of susy interactions in @xmath28s@xmath27sr@xmath29 is then : @xmath38 where @xmath26 and @xmath34 are implicitly functions of @xmath15 . accounting for the observation that only a fraction @xmath39 of the incident cosmic rays are protons , and using our stated assumption that the remaining cosmic rays contain on average @xmath12 nucleons , then the above discussion generalizes to : @xmath40\ ] ] where @xmath41 is the all - particle incident flux of cosmic rays . there is another , simpler way to understand the above calculation . since we only allow each incident cosmic ray one hard interaction ( and only 1 lsp per susy interaction ) , we can simply calculate the probability that that one interaction will be either qcd or susy . since @xmath42 , the probability of a susy interaction is then roughly @xmath43 . that probability is then integrated over the entire flux of incident cosmic rays to find the flux of lsps produced . the only unknown quantity that remains is @xmath34 . since this represents a total cross - section , summed over all possible susy final states , there is considerable room for model - dependence . however , we can make several simplifying observations / assumptions . first , because the cosmic rays and the atmospheric nuclei are baryonic , strong interactions should dominate the susy production processes . second , in most `` realistic '' susy models , there is an approximate degeneracy among the strongly - interacting sparticles , that is , among the squarks and the gluinos . these two statements allow us to greatly simplify the calculation by only considering superqcd interactions and by assuming that all squarks and gluinos have a common mass @xmath44 . independent of the details , the parton - level cross - section for susy production obviously goes like @xmath45 . to be slightly more realistic , we will use actual calculated cross - sections for the range of superqcd processes . the processes that we consider are : @xmath46 the relevant cross - sections have been calculated and tabulated in ref . @xcite ; we confine our calculation to tree level . to go to the proton - proton cross - section we use the cteq5 m parton distribution functions @xcite . the total @xmath47susy cross - section is shown in figure [ fig1 ] as a function of @xmath16 for several choices of @xmath44 . over the energy range of interest , the susy cross - section is dominated by @xmath48 and @xmath49 . there is a subtlety associated with exact versus approximate degeneracy of the squarks and gluinos and so we consider both the case in which @xmath50 and @xmath51 . we find a difference of about 50% in our calculated number densities as we vary over this range , and so we show a range of limits . = 3truein once a squark or gluino has been produced , it will decay in a cascade down to the the lsp . presumably , one lsp of positive charge will be produced for each of negative charge . searches for the latter are more difficult and we will concentrate only on the former . further , we will assume conservatively that only one positively charged lsp is produced per susy interaction , though the number can be significantly higher as the cascades of decays progress . one expects that these positively charged lsps form superheavy hydrogen by attracting a nearby electron and that this superheavy hydrogen eventually ( over the lifetime of the earth ) bonds into a superheavy water molecule in the earth s oceans . so as a final step we calculate the concentration of superheavy water in the oceans . considering the age of the oceans , @xmath52 , to be roughly 3 billions years , and assuming that the flux of cosmic rays has remained essentially unchanged over that time period , the number of superheavy water molecules per usual water molecule in the oceans ( the `` anomalous concentration '' , @xmath53 ) is @xmath54 where @xmath55 is the average depth of the oceans . the constraints on @xmath53 come from searches for superheavy molecules in large samples of water . experiments then place bounds on @xmath53 as a function of the mass of the stable , charged particle . in our approach , this mass in unknown , though it is bounded from above by @xmath44 . the strongest bound comes the experiment of smith @xcite , who find limits ranging from @xmath56 for @xmath57 to @xmath58 for @xmath59 . because their bound monotonically weakens as @xmath60 increases , we can place a very conservative bound on @xmath44 by setting @xmath61 . for that case , we find @xmath62 if the lsp is stable and charged . ( our bound has an uncertainty of roughly @xmath63 due to a @xmath64 uncertainty that comes from the subtleties for defining @xmath44 discussed above . ) these results are summarized in figure [ fig2 ] where we have shown the predicted anomalous concentration as a function of @xmath65 and the experimental bound of ref . @xcite . we have shown that there exist bounds from cosmic ray production in the upper atmosphere on charged stable relics ( a charged susy lsp in particular ) which are independent of cosmological constraints . under the assumption that the incident cosmic ray flux has remained constant over the last 3 billion years , we have calculated a conservative lower bound on the scale of new physics ( @xmath44 ) , using nothing more than standard particle physics . if susy has a stable and charged lsp , then we can place a lower bound on the mass scale of the squarks and gluinos at 230 gev . this procedure can easily be extended to other models of weak - scale physics . in such a case , limits similar to those found here could be placed on the masses of new strongly - interacting particles . importantly , these bounds will not change if the basic paradigms of cosmology at and below the weak scale are questioned , such as happens in models with large extra dimensions . for example , it is unclear how standard cosmological bounds can be used to constrain models such as that of ref . @xcite which predicts a light , stable top squark but becomes non - perturbative and higher - dimensional at the tev scale . the bound presented here should hold even in these highly non - standard cases . finally , if a charged particle is discovered which is stable on collider timescales , ruling out or verifying that it is stable on cosmological timescales will require that searches for superheavy water be examined again , though with larger initial samples . unfortunately , the steeply falling cosmic ray spectrum requires us to go to exponentially larger samples in order to significantly increase our sensitivity . for example , the method used in ref . @xcite would require an initial heavy water sample equal to that contained in the sno experiment in order to probe squark masses up to 700 gev . while such a large - scale search seems unnecessary at present , any future discovery of a new stable , charged particle might require just such an effort . otherwise there may be no other way to study the stability of that state on long timescales . ck would like to thank n. arkani - hamed , g. domokos , l. hall , h. murayama and j. poirier for enlightening conversations . this research was supported by the national science foundation under grant phy00 - 98791 . 99 a. de rujula , s. l. glashow and u. sarid , nucl . b * 333 * , 173 ( 1990 ) ; + s. dimopoulos , d. eichler , r. esmailzadeh and g. d. starkman , phys . d * 41 * , 2388 ( 1990 ) . p. f. smith and j. r. bennett , nucl . b * 149 * , 525 ( 1979 ) ; + t. k. hemmick _ et al . _ , phys . d * 41 * , 2074 ( 1990 ) ; + p. verkerk , phys . lett . * 68 * , 1116 ( 1992 ) ; + t. yamagata , y. takamori and h. utsunomiya , phys . d * 47 * , 1231 ( 1993 ) . f. smith , nucl . b * 206 * , 333 ( 1982 ) . n. arkani - hamed , s. dimopoulos , n. kaloper and j. march - russell , nucl . b * 567 * , 189 ( 2000 ) . d. j. chung , e. w. kolb and a. riotto , phys . d * 60 * , 063504 ( 1999 ) ; + g. f. giudice , e. w. kolb and a. riotto , phys . rev . d * 64 * , 023508 ( 2001 ) . a. kudo and m. yamaguchi , phys . b * 516 * , 151 ( 2001 ) . r. barbieri , l. j. hall and y. nomura , phys . d * 63 * , 105007 ( 2001 ) . d. j. bird _ [ hires collaboration ] , phys . lett . * 71 * , 3401 ( 1993 ) . data from a variety of original experiments has been compiled in p. greider , _ cosmic rays at earth _ , elsevier , amsterdam ( 2001 ) .
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supersymmetric models often predict a lightest superpartner ( lsp ) which is electrically charged and stable on the timescales of collider experiments .
if such a particle were to be observed experimentally , is it possible to determine whether or not it is stable on cosmological timescales ? charged , stable particles are usually considered to be excluded by cosmological arguments coupled with terrestrial searches for anomalously heavy water molecules .
but when the cosmology is significantly altered , as can happen in models with large extra dimensions , these arguments are in turn significantly weakened . in this paper we suggest an alternate way to use searches for superheavy water to constrain the lifetimes of long - lived , charged particles , independent of most cosmological assumptions . by considering susy production by cosmic rays in the upper atmosphere , we are able to use current bounds on superheavy water to constrain the mass scale of squarks and gluinos to be greater than about @xmath0 , assuming a stable
, charged lsp .
this bound can be increased , but only by significantly increasing the size of the initial water sample tested .
@=11 caption#1[#2]#3 @=12 .7ex .7ex gev mev ev kev tev m_z m_pl _ _ fc ^-1 ^-1 * d * * d^ * * u * * u^ * * y_d * * y_d^ * * y_u * * y_u^ * * v * * v^ * * v^0 * * v^0 * * x * b^0-|b^0 # 1#1 | # 1| # 1 * m * m_susy m_unif # 1#2#3nucl . phys .
* b#1 * ( 19#2 ) # 3 # 1#2#3phys . lett . * b#1 * ( 19#2 ) # 3 # 1#2#3phys . lett .
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# 1 * ( 19#2 ) # 3 _ i.e. _ _ et al . _
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_ /#1/ |#1 # 1#1 february 2002 + * bounds on charged , stable superpartners from cosmic ray production * + 0.5 cm _ department of physics , university of notre dame + notre dame , in 46556 , usa +
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You are an expert at summarizing long articles. Proceed to summarize the following text:
the quest for a natural way to break electroweak symmetry has long been the central motivation for constructing theories for physics beyond the standard model ( sm ) at the tev scale . however , the same logic applied to the even more severe fine - tuning problem associated with the cosmological constant would have predicted new physics near @xmath6 ev , which we have no evidence for . this suggests the possibility that our notions of naturalness are misleading , and that other fine - tuning mechanisms may be at work in nature . eliminating the use of naturalness as a guiding principle for model - building allows for some drastic changes to particle physics lore . arkani - hamed and dimopoulos have recently argued for a theory with `` split '' supersymmetry @xcite ( also @xcite ) . in this model the `` structure '' @xcite and `` atomic '' @xcite principles were used to explain the smallness of uv sensitive parameters ; namely the cosmological constant and the higgs vacuum expectation value . chiral symmetries keep all fermions light and a single fine tuning does the same for one higgs scalar , while all other scalars are at the high susy breaking scale . this framework , while preserving the successes of the mssm such as gauge coupling unification @xcite , also salvages some of its difficulties , giving a simple explanation for the absence of fcncs and cp violation ; the non - discovery of superpartners , and a light higgs ( see also @xcite ) , while simultaneously solving a variety of cosmological difficulties associated with the gravitino and moduli . an important quantitative prediction of the minimal model is the mass of the higgs , which has been computed to lie between 120 and 170 gev @xcite . as pointed out in @xcite , however , this prediction is sensitive to physics above @xmath0 and the presence of new matter beneath @xmath0 . in this paper we explore the bounds on the higgs mass in these more general versions of split susy which continue to conform to the essential philosophy of the model , in order to provide a falsifiable test of theories that are built on these principles . we analyse these models at one loop and examine the limits on the boundary value of the higgs quartic coupling at the susy breaking scale . in section [ sec : minmass ] we show that there exist mechanisms by which this can be made negative . how negative is determined by requiring stability of the sm vacuum . next , in analogy with the triviality bound in the sm , we use the requirement of perturbativity to the cutoff to put an upper limit on this boundary value . we take a totally agnostic viewpoint , assuming that there is some unknown physics in the region between @xmath0 and the unification scale , @xmath7 , that effectively decouples the higgs quartic from the electroweak gauge couplings . this could include new d - terms or f - terms such as in the nmssm . we then see how far this bound can be pushed in the large @xmath8 limit by : * varying @xmath0 and adding n su(5 ) @xmath9s at the weak scale , maximizing n at each @xmath0 in order to maintain perturbative gauge coupling unification * adding yukawa couplings between the higgs and new fermions that come in complete multiplets of su(5 ) we consider each of these in turn in section [ sec : maxmass ] , rge evolving the couplings down to the weak scale , where we can calculate the physical higgs mass . this gives us a firm prediction of this class of models , based on a minimal number of reasonable assumptions . we conclude in section [ sec : conclusion ] , with some discussion of possible interesting extensions of this work . we begin by discussing how we can _ decrease _ the higgs quartic coupling at @xmath0 . suppose that in the theory above this scale , we have an additional gauge singlet scalar field @xmath10 that picks up a mass term @xmath11 from susy breaking . like the higgses , it has an @xmath12-charge of zero so that the superpotential term @xmath13 is forbidden . however , the following @xmath14-term is permitted @xmath15 integrating out @xmath16 induces a term of the form @xmath17 , and our effective theory beneath @xmath0 now contains a negative contribution to the higgs quartic coupling of @xmath18 this can exceed the usual gauge d - term contribution proportional to @xmath19 , giving rise to the possibility that the sm vacuum state is not the true vacuum of the theory and allowing for the eventual decay of our vacuum to the true one by bubble nucleation . we use the methods in @xcite to calculate the decay rate per unit volume by approximating this to a pure @xmath20 theory ( see e.g @xcite ) : @xmath21 where @xmath12 is the size of the bubble by which this process takes place . in practice this integral is just dominated by the scale at which the integrand is maximized . as we will see , @xmath22 will turn out to be largest for @xmath23 , so the rate is dominated by @xmath24 . we musy have @xmath25 for this decay not to have occurred already , and solving this equation allows us to bound @xmath1 : @xmath26 saturating this bound we find the higgs mass shown in figure [ fig : minmass ] for two different values of @xmath8 . note the consistency of this lower limit on the higgs mass for @xmath27 gev with the lep - ii bound . we do not wish to lower @xmath0 any further since this will bring back the problems associated with the mssm that we were trying to alleviate . adding matter at the weak scale does not disrupt one - loop gauge coupling unification as long as this matter can be grouped into complete multiplets of su(5 ) . it does , however , increase the value of the couplings at @xmath7 since it contributes ( the same ) positive quantity to each rge . hence we need to ensure that we do not add so many particles that the couplings become non - perturbative before unification takes place . this limits n@xmath286 for susy breaking scale around @xmath29 gev , for example . we use these values to show in figure [ fig : higgsmassvslambda ] how the low energy physical higgs mass changes with @xmath1 . note that there is very little gain in mass for @xmath30 at high energies . next we take the higgs quartic close to its perturbative limit at the cutoff and vary @xmath0 , maximizing n at each scale to find an upper bound on the higgs mass . we expect the bound to increase substantially with n ; however as can be seen in figure [ fig : higgsmassnplusnbar ] this is not exactly the case ; at least for low cutoff there is no significant difference between the higgs mass in the theory with and without extra @xmath9s . this seems rather counterintuitive since increasing all gauge couplings increases the boundary values of the higgs - gaugino yukawa couplings ( @xmath31s ) at the susy breaking scale , which in turn should feed into the higgs mass . the reality of the story for @xmath32 is rather more complicated , however , and intimately involves three other couplings , @xmath33 , @xmath34 and @xmath35 , in the terms @xmath36 as well as @xmath37 . figure [ fig : contrib ] contains a graph of each of these contributions to the @xmath32 rge . notice that for low cutoff , the running of @xmath32 is dominated by itself and , since we have decoupled its boundary condition from the electroweak gauge couplings , it relies on none of the quantities that change on adding @xmath9s . even if @xmath32 was not the dominant coupling , increasing @xmath34 would actually decrease the higgs mass since it contributes via the positive @xmath38 term , partly undoing the effects of increasing @xmath35 and decreasing @xmath33 . for high cutoff on the other hand , the quartic runs down enough so that not only do the weak gauge coupling and gaugino yukawa start playing a much bigger part in determining its running ( although @xmath33 still does not ) , but the @xmath39 term actually becomes the dominant one . increasing n therefore increases the higgs mass through larger @xmath34 as well as @xmath35 . due to this property of the rges in this theory our method of adding @xmath9s will not increase the higgs mass much beyond 250 gev . as a check we can see in figure [ fig : higgsmassnplusnbarlowbc ] similar results using the boundary conditions for the mssm and the new fat higgs @xcite respectively . since these are much smaller than the perturbative limit used in the previous example , the results with and without new gut multiplets start to differ at a lower energy , confirming that @xmath32 now does not need to run down as much before the other couplings start becoming important . , although in reality the latter two need to be separated slightly in order for us to legitimately use the weak limit bound described in the paper . recall that this model already contains 4 @xmath9s of its own above the confinement scale - we need to take these into account in our perturbative unification constraint . ] returning to the effect of the top yukawa , we saw that this was negligible for all cutoffs since its value at low energies is fixed , and so it never becomes comparable in size to the other terms in the @xmath32 rge . this observation inspires an alternative approach in which n vector - like gut multiplets of quarks and leptons ( @xmath40s ) are added and coupled to the higgs in the usual fashion . these have vector - like masses , and the new top - like yukawa coupling , @xmath41 , plays exactly the same role as @xmath33 , except that it is not fixed at low energies and therefore can be more instrumental in determining the higgs mass . first we examine how the low energy value of this additional yukawa changes with boundary condition at the cutoff ( see figure [ fig : ytvsyt ] ) . as with @xmath32 , it is relatively insensitive to its boundary condition for @xmath422 . for such a large yukawa the higgs mass is also independent of the boundary value of @xmath32 , suggesting that its rge is now controlled by the new yukawa as intended . now we can analyse the higgs mass with changing cutoff in a similar manner to the ( @xmath43 ) case from earlier , maximizing n at each energy and comparing with our previous results in figure [ fig : maxhiggsmassxtend ] . if we examine the different contributions to the running of @xmath32 as before , we see that the larger @xmath44 indeed plays a leading role along the entire cutoff spectrum , giving rise to an overall increase in higgs mass with n , to a maximum of about 400 gev . it is possible that , as it stands , this modification with a dominant top - type yukawa coupling will give rise to undesirably large oblique parameters , especially a large positive contribution to t. this issue can be resolved in two ways , neither of which significantly affect our results . firstly we could increase the masses of these new fermions , which suppresses the higher - dimensional operators contributing to precision electroweak measurements like the t parameter . alternatively , we could impose an approximate custodial @xmath45 symmetry in the new matter sector , fixing the same boundary conditions on the bottom - type yukawa as the top - type . either way , these models can be made consistent with current experimental data . we see that it is possible to give limits on the physical higgs mass of @xmath46 in arbitrary extensions of split susy , including additional gut matter multiplets / gut singlets that are otherwise decoupled from the sm . adding new yukawa couplings pushes the upper bound to around @xmath47 gev . it is interesting that the _ lower _ limit on the higgs mass has already been excluded by lepii . however , we see that the possible hint for a 115 gev higgs can be accomodated in split susy models with @xmath48 gev and a slightly negative higgs quartic coupling near this scale . further analysis is required to determine how robust these predictions are to adding two - loop contributions to the running ( see @xcite for rges ) . from the corresponding sm results @xcite , we expect that this will decrease our limits by of the order of @xmath49 gev . it would also be interesting to see how nmssm - like boundary conditions for the higgs quartic , or even different uv completions of the nmssm , such as @xcite , affect these bounds . many thanks to nima arkani - hamed for suggesting the possibility of a negative quartic and also for numerous invaluable discussions on the vacuum stability bound among other interesting things . n. arkani - hamed and s. dimopoulos , arxiv : hep - th/0405159 . g. f. giudice and a. romanino , arxiv : hep - ph/0406088 . s. weinberg , phys . lett . * 59 * , 2607 ( 1987 ) . v. agrawal , s. m. barr , j. f. donoghue and d. seckel , phys . d * 57 * , 5480 ( 1998 ) [ arxiv : hep - ph/9707380 ] . s. dimopoulos and h. georgi , nucl . b * 193 * , 150 ( 1981 ) . s. dimopoulos , s. raby and f. wilczek , phys . d * 24 * , 1681 ( 1981 ) . w. j. marciano and g. senjanovic , phys . d * 25 * , 3092 ( 1982 ) . m. b. einhorn and d. r. t. jones , nucl . b * 196 * , 475 ( 1982 ) . l. e. ibanez and g. g. ross , phys . b * 105 * , 439 ( 1981 ) . n. sakai , z. phys . c * 11 * , 153 ( 1981 ) . p. langacker and n. polonsky , phys . rev . d * 52 * , 3081 ( 1995 ) [ arxiv : hep - ph/9503214 ] . j. d. wells , arxiv : hep - ph/0306127 . a. arvanitaki , c. davis , p. w. graham and j. g. wacker , arxiv : hep - ph/0406034 . s. coleman , `` aspects of symmetry : selected erice lectures '' . k. m. lee and e. j. weinberg , nucl . b * 267 * , 181 ( 1986 ) . g. isidori , g. ridolfi and a. strumia , nucl . b * 609 * , 387 ( 2001 ) [ arxiv : hep - ph/0104016 ] . s. chang , c. kilic and r. mahbubani , arxiv : hep - ph/0405267 . n. k. falck , z. phys . c * 30 * , 247 ( 1986 ) . m. e. machacek and m. t. vaughn , nucl . b * 249 * , 70 ( 1985 ) . m. e. machacek and m. t. vaughn , nucl . b * 236 * , 221 ( 1984 ) . m. e. machacek and m. t. vaughn , nucl . b * 222 * , 83 ( 1983 ) . g. altarelli and g. isidori , phys . b * 337 * , 141 ( 1994 ) . r. harnik , g. d. kribs , d. t. larson and h. murayama , arxiv : hep - ph/0311349 .
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we investigate the limits on the higgs mass in variations of split supersymmetry , where the boundary value of the higgs quartic coupling at the susy breaking scale ( @xmath0 ) is allowed to deviate from its value in the minimal model of arkani - hamed and dimopoulos .
we show that it is possible for @xmath1 to be negative and use vacuum stability to put a lower bound on this coupling , and hence on the mass of the physical higgs .
we also use the requirement of perturbativity of all couplings up to the cutoff to determine an upper limit for the higgs mass in models which are further modified by additional matter content . for @xmath2 gev
we find @xmath3 gev @xmath4 gev if the new matter is not coupled to any standard model field ; and @xmath3 gev @xmath5 gev if it has yukawa couplings to the higgs .
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You are an expert at summarizing long articles. Proceed to summarize the following text:
it has been known for some time that interactions on the 2.73 k blackbody cosmic microwave background ( cmb ) will severely degrade the energies of cosmic ray nucleons with energies beyond @xmath2 the greisen - zatsepin - kuzmin ( gzk ) cutoff @xcite . it was therefore very surprising when the fly s eye atmospheric fluorescence detector reported the observation of an extremely high energy cosmic ray ( ehecr ) event with an energy of @xmath3 @xcite . this was followed by the detection of a @xmath4 event by the agasa air shower array @xcite . these discoveries substantiated earlier claims from the volcano ranch @xcite , haverah park @xcite and yakutsk @xcite air shower arrays that cosmic rays do exist beyond the gzk cutoff . about a dozen such events are now known . detailed accounts of the data may be found in recent reviews @xcite . in figure [ fig1 ] we show the ehecr spectrum for energies exceeding @xmath5 @xcite ; note that the fluxes have been multiplied by @xmath6 . it is believed that cosmic rays with energies up to @xmath7 , the so - called ` ankle ' , are predominantly of galactic origin , possibly accelerated by the fermi mechanism in supernova remnants @xcite . above this energy , the spectrum flattens and the composition changes from being mostly heavy nuclei to mostly protons . such a correlated change in the spectrum and composition was first established by the fly s eye experiment @xcite and figure [ fig1 ] shows their suggested two - component fit to the data . the new component which dominates at energies beyond @xmath7 is isotropic and therefore can not possibly originate in the galactic disk @xcite . however it also extends well beyond the gzk cutoff raising serious problems for hypothetical extragalactic sources . because of the rapid energy degradation at these energies through photo - pion production on the cmb , such sources must exist within @xmath8 , in fact within @xmath9 for the highest energy fly s eye event @xcite . for heavy nuclei , the energy loss is less severe according to a revised calculation @xcite so the range may extend upto @xmath10 . general arguments @xcite provide correlated constraints on the magnetic field strength and spatial extent of the region necessary to accelerate particles to such high energies and these requirements are barely met by likely astrophysical sites such as active galactic nuclei and the ` hot spots ' of radio galaxies @xcite . moreover there are few such sources close to us and no definite correlations have been found between their locations and the arrival directions of the most energetic events @xcite . it has been speculated that gamma - ray bursts which too are isotropically distributed , may be responsible for ehecrs @xcite . however since these are at cosmological distances , one would expect to see the gzk cutoff in the cosmic ray spectrum contrary to observations ( cf . ref.@xcite ) . some of the above arguments may be evaded if the ehecr events are due not to nucleons but neutral particles such as photons and neutrinos . although high energy photons also suffer energy losses in traversing the cmb and the extragalactic radio background , there is no threshold effect which would cause a cutoff near the gzk value @xcite . however the observed shower profile of the highest energy fly s eye event @xcite argues against the primary being a photon since it would have interacted on the geomagnetic field and started cascading well before entering the atmosphere @xcite . the observed events are also unlikely to be initiated by neutrinos as they all have incident angles of less than @xmath11 from the zenith and thus too small a path length in the atmosphere for interactions @xcite . this argument may be evaded if neutrinos become strongly interacting at high energies due to new physics beyond the standard model @xcite , but such proposals are found not to be phenomenologically viable @xcite ( although this is disputed @xcite ) . ( alternatively , the propagating high energy neutrinos could annihilate on the relic cosmic neutrino background , assumed to have a small mass of @xmath12 ev , to make hadronic jets within the gzk zone @xcite . ) other exotic possibilities have been suggested , e.g. monopoles @xcite , stable supersymmetric hadrons @xcite and loops of superconducting cosmic string ( ` vortons ' ) @xcite . however these possibilities have many phenomenological problems @xcite and we do not discuss them further . thus one is encouraged to seek ` top - down ' explanations for ehecrs in which they originate from the decay of massive particles , rather than being accelerated up from low energies . the most discussed models in this connection are based on the annihilation or collapse of topological defects such as cosmic strings or monopoles formed in the early universe @xcite . when topological defects are destroyed their energy is released as massive gauge and higgs bosons which are expected to have masses of @xmath13 if such defects have formed at a gut - symmetry breaking phase transition . the decays of such particles can generate cascades of high energy nucleons , @xmath14-rays and neutrinos . a more recent suggestion is that ehecrs arise from the decays of metastable particles with masses @xmath15 which constitute a fraction of the dark matter @xcite . these authors suggest that such particles can be produced during reheating following inflation or through the decay of hybrid topological defects such as monopoles connected by strings , or walls bounded by strings . the required metastability of the particle is ensured by an unspecified discrete symmetry which is violated by quantum gravity ( wormhole ) effects . another suggestion is that the long lifetime is due to non - perturbative instanton effects @xcite . in ref.@xcite , a candidate metastable particle is identified in a @xmath16 gut . a generic feature of these ` top - down ' models is that the ehecr spectrum resulting from the decay cascade is essentially determined by particle physics considerations . of course the subsequent propagation effects have astrophysical uncertainties but since the decays must occur relatively locally in order to evade the gzk cutoff @xcite , they are relatively unimportant . thus although the proposal is speculative , it is possible , in principle , to make reliable calculations to confront with data . in this work we consider another possible candidate for a relic metastable massive particle @xcite whose decays can give rise to the observed highest energy cosmic rays . first we discuss ( [ crypton ] ) why this candidate , which arises from the hidden sector of supersymmetry breaking , is perhaps physically better motivated than the other suggested relics . we then undertake ( [ decay ] ) a detailed calculation of the decay cascade using a monte carlo event generator to simulate non - perturbative qcd effects . this allows us to obtain a more reliable estimate of the cosmic ray spectrum than has been possible in earlier work on both topological defect models @xcite and a decaying particle model @xcite . we confront our results with observations and identify the mass and abundance / lifetime required to fit the data . we conclude ( [ concl ] ) with a summary of experimental tests of the decaying particle hypothesis . soon after the discovery of the anomaly - free heterotic superstring theory in ten dimensions based on the gauge group @xmath17 , it was pointed out @xcite that in the physical low energy theory ( where a grand unified @xmath18 or @xmath19 group is broken by wilson lines ) , the minimum value of magnetic charge is not the dirac quantum @xmath20 but an integral multiple thereof . conversely , the minimum electric charge is smaller than the electron charge @xmath21 by the same ratio . where @xmath22 is four - dimensional minkowski space and @xmath23 is some compactified six - manifold . such fractionally charged states exist because @xmath23 is not simply connected these are states in which a closed string wraps around a non - contractible loop in @xmath23 . ] this was found to be a generic feature of all superstring models based on a level - one kac - moody algebra @xcite . in view of the severe experimental upper bounds on the relic abundance of fractional charges @xcite , this posed a potential embarrassment for superstring phenomenology @xcite . a simple solution to the problem of fractional charges ( with an obvious historical analogue in quarks and qcd ) is to confine them and it was shown that this can be done in the hidden sector of supersymmetry breaking in the framework of the @xmath24 unification model @xcite . in this model , all fractionally charged states have charges @xmath25 and are placed in * 4 * or * 6 * representations of a hidden @xmath26 gauge group which becomes strong at a scale @xmath27 and in * 10 * representations of a hidden @xmath28 group which becomes strong at a scale @xmath29 . this results in integer - charged particles ` cryptons ' which may be 2-constituent mesons , 3-constituent baryons or 4-constituent ` tetrons ' @xcite . some of these mesons could be light ( in analogy to the pion of qcd ) but most of the states should be heavy with masses of order the confinement scale @xmath30 . ( other possibilities for stable superstring relics have been discussed in ref.@xcite . ) the constituent fields have very few renormalizable ( @xmath31 ) superpotential interactions , so most of these states can only decay via higher - order ( @xmath32 ) superpotential terms . generically , crypton lifetimes are expected to be @xcite @xmath33 where , @xmath34 gev is the normalized planck scale , giving @xmath35 for @xmath26 and @xmath28 bound states respectively . thus @xmath36 for @xmath37 and @xmath38 for @xmath39 . detailed studies of the possible effects of decays of relic cryptons on primordial nucleosynthesis and the cmb spectrum @xcite , as well as on the diffuse @xmath14-ray background @xcite have established that such particles , if they survive as relics of the big bang , must either decay well before nucleosynthesis or have lifetimes longer than the age of the universe ( @xmath40 ) . in the latter case , if such particles make an interesting contribution to the dark matter , their lifetime is further required to exceed @xmath41 in order to respect experimental bounds on the flux of high energy neutrinos expected from their decays @xcite . it is seen from eq.([tau10tau4 ] ) that these constraints favour @xmath26 mesons over their @xmath28 counterparts as possible constituents of the dark matter . it is then natural to contemplate the possibility that such cryptons with a mass of @xmath42 and a lifetime @xmath43 are also responsible for the observed highest energy cosmic rays . recently the above discussion has been extended to other massive metastable particle candidates in superstring / m - theory @xcite . these authors discuss constructions with higher - level kac - moody algebras ( necessary to accommodate adjoint higgs representations in ( unified ) models other than @xmath24 ) and note that similar metastable bound states occur in such models . they go on to consider other candidate particles in m - theory such as kaluza - klein states associated with extra dimensions but find that these are not as attractive , being either too heavy or too unstable . they suggest that although the @xmath44 model @xcite discussed above was constructed in the weak coupling limit , it may be elevated to an m - theory model in the strong coupling limit . the @xmath26 tetrons are then still the most likely candidates for massive metastable dark matter with the modification that the planck scale @xmath45 in eq.([taucrypton ] ) may be replaced by a somewhat smaller scale . the main reason why this possibility was not seriously entertained earlier concerns the expected relic abundance of such massive particles . if cryptons were maintained in chemical equilibrium in the early universe through self - annihilations , their present energy density is given by the usual ` freeze - out ' calculation as inversely proportional to the ( velocity - averaged ) annihilation cross - section @xcite . estimating this to be @xmath46 we see that equilibrium would have been established if the annihilation rate exceeded the hubble expansion rate ( @xmath47 ) , i.e. at temperatures @xmath48 the relic abundance is then simply estimated as the equilibrium value at decoupling : @xmath49 this is the basis for the conclusion that no stable relic particle may have a mass in excess of @xmath50 without ` overclosing ' the universe , i.e. contributing @xmath51 @xcite . this does not necessarily apply to cryptons since a period of inflation should have diluted their abundance to essentially zero , along with monopoles and other such supermassive relics . if the reheating temperature following inflation is restricted to be @xmath52 in order not to produce too many gravitinos @xcite , cryptons would not have been generated afterwards . however it has been recently recognized that in supersymmetric cosmology , there is likely to be a late stage of ` thermal inflation ' @xcite due to symmetry breaking along flat directions at intermediate scales @xcite . this would adequately dilute the abundance of thermally generated gravitinos following inflation so the bound quoted above on the reheating temperature is no longer valid and the value of @xmath53 may be much higher . of the scalar ` inflaton ' field is constrained to be @xmath54 by the anisotropy in the cmb observed by cobe , where the slope parameter @xmath55 is required to be @xmath56 to permit inflation to occur @xcite . ( the number of e - folds of expansion until the end of inflation is just @xmath57 and this should exceed @xmath58 in order to solve the flatness and homogeneity problems of the standard cosmology . ) the reheat temperature @xmath59 can , in principle , have been as high as @xmath60 although it is usually considerably smaller since the inflaton field is very weakly coupled in most inflationary models . ] in that case cryptons even as massive as @xmath61 may well have been brought back into thermal equilibrium during reheating after inflation and survived with the huge relic abundance ( [ crypdens ] ) . however thermal inflation would also have diluted this to an acceptable level as was noted in ref.@xcite ; to obtain @xmath62 , the number of e - folds of thermal inflation required is just @xmath63 this fits in well with the expectation that @xmath64 for the intermediate scale @xmath65 in the range @xmath66 @xcite . of course given the uncertainty in the value of @xmath67 ( and indeed the possibility that there may be more than one such epoch ) , @xmath68 could well have been reduced to a negligibly small value . another possibility is that massive particles such as cryptons were never in thermal equilibrium but were created with a cosmologically interesting abundance due to the varying gravitational field during ( primordial ) inflation @xcite . a cosmologically interesting relic abundance then arises for @xmath69 where @xmath70 is the likely hubble parameter during inflation @xcite . this is certainly very encouraging but it should be remembered that a later stage of thermal inflation would dilute such an abundance to a negligible level , as discussed above . it is clear that the relic abundance of massive particles such as cryptons will necessarily be very uncertain given our ignorance of the thermal history of the universe prior to nucleosynthesis . however as the above discussion illustrates , there are two complementary ways in which a cosmologically interesting abundance may result so we may reasonably consider such particles as candidates for the dark matter . we now move on to discuss whether relic cryptons can indeed be the source of the ehecr by determining the expected spectrum of high energy particles from their decays . to calculate the expected flux of cosmic rays from decays of massive particles such as cryptons , we must consider the contribution from both decaying particles in the halo of our galaxy as well as those elsewhere in the universe . since such massive particles would behave as cold dark matter and cluster efficiently in all gravitational potential wells , their abundance in our galactic halo would be enhanced above their cosmological abundance by a factor @xmath71 note that @xmath72 where @xmath73 is the critical density in terms of the present hubble parameter @xmath74kmsec@xmath75mpc@xmath75 . if for simplicity we assume a spherical halo of uniform density , @xmath76 then @xmath77 and the number density of cryptons in the halo is @xmath78 the actual density of dark matter in the halo must of course fall off as @xmath79 to account for the flat rotation curve of the disk but we do not consider it necessary at this stage to investigate realistic mass models . thus the universal density of cryptons is smaller than the halo density by about the same numerical factor by which the distance to the horizon ( @xmath80 ) exceeds the halo radius , so the extragalactic contribution to the ehecr flux from decaying cryptons can not exceed the halo contribution . in particular , the gzk cutoff scale for protons @xcite or heavy nuclei @xcite are all much smaller than the horizon distance , so only the halo contribution need be considered , as was emphasized in ref.@xcite . only for neutrinos would the extragalactic component be comparable in magnitude @xcite . henceforth we restrict ourselves to considering crypton decays in the halo alone . now the injection spectrum from particle decay is , to a good approximation , @xmath81 for lifetimes longer than the age of the universe ( @xmath82 ) . here @xmath83 is a measure of particle energy ( assuming 2-body decays ) and the fragmentation function @xmath84 is the average number of particles @xmath85 released per decay , per unit interval of @xmath86 , at the value @xmath86 . the flux at earth is then @xmath87 the final state particles which interest us most are ` protons ' and neutrinos / antineutrinos where the former includes other nucleons , e.g. antiprotons and neutrons , since they all interact similarly in the earth s atmosphere . to compare with observations we multiply the fluxes by @xmath6 and define @xmath88 for photons and electrons / positrons , propagation energy losses are substantial even within the halo and we do not attempt to determine these . however their injection spectra from particle decay are given by the same computation as for protons and neutrinos , to which we now turn . heavy particles , whether gut - scale bosons ( in topological defect models ) , cryptons or other hypothetical massive particles , will decay into quarks and leptons . the quarks will hadronize producing jets of mostly pions with a small admixture of nucleons and antinucleons . the neutral pions will decay to give photons while charged pion decays will yield neutrinos and antineutrinos in addition to leptons . thus the final spectrum of the decay produced particles will be essentially determined by the ` fragmentation ' of quarks / gluons into hadrons . this is a non - perturbative qcd process and it has not been possible to calculate it by analytic means . usually phenomenologically motivated approximations are used to model experimental data on inclusive jet multiplicities and scaling violations @xcite . so far , authors of proposals involving heavy particle decay , e.g. in the context of topological defect models @xcite , have employed a hadronic fragmentation function suggested by hill @xcite @xmath89(1-x)^2}{x\sqrt{\ln(1/x ) } } .\ ] ] it is further _ assumed _ that 3% of the hadronic jets from massive relic particle decays turn into nucleons , while the other 97% are pions which decay into photons and neutrinos . this was based on the leading logarithm approximation of qcd @xcite applied to experimental data from petra on jet production in @xmath90 collisions at tens of gev . the estimated jet multiplicity from gluon fragmentation was convoluted with the gluon distribution to determine the total hadron yield ; to estimate the spectrum , it was assumed that the first moment of the distribution is normalized to unity and the large @xmath86 behaviour was guessed to be @xmath91 @xcite . as we shall see , the hill fragmentation function ( [ hillfrag ] ) significantly _ overestimates _ the yield of high @xmath86 final states from the decay of very massive particles and , moreover , photons and neutrinos are actually produced with a spectrum quite different from that of nucleons . thus the decay spectra derived using eq.([hillfrag ] ) for topological defect models @xcite are inaccurate . subsequently , another form called the modified leading logarithm approximation ( mlla ) which gives a better description of data at low @xmath86 has been proposed @xcite ; a gaussian approximation to this is @xmath92 , \ ] ] where @xmath93 is a constant and @xmath94^{3/2 } , \ ] ] with @xmath95 and @xmath96 . this fragmentation function is employed by the authors of ref.@xcite to compute the spectrum from relic particle decays ; they determine @xmath93 by requiring that the integral of @xmath97 over the range @xmath98 $ ] be equal to the fraction of the energy transferred to hadrons . however this procedure is not exact as the form ( [ mlla ] ) is inapplicable for large @xmath86 and therefore can not be normalized in this manner . thus the shape of the cosmic ray spectrum computed @xcite by this method for decaying particles is only reliable for small @xmath86 and its normalization uncertain . given the importance of determining the energy spectrum accurately , we decided to improve on these approximate formulations by using the standard tool employed by experimental high energy physicists to study qcd fragmentation , viz . a monte carlo event generator . here the non - perturbative hadronization process is simulated on a computer by a well tested phenomenological model @xcite . although this requires extensive numerical calculations , it is the only means by which successful contact can be made between theory and experimental data . we chose the programme herwig @xcite ( hadron emission reactions with interfering gluons ) which incorporates the cluster model for hadronization and is based on a shower algorithm of the ` jet calculus ' type @xcite . to check our results we also ran the jetset programme @xcite and found good agreement over the energy range where comparison was possible . however for the very high energies studied in this work , herwig proved to be more suitable for reasons of computing time @xcite . even so the calculations described here took several months on a digital alpha workstation . for definiteness , we assume the heavy particles to decay into a quark - antiquark pair with unit branching ratio . the quark and antiquark , each carrying away energy @xmath99 , form jets which lead to the generation of many particles through cascading , hadronization and decays of some of the generated particles . this can be simulated by herwig via the annihilation process @xmath100 with center - of - mass energy @xmath101 , where @xmath102 stands for all six kinematically allowed quark flavours . the event generator outputs kinematical details of all final state particles , e.g. protons , photons and leptons ( electrons , positrons and neutrinos ) . we divided the @xmath86-range into 100 bins of width @xmath103 . after each event simulation the number of protons , neutrinos , photons as well as electrons and positrons per energy bin was counted . we ran 10000 events for each of the masses @xmath104 and @xmath105 . after all events had been run , the particle numbers in the bins are divided by the bin width and the number of events , in order to obtain the fragmentation functions @xmath84 . apart from altering some relevant parameters in the computer code to allow it to run at the high energies studied here , we also switched off initial state radiation since it is not relevant for the present study . unfortunately , it was not feasible to study the high @xmath86 behaviour of the fragmentation functions for decaying particle masses higher than @xmath105 because of numerical convergence problems in the computer code . ( already for masses exceeding @xmath106 quadruple precision had to be used . ) hence we have had to extrapolate the fragmentation functions to high @xmath86 for very heavy masses as described later . first we show the proton fragmentation function obtained from the herwig runs in figure [ fig2 ] to illustrate that it depends on the decaying particle mass , contrary to the approximation ( [ hillfrag ] ) employed in previous work on topological defects @xcite . rather than being constant , it decreases with increasing @xmath107 for @xmath108 , while at very low @xmath86 it increases with increasing @xmath107 . the large fluctuations at @xmath109 are due to the fact that relatively few particles are produced with such high energies despite the 10000 events per simulation . we note also that the shape differs significantly at high @xmath86 from the approximation used in ref.@xcite . in figure [ fig3 ] the fragmentation functions for protons , photons , neutrinos and electrons are compared for @xmath110 . it is seen that at very low @xmath86 there are more photons and neutrinos generated by the particle decay than electrons and protons . in the regime @xmath111 , photons , neutrinos and protons are generated with roughly equal abundances . however , for @xmath109 , photons and neutrinos again outnumber protons , in particular protons cut off at @xmath112 whereas neutrinos and photons are generated in the cascades with energies up to @xmath113 . these differences will lead to different shapes of the expected fluxes @xmath114 as can be seen from eq.([fluxi ] ) . we now compare our proton fragmentation function with the commonly used hill approximation @xcite in figure [ fig4 ] . although his form provides a good fit for a low decaying particle mass , viz . @xmath115 , it no longer does so for a high mass , viz . this is understandable given that the numerical co - efficients in eq.([hillfrag ] ) were chosen to match relatively low energy collider data . however the functional form itself is well motivated and using our herwig runs we can determine new numerical co - efficients appropriate to heavier mass particles . another advantage of the present approach is that the spectrum of neutrinos and photons is determined separately from that of the protons and not simply assumed to be proportional as in previous work @xcite . to study the highest energy cosmic ray events we need to consider particle masses beyond @xmath105 but this is difficult to do directly with herwig for technical reasons as mentioned earlier . we therefore resort to an extrapolation procedure as follows . for the range @xmath116 $ ] the fragmentation functions are smooth and evolve monotonically with @xmath107 so the fragmentation functions for a @xmath117 particle is obtained from simple linear extrapolation of the lower energy fragmentation functions in each individual energy bin . for @xmath118 $ ] we first fit the calculated fragmentation functions to the form @xmath119(1-x)^2}{x\sqrt{\ln(1/x ) } } , \ ] ] for protons , and the form @xmath120(1-x)^2}{x\ln(1/x ) } , \ ] ] which proves more suitable for photons , neutrinos and electrons . the numerical co - efficients @xmath121 and @xmath122 are determined for particle masses less than @xmath105 by minimizing @xmath123 in the fit to the actual herwig runs . in figures [ fig5 ] and [ fig6 ] we show these fits for @xmath124 to the proton and neutrino fragmentation functions corresponding to masses of @xmath125 and @xmath126 . then we determine the appropriate co - efficients for heavier masses by extrapolation . an example , for @xmath127 , is shown in the figures . for @xmath128 , statistical fluctuations become too severe so we extrapolate the fitting functions between the value at @xmath129 and a cutoff which is taken to be @xmath130 for protons and @xmath131 for neutrinos , based on the observed behaviour for masses upto @xmath105 shown in figures [ fig2 ] and [ fig3 ] . finally , we mention the continuation of the proton fragmentation function for very low @xmath86 , viz . @xmath132 , which is relevant at high masses e.g. @xmath127 . since it proved impractical to have additional binning intervals at very small @xmath86 , we employ the fragmentation function ( [ mlla ] ) in this regime , normalized to our computations at @xmath133 . with the fragmentation functions obtained above , we can now calculate the expected fluxes of protons and neutrinos from decays of particles such as cryptons in the halo . we normalize the calculated proton flux ( [ fluxi ] ) to the observed cosmic ray flux at @xmath134 @xcite : @xmath135 = 24.32 .\ ] ] note that the corresponding neutrino flux @xmath136 is then a _ prediction _ as the fragmentation function for neutrinos is computed independently . the expected proton fluxes are shown in figure [ fig7 ] . we see that a crypton with @xmath110 fits the flat power law well but can not explain the events beyond @xmath137 . although this is easily achieved for @xmath127 , the decays of such a massive particle would overproduce protons for @xmath138 . thus a crypton with mass @xmath139 provides the best compromise although it too predicts a spectrum somewhat flatter than the one indicated observationally . ( the reader is reminded that all differential fluxes have been multiplied by @xmath6 in eq.([fluxi ] ) . ) an interesting signature for forthcoming experiments is the predicted ratio of the proton to neutrino flux @xcite . in figure [ fig8 ] we compare the expected flux of protons and neutrinos for @xmath140 . ( we also show the photon flux to illustrate the difference from the prediction in ref.@xcite but emphasize that this will be degraded through interactions with photon backgrounds during travel to earth . ) as can be seen , the neutrino flux exceeds the proton flux for @xmath141 and also for @xmath142 , as may have been anticipated from the comparison of their respective fragmentation functions . thus the ratio @xmath143 has a characteristic peak at about @xmath144 as shown in figure [ fig9 ] . this could be an useful diagonistic of the decaying particle hypothesis for future experiments such as the pierre auger project . note that taking the extragalactic contribution into account would boost the neutrino flux by a factor of @xmath145 over that shown in the figures . the abundance and lifetime of decaying particles such as cryptons are related through the spectrum normalization ( [ fenorm ] ) as : @xmath146 where @xmath147 for crypton masses @xmath148 respectively . for a given crypton mass , a higher lifetime must be compensated for by a higher relic abundance , as illustrated in figure [ fig10 ] . so for example , if @xmath149 , cryptons with a mass of @xmath140 are required to have a lifetime of @xmath150 if they are to explain the ehecr flux . if the enhancement in the halo is @xmath151 as expected for cold dark matter , then the lifetime may be increased to @xmath152 if @xmath153 ; alternatively , for the same lifetime one could tolerate a lower relic abundance @xmath154 . with regard to the fluxes of electrons and photons , both species would generate electromagnetic cascades on the prevalent radiation backgrounds through pair production and inverse compton - scattering . a thorough analysis of such propagation effects and the resulting modifications of the injected photon and electron spectra has been performed @xcite . it was found that the relic decaying particles with @xmath155 would contribute excessively to the diffuse @xmath14-ray background and are therefore ruled out . hence , the mass range we favour , viz . @xmath156 , does not lead to any conflict with observations . this conclusion is strengthened by the fact that according to our calculations the previous estimate @xcite of the @xmath14-ray flux from decaying particles was too high . although the positrons released in the decays may be accumulated in the galactic halo , the astrophysical uncertainty in the containment time does not allow a restrictive constraint to be derived from limits on the positron flux in cosmic rays @xcite . with regard to the neutrino background , the predicted flux at high energies is well below the upper limits derived from consideration of horizontal air showers @xcite , again because the decaying crypton mass is restricted to be less than about @xmath117 . it is also interesting to consider the flux at lower energies of @xmath157 where experiments such the forthcoming antares detector @xcite will be most sensitive . as seen in figure [ fig8 ] the predicted neutrino flux dominates over the proton flux at low energies , thus the bulk of the energy released by the decaying cryptons ends up as neutrinos . therefore we expect the neutrino flux at tev energies to be at least @xmath158 times larger than the ehecr flux at @xmath159 . moreover the neutrinos should be well correlated in both time and arrival direction with the cosmic rays since the path length in the galactic halo is @xmath160 . this is in contrast to the case of other suggested cosmologically distant sources such as gamma - ray bursts where the relative time delay can be upto @xmath161 @xcite . we have investigated the hypothesis that the highest energy cosmic rays , in particular those observed beyond the gzk cutoff , arise from the decay of massive metastable relic particles which constitute a fraction of the dark matter in the galactic halo . to simplify computations ( using the herwig monte carlo event generator ) we have considered only decays into @xmath162 pairs with unit branching ratio . comparison with experimental data indicates that a decaying particle mass of @xmath163 is required to fit the spectral shape while the absolute flux requires a lifetime of @xmath164 if such particles contribute the critical density . the predicted decay spectra may be somewhat altered if 3-body decays and other final states ( e.g. supersymmetric particles @xcite ) are considered . however our conclusions regarding the preferred mass and relic abundance / lifetime of the decaying particle are unlikely to be affected . in particular it would appear that the approximations used to calculate the particle spectra in previous studies of decaying topological defects @xcite and hypothetical massive particles @xcite were not sufficiently accurate . our work indicates that the topological defect model is disfavoured unless the mass of the decaying gauge bosons is less than about @xmath117 , which is well below the unification scale of @xmath165 . ( a similar conclusion is arrived at by independent arguments in refs.@xcite . ) by contrast , cryptons from the hidden sector of supersymmetry breaking have a mass of the required order , as well as a decay lifetime which is naturally suppressed . however their relic abundance is difficult to estimate reliably , although we have argued that it may be cosmologically interesting . the primary intention of this work is to attempt to quantify the decaying particle hypothesis in a manner which is of interest to experimentalists . we have therefore computed the expected neutrino to proton ratio as a function of energy since this is an important test of competing hypotheses for forthcoming experiments , in particular the pierre auger project @xcite . of course our cleanest prediction is that the cosmic ray spectrum should cut off just below the mass of the decaying crypton , at @xmath166 . moreover , with sufficient event statistics it should be possible to identify the small anisotropy which should result from the distribution of the decaying particles in the galactic halo @xcite . thus although the hypothesis investigated here is very speculative , it is nevertheless testable . perhaps nature has indeed been kind to us and provided a spectacular cosmic signature of physics well beyond the standard model . efimov et al . icrr symp . on astrophysical aspects of the most energetic cosmic rays , eds . m. nagano and f. takahara ( world scientific , 1990 ) p.20 ; + b.n . afanasiev et al . , proc . 24th intern . cosmic ray conf . 2 ( 1995 ) 756 . watson , nucl . suppl . ) 22a ( 1991 ) 116 ; proc . dpf summer study on high energy physics , snowmass , 1994 , eds . e.w . kolb and r.d . peccei ( world scientific , 1995 ) p.126 ; + p. sokolsky , p. sommers and b.r . dawson , phys . rep . 217 ( 1992 ) 225 ; + j.w . cronin , nucl . suppl . ) 28b ( 1992 ) 213 ; + s. yoshida and h. dai , astro - ph/9802294 . aharonian , b.l . kanewski and v.a . sahakian , j. phys . g17 ( 1991 ) 1989 ; + h.p . vankov and p.v . stavrev , phys . lett . b226 ( 1991 ) 178 ; + f. halzen , r. vazquez , t. stanev and h.p . vankov , astropart . phys . 3 ( 1995 ) 151 ; + t. stanev and h.p . vankov , phys . d55 ( 1997 ) 1365 . hill , d.n . schramm and t.p . walker , phys . d36 ( 1987 ) 1007 ; + p. bhattacharjee , c.t . hill and d.n . schramm , phys . 69 ( 1992 ) 567 ; + f.a . aharonian , p. bhattacharjee and d.n . schramm , phys . d46 ( 1992 ) 4188 ; + g. sigl , space sci . rev . 75 ( 1996 ) 375 . g. lazarides , c. panagiotakopoulos and q. shafi , phys . 56 ( 1986 ) 557 ; nucl . b307 ( 1988 ) 937 ; + k. enqvist , d.v . nanopoulos and m. quiros , phys . 169b ( 1986 ) 343 ; + o. bertolami and g.g . ross , phys . lett . 183b ( 1987 ) 163 ; + j. ellis , k. enqvist , d.v . nanopoulos and k. olive , phys . b188 ( 1987 ) 415 , b225 ( 1989 ) 313 . l. dokshitzer , v.a . khoze , a.h . mueller and s.i . troyan , _ basics of perturbative qcd _ ( editions frontieres , 1991 ) ; + r.k . ellis , w.j . stirling and b.r . webber , _ qcd and collider physics _ ( cambridge university press , 1996 ) ; + v.a . khoze and w. ochs , int . a12 ( 1997 ) 2949 .
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the expected proton and neutrino fluxes from decays of massive metastable relic particles are calculated using the herwig qcd event generator .
the predicted proton spectrum can account for the observed flux of extremely high energy cosmic rays beyond the greisen - zatsepin - kuzmin cutoff , for a decaying particle mass of @xmath0 gev .
the lifetime required is of @xmath1 yr if such particles constitute all of the dark matter ( with a proportionally shorter lifetime for a smaller contribution ) .
such values are plausible if the metastable particles are hadron - like bound states from the hidden sector of supersymmetry breaking which decay through non - renormalizable interactions .
the expected ratio of the proton to neutrino flux is given as a diagonistic of the decaying particle model for the forthcoming pierre auger project . -3cm^-3
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You are an expert at summarizing long articles. Proceed to summarize the following text:
the role of electronic interactions on the observed properties of semiconductor quantum dots has increasingly been found to be of vital importance , as many of the papers in these proceedings and elsewhere illustrate . more and more , transport and capacitance experiments , as well as detailed studies of far - infrared response , require the consideration of interactions in order to understand the observed experimental features . since a number of experiments explore specifically the electronic level spectrum , it is of interest to contrast these studies with a detailed theoretical analysis of the role of interactions . we present here an analysis of the energy level statistics of a quantum dot system as function of variable interaction strength , and then as function of an applied magnetic field . as the level spectrum is considered in detail , the correspondence with the dynamical integrability of a classical system is also of interest . it is anticipated that as the coulomb interaction is introduced , the dynamics would in general become chaotic and this would in turn be reflected in various statistical properties of the level spectrum . it has been known for some time now that as a classical system becomes non - integrable @xcite , the corresponding quantum system exhibits a level spacing distribution given by the ` wigner surmise ' derived in the context of random matrix theory ( rmt ) @xcite . in fact , this behavior has been verified in a number of theoretical and experimental systems , although typically the classical non - integrability is due to disorder or boundary ( geometrical ) effects . examples discussed in the literature include small disordered metallic particles @xcite , particles moving in a variety of ` stadia ' @xcite , and in two - dimensional antidot arrays @xcite . rmt also has been used to make definite predictions on the statistical distribution of coulomb blockade peak amplitudes @xcite . this behavior has in fact been shown recently to be a good description of quantum dot systems in beautiful experimental realizations @xcite . one should also mention that recent work on interacting systems , some without an obvious classical counterpart , has also shown that these exhibit the level repulsion characteristic of quantum versions of non - integrable classical systems @xcite . moreover , recent analysis of the level spectrum of excitons in quantum wells ( via photoluminescence excitation spectroscopy ) has shown evidence of level repulsion @xcite . a level structure described by rmt has been sought recently in the charging ( or addition ) spectrum of quantum dots @xcite . detailed analysis of experimental charging energies , after proper subtraction , would be expected to yield the single - particle like _ excitation _ spectrum described by rmt . unfortunately , it appears that the extraction of this excitation spectrum is obscured by the systematic shift in the charging energies , and the resulting level spacing is described by a normal distribution , rather than by the rmt functions . this would then suggest that a more direct measurement of the excitation spectrum ( via perhaps nonlinear tunneling currents ) would be desirable ( although clearly difficult experimentally beyond the first few excitations @xcite ) . one should also mention here that it is believed that the nonlinear transport experiments explore mainly the excitations of the center of mass of the system ( in the typically parabolic quantum dots ) @xcite , due to the strong electronic correlations suppressing most of the tunneling ` channels ' . this prevalence of the center of mass excitations is however expected theoretically to diminish as the energy of the excitation increases @xcite . this regime would be reached only as the bias voltage is raised in transport experiments , and makes it then difficult to achieve in practice . we hope , however , that the results presented here would motivate more experimental work in this direction . we have studied the effects of particle interaction on the classical integrability of a system of two masses moving inside a @xmath0-dimensional billiard . we find , in general , that the motion is strongly chaotic ( typically exhibiting ` soft ' chaos , with a mixed dynamics , where regions of phase space are still periodic or quasi - periodic ) , and with a strong dependence on the characteristic interaction length and strength . as perhaps one of the simplest examples ( see ref . @xcite for a description of the general case ) , consider two particles of equal masses moving in a 1d box defined in the interval @xmath1 ( we measure all lengths in terms of the box size ) . the particles are assumed to interact via a screened potential @xmath2 , where @xmath3 is the inverse screening length . notice that this potential goes to a hard - core @xmath4function when @xmath5 , and the particles behave then as non - interacting but impenetrable points . in that case , the dynamics can be integrable in special cases of the mass ratios @xcite . in this sense , @xmath3 plays the role of a perturbation parameter which changes the degree of integrability of the system , as it determines the effective ` radius ' of the particles for a given total energy . direct calculations show that this is indeed the case . the problem could in principle be solved by direct integration of the equations of motion , but we find convenient to transform it to a set of center - of - mass and relative coordinates @xmath6 , and @xmath7 , respectively , where the total mass @xmath8 , and the reduced mass @xmath9 . these equations define a two - dimensional space of coordinates @xmath10 . in this space , we have a new set of equations for the boundary of the billiard , say @xmath11 . the hamilton equations are transformed to @xmath12 , @xmath13 , and @xmath14 \nonumber \\ \dot { p } & = & \sum _ { j } { b}_j({p , p } ) \ , \delta[f_j({r , r } ) ] . \end{aligned}\ ] ] the functions @xmath15 and @xmath16 describe the change in the momenta @xmath17 and @xmath18 , due to the bounce on the @xmath19-th wall . notice that these equations describe the motion of a single ` hyperparticle ' in the two - dimensional @xmath20-space . this _ hyperbilliard _ description can be generalized to any number of dimensions @xcite . notice that bounces of the hyperparticle in the hyperbilliard correspond to bounces of the masses in the real / original dot . the walls of the billiard cause the breaking of translational symmetry of the system , and as a consequence , the center - of - mass ( cm ) momentum is no longer a constant of motion . in the case of non - interacting and equal - mass particles , the changes in the cm momentum are determined only by the geometry of the billiard . in our case , however , the interaction couples the cm and relative momenta after each bounce , which in turn depend on the momenta of each of the original masses . * dynamical map*. we should also mention that these equations in @xmath20-space provide an interesting description which is also extremely useful : in between bounces , the hyper - particle moves freely along the @xmath21-axis , whereas the interaction acts only along @xmath22 . these two motions are independent , and only become correlated at each bounce , when the different momentum components are changed , while keeping the total energy constant . understanding this fact allows one to describe the motion in terms of a _ dynamical map _ connecting the different bounces . if the coordinate of the hyperparticle is @xmath23 at the time of the @xmath24-th bounce , then the time spent until the next bounce on the @xmath19-th wall is obviously the same along _ both _ components , and one can then write @xmath25 . since the cm motion is that of a free particle , except for the collisions with the walls , @xmath26 can be calculated simply . similarly , for a pure coulomb potential ( @xmath27 ) , @xmath28 can be calculated analytically , so that the previous equation can be written in a more explicit form , @xmath29 , where @xmath30 is now the time elapsed going from @xmath31 to @xmath32 , expressed in terms of the time @xmath33 spent by the particle from the turning point to @xmath20 , @xmath34 . here @xmath35 , represents the energy left for the relative motion , as @xmath36 is the total energy of the two - mass system . for a weakly screened potential , @xmath37 , we can expand @xmath38 to first order and obtain a similar expression , where @xmath36 is scaled to @xmath39 . what follows , after these definitions is to characterize all the different possible trajectories in the triangular region in @xmath20-space . a simple algorithm can then be obtained to determine the poincar surfaces of section . this nontrivial ( and clearly nonlinear ) algebraic map provides then a full description of the dynamics . its use ( in lieu of the direct integration of the equations of motion ) simplifies calculations a great deal , and allows one to better characterize the system , as we describe below . * poincar sections*. in order to characterize the motion of the two interacting particles ( or the hyperparticle with its many degrees of freedom ) , we explore poincar sections of the resulting phase space . having a four - dimensional space in this case @xmath40 , we select to show those where one of the particles is at one end of the box ( notice that the energy is a constant of motion here ) . figure 1 shows a typical poincar section for @xmath41 and @xmath42 , obtained using the map described above ( and which is virtually identical to the one obtained directly from integration of the equations of motion @xcite ) . notice that chaotic trajectories nearly fill the available phase space ( for this given total energy ) . increasing values of @xmath3 give chaotic orbits that fill more of the available phase space . moreover , there are also a number of islands of stability , as expected from the kam theorem @xcite , near the fixed point corresponding to periodic symmetric motion in the non - interacting system ( and indicated in the figure with a cross on the right axis ) . in fact , for all values of @xmath36 and @xmath3 , the initial condition where @xmath43 , and @xmath44 gives rise to a periodic orbit . other islands also appear purely due to the interaction , as can be seen near @xmath45 , and are associated with a sort of correlated motion of the two masses . general values of @xmath36 and @xmath3 give rise to this type of mixed dynamics , with chaotic and regular trajectories sharing the available phase space . this increasing degree of ` soft chaos ' in the system ( as @xmath3 increases , for example ) , should be reflected in the level statistics of the corresponding quantum mechanical system , as we describe explicitly elsewhere in these proceedings . one naturally expects that this effect of interactions turning a regular system into a non - integrable one would be rather pervasive , regardless of the type of particle confinement and details of the interactions . in the following section we illustrate this effect in a somewhat different quantum mechanical model of a quantum dot . here , we will use a model of a quantum dot which has been very successful in the description of recent experiments in these structures @xcite . the quantum dot is modeled as a parabolic potential well with circular symmetry , and the frequency ( or curvature ) is chosen to fit characteristic single - particle excitation energies in these devices . in this case , the few - particle problem , which includes fully the effect of interactions , can be solved quite accurately ( numerically ` exactly ' ) even in the presence of magnetic fields . the approach is based on a canonical ( jacobi ) transformation to a set of auxiliary harmonic oscillator generating operators which allow one to write the interaction matrix elements in a closed analytical form , easily calculable @xcite . this , together with the ability to separate the center of mass ( cm ) degree of freedom from the ` relative ' ones ( a feature almost exclusive to the parabolic confinement potential ) , allows one to completely characterize and solve for the spectrum of up to three electrons in this parabolic well . correspondingly , the spectrum can be characterized by @xmath46 , where the first term gives the cm manifold , and @xmath47 is obtained from the relative - motion part of the hamiltonian . in the case of three particles , this latter part has diagonal elements given by @xmath48 , which are mixed by the coulomb interaction ( taken here as given by @xmath49 , with @xmath50 a background dielectric constant which would be assumed variable in the results section below ) . notice that in all these expressions , @xmath51 arise from the auxiliary harmonic oscillators introduced in the canonical transformation @xcite , with @xmath52 /2 $ ] . here , @xmath53 characterizes the single - particle harmonic confinement potential of the quantum dot ( with a typical value of @xmath54 12 mev ) , and @xmath55 is the cyclotron frequency of the electron . due to the rotational symmetry of the system , the coulomb interaction only couples states with the same relative angular momentum , given by @xmath56 . this is important , for apart from making the calculation simpler , it also allows us to carry out matrix diagonalizations with extremely high accuracy and for very many eigenstates ( typically one to two thousand levels ) . this complete convergence is obviously important if one is interested in the analysis of the level spacings , as we now proceed to do . * level statistics results*. the statistical analysis of the levels obtained as described above for three particles can be carried out according to the typical prescriptions in the literature . the level spectrum is used to extract a slowly - varying density of states with a smooth energy dependence , characteristic of the system at hand . this _ unfolding _ procedure then leaves one to study the structure of the level fluctuations of the spectrum , on which a number of statistical tests and statements can be made @xcite . the unfolding here is performed by defining the ` staircase ' function @xmath57 which gives the cumulative number of states below @xmath36 , and then fitting this to a smooth polynomial ( typically of fourth degree ) , @xmath58 . the ` linearized ' or unfolded level sequence is then obtained from @xmath59 , where @xmath60 is the original sequence . this process , although not unique , gives similar results to other unfolding procedures ( see @xcite for a good discussion ) . one can perform a number of statistical analyses . here we focus on the nearest - neighbor spacings ( nns ) , obtained from @xmath61 , and calculate the probability density function @xmath62 for a given sequence . it is useful to define the integrated probability function @xcite , @xmath63 . notice that @xmath64 is nothing but the total number of nns below a given @xmath65 , and can be uniquely calculated , without any dependence on the specific binning used to calculate the typical histogram representations of @xmath62 . in what follows , we use @xmath64 to make quantitative statements , but revert to showing the more conventional @xmath62 . figure 2 shows a sequence of @xmath57 staircase curves for different values of the background dielectric constant @xmath50 , defined above . the sequences analyzed have all the same value of relative angular momentum , @xmath66 , and total spin @xmath67 , and for a magnetic field value of @xmath68 t . increasing values of @xmath50 would produce a progressively weaker value of the coulomb interaction , and vice versa . we have varied @xmath50 such that the interaction varies by up to a factor of 50 . in the figure , the curve labeled 1 corresponds to the value of @xmath69 , found in gaas , where most quantum dots are defined . increasing the effective coulomb interaction ( with @xmath70 ) , produces the smoothest staircase function ( labeled 10 ) , completely devoid of the harmonic oscillator ` steps ' in weaker interactions ( as seen here for relative interaction strength 1/5 and 1/3 ) . one expects that for interaction 10 , the level mixing produced would be substantial , resulting in a level structure well described by the rmt distributions , as seen in other systems . we find that this is not the case here . in fact , if one analyzes the appropriate histograms for the nns distribution function @xmath62 , the interaction strength shows its effect quite clearly , as shown in fig . , we show the corresponding @xmath62 for three different interaction strength ( or @xmath50 ) values . the anticipated poisson distribution function one obtains for a ` generic ' integrable system @xcite is similar to the case for interaction 1/3 . however , notice that this @xmath18 here goes to zero even _ faster _ than the expected poisson form , reflecting the non - generic character of the delta - function like distribution of the pure harmonic oscillator system . moreover , as the interaction becomes stronger , notice that the distribution functions have a maximum not at zero @xmath65 , but rather at a finite value . this behavior is more in agreement with the rmt predictions , where @xmath18 would be expected to be given by a gue function ( given the finite magnetic field ) , where @xmath71 . one should notice , however , that in all cases we have studied , @xmath62 never fully reaches the anticipated gue for a fully chaotic system . the reason for this lack of full crossover into the gue distribution is perhaps associated with the mixed dynamics of the corresponding classical system . in such cases , it has been argued that the level distribution can be seen as a superposition of poisson and gue functions , with weights corresponding to the coverage in phase space for the integrable and non - integrable regions , respectively @xcite . at this point , however , we have not analyzed the classical system that directly models this quantum dot , and expect to report on this relation elsewhere . however , it may also be the case that this incomplete crossover to a gue arises from the peculiar non - generic non - poissonian distribution function of the integrable ( non - interacting ) system composed by overlapping harmonic oscillators , and/or remnant hidden symmetries . we have also fitted the @xmath72-functions to the well known brody distribution , @xmath73 , providing a ( phenomenological ) measure of the crossover ( @xmath74 for poisson , @xmath75 for goe ) . in fig . 2 and for unit interaction strength , we get @xmath76 , while @xmath77 for interaction 10 . furthermore , the non - typical character of the level distribution in the harmonic oscillator is also reflected in the magnetic field dependence @xcite . if one studies @xmath62 for a given interaction strength but increasing magnetic fields , the system has a non - monotonic evolution . examples of this behavior are shown in ref.@xcite . for low magnetic field ( @xmath78 t , and ` unit ' interaction , with @xmath79 ) , @xmath62 is not the harmonic oscillator delta - like function , but is somewhat shifted towards a gue ( brody - fit with @xmath80 ) . as the field increases , however , the @xmath18-distribution reaches a ` maximum ' crossover ( @xmath81 , for @xmath82 t ) , before going back towards a more delta - like function , as seen for @xmath83 t , which drops faster than a poissonian distribution . that this occurs is understandable , since for high magnetic fields one would expect to reach a regime where the interactions would be a weak perturbation ( for a given value of @xmath50 ) , and the spectrum would evolve towards a set of landau levels this non - monotonic behavior is seen in all the level sequences we have studied . we have shown that the interactions introduce classical non - integrability in a quantum - dot system , even if the geometry is integrable for the one - particle problem . moreover , this behavior , given the bohigas _ et al_. conjecture @xcite , would be expected to be reflected in the accompanying level structure of the corresponding quantum mechanical version of the system . we have found , that as we introduce interactions , indeed the nns fluctuation distribution function exhibits an apparent crossover towards one of the rmt functions , with a characteristic maximum away from zero spacing . this crossover , is found to be not complete , however , even for rather strong interaction , perhaps due to the mixed dynamics in the classical counterpart . the nns distribution exhibits non - monotonic magnetic field dependence , associated with the fact that in that regime the particle interactions are but a weak perturbation of the landau level spectrum . a number of theoretical questions still remain , however , as more direct comparison of the classical and quantum systems is needed , perhaps including a study of the associated wave functions and possible appearance of ` scars ' @xcite . nevertheless , the importance of interactions in the details of level structure and associated experiments , as discussed in the introduction , can not be ignored . we hope that this , as well as semiclassical treatments ( see richter s article in these proceedings ) , motivate further understanding of this fascinating problem . gorkov and g.m . eliashberg , sov . jetp * 21 * , 940 ( 1965 ) ; k.b . efetov , adv . phys . * 32 * , 53 ( 1983 ) ; b.l . altshuler and b.z . spivak , sov . jetp * 65 * , 343 ( 1987 ) ; y.h . zheng and r.a . serota , phys . b * 50 * , 2492 ( 1994 ) .
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the role of electronic interactions in the level structure of semiconductor quantum dots is analyzed in terms of the correspondence to the integrability of a classical system that models these structures .
we find that an otherwise simple system is made strongly non - integrable in the classical regime by the introduction of particle interactions .
in particular we present a two - particle classical system contained in a @xmath0-dimensional billiard with hard walls .
similarly , a corresponding two - dimensional quantum dot problem with three particles is shown to have interesting spectral properties as function of the interaction strength and applied magnetic fields .
+ keywords : quantum dots , chaos , level statistics + sergio e. ulloa + e - mail : [email protected] + fax : ( 614 ) 5930433
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You are an expert at summarizing long articles. Proceed to summarize the following text:
the magnetosphere of an accreting x - ray pulsar expands as the mass accretion rate decreases . as it grows beyond the co - rotation radius , centrifugal force prevents material from entering it . thus , accretion onto the magnetic poles ceases , and , consequently , x ray pulsations cease . this phenomenon has recently been observed , for the first time , in gx 1 + 4 and gro j1744 - 28 with rxte@xcite . here , we present further evidence to show that the phenomenon repeated itself for gro j1744 - 28 during the decaying phase of its latest x - ray outburst . the asm light curve ( as shown in the top panel of fig . 1 ) reveals that there have been two episodes of x - ray outburst in gro j1744 - 28 , separated by roughly one year . the source has been extensively monitored by the main instruments aboard rxte since its discovery@xcite . for detailed analyses , we have selected a number of pca observations , based on the asm light curve , to cover the decay phase of the outbursts . 1 ( bottom panel ) shows the pulsed fraction ( @xmath0 ) measured with each observation . for comparison , the published results@xcite for the first outburst are also presented here . a striking feature is the precipitous drop of the pulsed fraction as the source became `` quiescent '' both times . gro j1744 - 28 was generally not so quiet after the first outburst . in previous work@xcite , we happened to catch a brief period ( as indicated in fig . 1 ) when the pulsed emission became very weak or was not detected at all in some observations . following the latest ourburst , the source has shown little activity . its presence ( at about 20 - 30 mcrab ) has , however , been firmly established by the pca slew data . this provides a good opportunity to verify our previous interpretation of the phenomenon . we have searched for the known 2.14 hz pulse frequency , employing various techniques including ffts and epoch - folding , but have failed to detect it since the end of june 1997 ( as marked in fig . 1 ) . the results therefore argue strongly that the centrifugal barrier is active in this source during such faint period , as we have concluded previously@xcite . the source also shows interesting spectral evolution during the decay . the observed x - ray spectrum can be characterized by a simple power law with an exponential high - energy cutoff . as the quiescent state is approached , the spectrum softens significantly : the power - law becomes steeper , and more prominently , the cutoff energy decreases by roughly a factor of 2 ( see fig . 2 ) . at the end of the first `` quiescent '' period , the spectrum would recover to the bright - state shape . we have proposed before that the x - ray emission probably consists of two components : the emission from a large portion of the neutron star surface ( thus unpulsed ) , due to the `` leakage between field lines '' @xcite , and that from `` hot spots '' near the poles ( pulsed plus unpulsed ) . when the source was bright , the latter dominated , so the spectrum was hard ( corresponding to a much higher temperature of the hot spots ) . however , as soon as the centrifugal barrier took effect in the quiescent state , the observed x - rays were all due to the surface emission and their spectrum was therefore softer . it is interesting to note that the pileup of accreting matter on the neutron star surface might also cause unstable thermonuclear burning and produce type i bursts@xcite , like in x - ray bursters . the lack of such ( or does it ? ) in gro j1744 - 28 may be due to the suppression of this process by a significantly higher field@xcite . gro j1744 - 28 does produce x - ray bursts@xcite , unlike any other x - ray pulsars . the bursts are thought to be the product of accretion instability@xcite . they occurred at a rate of one to two dozen per hour near the peak of the outbursts@xcite , and the rate decreased as the x - ray flux decayed . at the start of the first quiescent period , the bursting activity ceased entirely@xcite for weeks before resuming again near the end@xcite . 3 ( the top panel ) shows an example of such activity ( with 7 major bursts ) on mjd 50260 ( @xmath1 26 june 1996 ) . we have separated the light curve of 26 june 1996 into burst and non - burst intervals . the x - ray pulsation is detected during the bursts but is _ not _ detected outside of them ( see fig . 3 ) . this is again consistent with the presence of the centrifugal barrier in gro j1744 - 28 . a sudden surge in the mass accretion rate that produces a burst would also momentarily push the magnetosphere inside the co - rotation radius and thus , the accretion to the poles would resume to produce the pulsed emission . as the system relaxes following a burst , the magnetosphere expands again ; the inhibition of accretion by the centrifugal barrier again suppresses the pulsation . we conclude by summarizing the main results as follows : * the results support our previous conclusion that the cessation of pulsed emission when the source becomes faint is a manifestation of the centrifugal barrier . * for gro j1744 - 28 , the x - ray emission in the quiescent state ( unpulsed ) likely comes from a large portion of the neutron star surface , due to the penetration of accretion flows through the magnetosphere . * accretion instability can still occur in the quiescent state ( less frequently ) , and produce type ii bursts . the pulsed emission was apparent during the bursts , presumably due to the resumption of accretion to the magnetic poles because of the momentary shrinkage of the magnetosphere . the pulsation stopped as the system recovered to the quiescent state .
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we present further observational evidence of the effects of a centrifugal barrier in gro j1744 - 28 , based on continued monitoring of the source with rxte .
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You are an expert at summarizing long articles. Proceed to summarize the following text:
cold gases of rydberg atoms are currently receiving a growing attention in the communities of quantum optics @xcite , quantum information @xcite , and many - body physics @xcite . this is rooted in the fact that they offer strong and long - ranged interactions and at the same time grant long coherent lifetimes . currently , considerable efforts are devoted to developing all - optical quantum information protocols @xcite with the rydberg - atom - mediated interaction between individual photons @xcite . fundamentally important optical devices that operate on the single photon level , such as phase shifters @xcite , switches @xcite and transistors @xcite , have been demonstrated experimentally in rydberg gases . single photon switchs might form a central building block of an all - optical quantum information processor @xcite . the prime function of such switches is to control the transmission of an incoming photon through a single gate photon . one promising way to realize this is to store the gate photon in form of a gate ( rydberg ) atom immersed in an atomic gas which is in a delocalized spinwave state @xcite . the gate atom then prevents transmission of incident photons through the gas , while ideally the coherence of the rydberg spinwave state is preserved @xcite . the latter property would permit the subsequent coherent conversion of the rydberg spinwave into a photon which would pave the way for gating the switch with superposition states that can also be subsequently retrieved . currently , there is only a basic understanding of how the coherence of the rydberg spinwave might be affected by the scattering of incoming photons and no systematic study of this important question exists . , excited state @xmath0 ( decay rate @xmath1 ) and rydberg state @xmath2 are resonantly coupled by a single photon field @xmath3 ( with collective coupling strength @xmath4 ) and a classical field of rabi frequency @xmath5 . initially a gate photon is stored as a spinwave in the rydberg state @xmath6 ( indicated by the green circle ) . ( b , c ) polarization profiles @xmath7 for a spinwave consisting of two possible gate atom positions @xmath8 ( @xmath9 ) and their dependence on the blockade radius @xmath10 and the system length @xmath11 . ( b ) for @xmath12 and @xmath13 the polarization profiles associated with the two gate atom positions are distinguishable . ( c ) when @xmath14 the polarization profile is independent of the gate atom position which leads to enhanced coherence of the stored spinwave . ] in this work we address this outstanding issue within a simple model system . we study the propagation of a single photon under conditions of electromagnetically induced transparency ( eit ) in a cold atomic gas in which a gate photon is stored as a rydberg spinwave . an incident photon subsequently experiences a rydberg mediated van der waals ( vdw ) interaction with this stored gate atom which lifts the eit condition and renders the atomic medium opaque . in this case the incident photon is scattered incoherently off the rydberg spinwave . we study the photon propagation and explore the dependence of rydberg spinwave coherence on the interaction strength ( parameterized by the blockade radius @xmath15 ) , the system length @xmath11 and bandwidth of the incident photon pulse . our findings confirm that strong absorption , i.e. high gain , can be achieved already for large systems ( @xmath12 ) while coherence of the spinwave is preserved only for sufficiently strong interactions , i.e. @xmath14 . intuitively , this can be understood by regarding the scattering of the incoming photon as a measurement of the position of the gate atom . when @xmath14 this measurement is not able to resolve the position of the excitation and hence coherence of the rydberg spinwave is maintained . our study goes beyond this simple consideration by taking into account propagation effects , a realistic interaction potential and a finite photon band width . the results can therefore be considered as upper bounds for the fidelity with which a rydberg spinwave can be preserved and re - converted into a photon in an experimental realization of a coherent cold atom photon switch . the paper is organized as follows . in section ii , we introduce a one - dimensional model system to study the propagation dynamics of single source photons in the atomic gas prepared in a rydberg spinwave state . in sec . iii , the model system is solved numerically with realistic parameters . we identify the working regime for a single photon switch where the source photon is scattered completely . in sec . iv , we numerically study the fidelity between the initial spinwave state and the final state after the source photon is scattered . our calculation shows that the coherence of the spinwave is preserved when @xmath16 while the final state becomes a mixed state when @xmath17 . in sec . v , we provide analytical results for a coherent single photon switch ( @xmath18 ) . we reveal that the transmission and switch fidelity depend nontrvially on the optical depth and bandwidth of the source photon field . we summarize in sec . vi . our model system is a one - dimensional , homogeneous gas consisting of @xmath19 atoms , whose electronic levels are given in fig . [ fig : illustration]a . the photon field @xmath20 and the eit control laser ( rabi frequency @xmath5 ) resonantly couple the groundstate @xmath21 with the excited state @xmath0 and @xmath0 with the rydberg state @xmath2 . following ref . @xcite , we use polarization operators @xmath22 and @xmath23 to describe the slowly varying and continuum coherence of the atomic medium @xmath24 and @xmath25 , respectively . all the operators @xmath26 are bosons and satisfy the equal time commutation relation , @xmath27=\delta(z - z')$ ] . initially , the atoms are prepared in a delocalized spinwave state with a single gate atom in the rydberg state @xmath6 , @xmath28 is the wavenumber of the spinwave and @xmath29 abbreviates many - body basis with the gated atom located at position @xmath30 and the rest in the groundstate . the rydberg spinwave state is created routinely in experiments @xcite . when interacting with the incoming single photon , the general many - body state of this one - dimensional system is expanded as @xcite @xmath31|\psi_n(0)\rangle , \label{eq : state}\end{aligned}\ ] ] where @xmath32 is probability amplitude of the initial spinwave state . in the weak field approximation , we will assume @xmath33 at any moment . we have defined @xmath34 , i.e. the expectation value of the operator @xmath35 . specifically one finds that @xmath36 is the probability amplitude in the one photon state , @xmath37 and @xmath38 are the amplitude of one atom in the @xmath0 and @xmath2 state , respectively . in order to develop a first intuition for the physics at work we first consider a spinwave that is delocalized merely over two atoms embedded in the atomic cloud ( see fig . [ fig : illustration]b , c ) . we assume furthermore that the interaction between atoms in state @xmath2 and the gate atom is infinite for distances smaller than the so - called blockade radius @xmath10 and zero otherwise . outside the blockade region , the photon propagates ( along the @xmath39 direction ) as a dark - state polariton by virtue of eit @xcite . inside the blockade region the medium behaves like an ensemble of two - level system . here the incoming photon is building up a non - zero polarization @xmath37 , whose modulus square is the probability density distribution for finding an atom in the decaying state @xmath0 according to eq . ( [ eq : state ] ) @xcite . eventually , this leads to the loss of the incoming photon and makes the medium opaque . in order to understand how such photon scattering affects the coherence of the properties of the spinwave one needs to analyze the shape of the polarization profile . as shown in fig . [ fig : illustration]b this in general depends on the position of the gate atom when the system length is larger than the blockade radius @xmath40 . here , since @xmath41 and @xmath13 , it is possible to distinguish the profiles @xmath7 which are associated with the two possible positions of the gate atom . conversely , the polarization @xmath7 becomes independent of the gate atom position when @xmath14 ( see fig . [ fig : illustration]c ) . in this case as discussed in detail later the coherence of the spinwave will be preserved as one can not distinguish gate atoms from the scattered photon . let us now consider the actual photon propagation together with a realistic interaction potential . the dynamics of the system follows the master equation @xcite @xmath42+\gamma\int_0^l dz\hat{p}(z , t)\hat{\rho}(t)\hat{p}^{\dagger}(z , t),\end{aligned}\ ] ] where the first term on the right - hand side ( rhs ) is the evolution of @xmath43 under the effective hamiltonian @xmath44 , and the spontaneous decay ( with rate @xmath1 ) from the state @xmath0 is governed by the second term . in the effective hamiltonian , the photon propagation in the medium is governed by the hamiltonian @xmath45 with the vacuum light speed @xmath46 . the atom - photon coupling is described by @xmath47 , \end{aligned}\ ] ] where @xmath48 with @xmath49 being the single atom - photon coupling strength . the vdw interaction between an atom in the state @xmath2 and the gate atom at position @xmath30 is @xmath50 the interaction potential depends on the gate atom position , @xmath51 where @xmath52 gives the vdw interaction with @xmath53 being the dispersion coefficient . for the case of a single incoming photon which we consider here the solution of the master equation ( [ eq : masterequation ] ) is @xcite @xmath54 where @xmath55 and @xmath56 . the first term on the rhs describes the unhindered photon propagation through the medium , while the second term accounts for the photon scattering , i.e. photon - loss from the medium . to calculate ( [ eq : densityevolution ] ) we first treat the dynamics under the effective hamiltonian in the heisenberg picture . to this end we obtain the equation of motion for the expectation values @xmath57 from the corresponding operator heisenberg equation @xcite . note , that due to the linearity of the equations we can moreover calculate the expectation value for each component @xmath58 of the rydberg spinwave , i.e. each of the possible positions of the gate atom , separately . this yields the set of equations [ eq : me ] @xmath59 where the index @xmath60 labels the respective spinwave component . alternatively , these equations can be obtained from a heisenberg - langevin approach @xcite . we solve the coupled equations ( [ eq : me ] ) through a fourier transform yielding the formal solution for the polarization @xmath61 here we have abbreviated @xmath62 and introduced the electric susceptibility @xmath63(\omega+i\gamma/2)}.\end{aligned}\ ] ] from @xmath64 one can actually extract the blockade radius as the critical distance at which the vdw interaction and the control laser are equally strong . this yields @xmath65 @xcite . the polarization ( [ eq : pfft ] ) depends on the fourier transform @xmath66 of the photon field at position @xmath67 . to be specific we take the photon pulse to be a gaussian at @xmath68 which is normalized in space , @xmath69.\ ] ] here @xmath70 is the temporal duration of the pulse and @xmath71 is the initial central position ( @xmath72 ) . the band width of the pulse is then given by @xmath73 . note , that it is generally not possible to evaluate the formal solution ( [ eq : pfft ] ) analytically . moreover , numerical calculations are challenging since the involved time and length scales span several orders of magnitude @xcite . let us now calculate the photon transmission as a function of the pulse duration @xmath70 , which to our knowledge has not been examined previously . we define the transmission of the photon pulse as @xmath74 . in fig . [ fig : transmission]a , we show @xmath75 as a function of the pulse width for two values of the atom - photon coupling strength @xmath4 . for fixed pulse length @xmath70 , we find that stronger couplings generally are accompanied by a lower transmission . furthermore , we observe that the transmission increases with decreasing pulse duration @xmath70 . this is due to the fact that the pulse contains increasingly more weight on frequency components , which are outside the absorption window of the medium . for the purpose of complete photon scattering , one thus has to utilize narrow frequency band pulses . as a function of the pulse duration @xmath70 for @xmath76 ( squares ) and @xmath77 ( circles ) . the medium becomes transparent when @xmath78 , i.e. the band width @xmath79 of the pulse is large . ( b ) photon transmission as a function of the coupling constant @xmath4 for @xmath80 ( squares ) and @xmath81 ( circles ) . the solid curve is the analytical result obtained from eq . ( [ eq : transmission ] ) . the dashed curve is plotted as a guide to the eye . note , that @xmath82 when @xmath83 for both @xmath80 and @xmath81 . the data is calculated for rubidium atoms with the parameters , @xmath84 m , @xmath85 , and @xmath86 mhz . the blockade radius can be changed through selecting different rydberg states . for example , @xmath87 m when @xmath88 and @xmath89 , where @xmath90 . ] next , we briefly discuss the dependence of the transmission @xmath75 on the strength of the atom - photon coupling @xmath4 . [ fig : transmission]b shows data for two choices of the blockade radius , @xmath80 and @xmath81 . as expected , @xmath75 decreases with increasing @xmath4 . however , for the system parameters chosen here , there is virtually no dependence of @xmath75 on the value of the blockade radius when @xmath76 , where @xmath82 . these findings indicate that one reaches the strong scattering regime when @xmath91 and @xmath92 . this is the working regime for the single photon switch where the medium becomes opaque for the incident photon . focusing on this regime , our next task is to investigate how the photon scattering influences the rydberg spinwave . we quantify the difference between the initial rydberg spinwave @xmath93 and the final state @xmath94 by the fidelity @xcite @xmath95 ^ 2.\ ] ] as the initial spinwave is a pure state , this simplifies to @xmath96 , where @xmath97 . this shows that a high fidelity can be obtained only if the polarization profiles @xmath98 for each spinwave component are essentially equal : only when @xmath99 and thus @xmath100 the fidelity is close to one . this is the formal version of the intuitive statement that we made earlier in conjunction with the discussion of fig . [ fig : illustration]b , c . for completeness we provide a numerical example for which we choose @xmath81 and select only two components of the spinwave , where the gate atom is located at either @xmath101 or @xmath102 . the resulting polarization profile @xmath103 is shown in fig . [ fig : psi2]a , b . for @xmath101 , non - vanishing polarization is built up within the blockade region as long as the photon is inside the medium ( fig . [ fig : psi2]a ) . integrating over time we obtain the intensity @xmath104 which clearly shows a decay to zero within a blockade distance @xmath10 ( see fig . [ fig : psi2]c ) . in contrast , for @xmath102 , appreciable polarization is built up also outside @xmath10 and the profile is peaked at approximately @xmath105 ( fig . [ fig : psi2]b , d ) . clearly , both polarization profiles are strikingly different which in turn causes a loss of fidelity when the blockade radius is smaller than the system length . we verify this by numerically calculating the fidelity as a function of the blockade radius . the data is displayed in fig . [ fig : fidelity]a , together with the corresponding transmission @xmath75 . as anticipated , the fidelity decreases significantly below unity when @xmath15 is decreased with respect to @xmath11 . note , that the transmission is close to zero throughout . and ( c , d ) the time - integrated intensity @xmath106 for @xmath81 and @xmath76 for two different positions of the gate atom . the gate atom ( green circle ) is located at @xmath101 in panels ( a ) and ( b ) and at @xmath102 in panels ( c ) and ( d ) . the dashed line marks the blockade radius with respect to the gate atom position . ] a fidelity smaller than unity directly indicates the formation of a mixed state after the photon scattering . the final state density matrix is @xmath107 } |z_j\rangle\langle z_k|.\ ] ] the final state can only be pure when @xmath108 and hence @xmath109 . the formation of a mixed state is a consequence of the actual measurement of the gate atom position @xcite which is performed by the photon scattering : when @xmath40 one in principle gains information on the position of the gate atom since the spatial uncertainty of its wave function is reduced from @xmath11 to the blockade region . the final state is then a mixture of all states compatible with this additional information . in the remainder of the paper we will focus on the case of a coherent photon switch , i.e. @xmath110 . here the expression for the susceptibility of the medium simplifies to that of an ensemble of two - level atoms , @xmath111 which permits the derivation of analytical results . for a narrow band width pulse we can derive explicit solutions to eq . ( [ eq : me ] ) that have no dependence on the position of the gate atom @xcite . for example , the polarization @xmath37 is given by @xmath112\nonumber\\ & & \times \text{ec}\left[\frac{c\gamma^3\tau^2 - 2\gamma^2(c\mathcal{t}+z_0)-8g^2z}{2\sqrt{2}c\tau\gamma^2 } \right],\end{aligned}\ ] ] where @xmath113 is the complementary error function . the corresponding time - integrated profile @xmath106 agrees perfectly with the numerical result from eq . ( [ eq : me ] ) ( see fig . [ fig : fidelity]b ) . the transmission @xmath75 is given by @xmath114^{-1}\left[1+\text{er}\left(\frac{l - z_0}{c\tau}-\frac{\alpha}{\gamma\tau}\right)\right],\ ] ] where @xmath115 is the error function and @xmath116 is the optical depth of a resonant two - level medium . the excellent agreement between the analytical and numerical calculation is shown in fig . [ fig : transmission]b . neglecting the finite band width of the photon pulse , i.e. when all the frequency components are in the absorption window , eq . ( [ eq : transmission ] ) reduces to the well - known form @xmath117 @xcite . with @xmath76 the transmission is already negligible when @xmath118 , but the fidelity approaches unity only when @xmath119 . ( b ) intensity @xmath106 in the strong blockade regime ( @xmath80 ) for @xmath76 ( square ) and @xmath120 ( circle ) . the squares and circles are numerical data . the solid and dotted curves are the analytical results obtained from eq . ( [ eq : polarization_response ] ) . ] finally , the fidelity can be expressed as a function of the optical depth and pulse band width @xmath121.\ ] ] this shows that indeed a small band width is a requirement for reaching a large fidelity . for example , the transmission is negligible ( @xmath122 ) when @xmath123 and @xmath76 according to the data in fig . [ fig : transmission]a . however , the fidelity is below unity ( @xmath124 ) due to non - negligible contributions from the terms accounting for the finite band width . in the limit of very long pulses one finds @xmath125 and thus the fidelity is solely determined by the transmission . in summary , we have studied the coherence of a rydberg spinwave in the operation of a signle photon switch . the current study is limited to a single gate atom and an incoming single - photon pulse , which permits the description of multi - photon scattering , however , only if the photons enter the switch sequentially . addressing this limitation and extending the discussion to correlated and entangled photon pulses that fall in the operation regime of single photon transistors will be subject to future studies . _ acknowledgements. _ we acknowledge helpful discussions with d. viscor , b. olmos and m. marcuzzi . the research leading to these results has received funding from the european research council under the european union s seventh framework programme ( fp/2007 - 2013 ) / erc grant agreement no . 335266 ( escquma ) , the eu - fet grants no . 295293 ( quilmi ) and no . 512862 ( hairs ) , as well as the h2020-fetproact-2014 grant no . 640378 ( rysq ) . wl is supported through the nottingham research fellowship by the university of nottingham . here we will show how to obtain the analytical solution to the coupled equations ( 3 ) in the main text in the strong blockade regime ( @xmath126 ) and for narrow band pulses . in the frequency domain , the solution to equations ( 3 ) is given by , @xmath127\tilde{\mathcal{e}}_0(\omega),\\ \label{eq : p } \tilde{p}(z,\omega)&=&\chi\tilde{\mathcal{e}}(z,\omega),\\ \label{eq : s } \tilde{s}(z,\omega)&=&-\frac{\omega}{\omega - v(z)}\chi\tilde{\mathcal{e}}(z,\omega).\end{aligned}\ ] ] due to the strong blockade condition , we have removed the dependence of @xmath128 , @xmath129 , and @xmath130 on the gate atom index @xmath60 and replaced the susceptibility by the one corresponding to two - level atoms , @xmath131 . moreover , we set @xmath132 , which is a good approximation as @xmath133\approx \omega / v(z)\approx 0 $ ] in eq . ( [ eq : s ] ) . our aim is to obtain analytical expressions of @xmath3 and @xmath37 . applying the inverse fourier transform on the both sides of eqns . ( [ eq : e ] ) and ( [ eq : p ] ) , we obtain the formal solution for @xmath3 and @xmath37 in time domain , @xmath134 the integration is in general difficult to carry out analytically due to the complicated form of the susceptibility . we overcome this difficulty by expanding the susceptibility in powers of @xmath135 , @xmath136.\ ] ] first let us calculate the approximate solution for @xmath3 . to carry out analytical calculations and at the same time take into account contributions due to the finite band width , we will keep terms up to the second order of @xmath135 in eq . ( [ eq : s_expansion ] ) . this yields the solution for @xmath3 @xmath137,\end{aligned}\ ] ] with @xmath138 . for the current problem , we always have @xmath139 as the photon travelling time through the medium is the shortest time scale . for example , @xmath140 second for @xmath141 m . with the solution for @xmath3 , we can calculate the transmission @xmath74 . we need to carry out the respective two integrals over time at @xmath67 and @xmath142 . this can be done analytically , @xmath143 and @xmath144\approx \frac{e^{-\alpha}}{2c\xi(l)}\left[1+\text{er}\left(-\frac{z_0}{c\tau\xi(l)}-\frac{\alpha}{\gamma\tau\xi(l)}\right)\right].\ ] ] this leads to the analytical form of the transmission ( 7 ) in the main text . with the analytical solution for @xmath3 at hand , there are two ways to calculate @xmath37 . we can directly calculate @xmath37 from eq . ( 3b ) by inserting the solution ( [ eq : efield ] ) and @xmath145 . this yields the linear response of the medium to the photon electric field , @xmath146 the integration over time can be carried out analytically , which gives @xmath147 \left\{1+\text{ec}\left[\frac{24g^2z+2\gamma^2(c\mathcal{t}+z_0)-c\gamma^3\tau^2}{2\sqrt{2\xi}c\tau\gamma^2 } \right]\right\}.\\ p(z , t ) & \approx & ig\sqrt{\frac{\sqrt{\pi}\tau}{2c}}\exp\left[\frac{c\gamma^2\tau^2 - 4\gamma(c\mathcal{t}+ z_0)}{8c}-\frac{6g^2z}{c\gamma}\right ] \left[1+\text{er}\left(\frac{1 - 3\xi^2(z)}{4\sqrt{2}\xi(z)}\gamma\tau + \frac{c\mathcal{t}+z_0}{\sqrt{2}\xi(z)c\tau } \right)\right].\end{aligned}\ ] ] however it is difficult to calculate the fidelity from eq . ( [ eq : p_response ] ) due to the presence of the error function . we thus calculate @xmath37 alternatively using the fourier transform method . we note that the susceptibility @xmath148 appears at two places in eq . ( [ eq : psi2ifft ] ) : one in front of @xmath149 and another one in the exponential function . in order to obtain an analytical result , we will expand the former susceptibility up to the second order of @xmath135 while the latter up to the linear order . after performing the inverse fourier transform , we obtain the expression for @xmath37 , @xmath150 ^ 2\right\}.\end{aligned}\ ] ] 43 ifxundefined [ 1 ] ifx#1 ifnum [ 1 ] # 1firstoftwo secondoftwo ifx [ 1 ] # 1firstoftwo secondoftwo `` `` # 1 '' '' @noop [ 0]secondoftwo sanitize@url [ 0 ] + 12$12 & 12#1212_12%12 @startlink[1 ] @endlink[0 ] @bib@innerbibempty in @noop _ _ , vol . ( , ) p. link:\doibase 10.1038/nature11361 [ * * , ( ) ] link:\doibase 10.1103/physrevlett.110.153601 [ * * , ( ) ] link:\doibase 10.1103/revmodphys.82.2313 [ * * , ( ) ] link:\doibase 10.1103/physrevlett.101.250601 [ * * , ( ) ] link:\doibase 10.1103/physreva.80.033418 [ * * , ( ) ] link:\doibase 10.1038/nphys1614 [ * * , ( ) ] link:\doibase 10.1103/physrevlett.104.043002 [ * * , ( ) ] link:\doibase 10.1103/physrevlett.106.025301 [ * * , ( ) ] link:\doibase 10.1103/physrevlett.107.060406 [ * * , ( ) ] link:\doibase 10.1103/physreva.86.033422 [ * * , ( ) ] link:\doibase 10.1103/physreva.87.053414 [ * * , ( ) ] link:\doibase 10.1103/physreva.88.053627 [ * * , ( ) ] link:\doibase 10.1038/35051009 [ * * , ( ) ] link:\doibase 10.1103/physrevlett.112.040501 [ * * , ( ) ] link:\doibase 10.1103/physrevlett.100.170501 [ * * , ( ) ] link:\doibase 10.1103/physrevlett.107.133602 [ * * , ( ) ] link:\doibase 10.1103/physrevlett.112.133606 [ * * , ( ) ] link:\doibase 10.1038/nature12512 [ * * , ( ) ] link:\doibase 10.1103/physrevlett.112.073901 [ * * , ( ) ] link:\doibase 10.1103/physrevlett.113.053602 [ * * , ( ) ] link:\doibase 10.1103/physrevlett.113.053601 [ * * , ( ) ] @noop * * , ( ) link:\doibase 10.1038/nphoton.2010.94 [ * * , ( ) ] link:\doibase 10.1038/nphoton.2012.181 [ * * , ( ) ] link:\doibase 10.1103/physrevlett.109.133602 [ * * , ( ) ] link:\doibase 10.1103/physrevlett.113.083601 [ * * , ( ) ] link:\doibase 10.1103/physrevlett.114.043602 [ * * , ( ) ] link:\doibase 10.1103/physreva.66.023818 [ * * , ( ) ] link:\doibase 10.1103/physreva.78.053816 [ * * , ( ) ] link:\doibase 10.1103/physreva.85.033811 [ * * , ( ) ] link:\doibase 10.1103/physreva.87.023821 [ * * , ( ) ] link:\doibase 10.1103/physrevlett.112.243601 [ * * , ( ) ] link:\doibase 10.1103/physreva.65.022314 [ * * , ( ) ] link:\doibase 10.1126/science.1217901 [ * * , ( ) ] link:\doibase 10.1038/nature12227 [ * * , ( ) ] @noop link:\doibase 10.1103/revmodphys.70.101 [ * * , ( ) ] link:\doibase 10.1103/physreva.76.033805 [ * * , ( ) ] @noop link:\doibase 10.1016/0034 - 4877(76)90060 - 4 [ * * , ( ) ] @noop _ _ ( , ) @noop
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we study coherence in a cold atom single photon switch where the gate photon is stored in a rydberg spinwave . with a combined field
theoretical and quantum jump approach and by employing a simple model description we investigate systematically how the coherence of the rydberg spinwave is affected by scattering of incoming photons . with large - scale numerical calculations
we show how coherence becomes increasingly protected with growing interatomic interaction strength . for the strongly interacting limit
we derive analytical expressions for the spinwave fidelity as a function of the optical depth and bandwidth of the incoming photon .
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You are an expert at summarizing long articles. Proceed to summarize the following text:
magnetic reconnection is the physical process of breaking and rearrangement of magnetic field lines , which changes the topology of the field . it is one of the most fundamental processes of plasma physics and is believed to be at the core of many dynamic phenomena in laboratory experiments and in cosmic space . unfortunately , in spite of being so important , magnetic reconnection is still relatively poorly understood from the theoretical point of view . the reason is that plasmas usually have very high temperatures and low densities . in such plasmas , the spitzer resistivity is extremely small and magnetic fields are almost perfectly frozen into cosmic plasmas . as a result , simple theoretical models , such as the sweet - parker reconnection model @xcite predict that the magnetic reconnection processes should be extremely slow and insignificant throughout the universe . on the other hand , astrophysical observations show magnetic reconnection tends to be fast and is likely to be the primary driver of many highly energetic cosmic processes , such as solar flares and geomagnetic storms . this contradiction between theoretical estimates and astrophysical observations triggered multiple attempts to build a theoretical model of fast magnetic reconnection . first , in 1964 petschek proposed a fast reconnection model @xcite , in which fast reconnection is achieved by introducing switch - off magnetohydrodynamic ( mhd ) shocks attached to the ends of the reconnection layer in the downstream regions and by choosing the reconnection layer length to be equal to its minimal possible value under the condition of no significant disruption to the plasma flow . however , later numerical simulations and theoretical derivations did not confirm the petschek theoretical picture for the geometry of a reconnection layer @xcite . second , numerical simulations studies of anomalous magnetic reconnection , for which resistivity is enhanced locally in the reconnection layer , were pioneered by ugai and tsuda @xcite , by hayashi and sato @xcite , and by scholer @xcite . third , lazarian and vishniac proposed that fast reconnection can occur in turbulent plasmas @xcite , although the back - reaction of magnetic fields can slow down reconnection in this case @xcite . finally , recently there have been number of attempts to explain the fast magnetic reconnection by considering non - mhd effects @xcite . most previous theoretical and numerical studies concentrated on reconnection processes in two - dimensions or in `` two - and - a - half - dimensions '' . the later is the term used for a problem in which physical scalars and all three components of physical vectors depend only on two spatial coordinates ( e.g. @xmath0 and @xmath1 ) and are independent of the third coordinate ( @xmath2 ) . in this paper we consider two - and - a - half - dimensional magnetic reconnection with anomalous resistivity in the classical sweet - parker - petschek reconnection layer , which is shown in the left plot in fig . [ fig_reconnection_layer ] . the reconnection layer is in the x - y plane with the y - axis being along the layer and the x - axis being perpendicular to the layer . the length of the layer is equal to @xmath3 . note that @xmath4 is approximately equal to or smaller than the global magnetic field scale , which we denote as @xmath5 . the thickness of the classical reconnection layer , @xmath6 , is much smaller than its length , i.e. @xmath7 . the classical sweet - parker - petschek reconnection layer is assumed to possess a point symmetry with respect to its geometric center point o and reflection symmetries with respect to the axes @xmath0 and @xmath1 ( refer to fig . [ fig_reconnection_layer ] ) . thus , for example , the x- and y - components of the plasma velocities @xmath8 and of the magnetic field @xmath9 have the following simple symmetries : @xmath10 , @xmath11 , @xmath12 and @xmath13 . there might be a pair of petschek shocks attached to each of the two reconnection layer ends in the downstream regions ( see fig . [ fig_reconnection_layer ] ) . because of the mhd jump conditions on the petschek shocks , the presence of these shocks requires the presence of a significant perpendicular magnetic field @xmath14 at the reconnection layer ends @xcite . the plasma outflow velocity from the reconnection layer is approximately equal to the alfven velocity @xmath15 ( if the plasma viscosity is not very large ) . the plasma inflow velocity @xmath16 outside of the reconnection layer , at point m in fig . [ fig_reconnection_layer ] , is much smaller than the outflow velocity , @xmath17 . finally , the magnetic field outside the reconnection layer is mostly in the direction of the layer ( i.e. in the y - axis direction ) . the problem of quasi - stationary anomalous magnetic reconnection in the classical sweet - parker - petschek reconnection layer was recently theoretically addressed by kulsrud @xcite for the special case of zero guide field ( @xmath18 ) , zero plasma viscosity and anomalous resistivity that is a piecewise linear function of the electric current [ see eq . ( [ linear_resistivity ] ) ] . there are two major results of the kulsrud anomalous reconnection model . first , in the case of the petschek geometry of the reconnection layer and a constant resistivity , one has to calculate the half - length of the reconnection layer @xmath4 from the mhd equations and the jump conditions on the petschek shocks , instead of treating @xmath4 as a free parameter ( as petschek did ) . when the layer half - length @xmath4 is calculated correctly , it turns out to be approximately equal to the global magnetic field scale , @xmath19 . in this case the petschek reconnection reduces to the slow sweet - parker reconnection @xcite . this theoretical result is in agreement with the results of numerical simulations of two - dimensional reconnection with constant resistivity @xcite . the second major result of the kulsrud reconnection model is that in the case when resistivity is anomalous and enhanced ( e.g. by plasma instabilities ) , the reconnection rate becomes considerably faster than the sweet - parker rate . in this paper we develop and use a new theoretical approach for calculation of the reconnection rate in the case of anomalous resistivity . this approach is based on application of the mhd equations in a small region , which is localized at the geometric center of a thin reconnection layer ( point o in fig . [ fig_reconnection_layer ] ) and has a size of order of the layer half - thickness @xmath20 . it turns out that by using local analytical calculations in a thin reconnection layer , we can derive an accurate and rather precise estimate for the reconnection rate . in particular , we find the interesting and important result that a quasi - stationary reconnection rate is fully determined by the anomalous resistivity function and by the magnetic field configuration just outside the reconnection layer ( at point m in fig . [ fig_reconnection_layer ] ) . the underlying physical foundations of our analytical derivations are , of course , the same as those previously used by others @xcite . however , our theoretical and computational approach is somewhat different from the conventional approach to the reconnection problem , and we explain the difference in the next section . there are three main benefits of our novel theoretical approach to magnetic reconnection . first , our approach allows us to extend the results for anomalous reconnection obtained by kulsrud in 2001 @xcite to a general anomalous reconnection case , when the guide field and plasma viscosity are arbitrary and anomalous resistivity @xmath21 is an arbitrary function of the electric current and the two spatial coordinates . second , it gives a new important insight into the reconnection problem , such as dependence of the reconnection rate on magnetic field configuration just outside the reconnection layer . third , our approach , based on local calculations , is applicable to cases when there is no well - defined global magnetic field structure , such as the case of multiple current sheets in a turbulent plasma . in our calculations we make only a few assumptions , which are described in detail in the beginning of sec . [ new_model ] . this paper is organized as follows . because our derivations are rather complicated in the general case of reconnection with anomalous resistivity , we find it useful and instructive to consider the sweet - parker reconnection first and to compare our theoretical approach to the classical sweet - parker calculations in the next section . in sec . [ new_model ] , we derive our general equations for magnetic reconnection with anomalous resistivity , including the equation for the reconnection rate . in sec . [ reconnection_limits ] we consider and analyze different special cases of magnetic reconnection , in which our equations simplify and become easier for analysis and comparison to the previous theoretical and simulation results . in sec . [ simulations ] we present the results of our numerical simulations of unforced anomalous magnetic reconnection . these simulations are intended for a demonstration of the predictions of our theoretical reconnection model for the special case of the piecewise linear resistivity function that was considered by kulsrud @xcite . finally , in sec . [ conclusions ] we give our conclusions and discuss our results . some derivations are given in the appendices of the paper . for simplicity and brevity , hereafter in the paper for all electromagnetic variables we use physical units in which the speed of light and four times @xmath22 are replaced unity , @xmath23 and @xmath24 . to rewrite our equations in the standard cgs units , one needs to make the following substitutions for electromagnetic variables in the equations : magnetic field @xmath25 , electric field @xmath26 , electric current @xmath27 , resistivity @xmath28 ( does not change ) , magnetic field vector potential @xmath29 . in this section we consider sweet - parker reconnection @xcite . we assume that resistivity is constant , @xmath30 , the plasma viscosity is zero , the guide field is zero ( @xmath18 ) , and the geometry of the reconnection layer is the classical sweet - parker geometry with the layer half - thickness @xmath31 and the layer half - length @xmath32 , as shown in fig . [ fig_reconnection_layer ] . the purpose of this section is to introduce and explain our new theoretical approach to calculation of the reconnection rate and to compare it with the classical sweet - parker calculations . for this purpose , we first present the classical , conventional derivation of the sweet - parker formula for the reconnection rate , and afterward we present our new derivation of the formula and explain the difference between the two derivations . in the case of a quasi - stationary reconnection the magnetic field in the reconnection region changes slowly in time , @xmath33 . therefore , in the two - and - a - half - dimensional geometry ( @xmath34 ) the faraday s law equation @xmath35 results in the z - component of the electric field being constant in the reconnection region , @xmath36 and @xmath37 is a function of time only . on the other hand , in two - and - a - half dimensions the @xmath0- and @xmath1-components of faraday s law equation @xmath38)$ ] reduce to the following equation for the z - component of the magnetic vector potential @xmath39 : -e_z(t)&=&a_z / t= -(*v*)a_z+^2 a_z + & = & v_xb_y - v_yb_x - j_z . [ mhd_equation ] ( the resistivity @xmath30 is constant in the sweet - parker reconnection case , but equation ( [ mhd_equation ] ) is general and is valid even if resistivity @xmath21 is anomalous and non - constant . ) now , the left- and right - hand - sides of equation ( [ mhd_equation ] ) are constant in space . therefore , the right - hand - side of equation ( [ mhd_equation ] ) is constant across the reconnection layer ( i.e. along the x - axis ) . we equate its values at points o and m , which are on the x - axis and are shown in fig . [ fig_reconnection_layer ] . as a result , we immediately obtain _ oj_o = v_rb_m , [ first_mhd ] where we use the following notations : point o is the geometric center of the reconnection layer , where the z - component of the current is @xmath40 and the plasma velocity is zero . point m is a point on the x - axis just outside the reconnection layer , where the resistivity term can be neglected in equation ( [ mhd_equation ] ) , see fig . [ fig_reconnection_layer ] . we also use the notations @xmath41 and @xmath42 for the y - component of the field and x - component of the plasma velocity at point m , and take the reconnection velocity @xmath16 positive . note that @xmath43 at point m because of the symmetry of the problem with respect to the x - axis . next , we estimate current @xmath44 at the central point o of the reconnection layer as j_ob_m/_o , [ j_o ] where @xmath20 is the reconnection layer half - thickness , and we use ampere s law , @xmath45 , in which we drop the @xmath46 term because the reconnection layer is thin , then obviously the length of the layer would be of the order of its thickness . ] . next , consider the x- and y - components of the equation of plasma motion . we will see below that the sweet - parker reconnection is slow , @xmath47 . therefore , in the equation of plasma motion along the x - axis ( i.e. across the reconnection layer ) the inertial term can be neglected , and this equation becomes the force balance equation , @xmath48 , resulting in @xmath49 , where @xmath50 and @xmath51 are the values of the plasma pressure at points o and m. here we use the fact that the magnetic field is zero at the central point o because of the symmetry of the problem . as far as the equation of plasma motion along the y - axis ( i.e along the layer ) is concerned , the y - components of the magnetic tension force and the pressure gradient force are approximately equal in the sweet - parker reconnection case . indeed , on the y - axis the tension force can be estimated as @xmath52 . here we estimate the x - component of the field as @xmath53 because @xmath14 is produced by the rotation of the @xmath54 field component in the reconnection layer @xcite , and in the sweet - parker model it is assumed that the downstream pressure is equal to the upstream pressure @xmath51 ( see @xcite ) . the pressure and magnetic tension forces accelerate plasma along the reconnection layer ( i.e. along the y - axis ) up to the downstream velocity @xmath55 , which can be estimated from the energy conservation equation ( 1/2)v_out^2b_m^2/2 v_outv_ab_m/. [ sp_energy ] this equation means that the work done by the pressure and magnetic tension forces along the entire reconnection layer is equal to the kinetic energy of the plasma in the downstream regions . finally , in the sweet - parker reconnection model the plasma is assumed to be incompressible . therefore , the mass conservation condition for the entire reconnection layer results in lv_r_ov_out , [ sp_mass ] where @xmath55 , the velocity of plasma outflow in the downstream regions , is given by equation ( [ sp_energy ] ) . using equations ( [ first_mhd])-([sp_mass ] ) , we obtain the formula for the sweet - parker reconnection velocity @xcite , v_rv_a(_o / v_al)^1/2 . [ sp_speed ] equations ( [ first_mhd])-([sp_speed ] ) are the classical sweet - parker equations . note that equations ( [ first_mhd ] ) , ( [ j_o ] ) and ( [ sp_speed ] ) are local , in the sense that they are written for a small region of space , which is localized at the geometric center of a thin reconnection layer ( point o in fig . [ fig_reconnection_layer ] ) and has a size of order of the layer half - thickness @xmath20 . all physical quantities that enter these three equations are defined in this small region of space . at the same time equations ( [ sp_energy ] ) and ( [ sp_mass ] ) are global , in the sense that they result from consideration of the entire reconnection layer and they include the plasma outflow velocity @xmath56 , which is a physical quantity in the downstream regions at the ends of the reconnection layer . in our new theoretical approach to the calculation of the reconnection rate we intend to use only local equations . our intent and our derivations will be justified by the new and important results that we obtain in the next section of this paper and discuss in sec . [ conclusions ] . at present , let us explain our theoretical approach for the simple case of sweet - parker reconnection that we consider in this section . in the derivation of our theoretical model for magnetic reconnection we keep equations ( [ first_mhd ] ) and ( [ j_o ] ) unchanged because these equations are local . however , we rewrite equations ( [ sp_energy ] ) and ( [ sp_mass ] ) because they are global . to rewrite these two global equations in a local form , we consider a point @xmath57 that is located on the y - axis in an infinitesimal vicinity of the reconnection layer central point o ( see fig . [ fig_reconnection_layer ] ) and has an infinitesimally small value of its y - coordinate @xmath58 . since @xmath59 , along the o@xmath60 interval we can use the up - to - the - first - order taylor expansions @xmath61 , @xmath62 and @xmath63 for the values of the perpendicular magnetic field @xmath14 , plasma velocity @xmath64 and z - component of the current @xmath65 . these expansions are along the y - axis , and , of course , @xmath66 and @xmath67 are the first - order partial derivatives of @xmath14 and @xmath64 at point o ( note that @xmath68 ) . now equation ( [ sp_energy ] ) for the plasma acceleration along the y - axis can easily be rewritten in a local form at point @xmath60 as ( ) ^2 & & _ 0^yb_x(0,y)j_z(0,y)dy= _ 0^yy j_o dy + & = & j_oy^2/2 ^2j_o , [ sp_local_energy ] where the left - hand - side of this equation is the plasma kinetic energy at point @xmath60 . on the right - hand - side of equation ( [ sp_local_energy ] ) we keep only the magnetic tension force for plasma acceleration because , as we found above , in the sweet - parker reconnection case the y - components of the magnetic tension and pressure gradient forces are approximately equal to each other ( note that in our general calculations in the next section we will take all forces into account ) . next , equation ( [ sp_mass ] ) for the mass conservation of an incompressible plasma can easily be rewritten in the local form inside the area om@xmath69@xmath60 shown in fig . [ fig_reconnection_layer ] as @xmath70 . thus , we have v_r/_o , ( v_y / y)_o= -(v_x / x)_o . [ upsilon_estimate ] this equation can also be viewed as the first order taylor expansion of @xmath71 along the x - axis ( i.e. across the reconnection layer ) , @xmath72 , where we use the plasma incompressibility condition @xmath73 . note that our first and second equations are local and are exactly the same as equations ( [ first_mhd ] ) and ( [ j_o ] ) in the sweet - parker reconnection model . our third and fourth equations ( [ sp_local_energy ] ) and ( [ upsilon_estimate ] ) are also local and are the analogues of the two global sweet - parker equations ( [ sp_energy ] ) and ( [ sp_mass ] ) . using local equations instead of global ones is the first major difference between the sweet - parker and our theoretical models . however , in our theoretical model we have an additional unknown parameter @xmath66 , which does not directly enter the classical sweet - parker reconnection model . as a result , we need one additional equation to be able to calculate the reconnection rate under the framework of our local model . this additional equation comes from the condition that the right - hand - side of equation ( [ mhd_equation ] ) is constant not only across the reconnection layer , but also along the layer that is along the y - axis . this condition is not explicitly used in the sweet - parker model , but it is used in our model , and this is the second major difference between the two models . to derive the additional equation , used in our reconnection model , we differentiate the left- and right - hand - sides of equation ( [ mhd_equation ] ) along the y - axis ( i.e. along the reconnection layer ) . the first order partial derivatives @xmath74 are identically zero because of the symmetry of the problem with respect to the x - axis . the second order partial derivatives @xmath75 of the left- and right - hand - sides of equation ( [ mhd_equation ] ) result in 0&=&-2(v_y / y)_o(b_x / y)_o -_o(^2 j_z / y^2)_o + & & -2 + 2_oj_o / l^2 , [ sp_third_mhd ] where we use the fact that @xmath76 on the y - axis , we use our definitions @xmath66 and @xmath67 , and we estimate the second derivative of the current @xmath65 as @xmath77 . now all five equations ( [ first_mhd ] ) , ( [ j_o ] ) , ( [ sp_local_energy])-([sp_third_mhd ] ) are local . combining them together , we easily obtain the sweet - parker formula for the reconnection velocity , given by equation ( [ sp_speed ] ) , which is naturally also local , see ref . can be estimated as @xmath78 , the parameter @xmath67 can be estimated as @xmath79 , and our equation ( [ sp_third_mhd ] ) reduces to the sweet - parker equation ( [ first_mhd ] ) , as one expects . ] . the reader of this paper could question why we develop and suggest a new theoretical approach to the problem of quasi - stationary magnetic reconnection if we obtain the same results in the sweet - parker reconnection case . the answer is that our approach , based on local calculations , allows us to calculate the reconnection rate in the case of anomalous resistivity and also provides an additional important understanding of the reconnection problem . our derivations for anomalous reconnection are given in the next section and we discuss our results in sec . [ conclusions ] . note that our local - equations approach to the reconnection problem and the more conventional global - equations approach are the same from the point of view of the underlying physics . indeed , it is well known that by using the gauss - ostrogradski theorem , most physical equations can be written in two equivalent forms , in the form of local differential equations and in the form of global integral equations . in this section we study reconnection with anomalous resistivity and derive a simple and accurate estimate of the reconnection rate in the classical sweet - parker - petschek two - and - a - half dimensional reconnection layer shown in fig . [ fig_reconnection_layer ] . we consider resistivity to be a given arbitrary function of the z - component of the electric current and the two - dimensional coordinates , @xmath80 , which has finite derivatives in @xmath1 up to the second order and in @xmath0 and @xmath65 up to the first order to be a function of @xmath65 instead of the total current @xmath81 . this is because the reconnection process proceeds due to the z - component of the electric field , see eq . ( [ mhd_equation ] ) , and it is reasonable to assume that the electrical conductivity in the z - direction can be reduced by plasma instabilities due to large values of @xmath65 . ] . let us list the assumptions that we make . first , we assume that the characteristic lundquist number of the problem is large , which ( by our definition ) is equivalent to the assumption that resistivity is negligible outside the reconnection layer and the non - resistive mhd equations apply there . second , we assume that the plasma flow is incompressible , @xmath82 . in the limit of very high lundquist numbers and slow reconnection rates the incompressibility condition is a very good approximation in a reconnection layer even for compressible plasmas @xcite . third , we assume that the reconnection process is quasi - stationary . this can only be the case if the reconnection rate is small , @xmath83 [ @xmath84 and see eq . ( [ first_mhd ] ) ] , and there are no plasma instabilities in the reconnection layer . note that in our model the reconnection rate can still be much faster than the sweet - parker rate . fourth , we assume that the reconnection layer is thin , @xmath85 . if plasma kinematic viscosity is small ( in comparison with resistivity ) , then the plasma outflow velocity in the downstream regions is equal to the alfven velocity @xmath15 , we have plasma mass conservation condition @xmath86 , and this assumption of a thin reconnection layer is fully equivalent to the previous assumption of a small reconnection rate . however , if plasma is very viscous , then the plasma outflow velocity is smaller than @xmath15 and assumption @xmath85 is stronger than assumption @xmath87 . finally , note that we make no assumptions about the values of the guide field @xmath88 and the plasma viscosity . [ however , below we will see that our assumption of a quasi - stationary reconnection process in a thin current sheet layer results in a necessary condition @xmath89 for the plasma kinematic viscosity @xmath90 . refer to eq . ( [ nu ] ) for more details . ] now note that several equations that we derived in the previous section for the case of the sweet - parker reconnection with constant resistivity stay the same in the case of anomalous resistivity . indeed , equation ( [ mhd_equation ] ) stays valid when the resistivity @xmath21 is not constant . therefore , equation ( [ first_mhd ] ) also stays valid , except that @xmath91 now is the value of resistivity at the reconnection layer central point o ( see fig . [ fig_reconnection_layer ] ) , i.e. @xmath92 . equations ( [ j_o ] ) and ( [ upsilon_estimate ] ) , which result from the ampere s law and the plasma incompressibility respectively , obviously stay valid too ) and ( [ upsilon_estimate ] ) are exact for the harris model reconnection sheet @xcite , which has @xmath93 , @xmath94 , @xmath95 , @xmath96 and @xmath97 . ] . at the same time , in the general case of anomalous resistivity that we consider in this section , we need to re - derive equations ( [ sp_local_energy ] ) and ( [ sp_third_mhd ] ) , which are the equations of the plasma acceleration and spatial homogeneity of the electric field z - component along the reconnection layer ( i.e. along the y - axis ) . however , before we re - derive these two equations , we would first like to derive the equation of magnetic energy conservation . using eqs . ( [ first_mhd ] ) , ( [ j_o ] ) and ( [ upsilon_estimate ] ) , we immediately obtain b_m^2=_oj_o^2 , 1 , [ b_energy ] where for the purpose of comparison of our theoretical results to our numerical simulations we introduce a coefficient @xmath98 , which is of order of unity . equation ( [ b_energy ] ) is the equation for magnetic energy conservation . the rate of the supply of magnetic energy into the reconnection layer is equal to the rate of its ohmic dissipation inside the layer . next we derive the equation of plasma acceleration along the reconnection layer ( i.e. along the y - axis ) , taking into consideration all forces acting on the plasma . the mhd equation for the y - component of the plasma velocity @xmath64 , assuming the quasi - stationarity of the reconnection ( @xmath99 ) and plasma incompressibility ( @xmath82 ) , is @xcite ( * v*)v_y & = & -(/y ) ( p+b_x^2/2+b_y^2/2 ) + & & + ( * b*)b_y+^2v_y , [ v_y_equation ] where @xmath100 is the plasma density , @xmath90 is the plasma kinematic viscosity ( assumed to be constant ) and @xmath101 is the sum of the plasma pressure and the guide field pressure @xmath102 . taking the first order partial derivative @xmath74 of equation ( [ v_y_equation ] ) at the central point o , we obtain ^2 = -(^2 p / y^2)_o+j_o+ _ o , [ v_y_equation_2 ] where we use parameter @xmath66 and the ampere s law @xmath103 at point o , and we also use the formulas @xmath104 on the y - axis , and @xmath43 at point o , which follow from the symmetry of the problem with respect to the x- and y - axes . the pressure term @xmath105 on the right - hand - side of equation ( [ v_y_equation_2 ] ) can be precisely calculated in analogy with the sweet - parker derivation of the pressure decline along the reconnection layer , which employs the force balance condition for the plasma across the reconnection layer and leads to equation ( [ sp_energy ] ) . the viscosity term @xmath106_o$ ] in equation ( [ v_y_equation_2 ] ) can be calculated approximately by using estimates for the @xmath64 velocity derivatives . in appendix [ appendix_a ] we carry out these calculations and show that the pressure and viscosity terms are equal to ( ^2 p / y^2)_o & = & b_m(^2 b_y / y^2)_m + & & + o\{^2,j_o,/_o^2 } , [ pressure_term ] + _ o & & -/_o^2 , [ viscosity_term ] where @xmath107 is calculated just outside the reconnection layer at point m ( see fig . [ fig_reconnection_layer ] ) , and expression @xmath108 denotes terms that are small compared to either @xmath109 or @xmath110 or @xmath111 in the limit of a slow reconnection rate in a thin reconnection layer , see ref . terms can still be much larger than the @xmath107 term in equation ( [ pressure_term ] ) , in which case the pressure term @xmath105 is unimportant and negligible in equation ( [ v_y_equation_2 ] ) . this happens when reconnection with anomalous resistivity is much faster than sweet - parker reconnection . ] . substituting equations ( [ pressure_term ] ) and ( [ viscosity_term ] ) into equation ( [ v_y_equation_2 ] ) and using equation ( [ j_o ] ) for @xmath20 , we obtain ^2&&-b_m(^2 b_y / y^2)_m + j_o - j_o^2/b_m^2 , [ second_mhd ] + & & ( b_x / y)_o . note that this equation is exact if plasma viscosity can be neglected ( @xmath112 ) . now we use the condition of spatial homogeneity of the electric field z - component along the reconnection layer , i.e. along the y - axis . we take the second order partial derivatives @xmath75 of the left- and right - hand - sides of equation ( [ mhd_equation ] ) at the central point o ( note that the first order partial derivatives are identically zero ) . taking into account the symmetry of the problem , so that @xmath104 on the y - axis , and @xmath113 at point o , we obtain & & -(^2 j_z / y^2)_o - j_o(^2/y^2)_o + & & = 2(v_y / y)_o(b_x / y)_o= 2 , [ third_mhd ] where we use formulas @xmath67 and @xmath66 . finally , we need to estimate the @xmath114 term , which enters the left - hand - side of equation ( [ third_mhd ] ) . this estimation can be done by taking the second order partial derivative @xmath75 of equation ( [ j_o ] ) , while keeping @xmath20 constant because the partial derivative in @xmath1 is to be taken at a constant value @xmath115 . in appendix [ appendix_b ] we give the detailed derivations and find that the y - scale of the current @xmath65 is about the same as the y - scale of the outside magnetic field , i.e. @xmath116 . however , for the purpose of comparison of our theoretical results to numerical simulations in sec . [ simulations ] , we find it convenient to write j_o^-1(^2 j_z / y^2)_o= b_m^-1(^2 b_y / y^2)_m , 1 , [ j_yy ] where we introduce the coefficient @xmath117 , which is of order unity . let us take the dimensionless coefficients @xmath98 and @xmath117 , which enter equations ( [ b_energy ] ) and ( [ j_yy ] ) , and are of order unity , to be exactly unity , @xmath118 and @xmath119 . now we have all the equations necessary to determine all unknown physical parameters . in particular , using eqs . ( [ b_energy ] ) , ( [ second_mhd ] ) , ( [ third_mhd ] ) and ( [ j_yy ] ) , we easily obtain the following approximate algebraic equation for the z - current @xmath44 at the reconnection layer central point o : & & 3 + + + & & -(1 + ) , [ rate ] where the alfven velocity @xmath15 is defined as @xmath84 and @xmath92 is the resistivity at point o. given the resistivity function @xmath80 , as well as the magnetic field @xmath120 and its second order derivative @xmath121 outside the reconnection layer , we can solve equation ( [ rate ] ) for the current @xmath44 and find the reconnection rate , which is the rate of destruction of magnetic flux at point o , equal to @xmath122 . using eq . ( [ mhd_equation ] ) , we find that the reconnection rate is equal to @xmath123 . note that for the classical reconnection layer that we consider ( see fig . [ fig_reconnection_layer ] ) the right - hand - side of equation ( [ rate ] ) is positive because @xmath124 , where @xmath5 is the global scale of the magnetic field outside the reconnection layer . once the current @xmath44 is calculated by means of equation ( [ rate ] ) , we can easily calculate all other reconnection parameters , using eqs . ( [ first_mhd ] ) , ( [ j_o ] ) , ( [ upsilon_estimate ] ) and ( [ second_mhd ] ) , v_r & & _ oj_o / b_mv_a , [ v_r ] + & & _ oj_o^2/b_m^2 , [ upsilon ] + & & j_oj_o , [ beta ] + _ o & & b_m / j_o v_r/. [ delta ] equations ( [ rate])-([delta ] ) are the most general result for magnetic reconnection that we obtain in this paper . restoring coefficients @xmath125 and @xmath126 , equation ( [ rate ] ) becomes & & ( + 2)+ + + & & = -^2(+ ) . [ rate_general ] hereafter we will consider the natural case when @xmath127 and @xmath128 because plasma conductivity decreases as the current increases and we are interested in fast anomalous reconnection ( i.e. faster than the sweet - parker reconnection ) . in this case the first , second and third terms on the left - hand - side of equation ( [ rate ] ) are all positive . it is easy to see that the first term is related to sweet - parker reconnection with constant resistivity equal to @xmath91 , the second term is related to fast reconnection associated with the dependence of anomalous resistivity on the current , and the third term is related to fast reconnection associated with an _ ad hoc _ localization of resistivity in space ( see the next section for details ) . also note that if the plasma kinematic viscosity @xmath90 is larger than the resistivity @xmath91 , then , according to equation ( [ rate ] ) , the current @xmath44 and reconnection rate @xmath129 become smaller as @xmath90 grows , i.e. the reconnection slows down for viscous plasmas as one expects . we postpone the analysis of equations ( [ rate])-([delta ] ) until sec . [ conclusions ] . let us now make an estimate of the half - length of the reconnection layer @xmath4 ( see fig . [ fig_reconnection_layer ] ) . note that @xmath4 is not needed for the calculation of the reconnection rate @xmath129 by means of equation ( [ rate ] ) . nevertheless , we are still interested in a rough estimate of @xmath4 , in particular , because we need to check our assumption that the reconnection layer is thin , i.e. that the condition @xmath130 is satisfied . it is clear that @xmath4 can not be much larger than the global scale of the magnetic field outside the layer @xmath5 . therefore we have the condition @xmath131 . however , @xmath4 can be much smaller than @xmath5 , in which case the reconnection layer has a pair of the petschek switch - off mhd shocks attached to each end of the layer in the downstream regions @xcite , as shown in fig . [ fig_reconnection_layer ] . in this case @xmath4 should be calculated as the y - coordinate of the point on the y - axis at which the perpendicular field @xmath14 is strong enough to support the shocks @xcite . following kulsrud @xcite , we use the jump condition on the petschek switch - off shocks to obtain ) was used by kulsrud @xcite for the case of a viscosity - free plasma . it can be shown from the full non - ideal mhd equations that this condition is unchanged in the case of a viscous plasma . ] v_rb_x(0,y = l)/l/= lv_a / b_m,[schock_balance ] where we use the first - order taylor expansion for an estimate @xmath132 and , as before , @xmath84 . now , using eqs . ( [ v_r ] ) , ( [ beta ] ) and ( [ schock_balance ] ) , we can easily find @xmath4 . before we write the explicit formula for @xmath4 , note that the absolute value of the @xmath133 term in equation ( [ beta ] ) is equal or smaller than the @xmath134 term . this follows from the fact that the left - hand - side of equation ( [ rate ] ) is equal or greater than unity [ see our comments in the paragraph that follows eq . ( [ rate_general ] ) ] . therefore , the @xmath133 term can be omitted in equation ( [ beta ] ) for the purpose of estimating @xmath4 . as a result , we obtain the following rough estimates for the reconnection layer half - length @xmath4 and the velocity of plasma outflow in the downstream regions @xmath55 : l & & ( v_ab_m^2/_oj_o^2)(1+/_o)^-1 + & & ( v_a/)(1+/_o)^-1 , _ o ll , [ l ] + v_out & & lv_a(1+/_o)^-1 v_a , [ v_out ] where we use equations ( [ v_r ] ) , ( [ upsilon ] ) , ( [ beta ] ) and ( [ schock_balance ] ) . note that the condition @xmath131 is always satisfied because @xmath135 , and the left- and right - hand - sides of equation ( [ rate ] ) are equal or greater than unity . however , the condition that the reconnection layer is thin , @xmath130 , is satisfied only if plasma viscosity is not too large , _ o(v_ab_m/_oj_o)_o(v_a / v_r ) , [ nu ] where , to derive this formula , we use eqs . ( [ v_r ] ) , ( [ delta ] ) and ( [ l ] ) . in other words , to be able to form a thin reconnection layer , the plasma should not be too viscous . note that in the case of constant resistivity and large viscosity , @xmath136 and @xmath137 , the reconnection velocity is @xmath138 ( see @xcite ) and condition ( [ nu ] ) reduces to @xmath139 . finally , if the plasma viscosity is small in comparison to the resistivity , @xmath140 , then from equation ( [ v_out ] ) we immediately find an important and well - known result that in this case the velocity of the plasma outflowing in the downstream regions at the ends of the reconnection layer is approximately equal to the alfven velocity , @xmath141 ( see fig . [ fig_reconnection_layer ] ) . at the end of this section we would like to discuss several assumptions that we used in our derivations . first , the solution of equation ( [ rate ] ) is valid only if it gives @xmath142 , which is our assumption of a slow quasi - stationary reconnection . because of equation ( [ v_r ] ) , condition @xmath142 is equivalent to @xmath17 , i.e. the reconnection velocity , which is the velocity of the incoming plasma , must be small in comparison to the alfven velocity in the upstream region . second , the coefficient @xmath66 , given by eq . ( [ beta ] ) , must be much smaller than the current @xmath143 because the reconnection layer is assumed to be thin . it is easy to see that this condition is satisfied . indeed , the first term in the brackets @xmath144 $ ] in equation ( [ beta ] ) is much smaller than unity because of the upper limit for plasma viscosity given by equation ( [ nu ] ) . the second term in the brackets in equation ( [ beta ] ) is also much smaller than unity because @xmath44 is much larger than the electric current outside the reconnection layer due to our assumption of a large characteristic lundquist number of the system . in this section we focus on three special cases for the reconnection rate , which arise when one of the three terms on the left - hand - side of equation ( [ rate ] ) dominates over the other two . we consider the classical sweet - parker - petschek reconnection layer shown in fig . [ fig_reconnection_layer ] and define the global scale of the magnetic field outside the reconnection layer as @xmath145 ( see we use formula @xmath146 for the field y - component along the interval m@xmath69 that is outside the reconnection layer as shown in fig . [ fig_reconnection_layer ] . ] ) . in addition , for the purpose of clarity , in this section we focus only on resistivity effects and neglect plasma viscosity , assuming that @xmath140 . first , consider the case when resistivity is constant , @xmath147 . in this case only the first term on the left - hand - side of equation ( [ rate ] ) is nonzero and equations ( [ rate])-([delta ] ) and ( [ l ] ) reduce to lcl 3 ^ 1/2_o j_o^2l / v_ab_m^2 & & j_o(b_m / l)s_o^1/2 , + s_ov_al/_o1 , & & v_rv_as_o^-1/2 , + v_a / l , & & ( b_m / l)s_o^-1/2 , + _ ols_o^-1/2 , & & ll , [ sp_reconnection ] where we set the plasma kinematic viscosity to zero ( @xmath112 ) , use our definition of the global field scale @xmath145 , introduce the lundquist number @xmath148 and assume for our estimates that @xmath149 . the above equations are the sweet - parker reconnection equations with constant resistivity equal to @xmath91 . thus , we find that if resistivity is constant , then the reconnection must be sweet - parker and not petschek @xcite . we discuss this important result in sec . [ conclusions ] . note that in this section , contrary to sec . [ sweet_parker_model ] , we do not assume the sweet - parker geometrical configuration for the reconnection layer , but derive it together with the reconnection rate from our general equations of the previous section . a typical configuration of the reconnection layer in the case of sweet - parker reconnection is shown on the left - bottom plot in fig . [ fig_rosette ] . this plot is marked by letters `` s - p '' . now let us consider the case when resistivity is anomalous and is a monotonically increasing function of the electric current only , @xmath150 . let us further assume that this dependence of resistivity on the current is very strong , so that @xmath151 . in this case the second term on the left - hand - side of equation ( [ rate ] ) is dominant , and equations ( [ rate ] ) and ( [ l ] ) reduce to ( d / d j_z)_o & & _ o^3j_o^3l^2/v_a^2b_m^4 + & & ^1/3 , [ pk_reconnection_rate ] + l & & l + & & l^-1/2 l. [ pk_reconnection_l ] here again we set @xmath112 , and we use the formula @xmath152 and eqs . ( [ v_r ] ) and ( [ delta ] ) . from equation ( [ pk_reconnection_l ] ) we see that the half - length of the reconnection layer @xmath4 is much less than the global field scale @xmath5 in the case of a strong dependence of resistivity on the current ( @xmath151 ) . this means that in this case the geometry of the reconnection layer is petschek with a pair of shocks attached to each end of the layer in the downstream regions , as shown in fig . [ fig_reconnection_layer ] . equation ( [ pk_reconnection_rate ] ) for the reconnection rate was first analytically derived by kulsrud @xcite for a special case when @xmath153 . in the petschek - kulsrud reconnection case the reconnection rate , given by eq . ( [ pk_reconnection_rate ] ) , can be considerably faster than the sweet - parker reconnection rate . this fast reconnection has been previously observed in many numerical simulations done with anomalous resistivity @xmath150 ( e.g. see @xcite ) . the typical configuration of the reconnection layer in the case of petschek - kulsrud reconnection is shown in the left - bottom plot in fig . [ fig_rosette ] , this plot is marked by letters `` p - k '' . finally let us consider the special case when resistivity is given by @xmath154 and it is spatially localized around the central point o of the reconnection layer ( see fig . [ fig_reconnection_layer ] ) , so that @xmath155 . in other words , we assume that resistivity is anomalous and is localized on scale @xmath156 that is much smaller than the global field scale @xmath5 . in this case the third term on the left - hand - side of equation ( [ rate ] ) is dominant , and equations ( [ rate])-([delta ] ) and ( [ l ] ) reduce to lcl l / l__o j_o^2l / v_ab_m^2 & & j_o(b_m / l_)s_l^1/2 , + s_lv_al_/_o1 , & & v_rv_as_l^-1/2 , + v_a / l _ , & & ( b_m / l_)s_l^-1/2 , + _ ol_s_l^-1/2 , & & ll_l , [ localized_reconnection ] where we again set @xmath112 . the above equations are the same as the sweet - parker equations ( [ sp_reconnection ] ) with the global field scale @xmath5 replaced by the resistivity scale @xmath157 . note that , when resistivity is localized , the reconnection rate becomes faster than the sweet - parker rate by a factor @xmath158 and the geometry of the reconnection layer is petschek with a pair of shocks attached to each end of the layer ( see fig . [ fig_reconnection_layer ] ) . these results are in agreement with many previous numerical simulations of reconnection with spatially localized resistivity @xcite . in this section we present the results of our numerical simulations of unforced reconnection of two cylindrical magnetic flux tubes . these simulations are not intended as a check or a proof of our theoretical results for magnetic reconnection . our equations ( [ rate])-([delta ] ) have been derived analytically and are very general . a comprehensive testing of them would require extensive computational work , which is beyond the scope of this paper . instead , we present our simulations as a demonstration of our reconnection model predictions . following kulsrud @xcite , we assume that plasma resistivity is given by the following piecewise linear function of the z - component of the electric current : ( j_z)=_s+_\{0,j_z - j_c}/j_c , [ linear_resistivity ] where @xmath159 is the spitzer resistivity , which is assumed to be very small , @xmath160 is the anomalous resistivity parameter and @xmath161 is the critical current parameter . the kulsrud model s prediction for the reconnection rate in the case @xmath162 , which is given by equation ( [ pk_reconnection_rate ] ) , has already been checked and confirmed numerically by breslau and jardin @xcite . here we simulate reconnection with anomalous resistivity given by equation ( [ linear_resistivity ] ) for a different computational setup , a higher lundquist number and a wider range of parameters @xmath160 and @xmath161 ( without the restriction @xmath163 ) . our intent is to see how our general formula ( [ rate ] ) for the reconnection rate works in this case . we consider an unforced reconnection of two cylindrical magnetic flux tubes with the initial z - component of the magnetic field vector potential equal to a_z(x , y)&=&a_0 , + a_0&=&b_0r_0 , r_^2=(xd)^2+y^2 , [ tubes ] see the top - left plot in fig . [ fig_rosette ] . this convenient computational setup was suggested to us by mikic and vainshtein @xcite . we choose the parameters in equation ( [ tubes ] ) as @xmath164 ( the global scale of the field is unity ) , @xmath165 ( @xmath166% of initially reconnected flux ) and @xmath167 ( the maximal initial field is unity ) . thus , @xmath168 . in addition , for convenience we choose the plasma density @xmath169 , so that the typical alfven velocity and time are unity , @xmath170 and @xmath171 . the guide field is chosen to be zero , @xmath18 , and the initial plasma velocities are zero . the initial gas pressure @xmath101 is chosen in such way that each of the two cylindrical magnetic flux tubes ( [ tubes ] ) would initially be in complete equilibrium , @xmath172 , if there were no magnetic forces from the other tube . the plasma kinematic viscosity is chosen to be equal to the spitzer resistivity , @xmath173 . the boundary conditions are placed at @xmath174 , which are virtually at infinity ( the magnetic vector potential ( [ tubes ] ) drops to less than @xmath175 at the boundaries ) . because of the symmetry of the problem , in the case of a quasi - stationary reconnection considered here , it is enough to run simulations only in the upper - right quadrangle of the full computational box . we use the flash code for our simulations . this is a compressible adaptive - mesh - refinement ( amr ) code written and supported at the asc center of the university of chicago . ( for a comprehensive description of the flash code see @xcite ) . the mhd module of the code uses central finite differences to properly resolve all resistive and viscous scales . comparing numerical results obtained by simulations done with a compressible code to our approximate theoretical formulas derived for incompressible fluids is fine in a case of a very high lundquist number . this is because in this case the incompressibility condition is a very good approximation in a reconnection layer even for compressible plasmas @xcite . indeed , in our simulations the plasma density varies by no more than 15% inside the reconnection layer . the biggest advantage of the flash code for our purposes is that it is an already existing , well tested code with the amr feature , which allows us to place the boundary conditions at infinity . the size of the smallest elementary grid cell in our two - dimensional simulations was chosen typically to vary from @xmath176 to @xmath177 , which was sufficient to resolve the resistive reconnection layer . the two cylindrical magnetic flux tubes , initially set up according to equation ( [ tubes ] ) , attract each other ( in a similar way as two wires with colliniar currents do ) . as a result , as time goes on , the tubes move toward each other , form a thin reconnection layer along the y - axis and eventually completely merge together by reconnection . this merging process is displayed in the three top plots in fig . [ fig_rosette ] , which show the field vector potential @xmath178 in a central region of the full computational box . the two bottom - left plots in fig . [ fig_rosette ] show the electric current @xmath65 in a central region that includes the reconnection layer . these plots clearly show the reconnection layer configuration which is formed during the reconnection process in the cases of the sweet - parker and petschek - kulsrud reconnection ( refer to secs . [ sweet_parker_reconnection ] and [ petschek_kulsrud_reconnection ] ) . the bottom - right plot in fig . [ fig_rosette ] demonstrates the functional dependence on time typical of the normalized reconnection rate @xmath179 at the reconnection layer central point o. ( this point is shown in fig . [ fig_reconnection_layer ] ) . next we compare the maximal ( peak ) reconnection rate observed in the numerical simulations with the theoretical rate predicted by our reconnection model . when resistivity is given by equation ( [ linear_resistivity ] ) and @xmath173 , our theoretical formula ( [ rate ] ) for the reconnection rate reduces to 3 + ( 1+_s/_o ) , [ linear_model_rate ] where , as explained above , in our numerical simulations we choose @xmath167 , @xmath164 , @xmath180 and @xmath169 , and by definition the global field scale is @xmath145 . we also assume that @xmath181 , which is the case that we consider in our numerical simulations . as we can see from equation ( [ linear_model_rate ] ) , the theoretical reconnection rate @xmath129 is rather sensitive to @xmath120 and @xmath5 , which are the strength and scale of the magnetic field at point m outside the reconnection layer ( see fig . [ fig_reconnection_layer ] ) . therefore , in order to compare the theoretical results with the results of our simulations , we need to accurately calculate the @xmath120 and @xmath5 observed in the simulations . as a result , the choice of the exact position of point m , at which @xmath120 and @xmath5 are calculated , is important . first , in our simulations we choose the point m to be the point on the x - axis at which the observed resistivity term @xmath182 is three times smaller than that observed at the central point o , i.e. @xmath183 . the three plots on the left in fig . [ fig_rates_1_3 ] demonstrate our results for this choice . the top plot on the left shows the reconnection rate . the crosses are a log - log plot of the maximal ( peak ) reconnection rate observed in our simulations as a function of parameter @xmath160 for fixed @xmath184 and @xmath185 . the boxes are the theoretical reconnection rate , which is given by equation ( [ linear_model_rate ] ) with the appropriate values of @xmath120 , @xmath5 and density @xmath100 observed in the simulations . the solid horizontal line ( simulations ) and the dashed horizontal line ( theory ) correspond to the @xmath186 case . the inclined dotted line demonstrates the @xmath187 scaling [ refer to eq . ( [ pk_reconnection_rate ] ) ] . the crosses and the boxes do not follow the @xmath187 scaling for large values of @xmath160 simply because in this case the reconnection rate becomes relatively fast and the magnetic field @xmath120 outside of the reconnection layer is not piled up as much as in the case when @xmath160 is small and the reconnection rate is relatively slow . as a result , our rate curves flatten at large values of @xmath160 . the central and the bottom plots on the left in fig . [ fig_rates_1_3 ] show the observed values of coefficients @xmath98 and @xmath117 , which are directly calculated from the simulation data by using equations ( [ b_energy ] ) and ( [ j_yy ] ) . as we can see , @xmath98 and @xmath117 are of order of unity , as one expects . the three plots on the right in fig . [ fig_rates_1_3 ] are the same as the three plots on the left except point m is now chosen as the point on the x - axis at which @xmath188 . comparing the plots on the left and the plots on the right , we see that the choice of point m is indeed important . note that for the choice @xmath183 for the position of point m the half - thickness of the reconnection layer @xmath20 , defined as the abscissa of point m , increases from @xmath189 to @xmath190 as @xmath160 increases from zero to its maximal value shown on the plots in fig . [ fig_rates_1_3 ] . at the same time , for the choice @xmath188 for the position of point m @xmath20 ranges from @xmath191 to @xmath192 , which is noticeably larger . perhaps simulations of forced reconnection with a strict control of position of point m together with control of the outside field @xmath120 and its scale @xmath5 , could be better suited for comparison to our theoretical model . such simulations are beyond the scope of this paper . however , see more discussion of forced reconnection in the next section . finally , figure [ fig_rates_2_3 ] shows the same results as fig . [ fig_rates_1_3 ] , except the former has plots for a smaller value of the critical current , @xmath193 . we believe that the results presented in figs . [ fig_rates_1_3 ] and [ fig_rates_2_3 ] generally confirm our theoretical model . in particular , in all cases the theoretical reconnection rates and the rates observed in the simulations do not differ by more than 33% . the observed relatively small discrepancy between the theoretical and simulated rates is mainly due to the coefficient @xmath98 not being precisely constant in the simulations , while the variations of coefficient @xmath117 are somewhat less important [ see eq . ( [ rate_general ] ) for the theoretical reconnection rate ] . this discrepancy can be due to two causes . first , our theoretical model is general , but approximate , and second , the plasma is compressible in the simulations , while it is assumed to be incompressible in the theoretical model . let us summarize our main results . in this paper we take a new theoretical approach to the calculation of the rate of quasi - stationary magnetic reconnection . our approach is based on analytical derivations of the reconnection rate from the resistive mhd equations in a small region of space that is localized about the center of a thin reconnection layer and has its size equal to the layer thickness . our local - equations approach turns out to be feasible and insightful . it allows us to consider magnetic reconnection with an arbitrary anomalous resistivity and to calculate the reconnection rate for this general case [ see eq . ( [ rate ] ) ] . we find the interesting and important result that if plasma is incompressible and reconnection is quasi - stationary , then the reconnection rate is determined by the anomalous resistivity function @xmath194 and by the strength and structure of the magnetic field just outside of the reconnection layer ( i.e. at point m in fig . [ fig_reconnection_layer ] ) . thus , we find that the global magnetic field and its configuration are not directly relevant for the purpose of calculation of a quasi - stationary reconnection rate , although , of course , the local magnetic field outside the reconnection layer depends on the global field . one of the major results of this paper is that in the case of constant resistivity , @xmath30 , the magnetic reconnection rate is the slow sweet - parker reconnection rate and not the fast petschek reconnection rate [ refer to eqs . ( [ sp_reconnection ] ) ] . this result agrees with numerical simulations and at the same time contradicts the result of the original petschek theoretical model . in the framework of our theoretical approach , based on local equations , the reason for this contradiction can be understood as follows : in the petschek model our parameter @xmath195 , which is equal to the first order partial derivatives of the incompressible plasma velocities at the reconnection layer center , is basically treated as a free parameter . this is because in the petschek model @xmath196 can be estimated as the ratio of the plasma outflow velocity ( equal to the alfven velocity for viscosity - free plasma ) and the reconnection layer length , @xmath197 , and the layer length @xmath4 is treated as a free parameter by petschek . in his model @xmath4 is taken to be equal to the minimal possible value , that the petschek shocks do not seriously perturb the magnetic field in the upstream region . this is @xmath198 , where @xmath199 is the lundquist number @xcite . in this case @xmath200 and , according to equations ( [ v_r ] ) and ( [ upsilon ] ) , the reconnection velocity in this case is equal to @xmath201 , which is the petschek result . on the other hand , in our theoretical model the parameter @xmath196 is not treated as a free parameter . in fact , our three physical parameters @xmath67 , @xmath66 and @xmath202 are connected to each other and must be calculated from equations ( [ b_energy ] ) , ( [ second_mhd ] ) and ( [ third_mhd ] ) . let us discuss the meaning of these equations . equation ( [ b_energy ] ) is the equation of magnetic energy conservation . it says that the rate of supply of the magnetic field energy into the reconnection layer @xmath203 must be equal to the rate of the resistive dissipation of this energy @xmath204 . in the petschek model @xmath196 and , accordingly , the rate of the magnetic energy supply @xmath203 are basically prescribed by hand ( resulting in an _ ad hoc _ fast reconnection ) , while in our model they are self - consistently calculated from the mhd equations . equation ( [ second_mhd ] ) is the equation of plasma acceleration along the reconnection layer . it says that the magnetic tension force @xmath110 must be large enough in order to be able to push out all the plasma along the layer that is supplied into the layer . finally , equation ( [ third_mhd ] ) is the equation of spatial homogeneity of the electric field z - component along the reconnection layer . this equation sets an upper limit on the product @xmath205 in the case of a quasi - stationary reconnection and is directly related to the calculations and arguments given by kulsrud @xcite in the framework of the global - equations theoretical approach , see ref . are directly related to the explanation of why the petschek reconnection model does not work , as shown by kulsrud @xcite in the framework of the global - equations . kulsrud s argument is that the length of petschek reconnection layer @xmath4 is not a free parameter , but must be determined by the condition that the perpendicular magnetic field component @xmath14 has to be regenerated by the rotation of the parallel field component @xmath54 at the same rate as it is being swept away by the downstream flow . it is easy to see that the integration of `` local '' equation ( [ mhd_equation ] ) without the resistivity term over the area of the contour o@xmath206m@xmath206@xmath69@xmath206@xmath60 shown in fig . [ fig_reconnection_layer ] will result in the same `` global '' equation for the balance of the @xmath14 field , which was used by kulsrud in his work @xcite . ] . as a result , none of parameters @xmath196 , @xmath207 and @xmath44 can be treated as free parameters , and all of them must be self - consistently determined from the mhd equations . it is very instructive to briefly examine our results from the two distinct points of view in connection with the reconnection problem , which are expressed in numerous papers on computer simulation of magnetic reconnection . these two points of view are : unforced ( free ) magnetic reconnection and forced magnetic reconnection . in the case of unforced reconnection , one should solve our main equation ( [ rate ] ) for the current @xmath44 at the reconnection layer center and then solve for the reconnection velocity @xmath16 by using equation ( [ v_r ] ) . the solution for @xmath44 and @xmath16 will depend on @xmath120 , which is the strength of the magnetic field outside the reconnection layer and which enters the right - hand - side of equation ( [ rate ] ) . on the other hand , in the case of forced magnetic reconnection the reconnection velocity @xmath16 is prescribed and fixed . in this case the magnetic field outside the reconnection layer @xmath120 should be treated as an unknown quantity , and equations ( [ rate ] ) and ( [ v_r ] ) should be solved together in order to find the correct quasi - stationary values of @xmath44 and @xmath120 . in other words , in the forced reconnection case an initially weak outside magnetic field @xmath120 gets piled up to higher and higher values until the resulting current @xmath44 in the reconnection layer becomes large enough to be able to exactly match the prescribed reconnection velocity @xmath16 and to be able to reconnect all magnetic flux and magnetic energy , which are supplied into the reconnection region in the quasi - stationary reconnection regime . finally , a couple of words about plasma viscosity and guide field and their effect on magnetic reconnection . first , according to our equations ( [ rate ] ) and ( [ v_r ] ) , in the case when the resistivity is constant , @xmath30 , and the plasma viscosity is much larger than resistivity , @xmath137 , the reconnection velocity becomes @xmath208 , which is @xmath209{\nu/\eta_o}$ ] times smaller than the sweet - parker reconnection velocity given by formula ( [ sp_reconnection ] ) , see @xcite . thus , we see that the reconnection rate becomes smaller when the plasma viscosity becomes large . however , note that in many astrophysical and laboratory applications plasmas are very hot and highly rarefied . under these conditions the ion gyro - radius becomes much shorter than the ion mean - free - path , and the plasma becomes strongly magnetized . as a result , the plasma viscosity becomes the braginskii viscosity , which is dominated by magnetized ions @xcite . in this case in all our equations above the isotropic viscosity @xmath90 , which is proportional to the ion mean - free - path , should be replaced by the braginskii perpendicular viscosity , which is proportional to the ion gyro - radius and is much smaller than the perpendicular viscosity in a strongly magnetized plasma . second , according to our results , in two - and - a - half dimensional geometry the guide field @xmath88 has no effect on the quasi - stationary reconnection rate . indeed , in our derivations the guide field appears only as magnetic pressure @xmath102 term in addition to the plasma pressure . the combined pressure @xmath101 enters equations ( [ v_y_equation ] ) and ( [ v_y_equation_2 ] ) of plasma acceleration along the reconnection layer and the value of spatial derivative of @xmath101 is given by equation ( [ pressure_term ] ) , which does not involve @xmath88 . thus , the guide field gets eliminated and does not enter into our final equation ( [ rate ] ) for the reconnection rate . however , if one assumes that the anomalous resistivity @xmath21 depends on x- and y - components of the current @xmath210 and @xmath211 in addition to its dependence on the z - component of the current @xmath65 , then the reconnection rate will depend on the guide field @xmath88 . in this paper we consider quasi - stationary magnetic reconnection in a thin reconnection layer . we leave a study of tearing modes instability in a reconnection layer and non - quasi - stationary reconnection for a future paper . it is our special pleasure to thank ellen zweibel and dmitri uzdensky for many stimulating discussions and useful comments . we are grateful to zoran mikic and samuel vainshtein for suggesting to us the convenient computational setup used in our simulations . we would also like to thank andrey beresnyak , amitava bhattacharjee , stas boldyrev , fausto cattaneo , jeremy goodman , hantao ji , alexander obabko , robert rosner and masaaki yamada for a number of valuable comments . this work was supported by the center for magnetic self - organization ( cmso ) grant . the numerical simulations were supported by a center for magnetic reconnection studies ( cmrs ) grant . the software used in this work was in part developed by the doe supported asc / alliances center for astrophysical thermonuclear flashes at the university of chicago . the simulations were carried out on doe computers at the oak ridge national laboratory . below , for brevity , we assume that spatial derivatives are to be taken with respect to all indexes that are listed after the comma signs in the subscripts , e.g. @xmath212 . we derive equation ( [ pressure_term ] ) first . the derivation is somewhat analogous to the sweet - parker derivation of the pressure decrease along the reconnection layer , which leads to equation ( [ sp_energy ] ) . namely , to find the pressure decrease and the pressure second derivative along the layer , we integrate the pressure gradient vector along the contour o@xmath206m@xmath206@xmath69@xmath206@xmath60 shown in fig . [ fig_reconnection_layer ] and use the force balance condition for the plasma across the reconnection layer . now we carry out these calculations in a mathematically precise way . for infinitesimally small values of the y - coordinate , taking into account the symmetry of the reconnection layer with respect to the x- and y - axes and plasma incompressibility , we use the following taylor expansions in @xmath1 for the plasma velocity @xmath213 and for the magnetic field @xmath214 : l v_x = v_x^(0)(x)+(y^2/2)v_x , yy^(0)(x)+(y^4/24)v_x , yyyy^(0)(x ) , + b_x = yb_x , y^(0)(x)+(y^3/6)b_x , yyy^(0)(x ) , + v_y = -yv_x , x^(0)(x)+(y^3/6)v_y , yyy^(0)(x ) , + b_y = b_y^(0)(x)+(y^2/2)b_y , yy^(0)(x ) , [ app_a_expansions ] where the variables with the superscripts @xmath215 are taken at @xmath216 and depend only on @xmath0 . assuming quasi - stationarity of reconnection ( @xmath99 ) and plasma incompressibility , the mhd equation for the plasma velocity @xmath8 is p+**(b_x^2+b_y^2)/2&= & -(*v***)*v*+ ( * b***)*b * + & & + ^2*v * , [ app_a_v_equation ] where @xmath101 is the sum of the plasma pressure and the z - component field magnetic pressure @xmath102 . let us calculate line integrals of the left- and right - hand - sides of equation ( [ app_a_v_equation ] ) along the contour o@xmath206m@xmath206@xmath69@xmath206@xmath60 shown in fig . [ fig_reconnection_layer ] . first , the line integral of the left - hand - side is obviously & & p(0,y)-p(0,0)+(1/2)b_x^2(0,y ) + & & = ( y^2/2)[p_,yy(0,0)+^2 ] , [ app_a_lhs_integral ] where @xmath58 is the y - coordinate of points @xmath69 and @xmath60 , and we use the formulas @xmath217 on the y - axis , @xmath94 at point o and @xmath61 for small @xmath1 [ see the definition of @xmath207 in eq . ( [ second_mhd ] ) ] . second , using expansion formulas ( [ app_a_expansions ] ) , we calculate the following line integrals along the contour o@xmath206m@xmath206@xmath69@xmath206@xmath60 , up to the second order in @xmath58 : & = & ( ^2- v_x^(m ) + & & + 2_o^m v_x , x^(0)v_x , yy^(0)dx ) , [ app_a_v_integral ] + & = & ( b_y^(m)b_y , yy^(m)+^2 + b_x , y^(m)j_z^(m ) + & & -_o^m dx ) , [ app_a_b_integral ] + & = & ( 2v_y , yyy^(m)-v_x , xxx^(m)-v_y , yyy^(0)(0 ) + & & -_o^m v_x , yyyy^(0)dx ) , [ app_a_nu_integral ] where the variables with the superscripts @xmath218 are calculated at point m ( see fig . [ fig_reconnection_layer ] ) . next we estimate the terms on the right - hand - sides of eqs . ( [ app_a_v_integral])-([app_a_nu_integral ] ) . recall the notations @xmath4 and @xmath20 for the half - length and half - thickness of the reconnection layer ( see fig . [ fig_reconnection_layer ] ) , and that @xmath85 ( which is our assumption of a thin reconnection layer ) . the z - current inside the reconnection layer is approximately equal to @xmath219 [ see eq . ( [ j_o ] ) ] , where @xmath220 is the field at point m , while the z - current outside the layer is @xmath221 . the last two terms on the right - hand - side of equation ( [ app_a_b_integral ] ) can be estimated as @xmath222 and @xmath223dx \sim\delta_o b_m\beta / l'^2\sim\beta j_o(\delta_o / l')^2\ll \beta j_o$ ] . next , from equation ( [ mhd_equation ] ) with the resistivity term dropped we see that outside the reconnection layer the typical scale of the plasma velocity @xmath8 is about the same as the typical scale of the magnetic field @xmath9 and , therefore , can not be smaller than @xmath4 . thus , the estimates of the three terms on the right - hand - side of equation ( [ app_a_v_integral ] ) are @xmath224 ^ 2\sim v_x^{(m)}[v_{x , xx}^{(m)}+v_{x , yy}^{(m)}]\sim v_r^2/l'^2\sim\upsilon^2(\delta_o / l')^2\ll \upsilon^2 $ ] [ see eq . ( [ upsilon_estimate ] ) ] and @xmath225 . the estimates of the four terms on the right - hand - side of equation ( [ app_a_nu_integral ] ) are @xmath226 , @xmath227 and @xmath228 . note that here we use @xmath4 for estimation of y - derivatives . in fact , using the global scale @xmath5 of the magnetic field outside the reconnection layer would have been more appropriate for some of the estimations ( as shown in appendix [ appendix_b ] ) . however , using @xmath229 for the upper estimates of the @xmath74 derivatives is perfectly fine for the purposes in this appendix . next , calculating the line integral of the right - hand - side of eq . ( [ app_a_v_equation ] ) by using formulas ( [ app_a_v_integral])-([app_a_nu_integral ] ) and our estimates made in the previous paragraph , and then equating the result to the right - hand - side of eq . ( [ app_a_lhs_integral ] ) , we easily obtain equation ( [ pressure_term ] ) . note that the first term on the right - hand side of equation ( [ pressure_term ] ) can also be written in terms of the second derivative of the magnetic pressure at point m : @xmath230 ( note that @xmath231 because the reconnection layer is thin ) . therefore , as noted by zweibel @xcite , equation ( [ second_mhd ] ) is similar to bernoulli s equation . now we derive equation ( [ viscosity_term ] ) , which gives an approximate estimate of the viscosity term @xmath106_o$ ] in equation ( [ v_y_equation_2 ] ) . we make this estimate as follows : note that @xmath232 on the y - axis because of the symmetry of the problem relative to this axis . therefore , from the second order taylor expansion of @xmath233 in @xmath0 , we obtain an approximate formula @xmath234/\delta_o^2 \approx - v_{y , y}(0,0)/\delta_o^2=-\upsilon/\delta_o^2 $ ] , where we take into account that @xmath235 . we can rewrite this approximate formula for @xmath236 as the following exact formula : @xmath237 , where @xmath238 is an unknown coefficient of order unity . our numerical simulations of reconnection with constant resistivity show that @xmath238 is indeed about unity if @xmath20 is estimated by equation ( [ j_o ] ) . thus we immediately find that @xmath106_o= \rho\nu[v_{y , yxx}(0,0)+v_{y , yyy}(0,0)]\approx \rho\nu v_{y , yxx}(0,0)\approx-\rho\nu\upsilon/\delta_o^2 $ ] , which is equation ( [ viscosity_term ] ) . here we also use @xmath239 . here as in appendix [ appendix_a ] , we assume that spatial derivatives are to be taken with respect to all indexes that are listed after the comma signs in the subscripts , e.g. @xmath240 . we derive equation ( [ j_yy ] ) in two steps . first , we estimate @xmath241 at points o and m ( see fig . [ fig_reconnection_layer ] ) , since we will need these estimates below . consider the formula @xmath242 , which represents the fact that the magnetic field is divergence - free . take the @xmath74 and @xmath243 derivatives of this formula and integrate the resulting equations along the interval om shown in fig . [ fig_reconnection_layer ] . we obtain b_x , y^(m)-&= & _ b_x , yxdx = -_o^m b_y , yydx + & & -_o b_y , yy^(m ) , [ app_b_bxy ] + b_x , yyy^(m)-b_x , yyy^(o ) & = & _ o^m b_x , yyyxdx = -_o^m b_y , yyyydx + & & -_o b_y , yyyy^(m ) , [ app_b_bxyyy ] where the variables with the superscripts @xmath244 and @xmath218 are taken at points o and m respectively and @xmath245 [ see eq . ( [ second_mhd ] ) ] . note that @xmath20 is the half - thickness of the reconnection layer , equal to the abscissa of point m ( see fig . [ fig_reconnection_layer ] ) . in making the estimates of the integrals on the right - hand - sides of eqs . ( [ app_b_bxy ] ) and ( [ app_b_bxyyy ] ) , we take into account that @xmath246 because @xmath217 on the y - axis . now , using eq . ( [ app_b_bxy ] ) , we estimate that @xmath247 . let @xmath5 be the global scale of the magnetic field outside the reconnection layer . then , since point m is located outside the reconnection layer , we have @xmath248 and @xmath249 . next , using these estimates , the above estimate for @xmath250 and eq . ( [ app_b_bxyyy ] ) , we obtain the following formula : b_x , yyy^(o)~b_x , yyy^(m)~/l^2(_o / l^2)b_y , yy^(m ) . [ app_b_bx_estimates ] second , we estimate @xmath251 at point o. consider ampere s law formula @xmath252 . we take the @xmath75 derivative of this equation and integrate the result along the interval om shown in fig . [ fig_reconnection_layer ] . we find that _ o^mj_z , yydx & = & _ o^mb_y , yyxdx-_o^mb_x , yyydx + & = & b_y , yy^(m)-_o^mb_x , yyydx . the integral @xmath253 can be estimated by using formula ( [ app_b_bx_estimates ] ) . the integral of @xmath251 can be estimated as @xmath254 . as a result , we obtain _ o j_z , yy^(o ) b_y , yy^(m)_o _ ob_y , yy^(m)b_y , yy^(m ) , [ app_b_app_b_jzyy ] where we use @xmath31 and latexmath:[$\delta_o\beta / l^2=(\delta_o j_o / l^2)(\beta / j_o)\approx ( b_m / l^2)(\beta / j_o)\approx \pm b_{y , yy}^{(m)}(\beta / j_o)\ll as suggested by uzdensky @xcite , there exists a nice graphical interpretation of the fact that the y - scale of the current @xmath65 is about the same as the scale of the outside magnetic field , i.e. @xmath258 and that equation ( [ j_yy ] ) holds . there can be two different cases of the reconnection layer geometry . first , the half - length of the reconnection layer @xmath4 can be approximately equal to the global scale @xmath5 of the outside field . in this case the reconnection is sweet - parker and @xmath19 is the only available scale in the y - direction . therefore , in this case @xmath259 . in the second case the reconnection layer half - length is much smaller than the global scale , @xmath260 , and the reconnection is fast ( relative to the sweet - parker reconnection ) . in this case , the z - current @xmath261 on the y - axis drops abruptly , as the y - coordinate passes value @xmath4 and point @xmath262 moves from the region inside the reconnection layer to the region of the outflowing plasma that is located between the petschek shocks ( see fig . [ fig_reconnection_layer ] ) . however , the z - current @xmath65 stays large inside the shocks . in other words , @xmath65 is a smooth function ( on the global scale @xmath5 ) along the lines that lie inside the reconnection layer and extend into the shock separatrices . thus , in this case , despite @xmath260 , the y - scale of the z - current at the reconnection layer central point o is still @xmath5 . this graphical interpretation is well demonstrated by the bottom - left plot of the current for the petschek - kulsrud ( `` p - k '' ) reconnection case in fig . [ fig_rosette ] .
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in this paper quasi - stationary , two - and - a - half - dimensional magnetic reconnection is studied in the framework of incompressible resistive magnetohydrodynamics ( mhd ) . a new theoretical approach for calculation of the reconnection rate
is presented .
this approach is based on local analytical derivations in a thin reconnection layer , and it is applicable to the case when resistivity is anomalous and is an arbitrary function of the electric current and the spatial coordinates .
it is found that a quasi - stationary reconnection rate is fully determined by a particular functional form of the anomalous resistivity and by the local configuration of the magnetic field just outside the reconnection layer .
it is also found that in the special case of constant resistivity reconnection is sweet - parker and not petschek .
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You are an expert at summarizing long articles. Proceed to summarize the following text:
young s experiment , originally the definitive proof of the wave nature of light , commands an essential role in the discussion of the foundations of quantum mechanics . for example , in the bohr - einstein - dialogue @xcite , the double - slit experiment was used as a gedanken experiment with individual quanta . in that discussion , einstein wanted to argue that quantum mechanics is inconsistent in the sense that one can have path information and observe the interference pattern at the same time , while bohr was always able to demonstrate that einstein s point of view was not correct . indeed , if one carefully analyzes any situation where it is possible to fully know the path the particle took , the interference pattern can not be observed . likewise , if one observes the full interference pattern , no path information is available . young s experiment today is considered the most beautiful demonstration of the concept of quantum superposition [ fig . 1 ] . whenever we do not know , not even in principle , which of the two paths the particle takes , the quantum state can be written as @xmath0 in that case , no information whatsoever is available about the slit the particle passes through . indeed , if one asked which path the particle takes in an experiment for a specific run , one would find the particle in either slit with equal probability . yet , obviously , this requires the use of detectors . if one places one detector each into each slit and if one describes the detector states by quantum mechanics , then , clearly , the quantum state of the whole system becomes @xmath1 ) only has the meaning that the property of the particle to take a definite path is related to a property of the detectors . the two detector states @xmath2 and @xmath3 describe the detector having registered the particle passing through the left and right slit , respectively . these can even be states of an internal degree of freedom of the interfering particle ( e.g. spin or polarization states or internal atomic states ) . a proposal for such an experiment has been made by scully _ @xcite performed a neutron interference experiment and drr _ et al . _ @xcite performed an atomic interference experiment where the disappearance of the interference pattern has to be attributed to the correlations between the internal neutron or atomic states , which serve as which - path detectors , and the paths taken inside the interferometer . in these experiments the loss of interference is due to the fact that path information is available , in principle , independent of the fact whether the experimentalist cares to read it out or not . if the two detector states are orthogonal , then the two particle states can not interfere , as eq . ( [ detectors ] ) describes then a maximally entangled state and thus one could determine the path of the particle by observing the detector state . only if the two detector states are not orthogonal @xcite or if they are projected by a measurement onto a state that is orthogonal to neither one of them @xcite then path interference of a certain contrast may reappear , as then the complete knowledge about the path is not available . technological progress in the times since the bohr - einstein - dialogue made it possible to realize quantum interference with many different particles all the way to massive molecules , like the fullerenes @xcite c-60 and c-70 . it is interesting to note that in the latter experiment , the fullerene molecules are at temperatures as high as 900 k. this implies that they are not completely decoupled from the environment . on the contrary , they typically emit a few photons on their path from the source to the detector @xcite . so why do interference fringes still appear [ fig . 2 ] ? could one not use the emitted photons to trace the path of the fullerene ? the reason can easily be understood by referring to eq . ( [ detectors ] ) . the wavelength of the emitted photons is typically of the order of a few micrometers , which has to be compared to the path separation , which is much lower . therefore , the states of the two photons emitted by a fullerene on either of the interfering paths are nearly identical , implying that the photons carry virtually no information into the environment . the modulus of the scalar product between the two states of the photons corresponding to the emission by a fullerene on either of the interfering paths can be used to quantify the information about the path of the fullerene , which can in principle be extracted if the photons were observed . only if the scalar product is non - zero , then an interference pattern of a certain contrast may appear , as then the path is not completely known . in general , the contrast ( visibility @xmath4 ) of the interference pattern is equal to the modulus of the scalar product between the two detectors states , @xmath5 . we now calculate the scalar product between the two photon states which serve as detector states in the fullerene experiment . for the reason of simplicity we consider the fullerene experiment as a double - slit experiment . suppose that the interfering fullerene emits @xmath6 photons at the moment it reaches the screen with the two slits . that is , the photons are emitted by the fullerene either at the left slit or at the right slit . then the visibility @xmath4 of the fullerene interference pattern at the observation screen is equal to the modulus of the following scalar product @xmath7 because the two possible states are the same for every of the n photons , one can transform eq . ( [ vuk ] ) into @xmath8 where @xmath9 are the two amplitudes ( spherical waves ) of a photon at observation point @xmath10 , which are emitted from the point source localized at the position @xmath11 of the left slit and @xmath12 of the right slit , respectively . here @xmath13 is the wave - number of the photon . to calculate the integral in eq . ( [ srce ] ) we use the substitution @xmath14 and @xmath15 , @xmath16 and @xmath17 . the integration volume is @xmath18 . using straightforward algebra one obtains @xmath19 where @xmath20 and @xmath21 is the separation between two slits . such dependence of the visibility on the number @xmath6 of emitted photons and their wave - number @xmath13 is in agreement with decoherence observed in an atom interferometry @xcite . it is now clear from eq . ( [ mrmor ] ) that in the extreme case of the wave length much smaller then the slit separation and/or sufficiently large number of emitted photons the visibility @xmath4 vanishes . yet , in the fullerene experiment another extreme case is reached . there the slit separation @xmath22 , the photons wave length is of the order of @xmath23 , and the estimated number of photons emitted during the entire time of flight of the fullerene are 1 - 2 . therefore @xmath24 and the high visibility remains preserved . the possible choice between path information and the observability of interference patterns is one of the most basic manifestations of quantum complementarity , as introduced by niels bohr . following our discussion , it is clear that it is the experimentalist who decides which observable to measure . he can decide , for example , whether to put a detector into the interfering paths or not . this role of the observer has led to numerous misunderstandings about the copenhagen interpretation of quantum mechanics . very often , and erroneously , a strong subjective element is brought into the discussion , implying that even the consciousness of the observer has a role in the quantum measurement process . one has to be very careful at this point . just to follow our example , the observer can decide whether or not to put detectors into the interfering path . that way , by deciding whether or not to determine the path through the two - slit experiment , he can decide which property can become reality . if he chooses not to put the detectors there , then the interference pattern will become reality ; if he does put the detectors there , then the beam path will become reality . yet , most importantly , the observer has no influence on the specific element of the world which becomes reality . specifically , if he chooses to determine the path , he has no influence whatsoever which of the two paths , the left one or the right one , nature will tell him is the one where the particle is found . likewise , if he chooses to observe the interference pattern he has no influence whatsoever where in the observation plane he will observe a specific particle . both outcomes are completely random . we therefore argue that the observer has a qualitative influence on nature by deciding via his choice of apparatus which quality can manifest itself as reality , but he has no quantitative influence in the sense of which specific result will be the outcome . it therefore appears that the objective randomness of quantum measurement provides a limit to the control any experimentalist has . bohr @xcite writes succinctly : ... a subsequent measurement to a certain degree deprives the information given by a previous measurement of its significance for predicting the future course of phenomena . obviously , these facts not only set a limit to the extent of the information obtainable by measurement , but they also set a limit to the meaning which we may attribute to such information . we meet here in a new light the old truth that in our description of nature the purpose is not to disclose the real essence of the phenomena but only to track down , so far as it is possible , relations between the manifold aspects of our experience . we will now argue that the impossibility of joint perfect observation of both path and the interference pattern is a natural consequence of the finiteness of the information content of a quantum system . on the basis of a specific measure of information we will define information content of a quantum system . that information can fully be contained either in the path or in the interference pattern . in both of them only partially to the extent defined by the fundamental limit on the information content . therefore we will give a quantitative information - theoretic formulation of quantum complementarity in young s experiment . in a double - slit experiment the path information is a dichotomic , i.e. a two - valued observable while the position in the interference pattern is a continuous one , which makes the consideration more complicated . for that reason we will modify our set - up to that of an interferometer [ fig . 3 ] where both path information and interference observation are dichotomic . afterwards we will extend our analysis to a double - slit experiment . if in fig . 3 the incoming state @xmath25 has amplitude @xmath26 and the incoming state @xmath27 has amplitude @xmath28 ( @xmath29 , then by the usual rules of a symmetric beam splitter @xcite , the outgoing states @xmath30 and @xmath31 become @xmath32 where we allow for an arbitrary , but constant , phase difference @xmath33 between amplitudes @xmath26 and @xmath28 . it now follows that the probabilities @xmath34 , @xmath35 , @xmath36 , and @xmath37 to find an individual particle in any of the four beams are : @xmath38 evidently , because of unitarity , @xmath39 and @xmath40 . how can we see now the complementarity between the path information and the interference phenomenon ? it is suggestive to assume that our ability to determine which path the particle takes is related to the modulus @xmath41 of the difference between the probabilities in path 1 and path 2 . this difference results in the minimal value of 0 if both probabilities are equal and in the maximal value of unity if one of the probabilities is 1 . in the same way as we assume the information available about the path to be proportional to the modulus of the difference @xmath41 , we may also assume the information in the interference pattern to be proportional to the modulus of the difference @xmath42 . there is some complementarity between @xmath41 and @xmath42 , and we will now express it quantitatively such that the total information is a constant . indeed , we find , if we introduce our new measure of information @xcite we are led to a quantitative statement of the complementarity principle . our new measure of information , which is suitable to define the information gain in a quantum experiment , takes probability squares as a quantitative statement of our knowledge . in @xcite it was shown that this particular measure of information is related to the estimation of the future number of occurrence of a specific outcome in a repetition of a binary experiment with two probabilistic outcomes . we now introduce the following quantitative amounts of information @xmath43 where we have introduced the probabilities @xmath44 and @xmath45 as those probabilities where we use an additional phase shifter of phase @xmath46 in , say , beam 2 , resulting in the probabilities @xmath47 the reason that we consider also the probabilities @xmath44 and @xmath45 is that for any specific phase shifts @xmath33 between the two incoming amplitudes , even without path information , our knowledge whether the particle will be found in beam 3 or 4 might not be maximal ( fig . [ fig2 ] ) . this knowledge however can then be re - established if an additional phase shift of @xmath46 is introduced between the two amplitudes . now , for the sum of the three individual measures of information , we obtain @xmath48 such a complementarity relation resulting in a constant is possible only if our new measure ( [ measures ] ) is used and could not be obtained if , for example , shannon s measure of information were used @xcite . an important property of the information content of a quantum system as defined by eq . ( [ content ] ) is that it neither depends on the incoming amplitudes @xmath26 and @xmath28 , nor on the phase factor @xmath33 between them . this means that the total information is invariant under unitary transformations and thus equal for all possible pure incoming states . therefore different pure incoming states might have different individual measures of information @xmath49 , @xmath50 and @xmath51 but their sum is always 1 bit of information . here @xmath49 describes the path information and @xmath50 and @xmath51 together describe the visibility of the interference effect . we may therefore introduce the new variables @xmath52 and @xmath53 , and we obtain the final result ( see also @xcite . ) @xmath54 which is a quantitative statement of the principle of complementarity in young s experiment . one may reinterpret eq . ( [ central ] ) such that a single particle in young s experiment is just the representative of one bit of information and the experimentalist has the choice by deciding whether to determine the path or not , whether this information resides in the path or in interference or in both of them partially to the extend defined by eq . ( [ central ] ) . we will now extend our consideration to the situation of a double - slit experiment [ fig . we assume that the amplitude of the interfering particle is @xmath26 in the left slit and @xmath28 in the right slit @xmath55 , where again we allow for an arbitrary phase difference @xmath33 between the two amplitudes . a typical interference pattern in the fraunhofer limit has a sinusoidal form with a periodicity of @xmath56 where @xmath57 is the de - broglie wave - number , @xmath21 is the separation between the two slits and @xmath58 is the distance between the plane with slits and the observation plane . consider now two pairs of points @xmath59 , @xmath60 and @xmath61 in the observation plane , as shown in fig . 1 . on the basis of our new measure of information we now introduce the amount of information @xmath62 for the pairs of points @xmath63 and @xmath64 , and similarly @xmath65 for @xmath66 and @xmath67 . here , for example , @xmath68 is the conditional probability to detect particle at @xmath63 given that the particle is to be found either in @xmath63 or @xmath64 . therefore @xmath69 is the measure of the information that the particle will be found in the specific point @xmath63 or in the specific point @xmath64 given that we know it will be found at @xmath63 or @xmath64 anyway . the probability density to detect the particle at point @xmath70 in the observation plane in the fraunhofer limit is given by @xmath71.\ ] ] here the probability distribution is normalized such that the total probability to find the particle somewhere within the interval @xmath72 $ ] of one period is unity . if we now use @xmath49 for the amount of information contained in the path and @xmath69 in the pair of observation points @xmath63 , @xmath64 and @xmath73 in the pair @xmath66 , @xmath67 , then we obtain again that @xmath74 . we notice that the four selected points @xmath63 , @xmath64 , @xmath66 and @xmath67 for which the probability is calculated are just separated by @xmath75 and can be selected for any choice of @xmath70 . like in the case of the interferometer , we will now summarize all individual measures of information @xmath69 and @xmath73 for all @xmath70 and thus obtain the information contained in the full interference pattern . we still use @xmath76 as given above for the measure of information contained in the path . yet now we suggest the information contained in the interfering path to be defined by the integral @xmath77 ^ 2 dy \label{integral}\ ] ] note that the integrand in eq . ( [ integral ] ) contains the combinations @xmath78 ^ 2 + [ p(y+y/4)-p(y+3y/4)]^2\ ] ] for every @xmath70 within the interval @xmath79 , which correspond exactly to the sum @xmath80 introduced above . one can easily calculate that @xmath81 . therefore we have again @xmath82 for the sum of the measures of information contained in the path and in the interference pattern . the discussion presented above obviously is just one specific example of quantum complementarity at work . it is obvious that this can be extended to much more complicated situations , as for example to the notion of quantum entanglement @xcite . from a fundamental perspective , this approach suggests that the most basic notion of quantum mechanics is information @xcite . n. bohr , atomic physics and human knowledge ( wiley , new york , 1958 ) . m. o. scully , b. g. englert and h. walther , nature * 351 * , 111 ( 1991 ) . j. summhammer , g. badurek , h. rauch , u. kischko , and a. zeilinger , phys . a * 27 * , 2523 ( 1983 ) . s. drr , t. nonn and g. rempe , nature * 395 * , 33 ( 1998 ) . w. k. wootters and w. h. zurek , phys . rev . d * 19 * , 473 ( 1979 ) . m. o. scully , and k. drhl , phys . rev . a * 25 * ( 1982 ) 2208 . m. arndt , o. nairz , j. voss - andreae , c. keller , g. van der zouw , and a. zeilinger , nature * 401 * , 680 ( 1999 ) . o. nairz , b. brezger , m. arndt , and a. zeilinger , phys . . lett . * 87 * , 160401 ( 2001 ) m. arndt , o. nairz , and a. zeilinger , in : `` quantum [ un]speakables '' , r. bertlmann , a. zeilinger eds . springer verlag ( spring 2002 ) . d. a. kokorowski , a. d. cronin , t. d. roberts , and d. e. pritchard , phys . * 86 * , 2191 ( 2001 ) . n. bohr , atomic theory and the description of nature ( cambridge university press , new york , 1934 ) . a. zeilinger , am . * 49 * , 882 ( 1981 ) . . brukner and a. zeilinger , phys . lett . * 83 * , 354 ( 1999 ) . . brukner and a. zeilinger , phys . a * 63 * , 022113 ( 2001 ) . b. g. englert , phys . lett . * 77 * , 2154 ( 1996 ) . brukner , m. zukowski , and a. zeilinger , lanl preprint ( 2001 ) quant - ph/0106119 . a. zeilinger , found . * 29 * , 631 - 643 ( 1999 ) .
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young s experiment is the quintessential quantum experiment .
it is argued here that quantum interference is a consequence of the finiteness of information .
the observer has the choice whether that information manifests itself as path information or in the interference pattern or in both partially to the extent defined by the finiteness of information .
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You are an expert at summarizing long articles. Proceed to summarize the following text:
rayleigh - taylor ( rt ) instability occurs when a heavy fluid is supported by a lighter fluid in a gravitational field , or , equivalently , when a heavy fluid is accelerated by a lighter fluid . rt instability and the related processes have found applications in various astronomical systems , such as the expansion of supernova remnants ( e.g. , ribeyre et al . 2004 ) ( where inertial acceleration plays the role of the gravitational field ) , the interiors of red giants ( e.g. , chairborne and lagard 2010 ) , the radio bubbles in galaxy clusters ( pizzolato and soker 2006 ) . the evolution of the rt instability is influenced by many different factors . for example , viscosity tends to reduce the growth rate and to stabilize the system ( e.g. , chandrasekhar 1961 ) . growth rate of the short - wavelength unstable perturbations decreases because of the compressibility ( e.g. , shivamoggi 2008 ) . a dynamically important radiation field affects rt instability as well ( jacquet and krumholz 2011 ) . however , the most important effects in the astrophysical context are probably those due to the presence of the magnetic fields . one can decompose the magnetic field lines into a component perpendicular to the interface and a component parallel to it . we will deal only with the effect of a tangential magnetic field . incompressible rt instability in a plane parallel to a uniform tangential magnetic field in both fluids has been studied analytically by chandrasekhar ( 1961 ) . the linear stability theory shows that a tangential magnetic field slows down the growth rate of the rt instability . the growth rate @xmath0 for the modes with wavenumber @xmath1 parallel to the magnetic field lines is given by @xmath2 here , we use cartesian coordinates and denote the quantities of the plasma below the discontinuity ( @xmath3 ) with a subscript 1 and those in above the discontinuity ( @xmath4 ) with a subscript 2 . the magnetic field permeating the plasma is uniform and tangent to the discontinuity , so @xmath5 , while gravity is perpendicular to it , so @xmath6 . if we set @xmath7 , the classical dispersion relation for rt instability is obtained , i.e. @xmath8 there are a number of astrophysical systems in which the magnetic rt instability is expected to be important , among them are the accretion onto the magnetized compact objects ( wang and nepveu 1983 ) , buoyant bubbles generated by the radio jets in clusters of galaxies ( robinson et al . 2005 ) , and the thin shell of ejecta swept up by a pulsar wind ( bucciantini et al . but one should note magnetic rt growth rate ( i.e. , equation ( 1 ) ) is based on the ideal mhd approximation , in which the multifluid nature is neglected for simplicity . however , a partially ionized plasma represents a state which often exists . thus , we are interested to know how the growth rate of rt instability is modified in a partially ionized medium , in particular when the coupling between the ionized and the neutral components is not complete . there are a few studies related to this issue . for example , chhajlani ( 1998 ) studied magnetic rt instability considering the surface tension and finite larmor radius correction ( flr ) in the absence of gravity and the pressure gradient for the neutral particles . it was found that an increase in the collision frequency causes a decrease in the growth rate of the system . just recently , diaz , soler & ballester ( 2012 ) ( hereafter dsb ) studied rt instability in a partially ionized compressible ( and also incomessible ) plasma . their purpose was to study the stability thresholds and the linear growth rate of the rt unstable modes in a two - fluid plasma consisting of ions and neutrals . they also studied the effects of compressibility and the collision between the particles . they calculated the growth rate as a function of the wavelength of the perturbations for different values of the gravity and concluded that collisions are not able to fully suppress the instability . but in the highly collisional regime the growth rate is significantly lowered by an order of magnitude in comparison to the classical result , specially for low values of gravity . also , the linear growth rate is significantly lowered by compressibility and ion - neutral collisions compared to the incompressible collisionless case . then , as an astrophysical implication , dsb applied their results to the solar prominences . in this article , although we follow a similar problem to dsb , not only our presentation of the results are different from that study , but also we apply the results to another astronomical object which is subject to rt instability . more specifically , we study rt instability in a two - fluid magnetized medium consisting of the ions and the neutrals and obtain the growth rate of the unstable modes for different wavenumbers ( not different gravity like dsb ) and compare them with the fully ionized or neutral cases . we also determine the most unstable mode and its relation to the wavenumber . then , possible effects of ionization and the collisions on the growth rates are analyzed . more important , we apply our results to an astronomical object different from dsb system , i.e. the interaction zone between loop1 and local bubble , which seems to be subject to rt instability and the maximum growth rate and the corresponding wavenumber are obtained . breitschwerdt , freyberg and egger ( 2000 ) showed the local clouds surrounding the solar system have been formed as a result of the growing magnetic rt instability in the interaction zone between the loop1 and local bubble . we now know that these clouds are partially ionized ( slavin 2008 , welsh 2009 ) . thus , we can apply our results to the interaction region between the loop i and the local bubble . in the second section , we will present the basic equations and the assumptions of the model . in the third section , we will analyze the growth rate of the unstable perturbations and finally in forth section we perform an application to interaction region between the loop i and the local bubble . we follow the analysis of chandrasekhar ( 1961 ) , but including neutrals and ions which are coupled via collisions . our basic equations are similar to the other related two - fluid studies ( e.g. , shadmehri and downes 2007 ) , but here there is a gravitational acceleration in the equations of motion for both the ions and the neutrals . in order to proceed analytically , it is assumed that the system is incompressible and the non - ideal dissipative process related to the evolution of the magnetic field are neglected . so , the convective term in the induction equation dominates the resistive one . we suppose the neutral and ion components have not velocity in the unperturbed state . the magnetic field is assumed to be parallel to the interface , i.e. @xmath9 . finally , all the unperturbed physics quantities are assumed to be spatially uniform in each medium . the basic equations are @xmath10 @xmath11 @xmath12 @xmath13 @xmath14 @xmath15 where @xmath16 is the collision rate coefficients per unit mass so that @xmath17 is the neutral - ion collision frequency . the collision frequency determines the coupling between each component and the magnetic field . here , @xmath18 is a uniform vertical gravitational acceleration . now , we perturb the physical variables as @xmath19 . thus , linearized equations for the neutrals become @xmath20 @xmath21 @xmath22 where @xmath23 and @xmath24 are the @xmath25 and the @xmath26 components of the perturbed velocity of the neutrals , respectively . also , the linearized equations for the ions are @xmath27 @xmath28 @xmath29 @xmath30 where @xmath31 and @xmath32 are the @xmath25 and the @xmath26 components of the perturbed velocity of the ions , respectively . after some mathematical manipulations , we can reduce the above differential equations to a set of two differential equations for @xmath24 and @xmath32 , i.e. @xmath33 @xmath34 where @xmath35 . up to this point , our basic linearized equations are similar to shadmehri and downes ( 2007 ) who studied two - fluid kelvin - helmholtz instability . however , the boundary conditions for rt instability are different from kelvin - helmholtz instability . having solutions of the above equations and by imposing appropriate boundary conditions , we can obtain a dispersion relation for rt instability . behavior of the flow at the upper and the lower layers is determined by the general solutions of the linear differential equations ( [ 8 ] ) and ( [ 9 ] ) . one can easily show that the general solution of the equations is a linear superposition of two independent solutions @xmath36 and @xmath37 . now , we must apply the following proper boundary conditions to obtain a physical solution : ( 1 ) the perturbations tends to zero as @xmath26 goes to the infinity ; ( 2 ) the @xmath26-component of the velocity is continuous at the interface ; ( 3 ) the total pressure is also continuous at the interface . thus , the general solutions become @xmath38 @xmath39 where @xmath40 , @xmath41 , @xmath42 and @xmath43 are constants to be determined from the above boundary conditions . continuity of the vertical displacement at @xmath44 ( the second boundary condition ) gives the following relations @xmath45 also , based on the continuity of the ions and neutrals pressures at the interface @xmath44 ( the third boundary condition ) , we have @xmath46 @xmath47 where @xmath48 is the perturbed magnetic pressure . having solutions ( [ 10 ] ) and ( [ 11 ] ) , we can simply obtain perturbed pressures and substitute them into the above equations . therefore , we obtain @xmath49 @xmath50 and after lengthy ( but straightforward ) mathematical manipulations , we then obtain @xmath51 @xmath52x^2+\big[2y^2\lambda(\alpha_i\alpha_n+1)-\end{aligned}\ ] ] @xmath53x -2y^3(\alpha_n-1)+\\\end{aligned}\ ] ] @xmath54 where @xmath55 @xmath56 where @xmath57 is alfven velocity for @xmath58 . dispersion relation ( [ eq : main ] ) is the main equation of our stability analysis . obviously , if we neglect collision between ions and neutrals ( i.e. , @xmath59 ) , equation ( [ eq : main ] ) simply reduces to a dispersion relation for the ions which are tied to the magnetic field lines and another dispersion relation for the neutral component , i.e. @xmath60 if we set the first parenthesis equal to zero , magnetic criterion for rt instability is obtained . also , the second parenthesis gives the classical condition of non - magnetic rt instability . although dispersion relation for rt instability within one fluid approximation gives analytical solutions , it is very unlikely to obtain roots of equation ( [ eq : main ] ) analytically . so , we follow the problem numerically by assuming some numerical values for the input parameters . we assume @xmath61 and the dispersion relation ( [ eq : main ] ) is numerically solved for different values for the parameters @xmath62 , @xmath63 and @xmath64 . obviously , we would have an unstable mode if @xmath25 has a positive real part . figure 1 shows non - dimensional growth rate @xmath65 of the unstable modes versus the non - dimensional wavenumber @xmath66 . parameter @xmath63 denotes the ratio of densities of the neutrals and ions in layer 1 . in figure 1 , we assume @xmath67 and each curve is labeled by the corresponding non - dimensional collision rate @xmath62 . figure 2 shows growth rate of the perturbations versus the wavenumber but for @xmath68 . in this case , the ions are stable and the neutrals determine growth rate of the perturbations . also , figure 3 and 4 show the growth rate of the neutrals and the ions for @xmath69 , respectively . again , we can see the stabilizing role of the collision between the ions and the neutrals . note that when there is no collision between the ions and neutrals ( i.e. , @xmath70 ) , each component of our two - fluid system behaves independently . but when collision between ions and neutrals is considered , we found two unstable modes up to a certain wavenumber that are related on the neutral and ion fluids separately , with the neutral ones having a tendency with much larger growth rate . curves with the same color are corresponding to the same value of @xmath62 . since magnetic field has a stabilizing role , we can see that unstable mode corresponding to the ions has a smaller growth rate in comparison to the neutrals unstable mode because of the coupling to the magnetic field lines . dsb obtained plots of dispersion relation of the unstable modes versus the non - dimensional gravity but we plot unstable modes versus wavenumber . therefor , we can conclude : + ( 1 ) the neutral mode is unstable for all wavenumbers but for ions we can find a critical wavenumber for which the instability becomes ineffective . this result is valid irrespective of the ionization fraction and the collision rate . + ( 2 ) growth rate of the unstable perturbations for the ions tends to become zero as the collision rate increases and thereby , behavior of the system is determined by the growth rate of the neutrals . moreover , in this case , the profile of the growth rate for the neutrals is similar to the ions without collision . + ( 3 ) however , as the collision frequency increases , not only growth rate of ions reduces but the unstable perturbations for the neutrals are significantly reduced in particular at short wavelengths . + in the next section we apply our results to an astronomical object . our solar system is embedded in an ionized cloud named local cloud . in vicinity of local clouds there are also other cloudlets of comparable size . winds and supernovae events that are associated with clusters of massive early - type stars have a profound effect on the surrounding interstellar medium ( ism ) , including the creation of large cavities . these cavities , which are often referred to as `` interstellar bubbles '' , are typically @xmath71 in diameter and have low neutral gas densities of @xmath72 ( weaver et al . the local clouds are inside a local x - ray emitting cavity which called the local bubble . breitschwerdt et . al ( 2000 ) presented observational evidences based rosat pspc data that manifest existence of an interaction shell between our local interstellar bubble and the adjacent loop i superbubble . they showed that due to the overpressure in loop i , a rayleigh - taylor instability would operate , even in the presence of tangential magnetic field . their calculations showed that the most unstable mode has a growth time about @xmath73 years ( depending on magnetic field strength ) which was in agreement with the interaction time between the two bubbles . moreover , the wavelength of the fastest growing mode was about 2.2 pc which was comparable to or less than the thickness of the interaction zone . also , breitschwerdt et al . ( 2006 ) have performed 3d high resolution hydrodynamic simulations of the local bubble ( lb ) and the neighboring loop i ( l1 ) and reproduced the observed sizes of the local and loop i superbubbles , the generation of blobs like the local cloud as a consequence of a dynamical instability ( breitschwerdt et al . nevertheless , reports suggest the local clouds are partially ionized ( e.g. , slavin 2008 , welsh et al . since the clouds are formed due to rt instability ( breitschwerdt et al . 2000 and 2006 ) , we conclude the interaction zone between loop i and local bubble is partially ionized and because loop i is hot , the interaction shell must be ionized . now we are interested to know what effect does the ionization may have on the growth rate of system . + we begin with breitschwerdt and slavin s assumptions for our system and obtain a dispersion equation from equations ( [ eq:13])and ( [ eq:14 ] ) for a case where @xmath74 and @xmath75 . thus , @xmath76 @xmath77 where @xmath78 @xmath79 where @xmath80 is alfven velocity for @xmath81 . we have the following input numbers ( breitschwerdt et al . 2000 ) @xmath82 now , we can estimate @xmath63 and @xmath62 . slavin ( 2008 ) found the ionization fraction is around @xmath83 for hydrogen and around @xmath84 for helium . also , welsh et al . ( 2009 ) presented an amount of 0.1 for ionized fraction . if we suppose the ionized fraction is 0.1 , then value of @xmath63 becomes 9 . we can write the collision frequency ( shadmehri et al . 2008 ) , @xmath85 where @xmath86 . we neglected ionization of helium . thus , @xmath87 having the above input numbers , we can solve equation ( @xmath88 ) to find growth time of the fastest growing mode . figure 5 shows the results of such a calculation for the specified parameters . then , the the most unstable mode of system is @xmath89 we found shorter growth time in comparison to the classical magnetic rt instability ( around @xmath90yrs ) . the wavenumber of the fastest growing mode is @xmath91 thus , the size of structures formed by rt instability reduces in comparison to the classical magnetic rt instability that is @xmath92 . when the nondimensional collision frequency lambda is large , the ionization fraction has also a vital role . ionized particles are coupled to the magnetic field lines , but their coupling to the neutral particles is determined via collisions ( i.e. , @xmath62 ) . now , we can consider two identical systems with the same input parameters except their ionization fractions . if both systems have a large collision rate , the system with a larger ionization fraction is more affected by the magnetic field lines in comparison to the system with a smaller ionization fraction . in other words , although the ionized and the neutral particles are coupled to the same level , but when the ionization fraction is larger the system is more under influence of the magnetic field lines . figure 5 clearly shows this effect . each curve is labeled by its ionization fraction . here , the non - dimensional collision frequency is @xmath93 . we can see that as the ionization fraction increases , the growth rate of the unstable mode decreases simply because more particles are affected by the magnetic field lines . it is because of the two - fluid nature of the system . the effect is more significant when the collision frequency decreases and the coupling between the ionized and the neutral particles is not complete . thus , it seems that one - fluid approach is not adequate even when the collision frequency is large but the ionization fraction is not large enough . however , it is difficult to determine a critical value for lambda so that beyond which the system tends to mhd case . because such a transition depends on the ionization fraction among the other input parameters . moreover , when two - fluid approach is adopted the maximum growth rate is modified . but compressibility does not lead to such an effect . in fact , compressibility becomes less effective when the density contrast of the layers increases . so , the compressibility correction in a two - fluid system subject to rt instability depends on the density contrast of the layers . and various non - dimensional collision rate @xmath62 . curves with the same color are corresponding to the same value of @xmath62.,width=264,height=226 ] .,width=264,height=226 ] .,width=264,height=226 ] and different ionization fractions.,width=264,height=226 ] in this study , we investigated the magnetic rt instability in a two - fluid medium consisting of the neutrals and the ions . a general dispersion relation is obtained . by analyzing the unstable roots of the dispersion equation , two unstable modes are found that are related to those of the neutral and ion fluids separately , with the neutral ones having a much higher growing rate . for each ionization fraction and collision rate , the stability of the system only depends on the behavior of the neutrals . for lower values of collision rate the curves are similar to the collisionless case . we found that the growth rate of the unstable perturbations decreases when the collision rate increases . also , the instability of the system strongly depends on the ionization fraction . when the ionization fraction increases , for a given collision rate , the growth rate of the perturbations decreases . finally , we apply our results for interaction zone between the loop i and the local bubble that is caused form local clouds . we obtained a shorter value for the growth time and size of the clouds . although classical magnetic rt instability has been applied to this system for explaining some of the observed structures , our analysis shows that rt instability may operates less effectively if the two - fluid nature of the system is considered . but we note that our results are valid within linear regime and non - linear numerical simulations are needed to confirm the linear results . we note that magnetic effects only suppress the linear growth rate for perturbations aligned with the magnetic field . those perpendicular to it are unaffected . indeed , as shown by stone & gardiner ( 2007 ) , in 3d the net effect of magnetic fields is actually to enhance the non - linear growth rate by suppressing secondary kh instabilities . thus in the real world , it seems like magentic rt instability never occur . the hydrodynamic modes perpendicular to the field always end up taking over . we also think that the hall effect is an interesting problem , but it is beyond the scope of the present study . our analysis is restricted to a two - fluid case , i.e. a system consisting of ion and neutral particles . we could also start from the one fluid mhd equations , but considering modified induction equation with resistivity , hall and ambipolar terms . in that framework we could study possible effects of non - ideal terms ( including hall term ) . but we think it deserves a separate analysis independent of the present study .
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we study linear theory of the magnetized rayleigh - taylor instability in a system consisting of ions and neutrals .
both components are affected by a uniform vertical gravitational field .
we consider ions and neutrals as two separate fluid systems where they can exchange momentum through collisions .
however , ions have direct interaction with the magnetic field lines but neutrals are not affected by the field directly .
the equations of our two - fluid model are linearized and by applying a set of proper boundary conditions , a general dispersion relation is derived for our two superposed fluids separated by a horizontal boundary .
we found two unstable modes for a range of the wavenumbers .
it seems that one of the unstable modes corresponds to the ions and the other one is for the neutrals .
both modes are reduced with increasing the collision rate of the particles and the ionization fraction .
we show that if the two - fluid nature is considered , rt instability would not be suppressed and also show that the growth time of the perturbations increases . as an example
, we apply our analysis to the local clouds which seems to have arisen because of the rt instability . assuming that the clouds are partially ionized , we find that the growth rate of these clouds increases in comparison to a fully ionized case .
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You are an expert at summarizing long articles. Proceed to summarize the following text:
when cosmic microwave background ( cmb ) photons pass through galaxy clusters , they compton up - scatter on hot electrons , making a small increment ( decrement ) above ( below ) the peak of the cmb primary spectrum . the size of this distortion in the cmb spectrum , the sunyaev - zeldovich ( sz ) effect @xcite , is proportional to the electron pressure integrated along the line - of - sight . besides being an important probe of the physics of the intracluster medium , the sz effect is a promising tool in cosmology : the signal produced by a cluster is practically redshift independent and can be observed to high redshifts . the planck surveyor satellite @xcite will produce a cluster catalogue with up to @xmath010.000 ( sufficiently massive ) clusters out to @xmath1 @xcite . by combining sz and x - ray observations of relaxed clusters the hubble constant @xmath2 can be derived , limits can be put on @xmath3 , and the gas mass fraction in clusters can be measured @xcite . the sz effect was first detected in three nearby clusters more than two decades ago by @xcite , but the last five years , due to advances in sub millimetre receiver technology , routine measurements have been done for @xmath4 clusters using facilities such as the ovro and bima telescopes . right now a large number of telescopes have started observing the sz signal or are under construction ( e.g. acbar , carma , suzie iii , spt , apex etc . ) , and in the near future alma will gradually become online enabling unprecedented resolution and sensitivity for making detailed observations of individual clusters . current sz surveys of galaxy clusters ( see e.g. * ? ? ? * for a set of current observations ) have observed the unresolved integrated sz signal from clusters , but with carma , alma and the spt also the sz signal as a function of radius from the cluster centre will be measured . @xcite has used the 3rd year wmap data to extract sz images from 193 clusters . stacking them they have obtained an averaged sz profile for clusters with @xmath5kev . spatially resolved sz images probe , in a way complementary to x - ray observations , the distribution of thermal energy in clusters which in turn is closely linked to the total cluster mass . furthermore , since the sz effect essentially depends on the electron pressure , the physics going into understanding it is simpler and more robust , than is the case for the x - ray emission . in this paper we use realistic high resolution n - body / sph simulations of galaxy clusters and groups together with observed temperature profiles of nearby groups and clusters to predict the radial sz profile of different types of systems of galaxies , and compare our results to the averaged profile of @xcite . we introduce a universal fitting formula , with only two free parameters , that can be employed in future observations . furthermore , we show that fitting this profile enables a precise estimate of the total mass in the system . in the next section we describe the computer experiments , that are used to construct the synthetic sz profiles . in section 3 we discuss the simulated data and present our results , and in section 4 we discuss and provide our conclusions . we use 24 galaxy group and galaxy cluster simulations to study the sz effect for systems of virial temperatures of about 1 to 6 kev . the models include 12 groups of approximately the same mass ( @xmath6 ) , 11 identical clusters with @xmath7 ( `` virgo '' clusters ) , but simulated with different gas physics , and a single large cluster of @xmath8 ( a `` coma '' cluster ) . the models are re - simulated from a low resolution cosmological dark matter simulation , where the halos are identified with a halo finder . the particles are traced back in time to a initial redshift @xmath9 , and the virial volume is repopulated with both gas and dark matter at a high resolution . the code includes radiative cooling , star formation , supernova feedback , chemical evolution and back reaction from a redshift dependent uv field . the different models are summarised in table [ tab : models ] . for a detailed description of the different gas physics going into the code in general , and the `` virgo '' cluster simulations in particular we refer to @xcite . the groups were all selected at random , the only criterion being their virial mass , and therefore they comprise a cosmological fair sample of groups with @xmath10 . the different masses of the clusters allow us to probe the mass dependence of the sz profile , while the `` virgo '' models are used to investigate how robust our predictions are for the sz profile with respect to assumptions on the underlying physics . the groups , being a statistically unbiased sample , give limits on cosmic variance for the given virial mass ( @xmath11 ) . we have divided the groups into three different classes according to their morphology and evolutionary history : groups with merging activity ( 186,231,239,262 ) , fossil groups ( 189,228,236,244 ) and normal groups ( 190,233,247,276 ) . a group is considered fossil if the difference in apparent r - band magnitude of the first and second brightest galaxy is greater than two @xcite , and a group is merging if it by visual inspection has significant merging activity in the core , or if the rms scatter in 3d radial shells of the temperature , pressure and density , is comparable to the average value in the shell . @c@c@c@c@c@ name&@xmath12\,\,$]&@xmath13\,\,$ ] & @xmath14$]&commentsgroups & 186 & 0.99 & 1.08 & @xmath15 & merging group189 & 1.07 & 1.09 & @xmath16 & cool - core , fossil group190 & 1.21 & 1.22 & @xmath17 & normal group 228 & 1.07 & 1.10 & @xmath18 & cool - core , fossil group231 & 0.99 & 1.08 & @xmath19 & merging group233 & 1.03 & 1.09 & @xmath20 & normal group 236 & 1.00 & 0.99 & @xmath21 & fossil group239 & 0.91 & 1.11 & @xmath22 & merging group244 & 1.13 & 1.08 & @xmath23 & fossil group247 & 1.01 & 1.15 & @xmath24 & normal group262 & 0.97 & 1.03 & @xmath25 & merging group276 & 1.02 & 1.07 & @xmath20 & cool - core groupclusters & ay - sw & 2.13 & 2.07 & @xmath26 & arimoto - yoshii imfay - sw-8 & 2.18 & 2.10 & @xmath27 & 8 times resolutionay - vol39 & 2.18 & 2.18 & @xmath27 & @xmath28 ay - swx2 & 2.18 & 2.04 & @xmath29 & 2 times snii feedback ay - swx4 & 2.17 & 2.09 & @xmath30 & 4 times snii feedback ay - ph0.75&2.23 & 2.10 & @xmath31 & preh . @xmath32part @ z=3 ay - ph1.5 & 2.27 & 2.14 & @xmath33 & preh . @xmath34part @ z=3 ay - ph50 & 2.19 & 2.07 & @xmath31 & preh . @xmath35part @ z=3 ay - cond & 2.26 & 2.08 & @xmath36 & thermal conduction sal - sw & 2.24 & 2.10 & @xmath37 & salpeter imf sal - wfb & 2.27 & 2.23 & @xmath31 & weak feedback coma & 5.57 & 5.27 & @xmath38 & relaxed massive cluster the sz profiles of the simulated systems can only be used as templates for an universal fitting formula , if the profiles are in accordance with observations . in fig . [ fig : szeprofs ] are shown the spherically averaged sz profiles compared to the only currently published sz profile ( @xcite ) , obtained using data from the wmap satellite . the observed points may indicate a slightly more bended profile than the simulations , but within the 1-@xmath39 error bars there is fairly good agreement . from the simulations there is a clear trend towards steeper profiles , for more massive and relaxed clusters and groups reflecting differences in the underlying dm potentials ( see fig . [ fig : dmpot ] ) . this should be recalled when constructing an average observational profile , because the average profile may not represent a true ( `` universal '' ) physical profile , but rather a smeared average of the real profiles . the central core of the sz profile ( @xmath40 can not be probed by wmap , due to its limited resolution of at most @xmath41 . to extend the dynamic range of the observations we combine gravitational potentials from the simulations with observed average x - ray temperature profiles . under the assumption of hydrostatic equilibrium in the core of the systems , we can then predict the inner part of the sz profile . the compton @xmath42-parameter , which determines the overall temperature decrement in the cmb radiation due to the sz effect , is proportional to the integrated pressure @xmath43 of the electrons along the line of sight @xmath44 assuming hydrostatic equilibrium and spherical symmetry @xmath45 we can use the proportionality of @xmath43 , and the compton @xmath42-parameter in a volume element , @xmath46 , to obtain @xmath47 we have constructed three different averaged gravitational potentials based on the normal groups , the fossil groups and the `` virgo '' clusters . for the temperature profiles we use observed clusters by @xcite , and make one average profile constructed from the galaxy groups with virial temperatures less than 2.5 kev , and one constructed from the massive clusters in the sample with virial temperatures between 3.5 and 8.5 kev . the dark matter distribution , derived from the simulations , is quite robust against the specific gas physics involved in the simulation . this is demonstrated in fig . [ fig : dmpot ] , where it is seen that all the `` virgo '' models , in spite of the different gas physics , have essentially the same gravitational potentials from @xmath48 and outwards . however , there are systematic differences between different types of systems with the same mass . fossil groups are more relaxed compared to normal groups , and their mass distribution is similar to relaxed clusters , like the `` virgo '' and `` coma '' models from @xmath48 and outwards ( see fig . [ fig : dmpot ] ) . therefore the above reconstruction procedure gives a good idea of future resolved sz observations of the cores of clusters of galaxies , and our six resolved profiles are a fair sample of what can be expected . from fig . [ fig : szeprofs ] it is clear , that albeit the sz profiles constructed from the observed temperature profiles and the dark matter density profiles are in approximate agreement with the profiles extracted from the simulations , they have a tendency to be more peaked at the centre . this can be traced to differences in the observed and simulated temperature profiles ( see e.g. eq . [ eq : hs ] ) , and to the typical masses of the observed systems . the different temperature profiles are normalised with a global temperature @xmath49 . to enable direct comparison to the observed temperature profiles by @xcite we compute @xmath49 as the average projected spectral - like temperature @xcite in the interval @xmath50 . all profiles converge to the same universal curve for @xmath51 , but in the inner part of the systems there are differences ( see fig . [ fig : tempprof ] ) . the simulated systems have all flat profiles at the core , which are in good qualitative agreement with observations of clusters of similar virial mass , but the average observed profiles have lower normalised temperatures towards the centre , with more or less the same offset between the `` coma '' cluster and the observed cluster profile ( constructed from clusters with 3.5 kev @xmath52 8.5 kev ) , and between the average simulated and observed group temperature profiles ( the latter constructed from groups with @xmath53 kev @xcite ) . the offset compared to observations and relatively flat temperature profiles are well known problems for simulations invoking radiative cooling and feedback processes ( e.g. * ? ? ? * ; * ? ? ? * ) , but it is further accentuated by an offset in mass between the observed and simulated groups and the observed clusters , and the simulated `` virgo '' clusters . as can be seen in fig . [ fig : szeprofs ] all the sz profiles have nearly the same form , but the overall normalisation , tension and slope of the different profiles depends on the mass , and the specific cluster . to construct a simple model we have tried to fit a variety of combinations of beta profiles and exponential profiles for the density combined with a polytropic or isothermal equation of state , but it does not yield a satisfactory fit . the correct temperature to use , when constructing a sz profile , is the mass weighted , and it does not necessarily agree with the temperature inferred from x - ray observations ( @xcite ) , which may explain why the above combination works well for x - ray observations , while not so for our sample of sz profiles . to circumvent this problem we are using a novel universal profile that directly fit the data , without assuming anything about the underlying temperature and density distributions , while using as few parameters as possible . the main morphological differences are in the slope / tension and in the normalisation of each profile , and it is indeed possible to construct a model , that with only two free parameters , a normalisation , and a characteristic scale can fit the full set of simulations and observations in detail . the profiles extracted from the simulated data do not per se contain any errors , but a measure of the natural scatter in a profile @xmath54 at a given radial distance , is the variance @xmath55 of @xmath56 inside the radial bin at @xmath57 . we have used this variance estimate to construct a goodness of fit parameter @xmath58 for each projected 2d and spherically averaged 3d profile @xmath59 , @xmath60 given by @xmath61 where @xmath62 is the model with parameters @xmath63 . the internal scatter in relaxed systems is much smaller , giving stronger constraints than from merging systems . this is sensible , since the observational scatter among relaxed clusters is smaller too . we have found that using a triple exponential model gives an excellent fit to the data : @xmath64\,,\end{aligned}\ ] ] where @xmath65 . while this model at first sight might appear complicated , it has two attractive features : ( i ) the 2d profile can be derived in a closed form from the 3d profile . ( ii ) most of the parameters can be fixed to global values , leaving only two free parameters for fitting all systems considered in this paper . with only two parameters in the model it can readily be applied to future observations , even if they only map the sz profile with a few observational points . by integrating along one of the axes we find the 2d profile to be @xmath66 r\,,\end{aligned}\ ] ] where @xmath67 is the modified bessel function of the second kind . minimising @xmath68 simultaneously for all models over the range @xmath69 , while varying @xmath70 and @xmath71 as global parameters , and @xmath72 , @xmath73 for each model , treating the @xmath74 , @xmath75 , and @xmath76projections as different clusters , we find the global best fit parameters to be @xmath77 in fig . [ fig : gof ] we see how these yield very reasonable @xmath58 for all projections , and the 3d profiles . the only systems that have a @xmath58 significantly larger than one are either merging , or characterised by a very smooth profile with almost no variation in the radial bin , and hence a very small @xmath55 ( see e.g. fig . [ fig : profiles ] ) . + + the two free parameters , the overall scale @xmath73 , and the overall normalisation @xmath72 scale with @xmath78 and with @xmath79 , the integrated sz effect inside @xmath78 , respectively , but there is a large scatter between different clusters , where the tightest relation is found for the fossil groups , that act as an extension of the clusters with almost the same scaling relation . they are related as @xmath80 to push the boundaries of our fitting formula for other systems we have also applied it to the @xcite data , and the profiles for the centre of the systems , derived from observed temperature profiles ( see fig . [ fig : fits ] ) . the average sz profile is described well by the formula , while there are problems with fitting the core ( @xmath81 ) of the cluster set of central profiles ( the lower right panel in fig . [ fig : fits ] ) . even though some of these combinations ( e.g. a normal group dm potential combined with a massive cluster temperature profile ) are extreme , the lack of agreement may be because in the central parts of the observed systems there are significant non - thermal contributions to the pressure balance from e.g. an agn @xcite or from cosmic rays @xcite , and hence the assumption of hydrostatic equilibrium does not apply . or it may be that the simulated systems do not include an adequate description of the physical processes in the centre of the systems . nonetheless , the contribution of the central @xmath82 to the total sz signal is approximately 5% , and the parameters are not much affected even if we can not reconstruct the innermost part of the profiles with perfection . the sz profile of the systems seems to be universally well described by only two parameters , except for possibly in the central parts of the clusters . it measures the distribution of thermal energy in the system , and is therefore related to the gravitational potential , if we assume the system is relaxed . using assumptions about hydrostatic equilibrium and using a phenomenological approach several `` fundamental plane '' relations have been constructed ( e.g. * ? ? ? * ; * ? ? ? the main ingredients for the fundamental plane has been the integrated sz effect and a characteristic scale in the system , for example @xmath83 , the radius enclosing half of the integrated signal . this has given a relation between the integrated sz effect , a characteristic scale , and the total mass with a rough scatter of 14% @xcite . with our fit to the profiles we get a very precise measurement of this characteristic scale , that take into account the different bending of the profiles . fitting the total mass as a function of @xmath79 and @xmath73 we find ( see fig . [ fig : massrel ] ) instead of @xmath79 , but it does not yield as good a fit . ] @xmath84 with only a 4% scatter . in @xcite a phenomenological model is used to construct a set of , essentially , one dimensional models for the clusters . it includes normal or tophat distributed controlling parameters , with broadening in agreement with observed clusters . to check if the smaller scatter we see for eq . [ eq : massestimate ] is due to more similarity in the simulated systems , we have tried to use the fundamental plane of @xcite , on the simulated systems . we find good agreement with a 15% scatter for our data ( see fig . [ fig : massrel ] ) . + the first resolved sz profiles will be obtained for nearby massive clusters . to make a crude test for how our fitting formula works on real data , we have made a set of mock observations taking as a starting point the most massive cluster in our sample , the `` coma '' cluster . we use five logaritmically spaced rings at a distance of @xmath85\,{r_{180}}$ ] . the resolution of the first generation of sz instruments is limited , and therefore we have chosen not to include observational points at smaller radii . to get a good measure of the scatter we generated @xmath86 mock observations , with normal distributed logarithmic error bars of 0.22 dex on the total in each ring @xmath87 , giving an relative error of @xmath88 on @xmath87 itself . for simplicity we have disregarded any correlation there may exist between the different radial bins , and artificially have fixed the relative errors to a uniform level . the five rings are then fitted ( see fig . [ fig : synth ] ) using eq . ( [ eq : profile ] ) giving @xmath73 . the integrated sz effect @xmath79 for each mock observation is found by integrating the fitted profile . inserting @xmath73 and @xmath79 derived from each mock observation into eq . ( [ eq : massestimate ] ) we find a reconstructed mass of @xmath89 , or an @xmath90 error on the reconstructed mass . there is a @xmath91 systematic offset compared to the real mass of @xmath92 . with five rings the error on the total signal is @xmath93 . the mass goes roughly as @xmath94 , and we would expect roughly a @xmath95 error on the mass , in agreement with what is found . in this paper we have constructed a simple empirical model for the radial profile of the sunyaev zeldovich effect in groups and clusters of galaxies . the model has been motivated by , and validated against a mixture of simulations and observations , and is characterised by only two parameters : the overall normalisation , and a typical length scale related to the slope of the profile . it gives a very good fit to the simulated systems , and there is a tight relation between the parameters and the total mass of the system . furthermore , the results are robust to the detailed gas physics employed in the simulations . this can be seen by considering the subset of the systems , the `` virgo '' clusters , that are started from the same initial conditions , but simulated with different implementations of the gas physics ( see table 1 ) . because of the different gas physics there is an appreciable scatter in @xmath79 , but still the mass ( @xmath96 ) of the clusters is well reconstructed ( see fig . [ fig : massrel ] ) . the simulations are in good agreement with an observed average profile extracted from wmap data for the outer part of the sz profiles @xmath97 . currently there are no published high resolution observations of the core sz profile , but we can get an indirect measure by using temperature profiles extracted from clusters observed in x - rays together with dark matter potentials from the simulations . reconstructing the sz profile , under the assumption of hydrostatic equillibrium , we see that these `` core profiles '' are relative peaked , compared to the simulations . this is traced to differences in the temperature profile of the simulated systems , compared to what is observed using x - rays . we have to await future observations of the resolved cluster cores , to determine to what extent this cusp in the central part is real , or a result of the hypothesis of hydrostatic equilibrium , which is known to be violated in the centre of clusters of galaxies , where non - thermal processes such as agn heating @xcite , and cosmic rays @xcite can play an important role for the pressure balance . we stress that the model we have presented here is readily applicable to future observations . this will give a good proxy for the mass of the observed system . it will also help in reconstructing the full sz profile from observations with low spatial resolution , to be used in conjunction with x - ray observations in the study of cluster dynamics . a fitting routine written in idl for the profile that can be applied to observed or simulated data can be found at + http://www.phys.au.dk/~haugboel/software.shtml . we thank the danish center for scientific computing for granting the computer resources that made this work possible . the dark cosmology centre is funded by the danish national research foundation . this research was supported by the dfg cluster of excellence `` origin and structure of the universe '' . kp acknowledges support from the instrument center for danish astrophysics .
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the sunyaev - zeldovich ( sz ) effect gives a measure of the thermal energy and electron pressure in groups and clusters of galaxies . in the near future sz surveys
will map hundreds of systems , shedding light on the pressure distribution in the systems .
the thermal energy is related to the total mass of a system of galaxies , but it is only a projection that is observed through the sz effect . a model for the 3d distribution of pressure is needed to link the sz signal to the total mass of the system . in this work
we construct an empirical model for the 2d and 3d sz profile , and compare it to a set of realistic high resolution sph simulations of galaxy clusters and groups , and to a stacked sz profile for massive clusters derived from wmap data .
furthermore , we combine observed temperature profiles with dark matter potentials to yield an additional constraint , under the assumption of hydrostatic equilibrium .
we find a very tight correlation between the characteristic scale in the model , the integrated sz signal , and the total mass in the systems with a scatter of only 4% .
the model only contains two free parameters , making it readily applicable even to low resolution sz observations of galaxy clusters .
a fitting routine for the model that can be applied to observed or simulated data can be found at http://www.phys.au.dk/~haugboel/software.shtml .
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You are an expert at summarizing long articles. Proceed to summarize the following text:
the search for the surface gravity effect of the free translational oscillations of the inner core , the so - called slichter modes @xcite , has been a subject of observational challenge , particularly since the development of worldwide data from superconducting gravimeters ( sgs ) of the global geodynamics project @xcite . indeed these relative gravimeters are the most suitable instruments to detect the small signals that would be expected from the slichter modes @xcite . a first claim by @xcite of a triplet of frequencies that he attributed to the slichter modes led to a controversy ( e.g. @xcite ) . this detection has been supported by @xcite and @xcite but has not been confirmed by other authors @xcite . @xcite have shown it is necessary to consider dynamic love numbers to calculate the slichter mode eigenperiods . latest theoretical computation predicts a degenerate ( without rotation or ellipticity ) eigenperiod of 5.42 h @xcite for the seismological reference prem @xcite earth model . a more recent study by @xcite states that the period could be shorter because of the kinetics of phase transformations at the inner - core boundary ( icb ) . the interest raised by the slichter modes resides in its opportunity to constrain the density jump and the viscosity in the fluid outer core at the icb . the density jump at the icb is a parameter that constrains the kinetic energy required to power the geodynamo by compositional convection . some discrepancies have been obtained for the value of this parameter . on the one hand , by analyzing seismic pkikp / pcp phases , @xcite found that it should be smaller than 450 kg / m@xmath0 , later increased to 520 kg / m@xmath0 @xcite . on the other hand , using normal modes observation , @xcite obtained 820 @xmath1 180 kg / m@xmath0 . such differences in the estimate of the icb density jump have been partially attributed to the uncertainties associated with the seismic noise @xcite . a model that satisfies both the constraints set by powering the geodynamo with a reasonable heat flux from the core , and pkp traveltimes and normal mode frequencies has been proposed by @xcite with a large overall density jump between the inner and outer cores of 800 kg / m@xmath0 and a sharp density jump of 600 kg / m@xmath0 at the icb itself . in the following we will adopt the prem value of 600 kg / m@xmath0 . the non - detection of the slichter modes raises the question of their expected amplitude , their damping and the possible mechanisms to excite them . a certain number of papers have considered the damping of the inner core oscillation through anelasticity of the inner core and mantle @xcite , through viscous dissipation in the outer core @xcite or through magnetic dissipation @xcite . @xcite and @xcite have summarized the theoretical q values expected for the slichter mode . @xcite have concluded that it should most probably be equal to or larger than 2000 . various sources of excitation have been previously considered . the seismic excitation has been studied by @xcite , @xcite and @xcite . they have shown that earthquakes can not excite the slichter modes to a level sufficient for the sgs to detect the induced surface gravity effect . for instance , even for the 1960 @xmath2 chilean event the induced surface gravity effect does not reach the nanogal level ( 1 ngal@xmath3 nm / s@xmath4 ) . surficial pressure flow acting at the icb and generated within the fluid outer core has been considered by @xcite and @xcite as a possible excitation mechanism . however , the flow in the core at a timescale of a few hours is too poorly constrained to provide reliable predictions of the amplitude of the slichter modes . @xcite have investigated the excitation of the slichter modes by the impact of a meteoroid , which they treated as a surficial seismic source . for the biggest known past collision associated to the chicxulub crater in mexico with a corresponding moment - magnitude @xmath2 , the surface excitation amplitude of the slichter mode was barely 0.0067 nm / s@xmath4 @xmath5 0.67 ngal . nowadays , a similar collision would therefore not excite the slichter modes to a detectable level . the degree - one surface load has also been investigated by @xcite . they showed that a gaussian - type zonal degree - one pressure flow of 4.5 hpa applied during 1.5 hour would excite the slichter mode and induce a surface gravity perturbation of 2 ngal which should be detectable by sgs @xcite . this determination was based on a purely analytical model of surface pressure . in this paper we will use hourly surface pressure data provided by two different meteorological centers and show that the surface atmospheric pressure fluctuations can only excite the slichter modes to an amplitude below the limit of detection of current sgs . 1.5pt in this section , we consider a spherical earth model , for which the frequencies of the three slichter modes degenerate into a single frequency , and establish a formula for the spectral energy of the amplitude of the mode when it is excited by a surface load . developed in a surface spherical harmonics expansion , a degree - one load @xmath6 contains three terms : @xmath7 where @xmath8 and @xmath9 are the colatitude and longitude , respectively . the green function formalism suited for surface - load problems @xcite has been generalized to the visco - elastic case by @xcite and has been established for the degree - one slichter mode by @xcite . the degree - one radial displacement due to load ( [ load ] ) is given by @xmath10 \nonumber \\ & & \lbrack \int_{-\infty}^{t}e^{i\nu t ' } ( \sigma_{10}(t ' ) \cos\theta + \sigma_{11}^c(t ' ) \sin\theta \cos\phi + { \sigma}_{11}^s(t ' ) \sin\theta \sin\phi ) dt ' \rbrack , \label{radialdisplacement(t)}\end{aligned}\ ] ] and the perturbation of the surface gravity is @xmath11 \nonumber \\ & & \lbrack \int_{-\infty}^{t}e^{i\nu t ' } ( \sigma_{10}(t ' ) \cos\theta + \sigma_{11}^c(t ' ) \sin\theta \cos\phi + { \sigma}_{11}^s(t ' ) \sin\theta \sin\phi ) dt ' \rbrack \nonumber \\ & & [ -\omega^2u(r_s)+\frac{2}{r_s}g_0 u(r_s)+\frac{2}{r_s}p(r_s)].\end{aligned}\ ] ] in the last two equations , @xmath12 and @xmath13 are , respectively , the radial displacement and perturbation of the gravity potential associated to the slichter mode , @xmath14 is the earth radius , and @xmath15 is the complex frequency . as in @xcite , we adopt a quality factor @xmath16 and a period @xmath17 h for a prem - like earth s model . the sources of excitation we consider are continuous pressure variations at the surface . a similar problem was treated by @xcite and @xcite for the atmospheric excitation of normal modes where the sources were considered as stochastic quantities in space and time . as we use a harmonic spherical decomposition of the pressure field , the correlation in space depends on the harmonic degree , here the degree - one component of wavelength @xmath18 . the correlation in time is performed in the spectral domain . as a consequence we introduce the energy spectrum of the degree - one pressure fluctuations @xmath19 and the energy spectrum of the radial displacement @xmath20 where @xmath21 is the fourier transform of @xmath22 , @xmath23 is the fourier transform of @xmath24 and @xmath25 denotes the complex conjugate . the fourier transform of eq . ( [ radialdisplacement(t ) ] ) is @xmath26 \int_{-\infty}^{+\infty } e^{-i\omega t } \lbrack \int_{-\infty}^{t}e^{i\nu_kt ' } \nonumber \\ & & ( \sigma_{10}(t ' ) \cos\theta + \sigma_{11}^c(t ' ) \sin\theta \cos\phi + { \sigma}_{11}^s(t ' ) \sin\theta \sin\phi ) dt ' \rbrack dt \nonumber \\ & = & \frac{r_s^2 u(r)}{\omega_0(1+\frac{1}{4q^2 } ) } [ u(r_s)g_0+p(r_s ) ] \frac{\omega_0(1-\frac{1}{4q^2})-\omega+\frac{i}{2q}(\omega-2\omega_0)}{\frac{\omega_0 ^ 2}{4q^2}+(\omega_0-\omega)^2 } \nonumber \\ & & ( { \hat \sigma_{10}}(\omega)\cos\theta + { \hat \sigma_{11}^c}(\omega)\sin\theta \cos\phi + { \hat \sigma_{11}^s}(\omega)\sin\theta \sin\phi ) , \nonumber\end{aligned}\ ] ] and , therefore , we have @xmath27 ^ 2 \nonumber \\ & & \frac{\lbrack \omega_0(1-\frac{1}{4q^2})-\omega \rbrack^2+\frac{(\omega-2\omega_0)^2}{4q^2 } } { \lbrack \frac{\omega_0 ^ 2}{4q^2}+(\omega_0-\omega)^2\rbrack ^2 } s_p(\theta,\phi;\omega ) . \end{aligned}\ ] ] from eq.([exdispl ] ) and ( [ deltag ] ) , we obtain the spectral energy of gravity @xmath28 for the excitation of the slichter mode by a surface fluid layer @xmath29 ^ 2 \nonumber \\ & & \frac{\lbrack \omega_0(1-\frac{1}{4q^2})-\omega \rbrack^2+\frac{(\omega-2\omega_0)^2}{4q^2 } } { \lbrack \frac{\omega_0 ^ 2}{4q^2}+(\omega_0-\omega)^2\rbrack ^2 } s_p(\theta,\phi;\omega).\end{aligned}\ ] ] the operational model of the european centre for medium - range weather forecasts ( ecmwf ) is usually available at 3-hourly temporal resolution , the spatial resolution varying from about 35 km in 2002 to 12 km since 2009 . this is clearly not sufficient to investigate the slichter mode excitation . however , during the period of cont08 measurements campaign ( august @xmath30 , 2008 ) , atmospheric analysis data were provided by the ecmwf also on an hourly basis . cont08 provided 2 weeks of continuous very long baseline interferometry ( vlbi ) observations for the study , among other goals , of daily and sub - daily variations in earth rotation @xcite . we take advantage of this higher - than - usual temporal resolution to compute the excitation of the slichter mode by the surface pressure fluctuations . to do so , we first extract the degree - one coefficients of the surface pressure during this period , considering both an inverted and a non - inverted barometer response of the oceans to air pressure variations @xcite . both hypotheses have the advantage to give simple responses of the oceans to atmospheric forcing , even if at such high frequencies , static responses are known to be inadequate . the use of a dynamic response of the ocean @xcite would lead to more accurate results but we would need a forcing ( pressure and winds ) of the oceans at hourly time scales which is not available . the degree-1 surface pressure changes contain three terms : @xmath31 from it , we can estimate the surface mass density by @xmath32 where @xmath33 is the mean surface gravity and compute the energy spectrum @xmath34 as defined in eq.([sp ] ) . the time - variations and fourier amplitude spectra of the harmonic degree - one coefficients @xmath35 , @xmath36 and @xmath37 are plotted in fig.[fig : time_fft_ib ] for the ib and non - ib hypotheses . the power spectral densities ( psds ) computed over the whole period considered here ( august 2008 ) are represented in fig.[fig : mapspsd ] for both oceanic responses . we also consider the ncep ( national centers for environmental prediction ) climate forecast system reanalysis ( cfsr ) model @xcite , for which hourly surface pressure is available with a spatial resolution of about 0.3@xmath38 before 2010 , and 0.2@xmath38 after . ncep / cfsr and ecmwf models used different assimilation schemes , e.g. 4d variational analysis for ecmwf , and a 3d variational analysis for ncep / cfsr every 6 hours . the temporal continuity of pressure is therefore enforced in the ecmwf model , whereas 6-hourly assimilation steps can sometimes be seen in the ncep / cfsr pressure series , for certain areas and time . the power spectral densities of the degree - one ncep / cfsr surface pressure field computed over august 2008 are represented in fig . [ fig : mapspsdncep ] for both oceanic responses . using equations ( [ deltag ] ) and ( [ exg ] ) we can compute the surface gravity perturbation induced by the slichter mode excited by the degree - one ecmwf and ncep / cfsr surface atmospheric pressure variations during august 2008 . the computation is performed with the eigenfunctions obtained for a spherical , self - gravitating anelastic prem - like earth model as in @xcite . we remove the 3 km - thick global ocean from the prem model because the ocean response is already included in the degree - one atmospheric coefficients . the power spectral densities of the surface gravity effect induced by the slichter mode excited by the ecmwf atmospheric data are plotted in figs [ fig : dgib - nib_maps_ecmwf ] for an inverted and a non - inverted barometer response of the oceans . the psd is given in decibels to enable an easy comparison with previous sg noise level studies ( e.g. @xcite ) . ncep / cfsr weather solutions give a similar surface excitation amplitude less than -175 db . we also consider the psds of the excitation amplitude at the sg sites djougou ( benin ) and bfo ( black forest observatory , germany ) for both oceanic responses in fig . [ fig : dg_ecmwf_b1dj_ngal ] . according to fig . [ fig : dgib - nib_maps_ecmwf ] the djougou site turns out to be located at a maximum of excitation amplitude in the case of an inverted barometer response of the oceans . bfo is the sg site with the lowest noise level at sub - seismic frequencies @xcite . the later noise level is also plotted in fig . [ fig : dg_ecmwf_b1dj_ngal ] . in that figure , we can see that the excitation amplitude at djougou reaches -175 db . for an undamped harmonic signal of amplitude @xmath39 the psd is defined by @xmath40 where @xmath41 is the number of samples and @xmath42 the sampling interval . consequently , assuming a 15-day time duration with a sampling rate of 1 min , a psd amplitude of -175 db corresponds to a harmonic signal of 0.3 ngal , which is clearly below the 1 ngal detection threshold and below the best sg noise level . a decrease of noise by a factor 3 would be necessary to be able to detect such a sub - nanogal effect . stacking @xmath43 worldwide sgs of low noise levels would improve the signal - to - noise ratio by a factor @xmath44 . supposing that we had a large number of sg sites with equal noise levels ( same as bfo sg noise level ) , then we would need to stack 10 datasets to improve the snr by 10 db , so as to reach the nanogal level . both ecmwf and ncep atmospheric pressure data lead to a similar , presently undetectable , excitation amplitude for the slichter mode during august 2008 . however , as we have at our disposal 11 years of hourly ncep / cfsr surface pressure field , we can look at the time - variations of the excitation amplitude of the slichter mode over the full period . ncep atmospheric pressure data are assimilated every 6 h introducing an artificial periodic signal . in order to avoid the contamination by this 6 h - oscillation in spectral domain on the slichter mode period of 5.42 h , we need a data length of 2.5 days at least . we consider time - windows of 15 days shifted by 7 days and compute the surface excitation amplitude of the slichter mode at the bfo and djougou superconducting gravimeter sites and at location on earth for which the excitation amplitude is maximum ( fig.[fig : dg_bfo - dj_ncep_11yr ] ) . note that this location of maximum amplitude is also varying in time . we can see that the excitation amplitude is larger than 0.4 ngal at bfo for instance between january and march 2004 and in november 2005 for both oceanic responses . there is also a peak of excitation at djougou in november 2005 and between january and march 2004 but only for an ib - hypothesis . however , during these 11 years between 2000 and 2011 , the maximum surface excitation amplitude stays below 0.7 ngal . as a consequence we can conclude that the degree - one surface pressure variations are a possible source of excitation for the slichter mode but the induced surface gravity effect is too weak to be detected by current sgs . using a normal mode formalism , we have computed the surface gravity perturbations induced by a continuous excitation of the slichter mode by atmospheric degree - one pressure variations provided by two meteorological centers : ecmwf and ncep / cfsr . both inverted and non - inverted barometer responses of the oceans to the atmospheric load have been employed . we have shown that the induced surface gravity signal does not reach the nanogal level , which is considered as being the level of detection of present sgs . the surficial degree - one pressure variations are a probable source of excitation of the slichter mode but the weak induced surface amplitude is one additional reason why this translational mode core has never been detected . an instrumental challenge for the future gravimeters would be to further decrease their noise levels . another source of possible excitation that has not been investigated yet is the dynamic response of the oceans . the oceans are known to be a source of continuous excitation of the fundamental seismic modes @xcite . so a further study would require to improve the response of the oceans . we would like to thank two anonymous reviewers for their comments on this work . we acknowledge the use of meteorological data of the ecmwf and ncep . courtier , n. , ducarme , b. , goodkind , j. , hinderer , j. , imanishi , y. , seama , n. , sun , h. , merriam , j. , bengert , b. , smylie , d.e . global superconducting gravimeter observations and the search for the translational modes of the inner core , _ phys . earth planet . _ , _ 117 _ , 320 . pagiatakis , s. d. , yin , h. and abd el - gelil , m. ( 2007 ) . least - squares self - coherency analysis of superconducting gravimeter records in search for the slichter triplet , _ phys . earth planet . _ , _ 160 _ , 108 - 123 . rosat , s. , hinderer , j. , crossley , d.j . , rivera , l. ( 2003 ) . the search for the slichter mode : comparison of noise levels of superconducting gravimeters and investigation of a stacking method . earth planet . _ , _ 140 _ ( 13 ) , 183 - 202 . rosat , s. , rogister , y. , crossley , d. et hinderer , j. ( 2006 ) . a search for the slichter triplet with superconducting gravimeters : impact of the density jump at the inner core boundary , _ j. of geodyn . _ , _ 41 _ , 296 - 306 . rosat , s. ( 2007 ) . optimal seismic source mechanisms to excite the slichter mode . int . assoc . of geod . symposia , dynamic planet , cairns ( australia ) , _ vol . 130 _ , 571 - 577 , springer berlin heidelberg new york . rosat , s. , sailhac , p. and gegout , p. ( 2007 ) . a wavelet - based detection and characterization of damped transient waves occurring in geophysical time - series : theory and application to the search for the translational oscillations of the inner core , _ geophys . j. int . _ , _ 171 _ , 55 - 70 . tkalcic , h. , kennett , b. l. n. and cormier , v. f. ( 2009 ) . on the inner - outer core density contrast from pkikp / pcp amplitude ratios and uncertainties caused by seismic noise , _ geophys . _ , _ 179 _ , 425 - 443 . . the best sg noise level and the levels corresponding to the 1 ngal and 0.3 ngal signals are indicated.,title="fig:",width=377 ] . the best sg noise level and the levels corresponding to the 1 ngal and 0.3 ngal signals are indicated.,title="fig:",width=377 ]
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using hourly atmospheric surface pressure field from ecmwf ( european centre for medium - range weather forecasts ) and from ncep ( national centers for environmental prediction ) climate forecast system reanalysis ( cfsr ) models , we show that atmospheric pressure fluctuations excite the translational oscillation of the inner core , the so - called slichter mode , to the sub - nanogal level at the earth surface .
the computation is performed using a normal - mode formalism for a spherical , self - gravitating anelastic prem - like earth model .
we determine the statistical response in the form of power spectral densities of the degree - one spherical harmonic components of the observed pressure field .
both hypotheses of inverted and non - inverted barometer for the ocean response to pressure forcing are considered .
based on previously computed noise levels , we show that the surface excitation amplitude is below the limit of detection of the superconducting gravimeters , making the slichter mode detection a challenging instrumental task for the near future .
slichter mode ; ecmwf atmospheric model ; ncep / cfsr atmospheric model ; superconducting gravimeters ; surface gravity ; normal mode
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You are an expert at summarizing long articles. Proceed to summarize the following text:
neutron stars are believed to form from the core collapse of massive stars and the accretion induced collapse of massive white dwarfs . if the stellar core or white dwarf is rotating , conservation of angular momentum implies that the resulting neutron star must rotate very rapidly . it has been suggested @xcite that such a rapidly rotating star may develop a non - axisymmetric dynamical instability , emitting a substantial amount of gravitational radiation which might be detectable by gravitational wave observatories such as ligo , virgo , geo and tama . rotational instabilities arise from non - axisymmetric perturbations having angular dependence @xmath4 , where @xmath5 is the azimuthal angle . the @xmath0 mode is called the bar mode , which is usually the strongest mode for stars undergoing instabilities . there are two types of instabilities . dynamical _ instability is driven by hydrodynamics and gravity , and it develops on a dynamical timescale , i.e. the timescale for a sound wave to travel across the star . a _ secular _ instability , on the other hand , is driven by viscosity or gravitational radiation reaction , and its growth time is determined by the relevant dissipative timescale . these secular timescales are usually much longer than the dynamical timescale of the system . in this paper , we focus on the dynamical instabilities resulting from the new - born neutron stars formed from accretion induced collapse ( aic ) of white dwarfs . these instabilities occur only for rapidly rotating stars . a useful parameter to characterize the rotation of a star is @xmath6 , where @xmath7 and @xmath8 are the rotational kinetic energy and gravitational potential energy respectively . it is well - known that there is a critical value @xmath9 so that a star will be dynamically unstable if its @xmath10 . for a uniform density and rigidly rotating star , the maclaurin spheroid , the critical value is determined to be @xmath11 @xcite . numerous numerical simulations using newtonian gravity show that @xmath9 remains roughly the same for differentially rotating polytropes having the same specific angular momentum distribution as the maclaurin spheroids @xcite . however , @xmath9 can take values between 0.14 to 0.27 for other angular momentum distributions @xcite ( the lower limit @xmath12 is observed only for a star having a toroidal density distribution , i.e. the maximum density occurs off the center @xcite ) . numerical simulations using full general relativity and post - newtonian approximations suggest that relativistic corrections to newtonian gravity cause @xmath9 to decrease slightly @xcite . most of the stability analyses to date have been carried out by assuming that the star rotates with an _ ad hoc _ rotation law or using simplified equations of state . the results of these analyses might not be applicable to the new - born neutron stars resulting from aic . recently , fryer , holz and hughes @xcite carried out an aic simulation using a realistic rotation law and a realistic equation of state . their pre - collapse white dwarf has an angular momentum @xmath13 . after the collapse , the neutron star has @xmath1 less than 0.06 , which is too small for the star to be dynamically unstable . however , they point out that if the pre - collapse white dwarf spins faster , the resulting neutron star could have high enough @xmath1 to trigger a dynamical instability . they also point out that a pre - collapse white dwarf could easily be spun up to rapid rotation by accretion . the spin of an accreting white dwarf before collapse depends on its initial mass , its magnetic field strength and the accretion rate , etc . @xcite . liu and lindblom @xcite ( hereafter paper i ) in a recent paper construct equilibrium models of new - born neutron stars resulting from aic based on conservation of specific angular momentum . their results show that if the pre - collapse white dwarfs are rapidly rotating , the resulting neutron stars could have @xmath1 as large as 0.26 , which is slightly smaller than the critical value @xmath9 for maclaurin spheroids . however , the specific angular momentum distributions of those neutron stars are very different from that of maclaurin spheroids . so there is no reason to believe that the traditional value @xmath14 can be applied to those models . the purpose of this paper is first to determine the critical value @xmath9 for the new - born neutron stars resulting from aic , and then estimate the signal to noise ratio and detectability of the gravitational waves emitted as a result of the instability . we do not intend to provide an accurate number for the signal to noise ratio , which requires a detailed non - linear evolution of the dynamical instability . instead , we use newtonian gravitation theory to compute the structure of new - born neutron stars . then we evolve the linearized newtonian hydrodynamical equations to study the star s stability and determine the critical value @xmath9 . relativistic effects are expected to give a correction of order @xmath15 , which is about 8% for the rapidly rotating neutron stars studied in this paper . here @xmath16 is a typical sound speed inside the star and @xmath17 is the speed of light . this paper is organized as follows . in sec . [ sec : eqm ] , we apply the method described in paper i to construct a number of equilibrium neutron star models with different values of @xmath1 . in sec . [ sec : stab ] , we study the stability of these models by adding small density and velocity perturbations to the equilibrium models . then we evolve the perturbations by solving linearized hydrodynamical equations proposed by toman et al @xcite . from the simulations , we can find out whether the star is stable , and determine the critical value @xmath9 . in sec . [ sec : gw ] , we estimate the strength and signal to noise ratio of the gravitational waves emitted by this instability . in sec . [ sec : mag ] , we estimate the effects of a magnetic field on the stability results . finally , we summarize and discuss our results in sec . [ sec : dis ] . in this section , we describe briefly how we construct new - born neutron star models from the pre - collapse white dwarfs . a more detailed description is given in paper i. we consider two types of pre - collapse white dwarfs : those made of carbon - oxygen ( c - o ) and those made of oxygen - neon - magnesium ( o - ne - mg ) . the collapse of a massive c - o white dwarf is triggered by the explosive carbon burning near the center of the star @xcite . the central density of the pre - collapse c - o white dwarf must be in the range @xmath18 in order for the collapse to result in a neutron star , rather than exploding as a type ia supernova @xcite . the collapse of a massive o - ne - mg white dwarf , on the other hand , is triggered by electron captures by @xmath19 and @xmath20 when the central density reaches @xmath21 @xcite . we construct three sequences of pre - collapse white dwarfs , with models in each sequence having different amounts of rotation . sequences i and ii correspond to c - o white dwarfs with central densities @xmath22 and @xmath23 respectively . sequence iii is for o - ne - mg white dwarfs with @xmath24 . all white dwarfs are assumed to rotate rigidly , because the timescale for a magnetic field to suppress differential rotation is much shorter than the accretion timescale ( see sec . [ sec : dis ] ) . the pre - collapse white dwarfs constructed in this section are described by the equation of state ( eos ) of a zero - temperature ideal degenerate electron gas with electrostatic corrections derived by salpeter @xcite . at high density , the pressure is dominated by the ideal degenerate fermi gas with electron fraction @xmath25 that is suitable for both c - o and o - ne - mg white dwarfs . electrostatic corrections , which depend on the white dwarf composition through the atomic number @xmath26 , contribute only a few percent to the eos for the high density white dwarfs considered here . equilibrium models are computed by hachisu s self - consistent field method @xcite , which is an iteration scheme based on the integrated euler equation for hydrostatic equilibrium : @xmath27 where @xmath28 is the rotational angular frequency of the star ; @xmath29 is a constant ; @xmath30 is the radius from the rotation axis ; @xmath31 is the specific enthalpy , which is related to the density @xmath32 and pressure @xmath33 by @xmath34 the gravitational potential @xmath35 satisfies the poisson equation @xmath36 where @xmath37 is the gravitational constant . the self - consistent field method determines the structure of the star for fixed values of two adjustable parameters . in ref . @xcite , the maximum density and axis ratio ( the ratio of polar to equatorial radii ) are the chosen parameters . however , it is more convenient to choose the central density @xmath38 and equatorial radius @xmath39 as the two parameters for the models studied here . sequence i : c - o white dwarfs with @xmath22 .properties of pre - collapse white dwarfs . here @xmath28 is the rotational angular frequency ; @xmath40 is the maximum rotational angular frequency of the white dwarf in the sequence without mass - shedding ; @xmath39 , @xmath41 , @xmath42 , @xmath43 and @xmath1 are respectively the equatorial radius , polar radius , mass , angular momentum and the ratio of rotational kinetic to gravitational potential energies . [ cols="^,^,^,^,^,^ " , ] the eigenfunctions of the most unstable bar mode for the other unstable equilibrium neutron stars are similar to those displayed above . table [ tab : omega ] summarizes the oscillation frequencies [ @xmath44 and e - folding time @xmath45 of the unstable models we have studied . the table also shows the ratio of the rotational frequency of the pre - collapse white dwarfs to the maximum frequency @xmath40 of the white dwarf in the sequence . we find that the oscillation frequencies are almost the same ( @xmath46 ) for all the cases . we do not observe any instability in our simulations for stars with @xmath47 . hence we conclude that @xmath9 is somewhere between 0.241 and 0.251 , and the pre - collapse white dwarf has to have @xmath48 in order for the collapsed star to develop a dynamical instability . in this section , we estimate the strength of the gravitational radiation emitted by neutron stars undergoing a dynamical instability . we also estimate the signal to noise ratio and discuss the detectability of these sources . the rms amplitude of a gravitational wave strain , @xmath49 , depends on the orientation of the source and its location on the detector s sky . when averaged over these angles , its value is given by @xcite @xmath50 where @xmath51 and @xmath52 are the rms amplitudes of the plus and cross polarizations of the wave respectively , and @xmath53 denotes an average over the orientation of the source and its location on the detector s sky . in the presence of perturbations , the density and velocity of fluid inside the star become @xmath54 where the perturbation functions @xmath55 and @xmath56 have angular dependence @xmath4 . the amplitude of the gravitational waves produced by time varying mass and current multipole moments can be derived from ref . the result is @xmath57 \label{eq : h0}\ ] ] where @xmath58 is the distance between the source and detector ; @xmath17 is the speed of light , and @xmath59 ^ 2 } \ ; \\ d_{lm}^{(l ) } & = & \frac{d^l}{dt^l } d_{lm } \ ; \\ s_{lm}^{(l ) } & = & \frac{d^l}{dt^l } s_{lm } \ .\end{aligned}\ ] ] for a newtonian source , the mass moments @xmath60 and current moments @xmath61 are given by @xmath62 where @xmath63 are the magnetic type vector spherical harmonics . the functions @xmath60 and @xmath61 have the property that @xmath64 and @xmath65 . hence it is sufficient to consider only positive values of @xmath66 and eq . ( [ eq : h0 ] ) becomes @xmath67 \ . \label{eq : h}\ ] ] the energy and angular momentum carried by the gravitational waves can also be derived from @xcite . the result is @xmath68 } \ ; \label{eq : edot } \\ \dot{j } & = & \sum_{l=2}^{\infty } \sum_{m=0}^l \frac{g}{c^{2l+1}}\ , 2imn_l \overline { \left [ d_{lm}^{(l)*}d_{lm}^{(l+1)}+s_{lm}^{(l)*}s_{lm}^{(l+1)}\right ] } \ , \label{eq : jdot0}\end{aligned}\ ] ] where the overline denotes time average over several periods . when a neutron star develops a dynamical instability and the bar mode ( @xmath0 ) is the only unstable mode , the values of @xmath31 , @xmath69 and @xmath70 will be dominated by the term involving @xmath71 . since the unstable bar mode has even parity under reflection about the equatorial plane , @xmath72 and the next leading term will involve @xmath73 and @xmath74 . these terms are expected to be smaller than the @xmath71 term by a factor of @xmath75 for @xmath69 and @xmath70 , and a factor of @xmath15 for @xmath31 . in our models , @xmath76 , so the contribution of higher order mass and current multipole moments are small and will be neglected . strictly speaking , the above analysis only applies when the amplitudes of the perturbations are small . when the amplitudes are large , however , the fluid motion does not separate neatly into decoupled fourier components , so all @xmath60 and @xmath61 will contribute . however , it is expected that the @xmath71 term will still be the most important term . since the detailed non - linear evolution of the dynamical instability is not known , the aim of this section is to provide an order of magnitude estimate of the gravitational radiation from these sources . hence we shall only consider the effect of the mass quadrupole moment and assume @xmath71 can be approximated by the bar - mode eigenfunctions computed in sec . [ sec : lsaresult ] . in this approximation , ( [ eq : h])([eq : jdot0 ] ) become @xmath77 where @xmath78 is the oscillation frequency of the bar mode . substituting the bar - mode eigenfunctions ( from sec . [ sec : lsaresult ] ) into eq . ( [ eq : dlm ] ) , we find that @xmath79 for all the unstable models we have studied . here @xmath80 is the amplitude of the bar mode defined in eq . ( [ def : alpha ] ) . the mass quadrupole moment @xmath71 has a time dependence @xmath81 , where @xmath82 is the angular frequency of the mode . hence the time derivative @xmath83 and we obtain @xmath84 the signal to noise ratio of these sources depends on the detailed evolution of the bar mode when the density perturbation reaches a large amplitude and non - linear effects take over . recently , new , centrella and tohline @xcite and brown @xcite perform long - duration simulations of the bar - mode instability . they find that the mode saturates when the density perturbation is comparable to the equilibrium density , and the mode pattern persists , giving a long - lived gravitational wave signal . here we assume that this is the case , and that the mode dies out only after a substantial amount of angular momentum is removed from the system by gravitational radiation . we then follow the method described in refs . @xcite to estimate the signal to noise ratio . in the stationary phrase approximation , the gravitational wave in the frequency domain @xmath85 is related to @xmath49 by @xmath86 combining eqs . ( [ eq : h2 ] ) , ( [ eq : jdot ] ) and ( [ eq : stapp ] ) , we obtain @xmath87 the signal to noise ratio is given by @xmath88 where @xmath89 is the spectral density of the detector s noise . if we assume that the oscillation frequency remains constant in the entire evolution , we obtain @xcite @xmath90 where @xmath91 is the total amount of angular momentum emitted by gravitational waves . to estimate @xmath91 , we assume that the mode dies out when the angular momentum of the star decreases to @xmath92 , which is the angular momentum of the marginally bar - unstable star . then we have @xmath93 for all the unstable stars , and the signal to noise ratio for ligo - ii broad - band interferometers @xcite is @xmath94 the timescale of the gravitational wave emission can be estimated by the equation @xmath95 where @xmath96 is the amplitude @xmath80 of the density perturbation at which the mode saturates . we have used eqs . ( [ eq : jdot ] ) and ( [ res : d22 ] ) to calculate the numerical value in the last equation . the detectability of this type of sources also depends on the event rate . the event rate for the aic in a galaxy is estimated to be between @xmath97 and @xmath98 per year @xcite . of all the aic events , only those corresponding to the collapse of rapidly rotating o - ne - mg white dwarfs can end up in the bar - mode instability , and the fraction of which is unknown . if a signal to noise ratio of 5 is required to detect the source , an event rate of at least @xmath3/galaxy / year is required for such a source to occur at a detectable distance per year . hence these sources will not be promising for ligo ii if the event rate is much less than @xmath3 per year per galaxy . the event rate of the core collapse of massive stars is much higher than that of the aic . the structure of a pre - supernova core is very similar to that of a pre - collapse white dwarf , so our results might be applicable to the neutron stars produced by the core collapse . if the core is rapidly rotating , the resulting neutron star might be able to develop a bar - mode instability . if a significant fraction of the pre - supernova cores are rapidly rotating , the chance of detecting the gravitational radiation from the bar - mode instability might be much higher than expected . as mentioned in sec . [ sec : ns ] , a new - born hot proto - neutron star is dynamically stable because its @xmath1 is too small . it takes about 20 s for the proto - neutron star to cool down and evolve into a cold neutron star , which may have high enough @xmath1 to trigger a dynamical instability . the proto - neutron stars , as well as the cold neutron stars computed in sec . [ sec : ns ] , show strong differential rotation ( paper i ) . this differential rotation will cause a frozen - in magnetic field to wind up , creating strong toroidal fields . this process will result in a re - distribution of angular momentum and destroy the differential rotation . if the timescale of this magnetic braking is shorter than the cooling timescale , the star may not be able to develop the dynamical instability discussed in secs . [ sec : stab ] and [ sec : gw ] . in this section , we estimate the timescale of this magnetic braking . in the ideal magnetohydrodynamics limit , the magnetic field lines are frozen into the moving fluid . the evolution of magnetic field @xmath99 is governed by the induction equation @xmath100 in our equilibrium models , @xmath101 . hence @xmath102 and eq . ( [ eq : bind ] ) becomes @xmath103 where @xmath104 is the time derivative in the fluid s co - moving frame . ( [ eq : bind2 ] ) can be integrated analytically ( see e.g. appendix b of @xcite ) . the magnetic field @xmath105 at the position @xmath106 of a fluid element at time @xmath107 is related to the magnetic field @xmath108 at the position @xmath109 of the same fluid at time @xmath110 by @xmath111 where @xmath112 is the coordinate strain between @xmath110 and @xmath107 . with @xmath113 , it is easy to show that the induced magnetic field has components only in the @xmath114 direction . its magnitude @xmath115 , after a time @xmath107 , is easy to compute from eq . ( [ eq : bevol ] ) . the result is @xmath116 where @xmath117 is the component of magnetic field in the @xmath118 direction . the induced magnetic field will significantly change the equilibrium velocity field when the energy density of magnetic field @xmath119 is comparable to the rotational kinetic energy density @xmath120 . this will occur in a timescale @xmath121 set by @xmath122 . using eq . ( [ eq : bi ] ) , we obtain @xmath123 where @xmath124 is the length scale of differential rotation , and @xmath125 is the speed of alfvn waves . observational data suggest that the magnetic fields of most white dwarfs are smaller than @xmath126 , although a small fraction of `` magnetic white dwarfs '' can have fields in the range @xmath127 . assuming flux conservation , the magnetic fields of the hot proto - neutron stars just after collapse would be @xmath128 for those @xmath126 white dwarfs . using the angular velocity distribution in paper i for the hot proto - neutron star , we find that the magnetic timescale in the dynamically important region ( @xmath129 ) is @xmath130 which is much longer than the neutrino cooling timescale ( @xmath131 ) . hence the angular momentum transport caused by the magnetic field is negligible during the cooling period . the magnetic timescale for the cold neutron stars can be calculated from the angular frequency distribution computed in sec . [ sec : ns ] . we find that @xmath121 for the cold models is about half of that given by eq . ( [ res : taub ] ) , which is still much longer than the timescale of gravitational waves @xmath132 calculated in the previous section . the instability results presented in the previous two sections remain unchanged unless the neutron star s initial magnetic field @xmath117 is greater than @xmath133 . in that case , a detailed magnetohydrodynamical simulation has to be carried out to compute the angular momentum transport . the magnetic timescale for these nascent neutron stars is significantly different from that estimated by baumgarte , shapiro and shibata @xcite and shapiro @xcite . they consider differentially rotating `` hypermassive '' neutron stars , which could be the remnants of the coalescence of binary neutron stars . those neutron stars are very massive ( @xmath134 ) and have much higher densities than the new - born neutron stars studied in this paper . they also use a seed magnetic field of strength @xmath135 , which is much larger than our estimate . these two differences combined make our magnetic braking timescale two orders of magnitude larger than theirs . it should be noted that it is the magnetic field just after the collapse that is relevant to our analysis here . the strong differential rotation of the neutron star will eventually generate a very strong toroidal field ( @xmath136 ) and destroy the differential rotation . the final state of the neutron star will be in rigid rotation , and its magnetic field will be completely different from the initial field . for this reason , the field strength @xmath137 observed in a typical pulsar is probably not relevant here . we have applied linear stability analysis to study the dynamical stability of new - born neutron stars formed by aic . we find that a neutron star has a dynamically unstable bar mode if its @xmath1 is greater than the critical value @xmath2 . in order for the neutron star to have @xmath10 , the pre - collapse white dwarf must be composed of oxygen , neon , magnesium and have a rotational angular frequency @xmath138 , corresponding to 93% of the maximum rotational frequency the white dwarf can have without mass shedding . the eigenfunction of the most unstable bar mode is concentrated within a radius @xmath139 . the oscillation frequency of the mode is @xmath140 . when the amplitude of the mode is small , it grows exponentially with an e - folding time @xmath141 for the most rapidly rotating star ( @xmath142 ) , which is about 5.5 rotation periods at the center of the star . the signal to noise ratio of the gravitational waves emitted by this instability is estimated to be 15 for ligo - ii broad - band interferometers if the source is located in the virgo cluster of galaxies ( @xmath143 ) . the detectability of these sources also depends on the event rate . the event rate of aic is between @xmath97 and @xmath144 . only those aic events corresponding to the collapse of rapidly rotating o - ne - mg white dwarfs can end up in the bar - mode instability . while it is likely that the white dwarfs would be spun up to rapidly rotation by the accretion gas prior to collapse @xcite , it is not clear how many of the aic events are related to the o - ne - mg white dwarfs . if the event rate is less than @xmath145 , it is not likely that ligo ii will detect these sources . however , the event rate of the core collapse of massive stars is much higher than that of the aic . a bar - mode instability could develop for neutron stars formed from the collapse of rapidly rotating pre - supernova cores . if a significant fraction of the cores are rapidly rotating , the chance of detecting the gravitational radiation from bar - mode instability would be much higher . if the pre - collapse white dwarf is differentially rotating , the resulting neutron star can have a higher value of @xmath1 . the bar - mode instability is then expected to last for a longer time . however , any differential rotation will be destroyed by magnetic fields in a timescale @xmath146 , where @xmath147 is the size of the white dwarf and @xmath148 . for a massive white dwarf with @xmath149 , @xmath150 which is much shorter than the accretion timescale . hence rigid rotation is a good approximation for pre - collapse white dwarfs . the magnetic field of a neutron star is much stronger than that of a white dwarf . the timescale for a magnetic field to suppress differential rotation depends on the initial magnetic field @xmath117 of the proto - neutron star . if the magnetic field of the pre - collapse white dwarf is of order @xmath126 , the initial field will be @xmath151 according to conservation of magnetic flux . in this case , the magnetic timescale is @xmath152 . this timescale is much longer than the time required for a hot proto - neutron star to cool down and turn into a cold neutron star , and go through the whole dynamical instability phase . if @xmath153 , a significant amount of angular momentum transport will take place during the cooling phase . a detailed magnetohydrodynamical simulation has to be carried out to study the transport process in this case . however , such a strong initial magnetic field is possible only if the pre - collapse white dwarf has a magnetic field @xmath154 . finally , we want to point out that the collapse of white dwarfs will certainly produce asymmetric shocks and may eject a small portion of the mass . we expect that our neutron star models describe fairly well the inner cores of the stars but not the tenuous outer layers . our stability results are sensitive to the region with @xmath155 . the results could change considerably if the structure in this region is very different from that of our models . this issue will hopefully be resolved by the future full 3d aic simulations . i thank lee lindblom for his guidance on all aspects of this work . i also thank kip s. thorne and stuart l. shapiro for useful discussions . this research was supported by nsf grants phy-9796079 and phy-0099568 , and nasa grant nag5 - 4093 . we see from figs . [ fig : cdeneq]-[fig : veleq ] that the bar - mode eigenfunction has peciliar structures at the corotation radius ( @xmath156 ) at which @xmath157 . the density perturbation has a small peak and the velocity perturbation is almost parallel to the @xmath5 direction . in this appendix , we shall show that these are caused by the resonance of the fluid driven by the mode . for simplicity , we only consider the fluid s motion on the equatorial plane . assume that the perturbations are dominated by a mode that goes as @xmath158 . we also assume that this mode is even under the reflection @xmath159 . hence we have @xmath160 and @xmath161 . in cylindrical coordinates , the linearized euler equation takes the form @xmath162 the density perturbation @xmath163 is related to the pressure perturbation @xmath164 by @xmath165 the @xmath30-component of the lagrangian displacement is given by @xmath166 our numerical simulations show that @xmath164 is well - behaved and smooth near the corotation radius at which @xmath167 . the perturbed gravitational potential @xmath168 is expected ( and is confirmed by our numerical simulations ) to be smooth since it depends on the overall distribution of the density perturbation . we can then use eqs . ( [ eq : euler1])-([eq : ximode ] ) to express all the other perturbed quantities in terms of @xmath164 and @xmath168 . near the corotation radius , the expressions are : @xmath169 \label{eq : vphi } \ , \\ \kappa^2 & = & \varpi \partial_{\varpi}\omega^2 + 4\omega^2 \ , \\ b & = & \frac{\partial_{\varpi } p \partial_{\varpi}\rho}{\rho^2}\left(1- \frac{\gamma_{\rm eq}}{\gamma_p } \right ) \ .\end{aligned}\ ] ] it follows from eqs . ( [ eq : ximode ] ) and ( [ eq : drhomode ] ) that if @xmath170 is not of order @xmath171 near the corotation radius , both @xmath172 and @xmath173 will be large . the large magnitude of the lagrangian displacement is caused by the fluid being driven in resonance by the mode . the large displacement of the fluid causes @xmath173 to be large due to the second term of eq . ( [ eq : drhomode ] ) . this term arises becuase of the different compressibilities of stationary and oscillating fluid ( i.e. @xmath174 ) . in the case of the bar mode ( @xmath0 ) , the corotation radius is located at @xmath175 . the equilibrium density on the equator @xmath176 and the stationary fluid is very compressible ( @xmath177 ) . the high compressibility of the stationary fluid make the background equilibrium density @xmath32 drop rapidly as @xmath30 increases , i.e. @xmath178 is large . the oscillating fluid is far less compressible ( @xmath179 ) . as a result , when the oscillating fluid moves to a new location , it does not expand or compress to an extent that can compensate for the difference between the background densities at the old and new locations . since both @xmath172 and @xmath178 are large , @xmath163 is dominated by the second term of eq . ( [ eq : drhomode ] ) near the corotation radius . this explains the narrow secondary peak of @xmath163 seen in fig . [ fig : cdeneq ] . we see from eq . ( [ eq : vphi ] ) that @xmath180 .
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the dynamical instability of new - born neutron stars is studied by evolving the linearized hydrodynamical equations .
the neutron stars considered in this paper are those produced by the accretion induced collapse of rigidly rotating white dwarfs .
a dynamical bar - mode ( @xmath0 ) instability is observed when the ratio of rotational kinetic energy to gravitational potential energy @xmath1 of the neutron star is greater than the critical value @xmath2 . this bar - mode instability leads to the emission of gravitational radiation that could be detected by gravitational wave detectors . however , these sources are unlikely to be detected by ligo ii interferometers if the event rate is less than @xmath3 per year per galaxy
. nevertheless , if a significant fraction of the pre - supernova cores are rapidly rotating , there would be a substantial number of neutron stars produced by the core collapse undergoing bar - mode instability .
this would greatly increase the chance of detecting the gravitational radiation .
0.5 cm pacs : 04.30.db , 95.30.sf , 97.60.jd 2
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You are an expert at summarizing long articles. Proceed to summarize the following text:
we consider a two - dimensional scalar field @xmath2 defined on the upper - half - plane , @xmath3 , which is free in the two - dimensional bulk but subject to a periodic boundary potential . the action is @xmath4 where @xmath5 is a complex parameter which dials the strength and the phase of the periodic boundary interaction . the period of the potential is chosen such that the interaction has dimension one under boundary scaling . this ensures that the boundary theory preserves half of the conformal symmetry of the bulk system . this model arises in various contexts , for example , in connection with critical behavior in dissipative quantum mechanics @xcite , quantum hall edge states @xcite , and open string theory in an on - shell tachyon background @xcite . more recently a @xmath6 boundary conformal field theory involving a scalar field with a ` wrong - sign ' kinetic term and an exponential boundary interaction has been applied to the rolling tachyon field of an s - brane or an unstable d - brane @xcite . such a theory is related by analytic continuation of the field to the boundary theory considered here . a conformal field theory analysis of the model ( [ action ] ) , carried out a decade ago @xcite , revealed an underlying @xmath1 symmetry which allows many exact results to be established , including exact boundary states and the one - loop partition function . in this lecture we report on ongoing work aimed at calculating correlation functions of both bulk and boundary fields . the general problem involving fields carrying arbitrary momenta remains unsolved but we give a prescription for correlation functions of an important sub - class of fields , the so - called discrete primaries . we take the scalar field to be non - compact , _ i.e. _ taking value in @xmath7 , although the compact case , where the field takes value on a circle of radius @xmath8 , is also of interest . in the absence of the boundary interaction , the field @xmath2 satisfies a neumann boundary condition at @xmath9 , and this is conveniently dealt with using the so - called doubling trick or method of images ( see f.ex . the theory on the upper half - plane is then mapped into a chiral theory on the full complex plane and the boundary eliminated . to see how this works we note that away from the boundary the free field can be written as a sum of left- and right - moving components , @xmath10 . for @xmath11 the neumann boundary condition on the real axis @xmath12 determines the right - moving field in terms of the left - moving one . we can therefore reflect the right - moving field through the boundary and represent it as a left - moving field in the lower - half - plane . the theory then contains only left - moving fields , but a left - moving field at @xmath13 in the unphysical lower - half - plane is to be interpreted as a right - moving field at @xmath14 in the physical region . and @xmath13 denote the complex conjugate of @xmath14 . we use @xmath15 for the argument of a right - moving field @xmath16 , but @xmath13 for that of a left - moving image field @xmath17 . ] more generally , any quasi - primary field in the bulk separates into left- and right - moving parts @xmath18 . using the doubling trick @xmath19 becomes a left - moving field @xmath20 , with holomorphic dimension @xmath21 . a bulk @xmath22-point function @xmath23 in the original theory on the upper - half - plane then becomes a @xmath24-point function of holomorphic fields @xmath25 on the infinite plane . it turns out that the doubling trick can be applied even when the boundary interaction in ( [ action ] ) is turned on . this is because the boundary potential can in fact be expressed in terms of the left - moving field alone @xmath26 the operators appearing in the interaction are currents of a left - moving @xmath1 algebra @xmath27 in the boundary action ( [ action ] ) the @xmath28 and @xmath29 currents are integrated along the real axis and in a perturbative expansion of a correlation function such integrals are repeatedly inserted into the amplitude . as usual , divergences arise when operator insertions coincide but , by a clever choice of regularization , callan @xcite were able to sum the perturbation series explicitly to obtain the exact interacting boundary state . it turned out to be remarkably simple , with the net effect of the interaction being a global @xmath1 rotation , @xmath30 , acting on the free neumann boundary state . the bulk theory is that of a free boson . it contains holomorphic primary fields @xmath31 with conformal weight @xmath32 for all @xmath33 and also the corresponding anti - holomorphic fields . at special values of the momentum , @xmath34 , where @xmath35 is an integer or integer - plus - half , some descendant states have vanishing norm and new primary fields appear , the so - called discrete primaries @xcite , which come in @xmath1 multiplets labelled by @xmath35 and @xmath36 , with @xmath37 . the discrete fields @xmath38 in a given @xmath1 multiplet are degenerate in that they all have conformal weight @xmath39 . they are composite fields made from polynomials in @xmath40 , @xmath41 , _ etc . _ accompanied by @xmath42 , and normal ordered with respect to the free holomorphic propagator @xmath43 . the discrete fields have the following representation @xcite , @xmath44 where the lowering current is integrated along nested contours surrounding @xmath14 . a discrete bulk primary is constructed from a pair of holomorphic and anti - holomorphic primaries , @xmath45 a priori , the left- and right - moving fields can carry different @xmath1 labels . however , the spin @xmath46 takes an unphysical value unless @xmath47 . we are considering a non - compact free boson , which has no winding states , so we must also require @xmath48 , which amounts to @xmath49 . finally we have @xmath50 where @xmath51 . when a bulk field approaches the boundary at @xmath52 , new divergences appear that are not removed by the bulk normal ordering . this is a general feature of conformal field theories with boundaries and signals the presence of so called boundary operators . the boundary conditions to either side of a boundary operator can be different , in which case it is called a boundary condition changing operator ( see for example @xcite ) . in a general boundary conformal field theory a bulk field approaching the boundary can be expanded in terms of boundary fields @xcite , @xmath53 where @xmath54 . the @xmath55 are boundary fields ( possibly boundary condition changing ) with boundary scaling dimensions @xmath56 and the @xmath57 are called bulk - to - boundary operator product coefficients . in addition to the bulk - to - boundary ope , the boundary fields form an operator product algebra amongst themselves , @xmath58 the boundary ope coefficients @xmath59 and the bulk - to - boundary ope coefficients @xmath57 , along with the boundary scaling dimensions @xmath56 , are characteristic data of a given boundary conformal field theory . in particular , the boundary scaling dimension @xmath56 of a boundary operator @xmath55 is given by the energy eigenvalue of the corresponding open string state , where the open - string hamiltonian is the @xmath60 generator of the virasoro algebra that is preserved by the conformally invariant boundary conditions . in the theory at hand , boundary conditions are labelled by the boundary coupling @xmath5 in ( [ action ] ) . a boundary condition changing operator that changes @xmath5 to @xmath61 at the insertion point corresponds to an open string state where the two string endpoints interact with boundary potentials of different strength @xmath5 and @xmath61 . to work out the open string spectrum we find it convenient to re - express the interacting boundary theory in terms of free fermions as shown in @xcite . the @xmath1 currents are bi - linear in the fermions and the boundary interaction may be viewed as a localized mass term . the open string spectrum is then found by solving a straightforward eigenvalue problem for the fermions . we will not repeat the construction here but simply quote the result . in @xcite both string endpoints were taken to interact with the same boundary potential , _ i.e. _ it was assumed that @xmath62 . in this case the partition function may be written @xmath63 where @xmath64 is the dedekind eta function , @xmath65 with @xmath66 the parameter length of the open string , and @xmath67 is related to the target space momentum @xmath68 by @xmath69 the value of @xmath67 is determined by continuity from @xmath70 at @xmath11 . the scaling dimensions of the corresponding boundary operators are given by @xmath71 with @xmath72 . the spectrum obtained from ( [ ggspectrum ] ) is shown in the left - most graph in . the free spectrum , @xmath73 , has split into bands with forbidden gaps in energy in between them . note that the energy eigenvalues are invariant under @xmath74 for any integer @xmath22 . the boundary potential breaks translation invariance in the target space to a discrete subgroup and target space momentum is only conserved mod @xmath75 in our units in the interacting system . the momentum of a given operator can therefore always be shifted into the so - called first brillouin zone , @xmath76 . the spectra in are displayed in an extended zone scheme , with the periodicity appearing explicitly . it is interesting to note that bands also appear in the open string channel of the boundary conformal field theory of a free boson compactified on a circle , when the radius of the circle is an irrational multiple of the self - dual radius @xmath77 @xcite . like the system we are considering here , those theories admit a one - parameter family of boundary states that interpolate between neumann and dirichlet boundary conditions , but it is unclear to us at present how deep the parallels between the systems run . the band spectrum described by ( [ ggspectrum ] ) is a special case of a more general structure that appears when we allow for open strings with different boundary coupling , @xmath5 and @xmath61 , at the two endpoints . for simplicity we take @xmath78 , with @xmath79 . the fermion eigenvalue problem solved in @xcite can easily be extended to cover this case also . the only modification is to change @xmath5 to @xmath61 in one of the boundary mass terms for the worldsheet fermions in equation ( 29 ) of that paper , leaving the other one unchanged . the eigenvalue equation for the spectrum then becomes @xmath80 where @xmath81 . clearly this reduces to the previous result ( [ ggspectrum ] ) as @xmath82 but when @xmath83 there are important new features . in particular , there are additional gaps in the spectrum as shown in . but the other one takes the values a ) @xmath84 , b ) @xmath85 , c ) @xmath86 . [ fig : genspectrum ] ] in this section we consider scattering amplitudes involving bulk fields . general bulk amplitudes are functions of the boundary coupling @xmath5 and our goal is to determine this dependence . some bulk amplitudes that are zero in the free theory are nonvanishing in the presence of the boundary interaction . the periodic boundary potential breaks translation invariance in the target space and can absorb momenta in @xmath75 . callan @xcite gave a simple prescription for scattering amplitudes that describe elementary string excitations reflecting off the interacting worldsheet boundary . their method , which can be called _ the method of rotated images _ , rests on the fact that the bulk operators @xmath87 and @xmath88 , which create and destroy left- and right - moving excitations , are in fact currents of the underlying @xmath1 algebra . the method can be generalized to deal with bulk scattering amplitudes involving operators , which carry well defined @xmath1 quantum numbers , _ i.e. _ the discrete bulk operators @xmath89 in equation ( [ discretebulkfield ] ) . consider some number of these states inserted in the upper half - plane and use the method of images so that for each insertion @xmath90 . this is possible even with the boundary interaction turned on because the anti - holomorphic field @xmath91 commutes with the holomorphic @xmath1 currents in the interaction and reflects through the underlying neumann boundary condition exactly as in the free theory . in a perturbative expansion of the bulk amplitude the @xmath1 currents that appear in the boundary interaction are repeatedly integrated along the real axis . their integration contours may be deformed away from the real axis into the lower - half - plane , where they will act on the image fields @xmath92 . the @xmath1 current algebra ensures that there will be no cuts generated , and so the integration contours can be closed around the image fields . the net effect is the global @xmath1 rotation @xmath93 , given at the end of section [ currentalgebra ] , acting on each image field insertion . the original right - moving discrete field was a component of a rank @xmath94 irreducible tensor operator , so the rotated image is @xmath95 with the rotation coefficient given by @xmath96 where @xmath97 are standard @xmath1 states . a bulk @xmath22-point function is then expressed in terms of @xmath24 point functions of free holomorphic fields and @xmath1 rotation coefficients , @xmath98 as a simple example of this prescription we consider the one - point function of a general discrete bulk primary , @xmath99 conformal invariance requires the scaling dimension of the two chiral operators to be the same , i.e. @xmath100 , and momentum conservation in the free chiral theory requires @xmath101 . the one - point function is therefore @xmath102 for comparison , note that in the free theory with neumann boundary conditions the only bulk operator that has a non - vanishing one - point function is the unit operator . higher - point bulk amplitudes involving discrete fields can be computed in an analogous fashion , but unfortunately our prescription can not be applied to all bulk amplitudes in the model . in general , bulk amplitudes will involve both the discrete bulk fields and fields carrying generic target space momenta . the only constraint from momentum conservation is that the total momentum of all the fields in a given correlator add up to an integer times @xmath103 in our units . the operator product of the currents in the boundary interaction and generic momentum fields is non - local , and so the effect of the interaction is no longer captured by a global @xmath1 rotation . the physics of the open string sector is encoded in correlation functions of boundary operators as discussed in section [ boundaryfields ] . it is straightforward to identify the boundary operators in the free theory . consider bringing a bulk primary operator at generic momentum @xmath104)$ ] close to the boundary . now replace the right - moving part by its left - moving image and apply the left - moving ope , @xmath105 , where @xmath106 and the boundary scalar field is related to the left - moving field by @xmath107 . for the discrete bulk fields one obtains @xmath108 the @xmath109 are primary fields on the boundary , which we refer to as _ discrete boundary fields_. the superscript on @xmath110 signals that these are boundary operators of the free theory . the bulk - to - boundary ope coefficients @xmath111 can be obtained by straightforward calculation . by momentum conservation they vanish unless @xmath112 . the discrete boundary fields inherit the @xmath1 structure from the chiral discrete fields , but since @xmath113 we find that only integer values of @xmath114 are allowed on the boundary . the boundary scaling dimension of @xmath110 is @xmath115 . we are interested in calculating amplitudes involving arbitrary boundary fields in the interacting theory , including boundary condition changing fields . this is delicate since operators are now inserted on the boundary where the non - linear self - interaction takes place . we can nevertheless anticipate the structure of low - order amplitudes from conformal symmetry . for two- and three - point functions one finds @xmath116 is the scaling dimension obtained from equation . it remains to determine the @xmath5-dependence of the coefficients @xmath117 and @xmath59 . explicit calculations for fields carrying arbitrary momenta are difficult because such fields are non - local with respect to the @xmath1 currents in the boundary interaction , but we can proceed further with amplitudes involving the discrete boundary fields . a key observation , that can be read off from equation , is that the boundary scaling dimension of @xmath118 is @xmath115 , independent of the coupling . as a result we can write @xmath118 as a linear - combination of free boundary operators within the same @xmath1 multiplet @xmath119 we now write the general bulk - to - boundary ope at non - zero boundary coupling , @xmath120 we then apply the method of rotated images as described in section [ bulkamplitudes ] on the left - hand side , plug in the expansion ( [ eq : psig ] ) on the right - hand side , and use equation ( [ eq : dbb ] ) . this results in a set of algebraic equations that relate the ope coefficients @xmath121 and the expansion coefficients @xmath122 . unfortunately , there are not enough equations to determine all the coefficients , so we need further input . we propose the following prescription for determining @xmath122 . first the free boundary operators @xmath110 are obtained as in equation . the effect of the interaction on these operators is then computed by letting the integration contours of the boundary currents approach the real axis , where @xmath110 is inserted , from above . the contours are then moved into the lower - half - plane , resulting in an @xmath1 rotation acting on @xmath110 , giving @xmath123 . this leads to boundary amplitudes of the form @xmath124 and bulk - to - boundary ope coefficients @xmath125 we stress that other prescriptions are possible . one alternative would be to deform the integration contours into the upper - half - plane in which case @xmath126 and the effect of the interaction is shifted entirely into the bulk - to - boundary coefficients @xmath121 . we note , however , that the boundary amplitudes ( [ boundaryamplitudes ] ) have the desirable feature that momentum is only conserved modulo @xmath103 which reflects broken translation invariance . we thank c. callan , d. friedan and i. klebanov for discussions . this work was supported in part by grants from the icelandic research council , the university of iceland research fund and the icelandic research fund for graduate students . thanks the new high energy theory center at rutgers university for hospitality .
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correlation functions of discrete primary fields in the @xmath0 boundary conformal field theory of a scalar field in a critical periodic boundary potential are computed using the underlying @xmath1 symmetry of the model .
bulk amplitudes are unambigously determined and we give a prescription for amplitudes involving discrete boundary fields .
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the basic questions faced by extreme value analysis consist in estimating the probability of exceeding a threshold that is larger than the sample maximum and estimating a quantile of an order that is larger than 1 minus the reciprocal of the sample size . in words , they consist in making inferences on regions that lie outside the support of the empirical distribution . in order to face these challenges in a sensible framework , extreme value theory ( ) assumes that the sampling distribution @xmath0 satisfies a regularity condition . indeed , in heavy - tail analysis , the tail function @xmath1 is supposed to be regularly varying that is , @xmath2 exists for all @xmath3 . this amounts to assume the existence of some @xmath4 such that the limit is @xmath5 for all @xmath6 . in other words , if we define the _ excess distribution above the threshold @xmath7 _ by its survival function : @xmath8 for @xmath9 , then @xmath10 is regularly varying if and only if @xmath11 converges weakly towards a pareto distribution . the sampling distribution @xmath0 is then said to belong to the _ max - domain of attraction _ of a frchet distribution with index @xmath12 ( abbreviated in @xmath13 ) and @xmath14 is called the _ extreme value index_. the main impediment to large exceedance and large quantile estimation problems alluded above turns out to be the estimation of the extreme value index . since the inception of extreme value analysis , many estimators have been defined , analysed and implemented into software . @xcite introduced a simple , yet remarkable , collection of estimators : for @xmath15 , @xmath16 where @xmath17 are the _ order statistics _ of the sample @xmath18 ( the non - increasing rearrangement of the sample ) . an integer sequence @xmath19 is said to be _ intermediate _ if @xmath20 while @xmath21 . it is well known that @xmath0 belongs to @xmath22 for some @xmath4 if and only if , for all intermediate sequences @xmath19 , @xmath23 converges in probability towards @xmath14 @xcite . under mildly stronger conditions , it can be shown that @xmath24 is asymptotically gaussian with variance @xmath25 this suggests that , in order to minimise the quadratic risk @xmath26 $ ] or the absolute risk @xmath27 , an appropriate choice for @xmath28 has to be made . if @xmath28 is too large , the hill estimator @xmath23 suffers a large bias and , if @xmath28 is too small , @xmath23 suffers erratic fluctuations . as all estimators of the extreme value index face this dilemma ( see * ? ? ? * ; * ? ? ? * ; * ? ? ? * and references therein ) , during the last three decades , a variety of data - driven selection methods for @xmath28 has been proposed in the literature ( see @xcite , @xcite , @xcite , @xcite , @xcite , @xcite , @xcite , @xcite to name a few ) . a related but distinct problem is considered by @xcite : constructing uniform and adaptive confidence intervals for the extreme value index . the rationale for investigating adaptive hill estimation stems from computational simplicity and variance optimality of properly chosen hill estimators @xcite . the hallmark of our approach is to combine techniques of with tools from concentration of measure theory . as up to our knowledge , the impact of the concentration of measure phenomenon in has received little attention , we comment and motivate the use of concentration arguments . talagrand s concentration phenomenon for products of exponential distributions is one instance of a general phenomenon : concentration of measure in product spaces @xcite . the phenomenon may be summarised in a simple quote : functions of independent random variables that do not depend too much on any of them are almost constant @xcite . the concentration approach helps to split the investigation in two steps : the first step consists in bounding the fluctuations of the random variable under concern around its median or its expectation , while the second step focuses on the expectation . this approach has seriously simplified the investigation of suprema of empirical processes and made the life of many statisticians easier @xcite . to point out the potential uses of concentration inequalities in the field of is one purpose of this paper . in statistics , concentration inequalities have proved very useful when dealing with estimator selection and adaptivity issues : sharp , non - asymptotic tail bounds can be combined with simple union bounds in order to obtain uniform guarantees of the risk of collection of estimators . using concentration inequalities to investigate adaptive choice of the number of order statistics to be used in tail index estimation is a natural thing to do . in the present setting , tail index estimators are functions of independent random variables . talagrand s quote raises a first question : in which way are these tail functionals smooth functions of independent random variables ? we do not attempt here to revisit the asymptotic approach described by @xcite which equates smoothness with hadamard differentiability . our approach is non - asymptotic and our conception of smoothness somewhat circular , smooth functionals are these functionals for which we can obtain good concentration inequalities . in this paper , we combine talagrand s concentration inequality for smooth functions of independent exponentially distributed random variables ( theorem [ bernstein : expo ] ) with three traditional tools of : the quantile transform , karamata s representation for slowly varying functions , and rnyi s characterisation of the joint distribution of order statistics of exponential samples . this allows us to establish concentration inequalities for the hill process @xmath29 ( theorem [ prp : hill : concentration ] ) in section [ sec : conc - ineq - hill ] . in section [ sec : adapt - hill - estim ] , we build on these concentration inequalities to analyse the performance of a variant of lepki s rule defined in sections [ sec : lepsk - meth - adapt ] and [ sec : adapt - hill - estim ] : theorem [ thm : adapt - hill - estim ] describes an oracle inequality and corollary [ cor : adapt - hill - estim-2ndrv ] assesses the performance of this simple selection rule under a mild assumption on the so - called von mises function . note that the condition is less demanding than the regular variation condition on the von mises function that has often been assumed when looking for adaptive tail index estimators ( notable exceptions being @xcite and @xcite ) . it reveals that the performance of hill estimators selected by lepski s method matches known lower bounds ( see section [ sec : lower - bound ] ) that is , they suffer the loss of efficiency which is inherent to this problem , but not more . proofs are given in section [ sec : proofs ] . finally , in section [ sec : simulations ] , we examine the performance of this adaptive hill estimator for finite sample sizes using monte - carlo simulations . the quantile function @xmath30 is the generalised inverse of the distribution function @xmath0 . the _ tail quantile function _ of @xmath0 is a non - decreasing function defined on @xmath31 by @xmath32 , or by @xmath33 in this text , we use a variation of the quantile transform that fits : if @xmath34 is exponentially distributed , then @xmath35 is distributed according to @xmath0 . moreover , by the same argument , the order statistics @xmath17 are distributed as a monotone transformation of the order statistics @xmath36 of a sample of @xmath37 independent standard exponential random variables . @xmath38 thanks to rnyi s representation for order statistics of exponential samples , agreeing on @xmath39 , the rescaled exponential spacings @xmath40 are independent and exponentially distributed . the quantile transform and rnyi s representation are complemented by karamata s representation for slowly varying functions . recall that a function @xmath41 is _ slowly varying at infinity _ if for all @xmath3 , @xmath42 . the von mises condition specifies the form of karamata s representation ( see * ? ? ? * corollary 2.1 ) of the slowly varying component @xmath43 of @xmath44 . [ dfn : vmises : cond ] a distribution function @xmath0 belonging to @xmath45 , satisfies the von mises condition if there exist a constant @xmath46 , a constant @xmath47 and a measurable function @xmath48 on @xmath31 such that , for @xmath49 @xmath50 with @xmath51 . the function @xmath48 is called the _ von mises function_. in the sequel , we assume that the sampling distribution @xmath52 , @xmath4 , satisfies the von mises condition with @xmath53 , von mises function @xmath48 and define the non - increasing function @xmath54 from @xmath55 to @xmath56 by @xmath57 . in the text , we assume that @xmath58 . combining the quantile transformation , rnyi s and karamata s representations , it is straightforward that , under the von mises condition , the sequence of hill estimators is distributed as a function of the largest order statistics of a standard exponential sample . [ hill : rep ] the vector of hill estimators @xmath59 is distributed as the random vector @xmath60 where @xmath61 are independent standard exponential random variables while , for @xmath62 , @xmath63 is distributed like the @xmath64th order statistic of an @xmath37-sample of the exponential distribution . for a fixed @xmath65 , a second distributional representation is available , @xmath66 where @xmath67 and @xmath68 are defined as in proposition [ hill : rep ] . this second , simpler , distributional representation stresses the fact that , conditionally on @xmath68 , @xmath69 is distributed as a mixture of sums of independent random variables approximately distributed as exponential random variables with scale @xmath14 . this distributional identity suggests that the variance of @xmath70 scales like @xmath71 , an intuition that is corroborated by analysis , see section [ sec : conc - ineq - hill ] . the bias of @xmath70 is connected with the von mises function @xmath48 by the next formula @xmath72= { \ensuremath{\mathbb{e}}}\left[\int_1^\infty \frac{\eta\left({\mathrm{e}}^{y_{(k+1 ) } } v\right)}{v^2 } \mathrm{d}v\right]\ , .\ ] ] henceforth , let @xmath73 be defined on @xmath31 by @xmath74 the quantity @xmath75 is the bias of the hill estimator @xmath70 given @xmath76 . the second expression for @xmath73 shows that @xmath73 is differentiable with respect to @xmath77 ( even though @xmath48 might be nowhere differentiable ) and that @xmath78 the von mises function governs both the rate of convergence of @xmath79 towards @xmath80 , or equivalently of @xmath81 towards @xmath5 , and the rate of convergence of @xmath82 towards @xmath83 . the difficulty in extreme value index estimation stems from the fact that , for any collection of estimators , for any intermediate sequence @xmath19 , and for any @xmath4 , there is a distribution function @xmath84 such that the bias @xmath85 decays at an arbitrarily slow rate . this has led authors to put conditions on the rate of convergence of @xmath79 towards @xmath80 as @xmath77 tends to infinity while @xmath3 , or equivalently , on the rate of convergence of @xmath81 towards @xmath5 . these conditions have then to be translated into conditions on the rate of decay of the bias of estimators . as we focus on hill estimators , the connection between the rate of convergence of @xmath79 towards @xmath80 and the rate of decay of the bias is transparent and well - understood @xcite : the theory of @xmath86-regular variation provides an adequate setting for describing both rates of convergence @xcite . in words , if a positive function @xmath87 defined over @xmath55 is such that , for some @xmath88 , for all @xmath89 , @xmath90 } g(tx)/g(t ) < \infty$ ] , @xmath87 is said to have _ bounded increase_. if @xmath87 has bounded increase , the class @xmath91 is the class of measurable functions @xmath92 on some interval @xmath93 , such that as @xmath94 @xmath95 for all @xmath96 for example , the analysis carried out by @xcite rests on the condition that , if @xmath84 , for some @xmath97 , @xmath98 and @xmath99 , @xmath100 this condition implies that @xmath101 with @xmath102 @xcite . thus , under the von mises condition , condition implies that the function @xmath103 belongs to @xmath91 with @xmath104 moreover , the abelian and tauberian theorems from @xcite assert that @xmath105 if and only if @xmath106 for any intermediate sequence @xmath19 . in this text , we are ready to assume that if @xmath84 and satisfies the von mises condition , then , for some @xmath97 and @xmath99 and @xmath107 , @xmath108 this condition is arguably more stringent than ( [ eq : kimcarpcond ] ) . however , we do not want to assume that @xmath48 satisfies a regular variation property . this would imply that @xmath109 is @xmath110-regularly varying . indeed , assuming as in @xcite and several subsequent papers that @xmath0 satisfies @xmath111 where @xmath112 are constants and @xmath113 , or equivalently,@xcite that @xmath114 satisfies @xmath115 ( which entails that @xmath48 is regularly varying ) makes the problem of extreme value index estimation easier ( but not easy ) . these conditions entail that , for any intermediate sequence @xmath19 , the ratio @xmath116|/(n / k_n)^\rho$ ] converges towards a finite limit as @xmath37 tends to @xmath117 @xcite . this makes the estimation of the second - order parameter a very natural intermediate objective ( see for example * ? ? ? the necessity of developing data - driven index selection methods is illustrated in figure [ fig : riskcomp - student ] , which displays the estimated standardised root mean squared error ( rmse ) of hill estimators @xmath118^{1/2}\ ] ] as a function of @xmath119 for four related sampling distributions which all satisfy the second - order condition with different values of the second - order parameters . for samples of size @xmath120 from student s distributions with different degrees of freedom @xmath121 . all four distributions satisfy condition with @xmath122 . the increasing parts of the lines reflect the values of @xmath110 . rmse is estimated by averaging over @xmath123 monte - carlo simulations . ] under this second - order condition , @xcite proved that the asymptotic mean squared error of the hill estimator is minimal for sequences @xmath124 satisfying @xmath125 with @xmath126 . since @xmath97 , @xmath98 and the second - order parameter @xmath99 are usually unknown , many authors have been interested in the construction of data - driven selection procedures for @xmath28 under conditions such as . a great deal of ingenuity has been dedicated to the estimation of the second - order parameters and to the use of such estimates when estimating first order parameters . as we do not want to assume a second - order condition such as condition , we resort to lepski s method which is a general attempt to balance bias and variance . since its introduction @xcite , this general method for model selection has been proved to achieve adaptivity and to provide one with oracle inequalities in a variety of inferential contexts ranging from density estimation to inverse problems and classification @xcite . very readable introductions to lepski s method and its connections with penalised contrast methods can be found in @xcite . in , we are aware of three papers that explicitly rely on this methodology : @xcite , @xcite and @xcite . the selection rule analysed in the present paper ( see section [ sec : adapt - hill - estim ] for a precise definition ) is a variant of the preliminary selection rule introduced in @xcite @xmath127 where @xmath128 is a sequence of thresholds such that @xmath129 and @xmath130 , and @xmath131 is the hill estimator computed from the @xmath132 largest order statistics . the definition of this stopping time " is motivated by lemma 1 from @xcite which asserts that , under the von mises condition , @xmath133 | = o_p \left ( \sqrt{\ln \ln n } \right ) { \enspace .}\ ] ] in words , this selection rule almost picks out the largest index @xmath119 such that , for all @xmath64 smaller than @xmath119 , @xmath70 differs from @xmath131 by a quantity that is not much larger than the typical fluctuations of @xmath131 . this index selection rule can be performed graphically by interpreting an alternative hill plot as shown on figure [ fig : cauchy - lepski ] ( see * ? ? ? * ; * ? ? ? * for a discussion on the merits of alt - hill plots ) . computed on a pseudo - random sample of size @xmath134 from student distribution with 1 degree of freedom ( cauchy distribution ) . hill estimators are computed from the positive order statistics . the grey ribbon around the plain line provides a graphic illustration of lepski s method . for a given value of @xmath64 , the width of the ribbon is @xmath135 . a point @xmath136 on the plain line corresponds to an eligible index if the horizontal segment between this point and the vertical axis lies inside the ribbon that is , if for all @xmath137 , @xmath138 . if @xmath139 were replaced by an appropriate quantile of the gaussian distribution , the grey ribbon would just represent the confidence tube that is usually added on hill plots . the triangle represents the selected index with @xmath140 . the cross represents the oracle index estimated from monte - carlo simulations , see table [ tab : ratios:1 ] . ] the goal of @xcite is not to investigate the performance of the preliminary selection rule defined in display but to design a selection rule @xmath141 , based on @xmath142 , that would asymptotically mimic the optimal selection rule @xmath143 under second - order conditions . our goal , as in @xcite , is to derive non - asymptotic risk bounds without making a second - order assumption . in both papers , the rationale for working with some special collection of estimators seems to be the ability to derive non - asymptotic deviation inequalities for @xmath70 either from exponential inequalities for log - likelihood ratio statistics or from simple binomial tail inequalities such as bernstein s inequality ( see * ? ? ? * section 2.8 ) . in models satisfying condition , the estimators from @xcite achieve the optimal rate up to a @xmath144 factor . @xcite prove that the risk of their data - driven estimator decays at the optimal rate @xmath145 up to a factor @xmath146 in models satisfying condition . we aim at achieving optimal risk bounds under condition using a simple estimation method requiring almost no calibration effort and based on mainstream extreme value index estimators . before describing the keystone of our approach in section [ sec : talagr - conc - phen ] , we recall the recent lower risk bound for adaptive extreme value index estimation . one of the key results in @xcite is a lower bound on the accuracy of adaptive tail index estimation . this lower bound reveals that , just as for estimating a density at a point @xcite , or point estimation in sobolev spaces @xcite , as far as tail index estimation is concerned , adaptivity has a price . using fano s lemma , and a bayesian game that extends cleanly in frameworks of @xcite and @xcite , @xcite were able to prove the next minimax lower bound . [ thm - kc - lower - bound ] let @xmath147 and @xmath148 $ ] . then , for any tail index estimator @xmath149 and any sample size @xmath37 such that @xmath150 , there exists a probability distribution @xmath151 such that a. @xmath152 with @xmath4 , b. @xmath151 meets the von mises condition with von mises function @xmath48 satisfying @xmath153 for some @xmath154 , c. @xmath155 + and @xmath156 \geq \frac{c_\rho}{4(1 + 2{\mathrm{e } } ) } \left(\frac{v\ln\ln n}{n}\right)^{|\rho|/(1 + 2|\rho| ) } \ , , \ ] ] with @xmath157 . using birg s lemma instead of fano s lemma , we provide a simpler , shorter proof of this theorem ( see appendix [ proof : lower : bound ] ) . the lower rate of convergence provided by theorem [ thm - kc - lower - bound ] is another incentive to revisit the preliminary tail index estimator from @xcite . however , instead of using a sequence @xmath128 of order larger than @xmath158 in order to calibrate pairwise tests and ultimately to design estimators of the second - order parameter ( if there are any ) , it is worth investigating a minimal sequence where @xmath139 is of order @xmath159 , and check whether the corresponding adaptive estimator achieves the carpentier - kim lower bound ( theorem [ thm - kc - lower - bound ] ) . in this paper , we focus on @xmath139 of the order @xmath158 . the rationale for imposing @xmath139 of the order @xmath158 can be understood by the fact that , even if the sampling distribution is a pure pareto distribution with shape parameter @xmath14 ( @xmath160 for @xmath161 ) , if @xmath162 the preliminary selection rule will , with high probability , select a small value of @xmath119 and thus pick out a suboptimal estimator . this can be justified using results from @xcite ( see appendix [ sec : calibr - prel - select ] for details ) . such an endeavour requires sharp probabilistic tools . they are the topic of the next section . deriving authentic concentration inequalities for hill estimators is not straightforward . fortunately , the construction of such inequalities turns out to be possible thanks to general functional inequalities that hold for functions of independent exponentially distributed random variables . we recall these inequalities ( proposition [ poincare : expo ] and theorem [ bernstein : expo ] ) which have been largely overlooked in statistics . a thorough and readable presentation of these inequalities can be found in @xcite . we start by the easiest result , a variance bound that pertains to the family of poincar inequalities . [ poincare : expo ] if @xmath87 is a differentiable function over @xmath163 and @xmath164 where @xmath165 are independent standard exponential random variables , then @xmath166 \ , .\ ] ] the constant @xmath167 can not be improved . the next corollary is stated in order to point the relevance of this poincar inequality to the analysis of general order statistics and their functionals . recall that the _ hazard rate _ of an absolutely continuous probability distribution with distribution @xmath0 is : @xmath168 where @xmath92 and @xmath1 are the density and the survival function associated with @xmath0 , respectively . [ var : os ] assume the distribution of @xmath169 has a positive density , then the @xmath119th order statistic @xmath170 satisfies @xmath171 \le \frac{c}{k } \left(1 + \frac{1}{k}\right){\ensuremath{\mathbb{e}}}\left [ \frac{1}{h(x_{(k)})^2 } \right]\ ] ] where @xmath172 can be chosen as @xmath167 . by smirnov s lemma @xcite , @xmath172 can not be smaller than @xmath173 . if the distribution of @xmath169 has a non - decreasing hazard rate , the factor of @xmath167 can be improved into a factor @xmath174 @xcite . @xcite show that smooth functions of independent exponential random variables satisfy bernstein type concentration inequalities . the next result is extracted from the derivation of talagrand s concentration phenomenon for product of exponential random variables in @xcite . the definition of sub - gamma random variables will be used in the formulation of the theorem and in many arguments . [ dfn : sub - gamma ] a real - valued centred random variable @xmath169 is said to be _ sub - gamma _ on the right tail with variance factor @xmath175 and scale parameter @xmath176 if @xmath177 we denote the collection of such random variables by @xmath178 . similarly , @xmath169 is said to be sub - gamma on the left tail with variance factor @xmath175 and scale parameter @xmath176 if @xmath179 is sub - gamma on the right tail with variance factor @xmath175 and tail parameter @xmath176 . we denote the collection of such random variables by @xmath180 and @xmath181 by @xmath182 . if @xmath183 , then for all @xmath184 , with probability larger than @xmath185 @xmath186 the entropy of a non - negative random variable @xmath169 is defined by @xmath187= { \mathbb{e}}[x \ln x]-{\mathbb{e}}x \ln{\mathbb{e}}x$ ] . [ bernstein : expo ] assume that @xmath87 is a differentiable function on @xmath163 with @xmath188 . let @xmath164 where @xmath61 are @xmath37 independent standard exponential random variables and @xmath189 . then , for all @xmath190 such that @xmath191 , @xmath192 \leq \frac{2\lambda^2}{1-c } { \ensuremath{\mathbb{e}}}\left [ { \mathrm{e}}^{\lambda ( z-{\ensuremath{\mathbb{e}}}z ) } \| \nabla g\|^2\right ] \ , .\ ] ] let @xmath175 be the essential supremum of @xmath193 , then @xmath194 is sub - gamma on both tails with variance factor @xmath195 and scale factor @xmath196 . again , we illustrate the relevance of these versatile tools on the analysis of general order statistics . this general theorem implies that if the sampling distribution has non - decreasing hazard rate , then the order statistics @xmath170 satisfy bernstein type inequalities ( see * ? ? ? * section 2.8 ) with variance factor @xmath197 $ ] ( the poincar estimate of variance ) and scale parameter @xmath198 ) . starting back from the efron - stein - steele inequality , the authors derived a somewhat sharper inequality @xcite . [ prp : hazard : conc : ineg ] assume the distribution function @xmath0 has non - decreasing hazard rate @xmath199 that is , @xmath200 is @xmath201 and concave . let @xmath202 be distributed as the @xmath119th order statistic of a sample distributed according to @xmath0 . then , @xmath194 is sub - gamma on both tails with variance factor @xmath203 $ ] and scale factor @xmath204 . this corollary describes in which way central , intermediate and extreme order statistics can be portrayed as smooth functions of independent exponential random variables . this possibility should not be taken for granted as it is non trivial to capture in a non - asymptotic way the tail behaviour of maxima of independent gaussians @xcite . in the next section , we show in which way the hill estimator can fit into this picture . in this section , the sampling distribution @xmath0 is assumed to belong to @xmath22 with @xmath4 and to satisfy the von mises condition ( definition [ dfn : vmises : cond ] ) with bounded von mises function @xmath48 . it is well known that , under the von mises condition , if @xmath19 is an intermediate sequence , the sequence @xmath205 converges in distribution towards @xmath206 , suggesting that the variance of @xmath23 scales like @xmath207 ( see * ? ? ? * ; * ? ? ? * ; * ? ? ? * ; * ? ? ? proposition [ prop : bound - vari - hill ] provides us with handy non - asymptotic bounds on @xmath208 - \gamma^2/k $ ] using the von mises function . [ prop : bound - vari - hill ] let @xmath70 be the hill estimator computed from the @xmath209 largest order statistics of an @xmath37-sample from @xmath0 . then , @xmath210 \leq { \operatorname{var } } [ \widehat{\gamma}(k ) ] - \frac{\gamma^2}{k } \leq \frac{2\gamma}{k } { \ensuremath{\mathbb{e}}}\left [ \overline{\eta}\left({\mathrm{e}}^{y_{(k+1 ) } } \right)\right ] + \frac{5}{k}{\ensuremath{\mathbb{e}}}\left [ \overline{\eta}\left({\mathrm{e}}^{y_{(k+1 ) } } \right)^2\right ] \ , .\ ] ] the next abelian result might help in appreciating these variance bounds . [ sec : abel - vari - hill ] assuming that @xmath48 is @xmath110-regularly varying with @xmath99 , then , for any intermediate sequence @xmath19 , @xmath211 we may now move to genuine concentration inequalities for the hill estimator . the exponential representation suggests that the rescaled hill estimator @xmath212 should be approximately distributed according to a @xmath213 distribution where @xmath119 is the shape parameter and @xmath14 the scale parameter . therefore , we expect the hill estimators to satisfy bernstein type concentration inequalities that is , to be sub - gamma on both tails with variance factors connected to the tail index @xmath14 and to the von mises function . representation actually suggests more . following @xcite , we actually expect the sequence @xmath214 to behave like normalized partial sums of independent square integrable random variables that is , we believe @xmath215 to scale like @xmath158 and to be sub - gamma on both tails ( see appendix [ sec : calibr - prel - select ] ) . the purpose of this section is to meet these expectations in a non - asymptotic way . proofs use the markov property of order statistics : conditionally on the @xmath216th order statistic , the first largest @xmath217 order statistics are distributed as the order statistics of a sample of size @xmath217 of the excess distribution . they consist of appropriate invocations of talagrand s concentration inequality ( theorem [ bernstein : expo ] ) . however , this theorem generally requires a uniform bound on the gradient of the relevant function . when hill estimators are analysed as functions of independent exponential random variables , the partial derivatives depend on the points at which the von mises function is evaluated . in order to get interesting bounds , it is worth conditioning on an intermediate order statistic . throughout this subsection , let @xmath218 be an integer larger than @xmath219 and @xmath217 an integer not larger than @xmath37 . we denote @xmath220 , @xmath37 independent standard exponential random variables and we work on the probability space where all @xmath221 are defined , and therefore consider the hill estimators defined by representation . as we use the exponential representation of order statistics , besides hill estimators , the random variables that appear in the main statements are order statistics of exponential samples . as before , @xmath222 will denote the @xmath119th order statistic of a standard exponential sample of size @xmath37 ( we agree on @xmath39 ) . the first theorem provides an exponential refinement of the variance bound stated in proposition [ prop : bound - vari - hill ] . however , as announced , there is a price to pay : statements hold conditionally on some order statistic . this is not an impediment to analyse lepski s rule using this theorem . indeed , when analysing lepki s rule it is sufficient to control the hill process @xmath223)\right)_i$ ] for indices @xmath64 ranging between @xmath224 ( that should not be smaller than @xmath144 ) and some upper bound @xmath28 that achieves a certain balance between bias and standard deviation ( the bias of @xmath23 should be of order @xmath139 times the standard deviation , that is approximately @xmath225 where @xmath226 ) . the second clause of next theorem is the cornerstone in the derivation of the risk bounds presented in the next section . in the sequel , let @xmath227 where @xmath228 may be chosen not larger than @xmath167 and @xmath229 not larger than @xmath230 . [ prp : hill : concentration ] let @xmath231 be a shorthand for @xmath232 . for some @xmath119 such that @xmath233 where @xmath234 , let @xmath235 \right|\ , .\ ] ] then , conditionally on @xmath231 , a. for @xmath236 , @xmath237\right ) \in \gamma_\pm\left(4 \left(\gamma + 3\overline{\eta}(t ) \right)^2,\left(\gamma+2\overline{\eta}(t)\right)\right ) \ , .\ ] ] b. let @xmath238 be such that @xmath239 where @xmath240 with @xmath241 . assume that @xmath242 , then @xmath243 and @xmath244 \leq \gamma \xi_n \left(1 + \frac{3r_n}{\sqrt{j } } \right)\ , .\ ] ] if @xmath0 is a pure pareto distribution with shape parameter @xmath4 , then @xmath245 is distributed according to a gamma distribution with shape parameter @xmath119 and scale parameter @xmath173 . tight and well - known tail bounds for gamma distributed random variables assert that @xmath246 \right| \geq \frac{\gamma}{\sqrt{k } } \left ( \sqrt{2\ln\left(2/\delta \right ) } + \frac{\ln \left(2/\delta \right)}{\sqrt{k}}\right ) \right\ } \leq 2 \delta \ , .\ ] ] first part of statement ii ) reads as : conditionally on @xmath247 , with probability larger than @xmath248 , @xmath249\right| \leq \gamma ( 1 + 3r_n /\sqrt{j } ) \left ( \sqrt{8 \ln \left(2/\delta \right ) } + \frac{\ln \left(2/\delta \right)}{\sqrt{\ell}}\right ) \ , . \,\ ] ] combining both parts of statement ii ) , we also get that , conditionally on @xmath247 , with probability larger than @xmath248 , @xmath250 the reader may wonder whether resorting to the exponential representation and usual chernoff bounding would not provide a simpler argument . the straightforward approach leads to the following conditional bound on the logarithmic moment generating function , @xmath251 \right ) \right ) \mid y_{(k+1)}\right]}\\ & \leq & \frac{\left(\gamma + \overline{\eta}({\mathrm{e}}^{y_{(k+1)}})\right)^2}{2k\left(1-\lambda(\gamma + \overline{\eta}({\mathrm{e}}^{y_{(k+1 ) } } ) ) \right ) } + \lambda \left(\overline{\eta}({\mathrm{e}}^{y_{(k+1)}})-b\left ( { \mathrm{e}}^{y_{(k+1)}}\right ) \right ) \ , . \end{aligned}\ ] ] a similar statement holds for the lower tail . this leads to exponential bounds for deviations of the hill estimator above @xmath252 + \overline{\eta}({\mathrm{e}}^{y_{(k+1)}})-b\left ( { \mathrm{e}}^{y_{(k+1)}}\right)$ ] that is , to control deviations of the hill estimator above its expectation plus a term that may be of the order of magnitude of the bias . attempts to rewrite @xmath253 $ ] as a sum of martingale increments @xmath254-{\ensuremath{\mathbb{e}}}[\widehat{\gamma}(k ) \mid y_{(i+1)}]$ ] , for @xmath255 , and to exhibit an exponential supermartingale met the same impediments . at the expense of inflating the variance factor , theorem [ bernstein : expo ] provides a genuine ( conditional ) concentration inequality for hill estimators . as we will deal with values of @xmath119 for which bias exceeds the typical order of magnitudes of fluctuations , this is relevant to our purpose . we are now able to characterise the performance of the variant of the selection rule defined by @xcite with @xmath256 where @xmath257 . let @xmath258 where @xmath259 is a constant to be defined below . the deterministic sequence of indices @xmath260 is defined ( for @xmath37 large enough ) by @xmath261 where @xmath262 the sequence @xmath263 is defined by choosing @xmath264 . the deterministic sequences @xmath265 and @xmath260 achieve specific balances between bias and variance . in full generality , because @xmath266 is just an upper bound on the conditional bias @xmath75 , it is difficult to precisely connect @xmath265 and @xmath260 with the oracle sequence @xmath267 . we call these two sequences the pivotal sequences . in the sequel , @xmath28 stands for @xmath268 . if the context is not clear , we specify @xmath269 or @xmath270 . let @xmath271 . recall , from section [ sec : conc - ineq - hill ] , that @xmath272 and agree on the shorthands @xmath273 and @xmath274 which is defined by replacing @xmath28 by @xmath224 in the definition of @xmath275 ( @xmath274 depends on @xmath276 but not on the sampling distribution ) . in the sequel , @xmath259 is assumed to be chosen so that @xmath277 for @xmath278 and @xmath279 ( @xmath259 may be chosen not larger than 100 ) . the index @xmath280 is selected according to the following rule : @xmath281 where @xmath282 . the quantity @xmath283 scales like @xmath284 . the tail index estimator is @xmath285 . as tail adaptivity has a price ( see theorem [ thm - kc - lower - bound ] ) , the ratio between the risk of the data - driven estimator @xmath285 and the risk of the pivotal index @xmath286 can not be upper bounded by a constant factor , let alone by a factor close to @xmath173 . this is why in the next theorem , we compare the risk of the empirically selected index @xmath285 with the risk of the pivotal index @xmath23 . recall , from section [ sec : conc - ineq - hill ] , that @xmath287 [ thm : adapt - hill - estim ] assume the sampling distribution @xmath288 satisfies the von mises condition with bounded von mises function @xmath48 , and @xmath289 let @xmath290 is large enough so that @xmath28 ( definition [ def : kn ] ) is well defined . then , for @xmath291 , with probability larger than @xmath292 , @xmath293 and , with probability larger than @xmath294 , @xmath295 where @xmath296 for @xmath297 , @xmath298 if the bias @xmath73 is @xmath110-regularly varying ( or equivalently , if the von mises function @xmath48 or even @xmath54 are regularly varying ) , then , elaborating on proposition 1 from @xcite , sequences @xmath299 and @xmath265 are connected by @xmath300 and their quadratic risk are related by @xmath301}{{\ensuremath{\mathbb{e}}}[(\gamma-\widehat{\gamma}(k_n^*))^2 ] } = \frac{2}{2|\rho|+1}(2|\rho|)^{2|\rho|/(1 + 2|\rho| ) } \ , .\ ] ] moreover , under the second - order assumption , the two pivotal sequences @xmath265 and @xmath260 are also connected . thus , if the bias is @xmath110-regularly varying , theorem [ thm : adapt - hill - estim ] provides us with a connection between the performance of the simple selection rule and the performance of the ( asymptotically ) optimal choice . recall that one of the main aims of this paper is to derive performance guarantees for the data - driven index selection method @xmath280 without resorting to second - order assumptions that is , without assuming that the von mises function is regularly varying . the next corollary upper bounds the risk of the preliminary estimator when we just have an upper bound on the bias . [ cor : adapt - hill - estim-2ndrv ] assume that , for some @xmath97 and @xmath99 , for all @xmath302 , @xmath303 then , there exists a constant @xmath304 depending on @xmath305 and @xmath110 such that , with probability larger than @xmath294 , @xmath306 where @xmath307 is defined in theorem [ thm : adapt - hill - estim ] . this meets the information - theoretic lower bound of theorem [ thm - kc - lower - bound ] . this proposition is a straightforward consequence of rnyi s representation of order statistics of standard exponential samples . as @xmath0 belongs to @xmath22 and meets the von mises condition , there exists a function @xmath48 on @xmath31 with @xmath308 such that @xmath309 and @xmath310 then , @xmath311 let @xmath312 . by the pythagorean relation , @xmath313 + { \operatorname{var}}\left ( { \ensuremath{\mathbb{e}}}[z \mid y_{(k+1)}]\right ) \ , .\ ] ] representation asserts that , conditionally on @xmath68 , @xmath194 is distributed as a sum of independent , exponentially distributed random variables . let @xmath34 be an exponentially distributed random variable . @xmath314 where we have used the cauchy - schwarz inequality and @xmath315 taking expectation with respect to @xmath68 leads to @xmath316 \leq k{\gamma^2 } + 2k\gamma{\ensuremath{\mathbb{e}}}\left [ \overline{\eta}\left({\mathrm{e}}^{y_{(k+1 ) } } \right)\right ] + k{{\ensuremath{\mathbb{e}}}\left [ \overline{\eta}\left({\mathrm{e}}^{y_{(k+1 ) } } \right)^2\right ] } \ , .\ ] ] the last term in the pythagorean decomposition is also handled using elementary arguments . @xmath317 = k \gamma + k\int_0^\infty{\mathrm{e}}^{-u } \eta\left({\mathrm{e}}^{u + y_{(k+1)}}\right ) \mathrm{d}u \ , .\end{aligned}\ ] ] as @xmath68 is a function of independent exponential random variables ( @xmath318 ) , the variance of @xmath319 $ ] may be upper bounded using poincar inequality ( proposition [ poincare : expo ] ) @xmath320\right ) \leq 4k { \ensuremath{\mathbb{e}}}\left [ \overline{\eta}\left({\mathrm{e}}^{y_{(k+1 ) } } \right)^2\right]\ , .\ ] ] in order to derive the lower bound , we first observe that @xmath321 \ , .\ ] ] now , using cauchy - schwarz inequality again , @xmath322 in the proof of theorem [ prp : hill : concentration ] , we will use the next maximal inequality ( see * ? ? ? * corollary 2.6 ) . recall the definition of @xmath178 ( definition [ dfn : sub - gamma ] ) . [ zlapcor ] let @xmath323 be real - valued random variables belonging to @xmath178 . then @xmath324 \leq \sqrt{2v\ln n}+ c\ln n~.\ ] ] proofs follow a common pattern . in order to check that some random variable is sub - gamma , we rely on its representation as a function of independent exponential variables and compute partial derivatives , derive convenient upper bounds on the squared euclidean norm and the supremum norm of the gradient and then invoke theorem [ bernstein : expo ] . at some point , we will use the next corollary of theorem [ bernstein : expo ] . [ sec : proof - prop - refprp : technique ] if @xmath87 is an almost everywhere differentiable function on @xmath325 with uniformly bounded derivative @xmath326 , then @xmath327 is sub - gamma with variance factor @xmath328 and scale factor @xmath329 we start from the exponential representation of hill estimators ( proposition [ hill : rep ] ) and represent all @xmath131 as functions of independent random variables @xmath330 where the @xmath331 , are standard exponentially distributed and @xmath332 is distributed like the @xmath216th largest order statistic of an @xmath37-sample of the standard exponential distribution . we consistently use the notation @xmath333 , for @xmath334 . @xmath335 let @xmath336 be such that @xmath337 , let us agree on @xmath338 let @xmath339 for @xmath340 , as @xmath341 for @xmath342 and @xmath83 otherwise , @xmath343 this entails that , for @xmath344 , @xmath345 for @xmath346 , @xmath347 this is enough to entail that , for @xmath348 , @xmath349 all in all , for @xmath350 , @xmath351 [ [ proof - of - i ] ] proof of i ) + + + + + + + + + + + an upper bound on the variance factor for @xmath352 , conditionally on @xmath231 , is obtained by specialising to the case @xmath353 and using and as well as the monotonicity of @xmath54 , @xmath354 using theorem [ bernstein : expo ] conditionally on @xmath355 , we realise that @xmath356)$ ] is sub - gamma on both sides with variance factor not larger than @xmath357 and scale factor not larger than @xmath358 . this yields & \ { ( i)-@xmath359 ( + ) t } 2 ^-s . & \ { ( i)-@xmath359 ( + ) } 2 ^-s . [ [ par : proof_of_ii _ ] ] proof of ii ) + + + + + + + + + + + + the proof of the upper bound on @xmath360 $ ] in statement ii ) from theorem [ prp : hill : concentration ] relies on standard chaining techniques from the theory of empirical processes and uses repeatedly the concentration theorem [ bernstein : expo ] for smooth functions of independent exponential random variables and the maximal inequality for sub - gamma random variables ( proposition [ zlapcor ] ) . for general @xmath336 , the variance factor for @xmath361 is upper bounded by @xmath362 let @xmath238 be such that @xmath363 where @xmath364 with @xmath257 . now , as we assume , in the sequel , that @xmath365 , we may use the next upper bound for the variance factor of @xmath366 ( conditionally on @xmath367),@xmath368 recall that @xmath369 \right|\ , .\ ] ] as it is commonplace in the analysis of normalised empirical processes ( see * ? ? ? * ; * ? ? ? * ; * ? ? ? * and references therein ) , we peel the index set over which the maximum is computed . let @xmath370 and , for all @xmath371 , @xmath372 . define @xmath373 as @xmath374 \right|\ , .\ ] ] then , @xmath375 & = & { \ensuremath{\mathbb{e}}}[\max_{j\in \mathcal{l}_n } z^a_j \mid y_{(k+1 ) } ] \\ & \leq & { \ensuremath{\mathbb{e}}}[\max_{j\in \mathcal{l}_n } ( z^a_j-{\ensuremath{\mathbb{e}}}[z^a_j\mid y_{(k+1 ) } ] ) \mid y_{(k+1 ) } ] ] + \max_{j\in \mathcal{l}_n } { \ensuremath{\mathbb{e } } } [ z^a_j\mid y_{(k+1 ) } ] ] \ , . \end{aligned}\ ] ] we now derive upper bounds on both summands by resorting to the maximum inequality for sub - gamma random variables ( proposition [ zlapcor ] ) . we first bound @xmath376 $ ] , for @xmath377 . note that direct invocation of lemma [ zlapcor ] and statement i ) shows that @xmath378 \leq 2 \gamma(1 + 3r/\sqrt{j})\left ( \sqrt{8 j \ln(2 ) } + j \ln(2)\right ) \ , .\ ] ] this bound will be useful for handling small values of @xmath379 . for @xmath380 , @xmath381 . we now handle generic @xmath379 using chaining . fix @xmath377 , @xmath382 \right| \leq \frac{1}{2^{j/2 } } \max_{i \in \mathcal{s}_j } \ , i\left| \widehat{\gamma}(i ) - { \ensuremath{\mathbb{e}}}[\widehat{\gamma}(i ) \mid y_{(k+1 ) } ] \right| \ , .\ ] ] in order to alleviate notation , let @xmath383 \right)$ ] , for @xmath384 . for @xmath384 , let @xmath385 be the binary expansion of @xmath64 . then , for @xmath386 , let @xmath387 be defined by @xmath388 so that @xmath389 , @xmath390 and @xmath391 . using the fact that @xmath392 does not depend on @xmath64 and that @xmath393=0\ , , \ ] ] we obtain @xmath394 \right)\mid y_{(k+1 ) } \right]}\\ & = & { \ensuremath{\mathbb{e}}}\left [ \max_{i \in \mathcal{s}_j } w(i ) \mid y_{(k+1 ) } \right ] \\ & = & { \ensuremath{\mathbb{e}}}\left [ \max_{i \in \mathcal{s}_j } w(\pi_j(i ) ) -w(\pi_0(i))\mid y_{(k+1 ) } \right ] \\ & = & { \ensuremath{\mathbb{e}}}\left [ \max_{i \in \mathcal{s}_j } \sum_{h=0}^{j-1 } ( w(\pi_{h+1}(i ) ) -w(\pi_h(i)))\mid y_{(k+1 ) } \right ] \\ & \leq & \sum_{h=0}^{j-1 } { \ensuremath{\mathbb{e}}}\left [ \max_{i \in \mathcal{s}_j } ( w(\pi_{h+1}(i ) ) -w(\pi_h(i ) ) ) \mid y_{(k+1 ) } \right ] \ , .\end{aligned}\ ] ] now , for each @xmath395 , the maximum is taken over @xmath396 random variables which are sub - gamma with variance factor @xmath397 and scale factor @xmath398 . by proposition [ zlapcor ] , since @xmath399 , @xmath394 \right)\mid y_{(k+1 ) } \right]}\\ & \leq & \gamma \sum_{h=0}^{j-1 } \left(\left(1 + \frac{r_n}{\sqrt{j}}\right ) \sqrt{8 h 2^{(j - h-1)}\ln 2 } + \sqrt{32 h \ln 2 } r_n+ \left(1 + \frac{2r_n}{\sqrt{j}}\right ) h \ln 2\right ) \\ & \leq & \gamma \left(1 + \frac{2r_n}{\sqrt{j}}\right ) \left ( 2^{(j-1)/2 } 4.15 \sqrt{8 \ln 2 } + \frac{2}{3 } \sqrt{32 c_2 } \ln(2 ) j^{2}+ \frac{j(j-1)}{2 } \ln 2\right ) \end{aligned}\ ] ] where we have used @xmath400 for @xmath401 , @xmath402 \right)\mid y_{(k+1 ) } \right ] } & \leq & 17 \gamma \ , 2^{j/2 } \left(1 + \frac{2r_n}{\sqrt{j}}\right)\ , . \end{aligned}\ ] ] finally , for all @xmath403 , @xmath404 \leq 34 \ , \gamma \ , \left(1 + \frac{3r_n}{\sqrt{j}}\right ) \,.\ ] ] in order to prove statement ii ) , we check that , for each @xmath403 , @xmath373 is sub - gamma on the right - tail with variance factor at most @xmath405 and scale factor not larger than @xmath406 . under the von mises condition ( definition [ dfn : vmises : cond ] ) , the sampling distribution is absolutely continuous with respect to lebesgue measure . for almost every sample , the maximum defining @xmath373 is attained at a single index @xmath384 . starting again from the exponential representation and repeating the computation of partial derivatives , we obtain the desired bounds . by proposition [ zlapcor ] , @xmath407 ) \mid y_{(k+1)}\right ] } \\ & \leq & \left ( \sqrt{8\ln |\mathcal{l}_n| } + \frac { \ln |\mathcal{l}_n|}{\sqrt{\ell}}\right)\left(\gamma + 3 \overline{\eta}(t ) \right ) \\ & \leq & 4 \sqrt{\ln |\mathcal{l}_n| } \left(\gamma + 3 \overline{\eta}(t ) \right ) \\ & \leq & 4 \ , \gamma\ , \sqrt{\ln |\mathcal{l}_n| } \left(1 + \frac{3r_n}{\sqrt{j } } \right ) \ , \end{aligned}\ ] ] where we have used @xmath408 , for @xmath409 . combining the different bounds leads to the upper bound on @xmath360 $ ] . throughout this proof , let @xmath410 let us define the events @xmath411 and @xmath412 as @xmath413\right|\leq \gamma z_\delta \big\}\ , , \\ e_2 & = & \big\ { t_n \geq \frac{n}{k_n^\delta } \big\ } \text { with } k_n^\delta= k_n+2{\ln \left(1/\delta \right)}+\sqrt{2k_n \ln ( 1/\delta)}{\enspace .}\end{aligned}\ ] ] the fact that @xmath414 follows from the following reformulation of proposition 4.3 from @xcite ( a proof is given in appendix [ proof : prop : right : tail : order : stat ] ) . [ prop : right : tail : order : stat ] for @xmath184 , with probability larger that @xmath248 , @xmath415 where @xmath68 is the @xmath209th largest order statistic of an exponential sample of size @xmath37 . by theorem [ prp : hill : concentration ] , @xmath416 . hence , the event @xmath417 has probability at least @xmath418 . under @xmath419 , a. @xmath420 . b. for all @xmath421 , @xmath422|\leq \overline{\eta}(t_n).$ ] the first step of the proof consists in checking that under @xmath417 , the selected index is not smaller than @xmath28 . it suffices to check that for all @xmath423 , @xmath424 for all @xmath425 , @xmath426\right| + \left|\widehat{\gamma}(i ) -{\ensuremath{\mathbb{e}}}[\widehat{\gamma}(i)\mid t_n]\right|\\ & \leq & \overline{\eta}(t_n ) + \frac{\gamma z_\delta}{\sqrt{i } } \\ & \leq & \frac{\gamma r_n}{\sqrt{k_n } } + \frac{\gamma z_\delta}{\sqrt{i } } \end{aligned}\ ] ] so that @xmath427 meanwhile , for all @xmath428 , @xmath429\big|}_{\textsc{(i)}}+ \underbrace{\left| { \ensuremath{\mathbb{e}}}[\widehat{\gamma}(i)-\widehat{\gamma}(k)\mid t_n]\right|}_{\textsc{(ii ) } } + \underbrace{\left| \widehat{\gamma}(k)-{\ensuremath{\mathbb{e}}}[\widehat{\gamma}(k)\mid t_n]\right|}_{\textsc{(iii ) } } \ , . \end{aligned}\ ] ] under @xmath417 , for @xmath430 , @xmath431 under @xmath412 , @xmath432\right| + \left| { \ensuremath{\mathbb{e } } } [ \gamma-\widehat{\gamma}(k)\mid t_n]\right| \\ & \leq & 2 \overline{\eta}(t_n ) \\ & \leq & 2 \gamma r_n /\sqrt{k_n } \ , . \end{aligned}\ ] ] plugging upper bounds on ( i ) , ( ii ) and ( iii ) , it comes that , under @xmath417 , for all @xmath433 and for all @xmath434 , @xmath435 in order to warrant that , under @xmath417 , for all @xmath436 and for all @xmath64 such that @xmath437 , @xmath438 , it is enough to have @xmath439 the last inequality holds because @xmath440 by definition of @xmath283 . hence , with probability larger than @xmath441 , @xmath417 is realised , and under @xmath417 , @xmath442 . we now check that if @xmath443 , the risk of @xmath285 is not much larger than the risk of @xmath23 . @xmath444 therefore , under @xmath445 , @xmath446 now , consider the event @xmath447 with @xmath448\right| \leq \left(\gamma+3 \overline{\eta}(t_n ) \right ) \big ( \sqrt{8 \ln \left(2/\delta \right ) } + \frac{\ln \left(2/\delta \right)}{\sqrt{k_n}}\big)\bigg\ } { \enspace .}\ ] ] since , @xmath449 , thanks to statement i ) from theorem [ prp : hill : concentration ] , the event @xmath447 has probability at least @xmath450 . then , by definition of @xmath28 , under @xmath412 , @xmath451\right|\leq \overline{\eta}(t_n ) \leq \gamma r_n/\sqrt{k_n } \ , .\ ] ] hence , under @xmath452 , @xmath453|+ | \widehat{\gamma}({k}_n ) -\mathbb{e}[\widehat{\gamma}({k}_n)\mid t_n]| \\ & \leq & \frac{\gamma}{\sqrt{k_n}}\left(r_n + \left(1 + \frac{3r_n}{\sqrt{k_n}}\right)\big ( \sqrt{8 \ln \left(2/\delta \right ) } + \frac{\ln \left(2/\delta \right)}{\sqrt{k_n}}\big ) \right ) \ , . \end{aligned}\ ] ] therefore , plugging this bound into , with probability larger than @xmath292 , @xmath454 where @xmath455 if , for some @xmath97 and @xmath99 , @xmath456 then , by the definition of @xmath28 , @xmath457 which entails that @xmath458 solving this inequality leads to @xmath459 and finally to @xmath460 thus , for sufficiently large @xmath37 , there exists a constant @xmath176 depending on @xmath461 such that @xmath462 starting from equation of theorem [ thm : adapt - hill - estim ] , with probability @xmath292 , @xmath463 and , there exists a constant @xmath464 , depending on @xmath305 and @xmath110 , such that @xmath465 hence , with probability larger than @xmath294 , @xmath466 risk bounds like theorem [ thm : adapt - hill - estim ] and corollary [ cor : adapt - hill - estim-2ndrv ] are conservative . for all practical purposes , they are just meant to be reassuring guidelines . in this numerical section , we intend to shed some light on the following issues : 1 . is there a reasonable way to calibrate the threshold @xmath283 used in the definition of @xmath280 ? how does the method perform if we choose @xmath283 close to @xmath467 ? how large is the ratio between the risk of @xmath285 and the risk of @xmath468 for moderate sample sizes ? the finite - sample performance of the data - driven index selection method described and analysed in section [ sec : adapt - hill - estim ] has been assessed by monte - carlo simulations . computations have been carried out in ` r ` using packages ` ggplot2 ` @xcite , ` knitr ` , ` foreach ` , ` iterators ` , ` xtable ` and ` dplyr ` ( see * ? ? ? * for a modern account of the r environment ) . to get into the details , we investigated the performance of index selection methods on samples of sizes @xmath469 and @xmath470 from the collection of distributions listed in table [ tab - risk - profiles ] . the list comprises the following distributions a. frchet distributions @xmath471 for @xmath3 and @xmath472 . b. student distributions @xmath473 with @xmath474 degrees of freedom . c. the log - gamma distribution with density proportional to @xmath475 , which means @xmath476 and @xmath477 . d. the lvy distribution with density @xmath478 , @xmath479 and @xmath480 ( this is the distribution of @xmath481 when @xmath482 ) . e. the @xmath483 distribution is defined by @xmath484 and von mises function equal to @xmath485 . this distribution satisfies the second - order regular variation condition with @xmath486 but does not satisfy condition . f. two pareto change point distributions with distribution functions @xmath487 and @xmath488 , @xmath489 , and thresholds @xmath7 adjusted in such a way that they correspond to quantiles of order @xmath490 and @xmath491 , respectively . frchet , student , log - gamma distributions were used as benchmarks by @xcite , @xcite and @xcite . table [ tab - risk - profiles ] , which is complemented by figure [ fig : risk - comp ] , describes the difficulty of tail index estimation from samples of the different distributions . monte - carlo estimates of the standardised root mean square error ( rmse ) of hill estimators @xmath492^{1/2}\ ] ] are represented as functions of the number of order statistics @xmath119 for samples of size @xmath470 from the sampling distributions . all curves exhibit a common pattern : for small values of @xmath119 , the rmse is dominated by the variance term and scales like @xmath493 . above a threshold that depends on the sampling distribution but that is not completely characterised by the second - order regular variation index , the rmse grows at a rate that may reflect the second - order regular variation property ( if any ) of the distribution . not too surprisingly , the three frchet distributions exhibit the same risk profile . the three curves are almost undistinguishable . the student distributions illustrate the impact of the second - order parameter on the difficulty of the index selection problem . for sample size @xmath494 , the optimal index for @xmath495 is smaller than @xmath496 , it is smaller than the usual recommendations . for such moderate sample sizes , distribution @xmath495 seems as hard to handle as the @xmath497-gamma distribution which usually fits in the horror hill plot gallery . the @xmath498-stable lvy distribution and the @xmath483-distribution behave very differently . even though they both have second - order parameter @xmath110 equal to @xmath499 , the @xmath483 distribution seems almost as challenging as the @xmath500 distribution while the lvy distribution looks much easier than the frchet distributions . the pareto change point distributions exhibit an abrupt transition . .estimated oracle index @xmath501 and standardised rmse @xmath502^{1/2}/\gamma$ ] for benchmark distributions . estimates were computed from @xmath123 replicated experiments on samples of size @xmath503 [ cols="<,^,^,^,^",options="header " , ] figure [ fig : pcp - risk - plot ] concisely describes the behaviour of the two index selection methods on samples from the pareto change point distribution with parameters @xmath504 and threshold @xmath7 corresponding to the @xmath490 quantile . the plain line represents the standardised rmse of hill estimators as a function of selected index . this figure contains the superposition of two density plots corresponding to @xmath505 and @xmath506 . the density plots were generated from @xmath123 points with coordinates latexmath:[$(\widehat{k}(r_n ) , points with coordinates latexmath:[$(\widehat{k}^{\textsc{dk}}_n , contoured and well - concentrated density plot corresponds to the performance of @xmath285 . the diffuse tiled density plot corresponds to the performance of @xmath505 . facing pareto change point samples , the two selection methods behave differently . lepski s rule detects correctly an abrupt change at some point and selects an index slightly above that point . as the conditional bias varies sharply around the change point , this slight over estimation of the correct index still results in a significant loss as far as rmse is concerned . the drees - 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verlag , 2009 . h. wickham . _ advanced r_. chapman & hall / crc , 2014 . let @xmath509 . then , @xmath510 and @xmath511 let @xmath189 , then for all @xmath512 , @xmath513\lambda^2 } { 2(1-c ) } \ , .\ ] ] now , start from the first statement in theorem [ bernstein : expo ] , @xmath514 & \leq & \frac{2\lambda^2}{1-c } { \ensuremath{\mathbb{e}}}\left [ { \mathrm{e}}^{\lambda ( z-{\ensuremath{\mathbb{e}}}z ) } \| \nabla f\|^2\right ] \\ & = & \frac{4\lambda^2}{2(1-c ) } \frac{1}{k } \left ( 1 + \frac{1}{k}\right ) { \ensuremath{\mathbb{e}}}\left [ \frac{{\mathrm{e}}^{\lambda ( z-{\ensuremath{\mathbb{e}}}z)}}{h(z)^2}\right ] \\ & \leq & \frac{4\lambda^2}{2(1-c ) } \frac{1}{k } \left ( 1 + \frac{1}{k}\right ) { \ensuremath{\mathbb{e}}}\left [ { { \mathrm{e}}^{\lambda ( z-{\ensuremath{\mathbb{e}}}z)}}\right ] { \ensuremath{\mathbb{e}}}\left [ \frac{1}{h(z)^2}\right]\end{aligned}\ ] ] where the last inequality follows from chebychev negative association inequality . hence , @xmath515 = \frac{\operatorname{ent } \left [ { \mathrm{e}}^{\lambda ( z-{\ensuremath{\mathbb{e}}}z)}\right]}{\lambda^2 { \ensuremath{\mathbb{e}}}\left [ { \mathrm{e}}^{\lambda ( z-{\ensuremath{\mathbb{e}}}z)}\right ] } \leq \frac{1}{2(1-c ) } \frac{4}{k } \left ( 1+\frac{1}{k}\right ) { \ensuremath{\mathbb{e}}}\left [ \frac{1}{h(z)^2}\right ] \ , .\ ] ] this differential inequality is readily solved and leads to the corollary . the proof proceeds by classical arguments . in the sequel , we use the almost sure representation argument . without loss of generality , we assume that all the random variables live on the same probability space , and that , for any intermediate sequence @xmath19 , @xmath516 converges almost surely towards a standard gaussian random variable . complemented with dominated convergence arguments , the next lemma will be the key element of the proof . [ lem : rvplus ] let @xmath517 and @xmath518 be the @xmath519th largest order statistic of a standard exponential sample , then , for any intermediate sequence @xmath19 and @xmath520 , @xmath521 note that @xmath522 then , the result follows since @xmath523 and the convergence @xmath524 is locally uniform on @xmath525 . in order to secure dominated convergence arguments , we will use drees s improvement of potter s inequality ( see * ? ? ? for every @xmath526 , there exists @xmath527 such that , for @xmath528 , @xmath529 to prove proposition [ sec : abel - vari - hill ] , we start from representation : @xmath530 by the pythagorean relation , @xmath531\right)+{\ensuremath{\mathbb{e}}}\left [ { \operatorname{var}}\left ( \widehat{\gamma}(k_n ) \mid y_{(k_n+1)}\right)\right ] \ , , \ ] ] so that @xmath532\right)}{\eta(n / k_n ) } + k_n { \ensuremath{\mathbb{e}}}\left [ \frac{{\operatorname{var}}\left ( \widehat{\gamma}(k_n ) \mid y_{(k_n+1)}\right)-\frac{\gamma^2}{k_n}}{\eta(n / k_n ) } \right ] \ , . \end{aligned}\ ] ] the second summand can be further decomposed using . @xmath533\right)}{\eta(n / k_n)}}_{\textsc{(i ) } } \\ & & + \underbrace{\eta ( \tfrac{n}{k_n}){\ensuremath{\mathbb{e}}}\left[{\operatorname{var}}\left [ \int_0^e \frac{\eta({\mathrm{e}}^{u+y_{(k_n+1 ) } } ) } { \eta ( n / k_n ) } { \mathrm{d}}u \mid y_{(k_n+1 ) } \right]\right]}_{(\textsc{ii})}\\ & & + \underbrace{2\gamma { \ensuremath{\mathbb{e}}}\left[{\operatorname{cov}}\left [ e,\int_0^e \frac{\eta({\mathrm{e}}^{u+y_{(k_n+1)}})}{\eta ( n / k_n)}{\mathrm{d}}u \mid y_{(k_n+1)}\right]\right]}_{(\textsc{iii } ) } . \end{aligned}\ ] ] we check that ( i ) and ( ii ) tend to @xmath83 and then that ( iii ) converges towards a finite limit . fix @xmath526 and define @xmath534 . + let @xmath535 denote the event @xmath536 . for @xmath37 such that @xmath537 , as @xmath518 is sub - gamma with variance factor @xmath538 , @xmath539 we first check that ( ii ) tends to @xmath83 . let @xmath37 be such that @xmath540 and @xmath541 denote the random variable @xmath542 . note that , for @xmath543 , @xmath544 using jensen s inequality and fubini s theorem , @xmath545\right]}\\ & \leq & { \ensuremath{\mathbb{e}}}\left[{\ensuremath{\mathbb{e}}}\left [ e \int_0^e \left ( \frac{\eta({\mathrm{e}}^{u+y_{(k_n+1)}})}{\eta ( n / k_n)}\right)^2{\mathrm{d}}u \mid y_{(k_n+1 ) } \right]\right ] \\ & = & \int_0^\infty { \mathrm{e}}^{-v } v \int_0^v { \ensuremath{\mathbb{e}}}\left[\left ( \frac{\eta({\mathrm{e}}^{u+y_{(k_n+1)}})}{\eta ( n / k_n)}\right)^2 \right ] { \mathrm{d}}u \mathrm{d}v \\ & = & \int_0^\infty { \mathrm{e}}^{-v } v \int_0^v { \ensuremath{\mathbb{e}}}\left[\left ( \frac{\eta({\mathrm{e}}^{u+w_n}n / k_n)}{\eta ( n / k_n)}\right)^2 \right ] { \mathrm{d}}u \mathrm{d}v \end{aligned}\ ] ] we now apply potter s inequality ( [ potter : drees ] ) on the event @xmath546 with @xmath547 and @xmath548 : @xmath549\right ] } \\ & \leq & \int_0^\infty { \mathrm{e}}^{-v } v \int_0^v { \ensuremath{\mathbb{e}}}\left [ \mathbb{1}_{a_{n } } { \mathrm{e}}^{2\rho ( u+w_n ) } \left ( 1+\epsilon { \mathrm{e}}^{\delta ( u+|w_n| ) } \right)^2 + \mathbb{1}_{a_n^c } \frac{m^2}{\eta(n / k_n)^2 } \right ] { \mathrm{d}}u \mathrm{d}v \\ & \leq & \int_0^\infty { \mathrm{e}}^{-v } v \int_0^v { \ensuremath{\mathbb{e}}}\left [ { \mathrm{e}}^{2\rho w_n } 2 \left ( 1+\epsilon^2 { \mathrm{e}}^{2\delta ( u+|w_n| ) } \right ) \right ] { \mathrm{d}}u \mathrm{d}v+ \frac{2m^2}{\eta(n / k_n)^2}{\ensuremath{\mathbb{e}}}\mathbb{1}_{a_n^c } \ , . \end{aligned}\ ] ] the first summand has a finite limit thanks to lemma [ lem : rvplus ] . the second summand converges to @xmath83 as @xmath550 tends to @xmath83 exponentially fast while @xmath551 tends to infinity algebraically fast . bounds on ( i ) are easily obtained , using jensen s inequality and poincar inequality . @xmath552\right)}{\eta(n / k_n ) } & = & \frac{k_n { \operatorname{var}}\left ( \int_0^\infty { \eta \left ( { \mathrm{e}}^{u+y_{(k_n+1 ) } } \right ) } { \mathrm{e}}^{-u } { \mathrm{d}}u\right)}{\eta(n / k_n ) } \\ & \le & 4 \eta(n / k_n ) { \mathbb{e}}\left [ \left ( \int_0^\infty \frac{\eta \left ( { \mathrm{e}}^{u+y_{(k_n+1 ) } } \right)}{\eta(n / k_n ) } { \mathrm{e}}^{-u } { \mathrm{d}}u \right)^2\right ] \\ & \leq & 4 \eta(n / k_n ) { \mathbb{e}}\left [ \int_0^\infty \left ( \frac{\eta \left ( { \mathrm{e}}^{u+y_{(k_n+1 ) } } \right)}{\eta(n / k_n)}\right)^2 { \mathrm{e}}^{-u } { \mathrm{d}}u \right ] \ , . \ ] ] using the line of arguments as for handling the limit of ( ii ) , we establish that ( i ) converges to @xmath83 . we now check that ( iii ) converges towards a finite limit . note that @xmath553\right ] } \\ & = & { \ensuremath{\mathbb{e}}}\left [ ( e-1)\int_0^e \frac{\eta({\mathrm{e}}^{u+y_{(k_n+1)}})}{\eta ( n / k_n)}{\mathrm{d}}u \right ] \ , . \end{aligned}\ ] ] by lemma [ lem : rvplus ] , for almost every @xmath520 , @xmath554 and @xmath555 the first term is finite as the integral of a continuous function on a compact . + thus , @xmath556 the expected value of the last random variable is @xmath557 . we check that , for sufficiently large @xmath37 , @xmath558}\\ & \leq & { \mathbb{e}}\left [ | e-1| \int_0^e { \mathrm{e}}^{\rho ( u+w_n ) } \left(1 + \epsilon { \mathrm{e}}^{\delta ( u+|w_n|)}\right ) + \mathbb{1}_{a_n^c } |e-1| \frac{m}{|\eta(n / k_n)| } { \mathrm{d}}u \right ] \\ & \leq & { \ensuremath{\mathbb{e}}}\left[{\mathrm{e}}^{\rho w_n } \left ( 2 + \frac{\epsilon}{\delta(1-\delta)^2 } { \mathrm{e}}^{\delta|w_n| } \right)\right ] + \frac{m}{|\eta(n / k_n)| } { \ensuremath{\mathbb{e}}}\mathbb{1}_{a_n^c } \\ & \leq & 4 { \mathrm{e}}^{\frac{\rho^2}{k_n } } + \frac{2 \epsilon}{\delta(1-\delta)^2 } { \mathrm{e}}^{\frac{(\delta-\rho)^2}{k_n } } + \frac{m}{|\eta(n / k_n)| } { \ensuremath{\mathbb{e}}}\mathbb{1}_{a_n^c } { \enspace .}\ ] ] we now way conclude by dominated convergence that @xmath559 the proof of proposition 4.3 from @xcite yields that , with probability larger than @xmath248 , for @xmath560 , @xmath561 we may choose @xmath562 and notice that @xmath563 . this yields @xmath564 lower bounds on tail index estimation error @xcite are usually constructed by defining sequences of local models around a pure pareto distribution with shape parameter @xmath565 . when deriving lower bounds for the estimation error under constraints like @xmath54 is regularly varying , the elements of the local model for sample size @xmath37 may be defined by @xmath566 where @xmath199 is square integrable over @xmath567 $ ] , @xmath568 , @xmath569 @xcite . the sequences @xmath570 and @xmath571 are chosen in such a way that @xmath572 satisfies the required constraint . if the local alternatives are pareto change point distributions as in @xcite and @xcite , @xmath573 , @xmath574 . @xcite explores a richer collection of local alternatives in order to fit into the theory of weak convergence of local experiments . in order to explore adaptivity as in @xcite , it is necessary to handle simultaneously a collection of sequences @xmath575 corresponding to different rates of decay of the von mises function . the difficulty of estimation is connected with the difficulty of distinguishing alternatives with different tail indices that is , with the hardness of a multiple hypotheses testing problem . in order to lower bound the testing error , @xcite chose to use fano s lemma ( * ? ? ? * see ) . using fano s lemma requires bounding the kullback - leibler divergence between the different local alternatives which is not as easy as bounding the divergence between a pareto change point distribution and a pure pareto distribution . the next lemma is from @xcite . it can be used in the derivation of risk lower bounds instead of the classical fano lemma . just as fano s lemma , it states a lower bound on the error in multiple hypothesis testing . however , as it only requires computing the kullback - leibler divergence to the localisation center , in the present setting , it significantly alleviates computations and makes the proof more concise and more transparent . [ thm - kc - lower - bound : appendix ] let @xmath12 , @xmath579 , and @xmath580 . then , for any tail index estimator @xmath149 and any sample size @xmath37 such that @xmath581 , there exists a collection @xmath582 of probability distributions such that a. @xmath583 with @xmath584 , b. @xmath585 meets the von mises condition with von mises function @xmath586 satisfying @xmath587 where @xmath588 , c. @xmath589 + and @xmath590 \geq \frac{c_\rho}{4(1 + 2{\mathrm{e } } ) } \left(\frac{v\ln\ln n}{n}\right)^{|\rho|/(1 + 2|\rho| ) } \ , , \ ] ] with @xmath157 . the center of localisation @xmath596 is the pure pareto distribution with shape parameter @xmath4 ( @xmath597 ) . the local alternatives @xmath598 are pareto change point distributions . each @xmath585 is defined by a breakpoint @xmath599 and an ultimate pareto index @xmath600 . if @xmath601 denotes the distribution function of @xmath585 , @xmath602 karamata s representation of @xmath603 is @xmath604 with @xmath605 the kullback - leibler divergence between @xmath585 and @xmath596 is readily calculated , @xmath606 if @xmath584 , the next upper bound holds , @xmath607 the breakpoints and tail indices are chosen in such a way that all upper bounds are equal ( namely @xmath608 does not depend on @xmath64 ) , @xmath609 so that @xmath610 , for all @xmath611 . now , let @xmath149 be any tail index estimator . define region @xmath614 , as the set of samples such that @xmath600 minimises @xmath615 , for @xmath616 . then , if the event @xmath614 is not realised , @xmath617 by birg s lemma , @xmath618 in order to make the whole construction useful , it remains to choose the `` second - order parameters '' @xmath619 s ( the true second - order parameter of each @xmath585 is infinite ! ) . we will need an upper bound on @xmath620 ( but we already have @xmath621 ) , as well as a lower bound on @xmath622 for @xmath623 that scales like @xmath624 . following @xcite , we finally choose @xmath619 as @xmath625 for @xmath611 . then , for @xmath626 , using that @xmath627 and @xmath628 , @xmath629 \\ & \geq & \frac{1}{2}\left(\frac{n}{v\ln m } \right)^{\rho_i/(1 + 2|\rho_i| ) } \left [ 1- \exp \left ( \frac{-(i - j)}{2(1 + 2|\rho_i|)(1 + 2|\rho_j|)}\right)\right ] \\ & \geq & \frac{c_\rho}{2 } \left(\frac{n}{v\ln m } \right)^{\rho_i/(1 + 2|\rho_i|)}\end{aligned}\ ] ] where @xmath630 may be chosen as @xmath631 .
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this paper presents an adaptive version of the hill estimator based on lespki s model selection method .
this simple data - driven index selection method is shown to satisfy an oracle inequality and is checked to achieve the lower bound recently derived by @xcite . in order to establish the oracle inequality , we derive non - asymptotic variance bounds and concentration inequalities for hill estimators . these concentration inequalities are derived from talagrand s concentration inequality for smooth functions of independent exponentially distributed random variables combined with three tools of extreme value theory : the quantile transform , karamata s representation of slowly varying functions , and rnyi s characterisation for the order statistics of exponential samples .
the performance of this computationally and conceptually simple method is illustrated using monte - carlo simulations .
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You are an expert at summarizing long articles. Proceed to summarize the following text:
in t hooft s large @xmath2 limit @xcite gauge theories are expected to be drastically simplified . thus , in this limit the gauge theory diagrams are organized in terms of riemann surfaces , where each extra handle on the surface suppresses the corresponding diagram by @xmath3 . the large @xmath2 expansion , therefore , resembles perturbative expansion in string theory . in the case of four - dimensional gauge theories this connection can be made precise in the context of type iib string theory in the presence of a large number @xmath2 of d3-branes @xcite . thus , we consider a limit where @xmath4 , @xmath5 and @xmath6 , while keeping @xmath7 fixed , where @xmath8 is the type iib string coupling . note that in this context a world - sheet with @xmath9 handles and @xmath10 boundaries is weighted with @xmath11 once we identify @xmath12 , this is the same as the large @xmath2 expansion considered by t hooft . note that for this expansion to make sense we must keep @xmath13 at a small value @xmath14 . in this regime we can map the string diagrams directly to ( various sums of ) large @xmath2 feynman diagrams . note , in particular , that the genus @xmath15 planar diagrams dominate in the large @xmath2 limit , then no matter how large @xmath2 is , for sufficiently many boundaries the higher genus terms become relevant , and we lose the genus expansion . in fact , in this regime one expects an effective supergravity description to take over as discussed in @xcite . ] . if the space transverse to the d3-branes in the setup of @xcite is @xmath16 , then we obtain the @xmath17 supersymmetric @xmath18 gauge theory on the d3-branes , which is conformal . on the other hand , we can also consider orbifolds of @xmath16 , which leads to gauge theories with reduced supersymmetry versions of these orbifold theories via the compactifications of type iib on ads@xmath19 ( where @xmath20 is the orbifold group ) were originally discussed in @xcite . ] . as was shown in @xcite , if we cancel all twisted tadpoles in such models , in the large @xmath2 limit the corresponding @xmath21 gauge theories are conformal . moreover , in the planar limit the ( on - shell ) correlation functions in such theories are the same as in the parent @xmath17 gauge theory . in this paper we discuss non - conformal gauge theories within the setup of @xcite . such theories can be obtained by allowing some twisted tadpoles to be non - vanishing . in particular , we can have consistent embeddings of non - conformal gauge theories if we allow logarithmic tadpoles , which correspond to the twisted sectors with fixed point loci of real dimension two . in particular , even though the corresponding string backgrounds are not finite ( in the sense that we have logarithmic ultra - violet divergences ) , they are still consistent as far as the gauge theories are concerned , and the divergences correspond to the running in the four - dimensional gauge theories on the d3-branes . regularization of the aforementioned divergences can be conveniently discussed in the context of what we refer to as the brane - bulk duality , which is a consequence of the open - closed string duality . in particular , in certain non - trivial @xmath0 cases in the planar limit the corresponding gauge theories perturbatively are not renormalized beyond one - loop . in fact , in this limit the ( on - shell ) correlation functions in these theories are the same as in the parent @xmath1 non - conformal gauge theories . in the @xmath1 as well as the aforementioned @xmath0 cases the brane - bulk duality is particularly simple , and implies that the quantum corrections in the corresponding gauge theories are encoded in classical higher dimensional field theories whose actions contain the twisted fields with non - vanishing tadpoles . in particular , various quantum corrections can be obtained via integrating out the bulk fields in the corresponding classical action , that is , by considering the self - interaction of the d3-branes via the bulk fields . we give explicit computations in various @xmath21 examples in this context , including the treatment of divergences . we also discuss whether the brane - bulk duality can be extended to the non - perturbative level in the aforementioned theories . in the @xmath1 cases we argue that , since we are working in the large @xmath2 limit , the low energy effective action does not receive non - perturbative corrections . we also conjecture that this should be the case for the corresponding @xmath0 theories as well . in the @xmath22 cases we verify that there are no non - perturbative corrections to the superpotential in these theories in the large @xmath2 limit . the remainder of this paper is organized as follows . in section ii we discuss our setup . in section iii we discuss non - conformal large @xmath2 gauge theories which can be constructed within this setup . in section iv we discuss the large @xmath2 limit and brane - bulk - duality . in sections v , vi and vii we give details of classical computations that in the context of the brane - bulk duality reproduce quantum results in the corresponding @xmath1 , @xmath22 and @xmath23 gauge theories , respectively . in section viii we comment on the non - perturbative extension of the brane - bulk duality . in section ix we give a few concluding remarks . in appendix a we compute the brane - bulk couplings used in sections v , vi and vii . in this section we discuss the setup within which we will consider four - dimensional large @xmath2 gauge theories in the context of brane - bulk duality . parts of our discussion in this section closely follow @xcite . thus , consider type iib string theory in the presence of @xmath2 coincident d3-branes with the space transverse to the d - branes @xmath24 . the orbifold group @xmath25 ( @xmath26 ) must be a finite discrete subgroup of @xmath27 . if @xmath28 , we have @xmath22 ( @xmath1 ) unbroken supersymmetry , and @xmath23 , otherwise . we will confine our attention to the cases where type iib on @xmath29 is a modular invariant theory . if @xmath30 , then modular invariance requires that the set of points in @xmath16 fixed under the @xmath31 twist has real dimension 2 . ] . the action of the orbifold on the coordinates @xmath32 ( @xmath33 ) on @xmath29 can be described in terms of @xmath34 matrices : @xmath35 . the world - sheet fermionic superpartners of @xmath32 transform in the same way . we also need to specify the action of the orbifold group on the chan - paton charges carried by the d3-branes . it is described by @xmath36 matrices @xmath37 that form a representation of @xmath20 . note that @xmath38 is an identity matrix , and @xmath39 . the d - brane sector of the theory is described by an oriented open string theory . in particular , the world - sheet expansion corresponds to summing over oriented riemann surfaces with arbitrary genus @xmath9 and arbitrary number of boundaries @xmath10 , where the boundaries of the world - sheet are mapped to the d3-brane world - volume . moreover , we must consider various `` twists '' around the cycles of the riemann surface . the choice of these `` twists '' corresponds to a choice of homomorphism of the fundamental group of the riemann surface with boundaries to @xmath20 . for example , consider the one - loop vacuum amplitude ( @xmath40 ) . the corresponding graph is an annulus whose boundaries lie on d3-branes . the annulus amplitude is given by @xmath41 the one - loop partition function @xmath42 in the light - cone gauge is given by @xmath43 where @xmath44 is the light - cone hamiltonian , @xmath45 is the fermion number operator , @xmath46 is the real modular parameter on the cylinder , and the trace includes sum over the chan - paton factors . the states in the neveu - schwarz ( ns ) sector are space - time bosons and enter the partition function with weight @xmath47 , whereas the states in the ramond ( r ) sector are space - time fermions and contribute with weight @xmath48 . the elements @xmath49 acting in the hilbert space of open strings act both on the left end and the right end of the open string . this action corresponds to @xmath50 acting on the chan - paton indices . the partition function ( [ partition ] ) , therefore , has the following form : @xmath51 where @xmath52 are characters corresponding to the world - sheet degrees of freedom . the `` untwisted '' character @xmath53 is the same as in the @xmath17 theory for which @xmath54 . the information about the fact that the orbifold theory has reduced supersymmetry is encoded in the `` twisted '' characters @xmath52 , @xmath55 . in this subsection we discuss one - loop tadpoles arising in the above setup . as was pointed out in @xcite , if all tadpoles are canceled , then the resulting theory is finite in the large @xmath2 limit . however , not all tadpoles need to be canceled to have a consistent four - dimensional gauge theory . in fact , we can obtain non - conformal gauge theories if we allow such tadpoles . the characters @xmath52 in ( [ gsq ] ) are given by @xmath56^{2+d_a } } \left[{\cal x}_a(e^{-2\pi t})-{\cal y}_a(e^{-2\pi t})\right]~,\ ] ] where @xmath57 is the real dimension of the set of points in @xmath16 fixed under the twist @xmath49 . the factor of @xmath58 in the denominator comes from the bosonic zero modes corresponding to four directions along the d3-brane world - volume . two of the @xmath59-functions come from the bosonic oscillators corresponding to two spatial directions along the d3-brane world - volume ( the time - like and longitudinal contributions are absent as we are working in the light - cone gauge ) . the other @xmath57 @xmath59-functions come from the bosonic oscillators corresponding to the directions transverse to the d - branes untouched by the orbifold twist @xmath49 . finally , the characters @xmath60 , @xmath61 correspond to the contributions of the world - sheet fermions , as well as the world - sheet bosons with @xmath49 acting non - trivially on them ( for @xmath55 ) : @xmath62~,\\ & & { \cal y}_a = { 1\over 2 } { \rm tr}^\prime \left[g_a ( -1)^f e^{-2\pi t l_0}\right]~,\end{aligned}\ ] ] where prime in @xmath63 indicates that the trace is restricted as described above . for the annulus amplitude we therefore have @xmath64~,\ ] ] where @xmath65^{2+d_a}}{\cal x}_a(e^{-2\pi t})~,\\ & & { \cal b}_a=\left({\rm tr}\left(\gamma_a\right)\right)^2 \int_0^\infty { dt\over t^3 } { 1\over \left[\eta(e^{-2\pi t})\right]^{2+d_a}}{\cal y}_a(e^{-2\pi t})~.\end{aligned}\ ] ] these integrals . ( the entire partition function vanishes as the numbers of space - time bosons and fermions are equal . ) for consistency , however , we must extract tadpoles from individual contributions @xmath66 and @xmath67 . thus , for instance , cancellation of certain tadpoles coming from @xmath67 is required for consistency of the equations of motion for the twisted r - r four - form which couples to d3-branes ( see below ) . ] are generically divergent as @xmath68 reflecting the presence of tadpoles . to extract these divergences we can change variables @xmath69 so that the divergences correspond to @xmath70 : @xmath71 the closed string states contributing to @xmath66 ( @xmath67 ) in the transverse channel are the ns - ns ( r - r ) states with @xmath72 ( and @xmath73 is the number of such states ) . the massive states with @xmath74 do not lead to divergences as @xmath70 . on the other hand , the divergence property of the above integrals in the @xmath70 limit is determined by the value of @xmath57 . given the orientability of @xmath20 the allowed values of @xmath57 are @xmath75 . for @xmath76 there is no divergence in @xmath77 , so we have no restriction for @xmath39 . for @xmath78 the corresponding twisted ns - ns closed string sector contains tachyons . this leads to a tachyonic divergence in @xmath66 unless @xmath79 for the corresponding @xmath49 twisted sector . finally , the ground states in the r - r sectors are massless , so we get divergences due to massless r - r states in the integral in @xmath67 for large @xmath80 for @xmath81 . in such sectors in non - supersymmetric cases we can also have tachyonic ns - ns divergences , while in supersymmetric cases we have massless ns - ns divergences . to avoid complications with tachyons , for now we will focus on supersymmetric theories ( we will discuss non - supersymmetric cases in section vii ) . we must therefore consider massless tadpoles arising for @xmath81 . for @xmath82 the corresponding integrals are linearly divergent with @xmath80 as @xmath70 . to cancel such a tadpole we must require that @xmath83 for the corresponding @xmath49 twisted sector . on the other hand , if such a tadpole is not canceled , in the four - dimensional field theory language this would correspond to having a _ quadratic _ ( in the momentum ) divergence at the one - loop order . this would imply that the corresponding four - dimensional background is actually inconsistent in the sense that either four - dimensional supersymmetry contains a twist with @xmath82 , we have @xmath22 supersymmetry . ] and/or poincar invariance must be broken propagate in four dimensions , in the presence of the tadpoles their equations of motion read @xmath84 ( @xmath85 , @xmath86 are the tadpoles , and the ellipses stand for terms of higher power in @xmath87 ) , so that we can _ a priori _ have solutions with broken four - dimensional poincar invariance . ] . this is related to the fact that in such cases the corresponding twisted closed string states , which propagate only in the d3-brane world - volume as they are supported at the orbifold fixed _ points _ in @xmath16 , have inconsistent field equations ( yet they couple to the gauge / matter fields describing the low energy limit of the d3-brane world - volume theory ) . in fact , as was pointed out in @xcite , generically uncancelled tadpoles arising in the @xmath82 cases lead to non - abelian gauge _ anomalies _ in the corresponding d3-brane gauge theories . thus , let us consider the following example . let @xmath88 , where the action of @xmath89 on the complex coordinates @xmath90 ( @xmath91 ) on @xmath29 is that of the z - orbifold : @xmath92 ( where @xmath9 is the generator of @xmath20 , and @xmath93 ) . next , let us choose the representation of @xmath20 when acting on the chan - paton charges as follows : @xmath94 ( where @xmath95 , and @xmath96 is an @xmath97 identity matrix ) . the massless spectrum of this model is the @xmath22 supersymmetric @xmath98 gauge theory with the matter consisting of chiral supermultiplets in the following representations : @xmath99 note that this spectrum is anomalous for non - vanishing @xmath100 ( the non - abelian gauge anomaly does not cancel ) unless @xmath101 . on the other hand , @xmath102 if and only if @xmath101 . here we should mention that not all the choices of @xmath37 that do not satisfy @xmath103 for @xmath82 lead to such apparent inconsistencies . thus , consider the same @xmath20 as above but with @xmath104 . the massless spectrum of this model is the @xmath22 supersymmetric @xmath18 gauge theory with no matter , so it is anomaly free . in fact , for any orbifold group @xmath20 containing a twist @xmath49 with @xmath82 we obtain an anomaly free theory only if @xmath79 or @xmath105 . in the latter case , however , as we discussed above , the corresponding background is nonetheless inconsistent , so we must require that for such twists @xmath79 . finally , let us discuss @xmath106 cases . for such twists the corresponding integrals are only logarithmically divergent as @xmath70 . if such a tadpole is not canceled , that is , if the corresponding @xmath107 , in the four - dimensional field theory language this corresponds to having a _ logarithmic _ divergence ( in the momentum ) at the one - loop order . as we will see in the following , these logarithmic divergences are precisely related to the running in the corresponding gauge theories , which are _ not _ conformal ( even in the large @xmath2 limit ) . note that the corresponding twisted closed string states now propagate in two extra dimensions , so that the four - dimensional backgrounds are perfectly consistent - the tadpoles for these fields simply imply that these fields have non - trivial ( logarithmic ) profiles in these two extra dimensions , while the four dimensions along the d - brane world - volume are still flat ( and the four - dimensional supersymmetry is unbroken ) . in fact , the presence of such tadpoles does not introduce any anomalies in @xcite in a somewhat more complicated way . ] . a simple way to see this for general @xmath20 is to note that ( in supersymmetric cases which we are focusing on here ) an individual twist with @xmath106 preserves @xmath1 supersymmetry , so that the corresponding gauge theory is anomaly free . this then immediately implies that we do not have any non - abelian gauge anomalies in a theory with multiple such twists either . indeed , such anomalies would have to arise at the one - loop level . in the string theory language the relevant diagram is a @xmath15 , @xmath108 diagram ( the annulus ) with three external lines corresponding to non - abelian gauge fields attached to a single boundary . thus , if we attach two external lines to one boundary , while the third one to another boundary , the corresponding diagram will vanish - indeed , the chan - paton structure of such a diagram is given by ( @xmath109 , @xmath110 , are the chan - paton matrices corresponding to the external lines ) @xmath111 which vanishes as for non - abelian gauge fields @xmath112 , so @xmath113 as well for by definition @xmath109 are invariant under the orbifold group action as @xmath109 correspond to the gauge bosons of the gauge group left unbroken by the orbifold ( in particular , note that @xmath109 commute with @xmath37 ) . as to the aforementioned diagram with all three external lines attached to one boundary , its chan - paton structure is given by @xmath114 for twists with @xmath106 , each diagram of this type is @xmath1 supersymmetric as it does not contain any information about further supersymmetry breaking ( that is , the characters corresponding to the world - sheet degrees of freedom multiplying the chan - paton trace ( [ anomaly ] ) are those of an @xmath1 theory ) , so such diagrams do not introduce any anomalies . we therefore conclude that a theory with multiple @xmath106 twists with @xmath107 is also anomaly free . thus , as we see , in supersymmetric cases we can have non - trivial twists with @xmath81 . consistency of the background requires that @xmath115 while for @xmath106 we do not have such a restriction . in this section we discuss certain large @xmath2 gauge theories arising in the above setup with some twists @xmath49 with @xmath106 such that @xmath107 . as we have already mentioned , such theories are not conformal . the simplest examples of such theories are those with @xmath1 supersymmetry . perturbatively such theories are not renormalized beyond the one - loop order . we can also construct non - conformal @xmath22 supersymmetric theories in this way . in general such theories are rather complicated . however , as we will see in the following , certain non - trivial @xmath22 theories of this type have the property that in the large @xmath2 limit the leading ( that is , planar ) diagrams do not renormalize the gauge theory correlators beyond the one - loop order . in fact , the corresponding correlation functions in such an @xmath22 theory are ( up to overall numerical coefficients ) the same as in a parent @xmath1 theory . the simplest example of an @xmath1 theory is obtained if we take @xmath116 , where the action of @xmath117 on the complex coordinates @xmath90 ( @xmath91 ) on @xmath29 is as follows : @xmath118 , @xmath119 ( where @xmath120 is the generator of @xmath20 ) . next , let us choose the representation of @xmath20 when acting on the chan - paton charges as follows : @xmath121 . the massless spectrum of this model is the @xmath1 supersymmetric @xmath18 gauge theory with no matter . the non - abelian part of this theory is not conformal , and is asymptotically free . as to the overall center - of - mass @xmath122 , it is free , and can therefore be ignored . more generally , if we have twists with @xmath123 , the gauge group generically is a product of @xmath124 factors , and we also have matter , which can be obtained using the quiver diagrams ( see @xcite ) . there is always an overall center - of - mass @xmath122 , which is free . other @xmath122 factors , however , run as the matter is charged under them . in the large @xmath2 limit , however , these @xmath122 s decouple in the infra - red , and can therefore also be ignored s are actually anomalous , and are broken at the tree - level via the generalized green - schwarz mechanism . in particular , in cases with twists with @xmath82 we have mixed @xmath125 anomalies . however , in the cases with matter we are interested in here we do have running @xmath122 s that are anomaly free . these @xmath122 s decouple in the infra - red in the large @xmath2 limit . ] . as we have already mentioned , within the above setup we can construct non - trivial @xmath22 supersymmetric theories which are not renormalized beyond the one - loop order in the planar limit . the idea here is similar to that in @xcite , where it was noticed that in theories with vanishing twisted tadpoles the planar diagrams are the same ( up to overall numerical factors ) as in the parent @xmath17 theory . this is because the diagrams that contain information about supersymmetry breaking always contain twisted chan - paton traces @xmath126 , which vanish in such theories . in this subsection we will consider theories where the planar diagrams are the same as in a parent @xmath1 theory in essentially the same way . to obtain such models , consider an orbifold group @xmath20 , which is a subgroup of @xmath127 but is not a subgroup of @xmath128 . let @xmath129 be a non - trivial subgroup of @xmath20 such that @xmath130 . we will allow the chan - paton matrices @xmath37 corresponding to the twists @xmath131 ( which have @xmath106 ) not to be traceless , so that the corresponding @xmath1 model is not conformal . however , we will require that the other chan - paton matrices @xmath37 for the twists @xmath132 be traceless . the resulting @xmath22 model is not conformal . however , in the planar limit the perturbative gauge theory amplitudes are not renormalized beyond one loop . the proof of this statement is straightforward . thus , consider a planar diagram with @xmath10 boundaries ( but no handles)-loop diagram in the field theory language . ] with all external lines attached to a single boundary , which without loss of generality can be chosen to be the outer boundary . such a diagram is depicted in fig.1 . next , we need to specify the twists on the boundaries . a convenient choice ( consistent with that made for the annulus amplitude ( [ partition ] ) ) is given by is non - abelian , and we have to choose base points on the world - sheet to define the twists . our discussion here , however , is unmodified also in this case . ] @xmath133 where @xmath134 corresponds to the outer boundary , while @xmath135 , @xmath136 , correspond to inner boundaries . let @xmath109 , @xmath137 , be the chan - paton matrices corresponding to the external lines . then the above planar diagram has the following chan - paton group - theoretic dependence : @xmath138 where the sum involves all possible distributions of the @xmath135 twists that satisfy the condition ( [ mono ] ) , as well as permutations of the @xmath109 factors ( note that @xmath109 here correspond to the states that are kept after the orbifold projections , so that they commute with all @xmath37 ) . note that the diagrams with all twists @xmath139 are ( up to overall numerical coefficients ) the same as in the parent @xmath1 theory , and therefore vanish beyond one loop . all other diagrams contain at least one twist @xmath140 with @xmath141 . this follows from the condition ( [ mono ] ) . this then implies that all such diagrams vanish as @xmath142 in particular , this implies that the non - abelian gauge couplings do not run in the large @xmath2 limit beyond one loop limit compared with the leading one - loop contribution . this is analogous to what happens in theories discussed in @xcite . in fact , the techniques used there to prove that the higher loop corrections are subleading are very similar to the one we are using here . ] . the anomalous scaling dimensions ( two - point functions corresponding to the wave - function renormalization ) for matter fields , the corresponding diagrams are ( up to overall numerical factors ) the same as in the parent @xmath1 theory , so they are not renormalized in the planar limit . finally , the yukawa ( three - point ) and quartic scalar ( four - point ) couplings are not renormalized in perturbation theory due to the @xmath22 non - renormalization theorem for the superpotential . let us now consider a simple example of the aforementioned gauge theories . thus , let @xmath143 , and let the corresponding twisted chan - paton matrices @xmath144 . let us assume that @xmath145 act non - trivially on the complex coordinates @xmath146 on @xmath147 , while leaving the coordinate @xmath148 untouched . moreover , let us assume that the orbifold group @xmath129 is abelian , and its action on the coordinates @xmath146 is diagonal . next , let @xmath120 be a generator of a @xmath31 group with the following action : @xmath149 , @xmath150 . note that @xmath120 commutes with @xmath151 . the full orbifold group is given by @xmath152 , which is also abelian , and @xmath153 . let the twisted chan - paton matrix @xmath154 . then the theory is the @xmath22 supersymmetric @xmath155 gauge theory with chiral matter multiplets , call them @xmath156 and @xmath157 , in @xmath158 and @xmath159 , respectively ( in the following we will ignore the @xmath122 factors ) . note that there is no tree - level superpotential in this theory : @xmath160 this implies that we can break the gauge group down to the diagonal @xmath161 subgroup by giving the appropriate vacuum expectation values to the fields @xmath156 and @xmath157 . the resulting theory in the ir is then the @xmath1 supersymmetric @xmath161 gauge theory without matter . in the string theory language this corresponds to moving the d3-branes off the @xmath31 orbifold singularity , hence the @xmath1 supersymmetry of the resulting theory . before we end this section , as an aside we would like to discuss an embedding of pure @xmath22 supergluodynamics in the setup discussed in section ii . thus , consider the orbifold group @xmath162 with the generators @xmath163 and @xmath164 of the two @xmath31 s acting on the complex coordinates @xmath90 on @xmath116 as follows : @xmath165 where @xmath166 . next , let the twisted chan - paton matrices be given by @xmath167 . the d3-brane gauge theory is then given by the @xmath22 supersymmetric @xmath161 super - yang - mills theory without matter ( plus an overall @xmath122 which can be ignored ) . note that the real dimensions of the points fixed under the twists @xmath168 are @xmath169 . the above embedding shows that at least perturbatively pure supergluodynamics possesses a discrete symmetry , which is not evident in the field theory language . thus , we have the @xmath170 symmetry as all the interactions involving twisted closed string states must be invariant under the orbifold group . in addition , we have a @xmath171 symmetry which permutes the closed string states coming from the @xmath168 twisted sectors . in particular , the generator @xmath172 of this @xmath171 group has the following action : @xmath173 thus , the @xmath171 subgroup does not commute with the @xmath174 subgroup . thus , @xmath163 and @xmath172 generate a non - abelian discrete subgroup of @xmath175 , namely , the _ tetrahedral _ group @xmath176 . thus , as we see , the perturbative large @xmath2 pure superglue theory possesses a discrete @xmath171 symmetry . in fact , this symmetry persists even at finite @xmath2 . indeed , even the diagrams with handles possess this symmetry as this symmetry is a symmetry of the underlying embedding of the pure supergluodynamics into string theory via the orbifold setup . in fact , within this setup the @xmath171 symmetry is a discrete _ gauge _ symmetry . indeed , instead of @xmath170 , the orbifold group can be chosen as the full tetrahedral group @xmath176 , where the generator @xmath172 of the @xmath171 subgroup of @xmath176 acts on the complex coordinates @xmath90 as follows : @xmath177 ( @xmath178 ) . if we now choose the twisted chan - paton matrix @xmath179 , then the d3-brane gauge theory is still pure superglue . this embedding makes it evident that the @xmath171 symmetry is indeed a gauge symmetry , so it might be an exact symmetry of pure superglue even non - perturbatively which is a subgroup of @xmath175 but not a subgroup of @xmath128 . the corresponding gauge theory has @xmath22 supersymmetry . note that the non - trivial twists in @xmath20 all have @xmath106 , and we can choose the corresponding twisted chan - paton matrices as @xmath105 , so that ( upon dropping the overall @xmath122 ) we obtain the @xmath161 pure super - yang - mills theory . an appropriate abelian subgroup of @xmath20 is then a symmetry of this theory . ] . we would like to end this subsection by pointing out one immediate consequence of the aforementioned @xmath171 symmetry - both the one - loop as well as the two - loop - function becomes gauge dependent . ] @xmath180-function coefficients in the pure supergluodynamics are multiples of 3 . as we mentioned in introduction , perturbative expansion of d3-brane gauge theories simplifies substantially in the large @xmath2 limit . it is advantageous to work in the full string theory framework , which in a way is much simpler than the feynman diagram techniques . ( at the end of the day we will take the @xmath181 limit , which amounts to reducing the theory to the gauge theory subsector . ) thus , in the string theory language there are two classes of diagrams we need to consider : ( _ i _ ) diagrams without handles ; ( _ ii _ ) diagrams with handles . the latter correspond to closed string loops , and can be neglected as they are subleading in the large @xmath2 limit . we will therefore focus on the diagrams without handles . the latter diagrams can be viewed as _ tree - level _ closed string graphs connecting various boundaries . this suggests that ( upon taking the @xmath181 limit ) in some cases ( which we will identify in a moment ) we might be able to rewrite the perturbative expansion in the large @xmath2 limit of the corresponding gauge theories in the language where various quantum corrections in the gauge theory are encoded in a _ classical _ higher dimensional _ field _ theory . the effective quantum action for the four - dimensional large @xmath2 gauge theory is then obtained by starting with the corresponding higher dimensional classical action , integrating out the bulk fields , and restricting to the gauge theory subsector . that is , we consider the d - branes in the background of the bulk fields that is created by the d - branes themselves . the field theory cut - off in this language arises precisely from the fact that , in the above setup , the twisted closed string states have tadpoles resulting in logarithmic profiles in the extra dimensions , and we need to regularize these profiles at the sources , that is , the d - branes , whose locations in the extra dimensions are given by points in @xmath29 . the aforementioned proposal that in the large @xmath2 limit various quantum corrections in the d - brane gauge theory should in some cases be encoded in a higher dimensional classical field theory is what we refer to as the _ brane - bulk duality_. in this subsection we would like to identify the cases where the brane - bulk duality is applicable as stated above . note that the brane - bulk duality is a consequence of the _ open - closed duality _ property of string theory . in particular , the diagrams without handles in the transverse closed string channel can indeed be viewed as tree - level closed string graphs connecting various boundaries . at the one - loop order the corresponding diagram ( up to external lines ) is an annulus , where two boundaries are connected by a single closed string tube . in particular , this diagram does _ not _ involve closed string interactions . moreover , in the field theory limit @xmath181 contributions due to the massive closed string states in the transverse closed string channel are always finite . that is , these contributions correspond to heavy string _ thresholds_. in the field theory language they translate into subtraction scheme dependent artifacts , which arise due to a particular choice of regularization . thus , at the one - loop order the contributions due to massive string modes can be absorbed into the subtraction scheme dependence , so that in the field theory limit the brane - bulk duality indeed holds - various one - loop corrections in the gauge theory can be computed by calculating the classical d - brane self - interaction via the massless bulk fields . beyond the one - loop order , however , that is , when the number of boundaries is greater than two , we must include closed string interactions in the transverse closed string channel . the massive closed string states are now expected to contribute in a non - trivial way . in particular , _ a priori _ there is no longer a clear interpretation of these contributions in terms of thresholds . another way of phrasing this is that at higher loop orders we can have mixed contributions coming from both massless as well as massive states . that is , in general the brane - bulk duality can still be formulated except that at higher loop orders it will involve massless as well as _ infinitely _ many massive states as far as the corresponding higher dimensional classical `` field theory '' is concerned . in particular , in general this classical `` field theory '' is nothing but the corresponding closed string theory , so we are back to the original open - closed string duality . there are , however , non - trivial cases where in the large @xmath2 limit the bulk closed string theory can be consistently truncated to a field theory containing only a finite number of massless fields . these cases are those where in the planar limit the gauge theory perturbatively is not renormalized beyond one loop . thus , this is clearly the case for @xmath1 theories . in theories without matter this , in fact , holds even at finite @xmath2 . in theories with matter , however , we have running @xmath122 s whose decoupling is ensured only in the large @xmath2 limit . in subsection iiia we discussed @xmath22 theories which perturbatively are not renormalized beyond the one - loop order in the planar limit . it is clear that the aforementioned formulation of the brane - bulk duality also holds in such theories , where , once again , various quantum corrections in the gauge theory are encoded in a higher dimensional classical field theory with a finite number of massless fields . in fact , in @xmath1 as well as @xmath22 theories the bulk fields entering the relevant part of the corresponding higher dimensional classical action are _ twisted _ closed string states . indeed , at the one - loop order the diagrams involving untwisted closed string states in the transverse closed string channel are @xmath17 supersymmetric , so that they do not contribute into the renormalization of the corresponding ( on - shell ) gauge theory correlators . to clarify the above discussion , let us give a schematic description of the above procedure for obtaining quantum corrections in the corresponding gauge theories via the brane - bulk duality . thus , we start from the classical action @xmath182+s_{\rm{\small bulk}}[\sigma]+ s_{\rm{\small int}}[\phi,\sigma]~,\ ] ] where @xmath183 is a collective notation for the ( fractional ) d3-brane world - volume fields , while @xmath184 is a collective notation for the _ massless twisted bosonic _ ( ns - ns and r - r ) bulk fields ; @xmath185 $ ] is the classical four - dimensional action for the brane fields @xmath183 , while @xmath186 $ ] is the classical higher dimensional action for the bulk fields @xmath184 ( note that some of these fields such as twisted closed string states propagate in less than 10 dimensions ) ; finally , @xmath187 $ ] , which is a classical four - dimensional functional ( with the integral over the ( fractional ) d3-brane world - volume ) , describes the coupling of the bulk fields to the brane fields , as well as to the brane itself ( in particular , it includes all tadpoles ) . since here we are interested in one - loop corrections in the gauge theory language , the action @xmath186 $ ] contains only the terms quadratic in @xmath184 ( that is , the kinetic terms ) , while @xmath187 $ ] contains only the terms linear in @xmath184 . moreover , the fields @xmath184 include only the twisted fields with non - vanishing ( logarithmic ) tadpoles . next , we solve the equations of motion for @xmath184 following from the above action : @xmath188 in particular , we are interested in the solution where the fields @xmath184 do _ not _ explicitly depend on the coordinates @xmath189 along the d3-brane world - volume , but only functionally via @xmath190 ; this solution , however , does in general explicitly depend on the coordinates @xmath191 transverse to the d - branes . let us denote this solution via @xmath192 . the effective quantum action for the fields @xmath183 in the large @xmath2 limit is then given by @xmath193= s_{\rm{\small brane}}[\phi]+ s_{\rm{\small int}}[\phi,{\overline\sigma}]~.\ ] ] note that @xmath194 are generically divergent , so we need to further clarify the meaning of the second term in ( [ quantum ] ) . thus , let @xmath195 note that @xmath196 are independent of @xmath189 , and , in fact , are nothing but the tadpoles for the fields @xmath184 . if all @xmath79 ( for @xmath81 ) , then there are no tadpoles for the corresponding twisted fields ( that is , the corresponding @xmath197 ) , and the large @xmath2 gauge theory is finite @xcite factors can still run , but , as we have already mentioned , in the large @xmath2 limit they decouple in the infra - red , and can therefore be ignored . ] . on the other hand , if some @xmath107 for twists with @xmath106 , some divergences no longer cancel , and we need to regularize @xmath194 in ( [ quantum ] ) . note , however , that the corresponding divergences are only logarithmic , and , in fact , correspond to the running in the four - dimensional gauge theory , which is no longer conformal . in this subsection we would like to address one technical point in the context of the brane - bulk duality , namely , the issue of regularizing the aforementioned divergences . here we will consider a simple toy model which possesses the main ingredients for illustrating the regularization procedure . in fact , as we will see in the following sections , this model actually captures all the key features of the @xmath1 models as well the @xmath22 models discussed in subsection iiia . thus , consider the six dimensional theory with the following action : @xmath198 here @xmath199 is a six dimensional scalar field , whose dimension is @xmath200 ; @xmath201 is a four - dimensional operator with dimension @xmath202 localized on the hypersurface @xmath203 ; the couplings @xmath204 ( which are assumed to be non - vanishing ) are dimensionless ; finally , @xmath205 is a parameter of dimension ( length ) . in the above action @xmath156 is an analog of a twisted closed string state with a non - vanishing tadpole ( the last term in ( [ toy ] ) ) ; @xmath203 plays the role of a 3-brane ; @xmath206 is an analog of a dimension - four gauge theory operator such as @xmath207 ; finally , @xmath208 is analogous to @xmath209 . following the procedure described in the previous subsection , we look for a solution to the equation of motion for @xmath156 : @xmath210 \delta^{(2)}(z)~.\ ] ] here @xmath211 are the coordinates transverse to the 3-brane ( whose location is chosen to be @xmath212 ) ; also , we use the notation @xmath213 . it is convenient to fourier transform to the momentum space @xmath214 where @xmath215 and @xmath216 are the momenta corresponding to @xmath217 and @xmath218 , respectively . thus , we have the following solution @xmath219~.\ ] ] this gives @xmath220~.\ ] ] the corresponding effective quantum action on the 3-brane is then given by @xmath221{\cal o}_4(p=0 ) + b^2 l^4\int { d^4 p\over(2\pi)^4}\int { d^2 k\over ( 2\pi)^2 } { { \cal o}_4(p){\cal o}_4(-p)\over { k^2+p^2}}+\dots~,\end{aligned}\ ] ] where the ellipses in the last line stand for a divergent constant piece ( which does not contain the operator @xmath206 , and is proportional to @xmath222 as it is solely due to the presence of the @xmath156 tadpole ) . the second term in the last line in ( [ toyqu ] ) is a non - local higher dimensional operator , and it disappears in the @xmath223 limit . the first term containing @xmath224 is the same as that in the classical 3-brane world - volume action except for the corresponding coupling - the quantum coupling in @xmath225 is a renormalized coupling : @xmath226 here @xmath227 is an ultra - violet ( uv ) cut - off , while @xmath228 is the infra - red ( ir ) cut - off . in the four - dimensional 3-brane world - volume field theory language the ir cut - off is interpreted as the renormalization group ( rg ) scale at which the renormalized coupling @xmath229 is measured . thus , if in the above toy model we adapt the interpretation of the previous subsection , from the classical dynamics of the bulk field @xmath156 we will obtain the `` one - loop '' renormalization of the 3-brane world - volume field theory . in fact , in this model the 3-brane world - volume theory is not renormalized beyond the `` one - loop '' order . in this section we would like to apply the brane - bulk duality discussed in the previous section to gauge theories arising in the orbifold construction discussed in section ii . here we will focus on the simplest examples of this type , in particular , those with @xmath1 supersymmetry . perturbatively such gauge theories are not renormalized beyond the one - loop order even for finite @xmath2 . thus , the brane - bulk duality in such examples can be understood in detail theories in the regime where the effective t hooft coupling is large were discussed in @xcite . ] . we described the simplest example of an @xmath1 gauge theory in the above context in the beginning of the previous section . in this example @xmath230 , and the twisted chan - paton matrix @xmath121 . the gauge group is @xmath18 , and we have no matter fields . in subsection a we discuss the brane - bulk duality in this model in the case of unbroken gauge symmetry . we will discuss the case of spontaneously broken gauge symmetry in subsection b. finally , in subsection c we discuss examples with matter . to discuss the brane - bulk duality in this model , we need the relevant part of the classical action containing the d3-brane as well as the bulk fields . this action has the following form @xcite ( we are not including terms containing @xmath231 for we are going to be interested in renormalization of the @xmath232 term ) . also , in @xmath233 $ ] only the @xmath122 contribution is non - vanishing . ] : @xmath234 + b~\sigma~{\rm tr}\left[f^{\mu\nu}f_{\mu\nu}\right]+ c~\sigma+ d~\epsilon^{\mu\nu\sigma\rho}~c_{\mu\nu}~ { \rm tr}\left[f_{\sigma\rho}\right]\big)\nonumber\\ & & -\int_{{\rm d3}\times { \bf r}^2 } \left({1\over 2}\partial^\mu\sigma~\partial_\mu\sigma+{1\over 12 } h^{\mu\nu\sigma } h_{\mu\nu\sigma}\right)~. \label{eq : actionu}\end{aligned}\ ] ] here @xmath184 is a twisted ns - ns scalar , while @xmath235 is a twisted two - form ( whose field strength is @xmath236 ) . we have normalized the kinetic terms of the bulk @xmath184 and @xmath235 fields in the standard way . once this normalization is fixed , the couplings @xmath237 can be determined . we discuss these couplings in appendix a. the combinations relevant for our discussion here are given by @xmath238 finally , the coupling @xmath239 is given by @xmath240 where @xmath241 is the tree - level yang - mills gauge coupling , and the @xmath18 generators @xmath242 are normalized as @xmath243 . the yang - mills gauge coupling is related to the string coupling @xmath8 via @xmath244 . as in section iv , we solve classical equations of motion for the fields @xmath184 and @xmath235 ( the latter is a gauge field , so we must use gauge fixing such as @xmath245 ) , and integrate them out of the classical action ( [ eq : actionu ] ) . the resulting effective quantum action is given by ( we are using the momentum representation , and drop higher dimensional terms as well as those independent of @xmath246 ) : @xmath247~ { \rm tr}\left[f^{\mu\nu}(p)~f_{\mu\nu}(-p)\right]+\right.\nonumber\\ & & \left.{d^2\over \pi^2}\int { d^2 k\over{k^2+p^2}}~ { \rm tr}\left[f^{\mu\nu}(p)\right]~{\rm tr}\left[f_{\mu\nu}(-p ) \right]\right)~.\end{aligned}\ ] ] since only the @xmath122 subgroup contributes into @xmath248 $ ] , we can rewrite the last term in the above expression in terms of the @xmath122 field strength @xmath249 ( note that the corresponding generator is @xmath250 ) : @xmath251~{\rm tr}[f_{\mu\nu}(-p)]= { \rm tr}[f^{\mu\nu}(p)]~{\rm tr}[f_{\mu\nu}(-p)]= n~{\rm tr}[f^{\mu\nu}(p)~f_{\mu\nu}(-p)]~.\ ] ] thus , we have @xmath247~ { \rm tr}\left[{\widehat f}^{\mu\nu}(p)~ { \widehat f}_{\mu\nu}(-p)\right]+\right.\nonumber\\ & & \left.\left[a-{bc\over 2\pi^2}\int { d^2 k\over k^2}+ { n d^2\over \pi^2}\int { d^2 k\over{k^2+p^2}}\right]~ { \rm tr}\left[f^{\mu\nu}(p)~f_{\mu\nu}(-p ) \right]\right)~,\end{aligned}\ ] ] where @xmath252 is the @xmath161 field strength . note that the integrals contributing to the @xmath253 coupling individually are uv divergent . however , since we have ( [ bcd ] ) , the total contribution is uv finite . on the other hand , the first integral is ir divergent , while the second integral is ir finite for @xmath254 ( for this latter integral the @xmath255 term in the denominator plays the role of the ir cut - off ) . we must therefore introduce an ir cut - off in the first integral . the fact that the @xmath122 gauge coupling should not be renormalized then dictates that the ir cut - off in the first integral must be chosen as follows gauge coupling can in general receive finite ( string ) threshold corrections . here , however , we ignore such corrections as we are interested in the field theory limit . ] : @xmath256 the effective quantum action therefore reads : @xmath257+a~ { \rm tr}\left[f^{\mu\nu}(p)~f_{\mu\nu}(-p ) \right]\right)~,\ ] ] where @xmath258 this is nothing but the one - loop renormalized yang - mills gauge coupling with the @xmath180-function coefficient @xmath259 . in this subsection we will discuss the brane - bulk duality in the above model in the case where the gauge symmetry is spontaneously broken : @xmath260 , @xmath261 . in the field theory language this corresponds to the complex adjoint scalar in the @xmath1 vector supermultiplet having an appropriate vacuum expectation value . in the string theory language this corresponds to splitting the @xmath2 d3-branes into two stacks of @xmath262 and @xmath263 d3-branes , and moving them apart by some distance @xmath264 in the two real directions transverse to the d - branes untouched by the @xmath31 orbifold action . the relevant part of the classical action in this case is given by : @xmath265 + b~\sigma~{\rm tr}[f_1^{\mu\nu}f_{1\mu\nu}]+c_1~\sigma+ d~\epsilon^{\mu\nu\sigma\rho}~c_{\mu\nu}~{\rm tr}[f_{1\sigma\rho}]\nonumber\\ & & -\int_{{\rm d3}_2 } a~{\rm tr}[f_2^{\mu\nu}f_{2\mu\nu}]+ b~\sigma~{\rm tr}[f_2^{\mu\nu}f_{2\mu\nu}]+c_2~\sigma+ d~\epsilon^{\mu\nu\sigma\rho}~c_{\mu\nu}~{\rm tr}[f_{2\sigma\rho}]\nonumber\\ & & -\int_{{\rm d3}\times { \bf r}^2 } \left({1\over 2}\partial^\mu\sigma~\partial_\mu\sigma+{1\over 12 } h^{\mu\nu\sigma } h_{\mu\nu\sigma}\right)~. \label{eq : action}\end{aligned}\ ] ] the couplings @xmath266 are the same as before , while the relevant combinations containing the couplings @xmath267 are now given by @xmath268 in the following we will assume that the d3@xmath269-branes ( that is , the stacks of @xmath270 d3-branes ) are located at @xmath271 , respectively , in the extra two dimensions . here @xmath191 , @xmath272 , are real coordinates , and in the following we will use the notation @xmath273 , @xmath274 . upon integrating out the bulk fields , we obtain : @xmath275~ { \rm tr}\left[f^{\mu\nu}_1(p)~f_{1\mu\nu}(-p)\right]+\right.\nonumber\\ & & \left[a-{bc_2\over 2\pi^2}\int { d^2 k\over k^2}- { bc_1\over 2\pi^2}\int { d^2 k\over k^2}~e^{ik\cdot x}\right]~ { \rm tr}\left[f^{\mu\nu}_2(p)~f_{2\mu\nu}(-p)\right]+\nonumber\\ & & { d^2\over \pi^2}\int { d^2 k\over{k^2+p^2}}~\big ( { \rm tr}\left[f^{\mu\nu}_1(p)\right]~{\rm tr}\left[f_{1\mu\nu}(-p)\right]+ { \rm tr}\left[f^{\mu\nu}_2(p)\right]~{\rm tr}\left[f_{2\mu\nu}(-p)\right ] \big)+\nonumber\\ & & \left . { 2d^2\over \pi^2}\int { d^2 k\over{k^2+p^2}}~e^{ik\cdot x}~ { \rm tr}\left[f^{\mu\nu}_1(p)\right]~{\rm tr } \left[f_{2\mu\nu}(-p)\right]\right)~.\end{aligned}\ ] ] as in the previous subsection , let us separate the @xmath122 contributions : @xmath276~ { \rm tr}\left[{\widehat f}^{\mu\nu}_1(p)~{\widehat f}_{1\mu\nu}(-p)\right]+ \right.\nonumber\\ & & \left[a-{bc_2\over 2\pi^2}\int { d^2 k\over k^2}- { bc_1\over 2\pi^2}\int { d^2 k\over k^2}~e^{ik\cdot x}\right]~ { \rm tr}\left[{\widehat f}^{\mu\nu}_2(p)~{\widehat f}_{2\mu\nu}(-p)\right ] + \nonumber\\ & & \left[a-{bc_1\over 2\pi^2}\int { d^2 k\over k^2}- { bc_2\over 2\pi^2}\int { d^2 k\over k^2}~e^{ik\cdot x}+ { n_1 d^2\over \pi^2}\int { d^2 k\over{k^2+p^2}}\right]~ { \rm tr}\left[f^{\mu\nu}_1(p)~f_{1\mu\nu}(-p)\right ] + \nonumber\\ & & \left[a-{bc_2\over 2\pi^2}\int { d^2 k\over k^2}- { bc_1\over 2\pi^2}\int { d^2 k\over k^2}~e^{ik\cdot x}+ { n_2 d^2\over \pi^2}\int { d^2 k\over{k^2+p^2}}\right]~ { \rm tr}\left[f^{\mu\nu}_2(p)~f_{2\mu\nu}(-p)\right ] + \nonumber\\ & & \left . { 2d^2\over \pi^2}\int { d^2 k\over{k^2+p^2}}~e^{ik\cdot x}~ { \rm tr}\left[f^{\mu\nu}_1(p)\right]~{\rm tr } \left[f_{2\mu\nu}(-p)\right]\right)~.\end{aligned}\ ] ] to further simplify this expression , let us explicitly take the traces over the @xmath122 generators . thus , we have @xmath277 where @xmath278 is a null @xmath97 matrix , and @xmath279 are the @xmath122 field strengths . we therefore obtain : @xmath276~ { \rm tr}\left[{\widehat f}^{\mu\nu}_1(p)~{\widehat f}_{1\mu\nu}(-p)\right]+ \right.\nonumber\\ & & \left[a-{bc_2\over 2\pi^2}\int { d^2 k\over k^2}- { bc_1\over 2\pi^2}\int { d^2 k\over k^2}~e^{ik\cdot x}\right]~ { \rm tr}\left[{\widehat f}^{\mu\nu}_2(p)~{\widehat f}_{2\mu\nu}(-p)\right ] + \nonumber\\ & & \left[a-{bc_1\over 2\pi^2}\int { d^2 k\over k^2}- { bc_2\over 2\pi^2}\int { d^2 k\over k^2}~e^{ik\cdot x}+ { n_1 d^2\over \pi^2}\int { d^2 k\over{k^2+p^2}}\right]~ { 1\over 2}{\overline f}^{\mu\nu}_1(p)~{\overline f}_{1\mu\nu}(-p ) + \nonumber\\ & & \left[a-{bc_2\over 2\pi^2}\int { d^2 k\over k^2}- { bc_1\over 2\pi^2}\int { d^2 k\over k^2}~e^{ik\cdot x}+ { n_2 d^2\over \pi^2}\int { d^2 k\over{k^2+p^2}}\right]~ { 1\over 2}{\overline f}^{\mu\nu}_2(p)~{\overline f}_{2\mu\nu}(-p ) + \nonumber\\ & & \left . { \sqrt{n_1 n_2 } d^2\over \pi^2}\int { d^2 k\over{k^2+p^2}}~e^{ik\cdot x}~ { \overline f}^{\mu\nu}_1(p)~ { \overline f}_{2\mu\nu}(-p)\right)~.\end{aligned}\ ] ] here it is convenient to rotate the above @xmath122 s to the following basis @xmath280 where @xmath281 here @xmath282 is the field strength of the overall center - off - mass @xmath122 in @xmath283 , while @xmath284 is the field strength of the @xmath122 in the breaking @xmath285 . indeed , we have @xmath286 the matrices multiplying @xmath287 and @xmath288 on the r.h.s . of this equation are nothing but the properly normalized generators of @xmath289 and @xmath290 , respectively . in the new basis we have @xmath291~ { 1\over 2}{\overline f}^{\mu\nu}_1(p)~{\overline f}_{1\mu\nu}(-p ) + \right.\nonumber\\ & & \left[a-{bc_2\over 2\pi^2}\int { d^2 k\over k^2}- { bc_1\over 2\pi^2}\int { d^2 k\over k^2}~e^{ik\cdot x}+ { n_2 d^2\over \pi^2}\int { d^2 k\over{k^2+p^2}}\right]~ { 1\over 2}{\overline f}^{\mu\nu}_2(p)~{\overline f}_{2\mu\nu}(-p ) + \nonumber\\ & & \left.{\sqrt{n_1 n_2 } d^2\over \pi^2}\int { d^2 k\over{k^2+p^2}}~e^{ik\cdot x}~ { \overline f}^{\mu\nu}_1(p)~ { \overline f}_{2\mu\nu}(-p)\right)=\nonumber\\ & & \int{d^4 p\over ( 2\pi)^4}~ \left({\sqrt{n_1 n_2}d^2\over \pi^2}~{{n_1-n_2}\over n}~ \left[\int{d^2 k\over { k^2+p^2}}~ e^{ik\cdot x}-\int{d^2 k\over k^2}~e^{ik\cdot x}\right]~ { \overline f}^{\mu\nu}_+(p)~{\overline f}_{-\mu\nu}(-p)+\right.\nonumber\\ & & \left(a+{2n_1 n_2\over n}~{d^2\over \pi^2}~\left[\int{d^2 k\over { k^2+p^2}}~ e^{ik\cdot x}-\int{d^2 k\over k^2}~e^{ik\cdot x}\right]\right)~ { 1\over 2}{\overline f}^{\mu\nu}_+(p)~{\overline f}_{+\mu\nu}(-p)+\nonumber\\ & & \left.\left(a-{d^2\over n\pi^2}~\left[2n_1 n_2 \int{d^2 k\over { k^2+p^2}}~ e^{ik\cdot x}+\left[n_1 ^ 2+n_2 ^ 2\right ] \int{d^2 k\over k^2}~e^{ik\cdot x}\right]\right)~ { 1\over 2}{\overline f}^{\mu\nu}_-(p)~{\overline f}_{-\mu\nu}(-p)\right)~. \label{u(1)s}\end{aligned}\ ] ] in arriving at this equation we have used ( [ couplings12 ] ) , as well as the fact that , as we discussed in the previous subsection , in the integral @xmath292 the ir cut - off is given by @xmath293 , so that @xmath294 in fact , the r.h.s . of ( [ u(1)s ] ) further simplifies once we go to the field theory limit . this limit is given by @xmath295 in this limit the two stacks of d3-branes come on top of each other , but the gauge symmetry is still @xmath296 - the original @xmath18 gauge group is broken by the adjoint higgs vacuum expectation value parametrized by @xmath297 . in particular , in this limit we have @xmath298 this implies that on the r.h.s . of ( [ u(1)s ] ) the @xmath299 term goes to zero , the coupling for the @xmath300 term goes to @xmath301 ( that is , this coupling is not renormalized , which is consistent with the fact that the overall center - of - mass @xmath122 should not run ) , while the coupling for the @xmath302 term is renormalized as follows : @xmath303 this coupling , as well as the non - abelian gauge couplings , need to be regularized . this regularization , however , depends on whether @xmath304 or @xmath305 . this is because the aforementioned @xmath181 limit must be taken differently depending on @xmath255 . however , ( [ vanish ] ) holds regardless of how the limit is taken . collecting the above results we see that the effective quantum action is given by : @xmath306 + { \widetilde a}_2(p^2,\phi^2)~ { \rm tr}\left[{\widehat f}^{\mu\nu}_2(p)~{\widehat f}_{2\mu\nu}(-p)\right ] + \nonumber\\ & & { 1\over 2 } a~{\overline f}^{\mu\nu}_+(p)~{\overline f}_{+\mu\nu}(-p ) + { 1\over 2}{\widetilde a}_-(p^2,\phi^2)~ { \overline f}^{\mu\nu}_-(p)~{\overline f}_{-\mu\nu}(-p)\big)~ , \end{aligned}\ ] ] where the @xmath290 coupling is given by ( [ u(1)_- ] ) , while the non - abelian couplings are given by : @xmath307 as before , the integral ( [ k^2 ] ) is regularized as follows : @xmath308 however , as we have already mentioned , the regularization of the integral @xmath309 depends upon @xmath255 . for @xmath310 we take the @xmath311 limit in the exponent @xmath312 and regularize the resulting integral as in ( [ k^2reg ] ) . for @xmath310 we therefore have @xmath313 the r.h.s . of this equation is nothing but the one - loop renormalized @xmath161 gauge coupling with the @xmath180-function coefficient @xmath259 . all three gauge couplings , that is , those for @xmath314 , @xmath315 and @xmath290 , run together at @xmath310 as at large momenta the effects of the @xmath316 breaking are negligible . now , at small momenta @xmath305 the @xmath181 limit must be taken differently . in particular , in the integral ( [ k^2x ] ) we first redefine the integration variables via @xmath317 . the resulting integral @xmath318 must then be regularized for small @xmath319 . note that @xmath319 has dimension of @xmath320 , so this is a uv divergence . the regularized integral ( [ zeta^2 ] ) is then given by @xmath321 where @xmath322 parametrizes the subtraction scheme dependence ( see below ) . for small momenta @xmath323 we therefore have : @xmath324 where @xmath325 and @xmath326 are the one - loop beta function coefficients for @xmath314 and @xmath315 , respectively . in the above expressions the terms proportional to @xmath327 correspond to the _ threshold _ corrections due to the massive gauge bosons ( that is , the gauge bosons that become heavy in the higgs mechanism ) . as usual , to connect the gauge coupling evolution above and below the threshold , we need to specify a subtraction scheme . thus , we can choose the subtraction scheme where the gauge couplings are matched at the scale @xmath328 , where @xmath329 is the mass of the heavy gauge bosons . in particular , this implies that @xmath330 we then have the following gauge coupling running . for @xmath331 @xmath332 where @xmath333 which is the @xmath161 gauge coupling at @xmath331 . for @xmath334 we have @xmath335 so that below the threshold the @xmath314 and @xmath315 gauge couplings run with the @xmath180-function coefficients @xmath325 and @xmath326 , respectively , while the @xmath290 gauge coupling does not run at all . thus , using the brane - bulk duality approach we reproduce the expected perturbative running of gauge couplings in the corresponding @xmath1 gauge theories . in the brane - bulk duality approach , however , we do not perform any loop computations . rather , the information about the loop corrections in gauge theory is encoded in the corresponding classical higher dimensional field theory . this is a consequence of the fact that the brane - bulk duality simply follows from the closed - open string duality . before we end this subsection , the following remark is in order . in the above computations we had to regularize various ( logarithmically ) divergent integrals . the corresponding regularizations depend upon the four - dimensional momentum squared . for instance , in this subsection we saw that the regularization of the integrals containing the information about the threshold depends on whether @xmath255 is above or below the threshold . in particular , the details of the corresponding regularization are somewhat different from what happens in the direct loop computation in the field theory language . this appears to be a common feature of string theory embeddings of gauge theories as far as computations of , say , gauge coupling running are concerned . in particular , to reproduce the gauge coupling running , it appears to be necessary to introduce an ( ir ) cut - off which is @xmath255 dependent . this appears to be a consequence of the fact that such computations are typically done within the _ on - shell _ formulation of string theory . for completeness in this subsection we would like to briefly discuss examples of @xmath1 gauge theories with matter . the simplest examples of this type are obtained via the aforementioned @xmath31 orbifold construction with the twisted chan - paton matrix given by @xmath336 , where @xmath261 . in this case the gauge group is @xmath296 , and we have matter hypermultiplets in @xmath337 and @xmath338 , where the @xmath122 charges are given in parentheses . note that all @xmath2 d3-branes are now coincident , and the gauge symmetry is broken due to the non - trivial orbifold action on the chan - paton factors . the relevant part of the classical action is given by ( for simplicity the matter field contributions are not shown , and we use the notation @xmath339 ) : @xmath234 + b~\sigma~{\rm tr}\left[\gamma_r~f^{\mu\nu}f_{\mu\nu}\right]+ { \rm tr}(\gamma_r)~{\widehat c}~\sigma+ d~\epsilon^{\mu\nu\sigma\rho}~c_{\mu\nu}~ { \rm tr}\left[\gamma_r~f_{\sigma\rho}\right]\big)\nonumber\\ & & -\int_{{\rm d3}\times { \bf r}^2 } \left({1\over 2}\partial^\mu\sigma~\partial_\mu\sigma+{1\over 12 } h^{\mu\nu\sigma } h_{\mu\nu\sigma}\right)=\nonumber\\ & & - \int_{\rm d3 } \big(a~\left[{\rm tr}\big[f^{\mu\nu}_1 f_{1\mu\nu}\right]+ { \rm tr}\left[f^{\mu\nu}_2 f_{2\mu\nu}\right]\big]+ b~\sigma~\big[{\rm tr}\left[f^{\mu\nu}_1 f_{1\mu\nu}\right]- { \rm tr}\left[f^{\mu\nu}_2 f_{2\mu\nu}\right]\big]\nonumber\\ & & + ( n_1-n_2)~{\widehat c}~\sigma+ d~\epsilon^{\mu\nu\sigma\rho}~c_{\mu\nu}~ \big[{\rm tr}\left[f_{1\sigma\rho}\right]- { \rm tr}\left[f_{2\sigma\rho}\right]\big]\big)\nonumber\\ & & -\int_{{\rm d3}\times { \bf r}^2 } \left({1\over 2}\partial^\mu\sigma~\partial_\mu\sigma+{1\over 12 } h^{\mu\nu\sigma } h_{\mu\nu\sigma}\right)~ , \label{eq : actionu2}\end{aligned}\ ] ] where the couplings @xmath266 are the same as before , while @xmath340 note that for @xmath341 or @xmath342 we recover the action ( [ eq : actionu ] ) . integrating out the bulk fields , we obtain : @xmath343~ { \rm tr}\left[f^{\mu\nu}_1(p)~f_{1\mu\nu}(-p)\right]+ \right.\nonumber\\ & & \left[a-{b{\widehat c}\over 2\pi^2}~(n_2-n_1)\int { d^2 k\over k^2 } \right]~ { \rm tr}\left[f^{\mu\nu}_2(p)~f_{2\mu\nu}(-p)\right]+\nonumber\\ & & \left.{d^2\over \pi^2}\int { d^2 k\over{k^2+p^2}}~ \left({\rm tr}[f^{\mu\nu}_1(p)]-{\rm tr}[f^{\mu\nu}_2(p)]\right)~ \left({\rm tr}[f_{1\mu\nu}(-p)]-{\rm tr}[f_{2\mu\nu}(-p)]\right)\right)~.\end{aligned}\ ] ] the @xmath122 contributions can be extracted as in the previous subsection , and , in fact , all the corresponding normalizations are exactly the same as before . we therefore obtain : @xmath344 + { \widetilde a}_2(p^2)~ { \rm tr}\left[{\widehat f}^{\mu\nu}_2(p)~{\widehat f}_{2\mu\nu}(-p)\right ] + \nonumber\\ & & { 1\over 2 } a~{\overline f}^{\mu\nu}_+(p)~{\overline f}_{+\mu\nu}(-p ) + { 1\over 2}{\widetilde a}_-(p^2)~ { \overline f}^{\mu\nu}_-(p)~{\overline f}_{-\mu\nu}(-p)\big)~ , \end{aligned}\ ] ] where the @xmath290 coupling is given by @xmath345 while the non - abelian couplings are given by @xmath346 where @xmath347 is the @xmath290 one - loop @xmath180-function coefficient , while @xmath348 and @xmath349 are the one - loop @xmath180-function coefficients for the @xmath314 and @xmath315 subgroups , respectively . thus , as we see , once again , we correctly reproduce the running of the gauge couplings for the @xmath350 subgroup ( the overall center - of - mass @xmath289 does not run as there is no matter charged under it ) . note that the twisted tadpole for @xmath184 vanishes for @xmath351 , that is , for @xmath352 . the non - abelian one - loop @xmath180-function coefficients in this case vanish . the @xmath290 one - loop @xmath180-function coefficient , however , is still non - vanishing , so the @xmath290 gauge coupling runs . in the large @xmath2 limit @xmath290 decouples in the ir , and we are left with an @xmath1 superconformal field theory . in this section we would like to discuss the brane - bulk duality in @xmath353 supersymmetric theories of the type discussed in subsection iiia . thus , let us consider the example where @xmath354 , and the action of the generators @xmath163 and @xmath164 of the two @xmath31 s on the complex coordinates @xmath90 on @xmath355 is given by ( here @xmath166 ) @xmath356 let us choose the twisted chan - paton matrices as follows : @xmath357 , @xmath358 , where @xmath261 . in this case we have the @xmath22 supersymmetric @xmath296 gauge theory with chiral matter in @xmath337 and @xmath338 , where the @xmath122 charges are given in parentheses . note that all @xmath2 d3-branes are coincident , and the gauge symmetry is broken due to the non - trivial orbifold action on the chan - paton factors . for @xmath351 we have an @xmath22 theory of the type discussed in subsection iiia . in particular , in the planar limit the correlation functions in this theory are the same as in the parent @xmath1 supersymmetric gauge theory with @xmath18 gauge group and no matter . we can therefore discuss this theory in the context of the brane - bulk duality as in the large @xmath2 limit we have the corresponding non - renormalization theorem beyond the one - loop order . for calculational convenience in the following we will keep @xmath262 and @xmath263 arbitrary . the calculation of the one - loop effective quantum action then gives a correct result even for @xmath359 , but only for @xmath351 do we have the non - renormalization theorem beyond the one - loop order . the relevant part of the classical action is given by ( for simplicity the matter field contributions are not shown , and we use the notation @xmath339 ) : @xmath234\nonumber\\ & & + { 1\over\sqrt{2 } } \sum_\alpha\big[b~\sigma_\alpha~{\rm tr}\left[\gamma_{r_\alpha}~f^{\mu\nu } f_{\mu\nu}\right]+ { \rm tr}(\gamma_{r_\alpha})~{\widehat c}~\sigma_\alpha+ d~\epsilon^{\mu\nu\sigma\rho}~c_{\alpha\mu\nu}~ { \rm tr}\left[\gamma_{r_\alpha}~f_{\sigma\rho}\right]\big]\big)\nonumber\\ & & -\sum_\alpha \int_{{\rm d3}\times { \cal f}_\alpha } \left({1\over 2}\partial^\mu\sigma_\alpha~\partial_\mu\sigma_\alpha+ { 1\over 12 } h_\alpha^{\mu\nu\sigma } h_{\alpha\mu\nu\sigma}\right)~ , \label{eq : actionu3}\end{aligned}\ ] ] where the couplings @xmath360 are the same as before , and the overall factor of @xmath361 is due to the fact that the @xmath168 twisted fields @xmath362 and @xmath363 now propagate in @xmath364 , where each fixed point set @xmath365 is an orbifold @xmath366 . integrating out the bulk fields , we obtain : @xmath367~ { \rm tr}\left[f^{\mu\nu}_1(p)~f_{1\mu\nu}(-p)\right]+ \right.\nonumber\\ & & \left[a-{b{\widehat c}\over 4\pi^2}~(3n_2-n_1)\int { d^2 k\over k^2 } \right]~ { \rm tr}\left[f^{\mu\nu}_2(p)~f_{2\mu\nu}(-p)\right]+\nonumber\\ & & { d^2\over 2\pi^2}\int { d^2 k\over{k^2+p^2}}~ \left({\rm tr}[f^{\mu\nu}_1(p)]+{\rm tr}[f^{\mu\nu}_2(p)]\right)~ \left({\rm tr}[f_{1\mu\nu}(-p)]+{\rm tr}[f_{2\mu\nu}(-p)]\right)\nonumber\\ & & \left.{d^2\over \pi^2}\int { d^2 k\over{k^2+p^2}}~ \left({\rm tr}[f^{\mu\nu}_1(p)]-{\rm tr}[f^{\mu\nu}_2(p)]\right)~ \left({\rm tr}[f_{1\mu\nu}(-p)]-{\rm tr}[f_{2\mu\nu}(-p)]\right)\right)~.\end{aligned}\ ] ] the @xmath122 contributions can be extracted as in the previous section , and , in fact , all the corresponding normalizations are exactly the same as before . we therefore obtain : @xmath344 + { \widetilde a}_2(p^2)~ { \rm tr}\left[{\widehat f}^{\mu\nu}_2(p)~{\widehat f}_{2\mu\nu}(-p)\right ] + \nonumber\\ & & { 1\over 2 } a~{\overline f}^{\mu\nu}_+(p)~{\overline f}_{+\mu\nu}(-p ) + { 1\over 2}{\widetilde a}_-(p^2)~ { \overline f}^{\mu\nu}_-(p)~{\overline f}_{-\mu\nu}(-p)\big)~ , \end{aligned}\ ] ] where the @xmath290 coupling is given by @xmath345 while the non - abelian couplings are given by @xmath346 where @xmath368 is the @xmath290 one - loop @xmath180-function coefficient , while @xmath369 and @xmath370 are the one - loop @xmath180-function coefficients for the @xmath314 and @xmath315 subgroups , respectively . thus , we correctly reproduce the running of the gauge couplings for the @xmath350 subgroup ( the overall center - of - mass @xmath289 does not run as there is no matter charged under it ) . note that the twisted tadpole for @xmath371 does not vanish , so that the non - abelian part of the gauge theory is non - conformal even for @xmath372 . in the large @xmath329 limit @xmath290 decouples in the ir , and we are left with the @xmath1 supersymmetric @xmath373 gauge theory with chiral matter in @xmath374 and @xmath375 . so far we have been focusing on @xmath1 and @xmath22 supersymmetric theories . however , in the large @xmath2 limit we can also discuss certain non - trivial non - supersymmetric cases as well . the large @xmath2 property is crucial here . the reason is that in the cases where the orbifold group @xmath376 , we always have twisted ns - ns closed string sectors with tachyons . their contributions to the corresponding part of the annulus amplitude ( [ nsns ] ) is then exponentially divergent unless we require that @xmath377 even if this condition is satisfied , we must take the t hooft limit - indeed , otherwise it is unclear , for instance , how to deal with the diagrams with handles , which contain tachyonic divergences . in fact , the same applies to some non - planar diagrams without handles , that is , diagrams where the external lines are attached to more than one boundaries ( such diagrams are subleading in the large @xmath2 limit ) . to obtain well - defined non - supersymmetric non - conformal models , consider an orbifold group @xmath378 , which is not a subgroup of @xmath127 . let @xmath129 be a non - trivial subgroup of @xmath20 such that @xmath130 . we will allow the chan - paton matrices @xmath37 corresponding to the twists @xmath131 ( which have @xmath106 ) not to be traceless , so that the corresponding @xmath1 model is not conformal . however , we will require that the other chan - paton matrices @xmath37 for the twists @xmath132 be traceless . the resulting non - supersymmetric model is not conformal . however , in the planar limit the perturbative gauge theory amplitudes are not renormalized beyond one loop ( as usual , various running @xmath122 s decouple in the ir in this limit ) . the proof of this statement is completely parallel to that we gave in subsection iiia for @xmath22 theories . let us consider a simple example of such a theory . let @xmath379 , where the action of the generators @xmath120 and @xmath172 of the @xmath31 respectively @xmath171 subgroups on the complex coordinates @xmath90 on @xmath116 is as follows : @xmath380 , @xmath381 , @xmath382 , @xmath383 , where @xmath384 . the twisted chan - paton matrices are given by : @xmath385 , @xmath386 . then the theory is a non - supersymmetric @xmath387 gauge theory with matter consisting of complex scalars in @xmath388 , @xmath389 and @xmath390 , as well as chiral fermions in the above representations plus their complex conjugates . the gauge coupling renormalization in this model can be discussed in complete parallel with the previous sections . since the @xmath122 s decouple in the ir in the large @xmath2 limit , we will ignore them in the following runnings at one loop receive contributions from non - planar diagrams with the two external lines corresponding to the @xmath122 gauge bosons attached to two different boundaries . note , however , that these contributions are due to r - r exchanges ( and the r - r sectors do not contain tachyons ) , so that tachyons do not contribute to this . ] . then the relevant part of the classical action is given by ( for simplicity the matter field contributions are not shown , and we use the notation @xmath391 ) : @xmath392 + { 1\over\sqrt{3 } } \big[b~\sigma~{\rm tr}\left[\gamma_r~{\widehat f}^{\mu\nu } { \widehat f}_{\mu\nu}\right]+ { \rm tr}(\gamma_r)~{\widehat c}~\sigma\big]\big)\nonumber\\ & & -\int_{{\rm d3}\times { \cal f } } { 1\over 2}\partial^\mu\sigma~\partial_\mu\sigma~ , \label{eq : actionu4}\end{aligned}\ ] ] where the couplings @xmath393 are the same as before , and the overall factor of @xmath394 is due to the fact that the @xmath120 twisted field @xmath184 now propagates in @xmath395 , where the fixed point set @xmath396 is an orbifold @xmath397 . integrating out the bulk fields , we obtain ( @xmath110 ) : @xmath398~ , \end{aligned}\ ] ] where the non - abelian couplings are given by @xmath399 here @xmath400 are the one - loop @xmath180-function coefficients for the @xmath161 subgroups theory has no matter hypermultiplets , regardless of whether the final model has @xmath23 or @xmath22 supersymmetry , in the above construction the one - loop @xmath180-function coefficients for the non - abelian subgroups are always given by @xmath401 , where @xmath402 is the one - loop @xmath180-function coefficient of the non - abelian subgroup in the parent @xmath1 theory . ] . in the previous sections we discussed the brane - bulk duality in the context of @xmath1 as well as certain @xmath22 and @xmath23 large @xmath2 non - conformal gauge theories . in particular , we saw that in the planar limit perturbatively ( on - shell ) correlators in the corresponding @xmath0 theories are the same as in the parent @xmath1 theories . moreover , the one - loop effective quantum action in such theories , which in the large @xmath2 limit is not perturbatively renormalized beyond one - loop , can be computed by performing a classical computation in a higher dimensional field theory . in this section we would like to discuss whether the non - perturbative corrections modify the brane - bulk duality picture in such theories . one of the key simplifying features here is the large @xmath2 limit . thus , let us consider the @xmath1 supersymmetric @xmath161 gauge theory without matter . in this theory the low energy effective action can be described in terms of a prepotential @xmath396 , which perturbatively does not receive corrections beyond one loop . the non - perturbative corrections come from instantons @xcite : @xmath403 where @xmath404 is the dynamically generated scale of the theory : @xmath405 here @xmath406 is the yang - mills gauge coupling at some high scale @xmath228 , and @xmath259 is the one - loop @xmath180-function coefficient . thus , the instanton corrections are weighted with @xmath407 where @xmath408 is the effective t hooft coupling . note that these weights go to zero in the t hooft limit , which implies that the low energy effective action is not renormalized beyond one loop in the large @xmath2 limit . next consider the @xmath22 and @xmath23 orbifold theories discussed in subsection iiia ( as well as section vi ) and section vii , respectively . due to their underlying @xmath1 structure , in these theories we might hope that the non - perturbative corrections to the low energy effective action also vanish in the large @xmath2 limit . if so , then we have non - trivial statements about infinitely many non - trivial @xmath0 gauge theories , in particular , that in such theories the low energy effective action is not renormalized beyond one loop in the planar limit . checking this conjecture in the non - supersymmetric case is rather non - trivial , but in @xmath22 cases we can perform some partial checks . in particular , this conjecture implies that in the large @xmath2 limit the superpotential should not receive non - perturbative corrections , so that the classical superpotential should be exact as the superpotential does not receive any loop corrections in @xmath22 supersymmetric theories . this statement can indeed be checked explicitly for such theories . instead of being most general , here we will consider the simplest example of such an @xmath22 theory ( other @xmath22 cases can be discussed in a similar fashion ) . thus , consider the example discussed in section vi . in this example we have @xmath22 supersymmetric @xmath409 gauge theory with chiral matter supermultiplets in @xmath158 , and @xmath159 . to simplify the discussion , let us take the gauge coupling of the second @xmath161 factor to be much smaller than that of the first one . then the second @xmath161 can be treated as the global symmetry group for the first @xmath161 , and we have the @xmath161 gauge theory with @xmath2 flavors of quarks @xmath410 , @xmath411 , where @xmath412 and @xmath413 transform in the fundamental @xmath414 respectively anti - fundamental @xmath415 of the gauge group @xmath161 . the low energy dynamics is described in terms of the gauge invariant degrees of freedom given by the mesons @xmath416 and baryons @xmath417 @xcite : @xmath418 the classical moduli space in this theory receives quantum corrections , which can be accounted for via the following superpotential ( here @xmath419 is a lagrange multiplier related to the `` glueball '' field via @xmath420 ) @xcite : @xmath421 where @xmath404 is the dynamically generated scale of the theory , which is given by : @xmath422 once again , @xmath423 goes to zero in the large @xmath2 limit , so that the classical constraint @xmath424 is unmodified in this limit . thus , in this theory the classical superpotential , which vanishes , is indeed exact in the large @xmath2 limit . the above discussion suggests that the brane - bulk duality discussed in the previous sections in the context of the aforementioned gauge theories might hold even non - perturbatively , so that the corresponding low energy effective quantum action is not renormalized beyond one loop in the large @xmath2 limit . we would like to end our discussion with a few concluding remarks . first , one natural generalization we can consider is to extend the above discussion to the cases containing @xmath425 gauge groups . this can be done via orientifolding , that is , by including orientifold planes in the setup of section ii in the spirit of @xcite . in the @xmath1 cases we expect no subtleties , but in the @xmath22 cases with twists with @xmath82 some caution is needed @xcite due to the subtleties discussed in @xcite . another point we would like to comment on is the following . recently , in the brane world context @xcite , it was pointed out in @xcite that the einstein - hilbert term is generically expected to be induced via loop corrections on a brane as long as the brane world - volume theory is not conformal . subsequently , it was argued in @xcite that this effect should arise in the context of non - conformal gauge theories from d3-brane , in particular , this is expected to be the case in theories discussed in this paper . it would be interesting to see whether the brane - bulk duality can be used for simplifying computation of such corrections on the d3-branes . we would like to thank gregory gabadadze , rajesh gopakumar , martin roek , tom taylor , henry tye , cumrun vafa and slava zhukov for valuable discussions . this work was supported in part by the national science foundation . z.k . would also like to thank albert and ribena yu for financial support . in section v we used various brane - bulk couplings in the @xmath31 orbifold examples . these couplings can be computed within the boundary state formalism , where one computes the couplings of the boundary states to the twisted closed string states in the presence of non - trivial d - brane field backgrounds @xcite . here , however , since we only need the couplings relevant for the one - loop corrections in the gauge theory language , we will take a shortcut and deduce these couplings using the annulus amplitude in the presence of d - brane field backgrounds . the annulus amplitude in the light - cone gauge is given by ( here @xmath120 is the generator of the @xmath31 orbifold twist ) : @xmath426~.\ ] ] here we assume that we have a non - trivial _ constant _ background gauge field along the brane . the annulus amplitude @xmath427 then is almost the same as in the case without the background field , which we discussed in section ii , with the difference that some of the open string oscillator modings are modified @xcite . for our purposes here it will suffice to consider the background field of the form ( @xmath246 , @xmath428 , is the d3-brane gauge field strength ) : @xmath429 where @xmath430 is a cartan generator of the @xmath18 gauge group , which we will take to be hermitian . let @xmath431 , @xmath432 , be the eigenvalues of @xmath430 . it is convenient to introduce complex combinations of the world - sheet bosonic and fermionic degrees of freedom corresponding to the directions @xmath433 ( so that instead of two real world - sheet bosons and two real world - sheet fermions corresponding to these directions we have one complex world - sheet boson and one complex world - sheet fermion ) . then the modings of these complex world - sheet degrees of freedom are modified as follows ( modings of other world - sheet degrees of freedom are unchanged ) : @xmath434 where @xmath435 is given by : @xmath436~. \label{eq : epsilon}\ ] ] here the indices @xmath437 label the two ends of the open string . the cylinder amplitude is given by @xcite : @xmath438 e^{-{t \over 2\pi\alpha^\prime}~x_{\alpha\beta}^2}~,\ ] ] where @xmath439 is square of the distance ( in the two real extra directions untouched by the orbifold ) between the d3-branes labeled by @xmath440 and @xmath180 . the factor of @xmath441 in the denominator comes from the bosonic zero modes in the directions @xmath442 . the factor @xmath443 can be understood from the requirement that in the @xmath444 limit we must reproduce the corresponding answer ( see below ) . in particular , the untwisted character @xmath445 and the twisted character @xmath446 are given by . thus , for instance , we could have chosen the phase of the last term in the square brackets in ( [ u ] ) as @xmath48 instead of @xmath47 . note , however , that these phases do not affect our results here as the terms they multiply are vanishing ( @xmath447 ) . ] ( for simplicity the indices @xmath448 are not shown ) : @xmath449 ^ 3 - z_{-1/2}^\delta ~ [ z_{-1/2}^0]^3 \right.\nonumber\\ & & \left.- z_0^{-1/2+\delta } ~[z_0^{-1/2}]^3 + z_{-1/2}^{-1/2+\delta } ~[z_{-1/2}^{-1/2}]^3\right]~,\\ \label{t } t=&&{1 \over \eta^2(q ) ~z_{-1/2}^{-1/2+\delta } ~[z_0^{-1/2}]^2}\left[z_0^\delta ~z_0 ^ 0 ~ [ z_{-1/2}^0]^2 - z_{-1/2}^\delta ~z_{-1/2}^0 ~[z_{0}^0]^2\right.\nonumber\\ & & \left .- z_0^{-1/2+\delta } ~z_{-1/2}^0 ~[z_{-1/2}^{-1/2}]^2 + z_{-1/2}^{-1/2+\delta } ~z_{-1/2}^{-1/2 } ~[z_{-1/2}^0]^2\right]~.\end{aligned}\ ] ] here the characters @xmath450 are the usual complex fermion characters ( @xmath451 , @xmath452 ) : @xmath453 in particular , note that the character @xmath454 in the denominators in ( [ u ] ) and ( [ t ] ) in the @xmath444 limit becomes : @xmath455 combining this with the aforementioned factor @xmath443 gives the following contribution @xmath456 which precisely corresponds to the bosonic zero modes plus oscillators in the @xmath433 directions in the absence of the gauge field background . here we would like to extract couplings of the massless closed string states to the d - branes . this can be done by extracting the leading behavior of the annulus amplitude for @xmath68 . using modular transformation properties of the above characters , we obtain : @xmath457\big|_{{\rm { \small ns - ns } } } - 2\cos(\pi\delta)\big|_{{\rm{\small r - r}}}\right\}~,\\ & & t\sim { 2t\over \sin(\pi\delta ) } \left\{1\big|_{{\rm { \small ns - ns } } } - \cos(\pi\delta)\big|_{{\rm{\small r - r}}}\right\}~,\end{aligned}\ ] ] where individual contributions due to the ns - ns and r - r exchanges are shown . here we are interested in the brane - bulk couplings involving at most quadratic terms in the gauge field strength @xmath246 . then the relevant terms in the small @xmath458 limit are given by : @xmath459 \left\{1\big|_{{\rm { \small ns - ns } } } - 1\big|_{{\rm{\small r - r}}}\right\}~,\\ & & t\sim { t\over \pi\delta } \left[1-{1\over 3}(\pi\delta)^2\right ] \left\{\left[2+(\pi\delta)^2\right]\big|_{{\rm { \small ns - ns } } } - 2\big|_{{\rm{\small r - r}}}\right\}~.\end{aligned}\ ] ] using the fact , which follows from ( [ delta ] ) , that to the relevant order @xmath460={{1+(2\pi\alpha^\prime b)^2 q_\alpha q_\beta}\over ( 2\pi\alpha^\prime b ) ( q_\alpha - q_\beta)}~,\ ] ] we obtain the following massless untwisted respectively massless twisted closed string contributions into the annulus amplitude @xmath461 \int_0^\infty dt~t~e^{-{t\over 2\pi\alpha^\prime}~x_{\alpha\beta}^2}~,\\ \label{t1 } { \widetilde { \cal c}}_t=&&{1\over2(4\pi^2\alpha^\prime)^2 } \sum_{\alpha,\beta } \left\{\left[2 + ( 2\pi\alpha^\prime b)^2 \left(q_\alpha^2 + q_\beta^2\right)\right ] \big|_{{\rm { \small ns - ns}}}- \left[2 + 2(2\pi\alpha^\prime b)^2 q_\alpha q_\beta\right ] \big|_{{\rm{\small r - r}}}\right\}\nonumber\\ & & \int_0^\infty { dt\over t}~e^{-{t\over 2\pi\alpha^\prime}~x_{\alpha\beta}^2}~.\end{aligned}\ ] ] note that the integrals in these expressions are related to the corresponding euclidean propagators @xmath462 and @xmath463 : @xmath464 here @xmath465 , @xmath466 , and @xmath467 are the momenta corresponding to the coordinates @xmath468 . as we discussed in section v , the relevant part of the classical action is given by ( note that for the field of the form ( [ eq : choicef ] ) terms containing @xmath231 are vanishing ) : @xmath234 + b~\sigma~{\rm tr}\left[\gamma_r~f^{\mu\nu}f_{\mu\nu}\right]+ { \rm tr}(\gamma_r)~{\widehat c}~\sigma+ d~\epsilon^{\mu\nu\sigma\rho}~c_{\mu\nu}~ { \rm tr}\left[\gamma_r~f_{\sigma\rho}\right]\big)\nonumber\\ & & -\int_{{\rm d3}\times { \bf r}^2 } \left({1\over 2}\partial^\mu\sigma~\partial_\mu\sigma+{1\over 12 } h^{\mu\nu\sigma } h_{\mu\nu\sigma}\right)~. \label{eq : actionua}\end{aligned}\ ] ] here @xmath184 is a twisted ns - ns scalar , while @xmath235 is a twisted two - form ( whose field strength is @xmath236 ) . the twisted chan - paton matrix @xmath121 . note , however , that for generic values of @xmath469 the @xmath18 gauge group is broken down to @xmath470 . so the d3-branes are not necessarily coincident in the two transverse dimensions untouched by the orbifold action . from the tree - level action ( [ eq : actionua ] ) we obtain the following twisted massless contributions quadratic in the gauge field strength ( here we are using @xmath471 ) : @xmath472 note that this expression contains an overall factor of 2 from exchanging the two ends of the string as we are discussing an _ oriented _ string theory . comparing this expression with ( [ t1 ] ) , we obtain : @xmath473 the coupling @xmath474 can be determined in a similar fashion . z. kakushadze , g. shiu and s .- h.h . tye , nucl . * b533 * ( 1998 ) 25 ; + z. kakushadze , phys . lett . * b455 * ( 1999 ) 120 ; int . j. mod * a15 * ( 2000 ) 3461 ; phys . lett . * b459 * ( 1999 ) 497 ; int . j. mod a15 * ( 2000 ) 3113 . dienes , e. dudas and t. gherghetta , phys . b436 * ( 1998 ) 55 ; nucl . phys . * b537 * ( 1999 ) 47 ; hep - ph/9807522 ; + z. kakushadze , nucl . phys . * b548 * ( 1999 ) 205 ; nucl . phys . * b552 * ( 1999 ) 3 ; + z. kakushadze and t.r . taylor , nucl . phys . * b562 * ( 1999 ) 78 .
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we discuss non - conformal gauge theories from type iib d3-branes embedded in orbifolded space - times .
such theories can be obtained by allowing some non - vanishing logarithmic twisted tadpoles . in certain cases with @xmath0 supersymmetry correlation functions in the planar limit
are the same as in the parent @xmath1 supersymmetric theories .
in particular , the effective action in such theories perturbatively is not renormalized beyond one loop in the planar limit . in the @xmath1 as well as
such @xmath0 theories quantum corrections in the d3-brane gauge theories are encoded in the corresponding classical higher dimensional field theories whose actions contain the twisted fields with non - vanishing tadpoles .
we argue that this duality can be extended to the non - perturbative level in the @xmath1 theories .
we give some evidence that this might also be the case for @xmath0 theories as well .
= 10000 epsf
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symmetry plays an important role in our understanding of nature and is the primary ingredient that goes into shaping of physical laws . symmetry dictates the dynamics in nature @xcite . this simple _ and _ powerful insight has shaped our current knowledge of physics , right from the galilean invariance of newtonian mechanics , through the lorentz invariance of special relativity , to the reparametrization or diffeomorphism invariance of general relativity . so understanding the role and systematic construction of symmetries through their generators is an important aspect in the study of any physical theory . given an action , a symmetry is some transformation of the fields of the theory involving arbitrary functions of time that leaves the action off - shell invariant . rotations and translations in global space - time consist two important examples of such symmetries that are observed in nature . together , these operations constitute the poincar group . to implement this poincar symmetry at the local level , whereby arbitrary functions of time and space are involved in the variations , new fields must be introduced which compensate for terms in the action not remaining invariant under this localization . these additional fields then are shown to represent the gravitational fields and the theory which describes this procedure and the corresponding dynamics is known as the poincar gauge theory ( hereafter referred to as pgt ) @xcite . a lot of work has been done since then till recent times studying various aspects of the theory . [ fig1 ] the symmetries involved in pgt are motivated and derived by considering changes occurring due to local lorentz rotations and infinitesimal coordinate changes ( i.e. diffeomorphism ) . however , no canonical procedure of generating this symmetry through a set of generators exists till now . this problem was initially addressed in @xcite by computing the hamiltonian ( gauss ) generator following the approach of @xcite . however the off - shell symmetries that were obtained as a result , were different from the pgt symmetries . the two sets matched only on - shell . this mismatch was thought to be a consequence of the approach @xcite which , strictly speaking , is not a completely off - shell approach . recently , an attempt to remedy this situation was given in @xcite . a systematic and completely off - shell analysis of this issue was done in 2 + 1 dimensions , taking the 3d gravity model with torsion ( modelled on the mielke - baekler action @xcite ) and the dirac hamiltonian generator was computed . alas , the original conclusions of @xcite remained unaltered . here in this article , we adopt a totally different ( lagrangian ) approach and show how to systematically construct generators of the pgt symmetries . the symmetries obtained from these generators reproduce the pgt symmetries without using any equations of motion . the task of finding the symmetries of a given action is , in general , not trivial . they can not always be found on inspection , as is possible , say , in electrodynamics . there exist two approaches to systematically construct the symmetries inbuilt in a given action ; the hamiltonian approach and the lagrangian approach . in the hamiltonian approach @xcite , one first undertakes the dirac classification of all constraints into first - class and second - class . then the generators of the symmetries are obtained as some suitable combination of first - class constraints containing time derivatives of arbitrary functions . the poisson bracket of the fields with these generators give us the symmetries as transformations . the lagrangian method @xcite , on the other hand , hinges on the condition that the existence of symmetries necessarily implies the existence of certain identities involving quantities given by variations of the action w.r.t . the basic fields the euler derivatives . the lagrangian generators can then be found through comparison with the general expressions of such identities derived from a theory involving general symmetry transformation of fields in terms of arbitrary functions of time . we have used this later method of constructing generators here , after modifying it suitably for our model with vielbeins and connections as basic fields , that are written both in holonomic ( global coordinate ) and an - holonomic ( local coordinate ) indices . having constructed the generators first in 2 + 1 dimensions , we then repeat the process in 3 + 1 dimensions . the construction of symmetries through the hamiltonian approach in 3 + 1 dimensions itself is a very difficult task and pure dirac analysis and classification of constraints is non - trivial for , say , just the einstein - cartan theory . however it is shown that the lagrangian version can be carried out more easily , leading to a systematic derivation of the symmetries even in 3 + 1 dimensions . to do this , we first lift the gauge identities corresponding to pgt symmetries from 2 + 1 to 3 + 1 dimensions and then carry out the lagrangian analysis . however , at the end of the day , the symmetries obtained are the same as that found by directly lifting the 2 + 1 pgt symmetries in 3 + 1 dimensions . this shows that the pgt symmetries , arising out of geometrical considerations of reparametrization and local lorentz symmetry , are beautifully consistent and useful in an extended sense . the organization of the paper is as follows . in section [ sec : genform ] we give a general formulation of the lagrangian method to find out generators , starting from known gauge identities of a model . next , in section [ sec:2 + 1laggen ] , we identify the gauge identities , calculate the generators in 2 + 1 dimensions and thus systematically reproduce the pgt symmetries in a mielke - baekler type model in 2 + 1 dimensions . for finding symmetry generators in 3 + 1 dimensions , we then lift the 2 + 1 dimensional gauge identities to 3 + 1 in section [ sec:3 + 1laggen ] and find out the lagrangian generators . these are shown to generate the usual pgt symmetries in 3 + 1 dimensions . subsequently , we also demonstrate that the 2 + 1 dimensional generators , when stripped of their dimension dependant form containing dual fields , have the same form as in 3 + 1 dimensions . this suggests that our results may be extrapolated to any higher dimensions . the whole scheme is summarized in figure [ fig1 ] . our conclusions are given in section [ sec : conclu ] . besides this , we have also included [ sec : appuseslaggen ] , where we comment on other possible applications of lagrangian generators . [ [ conventions ] ] conventions : + + + + + + + + + + + + the coordinate frame or holonomic indices are written in greek while latin indices refer to the an - holonomic local lorentz frame . the time and space bifurcation is indicated by choosing the beginning letters of both greek @xmath0 and latin @xmath1 indices to run over the space indices , i.e. @xmath2 and choosing the middle letters of both greek @xmath3 and latin @xmath4 indices to run over both time and space indices @xmath5 . the totally antisymmetric tensor densities @xmath6 are all normalized so that @xmath7 are unity . the spacetime signature chosen is @xmath8 . in specifying spacetime points , we have denoted by ` @xmath9 ' both the time and space parts together while ` @xmath10 ' indicates only the spatial part of ` @xmath9 ' . thus @xmath11 . the action functional for a typical first - order theory invariant under the pgt symmetries in 2 + 1 dimensions , may be written by taking the triad @xmath12 and connection @xmath13 as independent variables , while constructing the configuration space @xcite . so in general , we have the action functional @xmath14.\end{aligned}\ ] ] arbitrary variations of the basic fields give rise to variation of the action in the following form @xmath15 where @xmath16 and @xmath17 are the euler derivatives . the corresponding equations of motion are just @xmath18 now let us propose the following symmetries of the fields @xmath19\\ \delta \omega^i_{\ \mu}(x)=\displaystyle\sum_{s=0}^n ( -1)^s\int \mathrm{d}^2{\bf z}~\left[\frac{\partial^s\xi^\sigma(z)}{\partial \mathrm{t}^s}\bar{\rho}^i_{\ \mu\sigma(s)}(x , z ) + \frac{\partial^s\theta^k(z)}{\partial \mathrm{t}^s}\bar{\zeta}^i_{\ \mu k(s)}(x , z ) \right ] \end{aligned}\end{aligned}\ ] ] where the functions @xmath20 , @xmath21 , @xmath22 and @xmath23 are known as ` lagrangian generators ' that generate variations in the basic fields while the six quantities @xmath24 , @xmath25 are functions of space and time serving as infinitesimal gauge parameters . note that their number is governed by the poincar symmetry group , which in 2 + 1 dimensions has six independent symmetries . these variations of fields are symmetries in the sense that @xmath26=s\left[b^i_{\ \mu},\omega^i_{\ \mu}\right]$ ] , or equivalently @xmath27 under these variations . substituting the variations of @xmath28 and @xmath29 in yields the variation of the action as @xmath30\\ & \ & \!\!-\int \!\!\mathrm{d}^2{\bf x } \int \!\!\mathrm{d}^2{\bf z } \,\displaystyle\sum_{s=0}^n ( -1)^s \!\int \!\!\mathrm{d}\mathrm{t } \left[\frac{\partial^s\xi^\sigma(z)}{\partial \mathrm{t}^s } \bar{\rho}^i_{\ \mu\sigma(s)}(x , z ) \bar{l}_i^{\ \mu}(x ) + \frac{\partial^s\theta^k(z)}{\partial \mathrm{t}^s } \bar{\zeta}^i_{\ \mu k(s)}(x , z ) \bar{l}_i^{\ \mu}(x)\right ] . \end{aligned}\end{aligned}\ ] ] to simplify the above expression , let us take a representative term the first term in the first line from . the other terms , having similar structure , can then be handled by a similar technique . @xmath31 . \end{aligned}\end{aligned}\ ] ] we now interchange derivatives ( wherever applicable ) by using partial integrals , throwing away the boundary terms by assuming the fields to be well behaved at infinity . also note that we have precisely the same number of negative signs before each term through the @xmath32 factor , as required for carrying out partial integrals ` @xmath33 ' times . so , simplifying the above equation yields @xmath34\\ & \!\!= \;-\int \!\!\mathrm{d}^2{\bf z } \int \!\!\mathrm{d}\mathrm{t } ~\xi^\sigma(z)~\left[\displaystyle \sum_{s=0}^n\int \mathrm{d}^2{\bf x } ~\frac{\partial^s}{\partial\mathrm{t}^s } \left\lbrace \rho^i_{\ \mu\sigma(s)}(x , z ) l_i^{\ \mu}(x ) \right\rbrace\right ] . \end{aligned}\end{aligned}\ ] ] substituting this back in the variation of the action , we get @xmath35\end{aligned}\ ] ] where @xmath36 and @xmath37 are defined as : @xmath38 since each of the gauge parameters @xmath39 and @xmath40 are independent quantities , the invariance of the action @xmath41 implies the following conditions @xmath42 these are known as ` _ _ gauge identities _ _ ' . they are identities in the sense that on substituting the euler derivatives @xmath43 and @xmath44 in , all terms cancel out and we see that the relations are zero identically . note that until now we have only used the definition of the euler derivatives in terms of variation of fields @xmath45 , but we have not set the euler derivative to zero , i.e. we have not used any equations of motion . in fact , using the equations of motion trivializes the gauge identities as @xmath46 relations . now the algorithm for finding out the lagrangian symmetry generators is simple . given an action , we can easily find the euler derivatives by varying the action w.r.t . the basic fields . using the euler derivatives , we can then try to build a set of independent , identically vanishing equations - the gauge identities . alternatively , we can also explicitly check any set of identities proposed to hold as gauge identities from physical considerations . the number of these gauge identities is identical to the number of independent symmetries . in this particular case of pgt , they are six in number and are stated compactly in . once we obtain a set of _ independent _ gauge identities for the action , we then finally compare the given identities with the general form presented in , and find out the generators @xmath47 . gravity in 2 + 1 dimensions is widely studied , both as a simplified problem in comparison to 3 + 1 dimensions , and also as a field which is interesting in its own right . among various models studied , one is the mielke - baekler model which explicitly includes a torsion term , along with the chern - simons action and the usual einstein - cartan piece . the particular mielke - baekler type action of recent interest @xcite that we study here is : @xmath48,\end{aligned}\ ] ] where @xmath49 , @xmath36 , @xmath50 and @xmath51 are arbitrary parameters , @xmath52 is the curvature tensor defined as @xmath53 and @xmath54 is the torsion given by , @xmath55 the first term proportional to ` @xmath49 ' is the einstein - hilbert action written in three dimensions using the identity @xmath56 where @xmath57 and @xmath58 . the second term is the cosmological constant part , the third one is the chern - simons action while the fourth includes torsion . these terms can be manipulated with the help of the adjustable parameters @xmath49 , @xmath36 , @xmath50 and @xmath51 . the action is known to be invariant under the following pgt symmetries @xcite @xmath59 a set of independent gauge identities corresponding to pgt symmetries for the action is already known @xcite @xmath60 here @xmath43 and @xmath44 are the euler derivatives obtained from the action , and are given by , @xmath61\\ \bar{l}_i^{\ \mu } & : = -\frac{\delta \mathcal{l}}{\delta \omega^i_{\ \mu } } = -\epsilon^{\mu\nu\rho}\left[\alpha_3r_{i\nu\rho } + at_{i\nu\rho } - \lambda\epsilon_{ijk}b^j_{\ \nu}b^k_{\ \rho}\right ] . \end{aligned}\end{aligned}\ ] ] substituting these in it may easily be checked that all terms cancel and they are indeed identities . now the lagrangian symmetry generators which will give us a set of symmetries of the action may be found by comparing these identities with the general gauge identities derived before in . we have employed the following strategy in comparing the two relations in question . any sum over greek ( holonomic ) indices is broken into the time and space part , _ i.e. _ say , @xmath62 ; the gauge identity @xmath63 is also broken into sets @xmath64 and @xmath65 ; and finally coefficients ( in general field dependant ) of the euler derivatives @xmath66 , @xmath67 , etc . are matched between the two relations and . let us now illustrate the details for one particular term : coefficient of @xmath67 in @xmath65 . an inspection of reveals that there occur terms either with zero or a single time derivative . this implies that the summation over ` @xmath33 ' in is restricted to only two values , @xmath68 so , the relevant terms from are , @xmath69 while those from are @xmath70 the above expression may be recast in the form @xmath71 to facilitate comparison with . this comparison yields the generators @xmath72 and @xmath73 . the other generators may also be found in a similar manner . we list all the non - zero ones below : @xmath74 @xmath75 @xmath76 @xmath77 thus , having obtained all the lagrangian generators , it is now possible to generate the transformations of the basic fields @xmath28 and @xmath29 through . we illustrate the process for @xmath78 . @xmath79\\ & \qquad-\int \mathrm{d}^2{\bf z } ~\left [ \partial_0\xi^0(z)\,\rho^i_{\ \alpha 0(1)}(x , z ) + \partial_0\xi^\beta(z)\,\rho^i_{\ \alpha\beta(1)}(x , z ) + \partial_0\theta^k(z)\,\zeta^i_{\ \alpha k(1)}(x , z ) \right ] \end{aligned}\end{aligned}\ ] ] using the form of the generators @xmath80 given in and , one obtains , @xmath81 \ - \ \int \mathrm{d}^2{\bf z } ~\left[\ 0 + 0 + 0\ \right]\\ = & -\epsilon^i_{\ jk}\,b^j_{\ \alpha}\,\theta^k - \partial_\alpha\xi^\mu\,b^i_{\ \mu } - \xi^\mu\,\partial_\mu b^i_\alpha , \end{aligned}\end{aligned}\ ] ] which corresponds to the @xmath82 ( space ) component of the pgt symmetry @xmath83 given in . all other pgt symmetries given in are easily reproduced by this procedure . the same pgt symmetries , being constructed out of local lorentz and general diffeomorphism symmetries , are respected by a wide class of lagrangians @xcite . in 3 + 1 dimensions , the chern - simons term of 2 + 1 dimensions automatically drops out . the other terms have their counterparts in 3 + 1 dimensions , in addition to some other new possible terms . however , for simplicity , it suffices our aim of constructing the generators of pgt symmetries , to consider only the most important part of the gravitational action the einstein - cartan term . thus we take the following action in 3 + 1 dimensions , @xmath84 where @xmath85 and the curvature scalar @xmath86 . the curvature tensor and the torsion tensor are defined as : @xmath87 the corresponding euler derivatives can be found in the standard way @xmath88 to find the appropriate gauge identities here , we will now take help of the identities found previously for 2 + 1 dimensions . the 2 + 1 dimensional model was constructed using the basic fields @xmath12 and @xmath13 , where the latter was a dual construct of the field @xmath89 , valid only in 2 + 1 dimensions . we would now like to write the gauge identity in terms of the fields @xmath12 and @xmath89 , thus getting rid of the use of special 2 + 1 dimensional properties . the resultant identities will then be proposed for 3 + 1 dimensions and a lagrangian analysis will be carried out to find out the corresponding symmetries . now let us consider the @xmath90 identity in . contracting it with the levi - civita symbol , we find , @xmath91 next , introducing relations between the dual fields and their corresponding counterparts through @xmath92 and using the following identity for levi - civita symbols @xmath93 we are able to write the gauge identity completely in terms of the fields @xmath12 and @xmath89 . the other gauge identity @xmath63 in the set of the 2 + 1 dim identities can also be ridden of the duals through a similar procedure . the resultant set of identities are : @xmath94 since these are now written independent of any dimensionally dependant dual fields , we may propose that they also hold in 3 + 1 dimensions . an explicit check , using the expressions for the euler derivatives confirms the proposition . expressing the action in terms of basic fields , rather than the duals , we have @xmath95.\end{aligned}\ ] ] the symmetry transformations of this action are now given by @xmath96\\ \delta \omega^{ij}_{\ \ \mu}(x ) & = \displaystyle\sum_{s=0}^n ( -1)^s\int \mathrm{d}^3{\bf z}~\left[\frac{\partial^s\xi^\sigma(z)}{\partial \mathrm{t}^s}\bar{\rho}^{ij}_{\ \ \mu\sigma(s)}(x , z ) + \frac{\partial^s\theta^{lk}(z)}{\partial \mathrm{t}^s}\bar{\zeta}^{ij}_{\ \ \mu lk(s)}(x , z ) \right ] , \end{aligned}\end{aligned}\ ] ] which are the 3 + 1 dimensional versions of . now adopting identical steps as in section [ sec : genform ] , we obtain the analogues of the gauge identities with @xmath63 and @xmath97 given by , @xmath98 next , we compare these with the set of identities term by term as explained in the discussion above eq . to find out the relevant lagrangian generators . the non - zero ones are listed below @xmath99 ~\delta({\bf x}-{\bf z}),\\ \end{aligned}\end{aligned}\ ] ] @xmath102\delta({\bf x}-{\bf z})\\ \bar{\zeta}^{ij}_{\ \ \alpha lk(0)}(x , z ) & = -\frac{1}{2}\left [ \delta^i_l\omega^j_{\ k\alpha } - \delta^i_k\omega^j_{\ l\alpha } - \delta^j_l\omega^i_{\ k\alpha } + \delta^j_k\omega^i_{\ l\alpha } \right]\delta({\bf x}-{\bf z } ) - \frac{1}{2}\left [ \delta^i_l\delta^j_k - \delta^i_k\delta^j_l \right ] \,\partial_\alpha^{(\bf x)}\delta({\bf x}-{\bf z})\\ \bar{\zeta}^{ij}_{\ \ 0lk(1)}(x , z ) & = { \ \ } \frac{1}{2 } \left [ \delta^i_l\delta^j_k - \delta^i_k\delta^j_l \right ] ~\delta({\bf x}-{\bf z}).\\ \end{aligned}\end{aligned}\ ] ] these generators will yield the symmetries of the action through the transformation . an explicit calculation , leads to the symmetries @xmath103 it may be easily checked that these transformations are indeed symmetries of the einstein - cartan action in 3 + 1 dimensions @xcite . we would now like to make a comparative remark on the structure of the lagrangian generators in 2 + 1 and 3 + 1 dimensions . let us consider the 2 + 1 dimensional generator @xmath104 from @xmath105 multiplying appropriately with levi - civita symbols and using the map for @xmath106 from , we get @xmath107 finally using the identity for contraction of levi - civita symbols and rearranging terms yield the following relation @xmath108\,\delta(x - z ) \\ & - \left [ \delta^m_h\delta^n_p-\delta^m_p\delta^n_h \right]\,\partial_\alpha^{({\bf x})}\delta(x - z ) , \end{aligned}\end{aligned}\ ] ] where we have defined the map @xmath109 thus we have re - written the 2 + 1 dimensional generator @xmath104 in terms of the original fields , getting rid of all duals . the object @xmath110 defined in , however is functionally identical to the corresponding 3 + 1 dimensional generator . thus the map expresses the 2 + 1 dimensional generator in a form that remains structurally the same even in 3 + 1 dimensions . similarly , all the other generators from 2 + 1 dimensions , can be stripped off the dual fields @xmath111 . these dual fields were defined for the special case of 2 + 1 dimensions . once having removed them , and expressed all basic fields in terms of their dimension independent form , we see that the same generators also hold in 3 + 1 dimensions . below , we list all the non - trivial maps , in the sense described above , between the lagrangian generators . @xmath112 observation of this structural similarity of the generators across dimensions , from 2 + 1 to 3 + 1 , indicates that in higher dimensions , similar results are expected to hold . in this paper we have succeeded in systematically computing the generators of the pgt ( _ i.e. _ poincar gauge theory ) symmetries , which is a new result . the lagrangian method of finding generators was employed , and in the case of first - order formulations of gravity , was seen to be much simpler than its hamiltonian counterpart . to begin with , we took a 2 + 1 dimensional model of a pgt - invariant lagrangian which has been of recent interest @xcite the 3d gravity model with torsion and a cosmological term . the starting gauge identities involving the euler derivatives that were required for the lagrangian analysis , were taken from a recent analysis @xcite . the lagrangian generators were subsequently computed and pgt symmetries were recovered using the same . we next repeated the procedure for 3 + 1 dimensions , where we took only the representative and most important einstein - cartan term in the action . we lifted the 2 + 1 dimensional gauge identities to 3 + 1 dimensions . the validity of the lifted gauge identities in 3 + 1 dimensions was explicitly checked . then the same method was adopted to calculate the generators giving rise to pgt symmetries for this case . the lagrangian generators themselves were also shown to preserve their structure across the 2 + 1 to 3 + 1 dimension transition . the pgt symmetries were shown to be consistent throughout this process as has been shown in figure [ fig1 ] . finding the generators that yield the pgt symmetries systematically , through a canonical method , was necessary to put this much studied symmetry at a firm level as it was not even known whether it was possible to generate the pgt symmetries by a canonical and completely off - shell process . a recent attempt @xcite to generate the symmetries of a pgt invariant 2 + 1 dimensional mielke - baekler @xcite type model through pure dirac hamiltonian generator construction in a completely off - shell manner ( following @xcite ) had given symmetries which were distinctly different from pgt symmetries . the two sets only matched on - shell , _ i.e. _ after imposition of equations of motion . a similar conclusion was also noted in @xcite by following a modified dirac hamiltonian approach @xcite where second class constraints were accounted by solving for the lagrange multipliers rather than by using dirac brackets . however , in this work , we have successfully shown that based on the lagrangian generators we have constructed , one can now put this tetrad - connection formulation of symmetries on the same footing with the metric formulations @xcite , pertaining to the canonical method of constructing the symmetries . in this article , we demonstrated the role of lagrangian generators in investigating gauge symmetries . in particular , the _ off - shell _ poincar gauge theory symmetries were reproduced through lagrangian generators for the 2 + 1 and 3 + 1 dimensional mielke baekler type models of gravity . one might wonder about other possible applications of these generators . in this appendix we briefly comment on the issue . as is well known , the canonical generators derived in the hamiltonian formalism , apart from yielding the gauge symmetries , are also used to find the conserved charges and central terms in the poisson algebra of spacetime symmetries @xcite . since the hamiltonian and lagrangian formulations complement one another , it is expected that the lagrangian generators will also have a similar , though not necessarily identical , role . we now elaborate on this and related points . the crucial ingredient in abstracting the lagrangian generators are the gauge identities . construction of these identities can be made from physical considerations . however , there also exist systematic schemes for arriving at these gauge identities from algorithms employing lagrangian constraints @xcite . so , given a model with some lagrangian , we can arrive at the gauge identities and the lagrangian generators systematically . now , some insight into these identities is gleaned from their connection with the bianchi identities of a model @xcite . in what follows , we adopt the einstein - hilbert action in 3 + 1 dimensions for the demonstration of this connection . the calculation follows @xcite closely . the einstein - hilbert action in 3 + 1 dimensions is written in terms of the basic field @xmath113 the metric as : @xmath114 where the ricci tensor @xmath115 is defined in terms of the christoffel connections @xmath116 in the usual way as in einstein general relativity : @xmath117 varying the action with respect to the metric @xmath113 we get the euler derivative @xmath118 @xmath119 where , @xmath120 invariance of the action leads to the usual einstein s equation @xmath121 . the gauge identity may be subsequently defined as @xmath122 which may also be expressed as , @xmath123 now , the bianchi identity for einstein general relativity is well known and is written in terms of the riemann tensor @xmath124 as : @xmath125 contracting @xmath126 with @xmath127 and @xmath128 with @xmath129 in the above identity ( the metricity condition @xmath130 holds in einstein general relativity ) , we reproduce the gauge identity . this immediately shows that the gauge identity is nothing but a suitably contracted form of the bianchi identity in this model . the gauge identity , or the contracted version of the bianchi identity , plays a significant role in the obtention of the noether central charges . as in the hamiltonian description , here too surface terms are important . if these terms are not dropped , then takes the form , @xmath131.\end{aligned}\ ] ] explicitly using @xmath132 we obtain , @xmath133 = 0\,.\end{aligned}\ ] ] the first term in the integrand vanishes due to the gauge identity . this also implies the vanishing of the second term in the integrand . effectively , this leads to the covariant conservation of the noether current , @xmath134 corresponding to each vector @xmath39 it is now possible to construct a conserved noether charge from . this yields the standard komar s integral in general relativity @xcite . a comment on the surface terms might be useful . in the hamiltonian approach , these terms are determined by requiring the functional differentiability of the generators . the corresponding criterion in the present lagrangian formulation is to retain all surface terms in the variation of the action under a general coordinate transformation , eventually leading to the gauge identity . this is clearly manifested in where the first term in the integrand yields the gauge identity while the second is the cherished surface term . we thus observe how the gauge identity , which is directly connected with the lagrangian generators , leads to conserved noether charges . also , the complementary aspects of lagrangian and hamiltonian generators , vis-@xmath135-vis the construction of conserved charges gets illuminated . d. j. gross , `` the role of symmetry in fundamental physics , '' proc . usa * 93 * 14256 ( 1996 ) . r. utiyama , `` invariant theoretical interpretation of interaction , '' phys . * 101 * 1597 ( 1956 ) . t. w. b. kibble , `` lorentz invariance and the gravitational field , '' j. math . * 2 * 212 , ( 1961 ) . d. w. sciama , `` on the analog between charge and spin in general relativity , '' p-415 , _ recent developments in general relativity , festschrift for leopold infeld , pergamon press , new york ( 1962)_. f. w. hehl , p. von der heyde , g. d. kerlick and j. m. nester , `` general relativity with spin and torsion : foundations and prospects , '' rev . * 48 * , 393 ( 1976 ) . m. blagojevic , `` gravitation and gauge symmetries , '' _ bristol , uk : iop ( 2002 ) 522 p_. m. blagojevic and m. vasilic , `` asymptotic symmetries in 3d gravity with torsion , '' phys . rev . d * 67 * , 084032 ( 2003 ) [ arxiv : gr - 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we have systematically computed the generators of the symmetries arising in poincar gauge theory formulation of gravity , both in 2 + 1 and 3 + 1 dimensions .
this was done using a completely lagrangian approach .
the results are expected to be valid in any dimensions , as seen through lifting the results of the 2 + 1 dimensional example into the 3 + 1 dimensional one .
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dry granular systems are known to exhibit both liquid and solid - like properties . while there are many instances where a granular assembly exhibits liquid - like properties @xcite , examples of granular systems illustrating solid - like properties are rare with one exception : the typical conical sandpile shape @xcite . to craft more complex shapes , e.g. , sand art , it is necessary to add additional constraints . these constraints can be added in the bulk or at the boundary . adding a small quantity of water to the sand introduces constraints in the bulk in the form of capillary bridges @xcite . conversely , encapsulating dry grains in a container is an example of constraints that are applied at the boundary @xcite . in general , encapsulation requires isolating materials from their surroundings and it is achieved either by ( i ) introducing another material in the interfacial region or ( ii ) facilitating processes at the interface that create self - encapsulation . examples of the former range from the simple instance of a bag or a silo containing cereal grains to more complex examples of thin polymer films wrapping liquid droplets @xcite , texturing of liquid droplets ( ` liquid marbles ' ) by attaching hydrophobic powder to its surface @xcite and stabilizing emulsions by particles or surfactants @xcite . in contrast , a liquid that oxidizes on contact with atmosphere to develop a stress bearing skin is an example of self - encapsulation @xcite . in this paper we study the self - encapsulation of a granular system consisting of hydrophobic sand @xcite under water . this system self - generates a _ skin _ which encapsulates dry hydrophobic sand - grains and stabilizes the trapped air ( bubble ) against the force of buoyancy . removal of trapped air breaks down this encapsulation . the paper also explores the mechanical properties of this system at multiple scales : from the pinning of the three phase contact line at the roughness scale of the particle , plastic flow at the grain - scale , to sample - spanning mechanical responses . it may be noted here that while hydrophilic granular systems both dry and wet are widely studied @xcite , hydrophobic sand grains submerged in a non - wetting liquid like water remain largely unexplored @xcite even though such systems are of practical importance , especially in pharmaceutical , food and petroleum industries where newer encapsulation strategies are in great demand @xcite . mechanical response of a system is usually described in a small neighborhood of a reference state . for a dry granular assembly a ` state ' of a system is described in terms of the center of mass of the grains and the forces among them . for values of strain smaller than @xmath0 , dry granular assembles show elastic response , i.e. , the reference state can be restored by setting the applied forces to zero ( see p 92 of @xcite ) . this elastic response comes from the reversible deformation of the region of contact between the grains . however , to describe the state of the hydrophobic sand immersed in water the information regarding the center of mass of the grains is not sufficient . we need to augment it with additional information about the detailed layout of the three phase contact lines and the local contact angles of the _ skin_. the three phase contact lines formed at the grain - water - air interface are immobilized ( pinned ) by defects present in the system @xcite . any additional deformation of the _ skin _ causes the contact angles to change from their reference values . this generates restoring forces in the system , i.e. , if the contact angle changes from @xmath1 to @xmath2 the restoring force per unit length is @xmath3 , here @xmath4 is the surface energy at the water - air interface . the pinning and depinning of these contact lines determine the mechanical properties of the material and its mode of failure . hence , the present study is an example of the more general problem of studying statics and dynamics of systems with elastic interfaces in a random environment @xcite . as hydrophobic granular material , we use polyhedral shaped ` magic sand ' grains sourced from education innovation inc . they are made by coating polyhedral shaped sand grains with a hydrophobic material . alternatively , these hydrophobic particles can be made in the laboratory by coating similar sand particles with fluoropel pfc from cytonix llc silica @xcite . when grains of magic sand are freely poured into water , they spontaneously form a cohesive lump : a typical example of which is shown in fig.[fig : intro](a ) . the immediately evident features are : ( i ) the system retains its shape and ( ii ) the outer surface of the lump has a luster that originates from the total internal reflections of light at the pinned water - air interface @xcite . on degassing of the system achieved by creating a partial vacuum over the liquid ( fig.[fig : intro](b ) , the lump begins to lose its lustre ( fig.[fig : intro](c ) and when this luster - free system is mechanically perturbed , it slumps to form a flat sand - bed inside water ( fig.[fig : intro](d ) . a cross - sectional top - view of the granular lump in fig.[fig : intro](a ) is schematically shown in fig.[fig : intro](e ) . the grains ( red coloured with dots ) on the boundary together with the pinned water - interface constitute an encapsulating granular _ skin _ which provides the submerged lump its structural integrity . the force of buoyancy of the trapped air and the weight of dry sand grains in the interior ( colored with green ) exerts stress on the _ skin _ causing it to be under tension . obtained for a water - drop ( @xmath5 ) placed on a single hydrophobic grain surface . ( b ) advancing and receding contact angles are measured to be @xmath6 and @xmath7 respectively , for the same water droplet in a tilted configuration . ( c ) existence of isolated air bubbles on the topographic grain surface viewed at a larger magnification . ] the scanning electron micrographs of the grain surface are shown in fig.[fig : intro](f ) and ( g ) , for two different values of magnification . the micron - sized rough patches ( see fig.[fig : intro](g ) which come from the hydrophobic coating on the particle surface are crucial for the observed wetting properties of these sand grains @xcite . the equilibrium contact angle is measured to be about @xmath8 for a @xmath9 water drop placed on a single hydrophobic grain ( see fig.[fig : contact_angle](a ) . for the same droplet in a tilted configuration , the advancing and receding contact angles are found to be @xmath10 and @xmath11 respectively , i.e. , the contact angle hysteresis is @xmath12 ; see fig.[fig : contact_angle](b ) . the higher magnification image of water - grain interface in fig.[fig : contact_angle](c ) shows presence of small isolated air bubbles trapped between the grain and the water which have similar equilibrium contact angles @xmath13 , i.e. , the surrounding water is in partial contact with the hydrophobic sand grains which creates a wenzel type wetting scenario @xcite . washing these grains in acetone removes the hydrophobic coating and exposes the underlying smooth hydrophilic surface ( fig.[fig : intro](h ) . in order to obtain a more comprehensive understanding of the mechanical properties of the system , we choose a relatively simple geometry which is a free - standing submerged cylindrical column built with magic - sand , using the following experimental steps . at first , by selective sieving we obtain grains of different mean sizes @xmath14 , ranging from @xmath15 to @xmath16 . second , an acrylic tube of inner diameter @xmath17 is made to rest vertically on the flat base of an empty rectangular glass container . the grains are thereafter gently poured into the tube upto a desired height @xmath18 . this forms a _ wall - supported _ granular column shown in the left panel of fig.[fig : fig_modulation](b ) . the typical packing fraction @xmath19 of the grain assembly is about @xmath20 . third , and in a key step , the acrylic tube is gradually pulled out while simultaneously filling the glass container with water . consequently , the hydrostatic pressure partially compensates for the over - pressure due to weight of the grains . this unique protocol prevents grains from clogging particles which are in constant physical contact with the wall of the acrylic tube are immobilized by the frictional interactions . as the tube is withdrawn , it drags along particles that are in contact with it this exposes the next layer of particles to water , on which the _ skin _ forms . the resulting particle rearrangements reduce the diameter by @xmath21 and increase the height of the column which is now partially submerged in water by @xmath22 . the radial contraction happens within the first few seconds and does not evolve appreciably over time . however , the axial strain @xmath23 keeps increasing in time ; see fig.[fig : fig_modulation](c ) . as the tube is pulled out , the number of sand grains which are in contact with it decreases . this reduces the extent of particle rearrangements . as a result , the rate of change of @xmath22 decreases with time . the fully submerged column has a length @xmath24 which is about @xmath25 larger than the initial length @xmath18 of the wall - supported column . hence , the volume fraction of the submerged column is about @xmath26 lower than its value for the wall - supported case . this free - standing submerged column is shown in the middle panel of fig.[fig : fig_modulation](b ) . the column is anchored via capillary forces with the hydrophobic substrate at the bottom and is slightly flared close to the anchoring region . under an identical protocol , hydrophilic beach sand instead slumps to form a canonical sandpile @xcite . the submerged column can be thought of as a thin cylindrical shell ( made from the _ skin _ ) containing non cohesive dry grains in its interior . the _ skin _ is about a particle diameter thick and it comprises of the sand grains on the outer surface of the column and the pinned three phase contact line . using a du - noy ring based technique @xcite we measure the interfacial energy of the _ skin _ to be about . at a microscopic scale this interfacial energy arises from the pinning of the three phase contact line to the defects present on the surface of the grains ( see fig . [ fig : contact_angle](c ) @xcite . the cohesive coupling stiffness @xmath27 due to this pinning depends on the size of the defect @xmath28 and the typical separation @xmath29 between them . in the limit of strong but sparsely distributed defects , @xmath30 where @xmath31 is the surface energy at the water - air interface @xcite . experimentally we observe that polyhedral grains with smooth coverage of hydrophobic coating consistently fail to sustain submerged columns . this suggests that of the multiple scales present in the system , the micron - sized hydrophobic patches present on the surface of the grains are most effective in pinning the contact line . the process of making the column introduces localized geometric imperfections which evolve slowly in time . this is captured by the contour plots of the variation @xmath32 of the column diameter from its mean value along its length over four orders of magnitude in waiting time @xmath33 ( fig.[fig : fig_modulation](d ) . in fig.[fig : fig_modulation](d ) the blue regions correspond to regions with positive , @xmath34 , modulation and the orange regions correspond to regions with negative , @xmath34 , modulation . the maximum @xmath32 is found to be around @xmath35 , nearly half - a - particle size @xmath36 ; whereas the width of the modulation in @xmath17 along @xmath24 is much larger , approximately 40 particle - lengths . the slow temporal evolution of the column s deformation is indicative of creep in the presence of strong inter - grain friction , brought about by the competition between the stresses exerted on the _ skin _ by the dry grains in the interior @xcite and the interfacial energy density of the _ skin_. henceforth , we refer to it as the wrinkling of the _ skin_. in this section we make an order of magnitude estimation of an effective elastic constant for the column . to do so , we apply increasing body forces to the column by gradually tilting the vertical water - filled glass container at a fixed rate ; for experimental details see fig.[fig : fig_modulation](a ) . fig.[fig : fig_curvature](a ) displays five representative images of the column for increasing values of the tilt angle @xmath37 made by the base of the container with respect to the horizontal . images corresponding to @xmath38 and @xmath39 capture the process of the column breaking which resembles a mode - i type transverse rupture . as the column breaks from a finite column height @xmath40 , its upper - part traces out a distinct arc , @xmath41 ( the inset of fig.[fig : fig_curvature](b ) . here @xmath42 is the failure angle made by the axis of the falling part with a line whose slope is @xmath43 and @xmath44 is the length of the falling part of the column . the column for @xmath45 is about @xmath46 longer than its length for @xmath47 . the variation of failure angle @xmath42 with @xmath37 changes slope at @xmath48 in a semi - log plot ( see fig.[fig : fig_curvature](b ) . the large slope beyond @xmath49 indicates a rapid toppling of the upper part of the column . we use @xmath49 to mark the onset of mechanical failure . the broken part of the column sinks as a single object . one can not build an arbitrarily long and/or thin column . if the gravitational energy @xmath50 of the column exceeds its bending energy @xmath51 , the column will break , here @xmath52 is the effective young s modulus of the column , @xmath53 is its cross sectional area , @xmath54 is moment of inertia of the column and @xmath55 and @xmath56 are the densities of sand and water , respectively . to estimate the maximum height @xmath57 of a column we equate the gravitational energy to the bending energy to obtain , @xmath58 at this maximum height the column breaks in its upright position , i.e. , @xmath59 ; this allows us to construct an alternative estimate of @xmath57 from the experimentally observed linear variation @xmath60 where @xmath61 and @xmath62 are fitting parameters ; see fig.[fig : fig_scaling_b](a ) . for @xmath63 , we obtain @xmath64 for @xmath65 , @xmath66 . this estimate of @xmath67 is consistent with experimental observations that stable @xmath68 diameter columns could only be built to the height of @xmath69 . columns built greater than this height are unstable and fail in their upright position . equating the above two expressions for @xmath70 we obtain @xmath71 for the submerged column . this is about two orders of magnitude smaller than that observed for wet sand containing @xmath72 water by volume @xcite . [ [ heuristic - arguments - for - calculating - h_c ] ] heuristic arguments for calculating @xmath40 : + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + the falling column is visually similar to the well studied problem of a `` falling chimney '' that breaks mid - air due to tilting @xcite . for the case of a falling chimney the tensile stress at the leading - edge and the compressive stress at the trailing - edge at small tilt angles is maximum at @xmath73 ( for tilt angles greater than @xmath74 it is about @xmath75 ) , and is usually the point from where the chimney fails @xcite . though , one expects a similar spatial variation of the stress for the submerged column , when @xmath76 , we find the following ( i ) the height @xmath77 at which the column breaks varies quadratically with the column length @xmath78 ; see fig.[fig : fig_scaling_b](b ) and ( ii ) @xmath40 decreases with increasing column diameter ; see the inset of fig.[fig : fig_scaling_b](b ) . from a dimensional argument , for values of @xmath24 smaller than @xmath67 , we propose that the height @xmath77 is related to the column length @xmath79 by the following relation @xmath80 where @xmath81 is the ratio of the gravitational and the bending energies @xcite . since @xmath82 , @xmath83 must vary as @xmath84 , i.e. , @xmath85 a good agreement of this expression with a large set of experimental data , obtained for different lengths and diameters of the submerged column and for various grain sizes , are shown in fig.[fig : fig_scaling_b](c ) . here , we have taken the effective young s modulus of the column @xmath86 to be a constant , independent of the grain scale , which implies an approximate validity of a continuum mechanical description of the system in this range of deformation . + the functional dependence of the critical buckling stress on the geometrical parameters for a cylindrical column is different from that of a cylindrical shell . for the column the critical stress is proportional to @xmath87 @xcite whereas for a shell it is proportional to @xmath88 @xcite . here @xmath89 is the thickness of the shell . in the present experiments @xmath90 and @xmath91 , where the shell thickness @xmath89 is of the order of particle diameter @xmath92 . thus , there are two critical stress values ; one that corresponds to the buckling of the shell ( _ skin _ ) and the other that corresponds to the buckling of the column . since @xmath93 , for increasing values of stress , the buckling of the shell which causes the _ skin _ to wrinkle , precedes the failure of the column ( assuming that the elastic constant of the entire column is greater than or equal to the elastic constant of the _ skin _ alone ) . the _ skin _ is an integral part of the column and its wrinkling influences the buckling of the column in the following way . the wrinkles on the _ skin _ generates geometric imperfections of the column . these imperfections ( i ) lowers the critical stress at which the system fails @xcite and ( ii ) influence the location from where the system fails . to study the role played by the wrinkles ( geometric imperfections of the column ) in determining the height @xmath40 from which the column fails we track the curvature @xmath94 of the column along its length for the leading edge till the column breaks up into two pieces . to calculate @xmath94 , the boundaries of the leading and the trailing edges are passed through a low pass filter that suppresses features smaller than a single grain size @xcite . these edges are detected using an edge detection algorithm based on @xcite . fig.[fig : fig_curvature_b ] shows the curvature data for the advancing edge . the contour plot of the curvature clearly shows the existence of bands of high curvature along the length of the column . these bands correspond to the wrinkling of the _ skin _ ( fig.[fig : fig_modulation](d ) . as the column tilts the lowest wrinkle develops into the most prominent imperfection of the column and it acts as a seed from where the failure of the column is initiated . the critical height @xmath40 from where the column breaks varies quadratically with the length @xmath24 of the column . since the location of the lowest wrinkle coincides with @xmath77 , it is natural to expect the wavelength associated with the wrinkles to have the same @xmath24 dependence as @xmath40 . however , within the framework of shell buckling , the wavelength of wrinkles is independent of the length of the column @xcite . so , a model based solely on buckling produced by static loading is inadequate to describe the experimental scenario . we speculate that the observed length dependence of @xmath40 is related to the perturbations imparted to the column by the mechanical noise associated with the process of tilting this gives the loading a certain dynamic character which could in principle facilitate coupling of the various global length - dependent buckling modes to the wrinkling induced localized imperfections of the column @xcite . this conjecture needs to be examined in future in greater detail . the inset image in fig.[fig : fig_scaling_c ] displays an instantaneous distance @xmath95 at the apex of the _ visually identified wedge _ between the falling top and the anchored bottom parts of the column . this wedge opens up from the trailing ( left ) edge and progresses towards the advancing ( right ) edge of the column . the variation of this distance @xmath95 with time @xmath96 is plotted in fig.[fig : fig_scaling_c ] , here the time @xmath97 is measured onwards from the first observable instance of the wedge opening at the left edge between the falling top and the anchored bottom . as can be seen , the wedge advances with an average velocity of @xmath98 . neglecting the viscous drag , the time required for a non - anchored column to topple can be obtained by equating the toppling torque acting through the center of mass of the falling part of the column and the rate of change of the angular momentum . this ` shortest ' falling time is about a second which is the same for the wedge to move across the sample . accounting for the viscous drag will increase the estimate of the falling time . unlike conventional solids which break by developing cavities in the bulk , the interior of the submerged column is made of dry grains that can not support an open gap . the gap can be sustained only if water drains in and generates a confining _ skin _ which prevents individual particles from falling into it . hence , the speed at which the wedge opens sets a lower limit on the tearing speed of the _ skin_. the inset of fig.[fig : fig_scaling_c ] shows that the particle size influences the speed @xmath99 at which the apex of the visually identified wedge between the falling top and the anchored bottom moves suggesting that the wedge opening is influenced by tearing of the _ skin_. figure [ fig : fig_curvature ] ( a ) shows that tilting of the column alters the shape of the column . to study the elastic and plastic response of the column , we utilize an incremental stress - cycling protocol ; which is the mechanical equivalent of minor hysteresis loop in magnets @xcite . here , the column is recursively stressed such that the minor loops have three branches : ( i ) an increase@xmath100 branch where the tilt angle increases from 0 to @xmath101 at a fixed rate ( ii ) a clamp@xmath100 branch where the tilt angle is held constant at @xmath101 for a fixed period of clamping time and ( iii ) a decrease@xmath100 branch which corresponds to decreasing the tilt angle from @xmath101 to 0 , with the same rate as in ( i ) . they are accessed sequentially . for each successive cycle , the maximum tilt angle @xmath102 increases linearly with @xmath103 , where @xmath104 is the numerical index for the cycle . an instantaneous configuration of the submerged column is referred to as @xmath105 , where @xmath106 is the time spent by the system in the clamped state . fig.[fig : fig_hist_loops]a shows the cyclic variation in the relative displacement @xmath107 as a function of the tilt angle @xmath108 for selective values of @xmath109 1 , 7 , 11 and 13 respectively ; here @xmath110 and @xmath111 is the distance of the mid - point of the top of the column from the point @xmath112 on the @xmath113axis ( the inset of fig.[fig : fig_curvature](b ) . for a first few cycles ( @xmath114 ) , i.e. , for smaller values of @xmath115 , the _ skin _ deforms in a reversible manner . we limit our statement of the reversibility : it is entirely possible that while the _ skin _ may behave in a reversible manner , the dry grains in the interior of the column may not . for larger values of @xmath115 ( @xmath116 ) , the column shows noticeable irreversibility and hence , the hysteresis increases . the parameter @xmath117 is a measure of the strain developed in each cycle while the parameter @xmath118 determines the loop area of each stress cycle ; which is a measure of the accumulated hysteresis and dissipation in the system , here @xmath119 . the top and bottom panels of fig.[fig : fig_hist_loops]b show variation of these two parameters as a function of the maximum tilt angle @xmath120 , respectively . the following observations can be made from fig.[fig : fig_hist_loops ] ( a ) and ( b ) : ( i ) the elastic regime of the column extends for @xmath121 beyond which the area of the hysteresis loop @xmath118 begins to increase abruptly . this elastic regime ( non - shaded region of fig . [ fig : fig_hist_loops](b ) can accommodate strains of the order of @xmath122 which is much larger than that observed for dry granular systems . ( ii ) in the elastic regime , the recursive stress cycling shows smooth variation in @xmath123 with @xmath37 . however , beyond the elastic regime ( @xmath116 or @xmath124 , the shaded region of fig . [ fig : fig_hist_loops](b ) this variation becomes increasing jagged and clearly shows discrete ` jumps ' interspersed with smoothly varying sections . these jumps are signatures of mechanical instabilities associated with contact line slippages . they are related to the stress ( tilt ) induced reduction of the energy barriers associated with the underlying pinning potential which allows the ambient noise ( fluctuations ) to induce creep like motion of the contact line . anomalies of elastic constants associated with the break down of linear elastic response generates similar jumps in the flow curves of disordered materials @xcite . ( iii ) the maximum size of the jump height seen in the quantity @xmath123 is about @xmath125 , i.e. , the contact line on an average moves by a particle scale . further motion of the contact line is possible only by jumping to the next grain which is restricted by the presence of sharp edges at the grain corners @xcite . ( iv ) the increase in @xmath118 is caused by the cumulative effects of these jumps . these jumps progressively generate local overhangs along the direction of the tilt . this results in the column breaking forward unlike a rigid `` falling chimney '' which breaks backward @xcite . the effects of stress assisted creep are best seen in the deformation of the column for the various clamp @xmath126 branches ( here @xmath37 is held constant at @xmath120 , see fig.[fig : fig_hist_loops](c ) . the creep increases with increasing tilt angles ( forces ) , e.g. , see the @xmath127 branch in fig.[fig : fig_hist_loops](c ) . even during creep the column evolves in a similarly punctuated stick - slip manner with jumps in the quantity @xmath128 being limited by the value @xmath129 . as the occurrences of creep increase , the survival@xmath130time @xmath131 of a stationary stick phase decreases ; see the inset of fig.[fig : fig_hist_loops](c ) . the jumps themselves are abrupt in the experimental time scale : they occur over a period less than , an order of magnitude shorter than the measured shortest survival time ( ) . the breaking of the column at the macroscopic scale ( see fig.[fig : fig_scaling_c ] ) is visible only for values of @xmath132 greater than . this region shows a rapid growth of @xmath123 , its initiation is marked by an arrow in fig.[fig : fig_hist_loops](c ) . these results imply that the _ skin _ , comprising of the air - water - grain interface on the surface layer of grains , is primarily responsible for the mechanical response of the entire system and any modification of the _ skin _ will modify the behaviour of the system . one striking modification is shown in fig.[fig : fig_vcaccum_column ] where we create a partial vacuum over the water surface , so that the trapped air within the column , one key component of the _ skin _ , escapes out from the system as bubbles . the time - lapsed images in fig.[fig : fig_vcaccum_column ] ( a ) @xmath133 ( e ) demonstrate that here too the failure is initiated by bending at a finite critical height . but , in contrast to the single tear shown above , the column shows large - scale plastic deformations . its structural integrity is extended for values of @xmath42 greater than @xmath134 by forming a deformed neck , i.e. , a narrowing of the column s diameter , instead of forming a tear or a crack as above . here , the failure proceeds via this necking instability whose spatial location is marked by an arrow in fig.[fig : fig_vcaccum_column](e ) . such large - scale ( many - grains wide , in this case ) plastic flow and necking instabilities are typical of ductile mode of failure in solids @xcite , in complete contrast with the brittle failure described above . this change in failure - mode is brought about by the loss of entrapped air causing weaker cohesion in the column that results in conformational changes of a softer _ skin _ and finally leads to the ductile failure of the column . for completeness , we note that a third , and limiting , mode of failure occurs if degassing continues for a long time . in the absence of air , the _ skin _ begins to lose its strength ( see the lustre - less lump in fig.[fig : intro](c ) . the weak _ skin _ eventually fails to sustain the inward hydrostatic pressure and slumps under the slightest mechanical disturbance . the water quickly drains in as the structure disintegrates into a heap of particles , seen in both fig.[fig : intro](d ) and fig.[fig : fig_vcaccum_column](f ) ; the hydrophobic sand then behaves like its hydrophilic counterpart ( right panel of fig.[fig : fig_modulation](b ) . the removal of trapped air from the column demonstrates the existence of an underlying dynamical transition via which the failure - mode transforms from being brittle to being ductile , both of which are collective ( multi - particles ) in nature , and finally to a total disintegration of the _ skin _ that represents failure at the single - particle level . this is analogous to dynamical transitions from collective response to single - particle response in a wide class of systems @xcite . this paper provides a detailed study of mechanical properties of underwater granular structures made of hydrophobic sand , where the self - generated cohesive _ skin _ on the boundary of the structure encapsulates the dry grains inside it from the surrounding medium water . in our experimental results , three distinct length scales of the system are found . ( i ) the scale of the hydrophobic patches @xmath135 influences the strength of the pinning of air - water interface on the grain surfaces . depinning of this interface produces macroscopically observable plastic deformations of the contact lines around ( ii ) the grain - scale @xmath136 of the sand particles . ( iii ) at the system scale @xmath137 , the compressive traction forces due to the column s own weight drives the wrinkling of the _ skin_. the regions of large curvature of the structure are the seeds from where the system - sized failure modes nucleate . by partially removing the trapped air from this structure , we also see that the collective failure can be tuned from brittle to ductile . a more complete removal of air causes the _ skin _ to crumble completely into individual non - cohesive grains inside water . these experimental findings imply the existence of a tunable dynamical transition between a collective and an individual ( single - grain ) mode of failure in this system . we expect that these new results will help in engineering the granular encapsulation with desired material - properties in a variety of applications and , at the same time , it provides a deeper insight of the multi - scale mechanics , generic to granular materials . 49 ifxundefined [ 1 ] ifx#1 ifnum [ 1 ] # 1firstoftwo secondoftwo ifx [ 1 ] # 1firstoftwo secondoftwo `` `` # 1 '' '' @noop [ 0]secondoftwo sanitize@url [ 0 ] + 12$12 & 12#1212_12%12 @startlink[1 ] @endlink[0 ] @bib@innerbibempty @noop * * , ( ) link:\doibase 10.1103/physrevlett.99.188001 [ * * , ( ) ] link:\doibase 10.1146/annurev.fluid.40.111406.102142 [ * * , ( ) ] link:\doibase 10.1038/nphys834 [ * * , ( ) ] @noop * * , ( ) link:\doibase 10.1039/c2sm25883h [ * * , ( ) ] link:\doibase 10.1080/00018730600626065 [ * * , ( ) ] link:\doibase 10.1002/nag.476 [ * * , ( ) ] link:\doibase 10.1080/00018730500167855 [ * * , ( ) ] http://link.aps.org/doi/10.1103/physrevlett.114.018001 [ * * , ( ) ] @noop * * , ( ) @noop * * , ( ) @noop * * , ( ) link:\doibase 10.1021/la0103822 [ * * , ( ) ] link:\doibase 10.1126/science.1074868 [ * * , ( ) ] @noop * * , ( ) in @noop _ _ , vol . ( , ) pp . @noop * * , ( ) @noop * * , ( ) @noop * * , ( ) @noop * * , ( ) in \doibase http://dx.doi.org/10.1016/s0167-3785(07)80042-x[__ ] , , vol . , ( , ) pp . @noop _ _ ( , ) @noop * * , ( ) @noop * * , ( ) @noop * * , ( ) @noop _ _ , vol . ( , ) http://link.aps.org/doi/10.1103/physreve.86.031402 [ * * , ( ) ] link:\doibase 10.1364/josa.45.000572 [ * * , ( ) ] link:\doibase 10.1364/josa.69.001205 [ * * , ( ) ] @noop * * , ( ) @noop * * , ( ) link:\doibase 10.1103/revmodphys.57.827 [ * * , ( ) ] @noop * * , ( ) @noop * * , ( ) @noop * * , ( ) @noop * * , ( ) @noop _ _ ( , ) @noop * * , ( ) link:\doibase 10.1063/1.4812809 [ * * , ( ) ] @noop * * , ( ) @noop * * , ( ) @noop * * , ( ) in @noop _ _ , vol . ( , ) pp . @noop * * , ( ) @noop * * , ( ) @noop * * , ( ) @noop * * , ( ) link:\doibase 10.1103/physrevlett.105.154301 [ * * , ( ) ]
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magic sand , a hydrophobic toy granular material , is widely used in popular science instructions because of its non - intuitive mechanical properties .
a detailed study of the failure of an underwater column of magic sand shows that these properties can be traced to a single phenomenon : the system self - generates a cohesive _ skin _ that encapsulates the material inside . the _ skin _ , consists of pinned air - water - grain interfaces , shows multi - scale mechanical properties : they range from contact - line dynamics in the intra - grain roughness scale , plastic flow at the grain scale , all the way to the sample - scale mechanical responses . with decreasing rigidity of the _ skin _ , the failure mode transforms from brittle to ductile ( both of which are collective in nature ) to a complete disintegration at the single grain scale .
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the model of a crystal of point - like charges immersed in a uniform neutralizing background of opposite charge was conceived by wigner @xcite to describe a possible crystallization of electrons . these wigner crystals of electrons have much in common with coulomb crystals of ions with the uniform electron background . the model of the coulomb crystal is widely used in different branches of physics , including theory of plasma oscillations ( e.g. , @xcite ) , solid state physics , and works on dusty plasmas and ion plasmas in penning traps ( e.g. , @xcite ) . moreover , coulomb crystals of ions , immersed in an almost uniform electron background , are formed in the cores of white dwarfs and envelopes of neutron stars . the properties of such crystals are important for the structure and evolution of these astrophysical objects ( e.g. , @xcite ) . in particular , the coulomb crystal heat capacity @xcite controls cooling of old white dwarfs and is used to determine their ages ( e.g. , @xcite ) . crystallization of white dwarfs can influence their pulsation frequencies . it can thus be studied by powerful methods of asteroseismology @xcite . microscopic properties of coulomb crystals determine the efficiency of electron - phonon scattering in white dwarfs and neutron stars , and , hence , transport properties of their matter ( such as electron thermal and electric conductivities , and shear viscosity , e.g. , @xcite ) as well as the neutrino emission in the electron - ion bremsstrahlung process @xcite . many - body ion correlations in dense matter produce screening of ion - ion ( nucleus - nucleus ) coulomb interaction and affect nuclear reaction rates in the thermonuclear burning regime with strong plasma screening and in the pycnonuclear burning regime ( when the reacting nuclei penetrate through the coulomb barrier owing to zero - point vibrations in crystalline lattice ) . the description of various nuclear burning regimes and observational manifestations of burning in white dwarfs and neutron stars are discussed in refs . @xcite and references therein . the manifestations include type ia supernova explosions of massive accreting white dwarfs , bursts and superbursts , deep crustal heating of accreted matter in neutron stars . since the late 1990s certain astrophysical applications have been requiring a comprehensive study of coulomb crystals in strong magnetic fields . the topic has become important due to the growing observational evidence that some very intriguing astrophysical objects , soft - gamma repeaters ( sgrs ) and anomalous x - ray pulsars , belong to the same class of sources called magnetars ( see , e.g. , @xcite and reference therein ) . these are thought to be isolated , sufficiently warm neutron stars with extremely strong magnetic fields @xmath0 g. for instance , the magnetic field of sgr 180620 , inferred from measurements of its spin - down rate , is @xmath1 g @xcite . magnetars are observed in all ranges of the electromagnetic spectrum . they show powerful quasipersistent x - ray emission , bursts and giant bursts with enormous energy release . during giant bursts , one often observes quasi - periodic x - ray oscillations which are interpreted ( e.g. , @xcite ) as vibrations of neutron stars ( involving torsion vibrations of crystalline neutron star crust ) . it is likely that the activity of magnetars is powered by their superstrong magnetic fields . thus the magnetars can be viewed as natural laboratories of dense matter in magnetic fields . in order to build adequate models of magnetar envelopes and interpret numerous observations , it is crucial to know the properties of magnetized coulomb crystals . the main goal of this paper is to study in detail coulomb crystals in an external uniform magnetic field . ( the results reported here were partially presented in @xcite . ) the coulomb crystals in question consist of fully ionized ions with charge @xmath2 and mass @xmath3 arranged in a crystal lattice and immersed into the rigid electron background ( in this case , rigid means unpolarizable or incompressible , i.e. constant and uniform ) . the effect of the magnetic field @xmath4 on the ion motion can be characterized by the ratio @xmath5 where @xmath6 are the ion cyclotron frequency and the ion plasma frequency , respectively ; @xmath7 is the ion number density , while @xmath8 is the speed of light . it is expected that the magnetic field modifies the properties of the ion crystal at @xmath9 ( see , however , figs.[b - c ] and [ u1um1 ] below ) . in a strong magnetic field the approximation of rigid electron background is a bigger idealization of the real situation in neutron star crust matter , since higher densities are required to achieve full ionization and suppress the polarizability of the electron background . the higher densities ( and @xmath10 ) imply smaller @xmath11 . however , the effective ion charge approximation turns out to be successful for analyzing partially ionized systems ( e.g. , @xcite ) . the quantity @xmath11 ( also equal to @xmath12 , where @xmath13 is the alfvn velocity , @xmath14 being the mass density ) is , actually , independent of the ion charge . hence , one can consider large @xmath11 in coulomb crystals at not too high densities having in mind the effective ion charge approximation . despite that , the effect of the polarizability of the compensating electron background has to be studied separately . the closely related problem of magnetized wigner crystals of electrons was studied in the early 1980s by usov , grebenschikov and ulinich @xcite and by nagai and fukuyama @xcite . usov et al . @xcite obtained the equations for crystal oscillation modes , studied qualitatively the oscillation spectrum , and diagonalized the hamiltonian of the crystal for a proper quantum description of the oscillations in terms of phonons . in addition , the authors investigated asymptotic temperature and magnetic field dependences of the specific heat , rms electron displacement from the respetive lattice site , and magnetic moment of the crystal . they obtained a soft phonon mode with the frequency @xmath15 near the center of the brillouin zone . it resulted in a rather unusual low - temperature specific heat behavior ( @xmath16 ) instead of the standard debye law ( @xmath17 ) . also , the authors mentioned the dependence of the crystal energy on the magnetic field direction as well as the increased stability of the crystal due to a suppression of electron vibration amplitudes . in addition , usov et al . @xcite considered the dielectric function of the crystal and studied the effect of the electromagnetic field induced by the electron motion . their main results were , however , of semi - qualitative nature , limited to various extreme cases , whereas the present paper focuses mostly on quantitative results , pertaining to coulomb crystals of ions , with an eye to astrophysical implications . nagai and fukuyama @xcite calculated phonon spectra of magnetized body - centered cubic and face - centered cubic ( bcc and fcc ) wigner crystals and compared the energies of these crystals at zero temperature as a function of the magnetic field . the crystal energy was calculated as a sum of the electrostatic ( madelung ) energy , and the zero - point energy of crystal vibrations . the effect of the anharmonic and exchange terms was neglected . the electrostatic energy is independent of the magnetic field ( as long as the field does not alter the lattice structure ) . the vibration energy does depend on the field because the field modifies phonon modes . at zero field the energy minimum is realized by the bcc structure , both with and without zero - point vibrations ( in general , it is not true for polarizable background , e.g. , @xcite ) . the authors showed that for a sufficiently strong magnetic field and at relatively high densities [ @xmath18 , where @xmath19 is the standard density parameter , @xmath20 , and @xmath21 is the bohr radius , @xmath22 being the electron number density ] the full energy is minimized by the fcc structure . it is worth mentioning that the authors did not consider the dependence of the zero - point energy on the magnetic field direction and performed all calculations for a fixed direction ( along one of the high symmetry axes of the crystals ) . however , this choice of the field direction for the bcc lattice was not optimum , as far as the lattice energy was concerned . moreover , the energy gain obtained by choosing the optimum direction is of the same order of magnitude as the difference between the zero - point energies of bcc and fcc lattices . nagai and fukuyama @xcite investigated an analogous structural transition between bcc and hexagonal close - packed ( hcp ) electron wigner lattices . the hcp lattice was found energetically favorable for a sufficiently strong magnetic field at @xmath23 . this result seems more robust since the energy difference between bcc and hcp lattices is several times larger than the energy gain obtained by choosing the optimum direction of the magnetic field in the bcc lattice ( the field direction adopted in @xcite was the same as in @xcite ) . in addition , nagai and fukuyama @xcite described the behavior of all 6 phonon modes of the hcp lattice in the magnetic field . also , they analyzed quantitatively the behavior of transverse and longitudinal electron displacements from the equilibrium positions for bcc and hcp lattices ( at zero temperature ) and concluded ( qualitatively ) that in strong magnetic fields the crystals became significantly more stable in the transverse direction and somewhat more stable in the longitudinal direction . the present paper is organized as follows . section [ eqs ] discusses equations for coulomb crystal oscillations in the magnetic field . section [ spectrum ] focuses on the phonon spectrum of such a crystal with a simple lattice ( i.e. , one ion in the primitive cell ) . the hamiltonian of the system is diagonalized in sec.[diag ] , which allows one to express the ion displacement operator via phonon creation and annihilation operators . section [ thermodyn ] presents numerical calculations of thermodynamic functions of the coulomb bcc crystal for a wide range of densities , temperatures and magnetic fields . in sec . [ crust_ex ] these results are applied to the real physical system found in magnetar crust . section [ rms ] is devoted to an analysis of the rms amplitudes of ion vibrations and to the debye - waller factor of the magnetized coulomb crystal . finally , sec.[moments ] considers the dependence of phonon spectrum moments on the magnitude and direction of the magnetic field . the results will be parameterized by the quantum parameter @xmath24 ( where @xmath25 and @xmath26 are temperature and ion plasma temperature , respectively ) , and by the coulomb coupling parameter @xmath27 . in this case , @xmath28 is the ion sphere radius . the boltzmann constant is set equal to @xmath29 . let us consider a coulomb crystal of ions in a uniform magnetic field @xmath30 . the lagrangian of an ion is @xmath31 where @xmath32 is the ion velocity , @xmath33 is the vector potential , and @xmath34 is the potential energy of the ion ( @xmath35 being the scalar potential ) . choosing the vector potential acting on the @xmath36-th ion as @xmath37^\alpha/2 = \varepsilon^{\alpha \beta \gamma } b^\beta u_i^\gamma/2 $ ] ( where @xmath38 is the @xmath36-th ion displacement ) , the lagrangian of the ion system can be written as : @xmath39 in this case , @xmath40 is the unit vector along the magnetic field , @xmath41 is the total number of ions , and @xmath42 is the field - free lagrangian . introducing the same collective coordinates as at @xmath43 , @xmath44 one can rearrange the lagrangian as @xmath45 in this case , @xmath46 is a wavevector in the first brillouin zone , index l enumerates direct lattice vectors @xmath47 , @xmath48 is the number of ions in the primitive cell ( @xmath49 for simple lattices ) , and index @xmath50 goes from 1 to @xmath51 . summations in eqs . ( [ uofa-2 ] ) and ( [ aofu-2 ] ) are over all wavevectors in the first brillouin zone and over all direct lattice vectors , respectively . in the thermodynamic limit , with @xmath52 being interpreted as generalized coordinates of the system , one obtains the following euler equation : @xmath53 where @xmath54 is the dynamic matrix of the lattice at @xmath43 . the fourier - transform @xmath55 yields the algebraic equation @xmath56 that can be solved if the ion vibration frequency @xmath57 satisfies the dispersion equation below @xcite @xmath58 due to the presence of the third term on the left - hand side , eq.([omegeuler-2 ] ) does not represent the eigennumber problem for a hermitian matrix , and the respective polarization vectors are not orthogonal . this complicates the reduction of the hamiltonian to the sum of hamiltonians of independent oscillators . the proper procedure will be discussed in sec . there are @xmath59 oscillation modes for a given vector @xmath46 in the first brillouin zone . the frequencies of these modes satisfy the generalized kohn s sum rule @xmath60 @xcite . at small @xmath61 the behavior of phonon frequencies is more complex than without the field . it depends substantially on the direction of @xmath46 with respect to @xmath30 . we restrict ourselves to @xmath62 . it is possible to obtain the exact asymptotes of @xmath57 at small @xmath63 ( cf . ref . @xcite ) . in this limit , the dispersion equation ( [ secular-2 ] ) can be written as : @xmath64 where @xmath65~ , \label{bb}\end{aligned}\ ] ] @xmath66 , while the coefficients @xmath67 and @xmath68 are the same as in the field - free dispersion equation @xcite : @xmath69~ , \nonumber \\ 256 \pi^2 f_0 & = & ( ka_{l})^4 [ ( \beta + \gamma_2)^2 + 2 ( \beta + \gamma_2 ) ( \gamma_1 - 3 \gamma_2 ) ( \hat{k}_x^2 \hat{k}_y^2 + \hat{k}_y^2 \hat{k}_z^2 + \hat{k}_z^2 \hat{k}_x^2 ) \nonumber \\ & + & 3 ( \gamma_1 - 3 \gamma_2)^2 \hat{k}_x^2 \hat{k}_y^2 \hat{k}_z^2]~. \label{c0}\end{aligned}\ ] ] the constants @xmath70 , @xmath71 , and @xmath72 characterize the crystal at @xmath43 . they were defined and calculated in ref . @xcite , and were recalculated for bcc and fcc coulomb lattices in ref . they are reproduced in table [ constants ] , along with the lattice constant @xmath73 , for completeness . the subscripts @xmath74 , @xmath75 , and @xmath76 refer to the cartesian coordinate system aligned with the main cube of the respective reciprocal lattice . the asymptote of the smallest frequency @xmath77 at @xmath78 can be derived by dropping the @xmath79-term and choosing the smallest root of the remaining quadratic equation : @xmath80~. \label{ass - o1}\ ] ] at sufficiently small @xmath63 , @xmath81 . then , for the phonons , propagating strictly perpendicular to the magnetic field , @xmath82 , and the phonons are acoustic . if , on the other hand , @xmath83 , then @xmath84 at small @xmath63 , in contrast with the linear dependence at @xmath43 . as the angle between @xmath46 and @xmath30 decreases , the quadratic asymptote of @xmath77 becomes valid for wider range of @xmath63 . neglecting angular dependences and numerical factors in eqs . ( [ bb ] ) and ( [ c0 ] ) , one can estimate the lowest phonon frequency as @xmath85 for propagation perpendicular to the field , and as @xmath86 for @xmath87 . in both cases , at @xmath88 , the mode typical energy is inversely proportional to the field strength . @xmath89 [ constants ] the asymptote of the biggest frequency @xmath90 can be found by dropping the last term in eq . ( [ sec - ass-2 ] ) and choosing the maximum root of the remaining quadratic equation : @xmath91~. \label{ass - o3}\ ] ] at small @xmath63 this yields @xmath92 + o(k^2)~. \label{ass - o3-k0}\end{aligned}\ ] ] in general , at any @xmath46 and at @xmath93 , the biggest frequency @xmath94 . this corresponds to the conventional cyclotron ion motion . consider @xmath95 . then @xmath96 for phonons propagating perpendicular to the field . from the sum rule , it follows that the intermediate frequency @xmath97 becomes 0 at @xmath95 . because @xmath98 , it is clear that @xmath99 for @xmath100 . if , on the other hand , the phonon propagates along the magnetic field , then at @xmath95 one has @xmath101 , and , hence , @xmath102 . therefore both maximum and intermediate frequency modes are optical . as the angle between @xmath46 and @xmath30 decreases from @xmath103 to 0 , the value of @xmath97 at @xmath78 increases from 0 to @xmath104 , whereas the value of @xmath90 at @xmath78 decreases from @xmath105 to @xmath106 . obviously , eq . ( [ sec - ass-2 ] ) can be solved analytically without neglecting any terms . however , at very small @xmath63 analytical schemes become numerically unstable . the thermodynamic properties of coulomb crystals at low temperatures require integration of functions containing the phonon frequencies near the center of the brillouin zone . under these conditions and especially at high magnetic fields the asymptotes given here become helpful . in natural ( a ) and logarithmic ( b ) scales . slowly growing mode @xmath107 at @xmath108 is not shown in panel ( a ) . magnetic field and wavevector @xmath46 orientations are defined in the text ; @xmath109 is the ion sphere radius.,height=275 ] figure [ spectinb ] shows the phonon spectrum of the bcc lattice ( in natural and logarithmic scales ) in the direction of @xmath46 determined by the polar angle @xmath110 and the azimuthal angle @xmath111 [ angles @xmath112 and @xmath35 are defined with respect to the same cartesian reference frame as the @xmath113-subscripts in eqs . ( [ bb ] ) and ( [ c0 ] ) ] . the magnetic field is parallel to the direction from an ion towards one of its nearest neighbors ( @xmath114 , @xmath115 ) . for @xmath116 , analytic asymptotes discussed above are used ; at higher @xmath117 , exact calculations are performed . one can observe the quadratic dependence of the lowest frequency on @xmath63 near the brillouin zone center . the dependence becomes linear closer to the brillouin zone boundary ( at @xmath118 ) . the polarization of the electron background in the absence of the magnetic field converts the optical mode with a frequency @xmath119 into an acoustic mode in the vicinity of the brillouin zone center ( e.g. , @xcite ) . the background polarization effects in a magnetized crystal will be investigated elsewhere . a similar conversion effect can be expected in this case as well , except that one of the modes must remain optical close to the center of the zone , with a frequency @xmath120 . this is so , because the coefficient of the @xmath121-term in the dispersion equation ( [ secular-2 ] ) , that determines the sum of the squares of all frequencies , will continue to contain @xmath122 , whereas in the field - free case at @xmath78 the @xmath121-term tends to zero for polarizable background . in this section the hamiltonian of the coulomb crystal in the magnetic field is represented as a sum of hamiltonians of independent oscillators . the derivation follows that of ref.@xcite , but some additional details are provided . consider the cartesian reference frame with the axes directed along the eigenvectors of the matrix @xmath123 ( again @xmath124 ) . then the lagrangian ( [ lofagen-2 ] ) has the form @xmath125 in this case , @xmath126 is the phonon frequency at @xmath43 . omitting the constant equilibrium electrostatic energy term @xmath127 and turning to the hamiltonian @xmath128 , one has @xmath129 where @xmath130 . in the quantum formalism , @xmath131 and @xmath132 become operators with the usual commutation rules @xmath133 = i \hbar \delta^{\alpha \beta } \delta_{\bm{k}{\bm { k'}}}$ ] . however , the operators @xmath134 and @xmath135 do not commute at @xmath136 : @xmath137 = i \hbar \omega_b \varepsilon^{\beta \gamma \alpha } n^\gamma \delta_{{\bm { k } ' } , -\bm{k}}$ ] . consequently , the creation and annihilation operators , defined in the same way as at @xmath43 , but with @xmath134 in place of @xmath138 , do not satisfy the required commutation relationships . in this situation one constructs the creation and annihilation operators as linear combinations of the operators of generalized coordinates and momenta ( e.g. , ref . @xcite ) , @xmath139 where @xmath140 and @xmath141 are constant coefficients ( and summation over @xmath142 is implied ) . they can be determined from the equation @xmath143 = \hbar \omega_{\bm{k } } \hat{a}^\dagger_{\bm{k}}$ ] ( e.g. , ref . @xcite ) . using eqs . ( [ hb ] ) and ( [ anz - a+ ] ) , one arrives at @xmath144 or @xmath145 so that @xmath146 the system ( [ eq - for - al ] ) can be solved only if the quantities @xmath147 satisfy the dispersion equation ( [ secular-2 ] ) , i.e. , if they are the eigenfrequencies of the oscillation modes with the wavevector @xmath46 . therefore , there are three possible solutions for the operator @xmath148 , corresponding to the three generally different frequencies @xmath149 , @xmath150 : @xmath151 . evidently , the equations for the coefficients @xmath152 and @xmath153 coincide . thus from now on index @xmath46 will be omitted where possible . the solutions of the system ( [ eq - for - al ] ) are sets of the cofactors to any of the rows of the matrix @xcite @xmath154 ( these sets are proportional to each other ) . now one has to normalize these solutions together with the operators @xmath155 , so that @xmath156 = 1 $ ] . accordingly , @xmath157 & = & \alpha_s^{\lambda \ast } \alpha_{s'}^{\lambda ' } i \hbar \omega_b \varepsilon^{\lambda ' \gamma \lambda } n^\gamma \nonumber \\ & + & \alpha_s^{\lambda \ast } \alpha_{s'}^{\lambda } \hbar \omega^2_{\lambda } \left ( \frac{1}{\omega_{s } } + \frac{1}{\omega_{s ' } } \right ) \nonumber \\ & = & \alpha_s^{\lambda \ast } \alpha_{s'}^{\lambda } \hbar \left ( \frac{\omega^2_{\lambda}}{\omega_{s } } + \omega_{s ' } \right)~ , \label{ort - gen}\end{aligned}\ ] ] where eq . ( [ eq - for - al ] ) was employed . now consider the following sequence of equations , @xmath158 \nonumber \\ & = & \alpha_{s'}^{\lambda}\ , \alpha_s^{\lambda \ast } ( \omega^2_{s } - \omega^2_{\lambda } ) + \alpha_{s'}^{\nu}\ , \alpha_s^{\nu \ast } \frac{\omega_{s}}{\omega_{s ' } } \ , ( \omega^2_{\nu } - \omega^2_{s ' } ) \nonumber \\ & = & \alpha_{s'}^{\lambda}\ , \alpha_s^{\lambda \ast } \left(\frac{\omega_{s}}{\omega_{s ' } } -1 \right ) ( \omega^2_{\lambda } + \omega_{s } \omega_{s ' } ) \label{aux-2}\end{aligned}\ ] ] [ where eq . ( [ eq - for - al ] ) was used twice ] . thus the right - hand side of eq . ( [ ort - gen ] ) vanishes , if @xmath159 . this property is analogous to the orthogonality condition for the polarization vectors at @xmath43 . at @xmath160 , eq . ( [ ort - gen ] ) determines the normalization of the coefficients @xmath161 : @xmath162 all the other commutators of the operators @xmath163 and @xmath164 vanish automatically . owing to the equality @xmath165 = \hbar \omega \hat{a}^\dagger$ ] and the normalization relationships obtained above , the hamiltonian ( [ hb ] ) can be rewritten in the canonical form @xmath166 moreover , with the help of eqs . ( [ hb ] ) and ( [ anz - a+ ] ) , it is possible to prove the relationships : @xmath167 multiplying @xmath168 by @xmath169 , and @xmath170 by @xmath171 , summing over @xmath172 , subtracting the second expression from the first one , and also using ( [ o1 ] ) and ( [ o2 ] ) , one obtains @xmath173 the above formula is instrumental in calculations of such crystal properties as the debye - waller factor and the rms ion displacement from a lattice site in the magnetic field . these results are reported in sec . phonon thermodynamic functions in the magnetic field are calculated using the same general formulas ( e.g. , @xcite ) and numerical integration schemes ( e.g. , @xcite ) as in the field - free case . the phonon free energy ( with phonon chemical potential @xmath174 and neglecting zero - point contribution ) reads @xmath175 } \nonumber \\ & = & v \sum_s \int_{\rm bz } \frac{{\rm d } \bm{k}}{(2 \pi)^3 } \ln{\left[1-\exp{\left(-\frac{\hbar \omega_{\bm{k}s}}{t}\right ) } \right]}~ , \label{fph}\end{aligned}\ ] ] where @xmath176 is the volume , and the integral is over the first brillouin zone . the phonon thermal energy @xmath177 and heat capacity @xmath178 are then given by @xmath179 in the magnetic field , there are two new parameters , @xmath180 and the field direction . a study of frequency moments of the phonon spectrum in sec.[moments ] will show that the dependence of these moments on the field direction is rather weak . this allows one to expect that the dependence of thermodynamic functions on the field direction is also weak . thus , in the present section the consideration is restricted to the field direction , which corresponds to the minimum zero - point energy of the crystal . for the bcc lattice , it is the direction from a lattice site to one of its closest neighbors . the calculated phonon heat capacity per one ion is presented in fig . [ b - c ] in logarithmic and linear scales [ panels ( a ) and ( b ) , respectively ] as a function of @xmath181 for several values of @xmath11 . in fig . [ b - c](a ) one can clearly see the change of the low - temperature asymptote from @xmath17 to @xmath16 due to the appearance of the soft mode @xmath15 for non - zero @xmath4 . also , this easily excited mode ( e.g. , at @xmath182 ) is responsible for a relatively high specific heat @xmath183 all the way up to @xmath184 . at these temperatures the field - free specific heat is already down by 6 orders of magnitude . in strong magnetic fields , higher temperatures are required to achieve the classical regime as compared to the field - free case . this is so , because the classical regime occurs when many phonons are excited in _ all _ modes . in a strong magnetic field , there is the high - frequency ( cyclotron ) mode , @xmath94 ( sec.[spectrum ] ) . hence , the classical regime is realized if @xmath185 , in contrast to the conventional criterion @xmath186 at @xmath187 . this is illustrated in fig . [ b - c](b ) . for instance , for @xmath188 the classical value @xmath189 is reached only at @xmath190 . -@xmath4 plane . the point , where all straight lines cross , corresponds to @xmath191.,height=275 ] the energies of three phonon modes are spaced far away from each other in strong magnetic fields . the minimum frequency @xmath192 , the maximum frequency @xmath94 , and the intermediate frequency @xmath193 ( sec . [ spectrum ] ) . this gives rise to the pronounced staircase structure of the heat capacity seen in fig . [ b - c](b ) at @xmath194 and @xmath195 . this discussion is further illustrated by fig . [ tb - diag ] , where one can assess qualitatively the behavior of the specific heat in various domains of the @xmath25-@xmath4 plane . at high temperatures , the crystal is classic and the specific heat reaches its maximum value of 3 . in strong magnetic fields ( @xmath88 ) when the temperature drops below @xmath196 , the cyclotron phonon mode is frozen out and @xmath197 . as the temperature drops further , below @xmath198 , the intermediate mode @xmath97 freezes out in large portions of the brillouin zone , and the specific heat , now mainly due to the fully - excited soft mode , approaches 1 . by contrast , in a non - magnetized ( @xmath118 ) crystal , the two lower modes deviate from being acoustic only in the very vicinity of the brillouin zone center , while the third one is @xmath199 . hence , when @xmath25 drops below @xmath198 in such a crystal , the debye law @xmath200 is recovered . finally , at any @xmath11 , when the temperature is so low , that only the least energetic mode contains any heat and the @xmath15 law is probed , @xmath201 behavior results . ( a ) and phonon entropy per ion ( b ) for the bcc lattice as functions of temperature for several values of the magnetic field . inset in panel ( b ) shows @xmath202 in the linear scale at @xmath182 . the magnetic field is directed towards one of the nearest neighbors.,height=275 ] in fig . [ b - se ] phonon thermal energy and entropy @xmath203 are plotted . in the case of energy , plotted in the linear scale in fig . [ b - se](a ) , one observes again the staircase structure . though less pronounced , it occurs for the same reason as for the specific heat . the classical harmonic oscillator limit @xmath204 is naturally reproduced . at low temperatures , the magnetic field gives rise to the @xmath205 dependence of the energy instead of the @xmath206 field - free asymptote . for sufficiently low temperatures , the thermal energy is dominated by the zero - point energy , determined by the spectrum moment @xmath207 and discussed in sec . [ moments ] . the numerical calculations of the entropy are shown in fig . [ b - se](b ) . in this case , as for the specific heat , the familiar field - free @xmath17 asymptote is replaced by @xmath16 in the quantum magnetized crystal . in the classical regime , the entropy depends on temperature logarithmically and is insensitive to the magnetic field . this asymptote is discussed in some detail in sec . [ moments ] . when drawn in linear scale , the entropy , as a function of @xmath208 , also shows an atypical structure at @xmath88 [ inset in fig . [ b - se](b ) ] . in this case , the dependence becomes piece - wise linear with several slope changes , corresponding to the sequential excitation of the three phonon modes . clearly , magnetized crystal thermodynamics , constructed in this section , can not be described by the well - known in solid state physics debye model ( e.g. , @xcite ) . on the other hand , at @xmath88 , partial thermodynamic quantities due to the cyclotron mode @xmath90 , can be represented with high accuracy by the simple einstein model ( e.g. , @xcite ) . as a practical application of the previous section results , consider fully ionized iron plasma in neutron star crust ( sec . [ introduct ] ) . typical values of @xmath112 and @xmath11 at @xmath209 and @xmath210 g can be estimated from fig . [ fe - phys ] . the curve @xmath211 shows the melting line of the classical coulomb crystal at @xmath212 . the line @xmath213 represents typical electrostatic energy per ion ( and separates the regions of a free ion gas above the line , and a strongly coupled ion system below the line ) . the electron degeneracy temperature is shown by dotted lines marked @xmath214 , @xmath215 , and @xmath216 for three values of the magnetic field , @xmath187 , @xmath217 , and @xmath210 g , respectively . finally , @xmath218 and @xmath219 are the electron ground - state energies of one - electron iron ion for @xmath209 or @xmath210 g , respectively . these energies are calculated using rescaling of equivalent hydrogen energies @xmath220 ( e.g. , @xcite ) , and the fitting formula @xcite for the energy of the @xmath221 state in the hydrogen atom . the assumption of full ionization in the degenerate plasma can be used if @xmath222 ( for a given magnetic field ) . at @xmath223 , the electron gas is nearly incompressible , and the rigid electron background model is very well justified . at @xmath224 the plasma can not be treated as fully ionized , but even in this case one can use present results for estimates by employing the effective ion charge approximation . fe matter . @xmath211 is the classical crystal melting line . @xmath225 and @xmath226 mark electron degeneracy temperature and electron - ion binding energy ( see text for details ) . subscripts 0 , 15 , and 16 refer to the magnetic field values @xmath187 , @xmath217 , and @xmath210 g , respectively . note , that @xmath227 k.,height=275 ] figure [ iron_spec_heat ] demonstrates the phonon and electron specific heat ( per ion ) of fully ionized @xmath228fe plasma as a function of density ( a ) and temperature ( b ) at @xmath187 and @xmath229 g. the electron contribution is calculated using standard formulas for strongly degenerate relativistic fermi gas . for @xmath230 g under the displayed conditions the plasma electrons fill the lowest landau level only ; the next landau level would be occupied at @xmath231 g @xmath232 , and the plasma would become partly ionized at @xmath233 g @xmath232 ( cf . [ fe - phys ] ) . the temperature dependence of the electron specific heat is linear , and near coincidence of two electron lines in fig . [ iron_spec_heat](b ) is largely accidental . for the ions to crystallize , the temperature must be below the melting temperature , which is @xmath234 k in the density range considered ( cf . [ fe - phys ] ) . fe matter versus density ( a ) and temperature ( b ) at @xmath187 and @xmath235 g. for the @xmath236 case , @xmath237 in panel ( a ) ; @xmath238 k and @xmath239 in panel ( b).,height=275 ] the phonon contribution at @xmath187 , recalculated here , reproduces the results of @xcite . at @xmath240 g the results of the present work are plotted . as seen from fig . [ iron_spec_heat ] , phonons dominate electrons in the specific heat in a wide range of temperatures and densities . the magnetic field provides an extra boost to the phonon specific heat , especially at lower temperatures ( in the quantum regime ) due to the easily excited soft mode . using eqs . ( [ uofa-2 ] ) and ( [ aofa-2 ] ) , one can derive the expression for the operator of the ion displacement from its equilibrium lattice position : @xmath241 then the rms ion displacement @xmath242 can be calculated as @xmath243 [ cf . ( [ norma ] ) ] , where @xmath244 is the mean number of phonons in a mode @xmath245 . in addition , it is possible to find the rms ion displacement @xmath246 in the direction along an arbitrary unit vector @xmath247 , @xmath248 in the field - free case , @xmath249 is isotropic ( in the bcc crystal ) , but in the magnetic field it becomes anisotropic . -axis ( @xmath250 ) is parallel , @xmath251-axis ( @xmath252 ) is perpendicular to @xmath253 . @xmath250 and @xmath252 are in units of the distance to the nearest neighbor . the solid , dash - dotted , long - dashed , dotted , and short - dashed curves correspond to @xmath254 , and 0.01 , respectively . the distance between the origin and a point on a chosen curve is equal to the rms ion displacement @xmath255 in that direction . , height=340 ] amplitudes of ion oscillations are depicted in fig . the ion equilibrium position is at the origin . the @xmath256-axis is parallel to @xmath30 and to the direction towards one of the nearest neighbors . there is symmetry in the plane azimuthal with respect to @xmath257 ( the plane perpendicular to the @xmath256-axis ) . hence , the @xmath251-axis indicates an arbitrary direction orthogonal to @xmath257 . the distance between the origin and a point on a chosen curve is equal to the rms displacement @xmath249 in the given direction . the distance is expressed in units of the spacing between the nearest neighbors , @xmath258 ( for bcc lattice ) . solid , dash - dotted , long - dashed , dotted , and short - dashed curves correspond to @xmath254 , and 0.01 , respectively . the 6 panels of fig . [ dw ] are for the 6 values of the quantum parameter @xmath112 , from @xmath259 to @xmath260 . the ion displacement at a different @xmath261 ( for given @xmath112 and @xmath11 ) can be found by straightforward scaling per eq . ( [ r2 t ] ) . the quantity @xmath249 is related to the debye - waller factor @xmath262 : @xmath263 . the thermal averaging yields @xmath264}~. \nonumber \ ] ] thus , @xmath265 . as shown in fig . [ dw ] , the ion displacements at @xmath266 are insensitive to the magnetic field ( all 5 lines merge ) and isotropic . at @xmath267 the displacement along the field remains essentially the same as at @xmath187 , whereas the transverse displacements shrink . the net effect will be an increase of the crystal stability and melting temperature . the anisotropy of displacements in a quantum , strongly magnetized crystal , at @xmath268 , @xmath88 , becomes much larger . the ion displacements are strongly suppressed in the transverse direction and are mildly reduced along the field ( see also @xcite ) . these effects are important for accurate computations of nuclear reaction rates in a strongly magnetized crust of a neutron star ( sec . [ introduct ] ) . the strongest effect is expected to occur at sufficiently high densities and not too high temperatures in the pycnonuclear burning regime . in that regime ( e.g. , @xcite ) the main contribution to the reaction rate comes from the closest neighbors in a crystalline lattice due to their zero - point vibrations . in the field - free case , the rate depends exponentially on the squared ratio of the equilibrium distance between the closest neighbors and the average amplitude of their displacements . the ratio is typically large , and the reaction rates are exponentially small ( but increase with the growth of stellar matter density ) . although the reaction rates in magnetized crystals have not been studied yet , one can expect a similar result , involving now the average displacement ( [ u2q ] ) in the direction of closest neighbors . if so , the rates would become extremely sensitive to the orientation of the magnetic field . if the field is not directed towards one of the nearest neighbors , the reactions will be significantly slowed down by greater tunnelling lengths . even if the magnetic field is directed towards one of the nearest neighbors then , in addition to the reduction of the longitudinal displacement , there will be a geometrical effect . in a bcc lattice every ion has 8 nearest neighbors , of which only 2 will be located along the magnetic field line . the reactions with the other 6 neighbors will be quenched . according to the well - known bohr - van leeuwen theorem , the classical partition function is not affected by the magnetic field . for instance , the classical asymptote of the entropy @xmath269 has the form : @xmath270 therefore , the quantity @xmath271 , where average over all phonon modes in the first brillouin zone is implied , _ should not depend _ on the magnetic field . this statement is easy to prove directly . by inspecting eq . ( [ secular-2 ] ) one concludes that its @xmath272-term can not contain @xmath4 at any @xmath46 . on the other hand , this term is equal to @xmath273 , which proves the point . phonon frequency moments @xmath274 depend on both the field strength and direction @xmath40 . the moment @xmath275 diverges , while the moment @xmath276 . consider the moments @xmath207 and @xmath277 . their dependences on the field strength are shown in fig . [ u1um1](a ) for the field direction corresponding to the minimum of the zero - point energy . at strong magnetic fields both moments behave in the same way , proportional to @xmath11 , although under the effect of different modes . the main contribution to @xmath207 comes from the mode @xmath278 , whereas the major contribution to @xmath277 is due to the mode @xmath279 ( near the center of the brillouin zone ) . and @xmath277 on the strength of the magnetic field directed towards one of the nearest neighbors . ( b ) : the dependence of phonon spectrum moment differences @xmath280 and @xmath281 on the magnetic field ( see explanation in the text).,height=275 ] the dependence of the phonon frequency moments on the magnetic field direction turns out to be rather weak . in fig . [ u1um1](b ) the differences @xmath282 and @xmath283 are shown , where @xmath284 is the magnetic field direction towards one of the nearest neighbors , while @xmath285 is the direction towards one of the next order nearest neighbors . the behavior of @xmath280 is of special importance because it determines zero - point energy gain @xmath286 resulting from the crystal rotation with respect to the magnetic field . in weak fields @xmath287 , the difference @xmath280 scales approximately as @xmath288 . in strong fields , @xmath280 saturates at a value @xmath289 , depending on the field direction . the difference @xmath281 scales as @xmath290 for low magnetic fields , and as @xmath11 for strong magnetic fields . the coulomb crystal of ions with incompressible charge compensating background of electrons in constant uniform magnetic field has been studied . the phonon mode spectrum of the crystal with bcc lattice has been calculated for a wide range of magnetic field strengths and orientations ( fig . [ spectinb ] ) . the phonon spectrum has been used in 3d numerical integrations over the first brillouin zone for a detailed quantitative analysis of the phonon contribution to the crystal thermodynamic functions , debye - waller factor of ions , and the rms ion displacements from the lattice nodes for a broad range of densities , temperatures , chemical compositions , and magnetic fields . the characteristic parameter that determines the strength of the magnetic field effects in the crystal is the ratio of the ion cyclotron frequency to the ion plasma frequency , @xmath180 . even a very strong field ( @xmath88 ) does not alter the partition function ( hence , thermodynamic functions , melting etc . ) of a classical crystal , i.e. a crystal at a temperature significantly exceeding the energy of any phonon mode ( @xmath291 for strongly magnetized crystal ) . strong magnetic field dramatically changes various properties of quantum crystals , especially at @xmath292 . low - temperature phonon specific heat , entropy , thermal energy increase by orders of magnitude ( e.g. , figs . [ b - c ] , [ b - se ] ) . the thermodynamic functions exhibit peculiar staircase structures , brought about by the vastly different energy scales of the crystal phonon mode . ion displacements from the equilibrium positions become strongly anisotropic ( fig . [ dw ] ) , and so does the debye - waller factor . these results can be applied directly to real physical systems found in the crust of magnetars ( neutron stars with superstrong magnetic field ) . in particular , the heat capacity of the magnetized coulomb crystal composed of fully ionized @xmath228fe has been analyzed ( sec . [ crust_ex ] ) . it has been shown that in the magnetic field ( as in the field - free case ) the phonon heat capacity dominates the electron one in a wide range of densities and temperatures ( fig . [ iron_spec_heat ] ) . in accordance with sec . [ thermodyn ] , magnetic field is responsible for a significant boost of phonon heat capacity , which may be important for simulations of magnetar cooling . since ion displacements are suppressed , one can expect increased crystal stability in the magnetar crust associated with an increase of the melting temperature in the quantum regime . finally , one expects a reduction of the pycnonuclear reaction rates due to increased tunnelling lengths . the rates are likely to become very sensitive to the orientation of the magnetic field ( sec . [ rms ] ) . the dependence of thermodynamic functions on the magnetic field orientation within the crystal turns out to be weak , at least for the bcc lattice , but finite . the energy gain , achieved by reorienting the crystal with respect to the magnetic field , can be estimated using fig . [ u1um1 ] .
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the body - centered cubic coulomb crystal of ions in the presence of a uniform magnetic field is studied using the rigid electron background approximation .
the phonon mode spectra are calculated for a wide range of magnetic field strengths and for several orientations of the field in the crystal .
the phonon spectra are used to calculate the phonon contribution to the crystal energy , entropy , specific heat , debye - waller factor of ions , and the rms ion displacements from the lattice nodes for a broad range of densities , temperatures , chemical compositions , and magnetic fields .
strong magnetic field dramatically alters the properties of quantum crystals . the phonon specific heat increases by many orders of magnitude .
the ion displacements from their equilibrium positions become strongly anisotropic .
the results can be relevant for dusty plasmas , ion plasmas in penning traps , and especially for the crust of magnetars ( neutron stars with superstrong magnetic fields @xmath0 g ) .
the effect of the magnetic field on ion displacements in a strongly magnetized neutron star crust can suppress the nuclear reaction rates and make them extremely sensitive to the magnetic field direction .
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You are an expert at summarizing long articles. Proceed to summarize the following text:
in the following , we formulate a one - electron microcanonical distribution for triatomic molecules . we denote the positions of the nuclei by @xmath1 , @xmath2 and @xmath3 and the inter - nuclear distances by @xmath4 , @xmath5 and @xmath6 , see fig . one can show that the coordinates of the nucleus c are expressed in terms of the inter - nuclear distances as follows : we denote the position vector of the electron by @xmath8 and the distances of the electron from nuclei a , b and c by @xmath9 , @xmath10 and @xmath11 , respectively . we then define the confocal elliptical coordinates @xmath12 and @xmath13 using the nuclei a and b as the foci of the ellipse , that is , where @xmath15 and @xmath16}$ ] . the third coordinate @xmath17}$ ] is the angle between the projection of the position vector @xmath8 on the xy plane and the positive x axis ; it thus defines the rotation angle around the axis that passes through nuclei a and b. the potential of the electron in the presence of the nuclei a , b and c , which have charges @xmath18 , @xmath19 and @xmath20 , respectively , is given by @xmath21 this potential is expressed in terms of the confocal elliptical coordinates as follows the one - electron microcanonical distribution is given by @xmath23 where @xmath24 is the ionization energy of the one - electron triatomic molecule . note that the energy is given by @xmath25 . the electron momentum in terms of the confocal elliptical coordinates is expressed as follows the @xmath36 distribution goes to zero and is thus well - behaved when the electron is placed on top of either nucleus a or b. however , when @xmath37 , i.e. , the electron is placed on top of nucleus c , @xmath38 . we eliminate this singularity by introducing an additional transformation . setting @xmath39 , @xmath40 and expanding @xmath41 around @xmath42 we find @xmath43 where @xmath44 and @xmath45 are the values of @xmath12 and @xmath13 , respectively , when the electron is placed on top of the nucleus c. to eliminate the singularity in eq . ( [ eq : sing ] ) , we introduce a new variable @xmath46 , such that @xmath47 . the new distribution takes the form since @xmath16}$ ] , @xmath49 and @xmath46 take both negative and positive values and therefore , if we choose one @xmath50 for all values of @xmath13 , @xmath50 must be odd . moreover , to avoid the singularity when the electron is placed on top of nucleus c , @xmath50 must be such that @xmath51 , i.e. , @xmath52 . combining the above two conditions , yields @xmath53 . the new distribution @xmath54 goes to zero when the electron is placed on top of the nucleus c , i.e. , when @xmath55 , @xmath56 and @xmath57 . to set up the initial conditions we find @xmath58 so that @xmath59 and equivalently @xmath60 we then find the maximum value @xmath61 of the distribution @xmath54 , for the allowed values of the parameters @xmath12 , @xmath46 and @xmath62 . we next generate the uniform random numbers @xmath63}$ ] , @xmath64}$ ] , @xmath65}$ ] and @xmath66}$ ] , with @xmath67 and @xmath68 . if @xmath69 then the generated values of @xmath12 , @xmath46 and @xmath62 are accepted as initial conditions , otherwise , they are rejected and the sampling process starts again . following the above described formulation , we obtain the initial conditions of the electron with respect to the origin of the coordinate system . to obtain the initial conditions for the position of the electron with respect to the center of mass of the triatomic molecule , @xmath70 , in terms of the ones with respect to the origin , @xmath8 , we shift the coordinates by @xmath71 , where @xmath72 is given by @xmath73 with @xmath74 with @xmath75 , @xmath76 and @xmath77 the masses of the nuclei . as an example , we next obtain the probability densities of the position and the momentum of the electron that is initially bound in @xmath78 when the molecule is driven by an intense infrared laser field . we assume the other electron tunnel - ionizes in the initial state . we consider the @xmath78 triatomic molecule in its ground state , where the distance of the nuclei in the equilateral triangle arrangement is 1.65 a.u . and the first and second ionization energies are @xmath79 a.u . and @xmath80 a.u . , respectively . we find the ionization potentials and equilibrium distances of the initial state using molpro , which is a quantum chemistry package @xcite . for the microcanonical distribution the relevant ionization energy is @xmath81 , since @xmath82 is associated with the electron that tunnel - ionizes in the initial state . in fig . [ position ] ( b ) we plot the probability density of the position of the electron on the x - z plane for @xmath83 using the above described microcanonical distribution . to compare , in fig . [ position ] ( a ) we plot the quantum mechanical probability density of the position of the electron on the x - z plane . that is , we plot @xmath84 , where @xmath85 is the quantum mechanical wavefunction for the @xmath86 molecule , which we obtain using molpro . the two plots , fig . [ position ] ( a ) and ( b ) , show that the two probability densities for the electron position compare well . however , the microcanonical probability density underestimates the electron probability density between the nuclei and overestimates the electron probability density around the nuclei . [ ht ! ] plane . the middle panel shows the microcanonical probability density of the electron momentum plotted on the @xmath87 plane for all values of @xmath88 . the right panel shows the projections on the @xmath89 axis of the probability densities plotted in fig . [ momentum ] ( a ) and ( b ) . , title="fig:",scaledwidth=50.0% ] in addition , in fig . [ momentum ] ( b ) for all values of the electron momentum component along the y - axis , @xmath88 , we plot the probability density of the electron momentum on the @xmath90 plane using the microcanonical distribution . to compare , in fig . [ momentum ] ( a ) we plot @xmath91 , which we obtain by first computing the quantum mechanical wavefunction in momentum space using the quantum mechanical wavefunction @xmath85 computed from molpro and then by integrating over @xmath88 @xmath93 the two plots , fig . [ momentum ] ( a ) and ( b ) , show that the two probability densities for the electron momentum compare well . however , the microcanonical probability density overestimates the higher values of the electron momentum . this can be seen more clearly in fig . [ momentum ] ( c ) where we plot the probability density of the electron momentum along the @xmath89 axis both quantum mechanically and using our microcanonical distribution . to obtain the plots in fig . [ momentum ] ( c ) we project the probability densities of the electron momentum in fig . [ momentum ] ( a ) and ( b ) on the @xmath89 axis . [ momentum ] ( c ) clearly shows that the probability density of the electron momentum obtained from the microcanonical distribution overestimates the higher values of the momentum component @xmath89 . this is consistent with our previous finding that the microcanonical distribution overestimates values of the electron position around the nuclei resulting to higher values of the momentum . in conclusion , in the current work we have formulated a microcanonical distribution for a general one - electron triatomic molecule . this distribution can be used to describe the initial state of the bound electron in semiclassical models of strongly - driven two - electron triatomic molecules . 10 meckel m , comtois d , zeidler d , staudte a , pavii d , bandulet h c , ppin h , kieffer j c , drner r , villeneuve d m and corkum p b 2008 _ science _ * 320 * 1478 lefebvre c , lu h z , chelkowski s and bandrauk a d 2014 _ phys . rev . a _ * 89 * 023403 liu j , ye d f , chen j and liu x 2007 _ phys . _ * 99 * 013003 emmanouilidou a and staudte a 2009 _ phys . a _ * 80 * 053415 emmanouilidou a , lazarou c , staudte a and eichmann u 2012 phys . rev . a * 85 * 011402 ( r ) hugh h , lazarou c and emmanouilidou a 2014 _ phys . rev . a _ * 90 * 053419 landau l d and lifshitz e m 1977 _ quantum mechanics _ ( pergamon press , new york ) ; delone n b and krainov v p 1991 _ j. opt . b _ * 8 * 1207 murray r , spanner m , patchkovskii s and ivanov m y 2011 _ phys . lett . _ * 106 * 173001 meng l , reinhold c o and olson r e 1989 _ phys . a _ * 40 * , 3637 werner h j. knowles p j , knizia g , manby f , schtz r m , et al . 2012 _ molpro , version 2012.1 , a package of ab initio programs _ , see http://www.molpro.net
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we formulate a microcanonical distribution for an arbitrary one - electron triatomic molecule .
this distribution can be used to describe the initial state in strongly - driven two - electron triatomic molecules .
namely , in many semiclassical models that describe ionization of two - electron molecules driven by intense infrared laser fields in the tunneling regime initially one electron tunnels while the other electron is bound . the microcanonical distribution presented in this work can be used to describe the initial state of this bound electron .
the nonlinear response of multi - center molecules to intense laser fields is a fundamental problem .
for instance , understanding the break - up dynamics of strongly - driven molecules paves the way for controlling and imaging molecular processes @xcite .
semi - classical models are essential in understanding the dynamics during the fragmentation of multi - center molecules driven by intense infrared laser pulses .
one reason is that treating the dynamics of electrons and nuclei at the same time poses an immense challenge for fully ab - initio quantum mechanical calculations .
currently quantum mechanical techniques can only address one electron in triatomic molecules in two - dimensions @xcite .
semi - classical models have provided significant insights , for example , in double ionization of strongly - driven @xmath0 with fixed nuclei " @xcite and in frustrated " double ionization during the fragmentation of strongly - driven @xmath0 @xcite , where one electron eventually stays bound in a highly excited state of the h atom .
the initial state that is commonly employed by semi - classical models for strongly - driven two - electron atoms and molecules , for intensities in the tunneling regime , involves one electron that tunnel - ionizes in the field - lowered coulomb potential and another electron that remains bound .
the electron that tunnel - ionizes emerges from the barrier with a zero velocity along the direction of the laser field , while its velocity perpendicular to the laser field is given by a gaussian distribution @xcite .
the tunneling rate can be obtained using semi - classical treatments , for instance see @xcite for atoms and @xcite for molecules .
the electron that is initially bound is commonly described in semi - classical models by a microcanonical distribution . to our knowledge , in the literature ,
a microcanonical distribution is available only for diatomic molecules @xcite . in this work
, we formulate a microcanonical distribution for any one - electron triatomic molecule which can also describe the initial state of the bound electron in the above described semi - classical models .
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You are an expert at summarizing long articles. Proceed to summarize the following text:
ams-02 2014 measurements of cosmic rays ( cr ) leptons @xcite , which have confirmed the rise of the positron fraction for kinetic energy above @xmath3 , up to @xmath4 , have stimulated different interpretations of this excess of positrons as primary evidence of dark matter ( dm ) annihilation . the interpretation of dm as a new source of positrons , to explain the departure from the pure secondary positron fraction , can be tested with other complementary and independent indirect measurements , as the cosmic microwave background ( cmb ) anisotropies . in fact , if dm particles self - annihilate at a sufficient rate , the expected signal would be directly sensitive to their thermally averaged cross section and it could have drawn an imprint on cmb temperature and polarization anisotropies . the aim of the present letter is to discuss the link between the dm annihilation cross sections at freeze - out , recombination and in the milky way galactic halo , which is essential to compare different indirect constraints on dm itself . the three different physical quantities are defined by the environment in three different epochs . the relation between them is not trivial , not a simply decreasing function and so it must be carefully analyzed to put coherent constraints on dm properties . once defined a consistent framework , one can compare information from cr physics with cmb observations . + the need of a high thermally averaged annihilation cross section , not purely thermal , _ i.e. _ @xmath5 , comes from the above mentioned observation of a huge excess in the cr positron fraction , in contrast with the expected behavior of secondaries produced in the interstellar medium ( ism ) . in order to describe this experimental evidence , a very high annihilation cross section has to be invoked , compared with typical expectation for a s - wave annihilating thermal relic matching the observed dm abundance @xcite . + the main way to interpret the positron excess is the sommerfeld enhancement @xcite , a non - perturbative quantum effect which modifies the annihilation cross section in the regime of small relative velocity of the annihilating particles and in presence of an effectively long - range force between them . indeed , this well - known quantum mechanical effect can occur in dm annihilations in the galactic halo , if the two annihilating particles exchange an interaction mediated by a force carrier . from a feynman diagram point of view , the weakly interacting massive particles ( wimp ) interact with the new boson through a multi - box diagram , depicted in fig . [ fig : feynman ] , annihilate into @xmath6 , and the decay of @xmath7 produces more light leptons than expected . the structure of the paper is the following : in section [ sec : two ] the basic setup of the sommerfeld enhancement and the relation between the galactic halo cross section and the one at recombination are briefly reviewed ; in section [ sec : three ] we report a collection of cr best fits for tev - ish dm candidates capable to reproduce the observed positron excess and we discuss the main sources of uncertainties ( from cr physics and the dm sector ) , in order to compare predictions with cmb constraints and obtain some general remarks in section [ sec : conclusion ] . the sommerfeld enhancement is fundamental to interpret indirect dm searches . the thermally averaged dm annihilation cross section ( at any time ) can be generally decomponed into powers of the velocity @xmath8 @xcite : @xmath9 where @xmath8 is the relative velocity of the annihilating particles , that is @xmath10 , @xmath11 is the constant s - wave term , @xmath12 is the p - wave term and @xmath13 the sommerfeld term where @xmath14 stands for a proper asymptotic ( cut - off ) value for the low velocity regime . the quadratic and quartic terms in eq . are commonly neglected at a freeze - out description , whereas the so called sommerfeld term is suppressed in the relativistic velocities regime @xcite . + the enhancement of the s - wave for these velocities is defined by the ratio of the masses of the dm candidate and the sommerfeld boson @xmath7 which provides the boosted annihilation , and by the effective coupling @xmath15 . defining the dimensionless parameter @xmath16 , the condition for the enhancement is @xmath17 , that is @xmath18 and @xmath19 . ( @xmath8 in natural unit on the abscissa ) . see @xcite for insights.,height=226 ] in fig . [ fig : crosssection ] the different contributions to the total dm annihilation cross section are shown according to eq . . it can be noted that the sommerfeld contribution is a function of the redshift @xmath20 but , after a particular value of @xmath8 ( @xmath21 ) , which is very close to the one expected for our dark halo , the quantum effect reaches a plateau that is approximately 100 times the thermal annihilation cross section , _ i.e. _ @xmath22 . here one can notice that there should be necessarily a slow velocity limit , given by the finite range of the attractive force mediated by @xmath7 : once the de broglie wavelength of the particle @xmath23 exceeds the range of the interaction @xmath24 , the quantum effect saturates reaching the value @xmath25 , that is a constant and does not depend on @xmath8 @xcite . moreover , for specific values of @xmath26 and @xmath27 , where @xmath28 is the ratio of @xmath8 to the speed of light @xmath29 , resonant threshold states can be produced ( see fig . [ fig : sommerfeld ] ) which are capable of further boosting @xmath30 @xcite , inducing @xmath31 . plane @xcite.,height=226 ] after cosmological recombination , dm had a velocity proportional to the inverse of the scale factor of the expanding universe because of its free stream , justifying very small velocity values ; on the other hand , the virialized velocities in a today galactic halo might be at least three order of magnitude greater . + the status of the three cross sections in the previously mentioned regimes can be basically summarized as follows : 1 . _ freeze - out_. during this phase the cross section is usually assumed to be nearly constant , without the general contribution of other even powers of @xmath8 . this is an absolute minimum from which the annihilation cross section evolves during the cooling of the universe . as shown in fig . [ fig : crosssection ] , for too relativistic freeze - out velocities , the p - wave term would enhance the annihilation cross section of one order of magnitude , generating an incompatibility w.r.t . the expected relic density at redshift @xmath32 . for this regime the relative velocity is : |_z 1000 = _ 0.3 c and the following relation for the cross section holds : ^ _ ~310 ^ -26 cm^3 s^-1 . recombination_. here the annihilation cross section has a greater value than the freeze - out one , _ i.e. _ @xmath33 . this does not imply the relation @xmath34 , because the quantum sommerfeld enhancement saturates after a certain velocity @xmath35 , which is @xmath36 . so , for @xmath37 the quantum sommerfeld boost @xmath38 does not increase the annihilation cross section further . this is the annihilation cross section measured by cmb experiments . in this regime the relative velocity of the dm particles is very low : |_600<z<1000 = _ ~10 ^ -(86)c and the annihilation cross section is clearly dominated by the sommerfeld enhancement : ^ _ = s__. 3 . _ galactic halo_. as for recombination epoch , the dominant term is the sommerfeld one . in our galaxy the velocity of the dm particles is supposed to be of the order of ( @xmath39 , which implies @xmath40 . in the galaxy the velocity regime is : |_z=0 = _ ~510 ^ -4c with an annihilating cross section : ^ _ = s__. if we take into account that the current value of @xmath8 in our galaxy does not completely saturate the sommerfeld effect , a conservative numerical factor @xmath41 can be assumed for the ratio @xmath42 between the enhancement at recombination and the one today ( see fig . [ fig : crosssection ] ) . the relation between the galactic annihilation cross section measured by ams-02 and the one measured by planck at the recombination can be written as : [ eqn : crosssection ] ^_(12)^_. the cmb experiments can constrain the dm annihilation cross section from the quantity introduced in refs . @xcite : p_f _ as a function of the factor @xmath43 that encodes the efficiency of the energy absorption at the recombination . therfore , another parameter has to be taken into account for the comparison between cr and cmb experiments . in principle @xmath44 , but it has been demonstrated that this can be taken as a constant at @xmath45 @xcite , around the recombination epoch ; it lies in the theoretical range between @xmath46 and @xmath47 , with recombination values generally chosen between @xmath48 and @xmath49 as a function of the annihilation channel ( see refs . so , for small @xmath43 values the annihilation constraints are relaxed . in refs . @xcite have been obtained the following bounds for the saturated annihilation cross section and the sommerfeld factor from wmap 5 yr data : @xmath50 for @xmath51 dm candidate and @xmath52 eqs . - lead to @xmath53 and @xmath54 . + now , with the latest planck 2015 data @xcite this constraint can be improved of about one order of magnitude ( see in fig . [ fig : planckannihilation ] the comparison between wmap9 and planck constraints ) , leading , for the previous case , to @xmath55 and @xmath56 . + , @xmath57 , @xmath58 ) , ( @xmath59 , @xmath60 , @xmath61 ) and ( @xmath62 , @xmath63 , @xmath64 ) for @xmath65 , @xmath66 and @xmath67 respectively.,height=226 ] in fig . [ fig : planckannihilation ] the constraints from planck @xcite , obtained with the full temperature data and the inclusion of low- and high-@xmath68 polarization data , are shown . here , the allowed region of parameters space for charged cr measurements was taken from cholis and hooper @xcite , under the assumption that the cr positron excess was due to pure dm annihilation . + from fig . [ fig : planckannihilation ] it can be noted that , for a tev - ish particle , _ e.g. _ with a mass of @xmath69 , and a _ pessimistic case _ with an ideal absorption efficiency @xmath70 , the annihilation cross section at recombination from planck is @xmath71 , which implies @xmath72 at most . instead , in an _ optimistic case _ with @xmath73 and @xmath74 , the constraint for a tev - ish dm becomes @xmath75 . + several studies suggest values less than @xmath76 for @xmath43 at recombination , that is a rather optimistic scenario for indirect search constraints @xcite . consequently , constraints of the order of @xmath77 for a heavy dm candidate can be still in agreement with the boosted cross sections which are necessary to reproduce cr positrons excess @xcite ; as a first step , only resonant sommerfeld boosts , that induce an overall enhancement @xmath78 w.r.t @xmath79 , can be for certain excluded . it follows that it is interesting and necessary to discuss and update the small rectangle in the right upper corner of fig . [ fig : planckannihilation ] , related to ams-02 2013 @xcite , fermi @xcite and pamela @xcite data , in light of new cr results and speculations , to avoid falling into misleading conclusions . + in fact , the analysis in @xcite was performed with only 2013 ams-02 data along with pamela , fermi and ams-01 ones . it must be also noted that fermi all electrons channel @xmath80 is in tension with ams-02 positron fraction , as stressed in @xcite , and it has not been reprocessed and confirmed after the recent pass 8 calibration performed in @xcite . data ( and consequently of positron fraction and @xmath81 data ) at the ams - days at cern on april 2015 , see https://cds.cern.ch/record/2010841 at @xmath82 . ] + furthermore , putting all these cr data together could produce some inconsistencies and issues : since the data used for fits were collected by different instruments , the fits errors are hard to estimate . the most interesting constraints in literature for tev - ish dm candidates , _ i.e. _ @xmath83 , using ams-02 positron data are the following : 1 . from the previously discussed ref . @xcite it must be noted that the reduced @xmath84 for tev - ish dm fits is much better for fermi data rather than for ams-02 2013 ones . the good fits , with @xmath85 , for ams-02 data in @xcite , for a dm which annihilates into a pair of intermediate states @xmath7 are @xmath86 with @xmath87 , @xmath57 with @xmath88 , @xmath60 with @xmath59 and @xmath63 with @xmath62 for @xmath89 , @xmath90 and @xmath91 combinations . the overall result is a dm with a mass up to @xmath69 and an annihilation cross section in the range of few @xmath77 ; 2 . in ref . @xcite some fits to ams-02 2014 positron fraction data with tev - ish dm are obtained : @xmath92 with @xmath93 for @xmath94 channel , @xmath95 with @xmath96 , with little deviations as a function of the annihilation channels combinations . some fits are achieved using also fermi - lat data , which have a great uncertainty w.r.t . ams-02 2014 data . 3 . in ref . @xcite the privileged regions , using ams-02 2013 data , are about @xmath97 with @xmath98 . they also used fermi and hess data ; 4 . in @xcite , using ams-02 2013 data , along with pamela and fermi ones , they obtain @xmath97 with @xmath99 ; 5 . in ref . @xcite , using ams-02 2013 , the results for a sommerfeld boost @xmath100 from a scalar , pseudoscalar or vector particle are approximately the same : @xmath97 with @xmath101 and @xmath102 for the @xmath66 channel , @xmath103 with @xmath104 and @xmath105 for the @xmath106 channel , @xmath107 with @xmath108 and @xmath109 for the @xmath110 channel ; 6 . in ref . @xcite , from ams-02 2013 positron fraction data the upper limit of @xmath111 is achieved , as a function of the annihilation channels , associated to a @xmath112 . they also use ams-02 positrons from icrc 2013 , which were not finalized nor official data ; 7 . in ref . @xcite , the best - fits obtained from ams-02 2014 data are about @xmath113 with @xmath114 , for @xmath115 and @xmath116 channels ; 8 . in ref . @xcite , the ams-02 2014 data lead to a best - fit via sommerfeld boson that is of the order of @xmath117 with @xmath118 , for @xmath66 and @xmath116 channels . these studies share common properties and weaknesses . first of all the allowed and privileged leptonic annihilation channels for dm with mass @xmath119 are generally the muonic and tauonic ones . also annihilation into quarks @xmath120 @xcite and @xmath121 @xcite are suitable for heavy dm . these best - fits are usually computed with dm masses @xmath122 , however mass values in the @xmath123 range , the most interesting one for a heavy wimp scenario @xcite , are poorly tested in literature ; on the other hand , some studies are performed for very heavy dm with @xmath124 , which is disfavored by recent astrophysical observations @xcite . in addition , the greater the mass , the greater the uncertainty of the annihilation scheme and the greater the degrees of freedom to tune the annihilation chain and fit the data . + it must be stressed that all recent observations point toward a tev - ish paradigm and suggest to look above @xmath51 for wimp dm masses @xcite ; at the same time too high dm masses @xmath125 could introduce some conflicts and could be not too suitable to fit the ams-02 2014 positron fraction and its supposed flattening , due to the achievement of a maximum of the positron production . it is more advisable to avoid the @xmath126 annihilation scenarios associated to very high masses and too high annihilation cross sections @xcite . + the previous fits have one order of magnitude of span in the annihilation cross section ; the ensemble of the fits prescriptions could be approximately described as a rectangle in the @xmath127 plane : @xmath128 . this must be translated in the @xmath129 plane : if we take @xmath130 , the most general permitted window becomes @xmath131 . + but such a rectangle is very loose and it is based on standard assumptions which can be easily extended . in fact the constraints from ams-02 positron data greatly relax introducing a dark disk ( dd ) in addition to a dark halo : this ensures dynamical enhancements of the dm annihilation cross section within the high density dd , @xmath132 , without the need of @xmath133 , @xcite . for indirect detection , the dd scenario could easily accommodate a large boost factor from local density enhancement in the range @xmath134 , depending on the disk height . + in addition , the constraints become less stringent if one consider both dm and pulsars as sources of primary positrons @xcite . pulsars and dd hypothesis can only relax the dm bounds , enlarging the allowed region in the @xmath129 plane . + for what concerns the uncertainties which afflict cr propagation physics , we still do not have a complete and well - posed understanding of the cr lepton problematic : non - standard propagation models may be introduced to account for part or all of the positrons measured in space @xcite ) . hence the constraints obtained from the positron sector ( positron spectrum and positron fraction ) strongly depend on the underlying cr propagation model and certainly on the ams-02 measurements errors . thanks to ams-02 unprecedented precision it will be soon possible to fix an almost univocal scheme of propagation of cosmic rays in our galaxy , allowing to correctly compute the effective background for dm indirect searches : in fact , tiny variations of the most significant parameters , such as the diffusive halo thickness , the diffusion coefficient and exponent and the electrons spectral indices , may lead to misleading interpretations of ams-02 data , pointing to an incorrect scenario . in fig . [ fig : pppc4 ] a qualitative illustration of propagation and measurement uncertainties , performed with pppc4dmid @xcite , show how unstable these fits are : a best - fit point in the @xmath127 plain carries up to 50% uncertainty in the choice of the annihilation cross section and dm mass values . and a set with about half the annihilation cross section ( blue curve ) and the ratio between a reference positron fit with @xmath87 and one with @xmath135 ( purple curve ) . as long the two curves lie below the dark yellow line , which represent the combination of the two main uncertainties sources , from cr propagation ( the min - max sets span in @xcite ) and from ams-02 errors , the best - fit point can be adjusted in the @xmath127 plain.,height=226 ] from fig . [ fig : pppc4 ] one argues that dm fits to ams-02 positrons and positron fraction with standard propagation models , up to @xmath136 , suffer large uncertainties , of the order of @xmath137 , if taken individually . this is the degree of uncertainty from the most precise space experiment which measures cr fluxes : when a fit on pamela , fermi ( or ams-01 , heat , hess ) data is performed , an uncertainty at least three times the ams-02 ones should be addressed : it could imply almost one order of magnitude in the annihilation cross section and more than @xmath51 for tev - ish dm masses . + finally , when exploring the dm @xmath127 space , other underlying parameters are fixed , such as the dark matter halo shape and the next - to - leading order ( nlo ) corrections to primary positrons from dm annihilation . the choice of the dm radial profile is not too significant , whereas full calculations of the nlo and nnlo electroweak ( ew ) corrections may modify the primary dm fluxes up to one order of magnitude @xcite , producing more light final states than expected in lo calculations and allowing lower values of the annihilation cross section : they are relevant for spectra predictions especially when @xmath138 is much larger than the ew scale and ew bremsstrahlung is permitted . we consider the collection of fits discussed in section [ sec : three ] : applying the constraint @xmath139 , according to the up - to - date calculations derived in @xcite , where @xmath76 is preferred only for the electron channels for non tev - ish dm candidates , an overall cr allowed region is obtained ( fig . [ fig : planckannihilationrev ] ) , which is something like @xmath140 . the qualitative fraction of this region permitted by cmb observations is about 9% , 4% and 2% of the total area obtained combining cosmic rays fits , assuming @xmath141 respectively . + our analysis is compatible with the one presented in @xcite ; in @xcite the region allowed by planck is @xmath142% . plane from ams-02/fermi / pamela ( dot - dashed rectangle ) , along with planck 2015 bound ( blue line ) . only a cosmic variance limited ( cvl ) experiment ( black line ) could completely falsify the cr indirect search constraints for tev - ish candidates . the points corresponding to the best - fits are the ones presented in section [ sec : three ] for @xmath58 . the rectangles for @xmath74 ( yellow ) and @xmath143 ( purple ) are also drawn.,width=340 ] + planck 2015 cmb data appear not to completely exclude annihilating dm as primary source for ams-02 positron excess , regardless the choice of the recombination efficiency value . from fig . [ fig : planckannihilationrev ] it emerges that future improvement on cmb measurements @xcite could hardly falsify the dark matter interpretation of cosmic - rays antiparticles , for what concerns tev - ish dark matter candidates . in addition , fundamental uncertainties from cr propagation physics and from dd hypothesis , pulsars contributions and alternative cr propagation models , not reported in fig . [ fig : planckannihilationrev ] , could almost arbitrarily enlarges the window in the @xmath129 plane , up to @xmath144 particles and down to nearly thermal cross sections . incoming cr high statistics measurements of leptons and nuclei from ams-02 will allow us to deepen the examination of this important issue , narrowing cr uncertainties and dm properties . besides that , the information from the antiproton channel is mandatory to put more consistent and coherent constraints on the annihilation capability of the dm candidate , because cr antiprotons have less important backgrounds @xcite , so granting more precise estimations . a cross check based on a hadron - lepton channels comparison will improve our understanding of dm annihilation and probably enlarge the analysis up to the @xmath144 scale . we wish to thank fabio finelli and daniela paoletti for useful comments and suggestions on the draft . mb acknowledge support by the `` asi / inaf agreement 2014 - 024-r.0 for the planck lfi activity of phase e2 '' . l. accardo _ et al . _ [ ams collaboration ] , phys . lett . * 113 * , 121101 ( 2014 ) .
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in view of the current interest in combining different observations to constraint annihilating wimp dark matter , we examine the relation between the sommerfeld effect at the recombination epoch and in the galactic halo . by considering an up - to - date collection of interpolations of cosmic rays lepton data ( ams-02 2014 , fermi and pamela ) , as dark matter annihilation signals ,
we show that current cosmic rays measurements and recent planck 2015 constraints from cmb anisotropies almost overlap for dark matter masses of the order of few @xmath0 , although great theoretical uncertainties afflict cosmic rays and dark matter descriptions . combining cosmic rays
fits we obtain proper minimal regions allowed by cmb observations , especially for @xmath1 and @xmath2 annihilation channels , once assumed viable values of the efficiency factor for energy absorption at recombination : the results are consistent with those obtained by the planck collaboration but allow a slightly larger overlap between cosmic rays constraints from the lepton sector and cmb .
incoming ams-02 measurements of cosmic rays antiprotons will help to clarify the conundrum .
dark matter , ams-02 , planck
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You are an expert at summarizing long articles. Proceed to summarize the following text:
the active manipulation of spin - dependent electron transport is the principal task in spintronics @xcite . one of the effective way to achieve spin polarization of electrons injected into semiconductors is spin - dependent tunnelling through a barrier in ferromagnetic metal / insulator / semiconductor heterostructures @xcite . the maximum of the spin injection efficiency reaches 52% at 100 k and 32% at 290 k for a mgo barrier on gaas @xcite . high electrical injection of spin - polarized electrons from a fe film through an al@xmath0o@xmath1 tunnel barrier into si has been demonstrated in @xcite . however , in si the electron spin polarization was observed at low temperatures 30% at 5 k , with polarization extending to at least 125 k. although important results in the spin injection have been obtained , high values of the spin polarization of injected electrons at room temperature has not been achieved . therefore , it is crucial to find a new method of the spin polarization that allows us to achieve high spin - injection efficiency . in this paper , we present new method of the electron spin polarization in ferromagnetic metal / insulator / semiconductor heterostructure , which based on spin - dependent backward scattering on exchange - splitted levels in a @xmath2 quantum well ( qw ) and on the electron capture by this qw . the qw is formed in the semiconductor near the insulator / semiconductor interface . the insulator layer is thin in order to provide electron tunneling from the ferromagnetic metal and to split levels in the qw by the exchange interaction . for the case , when one of exchange - splitted levels lies in the top region of the qw , the backscattering process of injected electrons becomes dependent on their spins . the capture of backscattered spin - polarized electrons by the qw leads to an additional coulomb repulsion for electrons tunneling from the ferromagnetic metal and to a considerable spin - polarized decrease of the current flowing in the heterostructure . in this way , high values of the spin polarization and manipulation of injected electron spins by the charge of the qw can be achieved . by analogy with the spin polarization in ferromagnetic metal / insulator / semiconductor heterostructures , spin polarization of electron current can be observed in heterostructures consisted of semiconductor substrates and granular films with ferromagnetic metal nanoparticles in an insulator matrix . ones of these heterostructures are sio@xmath0(co)/gaas heterostructures , where the sio@xmath0(co ) is the granular sio@xmath0 film with co nanoparticles @xcite . the @xmath2 qw ( accumulation electron layer ) with exchange - splitted levels is formed at the interface in the gaas @xcite . in sio@xmath0(co)/gaas heterostructures extremely large magnetoresistance and the current reduction on the temperature dependence are observed at room temperature . the paper is organized as follows . in the next section , we study the electron backscattering process on exchange - splitted levels in @xmath2 qw . in sec . iii we consider the capture of backscattered electrons by the qw and the current reduction caused by the qw charging . the spin polarization of electron current caused by the backscattering process and , consequently , by the current reduction is described in sec . finally , we summarize the results in sec . v. let us consider the backward scattering of injected electrons on exchange - splitted levels of the qw in a ferromagnetic metal / insulator / semiconductor heterostructure ( fig . [ bar2 ] ) . exchange interaction between electrons in the ferromagnetic metal and electrons in the qw through the thin insulator layer splits electron levels in the qw . electrons on exchange - splitted sublevels ( sublevels @xmath3 and @xmath4 in fig . [ bar2 ] ) have opposite spin orientations . difference of their energies is equal to the exchange energy @xmath5 . for clarification of the main features of scattering dependencies we restrict our consideration on the backscattering process on one of exchange - splitted levels the top sublevel @xmath3 with certain spin orientation and neglect insignificant parts of the electron wavefunction on the sublevel @xmath3 outside the qw . . @xmath6 is the ferromagnetic metal , @xmath7 is the insulator layer , @xmath8 is the applied electric field . ] the electron wavefunction on the sublevel @xmath3 is the product of the spatial function @xmath9 and the spin function @xmath10 @xmath11 where @xmath12 is the electron spin . in the wkb ( wentzel - kramers - brillouin ) approximation @xcite the spatial function in the qw can be written as @xmath13 where @xmath14 is the wavevector of the electron on the sublevel @xmath3 in the zero approximation with respect to @xmath15 , @xmath16 is the electron mass , @xmath17 is the energy counted from the qw bottom , @xmath18 is the width of the qw , @xmath19 is the normalization coefficient , @xmath20 is the number of the sublevel @xmath3 . the wavefunction of injected electron flying over the qw has the form of the product of the spatial function @xmath21 and the spin function @xmath22 @xmath23 where @xmath24 @xmath25 , @xmath26 is the energy counted from the qw bottom , @xmath27 is the normalization coefficient . for the interaction @xmath28 between the injected electron and the electron localized on the sublevel @xmath3 , in the first approximation with respect to @xmath28 the probability of the backscattering per unit time is @xcite @xmath29 where @xmath30 is the density of final states at the energy @xmath31 , @xmath32 is the final wavefunction and @xmath33 is the initial wavefunction combined of injected and localized electrons . if electrons form the singlet spin configuration ( @xmath34 , @xmath35 or @xmath36 , @xmath37 ) , then spatial parts of wavefunctions have the symmetric combination @xmath38 @xmath39 where @xmath40 is the wavefunction of the backscattered electron described by eq . ( [ 2bar ] ) with the substitution @xmath41 . for the singlet spin state the backscattering probability ( [ 3bar ] ) is equal to @xmath42 where @xmath43 @xmath44 if electrons form the triplet spin configuration ( @xmath34 , @xmath37 or @xmath36 , @xmath35 ) , then spatial parts of wavefunctions are antisymmetric @xmath45 @xmath46 for the triplet state the probability ( [ 3bar ] ) can be written as @xmath47 magnitudes @xmath48 and @xmath49 in relations ( [ 4bar ] ) and ( [ 5bar ] ) are functions of wavevectors @xmath50 and @xmath51 . besides , the wavevector @xmath51 depends on the number @xmath52 of localized level : @xmath53 . taking into account wavefunction forms ( [ 1bar ] ) and ( [ 2bar ] ) , for the uniform interaction @xmath54 we obtain @xmath55\ ] ] @xmath56 @xmath57 ^ 2.\ ] ] probabilities @xmath58 ( [ 4bar ] ) and @xmath59 ( [ 5bar ] ) strongly depend on the difference of wavevectors @xmath60 and , consequently , on the difference @xmath61 between the energy of injected electron @xmath62 and the energy of localized electron @xmath17 in the qw . for @xmath63 the energy difference is @xmath64 singlet and triplet backscattering probabilities versus the normalized wavevector difference @xmath65 for @xmath66 are shown in fig . probabilities are normalized by the magnitude of the singlet probability @xmath67 with @xmath68 and @xmath69 . in accordance with relation ( [ x1 ] ) , the singlet and triplet probabilities 1s and 1 t ( @xmath68 ) are shown as functions of the variable @xmath61 ( upper axis ) . in this case the qw contains only one exchange - splitted level . calculations have been done for @xmath70 nm and @xmath71 mev . it is necessary to notice that the probability of the singlet backscattering @xmath58 ( curves 1s , 2s ) is higher than the triplet backscattering probability @xmath59 ( curves 1 t , 2 t ) the backward scattering becomes dependent on spins of injected electrons . for scattering of injected electrons on the level with @xmath68 and with the wavevector @xmath72 , the ratio of singlet and triplet probabilities leads to the relation @xmath73 on the qw versus the normalized wavevector difference @xmath74 . @xmath75 is the wavevector of localized electron , @xmath52 is the number of level . probabilities are normalized by the magnitude of the singlet probability @xmath67 with @xmath68 and @xmath69 . 1s , 1 t are singlet and triplet backscattering on the first level ( @xmath68 ) ; 2s , 2 t are singlet and triplet backscattering on the second level ( @xmath76 ) , respectively . for the qw with the width @xmath70 nm and the energy depth @xmath71 mev the singlet and triplet probabilities 1s and 1 t are shown as functions of the difference @xmath61 between the energy of injected and localized electrons ( upper axis ) . ] the backscattering probability strongly reduces with growth of @xmath77 and @xmath61 . the greatest magnitude of backscattering is achieved for the level with @xmath68 . thus , the backscattering process becomes important , if ( 1 ) the qw contains only exchange - splitted level with @xmath68 , ( 2 ) one sublevel of the exchange - splitted level with certain spin orientation lies at the top of the qw and ( 3 ) the energy of injected electrons is closed to the energy of localized electron on this sublevel . the capture of backscattered electrons by the qw leads to the considerable current reduction dependent on spin orientation of injected electrons . if the fermi level lies below localized electron levels in the qw , then at a finite temperature these levels are partially filled by electrons . backscattered electrons are captured by the qw and , in accordance with their spin orientation , they occupy different localized levels . we suppose that the spin relaxation time is greater than the storage time of additional electrons in the qw . then , for the singlet scattering process backscattered electrons occupy the sublevel @xmath4 with spin orientation opposite to spin orientation of the sublevel @xmath3 ( fig . [ bar2 ] ) . the sublevel @xmath4 lies below the sublevel @xmath3 . on the contrary , for the triplet case backscattered electrons fall on the sublevel @xmath3 . the storage time @xmath78 of the presence of additional electrons in the qw depends on the electron - hole recombination , on temperature activation processes , and on the electron tunneling into the conduction band . for underlying levels the storage time @xmath78 is greater than the storage time of electrons on overlying ones . the additional charge in the qw leads to the electrostatic blockade of injected electrons and to the current reduction . in this way , the current flowing in ferromagnetic metal / semiconductor heterostructure with qw , which contains exchange - splitted levels , is unstable . this current instability is accompanied by the charge accumulation in the qw and by the current reduction depended on spin orientations of injected electrons . let us calculate the reduction of the current . for clarity , we consider the current reduction caused by the singlet backscattering . in the triplet case , the consideration is analogous . the current density flowing over the qw is equal to @xmath79 where @xmath80 is the electron charge , @xmath81 is the electron mobility , @xmath8 is the electric field , @xmath82 is the electron concentration over the qw , @xmath83 is the electron concentration without an electric field , @xmath84 is the potential of the field of additional localized electrons in the qw , @xmath85 is the boltzmann constant , and @xmath86 is the temperature . in the singlet backscattering case , the additional charge accumulates on the sublevel @xmath4 ( fig . [ bar2 ] ) . the potential @xmath84 of the field caused by this additional charge is determined by the equation @xcite @xmath87 where @xmath88 is the dielectric permittivity of the semiconductor in the qw region ; @xmath89 and @xmath90 are electron concentrations on the sublevel @xmath4 in the electric field and without a field , respectively . if the additional concentration of the charge @xmath91 is uniformly distributed over the qw width , then the solution of eq . ( [ 7bar ] ) is given by @xmath92 injected electrons must surmount the additional barrier with the energy height @xmath93 taking into account relations ( [ 6bar ] ) and ( [ 8bar ] ) , we obtain the current density of electrons incoming on the sublevel @xmath4 @xmath94.\ ] ] release of additional electrons from the sublevel @xmath4 is determined by the time @xmath95 and the current density of outgoing electrons can be written as @xmath96 for the equilibrium process @xmath97 and @xmath98=\frac{(n_b - n_b^{(0)})d}{\tau_b}. \label{9bar}\ ] ] relation ( [ 9bar ] ) is the equation in the unknown additional electron concentration @xmath91 . taking into account relation ( [ 6bar ] ) , we find the current reduction caused by the singlet electron backscattering on the qw @xmath99.\label{10bar}\ ] ] current reductions @xmath100 versus the applied electric field @xmath8 for different times @xmath95 are shown in fig . calculations are performed for @xmath101 , width of the qw @xmath102 2 nm , permittivity @xmath103 1 , @xmath104 300 k , @xmath105 @xmath106/v@xmath107s , and @xmath108 @xmath109 . from the presented dependencies we can see that backscattering of injected electrons on exchange - splitted levels and accumulation of electrons in the qw leads to considerable reduction of the current depended on its spin orientation . caused by the singlet electron backscattering in ferromagnetic metal / insulator / semiconductor heterostructure with quantum well ( qw ) contained exchange - splitted levels versus the applied electric field @xmath8 for different storage time @xmath95 of additional electrons in the qw . the backscattering probability @xmath101 , width of the qw @xmath102 2 nm , temperature @xmath104 300 k , and the electron concentration over the qw @xmath108 @xmath109 . ] the current reduction depends on the electron concentration @xmath83 in the semiconductor . for small values of @xmath83 the additional concentration @xmath91 in eq . ( [ 9bar ] ) leads to zero and the reduction is small , @xmath110 . for great values of the concentration @xmath83 ( for example , close to metal concentrations ) the qw contains filled levels and the additional charge in the qw is impossible . as a result of this , there is no any reduction of the current . for the triplet backscattering case , backscattered electrons accumulate on the level @xmath3 . the current reduction @xmath111 is determined by relation ( [ 10bar ] ) , in which we must perform the substitution @xmath112 . the additional electron concentration @xmath113 is the solution of eq . ( [ 9bar ] ) with substitutions @xmath114 and @xmath115 . in comparison with the singlet case , for @xmath116 and @xmath117 the current reduction @xmath111 caused by the triplet backscattering and by the accumulation of electrons in the qw is insignificant . the qw with exchange - splitted levels can be regarded as spin filter for injected electrons . let us consider the spin current flowing over the qw with square modulation of the spin projection @xmath118 ( fig . [ bar5]a ) @xmath119 and @xmath120 are currents with spin polarization @xmath121 and @xmath122 , respectively , after electron backward scattering on the qw . ( b ) spin polarization @xmath123 of the electron current caused by the electron backscattering versus the electric field @xmath8 for different values of the storage time @xmath95 . ] where @xmath52 is the electron concentration , @xmath81 is the mobility , @xmath8 is the electric field , @xmath124 . we suppose that the modulation period @xmath125 is much greater than the storage times @xmath126 and @xmath95 : @xmath127 . without spin - dependent backscattering and charge accumulation in the qw the spin current @xmath128 is not modified . in the presence of singlet and triplet backscattering and accumulation of backscattered electrons in the qw , the spin current @xmath128 decreases to @xmath129 and the reduction becomes dependent on its spin projection @xmath118 . the magnitude of the reduction of the current @xmath130 caused by the singlet backscattering on the sublevel @xmath3 ( fig . [ bar2 ] ) is higher than the magnitude of the reduction of the current @xmath131 caused by the triplet scattering process . taking into account relations ( [ 6bar ] ) , ( [ 10bar ] ) and ( [ 11bar ] ) , for time regions far from pulse edges we can write the spin polarization as @xmath132 the spin polarization @xmath123 versus the electric field @xmath8 has been calculated for different values of the storage time @xmath95 for backscattering probability @xmath133 , width of the qw @xmath102 2 nm , permittivity @xmath103 1 , temperature @xmath104 300 k , electron mobility @xmath134 @xmath106/v@xmath107s , concentration @xmath108 @xmath109 , and time @xmath135 10 ns ( fig . [ bar5]b ) . one can notice that the spin polarization @xmath123 increases with growth of the electric field @xmath8 and the storage time @xmath95 . the backward scattering of injected electrons on exchange - splitted levels of quantum wells in ferromagnetic metal / insulator / semiconductor heterostructures can be used as the effective way of the spin polarization of the current . the necessary condition to obtain high values of the spin polarization is : one of the exchange - splitted levels must be in the top region of the qw . if the energy of injected electrons is close to the energy of localized electrons , the backward scattering becomes dependent on spins of injected electrons on singlet or triplet spin configurations . it is found that the probability of the singlet backscattering @xmath58 is higher than the triplet backscattering probability @xmath59 . the capture of backscattered electrons by the qw leads to an additional coulomb repulsion for electrons and to the considerable spin - dependent reduction of the current flowing in the heterostructure . the spin polarization @xmath123 of the current increases with growth of the applied electric field and the storage time of electrons in the qw and its high values can be achieved at room temperature . in this way , the qw with exchange - splitted levels in ferromagnetic metal / insulator / semiconductor heterostructures can be regarded as spin filter . this work was supported by the government of russia ( project no . 14.z50.31.0021 , leading scientist m. bayer ) . 777 s.a . wolf , d.d . awschalom , r.a . buhrman , j.m . daughton , s. von molnar , m.l . roukes , a.y . chtchelkanova and d.m . treger , science * 294 * , 1488 ( 2001 ) . g. schmidt , j. phys . phys . * 38 * , r107 ( 2005 ) . i. utic , j. fabian , and s. das sarma , rev . phys . * 76 * , 323 ( 2004 ) . hammar and m. johnson , appl . letters * 79 * , 2591 ( 2001 ) . x. jiang , r. wang , r.m . shelby , r.m . macfarlane , s.r . bank , j.s . harris , and s.s.p . parkin , phys . letters * 94 * , 056601 ( 2005 ) . li , g. kioseoglou , o.m.j . van t erve , m.e . ware , d. gammon , r.m . stroud , and b.t . jonker , appl . letters * 86 * , 132503 ( 2005 ) . x.y . dong , c. adelmann , j.q . xie , c.j . palmstrm , x. lou , j. strand , p.a . crowell , j .- barnes , and a.k . petford - long , appl . letters * 86 * , 102107 ( 2005 ) . jonker , g. kioseoglou , a.t . hanbicki , c.h . li , and p.e . thompson , nature physics * 3 * , 542 ( 2007 ) . li , g. kioseoglou , o.m.j . van t erve , p.e . thompson , and b.t . jonker , appl . letters * 95 * , 172102 ( 2009 ) . g. kioseoglou , a.t . hanbicki , r. goswami , o.m.j . van t erve , c.h . li , g. spanos , p.e . thompson , and b.t . jonker , appl . letters * 94 * , 122106 ( 2009 ) . h. saito , j.c . le breton , v. zayets , y. mineno , s. yuasa , and k. ando , appl . letters * 96 * , 012501 ( 2010 ) . lutsev , a.i . stognij , and n.n . novitskii , jetp letters * 81 * , 514 ( 2005 ) . lutsev , a.i . stognij , n.n . novitskii , and a.a . stashkevich , j. magn . magn . mater . * 300 * , e12 ( 2006 ) . lutsev , a.i . stognij , and n.n . novitskii , phys . b * 80 * , 184423 ( 2009 ) . l. lutsev , _ giant injection magnetoresistance . experimental and theoretical study , perspectives of applications_. ( lambert academic publishing , saarbrucken , germany , 2013 ) , isbn 978 - 3 - 659 - 43964 - 3 [ in russian ] . lutsev , j. phys . : condens . matter * 18 * , 5881 ( 2006 ) . davydov , _ quantum mechanics _ ( pergamon press , oxford , 1976 ) . l.d . landau and e.m . lifshitz , _ the classical theory of fields_. vol . 2 ( 4th ed . ) ( butterworth - heinemann , 1975 ) .
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the backscattering process of injected electrons on exchange - splitted levels of quantum well ( qw ) in ferromagnetic metal / insulator / semiconductor heterostructure is studied .
it is found that , if one of the exchange - splitted levels lies in the top region of the qw and the energy of injected electrons is close to the energy of localized electron on this level , the backward scattering becomes dependent on spins of injected electrons .
accumulation of backscattered electrons in the qw leads to considerable reduction of the current depended on its spin orientation .
the spin polarization increases with growth of the applied electric field and the storage time of electrons in the qw .
high values of the spin polarization can be achieved at room temperature . in this way , the qw with exchange - splitted levels in ferromagnetic metal / insulator / semiconductor heterostructure can be used as effective spin filter .
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it is by now well established that the physics associated with classical and quantum fields in curved spacetimes can be reproduced , within certain approximations , in a variety of different physical systems the so - called `` analogue models of general relativity ( gr ) '' @xcite . the simplest example of such a system is provided by acoustic disturbances propagating in a barotropic , irrotational and viscosity - free fluid . in the context of analogue models it is natural to separate the kinematical aspects of gr from the dynamical ones . in general , within a sufficiently complex analogue model one can reproduce any pre - specified spacetime and the kinematics of fields evolving on it independently of whether or not it satisfies the classical ( or semiclassical ) einstein equations @xcite . indeed , to date there are no analogue models whose effective geometry is determined by einstein equations . in this sense we currently have both analogue spacetimes and analogues of quantum field theory in curved spacetimes , but ( strictly speaking ) no analogue model for gr itself @xcite . in order to reproduce a specific spacetime geometry within an analogue model , one would have to take advantage of the specific equations describing the latter ( for example , for fluid models , the euler and continuity equations , together with an equation of state ) , plus the possibility of manipulating the system by applying appropriate external forces . in the analysis of this paper we will think of the spacetime configuration as `` externally given '' , assuming that it has been set up as desired by external means any back - reaction on the geometry is neglected as in principle we can counter - balance its effects using the external forces . in the context of analogue models this is not merely a hypothesis introduced solely for theoretical simplicity , but rather a realistic situation that is in principle quite achievable . specifically , in this paper we analyze in simple terms the issue of quantum quasi - particle creation by several externally specified @xmath1-dimensional analogue geometries simulating the formation of black hole - like configurations . ( in a previous companion paper @xcite we investigated the causal structure of these , and other , spacetimes . ) in this analysis we have in mind , on the one hand , the possibility of setting up laboratory experiments exhibiting hawking - like radiation @xcite and , on the other hand , the acquisition of new insights into the physics of black hole evaporation in semiclassical gravity . all the discussion holds for a scalar field obeying the dalembert wave equation in a curved spacetime . this means that we are not ( for current purposes ) considering the deviations from the phononic dispersion relations that show up at high energies owing to the atomic structure underlying any condensed matter system . we shall briefly comment on these modifications at the end of the paper . for simplicity , throughout the paper we adopt a terminology based on acoustics in moving fluids ( we will use terms such as acoustic spacetimes , sonic points , fluid velocity , etc . ) , but our results are far more general and apply to many other analogue gravity models not based on acoustics . we summarise the main conclusions below . first of all , we recover the standard hawking result when considering fluid flows that generate a supersonic regime at finite time . ( that is , we recover a stationary creation of quasi - particles with a planckian spectrum . ) we then analyze the quasi - particle creation associated with other types of configurations . in particular , we shall discuss in detail a `` critical black hole '' a flow configuration that presents an acoustic horizon without an associated supersonic region . from this analysis we want to highlight two key results : * the existence of a supersonic regime ( sound velocity @xmath2 strictly smaller than fluid velocity @xmath3 ) is not needed in order to reproduce hawking s stationary particle creation . we demonstrate this fact by calculating the quantity of quasi - particle production in an evolving geometry which generates only an isolated sonic point ( @xmath0 ) , but without a supersonic region , in a finite amount of laboratory time . * moreover , in order to produce a hawking - like effect it is not even necessary to generate a sonic point at finite time . all one needs is that a sonic point develops in the asymptotic future ( that is , for @xmath4 ) _ with sufficient rapidity _ ( we shall explain in due course what we exactly mean by this ) . from the point of view of the reproducibility of a hawking - like effect in a laboratory , the latter result is particularly interesting . in general , the formation of a supersonic regime in a fluid flow normally considered to be the crucial requirement to produce hawking emission is associated with various different types of instability ( landau instability in superfluids , quantized vortex formation in bose einstein condensates , etc . ) that could mask the hawking effect . to reproduce a hawking - like effect without invoking a supersonic regime could alleviate this situation . from the point of view of gr , we believe that our result could also have some relevance , as it suggests a possible alternative scenario for the formation and semiclassical evaporation of black hole - like objects . the plan of the paper is the following : in the next section we introduce the various acoustic spacetimes on which we focus our attention , spacetimes that describe the formation of acoustic black holes of different types . in section [ sec : creation ] we present separately the specific calculations of redshift for sound rays that pass asymptotically close to the event horizon of these black holes . by invoking standard techniques of quantum field theory in curved spacetime , one can then immediately say when particle production with a planckian spectrum takes place . finally , in the last section of the paper we summarise and discuss the results obtained . associated with the flow of a barotropic , viscosity - free fluid along an infinitely long thin pipe , with density and velocity fields constant on any cross section orthogonal to the pipe , there is a ( 1 + 1)-dimensional _ acoustic spacetime _ @xmath5 , where the manifold @xmath6 is diffeomorphic to @xmath7 . using the laboratory time @xmath8 and physical distance @xmath9 along the pipe as coordinates on @xmath6 , the _ acoustic metric _ on @xmath6 can be written as@xmath10 = \omega^2\left[-c^2\,d t^2 + ( \d x+v\,\d t)^2\right ] \;,\label{metric}\]]where @xmath2 is the speed of sound , @xmath3 is the fluid velocity , and @xmath11 is an unspecified non - vanishing function @xcite . in general , all these quantities depend on the laboratory coordinates @xmath12 and @xmath13 . here , we shall assume that @xmath2 is a constant . hence , it is the velocity @xmath14 that contains all the relevant information about the causal structure of the acoustic spacetime @xmath5 . we direct the reader to the companion paper @xcite for a detailed analysis of the causal structure associated with a broad class of @xmath1-dimensional acoustic geometries , both static and dynamic . the sonic points , where @xmath15 , correspond to the so - called acoustic apparent horizons apparent horizons for the lorentzian geometry defined on @xmath6 by the metric ( [ metric ] ) . the fact of having an underlying minkowski structure associated with the laboratory observer makes the definition of apparent horizons in acoustic models less troublesome than in gr ( see reference @xcite , pp . consider a monotonically non - decreasing function @xmath16 such that @xmath17 and @xmath18 for @xmath19 . if one chooses @xmath20 in ( [ metric ] ) , the corresponding acoustic spacetime represents , for observers with @xmath21 , a static black hole with the horizon located at @xmath22 ( in this case apparent and event horizon coincide ) , a black hole region for @xmath23 , and a ( right - sided ) surface gravity@xmath24we can , moreover , distinguish three cases : * @xmath25 and @xmath26 for @xmath23 : a non - extremal black hole ; * @xmath25 and @xmath27 for @xmath23 : a `` critical '' black hole ; * @xmath28 and @xmath27 for @xmath23 : an extremal black hole . now , taking the above @xmath16 , let us consider @xmath13-dependent velocity functions@xmath29with @xmath30 a monotonically decreasing function of @xmath13 , such that @xmath31 and @xmath32 . ( the first condition serves to guarantee that spacetime is flat at early times , whereas we impose the second one only for simplicity . all the analysis in the paper could be performed without adopting this assumption , leaving the physical results unchanged . however , that would require more case - by - case splitting , only to cover new situations without physical interest . ) there are basically two possibilities for @xmath30 , according to whether the value @xmath33 is attained for a finite laboratory time @xmath34 or asymptotically for an infinite future value of laboratory time . in the first case @xmath35 and the corresponding metric ( [ metric ] ) represents the formation of a non - extremal , critical , or extremal black hole , respectively . for small values of @xmath36 we have@xmath37 ^ 2)\;,\label{appxi - bh}\]]where @xmath38 is a positive parameter . hence the function @xmath30 behaves , qualitatively , as shown in figure [ f : xi - bh ] . apart from this feature , the detailed behaviour of @xmath30 is largely irrelevant for our purposes . if instead @xmath39 is attained only at infinite future time , that is @xmath40 , one is describing the asymptotic formation of either a critical black hole ( if @xmath41 ; obviously , in this case choosing the non - extremal or the critical @xmath42 profile is irrelevant ) or an extremal black hole ( if @xmath28 ) . now the function @xmath30 behaves , qualitatively , as shown in figure [ f : xi - cbh ] . the relevant feature of @xmath43 is its asymptotic behaviour as @xmath44 . in the following we shall consider two possibilities for this asymptotics , although others can , of course , be envisaged : exponential : @xmath45 , with @xmath46 a positive constant , in general different from @xmath47 , and @xmath48 ; power law : @xmath49 , with @xmath50 and @xmath51 . for all the situations considered so far , spacetime is minkowskian in the two asymptotic regions corresponding to @xmath52 , and to @xmath4 , @xmath19 ( @xmath53 and @xmath54 , respectively , adopting the notation of reference @xcite ) . starting with a quantum scalar field in its natural minkowskian vacuum at @xmath52 , we want to know the total quantity of quasi - particle production to be detected at the right asymptotic region at late times , @xmath4 , caused by the dynamical evolution of the velocity profile @xmath14 . in the geometric acoustic approximation , a right - going sound ray is an integral curve of the differential equation@xmath55we are interested in sound rays propagating from @xmath56 ( see figure [ f : conf - bh ] ) ; that is , in solutions of ( [ diffeq ] ) that satisfy an initial condition @xmath57 , with @xmath58 in the limit @xmath59 ( so @xmath60 can be thought of as an `` initial '' event corresponding to the emission of the acoustic signal ) . if such a ray ends up on @xmath61 , we can identify `` final '' events @xmath62 on it , with @xmath63 as @xmath64 . for a ray connecting @xmath53 to @xmath54 one can also find an event @xmath65 such that @xmath66 , which corresponds to the crossing of the `` kink '' in @xmath3 , located at @xmath67 according to equation ( [ velocity ] ) , by the sound signal . finally , we can define , for such a ray , two parameters @xmath68 and @xmath69 as follows:@xmath70@xmath71such parameters correspond to null coordinates in spacetime . if an acoustic event horizon @xmath72 is present in the spacetime , the coordinate @xmath68 is regular on it ( , @xmath68 attains some finite value on @xmath72 ) , whereas @xmath69 tends to @xmath73 as @xmath74 is approached . we can express both @xmath68 and @xmath69 in terms of the velocity profile ( shape and dynamics ) and of the crossing time @xmath75 . to this end , we can integrate equation ( [ diffeq ] ) , first between @xmath76 and @xmath77:@xmath78then between @xmath77 and @xmath79:@xmath80on replacing @xmath81 from equation ( [ xi - x ] ) into ( [ u ] ) , we find the value of @xmath68 for a generic right - moving ray that crosses the kink at laboratory time @xmath75:@xmath82similarly , substituting @xmath83 from equation ( [ t - t ] ) into ( [ u ] ) , then adding and subtracting the quantity @xmath84 , we find@xmath85 in the analysis below , our chief goal consists of finding the relation between @xmath68 and @xmath69 for a sound ray that is close to the horizon , , in the asymptotic regime @xmath86 . from such a relation it is then a standard procedure to find the bogoliubov @xmath87 coefficients and hence the total quasi - particle content to be measured , in this case , by an asymptotic observer at @xmath54 . ( see , for example , reference @xcite ) . in the case of an exponential relation between @xmath68 and @xmath69 it is a well established result that a planckian spectrum is observed at late times @xcite , so hawking - like radiation will be recovered . when the apparent horizon forms at a finite laboratory time , say at @xmath88 , an event horizon always exists , generated by the right - moving ray that eventually remains frozen on the apparent horizon , at @xmath22 . for such a ray @xmath89 , and since @xmath35 , the @xmath68 parameter has the finite value@xmath90for a ray with @xmath91 we then obtain , combining equations ( [ uu ] ) and ( [ uh - fin]):@xmath92this _ exact _ equation is now in a form suitable for conveniently extracting _ approximate _ results in the region @xmath93 , corresponding to sound rays that `` skim '' the horizon . on the other hand , when the trapping horizon consists of just one single sonic point located at @xmath94 , it is not obvious that an event horizon exists . loosely speaking , in this case it might happen that the trapping horizon form `` after '' every right - going ray from @xmath53 has managed to cross @xmath22 . since there is a competition between two infinite quantities the time at which the trapping horizon forms , and the time at which the `` last '' right - going signal that connects @xmath95 with @xmath96 crosses @xmath22 a careful case - by - case analysis is in order . this is essentially all that can be said without relying on specific features of @xmath16 . we now consider separately the various situations of interest , focussing first on the issue of the existence of the event horizon . in the case of a non - extremal black hole , the qualitative behaviour of the function @xmath16 is shown , graphically , in figure [ f : v - bh ] . note that , for small values of @xmath97 , one can write@xmath98the function @xmath43 behaves as already shown in figure [ f : xi - bh ] . a sketch of the worldlines of right - moving sound rays is presented in figure [ f : tx - bh ] . note that in the portion of the diagram to the right of the curve @xmath67 ( , to the right of the moving kink in the velocity profile ) , spacetime is static . for @xmath52 , the geometry is minkowskian and the worldlines tend to approach straight lines with slope @xmath99 . the sound ray that generates the event horizon corresponds to a finite ) , given the asymptotic behaviours of @xmath100 and @xmath30 . ] value @xmath101 of the coordinate @xmath68 . hence , in this situation an event horizon always exists . this is also clear from the fact that the vertical half - line @xmath22 , @xmath102 in figure [ f : tx - bh ] is an apparent horizon . the function @xmath16 behaves as shown in figure [ f : v - cbh ] . regarding the right side of the profile , @xmath21 , it is indistinguishable from the profile of a non - extremal black hole ( figure [ f : v - bh ] ) . when the function @xmath43 is of the form ( [ appxi - bh ] ) , that is , when the apparent horizon is formed at a finite amount of laboratory time , the situation is exactly the same as for the non - extremal black hole discussed above . consider now that the sonic point is approached in an infinite amount of time , so the function @xmath43 behaves as in figure [ f : xi - cbh ] . the worldlines of right - moving sound rays are shown in figure [ f : tx - cbh ] . as in the formation of the non - critical black hole , the portion of the diagram to the right of the curve @xmath67 ( , to the right of the moving kink in the velocity profile ) corresponds to a static spacetime , and for @xmath52 the geometry is minkowskian the worldlines tend to approach straight lines with slope @xmath99 . however , now the apparent horizon is just the asymptotic point located at @xmath22 , @xmath4 , and in order to establish whether an event horizon does , or does not , exist one must perform an actual calculation of @xmath101 for the `` last '' ray that crosses the kink . the expression for @xmath101 is again obtained from equation ( [ uu ] ) , noticing that now @xmath103 along the generator of the would - be horizon , so@xmath104the necessary and sufficient condition for the event horizon to exist is that the limit on right hand side of equation ( [ baru - cbh ] ) be finite . the integrand on right hand side of ( [ baru - cbh ] ) can be approximated , for @xmath105 , as @xmath106 , while for @xmath52 it just approaches zero . hence @xmath101 is , up to a finite constant , equal to @xmath107 times the integral of @xmath30 , evaluated at @xmath108 . here we must distinguish between the exponential behaviour and the power law cases ( i ) and ( ii ) . in the former @xmath101 is finite , trivially . for the power law , it turns out that @xmath101 is finite iff @xmath109 . the typical spatial profile function @xmath16 for an extremal black hole is plotted in figure [ f : v - ebh ] . for @xmath12 approaching zero from positive values we can write@xmath110where @xmath111 is a constant . as far as dynamics is concerned , we must distinguish the cases in which the apparent horizon is formed at finite laboratory time @xmath34 , and in an infinite time ( , for @xmath4 ) . the function @xmath43 is of the type shown in figure [ f : xi - bh ] , and the worldlines of right - going sound rays are sketched in figure [ f : rays - ex - fin ] . the event horizon always exists . the function @xmath43 is as shown in figure [ f : xi - cbh ] , and the worldlines of right - going signals are shown in figure [ f : tx - cbh ] . as in the case of the formation of a critical black hole , the apparent horizon forms only asymptotically , for @xmath22 and @xmath4 , so the event horizon exists iff @xmath101 , given by equation ( [ baru - cbh ] ) , has a finite value . using the expansion ( [ appv - ebh ] ) in equation ( [ baru - cbh ] ) , one finds that @xmath112 is always finite when @xmath43 is asymptotically exponential . on the other hand , for a power law , the event horizon exists iff @xmath113 . ( note that the critical value of the exponent , @xmath114 , is now _ not _ the same as for the critical black hole , @xmath115 . ) the configurations we have analyzed until now are the simplest from a purely mathematical point of view . however , having in mind acoustic analogue geometries reproducible in a one - dimensional pipe in the laboratory , it is more sensible to consider double - sided configurations . by this we mean that , after passing ( or approaching ) the sonic / supersonic regime at @xmath22 , and traversing an interval of width @xmath116 , the fluid again goes back to a subsonic regime as @xmath117 . consider for example functions @xmath16 such that @xmath27 for @xmath118,@xmath119and which outside the interval @xmath118 tend monotonically to zero as @xmath97 increases ( see figures [ f : flat - spike ] and [ f : spike ] ) . the corresponding fluid configuration represents what could be called a static `` double - sided critical black hole '' . the formation of such a configuration can be modelled by the velocity function@xmath120with @xmath30 a monotonically decreasing function of @xmath13 , and @xmath100 as above . accordingly , the differential equation for right - going sound rays also splits:@xmath121geometries associated with the formation of non - extremal and extremal black holes can be constructed in the same way ; see figures [ f : v - bh - wh ] and [ f : v - ex - bh ] for plots of the respective @xmath100 functions . the function @xmath43 is of the type illustrated in figure [ f : xi - bh ] . the behaviour of right - going sound rays is shown in figure [ f : rays - spike - fin ] . the apparent horizon is the half - line @xmath22 , @xmath122 , and the event horizon always exists . let us now consider a function @xmath43 of the type illustrated in figure [ f : xi - cbh ] . the behaviour of right - going sound rays is shown in figure [ f : rays - spike - inf ] . for these particular configurations , whether an event horizon does , or does not , actually exist is now a rather tricky issue . the asymptotic behaviour of the function @xmath43 at @xmath123 ensures that all the right - going rays start to the left of the right - moving kink ( , the one in the region @xmath23 ) , then catch up with it , and begin to propagate through the intermediate region at a velocity @xmath124 that depends only on @xmath13 . before they reach the point @xmath22 , such rays might be overtaken by the right - moving kink , but only to start the chase again . after several mutual overtakings ( if the function @xmath43 is sufficiently complicated ) , the rays will always make an ultimate overtaking of the right - moving kink , embarking upon a final encounter with the left - moving kink on the right ( , in the region @xmath21 ) . let us denote by @xmath125 the time of such a last crossing of the right - moving kink , so the corresponding event is @xmath126 . also , let us denote by @xmath127 the time at which the same ray crosses the kink on the right , so that the corresponding event is @xmath128 . from equation ( [ diffeq - dsc ] ) we directly obtain the relation@xmath129\;.\label{ziopollo}\]]between @xmath127 and @xmath125 . ( when the ray crosses the right - moving kink more than once , equation ( [ ziopollo ] ) will be satisfied by more than one value of @xmath125 for any given @xmath127 . in order to avoid cumbersome notation , we shall simply denote by @xmath125 the largest of these roots , corresponding to the last crossing . ) then a necessary and sufficient condition for the existence of an event horizon is that , for @xmath130 , @xmath125 tends to a finite value , say @xmath131 . this guarantees that any right - going ray that last crosses the left kink at a time greater than @xmath131 does not reach the region @xmath21 ( as ray - crossing can not occur under the working hypothesis of this paper ) . applying this condition straightforwardly in order to see whether the event horizon exists is not easy . indeed , that would require us to evaluate the integral in equation ( [ ziopollo ] ) for a generic , finite value of @xmath127 , then solve for @xmath125 as a function of @xmath127 . it is easier to use one of the following two alternative strategies : 1 . instead of asking whether the event horizon exists , one can ask whether the event horizon _ does not _ exist . a necessary and sufficient condition for this is that , for @xmath130 , also @xmath132 . in such a case , we can insert the asymptotic expansions ( [ appv - bh ] ) or ( [ appv - ebh ] ) into equation ( [ ziopollo ] ) to get@xmath133for a non - extremal black hole , and@xmath134for an extremal one . plugging into these expressions the different asymptotic behaviours of the function @xmath43 , one can explicitly solve for @xmath125 as a function of @xmath127 for large values of the latter , and check whether @xmath125 does , or does not , tend to infinity when @xmath130 . 2 . setting @xmath135 into ( [ ziopollo ] ) , one obtains@xmath136\;.\label{xihinf}\]]it is possible to show that the event horizon exists if and only if equation ( [ xihinf ] ) possesses an odd number of finite solutions . is the solution of equation ( [ xihinf ] ) with the largest value . ] in order to establish whether this is the case , it is convenient to define the function of @xmath13@xmath137\;,\label{integral}\]]whose points of crossing with @xmath138 correspond to the solutions of equation ( [ xihinf ] ) . of course , for @xmath139 to be well defined ( and therefore for solutions of ( [ xihinf ] ) to exist at all ) one needs the integral defining it to be convergent . for asymptotically ( @xmath140 ) exponential and power - law behaviours of @xmath43 this happens in the cases already described . now , whenever @xmath139 is well defined , it is clearly a monotonically decreasing function , because the integrand in equation ( [ integral ] ) is always strictly positive . for @xmath123 , @xmath141 is just equal to the integral of the function @xmath124 evaluated at @xmath13 , up to a finite constant . in this limit , @xmath142 so we can write , for @xmath52:@xmath143given the condition @xmath144 , it is clear that the function @xmath141 is always greater than @xmath138 for @xmath52 . on the other hand , for @xmath145 , the asymptotic behaviour of @xmath141 is obtained by expanding @xmath100 in ( [ integral ] ) , which gives@xmath146for critical ( and non - extremal ) black holes and@xmath147for extremal ones . if , for @xmath4 , @xmath141 is smaller ( greater ) than @xmath138 , then equation ( [ xihinf ] ) has an odd ( even ) number of finite solution , and the event horizon does ( does not ) exist . note that , if @xmath148 , one must analyse subdominant terms in the asymptotic behaviour of @xmath43 in order to draw any conclusion . with either method , we find that for @xmath149 the existence of an event horizon in double - sided configurations follows the same rules as in the previously analysed one - sided configurations . when @xmath150 however , it is more difficult to have an event horizon in a double sided configuration , and in general , one has to increase the _ rapidity _ with which one approaches the sonic regime . more specifically , for a critical black hole and an asymptotically exponential @xmath43 , the event horizon exists if @xmath151 , but not when @xmath152 , while for a power law there is no horizon . for an extremal black hole and an exponential @xmath43 the horizon always exists , but in the case of a power law it does not exist if @xmath153 , and it exists for @xmath154 , with the additional condition @xmath155 for the particular value @xmath115 . for a critical black hole with asymptotically exponential @xmath43 and @xmath156 , as well as for an extremal black hole with a power law and @xmath115 , @xmath157 , the asymptotic analysis is not sufficient and one must take into account also subdominant terms in the expansion of @xmath43 for @xmath145 . for those situations in which an event horizon exists , we now find the asymptotic relation between @xmath69 and @xmath68 for rays close to the horizon generator . we also briefly discuss the implications of such a relation for quasi - particle creation in the various cases of interest . consider a sound ray corresponding to a value @xmath91 . for @xmath68 very close to @xmath101 , @xmath75 is very close to @xmath34 , and we can use the approximation ( [ appxi - bh ] ) for @xmath43 . furthermore , we can approximate @xmath100 as in ( [ appv - bh ] ) , so equation ( [ u - fin ] ) gives@xmath158 ^ 2)\;.\label{ut0-bh}\]]this provides us with the link between @xmath68 and @xmath75 . in order to link @xmath75 with @xmath69 , consider the integral on the right hand side of equation ( [ uu ] ) . for @xmath19 , the integrand function vanishes , while near @xmath159 it can be approximated by @xmath160 . then the integral is just given by the difference of the corresponding integrals evaluated at @xmath96 and @xmath161 , respectively , up to a possible finite positive constant . this gives given in reference @xcite . using equation ( 4.2 ) from that paper we have@xmath162expanding , we find again equation ( [ porcoqua ] ) , to the leading order in @xmath163.]@xmath164together with equation ( [ ut0-bh ] ) , this leads to@xmath165this relation between @xmath68 and @xmath69 is exactly the one found by hawking in his famous analysis of particle creation by a collapsing star @xcite . it is by now a standard result that this relation implies the stationary creation of particles with a planckian spectrum at temperature @xmath166 @xcite . for a critical black hole , the results are very different according to whether the sonic regime is attained in a finite or an infinite laboratory time . the calculation of the relation between @xmath68 and @xmath69 is exactly equal to the one presented for the non - extremal black hole case . the two geometries coincide everywhere to the right of the apparent horizon and can not be distinguished by the quasi - particle production observed at @xmath167 . let us suppose that we are in a situation in which the event horizon exists , so @xmath101 is finite . for another right - moving sound ray that corresponds to a value @xmath91 we find , combining equations ( [ uu ] ) and ( [ baru - cbh]),@xmath168\;.\label{deltau - cbh}\]]in the integration interval , @xmath43 is close to zero , so equation ( [ deltau - cbh ] ) can be approximated as@xmath169where the expansion ( [ appv - bh ] ) has been used . equation ( [ deltau-cbh ] ) gives@xmath170for an asymptotically exponential @xmath30 , and@xmath171for a power law with @xmath109 . for the link between @xmath75 and @xmath69 we obtain@xmath172as one can easily check inserting the appropriate asymptotic expansions into equation ( [ uu ] ) . ) which , expanded , gives equation ( [ tu ] ) . the result holds , however , independently of the details of @xmath16 . ] using equation ( [ tu ] ) into equations ( [ deltauexp ] ) and ( [ deltaunu ] ) we find@xmath173for the exponential case , and@xmath174for a power law with @xmath109 . ( remember that for @xmath175 the event horizon does not form . ) it is interesting to compare equations ( [ uuexp ] ) and ( [ uunu ] ) with the corresponding one for the non - critical black hole , equation ( [ uu-bh ] ) . whereas the latter is basically independent of the details of the black hole formation ( which only appear in the multiplicative constant ) , the relation between @xmath68 and @xmath69 in the critical case is not universal , but depends on the dynamical evolution . even for an asymptotically exponential @xmath43 , which leads to an exponential dependence on @xmath69 , the coefficient in the exponent is not universal as in equation ( [ uu-bh ] ) , but depends on dynamics through the parameter @xmath46 . this is not difficult to understand looking back at the way in which equations ( [ uu-bh ] ) and ( [ uuexp ] ) have been derived . for equation ( [ uu-bh ] ) , the exponential dependence was introduced relating @xmath75 with @xmath69 , which only involves sound propagation in the final static region and can not , therefore , be affected by dynamics . on the other hand , when deriving equation ( [ uuexp ] ) it is sound propagation in the initial , dynamical , regime that introduces the exponential ( in the particular case of an asymptotically exponential @xmath30 ) ; hence , it is not surprising that the final result keeps track of the dynamical evolution . however , it is interesting to note that in the limit @xmath176 equations ( [ uu-bh ] ) and ( [ uuexp ] ) coincide . this limit corresponds to a very rapid approach towards the formation of an otherwise - never - formed ( in finite time ) apparent horizon . regarding the creation of quasi - particles , this situation is operationally indistinguishable from the actual formation of the sonic point . however , this `` degeneracy '' might be accidental , given that the origin of the exponential relation is very different in the two cases . as for the case of a critical black hole , we must distinguish between a finite and an infinite time of formation of the event horizon . for a sound ray close to the one that generates the horizon , equation ( [ u - fin ] ) still holds . however , now one must use the expansion ( [ appv - ebh ] ) when approximating the integrand thus obtaining@xmath177 ^ 3)\;.\label{deltaut0-ebh1}\]]using again the approximation ( [ appv - ebh ] ) in the evaluation of the integral on the right hand side of equation ( [ uu ] ) one finds@xmath178finally,@xmath179interestingly , this is the same relation that one finds for the gravitational case @xcite . in particular , this implies that finite time collapse to form an extremal black hole will _ not _ result in a planckian spectrum of quasi - particles @xcite . this is completely compatible with the standard gr analysis , and is one of the reasons why extremal and non - extremal black holes are commonly interpreted as belonging to completely different thermodynamic sectors @xcite . assuming that the event horizon exists , we can again apply equation ( [ deltau - cbh ] ) and use the approximation ( [ appv - ebh ] ) in order to find the relation between @xmath68 and @xmath75 . the results are , for an asymptotically exponential @xmath43:@xmath180for a power law with @xmath181:@xmath182for a power law with @xmath115:@xmath183for a power law with @xmath109:@xmath184using the appropriate expansions in equation ( [ t - t ] ) , one obtains the relation between @xmath75 and @xmath69:@xmath185for an asymptotically exponential @xmath43 this becomes@xmath186for a power law , one must again distinguish between three cases ; for @xmath181:@xmath187for @xmath115:@xmath188for @xmath109:@xmath189putting together equations ( [ utexp - ebh ] ) and ( [ tuexp - ebh ] ) one finds the relationship between @xmath68 and @xmath69 for the exponential case:@xmath190for the power law one finds from equations ( [ utnu<1])([utnu>1 ] ) and ( [ utnu<1])([utnu>1 ] ) , for @xmath181:@xmath191for @xmath115:@xmath192and finally , for @xmath109:@xmath193 in all these cases , quasi - particle production is neither universal , nor planckian . it is not difficult to prove that in the formation , in a finite amount of time , of double - sided non - extremal black holes , double - sided extremal black holes , and double - sided critical black holes , the asymptotic relation between @xmath68 and @xmath69 is identical to that calculated in the corresponding subsections above . the amount and features of quasi - particle creation are then the same . we will demonstrate this in detail for the case of a double - sided critical black hole , and then proceed to consider the situation in which the formation takes place in an infinite amount of time . using the same notation as in section [ paragraph:2cbh2 ] , let us call @xmath125 the largest of the @xmath125 s that satisfy equation ( [ ziopollo ] ) , so @xmath125 is the time at which a right - going ray last crosses the kink on the left . there will be some regular relationship between @xmath68 and @xmath125 , expressed by some differentiable function @xmath139 , so that we can write @xmath194 . for the event horizon to exist , the corresponding @xmath68 must be finite ( equal to some value @xmath101 , say ) , so also @xmath195 must be finite ( as already done in section [ paragraph:2cbh2 ] , we denote by a suffix `` h '' the quantities that correspond to the horizon generator ) . for a ray very close to the horizon generator we have@xmath196where a dot denotes the derivative with respect to @xmath13 . on the horizon , @xmath197 so equation ( [ ziopollo ] ) reduces to@xmath198\;.\label{ziogallo}\]]subtracting ( [ ziogallo ] ) from ( [ ziopollo ] ) we obtain@xmath199 -\int_{t_2}^{t_{\rm h}}\d t\left[c + \bar{v}\left(\xi(t)\right)\right]\;.\label{ziopollogallo}\]]for a ray close to the horizon generator , @xmath127 is close to @xmath200 , and @xmath125 close to @xmath131 , so equation ( [ ziopollogallo ] ) gives , keeping only terms to the leading order:@xmath201together , equations ( [ uf ] ) and ( [ eccola ] ) provide a linear link between @xmath68 and @xmath127 . since the relationship between @xmath127 and @xmath69 is exactly the same as the one between @xmath75 and @xmath69 in equation ( [ porcoqua ] ) , the final result is again the one expressed by ( [ uu-bh ] ) : @xmath202 assuming that we are in a situation for which the event horizon does indeed exist , we can subtract equation ( [ xihinf ] ) with @xmath203 from equation ( [ ziopollo ] ) , finding:@xmath204-\int_{t_2}^{+\infty}\d t\left[c + \bar{v}\left(\xi(t)\right)\right]\;.\label{xi - xi}\]]for a ray close to the horizon generator , @xmath125 is close to @xmath131 and @xmath127 is large , so@xmath205for an asymptotically exponential @xmath43 we find , performing the integral,@xmath206similarly , for a power law with @xmath109:@xmath207 in both cases , the same results as in section [ subsec : cbh ] , equations ( [ uuexp ] ) and ( [ uunu ] ) , follow . in short , the amount and characteristics of the quasi - particle production calculated with the double - sided configurations are exactly the same as those calculated with the simpler profiles in the previous subsections except in two specific situations : the double - sided critical black hole with @xmath150 ( see figure [ f : spike ] ) and the double - sided extremal black hole . in the critical case , only the asymptotically exponential behaviour with @xmath208 produces an event horizon and , therefore , only then we can talk about a stationary and planckian creation of quasi - particles . in the extremal case the results described in section [ subsubsec : ebh2 ] only apply for @xmath209 ( with the further condition @xmath210 in the particular case @xmath115 ) , because otherwise the event horizon itself does not exist . in the present paper we have analyzed different dynamical black hole - like analogue geometries with regard to their properties in terms of quantum quasi - particle production . we have taken several @xmath1-dimensional spacetimes ( considered as externally fixed backgrounds ) , and for each of them we ( i ) have calculated whether it possesses an event horizon or not , and if the answer is `` yes '' , ( ii ) have calculated the asymptotic redshift function that characterizes the amount and properties of the late - time quasi - particle production . in table [ table ] the reader can find a summary of all our results . .this is a summary of the results found for the different configurations analyzed in the paper . to the table we have to add the following comments : for double - sided extremal black hole with @xmath150 , with an infinite time of formation and an asymptotic power law for @xmath43 , the horizon forms in the case @xmath115 if the further condition @xmath155 holds . for the double - sided configurations with @xmath150 and an infinite time of formation , the asymptotic analysis is not sufficient for drawing conclusions when @xmath43 is asymptotically exponential and @xmath156 , and when @xmath43 is asymptotically a power law and @xmath115 , @xmath157 . for those cases , one need also consider subdominant terms in @xmath43 , so the results will depend on the details of formation . [ cols="^,^,^,^,^,^",options="header " , ] the above results are pertinent to a purely mathematical model . their physical relevance has to be assessed with respect to their application to both experimental reproduction of the analogue hawking radiation , and to the lessons they can provide concerning the possible behavior of black hole formation and evaporation in semiclassical gravity . we now turn to separately consider these two issues . the study carried on in this paper has identified several velocity profiles that are potentially interesting for experiments . in particular the critical black hole models seem worth taking into consideration in connection with the realizability of a hawking - like flux in the laboratory . the creation of supersonic configurations in a laboratory is usually associated with the development of instabilities . there are many examples of the latter in the literature ; in reference @xcite it was shown that in an analogue model based on ripplons on the interface between two different sliding superfluids ( for instance , @xmath211he - phase a and @xmath211he - phase b ) , the formation of an ergoregion would make the ripplons acquire an amplification factor that eventually would destroy the configuration . therefore , this analogue system , although very interesting in its own right , will prove to be useless in terms of detecting a hawking - like flux . however , by creating , instead of an ergoregion , a critical configuration one should be able , at least , to have a better control of the incipient instability , while at the same time producing a dynamically controllable hawking - like flux . nevertheless , the actual realization of a critical configuration could also appear as a problematic task for entirely different reasons . the corresponding velocity profiles are characterized by discontinuities in the derivatives , so one might wonder whether they would be amenable to experimental construction , given that the continuum model is only an approximation . let us therefore discuss in some detail the validity of the latter for realistic systems . the main difference between an ideal perfect fluid model and a realistic condensed matter analogue is due to the microscopic structure of the system considered . in particular , it is generic to have a length scale @xmath212 which characterizes the breakdown of the continuum model ( @xmath212 is of the order of the intermolecular distance for an ordinary fluid ; of the coherence length for a superfluid ; and of the healing length for a bose einstein condensate ) . in general , the viability of the analogue model requires one to consider distances @xmath213 of order of at least a few @xmath212 , depending on the accuracy of the experiment performed . in particular , wave propagation is well defined only for wavelengths larger than @xmath212 ( generally with an intermediate regime , for wavelengths between @xmath212 and @xmath213 , where the phenomena exhibit deviations with respect to the predictions based on the continuum model ) . in general , a mathematical description based on the continuum model contains details involving scales smaller than @xmath213 ( for example , in the velocity profile ) . these details should , however , be regarded as unphysical : they are present in the model , but do not correspond to properties of the real physical system . in particular , they can not be detected experimentally , because this would require using wavelengths smaller than @xmath213 , which do not behave according to the predictions of the model ( and for wavelengths smaller than @xmath212 do not even make physical sense ) . for the mathematical models considered in the present paper , all this implies that one will not be able to distinguish , on empirical grounds , between those cases for which the velocity profiles differ from each other only by small - scale details . in particular , double - sided configurations with @xmath150 should be equivalent to configurations with a small , but non - zero , thickness @xmath214 . also , one would not be able to distinguish between two velocity profiles that differ only in a neighborhood @xmath213 of @xmath22 , one of which corresponds to a critical black hole , while the other describes an extremal one . in particular hawking radiation will not distinguish between the models within each of these pairs . this fact would not be troublesome , had our analysis led to identical results for the acoustic black holes of each pair . however , this is not the case ( see table [ table ] ) . but then what shall we see if we realize these models in a laboratory ? in realistic situations , what is relevant for hawking radiation is a coarse - grained profile obtained by averaging over a scale of order @xmath213 , thus neglecting the unphysical small scale details in @xmath16 . this implies that as far as double - sided critical black holes are concerned , the reliable results are those pertinent to the non - zero thickness case ( @xmath215 ) . similarly , since these extremal black holes are never exactly realizable in a laboratory ( as this would require tuning the velocity profile on arbitrary small scales ) , only the predictions based on the critical black hole mathematical model will survive in an experimental setting . indeed , the relevant surface gravity will be defined by averaging the slope of the velocity profile over scales which are of order of @xmath213 . this averaged surface gravity will be non - zero for both the critical and the extremal black hole , but will be approximately equal to the surface gravity at the horizon of the critical black hole , while it will obviously not coincide with the one of the extremal ( which is zero ) . in the body of the paper we have used a terminology particularly suitable to dealing with analogue models based on acoustics . let us now discuss the most relevant features of our findings using a language more natural to gr . when the geometry associated with the formation of a spherically symmetric black hole through classical gravitational collapse ( as , for example , in the oppenheimer - snyder model @xcite ) is described in terms of painlev - gullstrand @xcite coordinates ( whose counterpart , in the context of acoustic geometries , are the natural laboratory coordinates @xmath12 and @xmath13 ) , the apparent horizon forms in a finite amount of coordinate time . in this regard , the painlev - gullstrand time behaves similarly to the proper time measured by a freely - falling observer attached to the surface of the collapsing star . the non - extremal , non - critical @xmath1-dimensional model analysed in this paper , captures the main features of the formation of a ( non - extremal ) black hole . the dynamical collapse is represented by the function @xmath43 in our calculations . in the language of gr , we can think of @xmath43 as the radial distance between the surface of a collapsing star and its schwarzschild radius ; @xmath35 corresponds to the moment in which the surface of the star enters its schwarzschild radius , and this moment corresponds to a finite time ( which we took to be @xmath34 ) . for this model we recovered hawking s result that the formation of ( non - extremal ) black holes causes the quantum emission towards infinity of a stationary stream of radiation with a planckian spectrum , at temperature @xmath166 . the mechanism for particle creation is somewhat `` more than dynamical '' as the characteristics of the stationary stream of particles are `` universal '' and only depend on the properties of the geometry at the horizon , @xmath47 , and not on any detail of the dynamical collapse . indeed , for @xmath43 given by equation ( [ appxi - bh ] ) apparent horizon formation in a finite amount of time we have seen that asymptotic quasi - particle creation does not depend even on the coefficient @xmath38 . that is , particle production does not depend on the velocity with which the surface of the collapsing star enters its schwarzschild radius . this picture leans toward the ( quite standard ) view that hawking s process is not just dynamical , but relies on the actual existence of an apparent horizon and an `` ergoregion '' beyond it , able to absorb the negative energy pairs @xcite . however , by analyzing alternative models , in this paper we have seen two unexpected things : i ) : : one can also produce a truly hawking flux with a temperature @xmath166 through the formation in a finite amount of time of either a single - sided critical black hole , or a double - sided critical black hole of finite thickness " , or even one of zero thickness " ( see figure [ f : spike ] ) . this is an intriguing result , as in none of these cases there is an `` ergoregion '' beyond the apparent horizon , and in the last case there is just a single sonic point . ( in the language of gr , this last configuration corresponds to stopping the collapse of a star at the very moment in which its surface reaches the schwarzschild radius . ) ii ) : : moreover , one can also produce a stationary and planckian emission of quasi - particles by , instead of actually forming the apparent horizon , just approaching its formation asymptotically in time with sufficient rapidity ( @xmath216 ) . in this case the temperature is not @xmath166 but @xmath217 , with@xmath218and ( at any finite time ) there is neither an apparent horizon nor an ergoregion within the configuration . explanations of particle production based on tunneling then seem not viable , and the phenomenon is closer to being interpreted as dynamical in origin . if fact , these configurations interpolate between situations in which the dynamics appears more prominently when @xmath219 we have that the temperature goes as @xmath220 and others in which the characteristics of the approached configuration are the more relevant and `` universality '' is recovered when @xmath221 we have that the temperature goes as @xmath166 , indistinguishable from hawking s result . by looking at our simple critical model , we can say that , in geometrical ( kinematical ) terms , in order to obtain a steady and universal flux of particles from a collapsing ( spherically symmetric ) star there is no need for its surface to actually cross the schwarzschild radius ; it is sufficient that it tend towards it asymptotically ( in proper time ) , with sufficient rapidity . our critical configurations could prove to be relevant also in the overall picture of semiclassical collapse and evaporation of black hole - like objects . our results based on critical configurations suggest an alternative scenario to the standard paradigm . at this stage we are only able to present it in qualitative and somewhat speculative terms . being aware of the various assumptions that could ultimately prove to be untenable , we still think it is worth to present this possible alternative scenario . imagine a dynamically collapsing star . the collapse process starts to create particles dynamically before the surface of the star crosses its schwarzschild radius ( this particle creation is normally associated with a transient regime and has nothing to do with hawking s planckian radiation ) . the energy extracted from the star in this way will make ( due to energy conservation ) its total mass decrease , and so also its schwarzschild radius . by this argument alone , we can see that a process is established in which the surface of the star starts to closely chase its schwarzschild radius while both collapse towards zero ( this situation was already described by boulware in reference @xcite ) . now , the question is : will the surface of the shrinking star capture its shrinking schwarzschild radius in a finite amount of proper time ? let us rephrase this question in the language of this paper . in an evaporating situation our function @xmath43 still represents the distance between the surface of the star and its schwarzschild radius . the standard answer to the previous question is that @xmath43 becomes zero in a finite amount of proper time . to our knowledge , this view ( while certainly plausible ) is not guaranteed by explicit systematic and compelling calculations but still relies on somewhat qualitative arguments . the standard reasoning can be presented as follows : for sufficiently massive collapsing objects , the classical behaviour of the geometry should dominate any quantum back - reaction at any and all stages of the collapse process , as hawking s temperature ( considered as an estimate of the strength of this back - reaction ) is very low ; quantum effects would be expected to become important only at the last stages of the evaporation process . however , in opposition to this standard view , stephens , t hooft , and whiting @xcite have argued for the mutual incompatibility of the existence of external observers measuring a hawking flux and , at the same time , the existence of infalling observers describing magnitudes beyond the apparent horizon . the reason is that the operators describing any feature of the hawking flux do not commute ( and this non - commutation blows up at late times ) with the infalling components of the energy - momentum tensor operator at the horizon . therefore , if we accept this argument , the presence of a hawking flux at infinity would be incompatible with the actual formation of the trapping horizon , which would be destroyed by the back - reaction associated to hawking particles . this fact leads these authors ( seeking for self consistency ) , to look for the existence of a hawking flux ( or at least a flux looking very much like it ) in background geometries in which the collapse process of the star is halted , just before crossing the schwarzschild radius , producing a bounce . ( in our language this could be represented by a function @xmath43 monotonically decreasing from @xmath222 to some @xmath223 , at which it reaches a very small positive value , and then monotonically increasing from @xmath223 to @xmath224 . ) in their analysis they found exactly that : an approximately planckian spectrum of particles present at infinity during a sufficiently long time interval . however , the modified behaviour that deviates the least from the classical collapse picture , and at the same time eliminates the trapping horizon , is that in which @xmath43 does not reach zero , but just `` asymptotically approaches zero '' at infinite proper time , and does that very quickly . this is represented in our critical configurations by the exponential behaviour @xmath225 with a very large @xmath46 . the interesting point is that the analysis in this paper suggests that with quasi - stationary configurations like this , one could expect quasi - stationary planckian radiation at a temperature very close to @xmath166 , just like in the hawking process . standard gr suggests that the surface gravity @xmath47 ( inversely proportional to the total mass of the star ) would increase with time through the back - reaction caused by the quantum dissipation . moreover , it is sensible to think that during the evaporation process @xmath46 would also depend on @xmath13 . as the evaporation temperature increases ( @xmath47 increases ) the back - reaction would become more efficient and therefore we might expect that @xmath46 decreases . then , one could arrive at a situation as the one portrayed in figure [ f : critical - collapse ] . the evolution of the evaporation temperature would interpolate between a starting temperature completely controlled by @xmath47 and a late time temperature completely controlled by @xmath46 , showing a possible semiclassical mechanism for regularizing the end point of the evaporation process . in this scenario the complete semiclassical geometry will have neither an apparent horizon nor an event horizon . in this circumstance there would be no trans - planckian problem , nor information loss associated with the collapse and evaporation of this black hole - like object . whether this scenario is viable or not will be the subject of future work . let us end by making a brief comment concerning modified dispersion relations . everything we said in this paper assumes strict adherence to lorentz symmetry . even if semiclassical gravity contained lorentz - violating traces in the form of modified dispersion relations at high energy , one would still expect that the resulting scenario for the collapse and evaporation of a black hole - like object would keep the quasi - stationary hawking - like flux of particles as a robust prediction @xcite . however , the complete conceptual scenario could be very different . in the presence of dispersion at high energies , the notion of horizon itself shows up only as a low - energy concept . for example , with superluminal modifications of the dispersion relations , high energy signals will be able to escape from the trapped region . the non - analytic behaviour of some sets of modes at the horizon become regularized . therefore , the stephenst hooft whiting obstruction described above forbidding the formation of a ( now approximate ) trapping horizon need no longer apply . we expect that by analyzing different analogue models in which the lorentz violating terms appear at different energy scales one would be able to explore the transition between all these alternative paradigms . the authors would like to thank vctor aldaya and ted jacobson for stimulating discussions . . has been funded by the spanish mec under project fis2005 - 05736-c03 - 01 with a partial feder contribution . c.b . and s.l are also supported by a infn - mec collaboration . the research of m.v . was funded in part by the marsden fund administered by the royal society of new zealand . m.v . also wishes to thank both isas ( trieste ) and iaa ( granada ) for hospitality . novello , m. visser and g. volovik ( eds . ) , _ artificial black holes _ ( singapore , world scientific , 2002).c . barcel , s. liberati and m. visser , `` analogue gravity , '' living rev . relativity * 8 * , 12 ( 2005 ) [ arxiv : gr - qc/0505065 ] . url ( cited on 22 march 2006 ) : http://www.livingreviews.org/lrr-2005-12c . barcel , s. liberati and m. visser , `` analogue gravity from bose - einstein condensates , '' class . quantum grav . * 18 * , 11371156 ( 2001 ) [ arxiv : gr - qc/0011026].m . visser , c. barcel and s. liberati , `` analogue models of and for gravity , '' gen . relativ . gravit . * 34 * , 17191734 ( 2002 ) [ arxiv : gr - qc/0111111].c . barcel , s. liberati , s. sonego and m. visser , `` causal structure of analogue spacetimes , '' new j. phys . * 6 * , 186 ( 2004 ) [ arxiv : gr - qc/0408022].s . w. hawking , `` black hole explosions , '' nature * 248 * , 3031 ( 1974 ) ; 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( paris ) * 173 * , 677680 ( 1921 ) . + a. gullstrand , `` allgemeine lsung des statischen einkrperproblems in der einsteinschen gravitationstheorie , '' ark . mat . * 16 * , 115 ( 1922).l . parker , `` the production of elementary particles by strong gravitational fields , '' in _ asymptotic structure of space - time _ , edited by f. p. esposito and l. witten ( new york , plenum , 1977 ) , pp . 107226.g . e. volovik , `` black - hole horizon and metric singularity at the brane separating two sliding superfluids , '' pisma zh . fiz . * 76 * , 296300 ( 2002 ) [ jetp lett . * 76 * , 240244 ( 2002 ) ] [ arxiv : gr - qc/0208020].d . g. boulware , `` hawking radiation and thin shells , '' phys . d * 13 * , 21692187 ( 1976).a . ashtekar and m. bojowald , `` black hole evaporation : a paradigm , '' class . quantum grav . * 22 * , 33493362 ( 2005 ) [ arxiv : gr - qc/0504029].c . r. stephens , g. t hooft and b. f. whiting , `` black hole evaporation without information loss , '' class . quantum grav . * 11 * , 621647 ( 1994 ) [ arxiv : gr - qc/9310006].w . g. unruh and r. schtzhold , `` on the universality of the hawking effect , '' arxiv : gr - qc/0408009 .
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we discuss the issue of quasi - particle production by `` analogue black holes '' with particular attention to the possibility of reproducing hawking radiation in a laboratory . by constructing simple geometric acoustic models
, we obtain a somewhat unexpected result : we show that in order to obtain a stationary and planckian emission of quasi - particles , it is _ not _ necessary to create an ergoregion in the acoustic spacetime ( corresponding to a supersonic regime in the flow ) .
it is sufficient to set up a dynamically changing flow _ either _ eventually generating an arbitrarily small sonic region @xmath0 , but without any ergoregion , _ or _ even just asymptotically , in laboratory time , approaching a sonic regime with sufficient rapidity .
pacs : 04.20.gz , 04.62.+v , 04.70.-s , 04.70.dy , 04.80.cc + keywords : analogue models , acoustic spacetime , hawking radiation
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You are an expert at summarizing long articles. Proceed to summarize the following text:
centaurus a ( ngc 5128 ) is the nearest ( d = 3.5 mpc ; 1 @xmath117 pc , hui et al . 1993 ) example of a giant elliptical galaxy associated with a powerful radio source . the large - scale radio morphology consists of twin radio lobes separated by @xmath1 5 degrees on the sky . the compact ( @xmath1 milliarcsecond ) radio nucleus is variable and has a strong jet extending @xmath1 4 arcminutes towards the northeast lobe . the spectacular optical appearance is that of a giant elliptical galaxy that appears enveloped in a nearly edge on , warped dust lane . there is also a series of faint optical shells . the stellar population in the dominant elliptical structure is old , whilst that of the twisted dust lane is young , sporadically punctuated by hii regions , dust and gas ( graham 1979 ) . the overall structure of cen a resembles that of a recent ( @xmath9 years , tubbs 1980 ) merger , between a spiral and a large elliptical galaxy . the dust lane is the source of most ( 90 % ) of the far - infrared luminosity ( l@xmath10 l@xmath11 ) and is thought to be re - radiated starlight from young stars in the dusty disk ( joy et al . 1988 ) . in sect . 2 we describe the observations and data analysis . sect . 3 looks at the general fir properties and proceeds to model the hii regions and the pdrs in the dust lane . 4 summarises the results and presents our conclusions . cen a was observed with the lws grating ( @xmath12 ) as part of the lws consortium s guaranteed time extragalactic programme . a full grating observation ( 43 - 196.7 @xmath0 m ) was taken of the nucleus at the centre of the dust lane and a series of line observations were taken at two positions in the se and nw regions of the dust lane . a short 157 @xmath0 m line observation was taken off - source at position # 4 ( see table 1 ) to estimate the galactic emission near the source . position # 1 was intended to provide a deeper integration coincident with position # 2 , but was accidently offset . a series of half - second integration ramps were taken at each grating position with four samples per resolution element ( @xmath13 m @xmath14 m and @xmath15 m @xmath16 m ) . the total integration time per resolution element and per pointing were : position # 1 88 s for the 52 @xmath0 m and 34 s for the 57 @xmath0 m ; position # 2 ( the centre ) , 30 s for the range 43196 @xmath0 m ; positions nw and se ( 2 point raster map ) 22 s for the the 63 @xmath0 m , 14 s for the 88 @xmath0 m , 12 s for the 122 @xmath0 m , 28 s for the 145 @xmath0 m and 12 s for the 158 @xmath0 m ; position # 4 12 s for the 158 @xmath0 m . the data were processed with ral pipeline 7 and analysed using the lia and isap packages . the lws flux calibration and relative spectral response function ( rsrf ) were derived from observations of uranus ( swinyard et al . 1998 ) . the full grating spectrum at the centre enabled us to estimate the relative flux uncertainty between individual detectors arising from uncertainties in the relative responsivity and the dark - current subtraction . the offsets between the detectors ( excluding detector sw1 ) was @xmath17 % . the 88 @xmath0 m line on detectors sw5 and lw1 had a 15 % systematic uncertainty and the line on detectors lw3 and lw4 had a 10 % systematic uncertainty . we therefore adopt a relative flux uncertainty of @xmath1 15% . because we only took spectra of individual lines at the nw and se positions there is no corresponding overlap in wavelength coverage at these positions . one indicator of relative flux uncertainty is a discrete step down in flux , of @xmath1 25 % , at @xmath1 125 @xmath0 m at the se position . the relative flux uncertainty is assumed to be @xmath18 25 % at these positions . the absolute flux calibration w.r.t . uranus for point like objects observed on axis is better than 15 % ( swinyard et al . 1998 ) . however , extended sources give rise either to channel fringes or to a spectrum that is not a smooth function of wavelength . this is still a calibration issue . for example , in fig . 2 , detectors sw5 , lw1 , lw2 have slopes that differ from those of their neighbours in the overlap region . this may account for the continuum shape , which is discussed in sect . the lws beam profile is known to be asymmetric and is still under investigation . we therefore adopt a value for the fwhm of 70 at all wavelengths , believing that a more sophisticated treatment would not significantly affect our conclusions . we also note that there is good cross calibration between the iso - lws results and the far - infrared imaging fabry - perot interferometer ( fifi ) ( madden et al . 1995 ) ; the peak fluxes agree to within @xmath1 10 % . . observation log [ cols= " < , > , > , < " , ] the results for the three regions are consistent with each other , having a gas density , n @xmath1 10@xmath19 @xmath7 , and an incident far - uv field , g @xmath1 10@xmath20 . at the nw position , only the combination of the 63 @xmath0 m / 158 @xmath0 m ratio and the ( 63 @xmath0 m + 158 @xmath0 m ) /fir continuum ratio gives a meaningful solution for g and n. the 146 @xmath0 m line is clearly detected but with a very rippled baseline due to channel fringes . the observed 146 @xmath0 m line flux would need to be reduced by @xmath1 60 % in order to obtain a consistent result with the 146 @xmath0 m / 63 @xmath0 m line ratio predicted by the pdr model . the lws results for the nucleus confirm those previously derived from ir , submm and co observations . the consistent set of derived pdr conditions for all three positions suggest that the observed fir emission in a 70 beam centred on the nucleus is dominated by star formation and not agn activity . joy et al . ( 1988 ) mapped cen a at 50 and 100 @xmath0 m on the kao . they concluded that the extended fir emission was from dust grains heated by massive young stars distributed throughout the dust lane , not the compact nucleus . hawarden et al . ( 1993 ) mapped cen a at 800 @xmath0 m and 450 @xmath0 m with a resolution of @xmath110 . they attribute the large scale 800 @xmath0 m emission to thermal emission from regions of star formation embedded in the dust lane . they note that the h@xmath21 emission within a few arcseconds of the nucleus , observed by israel et al . ( 1990 ) , indicates that significant uv radiation from the nucleus does not reach large radii in the plane of the dust lane i.e. the nuclear contribution to exciting the extended gas and dust disk is small . eckart et al . ( 1990 ) and wild et al . ( 1997 ) mapped cen a in @xmath22co j=1 - 0 , @xmath22co j=2 - 1 and @xmath23co j=1 - 0 . all three maps have two peaks separated by @xmath1 90 centred on the nucleus . it is interesting to note that our se position only clips the lowest contours of the co ( 1 - 0 ) and co ( 2 - 1 ) maps of wild et al . ( 1997 ) . in spite of this the derived pdr parameters are consistent with those encompassing the bulk of the molecular emission . there must be extended low level co ( 1 - 0 ) emission beyond the sensitivity limits of the wild et al . ( 1997 ) maps . the lowest contour is 17.5 k kms@xmath24 , corresponding to m@xmath25 @xmath1 10@xmath26 m@xmath11 if the material filled the lws beam . we present the first full fir spectrum from 43 - 196.7 @xmath0 m of cen a. we detect seven fine structure lines ( see table 2 ) , the strongest being those generated in pdrs . at the central position , the total flux in the far - infrared lines is @xmath1 1 % of the total fir luminosity ( l@xmath27 l@xmath11 for a distance of 3.5 mpc ) . the line flux is @xmath10.4 % fir and the line flux is @xmath1 0.2 % fir . these are typical values for starburst galaxies ( lord et al . the 52 @xmath0 m / 88 @xmath0 m line intensity ratio is @xmath1 0.9 , which corresponds to an electron density , n@xmath28 100 @xmath7 ( rubin et al . the _ thermal pressure _ of the ionized medium in the cen a dust lane is closer to that of starburst galaxies ( n@xmath29 250 @xmath7 in m82 ( colbert et al . 1999 ) and m83 ( stacey et al . 1999 ) ) than that of the milky way ( n@xmath29 3 @xmath7 ( pettuchowski & bennett 1993 ) ) . the / line intensity ratio is @xmath1 1.6 , giving an abundance ratio n++/n+ @xmath1 0.3 , which corresponds to an effective temperature , t@xmath30 35500 k ( rubin et al . 1994 ) . assuming a coeval starburst , then the tip of the main sequence is headed by o8.5 stars , and the starburst is @xmath1 6 @xmath4 years old . if the burst in cen a was triggered by the spiral - elliptical galaxy merger then its occurance was very recent . alternatively , the merger triggered a series of bursts of star formation and we are witnessing the most recent activity . we estimate that the n / o abundance ratio is @xmath1 0.2 in the hii regions in cen a. this value is consistent with the range of @xmath1 0.2 - 0.3 found for galactic hii regions ( rubin et al . n / o is a measure of the chemical evolution and we expect it to increase with time ( c.f . the solar value of @xmath1 0.12 ) . we estimate that @xmath1 10 % of the observed arises in the wim . the cnm contributes very little ( @xmath31 % ) emission at our beam positions . the bulk of the emission is from the pdrs . we derive the average physical conditions for the pdrs in cen a for the first time . there is active star formation throughout the dust lane and in regions beyond the bulk of the molecular material . the fir emission in the 70 lws beam at the nucleus is dominated by emission from star formation rather than agn activity . on scales of @xmath1 1 kpc the average physical properties of the pdrs are modelled with a gas density , n @xmath1 10@xmath19 @xmath7 , an incident far - uv field , g @xmath1 10@xmath20 and a gas temperature of @xmath1 250 k. many thanks to the dedicated efforts of the lws instrument team . the iso spectral analysis package ( isap ) is a joint development by the lws and sws instrument teams and data centers . contributing institutes are cesr , ias , ipac , mpe , qmw , ral and sron . colbert j.w . , malkan m.a . , clegg p.e . , et al . , 1999 , apj 511 , 721 + eckart a. , cameron m. , rothermel h. , et al . , 1990 , apj 363 , 451 + graham j. , 1979 , apj 232 , 60 + hawarden t.g . , sandell g. , matthews h.e . , et al . , 1993 , mnras 260 , 844 + heiles c. , 1994 , apj 436 , 720 + hui x. , ford h.c . , ciardillo r. , et al . , 1993 , apj 414 , 463 + israel f.p . , van dishoeck e.f . , baas f. et al . , 1990 , a&a 227 , 342 + joy m. , lester d.f . , harvey p.m. , et al . , 1988 , apj 326 , 662 + kaufman m.j . , wolfire m.g . , hollenbach d. , et al . , 1999 , apj in press + kulkarni s.r . , heiles c. , 1987 , in hollenbach , d. , thronson jr , h.a . interstellar processes . reidel , dordrecht , p. 87 + launay j.m . , roueff e. , 1977 , jphysb 10 , 879 + lester d.f . , dinnerstein h.l . , werner m.w . , et al . , 1987 , apj 320 , 573 + lord s.d . , malhotra s. , lim t.l . , et al . , 1996 , a&a 315 , l117 + madden s.c . , geis n. , genzel r. , et al , 1993 , apj 407 , 579 + madden s. , geis n. , townes c.h.et al . , 1995 , airbourne astronomy symposium on the galactic ecosystem , asp conf . series 73 , 181 . + meyer d.m . , jura m. , cardelli j.a . , 1998 , apj 493 , 222 + petuchowski s.j . , bennett c.l . , 1993 , apj 405 , 591 + rubin r.h . , 1985 , apjs 57 , 349 + rubin r.h . , simpson j.p . , erickson e.f . , et al . , 1988 , apj 327 , 377 + rubin r.h . , simpson j.p . , lord s.d . , et al . , 1994 , apj 420 , 772 + sofia u.j . , cardelli j.a . , guerin k.p . , et al . , 1997 , apj 482 , l105 + stacey g.j . , swain m.r . , bradford c.m . , et al . , 1999 , the universe as seen by iso , esa sp-427 p973 + swinyard b.m . , burgdorf m.j . , clegg p.e et al . , 1998 , spie 3354 + tubbs a.d . , 1980 , apj 241 , 969 + van gorkom j.h . , van der hulst j.m . , haschick a.d . , et al . , 1990 aj 99 , 1781 + wild w. , eckart a. , wilkind t. , 1997 , a&a 322 , 419 + wolfire m.g . , tielens a.g.g.m . , hollenbach d. , 1990 , apj 358 , 116 + wolfire m.g . , tielens a.g.g.m , hollenbach d. , 1995 , apj 443 , 152 +
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we present the first full fir spectrum of centaurus a ( ngc 5128 ) from 43 - 196.7 @xmath0 m .
the data was obtained with the iso long wavelength spectrometer ( lws ) .
we conclude that the fir emission in a 70 beam centred on the nucleus is dominated by star formation rather than agn activity .
the flux in the far - infrared lines is @xmath1 1 % of the total fir : the line flux is @xmath1 0.4 % fir and the line is @xmath1 0.2 % , with the remainder arising from , and lines .
these are typical values for starburst galaxies .
the ratio of the / line intensities from the hii regions in the dust lane corresponds to an effective temperature , t@xmath2 @xmath1 @xmath3 k , implying that the tip of the main sequence is headed by o8.5 stars and that the starburst is @xmath1 6 @xmath4 years old .
this suggests that the galaxy underwent either a recent merger or a merger which triggered a series of bursts .
the n / o abundance ratio is consistent with the range of @xmath1 0.2 - 0.3 found for galactic hii regions .
we estimate that @xmath5 5 % of the observed arises in the cold neutral medium ( cnm ) and that @xmath1 10 % arises in the warm ionized medium ( wim ) .
the main contributors to the emission are the pdrs , which are located throughout the dust lane and in regions beyond where the bulk of the molecular material lies . on scales of @xmath1 1
kpc the average physical properties of the pdrs are modelled with a gas density , n @xmath1 @xmath6 @xmath7 , an incident far - uv field , g @xmath1 @xmath8 times the local galactic field , and a gas temperature of @xmath1 250 k. 0h_0
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You are an expert at summarizing long articles. Proceed to summarize the following text:
the diffuse galactic emission ( dge ) arises from interactions of cosmic - rays ( crs ) with interstellar gas and radiation field in the galaxy . due to the smooth nature of the interstellar radiation field and the cr flux after propagation , the fine structure of the dge is determined by the structure of the interstellar gas . getting the distribution of the interstellar gas correct is therefore crucial when modeling the dge . it is generally assumed that galactic crs are accelerated in interstellar shocks and then propagate throughout the galaxy ( see e.g. * ? ? ? * for a recent review . ) . in this paper , cr propagation and corresponding diffuse emission is calculated using the galprop code ( see * ? ? ? * and references within . ) . we use the so - called conventional galprop model @xcite , where the cr injection spectra and the diffusion parameters are chosen such that the cr flux agrees with the locally observed one after propagation . the gas distribution is given as galacto - centric annuli and the diffuse emission is calculated for those same annuli . the distribution of h i is determined from the 21-cm lab line survey @xcite while distribution of molecular hydrogen , h@xmath1 , is found using the co ( @xmath2 ) survey of @xcite assuming @xmath3 . while converting observations of the 21-cm h i line to column density is theoretically possible , it is not practically feasible . to correctly account for the optical depth of the emitting h i gas , one must know its spin temperature , @xmath0 ( see e.g. * ? ? ? * ) . under the assumption of a constant @xmath0 along the line of sight , the column density of h i can be calculated from the observed brightness temperature @xmath4 using @xmath5 where @xmath6 is the background continuum temperature and @xmath7 @xmath8 k ( km / s)@xmath9 . the assumption of a constant @xmath0 along the line of sight is known to be wrong for many directions in the galaxy ( see e.g. * ? ? ? the @xmath0 values derived in this paper are therefore only a global average and should not be taken at face value . figure [ fig : tsratio ] shows how changing @xmath0 affects @xmath10 in a non - linear way , mainly affecting areas with @xmath4 close to @xmath0 in the galactic plane . this figure was created under the assumption of a fixed @xmath0 for the whole galaxy that is known to be wrong but has been used for dge analysis from the days of cos - b @xcite . note that for equation ( [ eq : opticaldepthcorrection ] ) to be valid the condition @xmath11 must hold . when generating the gas annuli , this condition is forced by clipping the value of @xmath4 . while the assumption of a constant spin temperature @xmath12 for the whole galaxy may have been sufficient for older instrument , it is no longer acceptable for a new generation experiment like fermi - lat @xcite . this has been partially explored for the outer galaxy in @xcite . in this paper we will show a better assumption for @xmath0 can be easily found and also show that direct observations of @xmath0 using absorption measurement of bright radio sources are needed for accurate dge modeling . in galactic coordinates . the figure clearly shows the non - linearity of the correction that can be as high as a factor of 2 in this case.,width=283 ] we assume the source distribution of cr nuclei and electrons are the same . cr propagation is handled by galprop and we use the conventional model so that after the propagation the cr spectra agree with local observations . the galprop diffuse emission is output in galacto - centric annuli , split up into different components corresponding to different processes ( bremsstrahlung , @xmath13-decay , and inverse compton scattering ) . to allow for radial variations in cr intensity we perform a full sky maximum likelihood fit , preserving the spectral shape of each component . we allow for one global normalization factor for the electron to proton ratio . additionally , we also allow for radial variation in the @xmath14 factor . this accounts for uncertainties in the cr source distribution and @xmath14 factor . the maximum likelihood fits were performed on the whole sky using the gardian package @xcite after preparing the fermi - lat data with the science tools . we use the same dataset as @xcite that has special cuts to reduce cr background contamination compared to the standard event selection @xcite . in addition to the dge model , we also include all sources from the 1 year fermi - lat source list @xcite and an isotropic component to account for egb emission and particle contamination . this fit is performed for different assumptions of @xmath0 and a likelihood ratio test is used to compare the quality of the fits . the simplest assumption is that of a constant @xmath0 for the whole galaxy and it deserves some attention for historical reasons . it will also serve as a baseline model for comparison with other assumptions . to get an approximation for the best model , we scan @xmath0 from 110 k to 150 k in 5 k steps . our results show that @xmath15 gives the maximum likelihood for this setup . one of the problems with the constant global @xmath0 approximation , apart from the fact that observations of the interstellar gas have shown it to be wrong , is that the maximum observed brightness temperature in the lab survey is @xmath16150 k which is greater than our best fit global @xmath0 . this is solved by clipping the observations when generating the gas annuli , which is not an optimal solution . a different possibility is to use the assumption @xmath17 here , @xmath18 is the maximum observed brightness temperature for each line of sight . this ensures @xmath0 is always greater than @xmath4 . scanning the values of @xmath19 and @xmath20 with a step size of 10 k and 5 k , respectively , gives us a maximum likelihood for @xmath21 and @xmath22 . while this assumption still does not account for the complexity of the interstellar medium , the log likelihood ratio between the best fit linear relation model and the best fit constant @xmath0 model is of the order of 1000 , a significant change . the most accurate @xmath0 estimates come from observations of h i in absorption against bright radio sources . we gathered over 500 lines of sight with observed @xmath0 from the literature @xcite . this covers about 0.2% of the pixels in the lab survey , allowing for accurate column density estimates only in those pixels . after taking our best fit linear relation model and correcting the pixels with known @xmath0 the fit was redone for the whole sky . note that we did not change the values of @xmath23 and @xmath20 . the log likelihood ratio of 105 tells us that this model is worse than the best fit linear relation . this is not unexpected , since the gamma rays are generated from cr interactions with the gas and if the gas distribution is wrong , we wo nt get the correct cr distribution from the fit . to limit the uncertainty involved with the linear relation assumption , we did another fit , limiting ourselves to the region @xmath24 , @xmath25 that covers the observations made in the canadian galactic plane survey ( cgps ) where the density of @xmath0 observations is the highest and is large enough to get a good fit to the lat data . the fit in this region results in a log likelihood ratio of 28 indicating a statistically significant improvement in the fit . this is despite the observed @xmath0 lines of sight only covering 25% of the fitted region and the values of @xmath19 and @xmath20 not being adjusted after correcting for known @xmath0 values . our small exercise here has shown that for accurate dge modeling we need to know more about the distribution of gas in the galaxy , especially the h i distribution . the standard constant @xmath0 assumption is not sufficient for current instruments and small adjustments cause large differences in the quality of the resulting model . we also show that direct observations of @xmath0 help in creating a better model of the dge . unfortunately , direct observations of @xmath0 are difficult since they require high resolution telescopes and bright radio continuum sources . some assumptions will therefore have to be made for the regions in between bright radio sources . it must be stated here that all of the above results are model dependent . the fermi - lat data can only provide us with the intensity of gamma - rays from a particular direction of the sky . uncertainties in our modeling of contribution other than those directly related to the h i distribution will affect the value obtained for @xmath0 . we are currently studying the systematic effects this will have on our results . we also note that even for the best fit models , the residuals show signs of structure , strongly indicating our models are less than perfect .
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the diffuse high - energy gamma - ray emission of the milky way arises from interactions of cosmic - rays ( crs ) with interstellar gas and radiation field in the galaxy . the neutral hydrogen ( h i ) gas component is by far the most massive and broadly distributed component of the interstellar medium .
using the 21-cm emission line from the hyperfine structure transition of atomic hydrogen it is possible to determine the column density of h i if the spin temperature ( @xmath0 ) of the emitting gas is known .
studies of diffuse gamma - ray emission have generally relied on the assumption of a fixed , constant spin temperature for all h i in the milky way .
unfortunately , observations of h i in absorption against bright background sources has shown it to vary greatly with location in the milky way
. we will discuss methods for better handling of spin temperatures for galactic diffuse emission modeling using the fermi - lat data and direct observation of the spin temperature using h i absorption .
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You are an expert at summarizing long articles. Proceed to summarize the following text:
can a local excitation ( source ) in classical field theories be invisible to observers outside the region of excitation ? this question has recently received renewed interest . berry _ et al . _ @xcite described a peculiar excitation case for the one - dimensional wave - equation of a perfectly elastic string under tension . they show that the response of the string can be made to be confined to a bounded region by carefully choosing a forced excitation of oscillatory type . this means that the excitation will not propagate away along the string . denardo gives a simple and intuitive explanation by using a wave interference argument @xcite . et al _ @xcite discuss conditions of finite string length and dissipation . other recent work investigated non - propagating excitations include marengo and ziolkowski @xcite who discuss the generalization of non - propagating conditions of dalembertian ( @xmath0 ) operators and its temporally reduced version the helmholtz operator ( @xmath1 ) on various related classical scalar and vector fields . marengo , devaney and ziolkowski @xcite give the condition for time - dependent but not necessarily time - harmonic non - radiating sources and for selective directional radiation for the inhomogeneous wave equation in three spatial dimensions . marengo and ziolkowski @xcite generalize these conditions to more general scalar and vector field dynamics . marengo , devaney and ziolkowski @xcite also give examples in one and three spatial dimension for the time - harmonic case . hoenders and ferwerda @xcite discuss the relationship of non - radiating and radiating parts of the case of the reduced helmholtz equation , which can be derived from the string equation by assuming general oscillatory time solutions ( see @xcite ) . denardo and miller @xcite discuss the related case of leakage from an imperfect non - propagating excitation on a string . gbur @xcite provides a comprehensive recent review of this topic and the reader is referred to this review for more detailed historical context . of the earlier work the following contributions are particularly relevent for the discussion here : schott @xcite gave the condition for non - radiation of a spherical shell on a circular orbit . bohm and weinberg @xcite extended this result to more general spherical charge distributions and goedecke @xcite showed how an asymmetrical charge distribution with spin is non - radiating . all of these works are concerned with the case of spatially moving sources . finally it is noteworthy , that non - radiating sources play an important role in inverse problems and have been investigated in a one - dimensional electrodynamic situation by habashy , chow and dudley @xcite . in this paper our purpose is to describe this phenomena in the case of a lattice string in one dimensions by discretizing dalembert s solution . this approach is used extensively to simulate vibrating strings and air tubes of musical instruments . see @xcite and references therein . this leads to explicit dynamical constructions of previously reported non - propagating excitations . its simplicity allows for additional insight into the mechanism that allows for the local confinements and the conditions under which they occur . i will show how the basic mechanisms that provide a time - harmonic stationary non - propagating excitation in one dimension as studied by berry _ et al . _ and gbur _ et al . _ @xcite allows for a much wider class of excitations . for instance can such an excitation be relieved from the time - harmonic assumption beyond one period allowing for non - propagating sources that are short - lived . directional excitations can easily be achieved using very simple bidirectional excitation patterns . these are explicit constructions of such waves in one spatial dimension whose general condition of existence in the three - dimensional case has been derived by marengo , devaney and ziolkowski @xcite . wave propagation can be virtually slowed down . in general i will show that non - propagating excitations can be extended to steered excitation regions with basic physical restrictions imposed by the underlying field dynamics . first i will give a quick derivation of the simple lattice model from the wave equation as can also be found in @xcite . then i will give a new argument and construction of the berry _ et al . _ type non - propagating excitation purely based on discrete string dynamics . this will then be compared to the original approach . then i will extend the discussion to examples of additional types of non - propagating waves , including directional and slowed waves . finally i will discuss very general constraints on such `` steered '' localized excitations . the lattice string model can easily be derived from the wave - equation by discretizing the dalembert solution . hence the continuous case will be discussed first . the one - dimensional homogeneous wave equation of the perfectly elastic string under tension is : @xmath2 where @xmath3 is derived from mass density @xmath4 and tension @xmath5 . the dalembert solution of the homogeneous `` free field '' case has the well known form ( * ? ? ? ( 4 ) ) : @xmath6 hence the solution of the general of the homogeneous wave - equation are two propagating waves whose content is restricted by initial and boundary conditions . as wave - equation is linear we have a connection between initial conditions and external driving forces . driving forces can be seen as infinitesimal time frames that act on the wave dynamics by imposing a new initial condition at each point in time . hence we need to consider the initial value problem to gain insight into both processes at once . at a given time frame @xmath7 let the following initial conditions hold : @xmath8 equation ( [ eq : in1 ] ) with ( [ eq : we ] ) gives a particular solution @xmath9 : @xmath10 taking the first temporal derivative of ( [ eq : we ] ) and satisfying equation ( [ eq : in2 ] ) we get : @xmath11 integrating with respect to @xmath12 we get ( * ? ? ? * eq . ( 10 ) p. 596 ) : @xmath13 from equations ( [ eq : satin1 ] ) and ( [ eq : satin2 ] ) we can solve for the traveling wave components : @xmath14 we see that forced displacement @xmath15 splits evenly between left and right traveling waves and the integrated forced velocity @xmath16 splits with a sign inversion . for our current discussion i will share the assumption of no initial velocity of berry _ et al . _ @xcite and hence the integral over @xmath16 will vanish . for the infinite string this is already the complete solution for any twice differentiable function of free solutions and external forced displacements . to arrive at lattice equations we discretize the solution of the wave - equation ( [ eq : we ] ) in time via the substitution @xmath17 where @xmath18 is the discrete time - step and @xmath19 is the discrete time index . this automatically corresponds to a discretization in space as well , because in finite time @xmath18 a wave will travel @xmath20 distance according to ( [ eq : we ] ) . the spatial index will be called @xmath21 . the free - field discrete dalembert solution : @xmath22 in general we can always express all discrete equations in terms of finite time steps or finite spatial lengths . we chose a temporal expression and substitute @xmath20 and suppress shared terms in @xmath23 to arrive at the index version of the discrete dalembert solution @xcite : @xmath24 by equations ( [ eq : force1 ] ) and ( [ eq : force2 ] ) we see that at an instance @xmath25 the discrete contribution of external forced displacements splits evenly between the traveling waves and we arrive at the discrete field equations including external forced displacements : @xmath26 next we will construct the non - propagating excitation from the lattice string dynamics directly . for simplicity and without loss of generality , we will assume a region aligning with the discretization domain throughout . we want to construct an excitation which is confined to a length @xmath27 . for now we will assume that the string should otherwise stay at rest . this implies that there are no incoming waves into the region @xmath28 $ ] from the outside . we are interested in a non - trivial excitation within the region . first we consider the contributions to the position @xmath29 . as there are no incoming external waves we get : @xmath30 we do expect non - trivial wave @xmath31 to reach the boundary but we require the total outgoing wave to vanish we have : @xmath32 the necessary external forced displacement contribution to for cancellation needs to be : @xmath33 the complete incoming wave ( [ eq : fullwg1 ] ) will see the same forced contribution ( [ eq : forceeq ] ) and with equation ( [ eq : noin ] ) we get : @xmath34 hence the matched forced displacement leads to a reflection with sign inversion at the region boundary at @xmath29 . following the same line of argument at point @xmath35 we get the related condition : @xmath36 with these two conditions we can study the permissible form of excitations . first we assume an initial forced displacement impulse from a position @xmath37 in the interior of the domain @xmath38 . hence @xmath39 and @xmath40 with @xmath41 . it will take half the impulse @xmath42 steps to reach the left boundary and the other half @xmath43 steps to reach the right one . at each boundary the respective condition ( [ eq : cond1],[eq : cond2 ] ) needs to be satisfied and we get : @xmath44 the impulse will then reflect back and create periodic matching conditions . @xmath45 with @xmath46 . hence we see that a single impulse will necessitate an infinite periodic series of forced external displacements at the boundaries to trap the impulse inside as each `` annihilation '' of a half - pulse reaching the boundary leads to a `` creation '' of a reflected one . the required impulse response of a boundary forced function @xmath47 can easily be observed from equations ( [ eq : boundf1][eq : boundf4 ] ) to be spatially periodic in @xmath48 with an initial phase factor dictated by the starting position @xmath37 . additionally the functional shape of the impulse responses @xmath47 is completely defined for all time steps as @xmath49 for all times that equations ( [ eq : boundf1][eq : boundf4 ] ) do nt apply . a condition for stopping a non - propagating excitation can be derived from the fact that a impulse will return to its initial position every @xmath48 time steps . additionally it is easy to see that the traveling impulses will occupy the same spatial position every odd multiple of @xmath50 with a sign inversion . hence an impulsive forced displacement @xmath51 with @xmath52 will cancel an initial impulse @xmath40 . from this we can immediately deduce the following property : the shortest possible single impulse finite non - propagating excitation takes @xmath50 time - steps.[th : shortest ] and more generally : the time of any single impulse excitation finite non - propagating excitation has to be @xmath53 . more importantly we observe the property : _ non - propagating excitations can be finite in duration_. this is an extension beyond berry _ et al . _ @xcite which assumes infinitely periodic temporal progressions in their derivations . the general solution for discrete non - propagating wave functions can be derived by observing that any initial `` phase '' @xmath54 is orthogonal to other phases @xmath55 for @xmath56 , i.e. @xmath57 for @xmath58 . within a @xmath50 period @xmath59 is well - defined by @xmath60 . interestingly though this provides the only restriction to the forced boundary functions . this can be seen by theorem [ th : shortest ] . after @xmath50 each @xmath54 will find constructive interference and can be annihilated or rescaled to an arbitrary other value @xmath61 . hence any arbitrary succession of @xmath62 force distributions with a @xmath50 termination is permissible . hence periodicity is not necessary . the time harmonic case can be derived if the initial force distribution within the domain is not modified over time . then a configuration will repeat after traveling left and right , being reflected at the domain boundary twice , traversing the length of the region twice . hence the lowest permissible wave - length is @xmath48 . by reflecting twice the wave will have gone through a @xmath63 phase shift , but we note that the periodicity condition is also satisfied if any number of additional @xmath63 shifts have been accumulated . hence we get for permissible wave - numbers : @xmath64 or @xmath65 by allowing only even @xmath19 we get the berry _ at al . _ condition @xcite for an even square distribution . the odd @xmath19 situation corresponds to the odd - harmonic out - of - phase construction proposed by denardo @xcite . many of these properties can be seen visually in the numerical simulation depicted in figure [ fig : nonprop ] . it is interesting to observe that two synchronous point - sources oscillating with the above phase condition will not be completely non - propagating . they will only be non - propagating after waves created at the wave onset have escaped . this is a refinement of the argument put forward by denardo @xcite and can intuitively be described as _ non - interference of the first trap period_. hence the first pairs of pulses will have half - amplitude components escaping in either direction but every subsequent period will be trapped . this behavior , which could be called imperfect trapping or trapping with transient radiation , is depicted in figure [ fig : pointpair ] . sources presented by berry _ et al . _ and denardo @xcite do not display this behavior because the force is assumed to be oscillatory at all times and hence has no onset moment . non - propagating excitations can be used as generic building blocks for other unusual excitation induced behavior on the string . in particular i will next describe how to construct an uni - directional emitter , and a virtually slowed propagation . in fact a non - propagating excitation can be seen as virtually stopping a wave at a particular position . a one - sided open trap immediately suggests another unusual excitation type , namely the directional excitation . the string is to be excited in such a way that a traveling wave in only one direction results . we start with a one - sided open trap . this is a trap that uses a reflection condition ( [ eq : cond1 ] ) and ( [ eq : cond2 ] ) only on one side of an initial excitation . evidently the wave then can only travel in the opposite direction . for the discussion we will describe a right - sided propagator ( i.e. a propagator traveling with increasing negative index ) . the trapping condition then reads : @xmath66 hence the trapping excitation point is a @xmath37 time - step lagging negative copy of the original excitation . the emitted wave will have the form @xmath67 the emitting wave will show self - interference at a phase of @xmath68 time - steps , as can be seen in the simulation depicted in figure [ fig : onesided ] . in general the self - interference phase can be chosen by the distance @xmath37 between the wave creation point and the trapping point . it is worth noting that it is possible to eliminate interference by trapping the lagging contribution and hence create a wave non - interference directional wave left of the trapping region . virtual slow waves can be achieved by alternating directional wave propagation with trapping . the slowness of the wave propagation can be controlled by the number and and duration times of the traps along a propagation . the propagation characteristics of the dynamic operator has not changed at all , hence we call the this state `` virtually slow '' as opposed to the case where the field itself induces a change in wave propagation speed . this also means that within a slowed or `` steered '' region the wave propagation is the one prescribed by the dynamic operator @xmath69 on the string @xmath70 . the amount of time spend in traps determines the overall slowness . one example of slow wave consists of an immediate alteration between one stage of trapping and one step of one - sided propagation is illustrated in figure [ fig : slowwave ] . the effective propagation speed of the wave can easily be read from the diagram to give @xmath71 . as is evident from theorem [ th : shortest ] , a unit @xmath72 trap will last @xmath73 time - steps and will not propagate spatially and one step of free propagation will last one time - step and and make one spatial step , hence resulting in a spatial to temporal ratio of @xmath74 . the trapping relations are : @xmath75 with @xmath76 . the generalized interpretation of the excitation interaction lead to the general dynamical confinement of waves by external excitation . for instance following very similar arguments as for virtual slow waves a construction is possible which gives a slowed `` cone of influence '' by successively widening the trap boundaries at a speed slower than the the wave speed @xmath77 . by this argument it is sufficient for the trap boundaries change to be less than @xmath77 for it to be trapping the wave . this is not a necessary condition by the following counter - example : let the trap width be @xmath35 and change rapidly by some slope @xmath78 to some new constant width @xmath79 at which it becomes constant . obviously the wave will then be able to reach the new boundary even though a local change of the boundary exceeded the dynamical speed @xmath77 . the necessary condition can be seen from our previous construction . at a trap boundary a wave is reflected and will propagate in the opposite direction of the domain following the linear characteristic @xmath77 . only if this characteristics intersects with the dynamic trapping boundary will there be another externally forced reflection as illustrated in figure [ fig : steering ] . these may in fact have regions where no trapping is necessary and possible . it is important to note that while we assumed that the incoming wave vanishes , see equation ( [ eq : noin ] ) , the outgoing wave condition ( [ eq : noout ] ) does not change if there is in fact an incoming wave . the `` reflection wave '' ( [ eq : cond1 ] ) and ( [ eq : cond2 ] ) can be rewritten for a non - zero incoming field without affecting the trapping : @xmath80 and @xmath81 these conditions are `` absorbing '' in the sense that an external field entering the trapping region will not leave it . the `` non - interacting '' property of a trap defined by the periodic matching conditions ( [ eq : boundf1][eq : boundf4 ] ) can be seen by assuming a non - zero incoming wave at one point of the trap boundary @xmath82 . then the total wave entering the trapping region the sum of the wave created by the trapping condition and the incoming wave value @xmath83 , where @xmath84 denotes the first trap boundary reached . when reaching the second trapping boundary @xmath85 the now outgoing wave will see a matching force @xmath86 leaving an outgoing wave contribution @xmath87 to escape the trapping region @xmath88 . in order to achieve selective radiation , only part of the content of a trapped region are trapped at the boundary as can be achieved by using a reduced force at the trapping boundary or by selectively omitting certain phases in the trapping force pattern . marengo and ziolkowski @xcite present ideas very much related to ideas presented here and in berry _ . however , they arrive at a definition of non - radiating ( nr ) sources that is not obviously similar to the traps presented here . in particular they define nr sources as being non - interacting . while @xcite note that a central property of nr sources is that they store non - trivial field energy , traps described here can not only store , but accumulate and selectively radiate waves . the difference can be understood by observing that for example berry _ _ assume a simple time - harmonic driver ( * ? ? ? * eq . ( 3 ) ) throughout their discussion : @xmath89 by our earlier discussion we see that the temporal progression of the boundary has to match the content of the interior domain . hence once the boundary is defined to be oscillatory the interior of the domain needs to be spatially harmonic as derived in @xcite and has been rederived here . hence a nr source as noted in literature , with exception of the general orthogonality formulation for time - varying sources given by marengo , devaney and ziolkowski @xcite , can be thought of as a time - oscillatory trap . the arguments made here use a formalism that is discrete in nature . however , the discreteness of the arguments are not necessarily restrictive . the continuous case can be imagined with the discrete time - step made small ( @xmath90 ) or alternatively , discrete pulses can be substituted with narrow distributions of compact support . in neither case are the results of interest derived here altered . as has already been derived in @xcite the critical condition for non - propagating waves lie at the boundary of the domain - range that the wave ought not to leave . in the discrete case it is easy to see how this insight can be used and generalized . in fact , the boundaries of the confining domain need not be static , nor need the condition be used in a two - sided fashion . in summary , this paper presented constructions of a broad class of non - propagating sources on a string lattice model using trapping conditions . in particular this includes numerical demonstrations of finite - duration non - propagating excitations , directional excitations , as well as virtually slowed waves . these examples help explain the extension of non - propagating sources beyond the time - periodic case and include treatment of onset , annihilation and spatial steering . these properties ought to be observable in experiments well - described by the wave equation . this equation often arises in problems in acoustics , elasticity , optics and electromagneticsm . and hence the results presented here apply to these domains of application . while here i discussed the forward problem , these results also relate to the inverse problem of finding source contributions from the one - dimensional field state as occur for example in acoustical , optical and electromagnetic detection problems . i am grateful for reprints provided by bruce denardo and greg gbur , who also brought relevant references to my attention . i m also thankful for helpful comments and pointers to relevant literature by an anonymous referee .
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using a lattice string model , a number of peculiar excitation situations related to non - propagating excitations and non - radiating sources are demonstrated .
external fields can be used to trap excitations locally but also lead to the ability to steer such excitations dynamically as long as the steering is slower than the field s wave propagation .
i present explicit constructions of a number of examples , including temporally limited non - propagating excitations , directional excitation and virtually slowed propagation . using these dynamical lattice constructions
i demonstrate that neither persistent temporal oscillation nor static localization are necessary for non - propagating excitations to occur .
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You are an expert at summarizing long articles. Proceed to summarize the following text:
vector mesons ( @xmath3 , @xmath4 , etc . ) play a significant role in hadronic physics . their interactions , though not constrained by low - energy theorems , apparently follow the broad pattern of vector meson dominance ( vmd ) @xcite . there have been numerous efforts to incorporate vector mesons into field - theoretical frameworks . historically , the yang - mills theory was discovered in an early attempt to treat the @xmath2 meson @xcite . more recently , interesting schemes based on `` hidden local symmetries '' ( hls ) were developed by bando _ et al . _ @xcite . in the original model @xcite , the @xmath2 meson is the boson of a spontaneously broken gauge group . the model has been extended to two hidden gauge groups @xcite ; then it also incorporates the lowest axial vector meson @xmath5 . with suitable parameters , these models can be quite successful phenomenologically , although they can not be systematically derived from qcd ( except in the limit of very light @xmath2 , if such a limit could be reached @xcite ) . in this paper we explore theories with very large , and even infinite number @xmath0 of hidden local symmetries . our motivation is twofold . first and most straightforwardly , there are excited states in the vector and axial vector channels ( @xmath6 , @xmath7 , @xmath8 , etc . @xcite ) , which must become narrow resonances in the limit of large number of colors @xmath9 . it is tempting to treat them as gauge bosons of additional broken gauge groups . , @xmath10 , etc . as a `` chain structure '' was made in ref . @xcite . ] the second motivation comes from recent theoretical developments . many strongly coupled gauge theories are found to have a dual description in terms of theories with gravity in higher dimensions @xcite . it was suggested that the string theory dual to large-@xmath9 qcd must have strings propagating in five dimensions , in which the fifth dimension has the physical meaning of the energy scale @xcite . in the framework of field theory , the fifth dimension can be `` deconstructed '' in models with a large number of gauge fields @xcite . we discovered that the continuum limit @xmath1 can lead to results that qualitatively , and in many cases even quantitatively , agree with phenomenology . most remarkably , the vector meson dominance , which in the hls theories required a tuning of parameters , becomes a natural consequence of the @xmath1 limit . another advantage of the limit @xmath1 is the possibility of matching to the asymptotic behavior of the current - current correlator known from perturbative qcd . as anticipated , a natural interpretation of this limit is a discretization , or deconstruction , of a 5-dimensional gauge theory . further , to our amusement , in the calculation of current - current correlators we found a relation very similar to the one employed in the ads / cft correspondence : the current - current correlator in 4d theory is expressed in terms of the variations of the classical 5d action with respect to the boundary values of the bulk gauge fields on the 4d boundaries . we limit our discussion to the isospin-1 sector of qcd . it is straightforward to extend the discussion to the isospin-0 sector ( @xmath11 , @xmath12 , and @xmath13 mesons ) . the detailed treatment of the @xmath14 problem , chiral anomaly , wess - zumino - witten term , and baryons is deferred to future work . the paper is organized as follows . in section [ sec : model ] we describe the open moose model . in section [ sec : observables ] we compute different physical observables : the vector meson mass spectrum , the decay constants of the pion and the vector mesons , the coupling between the vector mesons and the pions , and the pion electromagnetic form factor . we also check the validity of weinberg s spectral sum rules , and discover that the limit @xmath1 automatically leads to exact vmd for the pion formfactor . in section [ sec : continuum ] we take the limit of infinite number of the hidden groups @xmath1 . we show that the theory can be understood as a 5d yang - mills theory in an external metric and dilaton background . we establish an ads / cft - type prescription for calculating the current - current correlators . we consider two concrete realizations of the open moose in section [ sec : examples ] . we find that a `` cosh '' background metric interpolating between two ads boundaries leads to correct asymptotic behavior of the current - current correlator . this allows us to establish a relationship between hadron parameters such as @xmath15 , @xmath16 , and the qcd parameter @xmath9 . in section [ sec : baryon ] we show that the instanton , which is a quasiparticle in @xmath17 dimensions , becomes a skyrmion upon reduction to 4d , and thus describes the baryon . section [ sec : concl ] contains concluding remarks . the model under consideration is described by the following lagrangian , but write all indices as lower indices for simplicity , unless it could lead to a confusion . ] @xmath18 the covariant derivatives are defined as @xmath19 a shorthand notation is used for the product of the gauge field @xmath20 and its coupling constant : @xmath21 . if we assume @xmath22 , then eqs . ( [ dsigma1 ] ) and ( [ dsigma3 ] ) become special cases of eq . ( [ dsigma ] ) for @xmath23 and @xmath24 . the model contains @xmath25 nonlinear sigma model fields @xmath26 ( or , in general , @xmath27 ) , interacting via @xmath0 `` hidden '' gauge bosons @xmath28 . the model has a chiral @xmath29 symmetry and an @xmath30 local symmetry : @xmath31 in particular , the product @xmath32 is the pion field , which can be seen from its transformation properties , @xmath33 the parameters entering ( [ l ] ) are @xmath25 decay constants @xmath34 and @xmath0 gauge couplings @xmath35 . we shall assume they are invariant under a reflection with respect to the middle of the chain , @xmath36 which ensures parity is a symmetry in the theory ( [ l ] ) . in the case @xmath37 the model reduces to the chiral lagrangian . for @xmath38 it is the version of the hidden local symmetry realized in the limit of very light @xmath2 s @xcite . the model with @xmath39 and a particular choice of parameters @xmath40 , @xmath41 has been considered in ref . graphically , the model can be represented by a `` theory - space '' diagram shown in fig . [ fig : k ] . since this diagram is the usual `` moose diagram '' cut open , we shall call the model ( [ l ] ) the `` open moose '' theory . fig : k note that ( [ l ] ) is not the most general lagrangian satisfying all the symmetries and limited to lowest derivatives . in fact , terms of the following type are not forbidden : @xmath42 as well as analogous expressions containing products of more than two consecutive @xmath43 s . in order to restrict the lagrangian to the form ( [ l ] ) an additional condition of nearest neighbor locality in the @xmath44 space should be imposed . it is this condition that enables us later to interpret this theory as a dimensionally deconstructed 5d gauge theory in the limit @xmath1 . in this section we derive expressions for physical observables , such as pion and vector meson decay constants , mass spectrum , and pion - vector couplings in terms of the parameters of the moose @xmath34 and @xmath35 . for computations , the following gauge is the most convenient , @xmath45 where @xmath46 is a function of all @xmath34 , which we shall specify in a moment ( in eq . ( [ f ] ) ) . the advantage of this gauge is that the pion field @xmath47 does not mix with other fields : @xmath48 = 0.\ ] ] the value of @xmath46 is fixed by requiring that the kinetic term for @xmath49 is canonically normalized : @xmath50 therefore @xmath51 to determine @xmath15 we use noether s theorem to construct the axial current @xmath52 . let us consider an infinitesimal axial su(2 ) transformation . it acts only on @xmath53 and @xmath54 at the ends of the moose : @xmath55 if the parameter @xmath56 depends on coordinates , the lagrangian changes by @xmath57 . on the other hand from ( [ l ] ) one finds @xmath58 which means @xmath59 , i.e. , @xmath60 . equation ( [ f ] ) becomes @xmath61 it is a simple exercise to verify that for @xmath62 this general formula is in agreement with corresponding results in these theories . it is also perhaps useful to observe that sending @xmath63 on one of the links effectively sets gauge fields on the ends of this link equal to each other ( @xmath64 ) , effectively eliminating this link and reducing @xmath0 by one . the formula ( [ fpi ] ) obviously reflects this reduction the corresponding term @xmath65 drops out . in our gauge , the vacuum is @xmath67 for all @xmath44 . expanding to second order in @xmath28 , we find the terms that determine the masses of the vector mesons : @xmath68 ^ 2 \equiv \sum_{\substack{k=1\\k'=1}}^{k } ( m^2)_{kk ' } { { \rm tr } } a^k_\mu a^{k'}_\mu\,.\ ] ] the mass matrix can be diagonalized by using an orthogonal matrix @xmath69 satisfying @xmath70 in terms of new vector fields @xmath71 defined as @xmath72 the mass term ( [ lmass ] ) is diagonal . using eq . ( [ lmass ] ) and the orthogonality of @xmath73 , we can write the equation determining @xmath69 and @xmath66 as @xmath74 - f_k^2[(gb_n)^k-(gb_n)^{k-1 } ] = - \frac{m_n^2 b_n^k}{g_k}\,.\ ] ] this is essentially a discretized version of a sturm - liouville problem . we shall write the corresponding differential equation in section [ sec : continuum ] when we consider the continuum limit @xmath1 . we shall also use the discrete equation ( [ evdisc ] ) in section [ sec : vv ] . without solving eq . ( [ evdisc ] ) , we can conclude right away that there is a tower of eigenvalues @xmath66 , @xmath75 , corresponding to the masses of vector and axial vector mesons . the lowest @xmath76 and @xmath77 states correspond to the @xmath2 and @xmath78 mesons . moreover , states with opposite parity alternate in the spectrum : @xmath79 odd @xmath80 states correspond to vector mesons and even @xmath80 to axial vector mesons . in the real world , the trend of alternating parity can be seen in the hadronic spectrum for the few first vector and axial vector states . let us compute the coupling of @xmath80th vector meson to a pion pair , @xmath81 . expanding the lagrangian in @xmath82 and @xmath83 , isolating @xmath84 terms , and using eq . ( [ abalpha ] ) , we find @xmath85 [ ( ga_\mu)^{k-1 } + ( ga_\mu)^k ] \\ & = i \sum_{n=1}^k \sum_{k=1}^{k+1 } \frac{f_\pi^2 } { 4f_k^2 } [ ( gb_n)^{k-1 } + ( gb_n)^k ] { { \rm tr } } [ \partial_\mu \pi,\pi ] \alpha^n_\mu \ , . \end{split}\ ] ] recall that we use the matrix notation where @xmath86 and @xmath87 . if we normalize @xmath81 so that the relevant coupling is @xmath88 then @xmath89.\ ] ] note that this @xmath90 coupling vanishes for axial mesons @xmath91 , as required by parity , because their `` wave functions '' @xmath92 are odd under @xmath93 ( eq . ( [ parity ] ) ) . sec : vv we define the decay constants for the vector and axial vector mesons via the matrix elements of the vector and axial vector currents between the vacuum and the one - meson states , @xmath96 here @xmath97 is a single - particle state of the @xmath80-th vector boson ( @xmath98 , @xmath99 , etc . ) with isospin @xmath100 and polarization @xmath101 . both @xmath94 and @xmath95 have the dimension of [ mass@xmath102 . @xmath103 for axial vector mesons ( @xmath80 even ) and @xmath104 for vector mesons ( @xmath80 odd ) . it is convenient to compute @xmath94 by looking at the vector current - current correlator @xmath105 . the residues at the poles are easily related to @xmath94 . the correlator can be obtained by gauging the corresponding su(2)@xmath106 transformation and differentiating the action with respect to the gauge field @xmath107 : @xmath108}{\delta b^a_\mu(x)\delta b^b_\nu(y ) } \,,\ ] ] were @xmath109 $ ] is the vacuum energy functional in the presence of the external field @xmath110 . the su(2)@xmath106 transformation only affects the two @xmath43 links at the ends of the chain according to eq . ( [ usigmau ] ) : @xmath111 and therefore the terms containing the gauge field @xmath107 are only from @xmath23 and @xmath24 : @xmath112 keeping only terms bilinear in the fields we find ( the term @xmath113 is absent due to parity ) : @xmath114 + { \cal l}_{a^2 } + { \cal l}_{f^2}\,.\ ] ] extremizing the action with respect to @xmath82 at fixed @xmath110 and then taking the second derivative with respect to @xmath110 we find ( this is also equivalent to taking the gaussian integral over @xmath82 and then differentiating the logarithm of this integral ) @xmath115(x ) [ ( ga^b_\nu)^1 + ( ga^b_\nu)^k ] ( y ) \rangle,\ ] ] where the @xmath116 is the propagator of @xmath82 , i.e. , the inverse of the quadratic form found in @xmath117 . diagonalizing this expression using eq . ( [ abalpha ] ) and performing a fourier transformation with respect to the four - dimensional coordinate @xmath118 , we find @xmath119 where the decay constants @xmath94 are determined to be @xmath120 . \ ] ] note that @xmath103 for axial mesons for which @xmath121 . one can also use eq . ( [ evdisc ] ) together with eq . ( ( [ gnv ] ) ) to write a different representation for @xmath94 : @xmath122 = m_n^2\sum_{k=1}^k \frac{b_n^k}{g_k } \,.\ ] ] it is perhaps easier to look at this equation as a discretized version of integration by parts as we shall do in section [ sec : continuum ] . ) or ( [ gnvk ] ) can be also understood in the following way ( the reader might recognize the discussion given in refs . one should realize that because of the mixing of @xmath110 with @xmath82 the actual photon is not the field @xmath110 , but a linear combination of @xmath110 and all @xmath82 that leaves vacuum @xmath67 invariant . this is similar to the mixing of the standard model hypercharge boson and weak isospin vector boson to produce the photon . the corresponding linear combination of the @xmath82 fields has a `` wave function '' proportional to @xmath123 ( each @xmath124 enters with weight @xmath123 ) . the mixing of the actual photon is now entirely through derivative terms in the expansion of @xmath125 . the corresponding coefficients are given by the overlap of the photon `` wave function '' @xmath126 and the @xmath80th vector meson `` wave function '' @xmath127 . the factor @xmath128 is from the derivatives evaluated using the equation of motion for the @xmath80th meson . ] the calculation of the decay constants of the axial vector mesons @xmath95 is completely analogous . we introduce an auxiliary gauge field @xmath129 coupled to the axial current @xmath130 : @xmath131 in addition to the @xmath132 mixing and @xmath133 contact terms there is now also the mixing with pion field @xmath134 . differentiating the logarithm of the partition function twice , we obtain @xmath135 where @xmath136.\ ] ] note that , as expected , @xmath104 for vector mesons for which @xmath137 . weinberg @xcite has derived two sum rules for the weighted integrals of the difference of spectral functions of the vector and axial vector current correlators , @xmath138 and @xmath139 . we shall now verify that both sum rules hold by using transversality of the current - current correlators . from eq . ( [ gnvk ] ) it is easy to show that the @xmath140 correlator is transverse . transversality requires that the contact term in eq . ( [ vv ] ) is related to the pole terms through the following sum rule : @xmath141 this can be checked by using eq . ( [ gnvk ] ) and orthogonality of @xmath69 . we indeed find @xmath142\frac{b_n^k}{g_k}= 2 f_1 ^ 2.\ ] ] by using eq . ( [ gnvsum ] ) one can rewrite the correlator in a manifestly transverse form , @xmath143 the transversality of the @xmath144 correlator ( [ aa ] ) amounts to the following sum rule : @xmath145 by comparing eqs . ( [ gnvsum ] ) and ( [ gnasum ] ) we conclude that @xmath146 which is one of weinberg s sum rules . ] where @xmath147 is defined via @xmath138 in eq . ( [ vvpi ] ) . a similar equation defines @xmath148 . the sum rules state that @xmath149\mu^{-2}d\mu^2=f_\pi^2 ; \qquad ( ii ) \int [ \rho_v(\mu^2 ) - \rho_a(\mu^2)]d\mu^2 = 0.\ ] ] in our theory , according to eq . ( [ vvpi ] ) , @xmath150 . ] this sum rule holds for any @xmath0 . note that , for @xmath1 both sums ( [ gnvsum ] ) and ( [ gnasum ] ) must diverge , since @xmath34 must become infinite at the ends of the moose ( to ensure convergence in eq . ( [ fpi ] ) ) . however , their difference is finite . the second weinberg sum rule @xmath151 also holds . it is easy to prove by using the definitions ( [ gnv ] ) and ( [ gna ] ) and the orthogonality of @xmath69 : @xmath152 which vanishes for all @xmath153 . in the case @xmath38 , there is only one meson @xmath2 , and no axial mesons at all . the pion form factor ( defined to be the isovector part of the electromagnetic form factor ) , @xmath154 can be found isolating terms linear in @xmath110 in the lagrangian ( [ lb ] ) . there are two contributions to the form factor the direct interaction , given by the term @xmath155 in the lagrangian and the interaction mediated by vector mesons given by the @xmath156 mixing terms and the couplings @xmath157 . one finds @xmath158 using the expressions ( [ gnvk ] ) and ( [ gnpipik ] ) for @xmath94 and @xmath81 and the orthogonality of the matrix @xmath69 , the sum rule related to the total charge of pion can be verified , @xmath159 if we understand vmd as the statement that @xmath160 is saturated by a sum over resonances ( i.e. , dominance by the whole tower of mesons ) , then in our model vmd is valid when the contribution of the direct interaction is negligible , @xmath161 . thus vmd is a natural consequence of the @xmath1 limit ( due to eq . ( [ fpi ] ) ) . a stronger statement that @xmath160 is saturated by a single @xmath2 pole is not , in general , valid ( see , however , section [ sec : cosh ] ) . sec : continuum in the preceding section we derived formulas that are valid for an arbitrary @xmath0 . now we wish to consider the limit @xmath1 . in this limit the expressions that we found can be simplified , provided that @xmath34 and @xmath35 are sufficiently smooth functions of @xmath44 . in this case we can consider replacing the discrete variable @xmath44 by a continuum variable that we shall call @xmath162 : @xmath163 here @xmath164 plays the role of the `` lattice spacing . '' if the limit @xmath1 is performed in the following way , @xmath165 then @xmath162 becomes a continuum replacement for @xmath44 . if @xmath34 and @xmath35 are smooth functions of @xmath44 , we can also replace them by functions of @xmath162 , @xmath166 for the resonance `` wave functions '' @xmath69 that vary smoothly we can write @xmath167 so that orthogonality of the matrix @xmath69 translates into orthonormality of the functions @xmath168 , @xmath169 the wave functions of sufficiently high resonances with @xmath170 can not be expected to be smooth , so they must be treated discretely . we shall always be interested in a finite number of lowest resonances , while @xmath1 . sec : contform let us now write continuum limits for the main formulas we have derived in the preceding section . from eq . ( [ fpi ] ) , @xmath171 from eq . ( [ evdisc ] ) , @xmath172 with dirichlet boundary conditions @xmath173 ( since we set @xmath22 ) . from eq . ( [ gnpipik ] ) , @xmath174 from eq . ( [ gnv ] ) , using the fact that @xmath175 , @xmath176_{-u_0}^{+u_0}.\ ] ] by using eq . ( [ eigen ] ) , we find the continuum limit of eq . ( [ gnvk ] ) : @xmath177_{-u_0}^{+u_0}= m_n^2\int_{-u_0}^{+u_0 } \!du\ , \frac{b_n(u)}{g(u)}\,.\ ] ] analogously , eq . ( [ gna ] ) becomes @xmath178|_{+u_0}+[f^2(u ) ( g(u)b(u))']|_{-u_0}\,.\ ] ] it is very interesting that the physical observables we calculated are all well behaved in the continuum limit @xmath1 , @xmath179 ( provided the corresponding integrals over @xmath162 converge ) . for reference , the equations for other vertex couplings are presented in appendix [ app : mnp ] . our long - moose theory with @xmath181 can be also considered as a discretized ( or deconstructed ) five - dimensional continuum gauge theory in curved spacetime . the variable @xmath162 plays the role of the fifth , deconstructed , dimension . the smoothly varying fields @xmath43 s can be interpreted as the link variables along the fifth dimension @xmath182 , @xmath183 for this equation and for remainder of section [ sec : continuum ] we shall make a temporary switch of notations , absorbing the gauge coupling constants @xmath184 into the fields @xmath82 : @xmath185 . then the action ( [ l ] ) can be written in the 5d notations as @xmath186 we now compare this action to the action of a gauge field in a background of curved spacetime and a dilaton field . in the following , @xmath187 denotes the determinant of the metric tensor . the action is taken in the form : @xmath188 where @xmath189 are 5d lorentz indices . the coupling to the dilaton field is written so that the effective gauge coupling is @xmath190 . in our simple model we consider the metric and the dilaton as classical background fields with no dynamics of their own . taking the dilaton field to be dependent only on the fifth coordinate @xmath162 , @xmath191 , and the metric to be of the warped form , @xmath192 the action ( [ 5ds ] ) can be expanded as @xmath193 equation ( [ 5dsexp ] ) coincides with eq . ( [ sfgf ] ) if one makes the following identification : fgwarp @xmath194 notice that the warp factor @xmath195 is equal to @xmath196 , i.e. , @xmath197 it is also easy to see that the wave equation for the spin-1 mesons ( [ eigen ] ) is one of the yang - mills / maxwell equations for the 5d ( massive ) gauge field in the curved background . notice also that the pion field ( [ sigmaproduct ] ) is now the wilson line stretching between the two boundaries , @xmath198 we now show that correlators of conserved currents in our theory can be computed by using a prescription essentially identical to the ads / cft one . namely , the generating functional for the correlation functions of the currents is equal to the action of a solution to the classical field equations , with the sources serving as the boundary values for the classical fields . recall our calculation of the current - current correlators in section [ sec : vv ] . instead of the vector field @xmath107 and @xmath199 let us introduce two separate fields @xmath200 and @xmath201 , corresponding to gauging the su(2)@xmath202 and su(2)@xmath203 global symmetries of the theory . the appearance of these fields modify the first and last terms in the moose , @xmath204 in this section we also absorb the coupling @xmath35 into the field @xmath82 . remember that there are no dynamical fields associated with the ends of the moose @xmath205 and @xmath25 . we can treat the ends of the moose more equally with the other points by thinking that the values of the field @xmath206 at the ends of the moose , at @xmath205 and @xmath25 , are fixed at given values : @xmath207 if the field @xmath206 is smooth , we can translate this into the continuum limit by setting boundary conditions on the continuous 5d field @xmath208 : @xmath209 at tree level , the generating functional is thus equal to @xmath210 = e^{is_{\textrm{cl}}[a_\mu^{\textrm{cl}}]}\ ] ] where @xmath211 is the solution to the classical field equation that satisfies the boundary conditions ( [ bc ] ) . this formula is of the same form as the formula for ads / cft correspondence : the sources for the boundary theory ( in our case @xmath212 ) serve as the boundary values for the bulk field . in particular , in order to compute the correlation functions for the conserved currents @xmath213 or @xmath214 one just needs to differentiate the classical action with respect to the corresponding boundary values , e.g. , @xmath215 } { \delta a^l_\mu(x ) \delta a^l_\nu(y)}\ ] ] the fact that we have arrived at an ads / cft - like formula ( [ adscft ] ) makes one wonder if the hidden local symmetry models for the @xmath2 and @xmath78 vector mesons @xcite are ( very coarsely ) discretized versions of a 5d theory dual to qcd . this could explain why these models enjoy certain phenomenological success , and why in the @xmath39 model @xcite one is driven to choose the parameters so that it becomes a moose theory ( i.e. , nearest neighbor local ) . a difference from the usual ads / cft correspondence is that there are _ two _ boundaries in the open moose theory . however , if so desired , one can reformulate the 5d theory ( [ 5ds ] ) and ( [ warped ] ) in the spatial region @xmath216 ( which is one half of the original @xmath217 ) at the price of having two gauge fields obeying a matching condition at @xmath218 . then the spacetime will have only one boundary at @xmath219 . so far , our discussion has been general and valid for any choice of @xmath34 and @xmath35 . in this section we shall consider two concrete realizations of the open - moose theory . our goal is to illustrate the general formulas , and to compare the results with the phenomenology of vector mesons . the two examples are chosen because they are exactly solvable : the spectrum of the vector mesons and the coupling constants can be found in the closed form . the first example is also the simplest possible model , but it has a significant physical drawback that we point out at the end . we think nevertheless that it is a useful reference point for comparison and for understanding the robustness / sensitivity of the results towards the change of the background parameters @xmath220 and @xmath221 . consider a moose with parameters @xmath34 and @xmath35 independent of @xmath44 . , but we shall only consider @xmath181 . ] in the continuum limit @xmath181 the corresponding functions are therefore constant , does not affect the results ; it is equivalent to rescaling @xmath46 and @xmath184 . ] @xmath222 let us now apply general formulas from section [ sec : contform ] to determine the properties of this theory in terms of the parameters @xmath46 and @xmath184 . from ( [ fpiu ] ) @xmath223 the spectrum and wave functions of the spin-1 mesons are given by eq . ( [ eigen ] ) , which becomes @xmath224 this means @xmath225 the few first wave functions are plotted in fig . [ fig : sin ] . fig : sin from eq . ( [ gnpipiu ] ) , for @xmath226 , @xmath227 consider the @xmath2 meson , @xmath76 . the ratio of @xmath228 to @xmath229 is dimensionless and is equal to @xmath230 the coupling @xmath231 can be found from the width of the @xmath2 , which decays predominantly to two pions : @xmath232 , where @xmath233 is the velocity of the final - state pions . using @xmath234 mev , we find the ratio ( [ ksrf ] ) to be around 1.9 in nature . for comparison , the ksrf relation @xcite corresponds to this ratio being equal to 2 , and the value in georgi s vector limit ( i.e. , @xmath38 moose theory ) is 4 @xcite . therefore , our model would underpredict @xmath235 from experimental @xmath16 and @xmath15 . the decay constants @xmath94 and @xmath95 are given by eqs . ( [ gnvbg ] ) and ( [ gnabg ] ) and are equal to @xmath236 in eq . ( [ gnv1 ] ) @xmath237 refers to @xmath94 for odd @xmath80 s and @xmath95 for even @xmath80 s . for @xmath76 , we find @xmath238 . we can now predict the rate of the electromagnetic decay @xmath239 , using @xmath240 and the experimental values for @xmath15 and @xmath16 . we find @xmath241 kev , which is somewhat smaller than the measured value @xmath242 kev @xcite . it is also interesting to consider the contribution of the @xmath2 meson to pion form factor at @xmath243 ( eq . ( [ gvpipi0 ] ) ) , @xmath244 thus the single @xmath2-meson dominance holds to within 20% . the vmd is , however , exact if all vector mesons are included in the limit @xmath1 . indeed the direct pion - photon interaction in eq . ( [ gvpipi ] ) vanishes : @xmath245 . ) . each contribution is proportional to @xmath246 and @xmath247 . ] the drawback of this model is that it fails to satisfy the asymptotic condition on @xmath248 that follows from qcd : @xmath249 when @xmath250 . instead , @xmath248 in this model vanishes as @xmath251 when @xmath250 . indeed , according to eq . ( [ vvpi ] ) , with values of @xmath94 and @xmath66 found in eqs . ( [ gnv1 ] ) and ( [ mn1 ] ) , @xmath252 we shall now consider an exactly solvable model that will satisfy the condition @xmath253 at large @xmath254 . this model is given by [ coshbg ] @xmath255 according to eqs . ( [ fgwarp ] ) , this corresponds to a constant dilaton background and the following background metric , @xmath256 the two boundaries are located at @xmath257 . near the boundaries the metric becomes asymptotically ads@xmath258 . according to the ads / cft philosophy , @xmath162 has the physical meaning of the energy scale ; large @xmath162 s correspond to short distances . therefore one can expect that the current correlators has the conformal form at short distance , i.e. , as @xmath250 , @xmath259 the main reason for choosing the background ( [ coshbg ] ) is that @xmath260 is the simplest function interpolating between @xmath261 and @xmath262 , and that the mass spectrum can be found exactly ( see below ) . otherwise , we have no reason to prefer this background over any other that has two ads@xmath258 boundaries . ) coincides with the 5d part of the induced metric on a probe d7 brane in @xmath263 @xcite . ] applying eq . ( [ fpiu ] ) , one finds @xmath264 the wave equation for the vector mesons is @xmath265 which implies the following spectrum : @xmath266 in particular , @xmath267 and @xmath268 . taking @xmath16 as an input , this predicts @xmath269 mev , which is not far from the observed @xmath270 mev . however , the masses of higher excitations grow faster with @xmath80 than in the real world . the 5d eigenfunctions of the vector mesons are @xmath271 where @xmath272 are the associated legendre functions . the first few wave functions are ( see fig . [ fig : cosh ] ) : @xmath273 fig : cosh in order to establish eq . ( [ piexp ] ) , we compute the decay constants of vector mesons from eqs . ( [ gnvbg ] ) and ( [ gnabg ] ) , @xmath274 the correlation function for the vector current is found from eq . ( [ vvpi ] ) , @xmath275 at large @xmath276 , the sum can be replaced by an integral , which is logarithmically divergent . ) , subtracting a constant equal to @xmath277 for @xmath181 . of course , the equation is only valid for @xmath278 . ] one thus finds for large @xmath254 @xmath279 the asymptotic behavior of @xmath280 is the same . thus the current correlators have the correct asymptotics at large @xmath254 . moreover , they obey weinberg s sum rules , as proven in section [ sec : weinberg ] . the constraints imposed by the @xmath250 behavior and weinberg s sum rules on the masses @xmath66 and decay constants @xmath237 are quite nontrivial @xcite . it is remarkable that the open - moose construction generates examples that automatically satisfy these constraints . one can match the asymtotics ( [ pivlog ] ) with the result found from qcd , @xmath281 where @xmath9 is the number of colors , to obtain @xmath282 by using this relationship between @xmath283 and @xmath9 together with @xmath284 , we can now express all quantities in the model via a single mass @xmath16 and the number of colors @xmath9 . a short summary is given in appendix [ app : cosh ] . for example , for @xmath15 we find from eq . ( [ fpilambdag ] ) @xmath285 for @xmath286 eq . ( [ mrhofpinc ] ) predicts @xmath287 87 mev , rather close to the experimental value of 93 mev . interestingly , eq . ( [ mrhofpinc ] ) coincides with the one obtained from qcd sum rules @xcite . the large @xmath9 scaling in ( [ mrhofpinc ] ) also matches : @xmath288 , @xmath289 . another distinct feature of the model is that the pion form factor is dominated by a single @xmath2 pole . indeed , the coupling @xmath90 is found by substituting eq . ( [ bncosh ] ) into eq . ( [ gnpipiu ] ) , @xmath290 which vanishes for all @xmath291 . this is due to the orthogonality of legendre functions and the fact that @xmath292 therefore @xmath2 meson dominance is exact for the pion form factor : @xmath293 . for @xmath76 one finds @xmath294 the ksrf ratio in this model is equal to 3 . this means that the @xmath2 width is underpredicted by a factor of about 2/3 . however , it is still interesting to compute @xmath235 for arbitrary @xmath9 , in the chiral limit , @xmath295 the rate of the electromagnetic decay @xmath239 in this model , @xmath296 is equal to 6.5 kev at @xmath286 , which is rather close to the observed @xmath242 kev . interestingly , the prediction from qcd sum rules @xcite is close but different in this case : @xmath297 . the phenomenology of the @xmath78 meson in this model is discussed in appendix [ app : a1 ] . the excitations with @xmath298 have an unrealistic mass spectrum in our model , so we shall not discuss their phenomenology . the baryon appears in the framework of chiral lagrangians as a solitonic object : a skyrmion @xcite . one wonders : what is the corresponding object in 5d that can describe the baryon . an obvious candidate is the instanton , which can be `` lifted '' to become a quasiparticle in 5d . here we show that the instanton appears from the point of view of 4d as a skyrmion . we are interested only in topological aspects , and defer the question of stability of such a solution to future work . , @xmath2 , and @xmath78 mesons has been studied @xcite . it was determined that vector mesons not only stabilize the skyrmion , but also noticeably improve agreement with phenomenology . ] on an intuitive level , to see the relation between the instanton and the skyrmion , one can consider as an example the well - known instanton solution in the singular gauge ( in the flat background metric ) : @xmath300 in this solution we shall think of @xmath301 as a four - vector with coordinates @xmath302 running through 1 , 2 , 3 , and 5 ( i.e. , @xmath118 , @xmath303 , @xmath304 , and @xmath162 ) and @xmath305 . note that the metric signature for these 4 coordinates is euclidean @xmath306 ( see ( [ metric ] ) ) . the solution we wish to use to describe a baryon is static , i.e. , @xmath307 , and there is no dependence on @xmath308 . to see the behavior of the pion field we need to look at @xmath309 ( see ( [ sigmacont ] ) ) : @xmath310 we see that at every fixed @xmath162 the solution is a hedgehog , thus having the same topology as the skyrmion made of the pion field . we shall now show that for an arbitrary background metric the topological charge of the instanton is equal to the baryon charge of the pion - field skyrmion . our discussion is very similar to that of refs . the 5d yang - mills theory possesses a conserved topological current , @xmath311 here @xmath312 is defined so that its elements are @xmath313 . for simplicity we assume that @xmath283 is a constant and absorb it into the gauge field , and drop the hat in the 5d lorentz indices in subsequent formulas . the boundaries are assumed to be @xmath257 . that this current is conserved , @xmath314 , can be shown by using the bianchi identity @xmath315}=0 $ ] . the topological charge of a static solution is @xmath316 the numerical coefficient in eq . ( [ 5dtopcurr ] ) was chosen so that the static instanton has unit total charge . now consider a field configuration where the pion field , given by the wilson line along the @xmath162 coordinate ( [ sigmacont ] ) has a nontrivial winding , and @xmath317 goes to 0 at the boundaries . to compute the topological charge of this configuration , it is convenient to perform a gauge transformation to set @xmath318 . explicitly , @xmath319 according to ( [ sigmacont ] ) , @xmath320 . thus while @xmath321 remains 0 at the left boundary @xmath322 , it becomes nonzero at the right boundary : @xmath323 by using the identity @xmath324 one can rewrite the topological charge ( [ qff ] ) as @xmath325 since at @xmath322 @xmath326 and @xmath327 . by using eq . ( [ aibound ] ) one transforms this expression to @xmath328.\ ] ] it is now obvious that the topological charge becomes the winding number of the pion field . therefore , the instanton becomes a skyrmion , and corresponds to the physical baryon . we considered a theory of an `` open moose '' given by lagrangian ( [ l ] ) illustrated in fig . [ fig : k ] . this model describes a multiplet of massless goldstone bosons and a tower of vector and axial vector mesons . we developed a formalism for calculating the mass spectrum and the coupling constants in this theory for arbitrary parameters of the moose , @xmath34 and @xmath35 , and determine their values in the continuum limit , when the number of hidden symmetry groups @xmath0 tends to infinity . we applied this formalism to two exactly solvable realizations of the model and found that the physics of the lowest modes match quite well with the phenomenology of the @xmath83 , @xmath2 and @xmath78 mesons . we also find that the open - moose theory naturally incorporates the phenomenon of vector meson dominance . for example , the pion form factor is saturated by poles from a tower of vector mesons . moreover , since couplings between mesons are given by overlap integrals , the couplings of highly excited @xmath2 s to the pion are suppressed by the oscillations of their wave functions in the fifth dimension . this means that the pion form factor should be well approximated by the sum of contributions from a few lowest @xmath2 s . in the second example we considered ( the `` cosh '' background ) the situation is brought to an extreme : the pion form factor is saturated by a single pole @xmath2-meson dominance . we verified that both weinberg s spectral sum rules are automatically obeyed , in a nontrivial way , in any open - moose theory . one of our original motivations was to include the excited vector mesons beyond the lowest @xmath78 . with respect to that goal , we achieved only limited success , at least within the two exactly solvable models we considered . on the one hand , we do find that vector and axial vector mesons alternate in the spectrum , as it seems to be the case in qcd , at least for a few excited states . on the other hand , in both our simple models , the mass of an @xmath80-th state @xmath66 is @xmath329 at large @xmath80 , which seems to be in contradiction with the real world , and with the theoretical prejudice that @xmath330 . further study of different backgrounds might provide a model that reproduces desired features of excited mesons and help understand constraints that phenomenology and qcd theorems impose on functions @xmath220 and @xmath221 . alternatively , it is also possible that the excited vector mesons have `` stringy '' nature and can not , in principle , be incorporated into our field - theoretical scheme . by a suitable choice of background , even an exactly solvable one . but we did not find such models viable in other respects . ] the success that the model enjoys in describing the lowest states can be attributed to an apparent property of low - energy qcd : at intermediate distances correlation functions are reasonably well saturated by a single pole . in the `` cosh '' model the excited mesons ensure the correct behavior of the ( averaged ) spectral densities , thus playing the role of the continuum . this explains why some results of qcd sum rules are well reproduced . we hope that the study of the open moose theories will deepen our understanding of qcd at the fundamental level . one intriguing fact discovered in these theories is the similarity to the ads / cft correspondence . the procedure of calculating current - current correlators is essentially equivalent to the well - known ads / cft prescription : the correlators are given by the variational derivatives of the classical 5d action of the dual theory with respect to the sources living on the 4d boundary . there is overwhelming evidence that the @xmath331 supersymmetric yang - mills theory is described by a string theory . perhaps , an open moose theory is a low - energy limit of the string theory dual to qcd . in this regard , the result we found in the `` cosh '' model , @xmath332 is reassuring in view of a general expectation that such a dual theory should have a coupling proportional to @xmath333 in the t hooft limit . among the questions left for further study is the detailed phenomenology of isoscalar mesons ( @xmath11 , @xmath12 , @xmath13 , etc . ) . these mesons are described by an additional 5d abelian gauge field , which should be introduced into the action ( [ 5ds ] ) . most of our results should generalize straighforwardly to this case . however , there is an important new issue that the isoscalar sector brings into the theory . the global u(1)@xmath334 symmetry must be explicitly broken , e.g. , @xmath11 should not be massless . it is very encouraging that the 5d formulation of the theory provides a very natural mechanism for this . it is the topological 5d chern - simons term of the form @xmath335 where @xmath336 is the 5d vector field describing isoscalars . this term breaks the u(1)@xmath334 symmetry in the desired way . in particular , it is not invariant under u(1)@xmath334 transformations on the 4d boundary ( although it is invariant under local transformations in the bulk of 5d ) . it is easy to see that it also provides @xmath337 and other anomalous processes in qcd . the coefficient of the term ( [ aff ] ) can be fixed by matching to qcd chiral anomaly , and is therefore proportional to @xmath9 . the term ( [ aff ] ) also couples the @xmath12 meson field to the baryon current , providing a hard - core repulsion between baryons , and preventing the baryon / instanton from shrinking to zero size ( this effect is the origin of the stabilization of the skyrmion observed in ref . . it would be also interesting to see how the open - moose theory realizes di vecchia - veneziano - witten lagrangian @xcite and the corresponding phenomenology . other avenues for future study are the incorporation of finite quark masses , extension to three flavors and realization of the wess - zumino - witten topological term ( which does require a 5th dimension @xcite ) . * acknowledgments * the authors thank s. r. beane , g. gabadadze , w - y . keung , and especially t. imbo and m. strassler for discussions . the authors also wish to acknowledge the review talk by r. l. jaffe at the _ qcd and string theory _ workshop at the institute for nuclear theory in seattle , which provided much of the motivation for this work . m.a.s . acknowledges the hospitality of the institute for nuclear theory , university of washington , where part of this work has been done , and thanks riken bnl center and u.s . department of energy [ de - ac02 - 98ch10886 ] for providing facilities essential for the completion of this work . d.t.s . is supported , in part , by doe grant no.doe-er-41132 and the alfred p. sloan foundation . m.a.s . is supported , in part , by a doe oji grant and by the alfred p. sloan foundation . app : mnp for reference , we provide here some additional formulas for interaction vertices in the continuum limit of an arbitrary open - moose model . let us define @xmath338 , @xmath339 , @xmath340 , and @xmath341 so that the lagrangian contains @xmath342 then @xmath343,{\label}{pimn}\\ g_{\pi\pi mn } & = & -\frac{f_\pi^2}8\int\!\frac{du}{f^2(u)}\ , b_m b_n,{\label}{pipimn}\\ g_{mnp } & = & \int\!du\ , gb_m b_n b_p,{\label}{mnp}\\ g_{mnpq } & = & \int\!du\ , g^2 b_m b_n b_p b_q,{\label}{mnpq}\end{aligned}\ ] ] direct couplings to external currents are suppressed in the continuum limit ( this includes , in particular , vector - meson dominance by a whole tower of mesons ) . a simple qualitative interpretation of these couplings exists in terms of the overlaps of the wave functions in the @xmath162 space , which reflects the property of locality of the theory ( [ l ] ) or ( [ 5ds ] ) . it is straightforward for the last two , resonance - resonance , couplings ( [ mnp ] ) and ( [ mnpq ] ) . these terms come from the second , @xmath344 term in ( [ l ] ) . for the pion - resonance couplings ( [ pimn ] ) , ( [ pipimn ] ) and ( [ gnpipiu ] ) , one should bear in mind that the strength of the coupling is proportional to @xmath345 , and think of the pion wave function as being proportional to @xmath346 ( looking at ( [ sigmapi ] ) ) . the @xmath162 derivatives in ( [ pimn ] ) are necessary to account for the fact that , although the pion wave function is even in @xmath347 , the pion itself is a pseudoscalar . instead of expressing the results in terms of the parameters of the model @xmath348 and @xmath283 , we will use @xmath16 and @xmath9 . the relations are @xmath349 @xmath350 @xmath351 @xmath352 rho - meson dominance of the pion form factor : @xmath353 @xmath354 there is a `` @xmath355 rule '' for pion emission : @xmath356 @xmath357 there is also a `` triangle rule '' for triple resonance vertex : @xmath358 i.e. , the amplitude vanishes if a triangle ( even a degenerate one ) with sides @xmath359 , @xmath360 and @xmath361 does not exist . app : a1 let us discuss the phenomenology of the lowest axial vector meson ( the @xmath77 excitation in the open moose ) . from eq . ( [ mncosh ] ) , the mass of the @xmath78 meson in the `` cosh '' model is @xmath362 . the @xmath78 decays into @xmath363 with the coupling ( [ gnn+1 ] ) @xmath364 by using the formula @xcite @xmath365 we find @xmath366 experimentally , the total width of @xmath78 is 250 to 600 mev , of which about 60% comes from @xmath367 @xcite . the @xmath78 decay constant in our model is @xmath368 a lattice measurement of this constant yields ( in our normalization ) @xmath369 gev@xmath370 @xcite , while an analysis of hadronic @xmath371 decays gives @xmath372 gev@xmath370 @xcite . the agreement is fair , but not exceptionally good . it can be checked that the two terms cancel each other exactly , so the amplitude vanishes . on the other hand , the partial width of this decay is quoted to be @xmath376 kev @xcite . the simplest @xmath39 hidden local symmetry model also suffers from the same problem ; in ref . @xcite this was cured by adding higher - derivative terms to the action . it would be interesting to see if this rate can be made nonzero by adding more terms to the action ( [ 5ds ] ) . c. n. yang and r. l. mills , phys . * 96 * , 191 ( 1954 ) . m. bando , t. kugo , s. uehara , k. yamawaki and t. yanagida , phys . lett . * 54 * , 1215 ( 1985 ) . m. bando , t. fujiwara and k. yamawaki , prog . * 79 * , 1140 ( 1988 ) . for a review see m. bando , t. kugo and k. yamawaki , phys . rept . * 164 * , 217 ( 1988 ) . h. georgi , nucl . b * 331 * , 311 ( 1990 ) . k. hagiwara _ et al . _ [ particle data group collaboration ] , phys . rev . d * 66 * , 010001 ( 2002 ) . m. b. halpern and w. siegel , phys . d * 11 * , 2967 ( 1975 ) . j. m. maldacena , adv . theor . math . * 2 * , 231 ( 1998 ) [ int . j. theor . phys . * 38 * , 1113 ( 1999 ) ] [ arxiv : hep - th/9711200 ] . s. s. gubser , i. r. klebanov and a. m. polyakov , phys . b * 428 * , 105 ( 1998 ) [ arxiv : hep - th/9802109 ] . e. witten , adv . * 2 * , 253 ( 1998 ) [ arxiv : hep - th/9802150 ] . for a review see o. aharony , s. s. gubser , j. m. maldacena , h. ooguri and y. oz , phys . rept . * 323 * , 183 ( 2000 ) [ arxiv : hep - th/9905111 ] . a. m. polyakov , int . j. mod . phys . a * 14 * , 645 ( 1999 ) [ arxiv : hep - th/9809057 ] . n. arkani - hamed , a. g. cohen and h. georgi , phys . * 86 * , 4757 ( 2001 ) [ arxiv : hep - th/0104005 ] . c. t. hill , s. pokorski and j. wang , phys . d * 64 * , 105005 ( 2001 ) [ arxiv : hep - th/0104035 ] . n. m. kroll , t. d. lee and b. zumino , phys . rev . * 157 * , 1376 ( 1967 ) . s. weinberg , phys . lett . * 18 * , 507 ( 1967 ) . k. sfetsos , nucl . b * 612 * , 191 ( 2001 ) [ arxiv : hep - th/0106126 ] . a. falkowski and h. d. kim , jhep * 0208 * , 052 ( 2002 ) [ arxiv : hep - ph/0208058 ] . l. randall , y. shadmi and n. weiner , jhep * 0301 * , 055 ( 2003 ) [ arxiv : hep - th/0208120 ] . k. kawarabayashi and m. suzuki , phys . * 16 * , 255 ( 1966 ) . riazuddin and fayyazuddin , phys . rev . * 147 * , 1071 ( 1966 ) . m. kruczenski , d. mateos , r. c. myers and d. j. winters , jhep * 0307 * , 049 ( 2003 ) [ arxiv : hep - th/0304032 ] . s. r. beane , phys . rev . d * 64 * , 116010 ( 2001 ) [ arxiv : hep - ph/0106022 ] . m. a. shifman , a. i. vainshtein and v. i. zakharov , nucl . b * 147 * , 448 ( 1979 ) . t. h. skyrme , nucl . phys . * 31 * , 556 ( 1962 ) . g. s. adkins and c. r. nappi , phys . b * 137 * , 251 ( 1984 ) . y. igarashi , m. johmura , a. kobayashi , h. otsu , t. sato and s. sawada , nucl . b * 259 * , 721 ( 1985 ) . meissner and i. zahed , phys . lett . * 56 * , 1035 ( 1986 ) . for a review see also i. zahed and g. e. brown , phys . * 142 * , 1 ( 1986 ) . c. t. hill and p. ramond , nucl . b * 596 * , 243 ( 2001 ) [ arxiv : hep - th/0007221 ] . c. t. hill , phys . lett . * 88 * , 041601 ( 2002 ) [ arxiv : hep - th/0109068 ] . a. karch and e. katz , jhep * 0206 * , 043 ( 2002 ) [ arxiv : hep - th/0205236 ] . p. di vecchia and g. veneziano , nucl . b * 171 * , 253 ( 1980 ) . e. witten , nucl . b * 223 * , 422 ( 1983 ) . m. wingate , t. degrand , s. collins and u. m. heller , phys . lett . * 74 * , 4596 ( 1995 ) [ arxiv : hep - ph/9502274 ] . n. isgur , c. morningstar and c. reader , phys . d * 39 * , 1357 ( 1989 ) .
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motivated by phenomenological models of hidden local symmetries and the ideas of dimensional deconstruction and gauge / gravity duality , we consider the model of an `` open moose . ''
such a model has a large number @xmath0 of hidden gauge groups as well as a global chiral symmetry . in the continuum limit
@xmath1 the model becomes a 4 + 1 dimensional theory of a gauge field propagating in a dilaton background and an external space - time metric with two boundaries .
we show that the model reproduces several well known phenomenological and theoretical aspects of low - energy hadron dynamics , such as vector meson dominance .
we derive the general formulas for the mass spectrum , the decay constants of the pion and vector mesons , and the couplings between mesons .
we then consider two simple realizations , one with a flat metric and another with a `` cosh '' metric interpolating between two ads boundaries .
for the pion form factor , the single pole @xmath2-meson dominance is exact in the latter case and approximate in the former case .
we discover that an ads / cft - like prescription emerges in the computation of current - current correlators .
we speculate on the role of the model in the theory dual to qcd .
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recently , topological properties of time - reversal - invariant band insulators in two and three dimensions have been extensively studied@xcite . a class of insulators preserving the time reversal symmetry is called topological insulators characterized by non - trivial topological [email protected] topological insulators have been intensively studied because of the existence and potential applications of robust surface metallic states . both in two and three dimensions , the topological phases are typically realized in the systems with strong spin - orbit interaction@xcite . all the known topological insulators contain heavy or rare metal elements , such as bismuth or iridium , which poses constraints on the search for topological materials . irrespective of constitutents , ubiquitous mutual coulomb repulsions among electrons have been proposed to generate effective spin - orbit couplings @xcite . it has been proposed that an extended hubbard model on the honeycomb lattice can generate an effective spin - orbit interaction from a spontaneous symmetry breaking at the hartree - fock mean - field level leading to a topologically non - trivial phase@xcite . since the honeycomb - lattice system , which is dirac semimetals in the non - interacting limit , becomes a topologically nontrivial insulator driven by the coulomb interaction , this phase is often called a topological mott insulator ( tmi ) . this phenomenon is quite unusual not only because an emergent spin - orbit interaction appears from the electronic mutual coulomb interaction , but also it shows an unconventional quantum criticality that depends on the electron band dispersion near the fermi point@xcite . however , this proposed topological phase by utilizing the ubiquitous coulomb repulsions has not been achieved in real materials even though the tmi is proposed not only in various solids @xcite but also in cold atoms loaded in optical lattices @xcite . even in simple theoretical models such as extended hubbard models , it is not clear whether the tmis become stable against competitions with other orders and quantum fluctuations . reliable examination of stable topological mott orders in the extended hubbard model is hampered by competing symmetry breakings such as cdws . couplings driving the topological mott transitions are also relevant to formations of a cdw , which has not been satisfactorily discussed in the previous studies . since the emergence of the tmi in the honeycomb lattice requires the coulomb repulsion between the next nearest neighbor sites , the long - period cdw instability must be considered on equal footing , which is not captured in the small - unit - cell mean - field ansatz employed in the previous studies . examining charge fluctuations with finite momentum over entire brillouin zones is an alternative way to clarify the competitions among tmis and cdws , as studied by employing functional renormalization group methods @xcite . however , first order thermal or quantum phase transitions not characterized by diverging order - parameter fluctuations are hardly captured by such theoretical methods . the most plausible symmetry breking competing with tmis indeed occurs as a first order quantum phase transition as discussed later . the quantum many - body fluctuations beyond the mean - field approximation severely affects the stability of the tmi . the stability of the tmi and estimation of the critical value of interaction on the honeycomb lattice has mainly been considered by mean - field calculations which can not treat the correlation effect satisfactorily . however , there exists a reliable limit where the tmi becomes stable : for infinitesimally small relevant coulomb repulsions , the quadratic band crossing with vanishing fermi velocities cause the leading instability toward the tmi , as extensively examined by using perturbative renormalization group methods@xcite . however , examining the instabilities toward the tmi in dirac semimetals requires elaborate theoretical treatments . in this study , for clarification of the competitions among tmis and other symmetry breakings , we first examine the long - period cdw at the level of mean - field approximation that turns out to be much more stable compared to that of short period . indeed , this cdw severly competes the tmi on the honeycomb lattice . the tmi on the honeycomb lattice studied in the literatures is consequently taken over by the cdw . we , however , found a prescription to stabilize the tmis on the honeycomb lattice : by reducing the fermi velocity of the dirac cones , the tmi tends to be stabilized . we examine the realization of the tmis in the extended hubbard model on the honeycomb lattice by controlling the fermi velocity and employing a variational monte carlo method@xcite with many variational parameters@xcite , multi - variable variational monte carlo ( mvmc)@xcite , together with the mean - field approximation . finally , we found that , by suppressing the fermi velocity to a tenth of that of the original honeycomb lattice , the tmi emerges in an extended parameter region as a spontaneous symmetry breaking even when we take many - body and quantum fluctuations into account . this paper is organized as follows . in section [ sec : model and method ] , we introduce an extended hubbard model and explain the order parameter of tmi . we also introduce the mvmc method . in section [ sec : stability ] , we first show how the long - range cdw becomes stable over the tmi phase in standard honeycomb lattice models . then we pursue the stabilization of tmi by modulating fermi velocity at the dirac cone at the mean - field level . finally we study by the mvmc method the effect of on - site coulomb interaction which was expected to unchange the stability of the tmi phase at the level of mean - field approximation . section [ sec : dis ] is devoted to proposal for realization of tmis in real materials such as twisted bilayer graphene . in this section , we study ground states of an extended hubbard model on the honeycomb lattice at half filling defined by @xmath0 where the single particle parts of @xmath1 are defined as @xmath2 and @xmath3 is the spin - orbit interaction . here @xmath4 is a creation ( annihilation ) operator for a @xmath5- spin electron , @xmath6 is an electron density operator , @xmath7 represents the hopping of electrons between site @xmath8 and @xmath9 , and @xmath10 are on - site ( off - site ) coulomb repulsion . bracket @xmath11 denotes the next - neighbor pair , @xmath12 is the strength of the spin - orbit interaction and @xmath13 is the @xmath14=@xmath15-spin operator . in eq.([hso ] ) , the @xmath16-th site is in the middle of the next nearest neighboring pair @xmath8 and @xmath9 as shown in fig . [ fig : honeycomb ] , and @xmath17 is the vector from the site @xmath8 to @xmath9 . we start with the hopping matrix @xmath7 in eq.([eq : h0 ] ) for the bond connecting a pair of the nearest - neighbor sites @xmath18 , @xmath19 as the simplest extended hubbard model on the honeycomb lattice . later , we will examine the effect of third neighbor hoppings @xmath20 . we take @xmath21 as the unit of energy and set @xmath22 throughout the rest of this paper . for the non - interacting limit , @xmath23 , the system becomes a topological insulator when @xmath12 is nonzero , which is identical to the topological phase of the kane - mele model @xcite . for the off - site coulomb repulsion , we mainly consider the second neighbor interaction ( @xmath24 ) , which is necessary for the emergence of the correlation - induced topological insulator @xcite . the second neighbor coulomb repulsions @xmath25 effectively generate the spin - orbit interactions , which are identical to @xmath12 , and induce topological insulator phases even for @xmath26 . we note that the coulomb repulsion of on - site or the nearest neighbor site do not affect the stability of the tmi phase at the level of the mean - field approximation . indeed , our mvmc results show that this is essentially true beyond the level of mean - field approximation which will be discussed in the later section . therefore , for the moment , we focus only on the effect of @xmath27 for the consideration of interaction effects . . next nearest neighbor interaction @xmath25 and the loop current @xmath28 flowing between the next nearest neighbor are shown by red and blue arrows . , width=245 ] here the tmi is the broken symmetry phase characterized by the order parameter @xmath28 defined by @xmath29 where the self - consistent mean fields for the second neighbor bonds are given by @xmath30 here , @xmath31 denotes the expectation value for next nearest neighbor bonds and the order parameter @xmath28 is physically interpreted as spin dependent loop currents flowing within a hexagons constituting the honeycomb lattice . at the level of the mean - field approximation , this quantum phase transition is understood by decoupling two - body electron correlation term of the next nearest neighbor bond , @xmath32 , as @xmath33 we also note that this phase transition to tmi , which is proposed not only on the honeycomb lattice but also in several other lattice models , belongs to an unconventional universality class , which depends on the dimension of the system and the dispersion of the electron band@xcite . in this section , we pursue the topological mott phase transition by employing the mean - field analysis and the variational monte carlo method . for the latter method , we use a trial wave function of the gutzwiller - jastrow form , @xmath34 with a one body part , @xmath35 where @xmath36 is the variational parameters and @xmath37 is the number of the electrons in the system . though this form of the wave function restricts itself to the hilbert subspace with the zero total @xmath38-component of @xmath14=@xmath15 , @xmath39 , it can describe topological phases on the honeycomb lattice as long as we use complex variables for @xmath36 . here , @xmath40 and @xmath41 are the gutzwiller and jastrow factors defined as @xmath42 and @xmath43 respectively , with which the effects of electron correlations are taken into account beyond the level of mean - field approximation . the expectation value , @xmath44 , is minimized with respect to variational parameters , @xmath36 , @xmath45 , and @xmath46 , by using the monte - carlo sampling and using the stochastic reconfiguration method by calculating gradient of the energy and the overlap matrix in the parameter space@xcite . we optimize the parameters by typically 2000 stochastic reconfiguration steps . in the present implementation of the variational monte carlo method with complex variables , the feasible system size for the calculation is about up to 300 , from which we speculate properties in the thermodynamic limit . for this purpose , we perform the size extrapolation using the standard formula @xmath47 where @xmath48 is the linear dimension of the system size . we note that the order of taking the limit in the right hand side of eq.([zeta_limit ] ) is also important for the validity of the extrapolation , which is known as the textbook prescription for the defining spontaneous symmetry breakings . another practical way to determine the spontaneous symmetry breakings is in principle the finite size scaling of the correlation for the order parameter . however , the latter finite size scaling is not practically easy . the correlation of @xmath28 becomes too small because @xmath28 itself is about the order of @xmath49 and the correlation becomes the order of @xmath50 which becomes comparable to the statistical error of the monte carlo sampling . for the size extrapolation , we fit the data of the finite size calculations by a polynomial of @xmath51 , that is , we assume the size dependence by @xmath52 the above assumption for the finite size scaling is based on a practical observation and an analogy to the finite size scaling in the spin wave theory @xcite . as a practical observation , @xmath53 for the limit @xmath54 and @xmath55 is scaled by @xmath51 . we examine the ground states of the extended hubbard model on the honeycomb lattice by tuning the on - site coulomb repulsion @xmath56 and the second neighbor coulomb repulsion @xmath25 . even for the parameter sets favorable to the tmis that have been studied in the pioneering works on the tmi @xcite , we show that the tmi is not stabilized when we take into account other competing orders overlooked in the literature . however , here , we reveal that , by tuning the fermi velocities of the dirac cones , the tmi is indeed stabilized . in this section , we consider long - period cdw states and show that 6-sublattice order is stable when @xmath25 becomes dominat . this state is schematically shown in fig.[fig : cdw6sub](a ) , where the electron density per site is disproportionated into four inequivalent values . when we pick up very rich sites or very poor sites , they constitute triangular lattices . the moderately rich or poor sites constitute the honeycomb lattices . sites labelled by 1 ( red ) , 2 ( light red ) , 3 ( light blue ) and 4 ( blue ) correspond to the sites where the electron densities are very rich , rich , poor and very poor , respectively . ( b ) the growth of order parameter of the cdw state . ( c ) resulting phase diagram of honeycomb lattice for @xmath56 and @xmath25 at the level of mean - field approximation . we do not find the region of stable tmi phase . , width=302 ] figure [ fig : cdw6sub](b ) shows the growth of the order parameters of the 6-sublattice cdw state for several different parameters of @xmath56 . in this calculation , we have used the mean - field approximation , where we defined the mean field by @xmath57 where @xmath58 is defined through @xmath59 here , the condition of half filling is satisfied by the following identity , @xmath60 as can be seen from fig.[fig : cdw6sub](b ) , the order parameter grows above @xmath61 even if we set @xmath62 . this is a quite serious problem for the stabilization of the tmi since the critical value of @xmath25 for the tmi at the mean - field approximation is also about @xmath63 . in addition , the energy gain due to the formation of the topological mott order is much smaller than that for the cdw state . when we consider larger values of @xmath56 , then the system becomes an antiferromagnetic state . furthermore , the mean - field approximation often overestimates the ordered phase , because it does not take into account fluctuation effects seriously . actually , by using the mvmc method , we could not find the region where the topological insulator becomes the ground state in the parameter space of @xmath56 and @xmath27 . the resulting phase diagram by the mean - field approximation is shown in fig . [ fig : cdw6sub ] ( c ) . there the antiferromagnetic phase is denoted as nel , where the spin on different sites of the bipartite aligns in the anti - parallel direction . as examined above , we found that the cdw state dominates and could not find parameter regions where the tmi is stable . however , we find that the tmi becomes stable by modulating the fermi velocity at the dirac cones in the electronic band dispersion actually it is confirmed that , when the fermi velocity is 0 and then the dirac cones change to the quadratic band crossing points , the phase transition from a zero - gap semiconductors to a tmi occurs with infinitesimal coulomb repulsions @xcite . for the honeycomb lattice , it is possible to change dirac semimetals to quadratic band crossings by introducing the third neighbor hopping @xmath20 ( schematically shown in fig.[fig : t3 ] ( a ) ) . then the part of the hamiltonian @xmath64 is replaced with @xmath65 where the third neighbor hoppings are given as @xmath66 here the tnn stands for the third neighbor . we find that the fermi velocity linearly decreases by introducing @xmath20 and becomes @xmath67 at @xmath68 as @xmath69 , which is also shown in fig.[fig : t3 ] ( b ) . though tuning nominal value of @xmath70 in the graphene is difficult , tuning of the fermi velocity at the dirac cone has been proposed in bilayer graphene by changing the relative orientation angle between two layers@xcite , which is effectively equivalent to the tuning of @xmath70 . and fermi velocity @xmath71 at dirac point . here , @xmath71 is shown by the ratio to the fermi velocity at @xmath72 . , width=321 ] figure [ fig : mfphase ] shows the phase diagram calculated by the mean - field approximation for @xmath73 , @xmath74 , and @xmath75 . compared to the nel ordered phase and the tmi , the cdw phase is not largely affected by the fermi velocity . therefore , the region of tmi recovers . ( a ) 0.3 , ( b ) 0.4 and ( c ) 0.45 . sm denotes the semimetal . , width=321 ] next , we examine the stability by using the mvmc method . in our mvmc method , we impose the translational symmetry on the variational parameters to reduce the computational cost , as @xmath76 here , @xmath77 is taken as any bravais vector of the 6 sublattice unit cell illustrated in fig.[fig : cdw6sub](a ) , in order to examine possible spontaneous symmetry breakings including the cdw shown in fig.[fig : cdw6sub](a ) and the tmi on an equal footing . therefore the number of the sites for each calculation is taken as @xmath78 . we have calculated for @xmath79 , where about 2500 variational parameters are used for the calculation of the largest size . to perform the extrapolation to the small external filed limit , @xmath80 , we have also calculated for several different strengths of the spin - orbit interaction , @xmath81 , and @xmath82 . figure [ fig : vmcdata](a ) shows the numerical results for the order parameter of the tmi at @xmath83 and @xmath84 . the sudden drops in @xmath28 around @xmath85 signals the emergence of the 6-sublattice cdw state . indeed , when @xmath27 further increases , @xmath28 vanishes . this is physically quite natural because @xmath28 is interpreted as the loop current and hard to be stabilized inside the cdw phase where electrons are locked at specific sites . for @xmath86 at @xmath84 . ( b ) size extrapolation at @xmath87 for several different values of @xmath27 . here , four different sizes ( @xmath79 ) are calculated . lines are results of quadratic fittings . ( c)extrapolation of @xmath12 is shown for different values of @xmath27 . lines are results of quadratic fitting . , width=321 ] same calculations are carried out for @xmath88 , which is employed in the size extrapolation of @xmath28 . figure [ fig : vmcdata](b ) shows the size extrapolation at @xmath87 as a typical example . since the cdw becomes dominant at @xmath85 , the results for @xmath89 are shown in fig.[fig : vmcdata](b ) . when @xmath25 is small , @xmath28 empirically follows the size dependence @xmath90 with a constant @xmath91 as we see in fig.5(b ) . this extrapolation is performed for four different values of @xmath12 , and then we get the values for @xmath92 . these data are shown in fig . [ fig : vmcdata](c ) , where the second extrapolation to @xmath54 is performed . final results for @xmath93 are shown in fig . [ fig : vmcresult](a ) , where the results of @xmath94 are also shown . the errorbars are defined from the errors of the extrapolation , where the largest error of the first extrapolation is added to the errors of the second extrapolation . we note that the error bars arising from the statistical errors of the monte carlo sampling are much smaller . relatively large errorbars for small @xmath27 at @xmath94 is possiblly because of the existence of the critical point at a finite value of @xmath27 , which is about @xmath95 . when @xmath27 exceeds this critical value and @xmath28 in the thermodynamic limit remains nonzero , the error becomes smaller as can be seen from fig . [ fig : vmcresult](a ) . for @xmath83 , the order grows from small value of @xmath96 . however , at @xmath94 , non - zero @xmath28 can not be detected for small values of @xmath27 . though the estimate of the universality class from these data is difficult , theoretically it is expected to belong to that of the gross - neveu model@xcite , and our result does not contradict this criticality . for @xmath97 , we do not find the value of @xmath27 where @xmath28 in the thermodynamic limit remains nonzero , and therefore phase transitions is not expected . this is shown in fig . [ fig : vmcresult](b ) , where the size extrapolation at @xmath87 is shown . there , @xmath28 becomes @xmath67 at @xmath98 for all @xmath27 , which is completely different from the behavior at @xmath94 and @xmath83 . the resulting phase diagram is shown in fig . [ fig : ueffect](a ) . although we expect that the mvmc results show larger critical values @xmath99 for the transition at @xmath100 , the estimated results indicate @xmath101 slightly smaller than the mean - field results shown in fig.[fig : mfphase ] . the reason that @xmath101 becomes smaller at @xmath100 in the mvmc results is probably an artifact arising from a peculiar size dependence near the essential singularity at @xmath102 , as we see even in the mean - field calculation shown in fig . [ fig : ueffect](b ) where , the size dependence in the mean - field calculation are shown . the possible errors in in the estimate of @xmath28 is as large as @xmath103 and the resultant errors in the estimate of @xmath101 is around 0.1 . therefore , the stability of the tmi phase over the cdw and nel phases in the region @xmath104 for @xmath100 is robust . here we note that the boundary between the cdw and tmi phases around @xmath105 does not change when we take into account the quantum fluctuations ( by calculating with the vmc method ) , because it is a strong first - order transition . are shown . ( b)size extrapolation at @xmath106 and @xmath87 for several different values of @xmath27 . here , four different sizes ( @xmath79 ) are calculated . lines are results of quadratic fittings . , width=321 ] now we discuss the effect of the on - site coulomb repulsion . though the onsite coulomb repulsion does not affect the value of @xmath28 and therefore stability of the tmi in the mean - field approximation , our mvmc result shows that increasing @xmath56 decreases the value of @xmath28 if @xmath27 is fixed as shown in fig . [ fig : ueffect](c ) . while the increasing @xmath56 quantitatively decreases the value of @xmath28 , its effect does not destroy the stability of the tmi phase at @xmath107 at least if @xmath108 . it also suppresses the emergence of the cdw . therefore it may help to enlarge the region of the tmi phase . the same effect is expected when we consider the coulomb repulsion for the nearest neighbor sites @xmath109 . that is , it decreases the value of @xmath28 beyond the level of the mean - field approximation but does not essentially affect the phase transition . however , we also note @xmath109 may cause another type of cdw , and stabilization of tmi should be examined against this cdw phase when @xmath109 is large . and @xmath70 at @xmath110 obtained by mvmc calculations . ( b)size dependence of mean - field calculation for several different values of @xmath27 at @xmath83 . ( c ) @xmath27 dependence of order parameter of tmi for @xmath111 and @xmath112 at @xmath113 . decrease in @xmath28 is observed by introducing @xmath56 . , width=321 ] as a qualitative difference from the mean - field approximation , we found that effect of @xmath56 decreases the value of the order parameter of the tmi . in the case of the border between the semimetal and the nel ordered phases , it is expected that @xmath114 , namely the critical values for @xmath56 becomes larger by treating fluctuation effects carefully . furthermore , we expect that @xmath114 becomes a function of @xmath27 , which may enhance the fluctuation of the nel order and suppress the phase transition . in our calculation , we did not find nel ordered states or mixed states of tmi and nel ordered states . on the other hand , the cdw phase is expected to be much more stable against fluctuations and the mean - field solution gives a reasonably good description because of a large scale of the energy gain for the cdw phase in comparison to other phases as adequately shown even by the mean - field approximation . at the boundary of the tmi to nel ordered phases , the universality class may change as suggested in the kane - mele - hubbard model@xcite . here , we discuss the realization of the tmi in the real solids . a primary candidate of tmis is graphene . as a well - known fact , graphene is nothing but a two - dimensional honeycomb network of carbon atoms and hosts dirac electrons . however , it is also a well - known fact that , in free - standing graphene and graphene on substrates , significant single - particle excitation gaps have not been observed yet@xcite . below , we examine possible routes toward the realization of the tmi in graphene - related systems . first of all , as already studied above , the suppression of the fermi velocity of the dirac electrons is crucial for the stabilization of the tmis . as extensively studied in the literature @xcite , twisted bilayer graphene ( tblg ) offers dirac electrons with tunable fermi velocities . by choosing stacking procedures , the quadratic band crossing , in other words , the zero fermi velocity limit , is also achieved , which has been already observed in experiments@xcite . next , we need to clarify competitions with any other possible symmetry breakings in the tblg with the fermi velocity smaller than that of graphene . the suppressed fermi velocities may possibly cause instabilities towards not only the tmis but also other competing orders as discussed in this paper . for clarification of the competition we need an _ ab initio _ estimation of effective coulomb repulsions which directly correspond to the coulomb repulsions in the extended hubbard model@xcite . the _ ab initio _ study on the effective coulomb repulsions employs a many - body perturbation scheme called constrained random phase approximation ( crpa)@xcite . the crpa estimation gives the following values for the coulomb repulsions : the on - site and off - site coulomb repulsions are given as @xmath115 , @xmath116 , and @xmath117 with @xmath118 ev for free - standing graphene . if we neglect longer - ranged coulomb repulsions , we expect the nel or cdw orders by employing these crpa estimates of @xmath119 , and @xmath25 in graphene and tblg . therefore , the free - standing graphene and tblg do not offer a suitable platform for the tmis . however , by choosing dielectric substrates , the strength of the coulomb repulsions , @xmath120 , and @xmath121 , is suppressed due to enhancement of dielectric constant as @xmath122 , and @xmath123 , where @xmath124 is defined by dielectric constants of each materials as @xmath125 . ( here , we ignored the possible reduction of the effective dielectric constants at small distances . ) then , we may approach the parameter region , where tmis become stable , as shown in fig.[fig : ueffect](b ) . if we neglect @xmath126 , dielectric substrates with @xmath127 are enough to stabilize the tmis . even when the nearest - neighbor coulomb repulsion @xmath126 is taken into account , tmi is expected to be stable as long as @xmath126 is not strong . in the above discussion , we neglected further neighbor coulomb repulsions , namely , the third neighbor coulomb repulsions @xmath128 , the fourth neighbor ones @xmath129 , and so on . to justify the above discussion , we need to screen the further neighbor coulomb repulsions by utilizing dielectric responses of the substrates and/or adatoms . here we note that the screening from atomic orbitals on the same and neighboring sites effectively reduces the coulomb repulsions as extensively studied by using crpa@xcite . the relative strength of the on - site and second neighbor coulomb repulsions , @xmath130 , may also be controllable by utilizing the adatoms , which are expected to efficiently screen the on - site coulomb repulsions if the adatoms is located just on top of the carbon atoms . if we combine the control of the @xmath130 with the suppression of the further - neighbor coulomb repulsions , the above discussion may be relevant . by utilizing the adatoms , the nearest - neighbor coulomb repulsions @xmath126 are also expected to be well - screened by adatoms on nearest - neighbor carbon - carbon bonds . the suppression of @xmath126 is helpfull for suppressing the cdws competing with the tmi . finally , we estimate the single - particle excitations gap @xmath131 induced by the tmi . the excitation gap is crucial for actual applications of the tmi as a spintronics platform . if we set @xmath132 and expect the tblg with @xmath133 ev , we obtain the excitation gap up to 0.1 ev , where we use the mean - field estimation of the gap @xmath134@xcite with the value of the mvmc result for @xmath28 and assumed this formula is valid in the presence of electron correlation . the estimated gap scale is substantially larger than the room temperatures . our theoretical results support that topological insulators with such a large excitation gap @xmath135 ev are possibly obtained by using abundant carbon atoms . in this paper , we have studied the realization of the tmi phase for the electronic systems on a honeycomb lattice by using the mean - field calculation and mvmc method . we found that the cdw of the 6 sublattice unit cell is much more stable than the previously estimated cdw with smaller unit cells for the simplest case where the electronic transfer is limited to the nearest neighbor pair . for the stabilization of the tmi we need to suppress the fermi velocity at the dirac point than the standard dirac dispersion for the case with only the nearest neighbor transfer . in the case of the honeycomb lattice , this is realized by introducing the third neighbor hopping @xmath20 and we have given quantitative criteria for the emergence of the tmi . related real material is a bilayer graphene where the fermi velocity is tuned by changing the rotation angle between two parallel layers@xcite . actually , the quadratic band crossing is realized when the rotation angle is @xmath67 ( known as the ab stacking bilayer graphene@xcite ) , which is mimicked by @xmath136 . since smaller values of fermi velocity stabilizes the tmi at smaller values of @xmath27 , its effective control may offer a breakthrough in the realization of two dimensional tmis . we need further analyses for experimental methods of controlling the stability of the tmis and _ ab initio _ quantitative estimates of the stability for bilayer graphenes , which are intriguing future subjects of our study . the authors thank financial support by grant - in - aid for scientific research ( no . 22340090 ) , from mext , japan . the authors thank t. misawa and d. tahara for fruitful discussions . a part of this research was supported by the strategic programs for innovative research ( spire ) , mext ( grant number hp130007 and hp140215 ) , and the computational materials science initiative ( cmsi ) , japan .
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realization and design of topological insulators emerging from electron correlations , called topological mott insulators ( tmis ) , is pursued by using mean - field approximations as well as multi - variable variational monte carlo ( mvmc ) methods for dirac electrons on honeycomb lattices .
the topological insulator phases predicted in the previous studies by the mean - field approximation for an extended hubbard model on the honeycomb lattice turn out to disappear , when we consider the possibility of a long - period charge - density - wave ( cdw ) order taking over the tmi phase .
nevertheless , we further show that the tmi phase is still stabilized when we are able to tune the fermi velocity of the dirac point of the electron band . beyond the limitation of the mean - field calculation
, we apply the newly developed mvmc to make accurate predictions after including the many - body and quantum fluctuations . by taking the extrapolation to the thermodynamic and weak external field limit
, we present realistic criteria for the emergence of the topological insulator caused by the electron correlations . by suppressing the fermi velocity to a tenth of that of the original honeycomb lattice ,
the topological insulator emerges in an extended region as a spontaneous symmetry breaking surviving competitions with other orders .
we discuss experimental ways to realize it in a bilayer graphenesystem .
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the discontinuous galerkin method @xcite nowadays is a well - established method for solving partial differential equations , especially for time - dependent problems . it has been thoroughly investigated by cockburn and shu as well as hesthaven and warburton , who summarized many of their findings in @xcite and @xcite , respectively . concerning maxwell s equations in time - domain , the dgm has been studied in particular in @xcite . the former two apply tetrahedral meshes , which provide flexibility for the generation of meshes also for complicated structures . the latter two make use of hexahedral meshes , which allow for a computationally more efficient implementation @xcite . in @xcite the authors state that the method can easily deal with meshes with hanging nodes since no inter - element continuity is required , which renders it particularly well suited for @xmath2-adaptivity . indeed , many works are concerned with @xmath0- , @xmath1- or @xmath2-adaptivity within the dg framework . the first published work of this kind is presumably @xcite , where the authors consider linear scalar hyperbolic conservation laws in two space dimensions . for a selection of other publications see @xcite and references therein . the latter three are concerned with the adaptive solution of maxwell s equations in the time - harmonic case . in this article , we are concerned with solving the maxwell equations for electromagnetic fields with arbitrary time dependence in a three - dimensional domain @xmath3 . they read [ eq : maxwell ] @xmath4 with the spatial variable @xmath5 and the temporal variable @xmath6 subject to boundary conditions specified at the domain boundary @xmath7 and initial conditions specified at time @xmath8 . the vectors of the electric field and flux density are denoted by @xmath9 and @xmath10 and the vectors of the magnetic field and flux density by @xmath11 and @xmath12 . the electric current density is denoted by @xmath13 . however , we assume the domain to be source free and free of conductive currents ( @xmath14 ) . furthermore , we assume heterogeneous , linear , isotropic , non - dispersive and time - independent materials in the constitutive relations @xmath15 the material parameters @xmath16 and @xmath17 are the magnetic permeability and dielectric permittivity . at the domain boundary , we apply either electric ( @xmath18 ) or radiation boundary conditions ( @xmath19 ) , where @xmath20 denotes the local speed of light @xmath21 . we also introduce the electromagnetic energy @xmath22 contained in a volume @xmath23 obtained by integrating the energy density @xmath24 as @xmath25 this paper focuses on a general formulation of the dgm on non - regular hexahedral meshes as well as the projection of solutions during mesh adaptation . the issues of optimality of the projections and stability of the adaptive algorithm are addressed . special emphasis is put on discussing the computational efficiency . to the best of our knowledge , this is the first publication dealing with dynamical @xmath2-meshes for the maxwell time - domain problem employing the dg method in three - dimensional space . as they are key aspects of adaptive and specifically @xmath2-adaptive methods , we will also address the issues of local error and smoothness estimation . this includes comments on the computational efficiency of the estimates . as estimators are not at the core of this article the discussion is , however , rather short . we perform a tesselation of the domain of interest @xmath26 into @xmath27 hexahedra @xmath28 such that the tesselation @xmath29 is a polyhedral approximation of @xmath26 . the tesselation is not required to be regular , however , it is assumed to be derivable from a regular root tesselation @xmath30 by means of element bisections . the number of element bisections along each cartesian coordinate , which is required to an obtain element @xmath31 of @xmath32 is referred to as the refinement levels @xmath33 . as we allow for anisotropic bisecting the refinement levels of one element may differ . in case of isotropic refinement we simply use @xmath34 . the intersection of two neighboring elements @xmath35 is called their interface , which we denote as @xmath36 . as we consider non - regular grids , every face @xmath37 of a hexahedral element may be partitioned into several interfaces depending on the number of neighbors @xmath38 such that @xmath39 . the face orientation is described by the outward pointing unitary normal @xmath40 . the union of all faces is denoted as @xmath41 , and the internal faces @xmath42 are denoted as @xmath43 . finally , the volume , area and length measures of elements , interfaces , faces and edges are referred to as @xmath44 , @xmath45 , @xmath46 and @xmath47 , where @xmath48 denotes any of the cartesian coordinates . every element of the tesselation @xmath32 is related to a master element @xmath49 ^ 3 $ ] through the mapping @xmath50 @xmath51 where @xmath52 denotes the element center . multiplying maxwell s equations ( [ eq : maxwell ] ) by a test function @xmath53 , integrating over @xmath54 and performing integration by parts yields [ eq : weakmaxwell ] @xmath55 where the explicit dependencies of @xmath56 and @xmath6 have been omitted . equations ( [ eq : weakmaxwell ] ) constitute the generic weak formulation of the time - dependent maxwell s equations . in the following , we will replace the exact field solutions @xmath9 and @xmath11 by approximations using the discontinuous galerkin framework . the space and time continuous electromagnetic field quantities are approximated on @xmath32 as @xmath57 where @xmath58 . the element - local approximation @xmath59 reads @xmath60 with the polynomial basis functions @xmath61 and the time - dependent vector of coefficients @xmath62^\text{t } , \ ] ] representing the numerical degrees of freedom . the basis functions are defined with element - wise compact support , which is an essential property of dg methods @xmath63 we define the basis functions on the master element @xmath64 and obtain the element - specific basis through the mapping @xmath50 as @xmath65 we employ cartesian grids and tensor product basis functions of the form @xmath66 where @xmath1 is a multi - index obtained from all @xmath67 . we denote by @xmath68 the local maximum approximation orders of element @xmath28 . the finite element space ( fes ) @xmath69 spanned by the basis functions is given by the tensor product of the respective one - dimensional spaces @xmath70 the approximation may , thus , make use of different orders @xmath71 in each of the coordinate directions , where we drop the subscript if they are equal . the basis functions are legendre polynomials scaled such that @xcite @xmath72 in the following the dependence of the spatial and temporal variable is not written down explicitly . if now we were to substitute the exact electromagnetic field solution @xmath73 for its approximation the surface integral term of ( [ eq : weakmaxwell ] ) can not be evaluated straightforwardly at the internal faces @xmath43 . this is due to the ambiguity of the dg approximation at any interface as a result of ( [ eq : approx ] ) and ( [ eq : basisfcts ] ) . weak continuity at internal faces is obtained locally by introducing numerical interface fluxes as @xmath74 where @xmath75 is a unique interface value computed solely from @xmath76 and @xmath77 , where @xmath78 is a neighboring element . common choices include centered and upwind fluxes . the centered interface value is given as @xmath79 computing the upwind value is more involved . it is obtained as the exact solution of maxwell s equations for piece - wise constant initial data after an infinitesimal time span , which is referred to as the riemannian problem @xcite . for the @xmath80-component of the electric and magnetic field at an interface with normal @xmath81 they read [ eq : upw ] @xmath82 with the intrinsic impedance and admittance @xmath83 other components are obtained by cycling the component indices and signs . note that centered fluxes preserve the hamiltonian structure of maxwell s equations while this property does not carry over to the semi - discrete equations when applying upwind fluxes due to the mixing of electric and magnetic quantities in ( [ eq : upw ] ) . consequently , an energy conservation property @xcite can be obtained with the centered flux formulation only , determining the kind of time integration schemes to be used as well @xcite . our implementation includes both flux types . having resolved the ambiguity at interfaces , we insert the approximations ( [ eq : approx ] ) into the weak formulation ( [ eq : weakmaxwell ] ) and follow the galerkin procedure yielding the semi - discrete dg formulation [ eq : dgmaxwell ] @xmath84 @xmath85 , @xmath86 . the volume integrals are referred to as the mass and stiffness terms , the surface integrals represent face fluxes . note that no assumptions on the grid regularity have been made in the derivation . due to the strictly element - local support of the basis and test functions , the dgm is highly suited for the application on non - regular grids . the actual difference of the refinement levels @xmath87 and @xmath88 of neighboring elements , i.e. , the level of hanging nodes , plays a minor role as shown in the following . inspecting equations ( [ eq : dgmaxwell ] ) it is seen that the mass and stiffness terms are not affected by the grid regularity as they are strictly local to the element @xmath28 . the flux term , however , involves neighboring elements as well . decomposing the surface integral into the six contributing face integrals @xmath89 and considering centered fluxes for brevity each of these can be expressed as @xmath90.\ ] ] accounting for the kind of non - regular grids described above , i.e. grids obtained from a regular root tesselation , requires no more than summing up the contributions of all neighboring elements to the total flux . this is independent of the hanging node levels as well as the actual number of neighboring elements . inserting the approximation ( [ eq : elocal ] ) into ( [ eq : faceinterfaces ] ) yields @xmath91.\ ] ] again , the first integral term does not depend on the grid regularity . assuming @xmath92 to be aligned with the @xmath93-coordinate and to point towards positive direction it amounts to @xmath94 due to the basis function scaling ( [ eq : orthogonalbasis ] ) . the second integral term can be expressed as @xmath95 in this case the orthogonality property of the basis functions is lost due to non - identical supports of @xmath96 and @xmath97 . we gather the terms ( [ eq : faceinterfaceregular ] ) and ( [ eq : faceinterfaceirregular ] ) in the interior and exterior flux matrices @xmath98 and @xmath99 . following to the usual notation the sign indicates the evaluation from the interior and exterior side of the interface . any non - regularity of the grid is now concealed within @xmath100 , which reduces to the standard form on regular grids . for high level hanging nodes the number of integrals to compute quickly becomes large , imposing a heavy computational burden if integration is performed at run time . however , as the integrals @xmath101 in ( [ eq : faceinterfaceirregular ] ) do not include the actual approximation but basis functions only , they can be precomputed analytically ( making use of the master basis functions ) and stored in tabulated form in the code . this has to be done for all combinations of @xmath102 and @xmath103 as well as for each possible edge overlap according to the respective difference in the refinement levels @xmath104 ( cf . [ fig : nonregularinterface ] ) . the number of possible overlaps grows as @xmath105 . we tabulated the integrals up to @xmath106 and for basis functions up to order six , yielding 247 matrices @xmath107 of size @xmath108 . in the isotropic refinement case @xmath106 corresponds to one element interfacing with @xmath109 neighbors . in the case of even larger differences in the refinement levels of neighboring elements , which are unlikely to occur a numerical integration is invoked at run time . if the neighboring element has a smaller instead of higher refinement level the respective transposed matrix @xmath110 is applied . for upwind fluxes , the interior and exterior flux matrices do not change , however , they are applied to both , the electric and the magnetic field due to ( [ eq : upw ] ) . -axis for a better visualization . the left hand root element has been refined several times , the right hand element is at root level . the interface i connects an element of refinement levels @xmath111 with the root level element . the tick marks indicate possible locations for the imprint of elements of these refinement levels . the actual imprint on the root element face fills the first and sixth slab along the @xmath80- and @xmath112-axis , respectively . the interface ii fills the respective second slab along the @xmath80-axis.,scaledwidth=60.0% ] in order to further enhance computational performance , all combinations of @xmath113 and the integrals @xmath114 arising form the stiffness terms of ( [ eq : dgmaxwell ] ) are evaluated and tabulated as well . precomputing the interface integrals maintains the high computational efficiency of the dg methods also for non - regular grids . using matrix notation , the semi - discrete dg maxwell equations ( [ eq : dgmaxwell ] ) read @xmath115 where * s * and * z * denotes the stiffness and impedance matrix . the matrix operator on the right hand side of ( [ eq : maxwellsemimatrix ] ) represents a weak dg curl operator . choosing @xmath116 as either zero or one yields centered or upwind fluxes , respectively . by applying centered fluxes the hamiltonian structure of maxwell s equations in continuum is preserved , whereas upwind fluxes lead to a mixed form . symplectic explicit time integration can be applied in the former case but not in the latter one @xcite . for examples of symplectic time integration for maxwell s equations in the dg framework see , e.g. , @xcite . in @xcite upwind fluxes and runge - kutta schemes are applied for the time integration , where the latter one is concerned with maxwell s equations . the adaptation techniques presented in the following are based on projections between the finite element spaces introduced in ( [ eq : dgtensorspace ] ) . the projection operators have been introduced in @xcite , however , they are included for completeness . also , we address the issue of stability in depth and amended this section with examples . the approximation @xmath117 to a given function @xmath118 in the fes @xmath69 is obtained by performing an orthogonal projection . the projection is carried out in an element - wise manner , by means of the projection operator @xmath119 given by @xmath120 where @xmath121 denotes the inner product @xmath122 on the element @xmath28 with the associated 2-norm @xmath123 . when applied successively to all elements and all components of given initial conditions of the electric field , @xmath124 , and the magnetic field , @xmath125 , the respective dg approximations @xmath126 and @xmath127 are obtained . these approximations are optimal in the sense that the projection errors @xmath128 are orthogonal to the space of basis functions @xmath69 @xmath129,\ , \varphi_i^p = \hat{\varphi}^p \circ g_i^{-1},\ , \varphi^p \in \mathcal{v}^p\ ] ] as stated above @xmath0-refinement is achieved by means of element bisections along the coordinate directions , where we allow for anisotropic refinements . the refined elements are referred to as the left and right hand side element @xmath130 and @xmath131 with basis functions denoted as @xmath132 and @xmath133 spanning the spaces @xmath134 and @xmath135 in a full analogy to @xmath69 defined in ( [ eq : dgtensorspace ] ) . the approximation orders @xmath136 and @xmath137 in each child element do not have to be identical , neither are they required to be equal to the respective order @xmath71 of the parent element . the direct sum of the spaces @xmath138 and @xmath139 is denoted by @xmath140 @xmath141 in the following , the projection can be applied in order to project an approximation given in an element @xmath28 to the fes associated with an @xmath0-refined or @xmath0-reduced element . for @xmath0-refinement this yields @xmath142 due to the tensor product character of the basis , this can be expressed as @xmath143 for the left and right child , respectively . if refinement is carried out along one coordinate only , e.g. @xmath80 , this further simplifies to @xmath144 where @xmath145 denotes the kronecker delta . note that above we loop over @xmath146 , whereas in ( [ eq : projsolutionrefine2 ] ) the loop parameter is @xmath1 . as , moreover , @xmath147 vanishes for any @xmath148 , we can limit the above loop to the range @xmath149 $ ] , which reduces the number of addends to the minimum possible . for the merging of elements , the approximation within the parent element , @xmath28 , is considered to be given piece - wise within its child elements . the projection reads @xmath150 where the simplifications ( [ eq : projsolutionrefine2 ] ) and ( [ eq : projsolutionrefine3 ] ) apply . for the case of @xmath1-enrichments , the local fes are amended with the @xmath151 order basis functions @xmath152 where any ( non - zero ) number of the local maximum approximation orders @xmath71 may be increased . also , an enrichment by more than one higher order basis function is possible . formally , we perform the orthogonal projection ( [ eq : projectionoperator ] ) , however , due to the orthogonality property of the basis functions the coefficients @xmath153 remain unaltered under a projection from @xmath154 to @xmath155 . practically , we simply extend the local vectors of coefficients @xmath156 with the new coefficients @xmath157 , which are initialized to zero . conversely , for the case of a @xmath1-reduction , the local fes is reduced by discarding the @xmath71-order basis functions @xmath158 again , by virtue of the orthogonality , we find that the coefficients @xmath159 are deleted from the local vectors of coefficients while the coefficients @xmath160 remain unaltered . we denote by @xmath161 the projection of the global approximation @xmath162 from the current discretization to another one obtained by local @xmath0- and @xmath1-adaptations . an approximation @xmath163 with coefficients according to is optimal in the sense of ( [ eq : dgapproxerror ] ) . the approximations within refined and merged elements with coefficients obtained through the orthogonal projections and are , hence , optimal in the same sense . if @xmath164 and @xmath165 holds true for all @xmath48 , the fes @xmath166 is a subspace of @xmath140 ( cf . ( [ eq : legendrespaceunion ] ) ) and every function of @xmath166 is representable in @xmath140 but not vice versa . in this case , a given approximation is exactly represented within an element under @xmath0-refinement but not under @xmath0-reduction . see fig . [ fig : adaplegendreremarks ] for an example . [ cc](a ) [ cc](b ) [ cc]@xmath167 [ cc]@xmath168 [ cc]@xmath169 [ cc]@xmath170 [ cc]@xmath171 [ lb]@xmath80 [ cb]@xmath172 and @xmath173 , the approximations of the parent and child elements agree point - wise . the projection to a merged element shown in ( b ) can , in general , not be exact due to the discontinuity . , title="fig:",scaledwidth=95.0% ] since the projections ( [ eq : projsolutionrefine ] ) for performing @xmath0-refinement are independent of the actual approximation , we also tabulated the projection operators @xmath174 and @xmath175 ( expressed in master basis functions ) yielding the matrix operators @xmath176 and @xmath177 , where the superscript denotes that the refinement level @xmath178 is increased . accordingly , we make use of the matrix operators @xmath179 and @xmath180 for evaluating the projections of ( [ eq : projsolutionreduce ] ) in the case of element merging . the matrix operators are related as @xmath181 this allows for the computation of the approximations within adapted elements by means of efficient matrix - vector multiplications . as all projection matrices are triangular the evaluation can be carried out as an in - place operation requiring no allocation of temporary memory . the global approximation associated with an adapted grid is computed as @xmath182 . it can be considered as initial conditions applied on the new discretization obtained by performing the refinement operations . assuming stability of the time stepping scheme ( cf . @xcite ) , it is sufficient to show that the application of the projection operators at some time @xmath183 does not increase the electromagnetic energy associated with the approximate dg solution , i.e. , @xmath184 in this case it follows @xmath185 and , thus , stability of the adaptive scheme . following ( [ eq : energydensity ] ) the energy associated with element @xmath28 is given as @xmath186 as a consequence of ( [ eq : projsolutionrefine2 ] ) , it is sufficient to show that the energy ( [ eq : dgenergy ] ) is non - increasing during any adaptation involving one coordinate only . [ [ sec : h - refinement-1 ] ] @xmath0-refinement + + + + + + + + + + + + + + + + + + + + + + + + + + for the following discussion of stability it is assumed that refinement is carried out along the @xmath80-coordinate . also we assume the maximum approximation orders @xmath187 and @xmath188 to be identical . it is clarified later , that this does not pose a restriction to the general validity of the results . in the case of @xmath0-refinement , the operators @xmath189 and @xmath190 project from the space @xmath166 to the larger space @xmath140 . following the argument of paragraph [ sec : optimality ] on optimality , any function defined in the space @xmath166 is exactly represented in @xmath140 . the conservation of the discrete energy is a direct consequence as the approximation in the parent and child elements are point - wise identical . we find the following relation for the 2-norms of the respective local vectors of coefficients @xmath191 the exemplary parent element approximation plotted in fig . [ fig : adaplegendreremarks]a has a maximum order of @xmath192 with all coefficients equal to one . the coefficients of the child element approximations and the square values of their 2-norms are given in tab . [ tab : adapdgcoeffsr ] . if the vector @xmath193 is considered to be either the vector of coefficients of the electric field @xmath194 or the magnetic field @xmath195 the result agrees with ( [ eq : dgenergy1drefine ] ) . .parent and child element coefficients of the function plotted in fig . [ fig : adaplegendreremarks]a [ cols="^,^,^,^,^,^,^,^,>",options="header " , ] we chose a maximum @xmath0-refinement level of two and the local element order to vary in between zero and four . the local energy density , introduced in ( [ eq : energydensity ] ) , serves as the criterion for controlling the adaptation procedure . denoting by @xmath196 the average energy density of element @xmath31 and by @xmath197 the normalized energy density with @xmath198 , we assigned the local refinement levels according to @xmath199 and polynomial orders as @xmath200 with @xmath201 . the initial discretization consisted of @xmath202 elements . during the simulation the number of elements varies and grew strongly after scattering from the reflector took place , when it reached close to 800,000 elements corresponding to slightly more than 55 million dof . for comparison , we note that employing the finest mesh resolution globally as well as fourth order approximations uniformly would lead to approximately 7.5 billion ( @xmath203 ) dof . this corresponds to a factor of approximately 130 in terms of memory savings . we emphasize that the simulations were carried out on a single machine . the implementation takes full advantage of multi - core capabilities through openmp parallelization . the numerous run - time memory allocations and deallocations are handled through a specialized memory management library based on memory blocking , which we implemented for supporting the main code @xcite . figure [ fig : eygrids ] depicts cut - views of the @xmath112-component of the electric field and the respective @xmath2-mesh at three instances in time . note that the scaling of the electric field differs for every time instance , which is necessary to allow for a visual inspection . the enlargement shows details of the computational grid . all elements are of hexahedral kind , however , we make use of the common tensor product visualization technique ( cf . @xcite ) using embedded tetrahedra for displaying the three tensor product orders ( out of which only @xmath204 and @xmath205 are visible in the depicted @xmath206-plane ) . as we employed isotropic @xmath0- as well as @xmath1-refinement all tetrahedra associated with one element share the same color . figure [ fig : signals ] shows plots of the outgoing and reflected waveform recorded along the waveguide center . the blue dashed line was obtained with the commercial cst microwave studio software @xcite on a very fine mesh and serves as a cross comparison result . -component of the electric field ( top panel ) and the computational grid ( bottom panel ) at three instances in time . the enlargement shows details of the grid . we employ hexahedral elements for the computation but make use of embedded tetrahedra for displaying the tensor product orders in the grid view . as isotropic @xmath1-refinement was employed in this examples all tetrahedra associated with one element share a common color . note that different scalings are used for the time instances in the top panel.,scaledwidth=100.0% ] we presented a discontinuous galerkin formulation for non - regular hexahedral meshes and showed that hanging nodes of high level can easily be included into the framework . in fact , any non - regularity of the grid can be included in a single term reflecting the contribution of neighboring elements to the local interface flux . we demonstrated that the method can be implemented such that it maintains its computational efficiency also on non - regular and locally refined meshes as long as the mesh is derived from a regular root tesselation by means of element bisections . this is achieved by extensive tabulations of flux and projection matrices , which are obtained through ( analytical ) precomputations of integral terms . we also presented local refinement techniques for @xmath0- and @xmath1-refinements , which are based on projections between finite element spaces . these projections were shown to guarantee minimal projection errors in the @xmath207-sense and to lead to an overall stable time - domain scheme . local error and smoothness estimates have been addressed , both of them relate to the size of the interface jumps of the dg solution . we considered the simulation of a smooth and a non - smooth waveform in a one - dimensional domain for validating the error and smoothness estimates . as an application example in three - dimensional space the backscattering of a broadband waveform from a radar reflector was considered . in this example the total wave propagation distance corresponds to approximately sixty wavelengths involving thousand of local mesh adaptations . as the implementation of the derived error and smoothness estimates for three - dimensional problems is subject of ongoing work , we chose to drive the grid adaptation using the energy density as refinement indicator . crosschecking with a result obtained using a commercial software package showed good agreement . l. fezoui , s. lanteri , s. lohrengel , s. piperno , convergence and stability of a discontinuous galerkin time - domain method for the 3d heterogeneous maxwell equations on unstructured meshes , esaim - math model num 39 ( 6 ) ( 2005 ) 11491176 . d. wirasaet , s. tanaka , e. j. kubatko , j. j. westerink , c. dawson , a performance comparison of nodal discontinuous galerkin methods on triangles and quadrilaterals , int j numer meth fluids 64 ( 10 - 12 ) ( 2010 ) 13361362 . l. krivodonova , j. xin , j. remacle , n. chevaugeon , j. flaherty , shock detection and limiting with discontinuous galerkin methods for hyperbolic conservation laws , appl numer math 48 ( 3 - 4 ) ( 2004 ) 323338 . smove , a program for the adaptive simulation of electromagnetic fields and arbitrarily shaped charged particle bunches using moving meshes , technical documentation : _ - school - ce.de / files2/schnepp / smove/_.
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a framework for performing dynamic mesh adaptation with the discontinuous galerkin method ( dgm ) is presented .
adaptations include modifications of the local mesh step size ( @xmath0-adaptation ) and the local degree of the approximating polynomials ( @xmath1-adaptation ) as well as their combination .
the computation of the approximation within locally adapted elements is based on projections between finite element spaces ( fes ) , which are shown to preserve an upper limit of the electromagnetic energy .
the formulation supports high level hanging nodes and applies precomputation of surface integrals for increasing computational efficiency .
error and smoothness estimates based on interface jumps are presented and applied to the fully @xmath2-adaptive simulation of two examples in one - dimensional space .
a full wave simulation of electromagnetic scattering form a radar reflector demonstrates the applicability to large scale problems in three - dimensional space .
discontinuous galerkin method , dynamic mesh adaptation , @xmath2-adaptation , maxwell time - domain problem , large scale simulations 65m60 , 78a25
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, coupling strength @xmath0 ) . @xmath1 is the nel temperature , while the temperature below which superconductivity ( a pseudogap ) is observed is denoted by @xmath2 ( @xmath3 ) . [ from ref . @xcite . ] [ phase - diagram - exp ] ] after intensive investigations over more than two decades , it has become clear that cuprate superconductors are among the most complicated systems studied in condensed matter physics @xcite . the complications arise mainly from that the parent compounds of cuprate superconductors are a form of non - conductor called a mott insulator with an antiferromagnetic ( af ) long - range order ( aflro ) @xcite , where a single common feature in the layered crystal structure is the presence of one to several cuo@xmath4 planes in the unit cell @xcite . inelastic neutron scattering ( ins ) experiments show that the low - energy spin excitations in these parent compounds are well described by an af heisenberg model @xcite with the magnetic exchange coupling constant @xmath5 ev . when these cuo@xmath4 planes are doped with charge carriers , the aflro phase subsides and superconductivity emerges leaving the af short - range order ( afsro ) correlation still intact @xcite . although there are hundreds of cuprate superconducting ( sc ) compounds , they all fit into a universal phase diagram @xcite as schematically illustrated in fig . [ phase - diagram - exp ] , where the physical properties mainly depend on the extent of doping , and the regimes have been classified into the underdoped , optimally doped , and overdoped , respectively . after aflro is destroyed rapidly by doping , there are three apparent regions of the phase diagram : ( a ) a d - wave sc phase , where the maximal sc transition temperature @xmath2 occurs around the optimal doping , and then decreases in both the underdoped and overdoped regimes @xcite ; ( b ) a normal - state pseudogap metallic phase , where an energy gap called the normal - state pseudogap @xmath6 exists @xcite above @xmath2 but below the normal - state pseudogap crossover temperature @xmath3 . however , in contrast to the domelike shape of the doping dependence of @xmath2 , @xmath3 is much larger than @xmath2 in the underdoped and optimally doped regimes @xcite , and then monotonically decreases upon the increase of doping . in particular , measurements taken by using a wide variety of techniques demonstrate that the normal - state pseudogap is present in both the spin and charge channels @xcite ; ( c ) a _ normal - metal _ phase with largely transport properties . in the doped regime , charge carriers couple to spin excitations @xcite . the combined ins and resonant inelastic x - ray scattering ( rixs ) experimental data have identified spin excitations with high intensity over a large part of moment space , and shown that spin excitations exist across the entire range of the sc dome @xcite . however , the charge - carrier doping causes substantial changes to the low - energy spin excitation spectrum @xcite , while it has a more modest effect on the high - energy spin excitations @xcite . in particular , rixs experiments @xcite show that the high - energy spin excitations persist well into the overdoped regime and bear a striking resemblance to those found in the parent compounds , indicating that a local - moment picture accounts for the observed spin excitations at elevated energies even up to the overdoped regime @xcite . experimentally , a large body of data available from a wide variety of measurement techniques have provided rather detailed information on cuprate superconductors , where some essential agreements have emerged . we refer the readers to the more detailed summaries of experimental results available in the literatures @xcite . superconductivity , the dissipationless flow of electrical current , is a striking manifestation of a subtle form of quantum rigidity on the macroscopic scale @xcite , where a central question is what mechanism causes the loss of electrical resistance below @xmath2 ? it is commonly believed that the existence of electron cooper pairs is the hallmark of superconductivity @xcite , since these electron cooper pairs behave as effective bosons , and can form something analogous to a bose condensate that flows without resistance . this follows from a fact that although electrons repel each other because of the coulomb interaction , at low energies there can be an effective attraction that originates by the exchange of bosons @xcite . in conventional superconductors , as explained by the bardeen - cooper - schrieffer ( bcs ) theory @xcite , these exchanged bosons are phonons that act like a bosonic _ glue _ to hold the electron pairs together , and then these electron pairs condense into a coherent macroscopic quantum state that is insensitive to impurities and imperfections and hence conducts electricity without resistance @xcite . the excitation in the sc - state has an energy gap @xmath7 , which determines both the quasiparticle energy spectrum and the energy of the condensate @xcite . in this conventional electron - phonon sc mechanism @xcite , the resulting wave function for the pairs turns out to be peaked at zero separation of the electrons , which leads to that the sc - state has an s - wave symmetry @xcite . as a consequence , the pairs in conventional superconductors are always related to an increase in kinetic energy which is overcompensated by the lowering of potential energy @xcite . at the temperature above @xmath2 , the electron is in the standard landau fermi - liquid state , which is generally referred to as a _ normal - state _ , where the density of states near the fermi level is smooth and generally treated as featureless @xcite . as in conventional superconductors , superconductivity in cuprate superconductors results when charge carriers pair up into charge - carrier pairs , which is supported by many experimental evidences , including the factor of @xmath8 occurring in the flux quantum and in the josephson effect , as well as the electrodynamic and thermodynamic properties @xcite . however , the normal - state of cuprate superconductors in the pseudogap phase is not normal at all , since the normal - state of cuprate superconductors in the pseudogap phase exhibits a number of the anomalous properties @xcite in the sense that they do not fit in with the standard landau fermi - liquid theory . superconductivity is an instability of the normal - state . however , one of the most striking dilemmas is that the sc coherence of quasiparticle peaks in cuprate superconductors is described by a standard bcs formalism , although the normal - state is undoubtedly not the standard landau fermi - liquid on which the conventional bcs electron - phonon sc mechanism is based . angle - resolved photoemission spectroscopy ( arpes ) experiments reveal sharp spectral peaks in the excitation spectrum @xcite , indicating the presence of quasiparticle - like states , which is also consistent with the long lifetime of electronic state as it has been determined by the conductivity measurements @xcite . moreover , arpes experiments also observe the bogoliubov - type dispersion of the sc - state @xcite predicted by the standard bcs formalism . however , as a natural consequence of the unconventional sc mechanism that is responsible for the high @xmath2 , the charge - carrier pairs in cuprate superconductors have a dominant d - wave symmetry @xcite . this d - wave sc - state also implies that there is a strongly momentum - dependent attraction between charge carriers without phonons @xcite . after over more than two decades of the painstaking effort , people are still debating the very mechanism of superconductivity in cuprate superconductors , where the crucial issues in cuprate superconductors are ( a ) what is the nature of the _ glue _ binding charge carriers into charge - carrier pairs , such that they can travel macroscopic distances without resistance ? ( b ) whether the pseudogap has a competitive or collaborative role in engendering superconductivity ? very soon after the discovery of superconductivity in cuprate superconductors @xcite , anderson @xcite proposed that in the parent compounds of cuprate superconductors , the spins form a superposition of singlets . this spin liquid of singlets is so - called the resonating valence bond ( rvb ) state . upon the charge - carrier doping , these rvb singlets would become charged , resulting in a sc - state . the rvb state is fundamentally different from the conventional nel state in which the doped charge carrier can move freely among the rvb spin liquid and then a better compromise between the charge - carrier kinetic energy and spin exchange energy may be achieved . since then many elaborations of this idea followed @xcite . in particular , it was realized that essential aspects of the rvb concept can be formulated within the charge - spin separation ( css ) slave - particle approach @xcite , since the essential physics of cuprate superconductors is dominated by the short - range repulsive interaction which remains relevant and causes css @xcite . this css slave - particle approach also led to the development of the gauge theory for cuprate superconductors @xcite . however , in the framework of the original rvb theory @xcite , the af exchange coupling @xmath9 attracts electrons of opposite spins to be on neighboring sites . this is the result of states of very high energy with a spin gap , and the corresponding interaction has only high - energy dynamics @xcite . the normal - state pseudogap is identified as the spin gap in the rvb state with an energy scale set by @xmath9 , and therefore is associated with the breaking of the rvb singlets @xcite . in this case , anderson @xcite suggested that the sc - state in cuprate superconductors is determined by the need to reduce the frustrated kinetic energy of the system , where the strong frustration of the kinetic energy in the normal - state is partially relieved upon entering the sc - state , indicating that the kinetic energy causes superconductivity @xcite . on the other hand , it has been argued phenomenologically that superconductivity in cuprate superconductors could arise from a lowering of the kinetic energy rather than the potential energy @xcite . in this scenario , superconductors would exhibit qualitatively new features in their optical properties a violation of the low - energy optical sum rule , and a change in the high - energy optical absorption when the system becomes sc @xcite . later , the high - precision optical measurements on cuprate superconductors in the near - infrared and visible region indicate small changes in the spectral weight associated with the onset of superconductivity @xcite , therefore supporting this argument @xcite that changes in the kinetic energy are indeed occurring . in particular , the similar experimental results have by now been obtained by differently experimental groups @xcite . more importantly , the recent experimental results @xcite from the arpes measurements on cuprate superconductors indicate that @xmath2 is correlated with the charge - carrier kinetic energy , which supports the notion of the kinetic - energy driven superconductivity . motivated by these experimental results @xcite , several calculations based on the strongly correlated models have been done to show that superconductivity may be driven by a lowering of the kinetic energy upon the formation of the sc - state @xcite . by constructing an effective hamiltonian for spin polarons forming in weakly doped antiferromagnets , it has been demonstrated that the driving mechanism which gives rise to superconductivity in such system is the reduction of the kinetic energy @xcite . moreover , a numerical study of the two - dimensional hubbard model within the dynamical cluster approximation has shown the lowering of the kinetic energy below @xmath2 for different doping levels @xcite . these theoretical calculation @xcite shows that the paired charge carriers in the afsro background can be more mobile than the single charge carriers , and this can overcome the normal increase in the kinetic energy upon the pair formation @xcite . superconductivity in cuprate superconductors is something entirely new , a manifestation of the strong electron correlation , or mottness @xcite . in the early days of superconductivity , we @xcite have developed a fermion - spin theory to confront the strong electron correlation , where the constrained electron operator is decoupled as a product of a charge carrier and a localized spin , with the charge carrier represented the charge degree of freedom of the electron together with some effects of spin configuration rearrangements due to the presence of the doped charge carrier itself , while the spin operator represented the spin degree of freedom of the electron , and then the strong electron correlation can be treated properly in actual calculations . in particular , these charge carriers and spin are gauge invariant , and in this sense , the collective modes for these charge carriers and spins are real and can be interpreted as the physical excitations of the system . in this fermion - spin theory , the basic low - energy excitations are charge - carrier quasiparticles , the spin excitations , and the electron quasiparticles . in this case , the charge transport is mainly governed by the scattering of charge carriers due to spin fluctuations , and the scattering of spins due to charge - carrier fluctuations dominates the spin dynamics , while as a result of the charge - spin recombination , the electron quasiparticles are responsible for the electronic properties . within the framework of the fermion - spin theory @xcite , we have established a kinetic - energy driven sc mechanism @xcite , where charge carriers are held together in d - wave pairs at low temperatures by the attractive interaction in the particle - particle channel that originates directly from the kinetic energy by the exchange of spin excitations in the higher powers of the doping concentration , and then these charge - carrier pairs ( then electron cooper pairs ) condense to the d - wave sc - state . although the physical properties of cuprate superconductors in the normal - state are fundamentally different from these in the standard landau fermi - liquid state , the kinetic - energy driven sc - state still is conventional bcs - like with the d - wave symmetry , and then the obtained formalism for the charge - carrier pairing can be used to compute @xmath2 and the related sc coherence of the low - energy excitations in cuprate superconductors on the first - principles basis much as can be done for conventional superconductors . moreover , this kinetic - energy driven sc - state is controlled by both the charge - carrier pair gap and quasiparticle coherence , which leads to that @xmath2 takes a domelike shape with the underdoped and overdoped regimes on each side of the optimal doping @xmath10 , where @xmath2 reaches its maximum . on the other hand , the same charge - carrier interaction mediated by spin excitations that induces the sc - state in the particle - particle channel also generates the normal - state pseudogap state in the particle - hole channel @xcite . as a consequence , the sc gap and normal - state pseudogap coexist but compete in the whole sc dome however , the normal - state pseudogap crossover temperature @xmath3 is much larger than @xmath2 in the underdoped and optimally doped regimes , and then monotonically decreases upon the increase of doping , eventually disappearing together with superconductivity at the end of the sc dome . this kinetic - energy driven sc mechanism therefore provides a natural explanation of both the origin of the normal - state pseudogap state and pairing mechanism for superconductivity . it is beyond the scope of this article to provide an overview of various theories of superconductivity in cuprate superconductors that have been put forward in the literatures , and some earlier reviews and different perspectives appear in refs . @xcite . in this article , we only attempt to review comprehensively the general framework of the kinetic - energy driven sc mechanism in the context of our work @xcite and to summarize several calculated results of physical quantities obtained based on the kinetic - energy driven sc mechanism . the number of topics for this review article is listed as follows . in section [ cssfst ] , we @xcite give an overview of the fermion - spin theory , and show that within the _ decoupling _ scheme , the fermion - spin representation is a natural representation of the constrained electron defined in a restricted hilbert space without double electron occupancy . the kinetic - energy driven sc mechanism @xcite is introduced in section [ kedm ] . in the rest of sections , we show how this kinetic - energy driven sc mechanism yields many results that are in broad agreement with various key experimental facts observed on cuprate superconductors . superconductors are not only perfect conductors , but also exhibit the so - called meissner effect , where they expel magnetic fields . in section [ meissner - effect ] , we consider main features of the doping dependence of the electromagnetic response observed on cuprate superconductors by using the muon - spin - rotation measurement technique , and show that in analogy to the domelike shape of the doping dependence of @xmath2 , the maximal superfluid density @xmath11 occurs around the critical doping @xmath12 , and then decreases in both lower doped and higher doped regimes . in section [ spin - response ] , we turn to the comparison of the calculated result of the dynamical spin response with rixs - ins experimental data . it is shown that the low - energy spin excitations in the sc - state have an hour - glass - shaped dispersion , with commensurate resonance that appears in the sc - state _ only _ , while the low - energy incommensurate ( ic ) spin fluctuations can persist into the normal - state . the high - energy spin excitations in the sc - state on the other hand retain roughly constant energy as a function of doping , with spectral weights and dispersion relations comparable to those found in the parent compounds . a brief description of the interplay between superconductivity and normal - state pseudogap state is given in section [ sc - pseudogap ] , where we @xcite identify the normal - state pseudogap as being a region of the self - energy effect in the particle - hole channel in which the normal - state pseudogap suppresses the spectral weight of the low - energy excitation spectrum . this normal - state pseudogap disappears at @xmath3 , and then system crossovers to the normal - metal phase with largely transport properties at the temperatures @xmath13 . in section [ charge - transport ] , we discuss the effect of the normal - state pseudogap on the infrared response of cuprate superconductors , and show that in the underdoped and optimally doped regimes , the transfer of the part of the low - energy spectral weight of the conductivity spectrum to the higher energy region to form a midinfrared band is intrinsically associated with the emergence of the normal - state pseudogap . finally , the article concludes with the suggestions , in section [ conclusion ] , for future work . in cuprate superconductors , the single common feature in the layered crystal structure is the presence of one to several cuo@xmath4 planes in the unit cell @xcite , and it seems evident that the nonconventional behaviors of cuprate superconductors are dominated by the cuo@xmath4 plane . in this case , as originally emphasized by anderson @xcite , the essential physics of the doped cuo@xmath4 plane is contained in the one - band large-@xmath14 hubbard model @xcite on a square lattice , @xmath15 where the summation is over all sites @xmath16 , and the hopping integrals @xmath17 connect sites @xmath16 and @xmath18 . we will restrict to our attention to the nearest ( @xmath19 ) and next nearest ( @xmath20 ) neighbor hopping . @xmath21 and @xmath22 are electron operators that respectively create and annihilate electrons with spin @xmath23 , @xmath24 , and @xmath25 is the chemical potential . this large-@xmath14 hubbard model ( [ hubbard ] ) indicates that the interactions in cuprate superconductors are dominated by the on - site mott - hubbard term @xmath14 , which is very large as compared with the electron hopping integrals @xmath19 and @xmath26 , i.e. , @xmath27 , @xmath26 , and therefore leads to that electrons become strongly correlated to avoid double occupancy . in this case , the on - site mott - hubbard term must be dealt properly before bothering with relatively minor terms @xcite . it has been shown @xcite that the correct way to deal with this large-@xmath14 term in eq . ( [ hubbard ] ) is to renormalize it by means of a canonical transformation @xmath28 , which eliminates large-@xmath14 term from the block which contains no doubly occupied states , and which presumably contains all the low - energy eigenstates and thus the ground state , and then the transformed hamiltonian can be obtained as , @xmath29 with the nearest - neighbors @xmath30 , the next nearest - neighbors @xmath31 , the magnetic exchange coupling constant @xmath32 , the spin operators @xmath33 . the kinetic - energy term in eq . ( [ tjmodel ] ) describes mobile charge carriers in the af background , while the heisenberg term in eq . ( [ tjmodel ] ) describes af coupling between localized spins . in particular , the nearest - neighbor hopping integral @xmath19 in the kinetic - energy term is much larger than the magnetic exchange coupling constant @xmath9 in the heisenberg term , and therefore the spin configuration is strongly rearranged due to the effect of the charge - carrier hopping @xmath19 on the spins , which leads to strong coupling between the charge and spin degrees of freedom of the electron . this transformed hamiltonian ( [ tjmodel ] ) is so - called @xmath19-@xmath9 model acting on a restricted hilbert space without double electron occupancy , where there are three physical states _ only _ , latexmath:[\ ] ] where the spectral functions @xmath560 and @xmath561 are obtained in terms of the charge - carrier diagonal and off - diagonal green s functions in eq . ( [ hgf-1 ] ) as @xmath562 and @xmath563 , respectively . with @xmath248 for @xmath246 and @xmath247 . inset : the corresponding experimental data of the underdoped bi@xmath4sr@xmath4cacu@xmath4o@xmath257 taken from ref . [ from ref . @xcite . ] [ sc - conductivity ] ] in fig . [ sc - conductivity ] , we @xcite show the calculated result of the conductivity ( [ conductivity ] ) in the sc - state as a function of energy at @xmath250 for @xmath246 and @xmath247 with @xmath248 in comparison with the corresponding experimental result @xcite of the underdoped bi@xmath4sr@xmath4cacu@xmath4o@xmath257 ( inset ) . the result in fig . [ sc - conductivity ] shows clearly that the two - component feature of the conductivity spectrum in the sc - state is the same as in the normal - state case . in particular , in comparison with the result of the conductivity in the normal - state in subsection [ normal - conductivity ] , it is found that the spectral weight of the low - energy component of the conductivity in the sc - state is further suppressed by the sc gap , however , there is no depletion of the spectral weight of the higher energy midinfrared band of the conductivity in the sc - state , which is consistent with the experimental observation on cuprate superconductors @xcite . in this case , although there is a coexistence of the sc gap and normal - state pseudogap , the onset of the region to which the spectral weight is transferred is also close to the normal - state pseudogap @xmath6 , then in analogy to the evolution of the conductivity spectrum with doping in the normal - state , the positions of the gap in the conductivity spectrum and midinfrared peak gradually shift to the lower energies with increasing doping @xcite . all the calculated results @xcite of the conductivity spectrum in the sc - state are also qualitatively consistent with the corresponding experimental data of cuprate superconductors in the sc - state @xcite . in the sc - state , there are two parts of the charge - carrier quasiparticle contribution to the redistribution of the spectral weight in the conductivity spectrum in the sc - state : the contribution from the first term of the right - hand side in eq . ( [ conductivity ] ) comes from the spectral function obtained in terms of the charge - carrier diagonal green s function ( [ hdgf-1 ] ) , and therefore is closely associated with the normal - state pseudogap @xmath6 in the particle - hole channel , while the additional contribution from the second term of the right - hand side in eq . ( [ conductivity ] ) originates from the spectral function obtained in terms of the charge - carrier off - diagonal green s function ( [ hodgf-1 ] ) , and is closely related to the charge - carrier pair gap @xmath242 in the particle - particle channel . however , since @xmath564 in the underdoped and optimally doped regimes as we have mentioned in section [ sc - pseudogap ] , the charge - carrier pair gap only suppresses the spectral weight of the low - energy component , while the normal - state pseudogap related shift of the spectral weight from the low - energy to the higher energy midinfrared band in the sc - state conductivity spectrum becomes arrested . the effect of the normal - state pseudogap on the infrared response in cuprate superconductors has been also discussed based on the preformed pair theory @xcite and the phenomenological theory of the normal - state pseudogap state @xcite , and the results of the unusual two - component conductivity spectrum are qualitatively consistent with the above obtained result based on the kinetic - energy driven sc mechanism by considering the interplay between the sc gap and normal - state pseudogap . in particular , their results @xcite also indicate that in the underdoped and optimally doped regimes , the transfer of the part of the low - energy spectral weight of the conductivity spectrum to the higher energy region to form a midinfrared band is intrinsically associated with the presence of the normal - state pseudogap . in a standard landau fermi - liquid , the shape of the conductivity spectrum @xmath544 is normally well accounted for by the low - energy drude formula that describes the free charge carrier contribution to @xmath543 , and then when the temperatures @xmath565 , the spectral weight of the condensate in the sc - state comes from low energies @xcite . however , in cuprate superconductors , the part of the low - energy spectral weight in the conductivity spectrum in the underdoped and optimally doped regimes is transferred to the higher energy region to form the unusual midinfrared band , and then the width of the low - energy band is narrowing , while the onset of the region to which the spectral weight is transferred is close to the normal - state pseudogap @xmath6 . moreover , since the unusual midinfrared band is taken from the low - energy band , so that both the low - energy non - drude peak ( the conductivity decays as @xmath529 at low energies ) and unusual midinfrared band describe the actual charge - carrier density . in the framework of the kinetic - energy driven sc mechanism , the normal - state pseudogap state is the result of the strong electron correlation , and therefore the transfer of the part of the low - energy spectral weight of the conductivity spectrum in the underdoped and optimally doped regimes to the higher energy region to form the unusual midinfrared band is a natural consequence of the strongly correlated nature in cuprate superconductors . in particular , this strong electron correlation which induces a shift of the spectral weight from the low - energy to the higher energy midinfrared band in the conductivity spectrum , has been confirmed by the early numerical simulations based on the @xmath19-@xmath9 model in the normal - state @xcite and in the sc - state @xcite . in the normal - state , the @xmath566 perfect decay of the conductivity at low energies in the optimally doped regime is closely related with the linear temperature resistivity , since it reflects an anomalous frequency dependent scattering rate proportional to @xmath214 instead of @xmath567 as would be expected in the standard landau fermi - liquid . this linear temperature resistivity is one of the characteristically anomalous properties of cuprate superconductors in the normal - state , and has been also phenomenological discussed within the marginal fermi - liquid theory @xcite . in particular , based on the slave - boson gauge theory , it has been shown within the @xmath19-@xmath9 model that above the bose - einstein temperature , the boson inverse lifetime due to scattering by the gauge field is of order @xmath335 , which suppresses the condensation temperature and leads to a linear @xmath335 resistivity @xcite . however , in the sc - state , the large normal - state pseudogap in the underdoped and optimally doped regimes heavily reduces the fraction of the charge carriers that condense in the sc - state @xcite . the calculated result of the conductivity spectrum summarized in this section shows very clearly that if the effect of the normal - state pseudogap is taken into account in the framework of the kinetic - energy driven sc mechanism , the conductivity of the @xmath19-@xmath9 model calculated based on the linear response approach per se can correctly reproduce the main features found in infrared response measurements on cuprate superconductor in both the normal- and sc - states . the conductivity spectrum in the underdoped and optimally doped regimes contains the low - energy non - drude peak and unusual midinfrared band . however , the position of the midinfrared band shifts towards to the low - energy non - drude peak with increasing doping . in particular , the low - energy non - drude peak incorporates with the midinfrared band in the heavily overdoped regime , and then the low - energy drude behavior recovers . the qualitative reproduction of the main features of infrared response measurements on cuprate superconductors also shows that the transfer of the part of the low - energy spectral weight in the conductivity spectrum in the underdoped and optimally doped regimes to the higher energy region to form the unusual midinfrared band can be attributed to the effect of the normal - state pseudogap on the infrared response in cuprate superconductors . in this article , we have given a brief review of the kinetic - energy driven sc mechanism , where the main conclusions are summarized as : \(a ) in the fermion - spin theory ( [ css ] ) , the constrained electron is decoupled as a product of a charge carrier and a localized spin , and the charge carrier represents the charge degree of freedom of the constrained electron together with some effects of the spin configuration rearrangements due to the presence of the doped charge carrier itself , while the spin operator represents the spin degree of freedom of the constrained electron . in the _ decoupling scheme _ , this fermion - spin representation ( [ css ] ) is a natural representation of the constrained electron defined in a restricted hilbert space without double electron occupancy . the main advantage of the fermion - spin theory ( [ css ] ) is that the electron local constraint for single occupancy is satisfied in actual calculations . in particular , the charge carrier _ or _ spin _ itself _ is @xmath54 gauge invariant , and in this sense , the collective modes for the charge carrier and spin are real and can be interpreted as the physical excitations of cuprate superconductors . although both charge carriers and spins contribute to the charge and spin dynamics , the charge - carrier relaxation time is responsible to the charge transport , and the spin relaxation time is responsible to the dynamical spin response , while as a result of the charge - spin recombination , the the electronic properties are dominated by electron quasiparticles . this is an efficient calculation scheme which can provide very good results even at the mf level . \(b ) in the framework of the kinetic - energy driven sc mechanism developed based on the fermion - spin theory ( [ css ] ) , the charge - carrier pairing interaction originates directly from the kinetic energy of the @xmath19-@xmath9 model ( [ csstjmodel ] ) by the exchange of spin excitations in the higher powers of the doping concentration , and then these charge - carrier pairs ( then the electron cooper pairs ) condense to the d - wave sc - state , where the spin excitation in cuprate superconductors plays a similar role to that of the phonon in conventional superconductors . although the physical properties of cuprate superconductors in the normal - state are fundamentally different from these in the standard landau fermi - liquid state , the kinetic - energy driven sc - state still is conventional bcs - like with the d - wave symmetry , and then the obtained formalism for the charge - carrier pairing can be used to compute @xmath2 and the related sc coherence of the low - energy excitations in cuprate superconductors on the first - principles basis much as can be done for conventional superconductors . moreover , the kinetic - energy driven charge - carrier pair state is controlled by both the charge - carrier pair gap and quasiparticle coherence , which leads to that the maximal @xmath2 occurs around the optimal doping , and then decreases in both the underdoped and overdoped regimes . this kinetic - energy driven sc mechanism also indicates that the strong electron correlation favors superconductivity , since the main ingredient is identified into a charge - carrier pairing mechanism not from the external degree of freedom such as the phonon , but rather solely from the internal spin degree of freedom of the constrained electron . \(c ) the same charge - carrier interaction arising through the exchange of spin excitations that induces the d - wave sc - state in the particle - particle channel also generates the normal - state pseudogap state in the particle - hole channel , therefore there is a coexistence of the sc gap and normal - state pseudogap in the whole sc dome . consequently , this normal - state pseudogap is identified as being a region of the self - energy effect in the particle - hole channel in which the normal - state pseudogap suppresses the spectral weight of the low - energy excitation spectrum . this normal - state pseudogap vanishes at the normal - state pseudogap crossover temperature @xmath3 , with @xmath3 that is much larger than @xmath2 in the underdoped and optimally doped regimes , and monotonically decreases upon the increase of doping , eventually disappearing together with @xmath2 at the end of the sc dome . in particular , the normal - state pseudogap is directly related to the quasiparticle coherence , and therefore antagonizes superconductivity . moreover , both the normal - state pseudogap and charge - carrier pair gap are dominated by one energy scale , and they are the result of the strong electron correlation in cuprate superconductors , while the domelike shape of the doping dependence of @xmath2 , the monotonic decrease of @xmath3 with doping , and relatively anomalous normal - state properties are a natural consequence of the mott physics in which double occupancy is suppressed by strongly coulombic repulsion . the theory also indicates that the kinetic - energy driven sc mechanism provides a natural explanation of both the origin of the normal - state pseudogap state and the charge - carrier pairing mechanism for superconductivity . \(d ) within the framework of the kinetic - energy driven sc mechanism , a number of typical properties of cuprate superconductors have been studied . in this review article , the selected results are summarized , including the doping dependence of the electromagnetic response , the dynamical spin response from low - energy to high - energy , and the charge transport , and are qualitatively comparable to the corresponding experimental results observed in cuprate superconductors . furthermore , this kinetic - energy driven bcs - type formalism gives an explanation of the raman scattering spectra @xcite obtained in terms of the electronic raman response measurement technique @xcite . it also gives a consistent description of the thermodynamic properties @xcite observed from heat capacity measurements @xcite . in particular , it has been used to successfully describe a number of the sc - state properties in the presence of impurities @xcite , including the microwave conductivity @xcite , the scanning tunneling microscopic measurements of the coherent bogoliubov quasiparticle dispersion , and the related extinction of bogoliubov quasiparticle scattering interference at low temperatures @xcite . establishing these agreements between the calculated results obtained based on the kinetic - energy driven sc mechanism and the corresponding experimental data observed from a wide variety of measurement techniques are important to confirm the nature of the sc phase of cuprate superconductors to be the kinetic - energy driven d - wave sc - state . in this review article , we have restricted our attention to the hole - doped cuprate superconductors . however , superconductivity in cuprates also emerges when electrons are doped into mott insulators @xcite . both the hole - doped and electron - doped cuprate superconductors have the layered structure of the square lattice of the cuo@xmath4 plane separated by insulating layers @xcite . although the significantly different behaviors of the hole - doped and electron - doped cuprate superconductors are observed due to the electron - hole asymmetry , the symmetry of the sc order parameter is common in both case @xcite , manifesting that two systems have similar underlying sc mechanism . in particular , the strong electron correlation is common for both the hole - doped and electron - doped cuprate superconductors , and then it is possible that superconductivity in the electron - doped cuprate superconductors is also driven by the kinetic energy as in the hole - doped case . in this case , the charge asymmetry @xcite in superconductivity of the electron - doped and hole - doped cuprate superconductors and the electronic raman response @xcite in the electron - doped cuprate superconductors have been discussed based on the kinetic - energy driven sc mechanism , and the calculated results are in qualitative agreement with the experimental data observed on the electron - doped cuprate superconductors . besides the square lattice cuprate superconductors , some cuprate materials @xcite , such as sr@xmath568cu@xmath569o@xmath570 , do not contain cuo@xmath4 planes common to cuprate superconductors but consist of two - leg cu@xmath4o@xmath251 ladders and edge - sharing cuo@xmath4 chains . in particular , the doped two - leg ladder cuprates are a system in which the sc state is realized by applying a high pressure in the highly charge - carrier doped regime @xcite . these ladder cuprate materials also are natural extensions of the cu - o chain compounds towards the cuo@xmath4 sheet structures . within the framework of the kinetic - energy driven sc mechanism , some typical properties of the two - leg ladder cuprate superconductors have been studied , including the pressure dependence of @xmath2 @xcite , the charge dynamics @xcite , and the spin dynamics @xcite , and the calculated results are also qualitatively consistent with the experimental data obtained from the experimental measurements on the two - leg ladder cuprate superconductors . finally , we should be noted that much remains to be done . in particular , for the normal - state pseudogap , which grows upon underdoping , it seems natural to seek a connection to the physics of the af insulating parent compounds @xcite . however , at half - filling , the @xmath19-@xmath9 model is reduced as the af heisenberg model with an aflro . although a small density of charge carriers is sufficient to destroy aflro , this aflro remains until the extremely low - doped regime ( @xmath571 ) @xcite . as we have mentioned in eq . ( [ ecpair ] ) , the conduct is disrupted by aflro at the extremely low - doped regime , and then an important issue is how to extend the theory of the normal - state pseudogap state for the doped regime without aflro to the case at the extremely low - doped regime with aflro for a proper description of the connection between the finite doping normal - state pseudogap and the zero - doping quasiparticle dispersion . one of authors ( sf ) thanks professor z. b. su and professor l. yu for the early collaborations , and he also thanks li cheng , huaiming guo , zheyu huang , zhongbing huang , zhihao geng , mateusz krzyzosiak , ying liang , bin liu , tianxing ma , jihong qin , yun song , weifang wang , zhi wang , feng yuan , and jingge zhang for the collaborations . this work was supported by the funds from the ministry of science and technology of china under grant nos . 2011cb921700 and 2012cb821403 , the national natural science foundation of china under grant nos . 11274044 and 11447144 , and the science foundation of hengyang normal university under grant no . the charge - carrier operators @xmath295 and @xmath104 in the basis @xcite , @xmath572 of the charge - carrier states are given by , @xmath573 while the spin raising and lowering operators @xmath79 and @xmath78 in the spin 1/2 space , @xmath574 are given by , @xmath575 in the product space @xmath122 , the basis vectors are @xcite , however , as we have mentioned in eq . ( [ rhspace ] ) , the restricted hilbert space without double electron occupancy in the @xmath19-@xmath9 model ( [ tjmodel ] ) consists of three states , @xmath119 , @xmath120 , @xmath121 , namely , @xmath578 to remove the extra degrees of freedom in the @xmath122 space , we @xcite introduce a projection operator @xmath127 . by requiring @xmath579 , @xmath580 , and @xmath581 , we can easily obtain its matrix representation , @xmath582 and its hermitian conjugation , @xmath583 using this projection operator , the electron operators in the restricted hilbert space without double electron occupancy are given by @xcite , as quoted in eq . ( [ cep2 ] ) . it is then straightforward to verify the operator relations quoted in eq . ( [ sr3 ] ) . in particular , the charge - carrier number operator , @xmath585 the physical meaning of eq . 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superconductivity in cuprate superconductors occurs upon charge - carrier doping mott insulators , where a central question is what mechanism causes the loss of electrical resistance below the superconducting transition temperature ? in this review , we attempt to summarize the basic idea of the kinetic - energy driven superconducting mechanism in the description of superconductivity in cuprate superconductors .
the mechanism of the kinetic - energy driven superconductivity is purely electronic without phonons , where the charge - carrier pairing interaction in the particle - particle channel arises directly from the kinetic energy by the exchange of spin excitations in the higher powers of the doping concentration .
this kinetic - energy driven d - wave superconducting - state is controlled by both the superconducting gap and quasiparticle coherence , which leads to that the maximal superconducting transition temperature occurs around the optimal doping , and then decreases in both the underdoped and overdoped regimes .
in particular , the same charge - carrier interaction mediated by spin excitations that induces the superconducting - state in the particle - particle channel also generates the normal - state pseudogap state in the particle - hole channel . the normal - state pseudogap crossover temperature is much larger than the superconducting transition temperature in the underdoped and optimally doped regimes , and then monotonically decreases upon the increase of doping , eventually disappearing together with superconductivity at the end of the superconducting dome .
this kinetic - energy driven superconducting mechanism also indicates that the strong electron correlation favors superconductivity , since the main ingredient is identified into a charge - carrier pairing mechanism not from the external degree of freedom such as the phonon but rather solely from the internal spin degree of freedom of the electron .
the typical properties of cuprate superconductors discussed within the framework of the kinetic - energy driven superconducting mechanism are also reviewed .
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it has been a longstanding goal in semiconductor spintronics to inject , transport , manipulate , and detect spin - polarized carriers in silicon - based devices.@xcite despite great success in the field over the past ten years using other semiconductors such as gaas,@xcite the goal of achieving the same with si has been reached only recently , using all - electrical hot - electron methods with undoped single - crystal silicon transport layers.@xcite later , spin injection into silicon was realized as well in an epitaxially - grown silicon n - i - p diode structure using circular polarization analysis of weak electroluminescence spectra for spin detection across a transport layer of 80 - 140nm.@xcite although our previous studies demonstrate electron spin manipulation in undoped silicon - even over a very long distance ( 350 microns)@xcite - it is necessary to investigate magnetic- and electric - field control of electron spin in _ doped _ silicon for integration of spintronics into present - day silicon - based microelectronic technology , where impurity doping plays a critical role . in this report , we present spin injection , transport and detection in an n - type doped silicon device using our all - electrical methods . unlike previous studies with undoped si , the presence of ionized impurities in the depletion regions of these doped transport layers gives rise to conduction band bending that for sufficient biasing conditions confines injected electrons for long dwell times . by modeling transport with drift and diffusion in the inhomogeneous electric fields provided by the band bending with a monte - carlo method , we simulate both spin precession and spin decay , showing that the transit time distribution of spin - polarized electrons can be controlled over a very wide range with an applied voltage , and can yield a measurement of spin lifetime . fig . [ fig1 ] illustrates the structure of our device . fabrication consists of ultra - high vacuum metal film wafer bonding to assemble a semiconductor - metal - semiconductor hot - electron spin detector ; a silicon - on insulator ( soi ) wafer including a 3.3@xmath0 m single - crystal ( 100 ) nominally 1 - 20 @xmath1 cm phosphorus - doped n - type silicon spin transport layer is bonded to an n - type bulk silicon collector wafer with a ni@xmath2fe@xmath3 ( 4nm)/ cu ( 4 nm ) bilayer . conventional wet - etching techniques expose the soi device layer , onto which a ferromagnetic - emitter tunnel junction hot - electron spin injector is built . the final device structure is al ( 40nm)/co@xmath4fe@xmath5 ( 10nm)/al@xmath6o@xmath7/al ( 5nm)/cu ( 5nm)/n - si ( 3.3 @xmath0m)/ni@xmath2fe@xmath3 ( 4nm)/cu ( 4nm)/n - si substrate , as displayed in fig . 1 . further details on fabrication of similar devices can be found in previous reports@xcite . an applied emitter voltage @xmath8 on the tunnel junction ( larger than the cu / n - si injector schottky barrier ) injects hot electrons tunneling from the ferromagnetic co@xmath4fe@xmath5 cathode through the thin - film al / cu anode base and into the doped silicon transport layer conduction band . the first collector voltage ( @xmath9 ) controls the voltage drop across the transport layer and modifies the spatially nonlinear conduction band potential energy . electrons escaping the transport layer are ejected over a schottky barrier at the detector side into hot - electron states in a buried ni@xmath2fe@xmath3 thin film . the final spin polarization is detected by measuring the ballistic component of this hot electron current ( second collector current , @xmath10 ) in the n - type si wafer below ; spin - dependent scattering in the ferromagnetic ni@xmath2fe@xmath3 makes this current dependent on the projection of final spin angle on the ni@xmath2fe@xmath3 detector magnetization . the spin - detection current @xmath10 was first measured with an external magnetic field parallel to the device plane . a spin - valve effect , resulting from the different in - plane coercive fields of injector and detector ferromagnetic layers , is displayed in fig . the measurements were done with @xmath8 = -1.6v applied , using different values of @xmath9 between 4.5v and 8v at temperature @xmath11 = 152k . because of the @xmath10-@xmath9 dependence , we normalize the data for comparison between different @xmath9 values . after this normalization , it can be seen that the measurement is only weakly dependent on accelerating voltage @xmath9 over this range . the in - plane magnetic field was swept between -4 koe to + 4 koe for this measurement . since the coercive fields of both ferromagnetic ( fm ) layers are smaller than 200 oe , the data obtained from the @xmath9 = 5v measurement is magnified in the inset of fig . 2 and the field sweep direction is specified by correspondingly colored arrows . when the in - plane magnetic field reaches approximately + 20 oe from the negative saturation field ( below -300 oe ) , the ni@xmath2fe@xmath3 layer switches its magnetization , causing an anti - parallel ( ap ) configuration in the two fm layers , which lowers the @xmath10 current relative to a parallel ( p ) configuration , because in this case spin `` up '' is injected , but spin `` down '' is detected . if the magnetic field increases further , the co@xmath4fe@xmath5 layer reverses magnetization , resulting in a p configuration and restoration of the higher @xmath10 . this happens as well in the opposite sweeping field direction due to the symmetric but hysteretic coercive fields of each fm layer . the magnetocurrent ( mc ) ratio ( @xmath10@xmath12@xmath10@xmath13)/@xmath10@xmath13 calculated from the spin - valve plot , where the superscripts refer to p and ap magnetization configurations in the two fm layers , is approximately 6% . as the magnetic field reaches up to @xmath144 koe after the magnetization reversal of both fm layers , @xmath10 monotonically rises because of domain magnetization saturation in the direction of the external field . = -1.6v , with different @xmath9 values applied as indicated in the plot . inset : data measured with @xmath9 = 5v plotted over a smaller field range . field sweep directions are indicated by red ( increasing ) and blue ( decreasing ) arrows.,width=302,height=264 ] to unambiguously confirm spin transport through the doped silicon layer , we have performed measurements of @xmath10 in an external magnetic field perpendicular to the device plane , which allows us to examine spin precession and dephasing ( hanle effect ) during transport.@xcite depending on the magnitude of the applied magnetic field and the transit time ( subject to drift and diffusion through the conduction band from injector to detector ) , the polarized electron spin ( initially parallel to the injector fm layer magnetization ) can arrive at the detector having rotated through precession angle @xmath15 , where @xmath16 is the transit time , @xmath17 is the magnetic field , @xmath18 is the electron spin g - factor , @xmath0@xmath19 is the bohr magneton , and @xmath20 is the reduced planck constant . our measurements in a perpendicular magnetic field , using the same experimental conditions as were applied in the spin - valve effect measurement ( v@xmath21 = -1.6v and t = 152k ) , are shown in fig . [ fig3 ] for the same varied values of @xmath9 as in fig . the measured @xmath10 was normalized for data comparison at different accelerating voltages @xmath9 , as in the spin - valve effect experiment . again , the inset of fig . [ fig3 ] shows the data for @xmath9 = 5v with magnetic field sweep directions indicated by correspondingly colored arrows . when a perpendicular magnetic field sweeps from -4 koe ( or from + 4 koe ) , @xmath10 exhibits a minimum before the field reaches 0 oe and then it suddenly drops and slowly moves up between 0 and + 1koe . the former minima is induced by a full spin flip due to spin precession ( average @xmath22 rad rotation ) during transport through the doped silicon layer , and the latter is induced by the in - plane magnetization switching of the two fm layers by a residual in - plane component of the largely perpendicular magnetic field , causing an antiparallel injector / detector magnetization configuration and reduction in signal as seen in previously - discussed in - plane spin - valve measurements . this argument is further upheld by changing @xmath9 ; minima attributed to precession appear at higher magnitude of applied perpendicular magnetic field as @xmath9 increases due to the shorter transit time , while the fm switching fields clearly do not change.@xcite electron spin precession in doped si spin - transport devices . emitter voltage and temperature are same as in fig . minima , corresponding to @xmath22 rad precession angle appear at higher magnetic field as @xmath9 increases and transit time decreases . inset : data measured with @xmath9 = 5v . field sweep directions are indicated by red ( increasing ) and blue ( decreasing).,width=321,height=283 ] average spin transit times @xmath16 on the order of 45 - 180ps can be determined from the magnetic field values at @xmath22 rad precession minima @xmath17@xmath23 in fig . 3 ( @xmath24 1 koe 4 koe ) using@xcite @xmath25 , where @xmath26 is the planck constant . correlating spin polarization from spin - valve measurement to these transit times can , in principle , be used to determine spin lifetime . however , these transit times are very short , so direct correlation as in ref . @xcite is unable to independently determine the ( long ) spin lifetime of conduction electrons in doped si . we have previously measured spin lifetime of 73ns at similar temperature using a 350 micron - thick undoped si transport layer device ; this lifetime increases to over 500ns at 60k.@xcite in the undoped silicon transport layers used in previous works,@xcite the schottky depletion region was much larger than the layer thickness . therefore , the conduction band was linear , resulting in a spatially constant induced electric field , and relatively `` ohmic '' spin transport where the spin transit time was inversely proportional to the injector - detector voltage drop . in this work , however , carrier depletion of the doped silicon due to schottky contacts and the resulting space - charge from ionized impurities causes a nonlinear conduction band that can have a potential energy minimum between depletion regions unless the voltage drop is very large . since injected electrons may sit in this potential well for a long time before escaping over the detector barrier , their spins will depolarize and the observed mc ratio will be suppressed . to significantly reduce this dwell time , an accelerating voltage ( induced by applied voltage @xmath9 , which adds to approximately 0.3v of the applied emitter voltage due to resistive tunnel junction electrodes@xcite ) can be used to alter the confining potential energy . in particular , for sufficient voltage the confining potential can be eliminated . it is therefore expected that the spin signal is strongly sensitive to applied voltage and `` non - ohmic '' spin transport results . modeling this non - ohmic behavior is necessary . in previous works using undoped si transport layers where the electric field is constant from injector to detector , a modeling technique using the arrival - time distribution given by the green s function solution to the drift - diffusion equation can be easily implemented.@xcite however , the electric field in these doped si devices is highly inhomogeneous , making it difficult to implement the standard method here because the drift velocity is spatially dependent , requiring green s function solution of a nonlinear partial differential equation . in general , this procedure is non - trivial . monte - carlo simulated transit - time distributions for injector - detector voltage drops of ( a ) 3.0v ; ( b ) 4.46v ; and ( c ) 6.0v . note timescale changes over 4 orders of magnitude from ( a ) to ( c ) . , width=283,height=566 ] results of hanle effect simulation , eq . ( [ mchanle ] ) . similar to the experimental results in fig . [ fig3 ] , minima corresponding to @xmath22 rad precession angle appear at higher magnetic field as @xmath9 increases and transit time decreases . magnetization switching is simulated by signal sign reversal between the coercive field values of the injector and detector ferromagnets.,width=283,height=283 ] comparison of experimental and monte - carlo simulated voltage dependence of ( a ) hanle peak full width at half maximum ; and ( b ) spin polarization @xmath27 at 152k . the simulation shown in ( a ) is insensitive to a choice of @xmath28 over the range 10 - 100ns . , width=283,height=453 ] to overcome this problem and simulate spin transport behavior in these doped devices , we use a monte - carlo technique which translates electrons a distance @xmath29 ( due to drift ) , and @xmath30 ( due to diffusion ) in a timestep @xmath31 , where @xmath32 is the drift velocity at the position @xmath33 and @xmath34 is the diffusion constant . ( the sign on the latter expression is randomly chosen to simulate the stochastic nature of 1-dimensional diffusion . ) the spatially - dependent electric field is calculated within the depletion approximation . using a doping density of @xmath35 @xmath36 , injector schottky barrier height of 0.6ev ( for cu / si ) and detector schottky barrier height of 0.75ev ( for nife / si ) results in a band diagram whose dependence on injector - detector voltage drop is shown in fig . [ banddiagram ] . this figure illustrates that the voltage drop across the si transport layer can be used to alter the dominant transport mode : at low bias a wide neutral region exists between depletion regions and electrons must diffuse _ against an electric field _ to escape to the detector , whereas for biases greater than 6v , the potential minimum is annihilated by the boundary so the internal electric field carries electrons toward the detector everywhere and drift is expected to dominate . a realistic empirical mobility model using eq . 10 from ref . @xcite is used to evaluate @xmath37 . the diffusion coefficient at each point in space is then calculated from the einstein relation @xmath38 , where mobility @xmath39 and @xmath40 is electric field . we simulate transport for @xmath41 electrons at each value of injector - detector voltage drop and the arrival time at the detector for each is recorded . the distribution of arrival times @xmath42 is constructed from a histogram of this data and are used to calculate the expected output due to spin precession in a perpendicular magnetic field ( hanle effect ) : @xmath43 where @xmath44 is the angle between the injector / detector magnetization and the device plane , @xmath28 is effective spin lifetime and the spin precession angular frequency @xmath45 . the tilting angle @xmath44 is caused by the external magnetic field partially overcoming the finite geometric anisotropy of the magnetic thin films@xcite and is necessary to correctly model the experimental results . as in ref . @xcite , we use @xmath46 . in addition , the final spin polarization after transport can be calculated from @xmath47 as can be seen in fig . [ banddiagram ] , an electric field opposing transport to the detector is present at low voltage . electrons must therefore diffuse against this electric field to escape the confining potential in the bulk of the si transport layer . under these conditions of diffusion - dominated transport , the arrival - time distribution has a very wide exponential shape with average transit time of approximately 500ns , as shown in fig . [ distributions ] ( a ) . although the width of the distribution can be reduced significantly by increasing the voltage drop to the point where the si transport layer is fully depleted as shown in fig . [ distributions ] ( b ) , the confining electric field remains and the exponential shape is maintained . this indicates that diffusion is still strong . for sufficiently high voltage drops , the potential energy minimum is annihilated by the detector boundary as indicated in fig . [ banddiagram ] and drift - dominated transport occurs . this is reflected in the gaussian - like shape of the distribution in fig . [ distributions ] ( c ) for a voltage drop of 6v . at this voltage , the average transit time is only approximately 50ps , consistent with the analysis of experimental hanle effect measurements . therefore , as a result of our monte - carlo modeling we see that the average electron transit time in our doped si spin transport devices can be controlled over approximately 4 orders of magnitude by changing the injector - detector voltage drop by only several volts ( from 3v to 6v ) . using eq . ( [ mchanle ] ) , we simulate the hanle effect in our devices using @xmath48ns ( choice of this value will be discussed later ) . [ hanle ] shows hanle effect simulations for voltages corresponding to the same @xmath9 values as in fig . [ fig3 ] ( again , a shift of 0.3v due to a portion of the emitter bias dropping across the resistive tunnel junction base@xcite is accounted for to make a direct comparison ) in wide agreement to those experiments . in particular , the qualitative shape and precession minima positions are well modeled . the most salient feature of the hanle effect simulation is the magnetic - field width of the central ( zero precession - angle ) peak , plotted as a function of injector - detector voltage drop in fig . [ hanlewidthpol](a ) and compared to the experimental values . note that the width is constant for voltages greater than 6v ( due to drift velocity saturation at high electric field in si ) , and the presence of a threshold near 5v ( due to appreciable lowering of the confining potential barrier at the detector side of the transit layer once full - depletion occurs at approximately that voltage ) . this sudden collapse of the hanle peak width is not seen in the voltage dependence of spin precession measurements using undoped drift - dominated spin transport devices . in fig . [ hanlewidthpol ] ( b ) , we show the voltage dependence of the measured spin polarization @xmath27 from experimental data using in - plane magnetic field spectroscopy as described in fig . again , a threshold is seen in the experimental data . however , the position of the spin polarization threshold in fig . [ hanlewidthpol ] ( b ) near 3.5v is at much smaller bias voltage as compared to the hanle width collapse threshold near 5v shown in fig [ hanlewidthpol](a ) . this indicates that the electrons maintain their spin despite a long dwell time which causes strong spin dephasing in the confining conduction band potential minimum at low voltages . comparing this behavior to the model results from eq . ( [ mcpol ] ) with different values of spin lifetime @xmath28 shows that this discrepancy in threshold position is consistent with a long spin lifetime of 10 - 100ns . this can be compared to a spin lifetime of approximately 73ns measured in undoped si at the same temperature using a different technique.@xcite in summary , we have demonstrated spin transport through n - type doped si . using a monte - carlo algorithm to model drift and diffusion , we simulated electron transport through the inhomogeneous internal electric field and make quantitative comparisons to experimental values of spin polarization and hanle peak width without any free fitting parameters . analysis of the arrival - time distribution indicates that in doped transport layers , the spin - polarized electron transit time can be controlled over several orders of magnitude with applied voltage . the resulting non - ohmic behavior seen here is in contrast to spin transport measurements using undoped silicon transport layers , and is expected to influence future semiconductor spintronic device designs utilizing current - sensing spin detection methods in n - type doped semiconductors .
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we demonstrate the injection and transport of spin - polarized electrons through n - type doped silicon with in - plane spin - valve and perpendicular magnetic field spin precession and dephasing ( `` hanle effect '' ) measurements . a voltage applied across the transport layer is used to vary the confinement potential caused by conduction band - bending and control the dominant transport mechanism between drift and diffusion . by modeling transport in this device with a monte - carlo scheme , we simulate the observed spin polarization and hanle features , showing that the average transit time across the short si transport layer can be controlled over 4 orders of magnitude with applied voltage . as a result
, this modeling allows inference of a long electron spin lifetime , despite the short transit length .
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You are an expert at summarizing long articles. Proceed to summarize the following text:
in linear algebra it is frequently necessary to use non - linear objects such as minimal and characteristic polynomials since they encode fundamental information about endomorphisms of finite - dimensional vector spaces . it is well - known that if @xmath2 is a field and @xmath3 is a zero - dimensional affine @xmath2-algebra , _ i.e. _ a zero - dimensional algebra of type @xmath4/i$ ] , then @xmath3 is a finite - dimensional @xmath2-vector space ( see proposition 3.7.1 of @xcite ) . consequently , it not surprising that minimal and characteristic polynomials can be successfully used to detect properties of @xmath3 . this point of view was taken systematically in the new book @xcite where the particular importance of minimal polynomials ( rather greater than that of characteristic polynomials ) emerged quite clearly . the book also described several algorithms which use minimal polynomials as a crucial tool . the approach taken there was a good source of inspiration for our research , so that we decided to dig into the theory of minimal polynomials , their uses , and their applications . the first step was to implement algorithms for computing the minimal polynomial of an element of @xmath3 and of a @xmath2-endomorphism of @xmath3 ( see algorithms [ alg : minpolyquotelim ] , [ alg : minpolyquotmat ] , [ alg : minpolyquotdef ] ) . they are described in section 3 , refine similar algorithms examined in @xcite , and have been implemented in ( see @xcite and @xcite ) as indeed have all other algorithms described in this paper . sections [ reductions of ideals ] and [ modularapproach ] constitute a contribution of decisive practical significance : they address the problem of computing minimal polynomials of elements of a @xmath5-algebra using a modular approach . as always with a modular approach , various obstacles have to be overcome ( see for instance the discussion contained in @xcite ) . in particular , we deal with the notion of reduction of an ideal modulo @xmath6 , and we do it by introducing the notion of @xmath7-denominator of an ideal ( see definition [ reductionmodp ] and theorem [ thm : rgbp ] ) , which enables us to overcome the obstacles . then ugly , usable , good and bad primes show up ( see definition [ def : usableprime ] and [ def : goodprime ] ) . fortunately , almost all primes are good ( see theorem [ almostallgood ] and corollary [ cor : badprimes ] ) which paves the way to the construction of the fundamental algorithm [ alg : modular ] . section 6 presents non - trivial examples of minimal polynomials computed with , and section 7 shows how minimal polynomials can be successfully and efficiently used to compute several important invariants of zero - dimensional affine @xmath2-algebras . more specifically , in subsection [ isradical and radical ] we describe algorithms [ alg : isradical0dim ] and [ alg : radical0dim ] which show how to determine whether a zero - dimensional ideal is radical , and how to compute the radical of a zero - dimensional ideal . in subsection [ ismaximal ] we present several algorithms which determine whether a zero - dimensional ideal is maximal or primary . the techniques used depend very much on the field @xmath2 . the main distinction is between small finite fields and fields of characteristic zero or big fields of positive characteristic . in particular , it is noteworthy that in the first case frobenius spaces ( see section 5.2 of @xcite ) play a fundamental role . finally , in section [ primary decomposition ] a series of algorithms ( see [ alg : pdsplitting ] , [ alg : pdsplittingchar0 ] , [ alg : primarydecompositioncore ] , and [ alg : primarydecomposition0dim ] ) describe how to compute the primary decomposition of a zero - dimensional affine @xmath2-algebra . they are inspired by the content of chapter 5 of @xcite , but they present many novelties . as we said , all the algorithms described in this paper have been implemented in . their merits are also illustrated by the tables of examples contained in sections [ timings ] and at the end of section [ uses ] . the experiments were performed on a macbook pro 2.9ghz intel core i7 , using our implementation in 5 . here we introduce the notation and terminology we shall use and the definition of minimal polynomial which is the fundamental object studied in the paper . let @xmath2 be a field , let @xmath8 $ ] be a polynomial ring in @xmath9 indeterminates , and let @xmath10 denote the monoid of power products in @xmath11 . let @xmath0 be a zero - dimensional ideal in @xmath12 ; this implies that the ring @xmath13 is a zero - dimensional affine @xmath2-algebra , hence it is a finite dimensional @xmath2-vector space . then , for any @xmath14 in @xmath12 there is a linear dependency mod @xmath0 among the powers of @xmath14 : in other words , there is a polynomial @xmath15 $ ] which vanishes modulo @xmath0 when evaluated at @xmath16 . [ def : minpoly ] let @xmath2 be a field , let @xmath17 $ ] , and let @xmath0 be a zero - dimensional ideal . given a polynomial @xmath18 , we have a @xmath2-algebra homomorphism @xmath19 \to p / i$ ] given by @xmath20 . the monic generator of the kernel of this homomorphism is called the * minimal polynomial * of @xmath21 ( or simply `` of @xmath14 '' when the ideal @xmath0 is obvious ) , and is denoted by @xmath22 . the particular case of @xmath23 , where @xmath24 is an indeterminate , is a very important and popular object when computing : in fact @xmath23 is the lowest degree polynomial in @xmath24 belonging to @xmath0 , that is @xmath25 = { \langle { { \mu_{x_i , i}}}(x_i ) \rangle}$ ] . it is well known that this polynomial may be computed via elimination of all the other indeterminates @xmath26 ( see for example corollary 3.4.6 of @xcite ) . however the algorithm which derives from this observation is usually impractically slow . [ rem : nfsigma ] for the basic properties of grbner bases we refer to @xcite . let @xmath7 be a term ordering on @xmath10 , and let @xmath0 be an ideal in the polynomial ring @xmath12 . for every polynomial @xmath27 it is known that @xmath28 , the @xmath7-normal form of @xmath14 with respect to @xmath0 , does not depend on which @xmath7-grbner basis of @xmath0 is used nor on which specific rewriting steps were used to calculate it ( see proposition 2.4.7 of @xcite ) . if @xmath0 is clear from the context , we write simply @xmath29 . [ rem : zdelta ] following convention , for @xmath30 we use the symbol @xmath31 to denote the * localization * of @xmath32 by the multiplicative set generated by @xmath33 , _ i.e. _ the subring of @xmath5 consisting of numbers represented by fractions of type @xmath34 where @xmath35 , @xmath36 . observe that @xmath31 depends only on the radical @xmath37 , _ i.e. _ the product of all primes dividing @xmath33 . furthermore , if @xmath38 then @xmath39 divides @xmath40 if and only if @xmath41 is a subring of @xmath42 . if @xmath6 is a prime number we use the symbol @xmath43 to denote the * finite field * @xmath44 . note that some authors write @xmath45 to mean the field @xmath43 : this is clearly ambiguous . let @xmath2 be a field , let @xmath17 $ ] , let @xmath0 be a zero - dimensional ideal , and let @xmath18 . a well - known method for computing @xmath22 is by elimination . one extends @xmath12 with a new indeterminate to produce @xmath46 $ ] , then defines the ideal @xmath47 in @xmath3 , and finally eliminates the indeterminates @xmath48 . here we give a refined version of algorithm 5.1.1 of @xcite . * ( minpolyquotelim ) * [ alg : minpolyquotelim ] _ notation : _ : : @xmath49 $ ] input : : @xmath0 , a zero - dimensional ideal in @xmath12 , and a polynomial @xmath50 1 : : create the polynomial ring @xmath51 $ ] 2 : : define the ideal @xmath52 3 : : return * the monic , minimal generator of @xmath53 $ ] * output : : @xmath54 $ ] another way to compute @xmath22 is via multiplication endomorphisms on @xmath1 . let @xmath18 then we write @xmath55 for the endomorphism `` multiplication by @xmath56 '' . there is a natural isomorphism between @xmath1 and @xmath57 $ ] associating @xmath56 with @xmath58 ( see proposition 4.1.2 in @xcite ) . the minimal polynomial of @xmath14 with respect to @xmath0 is the same as the minimal polynomial of the endomorphism @xmath58 . thus , if the matrix @xmath59 represents @xmath58 with respect to some @xmath2-basis of @xmath1 , we can compute the minimal polynomial of @xmath59 ( and thus of @xmath58 ) using the following algorithm which is a refined version of algorithm 1.1.8 of @xcite . * ( minpolyquotmat ) * [ alg : minpolyquotmat ] _ notation : _ : : @xmath49 $ ] with term ordering @xmath7 input : : @xmath0 , a zero - dimensional ideal in @xmath12 , and a polynomial @xmath50 1 : : compute @xmath60 , a @xmath7-grbner basis for @xmath0 ; + from @xmath60 compute @xmath61 , the corresponding monomial quotient basis of @xmath1 2 : : compute @xmath59 , the matrix representing the map @xmath58 w.r.t . @xmath61 3 : : let @xmath62 and @xmath63 4 : : _ main loop : _ for @xmath64 do + 4.1 ; ; let @xmath65 ( hence @xmath66 ) 4.2 ; ; is there a linear dependency @xmath67 , @xmath68 ? + yes : : return * @xmath69 * no : : append @xmath70 to @xmath71 output : : @xmath54 $ ] note that in step minpolyquotmat-4.2 a linear dependency among the matrices @xmath72 is equivalent to a linear dependency among just their _ first columns_. the reason is that the first column contains the coefficients of @xmath73 assuming that the monomial @xmath74 appears as the first element of @xmath61 . there is a still more direct approach . it comes from considering the very definition of minimal polynomial : we look for the first linear dependency among the powers @xmath75 in @xmath1 . here we give a refined version of algorithm 5.1.2 of @xcite . * ( minpolyquotdef ) * [ alg : minpolyquotdef ] _ notation : _ : : @xmath49 $ ] with term ordering @xmath7 input : : @xmath0 , a zero - dimensional ideal in @xmath12 , and a polynomial @xmath50 1 : : compute @xmath60 , a @xmath7-grbner basis for @xmath0 ; + from @xmath60 compute @xmath61 , the corresponding monomial quotient basis of @xmath1 2 : : let @xmath76 3 : : let @xmath77 and @xmath78 4 : : _ main loop : _ for @xmath64 do + 4.1 ; ; compute @xmath79 ; consequently @xmath80 4.2 ; ; is there a linear dependency @xmath81 , @xmath68 ? + yes : : return * @xmath69 * no : : append @xmath82 to @xmath71 output : : @xmath54 $ ] notice that minpolyquotmat and minpolyquotdef essentially do the same computation : the first using a matrix ( dense ) representation , and the second a polynomial ( sparse ) representation . these two algorithms , minpolyquotmat and minpolyquotdef , are indeed quite simple and natural , but we want to emphasize that a careful implementation is essential for making them efficient . the reward is performance which is dramatically better than the well known elimination approach ( see the timings in section [ sec : timings - minpoly ] ) . there are two crucial steps for achieving an efficient implementation . the first is when computing the powers of @xmath72 ( in step minpolyquotmat-4.1 ) and @xmath83 ( in step minpolyquotdef-4.1 ) : the _ incremental approach _ we give , using the last computed value , is very important . the second is the search for a linear dependency ( in steps minpolyquotmat-4.2 and minpolyquotdef-4.2 ) . we implemented it creating a c++ object called ` lindepmill ` . this object accepts vectors one at a time , and says whether the last vector it was given is linearly dependent on the earlier vectors ; if so , then it makes available the representation of the last vector as a linear combination of the earlier vectors . internally ` lindepmill ` simply builds up and stores a row - reduced matrix and the linear relations as new vectors are supplied . the topic of section [ modularapproach ] is to show how to compute the minimal polynomial of an element of a zero - dimensional affine @xmath5-algebra using a modular approach . in this section we describe the necessary tools to achieve such goal . given an ideal @xmath0 in @xmath84 $ ] what does it mean to reduce @xmath0 modulo a prime number @xmath85 ? since there is no homomorphism from @xmath5 to @xmath86 , there is no immediate answer to this question . in this section we let @xmath87 $ ] , investigate the problem , and provide a useful answer . we shall also assume that @xmath12 comes with a term ordering @xmath7 . we start with the following lemma ( recall the notation in remark [ rem : zdelta ] ) . [ lemma : samelocaliz ] let @xmath88 , let @xmath0 be a non - zero ideal in @xmath12 , let @xmath89 be its reduced @xmath7-grbner basis , and let @xmath18 . assume that @xmath14 and @xmath89 have all coefficients in @xmath31 . 1 . every intermediate step of rewriting of @xmath14 via @xmath89 has all coefficients in @xmath31 . 2 . the polynomial @xmath90 has all coefficients in @xmath31 . if @xmath91 , the result is trivially true . so we now assume @xmath92 . if @xmath14 can be reduced by @xmath89 then there exists @xmath93 such that @xmath94 for some @xmath95 and some power - product @xmath96 . let @xmath97 be the coefficient of @xmath98 in @xmath14 ; by hypothesis @xmath99 . then the first step of rewriting gives @xmath100 which has all coefficients in @xmath31 . we can now repeat the same argument for rewriting @xmath101 , and so on . using remark [ rem : nfsigma ] , we deduce that the final result , when no further such rewriting is possible , is the normal form of @xmath14 . it is reached after a finite number of steps . these considerations prove both claims . the following example illustrates the lemma . [ ex : samelocaliz ] let @xmath102 $ ] , let @xmath103 where @xmath104 , @xmath105 , and let @xmath106 . the reduced @xmath7-grbner basis of @xmath0 is @xmath107 where @xmath108 . we let @xmath109 so that @xmath110 $ ] . we have @xmath111 , and it is easy to check that the explicit coefficients in the equality @xmath112 are the coefficients of a sequence of rewriting steps from @xmath14 to @xmath90 . as shown by the lemma , they all lie in @xmath113 . the following easy example shows that the number @xmath33 introduced in the above lemma depends on @xmath7 . [ ex : dependondelta ] let @xmath114 $ ] , let @xmath115 where @xmath116 . depending on the term ordering chosen , the number @xmath33 can be @xmath117 , @xmath118 or @xmath119 . [ reductionmodp ] let @xmath33 be a positive integer , and @xmath6 be a prime number not dividing @xmath33 . we write @xmath120 to denote both the canonical homomorphism @xmath121 and its natural `` coefficientwise '' extensions to @xmath122 \to { { \mathbb f}}_p[x_1 , \dots , x_n]$ ] ; we call them all * reduction homomorphisms modulo @xmath6*. now we need more definitions . [ def : densigma ] let @xmath123 $ ] . 1 . given a polynomial @xmath50 , we define the * denominator of @xmath14 * , denoted by @xmath124 , to be @xmath74 if @xmath91 , and otherwise the least common multiple of the denominators of the coefficients of @xmath14 . 2 . given a non - zero ideal @xmath0 in @xmath12 , with reduced @xmath7-grbner basis @xmath89 , we define the * @xmath7-denominator of @xmath0 * , denoted by @xmath125 , to be the least common multiple of @xmath126 . + the following theorem illustrates the importance of the @xmath7-denominator of an ideal . [ thm : rgbp]*(reduction modulo @xmath6 of grbner bases ) * + let @xmath0 be a non - zero ideal in @xmath127 $ ] with reduced @xmath7-grbner basis @xmath89 . let @xmath6 be a prime number which does not divide @xmath125 . 1 . the set @xmath128 is the reduced @xmath7-grbner basis of the ideal @xmath129 . the set of the residue classes of the elements in @xmath130 is an @xmath43-basis of the quotient ring @xmath131/{\langle \pi_p(g ) \rangle}$ ] . 3 . for every polynomial @xmath132 $ ] such that @xmath133 we have the equality @xmath134 . we start by proving claim ( a ) . every polynomial @xmath135 in @xmath89 is monic , so @xmath136 is monic and @xmath137 . next we show that @xmath128 is a reduced @xmath7-grbner basis . so assume @xmath138 , let @xmath139 and let @xmath140 be the @xmath141-polynomial of @xmath142 . it rewrites to zero via a finite number of steps of rewriting : @xmath143 for @xmath144 . let @xmath145 , then @xmath146 and every @xmath147 have all coefficients in @xmath31 . lemma [ lemma : samelocaliz ] implies that each @xmath148 is in @xmath31 and that all coefficients of each @xmath149 are in @xmath31 . we now show that the @xmath141-polynomial of the @xmath6-reduced pair @xmath150 rewrites to zero via the set @xmath128 . first we see that @xmath151 . now applying @xmath120 to each rewriting step we get @xmath152 . if @xmath153 , this is a rewriting step for @xmath154 , otherwise `` nothing happens '' and we simply have @xmath155 . this shows that all the @xmath141-polynomials of @xmath128 rewrite to zero , and hence that @xmath128 is a @xmath7-grbner basis . finally we observe that @xmath156 for all @xmath157 , hence @xmath128 is actually the reduced @xmath7-grbner basis of the ideal @xmath129 . as already observed , we have @xmath158 for all @xmath159 , hence claim ( b ) follows from ( a ) . for part ( c ) we let @xmath160 . we use the same method as in the proof of part ( a ) but starting with @xmath161 . once again all rewriting steps have coefficients in @xmath31 , and applying @xmath120 to them we get either a rewriting step for @xmath162 or possibly a `` nothing happens '' step . therefore the image of the final remainder @xmath163 is the normal form of @xmath162 . the following example illustrates some claims of the theorem . [ ex : samelocalizcontinued ] we continue the discussion of example [ ex : samelocaliz ] . we choose @xmath164 and get @xmath165 as the @xmath166-reduction of @xmath0 . from theorem [ thm : rgbp ] we know that @xmath167 is the reduced @xmath7-grbner basis of @xmath129 . theorem [ thm : rgbp ] , in particular claim ( c ) , motivates the following definition . if @xmath6 is a prime number which does not divide @xmath125 , then the ideal generated by the @xmath168 in the polynomial ring @xmath169 $ ] is called the * @xmath166-reduction of @xmath0 * , and will be denoted by @xmath170 . observe that if @xmath0 is zero - dimensional so is @xmath170 . the following example shows the necessity of considering the reduced grbner basis in theorem [ thm : rgbp ] . [ ex : badexample ] let @xmath171 $ ] , let @xmath172 be the product of many primes , for instance the product of the first @xmath173 prime numbers , and let @xmath174 . the set @xmath175 is a grbner basis of @xmath0 , while the set @xmath176 is the reduced grbner basis of @xmath0 . reducing @xmath141 modulo @xmath6 where @xmath177 produces the ideal @xmath178 , while reducing @xmath89 produces the ideal @xmath179 . more investigation about @xmath166-reductions is done in @xcite . the topic of this section is to show how to compute the minimal polynomial of an element of a zero - dimensional affine @xmath5-algebra using a modular approach . modular reduction is a very well - known technique , however there is no universal method for addressing the specific problems of bad reduction arising in every application . our problem is no exception as we shall explain shortly . for more details on this topic we recommend reading section 6 of @xcite . for this entire section the ideal @xmath0 will be zero - dimensional , and since in theorem [ thm : rgbp].(b ) we have seen that the set of power - products @xmath180 can be mapped both to a basis of @xmath84/i$ ] and also to a basis of @xmath131/i_{(p,\sigma)}$ ] , we are motivated to provide the following definition . [ f - matrix ] let @xmath181 $ ] with term ordering @xmath7 . let @xmath0 be a zero - dimensional ideal in @xmath12 . let @xmath182 and let @xmath183 with elements in increasing @xmath7-order . so the natural image of @xmath184 in @xmath1 is a @xmath5-basis of monomials for @xmath1 . we denote the natural image of @xmath184 in @xmath84 $ ] by @xmath185 and the natural image of @xmath184 in @xmath186 $ ] by @xmath187 . given @xmath18 we denote by @xmath188 the @xmath189 matrix whose @xmath190-th column ( for @xmath191 ) contains the coordinates of @xmath192 in the basis @xmath185 . similarly , we denote by @xmath193 the @xmath189 matrix whose @xmath190-th column contains the coordinates of @xmath194 in the basis @xmath187 . we observe that these matrices depend on both @xmath7 and the corresponding ideals . the following proposition contains useful information about reduction of matrices . [ pmultmat ] let @xmath18 be a polynomial and let @xmath195 . 1 . for every @xmath196 , all the entries of the matrix @xmath188 are in @xmath31 . 2 . for every @xmath196 , we have @xmath197 for any prime @xmath198 . claim ( a ) follows from lemma [ lemma : samelocaliz ] applied to @xmath199 , and claim ( b ) follows directly from theorem [ thm : rgbp].(c ) . we start this subsection with an elementary result which is placed here for the sake of completeness . [ usinggauss ] let @xmath200 $ ] be monic polynomials such that @xmath135 divides @xmath14 , and let @xmath88 . if @xmath14 has coefficients in @xmath31 then also @xmath135 has coefficients in @xmath31 . by hypothesis we have a factorization @xmath201 in @xmath202 $ ] . set @xmath203 , @xmath204 and @xmath205 ; so each of @xmath206 , @xmath207 and @xmath208 is a primitive polynomial with integer coefficients . by gauss s lemma @xmath209 is a primitive polynomial with integer coefficients . hence @xmath210 ; in particular @xmath211 , and consequently @xmath212 . since @xmath213 $ ] we have @xmath214 , hence also @xmath215 which implies that @xmath216 $ ] . we now give a proposition which tells us which primes could appear in the denominator of a minimal polynomial . [ goodminpoly ] let @xmath18 , and let @xmath217 . then the minimal polynomial @xmath22 has all coefficients in @xmath31 . let @xmath58 be the @xmath5-endomorphism of @xmath1 given by multiplication by @xmath56 . it is known that @xmath218 ( see remark 4.1.3.(a ) of @xcite ) . let @xmath219 be the characteristic polynomial of the endomorphism @xmath58 ; by definition @xmath220 . next , let @xmath221 , let @xmath222 , let @xmath223 be the identity matrix of size @xmath224 , and let @xmath225 be the matrix which represents @xmath58 with respect to the basis @xmath184 . then we have @xmath226 . the entries of @xmath225 are the coefficients of the representations of @xmath227 in the basis @xmath184 for all @xmath228 . they are in @xmath31 by lemma [ lemma : samelocaliz ] . so we have proved that @xmath229 $ ] . from the cayley - hamilton theorem we deduce that @xmath230 is a divisor of @xmath219 . it follows from lemma [ usinggauss ] that also @xmath231 $ ] . the conclusion of the proposition above motivates the following definition . [ def : usableprime ] let @xmath18 be a polynomial , and let @xmath6 be a prime number . then @xmath6 is called a * usable prime for @xmath14 with respect to @xmath232 * if it does not divide @xmath233 . if @xmath0 and @xmath7 are clear from the context , we say simply a * usable prime*. a prime which is not usable is called * ugly*. it follows from the definition that , for a given input @xmath234 , there are only finitely many ugly primes , and it is easy to recognize and avoid them . in this subsection we refine the definition of usable . [ def : goodprime ] let @xmath6 be a usable prime for @xmath14 with respect to @xmath232 ; consequently , by proposition [ goodminpoly ] , @xmath235 is well - defined . we say that @xmath6 is a * good prime for @xmath14 * if @xmath236 , in other words if the minimal polynomial of the @xmath6-reduction of @xmath14 modulo the @xmath166-reduction of @xmath0 equals the @xmath6-reduction of the minimal polynomial of @xmath14 modulo @xmath0 over the rationals . otherwise , it is called * bad*. the following simple example illustrates how a prime can be bad even if it is usable . let @xmath237 $ ] , let @xmath238 , and let @xmath239 . the set @xmath240 is a reduced grbner basis of @xmath0 for every ordering , @xmath241 is a quotient basis of @xmath242/i$ ] regardless of ordering . moreover we have @xmath243 regardless of ordering , and thus every prime number is usable . over @xmath5 we have @xmath244 . whence we deduce that @xmath245 . if we change the base field to the finite field @xmath246 , we get @xmath247 which shows that @xmath248 . it is easy to see that @xmath117 is the only bad prime in this case . next we show that there are only finitely many bad primes . [ almostallgood]*(finitely many bad primes ) * + let @xmath87 $ ] , let @xmath0 be a zero - dimensional ideal in @xmath12 , let @xmath7 be a term ordering on @xmath10 , let @xmath18 , and @xmath6 be a usable prime . 1 . then @xmath235 is a multiple of @xmath249 . 2 . there are only finitely many bad primes . the prime @xmath6 is good if and only if @xmath250 . to simplify the presentation we let @xmath251 and @xmath252 let @xmath253 , and set @xmath254 . by proposition [ goodminpoly ] we have @xmath255 $ ] . by the definition of minimal polynomial we have @xmath256 . therefore we have an equality @xmath257 for certain @xmath258 where @xmath259 is the reduced @xmath7-grbner basis of the ideal @xmath0 . by lemma [ lemma : samelocaliz ] we know that each @xmath260 . since @xmath6 is a usable prime , it follows from proposition [ goodminpoly ] that we can apply @xmath120 to get @xmath261 which shows that @xmath262 , and hence that @xmath263 it is a multiple of @xmath264 . so claim ( a ) is proved . to prove ( b ) and ( c ) it suffices to show that only a finite number of usable primes are such that @xmath263 is a non - trivial multiple of @xmath265 , and we argue as follows . since @xmath196 is the degree of @xmath266 we deduce that the matrix @xmath267 has rank @xmath196 , hence there exists an @xmath268-submatrix of @xmath267 with non - zero determinant ; moreover this determinant lies in @xmath269 , so can be written as @xmath270 for some non - zero @xmath271 and some @xmath272 . for any prime @xmath6 not dividing @xmath273 , the matrix @xmath274 has maximal rank : by proposition [ pmultmat ] we have @xmath275 . hence for these primes the degree of @xmath264 is @xmath196 , and the conclusion follows . we do not have an absolute means of detecting bad primes but , for two different usable primes @xmath276 and @xmath277 we can compute the minimal polynomials of @xmath278 with respect to @xmath279 , and by comparing degrees we can sometimes detect that one prime is surely bad , though without being certain that the other is good . * ( detecting some bad primes)*[cor : badprimes ] + let @xmath280 be two usable primes , and let @xmath281 and @xmath282 be the minimal polynomials of the corresponding modular reductions . 1 . if @xmath283 then @xmath276 is a bad prime . 2 . if @xmath284 then @xmath276 is a good prime . claim ( a ) follows from parts ( a ) and ( c ) of theorem [ almostallgood ] . claim ( b ) follows from theorem [ almostallgood].(c ) since @xmath285 is an upper bound for the degrees of the minimal polynomials . combining these results we get the following algorithm . [ alg : modular ] * ( minpolyquotmodular ) * _ notation : _ : : @xmath286 $ ] with term ordering @xmath7 input : : @xmath0 , a zero - dimensional ideal in @xmath12 , and a polynomial @xmath18 1 : : compute the @xmath7-reduced grbner basis of @xmath0 2 : : choose a usable prime @xmath6 see definition [ def : usableprime ] . 3 : : compute @xmath287 and @xmath170 . 4 : : compute @xmath288 $ ] , the minimal polynomial of @xmath289 . 5 : : let @xmath290 and @xmath291 . 6 : : _ main loop : _ + 6.1 ; ; choose a new usable prime @xmath6 . 6.2 ; ; compute the minimal polynomial @xmath292 $ ] . 6.3 ; ; if @xmath293 then + 6.3.1 : : if @xmath294 then let @xmath290 and @xmath291 . 6.3.2 : : continue with next iteration of _ main loop _ 6.4 ; ; let @xmath295 , and let @xmath296 be the polynomial whose coefficients are obtained by the chinese remainder theorem from the coefficients of @xmath297 and @xmath298 . 6.5 ; ; compute the polynomial @xmath299 $ ] whose coefficients are obtained as the fault - tolerant rational reconstructions of the coefficients of @xmath296 modulo @xmath300 . 6.6 ; ; were all coefficients `` reliably '' reconstructed ? + yes : : if @xmath301 and @xmath302 return * @xmath303 * no : : let @xmath304 and @xmath305 . output : : @xmath299 $ ] , the minimal polynomial @xmath306 . the correctness of this algorithm follows from theorem [ almostallgood ] and the termination from corollary [ cor : badprimes ] . termination of the _ main loop _ depends on the test @xmath307 in step minpolyquotmodular-6.6(yes ) ; however evaluating @xmath308 modulo @xmath0 is typically computationally expensive compared to the cost of a single iteration . for this reason , in step minpolyquotmodular-6.5 we use the fault - tolerant rational reconstruction implemented in cocoa ( see @xcite ) which gives also an indication whether the reconstructed rational is `` reliable '' ( _ i.e. _ heuristically probably correct ) . this is a computationally cheap criterion which surely indicates `` reliable '' almost as soon as @xmath300 becomes large enough to allow correct reconstruction , while also almost certainly indicating `` not reliable '' before then . once @xmath297 has the correct degree , the degree check in step minpolyquotmodular-6.3 ensures that only results from good primes are used ; in this situation our fault - tolerant reconstruction is equivalent to monagan s mqrr @xcite . a disadvantage of algorithm [ alg : modular ] is that it needs a grbner basis over @xmath5 , requiring a potentially costly computation . we can make a faster heuristic variant of the algorithm by working directly with the given generators for @xmath0 . let @xmath309 be the set of given generators . we shall skip all `` ugly '' primes which divide @xmath310 . in steps minpolyquotmodular-3 and 4 we use the ideal @xmath311 instead of @xmath170 . in the _ main loop _ we skip step minpolyquotmodular-6.3 , since there are no guarantees on the degrees of bad @xmath298 . in other words , we keep all the @xmath298 but when using the chinese remainder theorem to combine , we take only those polynomials having the same degree as the current @xmath298 . in step minpolyquotmodular-6.6(yes ) we return directly @xmath303 skipping the check that @xmath308 is in @xmath0 ( since we want to avoid computing its grbner basis ) . this heuristic algorithm may sometimes give a wrong answer : _ e.g. _ if given as input the generators in example [ ex : badexample ] in this instance the answer would be obviously wrong since it is reducible . there are only finitely many primes giving a bad @xmath298 . we can see this by picking some term ordering @xmath7 , and tracing through the steps to compute the reduced @xmath7-grbner basis from the generators @xmath309 . any prime which divides a denominator or a leading coefficient at any point in the computation may give a bad @xmath298 ; to these we add the ( finitely many ) bad primes for that reduced grbner basis . all remaining primes will give a good @xmath298 . in this section we illustrate the merits of the algorithms explained in the paper . each example is described by introducing a polynomial ring @xmath12 , an ideal @xmath0 in @xmath12 , and a polynomial @xmath14 in @xmath12 which is denoted either by @xmath312 if it is linear or by @xmath14 if it is not linear . the task is to compute @xmath22 , the minimal polynomial of @xmath56 in @xmath1 . the column * example * gives the reference number to the examples listed below . the column * gb * gives the times to compute the grbner basis ( in seconds ) ; the columns * def , mat , elim * give the times ( in seconds ) of the computation of the algorithms [ alg : minpolyquotdef ] , [ alg : minpolyquotmat ] , and [ alg : minpolyquotelim ] respectively . the column * deg * gives the degree of the answer , as an indication of the complexity of the output .
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given a zero - dimensional ideal @xmath0 in a polynomial ring , many computations start by finding univariate polynomials in @xmath0 . searching for a univariate polynomial in @xmath0
is a particular case of considering the minimal polynomial of an element in @xmath1 .
it is well known that minimal polynomials may be computed via elimination , therefore this is considered to be a `` resolved problem '' . but being the key of so many computations , it is worth investigating its meaning , its optimization , its applications .
minimal polynomial
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You are an expert at summarizing long articles. Proceed to summarize the following text:
motivated by their potential applications in spintronics and quantum computing @xcite , the search for three - dimensional topological insulators ( 3dti ) has attracted considerable theoretical and experimental interest @xcite . these materials are called `` topological '' because they are distinguished from ordinary insulators by the so - called @xmath0 topological invariants associated with the bulk band structure @xcite . on the theory side , the state - of - the - art first - principles calculations guided by topological band theory @xcite has provided a powerful tool for uncovering new families of 3dti . based on this approach , it has been recently predicted that ternary half - heusler compounds can realize the topological insulating phase with proper strain engineering @xcite . there are several ways to determine the band topology of an insulator . intuitively , one can count the number of band inversions within the entire brillouin zone an _ odd _ number indicates that the material may be a 3dti @xcite . this method depends on an accurate interpretation of the atomic origin of the bands and is better suited for crystals with a high - symmetry lattice . a more rigorous method is to directly evaluate the @xmath0 topological invariants . for materials with inversion symmetry , the parity criteria developed by fu and kane @xcite can be readily applied . on the other hand , if the inversion symmetry is absent , one must resort to the more elaborated lattice computation of the @xmath0 invariants @xcite . we have recently applied this method to study the distorted half - heuslers @xcite and the noncentrosymmetric chalcopyrite compounds @xcite . in practical calculations , both methods require an accurate knowledge of the bulk band structures . so far , previous works on 3dti have employed either the local density approximation ( lda ) @xcite or generalized gradient approximation ( gga ) @xcite for the exchange - correlation potential . however , it is well known that these approximations have the tendency to underestimate the bandgap . in particular , if the material has a small positive bandgap , this underestimation may yield a negative value . therefore , these approximations may falsely predict an inverted band structure when the band order is actually normal . recently , a new semilocal potential that combines the modified becke - johnson ( mbj ) exchange potential and the lda correlation potential , called mbjlda @xcite , was proposed to obtain accurate bandgaps and band order . the mbjlda potential are computationally as cheap as lda or gga , but it has similar precision compared with the more expensive hybrid functionals and gw method . it has been demonstrated @xcite that the mbjlda potential can be used to describe many types of solids , including wide bandgap insulators , small bandgap @xmath1 semiconductors , and strongly correlated @xmath2 transition - metal oxides . more importantly , the mbjlda potential can effectively mimic the behavior of orbital - dependent potential around the bandgap , so it is expected to obtain accurate positions of states near the band - edge , which are the keys to determine the band inversion and the band topology . in this work we perform a systematic investigation of the band topology of the half - heusler family using the mbjlda potential , and compare it with the lda result . the improved accuracy of the mbjlda potential over lda is tested in the ordinary insulator cdte and the topologically nontrivial 3d - hgte , whose band structures are known experimentally and can be used as a benchmark . we then focus on the ternary half - heusler compounds . by calculating the band inversion strength , we confirm our previous prediction that a large number of half - heuslers are possible candidates for 3dti or 3d topological metal @xcite . the difference between lda and the mbjlda potential is also discussed and a clear discrepancy is found between these two methods . the sensitivity of the calculated band structure due to the type of exchange - correlation potential calls for more detailed experimental works . , @xmath3 , and @xmath4 state are denoted by red , black , and blue color , respectively . the size of dots is proportional to the probability of s - orbit projection . ( b ) and ( e ) is schematically experimental band structure of cdte and hgte , the red solid dots denote the s - orbit originated @xmath5 state . the band inversion strength is defined as @xmath6 . the calculated and experimental energy of @xmath5 , @xmath3 , and @xmath4 state are listed in table [ tab : energy].,width=336 ] the band structure calculations in this work were performed using full - potential linearized augmented plane - wave ( fp - lapw ) method @xcite , implemented in the package wien2k @xcite . a converged ground state was obtained using 10,000 k points in the first brillouin zone with @xmath7 , where @xmath8 represents the muffin - tin radius and @xmath9 is the maximum size of reciprocal - lattice vectors . wave functions and potentials inside the atomic sphere are expanded in spherical harmonics up to @xmath10 and @xmath11 , respectively . spin - orbit coupling is included by a second - variational procedure @xcite , where states up to 9 ry above fermi level are included in the basis expansion , and the relativistic @xmath12 corrections were also considered for @xmath13 and @xmath14 orbit in order to improve the accuracy @xcite . we first test the mbjlda potential for the binary compounds cdte and hgte . the energy bands of both compounds at the @xmath15 point near the fermi level split into @xmath5 ( 2-fold degenerate ) , @xmath3 ( 2-fold degenerate ) , and @xmath4 ( 4-fold degenerate ) states due to the zinc - blende crystal symmetry and strong spin - orbit interaction . the experimental band order of cdte and hgte are ( from high to low energy ) @xmath5 , @xmath4 , @xmath3 ( ref . ) and @xmath4 , @xmath5 , @xmath3 ( ref . ) , respectively . they are schematically shown in fig . [ fig : order](b ) and [ fig : order](e ) . from the viewpoint of band topology , the main difference between these two compounds is that cdte possesses a normal band order , i.e. , the @xmath16-like @xmath5 state sits above the @xmath17-like @xmath4 state , while hgte possesses an inverted band order in which the @xmath5 state is occupied and sits below the @xmath4 state . we then define the band inversion strength @xmath18 as the energy differences between these two states , i.e. @xmath19 a negative @xmath18 typically indicates that the materials are in a topologically nontrival phase , while those with a positive @xmath18 are in a topologically trivial phase . figure [ fig : order ] shows the lda and mbjlda band structures of cdte and hgte at their experimental lattice constants 6.48and 6.46 , respectively @xcite . the calculated energy of the @xmath5 , @xmath3 , and @xmath4 states together with their experimental values are listed in table [ tab : energy ] . for cdte , the lda potential yields a serious underestimation of @xmath18 . for hgte , it obtains the wrong band order of the @xmath5 and @xmath3 states ( but it does predict an inverted band order ) . on the other hand , the mbjlda result shows an excellent agreement with experiments for both compounds . we therefore conclude that the mbdlda potential is better suited to calculate the topological band structure . ccddd & & & & + & @xmath20 & 0.221 & 1.549 & 1.475 + cdte & @xmath21 & 0.0 & 0.0 & 0.0 + & @xmath22 & -0.897 & -0.795 & -0.95 + & @xmath21 & 0.0 & 0.0 & 0.0 + hgte & @xmath20 & -1.191 & -0.234 & -0.29 + & @xmath22 & -0.822 & -0.721 & -0.91 + ( color online ) crystal structure of half - heusler compound xyz in the @xmath23 space group . green spheres at ( 0.5,0.5,0.5 ) are atom x , dark blue spheres at ( 0.25,0.25,0.25 ) are atom y , and pink spheres at ( 0,0,0 ) are atom z.,width=226 ] having established the improved accuracy of the mbjlda potential over lda , we now turn to the ternary half - heusler compounds described by space group @xmath23 . the chemical formula of these materials is @xmath24 , where @xmath25 and @xmath26 are transition or rare earth metals and @xmath27 a heavy element . it can be regarded as a hybrid compound of @xmath28 with rock - salt structure , and @xmath29 and @xmath30 with the zinc - blende structure ( see fig . [ fig : struct ] ) . in our band structure calculations , the lattice constants are taken from experimental data library @xcite . for compounds without experimental lattice constant , we use the value obtained by total energy minimization in first - principles calculations . the band structures of half - heuslers are very similar to cdte / hgte . in particular , the low - energy electron dynamics is dominated by energy bands at the @xmath15 point . therefore , the band topology of half - heuslers can be characterized by the band inversion strength @xmath18 defined in eq . . in order to systemically explore their topological phase , we have calculated @xmath18 by both lda and mbjlda for 24 half - heusler compounds , as shown in fig . [ fig : map](a ) and ( b ) , respectively . following our previous work ( ref . ) , we have also calculated the @xmath0 invariant for all half - heusler compounds investigated here . the calculated @xmath0 invariant agrees with the intuitive band inversion picture , that is , materials with negative @xmath18 are in topologically nontrivial phase . as shown in fig . [ fig : map](b ) , we have identified 9 half - heusler compounds , luptsb , laptbi , yptbi , scptbi , thptpb , laaupb , yaupb , lupdbi , luptbi as possible candidates for topological insulators or topological metal . ( note that scptbi , thptpb , laaupb here are virtual compounds . ) here , the band structures of the first 6 compounds are with zero band - gap similar to that of hgte , and the last 3 compounds possess the nontrivial metallic band structures similar to fig . [ fig : bands](e ) or the conduction bands droping below the fermi level near the x point . as shown in section iii.a , the lda result generally yields a smaller value of the band inversion strength @xmath18 when compared with the mbjlda result , which agrees much better with the experimental data . we emphasize that , unlike the usual energy gaps , here @xmath18 can take on both positive and negative values , hence a positive @xmath18 is always larger than a negative one . this trend is also confirmed for half - heuslers . as shown in fig . [ fig : map ] , @xmath18 by mbjlda is always larger than that by lda . in particular , if lda calculation predicts a small negative @xmath18 , it may become positive when using mbjlda potential , such as scaupb , and ypdbi . this suggests that when predicting topological phases , one must be careful with the types of exchange - correlation potentials . to illustrate the general difference between the lda and mbjlda results , we present four scenarios shown in fig . [ fig : bands ] using some typical examples . both lda and mbjlda calculations predict that laptbi [ fig . [ fig : bands](a ) and [ fig : bands](b ) ] has an inverted band structure , which is very similar to hgte . the main difference is the band order exchange between @xmath5 and @xmath3 state . although the band topology will not change , the lda result clearly underestimates the @xmath18 due to the downward movement of @xmath5 state . for ypdbi , the problem is much more severe as the two potentials give different band topology . as shown in fig . [ fig : bands](c ) and [ fig : bands](d ) , the lda result shows an inverted band order but the mbjlda result shows a normal band order . the two types of exchange - correlation potentials also affects the electronic behavior around the fermi level as shown for yptbi in fig . [ fig : bands](e ) and [ fig : bands](f ) . the metallic phase predicted by lda calculation becomes a zero - bandgap semiconductor in mbjlda calculation . the band order of @xmath5 and @xmath3 is also exchanged , just like laptbi . bands ( marked by red dots ) to jump below the valence band top , providing the necessary band inversion that leads to a nontrivial topological order . the other labels are the same as fig . [ fig : order].,width=302 ] as mentioned in our previous work @xcite , for half - heusler compounds with small band gaps , the nontrivial topologically phase can be generally realized by applying hydrostatic strain to change the band order . for example , lapdbi has normal band order in its native state , a @xmath31 change in the lattice constant converts the trivial topological phase into a nontrivial topological phase ( fig . [ fig : lapdbi ] ) . again , there are different effects on the band order under the same hydrostatic strain when using different exchange - correlation potentials . for example , when one stretches lattice constant to @xmath32 , lda calculation will yield an inverted band structure [ fig . [ fig : bands](g ) ] , while it still has a normal band order by mbjlda calculation[fig . [ fig : bands](h ) ] . in summary , we have systematically investigated the topological band structure of the half - heusler family using both lda and mbjlda exchange - correlations potential . our result shows that a large number of half - heusler compounds are candidates for three - dimensional topological insulators . we also discussed the main differences between the lda and mbjlda potentials this work was supported by nsf of china ( 10674163 , 10974231 ) and the most project of china ( 2007cb925000 ) , welch foundation ( f-1255 ) and doe ( de - fg02 - 02er45958 , division of materials science and engineering ) , and by supercomputing center of chinese academy of sciences(sccas ) and texas advanced computing center(tacc ) . 100 j. e. moore , nature * 464 * , 194 ( 2010 ) . l. fu and c. l. kane , phys . b * 76 * , 045302 ( 2007 ) . d. hsieh , d. qian , l. wray , y. xia , y. s. hor , r. j. cava , and m. z. hasan , nature * 452 * , 970 ( 2008 ) . h. zhang , c .- x . qi , x. dai , z. fang , and s .- c . zhang , nature physics * 5 * , 438 ( 2009 ) . y. xia , d. qian , d. hsieh , l. wray , a. pal , h. lin , a. bansil , d. grauer , y. s. hor , r. j. cava , and m. z. hasan , nature physics * 5 * , 398 ( 2009 ) . y. l. chen , j. g. analytis , j .- h . chu , z. k. liu , s .- k . mo , x. l. qi , h. j. zhang , d. h. lu , x. dai , z. fang , s. c. zhang , i. r. fisher , z. hussain , and z .- x . shen , science * 325 * , 178 ( 2009 ) . b. yan , c .- x . liu , h .- j . zhang , c .- y . qi , t. frauenheim , and s .- c . zhang , europhysics letters * 90 * , 37002 ( 2010 ) . h. lin , r. s. markiewicz , l. a. wray , l. fu , m. z. hasan , and a. bansil , phys . lett . * 105 * , 036404 ( 2010 ) . t. sato , k. segawa , h. guo , k. sugawara , s. souma , t. takahashi , and y. ando , phys . . lett . * 105 * , 136802 ( 2010 ) . y. chen , z. liu , j. g. analytis , j .- h . chu , h. zhang , s .- k . mo , r. g. moore , d. lu , i. fisher , s. zhang , z. hussain , and z .- x . shen , arxiv:1006.3843v1 ( 2010 ) . d. xiao , y.g . yao , w. feng , j. wen , w. zhu , x .- q . chen , g. m. stocks , and z. zhang , phys . lett . * 105 * , 096404 ( 2010 ) . h. lin , l. a. wray , y. xia , s. xu , s. jia , r. j. cava , a. bansil , and m. z. hasan , nature materials * 9 * , 546 ( 2010 ) . s. chadov , x. qi , j. kbler , g. h. fecher , c. felser , and s. c. zhang , nature materials * 9 * , 541 ( 2010 ) . c. l. kane and e. j. mele , phys . . lett . * 95 * , 146802 ( 2005 ) . l. fu and c. l. kane , phys . b * 74 * , 195312 ( 2006 ) . l. fu , c. l. kane , and e. j. mele , phys . rev . lett . * 98 * , 106803 ( 2007 ) . j. e. moore and l. balents , phys . b * 75 * , 121306 ( 2007 ) . r. roy , phys . b * 79 * , 195322 ( 2009 ) . t. fukui and y. hatsugai , j. phys . 76 , 053702 ( 2007 ) . w. feng , j. ding , d. xiao , and y. yao , arxiv:1008.0056 ( 2010 ) . w. kohn and l. j. sham , phys . rev . * 140 * , a1133 ( 1965 ) . j. p. perdew and y. wang , phys . b * 45 * , 13244 ( 1992 ) . j. p. perdew , k. burke , and m. ernzerhof , phys . lett . * 77 * , 3865 ( 1996 ) . f. tran , and p. blaha , phys . lett . * 102 * , 226401 ( 2009 ) . d. j. singh , _ planewaves , pseudopotentials and the lapw method _ ( kluwer academic , boston , 1994 ) . p. blaha , k. schwarz , g. madsen , d. kvaniscka , and j. luitz , _ wien2k , an augmented plane wave plus local orbitals program for calculating crystal properties _ ( vienna university of technology , vienna , austria , 2001 ) . p. larson , phys . b * 68 * , 155121 ( 2003 ) . p. lemasson , solid state commun . * 43 * , 627 ( 1982 ) . d. w. niles , and h. hchst , phys . b * 43 * , 1492 ( 1991 ) . n. orlowski , j. augustin , z. golacki , c. janowitz , and r. manzke , phys . b * 61 * , r5058 ( 2000 ) . j. c. woolley , and b. ray , j. phys solids * 15 * , 27 ( 1960 ) . p. villars , and l. d. calvert , _ pearson s handbook of crystallographic data for intermetallic phases _ ( amer soc metals , 1991 ) . springermaterials - the landolt - bnstein database , http://www.springermaterials.com / navigation/. s. h. vosko , l. wilk , and m. nusair , can . j. phys . * 58 * , 1200 ( 1980 ) .
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we systematically investigate the topological band structures of half - heusler compounds using first - principles calculations .
the modified becke - johnson exchange potential together with local density approximation for the correlation potential ( mbjlda ) has been used here to obtain accurate band inversion strength and band order .
our results show that a large number of half - heusler compounds are candidates for three - dimensional topological insulators .
the difference between band structures obtained using the local density approximation ( lda ) and mbjlda potential is also discussed .
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You are an expert at summarizing long articles. Proceed to summarize the following text:
the long - standing debate about the existence of the odderon ( @xmath9-odd partner of the pomeron ) can be resolved definitely only by a high - energy experiment involving particle and anti - particle scattering , e.g. @xmath0 and @xmath10 scattering , in the same kinematical region . there was a single experiment of that kind , at the isr @xcite , where the two cross sections were found to differ . the unique observation , however relies on a few data points only , and isr was shut down shortly after that experiment , leaving some doubts on the validity of the effect . moreover , the isr energies were not high enough to exclude the alternative explanation of the difference , namely due to @xmath11 exchange still noticeable at the isr in the region of the dip . this is not the case at the lhc , where the contribution from secondary trajectories can be practically excluded within the diffraction cone region . waiting for a possible future upgrade of the lhc energy down to that of the tevatron , which will enable a direct confrontation of @xmath0 and @xmath10 data , here we analyze the available lhc data on @xmath0 scattering in a model with and without the odderon contribution . anticipating the final result , let us mention that one should not dramatize the question of the ( non)existence of the odderon : in our opinion , it exists simply because nothing forbids its existence . the only question is its parametrization and/or relative contribution with respect e.g. to the pomeron . due to the recent experiments on elastic and inelastic proton - proton scattering by the totem collaboration at the lhc @xcite , data in a wide range , from lowest up to tev energies , both for proton - proton and antiproton - proton scattering in a wide span of transferred momenta are now available . the experiments at tev energies gives an opportunity to verify different pomeron and odderon models because the secondary reggeon contributions at these energies are small . however none of the existing models of elastic scattering was able to predict the value of the differential cross section beyond the first cone , as clearly seen in fig.4 of the totem paper @xcite . it should be noted that the predictions of regge - pole models are rather qualitative , so the new experimental data always stimulate their improvement . let us remind that the isr measurements stimulated the development of multipole pomeron models , including the dipole one , that successively described the dip - bump structure and both cones of the differential cross section of hadron - hadron scattering @xcite . the first attempt to describe high - energy diffraction peculiarities in the differential cross sections , was made by chou and yang in `` geometrical '' @xcite model , which qualitatively reproduces the @xmath5 dependence of the differential cross sections in elastic scattering , however it does not contain any energy dependence , subsequently introduced by means of regge - pole models . an example to examine the role of dipole pomeron ( dp ) , we performed the control fit for data of isr in the model of dipole pomeron ( see below ) . as result , we curtained of that the role of odderon headily grows with the height of energy . in recent paper @xcite we have used a simple dipole pomeron model that reproduces successfully the structure of first and second diffraction cones in @xmath0 and @xmath1 scattering . the simplicity and transparency of the model enables one to control of various contributions to the scattering amplitude , in particular the interplay between the c - even and c - odd components of the amplitude , as well as their relative contribution , changing with @xmath4 and @xmath5 . it was shown that , while the contribution from secondary reggeons is negligible at the lhc , the inclusion of the odderon is mandatory , even for the description of @xmath0 scattering alone . therefore the precise measurement of @xmath12 differential cross section gives a chance to distinguish various models of pomeron @xcite and especially odderon @xcite , @xcite . to do this one needs to compare the predictions of the models . such a comparison makes sense only if the same data set is used when the parameters of the models are determined . the possible extensions of dp model include : * the dip - bump structure typical to high - energy diffractive processes ; * non - linear regge trajectories ; * possible odderon ( odd-@xmath9 asymptotic regge exchange ) ; * compatible with @xmath13 and @xmath14 channel unitarity ; below we suggest a simple model that can be used as a handle in studying diffraction at the lhc . it combines the simplicity of the above models approach , and goes beyond their limitations . being flexible , it can be modified according to the experimental needs or theoretical prejudice of its user and can be considered as the `` minimal model '' of high - energy scattering while its flexibility gives room for various generalizations / modifications or further developments ( e.g. unitarization , inclusion of spin degrees of freedom etc . ) . to start with , we choose the model , successfully describing @xmath0 and @xmath1 scattering @xcite within the framework of the simple dipole pomeron . assuming that the role of the odderon in the second cone increases with energy , for more adequate definition of data we vary the form of the odderon . being limited in our choice , we will chose an odderon copying many features of the pomeron , e.g. its trajectory being non - linear . in this paper , we consider the spinless case of the invariant high - energy scattering amplitude , @xmath15 , where @xmath4 and @xmath5 are the usual mandelstam variables . the basic assumptions of the model are : \1 . the scattering amplitude is a sum of four terms , two asymptotic ( pomeron ( p ) and odderon ( o ) ) and two non - asymptotic ones or secondary regge pole contributions . where @xmath16 and @xmath17 have positive @xmath9-parity , thus entering in the scattering amplitude with the same sign in @xmath0 and @xmath1 scattering , while the odderon and @xmath11 have negative @xmath9-parity , thus entering @xmath0 and @xmath10 scattering with opposite signs , as shown below : @xmath18,\ ] ] where the symbols @xmath19 stand for the relevant regge - pole amplitudes and the super(sub)script , evidently , indicate @xmath20 scattering with the relevant choice of the signs in the sum ( [ eq : amplitude ] ) . we treat the odderon , the @xmath9-odd counterpart of the pomeron on equal footing , differing by its @xmath21 parity and the values of its parameters ( to be fitted to the data ) . we examined also a fit to @xmath0 scattering alone , without any odderon contribution . the ( negative ) result is presented in sec . [ sec : odderon ] ; \3 . the main subject of our study is the pomeron and the odderon , as a double poles , or dp @xcite ) lying on a nonlinear trajectory , whose intercept is not equal to one . this choice is motivated by the unique properties of the dp : it produces logarithmically rising total cross sections at unit pomeron intercept . by letting @xmath22 we allow for a faster rise of the total cross section , although the intercept is about half that in the dl model since the double pole ( or dipole ) itself drives the rise in energy . a supercritical pomeron trajectory , @xmath23 in the dp is required by the observed rise of the ratio @xmath24 or , equivalently , departure form geometrical scaling @xcite . the dipole pomeron produces logarithmically rising total cross sections and nearly constant ratio of @xmath25 at unit pomeron intercept , @xmath26 in addition this mild logarithmic increase of @xmath27 does not supported by the result of the last experiment at lhc for energy 7 tev @xmath28 @xcite . along with the rise of the ratio @xmath25 beyond the sps energies requires a supercritical dp intercept , @xmath29 where @xmath30 is a small parameter @xmath31 . thus dp is about `` twice softer '' then that of donnachie - landshoff @xcite , in which @xmath32 due to its geometric form ( see below ) the dp reproduces itself against unitarity ( eikonal ) corrections . as a consequence , these corrections are small , and one can use the model at the `` born level '' without complicated ( and ambiguous ) unitarity ( rescattering ) corrections . dp combines the properties of regge poles and of the geometric approach , initiated by chou and yang , see @xcite . regge trajectories are non - linear complex functions . this nonlinearity is manifest e.g. as the `` break '' i.e. a change the slope @xmath33 gev@xmath3 around @xmath34 gev@xmath3 and at large @xmath35 , beyond the second maximum we observe nonzero curvature at least for wide @xmath36 region . in spite of a great varieties of models for high - energy diffraction ( for a recent review see @xcite ) , only a few of them attempted to attack the complicated and delicate mechanism of the diffraction structure . in the 80-ies and early 90-ies , dp was fitted to the isr , sps and tevatron data , see @xcite and @xcite for earlier references . now we find it appropriate to revise the state of the art in this field , to update the earlier fits , analyze the ongoing measurements at the lhc and/or make further predictions . we revise the existing estimates of the pomeron and particularly odderon contributions to the cross sections as a functions of @xmath4 and @xmath5 and argue that while the contribution from non - leading trajectories in the nearly forward region is negligible ( smaller than the experimental uncertainties ) , the odderon may be important , especially beyond the first cone . 0.2 cm we use the normalization : @xmath37 neglecting spin dependence , the invariant proton(antiproton)-proton elastic scattering amplitude is that of eq . ( [ eq : amplitude ] ) . the secondary reggeons are parametrized in a standard way with linear regge trajectories and exponential residua , where @xmath38 denotes @xmath17 or @xmath11 - the principal non - leading contributions to @xmath0 or @xmath39 scattering : @xmath40 with handbook slopes @xmath41 and @xmath42 the values of other parameters of the reggeons are quoted in table [ tab : fitparam ] . as argued in the introduction , the pomeron is a dipole in the @xmath43plane @xmath44=\ ] ] @xmath45.\ ] ] since the first term in squared brackets determines the shape of the cone , one fixes @xmath46},\ ] ] where @xmath47 is recovered by integration , and , as a consequence , the pomeron amplitude eq . ( [ pomeron ] ) can be rewritten in the following `` geometrical '' form ( for the details of the calculations see @xcite and references therein ) @xmath48}-\varepsilon_p r_2 ^ 2(s){\rm e}^{r^2_2(s)[\alpha_p-1]}],\ ] ] where @xmath49 we use a representative example of the pomeron trajectory , namely that with a two - pion square - root threshold , eq . ( [ eq : tr2 ] ) , required by @xmath14channel unitarity and accounting for the small-@xmath5 `` break '' @xcite , @xmath50 where @xmath51 - pion mass . an important property of the dp eq . ( [ gp ] ) is the presence of absorptions , quantified by the value of the parameter @xmath52 . this property , together with the non - linear nature of the trajectories , justifies the neglect of the rescattering corrections . more details can be found e.g. in ref . @xcite . ) the unknown odderon contribution is assumed to be of the same form as that of the pomeron , eqs . ( [ pomeron ] ) , ( [ gp ] ) , apart from different values of adjustable parameters ( labeled by the subscript `` @xmath53 '' ) . @xmath54 } -\varepsilon_o r_2 ^ 2(s){\rm e}^{r^2_{20}(s)[\alpha_o-1]}],\ ] ] where @xmath55 and @xmath56 the form and properties of odderon trajectory is the same along with the scale value @xmath57 . the adjustable parameters are : @xmath58 for the pomeron and @xmath59 for the odderon . the results of the fitting procedure is presented below . to check the role of the odderon , we first fit only @xmath0 scattering without any odderon ( supposed to fill the dip in @xmath10 ) . the resulting fit is shown in fig [ fig : pomeron ] , demonstrating that , while the pomeron appended with sub - leading reggeons reproduces qualitatively the dip for low energies , namely 23 , 32 , 45 , 53 and 62 gev @xcite . the dipole pomeron model gives a good description of the first and second cones , but deteriorates with increasing energy in range of the second cone . it is special notable in the energy inerval @xmath60 tev . in fig . [ fig : pomeron ] ( b ) the @xmath10 differential cross section calculated with the same parameters is shown . apart for a shoulder instead of the dip in @xmath0 , the quality of the fit beyond this shoulder is comparable to that in @xmath0 . [ fig : pomeron2 ] the model contains ( at most ) 17 parameters ( depending on the choice of the trajectories ) to be fitted to about 1200 data points simultaneously in @xmath4 and @xmath5 . by a straightforward minimization one has little chances to find the solution , because of possible correlations between different contribution and the parameters , including the @xmath61 and @xmath62 mixing and the unbalanced role of different contributions / data points . to avoid false @xmath63 minima , we proceed step - by - step : we first fit the model to the forward data : the total cross section and the ratio @xmath64 , starting with the dominant pomeron contribution with the sub - leading reggeons , then we perform the fit for first cone and finally adding the odderon to the whole region of momentum transfer . by using the fitted parameters as inputs , we repeat the fit with the complete set of the data on elastic @xmath0 and @xmath10 elastic scattering differential and total cross sections . the data compiled in @xcite were used in our fitting procedure . the data are : total @xmath0 and @xmath65 cross section measurements spanning energy range from 5 to 7 tev and to 2.0 tev , respectively . another set of the data are those on the ratio of the real to the imaginary part of the forward amplitude . these sets contain measurements from both experiments at the tevatron . collection of single - differential elastic cross sections as functions of @xmath5 , measured at different energies were used for the fits . first of all we check the possible best fit for forward scattering , i.e. fitting the total cross section @xmath66 and @xmath67 for well the established set of this type of data @xcite plus the new measurement at 7 tev @xcite . the quality of the fit is not worse then the standard compete fit @xcite although we apply the best global fit ( minimal @xmath63 ) as a formal criterion for the valid description , we are primarily interested in the region beyond the first cone , critical for the identification of the assumed odderon at tev energies . as mentioned in the introduction , we perform also a fit to @xmath0 data alone , see the previous section , to see whether the observed dynamics of dip can be reproduced by the pomeron alone . the contribution to the global @xmath63 from tiny effects , such as the small-@xmath35 `` break '' in the first ( and second ) cone , possible oscillations in the slope of the cone(s ) etc . should not corrupt the study of the dynamics in the dip - bump region . the following kinematical regions and relevant datasets were involved in the fitting procedure : 23 , 32 , 45 , 53 , 62 gev and 7 tev for @xmath0 scattering @xcite and 31 , 53 , 62 , 546 , 630 gev , 1,8 tev and 1,96 tev for @xmath65 scattering @xcite . these datasets were compiled in a in @xcite . the differential elastic scattering cross sections were further constrained to cover the momentum transfer range @xmath680.05 15 gev@xmath3 . next , we included in the fit the differential cross sections in first cone chosen , somewhat subjectively for @xmath69 along with forward data , to determine the remaining parameters of the reggeons and the pomeron,@xmath70 for reggeons , @xmath71 and @xmath72 for the pomeron , important in first cone . among the parameters of the previous fit we fixed the parameters responsible for rise of the total cross section . we performed two series of fits : with linear pomeron trajectory ( @xmath73 ) and with a nonlinear one ( @xmath74 ) . for the grand total of 600 experimental points for the linear trajectory the quality of fit is better for about 70 percent in second case . it is obvious that the nonlinearity of pomeron trajectory plays a noticeable role . the presence of a non - negligible curvature in the first cone slope can be clearly seen with the help of the local slope procedure ( see , for example , fig.(8 ) in @xcite ) . the resulting fits are presented in table 1 . and fig . [ fig : dif ] . now , for the adequate study of the role of the pomeron , and especially that of the odderon outside the first cone , we must properly choose the parameters with account for their possible correlations . [ tab : fitparam ] 18.2pt .parameters , quality of the fit and predictions of @xmath27 obtained in the whole interval in @xmath4 and @xmath5 . [ cols="<,^,^",options="header " , ] finally , we note that the best fit to the data does not necessarily implies the best physical model , but the opposite statement is always true . a basic problem in studying the pomeron and odderon is their identification i.e. their discrimination from other contributions . although this procedure is model - dependent , we try to do this possibly in a general way . we try to answer the important question : where ( in @xmath4 and in @xmath5 ) and to what extent will be the elastic data from the lhc dominated by the pomeron and odderon contribution ? the answer to this question is of practical importance since , by regge - factorization , it can be used in other diffractive processes , such as diffraction dissociation . it is also of conceptual interest in our definition and understanding of the phenomenon of high - energy diffraction . it ensues from our analysis that the dipole model of pomeron and odderon unambiguously follows , that at high ( tev ) energies the pomeron prevails in the first cone , while in the second one the odderon is dominated , interference of which at least qualitatively describes the dip in @xmath0-differential cross section and accordingly the plateau in the @xmath75 ( see fig [ fig : htbp ] ) . the aim of the present paper was to trace the pomeron and odderon contribution under conditions accessible within lhc kinematics . this was feasible due to the simplicity of the model , which has the important property of reproducing itself ( approximately ) against unitarity ( absorption ) corrections , that are small anyway ( for more details see @xcite and references therein ) . we have presented the `` minimal version '' of the dp model . it can be further extended , refined and improved , while its basic features remain intact . the anticipated rescaling of the lhc energy down to that of the highest teavatron energy may provide a definite answer to the questions concerning the odderon in @xmath0 vs. @xmath10 scattering , raised in the present paper . we acknowledge fruitful discussions and useful remarks by tams csrg , lszl jenkovszky , denys lontkovskyi and mikola romanyuk . we thank frigyes nemes for his help in calculations . 5 cm 99 a. breakstone _ et al . _ , phys . lett . * 54 * , 2180 ( 1985 ) . totem collaboration et al , epl 95 ( 2011 ) 41001 . vall , l.l . jenkovszky and b.v . struminsky , echaya ( russian translation : pepan ) * 19 * ( 1988 ) 180 . chou , and c.n . yang , phys .rev.lett . * 20 * ( 1968 ) 1615 . l.l.jenkovszky , a.i.lengyel , d.i.lontkovskyi , int . j. mod . phys . a 26 ( 2011 ) 4577 . models of elastic diffractive scattering to falsity at the lhc . arxiv1203.6013v1 [ hep - ph ] 27 mar 2012 . basarab nicolescu . recent advances in odderon physics . arxiv9911334v1 [ hep - ph ] 12 nov 1999 . basarab nicolescu . the odderon at rhic and lhc . arxiv0707.2923v1 [ hep - ph ] 19 jul 2007 . p. desgrolard , m. giffon , l.l . jenkovszky , z. phys . c * 55*(1992 ) 643 . covolan , p. desgrolard , m. giffon , l.l . jenkovszky and e. predazzi , z.phys . c * 58 * ( 1993 ) 109 . k. kontros , a. lengyel , and z. tarics , _ @xmath0 and @xmath76 elasctic scattering in a multipole pomeron model _ , hep - ph/0011398 . http://qcd.theo.phys.ulg.ac.be/@xmath77cudell/ ; + http://qcd.theo.phys.ulg.ac.be/@xmath77cudell/data.html/ ; + http://www.theo.phys.ulg.ac.be/@xmath77cudell/data/ ; + http://pdg.lbl.gov/2002/ ; cudell et al . , phys . d65 , 074024 ( 2002 ) [ arxiv : hep - ph/0107219 ] ; j.r . cudell et al . , phys . ( 2002 ) 201801 ; [ arxiv : hep - ph/0206172 ] . m. bozzo _ et al . _ , phys . b * 155 * , 197 ( 1985 ) ; d.1 . bernard _ et al . _ , phys . b * 171 * , 142 ( 1986 ) ; n.a . amos _ et al . _ , phys . b * 247 * , 127 ( 1990 ) ; f. abe _ et al . _ , phys . d * 50 * , 5518 ( 1994 ) ; c.royon , d0 results on diffraction , 14thworkshop on elastic and diffractive scattering december 15 - 21 2011 , qio nhon vietnam .
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a simple multipole pomeron and odderon model for elastic hadron scattering , reproducing the structure of the first and second diffraction cones is used to analyze @xmath0 and @xmath1 scattering .
the main emphasis is on the delicate and non - trivial dynamics in the dip - bump region , at @xmath2 gev@xmath3 and at the second cone .
the simplicity of the model and the expected smallness of the absorption corrections enables one the control of various contributions to the scattering amplitude , in particular the interplay between the c - even and c - odd components of the amplitude , as well as their relative contribution , changing with @xmath4 and @xmath5 .
the role of the non - linearity of the regge trajectories is verified .
a detailed analysis of the lhc energy region , where most of the exiting models may be either confirmed or ruled out , is presented .
= 0.3 cm 0.5 cm * indirect evidence of the odderon from the lhc data on elastic proton - proton scattering * 0.3 cm a.i .
lengyel @xmath6 , z.z .
tarics @xmath7 _ institute of electron physics , nat .
ac .
sc . of ukraine ,
_ 0.1 cm @xmath8 0.1 cm
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You are an expert at summarizing long articles. Proceed to summarize the following text:
the phenomenon of radicalization @xcite is of central interest in the context of criminality and terrorism . it is currently spreading all over the world including european countries . the recent unprecedented terrorists attacks in paris ( november 13 , 2015 ) and brussels ( march 22 , 2016 ) took life of respectively 130 and 32 persons with over 300 wounded in each case @xcite . it puts at a very high level the burden on making substantial progress in the mastering of the issue . over the years sociologists and social - psychologists have contributed a good deal of work to the phenomenon @xcite . however an understanding , which could lead to some practical curbing of radicalization is unfortunately still lacking as dramatically demonstrated by the recent series of terrorist attacks in france ( 2015 @xcite ) and in brussels ( 2016 @xcite ) . one promising direction is the prospect to access the huge amount of data ( big data ) which exists in the world wide web . it could open a valuable source of surveillance and forecasting to prevent some aspects of radicalization spreading . however , efficient data - mining tools are still to be constructed yet within the constraints related to the preservation of individual privacies . accordingly , under the current risk of loosing control of the situation any new attempt to tackle the issue of radicalization is valuable in itself . to identify some hints to implement novel adequate policies towards at least the hindrance of radicalization spreading is of particular importance . along this line it happens that the modern field of sociophysics @xcite where models inspired from physics are developed to describe a large spectrum of social behaviors , may contribute to the challenge . among others , sociophysics includes the study of opinion dynamics @xcite , language dynamics @xcite , crowd behavior @xcite , criminal activities @xcite , and cultural dynamics @xcite . our work , focusing on a formal modeling of radicalization ( see also @xcite ) from the viewpoint of opinion dynamics , subscribes to this trend @xcite . therefore , according to the analytical approaches developed in sociophysics the proposed model adopts some assumptions that allow to simplify the scenario of reference . the complexity underlying terrorism phenomena is thus reduced to a series of more simple local interactions monitored by two parameters , which tune the global dynamics of the system . the focus on local interactions to reach the global equilibrium state constitutes one major trend of statistical physics , i.e. , the branch of physics from which sociophysics developed . more specifically we consider a mixed population made up of two subpopulations , each one sharing a peculiar way of life . first one is a core population locally rooted in the country . in contrast , the other one is an immigrant ( two , three generations ) subpopulation whose way of life is rooted in another territory . differences between the two ways of life may be strong , numerous and contradictory . however , in case of a disagreement about some specific cultural habit like for instance wearing the islamic veil , both subpopulations do not stand at the same level of resilience . core agents consider that it is not up to them to modify their way of life or accept from newcomers behaviors perceived as contrary to their long time country rooted cultural habits . core agents behave here as inflexible agents . for them it is up to newcomers including immigrants even at second or third generation to adjust to the country prevailing way of life . it is thus up to newcomers to either choose to live peacefully with the core population adjusting part of their habits to the local constraints or to maintain the integrality of their habits at a cost of creating conflicts with the core population . accordingly , the newcomers can be considered as sensitive agents . they can choose between two individual states either peaceful or opponent . sensitive agents are entitled to shift state from peaceful to opponent and vice - versa . in addition we make the assumption that being in an opponent state may lead the corresponding agent to take part or to support violent activities . in principle , the latter choice can be linked to the appearance of local terrorist groups . we are dealing with a mixture of inflexible and sensitive agents in given fixed proportions @xmath0 and @xmath1 with @xmath2 . however , @xmath1 is made up of two time dependent parts @xmath3 and @xmath4 , which are the respective proportions of peaceful and opponent sensitive agents . at any time @xmath5 @xmath6 . the time dependence is driven by an internal dynamics among sensitive agents . it is the result of pairwise interactions both among themselves between peaceful and opponent agents and with inflexible agents . an opponent may drive a peaceful agent to opponent and an inflexible may drive peaceful an opponent agent . the associated dynamics is studied using a lotka - volterra - like ordinary differential equation . given an initial tiny minority of opponents we investigate the role of their activeness @xcite in turning peaceful agents to opponents via pairwise interactions . the effectiveness of their activism is materialized in the degree of radicalization of the sensitive population against the core population . it creates a social basis for passive supporters @xcite to emerge in support to terrorists @xcite . in parallel , the mechanism behind the dynamics of radicalization enlightens by symmetry a potential role core inflexible agents could have in the launching of an eventual counter radicalization . by individual counter activeness core agents can contribute substantially to both curb the radicalization spreading and in certain conditions make it shrink down to an equilibrium state where inflexible , peaceful and opponent agents co - exist . the associated required minimum core individual involvement is calculated . it is found to be a function of both the majority or minority status of the sensitive subpopulation with respect to the core subpopulation and the degree of activeness of opponents . it is worth to stress that different mathematical frameworks could be used to describe our dynamics . for instance , approaches based on evolutionary game theory @xcite allow to perform both computational and analytical ( e.g. , @xcite ) investigations . it requires to define a payoff matrix and rules for local interactions to monitor the updating . in this work we use stochastic processes based on opinion dynamics @xcite . local interactions reduce to contact processes , which make updating rules to depend on the relative densities of the various agent states . the choice of the current approach in the modeling arises from the aim to evaluate to which extent the heterogeneity of a population in cultural and behavioral terms may lead to critical and complex social phenomena as radicalization . furthermore , it is important to emphasize that the attribute ` inflexible ' adopted to describe the core population stand , refers to cultural habits and traditions which allow to peacefully coexist with individuals coming from abroad provided they share the fundamental features of the local cultural frame . it happens that opinion dynamics constitutes one of the most investigated topics in sociophysics and in computational social science . for instance , its dynamics have been recently studied using the framework of multiplex networks @xcite considering different social behaviors @xcite . it allows to understand phenomena recorded in huge social network datasets @xcite . opinion dynamics allows to analyze and to model the spreading of ideas , opinions , and feelings by reducing the study of complex social scenarios to the analysis of few variables @xcite . even terrorism and criminal activities may be studied by the same approach , i.e. , reducing the related process to a problem of opinion dynamics . to conclude , our results may contribute to shed a new light on the instrumental role core agents could play to curb radicalization and establish a coexistence with the sensitive population . some hints at novel public policies towards social integration are obtained . in the last years several authors have worked on opinion dynamics models to analyze various underlying behaviors , which produce social phenomena , e.g. , group polarization , conformity and extremism . in this section , we briefly review some of these investigations , which are connected to our work along the topic of extreme social phenomena , especially radicalization . a computational model for tackling political party competitions is introduced in @xcite . the authors investigate different possible occurrences of fragmentation according to variations in the amount of important political issues and their current relevance . different interaction patterns among voters are considered using an analytical approach . the focus is on the role of extremism in opinion dynamics with a qualitative analysis of real scenarios . the complex social phenomenon of group polarization is described in @xcite in the context of politics . in particular , the authors propose a model based on probability theory to drive the emergence of group polarization . the emergence of risks is shown to be related to the group polarization in a wide range of scenarios related to political and economical issues ( e.g. , immigration , religion , welfare state , human rights ) . the results highlight the necessity to a better understanding of the emergence of extreme opinions . the connection between contradictory public opinions , heterogeneous beliefs and the emergence of extremism is analyzed in @xcite . an agent - based model considers a population with different socio - cultural classes to describe the process of opinion spreading with calculations performed on small groups of individuals ( e.g. , composed of @xmath7 and @xmath8 agents ) . the model constitutes a useful reference for defining models related to complex social phenomena . moreover , the related results suggest that the direction of the inherent polarization effect , which occurs in the formation of a public opinion driven by a democratic debate , is biased due to the existence of common beliefs within a population . opinion dynamics is also studied using computational approaches , e.g. , by agent - based models on continuous or discrete spaces . such approaches require a careful attention during the implementation phase . for instance , in the work @xcite authors focus on the role of activation regimes . more precisely they compare different asynchronous updating schemes ( e.g. , random and uniform ) . the activation regime refers to the order or scheme adopted to let agents express their opinion . as a result , the activation regime is found to affect opinion dynamics processes in some cases ( i.e. , @xcite ) . it is therefore of importance to clearly state which activation regime is selected to implement a dynamical model . the role of conformity in the q - voter model by arranging agents on heterogeneous networks has been also investigated @xcite . the authors showed that different steady states may be reached by tuning the ratio of conformists versus that of nonconformists in an agent population , which evolves according to the dynamics of the q - voter model . in our model we do not consider complex topologies . however , the influence that may arise from different interaction patterns may constitute the topic of future investigations . in order to study the emergence of radicalization in an heterogeneous population we consider a system with @xmath9 interacting agents distributed among inflexible ( @xmath10 ) , peaceful ( @xmath11 ) and opponent ( @xmath12 ) agents . each category refers to a different behavior or feeling . inflexible and opponent agents have behaviors mapped respectively to states @xmath13 . peaceful agents have a behavior mapped to the state @xmath14 . inflexible agents never change state ( see also @xcite ) while peaceful and opponent agents may shift state from one to another over time . opponents may become peaceful and peaceful may become opponents . hence , neither peaceful nor opponent agents may assume the state of inflexible agents . inflexible agents interact with sensitive agents both peaceful and opponents . during these pairwise interactions when an inflexible agent meets an opponent it may well turn the opponent to peaceful via different paths . among those paths most are spontaneous through normal social and friendship practices . but as it will appear latter , exchanges could become intentional as to promote coexistence with sensitive agents via monitored informal exchanges . to account for all interacting pairs a parameter @xmath15 is introduced to represent on average the rate per unit of time of encounters where opponents become peaceful agents . in parallel and in contrast we introduce the parameter @xmath16 to account on average for the rate of success of opponents in convincing peaceful agents to turn opponents . contrary to inflexible agents opponents are acting intentionally to increase the support to their radical view within the sensitive population . the value of @xmath16 is a function of the power of conviction of opponents . it also takes into account the activeness of opponent agents since opponents are activists . it is not the case of the core inflexible agents who interact spontaneously with sensitive agents without an a priori goal . it is worth to stress that both @xmath15 and @xmath16 may in principle vary over time . however , the corresponding time scale for variation is expected to be much longer than the time scale of the dynamics driven by pairwise interactions . this is why at the present stage of our work @xmath15 and @xmath16 are assumed to be fixed and constant . analyzing their time dependence , which might be of great interest to get further insights on equilibriums among people belonging to different cultures is left for future work . we emphasize that our analytical approach entails to consider the system as if it was continuous , i.e. , analyzing the relative densities of agents in the various states . a similar approach is usually followed in other contexts as epidemic dynamics @xcite . a compartmental approach to the studying of epidemics entails to analyze the spreading of a disease by modifying the state of agents . for instance , the sis @xcite model considers a two - state population where agents may vary their state from @xmath17 ( i.e. , susceptible ) to @xmath10 ( i.e. , infected ) and vice - versa over time . considering the probability to get infected or to heal the dynamics can be studied analytically defining odes as if the underlying system were continuous . going to the analytical details of our model we defined the following system of equations @xmath18 where @xmath0 is the constant density of inflexible agents , while @xmath4 and @xmath3 are the respective densities of peaceful and opponent agents at time @xmath5 . dealing with densities the third equation of system [ eq : evolution ] allows to reduce the number of odes to one equation . in particular , choosing the peaceful agents density @xmath3 we get @xmath19 the equilibrium state of the population can be obtained from @xmath20 , which reads @xmath21 the two solutions of equation [ eq : evolution_equilibrium ] read @xmath22^{2 } - 4 \beta \alpha \sigma_i ( 1 - \sigma_i)}}{2 \beta}\ ] ] where @xmath23 is the equilibrium value of peaceful agents . those values simplify to @xmath24 which implies the two associated equilibrium opponent values @xmath25 indeed equation [ eq : evolution_reduced ] can be solved analytically to yield @xmath26 fig 1 shows the evolution of the system on varying the initial conditions . we analyze the respective stability ranges for @xmath27 and @xmath28 : @xmath29 where @xmath30 and @xmath31 , we obtain @xmath32 + 2\beta \sigma_p\ ] ] therefore , for respective values @xmath33 we obtain @xmath34 stability being achieved for @xmath35 , equation [ eq : lambda_value_solutions ] shows that @xmath36 is stable when @xmath37 . accordingly we get two stable regimes : @xmath38 with @xmath39 . these two regimes yield the respective equilibrium values for peaceful and opponent agents as from [ eq : equilibrium_solution_value ] and [ eq : equilibrium_solution_value_op ] @xmath40 the first equation of system [ eq : final_stability ] highlights that in some conditions the amount of opponent agents is equal to zero . hence , we perform a further investigation to study under which conditions it is possible to avoid the phenomenon of radicalization ( i.e. , by reaching the equilibrium state @xmath41 ) . in terms of opinion dynamics these results indicate that under appropriate conditions it is possible to remove one opinion from the system . given the relevance of this outcome in the related context , i.e. , criminal activities and terrorism , we explore in more details this result . from the above results radicalization can be totally thwarted if @xmath42 . accordingly , given @xmath0 and @xmath16 the individual involvement for the inflexible population in striking up with individual opponents must be at least at a level @xmath43 therefore , as seen from equation [ eq : alpha ] the larger @xmath0 the less effort is required from the inflexible population . however , the more active are the opponents ( i.e. , larger @xmath16 ) the more involvement is required . to visualize the multiplicative factor by which @xmath15 must overpass @xmath16 it is worth to draw the curve @xmath44 as a function of @xmath0 as shown in fig 2 . from equation [ eq : alpha ] it is seen that to prevent radicalization inflexible agents s involvement must be either lower ( @xmath45 ) or larger ( @xmath46 ) than that of opponents depending on the magnitude of the multiplicative factor @xmath44 . when @xmath47 , i.e. , @xmath48 core agents do not need to much individual engagement as could be expected in the case of a coexistence of a core majority population with a sensitive minority subpopulation . more precisely , the engagement depends on the opponent activism but the core population benefiting from its majority status . in this case its requirement is always lower than the opponent involvement . however , the situation turns difficult when the initial sensitive minority turns to a majority status as it occurred in some specific urban areas . in that case to avoid a radicalization requires a very high individual engagement from the core agents , which may be rather hard to implement . in particular since no collective information is available about the situation . we thus have three different cases : * 1 ) * @xmath48 , * 2 ) * @xmath49 , and * 3 ) * @xmath50 to consider to determine the respective level of individual core involvement to avoid the phenomenon of radicalization . + * case 1 . * for @xmath51 core agents need little involvement to thwart totally the radicalization of the sensitive subpopulation with values of @xmath15 much lower than @xmath16 . indeed , opponent agents need to produce very high values of @xmath16 ( compared to @xmath15 ) to survive , precisely the condition @xmath52 must be satisfied . however , very large values of @xmath16 can shrink to zero the amount of peaceful agents yielding a fully radicalized sensitive population , which although in a small minority status may produce substantial violence against inflexible agents . + * case 2 . * for @xmath53 the opponent activism must be counter with an equal core counter activism since @xmath54 makes opponent agents to extinct . instead , for @xmath45 peaceful and opponent agents coexist and the former disappear for large values of @xmath16 with again a fully radicalized sensitive population with @xmath55 . + * case 3 . * for values @xmath50 , if @xmath56 the equilibrium condition entails that @xmath57 ( and @xmath58 ) . if @xmath46 , we can reach the extinction of opponent agents as @xmath59 . in contrast when @xmath45 opponent agents strongly prevail in the population . + in order to asses the degree of radicalization in a population we can introduce two parameters : @xmath60 and @xmath61 . the former is defined to evaluate the fraction of opponent agents among flexible agents while the latter ( i.e. , @xmath61 ) evaluates the ratio between opponent and inflexible agents . therefore , @xmath60 represents the relative ratio of opponents among flexible agents and @xmath61 gives a measure about the real power of opponents agents in a population . an high value of @xmath60 ( i.e. , close to @xmath62 ) in a population with @xmath63 indicates that strategies to fight radicalization are too weak but at the same time opponents are few . therefore , in this case governments should take an action even if the situation seems still under control . on the other hand , a low value of @xmath60 ( i.e. , close to 0 ) together with a high value of @xmath61 represent an alarming situation . indeed , even if there are only a few opponents among flexible agents their amount is bigger than that of inflexible ones @xcite . to evaluate these measures , @xmath60 and @xmath61 have been defined as follows @xmath64 hence , recalling that @xmath65 and having solved analytically @xmath3 ( see [ eq : solution_neutral ] ) we are able to compute values of both parameters @xmath60 and @xmath61 at equilibrium and on varying the initial conditions see fig 3 . it is worth to note that the parameter @xmath60 as defined in [ eq : radicalization ] has a range in @xmath66 $ ] . at equilibrium @xmath67 means that there are no opponent agents in the population while @xmath68 means that all flexible agents became opponents . on the other hand , the parameter @xmath61 has potentially an unlimited range from @xmath69 to @xmath70 ( in the case @xmath0 is very close to @xmath69 and @xmath71 to @xmath62 ) . to conclude , we want to emphasize the meaningful role of the two parameters @xmath60 and @xmath61 . they represent a way to quantify in which extent radicalization phenomena are taking place in a population . moreover , in more general terms we envision a further utilization of these parameters in opinion dynamics since they clearly indicate the prevalence of one opinion / state over another one . the recent anti - western terrorist attacks @xcite in europe have brought the question of radicalization at a top priority of policy maker agenda of the different european governments . in particular , most of the terrorists involved in the various killings which took place in several european capitals were either national citizens or legal residents . this very fact points to the direct link existing between terrorism and radicalization @xcite . indeed , various institutions are faced with the difficult issue to implement innovative procedures to stop if not eradicate radicalization . the task turns out to be rather hard since radicalization has been prospering quietly in different areas of european countries for now many years without any substantial barrier . it has been a sensitive political issue and most officials had preferred the laissez - faire instead of addressing the problem in solid terms . the dramatic scores of @xmath72 paris and @xmath73 brussels attacks have now prompted the necessity to face the problem and start implementing counter measures . the burden is on european governments to find ways to tackle the radicalization . almost everyone is expecting action from the states . but the states seem to have no solid scheme to apply . one direction has been along the education side with the setting up of so - called de - radicalization programs . however , such an approach concerns identified radicals who have been arrested . all efforts and thoughts are focused on acting on radicalized citizens . coercive measures are implemented against known associations and active leaders . even to contain radicalization appears to be a challenging task . our model , although rather simple puts light on the process by which the phenomenon of radicalization spreads over within a sensitive population . it articulates around the capacity of radicalized agents to turn radical otherwise peaceful agents who had chosen to coexist with the native population sharing their habitat . this capacity is embedded in the coefficient @xmath16 . in addition , the main novelty of our model is to account for the possible capacity of native agents to overturn radical agents in making them choose the peaceful state quantified with the coefficient @xmath15 . moreover , the ratio of native versus sensitive populations ( @xmath74 ) was found to be a critical parameter . in the past this ratio was rather stable over time with slow evolution . it made feasible to evaluate the activeness of radicals , which has been not meaningful for decades . however , rather quick changes may occur in the demography of the sensitive populations especially with the substantial increase of recent years immigration . on this basis , our results show how the overall situation can be totally put upside down ( @xmath75 ) with respect to the extent of current radicalization while not much seems to have happened with respect to radical activities . keeping the same level of activeness from radicals , a slow change in the population ratio may produce a sudden spreading of radicalization . therefore the knowledge of the evolution of the current ratio of populations is a key parameter to evaluate the associated potential of radicalization spreading . and yet , in many countries like france , ethnic statistics are forbidden . most of curbing radicalization still involves the state and diverse official institutions . in contrast the phenomenon of radicalization results from informal interactions among sensitive and radical agents . at this point our results unveiled a new and unexpected promising path to fight radicalization . an innovative strategy could be implemented by launching a citizen counter radicalization movement mapped from the path used by opponents to spread radicalization within the sensitive population . instead of being the sole prerogative of national authorities de - radicalization would become a citizen matter . the same way a radical tries to turn a peaceful sensitive agent to hostility towards the natives , natives can try to bring back opponents to peaceful coexistence . normal citizens would have to engage in personal interactions with sensitive agents to establish a solid ground for coexistence . the required degree of efficient citizen involvement can be clearly identified using the degree of activeness of the radicals and the ratio of subpopulations , native versus sensitive . while this ratio is at the exclusive hand of national authorities , the citizen involvement is a citizen prerogative . in addition , the centrality of the ratio of subpopulations within a given territory emphasizes the importance of avoiding a de - mixing of the subpopulations . in case of a different discriminating distribution of the subpopulations within distinct sub - territories , radicalization would be enhance at once with the same proportion of radical due the large value of @xmath74 within the sub - territory where the sensitive population is mostly confined . it is worth to notice that the proposed model may in principle be applied also to criminal and terrorist scenarios in homogeneous populations as it occurred in the cases of italian red brigades and french revolution . these two cases are concerned with homogeneous populations as both inflexible , peaceful and opponent agents belong to the local core population . in the former case ( i.e. , red brigades ) inflexible agents represent individuals who respect laws and believe in institutions and governments . individuals having a different behavior can fall in the mild category of peaceful agents or in the extreme category of opponents ( i.e. , criminals ) . instead , in the case of the french revolution inflexible agents represent the small proportion of french nobility . the remaining part of the population is represented by peaceful and opponent agents . there , the extremely difficult life conditions fed opponent ideals and the wide proportion of the sensible subpopulation became completely opponent giving rise to a revolution . to summarize we have identified the equilibrium state of a mixed population in terms of order or disorder phases . we have also identified the ratio between social strategies and the strength of opponents ideal . since we refer to the concept of social strategies it is worth to emphasize that although the considered scenario can be modeled in various ways as those based on evolutionary games . there the concept of `` strategy '' acquires a particular meaning . here we develop a model based on opinion dynamics processes . as a result social strategies are embodied in a parameter while updating rules depend on the density of different opinions in the population . moreover , in this context opinions refer to the different cultural extractions and behaviors that can be observed in an heterogeneous population . we remark that today the question of de - radicalization has became a key priority issue of internal security in european countries . yet the challenge is intact with no ready to use solution . different state agencies are launching a series of experimental treatments but all are concerned with institutional managing of the issue . given the acute current terrorist threat people are expecting and requesting policy makers to take initiative to curb the current phenomenon of radicalization within sensitive local populations . unlike this heavy policy trend our study has enlightened the crucial role so called `` normal citizens '' could play to stop the spreading of radicalism . it could even shrink it back with a serious perspective to eventually eradicate the actual growing threat set in european cities . from our results it appears that an efficient action should not be limited to state involvement but also to call on individual voluntary engagement within their respective neighborhoods towards the sensitive individuals . given an evaluation of radical activeness within some sensitive neighborhoods we were able to calculate the required degree of `` normal citizen '' counter - activeness to curb radicalization . this degree of engagement was also found to depend on the ration of native to sensitive populations . focusing on local interactions in the modeling of these dynamics underlines the instrumental role neighborhood compositions can have in the shaping of the social behavior of the corresponding subpopulations . today it often happens that people of different cultures are fostered to coexist together in the same district of a city occupying each a series of connected blocks . the connection with natives is thus drastically reduced jeopardizing opportunities of real integration even after a few generations . in addition , within some delimited urban areas the majority group is no longer the native subpopulation . at the same time it is still majority in a close by other area . accordingly given the same tiny proportion of opponents with the same degree of activeness in the two neighborhoods , one ends up highly radicalized while the other stays very peaceful . local interactions and the degree of mixing are key factors to undermine the spreading of radicalization . a free and uncontrolled ( by authorities ) settling of people often leads to a geographical concentration of sensitive subpopulations . as a result this process may spontaneously develop a natural ground for the emergence of hate towards native individuals . people are thus lead towards the strengthening of the initial culture differences , which results in the establishment of social distances with native individuals despite being physically very close to them . to conclude , we want to highlight that our work creates a first step to envision new policies to support campaigns promoting the daily life sharing among people from different cultural backgrounds . in particular , we focus on methods that potentially may lead `` radical neighbors '' to the choice of coexistence , i.e. , renouncing to fight against the native population . at least we hope our results will trigger more research along this path of individuals engaging to establish a peaceful coexistence with sensitive agents . at this stage further studies along this direction are required , in particular from a computational social science perspective . it should be possible to identify earlier traces ( i.e. , big data ) and seeds of radical behavior in social networks . suitable tools to quantify their strength are also required . last but not least , we would like to stress that although we have been mentioning criminal activities we are not judging neither the motivations nor the ideal of opponent agents . indeed , they can be considered negative ( as in the case of current anti - western terrorism ) or positive ( as today in the case of the french revolution ) depending both on the side taken and the chosen epoch . our aim was to study the conditions of emergence or vanishing of radicalization as a social phenomenon independently of a moral judgment . maj would like to thank fondazione banco di sardegna for supporting his work . this work was supported in part by a convention dga-2012 60 0013 00470 75 01 .
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the phenomenon of radicalization is investigated within a mixed population composed of core and sensitive subpopulations .
the latest includes first to third generation immigrants .
respective ways of life may be partially incompatible . in case of a conflict
core agents behave as inflexible about the issue . in contrast
, sensitive agents can decide either to live peacefully adjusting their way of life to the core one , or to oppose it with eventually joining violent activities .
the interplay dynamics between peaceful and opponent sensitive agents is driven by pairwise interactions .
these interactions occur both within the sensitive population and by mixing with core agents .
the update process is monitored using a lotka - volterra - like ordinary differential equation . given an initial tiny minority of opponents that coexist with both inflexible and peaceful agents , we investigate implications on the emergence of radicalization .
opponents try to turn peaceful agents to opponents driving radicalization . however , inflexible core agents may step in to bring back opponents to a peaceful choice thus weakening the phenomenon . the required minimum individual core involvement to actually curb radicalization
is calculated.it is found to be a function of both the majority or minority status of the sensitive subpopulation with respect to the core subpopulation and the degree of activeness of opponents .
the results highlight the instrumental role core agents can have to hinder radicalization within the sensitive subpopulation .
some hints are outlined to favor novel public policies towards social integration .
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as known , many key features of continuous - time chaotic dynamics can be treated in terms of two - dimensional or even one - dimensional discrete - time maps ( see e.g. @xcite ) . among them , most simple and perfect ones are the famous bernoulli map , @xmath0 @xmath1mod@xmath2 , and the tent map , @xmath0 @xmath3 , both defined at the interval @xmath4 . almost any trajectory of both maps is chaotic , that is dense in all this interval . in computer , however , any trajectory finishes at zero after @xmath5 steps only , where @xmath6 is number of bits under use . this contraction of the discretized phase space , from total @xmath7 points to single point @xmath8 , happens because the maps are non - invertible , i.e. relation between @xmath9 and @xmath10 is not an one - to - one correspondence . but a suitable slight distortion can make a discretized map invertible . evidently , any such map is nothing but a permutation of @xmath7 states ( @xmath11 @xmath12/n$ ] ) . in particular , the invertible discrete versions of the bernoulli map describe the mersenne digital auto - generators of pseudo - random 0 - 1-sequences ( see e.g. @xcite ) . for example , the mersenne map in fig.1 represents transitions between 16 states of a particular `` toy '' 4-bit mersenne generator ( @xmath13 ) . the general formula of such maps is @xmath14 where @xmath15 , @xmath16 , and any of @xmath17 , @xmath18 , equals to either zero or unit , @xmath19 ( hence this is class of @xmath20 different maps ) . to see corresponding unitary evolution matrix , it is necessary to turn the map picture headfirst and insert zeros and ones in place of the empty cells and crosses , respectively . the map in fig.1 has two immovable points ( @xmath8 and @xmath21 ) and two cycles ( periodic orbits ) with equal lengths @xmath22 . in applications of much greater mersenne maps ( with @xmath23 @xcite ) , the main issue is construction of those having most long cycles , with @xmath24 . here , we will be interested in modest problem about statistics of cycles of discrete maps , to be concrete , definite class of discrete tent maps . example of the discrete bernoulli map ( the mersenne map , see body text ) . ] first consider the variety of generally asymmetric continuous tent maps ( ctm ) described by @xmath25 where @xmath26 . the inversion of ( [ ctm ] ) reads as @xmath27 with @xmath28 and @xmath29 representing two variants of the inversion . let us write , with the help of ( [ ictm ] ) , @xmath30 as a function of @xmath31 and then equate @xmath30 to @xmath31 at various possible values @xmath32 @xmath33 . thus we obtain @xmath34 linear equations which determine @xmath34 points belonging to all possible periodic orbits ( cycles ) of length @xmath35 . that are either irreducible ( indivisible ) orbits or , if @xmath35 has a divider @xmath36 , composite orbits consisting of @xmath37 repetitions of an irreducible orbit with length @xmath36 . consequently , @xmath38 where @xmath39 is number of different irreducible periodic orbits with length @xmath36 , and the sum is taken over all the dividers of @xmath35 . in particular , if @xmath35 is a prime number then ( [ expan ] ) yields @xmath40 since @xmath41 ( there are two immovable points ) . in general , it is not hard to derive from ( [ expan ] ) that @xmath42}-[l/2]\right)\leq ln_l\leq 2^l-2\,\,\label{any}\ ] ] here @xmath43 $ ] @xmath44 if @xmath45 is even , and @xmath43 $ ] @xmath46 if @xmath45 is odd . at any @xmath47 , according to ( [ any ] ) , @xmath48 at relatively small @xmath45 , from ( [ expan ] ) one finds : @xmath49 , @xmath50 , @xmath51 , @xmath52 , @xmath53 , @xmath54 , @xmath55 , @xmath56 , @xmath57 , @xmath58 , ... next , consider invertible discrete tent maps ( dtm ) representing discrete analogues of the continuous tent maps ( [ ctm ] ) . to be concrete , let us introduce the class of discrete maps defined by @xmath59 & , & \,x < a\,\,;\\ y=\left\{\frac { n(n - x)}{n - a}\right\}-1 & , & \,x\geq a\,\ , \end{array } \label{dtm}\ ] ] here , like in ( [ dbm ] ) , @xmath60 and @xmath61 are integers , @xmath62 @xmath63 ; @xmath64 is analogue of @xmath65 in ( [ ctm ] ) , @xmath66 ; @xmath67\,$ ] is an integer closest to @xmath68 from below ( with @xmath67 $ ] @xmath69 if @xmath70 is integer ) , and @xmath71 means closest integer greater than ( or equal to ) @xmath68 ( that is @xmath72 @xmath69 if @xmath70 is integer ) . two examples of maps defined by formula ( [ dtm ] ) , at @xmath73 , are shown in fig.2 . let us prove that ( [ dtm ] ) prescribers invertible maps , i.e. establishes an one - to - one correspondence between @xmath74 and @xmath75 . it is sufficient to prove that the upper r.h.s . and lower r.h.s . in ( [ dtm ] ) can not produce one and the same number . indeed , in such a case we would have @xmath76=y=\left\ { n(n - x^{\prime\prime})/(n - a)\right\}-1\,\,,\ ] ] with some @xmath77 and @xmath78 . this would mean that @xmath79 ( @xmath80 and @xmath81 are some integers ) , therefore , @xmath82 but , obviously , the latter inequality contradicts the preceding equality . the proof is finished . it is easy to see also , that absolute value of a deviation of any dtm ( [ dtm ] ) from its ctm prototype ( [ ctm ] ) , with @xmath83 , ( as well as deviation of ( [ dbm ] ) from bernoulli map ) does not exceed @xmath84 lowest bit only ( @xmath85 , in terms of @xmath86 ) . in this sense , any dtm tends to corresponding ctm , when @xmath87 . however , the inverted map remains strongly discontinuous , which is a payment for its unambiguity . hence , in essence , the limit of dtm at @xmath87 does not coincide with corresponding ctm . naturally , there is no simple rule for the cycles ( periodic orbits ) of the dtms . a number of various irreducible cycles and lengths of these cycles are extremely irregular functions of @xmath64 and @xmath88 . in contrary to ( [ expan ] ) , now @xmath89 where @xmath90 is a number of different irreducible periodic orbits with length @xmath35 ( possibly , @xmath91 ) . for example , at @xmath92 and some three next values of @xmath93 this expansion looks as @xmath94 where the second multiplier ( if any ) in each term is number of different cycles of a length represented by the first multiplier . in other example , for @xmath95 , @xmath96 typically , a dtm have a long cycle whose length is comparable with the total number of points , @xmath88 , i.e. maximal possible length . this is illustrated by fig.3 relating to @xmath97 . quite similar pictures take place also at greater @xmath98 . we see that practically any dtm has a cycle with length @xmath99 . such long cycles , whose length @xmath45 is comparable with @xmath88 , can be treated as discrete analogues of the chaotic trajectories of ctm . the left plot in fig.3 demonstrates highly irregular dependence of the maximal cycle length , @xmath100 , on the asymmetry parameter @xmath93 . at the same time , the right - hand plot there shows that even slight smoothing of the data produces rather regular results . therefore , it is reasonable to describe the cycles of dtms in statistical language , considering some specific subclasses of dtms instead of individual maps . let us consider the family of nearly ( asymptotically ) symmetric dtms determined by the conditions @xmath101 since @xmath102 , we obtain the growing statistical ensemble , which is all the more representative one because all the maps defined by ( [ sym ] ) do tend to the same limit @xmath103 @xmath104 . it would be interesting to investigate such the family of asymptotically symmetric discrete tent maps ( asdtm ) , in comparison with the usual symmetric ctm . here , simplest statistical characteristics of cycles ( periodic orbits ) of the asdtm will be under our attention . let @xmath105 designates a density of probability distribution of the cycle lengths in this family of maps . in other words , practically , @xmath106 where @xmath90 is now * _ summary _ * number of periodic orbits of length @xmath45 in all the maps of the asdtm family . it is useful to introduce also the quantity @xmath107 that is relative ( probability ) measure of points which belong to all periodic orbits with lengths @xmath108 at all the maps . in reality , with the help of an ordinal pc only , it would take rather long time to obtain all the periodic orbits if @xmath109 . but our computations , performed at @xmath110 , showed that already @xmath111 are satisfactory values , because next @xmath98 s increases do not change the picture qualitatively ( although , of course , providing better numeric accuracy ) . therefore , it is reasonable to prefer calculations at not high @xmath98 ( @xmath112 ) but apply slight smoothing over @xmath45 s values . concretely , the plot @xmath113 in fig.3 represents the result of averaging of the exact @xmath113 ( defined by ( [ w ] ) ) over the intervals @xmath114\,$ ] , with @xmath115 . such smoothed probability density , @xmath113 , is represented by the curved line at right - hand side of fig.3 . it is easy to find that the best fitting for it is nothing but the inverse proportional law : @xmath116 where factor @xmath117 ensures the normalization , @xmath118 the dependence ( [ ip ] ) is shown by the straight line , and by smooth curve in the inset which demonstrates good quality of this fitting for short cycles too . the curved line at left side of fig.3 shows an example of the probability ( [ p ] ) . it is not smoothed , therefore , formed by many steps with very different heights ( like the famous devil s staircase ) . its closeness to the thinner straight line , which corresponds to @xmath119 , says that the total @xmath120 points are distributed approximately equally between cycles with different lengths . this is just about what the approximation ( [ ip ] ) says . if combining ( [ w]),([p]),([ip ] ) and the approximation @xmath119 , we obtain the estimate of mean number of cycles per one map of the family : @xmath121 ( let us recollect that @xmath90 is * _ summary _ * value for all the asdtm ) . at the same time , as the above examples of map expansions into cycles do show , fluctuations in number of cycles from one particular map to another are significantly greater than the mean value ( [ mnc ] ) . according to these examples , as well to the @xmath105 plots , especial contribution to the fluctuations comes from cycles whose lengths are powers of two . nevertheless , when rising @xmath98 from 12 to 16 a definite decrease of the fluctuations was noticed . this observation pushes us to the hypothesis that formula ( [ ip ] ) represents a true asymptotics of the cycles length distribution ( [ w ] ) in the limit ( [ sym ] ) . there is a simple naive explanation of the hypothetical asymptotics ( [ ip ] ) . in above derivation of the estimate ( [ asympt ] ) for the cycles numbers of continuous tent maps ( ctm ) , the factor @xmath34 ( with @xmath45 being cycle lengths ) arises from the two - valued property of their inverse maps ( i.e. due to their irreversibility ) . since the discrete tent maps ( dtm ) under consideration have univalent inversions , in their case this factor must disappear . thus one deduces the inverse proportional dependence of number of the cycles ( periodic orbits ) on their lengths , i.e. comes to ( [ ip ] ) ( in other words , cycles of different lengths involve approximately equal amounts of points of the discrete phase space ) . in view of such reasonings , the inverse proportional law can be expected in case of any sufficiently reach family of unitary dtm , not only the asdtm family defined by ( [ sym ] ) ( moreover , in unitary discrete analogues of multi - modal piecewise linear ctm , not unimodular tent maps only ) . in factual finite - precision computer simulations of more or less general ctm , distinguished from the `` pathological '' map @xmath0 @xmath3 ( about it see introduction ) , e.g. asymmetrical ctm ( [ ctm ] ) or the map @xmath0 @xmath122 with e.g. @xmath123 , it can be expected that the law ( [ ip ] ) must manifest itself sooner than ( [ asympt ] ) . indeed , if the latter rule was true , then the very short orbits with lengths @xmath124 only ( recollect that @xmath98 is number of bits under operations ) would take all @xmath125 points of the discrete phase space . in reality , with no doubts , almost any choice of an initial point results in a long orbit ( `` chaotic trajectory '' ) , whose length @xmath45 is comparable with @xmath88 , while it is hard to hit casually into a short orbit . to resume , we performed computer statistical analysis of periodic orbits of unitary ( reversible ) discrete tent maps , and found that probability distribution of the orbits lengths well obeys the inverse proportional law ( [ ip ] ) .
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the discrete unitary ( reversible ) analogues of the continuous ( irreversible ) tent maps are numerically investigated , in particular , the lengths probability distribution of their periodic orbits .
it is found that its density can be well approximated by the inverse proportional law .
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the recent literature offers several results on the properties of gas flows on networks . for instance , in @xcite the well posedness is established for the gas flow at a junction of @xmath6 pipes with constant diameters . the equations governing the gas flow in a pipe with a smooth varying cross section @xmath7 are given by ( see for instance @xcite ) : @xmath8 the well posedness of this system is covered in @xcite where an attractive unified approach to the existence and uniqueness theory for quasilinear strictly hyperbolic systems of balance laws is proposed . the case of discontinuous cross sections is considered in the literature inserting a junction with suitable coupling conditions at the junction , see for example @xcite . one way to obtain coupling conditions at the point of discontinuity of the cross section @xmath9 is to take the limit of a sequence of lipschitz continuous cross sections @xmath10 converging to @xmath9 in @xmath11 ( for a different approach see for instance @xcite ) . unfortunately the results in @xcite require @xmath5 bounds on the source term and well posedness is proved on a domain depending on this @xmath5 bound . since in the previous equations the source term contains the derivative of the cross sectional area one can not hope to take the limit @xmath12 . indeed when @xmath9 is discontinuous , the @xmath5 norm of @xmath13 goes to infinity . therefore the purpose of this paper is to establish the result in @xcite without requiring the @xmath5 bound . more precisely , we consider the cauchy problem for the following @xmath14 system of equations @xmath15 endowed with a ( suitably small ) initial data @xmath16 belonging to @xmath17 , the space of integrable functions with bounded total variation ( tot.var . ) in the sense of @xcite . here @xmath18 is the vector of unknowns , @xmath19 denotes the fluxes , _ i.e. _ a smooth function defined on @xmath20 which is an open neighborhood of the origin in @xmath21 . the system ( [ integrable : eq : bal ] ) is supposed to be strictly hyperbolic , with each characteristic field either genuinely nonlinear or linearly degenerate in the sense of lax @xcite . concerning the source term @xmath22 , we assume that it satisfies the following caratheodory type conditions : * @xmath23 is measurable with respect to ( w.r.t . ) @xmath24 , for any @xmath25 , and is @xmath26 w.r.t . @xmath27 , for any @xmath28 ; * there exists a @xmath11 function @xmath29 such that @xmath30 ; [ hyp_on_g ] * there exists a function @xmath31 such that @xmath32 . note that the @xmath11 norm of @xmath29 does not have to be small but only bounded differently from @xmath33 whose norm has to be small ( see theorem [ th : introduction ] below ) . furthermore condition @xmath34 replaces the @xmath5 bound of the @xmath26 norm of @xmath22 in @xcite . finally observe that we do not require any @xmath5 bound on @xmath35 . on the other hand we will need the following observation : if we define @xmath36 by absolute continuity one has @xmath37 as @xmath38 . moreover , we assume that a _ non - resonance _ condition holds , that is the characteristic speeds of the system ( [ integrable : eq : bal ] ) are bounded away from zero : | _ i(u)| c > 0 , u , i\{1, ,n}. [ hyp : non_resonance ] the following theorem states the well posedness of ( [ integrable : eq : bal ] ) in the above defined setting . [ th : introduction ] assume @xmath39@xmath40 and ( [ hyp : non_resonance ] ) . if the norm of @xmath35 in @xmath41 is sufficiently small , there exist a constant @xmath42 , a closed domain @xmath43 of integrable functions with small total variation and a unique semigroup @xmath44 satisfying a. @xmath45 for all @xmath46 and @xmath47 ; b. @xmath48 for all @xmath46 and @xmath47 ; c. for all @xmath49 the function @xmath50 is a weak entropy solution of the cauchy problem ( [ integrable : eq : bal])([init : data ] ) and satisfies the integral estimates ( [ eq : first_int_cond ] ) , ( [ eq : second_int_cond ] ) . conversely let @xmath51\rightarrow\cd$ ] be lipschitz continuous as a map with values in @xmath52 and assume that @xmath53 satisfies the integral conditions ( [ eq : first_int_cond ] ) , ( [ eq : second_int_cond ] ) . then @xmath54 coincides with a trajectory of the semigroup @xmath55 . the proof of this theorem is postponed to sections [ sec : h - riemann_solver ] and [ sec : uniqueness ] , where existence and uniqueness are proved . before these technical details , we state the application of the above result to gas flow in section [ application ] . here we apply theorem [ th : introduction ] to establish the existence and uniqueness of the semigroup related to pipes with discontinuous cross sections . furthermore , we show that our approach yields the same semigroup as the approach followed in @xcite in the special case of two connected pipes . the technical details of section [ application ] can be found at the end of the paper in section [ section5 ] . theorem [ th : introduction ] provides an existence and uniqueness result for pipes with lipschitz continuous cross section where the equations governing the gas flow are given by @xmath56 here , as usual , @xmath57 denotes the mass density , @xmath58 the linear momentum , @xmath59 is the energy density , @xmath9 is the area of the cross section of the pipe and @xmath60 is the pressure which is related to the conserved quantities @xmath61 by the equations of state . in most situations , when two pipes of different size have to be connected , the length @xmath62 of the adaptor is small compared to the length of the pipes . therefore it is convenient to model these connections as pipes with a jump in the cross sectional area . these discontinuous cross sections however do not fulfill the requirements of theorem [ th : introduction ] . nevertheless , we can use this theorem to derive the existence of solutions to the discontinuous problem by a limit procedure . to this end , we approximate the discontinuous function @xmath63 by a sequence @xmath64 with the following properties @xmath65\\ a^+ , & x>\frac{l}2 \end{array}\right.\ ] ] where @xmath66 is any smooth monotone function which connects the two strictly positive constants @xmath67 , @xmath68 . one possible choice of the approximations @xmath69 as well as the discontinuous pipe with cross section @xmath9 are shown in figure [ fig:1 ] . with the help of theorem [ th : introduction ] and the techniques used in its proof , we are now able to derive the following theorem ( see also @xcite for a similar result obtained with different methods ) . [ theorem2 ] if @xmath70 is sufficiently small , the semigroups @xmath71 related with the smooth section @xmath69 converge to a unique semigroup @xmath55 . the limit semigroup satisfies and is uniquely identified by the integral estimates ( [ eq : first_int_cond ] ) , ( [ eq : second_int_cond ] ) with @xmath72 substituted by @xmath73 ( see section [ section5 ] ) for the point @xmath74 . more precisely let @xmath51\rightarrow\cd$ ] be lipschitz continuous as a map with values in @xmath52 and assume that @xmath53 satisfies the integral conditions ( [ eq : first_int_cond ] ) , ( [ eq : second_int_cond ] ) with @xmath72 substituted by @xmath73 for the point @xmath74 . then @xmath54 coincides with a trajectory of the semigroup @xmath55 . observe that the same theorem holds for the @xmath75 isentropic system ( see section [ section5 ] ) @xmath76 in @xcite @xmath75 homogeneous conservation laws at a junction are considered for given admissible junction conditions . the situation of a junctions with only two pipes with different cross sections can be modeled by our limit procedure or as in @xcite with a suitable junction condition . if we define the function @xmath77 which describes the junction conditions as @xmath78 then it fulfills the determinant condition in ( * ? ? ? * proposition 2.2 ) since it satisfies lemma [ lemma : riemann_prob0 ] . here @xmath79 is the solution of the ordinary differential equation ( [ ode ] ) in section [ section5 ] . with these junction conditions one can show that the semigroup obtained in @xcite satisfies the same integral estimate ( see the following proposition ) as our limit semigroup hence they coincide . [ prop1 ] the semigroup defined in @xcite with the junction condition given by ( [ dynamics ] ) satisfies the integral estimates ( [ eq : first_int_cond ] ) , ( [ eq : second_int_cond ] ) with @xmath72 substituted by @xmath73 for the point @xmath74 . the proof is postponed to section [ section5 ] . note that proposition 1 justifies the coupling condition ( [ eq : psieq ] ) as well as the condition used in @xcite to study the riemann problem for the gas flow through a nozzle . throughout the next two sections , we follow the structure of @xcite . we recall some definitions and notations in there , and also the results which do not depend on the @xmath5 boundedness of the source term . we will prove only the results which in @xcite do depend on the @xmath5 bound using our weaker hypotheses . consider the stationary equations associated to ( [ integrable : eq : bal ] ) , namely the system of ordinary differential equations : f(v(x))_x = g(x , v(x ) ) . [ eq : ode ] for any @xmath80 , @xmath81 , consider the initial data v(x_o ) = v. [ eq : init - data ] as in @xcite , we introduce a suitable approximation of the solutions to ( [ eq : ode ] ) , ( [ eq : init - data ] ) . thanks to ( [ hyp : non_resonance ] ) , the map @xmath82 is invertible inside some neighborhood of the origin ; in this neighborhood , for small @xmath83 , we can define _ h(x_o , u)= f^-1 . [ def : phi ] this map gives an approximation of the flow of ( [ eq : ode ] ) in the sense that f(_h(x_o , u))-f(u)=_0^hg(x_o+s , u)ds . [ approx_ode ] throughout the paper we will use the landau notation @xmath84 to indicate any function whose absolute value remains uniformly bounded , the bound depending only on @xmath85 and @xmath86 . [ lemma0 ] the function @xmath87 defined in ( [ def : phi ] ) satisfies the following uniform ( with respect to @xmath88 and to @xmath27 in a suitable neighborhood of the origin ) estimates . @xmath89 the lipschitz continuity of @xmath90 and ( [ epsilonh ] ) imply @xmath91 next we compute @xmath92\\ & \qquad\cdot \left(df(u)+\int_0^{h}d_ug(x_o+s , u)\;ds\right ) \end{split}\ ] ] which together to the identity @xmath93 implies @xmath94-df^{-1}\left(f(u)\right)\right\|\\ & \qquad\qquad\cdot\left(\|df(u)\|+\int_0^{h}\|d_ug(x_o+s , u)\|\;ds\right)\\ & \qquad + \left\|df^{-1}\left(f(u)\right)\right\|\cdot\int_0^{h}\|d_ug(x_o+s , u)\|\;ds\\ & \le \o(1)\tilde\varepsilon_h\xrightarrow{h\to 0}0 . \end{split}\ ] ] finally , denoting with @xmath95 the partial derivative with respect to the @xmath96 component of the state vector and by @xmath97 the @xmath98 component of the vector @xmath99 , we derive @xmath100 so that @xmath101 for any @xmath80 we consider the system ( [ integrable : eq : bal ] ) , endowed with a riemann initial datum : u(0,x)= \ { ll u _ & if x < x_o + u_r & if x > x_o . . [ eq : riemann_data ] if the two states @xmath102 , @xmath103 are sufficiently close , let @xmath77 be the unique entropic homogeneous riemann solver given by the map u_r = ( )(u _ ) = _ n(_n) _1(_1)(u _ ) , where @xmath104 denotes the ( signed ) wave strengths vector in @xmath105 , @xcite . here @xmath106 , @xmath107 is the shock rarefaction curve of the @xmath108 family , parametrized as in @xcite and related to the homogeneous system of conservation laws u_t+f(u)_x=0 . [ eq : homo ] observe that , due to ( [ hyp : non_resonance ] ) , all the simple waves appearing in the solution of ( [ eq : homo ] ) , ( [ eq : riemann_data ] ) propagate with _ non - zero _ speed . to take into account the effects of the source term , we consider a stationary discontinuity across the line @xmath109 , that is , a wave whose speed is equal to 0 , the so called . now , given @xmath83 , we say that the particular riemann solution : u(t , x)= \ { ll u _ & if x < x_o + u_r & if x > x_o . . t0 [ zero - wave ] is admissible if and only if @xmath110 , where @xmath99 is the map defined in ( [ def : phi ] ) . roughly speaking , we require @xmath102 , @xmath103 to be ( approximately ) connected by a solution of the stationary equations ( [ eq : ode ] ) . [ def : h_r_solver ] given @xmath83 suitably small , @xmath80 , we say that @xmath53 is a @xmath111riemann solver for ( [ integrable : eq : bal ] ) , ( [ hyp : non_resonance ] ) , ( [ eq : riemann_data ] ) , if the following conditions hold * there exist two states @xmath112 , @xmath113 which satisfy @xmath114 ; * on the set @xmath115 , @xmath53 coincides with the solution to the homogeneous riemann problem ( [ eq : homo ] ) with initial values @xmath102 , @xmath112 and , on the set @xmath116 , with the solution to the homogeneous riemann problem with initial values @xmath113 , @xmath103 ; * the riemann problem between @xmath102 and @xmath112 is solved only by waves with negative speed ( i.e. of the families @xmath117 ) ; * the riemann problem between @xmath113 and @xmath103 is solved only by waves with positive speed ( i.e. of the families @xmath118 ) . [ initial_estimates ] let @xmath80 and @xmath119 be three states in a suitable neighborhood of the origin . for @xmath111 suitably small , one has [ eq : useful_est ] |_h(x_o , u)-u| & = & (1)_0^h(x_o+s)ds , + [ eq : useful_est2 ] [ lemma : riemann_prob0 ] for any @xmath120 there exist @xmath121 , depending only on @xmath122 and the homogeneous system ( [ eq : homo ] ) , such that the following holds . for all maps @xmath123 satisfying @xmath124 and for all @xmath125 , @xmath126 there exist @xmath127 states @xmath128 and @xmath6 wave sizes @xmath129 , depending smoothly on @xmath130 , such that with previous notations : a. @xmath131 ; b. @xmath132 ; c. @xmath133 ; d. @xmath134 . the next lemma establishes existence and uniqueness for the @xmath111riemann solvers ( see fig.[hrs ] ) . [ lemma : riemann_prob ] there exist @xmath135 such that the following holds : for any @xmath136 , @xmath137 $ ] , @xmath102 , @xmath138 , there exists a unique @xmath111riemann solver in the sense of definition [ def : h_r_solver ] . by lemma [ lemma0 ] if @xmath139 is chosen sufficiently small then for any @xmath140 $ ] , @xmath80 the map @xmath141 meets the hypotheses of lemma [ lemma : riemann_prob0 ] . finally taking @xmath142 eventually smaller we can obtain that there exists @xmath143 such that @xmath144 , for any @xmath137 $ ] . in the sequel , @xmath145 stands for the implicit function given by lemmas [ lemma : riemann_prob0 ] and [ lemma : riemann_prob ] : @xmath146 which plays the role of a wave size vector . we recall that , by lemma [ lemma : riemann_prob0 ] , @xmath145 is a @xmath26 function with respect to the variables @xmath147 and its @xmath26 norm is bounded by a constant independent of @xmath111 and @xmath148 . in contrast with the homogeneous case , the wave size @xmath149 in the @xmath111riemann solver is not equivalent to the jump size @xmath150 ; an additional term appears coming from the dirac source term " ( see the special case @xmath151 ) . [ lemma : equiv_tot_var ] let @xmath152 be the constants in lemma [ lemma : riemann_prob ] . for @xmath153 , @xmath137 $ ] , set @xmath154 $ ] . then it holds : rcl |u_-u_r|&=&(1)(||+ ) , + [ eq : equiv_tot_var ] note that as shown in @xcite we can identify the sizes of the zero waves with the quantity = _ 0^h ( jh+s)ds . [ size_0_wave ] with this definition all the glimm interaction estimates continue to hold with constants that depend only on @xmath85 and on @xmath86 , therefore all the wave front tracking algorithm can be carried out obtaining the existence of @xmath156-approximate solutions as defined below . [ decadix ] given @xmath157 , we say that a continuous map @xmath158 is an @xmath159approximate solution of ( [ integrable : eq : bal])([init : data ] ) if the following holds : * as a function of two variables , @xmath160 is piecewise constant with discontinuities occurring along finitely many straight lines in the @xmath161 plane . only finitely many wave - front interactions occur , each involving exactly two wave - fronts , and jumps can be of four types : shocks ( or contact discontinuities ) , rarefaction waves , non - physical waves and zero - waves : @xmath162 . * along each shock ( or contact discontinuity ) @xmath163 , @xmath164 , the values of @xmath165 and @xmath166 are related by @xmath167 for some @xmath168 and some wave - strength @xmath169 . if the @xmath170 family is genuinely nonlinear , then the lax entropy admissibility condition @xmath171 also holds . moreover , one has @xmath172 where @xmath173 is the speed of the shock front ( or contact discontinuity ) prescribed by the classical rankine - hugoniot conditions . * along each rarefaction front @xmath174 , @xmath175 , one has @xmath167 , @xmath176 for some genuinely nonlinear family @xmath177 . moreover , we have : @xmath178 . * all non - physical fronts @xmath179 , @xmath180 travel at the same speed @xmath181 . their total strength remains uniformly small , namely : @xmath182 * the zero - waves are located at every point @xmath183 , @xmath184 . + along a zero - wave located at @xmath185 , @xmath186 , the values @xmath165 and @xmath166 satisfy @xmath187 for all @xmath188 except at the interaction points . * the total variation in space @xmath189 is uniformly bounded for all @xmath190 . the total variation in time @xmath191 is uniformly bounded for @xmath192 , @xmath193 . finally , we require that @xmath194 . keeping @xmath83 fixed , we are about to let first @xmath195 tend to zero . hence we shall drop the superscript @xmath111 for notational clarity . [ first - convergence ] let @xmath196 be a family of @xmath159approximate solutions of ( [ integrable : eq : bal])([init : data ] ) . there exists a subsequence @xmath197 converging as @xmath198 in @xmath199 to a function @xmath27 which satisfies for any @xmath200 : & & _ 0^_dxdt + & & + _ 0^_j(t , jh)(_0^h gds)dt = 0.[eq : hequation ] moreover @xmath201 is uniformly bounded and @xmath27 satisfies the lipschitz property @xmath202 now we are in position to prove ( * ? ? ? * theorem 4 ) with our weaker hypotheses . as in @xcite we can apply helly s compactness theorem to get a subsequence @xmath203 converging to some function @xmath27 in @xmath204 whose total variation in space is uniformly bounded for all @xmath190 . moreover , working as in ( * ? ? ? * proposition 5.1 ) , one can prove that @xmath205 converges in @xmath11 to @xmath54 , for all @xmath190 . [ th : limit_in_h ] let @xmath203 be a subsequence of solutions of equation ( [ eq : hequation ] ) with uniformly bounded total variation converging as @xmath198 in @xmath11 to some function @xmath27 . then @xmath27 is a weak solution to the cauchy problem ( [ integrable : eq : bal])([init : data ] ) . we omit the proofs of theorem [ first - convergence ] and [ th : limit_in_h ] since they are very similar to the proofs of ( * ? ? ? * theorem 3 and 4 ) . we only observe that , in those proofs , the computations which rely on the @xmath5 bound on the source term have to be substituted by the following estimates . * concerning the proof of theorem [ first - convergence ] : @xmath206 * concerning the proof of theorem [ th : limit_in_h ] : @xmath207 and @xmath208\right\}\\ & \qquad\quad+\int_{jh}^{(j+1)h}\omega(x)\left|u^h(t , x)-u(t , x)\right| . \end{split}\ ] ] we observe that all the computations done in ( * ? ? ? * section 4 ) rely on the source @xmath22 only through the amplitude of the zero waves and on the interaction estimates . therefore the following two theorems still hold in the more general setting . [ tsoin - tsoin ] there exists @xmath209 such that if @xmath210 is sufficiently small , then for any ( small ) @xmath83 there exist a non empty closed domain @xmath211 and a unique uniformly lipschitz semigroup @xmath212 whose trajectories @xmath213 solve ( [ eq : hequation ] ) and are obtained as limit of any sequence of @xmath159approximate solutions as @xmath195 tends to zero with fixed @xmath111 . in particular the semigroups @xmath214 satisfy for any @xmath215 , @xmath47 p^h_0 u_o = u_o , p^h_t p^h_su_o= p^h_s+tu_o , [ eq : semi_prop_ph ] p^h_t u_o - p^h_s v_o _ 1 ( ) l[eq : lip_prop_ph ] for some @xmath42 , independent on @xmath111 . [ th : final_existence ] if @xmath210 is sufficiently small , there exist a constant @xmath42 , a non empty closed domain @xmath43 of integrable functions with small total variation and a semigroup @xmath216 with the following properties a. @xmath217 b. @xmath218 c. for all @xmath49 , the function @xmath219 is a weak entropy solution of system ( [ integrable : eq : bal ] ) . d. for some @xmath209 and all @xmath83 small enough @xmath220 . e. there exists a sequence of semigroups @xmath221 such that @xmath222 converges in @xmath11 to @xmath223 as @xmath224 for any @xmath225 . [ newrem ] looking at ( * ? ? ? * ( 4.6 ) ) and the proof of ( * ? ? ? * theorem 7 ) one realizes that the invariant domains @xmath226 and @xmath227 depend on the particular source term @xmath228 . on the other hand estimate ( * ? ? ? * ( 4.4 ) ) shows that all these domains contain all integrable functions with sufficiently small total variation . since the bounds @xmath84 in lemma [ lemma : equiv_tot_var ] depend only on @xmath85 and on @xmath86 , also the constant @xmath229 in ( * ( 4.4 ) ) depends only on @xmath85 and on @xmath86 . therefore there exists @xmath230 depending only on @xmath85 and on @xmath86 such that @xmath226 and @xmath227 contain all integrable functions @xmath231 with @xmath232 . the proof of uniqueness in @xcite strongly depends on the boundedness of the source , therefore we have to consider it in a more careful way . as in @xcite we shall make use of the following technical lemmas whose proofs can be found in @xcite . [ lip_on_intervals ] let @xmath233 a ( possibly unbounded ) open interval , and let @xmath234 be an upper bound for all wave speeds . if @xmath235 then for all @xmath190 and @xmath83 , one has [ eq : lip_on_intervals ] _ a+t^b - t |(p^h_t|u)(x)-(p^h_t|v)(x)|dx l_a^b ||u(x)-|v(x)|dx . [ distance_on_intervals ] given any interval @xmath236 $ ] , define the interval of determinacy [ eq : int_of_det ] i_t=[a+t , b - t],t < . for every lipschitz continuous map @xmath237\mapsto\cd_h(\d)$ ] and @xmath238 : [ eq : distance_on_intervals ] & & w(t)-p^h_tw(0)_1(i_t ) + & & l_0^t \{_0 } ds . [ rem : salvezza ] lemmas [ lip_on_intervals ] , [ distance_on_intervals ] hold also substituting @xmath214 with the operator @xmath55 . in this case we have obviously to substitute the domains @xmath211 with the domain @xmath43 of theorem [ th : final_existence ] . let now @xmath147 be two nearby states and @xmath239 ; we consider the function [ eq : def_di_v ] v(t , x)=\ { lll u_&&x < t+x_o + u_r&&xt+x_o . . [ lemma_stime_integrali_omogenee ] call @xmath240 the self - similar solution given by the standard homogeneous riemann solver with the riemann data ( [ eq : riemann_data ] ) . 1 . in the general case , one has [ eq : stima_senza_ipotesi ] 1 t_-^+ 2 . assuming the additional relations @xmath241 and @xmath242 for some @xmath243 , @xmath244 one has the sharper estimate [ eq : stima_rarefazioni ] 1 t_-^+ |v(t , x)-w(t , x)|dx = (1)^2 ; 3 . let @xmath245 and call @xmath246 the eigenvalues of the matrix @xmath247 . if for some @xmath96 it holds @xmath248 and @xmath249 in ( [ eq : def_di_v ] ) then one has [ item : stima_per_lin ] 1 t_-^+ |v(t , x)-w(t , x)|dx = (1)|u_-u_r| ( we now prove the next result which is directly related to our @xmath111-riemann solver . [ lemma_stime_integrali_sorgente ] call @xmath240 the self - similar solution given by the @xmath111riemann solver in @xmath148 with the riemann data ( [ eq : riemann_data ] ) . 1 . [ item : hstima_senza_ipotesi ] in the general case one has [ eq : hstima_senza_ipotesi ] 1 t_-^+ 2 . [ item : hstima_per_lin ] assuming the additional relation @xmath250^{-1}(u^*)\displaystyle{\int_{0}^{h}}g(x_o+s , u^*)ds\ ] ] + with @xmath251 in ( [ eq : def_di_v ] ) one has the sharper estimate & & 1 t_-^+ |v(t , x)-w(t , x)|dx + & = & (1 ) ( _ 0^h(x_o+s)ds+|u_- u^*| ) _ 0^h(x_o+s)ds . [ eq : hstima_per_lin ] _ proof . _ estimate _ ( [ item : hstima_senza_ipotesi ] ) _ is a direct consequence of lemma [ lemma : equiv_tot_var ] . let us prove now _ ( [ item : hstima_per_lin])_. since @xmath252 we derive & & 1 t_-^+ |v(t , x)-w(t , x)|dx + & = & 1 t_-t^0| u_-w(t , x)|dx + 1 t_0^t| u_r - w(t , x)|dx + & = & (1)=(1)|| . this leads to ||&=&|e- e| + & = & (1)|u_r-_h(x_o , u_)| . to estimate this last term , we define @xmath253 and compute for some @xmath254 : |u_+^-1(u^*)y_1-b(y_2,u _ ) | & & (1)|y_1||u^*-u_|+(1)|y_1-y_2| + & & + |u_+^-1(u_)y_2 -b(y_2,u_)| . the function @xmath255^{-1}(u_\ell)y_2 -b\left(y_2,u_\ell\right)$ ] satisfies @xmath256 , @xmath257 , hence we have the estimate substitute @xmath258 then , we get @xmath259 which proves ( [ eq : hstima_per_lin ] ) . in this section we are about to give necessary and sufficient conditions for a function @xmath260 to coincide with a semigroup s trajectory . to this end , we prove the uniqueness of the semigroup @xmath55 and the convergence of all the sequence of semigroups @xmath214 towards @xmath55 as @xmath261 . we begin by introducing some notations : given a @xmath262 function @xmath263 and a point @xmath264 , we denote by @xmath265 the solution of the homogeneous riemann problem ( [ eq : riemann_data ] ) with data u_=_x - u(x),u_r=_x+u(x),x_o=. [ eq : def_riemann_data ] moreover we define @xmath266 as the solution of the linear hyperbolic cauchy problem with constant coefficients w_t+aw_x = g(x ) , w(0,x)=u(x ) , [ eq : def_linear_problem ] with @xmath267 , @xmath268 . we will need also the following approximations of @xmath266 . let @xmath269 be a piecewise constant function . we will call @xmath270 the solution of the following cauchy problem : @xmath271 define @xmath272 and let @xmath273 , @xmath274 , @xmath275 be respectively the @xmath276 eigenvalue , the @xmath276 right / left eigenvectors of the matrix @xmath277 . as in @xcite @xmath278 and @xmath270 have the following explicit representation @xmath279 where the function @xmath280 is defined by [ eq : definition_gh ] g_i^h(t , x)= \ { lll & _ i>0 + & _ i<0 . . using ( [ epsilonh ] ) we can compute |g_i^h(t , x)-_x-_it^x l_i , g(x)dx| = (1)_h . hence , for any @xmath281 with @xmath282 , we have the error estimate @xmath283.\ ] ] from ( [ eq : happrox_lin_sol ] ) , ( [ eq : definition_gh ] ) , it is easy to see that @xmath284 is piecewise constant with discontinuities occurring along finitely many lines on compact sets in the @xmath285 plane for @xmath190 . only finitely many wave front interactions occur in a compact set , and jumps can be of two types : contact discontinuities or zero waves . the zero waves are located at the points @xmath286 , @xmath287 and satisfy w^h(t , jh+)-w^h(t , jh-)=^-1(u^ * ) _ jh^(j+1)hg(jh+s)ds . [ eq : abra ] conversely a contact discontinuity of the @xmath276 family located at the point @xmath288 satisfies @xmath289 and w^h(t , x_(t)+)-w^h(t , x_(t)-)=r_i(u^ * ) [ eq : cadabra ] for some @xmath290 . now , we can state the uniqeness result in our more general setting . [ th : characterisation ] let @xmath291 be the semigroup of theorem [ th : final_existence ] and let @xmath234 be an upper bound for all wave speeds . then every trajectory @xmath292 , @xmath293 , satisfies the following conditions at every @xmath294 . 1 . [ first_int_cond ] for every @xmath295 , one has [ eq : first_int_cond ] _ 0 1 _ -^+ |u(+,x)-u^_(u();)(,x)|dx = 0 . 2 . [ second_int_cond ] there exists a constant @xmath296 such that , for every @xmath297 and @xmath298 , one has [ eq : second_int_cond ] 1 _ a+^b- |u(+,x)-u^_(u ( ) ; ) ( , x)| dx + c^2 . viceversa let @xmath51\rightarrow\cd$ ] be lipschitz continuous as a map with values in @xmath52 and assume that the conditions ( [ first_int_cond ] ) , ( [ second_int_cond ] ) hold at almost every time @xmath299 . then @xmath54 coincides with a trajectory of the semigroup @xmath55 . [ rem : recall ] the difference with respect to the result in @xcite is the presence of the integral in the right hand side of formula ( [ eq : second_int_cond ] ) . if @xmath35 is in @xmath5 , the integral can be bounded by @xmath300 and we recover the estimates in @xcite . note also that the quantity @xmath301 is a uniformly bounded finite measure and this is what is needed for proving the sufficiency part of the above theorem . * part 1 : necessity * given a semigroup trajectory @xmath302 , @xmath303 we now show that the conditions _ ( [ first_int_cond ] ) _ , _ ( [ second_int_cond ] ) _ hold for every @xmath304 . as in @xcite we use the following notations . for fixed @xmath305 we define @xmath306 with j_t^-&=&(-(2-t+),-(t- ) ) ; + [ intervals ] j_t^o&= & ; + j_t^+&=&(+(t-),+(2-t+ ) ) . let @xmath307 be the piecewise constant function obtained from @xmath308 dividing the centered rarefaction waves in equal parts and replacing them by rarefaction fans containing wave fronts whose strength is less than @xmath309 . observe that : 1 t_-^+|u^,_(u();)(,x ) -u^_(u();)(,x)|dx = (1 ) . [ eq : compare_to_eps ] applying estimate ( [ eq : distance_on_intervals ] ) to the function @xmath310 we obtain [ eq : stimapiu ] & & _ j_+| u^,_(u();)(,x)-(p^h_u^,_(u();)(0))(x ) |dx + & & l _ ^+_0 dt . the discontinuities of @xmath310 do not cross the dirac comb for almost all times @xmath311 . therefore we compute for such a time @xmath312 : [ eq : uno_asterisco ] & & 1 _ j_t+ & = & 1 _ j^-_t+j^o_t+j^+_t+ |u^,_(u();)(t-+,x)-(p_^hu^,_(u ( ) ; ) ( t- ) ) ( x)|dx . define @xmath313 the set of points in which @xmath314 has a discontinuity while @xmath315 is the set of points in which the zero waves are located . if @xmath316 is sufficiently small , the solutions of the riemann problems arising at the discontinuities of @xmath314 do not interact , therefore @xmath317 note that the shock are solved exactly both in @xmath310 and in @xmath318 therefore they make no contribution in the summation . to estimate the approximate rarefactions we use the estimate ( [ eq : stima_rarefazioni ] ) hence @xmath319 concerning the zero waves , recall that @xmath312 is chosen such that @xmath310 is constant there , and @xmath214 is the exact solution of an @xmath111riemann problem , hence we can apply ( [ eq : hstima_senza_ipotesi ] ) with @xmath151 and obtain @xmath320 finally using ( [ eq:51 ] ) and ( [ eq:49 ] ) we get in the end [ eq : due_asterisco ] 1 _ j^o_t+ |u^,_(u();)(t-+,x)-(p_^hu^,_(u ( ) ; ) ( t- ) ) ( x)|dx + = (1)\{_j^0_t(x)dx+_h+}. moreover , following the same steps as before and using ( [ eq : stima_senza_ipotesi ] ) and ( [ eq : hstima_senza_ipotesi ] ) with @xmath151 we get [ eq : tre_asterisco ] 1 _ j^+_t+ = (1 ) \{_j^+_t(x)dx+_h } . note that here there is no total variation of @xmath310 since in @xmath321 it is constant . a similar estimate holds for the interval @xmath322 . putting together ( [ eq : uno_asterisco ] ) , ( [ eq : due_asterisco ] ) , ( [ eq : tre_asterisco ] ) , one has 1 _ j_t+ |u^,_(u();)(t-+,x)-(p_^h u^,_(u();)(t- ) ) ( x)|dx + = (1 ) ( _ j_(x)dx+_h+ ) . hence , setting @xmath323 by ( [ eq : stimapiu ] ) , we have [ eq : stimameno ] _ j_+| u^,_(u();)(,x)-(p^h_v)(x ) |dx = (1 ) ( _ j_(x)dx+_h+ ) . finally we take the sequence @xmath221 converging to @xmath55 . using ( [ eq : lip_on_intervals ] ) we have [ after51 ] p^h_i_u()-p^h_i_v _ 1(j_+ ) & & lu()-v _ 1(j _ ) + & = & _ -2^+2|u(,x)-v(x)|dx + & = & | _ , where @xmath324 tends to zero as @xmath325 tends to zero due to the fact that @xmath326 has right and left limit at any point : for any given @xmath327 if @xmath325 is sufficiently small @xmath328 for @xmath329 . therefore by ( [ eq : compare_to_eps ] ) , ( [ eq : stimameno ] ) , we derive : & & 1 _ -^+ |u(+,x)-u^_(u();)(,x)|dx + & = & + |_+(1 ) . the left hand side of the previous estimate does not depend on @xmath309 and @xmath330 , hence 1 _ -^+ @xmath331 depend on @xmath325 ( see [ intervals ] ) . so taking the limit as @xmath332 in the previous estimate yields ( [ eq : first_int_cond ] ) . to prove _ ( [ second_int_cond ] ) _ let @xmath333 and a point @xmath334 be given together with an open interval @xmath233 containing @xmath295 . fix @xmath335 and choose a piecewise constant function @xmath336 satisfying @xmath337 together with @xmath338 let now @xmath270 be defined by ( [ eq : happrox_lin_sol ] ) ( @xmath339 ) . from ( [ eq : distance_between_w_wh ] ) , ( [ eq : second_totvar_prop ] ) we have the estimate [ eq : propp ] _ a+^b- |u^_(u();)(,x)- w^h(,x ) |dx(1)(+_h(b - a ) ) . using ( [ eq : int_of_det ] ) , ( [ eq : distance_on_intervals ] ) we get [ eq : prip ] & & _ a+^b- |w^h(,x)-(p_^h w^h(0))(x ) |dx + & & l _ ^+_0 dt where we have defined @xmath340 . let @xmath341 be a time for which there is no interaction in @xmath270 ; in particular , discontinuities which travel with a non - zero velocity do not cross the dirac comb ( this happens for almost all @xmath312 ) . we observe that by the explicit formula ( [ eq : happrox_lin_sol ] ) : @xmath342 @xmath343 as before for @xmath316 sufficiently small we can split homogeneous and zero waves @xmath344 the homogeneous waves in @xmath270 satisfy ( [ eq : cadabra ] ) , with @xmath345 in place of @xmath346 , hence we can apply ( [ item : stima_per_lin ] ) which together with ( [ eq:3;35 ] ) , ( [ eq:3;36 ] ) leads to @xmath347 where @xmath348 denotes the jump of @xmath349 at @xmath24 . the zero waves in @xmath270 satisfy ( [ eq : abra ] ) , hence we can apply ( [ eq : hstima_per_lin ] ) which together with ( [ eq:3;36 ] ) leads to @xmath350 let now @xmath221 be the subsequence converging to @xmath55 . since @xmath351 using ( [ eq : propp ] ) , ( [ eq : prip ] ) , ( [ eq : second_totvar_prop ] ) , and the last estimates we get & & 1 _ a+^b- |u(+,x)-u^_(u();)(,x)| dx + & & + l + & & + (1 ) \{+ ( \{|v ; ( a , b ) } + _ a^b(x)dx+_h_i)^2}. so for @xmath352 we obtain the desired inequality . + * part 2 : sufficiency * by remark [ rem : salvezza ] we can apply ( [ eq : distance_on_intervals ] ) to @xmath55 and hence the proof for the homogeneous case presented in @xcite , which relies on the property recalled in remark [ rem : recall ] , can be followed exactly for our case , hence it will be not repeated here . @xmath353 [ [ proof - of - theorem - thintroduction ] ] proof of theorem [ th : introduction ] + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + it is now a direct consequence of theorems [ th : final_existence ] and [ th : characterisation ] . consider the equation @xmath354 for some @xmath355 . equation ( [ dynamics ] ) is comprised in this setting with the substitution @xmath356 . for this kind of equations we consider the exact stationary solutions instead of approximated ones as in ( [ def : phi ] ) . therfore call @xmath357 the solution of the following cauchy problem : @xmath358^{-1}g(u(a))\\ u(0)=\bar u \end{array } \right.\ ] ] if @xmath9 is sufficiently small , the map @xmath359 satisfies lemma [ lemma : riemann_prob0 ] . we call @xmath9-riemann problem the cauchy problem @xmath360 its solution will be the function described in definition [ def : h_r_solver ] using the map @xmath361 instead of the @xmath99 in there . observe that if @xmath362 the @xmath9-riemann solver coincides with the usual homogeneous riemann solver . since @xmath70 , hypothesis @xmath34 is satisfied uniformly with respect to @xmath62 , moreover the smallness of @xmath367 ensures that the @xmath11 norm of @xmath35 in @xmath40 is small . therefore the hypotheses of theorem [ th : introduction ] are satisfied uniformly with respect to @xmath62 . let @xmath71 be the semigroup related with the smooth section @xmath69 . by remark [ newrem ] , if @xmath368 is sufficiently small , @xmath27 belongs to the domain of @xmath71 for every @xmath369 . since the total variation of @xmath370 is uniformly bounded for a fixed initial data @xmath27 , helly s theorem guarantees that there is a converging subsequence @xmath371 . by a diagonal argument one can show that there is a converging subsequence of semigroups converging to a limit semigroup @xmath55 defined on an invariant domain ( see ( * ? ? ? * proof of theorem 7 ) ) . to zero . we will estimate ( [ eq : first_int_condb ] ) in several steps . first define @xmath372 and compute @xmath373 as in ( [ after51 ] ) . then we consider the approximating sequence @xmath374 corresponding to the source term @xmath375 and the semigroups @xmath376 which converge to @xmath374 in the sense of theorem [ th : final_existence ] . hence we have @xmath377 for notational convenience we skip the subscript @xmath96 in @xmath378 . as in ( [ eq : compare_to_eps ] ) we approximate rarefactions in @xmath379 introducing the function @xmath380 . then we define ( see figure [ fig : p ] ) @xmath381 where @xmath382 is piecewise constant with jumps in the points @xmath286 satisfying @xmath383 . furthermore @xmath384 and @xmath385 is defined as in ( [ def : phi ] ) using the source term @xmath386 . observe that the jump between @xmath387 and @xmath388 does not satisfy any jump condition , but as @xmath382 is an `` euler '' approximation of the ordinary differential equation @xmath389 , this jump is of order @xmath390 . since @xmath380 and @xmath391 have uniformly bounded total variation we have the estimate @xmath392 the bound @xmath84 not depending on @xmath111 . we apply lemma [ distance_on_intervals ] on the remaining term @xmath393 to estimate this last term we proceed as before . observe that @xmath394 does not have zero waves outside the interval @xmath395 $ ] since outside the interval @xmath396 $ ] the function @xmath397 is identically zero . if @xmath316 is small enough , the waves in @xmath398 do not interact , therefore the computation of the @xmath11 norm in the previous integral , as before can be splitted in a summation on the points in which there are zero waves in @xmath394 or jumps in @xmath399 . observe that the jumps of @xmath400 in the interval @xmath401 , are defined exactly as the zero waves in @xmath394 so we have no contribution to the summation from this interval . outside the interval @xmath395 $ ] , @xmath214 coincides with the homogeneous semigroup , hence we have only the second order contribution from the approximate rarefactions in @xmath399 as in ( [ eq:49 ] ) . furthermore we might have a zero wave in the interval @xmath402 $ ] and a discontinuity of @xmath403 in the point @xmath404 of order @xmath390 . using ( [ eq : hstima_senza_ipotesi ] ) for the zero wave and ( [ eq : stima_senza_ipotesi ] ) for the discontinuity ( since @xmath214 is equal to the homogeneous semigroup in @xmath404 ) , we get @xmath405 which completes the proof if we let first @xmath406 tend to zero , then @xmath111 tend to zero , then @xmath62 tend to zero and finally @xmath325 tend to zero . as in the previous proof , the sufficiency part can be obtained following the proof for the homogeneous case presented in @xcite . call @xmath407 the semigroup defined in @xcite . the estimates for this semigroup outside the origin are equal to the ones for the standard riemann semigroup see @xcite . concerning the origin we first observe that the choice ( [ eq : psieq ] ) implies that the solution to the riemann problem in ( * ? ? ? * proposition 2.2 ) coincides with @xmath408 . we need to show that [ eq : first_int_condbappo ] _ 01 _ -^+ |u(+,x)-|u^_u()(,x)|dx=0 . with @xmath409 . as before , we first approximate @xmath408 with @xmath380 and @xmath326 with @xmath410 then we apply lemma [ distance_on_intervals ] ( which holds also for the semigroup @xmath407 ) and compute & & 1 _ -^+ |(s_|v)(x)-|u^,_u()(,x)|dx + & & l1 _ ^+_0 the discontinuities of @xmath380 are solved by @xmath411 with exact shock or rarefaction for @xmath412 and with the @xmath9riemann solver in @xmath413 therefore the only difference between @xmath414 and @xmath415 are the rarefactions solved in an approximate way in the first function and in an exact way in the second . recalling ( [ eq : stima_rarefazioni ] ) we know that this error is of second order in the size of the rarefactions . + to show that ( [ eq : first_int_condbappo ] ) holds , proceed as in ( [ eq:49 ] ) . rinaldo m. colombo and mauro garavello . on the @xmath60-system at a junction . in _ control methods in pde - dynamical systems _ , volume 426 of _ contemp . _ , pages 193217 . soc . , providence , ri , 2007 .
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we consider the cauchy problem for a @xmath0 strictly hyperbolic system of balance laws @xmath1 each characteristic field being genuinely nonlinear or linearly degenerate . assuming that the @xmath2 norm of @xmath3 and @xmath4 are small enough
, we prove the existence and uniqueness of global entropy solutions of bounded total variation extending the result in @xcite to unbounded ( in @xmath5 ) sources .
furthermore , we apply this result to the fluid flow in a pipe with discontinuous cross sectional area , showing existence and uniqueness of the underlying semigroup . _
2000 mathematics subject classification : _
35l65 , 35l45 , 35l60 .
_ keywords : _ hyperbolic balance laws , unbounded sources , pipes with discontinuous cross sections .
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the use of symmetry methods to understand bound states has an ancient origin and has given many of its fruits in atomic and subatomic physics - the work of marcos moshinsky is an excellent example of this @xcite . over the years , however , other approaches to understand more complicated systems have given up symmetry in both classical and quantum mechanics @xcite , @xcite . here we present one of the simplest ways of destroying continuous symmetries with purely quantum - mechanical consequences : a corner . we will show that a sharply bent waveguide supports a finite number of bound states whose nature has no classical counterpart . due to the geometry , the traditional methods of integrability can not be used to obtain the corresponding solutions . this type of systems has been studied in the past , either with mathematical tools @xcite proving the existence of bound states or by employing numerical methods for computing the corresponding energies and eigenfunctions @xcite . the approach in our considerations will be completely analytical , albeit we use approximations to perform computations . we want to show two different approaches to the problem in order to exhibit an apparent complexity of the description given in other studies , mostly numerical @xcite . first we shall deal with a convenient basis of states which shows the binding as a consequence of the presence of corners : a sharp obstacle diffracts waves of an arbitrary wavelength . some limitations of this method , such as the coupling of an infinite number of modes , will be pointed out . finally , the use of conformal coordinates will be introduced as a natural description to find all the bound states for arbitrary bending angles . we would like to emphasize that interesting applications may come in the form of bent wave guides carved in crystals , as suggested in @xcite , using methods connected to the evolution of airy profiles - a recent example can be found in @xcite . the landscape looks promising , considering the many applications of controlling , bending and focusing waves . we show how to obtain a one dimensional schroedinger equation with an effective potential along the longitudinal direction of the guide . one of the challenges for describing our system is to find a suitable set of states . a natural way to introduce box eigenmodes in our system is by using oblique modes , as shown in the figure . . the slope @xmath0 is related to the bending angle @xmath1 . we show an oblique mode with @xmath2 [ fig1 ] let us take units such that @xmath3 with @xmath4 the mass of the particle and write the stationary schroedinger equation in the form ^2_x , y + e = 0 [ 1 ] for the configuration in figure [ fig1 ] with dirichlet conditions , we express the solution in the oblique basis as ( x , y ) = _ n=1^ _ n(x ) ( ( y - a|x|+l ) ) [ 2 ] and @xmath5 for @xmath6 . the matrix elements of the schroedinger operator in this basis can be obtained by using the expansion ( [ 1 ] ) and integrating over the @xmath7 variable . let us denote the vector wave function by @xmath8 . using the following definition for matrix elements _ nm = _ nm n , _ mn = ( 1-(-)^m+n ) [ 7 ] and employing a gauge transformation given by \(x ) = ( - ) ( x ) + ( x ) = ( - ) ( ) [ 8 ] we reduce the schroedinger equation to ( x)=0 [ 9 ] where we see an effective potential of the form v_eff = ( ) ^2 ( a^2 + 1 ) ( x)^2 + ( ) ^2 ^2 . [ 10 ] this matrix couples , in principle , all components of @xmath9 . the origin of @xmath10 and @xmath11 is related to the momentum and the energy of a particle in a box . the @xmath12 dependence of the potential @xmath13 comes completely from the non - commutability of @xmath14 and @xmath10 and is non - differentiable at the corner due to @xmath15 . another important feature is that a change of units @xmath16 in equation ( [ 9 ] ) leads to a scale - free equation , thus proving that the energy @xmath17 must scale as @xmath18 . therefore , the presence of bound states in this problem is independent of the scale . this is in contrast with other bent systems with smooth boundaries @xcite . in previous works by wheeler , bestle and schleich @xcite , the wkb approximation was corrected in the presence of potentials with corners . as a result , the schroedinger equation with a potential could be approximated by a wkb equation with a source term . by neglecting completely the effect of the potential except at the non - differentiable point , the wkb equation with a source could be interpreted as a free schroedinger equation with a @xmath19 potential . in our case , this would cause a binding effect depending on the energy - dependent amplitude of the @xmath19 . we shall apply this idea in order to cope with @xmath13 obtained in the last section , with the additional feature that @xmath13 is a matrix potential . we start with the schroedinger equation + ^-2(x ) ( x ) = 0 [ 1.1 ] where @xmath20 is now a matrix . the corner - corrected wkb formalism given in @xcite can be easily generalized to matrix potentials . a wkb scattering equation ( such as eq ( 13 ) in @xcite ) can be obtained for ( [ 1.1 ] ) . the resulting equation , in turn , can be treated by keeping first order terms in @xmath0 and considering a negligible potential except at @xmath21 . this gives the isolated effect of the corner in the form + ( e- ^2 ) ( x ) = _ sc ( x ) + _ sc - [ ^2 , ] ( e- ^2 ) ^-1 ( x ) . [ 1.12 ] the energies for bound states in this problem are obtained by i ) imposing the appropriate boundary conditions @xmath22 at @xmath23 , ii ) substituting the corresponding solutions in the condition for the jump in the derivative ( due to @xmath19 ) , namely _ 0 + - _ 0- = - [ ^2 , ] ( e- ^2 ) ^-1 _ 0 . [ 1.13 ] the solutions are found to be ( x)= ( ^2 - e ) ^-1/4 ( - |x| ) _ 0 [ 1.14 ] with @xmath24 a normalization factor . the vector @xmath25 is determined by the condition coming from the jump in the derivative ( [ 1.13 ] ) , namely ^-1 _ 0 = 0 [ 1.15 ] this is a linear equation for @xmath25 and the wave function can be obtained by choosing @xmath25 as a member of the null space of the operator in ( [ 1.15 ] ) . the scaled energies @xmath26 are obtained from ( [ 1.15 ] ) by using the determinant ( [ ^2 , ] + 2 ^3/2 ) = 0 [ 1.16 ] the @xmath27 case can be solved easily by inserting in ( [ 1.16 ] ) the diagonal matrix @xmath28 . the resulting energy for the ground state is e_0 = ( ) ^2 ( - ) [ 1.18 ] this energy is below the threshold @xmath29 . the level @xmath30 can also be obtained ; it is above @xmath31 . therefore , in this @xmath27 approximation , the only state which decays exponentially with the distance is the ground state . the wave function can be found straightforwardly once @xmath32 is obtained . we show two cases in figures [ fig2 ] [ fig2bis ] . the solutions thus obtained show that a bound state develops due to the corner alone , producing a @xmath19 potential and coupling the oblique modes - such coupling is essential , since a @xmath33 approximation makes the interaction vanish . yet , we may ask : where are the excited states below threshold due to an increasing value of @xmath0 ? in the following , we analyze this problem using conformal coordinates as a way to obtain a suitable basis . such a basis can not be obtained by means of perturbation theory in @xmath0 using the solutions of this section . . a peak in the relevant region is visible . the decay with distance is exponential , title="fig : " ] . a peak in the relevant region is visible . the decay with distance is exponential , title="fig : " ] . the peak becomes more pronounced as @xmath0 increases . for large values of @xmath0 , the description is no longer valid.,title="fig : " ] . the peak becomes more pronounced as @xmath0 increases . for large values of @xmath0 , the description is no longer valid.,title="fig : " ] our aim now is to show that a non - trivial boundary value problem such as the bent waveguide can be mapped smoothly to a straight guide with an effective potential ( equivalently , a problem with position - dependent mass ) . the simplest way to describe this process is by using complex variables . denote @xmath34 , @xmath35 and consider a transformation such that @xmath36 . we impose the following conditions on @xmath37 : \1 ) the image of the bent waveguide is an infinite straight strip described by @xmath38 . \2 ) the laplacian @xmath39 is transformed into a quadratic form without cross - terms ( the matrix @xmath10 will not appear ) . \3 ) the transformation is smooth except at the points @xmath40 ( upper corner ) and @xmath41 ( lower corner ) in figure [ map2 ] . the condition 1 ) allows the use of a complete set of functions @xmath42 for the dirichlet problem and @xmath43 for the neumann problem , given that the contour lines meet the boundaries orthogonally . the condition 2 ) precludes the appearance of terms proportional to @xmath44 : the integral @xmath45 would produce unbounded matrix elements due to well - known properties of the momentum operator for a particle in a one - dimensional box . requirement 3 ) ensures that any suspicious behavior of the transformation is due to the corners . . we denote the lower corner by @xmath41 and place the upper corner at the origin @xmath40 , see the text . ] the requirements described above are fulfilled by choosing @xmath37 analytic for all the points in the interior of the waveguide . orthogonality is reached by the cauchy - riemann conditions . we introduce the operators _ z = _ x - i _ y , _ z^ * = _ x + i _ y [ 2.1 ] _ = _ u - i _ v , _ ^ * = _ u + i _ v [ 2.2 ] obtaining thus @xmath46 , @xmath47 . using the analytic function @xmath36 , the laplacian operator is transformed as @xmath48 . using the cauchy - riemann conditions we have @xmath49 , the jacobian of the transformation as a function of @xmath50 . our boundary value problem with the schroedinger equation becomes ( _ _ ^ * + ||^2 e ) ( u , v ) = 0 , u , v * r [ 2.5 ] we give an explicit example of our conformal map in the following . consider the schwarz - christoffel ( sc ) transformation @xcite of the upper half - plane onto the semi - infinite strip . by using a composition of a sc map for a trigon with internal angles @xmath51 together with a sc map for a trigon with internal angles @xmath52 and @xmath53 , we arrive at the bijective transformation between one half of the bent waveguide and a semi - infinite straight strip . the expression results in * = f(z ) = _ 0^^2(z/2 ) dx ( 1-x)^- x^- = _ 0^z d((/2))^q + = ( ^2(z/2 ) , , ) [ 2.9 ] where we have used the definition of the incomplete beta function @xcite and we have defined @xmath54 as the complementary angle of @xmath55 in @xmath56 radians . therefore @xmath57 . in passing we note that @xmath58 . the jacobian is given by @xmath59 and one can approximate it in terms of @xmath60 ( or @xmath61 ) for some cases of interest : @xmath62 . it is important to note that as one moves away from the corner in the direction of the arms , the jacobian ( [ 2.14 ] ) becomes a hyperbolic cotangent of a real variable and approches to unity exponentially fast . therefore , interactions vanish for points far from the corners . in connection with the existence of other maps , we stress that any other transformation constructed in this way will have the same behavior near the corners : the fractional power law of the map is completely determined by the angle of the guides and therefore by our physical system . one can try many methods of solution for the transformed boundary value problem given above . here we give an example based on the fact that our careful construction allows certain approximations . let us reduce ( [ 2.5 ] ) to a one - dimensional problem by defining @xmath63 and v_nm(v ) n | ||^2 | m = _ 0^f(c ) du ( ) ( ) ||^2 [ 2.6 ] the set @xmath64 is obviously orthonormal . after multiplying ( [ 2.5 ] ) by our complete set of functions and performing the corresponding integrals over @xmath65 , we find that ( [ 2.5 ] ) is equivalent to _ m ( - _ nm _ v^2 - e v_nm(v ) ) _ m(v ) = _ n _ n(v ) [ 2.8 ] near the origin , our potential ( [ 2.6 ] ) becomes v_nm(v)= _ 0^f(c ) du ( ) ( ) ( u^2 + v^2 ) ^q/(q+1 ) [ 2.15 ] it is justified to approximate the potential by its diagonal entries @xmath66 , since @xmath67 decreases as we move away from the diagonal for fixed @xmath68 . this can be seen by noting that for @xmath69 the integrand can take negative values and oscillates rapidly as @xmath70 increases . we have ( -^2_v -e v_n(v ) ) _ n(v ) = _ n _ n(v ) . [ 2.16 ] we include some plots of @xmath66 and its derivative for different angles , figures [ pot ] , [ slope ] . , @xmath71 increases from @xmath72 to @xmath73 in @xmath74 steps . the transversal mode energy is shown at @xmath75 . ] . near the origin we see an abrupt change ( a discontinuity appears in the limit @xmath76 ) ] the potentials shown in the figure have zero derivative at @xmath77 ( except for @xmath76 ) but , apart from that , their behavior is that of a shallow potential well whose derivative suffers an abrupt change in a small region near the corner . in the plot showing the slopes , we see that there is almost a jump in @xmath78 around @xmath77 . the formalism developed by wheeler , schleich and bestle @xcite in order to deal with kinks should be helpful here again . although the only case in which the derivative is strictly discontinuous corresponds to @xmath76 , the strong variation of @xmath79 occurs in a region which looks effectively like a corner for long wavelengths ( we expect this approximation to work at low energies ) . as the simplest approximation , let us consider the corner - corrected wkb formula for quantization of bound states , namely s = _ 0^v_0 dv , [ 2.17 ] 4s=2(k+)- 4 ( ) , [ 2.18 ] = e ( v_n(0 + ) - v_n(0- ) ) ( 2(_n+e v_n(0 ) ) ) ^-3/2 , [ 2.19 ] remarkably , one can obtain good results by neglecting the classical orbit @xmath80 and keeping the corner effect alone . using ( [ 2.17 ] ) , the approximation implies = ( ) = 1 , k even [ 2.20 ] finding thereby that all even states obey this relation , but the excited states are not supported by such a weak potential . the excitations come from the solutions of ( [ 2.20 ] ) by varying @xmath81 and not @xmath82 . this results in a transversal node structure , instead of longitudinal . with these considerations , the solution to our problem ( [ 2.20 ] ) requires @xmath83 and @xmath84 as the only input . such quantities can be estimated by means of ( [ 2.15 ] ) for @xmath85 . as mentioned before , we replace the jump in the derivative @xmath86 by twice the maximum of @xmath78 around the origin ( see the figure ) . we denote such quantity as @xmath87 . squaring ( [ 2.20 ] ) we obtain a cubic relation for @xmath17 : ( n^2 ^2 ( q/2 ) - e v_n(0 ) ) ^3 = e^2 ( v_max)^2 [ 2.22 ] one can estimate @xmath83 and @xmath87 as v_n(0 ) ( ( q/2 ) ) ^2q/(q+1 ) , v_max q ( ( q/2 ) ) ^(2q+1)/(q+1 ) . [ 2.24 ] with this , the energies in ( [ 2.22 ] ) can be expressed in terms of @xmath81 and @xmath71 using cardano s formula . we present the results graphically in figure [ final ] . as a consequence of the corner . an unlimited number of bound states appear as @xmath71 increases . ] and @xmath88 ( ground state energy @xmath89 ) . the peak of the distribution is located near the upper corner , as expected . this is achieved with only one transversal mode in conformal coordinates ] the agreement of the estimated solutions with previous numerical results @xcite is qualitatively good , considering that the action has been completely neglected in favor of the phase shift . also , the off - diagonal elements of the matrix potential ( [ 2.15 ] ) have been ignored . the trend of the curves as functions of the bending angle is comparable with the numerical results obtained in @xcite for several modes : the energies decrease and more states appear below threshold as @xmath90 . the node structure of our wavefunctions is also compatible with the solutions obtained in @xcite both numerically and experimentally . remarkably , the fractional power behavior of the jacobian alone is responsible for the structure of the spectrum of this system . we include a plot of the resulting wave density for the ground state of an l - shaped guide . in what touches the corrections , we expect the following : the action appears to be important as @xmath71 increases ( deeper wells ) . therefore ( [ 2.22 ] ) receives corrections which increase the energy . as a consequence , we expect a shift of the critical angles towards the value @xmath76 . precisely at this value of @xmath71 we also expect a non - zero lower bound for the energy @xmath91 , as can be seen by identifying such an extreme case with a guide of double thickness . in the first part we have shown that the existence of bound states could be understood as a corner effect and the coupling of transversal modes . we could learn that diffraction was essential to the final result , but the coupling of the modes gave an extra complication . the second part , however , shows that an appropriate set of coordinates may keep the effect of the corner without the coupling of such modes . this makes contact with recent dicussions @xcite in which perturbation theory seems doomed in the presence of unbounded curvature - our example fits perfectly in such class of problems . one can follow either approach to solve the problem , but its apparent complexity - e.g. the many modes used in @xcite for the l - shape - dissapears using the second method . at the end we can conclude that a simple way of breaking a continuous symmetry such as a corner still allows for a simple description as long as the important information is used : diffraction and conformality . the next task is to find the correction to the energy levels due to the classical orbits entering through the action . but we can say that for now our goal has been reached . as a final comment , the problem of bound states in multiple junctions -e.g . x - shaped and @xmath92-shaped geometries - lends itself for this kind of treatment by using neumann boundary conditions . due to the properties of our conformal coordinate system , this can be achieved by replacing @xmath93 by @xmath94 in our discussion . e. sadurn is grateful to the organizers of the symposium _ symmetries in nature _ for their kind hospitality . h. j. stoeckmann , _ quantum chaos : an introduction _ , cambridge university press , 1999 . g. a. luna acosta , a. a. krokhin , m. a. rodrguez and p. h. hernndez tejeda _ phys . b _ * 54 * 11410 ( 1996 ) ; + g. a. luna acosta , k. na , l. e. reichl and a. krokhin , _ phys . e _ * 53 * 3271 ( 1996 ) j. goldstone and r. jaffe _ phys . b _ * 45 * 24 ( 1992 ) . e. n. bulgakov , p. exner , k. n. pichugin and a. f. sadreev , _ phys . b _ * 66 * 155109 ( 2002 ) p. exner and p. seba , _ phys . _ * 30 * 11 ( 1989 ) p. exner , _ phys . a _ * 141 * 5 ( 1989 ) p. exner , p. seba and p. stovicek , _ czech . b _ * 39 * 1181 ( 1989 ) r. schult , d. g. ravenhall and h. w. wyld , _ phys . rev . b _ * 39 * 5476 ( 1989 ) j. p. carini , j. t. londergan , k. mullen and d. p. murdock , _ phys b _ * 48 * 4503 ( 1993 ) . p. polynkin , m. kolesik , , _ science _ * 324 * , 229 - 32 ( 2009 ) ; + p. polynkin , m. kolesik and j. moloney , _ phys . _ * 103 * , 123902 ( 2009 ) ; + j. kasparian , m. rodrguez , , _ science _ * 301 * , 61 - 4 ( 2003 ) . e. kajari , n. l. harshman , , _ appl . . b _ 100 , 43 - 60 ( 2010 ) . j. bjorken and s. orbach _ phys . * 23 * 2243 ( 1981 ) r. schinzinger and p. laura , _ conformal mapping : methods and applications_. dover , new york , 2003 . i. gradshteyn and i. ryzhyk , _ tables of integrals , series and products _ , seventh edition . academic press , amsterdam , 2007 . m. miski - oglu and b. dietz , the numerical solution can be obtained using finite element methods . private communication , 2010 . s. teufel , _ adiabatic perturbation theory in quantum dynamics _ , springer , berlin 2003 .
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is it possible to trap a quantum particle in an open geometry ? in this work we deal with the boundary value problem of the stationary schroedinger ( or helmholtz ) equation within a waveguide with straight segments and a rectangular bending .
the problem can be reduced to a one dimensional matrix schroedinger equation using two descriptions : oblique modes and conformal coordinates .
we use a corner - corrected wkb formalism to find the energies of the one - dimensional problem .
it is shown that the presence of bound states is an effect due to the boundary alone , with no classical counterpart for this geometry .
the conformal description proves to be simpler , as the coupling of transversal modes is not essential in this case . + * pacs : * 37.10.gh , 42.25.gy , 03.65.ge + * keywords : * bent waveguides , bound states , corners , dirichlet conditions .
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You are an expert at summarizing long articles. Proceed to summarize the following text:
in 1969 stuart kauffman started to study random boolean networks as simple models of genetic regulatory networks @xcite . random boolean networks that consists of a set of boolean _ gates _ that are capable of storing a single boolean value . at discrete time steps these gates store a new value according to an initially chosen random boolean function , which receives its inputs from random chosen gates . we will give a more formal definition later . kauffman made numerical studies of random networks , where the functions are chosen from the set of all boolean functions with @xmath0 arguments ( the so called _ @xmath2-networks _ ) . he recognised that if @xmath3 , the random networks exhibit a remarkable form of ordered behaviour : the limit cycles are small , the number of _ ineffective gates _ , which are gates that can be perturbed without changing the asymptotic behaviour , and the number of _ freezing gates _ that stop changing their state is large . in contrast if @xmath4 , the networks do not exhibit this kind of ordered behaviour ( see @xcite ) . the first analytical proof for this _ phase transition _ was given by derrida and pomeau ( see @xcite ) by studying the evolution of the hamming distance of random chosen initial states by means of so called _ annealed approximation_. the first proof for the number of freezing and ineffective gates was given by james lynch ( see @xcite , although slightly weaker results appeared earlier @xcite ) . depending on a parameter @xmath1 , that depends on the probabilities of the boolean functions , he showed that if @xmath5 almost all gates are ineffective and freezing , otherwise not . although his analysis is very general , until now it was only applied to networks with connectivity @xmath6 and non - uniform probabilities for the boolean function : if the probability of choosing a constant function is larger or equal the probability of choosing a non - constant non - canalizing function ( namely the xor- or the inverted xor - function ) , @xmath1 is less or equal to one . but it turns out that in some cases @xmath1 is equal to the expectation of the average sensitivity . therefore we will first study the average sensitivity in section [ sec:02 ] . afterwards it will be shown in section [ sec:03 ] how to use the results from the previous section to apply lynch s analysis to classical @xmath2-networks and _ biased _ random boolean networks . but first we will give some basic definition used throughout the paper in section [ sec:01 ] . in the following @xmath7 denotes the galois field of two elements , where addition , denoted by @xmath8 , is defined modulo 2 . the set of vectors of length @xmath0 over @xmath9 will be denoted by @xmath10 . if @xmath11 is a vector from @xmath10 , its @xmath12th component will be denoted by @xmath13 . with @xmath14 we will denote the _ unit vector _ which has all components zero except component @xmath12 which is one . the hamming weight of @xmath15 is defined as @xmath16 and the hamming distance of @xmath17 as @xmath18 a boolean function is a mapping @xmath19 . a function @xmath20 may be represented by its _ truth table _ @xmath21 , that is , a vector in @xmath22 , where each component of the truth table gives the value of @xmath20 for one of the @xmath23 possible arguments . to fix an order on the components of the truth table , suppose that its @xmath12th component equals the value of the corresponding function , given the binary representation ( to @xmath0 bits ) of @xmath12 as an argument . in this section we will focus on the _ average sensitivity_. the average sensitivity is a known complexity measure for boolean functions , see for example @xcite . it was already used to study boolean and random boolean networks for example in @xcite . let @xmath20 denote a boolean function @xmath24 and @xmath25 a unit vector . 1 . the sensitivity @xmath26 is defined as : @xmath27 2 . the average sensitivity @xmath28 is defined as the average of @xmath26 over all @xmath29 : @xmath30 now consider the random variable @xmath31 , where @xmath32 denotes the set a all @xmath33 boolean function with @xmath0 arguments . the probability measure is given by @xmath34 . the expected value of the average sensitivity of this random variable is denoted by @xmath35 , and is given by @xmath36 the expected value was already derived in @xcite , and is given by : + let the random variable @xmath37 be defined as above , then @xmath38 [ theo:01 ] we will now concentrate on biased boolean functions . the bias of a boolean function @xmath19 is defined as the number of @xmath39 in the functions truth table divided by @xmath40 . to define the bias of a random boolean function two definitions are possible . first we can assumes that the truth tables of the boolean functions are produced by independent bernoulli trials with probability @xmath41 for a one ( this should be called _ mean _ bias , used for example in @xcite ) . therefore consider the random variable @xmath42 . the probability of choosing a function @xmath20 is given by @xmath43 for @xmath44 this is equivalent to the definition of @xmath37 . as a second possibility , we can only choose functions which have bias @xmath41 whereas to all other functions we assign probability 0 ( we will call this _ fixed _ bias ) . therefore consider the random variables @xmath45 . denote the truth table of a function @xmath20 by @xmath21 . further denote the set of all boolean functions @xmath20 with @xmath0 arguments and @xmath46 with @xmath47 . the probability for a certain function chosen according @xmath48 is given by @xmath49 both definitions ensure that the expectation to get a one is equal to @xmath41 if the input of a function is chosen at random ( with respect to uniform distribution ) . but it will turn out that these two different methods of creating biased boolean functions , have a major impact on the average sensitivity . the expectation of the average sensitivity of @xmath42 was derived in @xcite : let the random variable @xmath50 be defined as above : @xmath51 [ theo : expectedsensitivityb ] for the random variable @xmath48 we will now proof the following theorem : let the random variable @xmath48 be defined as above : @xmath52 [ theo : expectedsensitivity ] [ cor:01 ] to find @xmath53 we will first consider the random variable @xmath54 where @xmath55 and the probability of a function is given by @xmath56 consider the boolean functions as functions into @xmath57 by identifying @xmath58 with @xmath59 . then we get or the function @xmath20 : @xmath60 where @xmath25 again denotes the unit vector with @xmath12th component set to @xmath39 . hence by the linearity of the expectation @xmath61 now we form a matrix with the truth tables of all functions with hamming weight @xmath62 as column vectors : @xmath63 @xmath64 has exactly @xmath65 columns and @xmath23 rows . each entry @xmath66 in the @xmath12th row and @xmath67th column equals the value of function @xmath68 given the binary representation of @xmath12 as input . hence @xmath69 is determined by the number of @xmath39 in the row associated with @xmath70 divided by the length of the row . consider an arbitrary row @xmath12 . this row has a one at position @xmath67 if the corresponding column @xmath71 has a one at position @xmath12 . but there are @xmath72 column vectors with a @xmath39 at position @xmath12 . it follows : @xmath73 as this holds for all @xmath70 , we have @xmath74 to find an expression for @xmath75 we consider two arbitrary rows @xmath76 ( @xmath77 ) . define the following sum : @xmath78 obviously @xmath79 only if we have a @xmath39 in both rows at position @xmath12 . this means for the column vectors @xmath80 of @xmath64 , we have @xmath81 . but there are exactly @xmath82 such column vectors in @xmath64 . therefore we have @xmath83 as @xmath84 for all @xmath85 it follows : @xmath86 hence substituting equations , and into equation leads to @xmath87 finally the claimed expression for @xmath53 can be obtained from the above equation by a substitution of @xmath62 : @xmath88 . [ comment ] it should be noted , that the theorems [ theo:01 ] and [ theo : expectedsensitivityb ] can be proved using in a similar way . also worth noting is the fact , that if the functions are chosen according @xmath89 or @xmath42 the expectation of the sensitivity of a fixed vector @xmath70 ( namely the expectation of @xmath90 ) is independent of @xmath70 ( see equation , , and ) . hence the following lemma holds if @xmath91 or @xmath42 , then @xmath92 [ lemma:01 ] before proceeding to the next section , it should be noted , that using the same arguments as in the proof of theorem [ theo : expectedsensitivity ] , we can also prove the expectation of average sensitivity of order @xmath93 , defined as @xmath94 in this case , instead of summing up all unit vectors in equation , we sum up all vectors of hamming weight @xmath93 . as the equations and hold for all @xmath29 we conclude that @xmath95 and by similar arguments @xmath96 respectively @xmath97 as already mentioned james lynch gave a very general analysis of randomly constructed boolean networks ( see @xcite ) . before stating his results we give a formal definition for boolean networks a boolean network * b * is a 4-tuple @xmath98 where @xmath99 is a set of natural numbers , @xmath100 is a set of labeled edges on @xmath101 , @xmath102 is a ordered set of boolean functions such that for each @xmath103 the number of arguments of @xmath104 is the _ in - degree _ of @xmath105 in @xmath100 , these edges are labeled with @xmath106 , and @xmath107 . suppose that a vertex @xmath12 has @xmath108 in - edges from vertices @xmath109 . for @xmath110 we define @xmath111 the state of * b * at time 0 is called the _ initial state _ @xmath11 , so we define @xmath112 . for time @xmath113 the state is inductively defined as @xmath114 . hence we can in interpret @xmath101 as set of gates , @xmath100 and @xmath115 describes their functional dependence and @xmath11 is the networks initial state . assume some ordering @xmath116 on the set of all boolean functions @xmath117 , where each function @xmath118 depends on @xmath108 arguments . further a random variable @xmath119 with probabilities @xmath120 such that @xmath121 and @xmath122 . now a random boolean network consisting of @xmath123 _ gates _ is constructed as follows : for each gate a boolean function is chosen independently , where the probability of choosing @xmath118 is given by @xmath124 . suppose a function @xmath20 was chosen that has @xmath0 arguments , these arguments are chosen at random from all @xmath125 equally likely possibilities . at last an initial state is chosen at random from the set on all equally likely states . if the boolean functions are chosen according to our previously defined random variable @xmath37 we will call this networks @xmath2-networks with connectivity @xmath0 . if the functions are chosen according to @xmath48 or @xmath42 we will call this networks _ biased random boolean networks _ with connectivity @xmath0 and fixed bias @xmath41 respectively mean bias @xmath41 . let us now state lynch s results . his analysis depends on a parameter @xmath126 depending only on the functions and their probabilities . we will define @xmath1 later in definition [ def : lambda ] . first we have to state lynch s definition of _ freezing _ and _ ineffective _ gates : + let @xmath127 and @xmath103 . 1 . gate @xmath105 freezes to @xmath110 in @xmath62 steps on input @xmath11 if @xmath128 for all @xmath129 . 2 . let @xmath130 . + a gate @xmath105 is @xmath62-ineffective at input @xmath15 if @xmath131 . now we will state the main result . + let @xmath132 , @xmath133 be positive constants satisfying @xmath134 and @xmath135 where @xmath136 . 1 . there is a constant @xmath137 such that for all @xmath127 @xmath138 when @xmath5 , @xmath139 and when @xmath140 , @xmath141 . there is a constant @xmath137 such that for all @xmath127 @xmath142 when @xmath5 , @xmath139 and when @xmath140 , @xmath141 . the above theorem shows that if @xmath5 almost all gates are freezing and ineffective and otherwise not . the next corollary gives us more information what happens if @xmath140 : let @xmath140 . for almost all random boolean networks 1 . if gate @xmath105 is not @xmath143-ineffective , there is a positive constant @xmath144 such that for @xmath145 , the number of gates affected by @xmath105 at time @xmath62 is asymptotic to @xmath146 , 2 . if gate @xmath105 is not freezing in @xmath143 steps , there is a positive constant @xmath144 such that for @xmath145 , the number of gates that affect @xmath105 at time @xmath62 is asymptotic to @xmath146 . now we will state the definition of @xmath1 for boolean networks : [ def : lambda ] let @xmath20 be a boolean function of @xmath0 arguments . for @xmath147 , we say that argument @xmath12 directly affects @xmath20 on input @xmath29 if @xmath148 . now put @xmath149 as the number of @xmath12 s that directly affect @xmath20 on input @xmath70 . given a constant @xmath150 $ ] , we define @xmath151 obviously @xmath149 is identical to @xmath90 which will be used instead in the further discussion . the constant @xmath152 is the probability that a random gate is one ( at infinite time ) given that all gates at time @xmath153 have probability @xmath154 of being one . ( see ( * ? ? ? * definiton 2 ) ) . assume that we choose the functions according a random variable @xmath155 which should be either @xmath37 , @xmath48 or @xmath42 . the functions are chosen out the set @xmath156 , we denote a function s probability with @xmath157 . it follows that @xmath158 @xmath159 denotes the expectation of the sensitivity for a fixed @xmath70 , equation follows from lemma [ lemma:01 ] . therefore , together with theorem [ theo:01 ] and theorem [ cor:01 ] we proved the following : for random boolean networks , if 1 . the functions are chosen according random variable @xmath42 , it follows that @xmath160 2 . the functions are chosen according random variable @xmath48 , it follows that @xmath161 [ theo:02 ] as a special case of the above theorem we get ( or by using theorem [ theo:01 ] ) in random boolean networks , where the functions are chosen according to the random variable @xmath162 @xmath163 the results about @xmath2-networks are consistent with experimental results . in fact if @xmath164 almost all networks almost all gates are freezing and almost all gates are ineffective and otherwise not ( see @xcite ) . obviously , the border between the ordered and disordered phase is given by @xmath165 . the resulting phase diagram for biased random boolean networks , where the functions are chosen according to @xmath48 and @xmath42 is shown in figure [ fig:01 ] . it it interesting to note that if the functions are chosen with fixed bias , then also boolean networks with connectivity @xmath166 can become unstable . this conclusion can be drawn from lynch s original result already . as mentioned in the introduction , he showed for @xmath166 , that @xmath140 if the probability of choosing a non - constant non - canalizing function , namely the xor or the inverted xor function , is larger than the probability of choosing a constant function . for example if the bias is @xmath154 , the probability of choosing a constant function is zero , whereas both xor and inverted xor function have probability greater zero , hence @xmath140 . ( dashed ) and @xmath48 ( solid),title="fig : " ] it is interesting to compare our results with previous results obtained first by derrida and pomeau using the so called _ annealed approximation _ ( see @xcite ) . in their _ annealed model _ the functions and connections are chosen at random at each time step . considering two instances of the same annealed network starting in two randomly chosen initial states @xmath167 they show that @xmath168 where @xmath169 if @xmath170 and @xmath171 otherwise . it is remarkable that the two models behave similar , but it is unclear whether this holds in general . we would like to thank our colleges georg schmidt and stephan stiglmayr for proofreading and uwe schoening for useful hints .
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in this work we consider random boolean networks that provide a general model for _ genetic regulatory networks_. we extend the analysis of james lynch who was able to proof kauffman s conjecture that in the _ ordered phase _ of random networks , the number of _ ineffective _ and _ freezing _ gates is large , where as in the _ disordered phase _ their number is small .
lynch proved the conjecture only for networks with connectivity two and non - uniform probabilities for the boolean functions .
we show how to apply the proof to networks with arbitrary connectivity @xmath0 and to random networks with _ biased _ boolean functions .
it turns out that in these cases lynch s parameter @xmath1 is equivalent to the expectation of _ average sensitivity _ of the boolean functions used to construct the network .
hence we can apply a known theorem for the expectation of the average sensitivity . in order to prove the results for networks with biased functions , we deduct the expectation of the average sensitivity when only functions with specific connectivity and specific bias are chosen at random .
random boolean networks , phase transition , average sensitivity pacs numbers : 02.10.eb , 05.45.+b , 87.10.+e
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seasonal unit roots and seasonal heterogeneity often coexist in seasonal data . hence , it is important to design seasonal unit root tests that allow for seasonal heterogeneity . in particular , consider quarterly data @xmath5 , @xmath6 generated by @xmath7 where @xmath8 are seasonally varying autoregressive ( ar ) filters , and @xmath9 have seasonally varying autocovariances . for more information on seasonal time series , see ghysels and osborn ( 2001 ) , and franses and paap ( 2004 ) . now suppose @xmath10 is a weakly stationary vector - valued process , and for all @xmath11 , the roots of @xmath8 are on or outside the unit circle . if for all @xmath12 , @xmath8 have roots at @xmath0 , @xmath1 , or @xmath4 , then respectively @xmath13 has stochastic trends with period @xmath14 , @xmath15 , or @xmath16 . to remove these stochastic trends , we need to test the roots at 1 , @xmath1 , or @xmath4 . to address this task , franses ( 1994 ) and boswijk , franses , and haldrup ( 1997 ) limit their scope to finite order seasonal ar data and apply johansen s method ( 1988 ) . however , their approaches can not directly test the existence of a certain root without first checking the number of seasonal unit roots . as a remedy , ghysels , hall , and lee ( 1996 ) designs a wald test that directly tests whether a certain root exists . however , in their own simulation , the wald test turn out less powerful than the augmented hegy test . does hegy test work in the seasonally heterogeneous setting ? to the best of our knowledge , no literature has offered a satisfactory answer . burridge and taylor ( 2001a ) analyze the behavior of augmented hegy test when only seasonal heteroscadasticity exists ; del barrio castro and osborn ( 2008 ) put augmented hegy test in the periodic integrated model , a model related but different from model . no literature has ever touched the behavior of unaugmented hegy test proposed by breitung and franses ( 1998 ) , the important semi - parametric version of hegy test . since unaugmented hegy test does not assume the noise having an ar structure , it may suit our non - parametric noise in better . to check the legitimacy of hegy tests in the seasonally heterogeneous setting , this paper derives the asymptotic null distributions of the unaugmented hegy test and the augmented hegy test whose order of lags goes to infinity . it turns out that , the asymptotic null distributions of the statistics testing single roots at 1 or @xmath1 are standard . more specifically , for each single root at 1 or @xmath1 , the asymptotic null distributions of the augmented hegy statistics are identical to that of augmented dickey - fuller ( adf ) test ( dickey and fuller , 1979 ) , and the asymptotic null distributions of the unaugmented hegy statistics are identical to those of phillips - perron test ( phillips and perron , 1988 ) . however , the asymptotic null distributions of the statistics testing any combination of roots at 1 , @xmath1 , @xmath2 , or @xmath3 depend on the seasonal heterogeneity parameters , and are non - standard , non - pivotal , and not directly pivotable . therefore , when seasonal heterogeneity exists , both augmented hegy and unaugmented hegy tests can be straightforwardly applied to single roots at 1 or @xmath1 , but can not be directly applied to the coexistence of any roots . as a remedy , this paper proposes the application of bootstrap . in general , bootstrap s advantages are two fold . firstly , bootstrap helps when the asymptotic distributions of the statistics of interest can not be found or simulated . secondly , even when the asymptotic distributions can be found and simulated , bootstrap method may enjoy second order efficiency . for the aforementioned problem , bootstrap therefore serves as an appealing solution . firstly , it is hard to estimate the seasonal heterogeneity parameters in the asymptotic null distribution , and to simulate the asymptotic null distribution . secondly , it can be conjectured that bootstrap seasonal unit root test inherits second order efficiency from bootstrap non - seasonal unit root test ( park , 2003 ) . the only methodological literature we find on bootstrapping hegy test is burridge and taylor ( 2004 ) . their paper centers on seasonal heteroscadasticity , designs a bootstrap - aided augmented hegy test , reports its simulation result , but does not give theoretical justification for their test . it will be shown ( remark [ re : seasonal iid bootstrap ] ) that their bootstrap approach is inconsistent under the general seasonal heterogeneous setting . to cater to the general heterogeneous setting , this paper designs new bootstrap tests , namely 1 ) seasonal iid bootstrap augmented hegy test , and 2 ) seasonal block bootstrap unaugmented hegy test . to generate bootstrap replicates , the first test get residuals from season - by - season augmented hegy regressions , and then applies seasonal iid bootstrap to the whitened regression errors . on the other hand , the second test starts with season - by - season unaugmented hegy regressions , and then handles the correlated errors with seasonal block bootstrap proposed by dudek , lekow , paparoditis , and politis ( 2014 ) . our paper establishes the functional central limit theorem ( fclt ) for both bootstrap tests . based on the fclt , the consistency for both bootstrap approaches is proven . to the best of our knowledge , this result gives the first justification for bootstrapping hegy tests under . this paper proceeds as follows . section 2 formalizes the settings , presents the assumptions , and states the hypotheses . section 3 gives the asymptotic null distributions of the augmented hegy test statistics , details the algorithm of seasonal iid bootstrap augmented hegy test , and establishes the consistency of the bootstrap . section 4 presents the asymptotic null distributions of the unaugmented hegy test statistics , specifies the algorithm of seasonal block bootstrap unaugmented hegy test , and proves the consistency of the bootstrap . section 5 compares the simulation performance of the two aforementioned tests . appendix includes all technical proofs . recall the quarterly data @xmath5 , @xmath6 generated by the seasonal ar model , @xmath17 where @xmath18 , @xmath19 . if for all @xmath12 , @xmath8 has roots on the unit circle , we suppose that all @xmath8 share the same set of roots on the unit circle , this set of roots on the unit circle is a subset of @xmath20 , and @xmath21 ; otherwise , suppose our data is a stretch of the process @xmath22 , @xmath6 . let @xmath23 and @xmath24 be the regression errors and regression coefficients of , respectively . more specifically , @xmath23 is the distance between @xmath25 and the vector space generated by @xmath26 , @xmath27 , and @xmath24 is the coefficient of the projection of @xmath23 on the aforementioned vector space . let @xmath28 , @xmath29 . denote by ar@xmath30 an autoregressive process with order @xmath31 , by vma@xmath32 a vector moving average process with infinite moving average order , and by varma@xmath33 a vector autoregressive moving average process with autoregressive order @xmath31 and moving average order @xmath34 . let @xmath35 be the real part of complex number @xmath36 . let @xmath37 be the largest integer smaller or equal to real number @xmath38 , and @xmath39 be the smallest integer larger or equal to @xmath38 . assump [ assump 1a ] assume @xmath40 where @xmath41 , @xmath42 ; the @xmath43 entry of @xmath44 , denoted by @xmath45 , satisfies @xmath46 for all @xmath47 and @xmath48 ; the determinant of @xmath49 has all roots outside the unit circle ; @xmath50 is a lower diagonal matrix whose diagonal entries equal 1 ; @xmath51 is a vector - valued white noise process with mean zero and covariance matrix @xmath52 ; and @xmath52 is diagonal . assumption [ assump 1a ] assumes that @xmath53 is vma@xmath32 with respect to white noise innovation . this is equivalent to the assumption that @xmath53 is a weakly stationary process with no deterministic part in the multivariate wold decomposition . the assumptions on @xmath50 and the determinant of @xmath49 ensure the causality and the invertibility of @xmath53 and the identifiability of @xmath52 . [ assump 1b ] assume @xmath54 where @xmath55 ; @xmath56 ; determinants of @xmath57 and @xmath58 have all roots outside the unit circle ; @xmath59 is the identity matrix ; @xmath60 is a lower diagonal matrix whose diagonal entries equal 1 ; @xmath51 is a vector - valued white noise process with mean zero and covariance matrix @xmath52 ; and @xmath52 is diagonal . assumption [ assump 1b ] restricts @xmath53 to be varma@xmath33 with respect to white noise innovation . compared to the vma@xmath32 model in assumption [ assump 1a ] , varma@xmath33 s main restraint is its exponentially decaying autocovariance . again , the assumptions on @xmath59 , @xmath60 and the determinant of @xmath57 and @xmath58 in assumption [ assump 1b ] ensure the causality and the invertibility of @xmath53 and the identifiablity of @xmath52 . at this stage @xmath61 is only assumed to be a white noise sequence of random vectors . in fact , @xmath61 needs to be weakly dependent as well . assump [ assump 2a ] ( i ) @xmath61 is a fourth - order stationary martingale difference sequence with finite @xmath62 moment for some @xmath63 . ( ii ) @xmath64 , @xmath65 @xmath2 , @xmath47 , @xmath48 , and @xmath66 , @xmath67 . [ assump 2b ] ( i ) @xmath61 is a strictly stationary strong mixing sequence with finite @xmath62 moment for some @xmath63 . ( ii ) @xmath61 s strong mixing coefficient @xmath68 satisfies @xmath69 . notice the higher moment @xmath61 has , the weaker assumption we require on the strong mixing coefficient of @xmath61 in assumption [ assump 2b ] . the strong mixing condition in assumption [ assump 2b ] actually guarantees ( ii ) of assumption [ assump 2a ] ( see lemma [ boundedness ] ) . we tackle the following set of null hypotheses . the alternative hypotheses are the complement of the null hypotheses . indeed , the alternative hypotheses can be written as one - sided . recall we suppose that for all @xmath11 , the roots of @xmath8 are either on or outside the unit circle . since @xmath71 , by the intermediate value theorem , @xmath72 implies @xmath73 , @xmath74 implies @xmath75 , and @xmath76 implies @xmath77 . to further analyze the roots of @xmath8 , hegy ( hylleberg , engle , granger , and yoo , 1990 ) propose the partial fraction decomposition @xmath78 thus @xmath79 + substituting into , we get @xmath80 where @xmath81 indeed , @xmath82 relates to the root of @xmath83 , i.e. , @xmath84 ; hence the proposition below . [ prop : hegy ] @xmath85 by proposition [ prop : hegy ] , the test for the null hypotheses can be carried on by checking the corresponding @xmath82 . further , @xmath82 can be estimated by ordinary least squares ( ols ) . unfortunately , ols can not be readily applied to season by season , because @xmath86 in are not asymptotically orthogonal for any fixed @xmath12 . ( see also ghysels and osborn , 2001 , p. 158 . ) on the other hand , @xmath87 in non - periodic regression equations and are asymptotically orthogonal ( see lemma [ le : unaug real1 ] ) . so we wonder if the ols estimators based on and can be used to test the null hypotheses . when we regress @xmath13 with non - periodic regression equations and , the seasonally heterogeneous sequence @xmath9 is fitted in seasonal homogeneous ar models . consider , as an example , fitting @xmath9 in a misspecified ar(1 ) model @xmath88 . then @xmath89 , where @xmath90 . \label{eqn : tgamma}\ ] ] since @xmath91 is positive semi - definite , we can find a weakly stationary sequence @xmath92 with mean zero and autocovariance function @xmath93 . we call @xmath92 a misspecified constant parameter representation ( see also osborn , 1991 ) of @xmath9 , and will refer to this concept in later sections . in seasonally homogeneous setting @xmath94 where @xmath95 , the augmented hegy test detailed below copes with the roots of @xmath96 at @xmath0 , @xmath1 , and @xmath4 . by calculations similar to , hegy ( 1990 ) get @xmath97 where augmentations @xmath98 , @xmath99 , pre - whiten the time series @xmath100 up to an order of @xmath48 . as the sample size @xmath101 , let @xmath102 , so that the residual @xmath103 is asymptotically uncorrelated . let @xmath104 be the ols estimator of @xmath105 , @xmath106 be the t - statistics corresponding to @xmath104 , and @xmath107 be the f - statistic corresponding to @xmath108 and @xmath109 . other f - statistics @xmath110 , @xmath111 , @xmath112 , and @xmath113 can be defined similarly . in seasonally homogeneous configuration , hegy ( 1990 ) proposes to reject @xmath114 if @xmath115 is too small , reject @xmath116 if @xmath117 is too small , reject @xmath118 if @xmath107 is too large , and reject other composite hypotheses if their corresponding f - statistics are too large . now we apply the augmented hegy test to seasonally heterogeneous processes . namely , we run regression equation with @xmath13 generated by . our results show that when testing roots at 1 or @xmath1 individually , the t - statistics @xmath119 , @xmath120 , and the f - statistics have standard and pivotal asymptotic distributions . on the other hand , when testing joint roots at 1 and @xmath1 , and when testing hypotheses that involve roots at @xmath4 , the asymptotic distributions of the t - statistics and the f - statistics are non - standard , non - pivotal , and not directly pivotable . [ aug real ] assume that assumption 1.b and one of assumption 2.a or 2.b hold . further , assume @xmath101 , @xmath121 , @xmath122 , and @xmath123 for some @xmath124 , @xmath125 . then under @xmath126 , the asymptotic distributions of @xmath106 , @xmath127 , and f - statistics are given by @xmath128 where @xmath129 , @xmath130 , @xmath131 , and @xmath132 , @xmath133 , @xmath134 , @xmath135 is a four - dimensional standard brownian motion . the asymptotic distributions presented in theorem [ aug real ] degenerate to the distributions in burridge and taylor ( 2001b ) and del barrio castro , osborn and taylor ( 2012 ) when @xmath9 is a seasonally homogeneous sequence with homoscedastic noise , and to the distributions in burridge and taylor ( 2001a ) when @xmath9 is a seasonally homogeneous finite - order ar sequence with heteroscedastic noise . [ pm1 ] notice @xmath136 s are standard brownian motions . when @xmath9 is seasonally homogeneous ( burridge and taylor , 2001b , del barrio castro et al . , 2012 ) , @xmath136 s are independent , so are the asymptotic distributions of @xmath119 and @xmath120 . on the other hand , when @xmath9 has seasonal heterogeneity , @xmath136 s are in general independent , so @xmath119 and @xmath120 are in general dependent , even asymptotically . hence , when testing @xmath137 , it is problematic to test @xmath114 and @xmath116 separately and calculate the level of the test with the independence of @xmath119 and @xmath120 in mind . instead , the test of @xmath137 should be handled with @xmath110 . further , because of the dependence of @xmath119 and @xmath120 , the asymptotic distribution of @xmath110 under heterogeneity is different from its counterpart when @xmath9 is seasonally homogeneous . hence , the augmented hegy test can not be directly applied to test @xmath137 . [ heter ] when @xmath9 is only seasonally heteroscedastic ( burridge and taylor , 2001a ) , @xmath138 does not occur in the asymptotic distributions of the f - statistics . on the other hand , when @xmath9 has generic seasonal heterogeneity , @xmath138 impacts firstly the correlation between brownian motions @xmath139 and @xmath140 , and secondly the weights @xmath141 and @xmath142 . as burridge and taylor ( 2001a ) point out , the dependence of the asymptotic distributions on weights @xmath141 and @xmath142 can be expected . indeed , @xmath143 is the partial sum of @xmath144 , while @xmath145 is the partial sum of @xmath146 . since these two partial sums differ in their variances , both @xmath147 and @xmath148 involve two different weights @xmath141 and @xmath142 . [ root ] theorem [ aug real ] presents the asymptotics when @xmath149 has all roots at 1 , @xmath1 , and @xmath4 . when @xmath13 has some but not all roots at 1 , @xmath1 , and @xmath4 , we let @xmath150 , @xmath151 , and calculate @xmath152 such that @xmath153 . the asymptotic distributions can be expressed with respective to @xmath152 and end up having the same form with those given in theorem [ aug real ] , where @xmath13 has all roots . [ power ] the preceding results give the asymptotic behaviors of the testing statistics under the null hypotheses . under the alternative hypotheses , we conjecture the powers of the augmented hegy tests tend to one , as the sample size goes to infinity . to see this , we can without loss of generality assume that @xmath13 has root at none of @xmath0 , @xmath1 or @xmath4 . then @xmath13 is stationary , and thus for @xmath154 , the @xmath155 corresponding to ( the misspecified constant parameter representation of ) @xmath13 are negative , due to proposition [ prop : hegy ] . we conjecture that for @xmath154 , the ols estimators @xmath156 in converge in probability to @xmath155 , and as a result the powers of the tests converge to one . see also theorem 2.2 of paparoditis and politis ( 2016 ) . to accommodate the non - standard , non - pivotal asymptotic null distributions of the augmented hegy test statistics , we propose the application of bootstrap . in particular , the bootstrap replications are created as follows . firstly , we pre - whiten the data season by season to obtain uncorrelated noises . although these noises are uncorrelated , they are not white due to seasonally heteroscadasticity . hence secondly we resample season by season in order to generate bootstrapped noise , as in burridge and taylor ( 2001b ) . finally , we post - color the bootstrapped noise . the detailed algorithm of this seasonal iid bootstrap augmented hegy test is given below . [ seasonal iid bootstrap ] step 1 : calculate the t - statistics @xmath119 , @xmath120 , and the f - statistics @xmath157 , @xmath158 and @xmath159 from the augmented hegy test regression @xmath160 step 2 : record ols estimators @xmath161 , @xmath162 and residuals @xmath163 from the season - by - season regression @xmath164 step 3 : let @xmath165 . store demeaned residuals @xmath166 of the four seasons separately , then independently draw four iid samples from each of their empirical distributions , and then combine these four samples into the vector @xmath167 , with their seasonal orders preserved ; + + step 4 : set all @xmath161 corresponding to the null hypothesis to be zero . for example , set @xmath168 for all @xmath12 when testing roots at @xmath4 . let @xmath169 be generated by @xmath170 step 5 : get t - statistics @xmath171 , @xmath172 , and the f - statistics @xmath173 from the regression @xmath174 step 6 : run step 3 , 4 , and 5 for @xmath175 times to get @xmath175 sets of statistics @xmath171 , @xmath172 , and the bootstrapped f - statistics @xmath176 . count separately the numbers of @xmath171 , @xmath172 and @xmath176 than which @xmath119 , @xmath120 , and the f - statistics @xmath157 are more extreme . if these numbers are higher than @xmath177 , then we consider @xmath119 , @xmath120 , and the f - statistics @xmath157 extreme , and reject the corresponding hypotheses . it seems also reasonable to keep steps 1 , 2 , 3 , 5 , and 6 of the algorithm [ seasonal iid bootstrap ] , but change the generation of @xmath178 in step 4 to @xmath179 this new algorithm is in fact theoretically invalid for the tests of any coexistence of roots ( see remark [ pm1 ] , [ heter ] , and [ root ] ) , but it is valid for individual tests of roots at 1 or @xmath1 , due to the pivotal asymptotic distributions of @xmath119 and @xmath120 in theorem [ aug real ] . [ re : seasonal iid bootstrap ] if we keep steps 1 , 3 , 5 , and 6 of algorithm [ seasonal iid bootstrap ] , but run regression equations with seasonally homogeneous coefficients @xmath156 and @xmath180 in steps 2 and 4 , then this algorithm is identical with burridge and taylor ( 2004 ) . however , this algorithm can not in step 2 fully pre - whiten the time series , and it leaves the regression error @xmath103 serially correlated . when @xmath103 is bootstrapped by seasonal iid bootstrap , this serial correlation structure is ruined . as a result , @xmath181 differs from @xmath100 in its correlation structure , in particular @xmath138 , and the conditional distributions of the bootstrapped f - statistics @xmath176 differ from the distributions of the original f - statistics @xmath157 ( see remark [ pm1 ] and [ heter ] ) . now we justify the seasonal iid bootstrap augmented hegy test ( algorithm [ seasonal iid bootstrap ] ) . since the derivation of the real - world asymptotic distributions in theorem [ aug real ] calls on fclt ( see lemma [ le : unaug real1 ] ) , the justification of bootstrap approach also requires fclt in the bootstrap world . from now on , let @xmath182 , @xmath183 , @xmath184 , @xmath185 , @xmath186 be the probability , expectation , variance , standard deviation , and covariance , respectively , conditional on our data @xmath13 . [ iid fclt ] suppose the assumptions in theorem [ aug real ] hold . let @xmath187 @xmath188 where @xmath189 , & \sigma_{2}^{\star}&=std^{\circ}[\frac{1}{\sqrt{4t}}\sum _ { t=1}^ { 4 t } ( -1)^{t}{\epsilon}_{t}^{\star}],\\ \sigma_{3}^{\star}&=std^{\circ}[\frac{1}{\sqrt{4t}}\sum _ { t=1}^ { 4 t } \sqrt{2}\sin(\frac{\pi t}{2}){\epsilon}_{t}^{\star } ] , & \sigma_{4}^{\star}&=std^{\circ}[\frac{1}{\sqrt{4t}}\sum _ { t=1}^ { 4 t } \sqrt{2}\cos(\frac{\pi t}{2}){\epsilon}_{t}^{\star}].\end{aligned}\ ] ] then , no matter which hypothesis is true , @xmath190 in probability as @xmath101 , where @xmath191 is a four - dimensional standard brownian motion . by the fclt given by proposition [ iid fclt ] and the proof of theorem [ aug real ] , in probability the conditional distributions of @xmath192 , @xmath193 and @xmath176 converge to the limiting distributions of @xmath106 , @xmath193 and @xmath157 , respectively . since conditional on @xmath13 , @xmath178 is a finite - order seasonal ar process , the derivation of the conditional distributions of @xmath192 , @xmath193 and @xmath176 turns out easier than that of theorem [ aug real ] , and in particular does not involve the fourth moments of @xmath178 . hence the consistency of the bootstrap . suppose the assumptions in theorem [ aug real ] hold . let @xmath194 be the probability measure corresponding to the null hypothesis @xmath195 . then , @xmath196 @xmath197 in the proceeding section our analysis focuses on the augmented hegy test , an extension of the adf test to the seasonal unit root setting . an important alternative of the adf test is the phillips - perron test ( phillips and perron , 1988 ) . while the adf test assumes an ar structure over the noise and thus becomes parametric , its semi - parametric counterpart , phillips - perron test , allows a wide class of weakly dependent noises . unaugmented hegy test ( breitung and franses , 1998 ) , as the extension of phillips - perron test to the seasonal unit root , inherits the semi - parametric nature and does not assume the noise to be ar . given seasonal heterogeneity , it will be shown in theorem [ unaug real ] that the unaugmented hegy test estimates seasonal unit root consistently under the very general vma@xmath32 class of noise ( assumption 1.a ) , instead of the more restrictive varma@xmath33 class of noise ( assumption 1.b ) , which is needed for the augmented hegy test . now we specify the unaugmented hegy test . consider regression @xmath198 let @xmath156 be the ols estimator of @xmath155 , @xmath199 be the t - statistic corresponding to @xmath156 , and @xmath107 be the f - statistic corresponding to @xmath108 and @xmath109 . other f - statistics @xmath110 , @xmath111 , @xmath112 , and @xmath113 can be defined analogously . similar to the phillips - perron test ( phillips and perron , 1988 ) , the unaugmented hegy test can use both @xmath156 and @xmath199 when testing roots at 1 or @xmath1 . as in the augmented hegy test , we reject @xmath114 if @xmath115 ( or @xmath119 ) is too small , reject @xmath116 if @xmath117 ( or @xmath120 ) is too small , and reject the joint hypotheses if the corresponding f - statistics are too large . the following results give the asymptotic null distributions of @xmath104 , @xmath199 , @xmath200 , and the f - statistics . [ unaug real ] assume that assumption 1.a and one of assumption 2.a or assumption 2.b hold . then under @xmath126 , as @xmath201 , @xmath202 where @xmath129 , @xmath130 , @xmath131 , @xmath132 , @xmath203 , @xmath134 , @xmath135 is a four - dimensional standard brownian motion , @xmath204 are defined in , @xmath205 , @xmath206 , @xmath207 , and @xmath208 . the results in theorem [ unaug real ] degenerate to the asymptotics in burridge and taylor ( 2001ab ) when @xmath9 is uncorrelated , and degenerate to the asymptotics in breitung and franses ( 1998 ) when @xmath9 is seasonally homogeneous . when @xmath9 is seasonally homogeneous ( breitung and franses , 1998 ) , the asymptotic distributions of @xmath209 and @xmath210 are independent . on the other hand , when @xmath9 has seasonal heterogeneity , @xmath209 and @xmath210 are dependent , as what we have seen for augmented hegy test ( remark [ pm1 ] ) . hence , when testing @xmath137 , it is problematic to test @xmath114 and @xmath116 separately and calculate the level of the test with the independence of @xmath209 and @xmath210 in mind . instead , the test of @xmath137 should be handled with @xmath110 . the parameters @xmath211 have the same definition as in theorem [ aug hegy ] . since @xmath212 , and @xmath213 , the asymptotic distributions of @xmath104 and @xmath106 , @xmath127 , only depends on the autocorrelation function of @xmath92 , the misspecified constant parameter representation of @xmath9 . since @xmath92 can be considered as a seasonally homogeneous version of @xmath9 , we can conclude that the asymptotic behaviors of the tests for single roots at 1 or @xmath1 are not affected by the seasonal heterogeneity in @xmath9 . on the other side , the asymptotic distributions of the f - statistics do not solely depend on @xmath92 . hence , the test for the concurrence of roots at 1 and @xmath1 and the tests involving roots at @xmath4 are affected by the seasonal heterogeneity . to remove the nuisance parameters in the asymptotic distributions , we notice that the asymptotic behaviors of @xmath104 and @xmath106 , @xmath193 have identical forms as in phillips and perron ( 1988 ) . in light of their approach , we can construct pivotal versions of @xmath104 and @xmath106 , @xmath193 that converge in distribution to standard dickey - fuller distributions ( dickey and fuller , 1979 ) . more specifically , for @xmath193 we can substitute any consistent estimator for @xmath214 and @xmath215 below : @xmath216 however , there is no easy way to construct pivotal statistics for @xmath108 , @xmath217 , @xmath109 , @xmath218 , and f - statistics such as @xmath107 . the difficulties are two - fold . firstly the denominators of the asymptotic distributions of these statistics contain weighted sums with unknown weights @xmath219 and @xmath220 ; secondly @xmath139 and @xmath140 are in general correlated standard brownian motions as in theorem [ aug real ] . the result in theorem [ unaug real ] can be generalized . suppose @xmath13 is not generated by @xmath126 , and only has some of the seasonal unit roots . let @xmath150 , and @xmath221 . then we can find @xmath222 such that @xmath223 . the asymptotic distributions of @xmath104 , @xmath106 , @xmath193 and the f - statistics have the same forms as those in theorem [ unaug real ] , with @xmath138 substituted by @xmath224 , and @xmath93 based on @xmath225 . as for the asymptotic results under the alternative hyphothese , we conjecture that the powers of the unaugmented hegy tests converge to one as sample size goes to infinity . as in remark [ power ] , we can assume without loss of generality that @xmath13 has no root at @xmath0 , @xmath1 , or @xmath4 . then for @xmath154 , the coefficient @xmath155 corresponding to ( the misspecified constant parameter representation of ) @xmath13 are negative , according to proposition [ prop : hegy ] . we conjecture that for @xmath154 , the ols estimators @xmath156 in converge to @xmath155 , and as a result the power of the tests tend to one . since many of the asymptotic distributions delivered in theorem [ unaug real ] are non - standard , non - pivital , and not directly pivotable , we propose the application of bootstrap . since the regression error @xmath9 of is seasonally stationary , we in particular apply the seasonal block bootstrap of dudek et al . the algorithm of seasonal block bootstrap unaugmented hegy test is illustrated below . [ seasonal block boot ] step 1 : get the ols estimators @xmath115 , @xmath117 , t - statistics @xmath119 , @xmath120 , and the f - statistics @xmath157 , @xmath158 and @xmath159 from the regression of the unaugmented hegy test @xmath226 step 2 : record residual @xmath227 from regression @xmath228 step 3 : let @xmath229 , choose a integer block size @xmath230 , and let @xmath231 . for @xmath232 , let @xmath233 where @xmath234 is a sequence of iid uniform random variables taking values in @xmath235 with @xmath236 and @xmath237 + + step 4 : set the @xmath161 corresponding to the null hypothesis to be zero . for example , set @xmath168 for all @xmath12 when testing roots at @xmath4 . generate @xmath238 by @xmath239 step 5 : get ols estimates @xmath240 , @xmath241 , t - statistics @xmath242 , @xmath243 , and f - statistics @xmath173 from regression @xmath244 step 6 : run step 3 , 4 , and 5 for @xmath175 times to get @xmath175 sets of statistics @xmath240 , @xmath241 , @xmath242 , @xmath243 , and @xmath173 . count separately the numbers of @xmath240 , @xmath241 , @xmath242 , @xmath243 , and @xmath173 than which @xmath115 , @xmath117 , @xmath119 , @xmath120 , and @xmath157 are more extreme . if these numbers are higher than @xmath177 , then consider @xmath115 , @xmath117 , @xmath119 , @xmath120 and @xmath157 extreme , and reject the corresponding hypotheses . [ sbb fclt ] let @xmath245 @xmath246 where @xmath247 , & \sigma_{2}^{*}&=std^{\circ}[\frac{1}{\sqrt{4t}}\sum _ { t=1}^ { 4 t } ( -1)^{t}v_{t}^{*}],\\ \sigma_{3}^{*}&=std^{\circ}[\frac{1}{\sqrt{4t}}\sum _ { t=1}^ { 4 t } \sqrt{2}\sin(\frac{\pi t}{2})v_{t}^ { * } ] , & \sigma_{4}^{*}&=std^{\circ}[\frac{1}{\sqrt{4t}}\sum _ { t=1}^ { 4 t } \sqrt{2}\cos(\frac{\pi t}{2})v_{t}^{*}].\end{aligned}\ ] ] if @xmath248 , @xmath101 , @xmath249 , then no matter which hypothesis is true , @xmath250 in probability , where @xmath191 is a four - dimensional standard brownian motion . by the fclt given by proposition [ sbb fclt ] , the proof of theorem [ unaug real ] , and the convergence of the bootstrap standard deviation @xmath251 ( dudek et al . , 2014 ) , we have that the conditional distribution of @xmath252 , @xmath253 , @xmath193 and @xmath173 in probability converges to the limiting distribution of @xmath104 , @xmath106 , @xmath193 and @xmath157 , respectively . hence the consistence of the bootstrap . [ coro : unaug real ] suppose the assumptions in theorem [ unaug real ] hold . let @xmath194 be the probability measure corresponding to the null hypothesis @xmath195 . if @xmath248 , @xmath101 , @xmath249 , then @xmath254 @xmath255 @xmath256 we focus on the hypotheses test for root at 1 ( @xmath114 against @xmath257 ) , root at @xmath1 ( @xmath116 against @xmath258 ) , and root at @xmath4 ( @xmath118 against @xmath259 ) . in each hypothesis test , we equip one sequence with all nuisance unit roots at 1 , @xmath1 , and @xmath4 , and the other with none of the nuisance unit roots . the detailed data generation processes are listed in table [ dgp ] . to produce power curves , we let parameter @xmath2600 , 0.004 , 0.008 , 0.012 , 0.016 , and 0.020 . notice that @xmath261 is set to be seasonally homogeneous for the sake of simplicity . further , we generate six types of innovations @xmath9 according to table [ noise ] , where @xmath262 . the values of @xmath263 are assigned so that the misspecified constant parameter representation ( see section [ sec : settings ] ) of the `` period '' sequence has almost the same ar structure as the `` ar '' sequence . .data generation processes [ cols="^,^,^,^ " , ] now we present in figure [ fig : root1 ] , [ fig : root-1 ] , and [ fig : rooti ] the main simulation result of the seasonal iid bootstrap augmented hegy test and the seasonal block bootstrap unaugmented hegy test . this simulation includes two cases of nuisance roots ( see table [ dgp ] ) and six types of noises ( see table [ noise ] ) , and sets sample size @xmath264 , number of bootstrap replicates @xmath265 , number of iterations @xmath266 , and nominal size @xmath267 . when our data have a potential root at 1 , but no other nuisance roots at @xmath1 or @xmath4 , the power curves of the both bootstrap tests almost overlap , according to ( a)-(f ) in figure [ fig : root1 ] . further , both power curves start at the correct size , @xmath267 , and tend to one when @xmath261 departs from zero . hence both tests work well when no nuisance root occurs . when data have a potential root at 1 and all nuisance roots at @xmath1 and @xmath4 , the sizes of seasonal block bootstrap unaugmented hegy test are distorted in ( g ) , ( h ) , ( j ) , and ( l ) in figure [ fig : root1 ] . these distortions may result from the errors in estimating @xmath82 and the need to recover @xmath13 with the estimated @xmath82 . the size distortion in ( j ) is particularly serious , since the unit root filter @xmath268 is partially cancelled by the moving average ( ma ) filter @xmath269 , and this cancellation can not be handled well by block bootstrap ( paparoditis and politis , 2003 ) . in contrast , in ( l ) the filter @xmath268 is enhanced by the ar filters @xmath270 , thus the size is distorted toward zero . on the other hand , seasonal iid bootstrap augmented hegy test is free of the size distortions when data have nuisance roots . this is in part because the test recovers @xmath13 using the true values of @xmath82 , namely zero , instead of using the estimated values . moreover , when both hegy tests have almost the correct sizes as in ( i ) and ( k ) , seasonal iid bootstrap augmented hegy test attains equal or higher powers . therefore , when testing the root at 1 , seasonal iid bootstrap augmented hegy test is recommended . ( a)-(f ) have no nuisance roots ; ( g)-(l ) have all nuisance roots ; + blue dotted curve is for seasonal iid bootstrap ; red solid curve is for seasonal block bootstrap . [ fig : root1 ] now we come to the tests for root at @xmath1 . when none of the nuisance root at 1 or @xmath4 exists , the power curves of the two tests are very close to each other , as ( a)-(f ) in figure [ fig : root-1 ] indicate . this patterns of curves have been seen in ( a)-(f ) in figure [ fig : root1 ] , and indicate the nice performance of both tests . when nuisance roots are present , sizes of seasonal block bootstrap unaugmented hegy test are distorted in nearly all scenarios in ( g)-(l ) in figure [ fig : root-1 ] . in particular , the size distortion in ( i ) is the worst , because of the partial cancellation of the seasonal unit root filter @xmath271 and the ma filter @xmath272 . however , the power curves of seasonal iid bootstrap augmented hegy test start around the nominal size 0.05 in all of ( g)-(l ) . further , these curves tend to 1 , as @xmath261 grows larger . therefore , we recommend seasonal iid bootstrap test for testing root at @xmath1 . ( a)-(f ) have no nuisance roots ; ( g)-(l ) have all nuisance roots ; + blue dotted curve is for seasonal iid bootstrap ; red solid curve is for seasonal block bootstrap . [ fig : root-1 ] finally we discuss the tests for roots at @xmath4 . with none of the nuisance root at 1 or @xmath1 , ( a)-(f ) in figure [ fig : rooti ] illustrate that both tests achieve sizes that are close to the nominal size , and powers that tend to one . when all of nuisance roots show up , both tests suffer from some size distortions . the empirical sizes of seasonal iid bootstrap augmented hegy test are biased toward zero in ( g)-(l ) ; the sizes of seasonal block bootstrap unaugmented hegy test are biased toward zero in ( g ) and ( h ) , but are biased toward one in ( j)-(l ) . on the other hand , seasonal block bootstrap unaugmented hegy test s empirical powers prevail throughout ( g)-(l ) , and therefore shall be recommended for testing roots at @xmath4 . ( a)-(f ) have no nuisance roots ; ( g)-(l ) have all nuisance roots ; + blue dotted curve is for seasonal iid bootstrap ; red solid curve is for seasonal block bootstrap . [ fig : rooti ] in this paper we analyze the augmented and unaugmented hegy tests in the seasonal heterogeneous setting . given root at 1 or @xmath1 , the asymptotic distributions of the test statistics are standard . however , given concurrent roots at 1 and @xmath1 , or roots at @xmath4 , the asymptotic distributions are neither standard , pivotal , nor directly pivotable . therefore , when seasonal heterogeneity exists , hegy tests can be used to test the single roots at 1 or @xmath1 , but can not be directly applied to any combinations of roots . bootstrap proves to be an effective remedy for hegy tests in the seasonal heterogeneous setting . the two bootstrap approaches , namely 1 ) seasonal iid bootstrap augmented hegy test and 2 ) seasonal block bootstrap unaugmented hegy test , turn out both theoretically solid . in the comparative simulation study , seasonal iid bootstrap augmented hegy test has better performance when testing roots at 1 or @xmath1 , but seasonal block bootstrap unaugmented hegy test outperforms when testing roots at @xmath4 . therefore , when testing seasonal unit roots under seasonal heterogeneity , the aforementioned bootstrap hegy tests become competitive alternatives of the wald - test proposed by ghysels et al . further study will be needed to compare the theoretical and empirical efficiency of the two bootstrap hegy tests and the wald - test by ghysels et al . the appendix includes the proof of the theorems in this paper . we first present the proof for the asymptotics of the unaugmented hegy test , then the asymptotics of the augmented hegy test , then the consistency of the seasonal iid bootstrap augmented hegy test , and finally the consistency of the seasonal block bootstrap unaugmented hegy test . thoughout the appendix , let @xmath273 , @xmath274 , @xmath275 $ ] , @xmath276 be a four - dimensional standard brownian motion , @xmath277 denotes @xmath278 , and @xmath279 denotes @xmath280 . see hamilton ( 1994 , proposition 18.1 , pp . 547 - 548 ) for the proof with iid innovations , chan and wei ( 1988 ) for the proof under assumption 2.a , and de jong and davidson ( 2000 ) for the proof under assumption 2.b . [ le : unaug real2 ] let @xmath282 , and @xmath283 , where @xmath284 stands for unaugmented hegy , and @xmath285 is defined in . let @xmath286 , @xmath287 be the matrix generated by assigning zero to all entries of @xmath288 but those above the main diagonal . then , under @xmath126 , @xmath289 for the proof of part ( a ) , see the lemma 3.2(a ) of burridge and taylor ( 2001a ) and its proof . for part ( b ) , we only present the proof of the first statement . other statements are proven in similar ways . by lemma [ le : unaug real1 ] , @xmath290 let @xmath291 , @xmath292 , @xmath293 , and @xmath294 . then @xmath295^{-1}(\xi_{1},\xi_{2},\xi_{3},\xi_{4 } ) ' \text { by lemma \ref{le : unaug real2}},\\ \hat{\sigma}^{2 } & = ( 4t)^{-1}(\bm{v}'\bm{v}+2({\bm{x}}_{u}\bm{\hat{\pi}}-{\bm{x}}_{u}\bm{\pi})'(\bm{v}-{\bm{x}}_{u}\bm{\pi})\\ & \mathrel{\phantom{= } } + ( { \bm{x}}_{u}\bm{\hat{\pi}}-{\bm{x}}_{u}\bm{\pi})'({\bm{x}}_{u}\bm{\hat{\pi}}-{\bm{x}}_{u}\bm{\pi}))\\ & = ( 4t)^{-1}\bm{v}'\bm{v}+o_{p}(1 ) \text { by the consistency of } \bm{\hat{\pi}}\\ & \stackrel{p}\rightarrow \text{tr}(\bm{\gamma}_{0})/4,\\ \bm{t}&=\hat{\sigma}^{-1}[\text{diag}({\bm{x}}_{u}'{\bm{x}}_{u})^{-1}]^{-1/2}({\bm{x}}_{u}'{\bm{x}}_{u})^{-1}{\bm{x}}_{u}'\bm{v}\\ & \rightarrow ( \text{tr}(\bm{\gamma}_{0})/4)^{-1/2}[\text{diag}(\eta_{1},\eta_{2},\eta_{3},\eta_{4})]^{-1/2}(\xi_{1},\xi_{2},\xi_{3},\xi_{4})'.\end{aligned}\ ] ] further , the asymptotic distributions of f - statistics are identical with the asymptotic distributions of the averages of the squares of the corresponding t - statsitics , i.e. , @xmath296 , due to the asymptotic orthogonality indicated by lemma [ le : unaug real2 ] ( a ) . the proof follows the lines of said and dickey ( 1984 ) and contains two parts . firstly , we show when @xmath297 and @xmath298 simultaneously , the statistic of interest tends to a limit free of @xmath48 , and then we prove this limit tends to a certain distribution as @xmath297 . to begin with , notice that when @xmath102 , the error term of regression tends to a limit . surprisingly , this limit is in general not @xmath299 , because the regression falsely assumes seasonally homogeneous coefficients and thus in general can not find the correct residuals @xmath299 . to find the limit , recall that @xmath92 is defined as a misspecified constant parameter representation of @xmath9 . under assumption 1.b , the spectral densities of @xmath92 are finite and positive everywhere , so @xmath92 has ar@xmath32 and ma@xmath32 expressions @xmath300 where @xmath301 , @xmath302 . let @xmath303 , and @xmath304 , where @xmath305 are the ar coefficients defined in . since a misspecified constant parameter representation of @xmath306 is @xmath307 , which is exactly @xmath308 defined in , no ambiguity arises . the following lemma provides two properties of @xmath309 , whose proof is left to the readers . now we show when @xmath101 and @xmath311 simultaneously , the statistics of interest tend to certain limits . let @xmath312 be the design matrix of regression equation , @xmath313 be the estimated coefficient vector of regression equation , @xmath314 , @xmath315 , and @xmath316 . define the @xmath317 dimensional scaling matrix @xmath318 . then @xmath319 let @xmath320 be the @xmath321 induced norm of matrices . now we want to define a diagonal matrix @xmath322 such that @xmath323 converges to 0 in probability . by the multivariate beveridge - nielson decomposition ( see hamilton , 1994 , pp . 545 - 546 ) , since @xmath324 converges in probability to the seasonal average of autocovariance of @xmath325 of lag @xmath326 , we let @xmath327 where @xmath328 following the definition of @xmath322 , we make the following decomposition : @xmath329\bm{d}_{t}\bm{x}'\bm{\zeta}^{(k)}\\ & + \bm{r}^{-1}\bm{d}_{t}\bm{x}'(\bm{\zeta}^{(k)}-\bm{\zeta)}\\ & + \bm{r}^{-1}\bm{d}_{t}\bm{x}'\bm{\zeta}. \end{aligned } \label{said decomposition}\ ] ] notice the last term in the right hand side summation , @xmath330 , is free of @xmath48 . later we will find out its asymptotic distribution as @xmath201 . but now we need to prove the first two terms in the right hand side of converge to zero as @xmath201 and @xmath311 . indeed , @xmath331.\end{aligned}\ ] ] notice that @xmath332 . under assumption 1.b , @xmath9 is a varma sequence with finite orders , thus @xmath92 also has an arma expression with finite orders ( see osborn , 1991 ) , @xmath333 hence , @xmath334 has exponentially decaying coefficient @xmath335 . it follows straightforwardly that @xmath336 . for , notice that @xmath337.\ ] ] by lemma [ zeta ] and the stationarity of @xmath61 , @xmath338\\ & = \frac{1}{4}\sum_{s=-3}^{0}\sum_{h=-\infty}^{\infty}cov(v_{4t+s - i}\zeta_{4t+s},v_{4t+s - h - i}\zeta_{4t+s - h})+o(1)\\ & = \frac{1}{4}\sum_{s=-3}^{0}\sum_{h=-\infty}^{\infty}cov(v_{s - i}\zeta_{s},v_{s - h - i}\zeta_{s - h})+o(1).\end{aligned}\ ] ] without loss of generality we can focus on @xmath339 and @xmath340 . by writing @xmath325 and @xmath306 as linear combinations of @xmath299 , @xmath341 the right hand side of this inequality is assumed to be bounded under assumption 2.a . on the other hand , the right hand side is also bounded under assumption 2.b , by the lemma below . [ boundedness ] suppose ( i ) @xmath342 is a strictly stationary strong mixing time series with mean zero and finite @xmath62 moment for some @xmath63 , and ( ii ) @xmath343 s strong mixing coefficient @xmath344 satisfies @xmath345 . then @xmath64 such that for all @xmath346 and @xmath347 , @xmath348 we have proven that @xmath354=o(1)$ ] . similarly , it can be shown that @xmath355=o(1)$ ] . hence , follows . to justify , notice @xmath356 where @xmath357 indicates standard four - dimensional brownian motion . since @xmath358 @xmath359 such that @xmath360 . follows from the definition of @xmath361 . combining equations , , , and , we have @xmath362 . notice @xmath363 by . the consistency follows from @xmath364 and @xmath122 . further , the asymptotic distribution of @xmath365 can be derived with the asymptotic equivalence of @xmath366 and @xmath330 . notice @xmath330 is free of @xmath48 . as @xmath367 , @xmath368 converges in distribution to a functional of brownian motion , and the asymptotics of @xmath369 can be found with the following lemma . @xmath371^{2}}{\frac{1}{4}({\bm{c}}_{4}'{\bm{\theta}}(1){\bm{\omega}}{\bm{\theta}}(1)'{\bm{c}}_{4}+{\bm{c}}_{3}'{\bm{\theta}}(1){\bm{\omega}}{\bm{\theta}}(1)'{\bm{c}}_{3})}\\ & \mathrel{\phantom{\rightarrow}}+\frac { var({\tilde{\zeta}}_{t})[\sqrt{\frac{1}{4}{\bm{c}}_{4}'{\bm{\theta}}(1){\bm{\omega}}{\bm{\theta}}(1)'{\bm{c}}_{4}\frac{1}{4}{\bm{c}}_{3}'{\bm{\theta}}(1){\bm{\omega}}{\bm{\theta}}(1)'{\bm{c}}_{3}}(\int_{0}^{1 } w_{3}(r)d w_{4}(r)-\int w_{4}(r)dw_{3}(r))]^{2}}{\frac{1}{4}({\bm{c}}_{4}'{\bm{\theta}}(1){\bm{\omega}}{\bm{\theta}}(1)'{\bm{c}}_{4}+{\bm{c}}_{3}'{\bm{\theta}}(1){\bm{\omega}}{\bm{\theta}}(1)'{\bm{c}}_{3})}.\end{aligned}\ ] ] firstly we focus on the convergence of @xmath372 . the convergence of @xmath373 can be proven analogously . define @xmath374 such that @xmath375 . let @xmath376 , @xmath377 , @xmath378 , @xmath379 . then @xmath380 , and @xmath381\\ & \rightarrow\frac{1}{4}{\tilde{\theta}}(1){\bm{c}}_{1}'{\tilde{\bm{\psi}}}(1){\bm{\theta}}(1){\bm{\omega}}^{1/2}\int { \bm{w}}d { \bm{w } } ' { \bm{\omega}}^{1/2}{\bm{\theta}}(1)'{\tilde{\bm{\psi}}}(1)'{\bm{c}}_{1 } \ \text{(by lemma \ref{zeta } and fclt)}\\ & \mathrel{\phantom{\rightarrow}}+\frac{1}{4}{\tilde{\theta}}(1)[\sum_{s=-3}^{0}\sum_{k=-3}^{s-1}e\zeta_{4t+k}\zeta_{4t+s}+{\bm{c}}_{1}'\sum_{i=1}^{\infty}e{\bm{\zeta}}_{t - i}{\bm{\zeta}}_{t}'{\bm{c}}_{1}]\\ & = \frac{1}{4}{\tilde{\theta}}(1){\bm{c}}_{1}'{\tilde{\bm{\psi}}}(1){\bm{\theta}}(1){\bm{\omega}}^{1/2}\int { \bm{w}}d { \bm{w } } ' { \bm{\omega}}^{1/2}{\bm{\theta}}(1)'{\tilde{\bm{\psi}}}(1)'{\bm{c}}_{1 } \ \text{(since } \{{\tilde{\zeta}}_{t}\ } \text { is white noise)}\\ & = \frac{1}{4}{\tilde{\theta}}(1)({\tilde{\psi}}(1))^{2}{\bm{c}}_{1}'{\bm{\theta}}(1){\bm{\omega}}^{1/2}\int { \bm{w}}d { \bm{w } } ' { \bm{\omega}}^{1/2}{\bm{\theta}}(1)'{\bm{c}}_{1 } \ ( \text{since } { \bm{c}}_{1}'{\tilde{\bm{\psi}}}(1)={\tilde{\psi}}(1){\bm{c}}_{1}')\\ & = var({\tilde{\zeta}}_{t})\tilde{\theta}(1)\int_{0}^{1 } w_{1}(r)dw_{1}(r)\\ & \text{(by osborn ( 1991 , p. 378 ) , } \frac{1}{4}{\bm{c}}_{1}'{\bm{\theta}}(1){\bm{\omega}}{\bm{\theta}}(1)'{\bm{c}}_{1}= var({\tilde{\zeta}}_{t})\tilde{\theta}(1)^{2}).\end{aligned}\ ] ] secondly we show the convergence of @xmath382 . let @xmath383 , @xmath384 , @xmath385 , @xmath386 , and @xmath387 . then @xmath388\\ & \mathrel{\phantom{=}}-\frac{1}{4t}\sum_{t=1}^{t}{\tilde{\theta}}_{b}[{\bm{c}}_{3}'{\bm{\xi}}_{t-1}{\bm{\zeta}}_{t}'{\bm{c}}_{4}-{\bm{c}}_{4}'{\bm{\xi}}_{t-1}{\bm{\zeta}}_{t}'{\bm{c}}_{3}-\sum_{s=-3}^{-1}\zeta_{4t+s}\zeta_{4t+s+1}+\zeta_{4t-3}\zeta_{4t}]\\ & \rightarrow \frac{1}{4}{\tilde{\theta}}_{a}[{\bm{c}}_{3}'{\tilde{\bm{\psi}}}(1){\bm{\theta}}(1){\bm{\omega}}^{1/2}\int { \bm{w}}d { \bm{w } } ' { \bm{\omega}}^{1/2}{\bm{\theta}}(1)'{\tilde{\bm{\psi}}}(1)'{\bm{c}}_{3}\\ & \mathrel{\phantom{\rightarrow \frac{1}{4}{\tilde{\theta}}_{a}[}}+{\bm{c}}_{4}'{\tilde{\bm{\psi}}}(1){\bm{\theta}}(1){\bm{\omega}}^{1/2}\int { \bm{w}}d { \bm{w } } ' { \bm{\omega}}^{1/2}{\bm{\theta}}(1)'{\tilde{\bm{\psi}}}(1)'{\bm{c}}_{4}]\\ & \mathrel{\phantom{=}}-\frac{1}{4}{\tilde{\theta}}_{b}[{\bm{c}}_{3}'{\tilde{\bm{\psi}}}(1){\bm{\theta}}(1){\bm{\omega}}^{1/2}\int { \bm{w}}d { \bm{w } } ' { \bm{\omega}}^{1/2}{\bm{\theta}}(1)'{\tilde{\bm{\psi}}}(1)'{\bm{c}}_{4}\\ & \mathrel{\phantom{\mathrel{\phantom{=}}-\frac{1}{4}{\tilde{\theta}}_{b}[}}-{\bm{c}}_{4}'{\tilde{\bm{\psi}}}(1){\bm{\theta}}(1){\bm{\omega}}^{1/2}\int { \bm{w}}d { \bm{w } } ' { \bm{\omega}}^{1/2}{\bm{\theta}}(1)'{\tilde{\bm{\psi}}}(1)'{\bm{c}}_{3}]\\ & ( \text{by lemma \ref{zeta } and fclt , the covariances of } \zeta_{t } \text { cancel out since } \{{\tilde{\zeta}}_{t}\ } \text { is white noise})\\ & = \frac{1}{4}{\tilde{\theta}}_{a}|{\tilde{\psi}}(i)|^{2}[{\bm{c}}_{4}'{\bm{\theta}}(1){\bm{\omega}}^{1/2}\int { \bm{w}}d { \bm{w } } ' { \bm{\omega}}^{1/2}{\bm{\theta}}(1)'{\bm{c}}_{4}+{\bm{c}}_{3}'{\bm{\theta}}(1){\bm{\omega}}^{1/2}\int { \bm{w}}d { \bm{w } } ' { \bm{\omega}}^{1/2}{\bm{\theta}}(1)'{\bm{c}}_{3}]\\ & \mathrel{\phantom{=}}-\frac{1}{4}{\tilde{\theta}}_{b}|{\tilde{\psi}}(i)|^{2}[{\bm{c}}_{3}'{\bm{\theta}}(1){\bm{\omega}}^{1/2}\int { \bm{w}}d { \bm{w } } ' { \bm{\omega}}^{1/2}{\bm{\theta}}(1)'{\bm{c}}_{4}-{\bm{c}}_{4}'{\bm{\theta}}(1){\bm{\omega}}^{1/2}\int { \bm{w}}d { \bm{w } } ' { \bm{\omega}}^{1/2}{\bm{\theta}}(1)'{\bm{c}}_{3}]\\ & ( \text{since } { \bm{c}}_{3}'{\tilde{\bm{\psi}}}(1)={\tilde{\psi}}_{b}{\bm{c}}_{4}'+{\tilde{\psi}}_{a}{\bm{c}}_{3 } ' , \ { \bm{c}}_{4}'{\tilde{\bm{\psi}}}(1)={\tilde{\psi}}_{a}{\bm{c}}_{4}'-{\tilde{\psi}}_{b}{\bm{c}}_{3 } ' , \ \text{and } { \tilde{\psi}}_{a}^{2}+{\tilde{\psi}}_{b}^{2}=|{\tilde{\psi}}(i)|^{2}).\end{aligned}\ ] ] similarly , @xmath389\\ & \mathrel{\phantom{=}}+\frac{1}{4}{\tilde{\theta}}_{a}|{\tilde{\psi}}(i)|^{2}[{\bm{c}}_{3}'{\bm{\theta}}(1){\bm{\omega}}^{1/2}\int { \bm{w}}d { \bm{w } } ' { \bm{\omega}}^{1/2}{\bm{\theta}}(1)'{\bm{c}}_{4}-{\bm{c}}_{4}'{\bm{\theta}}(1){\bm{\omega}}^{1/2}\int { \bm{w}}d { \bm{w } } ' { \bm{\omega}}^{1/2}{\bm{\theta}}(1)'{\bm{c}}_{3}]\end{aligned}\ ] ] the lemma follows from @xmath390 and ( osborn , 1991 ) @xmath391 now we come to the asymptotic distribution of the t - statistics and the f - statistics . notice , @xmath392_{ii}]^{-1/2}[(\bm{x}'\bm{x})^{-1}\bm{x}'\bm{\zeta}^{(k)}]_{i}\\ & = \hat{\sigma}^{-1}[[[(4t - k)^{-2}(\bm{x}'\bm{x})^{-1}]_{ii}]^{-1/2}-[[\bm{r}^{-1}]_{ii}]^{-1/2}](4t - k)[(\bm{x}'\bm{x})^{-1}\bm{x}'\bm{\zeta}^{(k)}]_{i}\\ & \mathrel{\phantom{=}}+\hat{\sigma}^{-1}[[\bm{r}^{-1}]_{ii}]^{-1/2}((4t - k)(\bm{x}'\bm{x})^{-1}\bm{x}'\bm{\zeta}^{(k)}-\bm{r}^{-1}(4t - k)^{-1}\bm{x}'{\bm{\zeta}})_{i}\\ & \mathrel{\phantom{=}}+\hat{\sigma}^{-1}[[\bm{r}^{-1}]_{ii}]^{-1/2}(\bm{r}^{-1}(4t - k)^{-1}\bm{x}'{\bm{\zeta}})_{i}\\ & = \hat{\sigma}^{-1}[[\bm{r}^{-1}]_{ii}]^{-1/2}(\bm{r}^{-1}(4t - k)^{-1}\bm{x}'{\bm{\zeta}})_{i}+o_{p}(1).\end{aligned}\ ] ] by the consistency of @xmath365 , we have @xmath393 . the asymptotic distributions of the t - statistics follows straightforwardly from lemma [ asy zeta ] . further , the asymptotic distributions of the f - statistics are identical with the asymptotic distributions of the averages of the squares of the corresponding t - statistics because of the asymptotic orthogonality of the regression . hence , the proof of theorem [ aug real ] is complete . define @xmath394 and @xmath234 such that @xmath395 and @xmath396 . by algorithm [ seasonal iid bootstrap ] , @xmath394 is a sequence of independent but not identical random variables , while @xmath234 is a sequence of iid random variables . recall @xmath397 where @xmath398 is the regression error . let @xmath399 let @xmath400 be the partial sum of @xmath401 above . formally , @xmath402 to justify theorem [ iid fclt ] , it suffices to show @xmath403 @xmath404 because the unconditional convergence in implies that in probability the conditional distribution of @xmath405 given @xmath13 converges to zero . to prove , we can without loss of generality focus on the uniform convergence of the first coordinate , that is , uniformly in @xmath406 , @xmath407 notice that uniformly in @xmath406 , @xmath408 where @xmath409 and @xmath410 have obvious definitions . now we show @xmath411 , and @xmath412 , uniformly in @xmath406 . for @xmath409 , notice if @xmath413 , then @xmath285 is weakly stationary , so @xmath414 is @xmath415 ( see berk , 1974 ) , and it follows straightforwardly that @xmath411 uniformly in @xmath406 . on the other hand , if @xmath416 , then by theorem [ aug real ] , @xmath417 . let @xmath418 it suffices to show that @xmath419 . by continuous mapping theorem , it suffices to prove @xmath420 , where @xmath421 . it is straightforward to show the weak convergence of the finite dimensional distributions of @xmath422 . furthermore , @xmath422 is tight , since ( see billingsley , 1999 , pp . 146 - 147 ) @xmath423 , @xmath424\\ = & e[var^{\circ}[\frac{q_{t}(r_{2})}{t}-\frac{q_{t}(r)}{t}]var^{\circ}[\frac{q_{t}(r)}{t}-\frac{q_{t}(r_{1})}{t}]]\rightarrow 0 . \end{aligned } \label{tightness}\ ] ] hence @xmath420 , and @xmath411 uniformly in @xmath406 follows . for @xmath410 , in light of the derivation of theorem [ aug real ] , it can be shown that @xmath425 holds not only under alternative hypotheses but also under the null . hence , it follows that uniformly in @xmath406 , @xmath412 . therefore , recalling , we have @xmath426 further , it is straightforward to show @xmath427\stackrel{p}\rightarrow 0 $ ] , and @xmath428\stackrel{p}\rightarrow 0 $ ] . using the same decomposition as in , @xmath429 . hence we have proven . secondly we prove , notice that the standard deviations in the definition of @xmath430 are bounded in probability . for example , @xmath431 further , given @xmath13 , for fixed @xmath27 , @xmath432 are conditionally iid random variables . finally , for all @xmath433 , @xmath434\stackrel{p}\rightarrow u,\ ] ] @xmath435 the convergence @xmath400 of to @xmath436 follows by generalizing ( see kreiss and paparoditis , 2015 ) the real world result of helland ( 1982 , theorem 3.3 ) to the bootstrap world . without loss of generality , assume block size @xmath230 is a multiple of four . let @xmath437 . then the @xmath438th block of @xmath439 starts from @xmath440 . let @xmath441 be the rescaled aggregation of the @xmath438th block , defined by @xmath442 let @xmath443 be the partial sum of the block aggregations above . formally , @xmath444 to prove theorem [ sbb fclt ] , it suffices to show @xmath445 @xmath446 where @xmath320 denotes the @xmath321 norm . to show , without loss of generality we focus on the uniform convergence of the first coordinate , that is , uniformly in @xmath406 , @xmath447 . notice that , @xmath448 where @xmath449 denotes the total number of the blocks , and @xmath450 is the length of the @xmath438th block . it suffices to only consider the first term in , since @xmath451 by the definition of @xmath452 , @xmath453 now we show the second term on the right hand side of the equation above converges uniformly in @xmath406 to 0 in probability . here we only present the result for j=1 , s=0 . notice if @xmath454 for some @xmath12 , then @xmath455 . hence , the result follows the weakly stationarity of the vector sequence @xmath456 . on the other hand , if @xmath457 for all @xmath12 , then @xmath458 . hence , we only need to show that @xmath459 has @xmath460 , where @xmath461 . the convergence of finite dimensional distribution of @xmath462 can be proven by the line of politis and paparaditis ( 2003 , p. 841 ) . furthermore , it can be shown that @xmath462 is tight using . hence @xmath460 . therefore , @xmath463 uniformly in @xmath406 . since it is straightforward to show @xmath464)| \stackrel { p } \rightarrow 0\ ] ] uniformly in @xmath406 , and @xmath465 ) -\frac{1}{\sqrt{4t}}\sum_{m=1}^{\lfloor lu_{1 } \rfloor}\sum_{h=1}^{b}(v_{i_{m}+h-1}-e^{\circ}[v_{i_{m}+h-1}])| \stackrel { p } \rightarrow 0\ ] ] uniformly in @xmath406 , we have obtained that @xmath466)| \stackrel { p } \rightarrow 0 \label{sbb fclt 2}\ ] ] uniformly in @xmath406 . now we show that @xmath467 -var^{\circ}[\frac{1}{\sqrt{b}}\sum_{h=1}^{b}v_{i_{m}+h-1}]\stackrel { p}\rightarrow 0 $ ] . notice , @xmath468 ) + \frac{1}{\sqrt{4t}}\sum_{m=1}^{l}\sum_{s=-3}^{0}\sum_{t=1}^{b/4}(e^{\circ}v_{i_{m}+4t+s-1}-\frac{1}{t}\sum_{t=1}^{t}v_{4t+s})\\ & \mathrel{\phantom{=}}-\sum_{j=1}^{4}\sum_{s=-3}^{0}(\hat{\pi}_{j , s}-\pi_{j , s})\frac{1}{\sqrt{4t}}\sum_{m=1}^{l}\sum_{t=1}^{b/4}(j , y_{i_{m}+4t+s-1}-\frac{1}{t}\sum_{t=1}^{t}y_{4t+s})\\ & = a_{t}+b_{t}-\sum_{j=1}^{4}c_{t , j}\end{aligned}\ ] ] where @xmath469 and @xmath470 have obvious definitions . it is straightforward to show @xmath471\stackrel { p } \rightarrow 0 $ ] , @xmath472\stackrel { p } \rightarrow 0 $ ] for @xmath27 , and @xmath473=var^{\circ}[\frac{1}{\sqrt{b}}\sum_{h=1}^{b}v_{i_{m}+h-1}]$ ] . hence , we have @xmath474 -var^{\circ}[\frac{1}{\sqrt{b}}\sum_{h=1}^{b}v_{i_{m}+h-1}]\stackrel { p}\rightarrow 0 . \label{sbb fclt 3}\ ] ] by and , we have shown @xmath475 uniformly in @xmath406 , and thus @xmath476 uniformly in @xmath406 , @xmath477 , @xmath478 and @xmath479 . secondly we prove . given assumption b.1 , it is sufficient to show that the following three properties hold : @xmath480 \stackrel { p } \rightarrow u , \ : \forall u\geq 0 , \ : and \ : \forall \ : i=1, ... ,4 , \label{sbb fclt3 } \\ & \sum\limits_{m=1}^{\lfloor lu \rfloor}e^{\circ}[{\upsilon}_{l , m}^{(i)^{2}}1(|{\upsilon}_{l , m}|>\epsilon ) ] \stackrel { p } \rightarrow 0 , \ : \forall u\geq 0 , \ : \ : \forall \ : i=1, ... ,4 , \label{sbb fclt4 } \\ & \sum\limits_{m=1}^{\lfloor lu \rfloor}e^{\circ}[{\upsilon}_{l , m}^{(i)}{\upsilon}_{l , m}^{(j ) } ] \stackrel { p } \rightarrow 0 , \ : \forall u\geq 0 , \ \forall \ : i , j \in \{1,2,3,4\ } , \ : i\neq j. \label{sbb fclt5}\end{aligned}\ ] ] helland ( 1982 ) shows that if @xmath481 is a martingale difference array and the above three properties hold in real world , then @xmath482 . by beveridge - neilson decomposition ( hamilton , 1994 , proposition 17.2 , p. 504 ) , helland s result can be generalized to the case when @xmath481 is a convolution of a constant array and a martingale difference array . further , helland s result can be generalized to the bootstrap world ( see kreiss and paparoditis , 2015 ) . hence the sufficiency of the three properties above . to verify and , notice that @xmath483 , @xmath484 , @xmath485=\lfloor lt \rfloor / l \rightarrow t,\ ] ] and , by the dominated convergence theorem , @xmath486 \stackrel { p } \rightarrow 0.\ ] ] hence , it remains to verify the , which indicates asymptotic independence between coordinates of @xmath443 . note that the third property need to be proved for all @xmath487 . here we cite as an example the case @xmath339 and @xmath488 . the rest of cases can be shown by similar calculations . notice , @xmath489 = & \frac{e^{\circ}[\frac{1}{\sqrt{b}}\sum_{h=1}^{b}v_{i_{1}+h-1}\frac{1}{\sqrt{b}}\sum_{r=1}^{b}\sqrt{2}\sin(\pi r/2)v_{i_{1}+r-1 } ] } { std^{\circ}[\frac{1}{\sqrt{b}}\sum_{h=1}^{b}v_{i_{1}+h-1}]std^{\circ}[\frac{1}{\sqrt{b}}\sum_{r=1}^{b}\sqrt{2}\sin(\pi r/2)v_{i_{1}+r-1}]}\\ & -\frac{e^{\circ}[\frac{1}{\sqrt{b}}\sum_{h=1}^{b}v_{i_{1}+h-1}]e^{\circ}[\frac{1}{\sqrt{b}}\sum_{r=1}^{b}\sqrt{2}\sin(\pi r/2)v_{i_{1}+r-1}]}{std^{\circ}[\frac{1}{\sqrt{b}}\sum_{h=1}^{b}v_{i_{1}+h-1}]std^{\circ}[\frac{1}{\sqrt{b}}\sum_{r=1}^{b}\sqrt{2}\sin(\pi r/2)v_{i_{1}+r-1}]}.\end{aligned}\ ] ] since @xmath490\stackrel { p } \rightarrow 0,\ : e^{\circ}[\frac{1}{\sqrt{b}}\sum_{r=1}^{b}\sqrt{2}\sin(\pi r/2)v_{i_{1}+r-1}]\stackrel { p } \rightarrow 0,\ ] ] and both @xmath491 $ ] and @xmath492 $ ] converge in probability to constants ( dudek et al . , 2014 ) , we only need to show that @xmath493 \stackrel { p } \rightarrow 0.\ ] ] notice , @xmath494\\ & = \frac{\sqrt{2}}{b(t - b/4)}\sum_{i=1}^{t - b/4}\sum_{h=1}^{b}\sum_{r=1}^{b}\sin(\pi r/2)v_{4i+h-4}v_{4i+r-4}\\ & = -a+b+o_{p}(1),\end{aligned}\ ] ] where @xmath495 the proof under assumption [ assump 1b ] is complete after showing @xmath496 by lemma [ martingale difference ] below . now consider assumption [ assump 2b ] . let @xmath497 . let ( @xmath498,@xmath499,@xmath500,@xmath501 be the eigenvalues of @xmath502 . it is sufficient ( wooldridge and white , 1988 , corollary 4.2 ) to show that the following two properties hold : @xmath503 notice , to show , it suffices to show , which is ensured by lemma [ boundedness ] and lemma [ martingale difference ] . equation follows from the continuity of the eigenvalue function . hence we have completed the proof when block size @xmath230 is a multiple of four . when @xmath230 is not a multiple of four , it is straightforward to show . for , let @xmath504 since @xmath505 , @xmath6 are mutually independent with respect to @xmath182 , and @xmath506 in probability for all @xmath11 , we have @xmath507 in probability . [ martingale difference ] suppose ( i ) @xmath342 is a fourth - order stationary time series with finite @xmath62 moment for some @xmath63 . ( ii ) @xmath64 , @xmath65 @xmath2 , @xmath47 , @xmath48 , and @xmath66 , @xmath508 . suppose @xmath509 and @xmath510 . then , @xmath511\rightarrow 0.\ ] ] @xmath512\\ & = \frac{1}{b^{2}n^{2}}\sum_{t_1=1}^{n}\sum_{t_2=1}^{n}\sum_{j_1=1}^{b}\sum_{j_2=1}^{b}cov[z_{0}z_{-j_1},z_{t_2-t_1}z_{t_2-t_1-j_1}]\\ & = \frac{1}{b^{2}n^{2}}\sum_{h=1-n}^{n-1}(n-|h|)\sum_{j_1=1}^{b}\sum_{j_2=1}^{b}cov[z_{0}z_{-j_1},z_{h}z_{h - j_1}]\\ & < \frac{k}{n}\rightarrow 0 . & \mbox{\qedhere}\end{aligned}\ ] ] 99 berk , kenneth n. `` consistent autoregressive spectral estimates . '' the annals of statistics ( 1974 ) : 489 - 502 . billingsley , patrick . convergence of probability measures . new jersey : john wiley & sons , 1999 . breitung , jorg , and philip hans franses . `` on phillips - perron - type tests for seasonal unit roots . 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both seasonal unit roots and seasonal heterogeneity are common in seasonal data . when testing seasonal unit roots under seasonal heterogeneity , it is unclear if we can apply tests designed for seasonal homogeneous settings , i.e. the hegy test ( hylleberg , engle , granger , and yoo , 1990 ) . in this paper ,
the validity of both augmented hegy test and unaugmented hegy test is analyzed .
the asymptotic null distributions of the statistics testing the single roots at @xmath0 or @xmath1 turn out to be standard and pivotal , but the asymptotic null distributions of the statistics testing any coexistence of roots at @xmath0 , @xmath1 , @xmath2 , or @xmath3 are non - standard , non - pivotal , and not directly pivotable .
therefore , the hegy tests are not directly applicable to the joint tests for the concurrence of the roots . as a remedy ,
we bootstrap augmented hegy with seasonal independent and identically distributed ( iid ) bootstrap , and unaugmented hegy with seasonal block bootstrap .
the consistency of both bootstrap procedures is established .
simulations indicate that for roots at @xmath0 and @xmath1 seasonal iid bootstrap augmented hegy test prevails , but for roots at @xmath4 seasonal block bootstrap unaugmented hegy test enjoys better performance .
* keywords : * seasonality , unit root , ar sieve bootstrap , block bootstrap , functional central limit theorem .
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You are an expert at summarizing long articles. Proceed to summarize the following text:
during the past few decades , elaborate solar neutrino @xcite and atmospheric neutrino @xcite experiments have provided a wealth of convincing evidences for the existence of massive neutrinos and neutrino mixing , which could have an essential impact on particle physics , astrophysics and cosmology . attentions have been focused on solving the puzzles of unexpected discrepancies between calculated and observed neutrino fluxes . instead of the more difficult and unlikely solution from an improved solar model @xcite , the solar @xmath3 deficit could be reconciled with the prediction if neutrino oscillations occur either in vacuum or in the presence of solar matter . the flavor oscillation can be parameterized by the mass - squared differences of the neutrino mass eigenstates @xmath4 and @xmath5 , the mixing angles between weak eigenstates and mass eigenstates of the neutrinos ( @xmath6 is assumed ) . in terms of these parameters , the just - so vacuum oscillation @xcite requires @xmath7 ev@xmath8 and @xmath9 , while the msw resonant effect @xcite in the sun becomes important if @xmath10 ev@xmath8 @xmath11 ev@xmath8 , @xmath12 ( large angle solution ) , or @xmath13 ev@xmath8 @xmath14 ev@xmath8 , @xmath15 ( small angle solution ) @xcite . recent atmospheric neutrino data from the super - kamiokande @xcite further provide a strong evidence in support of neutrino oscillation as the cause to deficit of muon neutrinos , provided @xmath16 ev@xmath8 and @xmath17 . it is clear that this solution to the neutrino anomaly in the atmosphere represents quite a distinct area in the parameter space as compared to that of the solar neutrino deficit . based on the conclusive lep experiment at cern @xcite that there are three flavors of light , active neutrinos participating in the weak interaction , a direct - mass hierarchy @xmath0 , with @xmath18 ( @xmath19 and @xmath20 ) naturally accommodates the scales of both the two mass - squared differences and provides solutions to both puzzles : the conversion @xmath21 causes the observed deficit in the solar @xmath3 flux and the vacuum oscillation @xmath22 suppresses the @xmath23 flux in the atmosphere . in addition to the sun and the atmosphere , type - ii supernovae are also natural sources that emit neutrinos . despite the first - ever observation of sn neutrino signals from sn 1987a @xcite , detailed neutrino spectral shapes have not yet been determined with certainty due to low statistics and the physical processes that are not well understood . this difficulty is accompanied by , for instance , the uncertainties in the characteristic temperatures @xmath24 as neutrinos were emitted from the neutrino - spheres . consequently , the interpretation of future measurements of sn neutrinos would contain ambiguity in that the observed spectrum , which may have been deformed through conversion processes , could be simulated by different set of parameters at different temperatures . it is therefore worthwhile to investigate how the uncertainty in @xmath24 could impact the interpretation of events at terrestrial detectors . in this paper , the parameters that solve solar and atmospheric neutrino problems are taken as inputs , a natural choice as also adopted by some earlier works @xcite @xcite . in addition , with the uncertainty in @xmath24 considered , we study whether a particular set of parameters could be singled out by future observations of sn neutrinos . unlike solar neutrinos , the initial neutrino flux from a supernova contains all flavors of neutrino : @xmath25 and their anti - particles . under the direct - mass hierarchy of neutrinos , the original @xmath2 spectra will be modified by the msw effect as neutrinos propagate through the resonance . the @xmath26 spectra , on the contrary , is subject only to vacuum oscillation which yields an large averaged survival probability of @xmath27 . the high - energy @xmath28 would not be converted to the easily detectable @xmath29 ( for instance , at super - kamiokande ) through the msw effect unless neutrino masses are inverted , in which case the heavier mass eigenstate has a larger component in @xmath3 than in @xmath23 or @xmath30 . since the mixing angles are defined in the first octant , the weak eigenstates @xmath31 , and @xmath3 are predominant in the mass eigenstates @xmath32 , and @xmath33 , respectively . under the direct - mass hierarchy where @xmath34 , the mass eigenstates follow the hierarchy @xmath35 , while in the inverted - mass hierarchy , for instance , @xmath36 , the pattern @xmath37 follows . some models and phenomenological consequences involving inverted neutrino masses have been discussed @xcite . although current msw solutions to the solar neutrino problem ( snp ) have excluded @xmath3 as the heavier eigenstate , the inverse hierarchy could remain viable if the just - so vacuum oscillation is the solution for the snp . if the inverted masses do apply and the resonance conditions for the anti - neutrinos are met , this could lead to an effective conversion between @xmath29 and the higher - energy @xmath38 to yield copious @xmath29-type events in the earth - bound detector . with the uncertainty in @xmath24 considered , it is our second goal to investigate influences of both the direct and inverted mass hierarchies to future observations of sn neutrinos and how future measurements can play a role in this unsettled issue of direct versus inverted neutrino masses . this paper is organized as follows . in section ii we summarize the general features of stellar collapse and properties of the emitted neutrinos , and show how the uncertainty in neutrino temperature could affect the outcomes in the detector . section iii and section iv contain more general results expected from the future observations at both super - kamiokande and sno , for direct and inverted masses , respectively . based on the measurements , possible schemes which could provide discrimination among input parameters and between the two mass hierarchies are proposed . section v contains discussions and our concluding remarks . a massive star ( @xmath39 ) becomes unstable at the last stage of its evolution . when the mass of the iron core reaches the chandrasekhar limits ( @xmath40 ) , it begins to collapse into a compact object of extremely high density , and the gravitational binding energy is released in the form of neutrinos . et al . _ @xcite have pointed out that the total emitted energy , the averaged neutrino luminosity , and the mean neutrino energy are independent of the explosive mechanism but depend only on the mass of the initial iron core . regardless of the details of collapsed and bounce , it is well established that to form a typical neutron star after the collapse , an amount of @xmath41 erg , about 99% of the binding energy would be released in the form of neutrinos . each ( anti)neutrino species will carry away about the same amount of energy . neutrinos are emitted from a collapsed star through two different processes : neutronization burst during the pre - bounce phase and thermal emission in the post - bounce phase . the neutronization burst of a @xmath3 flux is produced by the electron capture on protons : @xmath42 . the thermal emission creates @xmath43 pairs of all three flavors via the annihilation of @xmath44 pairs : @xmath45 . the duration of neutronization burst lasts about a few millisecond and takes away 1%-10% of the total binding energy . the thermal emission phase has a much wider spread of time structure , in the order of 10 seconds . the initial neutrino spectrum is usually approximated by a fermi - dirac or a boltzmann distribution with a constant temperature and zero chemical potential . to reduce the high - energy tail of the fermi - dirac distribution , some elaborate models introduce a nonzero chemical potential @xcite . it is clear that the event numbers in a detector depend crucially on the @xmath2 temperature . however , the numerical calculations based upon various models and physical arguments give rise to relatively wide ranges of temperature for each neutrino species @xcite and this uncertainty in @xmath24 could complicate the signatures concerning the oscillation of neutrinos from a supernova . one may refer to ref . @xcite for a review of neutrino oscillations . although all the three flavors are emitted from a supernova , the phenomenon of sn neutrino oscillation can be well described through @xmath46 and @xmath47 @xcite . under the direct - mass hierarchy , the probability @xmath48 is nearly independent of energy and is approximated by the vacuum oscillation expression . the probability @xmath49 is energy - dependent and contains four parameters under a proper parameterization of the mixing matrix in the full 3-@xmath2 formalism : @xmath50 . in what follows , these four parameters will simply be denoted as @xmath51 , and @xmath52 , respectively . among the above four parameters , the angle @xmath52 is special in certain aspect . in addition to the limit @xmath53 at 90% c.l . set by the chooz @xcite long baseline reactor in the disappearance mode @xmath54 , an analysis in ref . @xcite also has given allowed ranges of @xmath55 and @xmath56 for @xmath57 . note that to zeroth order of @xmath58 , the probability becomes @xcite @xmath59 one may examine @xmath52 in more details through the iso - probability contours for sn neutrinos at several distinct scales of @xmath60 , as shown in figure 1 . within the interested range of @xmath61 , ( @xmath62 , or @xmath63 ) , the @xmath49 contours are almost independent of @xmath52 for @xmath64 ( @xmath65 ) . the parameter @xmath52 begins to show slight influence on @xmath49 contours only at very small @xmath61 and very large @xmath52 . hence , for our purpose the choice of @xmath52 within @xmath57 would only affect the results slightly . the value @xmath66 will be adopted for definiteness . as for other input parameters , the following are taken : large angle msw solution(la ) : @xmath67 ev@xmath68 small angle msw solution(sa ) : @xmath69 ev@xmath70 just - so vacuum solution(js ) : @xmath71 ev@xmath72 . in what follows , an initial flux described by a fermi - dirac spectrum with zero chemical potential will be assumed . the detailed time evolution during the cooling phase has been ignored , while the averaged magnitudes and effective temperatures of @xmath73 flux are used instead @xcite . the event numbers at the detectors for the neutrino of type @xmath74 are estimated by @xmath75 here @xmath76 is the number of targets in the detector , @xmath77 is the initial number of @xmath78 , @xmath79 is the distance between the supernova and the earth , @xmath80(e@xmath81 ) is the cross section for the corresponding reaction , @xmath82 is the surviving probability for @xmath78 , @xmath83 is the temperature for @xmath78 , and @xmath84}.\ ] ] in evaluating the surviving probability , the electron number per nucleon is assumed to remain a constant ( @xmath85 ) and the density profile outside the neutrino - sphere ( @xmath86 cm ) is described by the power - law @xmath87 . to grasp the picture on how uncertainties in neutrino temperatures could affect the interpretation of observed events , we may tentatively assume @xmath88=3 mev , @xmath89= 6 mev ( @xmath90 ) , and compare outcomes from @xmath91=3 mev and @xmath91=4.5 mev . at super - kamiokande @xcite , contributions from the inverse beta decay @xmath92 predominate due to the high cross section . events from other interactions : @xmath93 @xcite and @xmath94 @xcite , will also be included in our calculations although these events accumulate up to less than 5% of the @xmath95 events . the threshold energy is taken to be 5 mev and the detector efficiency is assumed to be 100% . for 32 kton of water , one expects roughly @xmath96 neutrino events for a type ii supernova at the center of our galaxy ( @xmath97 10 kpc away ) . since the cross section for @xmath98 is proportional to @xmath99 , and @xmath100 for the fermi - dirac distribution , a larger temperature gap between @xmath29 and @xmath101 would cause a more severely distorted spectra from the original one . hence the difference between @xmath91 and @xmath102 determines to what extend the events are enhanced by oscillation . for the direct masses where @xmath0 , possible results of the ratio @xmath103 ( @xmath104 indicates oscillation , and @xmath105 indicates the case of no oscillation ) using specific input parameters la , sa , and js are shown in figure 2 . the curves representing la and sa are due to msw effects of the @xmath2-type events and the vacuum oscillation of the @xmath26-type events , while the js curve is due to vacuum oscillations of both @xmath2- and @xmath26-type events . one observes that js parameters could raise event numbers most effectively , an increase of @xmath97 55% is possible at @xmath1063 mev ( @xmath107 ) . the enhancement decreases as @xmath91 approaches @xmath102 . near a particular point where @xmath108 , the conversion of @xmath29 to @xmath101 would not alter the original @xmath29 spectrum , all the scenarios yield @xmath109 and are indistinguishable among each other . the complication arises from the fact that if , for instance , @xmath110 is observed , this observation is then either due to la parameters at @xmath111 ev or the js parameters at @xmath112 ev . the uncertainties in neutrino temperatures would therefore render a wide range of predictions at the detectors . informations such as the clues for oscillation and neutrino parameters would be hard to understand or even lost due to this complication . for the observation of sn neutrinos , the extremely distinct time structures between the neutronization burst and the beginning of thermal emission would allow a clear separation at the @xmath113 cherenkov detector . these two groups of events are discussed separately . the spectral shape and the total energy of @xmath3 from the early pre - bounce burst is still poorly known . for the purpose of qualitative discussion , the spectrum is arbitrarily chosen to be the same as that of thermal @xmath3 ( fermi - dirac ) with the same mean energy and a total of 5% the binding energy of a typical neutron star ( @xmath114 ) . during this early phase , one expects to observe the forward directional events due to elastic scattering @xmath115 and the backward events from @xmath116 . these neutronization events are summarized in table i. we note that the forward events are relatively insensitive to the uncertainty in @xmath3 temperature . the oscillation signature manifests itself through the drastically reduced forward events as compared to the original one , although practically the separation among la , sa , and js using events observed during this early phase is difficult . the backward events on the other hand , are more sensitive to @xmath117 . the difficulty associated with the backward events comes from the extremely small numbers . if @xmath118 mev and the total neutrino energy at this stage is down to @xmath119 of @xmath114 , the backward events are practically unobservable . because of the rapidly increased cross section for @xmath116 at higher energy : @xmath120 , the situation could be improved if @xmath117 is higher or if the neutrinos emitted during this phase have larger energy partition , which is quite model - dependent . unlike the backward events , the forward event numbers are roughly in the order of 10 even if @xmath121 3 mev and the @xmath3 flux takes away as low as @xmath119 of @xmath114 . by using the numerous @xmath122 emitted from the inverse beta decay , the distorted @xmath29 spectrum would be determined with better statistics . to account for the uncertainties in @xmath117 and @xmath123 , we may let @xmath124 @xmath91 , @xmath125 @xmath91 and compare the outcomes for @xmath91= 4 mev , 5 mev , and 6 mev . the parameters @xmath126 and @xmath127 are allowed to vary within @xmath128 and @xmath129 to roughly include the temperature ranges given by current models . expected ranges for the ratios @xmath130 are summarized in table ii , where @xmath131 includes events from the inverse beta decay and the neutrino interactions with oxygen , @xmath132 represents the forward scattering events . for @xmath133 mev , there are two overlapped areas in @xmath134 : between no , sa and between js , la . the same overlapping structure remains for @xmath135 mev and @xmath136 mev . despite the wealthy information conveyed through the @xmath122 spectrum at super - kamiokande , from table ii it seems unlikely that a clear separation among input parameters could be achieved using the otherwise model - independent quantity @xmath134 unless the uncertainties in neutrino temperatures are reduced significantly . the neutral current@xmath137 breakup reactions of deuterium in sno @xcite are flavor - blind for neutrinos : @xmath138 @xmath139 where @xmath140 . the charged current reactions include two parts , @xmath141 @xmath142 with 1 kton of @xmath143 and a threshold energy @xmath97 5 mev ( 100% detection efficiency assumed ) , both @xmath144 and @xmath145 should roughly yield event numbers in the order of @xmath146 . the ratios @xmath147 at sno seem to provide a solution as to how a particular set of parameter could be singled out , as will be shown below . one may arbitrarily fix @xmath117 and parameterize other temperatures in a similar way : let @xmath148 and allow an uncertainty in @xmath91 as well : @xmath149 , with @xmath150 and @xmath151 . the ratio @xmath152 for @xmath153 and 5 mev are listed in table iii . we found that even if the uncertainties in @xmath123 and @xmath91 may complicate the interpretation of observed events , each of the candidates gives rise to a distinct region in @xmath154 . in practical , if uncertainties in @xmath155 and @xmath156 can be reduced in the future , it would enable a smaller spread in each @xmath157 for a better separation . in the light of msw effect , distinctions between direct and inverted masses would most likely appear in the observed neutrino spectra . if the just - so vacuum oscillation is favored over the msw oscillations as solution to the snp , both direct ( @xmath0 ) and inverted - mass schemes ( @xmath1 ) are allowed since @xmath3 flux can also be converted to @xmath158 through the vacuum oscillation if neutrino masses are inverted . the @xmath29 flux on the contrary , would go through the msw resonance if the mass hierarchy is inverted . this conversion would presumably enlarge the @xmath29-type event rates effectively at super - kamiokande . without conflicting current solar and atmospheric neutrino data , we would focus on the js parameters : @xmath159ev@xmath8 and large @xmath61 , for a further investigation under the inverted mass scheme @xmath1 . the possible sn @xmath29 spectra are shown in figure 3 . curve @xmath160 is the original @xmath29 spectrum and curve @xmath161 represents the distorted one through the just - so vacuum oscillation under the direct - mass scheme while curve @xmath162 is obtained from the msw conversion under the inverted - mass scheme . curves @xmath161 and @xmath162 nearly overlap , implying that the matter effect is not as prominent as expected , and that the msw effect under the inverted - mass hierarchy is almost identical to the vacuum oscillation under the direct - mass hierarchy for @xmath29 at this particular region of parameter space . furthermore , the extremely small observable difference at the detectors would make the identification between the two mass patterns very difficult . the reason becomes clear if the required conditions for a msw resonance and an adiabatic transition to occur are both considered @xcite : a density profile @xmath87 would yield @xmath163 ev@xmath8 relevant to the msw oscillation in the supernova . this mass scale is much larger than the mass scale of js parameters ( @xmath164 ev@xmath8 ) . therefore a very effective conversion of @xmath29 to @xmath101 in a supernova is unlikely for either direct or inverted masses if the js parameters are applied . the strong conversion of @xmath29 to @xmath101 is actually disfavored by some analyses based on the sn 1987a data @xcite . if either la or sa msw conversion is favored over the just - so vacuum oscillation , the case for the inverted hierarchy @xmath1 would then become shaky or can even be ruled out . an alternative approach might shed some clues to the inverted - mass scheme and its outcomes . we may characterize consequences for the oscillation @xmath165 by the surviving probability of @xmath29 in three limit cases : @xmath166 , and @xmath167 . the case @xmath168 1 indicates that no conversion occurs among @xmath29 and @xmath101 , which is equivalent to the outcome of having massless neutrinos , and the mass pattern would then unlikely be the main issue . the js parameters , as already discussed , yield @xmath169 for the msw conversion ( or equivalently , the vacuum oscillation under the direct - mass scheme ) . one is therefore motivated to further study the consequences for a complete conversion of @xmath29 flux to @xmath101 , where @xmath170 . as pointed out by totani _ et al . _ @xcite , due to statistical uncertainties in experiments and the inconsistency among current analyses , one can not completely exclude the possibility of full conversion . we may tentatively neglect details of the physical conditions and parameters that required for a complete conversion to occur , and assume that the probability @xmath48 remains approximately a constant within interested range of the neutrino energy . to reasonably account for the contributions from @xmath3 and @xmath158 fluxes to the total events when a complete conversion occurs in the anti - neutrinos sector , one may first consider the contours of @xmath103 for events from the inverse beta decay only . under the inverted - mass scheme , a wide range of @xmath171 and @xmath61 are shown in figure 4 . given @xmath172 4.5 mev , @xmath117=3 mev and @xmath173=6 mev , the js parameters ( @xmath164 ev@xmath8 and large @xmath61 ) roughly result in a 20% increment to the event number . after a full conversion of the anti - neutrinos in which @xmath170 , one would expect to observe a sizable increase in the ratio @xmath103 . therefore a larger @xmath174 and a smaller @xmath175 , at least several orders of magnitude , are required for a full conversion to occur . the smallness of @xmath61 , alone with the small @xmath52 , fix the surviving probability of @xmath3 very close to unity . hence , a full conversion of @xmath29 to @xmath101 would be accompanied by nearly unchanged @xmath3 and @xmath158 fluxes : @xmath176 . the expected ratios @xmath177 at the super - kamiokande are listed in table iv . despite the better statistic provided by the @xmath29-type events at super - kamiokande , the detection is however not unique to the @xmath29 flux , the uncertainties in @xmath88 and @xmath178 therefore make the separation between @xmath170 ( complete conversion ) and @xmath179 difficult . at the sno detector , the charged - current channel @xmath180 is unique to @xmath29 . this channel can be distinguished from the neutral - current events and the other charged - current channel induced by @xmath3 . therefore the measurement of @xmath181 events at sno should be sensitive to the full conversion of @xmath29 to @xmath101 . ratios of the neutral - current events to the charged - current events @xmath181 , denoted as @xmath182 , are shown in table v. we also present values of the same ratio under the direct - mass scheme in table vi as a comparison . we observe that for a particular @xmath117 , both direct and inverted schemes yield a nearly identical range of @xmath182 if the js parameters are applied , this verifies a previous argument . the valuable message from this ratio is that for a particular @xmath117 , a complete conversion of the @xmath29 flux through msw resonance represents a unique range of @xmath182 as compared to other scenarios , including that of the direct masses . typical @xmath182 contours for inverted masses are shown in figure 5 , where @xmath117= 3 mev , @xmath91= 4.5 mev , and @xmath183= 6 mev are assumed . since the neutral - current reaction is blind to the oscillation , a complete swap of @xmath29 and @xmath101 fluxes would definitely yield smaller @xmath182 . calculations show that @xmath184 for a complete conversion and indicate that @xmath185 ev@xmath8 and @xmath186 are required for a near complete conversion to occur . tables v and vi suggest that for the detection of sn neutrinos , the direct and inverted masses could be distinguishable if a nearly complete conversion of @xmath29 to @xmath101 occurs , which yields a low @xmath182 and signals the existence of inverted - mass pattern . since the msw effects become important for supernova neutrinos at @xmath187 ev@xmath8 , a future supernova would provide a test ground for @xmath188 ev@xmath8 . if the nearly full conversion @xmath189 is observed in the sn neutrino flux , the consequences may have certain implications in that the required parameter spaces for a full conversion are obviously disfavored by current solar neutrino data , while the future solar and atmospheric observations may not severely change the mass scales required to explain the solar and the atmospheric neutrino deficits . under the constraints of mass scale from solar and atmospheric neutrinos , the @xmath190 scenario naturally leads to four possible hierarchies ( one direct and three inverted ) : @xmath191 case 1 is the normal , direct mass scheme . for case 2 , @xmath174 in the mass scale of the msw solution has been discussed @xcite . in our analysis we have applied case 3 , in which the mass scale of @xmath174 is suitable for the vacuum solution of snp ( @xmath192 ev@xmath8 ) . case 3 and case 2 become equivalent if @xmath164 ev@xmath8 since the msw and the vacuum oscillations for the @xmath29 flux would be nearly identical at this mass scale , as shown in section iv . for case 4 to survive , @xmath174 needs to be in the order of @xmath192 ev@xmath8 for the vacuum solution to apply . therefore , consequences for case 4 and case 1 become equivalent in the detection of supernova neutrinos if @xmath164 ev@xmath8 . in this work , responses at both super - kamiokande and sno detectors to neutrino fluxes coming from a supernova are studied under the consideration of uncertainties in neutrino temperatures . in particular , some phenomenological consequences for direct - mass and inverted - mass patterns of neutrinos are compared . we may summarize our results as follows . ( a ) uncertainties in neutrino temperatures can allow various interpretations of neutrino parameters . we have shown this through the expected outcomes at super - kamiokande for sn neutrinos . ( b ) the three candidates la , sa , and js manifest differently in the ratio @xmath154 at sno even if the uncertainties in neutrino temperature are allowed . future detection of sn neutrinos at sno would be able to single out favored mass and mixing parameters from the three candidates . ( c ) in addition to the direct - mass pattern , the inverted - mass scenario @xmath1 is investigated since it can allow the vacuum solution to the solar neutrino problem . by using the event ratio @xmath182 in sno , the direct - mass(@xmath0 ) and the inverted - mass(@xmath1 ) could be distinguished if a nearly complete @xmath189 conversion occurs in the anti - neutrino section . s.c . would like to thank nien - po chen and sadek mansour for suggestions in preparing the manuscript . t. k. is supported in part by the doe , grant no . de - fg02 - 91er40681 . r. davis jr . , prog . in nucl . and part * 32 * ( 1994 ) ; sage collaboration , j. n. abdurashitov _ et al . _ , lett . b * 328 * , 234 ( 1994 ) ; gallex collaboration , p. anselmann _ et al . _ , lett . b * 357 * , 237 ( 1995 ) ; kamiokande collaboration , k. hirata _ et al . _ , d * 44 * , 2241 ( 1991 ) . g. g. raffelt and j. silk , phys . lett . b * 366 * , 429 ( 1996 ) ; d. o. caldwell and r. n. mohapatra , phys . lett . b * 354 * , 371 ( 1995 ) ; g. m. fuller , j. r. primack and y. -z . qian , phys . d * 52 * , 1288 ( 1995 ) .
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possible outcomes of neutrino events at both super - kamiokande and sno for a type - ii supernova are analyzed considering the uncertainties in sn neutrino spectra ( temperature ) at emission , which may complicate the interpretation of the observed events . with the input of parameters deduced from the current solar and atmospheric experiments , consequences of direct - mass hierarchy @xmath0 and inverted - mass hierarchy @xmath1
are investigated .
even if the @xmath2 temperatures are not precisely known , we found that future experiments are likely to be able to separate the currently accepted solutions to the solar neutrino problem ( snp ) : large angle msw , small angle msw , and the vacuum oscillation , as well as to distinguish between the direct and inverted mass hierarchies of the neutrinos .
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You are an expert at summarizing long articles. Proceed to summarize the following text:
one hundred years after einstein , the theory of general relativity ( gr ) is still our best theory of gravity . in the framework of gr , the standard model of cosmology ( @xmath4 ) provides a successful description of the universe . in this model , the same fluctuations which give rise to the observed small variations in the temperature of the cosmic microwave background ( cmb ) grow under the force of gravity , and eventually form observed galaxies and other nonlinear structures such as filaments , voids , groups and clusters of galaxies . according to the model , only @xmath5 of the density in the universe is provided by normal baryonic matter @xcite . the @xmath4 model requires two additional components : a non - baryonic cold dark matter ( cdm ) , which contributes about 30% of the average density of the universe , and an even more mysterious dark energy , which makes up the rest @xcite . the model is remarkably successful on scales larger than a few megaparsecs . it predicted the amplitude and the spectrum of angular fluctuations in the cmb and in the distribution of galaxies @xcite that were later confirmed by observations @xcite . however , the @xmath4 model faces challenges on smaller scales . the most difficult ones are related with the rotation in the inner parts of spiral galaxies . it seems that the theory predicts too much dark matter inside @xmath6 1kpc from the centers of galaxies @xcite . while there are some possible solutions of the problem @xcite , the problems on small scales are the strongest challenge the standard model has encountered . when compounded with the fact that there is no direct evidence of dark matter or dark energy , the current problems of the standard cosmological model have encouraged a small but growing community of physicists to propose alternative theories of gravity to avoid the need for dark matter . this is the case for modified newtonian dynamics ( mond ) , proposed by @xcite to explain the rotation of galaxies without dark matter . according to mond , the rotation curves in the outer regions of galaxies do not decline because the force of gravity is significantly stronger than for newtonian gravity . at early times mond s main appeal was its simplicity : there is no need to make the assumption that the universe is filled with particles that nobody has seen . additional motivation came later from difficulties with explaining anomalies in the trajectories of the pioneer 10 and 11 space missions @xcite . yet , for a long time mond was not more than a conjecture . only recently , bekenstein proposed a relativistic version named tensor vector scalar theory ( teves ) @xcite . this alternative theory of gravity provides a framework to make predictions of numerous important observational phenomena , which @xmath4 has already done : the temperature fluctuations seen in the cmb , gravitational lensing , and the large scale structure of the universe . with maturity came problems . rotation curves of some galaxies the initial strong argument for mond can not be explained by mond . in about 1/4 of galaxies considered by proponents of mond the predicted velocities are well above the observations in the very central regions @xcite . rms velocities of stars in some dwarf spheroidal galaxies @xcite also present problems . so far , the most severe challenges for mond are coming from clusters of galaxies . dynamics of galaxies in clusters can not be explained by mond and requires introduction of dark matter , possibly in the form of a massive ( @xmath7ev ) neutrino @xcite . we do not know whether this modification can explain some properties of clusters of galaxies such as the `` bullet cluster '' , where the baryonic mass ( galaxies and gas ) is clearly separated from the gravitational mass , as indicated by gravitational lensing @xcite . in any case , for mond to survive too it must invoke dark matter in the form of massive neutrinos and dark energy in the form of an arbitrary constant added to a combination of two scalar fields used in teves mond @xcite . there is no doubt that alternative theories of gravity represent a challenge to the standard model of cosmology and gr . any theory or model must respond to these challenges . here we present a number of observations to test gravity and dark matter in the peripheral parts of galaxies at distances 50 - 500 kpc from the centers of galaxies . these scales can be tested by studying the motion of satellites of galaxies . this is a relatively old field in extragalactic astronomy and historically it was one of main arguments for the presence of dark matter @xcite . the paper is organized as follows . in section 2 , we present the observational results drawn from the sdss and the predictions from the standard model of cosmology . predictions from mond are computed and discussed in section 3 . finally , conclusions are given in section 4 . we use the sloan digital sky survey ( sdss ; www.sdss.org ) the largest photometric and spectroscopic astronomical survey ever undertaken of the local universe to study the motion of satellites . as of data release four ( dr4 ) @xcite , imaging data are available over 6670 deg@xmath8 in five photometric bands . in addition to the ccd imaging , the sdss 2.5 m telescope on apache point , new mexico , measured spectra of galaxies , providing distance determinations . approximately half million of galaxies brighter than @xmath9 over 4700 deg@xmath8 have been targeted for spectroscopic observations as part of sdss and are included in dr4 . redshift accuracy is better than 30 km / s and the overall completeness is @xmath690% . for our study we compute rest frame absolute magnitudes in the g - band from the extinction - corrected apparent magnitudes assuming a @xmath10 cosmology with a hubble constant @xmath11 ( @xmath12 = @xmath13 ) . galaxies are split into red ( early - types ) and blue ( late - types ) populations based on the bimodality observed in the @xmath14 color distribution ( * ? ? ? the local minima between the peaks of the color distribution occur near @xmath15 . all magnitudes and colors are k - corrected to @xmath16 . because calculations of mond gravity for non - spherical objects are complicated , we restrict our analysis only to red galaxies , the vast majority of which are either elliptical galaxies or are dominated by bulges . our galaxy sample was selected from the full redshift sample by taking all galaxies with recession velocity 3000 km / s @xmath17 km / s . the total number of selected galaxies is about 215,000 . the sdss heliocentric velocities were converted to the local group standard of rest before computing distances . we select our host galaxies as galaxies with absolute g - band magnitude brighter than @xmath18 and isolated : a galaxy must be at least 4 times brighter than any other galaxy within a projected distance @xmath19 mpc and a line - of - sight velocity difference @xmath20 km / s . we define satellites as all galaxies being at least 4 times fainter than their hosts and found within a projected distance @xmath21 kpc and velocity difference @xmath22 km / s with respect to their hosts . typically we find about 1.5 satellites per host . in total we have 9500 satellites with a mean luminosity of about @xmath23 . we bin the host galaxies by luminosity and collect information about the distribution of relative velocities @xmath24 and the number of satellites as the function of projected distance @xmath25 . figure 1 presents the observational results for primaries , which are at least 6 times brighter than any satellite . we also used primaries , which are 4 times brighter , and primaries , which are 10 times brighter than their satellites . we do not find any trend with the primary - satellite magnitude gap : results are nearly the same ( see section 3 for details ) . the only difference is the statistics , and , thus , the error - bars . the distribution of line - of - sight velocities ( top right panel ) clearly shows a two - component structure : a homogeneous background of interloper galaxies ( dwarfs which happen to lie along the line - of - sight , with large physical distances from the host galaxies but small projected and velocity differences , but which are not associated with the host ) and a nearly gaussian component . the surface density of the satellites also shows the same structure : at large separations the number density goes to a constant due to interlopers . we subtract the background and plot the surface density of satellites in the bottom right panel . in order to study the velocity dispersion of satellites @xmath26 as a function of projected distance @xmath25 we use a maximum likelihood method to approximate the number of satellites @xmath27 per unit projected area and per unit velocity difference @xmath24 using an 9-parameter function in the form of a constant plus a gaussian distribution with variable velocity dispersion and normalization : @xmath28/\sqrt { 2\pi}\sigma+ n_0 , \label{eq : delta}\ ] ] where the surface number - density of satellites @xmath29 and the rms line - of - sight velocity @xmath30 are 3th order shifted chebyshev polynomials of the first kind . the parameter @xmath31 is a constant representing the background of interlopers . the left panel in figure 1 shows the resulting rms velocity of satellites for three magnitude bins . in order to estimate statistical uncertainties , we run monte carlo simulations using the same number of hosts and satellites as in the corresponding magnitude bin in figure 1 . note that the data points are correlated . we use monte carlo simulations to test statistical significance that the observed velocities are declining with distance : data for each magnitude bin reject a constant @xmath32 at about @xmath33 confidence level . we also studied blue galaxies and find declining velocities for them . these results are in agreement with previous estimates @xcite , but we now can exclude constant rms velocities much more reliably . in order to compare observational results with the @xmath4 predictions , we use high - resolution cosmological n - body simulations @xcite , which we then treat as if they are the observational data . the simulations were done for the standard @xmath4model with parameters @xmath34 , @xmath35 , @xmath11 , @xmath36 . we use two simulations . both simulation had @xmath37 particles . one simulation had the computational box 120 @xmath38mpc box , mass resolution @xmath39 and force resolution 1.8 @xmath38kpc . the second simulation has the computational box 80 @xmath38mpc box , mass resolution @xmath40 and force resolution 1.2 @xmath38kpc . we do not find any differences between the simulations and decided to present results of the larger simulation , which provides better statistics of halos . we select isolated halos as halos , which within projected distance of 715 kpc and within relative velocity difference 1000 km / s do not have halos or subhalos with maximum circular larger than 1/2 of the halo s circular velocity . this corresponds to the mass ratio of a factor ten for distrinct halos . circular velocities are better characteristics of subhalos , which have poorly defined masses . although we resolve subhalos , we dicided to use dark matter particles as proxies for satellites . at distances larger than a fraction ( 1/4- 1/3 ) of a virial radius from center of a halo satellites trace the motion and the spacial distribution of the dark matter ( e.g. * ? ? ? results presented in figures 1 and 2 show that all three characteristics of observed satellites are reproduced by the model . note that we actually do not make fits . results from simulations have only one parameter : the maximum circular velocity of the dark matter halo . we just plot what we obtain from the simulations . once we fix the maximum circular velocity , the results are fairly insensitive to parameters of the cosmological model . two factors may affect the shape of the theoretical v - r diagram : the halo concentration and the velocity anysotropy . the halo concentrations @xmath41 are in the range @xmath42 for halos ( @xmath43 ) and normalization @xmath44 cosidered in this paper @xcite . ( here @xmath45 is the virial radius and @xmath46 is the characteristic radius of the nfw profile ) . for virial radii in the range @xmath47 kpc , the exact value of the halo concentration affects only the central region @xmath48 kpc . the average halo concentration depends weakly on the amplitude of the perturbations @xmath44 . existing observational data leave only very narrow range for variation of @xmath49 , which also limits the range of concentrations . the velocity anisotropy is less certain , but it is not large . n - body simulations indicate that @xmath50 is slightly positive and has a tendency to increase with the distance for radii smaller than the virial radius ( e.g. * ? ? ? * ) with typical values @xmath51 for halo masses considered in this paper . at larger distances @xmath50 declines and goes to zero @xcite . in order to demonstrate that our results of n - body simulations are robust and reliable , we use solutions of the jeans equation , which we obtain for parameters compatible with numerous previous simulations . once a solution for the radial velocity dispersion is obtained , we integrate it along a line of sight with appropriate corrections for the velocity anisotropy . for the density use the nfw profiles with concentration @xmath52 . for the sake of completeness , we add the cosmological background density , which has only a small effect at large distances . for @xmath50 we use approximation given by eq . ( [ eq : betb ] ) with parameters chosen in such a way that @xmath53 and at @xmath54 kpc @xmath55 . we use the same range of circular velocities as in figure [ fig : two ] . for more massive halos we use @xmath56 and @xmath57 . for less massive halos we adopted @xmath58 and @xmath59 . results presented in figure [ fig : jeans ] clearly indicate that very simple stationary models can accurately reproduce results of simulations . the situation is different for mond because there are no predictions on the same level of sophistication as for @xmath4 . in principle , those predictions can be made , but at this moment they have not been yet made . thus , we have only one option : solve the jeans equation for spherical systems . when doing so , we have a freedom of choosing two functions : the number - density profile of satellites and the velocity anisotropy @xmath60 . we also have two free parameters : stellar mass @xmath61 and the magnitude of external force ( see below for details ) . the predictions are constrained by two functions @xmath29 and @xmath62 . the velocity distribution function also gives constraints , but those are relatively weak . only the models with a large @xmath63 can be excluded . the stellar luminosity and colors constrain the stellar mass . roughly speaking we have two arbitrary functions to fit two observed functions . it should not be difficult given that the model is viable . in our case the jeans equation for the radial velocity dispersion @xmath64 can be written in the form : @xmath65 where @xmath66 is the velocity anisotropy and @xmath67 is the gravitational acceleration . the formal solution of the jeans equation can be written in the form : @xmath68 @xmath69 . \label{eq : jeanschi}\ ] ] it is convenient to chose forms of @xmath70 and @xmath50 such that the integral in eq.([eq : jeanschi ] ) is taken in elementary functions . we used the following approximations : @xmath71 @xmath72 @xmath73^{2\beta_1 } \label{eq : betb}\ ] ] @xmath74^{2\beta_1 } \left[\frac{1}{1+(\frac{r}{r_b})^2}\right]^{\beta_1 } \label{eq : betc}\ ] ] in the case of newtonian gravity the acceleration @xmath75 , where the mass @xmath76 includes both normal baryonic mass and dark matter . for mond the acceleration of spherical systems @xmath77 is given by the solution of the non - linear equation @xmath78 where @xmath61 is the mass of only baryons and @xmath79 . the term @xmath80 is the external constant gravitational acceleration . it formally breaks the spherical symmetry . thus , the eq . ( [ eq : gmond ] ) is only an approximation valid for small external force . here we use the same approximation as eq . ( 10 ) in @xcite , which we average over angles between a constant external acceleration @xmath80 and internal radial acceleration @xmath77 . the function @xmath81 can have different shapes . we tried the originally proposed form @xcite @xmath82 , but accepted and used for all our analysis the function @xmath83 , which gives slightly better results . the function @xmath84 is limited by the observations presented in figure 1 . the velocity anisotropy @xmath60 is a free function , but there are constraints . asymptotically @xmath50 goes to zero at large distances where gravitational effect of the central galaxy diminishes . ideally the velocity anisotropy at small distances also should be declining . there are different arguments why this should be the case . ( 1 ) the tangential velocities of few satellites of the milky way , for which the proper motions are measured , strongly reject radial orbits @xcite . there is no reason to believe that our galaxy should be special in this respect . ( 2 ) experience with gravitational dynamical systems indicate that in dynamically relaxed systems @xmath85 . numerous simulations of cosmological models illustrate this : @xmath50 is small in the central region and increases to @xmath86 at the viral radius @xcite . note that we should distinguish the velocity anisotropy and the orbital eccentricity . for centrally concentrated objects , which we are dealing with here , already isotropic velocities imply typical peri-/apocenter ratios of 1:4 - 1:5 . if eccentricity is larger , a significant fraction of satellites comes too close to the central @xmath87 kpc region where the satellites are destroyed by tidal forces . the term @xmath88 in eq.([eq : gmond ] ) describes the effect of external gravitational field . it is specific for mond . in newtonian dynamics a homogeneous external gravity does not affect relative motions inside the object . because the mond gravity is nonlinear , the internal force is affected by the external field ( note that this is not the tidal force ) . this external effect is quite complicated . the magnitude of the external force is substantial for the motion of the satellites . @xcite point out that in mond numerous sources and effects generate about equal magnitude of @xmath89 . for example , 600 km / s motion of the local group relative to the cmb implies acceleration of 600 km / s@xmath90 gyrs@xmath91 . here we assume that the acceleration is constant over the whole age of the universe . if @xmath89 increases with time , as it may be expected , then the acceleration is even larger . infall with @xmath92 km / s in the direction of virgo gives about half of the value . m31 produces about the same magnitude of the acceleration . while the acceleration in mond does not add linearly , it is reasonable to assume that @xmath93 . we will explore the effect the external field later . in order to make mond predictions , we must estimate the stellar mass for galaxies in our analysis . for two subsamples presented in figure [ fig : tuned ] we use the magnitude bins @xmath94 ( 2400 satellites ) and @xmath95 ( 2700 satellites ) . the average luminosities of galaxies in the bins are @xmath96 and @xmath97 . when estimating the luminosities , we assume @xmath98 @xcite . using measured @xmath99 colors we estimate the stellar masses of galaxies @xcite : @xmath100 and @xmath101 for the two luminosity bins . this implies the mass - to - light ratios are nearly the same for the bins : @xmath102 . these @xmath103 estimates are close to predictions of stellar population models . in our case , the galaxies are red ( @xmath104 ) , old mostly ellipticals , for which we expect nearly solar metallicity . indeed , @xcite gives @xmath105 for the kroupa imf for stellar population 10 gyrs old . adjusting for the 0.4 mag difference between b and g bands , we get @xmath106 ( the salpeter imf gives @xmath107 ) . larger metallicity is highly unlikely ( the galaxies are not really massive ellipticals ) and it does not make much difference : 10 percent increase if we take twice the solar metallicity . using different functional forms for @xmath60 , we solve the equations numerically and then integrate the solution along the line - of - sight to get a prediction for @xmath108 . figure 2 presents results for the two different magnitude ranges and for different parameters @xmath109const and @xmath110const . mond definitely has problems fitting the data because for any constant @xmath50 it predicts a sharply declining rm velocity at small distances followed by nearly flat velocities at large distances . the data show just the opposite behavior . it is easy to understand why predictions of mond has this shape . at large distances the newtonian acceleration is very small and the acceleration is strongly dominated by the mond correction : @xmath111 . in this case , the solution of eq.(1 ) gives @xmath112const for _ any _ constant value @xmath50 and @xmath70 . we can try to salvage mond , but so far there is no simple way of doing it . @xcite find that the way to improve the situation is to have a model with variable slope @xmath70 and with extremely radial orbits . we also tried different functional forms and numerous combinations of parameters . while the parameters , which fit the surface density and the rms velocities vary , they all give the same answer : density slope in the central 40 kpc should be @xmath113 and the velocity anisotropy at large distances should be @xmath114 . figure [ fig : tuned ] shows ( dashed curves ) the results for the same set of parameters as in @xcite : @xmath115 , @xmath116 , @xmath117 kpc , @xmath118 kpc , @xmath119 and @xmath120 . here we use the approximation given in eq.([eq : betb ] ) . the solution is very contrived : relatively small deviations from the best behavior ( small slope in the center and radial orbits in the outer radii ) result in failed fits . the full curves in the figure show what happens when the velocity anisotropy gets less radial : @xmath121 . even this fine - tuned solution fails unless the mass - to - light ratio for the larger magnitude bin is arificially increased by a factor of two : @xmath122 , which gives @xmath123 . @xcite also found the same trend , but they made two mistakes , which did not allow them to clearly see the problem . first , the solar mass - to - light ratio was used for the b band instead of the g band . second , the width of the magnitude bin is presented as an error in m / l giving impression of very large uncertainties in m / l . this is not correct : the statistical uncertainty of the average luminosity of galaxies in each bin is very low and can be neglected . it should be noted that there is nothing special about the galaxies , which are used here . the average luminosities differ by a factor 1.7 . so , it is not a large difference . colors of the galaxies are practically the same , which then gives the same m / l if we use stellar population models . the rms velocities are also perfectly consistent with simple scaling . for example , the ratio of rms velocities at the same projected distance of 70 kpc is 1.3 implying simple scaling @xmath124 . roughly speaking , we double the luminosity and that doubles the stellar mass . this does not work for mond : it needs twice more stellar mass . this is definitely a problem because there is no justification why galaxies with the same colors , with the same old population and practically the same luminosity should have dramatically different imf . the differences are very large : most of the stellar mass in the more luminous bin should be locked up in dwarfs with @xmath125 , while there is relatively little of those in the lower bin , which is consistent with the kroupa imf . in order to make the argument even more clear , we make analysis of velocities in a different way . this time we split the sample by stellar mass , but we still keep only red primaries with @xmath126 . we make the analysis twice : for satellites 4 times and for satellites 10 times less bright than the primary galaxy . there is no systematic difference between the two isolation conditions : within @xmath127 the results are the same . figure [ fig : stmass ] illustrates this point . we select primary galaxies to have the stellar mass in the range @xmath128 . the average stellar mass is @xmath129 , and the average luminosity is @xmath130 . there is a hint that more isolated primaries have slightly _ larger _ velocities of satellites . still , the differences are not statistically significant : for radii larger than 50 kpc the differences are smaller than 10 km / s . then we take the observed stellar mass and use it for mond models and apply the best tuned parameters . we find that parameters suggested by @xcite ( @xmath131 kpc , @xmath132 , @xmath120 ) improve the fits as compared with a constant @xmath50 models . still , they are not acceptable . for example , at @xmath133 kpc the mond model is a @xmath134 deviation . we found a better solution ( @xmath135 kpc and other parameters the same ) , which places mond `` only '' at @xmath136 . in order to envestigate what stellar mass is needed for mond , we split the sample of red primaries into 5 mass bins ranging from @xmath137 to @xmath138 . we chose the less stringent isolation condition because it gives typically 2 - 2.5 time more satellites resulting in smaller statistical errors . each bin has a large number of satellites : @xmath139 . we then run a grid of mond models with different masses and select models , which make best fits . just as the mond models in figure [ fig : tuned ] , there are models , which marginally fit the data . for each model we get stellar mass required by mond and compare it with the stellar mass estimated by the stellar population models . results are presented in figure [ fig : mondmass ] . for smaller masses mond gives masses , which are compatible with actual stellar mass observed in the galaxies . the plot clearly shows the problem with massive galaxies : mond requires increasingly more mass than observed in the galaxies ending up in large ( @xmath140 ) disagreement with the observations . the external gravity force @xmath89 is another mond component . it must exist on the level of @xmath141 . figure [ fig : tunedext ] shows what happens to the models when we add the external force . as the starting models we use the best fits shown in the figure [ fig : tuned ] as dashed curves . the models with the external force make the fits much worse . taken at face value , the models with realistic @xmath142 can be rejected . again , we can make the model work if we increase m / l by a factor @xmath143 . yet , this will make the situation with stellar masses even worse , than what we already have . using the sdss dr4 data we study the distribution and velocities of satellites orbiting red isolated galaxies . we find that the surface number - density of the satellites declines almost as a power law with the slope @xmath144 . the distribution of the line - of - sight velocities is nearly a perfect gaussian distribution with a constant component due to interlopers . the rms velocities are found to gradually decline with the projected distance . the constant rms velocity ( isothermal solution ) can be rejected at a 10 @xmath145 level . observational data strongly favor the standard cosmological model : all three major statistics of satellites the number - density profile , the line - of - sight velocity dispersion , and the distribution function of the velocities agree remarkably well with the theoretical predictions . thus , the success of the standard model extends to scales ( 50 - 500 ) kpc , much lower than what was previously considered . mond fails badly in cases with any acceptable power- law approximation for the number - density of satellites and a constant velocity anisotropy by producing sharply declining velocities at small distances followed by nearly flat velocities at large distances just the opposite of what is observed in real galaxies . models may be made to fit the satellites data only when all the following conditions are fulfilled : 1 . the slope of the density in the central 50 kpc is less then -2 . satellite velocities are nearly radial in outer regions ( @xmath146 at @xmath3 kpc ) . mass - to - light ratio increases from @xmath147 at @xmath148 to @xmath149 ( @xmath150 ) at @xmath151 . external force of gravity is smaller than the expected value of @xmath89 @xmath91 . the negative effect of the external force can be removed by additionally increasing the m / l ratios . we find that the later three conditions are difficult to realize in nature . satellites do not fall in to their parent galaxies with zero tangential velocities . the velocities are induced by other neighboring galaxies and by large - scale structures such as filaments . to some degree , it is similar to the tidal torque , which is responsible for the origin of the angular momentum of galaxies . yet , the interactions are more efficient for providing random velocities . measurements of peculiar velocities of galaxies the local volume ( 3 - 5 mpc around the milky way ) found deviations from the hubble flow that about 70 - 80 km / s @xcite . at these distances the gravitational pull of other large galaxies in the aria is larger than that of the milky way and we expect that the same deviations exist for the perpendicular velocity components . when a satellite with 70 - 80 km / s falls falls from 1 mpc and gets to 400 kpc from mw its tangential velocity increases few times ( conservation of the angular momentum ) . so , we expect @xmath152 . we have the same measurements for the satellites in sdss . when we look at distances 0.81 mpc , we find that the rms velocities hardly correlate with the luminosity and are about 100 km / s . note that @xmath4perfectly fits the constraint : at these distances it predicts @xmath153 . thus , mond assumption of very radial orbits does not agree with the observations . the large m / l ratios is another problem for mond . so far m / l was treated almost as a free parameter @xcite . this is not correct . it should be reminded that observationally there are no indications of large variability of imf for vastly different stellar ages and metallicities . the imf shows similar flattening for masses smaller than @xmath154 in the solar neighborhood and in the spheroid @xcite , in the galactic bulge @xcite , and in the ursa minor dwarf spheroidal galaxy @xcite . the galactic bulge has parameters similar to the red galaxies studied in this paper : very old ( on average ) stellar population with nearly solar metallicity . @xcite give the @xmath155 , which corresponds to @xmath156 . this is close to what we found for the red galaxies in sdss . in the case studied here , mond requires that m / l changes by a factor 2 - 2.5 when there is no change in colors and the absolute magnitude changes only by @xmath157 . there is no justification for this to happen . the situation is similar to what @xcite found for early type large spirals where mond models fail for any imf discussed in recent 20 years . only the salpeter imf gives better results . even in this case the models deviate by a factor of two around the stellar population predictions . the combination of all these problems makes mond a very implausible solution for the observational data on satellites of galaxies . there is one significant difference between mond and @xmath4 . the latter makes predictions for the distribution of mass and velocities for isolated galaxies and those predictions match the observational data . there are no theoretical predictions for mond . what we conveniently called `` mond predictions '' were actually _ requirements_. for example , mond must produce the radial velocities of satellites or shallow slope of the number - density in the central region . we thank s. mcgaugh for challenging us to do mondian analysis of the satellite motion . we are grateful to j. holtzman for numerous comments and suggestions and thank j. betancort - rijo for comments . we are especially grateful to angus et al for providing us with the draft of their paper and for extensive discussions . we still have different ways of looking at the situation with mond . still , these productive discussions resulted in improvements in their and our papers . we acknowledge support by the nsf grant ast-0407072 to nmsu and thank the spanish mec under grant pnaya 2005 - 07789 for their support . computer simulations used in this research were conducted on the columbia supercomputer at the nasa advanced supercomputing division and on seaborg at the national energy research scientific computing center ( nersc ) .
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the motion of satellite galaxies around normal galaxies at distances 50 - 500 kpc provides a sensitive test for the theories .
we study the surface density and the velocities of satellites around isolated red galaxies in the sloan digital sky survey .
we find that the surface number - density of satellites declines with the projected distance as a power law with the slope @xmath0 .
the rms velocities gradually decline : observations exclude constant velocities at a @xmath1 level .
we show that observational data strongly favor the standard model : all three major statistics of satellites the number - density profile , the line - of - sight velocity dispersion , and the distribution function of the velocities agree remarkably well with the predictions of the standard cosmological model .
thus , that the success of the standard model extends to scales ( 50 - 500 ) kpc , much lower than what was previously considered .
mond fails on these scales for models which assume any single power - law number - density profile of satellites and any constant velocity anisotropy by predicting nearly constant rms velocities of satellites .
satellite data can be fit by fine - tuned models , which require ( 1 ) specific non - power - law density profile , ( 2 ) very radial orbits at large distances ( velocity anisotropy @xmath2 at @xmath3 kpc ) , and ( 3 ) 2 - 2.5 times more stellar mass than what is found in the galaxies .
the external gravity force a necessary component for mond makes the situation even worse .
we argue that a combination of satellite data and observational constraints on stellar masses make these models very problematic . #
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You are an expert at summarizing long articles. Proceed to summarize the following text:
many natural and technological complex systems can be described as networks connecting a large number of elements whose states evolve in time @xcite . the collective behavior of a system resulting from interactions between its components can exhibit non - trivial features , including critical phenomena @xcite . for instance , a system of binary - state elements ( e.g. , representing individuals having opposing opinions on an issue ) connected through a network having modular organization can exhibit ordering dynamics at very distinct time - scales @xcite and under certain circumstances , self - organize into locally aligned clusters that correspond to the communities of the network ( i.e. , subnetworks characterized by a significantly higher connection density compared to the overall density of the network ) @xcite . in many situations , the links of the network ( representing the interactions ) can also evolve over time as a result of the changes in the states of the components that they connect . such connections may not only be characterized by weights ( indicating the strength of interaction ) but also sign ( representing the nature of the interaction ) . for instance , in the context of a network of synaptically - connected model neurons , positive links may correspond to excitatory interactions ( whereby activation of one element can result in activation of other connected elements ) while negative links can give rise to inhibition ( i.e. , activation of an element tends to suppress subsequent activation of neighboring elements ) @xcite . the occurrence of negative links can lead to the emergence of _ frustration _ because of the existence of inconsistent relations within cycles in the network @xcite . a network is said to be structurally balanced if its positive and negative links are arranged such that frustration is absent . such a network can always be represented as two communities , with interactions within each community being exclusively positive while those between communities can only be negative @xcite . this is of interest not only for social systems in the context of which the concept of balance was first introduced @xcite , but also for biological systems . for instance , it has been recently observed that the resting human brain is organized into a pair of dynamically anti - correlated subnetworks @xcite . this suggests that the network responsible for this behavior may be structurally balanced . this possibility is intriguing in view of the observation of non - trivial collective behavior in balanced networks of binary - state dynamical elements , such as , the coexistence of ordered and disordered regions referred to as `` chimera '' order , in the presence of an external field @xcite . the process by which networks can evolve to a balanced configuration has been explored by considering a link adaptation process inspired by hebb s principle , the basis behind neural plasticity @xcite . according to this mechanism , weights associated with each link change in proportion to the correlation in the activity of the connected elements . therefore , any initial frustration in the network can be removed by modifying the connection weights in accordance with the dynamical states of the elements . systems undergoing such link adaptation in the presence of environmental noise show high variability in the convergence time required to reach the balanced state @xcite . in this paper we consider how the rate of dynamical evolution of interactions in accordance with the hebbian adaptation principle affects the collective behavior of the network . starting from a fully connected network that is structurally balanced and introducing noise and external field so that the system exhibits chimera order , we show that different rates of adaptation can result in distinct outcomes . fast learning rates result in persistence of the chimera regime , although the balanced network now comprises communities with asymmetric sizes , viz . , the ordered sub - network containing a much larger fraction of elements of the network . slow learning , on the other hand , results in a completely ordered state with the interactions becoming only of positive nature . on removing the external field , fast learning results in retrieving a network that is similar to the original one , i.e. , structurally balanced with communities of almost equal size . however , slow learning gives rise to a fully frustrated network exhibiting disorder . we observe that there exists a critical interval for values of the learning rate , around which small changes result in the system converging to distinct final states . in the following sections we first discuss the model and adaptation dynamics , followed by description of the results of numerical simulations . we conclude with a brief summary of our results and a discussion of their implications . for the purpose of investigating how collective behavior of a complex system is affected by adaptive dynamics of interactions , we use one of the well - known spin models of statistical physics which are generic systems for analyzing cooperative phenomena . in particular , we use the binary - state ising spin , the spin orientations ( `` up '' or `` down '' ) representing a pair of mutually exclusive choices . the simplicity of the model makes it applicable to not just magnetic materials ( in the context of which it was originally proposed ) but any system where the elements choose between the two competing states based on interactions with neighboring elements , noise ( represented by thermal fluctuations characterized by a temperature @xmath0 ) and external stimulus ( often represented by a magnetic field @xmath1 ) . the interactions @xmath2 between any pair of spins ( @xmath3 , @xmath4 ) can be either positive ( promoting connected spins to have the same state ) or negative ( promoting connected spins to have opposite states ) in nature . for instance , in neuronal networks , one can view the neurons as binary - state devices that are either firing ( active ) or quiescent ( inactive ) . correspondingly , as pointed out earlier , the excitatory and inhibitory connections between neurons can be represented by positive and negative interactions , respectively . at a different scale , one can view genetic regulatory networks in a similar vein , with genes being in either of two states , viz . , getting expressed ( active ) or not(inactive ) . in this setting , genes promoting the expression of other genes correspond to positive interaction , while inhibiting the expression of other genes correspond to negative interactions . in a different context , the interactions between ising spins can also be used to represent social intercourse @xcite . here , the spin states are considered to be analogous to two competing opinions , while the interactions may represent the nature of the relation between a pair of individuals - positive corresponding to affiliative and negative corresponding to antagonistic relations . we consider a system of @xmath5 globally coupled ising spins arranged into two sub - populations ( called modules or communities ) each having @xmath6 spins , at a constant temperature @xmath0 . for the simulations reported below we have chosen @xmath7 . the interaction between a pair of spins belonging to the same module is positive having strength @xmath8 ( @xmath9 ) , while that between spins belonging to different modules is negative with strength @xmath10 ( where @xmath11 ) . in the absence of an external field and thermal fluctuations ( i.e. , @xmath12 , @xmath13 ) , the two modules will be completely ordered in opposite orientations . when subjected to an external field of strength @xmath1 , the energy of the system is described by @xmath14 where @xmath15 is the ising spin on the @xmath3-th node ( @xmath16 ) in the @xmath17-th module ( @xmath18 ) . as every spin interacts with every other spin , a mean - field treatment should describe the system accurately . for convenience we define @xmath19 and @xmath20 as system parameters . the state of all spins in the system are updated stochastically at discrete time - steps using the metropolis monte carlo ( mc ) algorithm with temperature @xmath0 expressed in units of @xmath21 where @xmath22 is the boltzmann constant . the initial state of the system we have chosen is one exhibiting chimera order @xcite , to obtain which we use @xmath23 , @xmath24 and @xmath25 . we also allow the possibility that the interaction strengths can change over time through an adaptation process inspired by the principle of hebbian learning , a classical concept in the area of neural networks @xcite . colloquially often described as neurons that fire together , wire together " , in this mechanism the strength of synaptic connection between a pair of neurons is incremented when the two exhibit correlated activation . in ref . @xcite , this idea was used to propose a link adaptation rule that applies to a system of spins coupled via positive , as well as , negative interactions as : @xmath26 which is applied after every mc step . the adaptation rate , @xmath27 , decides the time - scale over which the interaction strength changes relative to the spin dynamics . starting with an initially frustrated spin system , implementing the above @xmath2 dynamics in absence of thermal fluctuations ( i.e. , @xmath28 ) results in the system converging to a structurally balanced state @xcite . this can be intuitively understood in terms of changes in the energy landscape over which the state of the spin system evolves . frustrated systems have rugged energy landscapes comprising a large number of local minima ( a fact which is exploited in neural network models of associative memory where these minima are used to embed desired patterns one wishes the network to memorize " @xcite . thus , beginning at any randomly chosen initial state of spin configurations , the spin dynamics drives the system into the nearest local energy minimum . the subsequent evolution of the @xmath2s converts this into the global minimum of the system . however , in the presence of noise ( @xmath29 ) , fluctuations in the spin dynamics can prevent the system from being trapped in any state for sufficiently long duration . thus , the adaptation dynamics fails to alter the energy landscape sufficiently so as to turn the configuration into the global minimum . therefore , with increasing temperature , an extremely long time may be required to reach structural balance . in this paper we use a modified form of the hebb - inspired adaptation rule for interaction strengths that was introduced in ref . this is to take into account the asymmetry in the strengths of positive and negative interactions necessary for observing chimera order ( typically @xmath30 ) @xcite . we change the adaptation rule such that the upper and lower bounds for the evolving interactions have the same values as that of the positive and negative interactions ( respectively ) which give rise to the chimera ordered state . thus after each mc step , interaction strengths are changed according to : @xmath31 . \label{hebbrulenew}\ ] ] the state of the system at any time is characterized by two quantities . one of these is the frustration associated with the interactions , measured by calculating the fraction of frustrated triads ( i.e. , connected sets of 3 spins with an odd number of negative interactions ) in the system : @xmath32 which varies between @xmath33 ( no frustration ) and @xmath34 ( completely frustrated ) . and @xmath35 , in ( a ) chimera ordered state and ( b ) disordered state , shown for mc simulations with @xmath36 at ( a ) @xmath37 and ( b ) @xmath38 , respectively . ] shown when the spin system exhibits ( a ) chimera order and ( b ) disorder . the kernel smoothened distribution function is represented by a thick curve . results shown for mc simulations with @xmath36 at ( a ) @xmath37 and ( b ) @xmath38 , respectively . ] the other quantity is an order parameter @xmath39 that is used to identify a chimera state in a spin system with two modules that have magnetizations @xmath40 and @xmath35 , respectively . to numerically estimate the order parameter @xmath41 , we are confronted with a few potential complications . first , as the magnetizations of the two modules are stochastically fluctuating ( especially the one which is disordered , i.e. , having lower magnetization ) , we can use their average values . however , this gives rise to an additional problem as the modules can frequently exchange their order / disorder status , especially when the modules are of the same size [ fig . [ mag_time ] ( a ) ] . thus , a method is required to measure @xmath41 that is unaffected by the modules switching their magnetizations , while being able to determine if the time - averaged magnetizations of the two modules are different ( the signature of chimera order ) . for instance , a simple time average of the absolute value of differences between @xmath40 and @xmath35 will not give correct results as one obtains a finite value ( because of stochastic fluctuations ) even when the time averaged magnetizations are same for both modules [ fig . [ mag_time ] ( b ) ] . therefore , to estimate @xmath41 we have used the following algorithm . first , the frequency distribution of @xmath42 is computed . in the absence of chimera , the distribution is unimodal [ fig . [ o_distrib](a ) ] which is approximated as a gaussian function , while for chimera ordering the distribution is bimodal [ fig . [ o_distrib](b ) ] which is approximated as a superposition of two gaussian functions . for a unimodal distribution , the value of @xmath42 corresponding to the peak is used to calculate the order parameter @xmath41 . for bimodal distributions , a weighted average of the values corresponding to the peaks for example , if one of the peaks occurs at @xmath43 with value @xmath44 while the other is at @xmath45 with value @xmath46 , the order parameter is calculated as @xmath47 . in order to make the automated detection of the peak locations accurate , the frequency distributions need to be smoothed of all fluctuations in the frequencies so that the local maxima at the peaks can be uniquely determined . for this purpose we have used a kernel smoothing technique @xcite that gives a single peak location for unimodal distributions and locations for two peaks in the case of bimodal distributions . starting from a structurally balanced network comprising two modules , we first introduce field and thermal fluctuations so as to drive the system into chimera order - i.e. , one of the modules becomes ordered while the other is disordered . once this is achieved we allow adaptation dynamics of the interactions to take place . as the evolution of the interactions are related to the degree of fluctuations in the spin states , these would occur mostly in the disordered module where the spins are subject to the competing forces of negative interactions with the spins belonging to the ordered module and the influence exerted by the uniform external magnetic field which tries to align the spins in parallel with those of the ordered module . thus , at any instant , a few spins in the disordered module will get aligned with the spins in the other module and consequently may develop positive interactions with the latter as a result of link adaptation . this means that they will now no longer be part of the disordered module but become part of the ordered module . as a result , the size of the modules would change over time - the ordered module increasing at the expense of the disordered one . this has to be taken into account when measuring system properties , such as , magnetization per spin of the two modules . in order to have a consistent definition of module size that would be valid even when the system is no longer in structural balance , we heuristically measure it as follows . at each update of the @xmath2s , we randomly choose a spin from the ordered module and consider all spins that have positive interactions with it to belong to the ordered module , with the remaining spins comprising the disordered module . following this process , we can follow the time - evolution of modular membership of individual spins , as well as , that of the module sizes . however , this measure becomes less useful as the system becomes increasingly frustrated . mc steps ( indicated by broken line ) . the ordered module expands in size ( @xmath48 ) while the disordered one shrinks . learning rate @xmath27 = 0.1 ] and @xmath35 of the two modules in the network ( having sizes @xmath48 and @xmath49 , respectively ) before and after link adaptation dynamics is introduced at time @xmath50 mc steps ( indicated by broken line ) . in absence of adaptation , magnetizations show the characteristic signature for chimera order . once adaptation dynamics is operational , the ordered module with magnetization @xmath40 is seen to increase in size . ( bottom ) in absence of learning , the network exhibits no frustration as it is structurally balanced . following the introduction of adaptation dynamics , there is a transient increase in frustration , before the system once again achieves balance , but with modules of asymmetric sizes . ] [ hebb_module_size ] illustrates the evolution of module sizes with time following the introduction of link adaptation dynamics in the system . we can now measure the magnetizations per spin , @xmath40 and @xmath35 , of the two modules by taking into account the modular membership of each spin and the module sizes . fig . [ hebb_m1m2 ] shows these order parameters , as well as , the frustration in the network , as a function of time before and after the link adaptation dynamics is introduced . as expected , we observe that following the start of link adaptation dynamics , the ordered module begins to grow in size , which is also observed in the time - evolution of the interaction matrix ( shown as snapshots at regular intervals in fig . [ jsnaps ] ) . note that the time interval shown here corresponds to the same period over which the frustration in the system initially rises and then decreases again as the system once more reaches a balanced state , as indicated in fig . [ hebb_m1m2 ] . ] is introduced ( the time at which the link adaptation is initiated is indicated by the broken line ) . following a brief transient rise in the frustration , the system again attains a balanced state although with asymmetric module sizes . ] is introduced ( the time at which the link adaptation is initiated is indicated by the broken line ) . the system eventually becomes completely ordered with all interactions becoming positive ( resulting in merging of the two modules into a single one ) . ] is introduced ( the time at which the link adaptation is initiated is indicated by the broken line ) . initially , the two modules both become ordered . however , eventually the two modules merge into a single one as all interactions become positive . the time of merging coincides with the transient rise in frustration . ] is introduced ( the time at which the link adaptation is initiated is indicated by the broken line ) . we observe that while one of the modules remains ordered , the initially disordered modules is gradually also tending towards complete order . eventually one expects the two modules to merge into a single one . ] the time - evolution of the initial chimera - ordered network subjected to link adaptation dynamics with various learning rates ranging from @xmath51 to @xmath52 are indicated in figs . [ eps1e-1]-[eps1e-4 ] . for fast learning rates ( e.g. , @xmath53 ) , the system immediately converges to a balanced state characterized by a large ordered module ( comprising about @xmath54 of all the elements ) and a small disordered one . by observing the magnetization time - series of the two modules in fig . [ eps1e-1 ] it is easy to infer that the system still exhibits chimera order . we note that a rise in frustration is seen for a very brief duration during the initial period immediately following initiation of the link adaptation dynamics . for a slower learning rate , viz . @xmath55 , the process of expansion in size of the ordered module is different ( fig . [ eps1e-2 ] ) . we notice that soon after the beginning of link adaptation , the chimera ordering is lost as the disordered module becomes ordered . however , the modules still retain their different identities as they are defined based on the sign of interactions of the spins within them ( which should be positive ) and that with spins in the other module ( which should be negative ) . for even slower learning rates such as @xmath56 or @xmath52 ( figs . [ eps1e-3]-[eps1e-4 ] ) the duration over which the two modules exist while both being ordered is seen to increase . during this period the modular membership of the spins do not change significantly ( apart from small fluctuations ) . chimera ordering is lost as the interaction strengths for spins in the disordered module becomes weaker as a result of frequent switchings of their orientations . as a result the effect of the external field becomes dominant , which causes the spins in the disordered module to align with it . as spins in both modules are now aligned parallel to each other , adaptation of their interactions with time would eventually make all interactions positive , thereby merging the two modules into one . the time at which this happens is indicated by a transient rise in the frustration ( figs . [ eps1e-2 ] and [ eps1e-3 ] ) , occurring much later for slower rates of adaptation . the different scenarios we observe at fast and slow learning rates suggest that there is a competition between two processes : if the modular membership of the spins can change rapidly following initiation of link adaptation so that the system adopts a new structurally balanced configuration , the interaction strengths of the spins in the disordered module do not decrease significantly - thereby allowing the two modules to coexist with the chimera ordering intact ( although the modules now have very different sizes ) . note that the modular membership of the spins will change rapidly as they continually change their interaction strengths because of the fast adaptation rate . however , the average number of spins in each module will remain fairly steady . if the learning rate is slow , the interaction strengths will not have time to change sufficiently rapidly to allow the system to reach a new structurally balanced state following the initiation of link adaptation dynamics . as a result the field dominates over the weakened interactions of the spins in the disordered module , thereby eliminating chimera order and eventually causing the two modules to merge . the existence of chimera order even in situations where the two modules have very different sizes is a novel observation as earlier it had only been observed in a system with symmetric module sizes @xcite . in order to see how the region in the field - temperature parameter space where we observe chimera ordering varies with asymmetry in the module sizes , we show in fig . [ ratio ] the dependence of the order parameter @xmath57 on the strength of the external field ( @xmath1 ) , the scaled temperature ( @xmath58 ) and the ratio of sizes of the two modules ( @xmath59 ) in the absence of link adaptation dynamics . we observe that the region in parameter space where we find chimera order ( higher values of @xmath57 ) increases significantly as we go towards higher module size ratios ( i.e. , @xmath59 ) especially for lower values of temperature . ) parameter space shown for different ratios of the sizes of the two modules . chimera is indicated in terms of high values of the order parameter @xmath57 ( represented by different colors ) . ] we now consider the situation when the external field is withdrawn while allowing the link adaptation dynamics to continue . a difference is expected with the results shown in the earlier work @xcite where the effect of link adaptation on the evolution of a system subject to thermal fluctuations was investigated , as the learning rule is different on account of the asymmetry in the strengths allowed for negative and positive interactions . as the adaptation dynamics used here allows for much stronger negative interactions ( as compared to positive ) , we would expect the system to become more frustrated as negative interactions are much more likely to contribute frustrated triads . this is indeed what is observed for slow learning rates , e.g. , @xmath60 and @xmath55 ( fig . [ off_eps1e-1 - 2 ] ) . note that in a frustrated system , one can not define modules in a meaningful way , and thus the distinction between modules indicated in the time - series for magnetization is an artifact of the measurement method . for both of the slow learning rates we observe that the system converges to a fully frustrated state corresponding to the maximum value of the measure of frustration in the system ( @xmath61 ) . , before and after an external field is withdrawn ( the time at which the field is switched off is indicated by the broken line ) . we observe that the system becomes completely frustrated on withdrawal of the field . ( b ) the corresponding time - evolution shown for link adaptation rate @xmath55.,title="fig : " ] , before and after an external field is withdrawn ( the time at which the field is switched off is indicated by the broken line ) . we observe that the system becomes completely frustrated on withdrawal of the field . ( b ) the corresponding time - evolution shown for link adaptation rate @xmath55.,title="fig : " ] for faster link adaptation rates , however , we observe unexpected behavior in the system . for a learning rate @xmath62 [ fig . [ off_eps1.3 ] ] , which is only slightly higher than @xmath53 [ shown in fig . [ off_eps1e-1 - 2 ] ( a ) ] , the system initially becomes fully frustrated after the external field is withdrawn . surprisingly , after a period of @xmath63 mc steps during which the system remains frustrated , it suddenly converges to a balanced state . this intervening period between the field being switched off and the system spontaneously splitting into two modules having positive interactions among all elements within them ( and correspondingly , negative interactions between elements belonging to different modules ) becomes even shorter with increasing learning rates ( e.g. , see fig . [ off_eps1.5 ] for @xmath64 ) . thus , we can identify a critical value of around @xmath65 for the link adaptation rate @xmath27 , below which the system remains frustrated and above which the system converges to a balanced state , on withdrawal of the external field . indeed our simulations show that at @xmath62 there is a wide variation in the time required for the system to converge to balance after the field is switched off . , before and after an external field is withdrawn ( the time at which the field is switched off is indicated by the broken line ) . we observe that the system initially becomes completely frustrated on withdrawal of the field . however after about @xmath66 mc steps the system suddenly becomes balanced . ] , before and after an external field is withdrawn ( the time at which the field is switched off is indicated by the broken line ) . we observe that the system quickly becomes balanced after a brief transient rise in frustration following the withdrawal of the field . ] in the previous study of structural balance @xcite , it had been observed that the time required to converge to a structurally balanced state exhibits a bimodal distribution for a range of temperatures . we observe similar features in our model also - for instance , for a link adaptation rate of @xmath62 . for slightly slower learning rates ( e.g. , @xmath67 ) , the convergence time increases significantly . in our simulations , many realizations did not converge to the balanced state within the time of observation ( @xmath68 mc steps ) . for higher learning rates , the system rapidly converges to balance . however , unlike the convergence to balance seen in the previous study @xcite , the modules in the structurally balanced state that our model system reaches are of almost equal size ( see figs . [ off_eps1.3 ] and [ off_eps1.5 ] ) independent of the initial state of the network ( including the initial distribution of interactions which was seen to affect the nature of the balanced state that the system converges to in the earlier work ) . this may appear surprising as starting from a balanced state , one expects that the network will remain in that state as the interactions adapt so as to make the corresponding energy minimum even deeper . however , what we observe is that , on withdrawing the field the balanced state characterized by asymmetric module sizes is destabilized and the energy minimum moves in the configuration space eventually reaching the region corresponding to balanced state with modules having equal size . this difference between our results and that of the previous study suggests that the adaptation rule used here ( which is biased in favor of negative interactions ) is responsible for an intriguing meta - dynamics of the energy landscape itself . we have also observed how the structure of the energy landscape underlying our system evolves as the interactions change through the adaptation rule . in the model studied earlier @xcite , the frustration of the initially disordered system is around 0.5 . on initiating link adaptation , this value decreases as the system tends towards balance . when frustration reaches a value around 0.49 , we observe that the energy landscape becomes such that starting from any spin configuration one can reach the same minimum energy spin arrangement . this suggests that a small decrease in the frustration can accompany a a very large change in the basin of attraction of an energy minimum ( which now encompasses a significant fraction of the configuration space ) . we observe similar behavior in our model system where the adaptation occurs in the presence and subsequent absence of an external field . for example , for an adaptation rate of @xmath69 , when the field is withdrawn the system initially becomes completely frustrated with the value of frustration measured as around 0.97 . the corresponding energy landscape has a very large number of minima , each having very small basins of attraction . as the system evolves towards a balanced state , we note that even when the frustration decreases by a very small amount , e.g. , to 0.93 , the energy landscape transforms to one having an extremely deep energy minimum with a very large basin of attraction . we believe that this radical transformation of the energy landscape of the system at the initial stages of approach to balance points to features of landscape evolution that deserve further study . in this paper we have explored the behavior of a spin system that is subjected to an external field , while the interactions adapt in response to the system dynamics . the results obtained through our simulations that is described above is summarized in fig . [ schematic ] . from an earlier study , we had known that introducing link adaptation inspired by the hebbian principle in a frustrated system can result in it converging to a structurally balanced state - corresponding to evolution of the underlying energy landscape - that depends on the initial state of the system . applying this adaptation dynamics to a system exhibiting chimera order ( as done in here ) allows us to explore how the dynamics of the energy landscape is affected by an external field , as well as , asymmetry in the adaptation rule regarding negative and positive interactions . we observe that when system is adapting through faster learning rates , the chimera order is maintained throughout its evolution , with the ordered module increasing in size at the expense of the disordered module . for slower learning rates , first the interactions of the elements belonging to the disordered module become weak which results in loss of chimera order . this is followed by the system converging to a fully ordered state where almost all the interactions are positive . the sizes of the two modules defining the system where one observes chimera order can be varied to see how the ratio of module sizes can affect the subsequent evolution of the system . indeed , we see that the region in the field - temperature parameter space over which chimera order is seen increases as the ratio increases from 1 . when the external field is withdrawn , the system returns to a structurally balanced state with modules of similar sizes if the adaptation rate is high . however , for slower learning rates , the system becomes completely frustrated . we have also tried to explore how the structure of the energy landscape of the system evolves during learning . starting from a chimera ordered state that is subjected to adaptation rate and external field , we observe changes in the landscape after the field is switched off . initially the system becomes completely frustrated . however , after an interval ( which decreases with increasing learning rate ) the system shows a sudden large increase in the basin of attraction of an energy minimum , although the frustration of the system has decreased only negligibly . this surprisingly radical change in the landscape resulting from relatively small degree of change in the interaction structure of the network is a question that needs to be explored further . it is of interest to consider the implications of the results reported here for adaptation in biological systems . as connections in the brain evolve according to long - term potentiation that embodies the hebbian principle that has inspired the link adaptation dynamics used in our study , it is natural to expect that the brain may exhibit at least some of the features observed here . indeed , as mentioned earlier , the experimental observation of two dynamically anti - correlated subnetworks in the resting human brain @xcite strongly suggests that in the absence of strong external stimulation the underlying network is structurally balanced . on the other hand , when exposed to stimuli , correlated brain activity is indeed observed - although not encompassing the entire network . this is , to some extent , reminiscent of the chimera ordered state that is seen in our model system . thus , the brain may be seen as corresponding to a system subject to relatively fast adaptation rate . however , the hebbian principle can apply to a much broader class of biological systems , e.g. , gene regulation networks where the co - expression of genes has been suggested to result in their co - regulation over evolutionary time - scales @xcite . as the adaptation in this case is provided by natural selection , which is orders of magnitude slower than the learning process in the brain mentioned earlier , it is probably not unreasonable to conclude that this can be seen as one corresponding to our model system subject to slow rate of adaptation . it is easy to see that the frustrated system with a large number of energy minima can be considered to be analogous to the cellular differentiation process that allows convergence to any one of a large number of possible cell fates dependent upon initial conditions . indeed this analogy has been used earlier by kauffman to motivate boolean network models for explaining differentiation during biological development @xcite . extending this analogy , one may wonder whether the system has a state corresponding to the ordered state seen on exposing it to an external stimulus . as this state is unique and will be attained by the system independent of all initial conditions , it is tempting to suggest that induced pluripotency in cells exposed to chemical stimuli @xcite may be the biological analogue . therefore , it may be of interest to study the phenomena reported here in biologically realistic models of networks adapting at different time scales . we thank rajeev singh , shakti n. menon and r. janaki for helpful discussions . this work was supported in part by imsc complex systems project ( xii plan ) funded by the department of atomic energy , government of india . the numerical work was carried out in machines of the imsc high - performance computing facility , including `` satpura '' which is partially funded by dst ( grant no . sr / nm / ns-44/2009 ) .
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many complex systems can be represented as networks of dynamical elements whose states evolve in response to interactions with neighboring elements , noise and external stimuli .
the collective behavior of such systems can exhibit remarkable ordering phenomena such as _ chimera order _ corresponding to coexistence of ordered and disordered regions .
often , the interactions in such systems can also evolve over time responding to changes in the dynamical states of the elements . link adaptation inspired by hebbian learning
, the dominant paradigm for neuronal plasticity , has been earlier shown to result in structural balance by removing any initial frustration in a system that arises through conflicting interactions . here
we show that the rate of the adaptive dynamics for the interactions is crucial in deciding the emergence of different ordering behavior ( including chimera ) and frustration in networks of ising spins .
in particular , we observe that small changes in the link adaptation rate about a critical value result in the system exhibiting radically different energy landscapes , viz . , smooth landscape corresponding to balanced systems seen for fast learning , and rugged landscapes corresponding to frustrated systems seen for slow learning .
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in 1990 , carroll - field - jackiw @xcite have proposed a version of the maxwell electrodynamics corrected by a chern - simons - like term @xmath0 in order to incorporate a lorentz - violating background @xmath1 into the usual electrodynamics . this term implies a modified theory in which photons with different polarizations propagate with distinct velocities ( birefringence ) . some years later , colladay & kostelecky @xcite-@xcite have constructed an extension of the minimal standard model , the extended standard model ( sme ) , in which lorentz - violating tensor terms , stemming from a spontaneous symmetry breaking ( ssb ) of a more fundamental theory ( defined at the planck scale ) are properly incorporated in all interaction sectors . the construction of the sme was in part motivated by works demonstrating the possibility of lorentz and cpt spontaneous violation in the context of string theory @xcite-@xcite . recently , the sme has motivated innumerous interesting works @xcite-@xcite . one of the most remarkable controversies involving lorentz violation deals with the radiative generation of the carroll - field - jackiw term from the integration on the fermion fields @xcite-@xcite . lorentz violating theories investigations have also been concerned with the consistency aspects of the carroll - field - jackiw electrodynamics @xcite-@xcite , study of synchrotron radiation , electrostatics and magnetostatics in lorentz - violating electrodynamics @xcite-@xcite , influence of lorentz violation on the dirac equation @xcite-@xcite , cpt - probing experiments @xcite-@xcite , cerenkov radiation @xcite , and general aspects @xcite-@xcite . a theoretical model which provides an attractive electron - electron interaction could work , in principle , as a good framework to properly address the electron - electron pairing in planar systems . in fact , if an attractive electron - electron interaction is obtained in the context of a particular model , it may be seen as a first connection between such theoretical models and the attainment of electron pairing . in practice , this interplay has begun with the application of the maxwell - chern - simons ( mcs ) theory @xcite-@xcite for evaluating the electron - electron interaction in a planar model . however , it was soon established that the mcs model does not imply an attractive interaction for small topological mass @xmath2 regime compatible with low - energy excitations . currently it is well known that by including the higgs sector @xcite , @xcite , an attractive interaction can be got , assured it is suitably coupled to the fermion field by a quartic order term - @xmath3 ( that gives rise to the yukawa coupling with the higgs field after ssb ) . on the other hand , it has been recently verified that the mcs theory may also yield an attractive @xmath4 potential ( in the absence of higgs sector ) provided it is considered in the presence of a fixed lorentz - violating background . specifically , it has been evaluated the electron - electron interaction potential in the context of a planar lorentz - violating maxwell - chern - simons model ( arising from the dimensional reduction of the maxwell - carroll - field - jackiw model ) . such calculation was carried out both for the case of a purely timelike background @xcite and for a purely spacelike background @xcite , leading to interacting potentials with a well ( attractive ) region . some years ago , it was argued that there is a relation between lorentz violation and noncommutativity @xcite . the introduction of noncommutativity in the mcs model @xcite has appeared as a new mechanism able to provide @xmath4 attraction . in fact , the noncommutative extension of the minimal mcs model has shown to be a suitable framework to provide an attractive electron - electron potential . specifically , this model yields the same interaction potential attained by georgelin & wallet @xcite considering a non - minimal pauli magnetic coupling . this puts in evidence the relevance of non usual mechanisms for the attainment of electron - electron attractiveness . it should be also mentioned that noncommutative chern - simons theories have been applied successfully to describe properties of planar hall systems @xcite-@xcite , one of the points of clear connection of noncommutativity with condensed matter physics . the large number of applications of noncommutativity to condensed matter physics and the general relation between these mechanisms indicate that applications of lorentz violation to condensed matter systems should be a sensible and feasible issue as well . lorentz violation in the presence of the higgs sector and spontaneous symmetry breaking ( ssb ) was first investigated in the context of the 4-dimensional abelian - higgs carroll - field - jackiw electrodynamics @xcite . this model , by means of a dimensional reduction procedure , has originated a planar electrodynamics composed of the mcs sector with a higgs field , coupled to a massless klein - gordon mode - @xmath5 ( stemming from the dimensional reduction , @xmath6 ) and to the lorentz - violating background ( @xmath7 @xcite . the consistency of this model was also set up ( it turned out to be totally causal , unitary and stable ) . once ssb takes place , the gauge and the klein - gordon fields acquire mass , giving rise to a mcs - proca electrodynamics coupled to the lorentz - violating background . such a model was already used to perform an investigation concerned with condensed matter physics : the study of vortex - like solutions in a planar lorentz - violating environment @xcite . as a result , it was shown that it provides charged vortex solutions that recover the usual nielsen - olesen configuration in the asymptotic regime . in the present work , the aim is to investigate the electron - electron interaction in the context of the abelian - higgs lorentz - violating planar model previously defined , another issue with possible connection with condensed matter physics . the lagrangian of ref . @xcite does not stand for the most general neither the most simple model to perform such a task . however , there are two good reasons to adopt it : its consistency has been already established ; it is expected that it will provide shielded versions of the mcs lorentz - violating potentials derived in ref . @xcite ( which present a logarithmically asymptotic behavior ) . having stated the gauge model , the dirac sector is properly incorporated in it by exhibiting the minimal and the yukawa couplings with the gauge ( @xmath8 and the scalar ( @xmath9 fields , respectively . one then proceeds to carry out the electron - electron potential for the case of a purely timelike background and to discuss its possible attractiveness . the procedure is much similar to the one adopted in refs . @xcite , @xcite : starting from a known planar lagrangian , the mller scattering amplitudes ( for the gauge and scalar intermediations ) are constructed ; next , its fourier transforms are evaluated , leading to the interacting potentials . in the present case , the result is a totally screened potential * * ( * * due to the presence of the higgs sector * * ) * * composed of the sum of a scalar and a gauge contribution . this potential exhibits an attractive behavior even in the presence of the centrifugal barrier and the @xmath10gauge invariant term , thereby confirming its attractive character ( even under a more rigorous analysis ) and its possible relevance to the formation of electron - electron pairs in planar systems . the results obtained here are compared with the one of ref . @xcite in order to emphasize the role played by the higgs sector : it transforms logarithmically divergent solutions in entirely shielded ones . this comparison will be accomplished throughout this work . another point to be remarked is that the gauge potentials here derived may be attractive while the ones of refs . @xcite , @xcite are always repulsive , which constitutes a sensitive difference between the results of these works . this work is outlined as follows . in sec . ii , the reduced model derived in ref . @xcite , supplemented by the fermion field , is briefly described . in sec . iii are presented the spinors which fulfill the two - dimensional dirac equation , which , in its turn , are used to evaluate the mller scattering amplitude associated with the scalar and gauge intermediations . in sec . iv , the interaction potentials are evaluated by performing the fourier transform of the scalar and gauge scattering amplitudes . the results are properly discussed . in sec.v are presented the concluding remarks . at first , one ought to present the planar model which sets up the theoretical framework for the calculations realized in this work . the starting point is the ( 1 + 3)-dimensional abelian - higgs maxwell - carroll - field - jackiw ( mcfj ) model @xcite , consisting of the mcfj electrodynamics supplemented with the higgs sector . @xmath11 where @xmath12 is the lorentz - violating fixed background , @xmath13 runs from @xmath14 to @xmath15 @xmath16 is the covariant derivative and @xmath17 represents the scalar potential responsible for ssb ( @xmath18 and @xmath19 in a previous work , this model had undergone a dimensional reduction procedure in which the third spatial coordinate is frozen , implying : @xmath20 at the same time , the third component of the vector potential becomes a scalar field , @xmath21 whereas the third component of background becomes the topological mass:@xmath22 @xmath23 this process yields a lorentz - violating planar model incorporating the higgs sector @xcite , given as below : @xmath24 where the greek letters now run from @xmath14 to @xmath25 once the planar lorentz - violating model has been established , it is considered the spontaneous symmetry breaking process which provides mass to the gauge and scalar fields @xcite . relying on a tree - level analysis , one should retain only the bilinear terms , so that the planar lagrangian takes the form : @xmath26 where @xmath27and @xmath28is the vacuum expectation value of the @xmath29 field . here , one explains the reason for which the higgs field does not appear in the above equation : it only keeps high order couplings with the other fields . as it is well - known , these terms are not taken into account in a tree - level evaluation . the classical solutions of this planar model were achieved in ref . @xcite , where the effects of the fixed background on the mcs - proca electrodynamics were exhaustively analyzed . it was also reported that the scalar potential @xmath30 for a purely timelike background exhibits an attractive behavior , which may be seen as a cue indicating that a similar result may be shared by the gauge electron - electron potential to be evaluated in a dynamic configuration . now , it is necessary to introduce the spinor field suitably coupled to the gauge @xmath31 and scalar @xmath32 ones . in the absence of sources , the interaction lagrangian is read as : @xmath33 the term @xmath34 sets up the minimal coupling , whereas @xmath35 reflects the yukawa coupling with the scalar field . the fermion field @xmath36 is a two - component spinor with up spin - polarization , representing the positive energy solution of the dirac equation , @xmath37 here written in momentum space . the mass dimension of the fields and parameters involved in eq . ( [ l1 ] ) are the following : @xmath38 = \left [ a^{\mu}\right ] = 1/2,\left [ \psi\right ] = 1,\left [ s\right ] = \left [ v^{\mu}\right ] = 1,\left [ e_{3}\right ] = \left [ y\right ] = 1/2;$ ] it is noticeable that both coupling constants , @xmath39 and @xmath40 exhibit @xmath41 ^{1/2}$ ] dimension , a usual result in ( 1 + 2 ) dimensions . in ref . @xcite , the propagators of the scalar @xmath42 and gauge @xmath31 fields were properly evaluated in the following form : @xmath43 \nonumber\\ & + \frac{i(v\cdot k)s^{2}k^{2}}{(k^{2}-m_{a}^{2})\boxtimes(k)\boxplus ( k)}\left [ \sigma^{\mu\nu}+\sigma^{\nu\mu}\right ] -\frac{is(v\cdot k)}{\boxtimes(k)\boxplus(k)}\left [ \phi^{\mu\nu}-\phi^{\nu\mu}\right ] \biggr\ } , \label{prop_gauge}\]]@xmath44 with : @xmath45 @xmath46 the projector operators are defined as : @xmath47 @xmath48 @xmath49 @xmath50 @xmath51 @xmath52 @xmath53 while it is adopted the ( 1 + 2 ) metric convention : @xmath54 naturally , these expressions are essential for the evaluation of the amplitudes associated with the mller scattering , task developed in the next section . in the context of a low - energy interaction , the born approximation holds as a good approximation . consequently , the interaction potential arises as the fourier transform of the two - particle scattering amplitude . another point is that , in the case of the nonrelativistic mller scattering , it should be considered only the direct scattering process @xcite ( even for indistinguishable electrons ) , since in this limit they recover the classical notion of trajectory . from eq . ( [ l1 ] ) , one extracts the feynman rules for the interaction vertices involving fermions : @xmath55 . therefore , the @xmath4 scattering amplitudes are read as : @xmath56 \overline{u}(p_{2}^{\prime})(iy)u(p_{2}),\label{a1}\\ -i\mathcal{m}_{a } & = \overline{u}(p_{1}^{\prime})(ie_{3}\gamma^{\mu})u(p_{1})\left [ \langle a_{\mu}a_{\nu}\rangle\right ] \overline{u}(p_{2}^{\prime})(ie_{3}\gamma^{\nu})u(p_{2 } ) , \label{a2}\ ] ] here , @xmath57 and @xmath58 are obviously the scalar and photon propagators given in eqs . ( [ prop_gauge ] ) , ( [ prop_scalar ] ) . the scattering amplitudes , of eqs . ( [ a1 ] ) and ( [ a2 ] ) , are written for electrons of equal polarization mediated by the scalar and gauge particles , respectively . the spinors @xmath59 stand for the positive - energy solution of the dirac equation @xmath60 . the @xmath61 matrices satisfy the @xmath62 algebra , @xmath63 = 2i\epsilon^{\mu\nu\alpha}\gamma_{\alpha}$ ] , and correspond to the pauli matrices : @xmath64 considering all it , the following spinors @xmath65{c}e+m\\ -ip_{x}-p_{y}\end{array } \right ] , \text { \ \ } \overline{u}(p)=\frac{1}{\sqrt{n}}\left [ \begin{array } [ c]{cc}e+m & -ip_{x}+p_{y}\end{array } \right ] , \label{spinor}\ ] ] are explicitly obtained . they satisfy the normalization condition @xmath66 for @xmath67 . the mller scattering is easily attained in the frame of the center of mass , where the momenta of the incoming and outgoing electrons are read in the form:@xmath68 the transfer 4-momentum , carried by the gauge or scalar mediators , is : @xmath69 whereas @xmath70 is the scattering angle ( in the cm frame ) . starting from eqs . ( [ prop_scalar ] ) , ( [ a1 ] ) and from the definitions above , the scattering amplitude associated with the scalar intermediation is readily written : @xmath71 in the case of a purely timelike background , @xmath72v@xmath73 , @xmath74 and considering the general expression for the transfer momentum , @xmath75 , this amplitude takes the following form : @xmath76}{[\mathbf{k}^{4}+\left ( 2m_{a}^{2}+s^{2}-\text{v}_{0}^{2}\right ) \mathbf{k}^{2}+m_{a}^{4}][\mathbf{k}^{2}+m_{a}^{2 } ] } , \label{mscalar2}\ ] ] whose fourier transform will lead to the potential that reflects the scalar interaction carried by the @xmath77field . one the other hand , in the case of the gauge intermediation , the situation is more complicated as a consequence of the eleven terms present in the propagator ( [ prop_gauge ] ) . however , as a consequence of the current - conservation law ( @xmath78 only six terms of the gauge propagator contribute to the scattering amplitude ( @xmath79 the first two terms provide , in the non - relativistic limit , the maxwell - chern - simons - proca ( mcsp ) scattering amplitude , which leads to mcsp potential . the non - relativistic current - current amplitudes involving these two terms , @xmath80 are evaluated in refs.@xcite , @xcite,@xcite . the corresponding scattering amplitude is then given by : @xmath81 \right\ } .\label{mcsp}\ ] ] the current - current amplitude associated with the other terms of the gauge potential are also carried out , assuming the form below : @xmath82;\\ j^{\mu}(p_{1})\text { } ( \lambda_{\mu\nu})j^{\nu}(p_{2 } ) & = \text{v}_{0}^{2};\\ j^{\mu}(p_{1})\text { } ( q_{\mu\nu}-q_{\nu\mu})j^{\nu}(p_{2 } ) & = 2\frac { \mathbf{p}^{2}}{m_{e}}\text{v}_{0}^{2}[1-\cos\theta - i\sin\theta].\end{aligned}\ ] ] these terms lead to the following scattering amplitudes : @xmath83 where @xmath84 @xmath85 is defined in terms of the momenta @xmath86 of the incoming electrons and @xmath87.$ ] the total current - current amplitude mediated by the massive gauge particle corresponds to the sum of four contributions , @xmath88 where the terms @xmath89 and @xmath90 lead tobackground - depending corrections to the mcs - amplitude . notice that the amplitude @xmath91 was taken as null due to its dependence on @xmath92(working in the nonrelativistic approximation , @xmath93 . firstly , it is necessary to evaluate the interaction related with the scalar intermediation . according to the born approximation , the scalar interaction potential is given by the fourier transform of the scattering amplitude ( [ mscalar2 ] ) , that is:@xmath94}{[\mathbf{k}^{4}+\left ( 2m_{a}^{2}+s^{2}-\text{v}_{0}^{2}\right ) \mathbf{k}^{2}+m_{a}^{4}][\mathbf{k}^{2}+m_{a}^{2}]}\right ] e^{i\mathbf{k}\cdot\mathbf{r}}d^{2}\mathbf{k}\ ] ] in this form , this integral can not be exactly solved . however , it is possible to factorize the integrand in small factors so that an exact integration becomes feasible . in this sense , it is important to note that : @xmath95=[\mathbf{k}^{2}+m_{+}^{2}][\mathbf{k}^{2}+m_{-}^{2}],$ ] where the constants @xmath96 are given as below . after some algebraic calculations , one obtains @xmath97}{[\mathbf{k}^{4}+\left ( 2m_{a}^{2}+s^{2}-\text{v}_{0}^{2}\right ) \mathbf{k}^{2}+m_{a}^{4}][\mathbf{k}^{2}+m_{a}^{2}]}=-\frac{(g+d)}{[\mathbf{k}^{2}+m_{+}^{2}]}+\frac{(g+e)}{[\mathbf{k}^{2}+m_{-}^{2}]}+\frac{(1+d - e)}{[\mathbf{k}^{2}+m_{a}^{2}]},\ ] ] with coefficients and mass parameters given as:@xmath98 , g=\text{v}_{0}^{2}c,\text { \ } d = g\alpha_{+},\text { \ } e = g\alpha_{-},\label{b1}\\ \alpha_{\pm } & = \frac{2m_{a}^{2}}{(s^{2}-\text{v}_{0}^{2})\pm\sqrt { ( s^{2}-\text{v}_{0}^{2})(s^{2}-\text{v}_{0}^{2}+4m_{a}^{2})}},\label{t1}\\ m_{\pm}^{2 } & = \frac{1}{2}\left [ ( s^{2}-\text{v}_{0}^{2}+2m_{a}^{2})\pm\sqrt{(s^{2}-\text{v}_{0}^{2})(s^{2}-\text{v}_{0}^{2}+4m_{a}^{2})}\right ] . \label{m1}\ ] ] performing the fourier transforms , the expression @xmath99 is straightforwardly obtained . this result reveals a totally screened potential as a consequence of the massive character of the scalar intermediation . near the origin , this potential behaves as a pure logarithm , that is : @xmath100 whence one reaffirms its attractive character at the origin . far from the origin this potential vanishes exponentially , and is the point where it differs from the scalar potential obtained in the lorentz - violating mcs k_{0}(sr)-\frac{s^{2}}{w^{2}}\ln r\right\ } , $ ] whereas the gauge potential derived in this paper is : @xmath101k_{0}(wr)+\left ( \text{v}_{0}^{2}/w^{2}\right ) \ln r$ ] @xmath102 \biggr\}.$ ] ] case @xcite , which exhibits an asymptotic confining logarithmic behavior . it is well - known that such kind of confining potential can not describe a physical interaction in ( 1 + 2 ) dimensions . hence , the first advantage arising from the introduction of the higgs sector in this theoretical framework is the transformation of the non physical confining potential of ref . @xcite in a bessel @xmath103 potential ( entirely suitable for describing a planar interaction ) . the graphic in fig . [ scalarplot1 ] illustrates such a change of asymptotic behavior . [ ptb ] scalarplot1.eps the starting point for the evaluation of the gauge potential is the attainment of the @xmath104potential from the fourier transform of the @xmath105-amplitude . such fourier transform can not be directly solved from eq . ( [ mcsp ] ) , which then is properly factorized in the form @xmath106}+\frac{a_{-}}{[\mathbf{k}^{2}+m_{-}^{2}]}+\left [ \frac { b}{[\mathbf{k}^{2}+m_{+}^{2}]}-\frac{b}{[\mathbf{k}^{2}+m_{-}^{2}]}\right ] \left ( \frac{\mathbf{k}^{2}}{m_{e}}-\frac{2i}{m_{e}}i\mathbf{k}\times\mathbf{p}\right ) \right\ } , \label{mcsp2}\ ] ] where : @xmath107 .\ ] ] carrying out the fourier transform of eq . ( [ mcsp2 ] ) , the following potential turns out : @xmath108k_{0}(m_{+}r)+[a_{-}+\frac{b}{m_{e}}m_{-}^{2}]k_{0}(m_{-}r)-\frac{2bl}{m_{e}r}[m_{+}k_{1}(m_{+}r)-m_{-}k_{1}(m_{-}r)]\biggr\},\ ] ] where @xmath109 is the angular momentum ( a scalar in a two - dimensional space ) . it is interesting to point out that @xmath110 is exactly the electron - electron mcs - proca potential , obtained firstly in ref . near the origin , one has : @xmath111 @xmath112 implying the following result:@xmath113\ln r. \label{mcsp_limit}\ ] ] once one works in the limit of small topological mass , @xmath114 this potential exhibits a repulsive behavior near the origin . the interaction potential associated with the amplitudes @xmath115may be obtained from the fourier transform of the scattering amplitude given in eq . ( [ m2 ] ) ; however , such amplitude must be previously factorized as @xmath116 \frac{1}{[\mathbf{k}^{2}+m_{a}^{2}]}-(cn_{+}-hn_{+})\frac{1}{[\mathbf{k}^{2}+m_{+}^{2}]}+(cn_{-}-hn_{-})\frac { 1}{[\mathbf{k}^{2}+m_{-}^{2}]}\biggr\},\ ] ] with the coefficients given as : @xmath117 the fourier transforms are then performed , leading to a combination of @xmath103 functions , namely : @xmath118 k_{0}(m_{a}r)-(cn_{+}-hn_{+})k_{0}(m_{+}r)+(cn_{-}-hn_{-})k_{0}(m_{-}r)\biggr\},\ ] ] near the origin , this potential vanishes identically , @xmath119 far from the origin , it decays exponentially . applying the same procedure to @xmath90 , after some algebra , it ends up in : @xmath120}+\frac{c}{[\mathbf{k}^{2}+m_{-}^{2}]}+\frac { h}{[\mathbf{k}^{2}+m_{+}^{2}]}-\frac{h}{[\mathbf{k}^{2}+m_{-}^{2}]}\right ] ( \mathbf{k}^{2}-2i\mathbf{k}\times\mathbf{p})\biggr\},\ ] ] so that the resulting potential is : @xmath121\nonumber\\ & -c[m_{+}^{2}k_{0}(m_{+}r)-m_{-}^{2}k_{0}(m_{-}r)]+h[m_{+}^{2}k_{0}(m_{+}r)-m_{-}^{2}k_{0}(m_{-}r)]\biggr\}.\end{aligned}\ ] ] this latter potential exhibits the same behavior of @xmath122 near and far from the origin , that is : @xmath123 the total gauge interaction potential , @xmath124 after some simplifications , assumes the explicit form : @xmath125k_{0}(m_{+}r)\nonumber\\ & + [ s^{2}(cn_{-}-hn_{-})+cm_{-}^{2}s / m_{e}]k_{0}(m_{-}r)+s^{2}\left [ -c(l_{+}-l_{-})+h(l_{+}-l_{-})\right ] k_{0}(m_{a}r)\nonumber\\ & + \frac{2l}{r}\frac{s}{m_{e}}c[m_{+}k_{1}(m_{+}r)+m_{-}k_{1}(m_{-}r)]\biggr\}. \label{vgauge1}\ ] ] this full expression corresponds to the mcs - proca potential ( @xmath126 corrected by the lorentz - violating terms arising from @xmath127 in the limit of a vanishing background ( v@xmath128 one has @xmath129 @xmath130 , remaining only the @xmath110 potential , which shows the consistency of the obtained results . obviously , this is an expected outcome , since both @xmath129 are potential contributions induced merely by the presence of the background . far from the origin , this potential vanishes exponentially ( according to the asymptotic behavior of the bessel functions ) , a consequence of the massive character of the physical mediators . in this point , this outcome differs from the asymptotic logarithmically divergent gauge potential attained in ref . @xcite ( which is written in footnote 1 ) . the graph of fig . [ [ gauge_gauge ] ] shows a simultaneous plot of the gauge potential of eq . ( [ vgauge1 ] ) ( continuos line ) and the one of ref . @xcite ( dotted line ) , which has a deeper minimum , compared with the former ( for each set of parameters ) . [ ptb ] gauge_gaugeplot1.eps near the origin , the lorentz - violating contributions of eq . ( [ vgauge1 ] ) tend to zero , so that in this limit the gauge potential is entirely ruled by the @xmath110 contribution , namely : @xmath131\ln r\biggr\}. \label{vgauge2}\ ] ] it is interesting to note that this is the same behavior of the gauge potentials achieved in refs . @xcite , @xcite . as already claimed , this potential will always exhibit a repulsive behavior near the origin . this general behavior is illustrated in fig.[[gauge1plot1 ] ] for four sets of parameters . [ ptb ] gauge1plot1.eps concerning such a picture , it presents a comparison of the electron - electron mcs - proca potential ( corresponding to the case for which v@xmath132 with the gauge one for three different values of v@xmath73 . it shows that the gauge potential appreciably deviates from the mcs - proca behavior as the larger is the magnitude of the background ( v@xmath133 , that is , the larger the background modulus the deeper the attractive region of the potential . the attractiveness of the gauge potential is ascribed to the presence of the well - shaped region ( constituted by a part of decreasing behavior followed by a part of increasing behavior ) , exhibited in the graphics of fig . 2 . in a dynamic perspective , such a well - shaped curve may be described in terms of a region in which the gradient potential is negative followed by a positive gradient region in the sequel . another kind of potential curve which implies an attractive behavior is one whose potential gradient is always negative ( but with a decreasing modulus with increasing distance , @xmath134 for @xmath135 this is the behavior exhibited by the scalar potential in fig.[[scalarplot1 ] ] . the discussion on the attractiveness of the gauge potential must be conducted with caution and can not be based only on the expression contained in eq . ( [ vgauge1 ] ) . it happens that , specifically in ( 1 + 2 ) dimensions , a tree - level result may be altered by the 1-loop contributions associated with 2-photon diagrams . this fact was put in evidence by the controversy involving the attractive / repulsive character of the mcs potential @xcite , which has shown that this potential turns out truly repulsive ( instead of attractive ) whenever the 2-photon diagrams that assure its gauge invariance are taken into account . in short , such a discussion has shown that the correct behavior of a ( 1 + 2)- potential can only be achieved if the 2-photons diagrams are considered . nevertheless , there is a way to circumvent the awkward calculation of such diagrams , which consists in requiring the gauge invariance of the pauli equation , @xmath136 \psi(r,\phi)=e\psi(r,\phi ) , \label{pauli}\ ] ] and keeping it in the non - relativistic limit ( governed by the schrdinger equation ) . the gauge invariance of pauli equation is assured by the presence of the @xmath10term , which obviously does not appearin the context of a nonperturbative low - energy evaluation ( once it is associated with 2-photon exchange processes ) , but may come to be as relevant as the tree - level ones ( see hagen and dobroliubov @xcite ) in ( 1 + 2 ) dimensions . therefore , both this term and the centrifugal barrier must be kept as correction terms of the effective potential for the schrdinger equation derived from eq . ( [ pauli ] ) . it is this effective potential that represents the true electron - electron interaction in the non - relativistivistic limit . in order to obtain such a potential explicitly , one writes the laplacian operator , @xmath137 , $ ] corresponding to the @xmath138term , whose action on the total wavefunction , @xmath139 generates the repulsive centrifugal barrier term , @xmath140 such a term is then added to @xmath141 , already presented in eq . ( [ pauli ] ) , thus yielding the effective potential : @xmath142 . the vector potential , @xmath143 , stemming from the planar model described by lagrangian ( [ lagrange3 ] ) , has been already evaluated in ref . @xcite for the timelike case:@xmath144 \overset{\wedge}{r^{\ast}},\ ] ] whence the effective potential takes the form:@xmath145^{2}. \label{veff}\ ] ] this is the gauge invariant effective potential that comprises the two - photon contribution ( @xmath146term ) and the centrifugal barrier term , leading to the correct low - energy electron - electron interaction . based upon such full expression , one proceeds to verify whether the electron - electron interaction may come to be attractive in some region by means of the graphical analysis of fig . [ veffplot1 ] . [ ptb ] veffplot1.eps the graph in fig.[[veffplot1 ] ] shows that the effective and the gauge potential differ from each other , for the adopted parameter values , by an absolutely negligible amount , because in this case , the graphics come out perfectly superimposed ( revealing that they correspond numerically to the same value ) . thus , the effective potential also exhibits an attractive behavior , showing that the gauge interaction is really endowed with attractiveness . in the framework of this work , the total electron - electron interaction encompasses the gauge and the scalar contributions : * * @xmath147 * * this total potential is attractive whenever * * it presents a well - shaped region . the shape of the total potential near the origin depends on the value of the constants @xmath148 if @xmath149 the total potential will present a similar behavior to that of the scalar potential of fig . [ 1 ] , while in the case @xmath150 it will be approximately as the gauge potential of figs.[2 , 3 ] . as both these forms of potential are endowed with well regions , we conclude that the total potential may always be attractive . this constitutes a relevant result provided it ensures the possibility of obtaining @xmath4 bound states in the framework of this particular model . in this work , it was considered the mller scattering in the context of the planar lorentz - violating maxwell - chern - simons - proca electrodynamics , obtained from the dimensional reduction of the abelian - higgs maxell - carroll - field - jackiw model @xcite . for the case of a purely timelike background , the interaction potential was calculated as the fourier transform of the mller amplitude ( born approximation ) , carried out in the non - relativistic regime . the attained total potential presents two distinct contributions : the attractive scalar potential ( stemming form the yukawa exchange ) and the gauge one ( mediated by the mcs - proca gauge field ) . the scalar potential , as expected , is always attractive no matter if it is near or far from the origin , and represents a totally shielded interaction . such an interaction may be identified with phonon exchange processes , which represent physical excitations in several systems of interest . as for the gauge interaction , it is composed of the repulsive mcs - proca potential @xmath151 corrected by background - depending contributions , which impose relevant physical modifications . indeed , for larger values of v@xmath152 the gauge potential exhibits a pronounced attractive region . both the scalar and gauge potentials are entirely screened interactions , a consequence of the massive mediators generated by the higgs mechanism . this feature , in principle , may turn these potentials suitable for describing real planar systems of condensed matter physics . this is the main difference between the interaction potentials of this work and the potentials derived in ref . @xcite , which present a logarithmic asymptotic behavior and are unsuitable for representing a physical interaction in a low - energy planar system . hence , in an attempt of accounting for a real interaction in a lorentz - violating planar system , one should adopt the potentials here derived instead of the ones of ref . @xcite . in this work , one has argued that the total interaction potential exhibits an attractive region , able to bring about the formation of @xmath153 pairs . as a forthcoming feasible application , one can explicitly evaluate the @xmath4 binding energies by means of the numerical solution of the schrdinger equation written for the potentials here derived . this may be done by ascribing reasonable values for the free parameters of this model , in a similar procedure to the one of refs . @xcite , @xcite . it is expected that a fine tune of the parameters would yield binding energies in the scale of @xmath154 , a typical energy for electron - electron pairing in planar systems . it is also important to emphasize the difference between the total potential obtained in this work and the ones of ref . @xcite , which consist of an always repulsive mcs - proca contribution added to an attractive scalar potential . in this latter case , the attractive scalar potential arises from the intermediation played by the higgs field , and the total potential exhibits attractiveness only if this scalar contribution overcomes the mcs - proca one . therefore , the possibility of attaining an attractive interaction depends entirely on the presence of the higgs mode . this is not the case of the present work , in which the gauge potential itself may be attractive ( even for small values of v@xmath73 compared to the electron rest mass ) - see figs . [ [ gauge_gauge ] ] , [ [ gauge1plot1 ] ] - an effect of the background on the system . furthermore , * * the scalar intermediation field is the one stemming from the dimensional reduction ( @xmath155 instead of the higgs field , which now accounts for the screened character of the interactions . * * these are particularities that distinguish the present model from the ones of refs . @xcite , @xcite,@xcite . the general connection between noncommutativity and lorentz violation turns a sensible matter the comparison of the ghosh potentials @xcite and the results of ref . @xcite , in which it was evaluated the lorentz - violating version of the mcs potential . yet , the expressions of these potentials result to be different . indeed , while the lorentz - violating potentials increase logarithmically with distance , the noncommutative potentials exhibit a @xmath156 asymptotic behavior . so , it is clear that these potentials differ from each other substantially , which justifies this investigation in the presence of both noncommutativity and lorentz violation . it is instructive to clarify the reason to have adopted a purely timelike background . this was done for a simplicity issue , since in this case the interaction potential may be exactly solved ( without approximations ) . the physical interpretation of this background , however , is not a straightforward matter , since v@xmath73 may not be easily associated with any parameter of the system . some felling can be got observing the effect of the background on the behavior of the system . as example , it was reported in ref . @xcite that a purely timelike background modifies drastically the asymptotic behavior of the electrical field of the mcs electrodynamics . in fact , while the pure mcs solution presents an exponentially decaying solution , the lorentz - violating mcs electric field exhibits an increasing logarithmic behavior . in this case , the background may be seen as a constant field that annihilates the screening of the electric sector of the theory , changing its asymptotic behavior . this property justifies the asymptotic logarithmic behavior of the potentials of refs . @xcite , @xcite . a possible continuation of this work consists in evaluating the electron - electron potential in the case of a purely spacelike background , @xmath157 standing for a privileged direction in space able to bring about anisotropy for the solutions . the presence of anisotropy in ( 1 + 2 ) dimensions is a factor that can be properly described in the framework of a lorentz - violating background . * * in general , this is a more complicated case in which the potentials may be only obtained within the approximation for * * v**@xmath158 as done in ref . @xcite . in principle , this is a case where the background may be more clearly interpreted as an active feature of an anisotropic condensed matter system , whose solutions exhibit an explicit dependence on the direction stated by * v*. h. belich , o. m. del cima , m.m . ferreira jr . and j. a. helayl - neto , int . a * 16 , * 4939 ( 2001 ) ; h. belich , o. m. del cima , m.m . ferreira jr . and j. a. helayl - neto , eur . j. b * 32 * , 145 ( 2003 ) . h. christiansen , o. m. del cima . ferreira jr . and j. a. helayl - neto , int . a * 18 * , 725 ( 2003 ) ; h. belich , o. m. del cima , m.m . ferreira jr . and j. a. helayl - neto , j. phys . g * 29 * , 1431 ( 2003 ) .
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a planar maxwell - chern - simons - proca model endowed with a lorentz - violating background is taken as framework to investigate the electron - electron interaction .
the dirac sector is introduced exhibiting a yukawa and a minimal coupling with the scalar and the gauge fields , respectively .
the the electron - electron interaction is then exactly evaluated as the fourier transform of the mller scattering amplitude ( carried out in the non - relativistic limit ) for the case of a purely time - like background .
the interaction potential exhibits a totally screened behavior far from the origin as consequence of massive character of the physical mediators . the total interaction ( scalar plus gauge potential )
can always be attractive , revealing that this model may lead to the formation of electron - electron bound states .
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over the last three decades the orbital physics of @xmath6 systems and its interplay with spin has focused primarily on transition metal oxides , especially 3@xmath7 transition metals ( tms ) . while the magnetic @xmath6 configuration has been studied mostly in @xmath8 systems , it can also occur in mid- to late-5@xmath7 tm ionic systems . for these heavy ions spin - orbit coupling ( soc ) becomes a competing factor , mixing the various spin , orbital , charge , and lattice degrees of freedom . the interplay of strong electron correlation and large soc is relatively less explored , and certainly not well understood at all , because the behavior involves so many comparable energy scales . this situation arises in a broad family of magnetic mott insulating systems in which three - fold degenerate @xmath9 orbitals are partially filled.@xcite in such systems orbital degeneracy is protected only by cubic lattice symmetry , and typically the crystal field splitting is large enough that @xmath10 orbitals are out of the picture . in @xmath11 @xmath9 subshells where soc remains unquenched ( cubic symmetry ) , the six one - electron levels split into an upper @xmath12=1/2 doublet and a lower @xmath12=3/2 quartet . in this category ir - based based magnets have been studied actively.@xcite a prominent class of such systems is the ordered double perovskites , with chemical formula a@xmath0bb@xmath13o@xmath1 . we are interested in the case where b is a closed shell cation and b@xmath13 is a magnetic ion ; in such cases unusually high formal valence states can arise . a few examples attracting recent interest are b@xmath13=ru@xmath14 and os@xmath14 in a@xmath0nab@xmath13o@xmath1 ( a = la and nd),@xcite mo@xmath14 in ba@xmath0ymoo@xmath1 , os@xmath15 in ba@xmath0caoso@xmath16@xcite and heptavalent os in ba@xmath0boso@xmath1 ( b = li , na ) . if we narrow our focus to @xmath6 b@xmath13 ions only , the possibilities are practically confined to mo@xmath14 , re@xmath15 and os@xmath17 ) . et al . _ have reported a theoretical study@xcite of koso@xmath18 with heptavalent os , where large soc , strong correlations , and structural symmetry breaking conspire to produce an unexpectedly large orbital moment in the @xmath19 shell that nominally supports no orbital moment . the recently studied compounds@xcite ba@xmath0boso@xmath1 ( b = li , na ) show many features to make them of current interest . besides the double - perovskite structure , and being a rare example of a heptavalent osmium compound , ba@xmath4naoso@xmath5 ( bnoo ) is exotic in being a _ ferromagnetic mott insulator_,@xcite with order appearing at t@xmath20 = 6.8k with curie - weiss temperature @xmath21 -10k . although its single @xmath9 electron orders magnetically , it shows no evidence of the anticipated orbital order that causes jahn - teller distortion and should destroys its cubic symmetry . the sister compound la@xmath0naoso@xmath1 , on the other hand , with high - spin @xmath22 os configuration and a nominally cubic symmetry , is observed to be highly distorted.@xcite this distortion is ascribed to geometrical misfit arising from incompatible ionic radii . there is a recent example of an os - based based @xmath23 perovskite compound baoso@xmath24 that remains cubic;@xcite on the other hand a related perovskite @xmath25 naoso@xmath24 does distort.@xcite the question of origin of the magnetic ordering in bnoo is surely a delicate one , since isostructural , isovalent , and also mott insulating ba@xmath0lioso@xmath1 ( bloo ) orders _ antiferromagnetically _ in spite of a very similar curie - weiss susceptibility@xcite and similar volume . lee and pickett demonstrated@xcite that , before considering magnetism and on - site interaction effects , soc splits the @xmath9 bands into a lower @xmath12=@xmath26 quartet and an upper @xmath12=@xmath27 doublet , as expected . since bnoo is observed to be insulating and effects of spin - orbit coupling drive the behavior , it provides the first `` j@xmath28=@xmath26 '' mott insulator at quarter - filling , analogous to the `` j@xmath28=@xmath27 '' mott insulators at half - filling that are being studied in @xmath25 systems.@xcite including spin polarization and on - site hubbard u repulsion beyond the semilocal density approximation ( dft+u+soc ) with both the wien2k and fplo codes gave essentially full spin polarization but was not able to open a gap@xcite with a reasonable value of u. the complication is that the occupied orbital is a oso@xmath5 cluster orbital with half of the charge on os and the other half spread over the neighboring o ions . u should be a value appropriate to this cluster orbital and should be applied to that orbital , however the codes applied u only to the os @xmath11 orbitals . xiang and whangbo@xcite neglected the hund s rule @xmath29 in the dft+u method , and did obtain a gap . however , neglecting @xmath29 omits both the hund s rule exchange energy and the anisotropy ( orbital dependence ) of the hubbard interaction @xmath30 , whereas one of our intentions is to include all orbital dependencies to understand the anisotropy on the os site . in this paper we first establish how to model this system faithfully including all anisotropy , then address the interplay of soc coupling with correlation effects and crystal field splitting . the density functional extension to include some fraction of hartree - fock exchange the hybrid functional open the gap , but only when soc is included . the inclusion of nonlocal exchange and soc provides an understanding of the mott insulating ground state and a [ 110 ] easy axis , both in agreement with experimental data.@xcite we conclude that bnoo provides an example of a @xmath31 , quarter - filled mott insulator . some comparison is made to isovalent bloo , which had 6% smaller volume and aligns antiferromagnetically rather than ferromagnetically . two density functional theory based studies , mentioned briefly above , have been reported for bnoo . one was performed within a fully anisotropic implementation of the dft+u method@xcite while the other was dft+u@xcite ( gga+u ) , but neglecting anisotropy of the interaction . an overriding feature of this system is a strong hybridization of os 5@xmath32 orbitals with o 2@xmath33 states , with the result that the `` @xmath9 '' bands have half of their density on the four neighboring oxygen ions in the plane of the orbital . in keeping with this strong hybridization , the `` o @xmath34 '' bands have considerable os @xmath11 charge , such that the @xmath11 occupation of the nominally @xmath2 ion is actually 4 - 5 electrons , still leaving a highly charged ion but less than half of the formal 7 + designation . lee and pickett reported@xcite that fully anisotropic dft+u could not reproduce a mott insulating state because u is applied on the os ion whereas half of the occupied local orbital ( cluster orbital ) density lies on neighboring oxygen ions . a model treatment in which u is applied to the cluster orbital did produce the mott insulating state.@xcite whangbo s results indicate that part of the complication in this system involves the anisotropy of the repulsion within the os ion . the study reported by whangbo _ et . al._@xcite addressed three spin directions ( [ 001 ] , [ 110 ] , and [ 111 ] ) . within their treatment , the [ 111 ] spin direction led to the minimum energy , indicated a calculated band gap of 0.3 ev for u=0.21 ry = 2.85 ev ( j@xmath35=0 ) . using dft+u+soc and the same code but including anisotropy of the interaction @xmath30 , we have not reproduced this gap , indicating their gap is due to the neglect of hund s rule coupling and anisotropy of the interaction . because of the need to include all interactions and all anisotropy , we have adopted a different approach based on the hybrid exchange - correlation functional , described in the next section . this approach seems to be more robust , allowing us to probe the interplay of soc and strong correlation of bnoo , and also obtain results of the effect of pressure on the ground state of bnoo . our first challenge was to obtain a mott insulating state in bnoo when all interactions ( correlation and soc ) are accounted for . with large soc the result depends on the assigned direction of the moment . the hybrid functional approach plus soc leads directly to a ferromagnetic ( fm ) mott insulating ground state , as observed.@xcite in our studies we observed a strong preference for fm alignment , versus the commonplace antiferromagnetic ( afm ) alignment that often arises on the simple cubic lattice of perovskite oxides . ba@xmath4lioso@xmath5 however orders antiferromagnetically , which for nearest neighbor antialignment exchange coupling leads to frustration of ordering . all calculations reported here for either compound are for fm orientation . we study specifically the effects of spin orientation on the electronic structure , and initiate a study of the pressure dependence of bnoo considering the zero pressure lattice constant and at 1% , 2% , and 5% reduced lattice constants . the present first - principles dft - based electronic structure calculations were performed using the full - potential augmented plane wave plus local orbital method as implemented in the wien2k code.@xcite the structural parameters of bnoo with full cubic symmetry of the double perovskite structure were taken from experimental x - ray crystallographic data:@xcite @xmath36=8.28 , @xmath37=0.2256 . non - overlapping atomic sphere radii of 2.50 , 2.00 , 1.80 , and 1.58 a.u . are used for the ba , na , os , and o atoms , respectively . the brillouin zone was sampled with a minimum of 400 k points during self - consistency , coarser meshes were sometimes found to be insufficient . for the exchange - correlation energy functional for treating strongly correlated insulators , a variety of approaches in addition to dft+u exist and have been tested and compared for a few selected systems.@xcite as mentioned above , for technical reasons the relevant orbital is an octahedron cluster orbital rather than the standard localized , atomic - like orbital encountered in @xmath8 oxides the lda+u method is problematic . we have chosen to apply the onsite exact exchange for correlated electrons ( eece ) functional introduced and evaluated by novak and collaborators.@xcite , implemented similarly to common use in hybrid ( mixture of hartree - fock and local density exchange ) . this oeehyb functional is an extension of the dft+u method to parameter - free form : exact exchange with full anisotropy is evaluated for correlated orbitals ( os @xmath11 orbitals here ) without explicit reference to any ( screened or unscreened ) hubbard u repulsion or hund s exchange interaction j@xmath35 . the double counting term is evaluated directly from the density of the occupied correlated orbitals , again without any input parameters . the exact exchange is calculated within the atomic sphere in atomic - like fashion ( hence `` onsite '' ) . the onsite exact exchange replaces 25% of the local density exchange . this functional is implemented in the wien2k code,@xcite and we refer to it here as oeehyb . for the semilocal exchange - correlation functional , the parametrization of perdew , burke , and ernzerhof@xcite ( generalized gradient approximation ) is used . soc was included fully in core states and for valence states was included in a second - variational method using scalar relativistic wave functions,@xcite a procedure that is non - perturbative and quite accurate for @xmath32 orbitals even with large soc . this oeehyb method has some kinship with hybrid exchange - correlation functionals ( see tran _ et al._@xcite for a comparison of several hybrids ) . hybrids replace some fraction @xmath38 , typically 25% , of local density exchange with hartree - fock exchange , which then is approximated in various ways to reduce the expense to a reasonable level . the oeehyb approach deals with exact exchange only for correlated orbitals , however , making it appropriate for correlated materials but it will not increase bandgaps of ionic or covalent semiconductors . we note that as an alternative to the commonly used dft+u approach , the more conventional hybrid exchange - correlation functional as implemented in the vasp code@xcite has been applied to the iridate na@xmath4iro@xmath39 by kim _ et al._@xcite to obtain the magnetic insulating ground state . projected densities of states ( pdoss ) are presented in fig . [ dos_fig1 ] for bnoo at the experimental volume using oeehyb , initially without soc and with full cubic symmetry . the fermi level ( taken as the zero of energy ) lies in a deep pseudogap , due to small band overlap . all @xmath9 orbitals participate equally at this level , evident from the observation that the pdos is distributed almost equally over the os @xmath9 orbitals and the @xmath34 orbitals of the six neighboring o ions , a very strong hybridization effect that has been emphasized before.@xcite the conventional picture of a mott insulating state in a @xmath40 shell is that a single orbital , say @xmath41 , is occupied , and the crystal symmetry is broken ( and must be broken in the calculation ) to sustain , indeed to allow , occupation of a single orbital . however , due to the large soc which is expected to produce a substantial orbital moment which requires occupation of a complex ( _ viz . _ @xmath42 ) orbital , we have foregone this intermediate step of obtaining orbital - ordering broken symmetry , which would typically be the final result for a @xmath8 ion with negligible soc . adding soc to oeehyb , the result mentioned above , which necessarily lowers the symmetry , leads to a soc - driven mott insulating state with a 0.2 - 0.3 ev gap , depending on the direction assumed for the magnetization . it is for this reason that we use the oeehyb functional to model bnoo , as we have previously reported that the lda+u approach was unable to open a gap.@xcite the gga+u method is more promising , but it requires an unphysically large value of @xmath43 = 6 ev to open a gap for [ 001 ] orientation . one can obtain a band gap along for [ 110 ] and [ 111 ] orientations in gga+u with u values of 3.2 ev and 2.0 ev , respectively.@xcite our calculated band gaps , provided in table [ table1 ] , depend on direction of the magnetization , which experimentally can be manipulated with an applied field . it should be kept in mind that symmetry is lowered by including soc , and the resulting symmetry depends on the direction of magnetization . for spin along [ 001 ] the band gap is lower by 70 mev than for [ 110 ] and [ 111 ] . .calculated spin , orbital , and total ( @xmath44=@xmath45+@xmath46 ) moments of os , and band gap of ba@xmath4naoso@xmath5 , for four values of lattice parameter and for the three high symmetry directions of the magnetization . note : the total moment per f.u . will include an @xmath470.5@xmath3 spin moment not included within the os sphere , thus primarily on the oxygen ions . [ cols="^,^,^,^,^,^,^,^,^",options="header " , ] the hybrid density functional has been used for the modeling of ba@xmath4naoso@xmath5 with considerable success . unlike our attempts with gga+u+soc , when soc is included , and only then , it provides a robust fm mott insulating ground state for the moment aligned along any of the three symmetry directions . thus bnoo ( and bloo ) is a @xmath48=@xmath26 mott insulator at quarter filling , driven by the combination of strong exchange - correlation and large soc . though the spin magnetization remains completely spin - up when soc is included , the large changes in the composition of the occupied orbital are driven by soc . on the scale of 0.1 - 0.2 ev which is important considering the small gap , the band structure is strongly dependent of the orientation of the magnetization . in addition , our approach predicts the [ 110 ] direction as the easy axis , as observed , though [ 111 ] is close in energy . the character of the net moment is unexpected . on the one hand , the spin and orbital moments on os are 0.5 and -0.4@xmath3 , respectively , leaving a net moment on os of 0.1@xmath3 , similar to earlier indications.@xcite however this value leaves most of the total moment of 0.6@xmath3 arising from the oxygen ions in the oso@xmath5 cluster . the observed ordered moment is @xmath470.2@xmath3 thus indicates somewhat larger compensation , or small spin contribution , than our results provide . another unusual feature found here is that for spin along [ 001 ] , there is a 0.1@xmath3 orbital moment on the apical oxygen ion . this surprisingly large value can not arise from the very small soc on oxygen ; rather it is the hybridization of the os @xmath49 combination with the @xmath50 orbital that transfers orbital angular momentum to o. this effect does not operate for other spin orientations . we acknowledge many useful conversations with k .- w . lee , discussions with i. r. fisher , r. t. scalettar , and n. j. curro , and comments on the manuscript from r. t. scalettar . our research used resources of the national energy research scientific computing center ( nersc ) , a doe office of science user facility supported by the office of science of the u.s . department of energy under contract no . de - ac02 - 05ch11231 . this research was supported by doe stockpile stewardship academic alliance program under grant de - fg03 - 03na00071 . 27 g. chen , r. pereira , and l. balents , phys . b. * 82 * , 174440 ( 2010 ) . g. jackeli and g. khaliullin , phys . lett . * 102 * , 017205 ( 2009 ) . b. j. kim , h. jin , s. j. moon , j .- y . park , c. s.leem , j. yu , t.w . noh , c. kim , s .- oh , j. h. park , v. durairaj , g. cao , and e. rotenberg , phys . * 101 * , 076402 ( 2008 ) . s. j. moon , h. jin , k.w . kim , w. s. choi , y. s. lee , j. yu , g. cao , a. sumi , h. funakubo , c. bernhard , and t.w . noh , phys . rev . lett . * 101 * , 226402 ( 2008 ) . b. j. kim , h. ohsumi , t. komesu , s. sakai , t. morita , h. takagi , and t. arima , science * 323 * , 1329 ( 2009 ) . g. cao , t. f. qi , l. li , j. terzic , s. j. yuan , l. e. delong , g. murthy , and r. k. kaul , phys . lett . * 112 * , 056402 ( 2014 ) . a. a. aczel , d. e. bugaris , j. yeon , c. de la cruz , h .- c . zur loye , and s. e. nagler , phys . b. * 88 * , 014413 ( 2013 ) . a. a. aczel , p.j.baker , d. e. bugaris , j. yeon , h .- c . zurloye , t. guidi , and d. t. adroja , phys . . lett . * 112 * , 117603 ( 2014 ) . a. a. aczel , d. e. bugaris , l. li , j .- q . yan , c. de la cruz , h .- c . zur loye , and s. e. nagler , phys . b. * 87 * , 014435 ( 2013 ) . k. yamamura , m. wakeshima , and y. j. hinatsu , solid state chem . * 179 * , 605 ( 2006 ) . song , k .- h . ahn , k .- w . lee , and w. e. pickett , arxiv:14087.4078 . a. j. steele , p. j. baker , t. lancaster , f. l. pratt , i. franke , s. ghannadzadeh , p. a. goddard , w. hayes , d. prabhakaran , and s. j. blundell , phys . b. * 84 * , 144416 ( 2011 ) . a. s. erickson , s. misra , g. j. miller , r. r. gupta , z. schlesinger , w. a. harrison , j. m. kim , and i. r. fisher , phys . . lett . * 99 * , 016404 ( 2007 ) . k. e. stitzer , m. d. smith , and h .- c . z. loye , solid state science * 4 * , 311 ( 2002 ) . w. r. gemmill , m. d. smith , r. prozorov , and h .- c . zur loye , inorg . chem . * 44 * , 2639 ( 2005 ) . y. shi , y. guo , y. shirako , w. yi , x. wang , a. a. belik , y. matsushita , h. l. feng , y. tsujimoto , m. arai , n.wang , m. akaogi , and k.yamaura , j. am . soc . * 135 * , 44 ( 2013 ) . jung , y - j . song , k - w . lee , and w. pickett , phys . b. * 87 * , 115119 ( 2013 ) . k. w. lee and w. e. pickett , epl * 80 * , 37008 ( 2007 ) . h. j. xiang and m .- h . whangbo , phys . b. * 75 * , 052407 ( 2007 ) . p. blaha , k. schwarz , g. madsen , d. kvasicka , and j. luitz , wien2k , an augmented plane wave + local orbitals program for calculating crystal properties , technical university of vienna , vienna , ( 2001 ) . we have used version 12 of wien2k , because we have found that soc with the hybrid functional does not always work correctly in version 13 . f. tran , p. blaha , k. schwarz , and p. novak , phys . b * 74 * , 155108 ( 2006 ) . p. novk , j. kune , l. chaput , and w. e. pickett , phys . b * 243 * , 563 ( 2006 ) . g. kresse and j. furthmller , phys . b. * 54 * , 11169 ( 1996 ) . kim , j .- h . lee , and j .- h . cho , sci . rep . * 4 * , 5253 ( 2014 ) doi:10.1038/srep05253 . a. h. macdonald , w. e. pickett , and d. d. koelling , j. phys . c * 13 * , 2675 ( 1980 ) .
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the double perovskite ba@xmath0naoso@xmath1 ( bnoo ) , an exotic example of a very high oxidation state ( heptavalent ) osmium @xmath2 compound and also uncommon by being a ferromagnetic mott insulator without jahn - teller ( jt ) distortion , is modeled using the density functional theory ( dft ) hybrid functional based exact exchange for correlated electrons ( oeehyb ) method and including spin - orbit coupling ( soc ) .
the experimentally observed narrow gap ferromagnetic insulating ground state is obtained , with easy axis along [ 110 ] in accord with experiment , providing support that this approach provides a realistic method for studying this system . the predicted spin density for [ 110 ] spin orientation is nearly cubic ( unlike for other directions ) , providing an explanation for the absence of jt distortion . an orbital moment of -0.4@xmath3 strongly compensates the + 0.5@xmath3 spin moment on os , leaving a strongly compensated moment more in line with experiment .
remarkably , the net moment lies primarily on the oxygen ions .
an insulator - metal transition by rotating the magnetization direction with an external field under moderate pressure is predicted as one consequence of strong soc , and metallization under moderate pressure is predicted .
comparison is made with the isostructural , isovalent insulator ba@xmath4lioso@xmath5 which however orders antiferromagnetically .
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You are an expert at summarizing long articles. Proceed to summarize the following text:
in this paper we are concerned with the problem of boundary stabilization for a @xmath0 system of first - order hyperbolic _ quasilinear _ pdes , with actuation at only one of the boundaries . the quasilinear case is of interest since many relevant physical systems are described by @xmath0 systems of first - order hyperbolic quasilinear pdes , such as open channels @xcite , transmission lines @xcite , gas flow pipelines @xcite or road traffic models @xcite . this problem has been considered in the past for @xmath0 systems @xcite and even @xmath3 systems @xcite , using the explicit evolution of the riemann invariants along the characteristics . more recently , an approach using control lyapunov functions has been developed , for @xmath0 systems @xcite and @xmath3 systems @xcite . these results use only static output feedback ( the output being the value of the state on the boundaries ) . however , they do not deal with the same class of systems considered in this work ( which includes an extra term in the equations ) ; with this term , it has been shown in @xcite that there are examples ( even for linear @xmath0 system ) for which there are no control lyapunov functions of the `` diagonal '' form @xmath4 ( see next section for notation ) which would allow the computation of a static output feedback law to stabilize the system , even if feedback is allowed on both sides of the boundary . several other authors have also studied this problem . for instance , the linear case has been analyzed in @xcite ( using a lyapunov approach ) and in @xcite ( using a spectral approach ) . the nonlinear case has been considered by @xcite and @xcite using a lyapunov approach , and in @xcite , @xcite , and @xcite using a riemann invariants approach . the basis of our design is the backstepping method @xcite ; initially developed for parabolic equations , it has been used for first - order hyperbolic equations @xcite , delay systems @xcite , second - order hyperbolic equations @xcite , fluid flows @xcite , nonlinear pdes @xcite and even pde adaptive designs @xcite . the method allows us to design a full - state feedback law ( with actuation on only one end of the domain ) making the closed - loop system locally exponentially stable in the @xmath1 sense . the gains of the feedback law are the solution of a 4 x 4 system of first - order hyperbolic linear pdes , whose well - posedness is shown . the proof of stability is based on @xcite ; we construct a strict lyapunov function , locally equivalent to the @xmath1 norm , and written in coordinates defined by the ( invertible ) backstepping transformation . the paper is organized as follows . in section [ sect - prob ] we formulate the problem . in section [ sect - linear ] we consider the linear case and formulate a backstepping design that globally stabilizes the system in the @xmath5 sense . in section [ sect - main ] we present our main result , which shows that the linear design locally stabilizes the nonlinear system in the @xmath1 sense . the proof of this result is given in section [ sect - proof ] . we finish in section [ sect : conclusions ] with some concluding remarks . we also include an appendix with the proof of well - posedness of the kernel equations , and some technical lemmas . consider the system @xmath6,\ , t\in[0,+\infty),\label{eqn - z}\ ] ] where @xmath7\times[0,\infty)\rightarrow \mathbb{r}^2 $ ] , @xmath8 \rightarrow \mathcal{m}_{2,2}(\mathbb{r})$ ] , @xmath9 \rightarrow \mathbb{r}^2 $ ] , with @xmath10 denoting the set of @xmath0 real matrices . we assume that @xmath11 is twice continuously differentiable with respect to @xmath12 and @xmath13 , and we assume that ( possibly after an appropiate state transformation ) @xmath14 is a diagonal matrix with nonzero eigenvalues @xmath15 and @xmath16 which are , respectively , positive and negative , i.e. , @xmath17 where @xmath18 denotes the diagonal matrix with @xmath19 in the first position of the diagonal and @xmath20 in the second . we also assume that @xmath21 , implying that there is an equilibrium at the origin , and that @xmath22 is twice continuously differentiable with respect to @xmath12 . denote @xmath23,\ ] ] and assume that @xmath24\right)$ ] . denoting @xmath25^t$ ] , we study classical solutions of the system under the following boundary conditions @xmath26 which are consistent ( see @xcite ) with the signs of ( [ eqn - speeds ] ) , at least for small values of @xmath12 . we assume that @xmath27 is twice differentiable and vanishes at the origin . in ( [ eqn - bcz12 ] ) , @xmath28 is the actuation variable , and our task is to find a feedback law for @xmath28 to make the origin of system ( [ eqn - z]),([eqn - bcz12 ] ) locally exponentially stable . _ the case with @xmath29 in ( [ eqn - z ] ) was addressed in @xcite and @xcite by using control lyapunov functions to design a static output feedback law ; this approach has been shown to fail in @xcite for some cases with @xmath30 , at least for a `` diagonal '' lyapunov function of the form @xmath31 . _ next , we present a new design , based on the backstepping method , to stabilize a @xmath0 hyperbolic linear system ; this procedure will be used later to _ locally _ stabilize system ( [ eqn - z ] ) , ( [ eqn - bcz12 ] ) . consider the system @xmath32,\ , t\in[0,+\infty),\ ] ] where @xmath33\times[0,\infty)\rightarrow \mathbb{r}^2 $ ] , @xmath34 \rightarrow \mathcal{m}_{2,2}(\mathbb{r})$ ] , where the matrices @xmath35 and @xmath36 are respectively diagonal and antidiagonal , as follows : @xmath37 where @xmath38 are @xmath39)$ ] and @xmath40 are @xmath41)$ ] functions , verifying that @xmath42 , and with boundary conditions @xmath43 where @xmath44 and the components of @xmath45 are @xmath46 ^t$ ] . our objective is to design a full - state feedback control law for @xmath28 to ensure that the closed - loop system is globally asymptotically stable in the @xmath5 norm , which is defined as @xmath47 . there are two cases , depending on whether @xmath48 in ( [ eqn - bculinear ] ) is nonzero or @xmath49 . we first analyze the first case , thus assuming @xmath50 . our approach to designing @xmath28 , following the backstepping method , is to seek a mapping that transforms @xmath45 into a _ target _ variable @xmath51 with asymptotically stable dynamics as follows : @xmath52 with boundary conditions @xmath53 where the components of @xmath51 are denoted as @xmath54^t.\end{aligned}\ ] ] system ( [ eqn - gammalinear ] ) , ( [ eqn - bc1beta ] ) verifies the properties expressed in the following proposition . [ pr - target ] consider system ( [ eqn - gammalinear ] ) , ( [ eqn - bc1beta ] ) with initial condition @xmath55)$ ] . then , for every @xmath56 , there exists @xmath57 such that @xmath58 in fact , the equilibrium @xmath59 is reached in finite time @xmath60 , where @xmath61 is given by @xmath62 define @xmath63,\ ] ] where @xmath64 will be computed later . select : @xmath65 notice that @xmath66 defines a norm equivalent to @xmath67 . computing the derivative of @xmath68 and integrating by parts , we obtain @xmath69_0 ^ 1,\label{eqn - udotlinear1}\end{aligned}\ ] ] where we have used that @xmath70 and @xmath71 commute . since @xmath72>0,\ ] ] and , on the other hand , @xmath73_0 ^ 1 = -a \alpha^2(1,t)\mathrm{e}^{-\mu } -(b - q^2a)\beta^2(0,t),\end{aligned}\ ] ] choosing @xmath74 , @xmath75 , and @xmath76 , where @xmath77}\left\{\frac{1}{\epsilon_1(x ) } , \frac{1}{\epsilon_2(x ) } \right\}$ ] , we get that @xmath78 , therefore : @xmath79 where @xmath80 can be chosen as large as desired . this shows exponential stability of the origin for the @xmath51 system . to show finite - time convergence to the origin , one can find the explicit solution of ( [ eqn - gammalinear ] ) as follows . define first @xmath81 noting that they are monotonically increasing functions of @xmath13 , and thus invertible . note that the components of @xmath51 verify the differential equations @xmath82 which can be rewritten as follows @xmath83 the solution of these equations is @xmath84 and @xmath85 , where @xmath86 and @xmath87 are arbitrary functions . now , if @xmath88 , @xmath89 are the initial condition for the states , one obtains @xmath90 ( valid for @xmath91 ) and @xmath92 ( valid for @xmath93 ) . using the boundary conditions ( [ eqn - bc1beta ] ) one finds the remaining values of @xmath86 and @xmath87 , and thus the solution of the system , as follows : @xmath94 thus , after @xmath60 , where @xmath95 one has that @xmath96 . to map the original system ( [ eqn - wlinear ] ) into the target system ( [ eqn - gammalinear ] ) , we use the following transformation : @xmath97 where @xmath98 is a matrix of kernels . defining @xmath99 the original and target boundary conditions ( respectively ( [ eqn - bculinear ] ) and ( [ eqn - bc1beta ] ) ) can be written compactly ( omitting dependences in @xmath13 and @xmath100 ) as @xmath101 introducing ( [ eqn - tran ] ) into ( [ eqn - gammalinear ] ) , applying ( [ eqn - wlinear ] ) , integrating by parts and using the boundary conditions , we obtain that the original system ( [ eqn - wlinear ] ) is mapped into the target system ( [ eqn - gammalinear ] ) if and only if one has the following three matrix equations : @xmath102 expanding ( [ eqn - k1 ] ) , we get the following kernel equations : @xmath103 with boundary conditions obtained from ( [ eqn - k2])([eqn - k3 ] ) @xmath104 the equations evolve in the triangular domain @xmath105 . notice that they can be written as two separate @xmath0 hyperbolic systems , one for @xmath106 and @xmath107 and another for @xmath108 and @xmath109 . by theorem [ th - wp ] ( see the appendix ) , one finds that , for @xmath50 , under the assumption that @xmath38 are @xmath39)$ ] , @xmath40 are @xmath41)$ ] and that @xmath42 , there is a unique solution to ( [ eqn - kuu])([eqn - bc4 ] ) , which is in @xmath110 . to study the invertibility of transformation ( [ eqn - tran ] ) , we look for a transformation of the the target system ( [ eqn - gammalinear ] ) into the original system ( [ eqn - wlinear ] ) as follows : @xmath111 where @xmath112 introducing ( [ eqn - traninv ] ) into ( [ eqn - wlinear ] ) , applying ( [ eqn - gammalinear ] ) , integrating by parts and using the boundary conditions , we obtain as before a set of kernel equations : @xmath113 with boundary conditions @xmath114 again by theorem [ th - wp ] ( see the appendix ) , one finds that there is a unique solution to these equations , which is @xmath110 . from the transformation ( [ eqn - tran ] ) evaluated at @xmath115 , one gets @xmath116 with control law ( [ eqn - linearcontrol ] ) the following result holds . [ th - control ] consider system ( [ eqn - wlinear ] ) with boundary conditions ( [ eqn - bculinear ] ) , control law ( [ eqn - linearcontrol ] ) , and initial condition @xmath117)$ ] . then , for every @xmath56 , there exists @xmath57 such that @xmath118 in fact , the equilibrium @xmath119 is reached in finite time @xmath60 , where @xmath61 is given by ( [ eqn - tf ] ) . since the transformation ( [ eqn - tran ] ) is invertible , when applying control law ( [ eqn - linearcontrol ] ) the dynamical behavior of ( [ eqn - wlinear ] ) is the same as the behavior of ( [ eqn - gammalinear ] ) , which is well - posed from standard results and whose explicit solution and stability properties we know from proposition [ pr - target ] . thus , we obtain the explicit solutions of @xmath45 from the direct and inverse transformation , as follows : @xmath120 where @xmath121 is the explicit solution of the @xmath122 , @xmath123 system , given by ( [ eqn - expalpha])([eqn - expbeta ] ) , with initial conditions : @xmath124 in particular , we know that @xmath51 goes to zero in finite time @xmath60 , therefore @xmath45 also shares that property . finally , since the origin of the @xmath51 system is @xmath5 exponentially stable with an arbitrary large exponential decay rate , we conclude , using the inverse transformation , that the origin of the @xmath45 system is also @xmath5 exponentially stable with an arbitrary large exponential decay rate . equation ( [ eqn - l2normth ] ) follows by using the inverse and direct transformations to relate the @xmath5 norms of @xmath45 and @xmath51 ( using the fact that the kernels of the transformations are continuous , and thus bounded , functions ) . if the coefficient @xmath48 is zero in ( [ eqn - bculinear ] ) , the method presented in the paper is not valid since ( [ eqn - bc1 ] ) would require the value of one of the control kernels to be infinity at the boundary of the domain @xmath125 . similarly , if the coefficient is close to zero one still gets very large values for the kernels close to the boundary , resulting in potentially large control laws . the method can be modified to accommodate zero or small values of @xmath48 by setting a slightly different target system ( [ eqn - alpha])([eqn - beta ] ) , as follows : @xmath126 where @xmath127 is to be obtained from the method ; regardless of the value of @xmath127 , this is a cascade system which is still @xmath5 exponentially stable and converges in finite time by the same arguments of proposition [ pr - target ] , since now , using the same lyapunov function @xmath68 defined in ( [ eqn - lyaplinear ] ) , we obtain @xmath69_0 ^ 1 \nonumber \\ & & + 2 \beta(0,t ) \int_0 ^ 1\alpha(x , t ) a \frac{\mathrm{e}^{-\mu x}}{\epsilon_1(x ) } g(x ) dx , \label{eqn - udotlinearq0}\end{aligned}\ ] ] the new term ( which is the last one ) can be controlled by slightly modifying the coefficients of @xmath71 in the proof of proposition [ pr - target ] , obtaining the same result as before . the kernel equations resulting from the transformation are still the same ( [ eqn - kuu])([eqn - kvv ] ) , with the same boundary conditions ( [ eqn - bc2])([eqn - bc4 ] ) for @xmath107 , @xmath108 , and @xmath109 ( which reduces to @xmath128 when @xmath49 ) , but one obtains an _ undetermined _ boundary conditions for @xmath106 : @xmath129 where @xmath130 can be chosen as desired ; by choosing at least a continuous function , one can apply theorem [ th - wp ] and thus the kernel equations are well - posed . after @xmath130 has been chosen and the kernels have been computed , one obtains the value of @xmath127 as @xmath131 invertibility of the transformation follows as before , thus one obtains the same result of theorem [ th - control ] . the non - uniqueness in ( [ eqn - bc1q0 ] ) gives the designer some freedom in shaping the input function @xmath127 from @xmath123 to @xmath122 . also note that this has no impact in the feedback law as the kernels @xmath108 and @xmath109 ( which are the ones appearing in ( [ eqn - linearcontrol ] ) ) are uniquely defined and independent of the non - unique @xmath106 and @xmath107 . we wish to show that the linear controller ( [ eqn - linearcontrol ] ) designed using backstepping works _ locally _ for the nonlinear system , in terms that will be made precise . for that , we write our quasilinear system ( [ eqn - z ] ) in a form equivalent ( up to linear terms ) to ( [ eqn - wlinear ] ) . define @xmath132 we obtain a new state variable @xmath45 from @xmath12 using the following transformation : @xmath133=\left [ \begin{array}{cc } \varphi_1(x ) & 0\\ 0 & \varphi_2(x ) \end{array } \right ] \left [ \begin{array}{c } z_1(x , t)\\ z_2(x , t ) \end{array } \right]=\phi(x ) z(x , t),\label{eqn - zw}\ ] ] so that @xmath134 w(x , t)=\phi^{-1}(x ) w(x , t).\ ] ] it follows that @xmath45 verifies the following equation : @xmath135 where @xmath136 w.\end{aligned}\ ] ] it is evident that @xmath137 and that @xmath138 . also , @xmath139.\label{eqn - nonlinearc}\ ] ] thus , it is possible to write ( [ eqn - w ] ) as a linear system with the same structure as ( [ eqn - wlinear ] ) plus nonlinear terms : @xmath140 where @xmath141 and @xmath142 computing the boundary conditions of ( [ eqn - w2 ] ) by combining ( [ eqn - bcz12 ] ) with the transformation ( [ eqn - zw ] ) , and defining @xmath143 and @xmath144 , one obtains @xmath145 where @xmath146 . in what follows we will consider the case @xmath147 ; the case @xmath49 is analogous ( see remark [ rem - qzero ] ) . notice that the linear parts of ( [ eqn - w2 ] ) and ( [ eqn - bcu2 ] ) are identical to ( [ eqn - wlinear ] ) and ( [ eqn - bculinear ] ) , and that the coefficients @xmath148 and @xmath70 verify the assumptions of section [ sect - linear ] . also , it is clear that the nonlinear terms verify @xmath149 , @xmath150 , and @xmath151 therefore , we consider using the feedback law : @xmath152 which implies , in terms of the original @xmath12 variable : @xmath153 where the kernels are computed from ( [ eqn - kuu])([eqn - bc4 ] ) using the coefficients @xmath148 and @xmath70 from ( [ eqn - nonlinearc ] ) and ( [ eqn - nonlinearsigma ] ) . next , we show that the control law ( [ eqn - nonlinearcontrol2 ] ) , which is computed for the linear part of the system , asymptotically stabilizes the nonlinear system , although locally . however , the right space to prove stability of the closed - loop system is @xmath1 , instead of the space @xmath5 that was used in section [ sect - linear ] for the linear system . denoting : @xmath154,g(z)=g_0(z_2),q_1=\left[\begin{array}{c } 0 \\ 1 \end{array } \right],k(x)= \left[\begin{array}{c } \frac{\varphi_1(x)k^{vu}(1,x)}{\varphi_2(1 ) } \\ \frac{\varphi_2(x)k^{vv}(1,x)}{\varphi_2(1 ) } \end{array } \right],\end{aligned}\ ] ] the boundary conditions of the closed loop system would be written as : @xmath155 a necessary condition for system ( [ eqn - z ] ) with boundary conditions ( [ eqn - bcz ] ) to be well - posed in the space @xmath1 is that the initial conditions verify the corresponding second - order compatibility condition . these are @xmath156 while ( [ eqn - cc1 ] ) and ( [ eqn - cc3 ] ) are natural compatibility conditions , the conditions ( [ eqn - cc2 ] ) and ( [ eqn - cc4 ] ) are artificial ( since they show up due to the feedback law that has been designed ) and rather stringent , as they require very specific values of the initial conditions . thus , we modify our control law in a way that , without losing its stabilizing character , does not require any specific values in the initial values beyond the natural conditions ( [ eqn - cc1 ] ) and ( [ eqn - cc3 ] ) . the modification in the boundary conditions consists in adding a dynamic extension to the controller as follows : @xmath157 where @xmath158 is one of the states of the following system : @xmath159 where the constants @xmath160 and @xmath161 can be chosen as desired with the only conditions that @xmath162 and @xmath163 . it is evident that with positive values of these constants , ( [ syst - a - b ] ) is always stable . the initial conditions of @xmath158 and @xmath164 are an additional degree of freedom that can be used to eliminate the compatibility conditions ( [ eqn - cc2 ] ) and ( [ eqn - cc4 ] ) . with the modification of the control law , these compatibility conditions are now @xmath165 call @xmath166 selecting @xmath167 the compatibility conditions are automatically verified . we are now ready to state our main result . define the norms @xmath168 and @xmath169 . [ thm - main ] consider system ( [ eqn - z ] ) and ( [ syst - a - b ] ) with boundary conditions ( [ eqn - bcz2 ] ) and initial conditions @xmath170^t \in h^2([0,1])$ ] , and @xmath171 and @xmath172 verifying ( [ eqn - b0 ] ) , with the kernels @xmath108 and @xmath109 obtained from ( [ eqn - kuu])([eqn - bc4 ] ) where the coefficients @xmath148 and @xmath70 are computed from ( [ eqn - nonlinearc ] ) and ( [ eqn - nonlinearsigma ] ) . then , under the assumptions of smoothness for the coefficients stated in section [ sect - prob ] , for every @xmath56 , there exist @xmath173 and @xmath57 such that such that , if @xmath174 and if the compatibility conditions ( [ eqn - cc1 ] ) and ( [ eqn - cc3 ] ) are verified , then : @xmath175 we first establish some definitions and notation . for @xmath176 with components @xmath177 and @xmath178 denote @xmath179 , and @xmath180 } \vert \gamma(x ) \vert,\quad \vert \gamma \vert_{l^1}= \int_0 ^ 1 \vert \gamma(\xi ) \vert d\xi.\ ] ] in what follows , for a time - varying vector @xmath181 , we denote @xmath182 and @xmath183 to simplify our notation . for a @xmath184 matrix @xmath185 , denote : @xmath186 for the kernel matrices @xmath187 and @xmath188 denote @xmath189 for @xmath190)$ ] , recall the following well - known inequalities , that will be used later : @xmath191\leq c_4 \vert \gamma \vert_{h^1},\label{eqn - sobolev1}\\ \vert \gamma_x \vert_{\infty } & \leq & c_5\left [ \vert \gamma_x \vert_{l^2}+\vert \gamma_{xx } \vert_{l^2}\right]\leq c_6 \vert \gamma \vert_{h^2}.\quad\,\,\label{eqn - sobolev2}\end{aligned}\ ] ] define the following linear functionals , the first two of which are , respectively , the inverse and direct transformations ( [ eqn - tran ] ) and ( [ eqn - traninv ] ) : @xmath192(x ) & = & \gamma(x , t)-\int_0^x k(x,\xi ) \gamma(\xi , t ) d\xi,\\ \mathcal l[\gamma](x ) \label{eqn - call } & = & \gamma(x , t)+\int_0^x l(x,\xi ) \gamma(\xi , t ) d\xi,\\ \mathcal k_{1}[\gamma](x ) & = & -k(x , x)\gamma(x , t)+\int_0^x k_{\xi}(x,\xi ) \gamma(\xi , t ) d\xi,\qquad\\ \mathcal k_{2}[\gamma](x ) & = & -k(x , x)\gamma(x , t)-\int_0^x k_{x}(x,\xi ) \gamma(\xi , t ) d\xi,\qquad\\ \mathcal l_{1}[\gamma](x ) & = & l(x , x)\gamma(x , t)+\int_0^x l_x(x,\xi ) \gamma(\xi , t ) d\xi,\qquad\\ \mathcal l_{11}[\gamma](x ) & = & ( l_x(x , x)+l_\xi(x , x))\gamma(x , t)+l_x(x , x)\gamma(x , t ) \nonumber \\ & & + \int_0^x l_{xx}(x,\xi ) \gamma(\xi , t ) d\xi.\label{eqn - callx}\end{aligned}\ ] ] for simplicity , in what follows we drop writing the @xmath13 dependence in functionals and the @xmath100 dependence in the variables . using ( [ eqn - calk ] ) and ( [ eqn - call ] ) , we define @xmath193 $ ] and @xmath194 $ ] as : @xmath195,x\right),\quad f_2=f_{nl}\left(\mathcal l[\gamma],x\right).\end{aligned}\ ] ] to prove theorem [ thm - main ] , we notice that if we apply the ( invertible ) backstepping transformation ( [ eqn - tran ] ) to the nonlinear system ( [ eqn - w2 ] ) we obtain the following transformed system : @xmath196 and using the inverse transformation ( [ eqn - traninv ] ) the equation can be expressed fully in terms of @xmath51 as : @xmath197+f_4[\gamma]=0,\end{aligned}\ ] ] where the functionals @xmath198 and @xmath199 are defined as @xmath200\gamma_x\right ] , \label{eqn - f3 } \\ f_4&=&\mathcal k\left [ f_1[\gamma ] \mathcal l_{1 } \left[\gamma \right ] + f_2[\gamma]\right].\label{eqn - f4}\qquad\,\end{aligned}\ ] ] the boundary conditions are @xmath201 by the assumptions on the coefficients and applying theorem [ th - sm ] , the direct and inverse transformations ( [ eqn - tran ] ) and ( [ eqn - traninv ] ) have kernels that are @xmath202 functions . differentiating twice with respect to @xmath13 in these transformations , it can be shown that the @xmath1 norm of @xmath51 is equivalent to the @xmath1 norm of @xmath12 ( see for instance @xcite ) . thus , if we show @xmath1 local stability of the origin for ( [ eqn - gammanl2])([eqn - gammabc ] ) , the same holds for @xmath12 . we proceed by analyzing ( using a lyapunov function ) the growth of @xmath203 , @xmath204 and @xmath205 . relating these norms with @xmath206 , we then prove @xmath1 local stability for @xmath51 . define @xmath207 for @xmath71 as in ( [ eqn - ddef ] ) . proceeding analogously to ( [ eqn - udotlinear1])([eqn - udotlinear2 ] ) , we get some extra nonlinear terms : @xmath208_0 ^ 1 \nonumber \\ & & - 2 \int_0 ^ 1\gamma^t(x , t ) d(x)\left(f_3[\gamma,\gamma_x]+ f_4[\gamma ] \right ) dx .\end{aligned}\ ] ] let us analyze first the last term : @xmath209 + f_4[\gamma ] \right ) dx \right| \leq k_1 \int_0 ^ 1 \vert \gamma \vert \left ( \vert f_3[\gamma,\gamma_x]\vert + \vert f_4[\gamma ] \vert\right ) dx.\ ] ] applying lemma [ lem - f3f4bound ] ( see the appendix ) , we obtain that there exists a @xmath210 , such that for @xmath211 , @xmath212 \vert dx & \leq & k_2 \vert \gamma_x \vert_{\infty}\vert \gamma \vert^2_{l^2},\quad\\ \int_0 ^ 1 \vert \gamma \vert \vert f_4[\gamma ] \vert dx & \leq & k_3 \vert \gamma \vert_{\infty } \vert \gamma \vert_{l^2}^2,\end{aligned}\ ] ] and using inequality ( [ eqn - sobolev1 ] ) and noting that @xmath213 , we obtain @xmath212 \vert dx & \leq & k_5 \vert \gamma_x \vert_{\infty}v_1,\quad\\ \int_0 ^ 1 \vert \gamma \vert \vert f_4[\gamma ] \vert dx & \leq & k_6 \vert \gamma_x \vert_{\infty}v_1+k_7 v_1^{3/2}.\end{aligned}\ ] ] now , @xmath73_0 ^ 1 & = & b a^2(t)\mathrm{e}^{\mu } -a \alpha^2(1,t)\mathrm{e}^{-\mu } -b \beta^2(0,t ) \nonumber \\ & & + a(q\beta(0,t)+g_{nl}(\beta(0,t)))^2,\end{aligned}\ ] ] and for @xmath211 , @xmath214 , and @xmath215 , we obtain @xmath73_0 ^ 1 & \leq & -a \alpha^2(1,t)\mathrm{e}^{-\mu } + ( a(\vert q\vert + k_8)^2-b ) \beta^2(0,t ) \nonumber \\ & & + b a^2(t)\mathrm{e}^{\mu}\end{aligned}\ ] ] thus , choosing @xmath216 and @xmath217 and @xmath218 as in the proof of proposition [ pr - target ] , we obtain the following proposition : [ pr - gamma ] there exists @xmath210 such that if @xmath211 then @xmath219 where @xmath220 , @xmath221 , @xmath222 , @xmath223 and @xmath224 are positive constants . define @xmath225 . notice that the norms of @xmath226 and @xmath227 are related ( see lemma [ lem - gammaetaequiv ] in the appendix ) . taking a partial derivative in @xmath100 in ( [ eqn - gammanl ] ) we obtain an equation for @xmath226 as follows : @xmath228-\sigma(x)\right ) \eta_x + f_5[\gamma,\gamma_x,\eta]+f_6[\gamma,\eta]=0,\end{aligned}\ ] ] where @xmath229 and @xmath230 are defined as @xmath231\eta \right ] + \int_0^x k(x,\xi ) f_{12}[\gamma,\gamma_x]\eta(\xi ) d\xi + k(x,0 ) \lambda_{nl}\left(\gamma(0),0\right ) \eta(0 ) \nonumber \\ & & + \mathcal k \left [ f_{11}[\gamma,\eta]\gamma_x\right ] , \label{eqn - f5 } \\ f_6&=&\mathcal k\left[f_{11}[\gamma,\eta ] \mathcal l_x \left[\gamma\right ] \right ] + \mathcal k\left[f_{1}[\gamma]\mathcal l_x \left[\eta\right]\right ] + \mathcal k[f_{21 } [ \gamma,\eta]],\label{eqn - f6}\end{aligned}\ ] ] where @xmath232,x\right)\mathcal l[\eta],\quad\,\,\ , \\ f_{12}&= & \frac{\partial \lambda_{nl}}{\partial \gamma } \left(\mathcal l[\gamma],x\right ) \left(\gamma_x+ \mathcal l_x [ \gamma ] \right ) + \frac{\partial \lambda_{nl}}{\partial x } \left(\mathcal l[\gamma],x\right ) , \quad\,\,\ , \\ f_{21}&=&\frac{\partial f_{nl}}{\partial \gamma } \left(\mathcal l[\gamma],x\right)l[\eta].\quad\end{aligned}\ ] ] the boundary conditions for @xmath233^t$ ] are @xmath234 to find a lyapunov function for @xmath226 , we use the next lemma : [ lem - lyapeta ] there exists @xmath173 such that , for @xmath235 , there exists a symmetric matrix @xmath236>0 $ ] verifying the identity : @xmath237 \left(\sigma(x)-f_1[\gamma]\right)- \left(\sigma(x)-f_1[\gamma]\right)^t r[\gamma]=0,\label{eqn - symm}\ ] ] and the following bounds : @xmath238(x)&\leq&c_1+c_2 \vert \gamma \vert_{\infty},\label{eqn - rpos2}\\ \vert \left ( \left(r[\gamma]-d(x)\right ) \sigma(x)\right)_x \vert&\leq & c_2 \vert \gamma \vert_{\infty } \left(1 + \vert \gamma _ x \vert_\infty \right),\quad\label{eqn - thetax}\\ \vert \left ( r[\gamma ] \right)_t \vert & \leq & c_3 \left(\vert \eta\vert+ \vert \eta \vert_{l^1 } \right),\label{eqn - thetat}\end{aligned}\ ] ] where @xmath239 are positive constants . we explicitly construct @xmath236 $ ] as @xmath240=d(x)+\theta[\gamma],\ ] ] with @xmath241=\left [ \begin{array}{cc } 0 & \psi[\gamma ] \\ \psi[\gamma ] & 0 \end{array } \right],\label{eqn - thetadef}\ ] ] where @xmath242 $ ] is defined as : @xmath243=\frac{d_{11}(x)\left(f_1[\gamma]\right)_{12}-d_{22}(x)\left(f_1[\gamma ] \right)_{21}}{\epsilon_2(x)+\epsilon_1(x)+\left(f_1[\gamma]\right)_{11}-\left(f_1[\gamma]\right)_{22}},\label{eqn - psidef}\ ] ] where @xmath244\right)_{ij}$ ] denotes the coefficient in row @xmath245 and column @xmath246 in the matrix @xmath193 $ ] . identity ( [ eqn - symm ] ) follows by using the construction of @xmath236(x)$ ] in ( [ eqn - rdef])([eqn - psidef ] ) , and the fact that @xmath71 and @xmath70 are diagonal and commute . to ensure that the denominator of ( [ eqn - psidef ] ) is different from zero , denote @xmath247 } \left(\epsilon_1(x)+\epsilon_2(x)\right)>0 $ ] . applying ( [ eqn - f1bound ] ) from lemma [ lem - f3f4bound ] , there exists @xmath210 for which , if @xmath248 , one gets : @xmath249\right)_{22}-\left(f_1[\gamma]\right)_{11 } \geq k_1 - k_2 \vert \gamma \vert_{\infty},\ ] ] thus if @xmath250 with @xmath251 , we obtain @xmath249\right)_{22}-\left(f_1[\gamma]\right)_{11 } \geq \frac{k_1}{2},\label{dem - bound}\ ] ] thus @xmath242 $ ] is well - defined . applying again ( [ eqn - f1bound ] ) in the numerator of ( [ eqn - psidef ] ) to bound ( [ eqn - thetadef ] ) , we obtain : @xmath252 \vert \leq k_3 \vert \gamma \vert_{\infty},\label{eqn - thetabound}\ ] ] and noting @xmath253 , we obtain directly the bound ( [ eqn - rpos2 ] ) , and by choosing @xmath254 , with @xmath255 , we show @xmath236>0 $ ] . inequality ( [ eqn - thetax ] ) is equivalent to showing : @xmath256 \sigma(x)\right)_x \vert \leq c_2 \vert \gamma \vert_{\infty } \left(1 + \vert \gamma _ x \vert_\infty \right).\ ] ] we first use ( [ eqn - thetabound ] ) to bound @xmath257(x ) \sigma_x(x ) \vert$ ] , and for @xmath258(x ) \sigma(x ) \vert$ ] we take a derivative in @xmath13 in ( [ eqn - psidef ] ) , use the bound ( [ dem - bound ] ) and use the fact that @xmath259= f_{12}[\gamma,\gamma_x]$ ] and using lemma [ lem - f5f6bound ] , there exists @xmath260 such that if @xmath261 , @xmath262 \vert & \leq & k_1 \left ( \vert \gamma \vert_{\infty}+\vert \gamma_x \vert_{\infty}\right).\end{aligned}\ ] ] to show ( [ eqn - thetat ] ) we use @xmath263= \vert f_{11}[\gamma,\eta]$ ] and apply lemma [ lem - f5f6bound ] . setting @xmath264 the lemma follows . define : @xmath265(x ) \eta(x , t)dx.\ ] ] computing @xmath266 , applying lemma [ lem - lyapeta ] , and integrating by parts , we find @xmath267\left(\sigma(x ) -f_1[\gamma ] \right)\right)_x \eta(x , t)dx \nonumber \\ & & + \left[\eta^t(x , t)r[\gamma](x)\left(\sigma(x ) -f_1[\gamma](x ) \right ) \eta(x , t ) \right]_{x=0}^{x=1 } + \int_0 ^ 1\eta^t(x , t ) \left(r[\gamma]\right)_t \eta(x , t)dx \nonumber \\ & & -2\int_0 ^ 1\eta^t(x , t ) r[\gamma]f_5[\gamma,\gamma_x,\eta,\eta_x,]dx -2\int_0 ^ 1\eta^t(x , t ) r[\gamma]f_6[\gamma,\eta ] dx.\label{eqn - vdot}\end{aligned}\ ] ] the first three terms of ( [ eqn - vdot ] ) are analyzed using lemma [ lem - lyapeta ] . thus , there exists @xmath210 such that , for @xmath235 , we find , for the first term : @xmath268\left(\sigma(x ) -f_1[\gamma ] \right)\right)_x \eta(x , t)dx \nonumber \\ & \leq & -\lambda_1 v_2 + k_1 \vert \eta \vert_{l^2}^2\left ( \vert \gamma \vert_{\infty } + \vert \gamma _ x \vert_\infty\right ) . \quad \,\,\label{eqn - vdott1}\end{aligned}\ ] ] the second term of ( [ eqn - vdot ] ) is bounded using the boundary conditions , ( [ eqn - udotlinear1])([eqn - udotlinear2 ] ) , and lemma [ lem - lyapeta ] , as : @xmath269(x)\left(\sigma(x ) -f_1[\gamma](x ) \right ) \eta(x , t ) \right]_{x=0}^{x=1 } \nonumber \\ & \leq & -\lambda_2 \left ( \eta_1 ^ 2(1,t)+ \eta_2 ^ 2(0,t ) \right ) + k_2 \vert \gamma \vert_{\infty } \left ( \eta_2 ^ 2(0,t ) + \eta_1 ^ 2(1,t)\right ) \nonumber \\ & & + k_3(1 + \vert \gamma \vert_{\infty})(a(t)^2+b(t)^2).\end{aligned}\ ] ] finally , we bound the third term of ( [ eqn - vdot ] ) applying lemma [ lem - lyapeta ] as follows : @xmath270\right)_t \eta(x , t)dx & \leq & k_3\int_0 ^ 1 \vert \eta\vert^2 \left(\vert \eta\vert+ \vert \eta \vert_{l^1 } \right ) dx \leq k_4\vert \eta \vert^2_{l^2 } \vert \eta \vert_{\infty}.\qquad\label{eqn - vdott2}\end{aligned}\ ] ] applying lemmas [ lem - lyapeta ] and [ lem - f5f6bound ] to the last terms of ( [ eqn - vdot ] ) , we get , for @xmath271 , @xmath272f_5[\gamma,\gamma_x,\eta,\eta_x]dx \right| \leq k_5 \int_0 ^ 1 \vert \eta \vert \vert f_5[\gamma,\eta ] \vert dx \nonumber \\ & \leq & k_6 \vert \eta \vert^2_{l^2 } \left ( \vert \gamma \vert_{\infty}+\vert \gamma_x \vert_{\infty}\right ) + k_{7 } \vert \eta \vert_{l^2}\vert \eta(0,t)\vert \vert \gamma(0,t ) \vert , \end{aligned}\ ] ] and @xmath272f_6[\gamma,\eta]dx \right| \leq k_{8 } \int_0 ^ 1 \vert \eta \vert \vert f_6[\gamma,\eta ] \vert dx \leq k_{9 } \vert \eta \vert^2_{l^2 } \vert \gamma \vert_{\infty}.\end{aligned}\ ] ] thus , it is clear that by choosing @xmath273 small enough , using lemma [ lem - gammaetaequiv ] to bound @xmath274 by @xmath275 , and noting @xmath276 , we obtain the following proposition : [ pr - gammat ] there exists @xmath277 such that if @xmath278 @xmath279 for @xmath280 positive constants . define @xmath281 . notice that the norms of @xmath282 and @xmath283 are related ( see lemma [ lem - thetaetaequiv ] in the appendix ) . taking a partial derivative in @xmath100 in ( [ eqn - etanl2 ] ) we obtain an equation for @xmath282 : @xmath284-\sigma(x)\right ) \theta_x + f_7[\gamma,\gamma_x,\eta,\eta_x,\theta]+f_8[\gamma,\eta,\theta]=0,\label{eqn - thetanl2}\end{aligned}\ ] ] where @xmath285 and @xmath286 are defined as @xmath287\eta \right]+ \int_0^x k(x,\xi ) f_{12}[\gamma,\gamma_x]\theta(\xi ) d\xi + \mathcal k_{1 } \left [ f_1[\gamma]\theta \right ] \nonumber \\ & & + \int_0^x k(x,\xi ) f_{14}[\gamma,\gamma_x,\eta,\eta_x]\eta(\xi ) d\xi + k(x,0)\frac{\partial \lambda_{nl}}{\partial \gamma}\left(\gamma(0),0\right)\eta(0 ) \eta(0 ) \nonumber \\ & & + k(x,0 ) \lambda_{nl}\left(\gamma(0),0\right ) \theta(0 ) + \mathcal k \left [ f_{11}[\gamma,\eta]\eta_x\right ] + \mathcal k \left [ f_{13}[\gamma,\eta,\theta]\gamma_x\right ] , \label{eqn - f7 } \\ f_8&=&2\mathcal k\left[f_{11}[\gamma,\eta ] \mathcal l_x \left[\eta\right ] \right ] + \mathcal k\left[f_{1}[\gamma]\mathcal l_x \left[\theta\right]\right ] + \mathcal k\left[f_{13}[\gamma,\eta,\theta ] \mathcal l_x \left[\gamma\right ] \right ] \nonumber \\ & & + \mathcal k[f_{22 } [ \gamma,\eta,\theta]],\label{eqn - f8}\end{aligned}\ ] ] where @xmath288,x\right ) \mathcal l[\eta ] \mathcal l[\eta ] + \frac{\partial \lambda_{nl}}{\partial \gamma } \left(\mathcal l[\gamma],x\right ) \mathcal l[\theta],\\ f_{14}&= & \frac{\partial^2 \lambda_{nl}}{\partial \gamma^2 } \left(\mathcal l[\gamma],x\right ) \mathcal l[\eta ] \left(\gamma_x+\mathcal l_{1}[\gamma]\right ) + \frac{\partial \lambda_{nl}}{\partial \gamma } \left(\mathcal l[\gamma ] , x\right ) \left(\eta_x+ \mathcal l_{1}[\eta]\right ) \nonumber \\ & & + \frac{\partial^2 \lambda_{nl}}{\partial x \partial \gamma } \left(\mathcal l[\gamma],x\right ) \mathcal l[\eta ] , \quad\,\,\ , \\ f_{22}&=&\frac{\partial^2 f_{nl}}{\partial \gamma^2 } \left(\mathcal l[\gamma],x\right)\mathcal l[\eta]\mathcal l[\eta ] + \frac{\partial f_{nl}}{\partial \gamma } \left(\mathcal l[\gamma],x\right)\mathcal l[\theta].\quad\end{aligned}\ ] ] the boundary conditions for @xmath289^t$ ] are @xmath290 since ( [ eqn - thetanl2 ] ) has the same structure as ( [ eqn - etanl2 ] ) , we define : @xmath291(x ) \theta(x , t)dx,\ ] ] where @xmath236(x)$ ] was defined in lemma [ lem - lyapeta ] . computing @xmath292 , and proceeding exactly as in ( [ eqn - vdot ] ) , we find : @xmath293\left(\sigma(x ) -f_1[\gamma ] \right)\right)_x \theta(x , t)dx \nonumber \\ & & + \left[\theta^t(x , t)r[\gamma](x)\left(\sigma(x ) -f_1[\gamma](x ) \right ) \theta(x , t)\right]_{x=0}^{x=1 } + \int_0 ^ 1\theta^t(x , t ) \left(r[\gamma]\right)_t \theta(x , t)dx \nonumber \\ & & -2 \int_0 ^ 1\theta^t(x , t ) r[\gamma]f_7[\gamma,\gamma_x,\eta,\eta_x,\theta]dx -2 \int_0 ^ 1\theta^t(x , t ) r[\gamma]f_6[\gamma,\eta,\theta]dx.\qquad \label{eqn - wdot}\end{aligned}\ ] ] the first three terms of ( [ eqn - wdot ] ) are analyzed as in ( [ eqn - vdott1])([eqn - vdott2 ] ) : @xmath294f_6[\gamma,\eta,\theta]dx\right| + 2 \left|\int_0 ^ 1\theta^t(x , t ) r[\gamma]f_7[\gamma,\gamma_x,\eta,\eta_x,\theta]dx\right| \nonumber \\ & & + k_3\vert \theta \vert^2_{l^2 } \vert \eta \vert_{\infty } + k_4(\eta^4_2(0,t)+(1+\vert \gamma \vert_{\infty})(a^2(t)+b^2(t ) ) ) .\qquad\label{eqn - wdot2}\end{aligned}\ ] ] finally , applying lemmas [ lem - lyapeta ] and [ lem - f7f8bound ] in the last two terms of ( [ eqn - wdot2 ] ) , there exists a @xmath295 , such that for @xmath271 , @xmath296f_7[\gamma,\gamma_x,\eta,\eta_x,\theta]dx \right| \leq k_5 \int_0 ^ 1 \vert \theta \vert \vert f_7[\gamma,\gamma_x,\eta,\eta_x,\theta ] \vert dx \nonumber \\ & \leq & k_6 \vert \theta \vert_{l^2}^2 \left ( \vert \gamma \vert_{\infty}+\vert \gamma_x \vert_{\infty}\right)+ k_7\vert \theta \vert_{l^2 } \vert \eta \vert_{l^2}^2 + k_8\vert \theta \vert_{l^2 } \vert \eta_x \vert_{l^2}\vert \eta \vert_{\infty } + k_9 \vert \theta \vert_{l^2 } \nonumber \\ & & + k_{10 } \vert \theta \vert_{l^2 } \left ( \vert \eta \vert_{l^2}\vert \eta \vert_{\infty}^2+\vert\eta(0,t)\vert^2 + \vert \gamma(0,t)\vert \vert \theta(0,t ) \vert\right),\qquad\end{aligned}\ ] ] and @xmath297f_8[\gamma,\eta,\theta]dx \right| \leq k_{11 } \int_0 ^ 1 \vert \theta \vert \vert f_8[\gamma,\eta,\theta ] \vert dx \nonumber \\ & \leq & k_{11 } \vert \theta \vert^2_{l^2 } \vert \gamma \vert_{\infty}+k_{12 } \vert \eta \vert_{l^2 } \vert \theta \vert_{l^2 } \vert \eta \vert_{\infty } + k_{12 } \vert \eta \vert_{l^2 } \vert \theta \vert_{l^2}^2 + k_{13 } \vert \eta \vert_{l^2}^2 \vert \theta \vert_{l^2}.\qquad\,\,\end{aligned}\ ] ] thus , by choosing @xmath273 and @xmath298 small enough to apply lemma [ lem - thetaetaequiv ] , we finally obtain the following proposition : [ pr - gammatt ] there exists @xmath299 such that if @xmath300 then @xmath301 where @xmath302 are positive constants . defining @xmath303 , and combining propositions [ pr - gamma ] , [ pr - gammat ] , and [ pr - gammatt ] , there exists @xmath295 such that if @xmath304 @xmath305 for @xmath306 . to compensate the last term , we augment this lyapunov function and define @xmath307 . then , @xmath308 and choosing @xmath309 , one obtains @xmath310 for some positive @xmath221 . following @xcite and noting @xmath311 , then for sufficiently small @xmath312 , it follows that @xmath313 exponentially . given that @xmath314 ( by proposition [ prop - equiv ] ) is equivalent to the @xmath1 norm of @xmath51 when @xmath315 is sufficiently small , and since by construction @xmath316 verifies the required second - order compatibility conditions , there exists @xmath173 and @xmath57 such that if @xmath317 , then : @xmath318 since , as we argued , for small enough @xmath319 the @xmath1 norms of @xmath12 and @xmath51 are equivalent , this proves theorem [ thm - main ] . _ [ rem - qzero ] the proof has been carried out for the case @xmath50 . if @xmath49 , we have to modify the target system following section [ sect - q0 ] and this implies the appearance of a linear boundary term ( a coefficient times @xmath320 ) in the @xmath51 system ; similarly , in the @xmath226 and @xmath282 systems , @xmath321 and @xmath322 terms will appear . these terms can be controlled using the same lyapunov function by following the strategy outlined in section [ sect - q0 ] _ we have solved the problem of full - state boundary stabilization for a @xmath0 system of first - order hyperbolic quasilinear pdes with actuation on only one boundary . we have shown , using a strict lyapunov function , @xmath1 local exponential stability of the state . it is possible to extend this result to design an observer , as shown in @xcite , and combining both results one obtains an output - feedback controller with similar properties ( see @xcite ) . it would be of interest to extend the method to @xmath3 systems . for instance , a @xmath323 first - order hyperbolic system of interest is the saint - venant - exner system , which models open channels with a moving sediment bed @xcite ; the extension is shown ( for the linear case ) in @xcite . while extending the lyapunov analysis to @xmath324 systems has been done @xcite , considerable extra effort is required to extend backstepping to a general @xmath324 system , even in the linear case . in general , the method needs @xmath325 kernels resulting in a @xmath326 system of coupled first - order hyperbolic equations , whose well - posedness depends critically on the exact choice of the transformation and target system . the extension has been shown possible , for the linear case , if the system has @xmath327 positive and one negative transport speeds , with actuation only on the state corresponding to the negative velocity @xcite . we show well - posedness of the following hyperbolic @xmath2 system , which is generic enough to contain all the kernel equation systems that appear in the paper : @xmath328 evolving in the domain @xmath329 , with boundary conditions : @xmath330 this type of system has been called `` generalized goursat problem '' by some authors @xcite . however the boundaries of the domain @xmath125 are characteristic for ( [ eqn - hypf1 ] ) and ( [ eqn - hypf4 ] ) , thus the general results derived in @xcite can not be applied . the following theorems discusses existence , uniqueness and smoothness of solutions to the equations . [ th - wp ] consider the hyperbolic system ( [ eqn - hypf1])([eqn - bcf4 ] ) . under the assumptions @xmath331),\,g_i , c_{ji}\in \mathcal c(\mathcal t),\,\,i , j=1,2,3,4 $ ] and @xmath332)$ ] with @xmath42 , there exists a unique @xmath110 solution @xmath333 , @xmath334 . [ th - sm ] consider the hyperbolic system ( [ eqn - hypf1])([eqn - bcf4 ] ) . under the assumptions of theorem [ th - wp ] , and the additional assumptions @xmath335),\,g_i , c_{ji}\in \mathcal c^n(\mathcal t)$ ] , there exists a unique @xmath336 solution @xmath333 , @xmath334 . next we prove the theorems ; the proof is based on transforming the equations into integral equations and then solving them using a successive approximation method . the equations can be transformed into integral equations by the method of characteristics . for that , it is necessary to define : @xmath337 and @xmath338 . note that all the @xmath339 functions are monotonically increasing and thus invertible , due to positivity of the @xmath340 coefficients . under the assumptions of theorem [ th - wp ] , it also holds that @xmath341)$ ] . define , for @xmath342 , the characteristic lines along which ( [ eqn - hypf1])([eqn - hypf4 ] ) evolve : @xmath343 where the argument @xmath344 that parameterizes @xmath345 and @xmath346 belongs to the interval @xmath347 $ ] , with @xmath348 defined as @xmath349 the following holds if @xmath342 and @xmath350 $ ] , it holds that @xmath351 , for @xmath352 . also , under the assumptions of theorem [ th - wp ] , @xmath345 , @xmath346 , and @xmath348 are continuous in their domains of definition since they are defined as compositions of continuous functions . moreover , the following inequalities are verified @xmath353 using these definitions , ( [ eqn - hypf1])([eqn - hypf4 ] ) are integrated to : @xmath354(x,\xi),\end{aligned}\ ] ] where we have denoted @xmath355,\quad g_j(x,\xi)=\int_0^{s^f_j(x,\xi ) } g_j\left(x_j(x,\xi , s),\xi_j(x,\xi , s)\right ) ds , \\ i_j[f](x,\xi)&= & \sum_{i=1}^4 \int_0^{s^f_j(x,\xi ) } \hspace{-10pt } c_{ji}\left(x_j(x,\xi , s),\xi_j(x,\xi , s)\right ) f^i\left(x_j(x,\xi , s),\xi_j(x,\xi , s)\right ) ds,\label{eqn - integrat } \qquad\end{aligned}\ ] ] for @xmath356 . substituting the boundary conditions ( [ eqn - bcf1])([eqn - bcf4 ] ) and expressing the terms in ( [ eqn - bcf1 ] ) and ( [ eqn - bcf4 ] ) containing @xmath357 and @xmath358 in terms of the solution ( [ eqn - integrat ] ) we get four integral equations which have the following structure : @xmath359(x,\xi)+i_j[f](x,\xi),\qquad\end{aligned}\ ] ] where @xmath360 , @xmath361 is has the values @xmath362 and @xmath363 and the values of the @xmath364(x,\xi)$ ] are @xmath365 and @xmath366&=&q_1(x_1(x,\xi,0))i_2[f](x_1(x,\xi,0),0 ) + q_2(x_1(x,\xi,0))i_3[f](x_1(x,\xi,0),0),\qquad \,\,\\ q_4[f]&=&q_3(x_4(x,\xi,0))i_2[f](x_4(x,\xi,0),0 ) + q_4(x_4(x,\xi,0))i_3[f](x_4(x,\xi,0),0).\qquad \,\,\end{aligned}\ ] ] in this form , the equations are amenable to be solved using the successive approximation method . this is explained next . the successive approximation method can be used to solve the integral equations . define first the following functional acting on @xmath367 : @xmath368(x,\xi)=q_j[f](x,\xi)+i_j[f](x,\xi),\ ] ] and the vectors : @xmath369,\ , \phi[f]=\left[\begin{array}{c}\phi_1[f]\\\phi_2[f]\\\phi_3[f]\\\phi_4[f]\end{array}\right].\,\ ] ] define then @xmath370(x,\xi).\quad \label{eqn - fn}\end{aligned}\ ] ] finally define for @xmath371 the increment @xmath372 , with @xmath373 by definition . it is easy to see that , since @xmath374 is a linear functional , the equation @xmath375(x,\xi)$ ] holds . if @xmath376 exists , then @xmath377 is a solution of the integral equations ( and thus solves the original hyperbolic system ) . using the definition of @xmath378 , it follows that if @xmath379 converges , then @xmath380 first , define : @xmath381 } \left| q_i(x ) \right| , \bar c= \left(1+\sum_{i=1}^4\bar q_{i } \right)\left(\sum_{j=1}^4\sum_{i=1}^4\bar c_{ji } \right).\end{aligned}\ ] ] next , we prove the following two lemmas : [ lem - int ] for @xmath334 , @xmath371 , @xmath342 , and @xmath382 , @xmath383 defined as in ( [ eqn - x1])([eqn - s4f ] ) , it follows that @xmath384 we show the result for @xmath385 . it follows for @xmath386 by switching @xmath387 and @xmath388 , respectively , for @xmath389 and @xmath390 . for @xmath391 we can write : @xmath392^n ds.\ ] ] to prove the inequality , change the variable of integration to @xmath393 . then , taking into account @xmath394=\frac{1}{\phi_1'(z)}=\epsilon_1(z),\ ] ] the integral can be bounded as follows : @xmath395^n ds \nonumber \\ & = & \int_{\phi_1^{-1}\left(\phi_1(x)-\phi_1(\xi)\right)}^{x } z^n/ \epsilon_1(z ) dz \leq k_{\epsilon } \int_{0}^{x } z^n dz = k_{\epsilon } \frac{x^{n+1}}{n+1}.\end{aligned}\ ] ] for @xmath396 the integral can be written as : @xmath397^n ds . \ ] ] as before , change the variable of integration to @xmath398 . then one has that @xmath399 , thus the integral can be bounded as follows : @xmath400^n ds \nonumber \\ & = & \int_{\phi_3^{-1}\left(\phi_1(x)+\phi_2(\xi)\right)}^{x } z^n/ \epsilon_1(z ) dz \leq k_{\epsilon } \int_{0}^{x } z^n dz = k_{\epsilon } \frac{x^{n+1}}{n+1},\qquad\,\end{aligned}\ ] ] which concludes the proof . [ lemma - bound ] for @xmath334 , @xmath371 and @xmath342 , assume that @xmath401 then it follows that @xmath402 . we show it for @xmath385 ; the structure of the equations is the same for @xmath386 . for @xmath396 : i_2[\delta f^n](x,\xi)\right| \leq \sum_{i=1}^4 \bar c_{2i } \int_0^{s^f_2(x,\xi ) } \left| \delta f^n_i\left(x_2(x,\xi , s),\xi_2(x,\xi , s ) \right ) ds \right| \nonumber \\ & \leq & \bar \phi \frac{k_{\epsilon}^n \bar c^{n}}{n ! } \sum_{i=1}^4 \bar c_{2i } \int_0^{s^f_2(x,\xi ) } x_2^n(x,\xi , s ) ds \leq \bar \phi \frac{k_{\epsilon}^{n+1 } \bar c^{n+1 } x^{n+1 } } { ( n+1 ) ! } , \end{aligned}\ ] ] where lemma [ lem - int ] has been applied . similarly , for @xmath391 : @xmath404(x,\xi ) & \leq & \left| q_1[\delta f^n](x,\xi)\right|+\left| i_1[\delta f^n](x,\xi)\right| \nonumber \\ & \leq & \bar q_1 \bar \phi \frac{\bar c^n k_{\epsilon}^n}{n ! } \sum_{i=1}^4 \bar c_{2i } \int_0^{s^f_2(x_1(x,\xi,0),0 ) } x_2^n(x_1(x,\xi,0),0,s ) ds \nonumber \\ & & + \bar q_2 \bar \phi \frac{\bar c^n k_{\epsilon}^n}{n ! } \sum_{i=1}^4 \bar c_{3i } \int_0^{s^f_3(x_1(x,\xi,0),0 ) } x_3^n(x_1(x,\xi,0),0,s ) ds \nonumber \\ & & + \bar \phi \frac{\bar c^n k_{\epsilon}^n}{n ! } \sum_{i=1}^4 \bar c_{1i } \int_0^{s^f_1(x,\xi)}x_1^n(x,\xi , s)ds \nonumber \\ & \leq & \bar q_1 \bar \phi \frac{\bar c^n k_{\epsilon}^{n+1}}{n ! } \sum_{i=1}^4 \bar c_{2i } \frac{x_1(x,\xi,0)^n}{n+1 } + \bar q_2 \bar \phi \frac{\bar c^n k_{\epsilon}^{n+1}}{n ! } \sum_{i=1}^4 \bar c_{3i } \frac{x_1(x,\xi,0)^n}{n+1 } \nonumber \\ & & + \bar \phi \frac{\bar c^n k_{\epsilon}^{n+1}}{n ! } \sum_{i=1}^4 \bar c_{1i } \frac{x^n}{n+1 } \leq \bar \phi \frac{\bar c^{n+1 } k_{\epsilon}^{n+1}x^{n+1}}{(n+1 ) ! } , \end{aligned}\ ] ] since @xmath405 . thus the lemma is proved . next we show that ( [ eqn - fseriesdef ] ) converges . [ prop - conv ] for @xmath406 , @xmath334 , one has that @xmath407 the result follows if we show that @xmath408 . we prove the bound by induction . for @xmath409 , the result follows from ( [ eqn - fn ] ) . assume that the bound is correct for all @xmath245 in @xmath410 . then , we get for @xmath411 that @xmath412 ( x,\xi ) \right| \leq \bar \phi \frac{\bar c^{n+1 } k_{\epsilon}^{n+1 } x^{n+1}}{(n+1)!},\ ] ] where we have used lemma [ lemma - bound ] . thus the proposition follows . from proposition [ prop - conv ] we conclude that the successive approximation series is bounded and converges uniformly . thus , a bounded solution to equations ( [ eqn - hypf1])([eqn - bcf4 ] ) exists . this proves the existence part of theorem [ th - wp ] . to prove uniqueness , let us denote by @xmath413 and @xmath414 two different solutions to ( [ eqn - hypf1])([eqn - bcf4 ] ) . defining @xmath415 . by linearity of ( [ eqn - hypf1])([eqn - bcf4 ] ) , @xmath416 also verifies ( [ eqn - hypf1])([eqn - bcf4 ] ) , with @xmath417 for all @xmath245 . then @xmath418 for @xmath419 , and proposition [ prop - conv ] we conclude @xmath420 , which implies that @xmath421 . to prove that the solution is continuous , note that since ( [ eqn - fseriesdef ] ) converges uniformly , one only needs to prove continuity of each term . first , @xmath422 since the @xmath423 are defined as a sum of compositions of continuous functions . similarly , since @xmath424 is defined as the integral ( with continuous limits ) of continuous functions times the previous @xmath425 composed with continuous functions , by induction it can be shown that @xmath426 . thus @xmath427 and theorem [ th - wp ] is proved . next we sketch the proof of theorem [ th - sm ] . we only consider @xmath428 ; for @xmath429 the result can be proven by induction . denote @xmath430 and @xmath431 . by differentiating with respect to @xmath13 and @xmath432 in ( [ eqn - hypf1])([eqn - hypf4 ] ) , we find two uncoupled @xmath433 hyperbolic systems ( for @xmath434 and for @xmath435 ) @xmath436 now , differentiating the boundary conditions in ( [ eqn - bcf2 ] ) it is found that : @xmath437 and setting @xmath438 in ( [ eqn - hypf2])([eqn - hypf3 ] ) @xmath439 we can find a set of boundary conditions for @xmath440 and @xmath441 , @xmath442 , at the boundary @xmath438 : @xmath443 similarly , differentiating boundary conditions ( [ eqn - bcf1 ] ) and ( [ eqn - bcf4 ] ) we find boundary conditions for @xmath444 and @xmath445 at @xmath446 : @xmath447 and setting @xmath446 in ( [ eqn - hypf1])([eqn - hypf4 ] ) we can also find two sets of boundary conditions for @xmath441 , @xmath448 , at the boundary @xmath446:@xmath449 thus , both the @xmath450 s and @xmath451 s verify equations formally equivalent to ( [ eqn - hypf1])([eqn - bcf4 ] ) , with derivatives of the old equations coefficients as new coefficients , and the @xmath333 s as additional terms . if these equations have solutions , then the solutions must be the partial derivatives of the @xmath333 functions . now , under the assumptions of theorem [ th - sm ] , by theorem [ th - wp ] there is a ( at least ) continuous solution @xmath452 . plugging that solution into the equations we just derived for the @xmath450 s and @xmath451 s , one obtains equations whose coefficients and boundary conditions are ( at least ) continuous . hence theorem [ th - wp ] can be applied implying that the @xmath450 s and @xmath451 s are continuous . thus @xmath453 , proving the result . next we give some technical lemmas used throughout the paper . the first lemma follows from the fact that the control direct and inverse kernels are @xmath202 functions . @xmath454 \vert & \leq & c_1 \left ( \vert \gamma\vert+\vert \gamma \vert_{l^1 } \right),\\ \vert \mathcal l[\gamma]\vert & \leq & c_2 \left ( \vert \gamma\vert+\vert \gamma \vert_{l^1 } \right),\\ \vert \mathcal k_{1}[\gamma]\vert & \leq & c_3 \left ( \vert \gamma\vert+\vert \gamma \vert_{l^1 } \right),\\ \vert \mathcal k_{2}[\gamma]\vert & \leq & c_4 \left ( \vert \gamma\vert+\vert \gamma \vert_{l^1 } \right),\\ \vert \mathcal l_{1}[\gamma]\vert & \leq & c_5 \left ( \vert \gamma\vert+\vert \gamma \vert_{l^1 } \right),\\ \vert \mathcal l_{11}[\gamma]\vert & \leq & c_6 \left ( \vert \gamma\vert+\vert \gamma \vert_{l^1 } \right).\end{aligned}\ ] ] the next lemma is based on the fact that , since @xmath455 is twice differentiable with respect to @xmath456 and @xmath13 , and since we have @xmath149 , it follows that there exists a @xmath457 and @xmath458 , @xmath459 , @xmath460 such that if @xmath461 , then , for any @xmath462 , it holds that @xmath463 similarly , since @xmath464 is twice differentiable with respect to @xmath456 and once with respect to @xmath13 , and @xmath465 , there exists a @xmath466 and @xmath467 , @xmath468 , @xmath469 such that if @xmath470 , then for any @xmath471 , @xmath472 then , the following lemma holds . first , from ( [ eqn - gammanl2 ] ) we see that @xmath481(x)+f_4[\gamma](x)=0.\ ] ] therefore , calling @xmath210 the value of @xmath295 in lemma [ lem - f3f4bound ] and assuming @xmath211 , we can compute a bound on @xmath482 as follows : @xmath483\vert_{\infty}+\vert f_4[\gamma]\vert_{\infty } \nonumber \\ & \leq & k_2 \left ( \vert \gamma_x \vert_{\infty } + \vert \gamma_x \vert_{\infty } \vert \gamma \vert_{\infty } + \vert \gamma \vert_{\infty}^2 \right ) \leq k_3 \left ( \vert \gamma_x \vert_{\infty } + \vert \gamma \vert_{\infty } \right).\end{aligned}\ ] ] proceeding similarly with the @xmath5 norm , @xmath484\vert_{l^2}+\vert f_4[\gamma]\vert_{l^2 } \nonumber \\ & \leq & k_2 \left ( \vert \gamma_x \vert_{l^2 } + \vert \gamma_x \vert_{l^2 } \vert \gamma \vert_{\infty } + \vert \gamma \vert_{\infty } \vert \gamma \vert_{l^2 } \right ) \leq k_3 \left ( \vert \gamma_x \vert_{l^2 } + \vert \gamma \vert_{l^2 } \right).\end{aligned}\ ] ] for the last two inequalities , we solve for @xmath227 : @xmath485(x)+f_4[\gamma](x)\right).\ ] ] remembering the definition of @xmath70 , @xmath77}\left\{\frac{1}{\epsilon_1(x ) } , \frac{1}{\epsilon_2(x ) } \right\}>0 $ ] , and assuming that @xmath211 we obtain@xmath486 therefore if we choose @xmath487 , we reach the third inequality . proceeding similarly with the @xmath5 norm : @xmath488 so choosing @xmath489 , we reach the fourth inequality . therefore , choosing @xmath490 , all inequalities are verified and the lemma is proven . taking an @xmath13-derivative in ( [ eqn - gammanl2 ] ) : @xmath494 \gamma_{xx}+ f_{32}[\gamma,\gamma_x]+f_{42}[\gamma,\gamma_x]=0,\ ] ] where @xmath495 and @xmath496 are defined as : @xmath497\gamma_x\right ] + f_{12}[\gamma,\gamma_x]\gamma_x,\\ f_{42}&=&\mathcal k_{2 } \left [ f_1[\gamma ] \mathcal l_{1 } \left[\gamma \right ] + f_2[\gamma ] \right ] + f_{12}[\gamma,\gamma_x]\mathcal l_{1 } \left[\gamma \right ] \nonumber \\ & & + f_1[\gamma ] \mathcal l_{11 } \left[\gamma \right]+f_1[\gamma ] l(x , x ) \gamma_x+f_{23}[\gamma,\gamma_x ] \gamma_x , \end{aligned}\ ] ] where @xmath498 = \frac{\partial f_{nl}}{\partial x}\left(\mathcal l[\gamma],x\right)+\frac{\partial f_{nl}}{\partial \gamma}\left(\mathcal l[\gamma],x\right)\left(\gamma_x+\mathcal l_{1}[\gamma]\right).\ ] ] these functionals verify the following bound , similar to the bounds developed in lemma [ lem - f3f4bound ] , if @xmath473 : @xmath499 therefore , using lemma [ lem - gammaetaequiv ] , and inequality ( [ eqn - f1bound ] ) , and making @xmath492 small enough , we can compute the bounds as in the proof of lemma [ lem - gammaetaequiv ] . we can write ( [ eqn - etanl2 ] ) analogously to ( [ eqn - gammanl2 ] ) : @xmath501(x)+f_6[\gamma,\eta](x)=0,\ ] ] where @xmath502 is defined as : @xmath503\eta_x + f_{11}[\eta]\gamma_x \right].\end{aligned}\ ] ] the functional @xmath502 verifies the following bound , similar to the bounds developed in lemma [ lem - f3f4bound ] , if @xmath473 : @xmath504 therefore , using lemma [ lem - f5f6bound ] , lemma [ lem - gammaetaequiv ] , and inequality ( [ eqn - f1bound ] ) , and making @xmath315 small enough , we can compute the bounds as in the proof of lemma [ lem - gammaetaequiv ] .
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in this work , we consider the problem of boundary stabilization for a quasilinear @xmath0 system of first - order hyperbolic pdes . we design a new full - state feedback control law , with actuation on only one end of the domain , which achieves @xmath1 exponential stability of the closed - loop system .
our proof uses a backstepping transformation to find new variables for which a strict lyapunov function can be constructed .
the kernels of the transformation are found to verify a goursat - type @xmath2 system of first - order hyperbolic pdes , whose well - posedness is shown using the method of characteristics and successive approximations .
once the kernels are computed , the stabilizing feedback law can be explicitly constructed from them .
nonlinear hyperbolic systems , boundary conditions , stability , lyapunov function , backstepping , method of characteristics , integral equation , goursat problem tbd
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You are an expert at summarizing long articles. Proceed to summarize the following text:
in recent years low surface brightness ( lsb ) galaxies have received an increasing amount of attention . with the advent of deep , large - field surveys many of these galaxies have been discovered , both in clusters and in the field ( schombert et al . 1988 , 1992 [ hereafter sbsm ] , turner et al . 1993 , davies et al . 1994 , schwartzenberg et al . 1995 , sprayberry et al . 1995a ) . the main property which distinguishes lsb galaxies from normal `` classical '' galaxies that define the hubble sequence is their low surface brightness ( @xmath4 ) and not their size . lsb galaxies are not necessarily dwarfs . many of them are disk galaxies that typically have scale lengths of several kpc , ( mcgaugh & bothun 1994 , de blok et al . 1995 [ hereafter bhb95 ] ) i.e. , in the same size range as typical high surface brightness ( hsb ) galaxies . the faintest lsb disk galaxies hitherto discovered have central surface brightnesses of @xmath5 mag arcsec@xmath6 ( turner et al . 1993 ; irwin et al . 1990 ; davies , phillips & disney 1988 ) . this is almost two orders of magnitude fainter than those of the disk galaxies that obey freeman s law ( 1970 ) , which states that most disk galaxies have a central @xmath7-band surface brightness in the range @xmath8 mag arcsec@xmath6 . the existence of lsb galaxies with central surface brightnesses deviating many @xmath9 from the freeman value indicates that this law is not a complete description of the central surface brightness distribution of disk galaxies . attempts to construct the proper distribution from extant data ( mcgaugh , bothun & schombert 1995 , mcgaugh 1996 ) suggest that lsb galaxies are numerically common , though considerable uncertainty remains . de jong ( 1995 , 1996 ) investigated the central surface brightness distribution of spiral galaxies for a range of hubble types . he concluded , on the basis of a diameter - limited , statistically complete sample of 86 face - on spiral galaxies from the ugc ( including a few lsb galaxies ) , that freeman s law should be restated as that there is no preferred single value for the central surface brightness of disks in galaxies , but only an upper limit in surface brightness which galaxies can have ( corresponding approximately to the freeman value ) ( see also allen & shu , 1979 ) . towards later types the central surface brightness systematically decreases . in general the range of properties exhibited by lsb galaxies is much larger than that of the classical freeman galaxies . the sizes and luminosities of lsb galaxies range from the very small and faint dwarfs like gr8 ( hodge 1967 ) , to the giant and luminous lsb galaxy malin 1 ( impey & bothun 1989 , sprayberry et al . lsb galaxies _ in general _ are therefore likely to be not so much a single group , different from all other types of galaxies , but rather the dim counterparts of the known `` classical '' galaxies . we have drawn our sample from the list of lsb galaxies by sbsm . the sbsm list was made by scanning by eye selected fields of the new poss - ii survey , and applying the ugc diameter selection criteria to the objects discovered . the increased depth of these new plates resulted in the discovery of many lsb galaxies that would have been in the ugc had it been compiled at present using the poss - ii plates . sbsm make no claims about completeness in their survey , but rather that `` our purpose ... is simply to increase our known sample of lsb galaxies and to offset the clear bias in apparent magnitude catalogs for the discovery of only hsb systems . '' the present paper is written in this spirit . with this in mind it is striking that lsb galaxy surveys that have concentrated on detecting field lsb galaxies ( like e.g. sbsm ) have in general found that the large majority of the galaxies ( @xmath10 85% in the case of sbsm ) in these surveys are late - type spirals . ellipticals and early - type spiral galaxies form only a small minority . in our work we will therefore restrict ourselves to describing the properties of a subset of lsb galaxies : those of the late - type lsb disk galaxies . in the rest of the paper when we refer to lsb galaxies we will always implicitly mean this sub - group , unless noted otherwise . not included are dwarf ellipticals and spheroids , like those found in the local group . the group of late - type lsb galaxies is in many ways an extension of the hubble - type sequence towards very late types . they continue ( and are in many cases the most extreme representatives known of ) the trends defined by the hsb galaxies along the hubble sequence towards lower surface brightnesses . such trends are increasingly bluer colours ( bhb95 , mcgaugh & bothun 1994 [ hereafter mb94 ] , rnnback 1993 ) , decreasing oxygen abundances in the gas ( mcgaugh 1994 , rnnback & bergvall 1995 ) , and decreasing hi surface densities ( van der hulst et al.1993 [ hereafter vdh93 ] ) . a detailed investigation of these late - type lsb disk galaxies will therefore provide an insight into systematic changes of properties of spiral galaxies along the hubble sequence as an explicit function of surface brightness . one particular example of the trends with surface brightness was presented in zwaan et al ( 1995 ) ( cf . romanishin et al . it was shown that late - type lsb disk galaxies also lie on the tully - fisher relation ( see also sprayberry et al . 1995b ) , requiring a systematic and fine - tuned relation between the surface brightness of a galaxy and its overall mass - to - light ratio . in order to hope to understand this systematic relation , and perhaps to uncover other relations important to our understanding of galaxy properties and their history , resolved hi imaging , providing detailed rotation curves , is required . we have obtained such hi imaging data for 19 lsb galaxies with the vla and wsrt synthesis radio telescopes . in this paper we present the data and a global analysis . in forthcoming papers we will present a full decomposition of the rotation curves into disc and halo components , and , in combination with optical properties , discuss the processes that regulate the rate of evolution and star formation in these dim galaxies . in section 2 we discuss the sample selection . section 3 contains technical details on the observations . in section 4 we discuss the reduction of the data . in section 5 the derivation of parameters from these data are given . in section 6 we present the data in tabular and graphical form . section 7 contains a discussion of the hi surface densities . in section 8 the rotation curves are discussed . section 9 is devoted to the mass - to - light ratios , and finally in section 10 the main points of this paper are summarized . we present 21-cm hi synthesis radio observations of a sample of 19 lsb galaxies , taken from the catalogues of lsb galaxies by sbsm . the sbsm catalogue is not dominated by small , faded dwarf galaxies . lsb galaxies occupy the full range of types as defined by the hubble sequence , however with a strong preference towards later types . the redshift distribution is similar to those of the ugc galaxies , with only small deviations at the low and high velocity tails , where the lsb survey picks up lsb dwarfs , and lsb giants respectively . the spatial distribution shows that lsb galaxies are good tracers of the large - scale structure on scales of @xmath11 mpc ( cf . bothun et al . 1993 ; mo , mcgaugh & bothun 1994 ) . they do not fill the voids . the same applies to the distribution of hi masses : apart from a slightly smaller mean for the lsb galaxies , they are similar . the large fraction of lsb galaxies that has been detected in hi shows that these galaxies are just as likely to contain hi as their hsb counterparts of similar hubble - type . we refer to sbsm for a complete discussion of the properties of the galaxies in their catalogue . the typical lsb galaxy , as listed in sbsm , is therefore a late - type spiral galaxy in our local universe , with luminosity and hi mass not significantly different from late types already present in the ugc , but typically more extreme in surface brightness . in order to select a number of representative galaxies from the sbsm catalogue , we have applied the following criteria . 1 . the galaxy should have been detected in hi . this necessarily limits the sample to the 171 galaxies that have been observed with arecibo , but as explained in sbsm this does not bias the sample towards objects containing much hi , as the success rate of detections for these galaxies was at least 80% ( and would have been higher if not for the limits imposed by the observable velocity range ) . 2 . as the hi masses of galaxies in sbsm are strongly peaked around @xmath12 with a spread of 0.5 dex to both sides ( cf . fig . 5 in sbsm ) , we have selected only galaxies with hi masses in the range @xmath13 . this leaves 132 galaxies . 3 . in order not to confuse true dwarfs with lsb disk galaxies we have furthermore restricted the sample to those galaxies with redshifts between 3000 and 8000 km s@xmath1 , which sbsm define as `` lsb disks '' ( cf . their fig . this leaves 95 galaxies . 4 . from this sample of 95 galaxies , we selected 16 galaxies at random . this subsample represents in our opinion a typical cross - section of the larger sample of late - type lsb disk galaxies . the sample is listed in tables 13 . in order to compare the properties of these lsb galaxies with those of dwarf lsb galaxies and large lsb galaxies , we further supplemented the sample with examples of these two classes . the dwarf sample consists of 3 galaxies with hi masses smaller than @xmath14 @xmath15 and redshifts smaller than 3000 km s@xmath1 . these are f564-v3 , f571-v2 and f583 - 1 . we used the sample of vdh93 to represent the large lsb galaxies . although their hi masses are slightly larger , they set themselves apart from our main lsb sample mostly because of their larger optical scale lengths , and because of the fact that despite their low central surface brightness , they were included in the ugc catalogue . although our sample is not complete in a statistical sense , we have obtained a sample of galaxies which is representative of the late - type lsb disk galaxies found in the sbsm catalogue . we can use this sample to extend the range in surface brightness over which properties of disk galaxies are investigated towards lower surface brightnesses . it should also be noted that , to the best of our knowledge , no hi - rotation curves of galaxies with central surface brightnesses fainter than 23.5 @xmath7-mag arcsec@xmath6 ( the average surface brightness of our sample ) have been published . there are only a handful of rotation curves available in the literature of lsb dwarfs ( @xmath16 ) , most of them ddo dwarfs ( carignan & beaulieu 1988 , hoffman et al . this is surprising , given the fact that these galaxies are dark - matter dominated , providing ideal cases for the study of the dynamics and density distributions of the dark matter . the discussion on whether halos are isothermal or singular ( see flores & primack 1994 ) depends very much on evidence provided by this same handful of rotation curves . the sample presented here thus greatly extends the available data on galaxies in this still ill - charted lsb area of parameter space . this is currently the largest sample of lsb galaxies for which detailed optical , radio and abundance data are available ( vdh93 , mcgaugh 1994 , mb94 , bhb95 ) . we observed 19 lsb galaxies in the 21-cm hi line . observations of 10 galaxies were performed with the vla in its 3-km ( c ) configuration using two if systems , 27 antennas and 64 channels over a 3.125 mhz bandwidth . the spectra were hanning - smoothed on - line providing a velocity resolution of 20.6 km s@xmath1 . each galaxy was observed for 5 times 40 minutes interspersed with 5 minute observations of a phase calibrator . for bandpass calibration and the flux density scale we observed 3c286 with the same correlator settings as each galaxy . the observations were performed on 10 , 16 , 17 and 21 december 1990 . further details are given in table 1 . the other 9 galaxies were observed with the wsrt . each galaxy was observed for 12 hours with 38 baselines ranging from 72 m to 2736 m with 72 m increments . both if systems were used and the correlator was set up to provide 64 channels over a bandwidth of 2.5 mhz . no on - line spectral smoothing was performed , so the highest possible velocity resolution is 9.9 km s@xmath1 . off - line hanning smoothing reduces the velocity resolution to 16.5 km s@xmath1 . amplitude / phase and bandpass calibration was performed using observations of 3c147 and 3c286 . the observations were done in june and september 1990 and may and august 1991 . + _ columns : _ + ( 1),(3 ) name of the galaxy . + ( 2),(4 ) source of the data . + sources : + [ 1 ] carignan & freeman ( 1985 ) + [ 2 ] carignan & beaulieu ( 1989 ) + [ 3 ] broeils ( 1992 ) + [ 4 ] puche , carignan & wainscoat ( 1991 ) + [ 5 ] carignan & puche ( 1990 ) + [ 6 ] carignan ( 1985 ) + [ 7 ] puche , carignan & bosma ( 1990 ) + [ 8 ] kent ( 1986 ) + [ 9 ] begeman ( 1987 ) + [ 10 ] kent ( 1987 ) + [ 11 ] jobin & carignan ( 1990 ) + [ 12 ] begeman ( 1989 ) + [ 13 ] c^ ot ' e , carignan & sancisi ( 1991 ) + [ 14 ] carignan , sancisi & van albada ( 1988 ) + [ 15 ] roelfsema & allen ( 1985 ) + [ 16 ] bhb95 + [ 17 ] this work + [ 18 ] mb94 + [ 19 ] vdh94 + table a3 contains the identification of the galaxies plotted in fig . [ large ] . also given are the sources for the data . we retrieved the surface brightnesses from the original papers ( see notes to table a3 for a complete list ) , and where necessary converted to the @xmath7-band .
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we present very large array ( vla ) and westerbork synthesis radio telescope ( wsrt ) 21-cm hi observations of 19 late - type low surface brightness ( lsb ) galaxies .
our main findings are that these galaxies , as well as having low surface brightnesses , have low hi surface densities , about a factor of @xmath0 lower than in normal late - type galaxies .
we show that lsb galaxies in some respects resemble the outer parts of late - type normal galaxies , but may be less evolved .
lsb galaxies are more gas - rich than their high surface brightness counterparts .
the rotation curves of lsb galaxies rise more slowly than those of hsb galaxies of the same luminosity , with amplitudes between 50 and 120 km s@xmath1 , and are often still increasing at the outermost measured point .
the shape of the rotation curves suggests that lsb galaxies have low matter surface densities .
we use the average total mass surface density of a galaxy as a measure for the evolutionary state , and show that lsb galaxies are among the least compact , least evolved galaxies .
we show that both @xmath2 and @xmath3 depend strongly on central surface brightness , consistent with the surface brightness mass - to - light ratio relation required by the tully - fisher relation .
lsb galaxies are therefore slowly evolving galaxies , and may well be low surface density systems in all respects .
galaxies : evolution galaxies : fundamental parameters galaxies : structure galaxies : spiral galaxies : kinematics and dynamics dark matter
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You are an expert at summarizing long articles. Proceed to summarize the following text:
galaxy mergers are a natural and crucial ingredient of the @xmath0cdm hierarchical galaxy formation paradigm . although the fraction of galaxies undergoing a merger at any given time is relatively small , nearly all galaxies will experience a merger at some point in their histories ( e.g. , * ? ? ? * ; * ? ? ? * ; * ? ? ? particularly significant are ` major ' mergers , which can be transformative . in these cases , mergers violently alter the orbits of the stars in the galaxies and can transform rotationally supported discs into dispersion - supported spheroids ( e.g. , * ? ? ? * ; * ? ? ? * ; * ? ? ? * ; * ? ? ? * ; * ? ? ? * ; * ? ? ? * ; * ? ? ? furthermore , tidal torques exerted by the galaxies upon one another drive gas inwards ( e.g. , * ? ? ? * ; * ? ? ? * ) , thereby resulting in powerful starbursts ( e.g. , * ? ? ? * ; * ? ? ? * ) , triggering active galactic nuclei ( agn ) ( e.g. , * ? ? ? * ) , altering metallicity gradients @xcite , and leaving behind signatures of the starbursts and agn activity in the form of compact stellar cores ( e.g. , * ? ? ? * ; * ? ? ? * ) and supermassive black holes ( bhs ; e.g. , * ? ? ? also , mergers may drive the size evolution of quiescent galaxies ( e.g. , * ? ? ? * ; * ? ? ? it has thus been proposed that various seemingly different observational classes of objects in the universe including blue star - forming disc galaxies , irregular galaxies of a variety of morphologies , heavily dust - obscured ( ultra-)luminous infrared galaxies ( ( u)lirgs ) , both obscured and unobscured agn , ` post - starburst ' ( aka ` k+a ' ) galaxies , and ` red and dead ' elliptical galaxies may be related in a merger - driven evolutionary sequence ( e.g. , * ? ? ? * ; * ? ? ? * ; * ? ? ? * ; * ? ? ? * ; * ? ? ? * ; * ? ? ? galaxy mergers have been studied using numerical simulations for more than forty years ( @xcite ) , and for more than seventy years if one considers the pioneering laboratory method of @xcite . although the early simulations included only gravity , they provided much insight into the effects of mergers on galaxy morphologies , the formation of tidal tails and shells , and the kinematics of merger remnants . more sophisticated simulations ( e.g. , * ? ? ? * ; * ? ? ? * ; * ? ? ? * ; * ? ? ? * ; * ? ? ? * ) included also gas dynamical processes , which can be important for many galaxies because a significant fraction of the baryonic mass is in gas , and , unlike the stars , the gas is dissipational . hydrodynamic simulations of galaxy mergers have helped us to understand the driving mechanism of starbursts ( e.g. , * ? ? ? * ; * ? ? ? * ; * ? ? ? * ; * ? ? ? * ; * ? ? ? * ; * ? ? ? * ) , supermassive bh fueling and feedback ( e.g. , * ? ? ? * ; * ? ? ? * ; * ? ? ? * ; * ? ? ? * ; * ? ? ? * ; * ? ? ? * ; * ? ? ? * ) , feedback from supernovae ( e.g. , * ? ? ? * ) , the kinematics of merger remnants ( e.g. , @xcite ; @xcite ) , the survivability of discs during mergers ( e.g. , * ? ? ? * ; * ? ? ? * ; * ? ? ? * ; * ? ? ? * ; * ? ? ? * ) , the sizes of merger remnants ( e.g. , @xcite ; @xcite ; @xcite ) , and the formation of local ( e.g. , * ? ? ? * ) and high - redshift ulirgs ( e.g. , * ? ? ? * ; * ? ? ? * ; * ? ? ? * ; * ? ? ? * ; * ? ? ? * ; * ? ? ? * ; * ? ? ? * ; * ? ? ? * ) , among many other topics . whereas cosmological simulations are routinely performed using both grid - based eulerian and particle - based pseudo - lagrangian methods , idealised isolated ( i.e. , non - cosmological ) galaxy merger simulations have almost always been performed using pseudo - lagrangian smoothed particle hydrodynamics ( sph ; @xcite ; see @xcite , @xcite , and @xcite for recent reviews ) . sph is well - suited to simulating galaxy mergers because it naturally treats the large bulk velocities present before the discs coalesce and it concentrates resolution elements in regions where the mass is located ( the galaxy centre(s ) ; see section 9.4 of @xcite for further discussion ) . furthermore , the other primary method used for galaxy formation studies , adaptive mesh refinement ( amr ) , does not treat self - gravity as accurately as particle - based approaches @xcite . for these reasons , relatively few galaxy merger simulations have been performed using amr ( to our knowledge , the only published examples of such simulations are @xcite ) . recent studies that used simple , idealised test problems and cosmological simulations have highlighted potentially significant problems that are inherent in the standard formulation of the sph technique . @xcite demonstrated that the standard implementation of sph artificially suppresses the kelvin helmholtz ( kh ) and raleigh taylor ( rt ) instabilities because the method introduces spurious pressure forces near steep density gradients that allow a gap to be created between particles , thereby reducing their interactions . furthermore , sph can artificially damp subsonic turbulence @xcite and restricts gas stripping from substructures falling on to haloes @xcite . however , fixed grid - based methods also have drawbacks ; for example , these codes are not galilean invariant and can produce over - mixing @xcite . partially motivated by the limitations inherent in the traditional sph technique , @xcite developed a novel moving - mesh hydrodynamics code known as arepo . although similar approaches have been proposed earlier , such techniques have not yet seen wide - spread use in astrophysical applications . arepo is quasi - lagrangian because the unstructured mesh is advected with the flow . whereas the motion of the grid reduces mass exchange between cells compared with a static mesh ( e.g. , * ? ? ? * ) , arepo is not strictly lagrangian because the mass in a given cell can change with time . nevertheless , arepo offers a number of advantages over other methods that make it attractive for astrophysical applications . for example , it is galilean invariant and naturally concentrates resolution elements in dense regions , similar to particle - based techniques . because arepo uses a finite - volume method to solve the euler equations , it is better than traditional sph at capturing shocks and contact discontinuities and does not artificially suppress fluid instabilities ( see also * ? ? ? * ; * ? ? ? its ability to better capture weak shocks than sph is potentially significant for cosmological problems because these features are ubiquitous in the cosmic web ( e.g. , * ? ? ? because of its hybrid nature , arepo performs better than ( or at least as well as ) both traditional sph and grid - based approaches for idealised test problems . thus , it is useful to compare the results of arepo simulations to those of simulations performed using other techniques to investigate what effects , if any , the shortcomings of the traditional methods have on the results of cosmological and idealised galaxy merger simulations . some comparisons between cosmological simulations performed with arepoand gadget-3 in which all other ingredients , including the gravity solver and sub - resolution models , are identical have already been made . in some situations , the baryonic properties of the simulations performed with the two codes differ strikingly . for example , @xcite demonstrated that gas cooling is more efficient in the arepo simulations , which results in more star formation at late times . they attribute the difference to spurious heating in the outer regions of virialized haloes in the gadget-3 simulations and the inability of conventional sph to correctly develop a turbulent cascade to smaller scales . the arepo simulations produce galaxies with extended , relatively smooth gas discs , whereas in the gadget-3 simulations , the discs are more compact and clumpy @xcite . @xcite demonstrated that the relative fraction of gas supplied to galaxies in halos of moderate to large size through ` cold - mode accretion ' ( aka ` cold flows ' ) is dramatically less in the arepo simulations because with traditional sph , much of the cold gas that reaches the central galaxies does so because it is locked in ` blobs ' of purely numerical origin and because the rate at which gas cools from galaxies hot haloes is higher in arepobecause it allows for a proper cascade of turbulent energy to small scales . because of the nature of the aforementioned comparisons , the differences can only originate from differences in the hydrodynamical solver . it is sometimes argued ( e.g. , * ? ? ? * ) that the differences caused by inaccuracies in the numerical technique employed are subdominant to the effects caused by varying the prescriptions for star formation and stellar and agn feedback and hence may be ignored . however , the results of @xcite caution against this because they find differences in the accretion rate of hot gas on to galaxies between arepo and gadget-3 that in some cases approach two orders of magnitude , which is far greater than the discrepancies between simulations and observations that feedback effects are invoked to resolve ( see @xcite and @xcite for comparisons of arepocosmological hydrodynamical simulations with observations ) . with the exception of a single modest - resolution merger simulation presented in @xcite , the moving - mesh approach has not yet been used to simulate galaxy mergers . here , we present the first detailed study of idealised galaxy merger simulations performed using moving - mesh hydrodynamics . we present a small suite of simulations of equal - mass mergers simulated with both gadget-3 and arepo . to isolate differences caused by the different hydrodynamical solvers , we have kept all other components of the simulations namely , the gravity solver and the sub - resolution models for star formation , the interstellar medium ( ism ) , bh accretion , and agn feedback as similar as possible . other , more - comprehensive comparisons , such as the _ aquila _ @xcite and agora @xcite projects , allow multiple ingredients of the simulations to vary simultaneously , thereby yielding a general characterization of the systematic uncertainties due to different numerical methods and sub - resolution models . our goal is much more specific : we aim to isolate effects of the hydrodynamical solver from those caused by differences in the gravity solver or sub - resolution models , which is not readily possible in these other comparisons . in our work , we have chosen to use a ` standard ' implementation of sph , as incorporated in the gadget-3 code . we have not explored recent variants of sph that are designed to improve its reliability in some circumstances ( e.g. , * ? ? ? * ; * ? ? ? * ; * ? ? ? * ; * ? ? ? * ; * ? ? ? * ; * ? ? ? * ; * ? ? ? * ; * ? ? ? * ; * ? ? ? * ; * ? ? ? * ; * ? ? ? * ) for several reasons . the most important is that we wish to assess the reliability of previous simulations of gas dynamics in galaxy mergers that were performed using traditional formulations of sph . we also note that many of the sph modifications that have recently been proposed have not yet been tested under a wide range of conditions . it is presently thus still unclear which of the numerous modification should eventually be adopted in a new ` best sph ' variant . also , a number of problems with sph are still unresolved even in the most recent proposed revisions of the method . we discuss these issues further below . the remainder of this paper is organised as follows . in section [ s : methods ] , we describe the two different hydrodynamical methods used and the sub - resolution models for star formation , bh accretion , and supernova and agn feedback . in section [ s : results_no_bhs ] ( [ s : results_bhs ] ) , we compare gadget-3 and arepo simulations that do not ( do ) include bh accretion and agn feedback . section [ s : tests ] presents tests of different methods for treating bh accretion and agn feedback that we used to inform our choice of a fiducial treatment . in section [ s : discussion ] , we review some of the major issues with sph , discuss why sph works reasonably well for some applications but not others , and outline which previous work is likely to be robust to the hydrodynamical solver used and which may need to be reconsidered . section [ s : conclusions ] presents our conclusions . as noted above , we use two different codes , gadget-3 and arepo , because we wish to investigate differences in the outcome that are driven by variations in the method used to solve the hydrodynamics . gadget-3 uses sph @xcite , a pseudo - lagrangian method . in sph , the gas is discretised into particles , which are typically of fixed mass . the density field and other continuous quantities are calculated by taking the mean of the values of some number of nearest particles ( we use 32 ) weighted by the smoothing kernel . to derive the equations of motion , the lagrangian can be discretised and then the variational principle used @xcite . the formulation of sph used in gadget-3 is explicitly conservative even if the smoothing lengths vary @xcite . the advantages of modern sph include explicit conservation of mass , energy , entropy , and linear and angular momentum ; galilean invariance ; resolution that naturally becomes finer in regions of high gas density ; and accurate treatment of self - gravity . arepo adopts a novel version of the other primary technique used in astrophysical hydrodynamics , the finite - volume ( i.e. , eulerian ) grid - based approach . in traditional amr codes , cubic cells at fixed spatial locations are employed , and the cells are refined and de - refined according to some criteria ( e.g. , the mass of cells can be kept approximately constant ) . in arepo , the grid cells are not fixed in space ; rather , mesh - generating points are advected with the flow and a voronoi tesselation is used to generate an unstructured grid from the points . the euler equations are solved using a finite - volume approach . specifically , arepo uses a second - order unsplit godunov scheme with an exact riemann solver . advantages of this moving - mesh approach compared with sph include the following : it is better at resolving shocks and contact discontinuities ; it does not suppress fluid instabilities ; and the density field across a resolution element ( a cell ) can be reconstructed to first order ( unlike in sph , for which the density can only be constructed to zeroth order ) . both codes use the same tree - based gravity solver , which is a modified version of that used in gadget-2@xcite . collisionless particles are used to represent dark matter and stars ; these particles are assigned fixed gravitational softening lengths . in gadget-3 , the gas particles also have fixed gravitational softening . this treatment guarantees that , by using suitably small softening , one can have sufficient force resolution at all times . in contrast , the gravity solvers traditionally used in amr codes have force resolution that depends on the cell size and thus varies in time and space . even in collisionless simulations , this treatment can lead to the suppression of small - scale structure ( i.e. , dwarf galaxies ) if cells around forming haloes are not refined sufficiently early @xcite . because arepo treats the collisionless component using a tree - based method , it does not suffer from this problem . the quasi - lagrangian nature of the moving mesh cells also enables a superior treatment of gas self - gravity . arepo treats each cell as if the mass were concentrated at the cell centre and calculates the softened gravitational force using a softening that is of order the cell radius . for all components , gravitational interactions are softened using a cubic spline that has compact support ( e.g. , * ? ? ? full details of the treatment of self - gravity used in arepo can be found in section 5 of @xcite . in both gadget-3 and arepo , star formation and supernova feedback are implemented via the effective equation of state ( eos ) method of @xcite . only gas particles with density greater than a low - density cutoff ( @xmath1 @xmath2 ) are assumed to have an eos governed by the sub - resolution model . in the @xcite model , the ism is considered to consist of two phases in pressure equilibrium : cold dense clouds and a hot diffuse medium in which the cold clouds are embedded . the instantaneous sfr for each particle is calculated using a volume density - dependent kennicutt schmidt @xcite prescription , @xmath3 , with @xmath4 . star particles are spawned from gas particles or cells probabilistically according to their sfrs . feedback from supernovae is included as an effective pressurization of the ism such that the equation of state of the gas is stiffer than an isothermal eos . see @xcite and @xcite for further details . we stress that the sub - resolution models for star formation and feedback in gadget-3 and arepo are as similar as possible , which enables us to investigate differences caused solely by variations in the calculation of the hydrodynamics . as for the star formation and stellar feedback sub - resolution models , we have attempted to keep the sub - resolution models for bh accretion and agn feedback in the two codes as similar as possible . however , for reasons we discuss in section [ s : tests ] , the default sub - resolution models for bh accretion and feedback used in gadget-3 and arepo differ slightly . because we shall explore different numerical implementations of the bh accretion and feedback model , we describe the model in some detail here , but see @xcite for a more thorough presentation . each disc galaxy is initialised with a @xmath5 central sink particle that undergoes modified eddington - limited bondi lyttleton accretion @xcite . the accretion rate is the bondi lyttleton accretion rate multiplied by a dimensionless factor @xmath6 : @xmath7 where @xmath8 is the gravitational constant , @xmath9 is the bh mass , @xmath10 is the local gas density , @xmath11 is the local sound speed , and @xmath12 is the velocity of the bh relative to the gas . because we typically do not resolve the bondi radius , the parameter @xmath6 , which we set equal to 100 , is used to account for the fact that we underestimate the density at the bondi radius . in practice , the choice of @xmath6 matters only at early times because most of the mass growth occurs during times of eddington - limited accretion , and the mass of the final bh is insensitive to the value of @xmath6 . the eddington - limited accretion rate is @xmath13 where @xmath14 is the proton mass , @xmath15 is the thomson cross - section , and @xmath16 is the radiative efficiency , which is defined by @xmath17 where @xmath18 is the luminosity of the bh . we assume @xmath19 @xcite . the bh particles accrete mass at a rate @xmath20 in which the @xmath21 factor accounts for the rest - mass of the energy radiated away by the agn ( this factor was not included in the original @xcite treatment ) . in gadget-3 , gas particles are swallowed stochastically . in arepo , the more natural treatment is to continuously drain gas from the cell in which the bh is located . we have compared this treatment with one analogous to that in gadget-3 , in which cells are swallowed stochastically ; we found that for the mass and time resolutions of our simulations , the differences in the results are negligible ( but lower - resolution simulations can exhibit differences ) . to calculate the accretion rate given by equation ( [ eq : bondi ] ) , we must determine @xmath10 and @xmath22 near the bh . in gadget-3 , the sph estimates for these quantities are used . in arepo , we can adopt analogous estimates , but we can instead also employ the quantities for the cell in which the bh is located . in principle , the latter should better represent the properties of the gas around the bh . however , the individual cell values can also be more noisy , so it is not clear a priori which is preferred . we compare the results obtained using the different treatments in section [ s : accretion_test ] . based on those tests , we chose to use the cell density and sph - like estimate of the sound speed in our default treatment . one potential issue when calculating the accretion rate using either the sph density estimate or the cell density is that as the bh consumes or expels the nearby gas , the region used for the density estimate can grow in size . consequently , the bh can continue to accrete gas from larger and larger scales , whereas in reality , bh accretion should terminate once the gas near the bh is consumed or expelled . in sph , this problem is unavoidable unless one reduces the number of neighbours used to estimate the density ( which is problematic because the noise in the density estimate will be increased significantly ) or a scheme that is more complicated than the simple bondi lyttleton approach is employed ; one consequence is that lower - resolution simulations can sometimes exhibit more bh growth ( e.g. , * ? ? ? * ) . in arepo , this problem may be avoided by preventing cells near the bh from becoming too large . we do this by forcing the cells within some radius of the bh to be refined if they have a radius greater than some maximum value . we explore the effects of this refinement in section [ s : refinement_test ] . based on our tests , we decided to force cells within 500 pc of a bh to have a maximum size of 50 pc . bh accreting from a gas with sound speed @xmath23 km @xmath24 . thus , an even stricter refinement criterion could be justified , but because of the resolution of the simulations , there is no structure on smaller scales . ] finally , because the initial bh particles are similar in mass to the stellar and dark matter particles , two - body interactions can cause the bh to stray from the centre of the potential well . however , in reality , dynamical friction would cause the bhs to rapidly sink to the potential minimum . thus , we pin the bh to the halo potential minimum . for this reason , we also neglect the @xmath12 term in the denominator of equation ( [ eq : bondi ] ) . we also include a simple model for thermal feedback from the agn @xcite . the bh particles deposit some fraction @xmath25 of their luminosity ( as thermal energy ) to the surrounding gas . we use @xmath26 because in previous gadget-2 simulations , this value yielded an @xmath27 relation normalization consistent with that observed @xcite . we scale the number of gas particles over which we distribute the feedback energy with resolution such that the total mass of the particles is constant . as we demonstrate in section [ s : fb_test ] , this scaling minimizes the resolution dependence of the sub - resolution model . it is desirable to have sub - resolution models that do not depend significantly on numerical resolution , at least as long as the physics that the sub - resolution treatment is meant to represent remains inaccessible . otherwise , the problem is not well - posed numerically and the interpretation of the sub - resolution model becomes unclear . the initial disc galaxies are created following the procedure described in @xcite . the galaxies consist of dark matter haloes described by a @xcite profile with virial velocity @xmath28 and concentration @xmath29 , an exponential stellar disc with scalelength @xmath30 and scaleheight @xmath31 , an exponential gaseous disc with scalelength @xmath32 and scaleheight determined by requiring the disc to be rotationally supported , and a bulge described by a @xcite profile with scalelength @xmath33 . note that , unlike in @xcite , the gaseous and stellar discs do not have the same scalelength ; rather , the gaseous discs can be significantly more extended than the stellar discs . unlike sph , grid - based methods can not treat empty space . thus , for the arepo simulations , we must add a background grid of low - density cells to the initial conditions used for the gadget-3 simulations such that the entire simulation volume has positive gas density . see section 9.4 of @xcite for details of how the background mesh is added . our intention is not to present a comprehensive suite of merger simulations that addresses the full parameter space but rather to directly compare the results obtained using arepo and gadget-3 for a set of simulations based on galaxy models that are representative of a variety of actual galaxies . for definiteness , we use galaxy models that are similar to those of @xcite and the same merger parameters as @xcite . two of the isolated disc galaxies are intended to be milky way ( ) and small magellanic cloud ( ) analogues . the other two represent a dwarf starburst ( ) and a redshift @xmath34 disc galaxy ( ) . thus , the initial discs capture much of the diversity of real disc galaxies , but the sampling is by no means complete . the properties of the disc galaxies are given in table [ tab : galaxy_models ] . we simulate each disc galaxy with two different resolutions , which are specified in table [ tab : resolutions ] . we evolve each disc in isolation for 3 gyr . for the mergers , two identical disc galaxies are placed on parabolic orbits with initial separation and pericentric passage distance as specified in table [ tab : orb_params ] . as in @xcite , two orbits , the and orbits of @xcite , are used . for the orbit , the directions of the spin axes of the discs given in spherical coordinates are @xmath35 . for , @xmath36 . note that neither orbit is coplanar ; thus , our general conclusions should be relatively insensitive to orbit ( mergers with perfectly coplanar orbits , which are of course highly unlikely in nature , can exhibit pathological behavior that is not characteristic of the behaviour for other orbits ) . because our focus is to compare the results of otherwise - identical gadget-3and areposimulations , two orbits is sufficient ; a comprehensive suite of simulations would ideally include a much wider variety of orbits ( see , e.g. , * ? ? ? * ; * ? ? ? we run each merger simulation for @xmath37 gyr , depending on the simulation , which is sufficient time for the galaxies to coalesce . we typically simulate each merger at two different resolutions ( see table [ tab : resolutions ] for details ) . furthermore , to strengthen our conclusions regarding whether any resolution dependence is systematic , we performed the simulation with bh accretion and agn feedback included at a third , higher resolution . except where otherwise noted , we always plot the results of the highest - resolution run . we first present a comparison of gadget-3 and arepo simulations for which we disabled the bh accretion and agn feedback treatments in the codes . gyr , the absolute values of the sfrs for , , , and are @xmath38 , @xmath39 , @xmath40 , and @xmath41 , respectively . the sfhs are almost identical regardless of the resolution or code used.,title="fig : " ] + gyr , the absolute values of the sfrs for , , , and are @xmath38 , @xmath39 , @xmath40 , and @xmath41 , respectively . the sfhs are almost identical regardless of the resolution or code used.,title="fig : " ] + a comparison of the star formation histories ( sfhs ) and cumulative stellar mass formed versus time for the isolated disc simulations is shown in fig . [ fig : iso_nb_sfhs ] . the solid ( dashed ) lines indicate the higher ( lower ) resolution runs , and blueish ( reddish ) colours indicate arepo ( gadget-3 ) simulations . the isolated discs ( fig . [ fig : iso_nb_sfhs ] ) evolve in the expected manner : after some initial settling into equilibrium , as the gas is consumed ( in these idealised simulations , no additional gas is supplied during the simulations ) , the sfr decreases and the stellar mass formed increases . for the simulations , there are some minor resolution- and code - dependent differences in the sfhs at @xmath42 gyr . the different runs are almost identical . the simulations exhibit minor resolution - dependent differences in the sfhs throughout the simulation . for the model , there are minor differences in the sfrs of the gadget-3 and arepo simulations at @xmath43 gyr . in all cases , the curves of cumulative stellar mass formed versus time are almost indistinguishable . these results demonstrate that for the isolated disc simulations , the two codes agree very well and the simulations are converged with respect to particle number ( at least in terms of their sfhs ) . [ fig : merger_nb_sfhs ] shows the sfhs for the merger simulations without bh accretion and agn feedback . as expected from much previous work , the mergers exhibit the following generic evolution . the sfr initially oscillates for a short time as the discs settle into equilibrium . then , there is a slight elevation when the discs are at first pericentric passage , but the bulges prevent the discs from becoming very unstable . as the discs approach final coalescence , in many cases , strong tidal torques drive gas into the nucleus and fuel a starburst . however , the strength and shape of the starburst depend on the progenitor properties and orbit . after the strong starbursts ( , , , and ) , the sfr decreases to the pre - merger level or significantly below it . note that the decrease is driven solely by gas consumption and shock - heating of the gas because these simulations do not include agn feedback . in the other merger simulations , in which a strong starburst is not induced , the sfr can remain elevated for the duration of the simulation ( see especially and ) . as can be inferred from fig . [ fig : merger_nb_sfhs ] , the agreement for the merger simulations is also excellent , although there are more noticeable differences than for the isolated disc cases . the most prominent difference is that for , , , and , the time at which the starburst occurs depends on resolution . this is likely caused by resolution - dependent variations in the hydrodynamical drag experienced by the galaxies as they collide , which can slightly alter the merging time - scales . however , for a given resolution , the sfhs during the starburst are similar for the two codes . the most significant , albeit still relatively minor , differences between the arepo and gadget-3 simulations occur after the starbursts , e.g. , at @xmath44 , 2.9 , and 2.6 gyr for , , and , respectively . the sfhs do not vary systematically depending on the code used , e.g. , for the post - starburst phase of , the arepo sfrs tend to be lower , whereas for , the opposite is true . as for the isolated discs , the differences in the cumulative stellar mass formed versus time are negligible . + the agreement of the sfhs of the gadget-3 and arepo simulations for both the isolated disc and merger simulations indicates that the differences in the numerical schemes do not dramatically affect the global evolution of the simulated systems . however , it is possible that the detailed properties of the simulated mergers differ despite the good agreement for the integrated quantities , and this is indeed the case . [ fig : nb_gas_comparison ] presents gas surface density plots for four different simulation snapshots . ( the interested reader is invited to visit the aforementioned url to view animations that compare the time - evolution of the gas surface density for the highest - resolution runs of all simulations presented in this work . ) the top row shows the arepo result , and the bottom row shows the gadget-3 result for the same snapshot and resolution . the first column shows the isolated disc at @xmath45 gyr . the arepo and gadget-3 results are qualitatively similar , but the gas distribution appears slightly smoother in the arepo simulation and the orientation of the bar differs . the second column shows the merger at @xmath46 gyr , which is near final coalescence . again , the morphologies are very similar , but the spatial extent of the tidal tails differs slightly . the third column shows the merger near the peak of the starburst ( @xmath47 gyr ) . here , the gas distribution is again smoother in the arepo run , and the gas morphology of the nuclear region differs quite dramatically . finally , the fourth column shows the merger at @xmath48 gyr ( @xmath49 gyr after the peak of the starburst ) . here , the differences are the most dramatic : the nuclear disc that has re - formed is oriented edge - on in the arepo simulation but face - on in the gadget-3 simulation this is likely driven by the stochastic nature of the torques acting on the gas that accumulates in the centre of the remnant ( e.g. , * ? ? ? furthermore , the gadget-3 simulation features a clumpy , extended hot halo that is not present in the arepo simulation . in general , the arepo morphologies tend to be smoother than those yielded by gadget-3 , and the clumps that are often observed in the gadget-3 simulations ( and are spurious ; e.g. , @xcite ) are not present in the arepo simulations . the differences between the two codes are most pronounced during the starburst and post - starburst phases . we discuss the physical reasons for these differences in detail in section [ s : reasons_for_agreement ] . we shall now compare the gas phase structure in the gadget-3 and arepo simulations . animations showing the evolution of the gas phase structure for all simulations are available at the aforementioned url . for brevity , we will only discuss the general trends and present an illustrative example . throughout the simulations , the evolution of the gas phase structure is very similar in the arepo and gadget-3 simulations when bh accretion and agn feedback are not included . given the results presented above , this result is not surprising : if the phase structure differed significantly , then the sfhs would not agree so well . as for the gas morphology , the differences are more pronounced in the starburst and post - starburst periods of the simulations . specifically , the arepo simulations tend to exhibit less low - density , hot halo gas . in both the arepo and gadget-3 simulations , a hot halo forms when gas is shock - heated during final coalescence of the discs . in the arepo simulations , however , the hot halo gas cools more effectively for the reasons discussed in section [ s : reasons_for_agreement ] . [ fig : nb_pd_comparison ] demonstrates this difference . this figure shows example phase diagrams for the post - starburst phase ( @xmath48 gyr , @xmath50 myr after the starburst ) of the merger simulation performed using arepo ( left ) and gadget-3 ( right ) . note that these correspond to the same simulation and output time as the gas surface density plots shown in the fourth column of fig . [ fig : nb_gas_comparison ] . in the arepo simulation , there is less hot halo gas . the enhanced cooling of hot halo gas in the arepo simulation explains why the sfr is somewhat higher in the arepo simulation at this time ( see fig . [ fig : merger_nb_sfhs ] ) , which is also the case in some of the other merger simulations ( e.g. , ) . we now compare simulations that include bh accretion and agn feedback . here , we use the default accretion and feedback schemes discussed above , but we explore the implications of different choices in section [ s : tests ] . + + + + + + + + + + + fig . [ fig : iso_sfhs ] shows the sfhs and cumulative stellar mass formed versus time for the isolated disc simulations . blue ( red ) indicates arepo ( gadget-3 ) simulations with bh accretion and agn feedback ; the corresponding simulations without bh accretion and agn feedback are indicated in cyan ( magenta ) for comparison . solid ( dashed ) lines indicate higher ( lower ) resolution simulations . in general , for the isolated disc simulations , the agreement among the sfhs for the two codes and different resolutions is still relatively good , but the differences are clearly more significant than when bh accretion and agn feedback are disabled . for the case ( first row of fig . [ fig : iso_sfhs ] ) , the arepo simulations tend to have slightly higher sfrs , but the differences between the simulations sfrs are less than @xmath51 dex at all times . the cumulative stellar mass formed is indistinguishable . the sfhs for the simulations ( second row ) agree similarly well , and the cumulative stellar mass formed is nearly the same . for the simulations ( third row ) , the sfrs differ by as much as @xmath52 dex , but the resolution - dependent variations are as significant as those between the codes and the differences for simulations that vary only in whether they include bh accretion and agn feedback . in this case , the total stellar mass formed over the course of the simulations ( 3 gyr ) is @xmath51 dex less in the gadget-3 simulations with agn feedback , but all other simulations agree very well . finally , the sfrs of the isolated disc simulations ( fourth row ) can vary by as much as @xmath53 dex , and the primary cause of the difference is whether bh accretion and agn feedback are included ( in such simulations , the sfrs are systematically lower , as expected ) . however , the stellar mass formed is the same to within @xmath54 dex . figs . [ fig : e_sfhs ] and [ fig : f_sfhs ] show the sfhs and cumulative stellar mass formed versus time for the -orbit and -orbit merger simulations , respectively . for brevity , we will not discuss each panel individually but rather highlight general trends . for a given progenitor combination and orbit , the shapes of the sfhs are qualitatively similar for both codes and all resolutions ( except in the cases for which agn feedback significantly impacts the post - starburst sfr ) . however , there are significant quantitative code- and resolution - dependent differences . for many of the simulations ( i.e. , , , , , and ) , the sfhs are almost identical up to final coalescence . if a strong starburst is triggered , the precise time and amplitude of the starburst can vary depending on the code and resolution . in some but not all cases ( and , in particular ) , inclusion of agn feedback causes the post - starburst sfr to decrease more rapidly compared with the corresponding simulations without agn feedback . although the sfhs differ significantly at some times , the cumulative stellar mass formed versus time is often very similar for the different codes and resolutions , as we saw above for the simulations without agn feedback and the isolated disc simulations with agn feedback . in most cases , the cumulative stellar mass formed differs by less than @xmath51 dex at all times . the merger simulations exhibit the most significant differences in the cumulative stellar mass formed : when agn feedback is included , the cumulative stellar mass formed in the arepo simulations is @xmath55 dex greater than in the gadget-3 simulations . + + + + another quantity of interest is the bh mass versus time because differences in the bh masses would alter the strength of the agn feedback and thereby influence whether or not the merger remnants lie on the @xmath27 relation @xcite . it is natural to expect that the bh mass evolution is more code- and resolution - dependent than the sfh because the bh growth depends sensitively on the gas conditions in the nuclear region(s ) and could thus be affected by small - scale variations that would not significantly alter the integrated sfh . [ fig : iso_bh_mass ] shows the bh mass versus time for the isolated disc simulations . in the and simulations , the bhs grow by only a modest amount ( @xmath56 and @xmath57 dex , respectively ) over the 3.0 gyr of the simulations . in the and simulations , the bhs rapidly increase in mass by 2 - 3 orders of magnitude . this strong bh growth in the absence of a merger is driven by bar instabilities , which is clear from examination of the gas morphologies . the bh growth terminates once the gas near the bh is consumed or expelled . for a given code , the final bh masses in the isolated disc simulations differ with resolution by @xmath58 dex . however , the code - dependent variations can be more significant . in particular , note that the final bh masses in the arepo simulations are a factor of @xmath59 greater than those in the gadget-3 simulations . the bh mass evolution for the merger simulations is shown in fig . [ fig : merger_bh_mass ] . in these plots , the total mass of all bhs recall that each progenitor disc is seeded with a bh is plotted . for a given progenitor , the bhs grow significantly more in the merger simulations than in the isolated disc simulations . the merger exhibits the weakest growth ( @xmath60 dex ) , and the merger simulations exhibit the strongest bh growth ( almost four orders of magnitude ) . as for the sfhs , the code- and resolution - dependent differences amongst the bh masses in the merger simulations are more significant than for the isolated disc simulations . in many but not all examples , the bh masses are greater in the arepo simulations than in the gadget-3 runs . the resolution - dependent variations ( which are at most @xmath40 dex and usually significantly less ) are typically less than the code - dependent differences ( for a given resolution , these can be as great as an order of magnitude ) , and the resolution dependence is not systematic . consequently , for a given code , the final bh masses should be considered uncertain by as much as a factor of a few . this reflects the high degree of non - linearity in the feedback - regulated bh growth . any small variation in the local gas conditions at the bh s position can influence its exponential growth rate and hence become strongly amplified with time . note also that the bh accretion histories can differ significantly depending on the resolution and code ; thus , during the merger , the bh masses at a given time can differ more significantly than the final bh masses ( i.e. , the bh masses after the bh growth has terminated , which can occur at different times for different resolutions and codes ) . interestingly , for the merger , which was simulated at three resolutions , the bh mass evolution in the two higher - resolution ( resolutions r3 and r4 ) simulations performed with a given code is almost identical , but the @xmath49 dex difference in the final bh masses yielded by the two different codes persists . thus , it is possible that the bh masses yielded by a given code would converge if all simulations were performed at even higher resolution , but we have not performed such simulations because of the computational expense and because the code - dependent differences , which are the focus of this work , remain even for the highest - resolution simulations . + fig . [ fig : gas_comparison ] shows example gas surface density plots for the arepo ( top row ) and gadget-3 ( bottom row ) simulations with bh accretion and agn feedback . as for the simulations without bh accretion and agn feedback , the morphologies are qualitatively similar , but the details differ . furthermore , as for the sfhs , the code - dependent differences in the gas morphologies are more significant when bh accretion and agn feedback are included . the first column shows the isolated disc at @xmath61 gyr . at this time , a bar is evident in the arepo simulation and much of the gas in the central region has been consumed because of the bar instability . in the gadget-3 simulation , a bar is not evident ; rather , a large , irregularly shaped cavity , which contains some small clumps of dense gas , has formed . part of the reason for the significant differences between the arepo and gadget-3 results is that in the arepo simulations , the bh can only consume gas that is within 50 pc , whereas in gadget-3 , the region from which gas is accreted grows as the gas near the bh is depleted because a fixed number of neighbours is used to calculate the sph density estimate . note , however , that generically , holes tends to form around the bhs for the following physical reason : in the quiescent state that is eventually reached in isolation or at the end of a merger ( when the bh growth has effectively shut off ) , a small bubble of hot , low - density gas around the bh is created by the pressure that is sustained in our feedback model by the residual accretion . the second column shows the merger simulation at @xmath62 gyr ( after first pericentric passage ) . the results of both codes exhibit extended , smooth tidal features , and the morphologies are almost indistinguishable . this column exemplifies the good agreement that is characteristic of the pre - starburst phase of the merger simulations . the third column of fig . [ fig : gas_comparison ] shows the merger near the peak of the starburst ( @xmath63 gyr ) . some of the filamentary structure is similar in the arepo and gadget-3 simulations , but the detailed morphologies differ significantly . as noted above , the gadget-3 result exhibits many spurious clumps of gas that are not present in the arepo simulation . finally , the fourth column shows the merger at @xmath48 gyr , @xmath50 myr after the starburst . note that for comparison , this is the same simulation and time as shown in the fourth column of fig . [ fig : nb_gas_comparison ] , except that bh accretion and agn feedback are included here . for a given code , the gas morphologies are similar to those of the simulations for which bh accretion and agn feedback were not included ( fig . [ fig : nb_gas_comparison ] ) . one notable difference is that in the gadget-3 simulation , the gas disc is less pronounced and the hot halo is more prominent . once again , the gadget-3 simulation features an extended halo of hot gas and spurious clumps , both of which are not present in the arepo simulation . note that in this example , the code - dependent differences in the gas morphologies are more significant than those caused by the inclusion of agn feedback . this result is a counterexample to the conclusion of @xcite regarding the effects of various star formation and stellar feedback models compared with differences between codes and implies that it is highly desirable to use the most accurate hydrodynamical solver possible . as for the simulations without bh accretion and agn feedback , the gas phase structure in the gadget-3 and arepo simulations is similar in the pre - starburst phase but can differ significantly during and after the starburst . again , we only present one example to illustrate the characteristic differences here , but the interested reader can visit the aforementioned url to examine the evolution of the gas phase structure for all simulations . [ fig : pd_comparison ] shows gas phase diagrams for the simulations with bh accretion and agn feedback at the same time as in fig . [ fig : nb_pd_comparison ] ( @xmath48 gyr , @xmath64 myr after the peak of the starburst and agn activity ) . for a given code , the inclusion of agn feedback causes there to be more gas in the hot halo and correspondingly less gas on the eos ( the thin line in the lower - right corner ) . as for the simulations that did not include agn feedback , the gadget-3 simulation features more hot halo gas than the arepo simulation , in which the hot halo gas cools more efficiently . here , we present various tests that demonstrate the effects of the different treatments of bh accretion and agn feedback discussed above . we use the analogue as one test case . whereas for this simulation , the differences in the results are small in an absolute sense , they are systematic and can have more significant effects in other simulations . we chose to use the analogue as a test case because the sfh and bh growth are comparatively simple ; thus , differences can be more easily understood . furthermore , the bh grows very little ( @xmath65 , the seed mass , throughout the simulation ) and should have a negligible effect on the sfh of the galaxy . thus , the ` no bh ' case can be used as the baseline with which to compare the other runs ; ideally , the sfhs should be the same for the bh and ` no bh ' cases . we also investigate the effects of different bh accretion and agn feedback for the merger simulation . in this significantly more complicated case , the interpretation of the comparisons among the different treatments of bh accretion and agn feedback is less straightforward , but many of the effects observed for the case are also observed here . the tests presented here justify our fiducial choices for the bh accretion and feedback implementations and demonstrate some important numerical effects that are not always appreciated in the literature . + , but for the merger . the results are almost independent of the @xmath66 value for the higher resolution because the resolution around the bh is sufficiently high without the forced refinement . the final bh masses for the two resolutions agree better when @xmath67 pc , but this agreement does not indicate better convergence . see the text for details.,title="fig : " ] + , but for the merger . the results are almost independent of the @xmath66 value for the higher resolution because the resolution around the bh is sufficiently high without the forced refinement . the final bh masses for the two resolutions agree better when @xmath67 pc , but this agreement does not indicate better convergence . see the text for details.,title="fig : " ] in sph , as the gas density in the immediate vicinity of a bh decreases because of gas consumption and expulsion , the radius over which the density is calculated increases because the number of neighbours used for the density estimate and the particle masses are fixed . thus , in some situations , the bh accretion rate can be overestimated and gas fuels the bh from unphysically large scales ; this is especially problematic in cosmological simulations , for which the resolution is often not better than a kiloparsec . in arepo , the standard cell refinement scheme attempts to keep cell masses comparable ; thus , a similar effect , in which cells near the bh grow large , can occur . however , unlike in sph , we can overcome this potential problem by preventing cells near the bh from becoming too large and using the gas density of the cell that contains the bh to calculate the accretion rate . consequently , if the gas density in the vicinity of the bh decreases , the accretion rate decreases concomitantly . [ fig : refinement_test ] demonstrates the effects of limiting the maximum cell size near the bh particle for the isolated disc case . we force cells located within 500 pc of the bh to be refined if their size is greater than some value @xmath66 . we show the results for @xmath68 and 200 pc in fig . [ fig : refinement_test ] . the top panel indicates that the choice of @xmath66 has no effect on the sfh , and the slight difference between the different resolutions at late times is independent of @xmath66 . however , the growth of the bh differs systematically . for both resolutions , the bh grows more when @xmath67 pc because of the effect described above , and the consequences are more severe for the lower - resolution simulation . when @xmath68 pc , the final bh mass is slightly less and the two different resolutions agree perfectly . [ fig : refinement_test_merger ] shows the results of changing @xmath66 for the merger simulation . in this case , the sfh is better converged when a maximum cell when @xmath68 pc . for the higher - resolution simulations , the bh growth history is unaffected by the choice of @xmath66 . thus , for this resolution ( @xmath69 pc ) , the refinement near the bh is sufficiently high even without forcing refinement . interestingly , the final bh mass of the lower - resolution simulation with @xmath67 pc agrees better with that of the higher - resolution simulations than when @xmath68 pc . in this case , allowing larger cells near the bh by setting @xmath67 pc serves to mitigate some of the resolution dependence : in the lower - resolution simulation with @xmath68 pc , during the starburst at final coalescence , the central gas density is greater than in the higher - resolution runs . consequently , the bh grows more rapidly during that time . using @xmath67 pc results in a decreased central gas density and thus less - massive bh . however , this result should not be taken as an indication that the bh mass is better - converged when @xmath67 pc : if the bh mass were truly converged , the @xmath68 pc simulation should agree at least as well because in this simulation , the number of resolution elements is greater than or equal to that of the lower - resolution simulation . + + + + to calculate the bh accretion rate using equation ( [ eq : bondi ] ) , we require the gas density and sound speed near the bh . in arepo , both quantities can be estimated in a manner analogous to that in sph , i.e. , by averaging over some number of nearest neighbour cells ( typically , 32 ) . however , it is also possible to use the gas density and sound speed for the cell in which the bh is located . in principle , the cell values should be more representative of the conditions near the bh and thus ensure a more physical calculation of the accretion rate , but they may be too noisy to be useful . we will explore the effects of these choices now . [ fig : accretion_test ] demonstrates the variations that arise from using different methods to calculate the bh accretion rate in arepofor the isolated disc case . the gadget-3 results are shown for comparison . in all cases , the mass over which the feedback energy is distributed is kept constant , i.e. , the number of cells or particles over which the energy is distributed is increased as the cell mass is decreased . furthermore , cells within 500 pc of the bh are forced to have size less than @xmath68 pc . the differences in the sfh ( upper - left panel ) for the various treatments are small and comparable with the resolution - dependent differences . the bh growth history , in contrast , can be affected significantly ; the upper - right panel indicates that the final bh mass can differ by as much as 0.4 dex , and the effect in merger simulations can be even larger . the bh grows most in arepo when the sph density and sound speed are used ( the blue curve ) . the reason is that , as shown in the lower - left panel , in arepo , the sph density estimate is systematically greater than the cell density ( and similar to the gadget-3 density ) . the gadget-3 simulations ( red curve ) exhibit the next highest bh growth ; the reason that they feature less growth compared with the analogous arepo simulations is that the sph sound speed estimate ( lower - right panel ) is systematically higher in gadget-3than arepo . when the cell density is used , the bh grows less because the cell density is systematically lower than the sph density . the growth is most suppressed when the cell sound speed is used ( cyan curve ) because the cell sound speed is systematically higher than the sph sound speed . furthermore , the cell sound speed is very noisy , with variations of @xmath70 km@xmath24 from the mean value , and the cell density is significantly more noisy when the cell sound speed is used to calculate the bh accretion rate ( compare the cyan and green curves in the lower - left panel of fig . [ fig : accretion_test ] ) . fig . [ fig : accretion_test_merger ] compares the different bh accretion rate treatments for the merger simulation . as for the isolated disc case , the bh grows most when the sph estimates for the gas density and sound speed are used to calculate the accretion rate . using the cell density rather than the sph estimate results in a very similar final bh mass . furthermore , using both the cell density and sound speed results in the smallest final bh mass of the three lower - resolution simulations because the cell sound speed can be more than a factor of three greater than the sph estimate . although the differences in the bh masses caused by varying the sub - resolution model accretion rate calculation are significant ( @xmath52 dex ) , they are less than the differences between the two resolutions for a fixed code and those between the two codes for a fixed number of particles / cells . based on these results , we chose to use the cell density for our production runs because this is more representative of the density near the bh than is the sph density , yet the two have comparable amounts of noise . however , we chose to use the sph estimate for the sound speed because the cell value is too noisy . + + + + the manner in which the agn feedback energy is distributed can also affect the bh growth . as explained in section [ s : bh_fb_method ] , our preferred method is to keep the mass over which the feedback energy is distributed ( the ` feedback mass ' ) constant . thus , for higher - resolution runs , in which the particle / cell mass is lower , the number of particles / cells over which the feedback energy is distributed should be increased . however , this is not the only possible approach ; an alternative approach is to keep the number of particles / cells over which the feedback energy is distributed fixed . but , as we will show now , this causes a stronger resolution dependence , which is undesirable for a sub - resolution model . [ fig : fb_test ] demonstrates the differences between these two choices for the isolated disc case . as in the previous test , there are minor variations in the sfhs at late times , but the magnitude of the difference is similar to that caused by varying the resolution . however , varying the method for distributing the feedback energy causes significant and systematic changes in the bh growth history . first , note that the dashed and solid blue lines , which correspond to the lower - resolution arepo simulation and the higher - resolution simulation in which the feedback mass is kept constant , agree perfectly . the corresponding curves for the density and sound speed are ( necessarily ) in good agreement . in contrast , the arepo run in which the number of cells over which the feedback is distributed is kept constant ( the cyan line ) has systematically less bh growth . the reason for this discrepancy is that the latter simulation has systematically lower gas density ( bottom left ) and higher sound speed ( bottom right ) because the agn feedback heats a smaller mass of gas compared with the lower - resolution run . consequently , the steady state that is reached is at a higher temperature and lower density . the analogous effect can be observed for gadget-3 , but in this case , there is still some resolution dependence even when the feedback mass is kept constant . [ fig : fb_test_merger ] shows a comparison of the different methods for distributing the feedback energy for the merger . for this model , the sfh differs most significantly in the areporun in which an equal cell number is used ; in this case , star formation is quenched more effectively by the agn after the starburst than in any of the other simulations , and the sfh in this regime differs most significantly from that of the lower - resolution run . for the gadget-3simulations , the final bh mass of the lower - resolution run agrees slightly better with that of the higher - resolution run in which the mass over which the feedback energy is distributed is kept constant , as was observed for the case . however , for the areposimulations , the final bh mass in the higher - resolution run in which the number of cells over which the feedback energy is distributed is kept constant agrees better with the final bh mass of the lower - resolution simulation . the reason for this behaviour is that in the equal - mass run , the initial stage of rapid bh growth ( @xmath71 gyr ) is terminated slightly earlier and at a slightly lower bh mass than in the equal - cell - number run , primarily because the rapid decline in the gas density occurs earlier in the equal - mass run . consequently , during the final - coalescence phase , in which the bh undergoes eddington - limited accretion , the less - massive bh grows less . this unexpected behaviour demonstrates the difficulty of predicting the effects of the sub - resolution model in the significantly more complex context of galaxy mergers and highlights the aforementioned conclusion that the final bh masses should only be considered robust to within a factor of a few . furthermore , the difference in the final bh masses of the two higher - resolution simulations is less significant than the difference between either of these masses and that of the lower - resolution run . the purpose of the sub - resolution model is to encapsulate in a simple manner physics that is not included in the simulations . the implicit assumption in our model is that the feedback energy deposited by agn is thermalised over some physical scale , and this physical scale does not and should not depend on the resolution of our simulations . thus , in our production runs , we kept the feedback mass constant . in earlier work @xcite , we have compared results from gadget-3and arepoand , in some cases , identified significant differences that we attributed to numerical issues with the conventional formulation of sph . in the present paper and in studies of the ly-@xmath6 forest ( e.g. , * ? ? ? * ; * ? ? ? * ) , it has been found that sph can produce results in certain regimes that agree well with grid - based codes . to understand this situation , we first briefly review the primary limitations of the sph approach that have become clear in recent work . this then allows us to discuss why sph can be expected to work reasonably well in some applications but not in others . finally , we comment on what this implies for the numerical robustness of different types of previous work . the traditional formulation of sph used in this work has been and still is being used for many astrophysical simulations . it is thus important to understand which of these results may be influenced by numerical artefacts of the type discussed below . whereas we acknowledge that there have been impressive efforts to address at least some of these issues ( e.g. , * ? ? ? * ; * ? ? ? * ; * ? ? ? * ; * ? ? ? * ; * ? ? ? * ; * ? ? ? * ; * ? ? ? * ; * ? ? ? * ; * ? ? ? * ; * ? ? ? * ) , we feel that there remain many lingering misconceptions about sph that muddle the interpretation of the reliability of simulations performed even with updated versions of this algorithm . the transition from the continuum equations of fluid dynamics to the discrete form used by sph involves a two - step procedure ( e.g. , * ? ? ? first , the exact fluid quantities are replaced by smoothed versions via a convolution with the smoothing kernel . second , the integral forms of these convolutions are replaced by discrete sums over the sph particles such that they can be evaluated numerically . the error made in the discretization step depends not on the number of sph particles , @xmath72 , but instead on the number of neighbours in the discrete sums , @xmath73 . if , as is typically done , @xmath73 is held fixed as @xmath72 is increased , there will be a _ constant _ source of error in the local estimates even as the resolution of the simulation is nominally increased @xcite . consequently , local fluid estimates are often noisy and are not guaranteed to approach their continuum values as @xmath72 is increased . this source of noise , although small , may have a significant impact on flows in which the energy content is dominated by internal energy rather than kinetic or gravitational energy . the noise is particularly strong in gradients of interpolated quantities , most notably in the pressure force @xcite . furthermore , if @xmath72 is increased without simultaneously increasing @xmath73 , the solution may asymptote to a fixed result that is different from the true solution because of this constant source of error @xcite . tests by @xcite demonstrated that conventional implementations of sph do not accurately describe jumps in physical quantities across contact discontinuities because the pressure effectively becomes multi - valued at the interface , thereby resulting in artificial repulsive forces that act as a macroscopic surface tension . if two fluid phases in pressure equilibrium shear relative to one another , the spurious surface tension inhibits the proper growth of kelvin - helmholtz instabilities and the two phases will not mix together correctly . instead , the colder , more dense phase can fragment into clumps that retain their identity because of the presence of the spurious surface tension ( e.g. , * ? ? ? * ; * ? ? ? various new formulations of sph have shown promise in alleviating this problem ( e.g. , * ? ? ? * ; * ? ? ? * ; * ? ? ? * ; * ? ? ? * ; * ? ? ? * ) ; thus , we will not dwell on it here . however , we mention it for completeness and because nearly all previous studies of galaxy mergers using sph have employed traditional formulations of sph that are susceptible to this surface tension issue . in most implementations of sph , the mass continuity equation is not integrated in detail ; instead , estimates of the fluid density at any given simulation time are made using kernel interpolation ( e.g. , * ? ? ? * ) applied to the current particle positions . partly for this reason , sph is often referred to as a ` lagrangian ' method . however , sph is only ` pseudo - lagrangian ' because the particle mass is fixed in time and particle shapes are not allowed to become distorted arbitrarily by the flow ( see @xcite for a detailed discussion ) . consequently , sph can not properly describe fluid mixing on small scales , whereas in arepo , mixing is not suppressed because the implied mass exchange between cells is computed correctly according to the continuity equation . in essentially all widely used sph codes , shocks are captured using some form of artificial viscosity . ideally , this viscosity should operate only in and near shocks because it can have unintended consequences for the properties of the flow in other regions if it is not accurately controlled ; for example , it can cause spurious cooling in cosmological applications ( e.g. , * ? ? ? moreover , the action of the viscosity within shocks is to locally broaden them over several smoothing lengths , thereby degrading the effective spatial resolution in shocked regions . many grid - based codes , including arepo , instead treat shocks by solving the riemann problem across all cell - cell interfaces . this treatment has several advantages because it implies that such codes minimize the additional source of unphysical diffusion that would arise from artificial viscosity and because shocks can be spatially resolved more precisely than in sph . from the discussion in section [ s : limitations_of_sph ] , we can now provide arguments as to why sph yields results that are reliable in some situations and why it fails in others . for definiteness , we consider four applications : ( 1 ) idealized tests of driven supersonic and subsonic turbulence , ( 2 ) the intergalactic medium ( igm ) , ( 3 ) gas accretion on to galaxies , and ( 4 ) starbursts and agn activity in galaxy mergers , as described in this paper . @xcite compared gadget-3and arepofor idealized simulations of isothermal turbulence in periodic boxes subject to large - scale forcing . their tests demonstrate that the traditional formulation of sph , as incorporated in gadget-3 , does not yield a proper cascade of energy to small scales when the turbulence is subsonic , as illustrated in their fig . however , otherwise identical simulations performed with arepoand the navier - stokes version of arepo@xcite produced a well - developed turbulence spectrum , not only in the inertial range but also through the dissipational range . in contrast , sph was found to perform significantly more reliably for supersonic turbulence @xcite , yielding results that are in reasonable agreement with arepoindependent of the motion of the mesh . these apparently contradictory conclusions can be readily explained by the limitations discussed above in section [ s : local_noise ] and section [ s : shock_capturing ] . in the supersonic limit , the energy density of the fluid is dominated by kinetic energy . although noise is still present in the local fluid quantities ( see section [ s : local_noise ] ) , its influence is subdominant in this regime . similarly , spurious entropy generation from the artificial viscosity ( see section [ s : shock_capturing ] ) is also a minor source of error . in contrast , for subsonic turbulence , the internal energy is comparable in magnitude to the kinetic energy ; thus , the force errors from gradient noise and excessive spurious dissipation corrupt the solution on small scales such that the correct cascade of turbulent energy is not reproduced . early simulation results for the ly-@xmath6 forest obtained using both sph @xcite and grid codes @xcite agreed well . more refined comparisons ( e.g. , * ? ? ? * ; * ? ? ? * ) have demonstrated that when applied to the same initial conditions , sph and grid codes yield statistical measures for the ly-@xmath6 forest , such as flux probability distribution functions and power spectra , that agree at the @xmath74 level . the reasons for this agreement can be understood based on the discussion in section [ s : limitations_of_sph ] . the energy density of the igm gas responsible for the ly-@xmath6 forest is dominated by kinetic and gravitational energy ; thus , errors in the local fluid quantities due to noise are subdominant . furthermore , the physical state of the gas is simple , in the sense that different phases of gas are not in close proximity . thus , errors such as those discussed in sections [ s : inaccurate_instabilities ] and [ s : inaccurate_continuity ] will not greatly affect the igm , and hence the sph results are fairly reliable . whether this conclusion also extends to metal lines , for which issues of mixing of galactic outflows with pristine igm gas become important , has not been investigated thus far . within galaxy haloes , however , the physical state of the gas can differ significantly between , e.g. , gadget-3and arepo@xcite . why should sph predict the physical state of the gas responsible for the ly-@xmath6 forest so reliably yet fail so spectacularly within the haloes of galaxies ? this can be understood by realizing that the internal energy in approximately hydrostatic gas is no longer negligible compared with the kinetic and gravitational energy . noise in the local sph estimates can then significantly affect the physical state of the halo gas by , for example , producing spurious viscous heating effects that reduce cooling flows ( e.g. , * ? ? ? moreover , the gas within haloes can exhibit complex phase structure , with cold , dense gas in close proximity to and shearing relative to shock - heated diffuse gas . when simulated with traditional sph , the different gas phases will not mix correctly , as demonstrated by @xcite , because fluid instabilities ( section [ s : inaccurate_instabilities ] ) and mixing due to fluid motions ( section [ s : inaccurate_continuity ] ) are not treated properly . instead , the cold gas can fragment into clumps that remain intact ( e.g. , * ? ? ? * ; * ? ? ? * ) , thereby delivering an artificial supply of cold , low - angular - momentum gas to the central galaxy , which in turn can inhibit the formation of a rotationally supported disc . in this paper , we have demonstrated that the results of idealised ( i.e. , non - cosmological ) numerical experiments involving isolated disc galaxies and galaxy mergers are relatively similar between sph and the moving - mesh approach , in contrast with cosmological simulations of forming galaxies . this finding especially holds for simulations in which agn feedback is not included or , more generally , during early stages of mergers when the gas structure is relatively simple . later , once the gas becomes virialized and feedback from bh growth drives large - scale outflows , some detailed differences do however appear . the discussion in section [ s : limitations_of_sph ] can again be used as a guide to understand this behaviour . in the simulations presented here , we construct models of disc galaxies that evolve in isolation or merge . in these models , the gas is initially rotationally supported in the galaxy potential ; thus , its internal energy is small compared with its kinetic and gravitational energy and the flows are effectively supersonic . in this regime , similar to the gas that produces the ly-@xmath6 forest , we expect that noise in the sph estimates ( see section [ s : local_noise ] ) should not be a significant source of error . moreover , in the approach adopted here , as formulated originally in @xcite , there is no effort made to resolve the multiphase structure of the star - forming gas . instead , the ism is described using a sub - resolution model . because of this , as for the gas in the igm , the gas locally has a simple structure and different phases of gas do not exist in close proximity . as long as the gas can be characterised well in this manner , we do not expect errors associated with an inaccurate treatment of fluid instabilities ( section [ s : inaccurate_instabilities ] ) or mixing ( section [ s : inaccurate_continuity ] ) to affect our model galaxies . these conditions can start to be violated in a galaxy merger as gas is shock heated and virialized and as different gas phases start to shear relative to one another because of , e.g. , the action of agn feedback . we note that these conditions would also be violated in simulations that have enough resolution to truly resolve the multi - phase structure of the ism . in this case , extremely strong local density constrasts would be prevalent . furthermore , in contrast with cosmological simulations , in which gas cooling is a crucial determinant of how and when gas is supplied to galaxies , the gas flows in galaxy merger simulations are driven primarily by gravitational torques . in both the gadget-3 and arepo simulations presented here , we use the same accurate tree - based gravity solver . consequently , the gravitational forces are treated equally accurately in both codes . this is another reason that the results agree relatively well . for the same reason , the dark matter halo mass functions of arepo and gadget-3 cosmological simulations agree very well despite the significant differences in the distributions of baryons @xcite . as stressed above , the agreement is less good during the starburst and post - starburst phases of the merger simulations . during the starburst , gas is shock - heated and a hot halo forms . if agn feedback is included , the amount of gas in the hot halo is increased . once this hot halo forms , cooling becomes important for the post - starburst sfr and gas morphology and thus the relevant differences between sph and grid - based approaches become manifest . as in cosmological simulations @xcite and idealised simulations of ` inside - out ' disc formation @xcite , gas cools more effectively from the hot haloes formed in the mergers in arepo than in gadget-3 . furthermore , some gas that is ejected during the starburst and agn activity falls back on to the remnant during the post - starburst phase . in the arepo simulations , infalling clumps and filaments are effectively disrupted , whereas in the gadget-3 simulations they survive ( see section [ s : inaccurate_instabilities ] ) ; the differences in this regard are the same that explain why ` cold flows ' are less prominent in cosmological simulations performed with arepo @xcite , as discussed above . it is worthwhile considering the implications of the differences we have demonstrated , regarding both the robustness ( or lack thereof ) of previous work and comparisons of simulations with observations . we restrict ourselves to comments on the rich literature of simulations of isolated galaxies and galaxy mergers , which forms the central topic of this paper . for the reasons discussed above , we in general expect many of these past works to be at most weakly affected by the inaccuracies of traditional sph provided they use cold gas from the start and do not have sufficient resolution to resolve a truly multi - phase ism . however , the differences that we found are much more significant once hot haloes are present in the simulations . hence , we expect that the results of simulations that initially include hot haloes in the progenitor galaxies ( e.g. , * ? ? ? * ; * ? ? ? * ) could differ significantly if a moving - mesh technique rather than traditional sph were used . furthermore , conclusions regarding hot haloes produced by gas shocking in mergers ( e.g. , * ? ? ? * ) , the x - ray emission from hot haloes ( e.g. , * ? ? ? * ) , and the properties of starburst- and agn - driven winds ( e.g. , * ? ? ? * ; * ? ? ? * ) may depend on the numerical method employed . also , because the re - formation of discs in mergers depends on gas cooling from the hot halo of the remnant , studies of disc re - formation ( e.g. , * ? ? ? * ; * ? ? ? * ; * ? ? ? * ; * ? ? ? * ; * ? ? ? * ) might also be affected by the spurious suppression of cooling that is inherent in sph . indeed , we have already noted that the discs that re - form in the mergers presented here are very different in the gadget-3 and arepo simulations ( see the fourth columns of figs . [ fig : nb_gas_comparison ] and [ fig : gas_comparison ] ) . the differences between traditional sph and moving - mesh simulations may also have important implications for studying agn feedback with merger simulations . in particular , the differences in the bh masses yielded by arepo and gadget-3can affect the strength of the feedback , the agn luminosity and duty cycle ( e.g. , * ? ? ? * ; * ? ? ? * ; * ? ? ? * ) , and the @xmath75 relation of the merger remnants ( e.g. , * ? ? ? * ) . however , we caution that for the thermal agn feedback model presented here , the final bh masses depend primarily on the feedback coupling efficiency @xmath25 @xcite . because this number is very uncertain , it may be possible and reasonable to simply use a slightly reduced value of @xmath25 in arepo to make the final bh masses more consistent with those yielded by gadget-3 . furthermore , the scatter in the bh masses for different resolutions suggests that the values of the final bh masses are only robust to within a factor of a few ( see also * ? ? ? recently , @xcite have presented idealised isolated disc and galaxy merger simulations performed with gadget-3that include models for star formation and stellar feedback that are significantly more sophisticated than the eos approach used here . this work attempts to directly resolve the ism multi - phase structure . hence , one possible concern is the use of traditional sph . however , @xcite also demonstrate that the results of their simulations agree rather well with the results of gadget-3simulations that employ the much simpler eos approach . also , @xcite compared simulations performed using the traditional density - entropy formulation of sph used in gadget-3with simulations performed using an alternative pressure - entropy formulation of sph @xcite that is designed to overcome the inaccurate treatment of fluid instabilities ; they found that the spurious cold clumps present in the outflows in the standard gadget-3simulations were not formed when the new pressure - entropy flavour of sph was used . thus , many of the conclusions of @xcite are likely robust to the hydrodynamical solver employed because the rotational support is still dominant over pressure forces , although some details , such as the post - starburst gas morphologies , are clearly affected . we have compared a suite of idealised isolated disc and galaxy merger simulations performed using the sph code gadget-3 and the moving - mesh hydrodynamics code arepo , both with and without bh accretion and agn feedback . to isolate the effects of the hydrodynamical solver used , we have kept all other aspects of the simulations ( i.e. , the gravity solver , the treatment of cooling , and the sub - resolution models for star formation , supernova feedback , bh accretion , and agn feedback ) as similar as possible . our principal conclusions are the following : 1 . unlike for cosmological hydrodynamical simulations , the results of idealised ( non - cosmological ) isolated disc and merger simulations performed with arepo and gadget-3 are similar because ( 1 ) in these simulations , the gas is already initialised by hand in a rotationally supported disc , and thus the flow is supersonic in character and noise in the sph estimates is not a significant source of error ; ( 2 ) the gas phase structure is relatively simple ; and ( 3 ) gravitational torques ( rather than cooling ) are the primary determinant of the gas inflows . 2 . when bh accretion and agn feedback are not included , the results are quite insensitive to the code used . the sfhs and cumulative stellar mass formed versus time are remarkably similar . 3 . when bh accretion and agn feedback are included , the results of the two codes are qualitatively similar , but the quantitative differences are more significant . in particular , the bh masses can differ by as much as an order of magnitude . the arepo simulations typically yield larger bh masses . 4 . the gas morphologies and phase structures are also similar but differ in detail . primarily , arepoyields smoother , less clumpy morphologies and less prominent hot gaseous haloes in the post - starburst phase . 5 . as for cosmological simulations , the differences between the sph and moving - mesh results are primarily caused by more efficient cooling of hot halo gas and stripping of gas clumps and filaments in arepo . much of the previously published results of idealised isolated disc and galaxy merger simulations are likely robust to the inaccuracies that are inherent in traditional sph . however , for some studies , such as simulations in which the progenitors are initialised with hot haloes and studies of disc re - formation in mergers , the arepo results may differ qualitatively from those yielded by traditional sph . consequently , it would be interesting to revisit such studies using arepo . it is certainly reassuring that the bulk of our results for merger simulations exhibit good quantitative agreement between sph and the very different moving - mesh technique . interpreting this as a vindication of using sph for merger simulations would nevertheless be incorrect . as we have discussed , unsolved conceptual problems with the accuracy of sph and its convergence rate remain even in the most recent incarnations of the proposed improved versions of sph . we therefore think that more accurate numerical techniques , such as our moving - mesh approach , should clearly be preferred over traditional sph to perform such simulations , especially in future calculations for which a higher numerical precision and the resolution of multi - phase media is desired . we thank shy genel , phil hopkins , federico marinacci , rdiger pakmor , ewald puchwein , and debora sijacki for useful discussion ; jorge moreno and franois schweizer for detailed comments on the manuscript ; and the anonymous referee for a critical report that helped to improve the paper . cch is grateful to the klaus tschira foundation for financial support , acknowledges the hospitality of the aspen center for physics , which is supported by the national science foundation grant no . phy-1066293 , and heartily thanks andreas bauer for many python tips , which were very helpful for ( almost completely ) curing a long - standing idl dependency . vs acknowledges support from the european research council under erc - stg grant exagal-308037 . this research has made use of nasa s astrophysics data system bibliographic services .
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galaxy mergers have been investigated for decades using smoothed particle hydrodynamics ( sph ) , but recent work highlighting inaccuracies inherent in the traditional sph technique calls into question the reliability of previous studies .
we explore this issue by comparing a suite of gadget-3 sph simulations of idealised ( i.e. , non - cosmological ) isolated discs and galaxy mergers with otherwise identical calculations performed using the moving - mesh code arepo .
when black hole ( bh ) accretion and active galactic nucleus ( agn ) feedback are not included , the star formation histories ( sfhs ) obtained from the two codes agree well . when bhs are included , the code- and resolution - dependent variations in the sfhs are more significant , but the agreement is still good , and the stellar mass formed over the course of a simulation is robust to variations in the numerical method . during a merger ,
the gas morphology and phase structure are initially similar prior to the starburst phase .
however , once a hot gaseous halo has formed from shock heating and agn feedback ( when included ) , the agreement is less good . in particular , during the post - starburst phase , the sph simulations feature more prominent hot gaseous haloes and spurious clumps , whereas with arepo , gas clumps and filaments are less apparent and the hot halo gas can cool more efficiently .
we discuss the origin of these differences and explain why the sph technique yields trustworthy results for some applications ( such as the idealised isolated disc and galaxy merger simulations presented here ) but not others ( e.g. , gas flows onto galaxies in cosmological hydrodynamical simulations ) .
[ firstpage ] hydrodynamics methods : numerical galaxies : interactions galaxies : starburst galaxies : active galaxies : formation .
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You are an expert at summarizing long articles. Proceed to summarize the following text:
a promising way to explain the late - time accelerated expansion of the universe is to assume that at large scales general relativity ( gr ) breaks down , and a more general action describes the gravitational field . thus , in the latter context , infra - red modifications to gr have been extensively explored , where the consistency of various candidate models have been analysed ( see @xcite for a review ) . note that the einstein field equation of gr was first derived from an action principle by hilbert , by adopting a linear function of the scalar curvature , @xmath0 , in the gravitational lagrangian density . the physical motivations for these modifications of gravity were related to the possibility of a more realistic representation of the gravitational fields near curvature singularities and to create some first order approximation for the quantum theory of gravitational fields , and more recently in an attempt to explain the late - time cosmic acceleration . in this context , a more general modification of the hilbert - einstein gravitational lagrangian density involving an arbitrary function of the scalar invariant , @xmath1 , has been extensively explored in the literature , and recently a maximal extension of the hilbert - einstein action has been proposed @xcite . the action of the maximal extension of the hilbert - einstein action is given by @xcite @xmath3 where @xmath4 is an arbitrary function of the ricci scalar @xmath0 , and of the lagrangian density corresponding to matter , @xmath5 . the energy - momentum tensor of matter is defined as @xmath6 . varying the action with respect to the metric @xmath7 , the gravitational field equation of @xmath8 gravity is provided by @xmath9 g_{\mu \nu } = \frac{1}{2 } f_{l_{m}}\left ( r , l_{m}\right ) t_{\mu \nu } \,.\end{aligned}\ ] ] for the hilbert - einstein lagrangian , @xmath10 , we recover the einstein field equations of gr , i.e. , @xmath11 . for @xmath12 , where @xmath13 , @xmath14 and @xmath15 are arbitrary functions of the ricci scalar and of the matter lagrangian density , respectively , we obtain the field equations of modified gravity with an arbitrary curvature - matter coupling @xcite . an interesting application was explored in the context of @xmath16 gravity@xcite . the @xmath2 models possess extremely interesting properties . first , the covariant divergence of the energy - momentum tensor is non - zero , and is given by @xmath17 \frac{\partial l_{m}}{% \partial g^{\mu \nu } } \ , . \label{noncons}\end{aligned}\ ] ] the requirement of the conservation of the energy - momentum tensor of matter , @xmath18 , provides the condition given by @xmath19 \partial l_{m}/ \partial g^{\mu \nu } = 0 $ ] . secondly , the motion of test particles is non - geodesic , and takes place in the presence of an extra force . as a specific example , consider the case in which matter , assumed to be a perfect thermodynamic fluid , obeys a barotropic equation of state , with the thermodynamic pressure @xmath20 being a function of the rest mass density of the matter @xmath21 only , i.e. , @xmath22 , and consequently , the matter lagrangian density , becomes an arbitrary function of the energy density @xmath21 only , i.e. , @xmath23 ( for more details , we refer the reader to @xcite ) . thus , the equation of motion of a test fluid is given by @xmath24 , where the extra - force @xmath25 is defined by @xmath26 \left ( u^{\mu } u^{\nu } -g^{\mu \nu } \right ) \,.\ ] ] note that @xmath25 is perpendicular to the four - velocity , @xmath27 , i.e. , @xmath28 . the non - geodesic motion , due to the non - minimal couplings present in the model , implies the violation of the equivalence principle , which is highly constrained by solar system experimental tests . however , it has recently been argued , from data of the abell cluster a586 , that the interaction between dark matter and dark energy implies the violation of the equivalence principle @xcite . thus , it is possible to test these models with non - minimal couplings in the context of the violation of the equivalence principle . it is also important to emphasize that the violation of the equivalence principle is also found as a low - energy feature of some compactified versions of higher - dimensional theories . in the newtonian limit of weak gravitational fields @xcite , the equation of motion of a test fluid in @xmath4 gravity is given by @xmath29 where @xmath30 is the total acceleration of the system ; @xmath31 is the newtonian gravitational acceleration ; the term @xmath32 $ ] is identified with the hydrodynamic acceleration term in the perfect fluid euler equation . now , by assuming that in the newtonian limit the function @xmath33 can be represented as @xmath34 , where @xmath35 , so that @xmath36 given by @xmath37\,,\ ] ] is a supplementary acceleration induced due to the modification of the action of the gravitational field . in conclusion , the maximal extensions of gr , namely the @xmath2 gravity models open the possibility of going beyond the algebraic structure of the hilbert - einstein action . on the other hand , the field equations of @xmath2 gravity are equivalent to the field equations of the @xmath1 model in empty space - time , but differ from them , as well as from gr , in the presence of matter . thus , the predictions of @xmath2 gravitational models could lead to some major differences , as compared to the predictions of standard gr , or other generalized gravity models , in several problems of current interest , such as cosmology , gravitational collapse or the generation of gravitational waves . the study of these phenomena may also provide some specific signatures and effects , which could distinguish and discriminate between the various gravitational models . in addition to this , in order to explore in more detail the connections between the @xmath2 theory and the cosmological evolution , it is necessary to build some explicit physical models . fsnl acknowledges financial support of the fundao para a cincia e tecnologia through the grants cern / fp/123615/2011 and cern / fp/123618/2011 . f. s. n. lobo , arxiv:0807.1640 [ gr - qc ] . t. harko and f. s. n. lobo , eur . j. c * 70 * , 373 ( 2010 ) . t. harko , phys . b * 669 * , 376 ( 2008 ) . o. bertolami , c. g. boehmer , t. harko and f. s. n. lobo , phys . d * 75 * , 104016 ( 2007 ) . t. harko , t. s. koivisto and f. s. n. lobo , mod . a * 26 * ( 2011 ) 1467 . t. harko , f. s. n. lobo , s. i . nojiri and s. d. odintsov , phys . rev . d * 84 * , 024020 ( 2011 ) . o. bertolami , f. s. n. lobo and j. paramos , phys . d * 78 * , 064036 ( 2008 ) . o. bertolami , j. paramos , t. harko and f. s. n. lobo , arxiv:0811.2876 [ gr - qc ] . o. bertolami , f. gil pedro and m. le delliou , phys . b * 654 * , 165 ( 2007 ) .
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we consider a maximal extension of the hilbert - einstein action and analyze several interesting features of the theory .
more specifically , the motion is non - geodesic and takes place in the presence of an extra force .
these models could lead to some major differences , as compared to the predictions of general relativity or other modified theories of gravity , in several problems of current interest , such as cosmology , gravitational collapse or the generation of gravitational waves .
thus , the study of these phenomena may also provide some specific signatures and effects , which could distinguish and discriminate between the various gravitational models .
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You are an expert at summarizing long articles. Proceed to summarize the following text:
spinning objects have historically been interesting subjects to study . the spin reversal of the rattleback @xcite ( also called a celt or wobblestone ) and the behavior of the tippe top are typical examples . in the latter case , when a truncated sphere with a cylindrical stem , a so - called ` tippe top ' , is spun sufficiently rapidly on a table with its stem up , it will flip over and rotate on its stem . this inversion phenomenon has fascinated physicists and has been studied for over a century @xcite . in the present paper we revisit and study this very classical problem from a different perspective . recently the riddle of spinning eggs has been resolved by moffatt and shimomura [ ms ] @xcite . they discovered that if an axisymmetric body , such as a hard - boiled egg , is spun sufficiently rapidly , a ` _ gyroscopic balance _ ' condition ( gbc ) holds and that under this condition the governing equations of the system are much simplified . in particular , they derived a first - order ordinary differential equation ( ode ) for @xmath8 , the angle between the axis of symmetry and the vertical axis , and showed for the case of a prolate spheroid that the axis of symmetry indeed rises from the horizontal to the vertical . then the spinning behavior of egg - shaped axisymmetric bodies , whose cross sections are described by several models of oval curves , was studied under the gbc by one of the present authors @xcite . the tippe top is also an axisymmetric body and shows the similar behavior as the spinning egg . then one may ask : does the gbc also hold for the tippe top ? if so , how is it related to the inversion phenomenon of the tippe top ? in the first half of this paper we analyze the spinning motion of the tippe top in terms of the gbc . actually the gbc is not satisfied initially for the tippe top , contrary to the case of the spinning egg . the difference comes from how we start to spin the object : we spin the tippe top with its stem up , in other words , with its symmetry axis vertical while the egg is spun with its symmetry axis horizontal . in this paper we perform our analysis taking an eccentric sphere version of the tippe top instead of a commercially available one , a truncated sphere with a cylindrical stem . in order to examine the gbc of the tippe top more closely , we introduce a variable @xmath2 so that @xmath3 corresponds to the gbc , and study the behavior of @xmath2 . numerical analysis shows that for the tippe tops which will turn over , the variable @xmath2 , starting from a large positive value @xmath9 , soon takes negative values and fluctuates around a negative but small value @xmath10 such that @xmath11 . thus for these tippe tops , the gbc , which is not satisfied initially , will soon be realized but approximately . on the other hand , in the case of the tippe tops which will not turn over , @xmath2 remains positive around @xmath9 or changes from positive @xmath9 to negative values and then back to positive values close to @xmath9 again . we find that the behavior of @xmath2 is closely related to the inversion phenomenon of the tippe top . once @xmath2 fluctuates around the value @xmath12 , the system becomes unstable and starts to turn over . under the gbc the governing equations for the tippe top are much simplified and we obtain a first - order ode for @xmath8 , which has the same form as the one derived by ms for the spinning egg . then , this equation for @xmath8 and the geometry lead to the classification of tippe tops into _ three _ groups , depending on the values of @xmath13 and @xmath14 , where @xmath15 and @xmath16 are two principal moments of inertia , and @xmath17 is the distance from the center of sphere to the center of mass and @xmath18 is the radius of sphere . the tippe tops of group i never flip over however large a spin they are given . those of group ii show a complete inversion and the tippe tops of group iii tend to turn over up to a certain inclination angle @xmath4 such that @xmath5 , when they are spun sufficiently rapidly . this classification of tippe tops into three groups and its classificatory criteria totally coincide with those obtained by hugenholtz @xcite and leutwyler @xcite , both of whom resorted to completely different arguments and methods . in the latter half of this paper we study the steady states for spinning motion of the tippe top and examine their stability ( or instability ) . it is well understood that the main source for the tippe top inversion is sliding friction @xcite , which depends on the slip velocity @xmath0 of the contact point between the tippe top and a table . often used is coulomb friction ( see eq.([coulomb ] ) ) . in fact , coulomb friction is practical when @xmath19 is away from zero , but it is undefined for @xmath20 . however , we learn that at the steady state of the tippe top , the slip velocity @xmath0 necessarily vanishes . in order to facilitate a linear stability analysis of steady states and also to study the motion of the tippe top as realistically as possible , we adopt in our analysis a modified version of coulomb friction ( see eq.([friction ] ) ) , which is continuous in @xmath0 and vanishes at @xmath1 . actually the steady states of the tippe top and their stability were analyzed by ebenfeld and scheck [ es ] @xcite , who assumed a similar frictional force which is continuous at @xmath1 . they used the total energy of the spinning top as a liapunov function . the steady states were found as solutions of constant energy . and the stability or instability of these states were judged by examining whether the liapunov function assumes a minimum or a maximum at these states . also recently , bou - rabee , marsden and romero [ bmr ] @xcite analyzed the tippe top inversion as a dissipation - induced instability and , using the modified maxwell - bloch equations and an energy - momentum argument , they gave criteria for the stability of the non - inverted and inverted states of the tippe top . we take a different approach to this problem . first , in order to find the steady states for spinning motion of the tippe top , we follow the method used by moffatt , shimomura and branicki [ msb ] for the case of spinning spheroids @xcite . then the stability of these steady states is examined as follows : once a steady state is known , the system is perturbed around the steady state . particularly we focus our attention on the variable @xmath8 , which is perturbed to @xmath21 , where @xmath22 is a value at the steady state and @xmath23 is a small quantity . using the equations of motion , we obtain , under the linear approximation , a first - order ode for @xmath23 of the form , @xmath24 , where @xmath25 is expressed by the values of dynamical variables at the steady state . thus the change of @xmath23 is governed by the sign of @xmath25 . if @xmath25 is positive ( negative ) , @xmath26 will increase ( decrease ) with time . therefore , we conclude that when @xmath25 is negative ( positive ) , then the state is stable ( unstable ) . using this new and rather intuitive criterion we argue about the stability of the steady states in terms of the initial spin velocity @xmath6 given at the position near @xmath27 . we observe that our results on the stability of the steady states are consistent with ones obtained by es and msb . then we obtain a critical value @xmath7 of the initial spin which is required for the tippe top of group ii to flip over up to the completely inverted position at @xmath28 . finally we confirm by simulation our results on the relation between the initial spin @xmath6 and the stability of the steady states . the paper is organized as follows : in sec . 2 we explain the notation and geometry used in this paper , and give all the necessary equations for the analysis of the spinning motion of the tippe top . in sec . 3 we discuss about the gbc and its relevance to the inversion phenomenon of the tippe top . we also show that the assumption of the gbc leads to the classification of tippe tops into three groups . then in sec . 4 we study the steady states for the spinning motion of the tippe top and examine their stability . is devoted to a summary and discussion . in addition , we present four appendices . in appendix a , the equations of motion which are used to analyze the spinning motion of the tippe top are enumerated . in appendix b , it is shown that intermediate steady states for the tippe tops of group ii and group iii are stable when an initial spin @xmath29 falls in a certain range . in appendix c we demonstrate that our stability criterion for the steady state is equivalent to the one obtained by es . and finally , in appendix d , we show that our results on the stability of the vertical spin states are consistent with the criteria derived by bmr . a commercially available tippe top is usually a truncated sphere with a cylindrical stem . instead we perform our analysis taking a loaded ( eccentric ) sphere version of the tippe top . the center of mass is off center by a distance @xmath17 . there are no qualitative differences between the two . but if applied to the case of a commercial tippe top with a stem , our assertions would be valid up to the point when the stem touched the table surface . a loaded sphere ( eccentric ) version of the tippe top . the center of mass @xmath30 is off center ( @xmath31 ) by distance @xmath17 . the tippe top spins on a horizontal table with point of contact @xmath32 . its axis of symmetry , @xmath33 , and the vertical axis , @xmath34 , define a plane @xmath35 , which precesses about @xmath34 with angular velocity @xmath36 . @xmath37 is a rotating frame of reference with @xmath38 horizontal in the plane @xmath35 . the height of @xmath30 above the table is @xmath39 , where @xmath18 is the radius . the position vector of @xmath32 from @xmath30 is @xmath40 , where @xmath41 and @xmath42 . , scaledwidth=60.0% ] fig . [ loadedsphere ] shows the geometry . an axisymmetric tippe top spins on a horizontal table with point of contact @xmath32 . we will work in a rotating frame of reference @xmath37 , where the center of mass is at the origin , @xmath30 . the center @xmath31 of the sphere with radius @xmath18 is at a distance @xmath17 from the origin . the symmetry axis of the tippe top , @xmath33 , and the vertical axis , @xmath34 , define a plane @xmath35 , which precesses about @xmath34 with angular velocity @xmath36 . let @xmath43 be the euler angles of the body relative to @xmath44 . then we have @xmath45 , where the dot represents differentiation with respect to time , and @xmath8 is the angle between @xmath34 and @xmath33 . we choose the horizontal axis @xmath38 in the plane @xmath35 and thus @xmath46 is vertical to @xmath35 and inward . in a rotating frame of reference @xmath47 , where @xmath48 is in the plane @xmath35 and perpendicular to the symmetry axis @xmath33 and where @xmath49 coincides with @xmath46 , the tippe top spins about @xmath33 with the rate @xmath50 . since @xmath51 is expressed as @xmath52 in the frame @xmath47 , the angular velocity of the tippe top , @xmath53 , is given by @xmath54 . here @xmath55 , @xmath56 , and @xmath57 are unit vectors along @xmath48 , @xmath49 , and @xmath33 , respectively , @xmath58 is given by @xmath59 . the @xmath48 and @xmath49 are not body - fixed axes but are principal axes , so that the angular momentum , @xmath60 , is expressed by @xmath61 , where @xmath62 are the principal moments of inertia at @xmath30 . using the perpendicular axis theorem and the parallel axis theorem , we see that @xmath63 for any axisymmetric density distribution . the coordinate system @xmath47 is obtained from the frame @xmath37 by rotating the latter about the @xmath46 ( @xmath49 ) axis through the angle @xmath8 . hence , in the rotating frame @xmath37 , @xmath53 and @xmath60 have components @xmath64 respectively . the evolution of @xmath60 is governed by euler s equation @xmath65 where @xmath66 is the position vector of the contact point @xmath32 from @xmath30 , @xmath67 is the normal reaction at @xmath32 , @xmath68 , with @xmath69 being of order @xmath70 , the weight , and @xmath71 is the frictional force at @xmath32 . we consider only _ the situation in which the tippe top is always in contact with the table _ throughout the motion . since the point @xmath32 lies in the plane @xmath35 , @xmath66 has components @xmath72 , which are given by @xmath73 where @xmath74 is the height of @xmath30 above the table . the components of ( [ euler ] ) are expressed , respectively , as @xmath75 in terms of @xmath8 , @xmath76 , and @xmath77 the above equations are rewritten as @xmath78 now it is easily seen from ( [ angulareqs ] ) , ( [ eulerx ] ) and ( [ eulerz ] ) that there exists an exact constant of motion , @xmath79 which is valid irrespective of the reaction force @xmath80 at the contact point @xmath32 , in other words , whether or not slipping occurs . this so - called jellett s constant " @xcite is typical for the tippe top whose portion of the surface in contact with the table is spherical . the velocity , @xmath81 , of the contact point @xmath32 with respect to the center of mass @xmath30 is given by @xmath82 , and thus has components , @xmath83 the center of mass @xmath30 is not stationary . let @xmath84 represent the velocity of @xmath30 , then the slip velocity of the contact point @xmath32 , @xmath85 , is @xmath86 since @xmath87 , we have @xmath88 as was expected . the equation of motion for the center of mass @xmath30 is given by @xmath89 where @xmath90 is the mass of the tippe top and @xmath91 is the force of gravity . in components , eq.([eqcm ] ) reads @xmath92 since @xmath93 , eq.([eqcmz ] ) gives @xmath94 which shows that the normal force @xmath69 is of order @xmath95 when @xmath96 . we need an information on the frictional force @xmath97 . it is well understood that the sliding friction is the main source for the tippe top inversion @xcite . so we will ignore other possible frictions , such as , rolling friction @xcite and rotational friction which is due to pure rotation about a vertical axis . concerning the sliding friction , often used is a coulomb law , which states that @xmath98 where @xmath99 is a coefficient of friction . another possibility is a viscous friction law , which states that the friction is linearly related to @xmath0 . coulomb friction is practical when @xmath19 is away from zero but it is undefined at @xmath100 . the slip velocity of the contact point @xmath32 necessarily vanishes at the steady state of the tippe top . in order to study the motion of the tippe top as realistically as possible and also to facilitate a linear stability analysis of steady states , we modify the expression of coulomb friction ( [ coulomb ] ) as @xmath101 so that @xmath97 is continuous in @xmath0 and vanishes at @xmath102 . here we choose @xmath103 as a sufficiently small number with dimensions of velocity . note that @xmath88 and thus the @xmath104-component of @xmath97 is 0 . this completes the presentation of all the necessary equations for the analysis of the motion of tippe tops . we enumerate all these equations in appendix a. we need further the initial conditions . when we play with a tippe top , we usually give it a rapid spin with its axis of symmetry nearly vertical . so let us choose the following initial conditions for @xmath8 and other angular velocities : @xmath105 we take @xmath106 rad and @xmath107 . recall that the spin @xmath58 is given by @xmath59 , and thus we have @xmath108 . as for the initial condition for the velocity of the center of mass @xmath30 , we take @xmath109 since we usually do not give a large translational motion to the tippe top at the beginning . with the above initial conditions ( [ initialrotation ] ) and ( [ initialtrans ] ) , we analyze the behaviors of the tippe top using three angular ( [ euleromega]-[eulern ] ) and three translational ( [ eqcmx]-[eqcmz ] ) equations of motion , together with the knowledge of the frictional force , a modified version of the coulomb law ( [ friction ] ) , and the velocities ( [ vpx]-[vpz ] ) and ( [ velp ] ) . when we perform simulations we use the adaptive runge - kutta method . we define a variable @xmath2 as @xmath110 in terms of @xmath2 , the @xmath111- and @xmath104- components of @xmath60 in ( [ angulareqs ] ) and jellett s constant @xmath112 , ( [ jellettconstant ] ) , are expressed , respectively , as @xmath113 the condition @xmath3 has been introduced by ms @xcite in their analysis of spinning hard - boiled eggs , and referred to as the gbc . they discovered that the gbc , @xmath3 , is approximately satisfied for the spinning egg and , using this gbc , they resolved a long standing riddle : when a hard - boiled egg is spun sufficiently rapidly on a table with its axis of symmetry horizontal , the axis will rise from the horizontal to the vertical . we outline how ms found the gbc for the spinning egg @xcite . the system of the spinning egg obeys essentially the same equations of motion as the case of the tippe top , to be specific , eqs . ( [ euler ] ) and ( [ eqcm ] ) . the @xmath114-component of ( [ euler ] ) for the spinning egg is given by ( [ eulertheta ] ) , with the factor , @xmath115 , being replaced by @xmath116 . because the secular change of @xmath8 is slow and thus @xmath117 , the term @xmath118 can be neglected . furthermore , in a situation where @xmath119 is sufficiently large so that the terms involving @xmath76 in ( [ eulertheta ] ) dominate the terms @xmath120 and @xmath121 , eq . ( [ eulertheta ] ) is reduced , in leading order , to @xmath122 . hence , for @xmath123 , we arrive at the condition @xmath124 . the tippe top shows the similar behavior as the spinning egg . then one may ask : does the gbc also hold for the tippe top ? we will show that the answer is partly no " and partly yes " . partly no " means that the gbc is not satisfied initially . tippe tops are usually spun with @xmath125 , @xmath126 , and large @xmath127 and , therefore , @xmath128 is large , from which we find that @xmath129 is large[multiblock footnote omitted ] . thus the gbc does not hold at the beginning . however , we will see later that the gbc does approximately hold whenever the tippe top rises , which is the meaning of partly yes " . in fact , the argument of ms to derive the gbc for the spinning egg can also be applied to the tippe top . thus in a situation where @xmath76 is sufficiently large and for @xmath123 , the gbc is expected to be satisfied . on the other hand , in the case of the spinning egg , the gbc is approximately satisfied initially . we start to spin an egg with its symmetry axis horizontal , that is , with @xmath130 , @xmath131 and large @xmath132 . hence we find @xmath133 and @xmath134 for the spinning egg . we emphasize that the variable @xmath2 initially takes a large positive value for the tippe top . but our numerical analysis will show that when a tippe top turns over , @xmath2 soon makes a rapid transition from large positive values to negative values and starts to oscillate about a small negative value . before proceeding with a discussion of this transition of @xmath2 , let us consider the consequences when the gbc is exactly satisfied for the tippe top . in a situation where @xmath76 is sufficiently large and @xmath8 is not in the vicinity of 0 or @xmath135 , the gbc is realized for the tippe top . let us consider the case that the exact gbc , @xmath136 , is satisfied for the tippe top . then , we have @xmath137 from ( [ jellettconstant2 ] ) , and @xmath138 from the second equation in ( [ angulareqs2 ] ) . if the angular velocity @xmath76 around the vertical axis is reduced and , therefore , @xmath139 decreases , eq.([gbcjellett ] ) tells us that the height @xmath74 of the center of mass from the table increases since @xmath112 is a constant , which means the turning over of the tippe top . differentiating both sides of ( [ gbcjellett ] ) by time and using ( [ height.b ] ) and ( [ eulerz ] ) , we obtain a first - order ode for @xmath8 , @xmath140 we assume also that the @xmath114-component of @xmath141 , the translational velocity of the center of mass @xmath30 , in ( [ velp ] ) is negligible in the first approximation as compared with that of @xmath81 , and we set @xmath142 . we see that numerical simulation supports this assumption then , one can use eq . ( [ vpy ] ) and the gbc to eliminate @xmath77 and @xmath76 , and obtain @xmath143 as only a function of the dynamical variable @xmath8 as follows : @xmath144 since the frictional force @xmath145 is proportional to @xmath143 , we obtain from eqs.([equfortheta]-[vpsecond ] ) , @xmath146 with a _ positive _ proportional coefficient and @xmath147 equation ( [ equtheta ] ) implies that the change of @xmath8 is governed by the sign of @xmath148 . if @xmath148 is positive ( negative ) , then @xmath8 will increase ( decrease ) with time . therefore a close examination of the behavior of @xmath148 as a function of @xmath8 will be important . ) to a renormalization group equation which appears in quantum field theories for critical phenomena and high energy physics is emphasized in sec . we observe from ( [ vpytilde ] ) that @xmath149 at @xmath150 and @xmath135 , since @xmath151 at these angles . moreover , @xmath148 may vanish at an other angle , which is given by solving @xmath152 equation ( [ vpzero ] ) has a solution for @xmath8 if @xmath153 or @xmath154 and no solution otherwise . accordingly , tippe tops are classified into three groups , depending on the values of @xmath13 and @xmath14 : group i with @xmath153 ; group ii with @xmath155 ; and group iii with @xmath154 . we now examine the behaviors of tippe tops belonging to each group . [ cols="^,^ " , ] from these numerical analyses we see that the behavior of @xmath2 is closely related to the inversion phenomenon of the tippe top . as the tippe top turns over , simulation shows that @xmath2 becomes small ( in the sense @xmath156 ) and takes values close to @xmath157 , which implies that the relation ( [ gbcjellett ] ) is approximately satisfied . conversely , when the relation ( [ gbcjellett ] ) holds , it means that the center of mass of the tippe top goes up as @xmath139 decreases . in sec.3.2 we have studied the behaviors of the spinning tippe top when the gyroscopic balance condition @xmath3 is exactly satisfied . the situation corresponds to the one in which the tippe top is given an infinitely large initial spin . actually the initial spin given to the tippe top is finite and we know empirically that a tippe top with a small spin is stable and does not turn over . we will now consider how large an initial spin should be for the tippe top to turn over . for that purpose we will study the steady states of the tippe top and examine their stability . actually the steady states ( or the asymptotic states ) of the tippe top and their stability were analyzed by ebenfeld and scheck [ es ] @xcite . they used the total energy of the spinning top as a liapunov function . then the steady states were found as solutions of constant energy . the stability or instability of these states was determined by examining whether the liapunov function assumes a minimum or a maximum at these states under the constraint of jellett s constant . the tippe top inversion was also analyzed recently by bou - rabee , marsden and romero [ bmr ] @xcite as a dissipation - induced instability . bmr used the modified maxwell - bloch equations and an energy - momentum argument to determine the stability of the non - inverted and inverted states of the tippe top . here we take a different approach to this problem . and we discuss the stability of the steady states in terms of the initial spin velocity @xmath77 given at the non - inverted position @xmath27 . recently , moffatt , shimomura and branicki [ msb ] made a linear stability analysis of the spinning motion of spheroids @xcite . they identified the steady states , and then discussed their stability and found the critical angular velocity needed for the rise of the body . in order to find the steady states of the spinning tippe top , we adopt the method taken by msb for the case of spheroids . but for the stability analysis of the steady states , we develop a new stability criterion which is different from the ones used by es , bmr and msb . our approach to the stability problem of the tippe top is as follows . once a steady state is known , the system is perturbed around the steady state . particularly we focus our attention on the variable @xmath8 , which is perturbed to @xmath158 where @xmath22 is a value at the steady state and @xmath23 is a small quantity . using the equations of motion , we obtain , under the linear approximation , a first - order ode for @xmath23 of the following form : @xmath159 where @xmath160 and @xmath161 are values taken at the steady state . equation ( [ eqforstability ] ) implies that the change of @xmath23 is governed by the sign of the function @xmath162 . if @xmath162 is positive ( negative ) , @xmath26 will increase ( decrease ) with time . therefore , we conclude that _ when @xmath162 is negative ( positive ) , then the state is stable ( unstable)_. this is the criterion for stability of the steady state , which we will use in this paper . the stability criterion in this work is derived from an intuitive analysis of the equations of motion . we check in appendices c and d that they are consistent with those derived by es and bmr which are based on mathematically rigorous methods . superficially the above criterion ( [ eqforstability ] ) seems quite different from the one used by es @xcite , but actually we have found that both are equivalent and , therefore , our results are consistent with theirs . in appendix c we will show the equivalence of both criteria and that the stability conditions of the steady states which we will obtain coincide with the ones found by es . after all , es utilized the total energy ( an integral form ) @xcite , while we will use equations of motion ( differential forms ) . the criterion ( [ eqforstability ] ) for the stability of the tippe top also seems different from the ones used by bmr @xcite , which were derived from the tippe top modified maxwell - bloch equations . in order to obtain the stability criteria , both bmr and we linearize equations of motion about the steady states and use sliding friction , which is assumed to be an analytic function of the slip velocity , as the main mechanism behind tippe top inversion . thus it is well expected that both criteria lead to the consistent results on the stability of the non - inverted and inverted states . ( the stability of the intermediate states have not been analyzed yet by means of the modified maxwell - bloch equations ) . in appendix d we will show that the expressions of the criteria provided in bmr become more transparent when they are rewritten in terms of the parameters and classification criteria used in this paper , and that they lead to the same stability conditions for the vertical spinning states which will be obtained later by using the criterion ( [ eqforstability ] ) . besides , although bmr did not mentioned , the classification of tippe tops into three groups , group i , ii , and iii , is shown to be possible through the close examination of the criteria in bmr . the steady states of the spinning motion of the tippe top are obtained from the equations of motion ( [ euleromega]-[eulern ] ) and ( [ eqcmx]-[eqcmz ] ) by setting @xmath163 @xcite . since we assume that the sliding friction ( [ friction ] ) , i.e. , a modified version of coulomb law ) is non - analytic at @xmath102 and a nonlinear friction law that would not appear in the linear approximation @xcite . ] , is the only frictional force present , the energy equation @xmath164 shows that @xmath102 and @xmath165 at the steady states @xcite . thus we obtain for the steady states of the tippe top , @xmath166 where @xmath167 and the velocity equations ( [ vpx]-[vpy ] ) and ( [ velp ] ) have been used . the solutions for eqs.([uoxzero]-[vpyzero ] ) are : \i ) vertical spin state at @xmath168 : @xmath169 which is a spinning state about the axis of symmetry with the center of mass below the sphere center . \ii ) vertical spin state at @xmath28 : @xmath170 which is an overturned spinning state about the axis of symmetry with the center of mass above the sphere center . \iii ) intermediate states : @xmath171 the elimination of @xmath77 from ( [ intera ] ) and ( [ interb ] ) gives @xmath172 the necessary ( but not sufficient ) condition for the existence of such states is @xmath173 recall ( [ vpzero ] ) which was used for the classification of tippe tops into three groups in sec . 3.2 . thus , intermediate states may exist at @xmath174 for the tippe top of group i ( @xmath153 ) , at @xmath8 between 0 and @xmath135 for group ii ( @xmath155 ) , and at @xmath175 for group iii ( @xmath154 ) . there appear , in total , three categories of steady states for a loaded sphere version of the tippe top . the turnover of the tippe top is associated with the effect of the sliding friction ( with a coefficient @xmath99 ) at the point of contact @xmath32 . near the steady states , we know that @xmath176 , which is equivalent to the situation where @xmath177 . thus for the stability analysis of the steady states , we consider the limiting case of @xmath178 @xcite . since we expect @xmath179 near the steady states , we have @xmath180 and @xmath181 . eq.([vpy ] ) shows @xmath182 , which leads to @xmath183 and , hence , @xmath184 . then ( [ eqcmy ] ) gives @xmath185 , and @xmath186 from ( [ vpx ] ) and ( [ velp ] ) , and thus we have @xmath187 and @xmath188 . the above order estimation in @xmath99 near the steady states leads to the primary balance in ( [ eulertheta ] ) which holds at leading order in @xmath99 @xcite , @xmath189 note that with a sufficiently large @xmath76 , eq.([stabilityomega ] ) reduces to @xmath190 , the gbc . the angle @xmath8 is perturbed from @xmath150 , and we take @xmath191 . in the linear approximation we may take @xmath192 , since eq.([eulern ] ) implies that @xmath193 is quadratic in small quantities ( note @xmath194 ) . with @xmath195 , the primary balance ( [ stabilityomega ] ) gives @xmath196 in this approximation @xmath76 is also a constant . then eq.([euleromega ] ) gives @xmath197 and we may take @xmath198 since @xmath199 . hence , we require for the stability at @xmath168 @xmath200 using the expressions of both `` @xmath201 '' and `` @xmath202 '' solutions for @xmath76 in ( [ omegaat0 ] ) , the above condition is rewritten as @xmath203 which gives @xmath204 it is easily seen that the requirement ( [ zerostability2 ] ) is always satisfied for any spin velocity @xmath77 by the tippe top of group i ( @xmath205 . as for the tippe top of group ii or iii with @xmath206 , the requirement ( [ zerostability2 ] ) is rewritten as @xmath207 the stability of the vertical spin state at @xmath168 is summarized as follows : for the tippe top of group i with @xmath208 , the spinning state at @xmath168 is stable for any spin @xmath77 , while for the tippe top of group ii or iii with @xmath206 we require @xmath209 for its stability . in other words , the tippe top of group ii or iii becomes unstable at @xmath168 if @xmath210 a similar analysis can be made for the stability of the spinning state at @xmath28 . now put @xmath211 with @xmath212 . again we may take @xmath192 , but note that @xmath77 may be negative near @xmath213 . with @xmath214 , the primary balance ( [ stabilityomega ] ) gives @xmath215 in order for @xmath76 to have a real solution , we require @xmath216 for @xmath217 , the spin is insufficient to overcome the effect of gravity and the orientation becomes unstable @xcite . with @xmath218 , ( [ euleromega ] ) gives @xmath219 and we may take , @xmath220 hence we require for the stability at @xmath28 , @xmath221 using the expressions of both `` @xmath201 '' and `` @xmath202 '' solutions for @xmath76 in ( [ omegaatpi ] ) , the above condition gives @xmath222 first , the requirement ( [ pistability2 ] ) is never satisfied by the tippe top of group iii @xmath223 . so the tippe top of group iii is unstable at @xmath28 . actually it never turns over to the position with @xmath28 . for the tippe top of group i or ii which satisfies @xmath224 , the requirement ( [ pistability2 ] ) becomes @xmath225 note that @xmath226 . the stability of the vertical spin state at @xmath28 is summarized as follows : for the tippe top of group iii ( @xmath227 ) , the spinning state at @xmath213 is unstable for any spin @xmath77 , while for the tippe top of group i or ii with @xmath228 , the state at @xmath28 is stable if @xmath229 we have learned in sec.4.2.1 that the spinning state of group i at @xmath150 is stable . we also know from the discussion in sec.4.1 that the intermediate steady states of group i , if they exist , must occur at @xmath230 . this implies that the spinning motion of group i near @xmath150 does not shift to a possible intermediate steady state . on the other hand , the tippe tops of group ii and iii become unstable at @xmath150 when they are spun with a sufficiently large initial spin @xmath231 , where @xmath232 is given by ( [ zerostability4 ] ) , and they will start to turn over . here we are interested in the intermediate steady states of the tippe top which are reached from the initial spinning position near @xmath150 . therefore , in this subsection , we focus on the possible steady states only for the tippe tops of group ii and iii , and examine their stability . the jellett s constant given by ( [ jellettconstant ] ) or ( [ jellettconstant2 ] ) is rewritten as @xmath233 now eqs.([intera ] ) and ( [ interb ] ) and the above expression of j completely determine the intermediate steady states . they are derived by solving @xmath234= \bigl\{(\cos\theta-\frac{a}{r})^2+\frac{a}{c}\sin^2\theta\bigr\}^2 ~,\label{eqforintermediate}\ ] ] where @xmath235 define the following function : @xmath236 where @xmath237 and @xmath238 then , eq.([eqforintermediate ] ) is rewritten as @xmath239 since @xmath240 , we obtain @xmath241 ^ 2}(\frac{a}{c}-1)~ , \label{derivg}\\ f''(x ) & = & \frac{2}{[f_1(x)]^3}\bigl\ { \bigl([f_1(x)]^2+(\frac{a}{c}-1 ) \sqrt{f_2(x ) } \bigr ) ^2 + 3 [ f_1(x)]^4 \bigr\ } > 0~. \label{derivderivg}\end{aligned}\ ] ] the condition for the initial spin @xmath231 means @xmath242 . using ( [ zerostability4 ] ) , we find @xmath243 , which leads to @xmath244 . so we are looking for solutions of @xmath245 with @xmath244 . ( i)group ii @xmath246 when @xmath247 , @xmath248 and @xmath249 is a monotonically decreasing function for @xmath250 . hence , there is one and only one solution of @xmath245 at @xmath251 between @xmath252 and 1 , provided @xmath253 . otherwise , there is no solution , which means that there exists no intermediate steady state . expressing @xmath112 with the initial spin at @xmath150 as @xmath254 , we find that the condition @xmath255 gives @xmath256 thus , in the case @xmath257 , one intermediate steady state exists at @xmath251 , provided that @xmath258 we know @xmath259 from ( [ derivderivg ] ) , and so @xmath249 is concave upward for @xmath250 . if @xmath260 , then @xmath249 may have a local minimum at a certain @xmath261 between @xmath252 and 1 . recall that we are looking for the steady states which are reached from the position near @xmath168 and that the requirement for this is @xmath244 . hence , for the existence of such a steady state we need @xmath262 the first condition @xmath263 gives @xmath264 and the second one @xmath265 leads to @xmath266 . some tippe tops of group ii with @xmath267 satisfy @xmath268 as well as the conditions ( [ condsmallac ] ) , and thus @xmath269 . for such tippe tops , the corresponding @xmath249 has a local minimum between @xmath270 and 1 , where @xmath270 is a solution of @xmath271 . these tippe tops , therefore , have one intermediate steady state at @xmath251 between @xmath272 and @xmath270 when the condition @xmath273 is satisfied . see the discussion of case ( c ) in fig.[thetafgroupii ] . for the tippe tops of group ii with @xmath274 , there exists no intermediate state . we will see later , in the discussion of case ( d ) in fig.[thetafgroupii ] , that these tippe tops will turn over to @xmath28 once given a spin @xmath275 , since @xmath276 and , hence , @xmath277 for these tops . ( ii)group iii @xmath278 since @xmath279 should be positive , the allowed region of @xmath261 is @xmath280 with @xmath281 . eq.([derivg ] ) together with @xmath282 shows that @xmath249 is a monotonically decreasing function for @xmath280 . note that @xmath249 positively diverges when @xmath261 approaches @xmath283 from larger @xmath261 . hence , once @xmath244 is satisfied , @xmath245 has one and only one solution at @xmath251 such that @xmath284 . in other words , one intermediate steady state always exists at @xmath285 between 0 and @xmath286 for the tippe top of group iii , if the condition @xmath275 is satisfied . when @xmath287 gets larger , the angle @xmath22 gets closer to @xmath4 but never crosses @xmath4 . in order for @xmath22 to reach @xmath4 , @xmath287 should be infinite . now we know that there exists an intermediate steady state for the tippe top of group ii with property @xmath288 , when @xmath289 satisfies @xmath290 . also there is an intermediate steady state for the tippe top of group iii with @xmath291 if @xmath292 . let ( @xmath293 ) represent such a steady state so that ( @xmath293 ) are related by eqs.([intera ] ) and ( [ interb ] ) , and suppose this state to be perturbed to @xmath294 noting that @xmath295 and @xmath296 , we find that the perturbed state satisfies @xmath297 where @xmath298 ^ 2+\bigl(\frac{a}{c } -1 \bigr ) \sqrt{f_2(x_s ) } ~. \label{dxs}\ ] ] the details of the derivation of ( [ eqthetastability2 ] ) are given in appendix b. if @xmath299 , then @xmath300 . also when @xmath269 , we find that @xmath301 is still positive ( see appendix b ) . thus we observe from ( [ eqthetastability2 ] ) that @xmath302 with a negative constant at the intermediate steady state , which means that this state is indeed stable . finally it is emphasized that the spinning state of the tippe top of group i is stable at @xmath150 and the top will not turn over from the position near @xmath150 . on the other hand , the tippe top of group iii , when given a sufficiently large spin near the position @xmath150 , will tend to turn over and approach the steady state at @xmath22 but never up to the inverted position at @xmath28 . the tippe top of group ii will turn over to the inverted position at @xmath213 when it is given a sufficient initial spin . let us estimate the critical value @xmath7 so that the spinning top with @xmath303 reaches the inverted position . recall that jellett s constant ( [ jellettconstant3 ] ) is invariant during the turnover from @xmath150 to @xmath213 . from the relation @xmath304 , we obtain @xmath305 we already know that we need @xmath306 for the stability at @xmath213 , where @xmath307 is given in ( [ pistability4 ] ) . thus we find @xmath308 also from the instability condition of the tippe top of group ii at @xmath150 , we need @xmath231 , where @xmath232 is given by ( [ zerostability4 ] ) . hence the condition for the tippe top of group ii to turn over up to @xmath28 is that the initial spin @xmath289 should be larger than both @xmath309 and @xmath232 . in fact , we observe @xmath310 for the tippe top with @xmath311 , while @xmath312 for the tippe top with @xmath313 , where @xmath314 is given by ( [ rc ] ) . therefore , we obtain @xmath315 we now study the time evolution of the inclination angle @xmath8 from a spinning position near @xmath150 . simulations are made with various values of @xmath13 and @xmath14 , changing the input parameters @xmath15 and @xmath17 . other input parameters are the same as those given in ( [ inputparameter ] ) . initial conditions are @xmath316 rad , @xmath317 , and @xmath318 , and the initial value of the spin velocity @xmath6 is varied . since we have chosen a very small @xmath319 , we may consider @xmath6 as @xmath289 . the asymptotic value @xmath320 as a function of the initial spin velocity @xmath6 for tippe tops of group ii with various values of @xmath13 and @xmath14 ; ( a ) the one with @xmath321 and @xmath322 ; the others have @xmath323 but different @xmath14 such as ( b ) @xmath322 , ( c ) @xmath324 and ( d ) @xmath325 . ] figure [ thetafgroupii ] shows the asymptotic ( final ) angle of inclination , @xmath320 , as a function of @xmath6 for several types of tippe tops of group ii with different values of @xmath13 and @xmath14 ; ( a ) the one with @xmath321 and @xmath322 ; the others have @xmath323 but different @xmath14 such as ( b ) @xmath322 , ( c ) @xmath324 , and ( d ) @xmath325 . the asymptotic angle @xmath320 may be 0 or @xmath135 , or @xmath22 , the angle of a possible intermediate steady state . the symbols @xmath326 , @xmath327 , @xmath328 and @xmath329 represent the results for the tippe tops ( a ) , ( b ) , ( c ) and ( d ) , respectively , and the thin solid curves ( a ) , ( b ) and ( c ) are the trajectories obtained by solving ( [ eqforintermediate ] ) . we observe that the numerical results fall on the predicted curves . the values of @xmath330 , in units of rad / sec , for the tops ( a ) , ( b ) , ( c ) and ( d ) are 34.4(62.9 ) , 42.1(54.5 ) , 45.7(49.4 ) and 51.4(44.4 ) , respectively . in each case we see that the spinning state near @xmath150 is stable when @xmath331 . once @xmath6 gets larger than @xmath232 , the state becomes unstable and the tippe top turns over up to the asymptotic angle @xmath320 . for the tippe tops ( a ) and ( b ) the values of @xmath320 grow with @xmath6 from 0 to @xmath135 . on the other hand , the tippe top ( c ) satisfies @xmath269 with @xmath332 , and thus the intermediate steady state exists only at @xmath333 with @xmath334 , where @xmath335 is a solution of @xmath336 . we find @xmath337 . thus when @xmath6 gets larger than @xmath232 for the case of the tippe top ( c ) , the asymptotic angle @xmath320 jumps from 0 to @xmath335 . when @xmath338 , @xmath339 for the tops ( a ) , ( b ) and ( c ) . in the case of the tippe top ( d ) , we find @xmath340 and thus @xmath341 , which leads to @xmath342 . therefore , there is no intermediate steady state , and the asymptotic angle @xmath320 is 0 or @xmath135 depending on @xmath343 . the asymptotic value @xmath320 as a function of the initial spin velocity @xmath6 for tippe tops of group iii : ( a ) with @xmath344 and @xmath345 ; and ( b ) with @xmath344 and @xmath322 . ] we plot in fig.[thetafgroupiii ] the asymptotic angle @xmath320 as a function of @xmath6 for the tippe tops of group iii ; ( a ) with @xmath346 and @xmath347 and ( b ) with @xmath346 and @xmath348 . the symbols @xmath326 and @xmath327 represent the results of simulation for the tippe tops ( a ) and ( b ) , respectively , and the thin solid curves ( a ) and ( b ) are the trajectories obtained by solving ( [ eqforintermediate ] ) . we observe again that the numerical results on @xmath320 for both tops ( a ) and ( b ) fall on the predicted curves . the values of @xmath232 for the tops ( a ) and ( b ) are 13.3 and 23.5 rad / sec , respectively . in both cases the spinning position near @xmath168 is stable when @xmath6 is below @xmath232 . above @xmath232 , the value of @xmath320 grows with @xmath6 and approaches the fixed point @xmath4 . the values of @xmath4 for the tops ( a ) and ( b ) are 1.67 and 2.21 rad , respectively . for simulations we have used a modified version of the coulomb friction @xmath97 given in ( [ friction ] ) . the value @xmath320 is not affected by the strength of the coefficient @xmath99 . the strength of @xmath99 instead has an effect on the rate of rising of the tippe top . if we use another form than ( [ friction ] ) for the sliding friction , and moreover , it is expressed as a continuous function of @xmath0 and vanishes at @xmath102 , then we still expect that we get the same numerical results on @xmath320 _ vs. _ @xmath6 as shown in fig.[thetafgroupii ] and fig.[thetafgroupiii ] . this is due to the observation that the numerical value @xmath320 has fallen on the predicted curves which are derived from ( [ eqforintermediate ] ) and that we have obtained ( [ eqforintermediate ] ) using the property of @xmath97 which vanishes at the steady states together with @xmath0 . the time evolution of the angle @xmath8 for a tippe top of group ii from a spinning position near @xmath168 . ] figure [ thetaevolution ] shows the time evolution of the inclination angle @xmath8 for a tippe top of group ii from a spinning position near @xmath168 for various values of the initial spin velocity @xmath6 . input parameters and initial conditions are the same as before and we take @xmath321 and @xmath322 . the asymptotic angles @xmath320 which will be reached are 0 , 0.92 , 1.67 , 2.49 , @xmath135 and @xmath135 rad for @xmath34930 , 40 , 50 , 60 , 70 and 80 rad / sec , respectively . simulations with a modified version of the coulomb friction ( [ friction ] ) show that the larger value of @xmath6 is given , the faster the rate of rising becomes . we have examined an inversion phenomenon of the spinning tippe top , focusing our attention on its relevance to the gyroscopic balance condition ( gbc ) , which was discovered by moffatt and shimomura in the study of the spinning motion of a hard - boiled egg . in order to analyze the gbc in detail for the case of the tippe top , we introduce a variable @xmath2 given by ( [ xi ] ) so that @xmath3 corresponds to the gbc , and study the behavior of @xmath2 . contrary to the case of the spinning egg , the gbc is not satisfied initially for the tippe top . the simulation shows that , starting from a large positive value @xmath9 , the variable @xmath2 for the tippe tops which rise , soon fluctuates around a negative but small value @xmath10 such that @xmath11 . thus for these tippe tops , the gbc , though it is not fulfilled initially , will soon be satisfied approximately . once @xmath2 fluctuates around the value @xmath12 , these tops become unstable and start to turn over . on the other hand , in the case of the tippe tops which do not turn over , @xmath2 remains positive around @xmath9 or changes from positive @xmath9 to negative values and then back to positive values close to @xmath9 again . under the gbc the governing equations for the tippe top are much simplified and , together with the geometry of the tippe top , we obtain a first - order ode for @xmath8 in the following form @xcite ( see ( [ equfortheta ] ) or ( [ equtheta ] ) ) : @xmath350 it is noted that this equation has a remarkable resemblance to the renormalization group ( rg ) equation for the effective coupling constant @xmath351 , @xmath352 which appears in quantum field theories for critical phenomena @xcite and high energy physics @xcite . here in ( [ rgequation ] ) , @xmath353 is expressed as @xmath354 with a dimensionless scale parameter @xmath355 . provided that @xmath356 has a zero at @xmath357 , we find that , if @xmath358 , then @xmath359 as @xmath360 ( @xmath361 ) , and while if @xmath362 , @xmath359 as @xmath363 ( @xmath364 ) . the limiting value @xmath365 of @xmath366 is known as the ultraviolet ( infrared ) fixed point in the former ( latter ) case . similarity between the two equations , ( [ equthetalast ] ) and ( [ rgequation ] ) , and the notion of the rg equation brought us to a consequence that tippe tops are classified into three groups , depending on the values of @xmath13 and @xmath14 . a resemblance of eq.([eqforstability ] ) to the rg equation also gave us a hint that eq.([eqforstability ] ) might serve as a criterion for stability of the steady state in sec . the criterion ( 4.2 ) is a first - order ode for the ( perturbed ) inclination angle @xmath23 , and the results derived from this criterion coincide with those by es and bmr which are obtained by mathematically rigorous methods . the key ingredients in the process of arriving at this first - order ode are the order estimation in @xmath99 near the steady states and an intuitive analysis of the equations of motion . the criterion ( 4.2 ) can also be applied to the stability analysis of other spinning objects . in fact we have applied ( [ eqforstability ] ) to the spinning motion of spheroids ( prolate and oblate ) which was recently examined in detail by msb @xcite , and we have obtained consistent results with theirs . finally we have assumed , in the present work , a modified version of coulomb law ( [ friction ] ) for the sliding friction , since coulomb friction ( [ coulomb ] ) is non - analytic and undefined at @xmath100 . on the other hand , cohen used coulomb friction in his pioneering work on the tippe top @xcite , and analyzed its spinning motion numerically for the first time . he reported the result of a sample simulation in fig.5 of his paper @xcite . the coulomb friction is realistic provided that @xmath19 is away from zero , but its application to the spinning motion of the tippe top is very delicate . near steady states ( i.e. , near @xmath150 or @xmath135 or @xmath22 ) , @xmath0 almost vanishes ( see , for example , fig.[travelocity ] ( a ) and ( b ) ) . and there the @xmath111- and @xmath114-components of @xmath367 are changing signs rapidly and moreover non - analytically , and so are the components of friction , @xmath368 and @xmath145 . coulomb friction may not be adequate to be applied to such a situation . in fact , kane and levinson @xcite argued against the work of cohen , because it did not include adequate provisions for transitions from sliding to rolling and vice versa . they reanalyzed the simulation of cohen , assuming coulomb law for sliding friction , but also providing an algorithm that rolling begins when @xmath369 ( with @xmath370 1m / sec ) is satisfied , together with another algorithm for the transition from rolling to sliding . they found that a transition from sliding to rolling occurs soon after the motion has begun and that values of @xmath8 remain below 0.077 rad thereafter . or @xcite adopted a hybrid friction law adding viscous friction , which is linearly related to @xmath0 , to coulomb friction . other frictional forces such as the one which is due to pure rotation about the normal at the point of contact might have some effect . after all it is safe to say that we have understood general features of the tippe top inversion . but it would be not until we have had thorough knowledge of frictional force that we completely understood the inversion phenomena of the tippe top . _ and yet , it flips over_. we thank tsuneo uematsu for valuable information on the spinning egg and the tippe top . we also thank yutaka shimomura for introducing us to the paper @xcite and for helpful discussions . * we enumerate the equations of motion which are used to analyze the spinning motion of the tippe top : @xmath371 there exists an intermediate steady state for the tippe top of group ii with property @xmath288 , when an initial spin @xmath289 satisfies @xmath266 . there is also an intermediate steady state for the tippe top of group iii if @xmath231 . in this appendix we show that these steady states are stable . near the steady states the primary balance condition ( [ stabilityomega ] ) holds at leading order in @xmath99 . differentiating both sides of ( [ stabilityomega ] ) with respect to @xmath353 , we obtain @xmath372 using ( [ euleromega ] ) and ( [ eulern ] ) , and eliminating @xmath373 and @xmath193 , we find @xmath374 let ( @xmath293 ) represent an intermediate steady state so that @xmath160 , @xmath161 and @xmath22 are related by ( [ intera ] ) and ( [ interb ] ) , and suppose this state to be perturbed to @xmath294 since @xmath295 and @xmath296 , the perturbed state satisfies @xmath375 where @xmath376 at leading order in @xmath99 , we have @xmath377 ( recall @xmath378 ) , and thus we obtain from ( [ vpy ] ) , @xmath379 now we expect that the perturbed state still satisfies the primary balance condition ( [ stabilityomega ] ) , since @xmath380 and @xmath368 are @xmath381 . then a variation around the steady state gives @xmath382 where @xmath383 is given by ( [ ss ] ) . also taking a variation of jellett constant ( [ jellettconstant3 ] ) around the steady state ( and then , of course , we have @xmath384 ) , we obtain @xmath385 from ( [ varia ] ) and ( [ varib ] ) , @xmath386 and @xmath387 are expressed in terms of @xmath23 as @xmath388 inserting these expressions into ( [ deltafy ] ) , and then we obtain from ( [ eqthetastability1 ] ) @xmath389 where @xmath298 ^ 2+\bigl(\frac{a}{c } -1 \bigr ) \sqrt{f_2(x_s ) } ~,\ ] ] and @xmath390 , and eqs.([f1 ] ) and ( [ f2 ] ) have been used . if @xmath391 , then @xmath300 . also when @xmath269 , @xmath301 is still positive , which is explained as follows : the expression of ( [ derivg ] ) shows that the function @xmath392 is related to @xmath393 as @xmath394 ^ 2}d(x)~.\ ] ] when @xmath269 , an intermediate steady state at @xmath395 exists provided @xmath263 and @xmath265 . at that point @xmath396 is negative , and thus @xmath301 is positive . ebenfeld and scheck @xcite analyzed the stability of the spinning tippe top using the total energy as a liapunov function and gave the stability criteria for the steady states . we take a different approach to this stability problem . first the system is perturbed around the steady state . then , using the equations of motion and under the linear approximation , we obtain a first - order ode for @xmath23 of the form given in ( [ eqforstability ] ) . we make use of this equation and give a different stability criterion . in this appendix we show that both approaches are equivalent and thus they lead to the same conclusions on the stability conditions of the steady states . es wrote the total energy of the spinning top as the sum of two terms ( es-(33 ) ) ) of ref.@xcite as es-(@xmath397 ) . the jellett constant @xmath355 defined by es is related to our @xmath112 as @xmath398 . ] @xmath399 the second of which contains all the terms that will vanish at the steady states , while the first depends on @xmath400 and jellett s constant @xmath112 . in terms of the parameters used in this paper , @xmath401 and @xmath402 are expressed as follows : @xmath403}{rg(\eta_3 ) } \biggr\}^2~ , \label{e2}\end{aligned}\ ] ] with @xmath404 note that es set @xmath405 . the condition @xmath406 together with @xmath407 leads to the three solutions of the steady states : ( i ) vertical spin state at @xmath150 ( [ steadythetazero ] ) , ( ii ) vertical spin state at @xmath213 ( [ steadythetapi ] ) , and ( iii ) intermediate states ( [ eqforintermediate ] ) , or equivalently , ( [ intera]-[interb ] ) . it is recalled that we have obtained these solutions starting from equations of motion . at these steady states @xmath402 vanishes . for intermediate steady states , the factor @xmath408/rg(\eta_3 ) \}$ ] in ( [ e2 ] ) reduces to zero , due to ( [ intera]-[interb ] ) and jellett s constant given in ( [ jellettconstant3 ] ) . now we show that the criterion , eq.([eqforstability ] ) , for the stability of the steady states is equivalent to the one derived by es @xcite . for the stability analysis of the steady states , the order estimation in @xmath99 near the steady states is important , which has been pointed out by msb in their work on the linear stability analysis of the spinning motion of spheroids @xcite . as explained at the beginning of sec . 4.2 , near the steady states we have @xmath179 , @xmath409 , @xmath410 , and @xmath188 . since @xmath402 is already @xmath411 ( recall that it vanishes at the steady states ) , we have @xmath412 , while @xmath413 . thus near the steady states , the energy equation ( [ energyeq ] ) is written at leading order in @xmath99 as @xmath414 suppose the steady states to be perturbed to @xmath415 . since @xmath416 , we have @xmath417 , and @xmath418 is expanded as @xmath419 where we have used the fact @xmath420 . meanwhile @xmath421 is shown to be expressed as @xmath422 actually we have already obtained the expressions ( [ vpythetazero ] ) and ( [ vpythetapi ] ) for @xmath423 near the steady states at @xmath150 and @xmath213 , respectively . also near the intermediate steady states , @xmath386 and @xmath387 are expressed in terms of @xmath23 as ( [ deltans ] ) and ( [ deltaomegas ] ) , respectively , and thus we obtain ( [ vrotpydelta ] ) . now using eqs.([energyeqleading])-([vrotpydelta ] ) we find @xmath424 which means that we can identify @xmath162 in ( [ eqforstability ] ) as @xmath425 hence we conclude that the following assertions are equivalent : _ a steady state is stable ( unstable ) _ @xmath426 _ @xmath162 is negative ( positive ) _ @xmath427 is positive ( negative)_. in fact , es showed that if the quantity ( es-(39 ) ) with the upper sign is positive , then @xmath427 is positive at @xmath428 and the non - inverted rotating motion is liapunov stable . on the other hand , starting from the equations of motion we derived @xmath162 and obtained the condition ( [ zerostability2 ] ) for the stability of the rotating motion at @xmath428 . it is easily seen that the statement that the quantity ( es-(39 ) ) with the upper sign is positive is equivalent to the inequality given in ( [ zerostability2 ] ) , once we know that jellett s constant at @xmath428 is given by @xmath429 . similarly , if the quantity ( es-(39 ) ) with the lower sign is positive , then @xmath427 is positive at @xmath430 and the completely inverted rotating motion is liapunov stable . the condition that the quantity ( es-(39 ) ) with the lower sign is positive is equivalent to the inequality given in ( [ pistability2 ] ) . note , this time , @xmath431 . as for the intermediate steady state @xmath432 , es stated that if the steady state exists and the quantity ( es-(40 ) ) is negative , then @xmath427 is positive and the state is liapunov stable . in sec.4.2.3 we have shown that the stability of the intermediate steady state is determined by the sign of @xmath301 given in ( [ dxs ] ) . now it is interesting to note that @xmath301 is related to ( es-(40 ) ) as follows : @xmath433 ^ 2}{ac^2}\times ( { \rm es}.(40))~.\ ] ] hence the condition that the quantity ( es-(40 ) ) is negative is equivalent to @xmath434 . we have seen in sec.4.2.3 that there exists an intermediate steady state for the tippe top of group ii and also of group iii . ( we have not considered a possible intermediate steady state for group i , since such a state , even if it exists , can not be reached from the initial spinning position near @xmath168 . ) for these steady states , we have shown , in appendix b , that @xmath301 is positive and , therefore , the states are stable . recently bou - rabee , marsden and romero [ bmr ] treated tippe top inversion as a dissipation - induced instability . they showed that the modified maxwell - bloch ( mmb ) equations are a normal form for tippe top inversion and , using the mmb equations and an energy - momentum argument , they gave criteria for the stability on the non - inverted and inverted states of the tippe top @xcite . although we have not explored the connections between the mmb equations and the first - order ode ( [ eqforstability ] ) for @xmath23 , we show in appendix d that our results on the stability of the vertical spin states are consistent with the criteria provided by bmr . actually , rewritten in terms of dimensional parameters and classification criteria used in this paper , the expressions of those criteria become more transparent and they lead to the same stability conditions as ours for the vertical spinning states . besides , although bmr did not mentioned , the classification of tippe tops into three groups , group i , ii , and iii , according to the behaviors of spinning motion , is possible from the close examination of those criteria . bmr used the moments of inertia defined as the ones about the principal axes _ attached to the center of sphere instead of the center of mass_. the correspondence between the parameters used by bmr and ones in this paper are as follows : @xmath435 where @xmath436 is the spin rate of the initially standing equilibrium solution ( we added a subscript bmr to distinguish from our @xmath76 ) , and the dimensionless bmr s jellett " constant , @xmath437 , is restricted to have a certain value , i.e. , @xmath438 . also bmr expressed the vector from the center of sphere to the center of mass @xmath439 ( in the bmr notation @xmath440 ) as @xmath441 , where @xmath442 is a unit vector along the symmetry axis . using the tippe top modified maxwell - bloch equations , bmr obtained the stability criteria for the non - inverted state which are given by the three inequalities in ( bmr-(5.3 ) ) ) of ref.@xcite as bmr-(@xmath397 ) . ] . they took @xmath443 ( upward ) in ( bmr-(5.3 ) ) . since the non - inverted state has the center of mass below the center of sphere , we have @xmath444 , and thus @xmath445 . the first inequality of ( bmr-(5.3 ) ) is rewritten as @xmath446 , which is always satisfied . apart from some irrelevant positive constants , the second and third inequalities are expressed , respectively , as @xmath447 ^ 2c}\bigl(1-\frac{a}{r } \bigr)\frac{a}{c}+\bigl(1-\frac{a}{r } \bigr)^5 \frac{\nu^2}{\sigma^2}-\frac{a}{c}+\bigl(1-\frac{a}{r } \bigr)>0~ , \label{2ndineq } \\ & & -\bigl\ { \frac{a}{c}- \bigl(1-\frac{a}{r } \bigr ) -\frac{mga}{[n(\theta=0)]^2c } \bigl(1-\frac{a}{r } \bigr)^2\bigr\}>0~. \label{3rdineq}\end{aligned}\ ] ] from these inequalities , we find : * in the case @xmath448 , i.e. , for the tippe top of group i , the above inequalities are always satisfied . in other words , the non - inverted states ( @xmath150 ) of group i are always stable . * in the case @xmath449 , i.e. , for the tippe tops of group ii or iii , the inequality ( [ 3rdineq ] ) is satisfied if @xmath450 ^ 2<\frac{mga}{c\ { \frac{a}{c } -(1-\frac{a}{r})\}}\bigl(1- \frac{a}{r } \bigr)^2 , \label{d4}\ ] ] which is the same requirement given in ( [ zerostability3 ] ) for the stability of the tippe top of group ii or iii . note that the inequality ( [ 2ndineq ] ) is automatically satisfied when both @xmath449 and inequality ( [ 3rdineq ] ) hold . the inequalities ( bmr-(5.3 ) ) , which were derived as the stability criteria for the non - inverted state , can also be used for the stability criteria for the inverted state , but with some replacements . since @xmath443 ( upward ) , the inverted state has the center of mass above the center of sphere . thus we have @xmath451 and @xmath452 . changing variables in inequalities ( [ 2ndineq ] ) and ( [ 3rdineq ] ) as @xmath453 , @xmath454 , and @xmath455 ^ 2\rightarrow [ n(\theta\!=\!\pi)]^2 $ ] , we obtain @xmath456 ^ 2c}\bigl(1+\frac{a}{r } \bigr)\frac{a}{c}+\bigl(1+\frac{a}{r } \bigr)^5 \frac{\nu^2}{\sigma^2}-\frac{a}{c}+\bigl(1+\frac{a}{r } \bigr)>0~ , \label{2ndineqpi } \\ & & -\bigl\ { \frac{a}{c}- \bigl(1+\frac{a}{r } \bigr ) + \frac{mga}{[n(\theta=\pi)]^2c } \bigl(1+\frac{a}{r } \bigr)^2\bigr\}>0~ , \label{3rdineqpi}\end{aligned}\ ] ] for the stability for the inverted state . from the above two inequalities , we see : * in the case @xmath457 , i.e. , for the tippe top of group iii , the inequality ( [ 3rdineqpi ] ) is never satisfied . therefore , the inverted states ( @xmath213 ) of group iii are always unstable . * in the case @xmath458 , i.e. , for the tippe top of group i or ii , the inequality ( [ 3rdineqpi ] ) is satisfied if @xmath459 ^ 2>\frac{mga}{c\ { ( 1+\frac{a}{r } ) -\frac{a}{c}\}}\bigl(1 + \frac{a}{r } \bigr)^2 , \label{d7}\ ] ] which is the same requirement given in ( [ pistability3 ] ) for the stability of the tippe top of group i or ii at @xmath213 . the inequality ( [ 2ndineqpi ] ) is automatically satisfied when both @xmath458 and inequality ( [ 3rdineqpi ] ) hold . actually , bmr derived also the stability criteria for the inverted state , taking @xmath460 , which are given by the three inequalities in ( bmr-(5.4))\!+\ ! \nu^2(1\!-\ ! e^\star)^7\!+\ ! ( 1\!-\ ! e^{\star})^3\mu e^{\star}fr^{-1 } ( 1\!-\!\mu { e^{\star } } ^2)>0 $ ] . the error is traced back to the missing factor of @xmath461 in the expression of @xmath462 in bmr-(4.2 ) . ] . of course , we can use them to obtain the stability conditions for the inverted state . taking now @xmath463 and @xmath464 in the second and third inequality in ( bmr-(5.4 ) ) , we reach the same conclusions , ( bi ) and ( bii ) . bmr discussed in ref . @xcite about the heteroclinic connection between the non - inverted and inverted states of the tippe top . they used an energy - momentum argument to determine the asymptotic states of the tippe top and obtained the explicit criteria for the existence of a heteroclinic connection , which are given in theorem 6.2 and the appendix of ref . @xcite . in terms of the classification criteria and conditions obtained in this paper , the statement in bmr on the existence of a heteroclinic connection can be restated as follows : ( i ) a tippe top must belong to group ii in order to have a heteroclinic connection . ( ii ) further more , the initial spin @xmath289 should be larger than @xmath232 ( eq.([zerostability4 ] ) ) and @xmath309 ( eq.([critical1 ] ) ) so that a tippe top becomes unstable at @xmath150 and reaches the inverted position . the requirements @xmath465 and @xmath466 , respectively , correspond to the criteria @xmath467 and @xmath468 in theorem 6.2 in bmr . h. k. moffatt , y. shimomura and m. branicki , dynamics of an axisymmetric body spinning on a horizontal surface . part i : stability and the gyroscopic approximation , " proc . a * 460 * , 3643 - 3672 ( 2004 ) .
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we reexamine a very classical problem , the spinning behavior of the tippe top on a horizontal table .
the analysis is made for an eccentric sphere version of the tippe top , assuming a modified coulomb law for the sliding friction , which is a continuous function of the slip velocity @xmath0 at the point of contact and vanishes at @xmath1 .
we study the relevance of the gyroscopic balance condition ( gbc ) , which was discovered to hold for a rapidly spinning hard - boiled egg by moffatt and shimomura , to the inversion phenomenon of the tippe top .
we introduce a variable @xmath2 so that @xmath3 corresponds to the gbc and analyze the behavior of @xmath2 .
contrary to the case of the spinning egg , the gbc for the tippe top is not fulfilled initially .
but we find from simulation that for those tippe tops which will turn over , the gbc will soon be satisfied approximately .
it is shown that the gbc and the geometry lead to the classification of tippe tops into three groups : the tippe tops of group i never flip over however large a spin they are given .
those of group ii show a complete inversion and the tippe tops of group iii tend to turn over up to a certain inclination angle @xmath4 such that @xmath5 , when they are spun sufficiently rapidly .
there exist three steady states for the spinning motion of the tippe top . giving a new criterion for stability
, we examine the stability of these states in terms of the initial spin velocity @xmath6 . and we obtain a critical value @xmath7 of the initial spin which is required for the tippe top of group ii to flip over up to the completely inverted position .
( 5,2)(-270,-520 ) ( 2.3,35)ynu - hepth-05 - 102 ( 2.3,20)july 2005
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@xmath3-deuteron compton scattering probes the structure of the deuteron and provides necessary information that may allow the extraction of neutron properties , such as neutron electric and magnetic polarizabilities . the differential cross section of unpolarized @xmath3-deuteron compton scattering has been measured at incident photon energies of 49 mev and 69 mev with 10% uncertainty @xcite . theoretical calculations based on potential models and taking the nucleon polarizabilities as inputs find reasonably good agreement with data @xcite but are not sufficient to give tight constraints on the nucleon polarizabilities . in comparison to potential model calculations , a model independent , parameter free and analytic computation of unpolarized @xmath3-deuteron compton scattering based on a recently developed nucleon - nucleon effective field theory @xcite was presented in @xcite . contributions up to next - to - leading order ( nlo ) in the effective field theory expansion including diagrams contributing to the nucleon polarizabilities give excellent agreement with the data at 49 mev and reasonable agreement with the data at 69 mev . at this order ( nlo ) , the agreement with the data is comparable to the agreement between the data and potential model calculations @xcite , while inclusion of the next - next - to - leading order ( nnlo ) contributions should reduce the theoretical uncertainty from 10% to a few percent level . in addition to @xmath3-deuteron compton scattering , the effective field theory technique also successfully describes the @xmath4 scattering phase shifts up to center - of - mass momenta of @xmath5 per nucleon @xcite in all partial waves . the electromagnetic moments , form factors @xcite and polarizability @xcite of the deuteron as well as parity violation in the two - nucleon sector @xcite also have been explored with this new effective field theory . the results all agree with data ( where available ) within the uncertainty from neglecting higher order effects . in the case of polarized @xmath3-deuteron compton scattering , no experiments have been performed so far . planned experiments at tunl to examine the gerasimov - drell - hearn ( gdh ) sum rule using a circularly polarized photon beam will be the first attempt to study vector polarized @xmath3-deuteron compton scattering . experiments using tensor polarized targets are also technically feasible . existing technologies like spin - exchange optical pumping @xcite , which will be applied in the blast polarized @xmath6h and @xmath7h laser driven sources , and free electron lasers can provide high quality polarized deuteron targets and photon beams . the question is , what kind of physics can be measured in such an experiment ? is it interesting enough to motivate the experiments ? theoretically , four vector form factors ( corresponding to the @xmath8 interaction of the deuteron field ) and six tensor form factors ( corresponding to @xmath9 ) are identified in @xcite with the lower multipole contributions calculated using dispersion relations and potential models . in this paper , we perform an analytic calculation of the differential cross section of the tensor polarized @xmath3-deuteron compton scattering cross section to first non - vanishing order in the effective field theory expansion . at this order , the contributions to the cross section from the @xmath9 amplitude are the interference terms between the leading electric coupling diagrams and the subleading single potential pion exchange diagrams or the subleading magnetic moment coupling diagrams . thus the pion effects are of leading order . we can further isolate the pion contribution by setting the photon scattering angle to be 90@xmath10 at this angle , only the pion term contributes at this order to the tensor polarized differential cross section . this provides a clean way to study the photon pion dynamics in the two nucleon sector . the experimental precision required to measure these effects will also be discussed in the following sections . the process we will focus on is the low energy ( below pion production threshold ) tensor polarized compton scattering @xmath11 where the incident photon of four momentum @xmath12 in the deuteron rest frame ( lab frame ) scatters off the polarized deuteron target to an outgoing photon of four momentum @xmath13 . the polarization of photons and final state deuteron are not detected . for convenience , we will work in the lab frame and choose the @xmath14 direction to be the ( 0,0,1 ) direction . the scattering amplitude can be written in terms of scalar , vector and tensor form factors @xmath15 and @xmath16 corresponding to the @xmath17 interactions of the deuteron field , @xmath18 where @xmath19 and @xmath20 are the polarization vectors of the initial and final state deuterons . using the power counting described in @xcite , the scalar form factor @xmath21 contributes to the amplitude starting at leading order ( lo ) in the effective field theory expansion , while the tensor form factor @xmath16 and the vector form factor @xmath22 contribute starting at next - to - leading order ( nlo ) and next - next - to - leading order ( nnlo ) respectively . squaring the amplitude to form the cross section , the lo contribution to the cross section only comes from the @xmath23 term which is independent of the deuteron target polarization , while the target polarization dependent @xmath21 and @xmath16 interference term contributes at nlo . note that the vector form factor @xmath22 does not contribute to the cross section through the @xmath21 and @xmath22 interference term since the final state deuteron polarization is not detected . it is useful to define the tensor polarized differential cross section @xmath24 as a tensor combination of polarized differential cross sections to eliminate the lo polarization independent effect and make the @xmath21 and @xmath16 contribution become leading . @xmath24 is defined by @xmath25 \ \quad , \ ] ] where @xmath26 indicates the polarization of the deuteron target . we have @xmath27 = 4.0 in 0.8 in 0.1 in .2 in to predict @xmath24 , we need expressions for @xmath21 and @xmath16 . the calculation of the scalar form factor @xmath21 was carried out up to nlo in @xcite . two distinct structures contributing to @xmath21 can be parameterized by electric and magnetic form factors @xmath28 and @xmath29 as @xmath30 where @xmath31 and @xmath32 are the polarization vectors of the initial and final state photons , @xmath33 and @xmath34 are unit vectors in the direction of @xmath14 and @xmath35 . at lo ( denoted by a superscript on the form factors ) , @xmath28 receives contributions from the electric coupling of the @xmath36 operator ( minimal coupling ) , as shown in fig . [ fig : gdeleclead ] , to be @xmath37 \nonumber \\ & + & \left[\omega \rightarrow -\omega \right]\quad , \label{f0}\end{aligned}\ ] ] where cos @xmath38 is the cosine of the angle between the incident and outgoing photons and @xmath39 is the deuteron binding momentum . in eq . ( [ f0 ] ) terms suppressed by additional factors of @xmath40 ( i.e. , recoil effects ) are neglected since they only contribute to nnlo differential cross sections . at lo , the magnetic contribution vanishes , @xmath41 we do not show the nlo results for @xmath28 and @xmath29 here since we only calculate @xmath24 to its first non - vanishing order . the tensor form factor @xmath16 has more distinct structures . however , up to nlo , after neglecting pion contributions suppressed by factors of @xmath42 , @xmath16 can be parameterized by electric and magnetic form factors @xmath43 and @xmath44 as @xmath45 @xmath43 and @xmath44 are related to the form factors @xmath46 and @xmath47 defined in @xcite by a normalization factor . at lo , @xmath43 and @xmath44 vanish , @xmath48 = 4.0 in .2 in at nlo @xmath43 receives contributions from the electric coupling of the one potential pion exchange diagrams shown in fig . [ fig : polsubpi ] and gives a renormalization scale @xmath49 independent contribution @xmath50 \log \left [ { m_\pi + 2\gamma \over \mu } \right ] \right.\nonumber\\ & & \left . \ m_\pi ^2(m_\pi ^2 + 4\gamma ^2 - 4m_n\omega ) \log \left [ { \displaystyle { m_\pi + 2\sqrt{\gamma ^2-m_n\omega -i\epsilon } \over \mu } } \right ] \right.\nonumber\\ & & \left . \ -\ 8\left [ m_\pi ^2(2m_\pi ^2 - 4\gamma ^2+m_n\omega ) -3m_n^2\omega ^2\right ] \log \left [ { \displaystyle { m_\pi + \gamma + \sqrt{\gamma ^2-m_n\omega -i\epsilon } \over \mu } } \right ] \right.\nonumber\\ & & \left . \ + \ { 32m_n\omega \left [ 2m_\pi \gamma ^2-m_n\omega ( m_\pi -\gamma ) \right ] \over m_\pi + \gamma + \sqrt{\gamma ^2-m_n\omega -i\epsilon } } \right.\nonumber\\ & & \left . \ + \ { \displaystyle { 4m_n^2\omega ^2 ( 5m_\pi ^2 - 12\gamma ^2 ) \over ( m_\pi + 2\gamma ) ^2\ } } \right.\nonumber\\ & & \left . \ -\ 2\left [ m_\pi ^3 + 4\left ( 2m_\pi ^2-m_n\omega \right ) ( m_\pi -\gamma ) \right ] \left ( \gamma -\sqrt{\gamma ^2-m_n\omega -i\epsilon } \right ) \right\}\nonumber\\ & & \left . \ + \ \left\{\omega \rightarrow -\omega \right\}\right . \ \ \ , \label{fpi}\end{aligned}\ ] ] where @xmath51 mev is the pion decay constant . in the computation of the pion diagrams , the pion propagators with photon momentum dependence have the form @xmath52 where @xmath53 is the loop momentum . @xmath54 is of higher order in the power counting compared with the other scales in the propagators and can be neglected . we keep the first term in the @xmath14 expansion of the pion propagators to get the result of eq . ( [ fpi ] ) . the error of this approximation is of order @xmath55 of eq . ( [ fpi ] ) and of even higher order in powers of @xmath55 at @xmath56 and @xmath57 . because the error is formally of nlo in power counting , strictly speaking we have not presented the complete calculation at nlo . however , numerically the terms omitted are nnlo . = 1.2 in 0.1 in .2 in the magnetic moment coupling diagrams shown in fig . [ fig : maglead ] contribute to @xmath44 at nlo as @xmath58 where @xmath59 and @xmath60 are isoscalar and isovector nucleon magnetic moments in nuclear magnetons , with @xmath61 and @xmath62 . the leading order nucleon nucleon scattering amplitudes contribute to the magnetic moment diagrams through the @xmath63 bubble chains and has the expression @xcite @xmath64 the renormalization scale @xmath49 dependence in the denominator is canceled by the @xmath49 dependence of @xmath65 and @xmath66 , as required . values of @xmath65 and @xmath66 have been determined from nucleon nucleon scattering in the @xmath67 and @xmath68 channels @xcite to be @xmath69 and @xmath70 at @xmath71 having obtained the leading non - vanishing contributions for @xmath21 and @xmath16 , we now give the leading non - vanishing expressions for @xmath24 . at lo , @xmath24 vanishes , @xmath72 while at nlo , @xmath73(1+\cos ^2\theta ) + 2re[f_0^{lo}g_2^{nlo^{*}}]\cos \theta \right ] \quad .\ ] ] = 3.7 in .2 in there are two terms in @xmath74 the first term comes from the interference between the lo electric coupling diagrams and the nlo one potential pion exchange diagrams . the second term comes from the interference between the lo electric coupling diagrams and the nlo magnetic moment coupling diagrams . in fig . [ fig : dsig24969 ] , the one pion interference term contributing to @xmath75 are plotted as the dashed curves at incident photon energies of 49 and 69 mev . the angular distributions are dominated by the @xmath76 factor that comes from the sum of the inner product squares of the incident and outgoing electric fields over the two different modes ( electric fields parallel or perpendicular to the scattering plane ) . the small asymmetry comes from the finite size of the deuteron . the magnetic moment interference terms are plotted as the dotted curves . the @xmath77 factor comes from the sum of the inner product of the incoming and outgoing electric fields times the inner product of the incoming and outgoing magnetic fields over the two different modes . the sum of the two terms that form the nlo contributions to @xmath78 are plotted as solid lines . the result is dominated by the pion interference terms . higher order corrections could be as large as 30% of the contributions we show in this figure . = 3.7 in .2 in = 3.7 in .2 in = 4.0 in .2 in to indicate the relative size of @xmath79 we write the @xmath80 and @xmath81 differential cross sections in terms of the unpolarized and tensor polarized differential cross sections , @xmath82 and @xmath24 , as @xmath83 the unpolarized differential cross section , @xmath84 \cos \theta \right ] \quad , \ ] ] up to nlo can be found in @xcite . note that we have used @xmath85 @xmath86 = @xmath85 @xmath87 in eq . ( [ poldsig ] ) as photons are not circularly polarized . this is a consequence of parity conservation . in fig . [ fig : polrd4969 ] , we plot the @xmath80 and @xmath81 differential cross sections as dotted and dashed curves respectively at photon energies of 49 and 69 mev . the difference between each set of dotted and dashed curves gives @xmath88 while the weighted sum gives @xmath89 at both energies , @xmath78 at backward angles can be extracted if the error bars in @xmath80 and @xmath81 differential cross sections are @xmath90 , as that of the unpolarized experiment @xcite performed four years ago . with error bars of 4% , the forward angle part of @xmath91 would also be measurable . in these estimates , we have assumed that the higher order corrections to @xmath92 are 30% . we can isolate the one pion interference term from @xmath93 by going to @xmath94 . as can easily be seen from eq . ( [ sig2 ] ) , at @xmath94 the magnetic interference term vanishes . this provides a clean way to study the photon pion dynamics in the two nucleon sector . in fig . [ fig:90deg4969 ] , the nlo differential cross sections at @xmath94 for deuteron polarizations @xmath80 and @xmath81 are plotted as the dotted and dashed curves while @xmath78 is plotted as the solid curve . again , with measurements of the @xmath80 and @xmath95 polarization made to within 7% , the pion interference term effects would be observable for photon energies between 40 and 80 mev with the higher order effect estimated to be 30% . note that the effective field theory expansion power counting is expected to break down at photon energy of about 100 mev ( recent works by mehen and stewart @xcite suggest that in fact the breakdown scale may be much higher ) . but the neglected nlo terms which are suppressed by factors of @xmath42 could contribute a @xmath96 effect at 80 mev . thus the 80 mev photon energy sets an upper bound for the calculations performed in this paper . it is also interesting to compare our results with those from potential model calculations @xcite . in fig . [ fig : f2g2 ] , we show the fist non - vanishong order ( nlo ) results of @xmath43 and @xmath44 multiplied by a constant to compare with the form factors @xmath46 and @xmath47 calculated and plotted in @xcite . the agreement on @xmath44 is @xmath97 while the agreement on @xmath43 is @xmath98-@xmath99 . we have presented analytic expressions for the differential cross section of tensor polarized @xmath3-deuteron compton scattering in an effective field theory expansion . the first non - vanishing contributions are the interference terms between the leading electric coupling diagrams and the subleading single potential pion exchange diagrams , and between the leading electric coupling diagrams and the subleading magnetic moment coupling diagrams . at 90@xmath1 photon scattering angle , only the pion term contributes at this order to the tensor polarized differential cross section . this provides a clean way to study the photon pion dynamics in the two nucleon sector . for photon energy between 40 and 80 mev , the one pion interference term contributions would be measurable at 90@xmath1 provided the uncertainty in the @xmath3-deuteron compton scattering experiments are @xmath2 . at backward angles , the magnetic interference term adds to the pion term to give an effect which could be seen if the uncertainty in the measurements are @xmath100 . the author would like to thank martin savage , roxanne springer , harald griesshammer , jerry miller , jon karakowski , and daniel phillips for helpful discussions and hartmuth arenhovel for correspondence . this work is supported in part by the u.s . dept . of energy under grants de - fg03 - 97er41014 .
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the differential cross section for @xmath0-deuteron compton scattering from a tensor polarized deuteron is computed in an effective field theory .
the first non - vanishing contributions to this differential cross section are the interference terms between the leading electric coupling diagrams and the subleading single potential pion exchange diagrams or the subleading magnetic moment coupling diagrams . at 90@xmath1 photon scattering angle
, only the pion term contributes at this order to the tensor polarized differential cross section .
this provides a clean way to study the photon pion dynamics in the two nucleon sector .
the effect is measurable for photon energies between 40 and 80 mev provided the uncertainty in the measured cross sections are @xmath2 .
# 1#1 # 1#1 # 1| # 1| # 1#2 # 11.5ex-16.5mu # 1 # 1#2#3#4#1 * # 2 * , # 3 ( # 4 )
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You are an expert at summarizing long articles. Proceed to summarize the following text:
the superconducting transition in conventional superconductors is rather well described by mean field theories , essentially because in the coherence volume @xmath9 a large number of pairs is present . on the contrary , in cuprate superconductors the small coherence length @xmath10 , the reduced carrier density , the marked anisotropy and the high transition temperature @xmath6 strongly enhance the superconducting fluctuations ( sf ) . in a wide temperature range , which can extend up to @xmath11 or @xmath12 k above @xmath6 , a variety of phenomena related to sf @xcite can be detected by several experiments such as specific heat @xcite , thermal expansion @xcite , penetration depth @xcite , conductivity @xcite and magnetization @xcite measurements . also nmr - nqr relaxation has been used to detect sf @xcite . recently the field dependence of the nmr relaxation rate @xmath13 and of the knight shift have been studied , by varying the magnetic field from @xmath14 up to @xmath15 tesla @xcite . mitrovic et al . @xcite attributed the field dependence of @xmath16 to the corrections in the density of states ( dos ) contribution to the spin - lattice relaxation . on the other hand , gorny et al @xcite have observed @xmath16 field independent , up to 14 tesla , from 150 k to 90 k , in a sample showing a decrease in @xmath17 starting around @xmath18 k. an analysis of the role of sf in nmr experiments and of the field dependence of @xmath13 has been recently carried out by eschrig et al . @xcite , by extending the analitical approach of randeria and varlamov @xcite to include short wave - length and dynamical fluctuations . of particular interest is the role of sf in underdoped high temperature superconductors . in fact , in these compounds one has the peculiar phenomena of the opening , at @xmath19 , of a spin gap ( as evidenced by nmr and neutron scattering ) and of a pseudo - gap ( as indicated by transport and arpes experiments ) @xcite . these gaps , which are possibly connected @xcite , have been variously related to sf of particular character @xcite . in this paper magnetization and @xmath0cu(2 ) nmr - nqr relaxation measurements in underdoped ybco are reported and compared with the ones in the optimally doped compound . the magnetization data yield information on the fluctuating diamagnetism ( fd ) , while the nuclear relaxation rates @xmath13 convey insights on the effects of sf on the @xmath20-integrated spin susceptibility in the zero frequency limit . the paper is organized as follows . in sect . ii some basic equations describing sf and fd are recalled . after some experimental details ( sect . ii ) , in sect . iii the experimental findings and their analysis are presented , with emphasis on the effect of the field on @xmath13 and on the behavior of the diamagnetic magnetization in chain - ordered underdoped ybco . the main results and conclusions are summarized in sect . because of sf the number of cooper pairs per unit volume , which is given by the average value of the square of the modulus of the order parameter @xmath21 , is different from zero above @xmath6 . in the time - dependent ginzburg - landau ( gl ) description @xcite the collective amplitudes and the correspondent decay times of sf are given by @xmath22 where @xmath23 and @xmath24 , with @xmath25 . @xmath26 plays the role of the fourier components of the average number of pairs per unit volume , while @xmath27 is the correspondent relaxation time . fluctuating pairs can give rise to a diamagnetic magnetization above @xmath6 . the diamagnetic magnetization @xmath7 in general is not linear in the field @xmath28 . the curves @xmath7 vs. @xmath28 and @xmath7 vs. @xmath29 can be tentatively analized by generalizing the lawrence - doniach model , using the free energy functional @xcite @xmath30=\sum_n\int d{\bf r}\biggl [ a\vert \psi_n\vert^2 + { b\over 2}\vert \psi_n\vert^4 + \nonumber \\ + { \hbar^2\over 4 m_{\parallel}}\vert[{\bf \nabla_{\parallel}}- { 2ie\over\hbar c}{\bf a_{\parallel } } ] \psi_n\vert^2 + t \vert \psi_{n+1}- \psi_n\vert^2 \biggr]\end{aligned}\ ] ] where the last term takes into account the tunneling coupling between adjacent layers . from eq . 2 , deriving the free energy in the presence of the field and by means of a numerical derivation ( @xmath31 ) , one can obtain the fluctuating magnetization . according to scaling arguments @xcite for moderate anisotropy ( quasi-3d case ) one expects that the magnetization curves , at constant field , cross at @xmath8 when the magnetization is scaled by @xmath32 . the amplitude @xmath33 at @xmath34 departs from the prange s result @xcite by a factor @xmath35 , corresponding to the anisotropy ratio @xmath36 . accordingly @xcite , the data at different fields collapse onto an universal curve when @xmath37 is reported as a function of @xmath38 , for a critical exponent for the coherence length corresponding to @xmath39 . for strong anisotropies , namely quasi-2d systems , the @xmath7 curves at constant field cross each other at @xmath40 . @xmath41 is larger than the value obtained from the gaussian approximation by a factor @xcite around @xmath14 . collapse of the data onto a universal curve occurs for @xmath42 well above @xmath43 , usually in the range @xmath44 . the contributions to the relaxation rates w due to sf can be derived within a fermi liquid scenario , withouth specifying the nature of the interactions @xcite . the direct and most singular contribution , equivalent to the aslamazov - larkin paraconductivity , is not effective as nuclear relaxation mechanism . the positive maki - thompson ( mt ) contribution @xmath45 results from a purely quantum process , involving pairing of the electron with itself at a previous stage of motion , along intersecting trajectories . a negative sf contribution @xmath46 comes from the density of states reduction when electrons are subtracted to create pairs . approximate expressions for @xmath46 and @xmath45 can be derived by resorting to simple physical arguments as follows . the relaxation rates can be approximated in the form @xmath47 where @xmath48 is an average form factor for cu nuclei and @xmath49 an effective spectral density @xcite . sf modify the static spin susceptibility @xmath50 and the effective correlation time @xmath51 in eq . 3 . from eq.1 the in - plane density of pairs is @xmath52 and therefore the sf imply @xmath53\end{aligned}\ ] ] for the dynamical part in eq . 3 , the dos contribution can be obtained by averaging over the bz the collective correlation time in eq . 1 : @xmath54 in the dirty limit , differing from the exact calculation @xcite only by a small numerical factor . @xmath55 is the electron collision time . the mt contribution has to be evaluated in the framework of diagrammatic theories @xcite . the final result corresponds to the 2d - average over the bz of a decay rate of diffusive character @xmath56 ( with @xmath57 carrier diffusion constant ) , phenomenologically accounting for phase breaking processes by adding in the decay rate a frequency @xmath58 : @xmath59 where @xmath60 is a dimensionless pair breaking parameter . by indicating with @xmath61 the relaxation rate in the absence of sf and by neglecting the correction to @xmath62 ( eq . 4 ) , from eqs . 3 , 5 and 6 the relaxation rate in 2d systems turns out @xmath63 \label{ram}\end{aligned}\ ] ] to extend this equation to a layered system , when dimensionality crossover ( 2d@xmath64 3d ) occurs , one has to substitute @xcite @xmath65 by @xmath66 $ ] and @xmath67 by @xmath68 $ ] , with @xmath69 anisotropy parameter ( @xmath70 is the interlayer distance ) . it is noted that the mt contribution ( first term in eq . 7 ) is present only for s - wave orbital pairing , while it is averaged to almost zero for d - symmetry . the measurements have been carried out in oriented powders of optimally doped ybco in one chain - disordered underdoped and in two chain - ordered underdoped ybco samples @xcite the oxygen content in the underdoped compounds was close to 6.66 , with sligth differences in @xmath6 . electron diffraction microscopy evidenced the expected tripling of the a - axis , while resistivity measurements show a sharp transition with zero resistivity at t= 62 k and occurrence of paraconductivity below about 75 k @xcite . table i collects the main properties of the samples , as obtained from a combination of measurements . the @xmath0cu relaxation rates @xmath71 have been measured by standard pulse techniques . in nqr the recovery towards the equilibrium after the saturation of the @xmath72 line is well described by an exponential law , directly yielding @xmath73 . for the nmr relaxation a good alignment of the @xmath74 axis of the grains along the magnetic field is crucial to extract reliable values of w ( h ) . the nmr satellite line , corresponding to the @xmath75 transition can be used to adjust the alignment and to monitor the spread in the orientation of the c - axis , the resonance frequency being shifted at the first order by the term @xmath76 , due to the quadrupole interaction ( @xmath77 angle of the @xmath78-axis with the field ) . from the width of the spectrum ( fig . 1a ) the spread in the orientation of the c - axis appears within about two degrees . the recovery law for the nmr satellite transition is @xmath79 this law has been checked to fit well the experimental data ( fig . 1b ) , proving the lack of background contamination due to other resonance lines . the magnetization has been measured by means of a metronique ingegnerie squid magnetometer , on decreasing the temperature at constant field and , at selected temperatures , by varying @xmath28 . the paramagnetic contribution to m was obtained from m vs @xmath28 at t @xmath80 110 k , where practically no fluctuating magnetization is present . then @xmath7 was derived by subtraction , singling out a small contribution from paramagnetic impurities . the paramagnetic susceptibility turns out little temperature dependent around @xmath6 and this dependence will be neglected in discussing the much stronger diamagnetic term . in optimally doped ybco the measurements have been carried out at the purpose to confirm the results recently obtained in single crystals by other authors@xcite . the isothermal magnetization curves are satisfactorily described on the basis of the anisotropic gl functional ( eq . 2 ) , for @xmath81 . only close to @xmath6 and for @xmath82 tesla an observable departure is detected , indicating crossover to a region of non - gaussian sf , in agreement with recent thermal expansion measurements @xcite . the @xmath7 vs. @xmath29 curves cross at @xmath40 when @xmath7 is scaled by @xmath83 , as expected for moderate anisotropy @xcite . let us first comment the data in optimally doped ybco ( fig . the comparison between the nmr and nqr @xmath13 s has been already discussed in previous papers @xcite . here we only add a few comments motivated by more recents works @xcite involving the remarkable aspect of the field dependence . the nqr @xmath13 can be compared to eq . 7 , by using the values @xcite @xmath84 , @xmath85 and a dephasing time parameter @xmath86 . a quantitative fitting is inhibited by the fact that the background contribution to w does not follow the korringa law , in view of correlation effects among carriers @xcite . a firm deduction , however , is that an anisotropy parameter @xmath87 , and namely a crossover to a 3d regime , is required to avoid unrealistic values ( 2d line in figure 2a ) , as indicated also by magnetization measurements ( see later on ) . mitrovic et al . @xcite have discussed the field dependence of @xmath88 in terms of quenching of the dos term only , by resorting to the theory of eschrig et al . @xcite . the field dependence of the sf contribution to the relaxation rate is a delicate issue , because of the nontrivial interplay of many parameters which include reduced temperature @xmath89 , reduced field @xmath90 , anisotropy parameter @xmath91 , elastic and phase breaking times @xmath55 and @xmath92 . eschrig et al . @xcite have extended the analitical approach @xcite by taking into account arbitrary values of @xmath93 , short - wave fluctuations and dynamical fluctuations . the price of this generalization is the restriction to a 2d spectrum of sf fluctuations . in view of our experimental findings the analysis of the field effect in a purely 2d framework is questionable . on the other hand , a re - examination of the field effect for a layered system is now possible by applying the method of the dos term regularization devised by budzin and varlamov @xcite , which has indicated how the divergences can be treated . since the dynamical fluctuations could be relevant only for fields of the order of @xmath94 , up to @xmath95 @xmath96 the fluctuations can be safely treated in the nearly static limit . the field dependence for weak fields ( @xmath97 ) turns out @xcite @xmath98^{3/2}}\beta^2\end{aligned}\ ] ] where @xmath99 + \psi ^{\prime } ( 1/2)}= \bigg\ { \begin{tabular}{cc } $ ] 14(3)/^3@xmath100t1@xmath1014t@xmath1021t1/@xmath103 ( here @xmath104 ) . so both effects of increase and of decrease of @xmath13 on increasing the magnetic field are possible , depending on the mean free path in the specific sample . if @xmath105 the main correction to @xmath13 is due to the mt term and one should observe @xmath13 decreasing with increasing field , while if @xmath106 the dos correction becomes dominant and @xmath13 is expected to increase . as it appears from fig . 2 the effect of the field for @xmath107 is small , if any . it is noted that if eq . 7 is applied to the underdoped compounds the reduction in the fermi energy @xmath108 and the increase in the anisotropy ( i.e. decrease of @xmath91 ) would imply a sizeable increase in @xmath109 . this enhancement in the underdoped compounds does not occur ( see fig . 2b and 2c for the chain - disordered and chain - ordered ybco s , respectively ) , unless the decrease of @xmath13 over a wide temperature range , usually related to the spin - gap opening @xcite , should be attributed to a field - independent dos term . on the other hand , the conventional sf of gl character should occur only close to @xmath6 , where , on the contrary , @xmath50 is little temperature dependent . eq . 9 , in principle , does predict a field dependence . however , if typical values @xmath110 k , @xmath111 , @xmath112 and @xmath113 are used , for @xmath114 one has @xmath115 , hardly to evidence for @xmath116 . a comprehensive discussion of the field dependence of the sf contribution to the relaxation rate is given elsewhere @xcite . here we only remark that the results in strong fields ( @xmath117 ) from mitrovic et al . @xcite can hardly be justified on the basis of eq . as regards the magnetization measurements in optimally doped ybco , as already mentioned , our data indicate a 3d regime crossing from gaussian to critical fluctuations close to @xmath6 . the value of @xmath118 @xcite at the crossing point of the curves @xmath119 vs. @xmath29 turns out around @xmath120 , corresponding to an anisotropy factor @xmath121 . a collapse onto a common function is obtained when @xmath122 is plotted as a function of @xmath123 . the collapse fails in magnetic fields less than about 0.3 tesla . a small - field departure from the universal function has been already observed in underdoped compounds @xcite , but no mention about it is found in the literature for optimally doped ybco . in the underdoped compounds a marked enhancement of the fluctuating magnetization is observed ( figs . 3 and 4 ) . in fig . 4 the susceptibilty , defined as @xmath124 for @xmath125 tesla , is reported . the inset is a blow up of the results around the temperature where reversing of the sign of the magnetization occurs . the susceptibilities for chain - ordered and chain - disordered underdoped samples are compared with the one in optimally doped ybco in fig . 5 . in underdoped ybco the magnetization curves dramatically depart from the ones expected on the basis of eq . 2 and numerical derivative . no crossing point is observed in the curves @xmath7 vs @xmath29 and no collapse is found on anisotropic 3d or 2d - like curves for @xmath126 or @xmath7 vs @xmath127 ( fig . these anomalies are more manifested in the chain ordered compound , as it can be realized from the peculiar shape of the isothermal magnetization vs. @xmath28 ( fig . one could remark that a fraction of a few percent of non - percolating local superconducting regions can account for the absolute value of the diamagnetic susceptibility and for the shape of the magnetization curves . a trivial chemical inhomogeneity could be suspected . the fact that the anomalous diamagnetism has been observed in three samples differently prepared and since it is strongly enhanced in the chain - ordered ones , supports the intrinsic origin of the phenomenon . recent theories @xcite have discussed how a distribution of local superconducting temperatures , related to charge inhomogeneities , can cause anomalous diamagnetic effects . mesoscopic charge inhomogeneities have been predicted on the basis of various theoretical approaches @xcite . in underdoped compounds one could expect the occurrence of preformed cooper pairs causing `` local '' superconductivity lacking of long - range phase coherence . an inhomogeneous distribution of carriers , on a mesoscopic scale , is supported by @xmath128 plane optical conductivity @xcite and it is the basic ingredient of the stripes model @xcite . since the anomalous diamagnetism is enhanced in chain - ordered underdoped ybco one is tempted to relate it to the presence of stripes . the insurgence of phase coherence among adjacent charge - rich regions can qualitatively be expected to yield strong screening above the bulk superconducting temperature @xcite . therefore , one could phenomenologically describe the role of sf in underdoped compounds , as follows . at @xmath129 , just below @xmath130 , fluctuating pairs are formed , without long - range phase coherence . below @xmath131 , about @xmath132 k above @xmath6 , phase coherence among adjacent charge - rich drops @xcite start to develop , yielding the formation of superconducting loops with strong screening and causing the onset of the anomalous diamagnetism . close to @xmath6 , sf of gl character occur and long - range phase coherence sets in . by combining magnetization and nmr - nqr relaxation measurements , an attempt has been done to clarify the role of superconducting fluctuations and of fluctuating diamagnetism in underdoped ybco , vis - a - vis the optimally doped compound . in the latter case the fluctuations above @xmath6 are rather well described by an anisotropic ginzburg - landau ( gl ) functional and by scaling arguments for slightly anisotropic systems . a breakdown of the gaussian approximation for small magnetic fields has been observed close to @xmath6 . the @xmath0cu relaxation rates @xmath13 around @xmath6 show little field dependence , if any . one can not rule out the presence of a mt contribution , and then of a small @xmath133wave component in the spectrum of the fluctuations , which should be sample - dependent in view of the role of impurities in the pair - breaking mechanisms . in underdoped ybco an anomalous diamagnetism is observed , on a large temperature range . the diamagnetic susceptibility at @xmath6 is about an order of magnitude larger than the one in the optimally doped sample and the isothermal magnetization curves can not be described by the anisotropic gl functional . scaling arguments , such as the search of an universal function in terms of @xmath134 , appear inadequate to justify the experimental findings . the anomalies are more marked in the chain - ordered samples . also in the underdoped ybco @xmath0cu @xmath13 in nqr almost coincide with the nmr ones at @xmath135 tesla , in agreement with a theoretical estimate of the effect of the magnetic field . another conclusion drawn from the field and temperature dependences of the @xmath0cu relaxation rate is that the spin gap does not depend on the magnetic field and that the behavior of the spin susceptibility for @xmath129 can not be ascribed to sf of gl character . the fluctuating diamagnetism observed in underdoped ybco can be phenomenologically accounted for by the presence of charge inhomogeneities at mesoscopic level . the anomalies in the magnetization curves can be attributed to `` locally superconducting '' non - percolating drops , present above the bulk @xmath6 and favoured by chain ordering . a. bianconi , f. borsa , j. r. cooper and m. h. julien are thanked for useful discussions . the collaboration of paola mosconi in processing the squid magnetization data is gratefully acknowledged . thanks are due to b.j . suh and p. manca , for having provided well characterized underdoped ybco compounds . the availability of the metronique ingegnerie squid magnetometer from the departement of chemistry , university of florence ( prof . d. gatteschi ) is gratefully acknowledged . the research has been carried out in the framework of a `` progetto di ricerca avanzata ( pra ) '' , sponsored by istituto nazionale di fisica della materia ( pra - spis 1998 - 2000 ) . for a recent collection of papers , see fluctuations phenomena in high temperature superconductors , edited by m. ausloos and a.a . varlamov , kluwer academic publisher , ( 1997 ) a.a . varlamov , g.balestrino , e. milani and d.v . livanov , advances in physics 48 , 655 ( 1999 ) m.thinkham `` introduction to superconductivity '' , mc graw - hill n.y . ( 1996 ) chapters 8 - 9 see a.junod , in `` studies of high temperature superconductors '' , edited by a.v . narlikar ( nova science publishers , inc . new york , 1996 ) , vol . 18 v.pasler et al . phys . rev . 81 , 1094 ( 1998 ) z.h . lin et al . letters 32 , 573 ( 1995 ) and references therein m.a . howson et al . phys . rev . b 51 , 11984 ( 1995 ) and references therein see the review paper by l.n.bulaevskii , int . j. of modern physics b4 , 1849 ( 1990 ) and several papers in ref . 1 p.carretta , d. livanov , a.rigamonti and a.a.varlamov , phys.rev.b 54 , r9682 ( 1996 ) p.carretta , a.rigamonti , a.a.varlamov , d.livanov , nuovo cimento 19 , 1131 ( 1997 ) v. mitrovic et al . , phys . 82 , 2784 ( 1999 ) h.n.bachman et al . b 60 , 7591 ( 1999 ) k. gorny et al . , phys . rev . 82 , 177 ( 1999 ) m. eschrig et al . b 59 , 12095 ( 1999 ) m.randeira and a.a.varlamov , phys b 50 , 10401 ( 1994 ) a.rigamonti , f.borsa and p.carretta , rep.prog.phys.61 , 1367 ( 1998 ) a.g . loeser et al . science 273 , 325 ( 1996 ) h.ding et al . nature 382 , 51 ( 1996 ) g. v. m. williams , j. l. tallon , r. michalak and r. dupree , phys . b 54 , r6909 ( 1996 ) t. dahm , d. manske and l. tewordt , phys . b 55 , 15274 ( 1997 ) v.j.emery , s.a.kivelson and o.zachar phys.rev . b 56 , 6120 ( 1997 ) ; see also v. j. emery and s. a. kivelson , j. phys . 59 , 1705 ( 1998 ) c. castellani , c. di castro and m. grilli z.phys . b 103 , 137 ( 1997 ) c.baraduc , a. budzin , j - y henry , j.p . brison and l. puech , physica c 248 138 ( 1995 ) a.budzin and v.dorin , ref . 1 , p.335 m. b. salamon et al . b 47 , 5520 ( 1993 ) t.schneider and h. keller , inter . jornal of modern physics b 8 , 487 ( 1993 ) a.junod , j.y . genoud , g. triscone and t. schneider , physica c 294 , 115 ( 1998 ) r.e . prange , phys . b 1 , 2349 ( 1970 ) m.a.hubbard , m.b . salamon and b.w.veal , physica c 259 , 309 ( 1996 ) c.p.slichter `` principles of magnetic resonance '' , springer verlag , berlin - heidelberg ( 1990 ) p.manca , p.sirigu , g.castellani and a.migliori , il nuovo cimento 19 , 1009 ( 1997 ) ; g.castellani , a.migliori , p.manca and p.sirigu , il nuovo cimento 19 , 1075 ( 1997 ) u.welp et al . , phys.rev.lett . 67 , 3180 ( 1991 ) a.i.buzdin and a.a.varlamov , phys.rev.b 58 , 14195 ( 1998 ) p.mosconi , a.rigamonti and a. a. varlamov , appl . mag . reson . to be published yu . n. ovchinikov , s. a. wolf and v. z. kresin , phys . b 60 , 4329 ( 1999 ) e. z. kuchinskii and m. v. sadovskii , cond - mat/9910261 d.mihailovic , t.mertelj and k.a.muller , phys . rev . b 57 , 6116 ( 1998 ) ; see also kabanov et al . phys.rev.b 59 , 1497 ( 1999 ) m. r. norman , cond - mat/9904048 a.bianconi et al . phys.rev.lett . 76 , 3412 ( 1996 ) c. bergemann et al . , phys . rev . b 57 , 14387 ( 1998 )
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magnetization and @xmath0cu nmr - nqr relaxation measurements are used to study the superconducting fluctuations in yba@xmath1cu@xmath2o@xmath3 ( ybco ) oriented powders . in optimally doped ybco
the fluctuating negative magnetization @xmath4 is rather well described by an anisotropic ginzburg - landau ( gl ) functional and the curves @xmath5 cross at @xmath6 . in underdoped ybco , instead , over a wide temperature range an anomalous diamagnetism is observed , stronger than in the optimally doped compound by about an order of magnitude .
the field and temperature dependences of @xmath7 can not be described either by an anisotropic gl functional or on the basis of scaling arguments .
the anomalous diamagnetism is more pronounced in samples with a defined order in the cu(1)o chains .
the @xmath0cu(2 ) relaxation rate shows little , if any , field dependence in the vicinity of the transition temperature @xmath8 .
it is argued how the results in the underdoped compounds can be accounted for by the presence of charge inhomogeneities , favoured by chains ordering .
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the advent of high - resolution optical imaging and follow - up spectroscopic surveys in the past two decades , has led to the recognition of a morphologically distinct class of stellar assemblies in and around external galaxies , the first few cases of which were found in the fornax cluster @xcite . these so - called ultra - compact dwarfs ( ucds ; @xcite ) manifest themselves as compact objects with typical effective radii of 10 @xmath5 pc and absolute v - band magnitudes of -14 @xmath6 -9mag , just intermediate between the classical globular clusters ( gcs ) and dwarf elliptical galaxies . when spectroscopic information is available , ucds appear to harbor a predominantly old stellar population ( e.g. , @xcite ; janz et al . 2016 ) , in some cases with an extended star formation history ( e.g. , @xcite ) . candidate ucds are now routinely found in dense environments such as galaxy clusters ( e.g. , fornax , @xcite ; virgo , @xcite ; centaurus , @xcite ; coma , @xcite ; perseus , @xcite ) and galaxy groups ( e.g. , hcg22 and hcg90 , @xcite ) , but also in relatively isolated galaxies such as the sombrero (= m104 ; @xcite ) , ngc3923 and ngc4546 @xcite . the nature of ucds , however , is far less clear than their names might have indicated . viable formation scenarios proposed for ucds include : ( i ) they are the residual cores of tidally - stripped nucleated dwarf galaxies in dense environments @xcite ; ( ii ) they are the end - product of the aggregation of young massive star clusters formed during violent gas - rich galaxy mergers @xcite ; and ( iii ) they are ultra - massive gcs , as expected by extension of the gc luminosity function @xcite . adding to the mystery of ucds is a closely - relevant , growing class of stellar systems , namely , extended stellar clusters ( escs ) . with visual luminosities largely overlapping the typical range of gcs ( -9 @xmath6 -5 ) , the escs earn their names for their characterisic sizes ( 10 @xmath7 pc ) that are much larger than the classical gcs . a number of escs have also been found in galactic disks ( e.g. , ngc1023 and ngc3384 , @xcite ) , which are not easily explained by the scenario of stripped satellites . recently , brns & kroupa ( 2012 ; hereafter bk12 ) compiled from the literature a catalog of 813 confirmed and candidate ucds and escs . in reality , the known ucds and escs may represent a heterogeneous family of stellar systems that the above formation scenarios conspire to form . studies of ucds and escs have been concentrated on optical observations and theoretical investigations . up till now we know very little about their properties at other wavelengths , in particular the x - ray band . for gcs , it has long been established that they exhibit an over - abundance of low - mass x - ray binaries ( lmxbs ) with respect to the field ( @xcite ; @xcite ) , which is attributed to the efficient dynamical formation of neutron star binaries in the dense core of gcs ( @xcite , @xcite , @xcite ) . x - ray surveys of extragalactic gcs , mostly accomplished by the _ chandra x - ray observatory _ , have revealed that on average @xmath05% of gcs exhibit an x - ray counterpart ( presumably lmxbs ) at a limiting luminosity of @xmath8 , and that more massive gcs have a higher probability of hosting lmxbs @xcite . if some ucds ( and escs ) are indeed giant versions of gcs , they are naturally expected to host lmxbs . the first ucd reported to produce x - ray emission lies in m104 ( sucd1 ; @xcite ) , which has an x - ray luminosity of @xmath9 @xcite . in view of the rapidly growing inventory of ucds and escs , it would be interesting to examine the incidence rate of x - ray sources in these dense stellar systems , which should provide clues about their internal structure . on the other hand , many ucds found in galaxy clusters can well be the remnant of stripped nucleated galaxies . the recent discovery of a candidate super - massive black hole ( with a mass of @xmath0@xmath10 ) in the ucd of m60 ( m60-ucd1 ; @xcite ) , based on spatially - resolved stellar kinematics , lends strong support to this hypothesis . strader et al . ( 2013 ) identified a variable x - ray source with m60-ucd1 . x - ray emission could be an important tracer of this and other putative massive black holes embedded in ucds . in this work , we conduct a systematic survey of x - ray emission from ucds and escs using archival _ observations , in order to shed light on the origin and nature of these intriguing objects . in section 2 we describe the sample selection , data reduction and x - ray source detection . in section 3 we present the x - ray counterparts of the ucds and escs and analyze their properties in close comparison with the x - ray population found in gcs . discussion and summary of our results are given in sections 4 and 5 , respectively . quoted errors are at the 1@xmath11 confidence level throughout this work unless otherwise noted . we adopt the bk12 catalog of ucds and escs as our primary sample , which results from an exhaustive search of the literature available at the time . we note that only a ( growing ) fraction of the cataloged sources have been spectroscopically confirmed . nevertheless , bk12 showed that the spectroscopically confirmed sources are statistically representative of the full sample , in particular in the @xmath12 plane . the ucds and escs were collectively dubbed extended objects ( eos ) in bk12 to reflect the unclear physical distinction between the two groups , a naming convention we follow in this work . we caution that the term eo " should only be treated as a technical , rather than physical , distinction from the more compact gcs . the 813 cataloged sources were tentatively associated with 65 host galaxies ranging from dwarfs to milky way - like normal galaxies , and to giant ellipticals embedded in galaxy clusters . we dropped sources without explicit celestial coordinates . we also neglected sources associated with local group galaxies or with m87 , the cd galaxy of the virgo cluster . sources within the local group are essentially well - studied escs , while m87 hosts a large and still growing population of ucds @xcite that deserve a joint optical / x - ray investigation elsewhere . lastly , we excluded sources located in the perseus and coma clusters , chiefly to avoid potential bias introduced by the relatively poor x - ray source detection sensitivity , which is expected for the large distances and substantial local background caused by the hot intra - cluster medium in these two massive galaxy clusters . we cross - correlated the remaining sources with the _ chandra _ archive , requiring that all sources of interest fall within 8 arcmin from the aimpoint of an acis - i or acis - s observation , to ensure good sensitivity and accurate source positioning . in this way we selected 449 sources from the bk12 catalog . we then added to the bk12 sources 60 escs recently identified in ngc1023 from _ hst _ imaging ( @xcite ) , all of which have an effective radius @xmath13 10 pc , including an _ @xmath14cen_-like object with @xmath15 of 10 pc and @xmath16 of -8.9 mag ( hereafter ngc1023-eo1 ) . we have converted the @xmath17-band magnitudes given in forbes et al . ( 2014 ) into v - band magnitudes , assuming an underlying simple stellar population with age of 10 gyr , half - solar metallicity and a kroupa initial mass function ( imf ) . we also added two recently discovered ucds that are among the most massive ucds ever found ( m60-ucd1 : @xcite , @xcite ; m59-ucd3 : @xcite , @xcite ) . our final sample consists of 511 eos , which are distributed in 27 host galaxies . among them , 5 galaxies had only one observation and the other 22 with multiple exposures . data of the queried observations are publicly available by june 2015 . we reprocessed the _ chandra _ data with ciao v4.7 and the corresponding calibration files , following the standard procedure . briefly , we produced count and exposure maps in the 0.5 - 2 ( @xmath18 ) , 2 - 8 ( @xmath19 ) , and 0.5 - 8 ( @xmath20 ) kev bands for each observation . the exposure maps were weighted by an absorbed power - law spectrum , with a photon index of 1.7 and an absorption column density @xmath21 = @xmath22 . for galaxies with multiple observations , we calibrated their relative astrometry by matching the centroid of commonly detected point sources , using the ciao tool _ reproject_aspect_. the count and exposure maps of individual observations were then reprojected to a common tangential point , i.e. , the optical center of the putative host galaxy , to produce combined images of optimal sensitivity for source detection . the energy - dependent effective area among the acis ccds was taken into account , assuming the above incident spectrum , so that the quoted count rates throughout this work refer to acis - s3 . table 1 presents basic information of the host galaxies , including position , distance , effective exposure , and the number of eos within the _ chandra _ field - of - view ( fov ) . at distances of a few mpc and beyond , ucds and escs should remain unresolved even under the superb angular resolution afforded by _ chandra_. to form the basis of identifying x - ray counterpart of the ucds / escs , we ran source detection in the 0.5 - 2 ( @xmath18 ) , 2 - 8 ( @xmath19 ) and 0.5 - 8 ( @xmath20 ) kev bands for each host galaxy , following the procedure detailed in @xcite and @xcite . the source detection algorithm has been successfully applied for detecting x - ray sources in m104 ( @xcite ) , which is also included in the present study . for each detected source , background - subtracted , exposure map - corrected count rates in the individual bands are derived from within the 90% enclosed energy radius . according to the assumed incident spectrum , we have adopted a @xmath20-band count - rate - to - luminosity conversion factor of @xmath23 , where @xmath24 is the distance to a given galaxy ( table 1 ) . the 0.5 - 8 kev detection limit for each galaxy field , ranging between a few @xmath25 to a few @xmath26 ( with a median of @xmath27 ) , is given in table 1 . for the galaxies with multiple observations , we repeated the above procedure in individual exposures to probe long - term source variability . we search for x - ray counterpart of the ucds and escs from the list of detected sources in each galaxy field , by adopting a matching radius of @xmath28 , which corresponds to a physical scale of @xmath0156 pc at a distance of 16.1 mpc , the median of our sample galaxies . this choice allows for position uncertainty of sources found at relatively large off - axis angles 50 and off - axis angles of @xmath29-@xmath30 have 95% positional uncertainties of 05 - 18 . ] and the ( typically unknown ) relative astrometry between the x - ray and optical observations . a total of 17 x - ray counterparts are thus identified , including 1 in ngc1023 , 1 in ngc1399 , 6 in ngc4365 , 1 in m89 , 1 in m59 , 1 in m60 , 2 in m104 ( sombrero ) , and 4 in ngc5128 ( cen a ) . notably , all eight host galaxies are early - type galaxies , although the _ chandra _ pointed observations might have been biased against late - type galaxies . to assess the probability of random matches , we artificially shift the positions of all detected sources in each field by @xmath31 in ra and dec and average the number of coincident matches in the four directions . we find essentially zero random matches in all the fields except for ngc4365 , in which the above exercise results in 3.25 random matches . this can be understood , since ngc4365 has both a large population ( 216 ) of eos and a large number ( 369 ) of detected x - ray sources . by reducing the matching radius to @xmath32 , we find 11 x - ray counterparts , indicating that some of the eliminated matches could indeed be interlopers in particular , 4 eliminated cases are in ngc4365 . nevertheless , we consider all 17 sources as true x - ray counterparts and provide their basic properties in table 2 , in which we also quote the bk12 catalog for their v - band absolute magnitude and effective radius . from the literature we find that at least 7 of the 17 sources ( ngc1399-eo12 , m104-eo01 , m59-ucd3 , m60-ucd1 , ngc5128-eo03 , ngc5128-eo05 , ngc5128-eo01 ) are confirmed ucds / escs , for which we quote the spectroscopically - derived metallicity in table 2 ; for the remaining sources , we have estimated the metallicity from their optical color , assuming a simple stellar population with age of 10 gyr and a kroupa imf . roughly an equal number of x - ray counterparts are found in the metal - rich ( [ m / h ] @xmath13 0.3 ) and metal - poor ( [ m / h ] @xmath33 0.3 ) ucds / escs . figures 1 and 2 show the _ chandra _ 0.5 - 8 kev images of the eight fields , with the positions of the identified x - ray sources marked . among the eight host galaxies , some have a well - documented population of gcs , which are valuable for a direct comparison with the ucds / escs in the observed x - ray properties . for this purpose , we adopt the published gc catalogs of ngc1399 @xcite , ngc4365 @xcite , m104 @xcite and ngc5128 @xcite . again by adopting a matching radius of @xmath28 , we find 101 , 184 , 65 and 10 x - ray counterparts for the gcs in ngc1399 , ngc4365 , m104 and ngc5128 , respectively ( marked by green circles in figures 1 and 2 ) . the number of gcs within the individual fov ( 5140 in total ) and the number of their identified x - ray counterparts are listed in table 1 . the total incidence rate of x - ray sources in gcs ( hereafter xgcs ) is @xmath34% . this is to be contrasted with the @xmath2% incidence rate of the x - ray - emitting ucds / escs ( hereafter collectively called xeos ) found in all _ chandra _ fields ( or @xmath35$]% , if only sources in the bk12 catalog are taken into account ) . we caution that our parent sample of ucds and escs , resulted from literature compilation , is likely more heterogeneous than the gc sample , in terms of completeness . a comparison between our identifications and the x - ray identifications by pandya et al . ( 2016 ) is warranted . from their own literature compilation of ucd candidates , which are distributed primarily in galaxy clusters , pandya et al . ( 2016 ) identified 21 x - ray counterparts by adopting a matching radius of @xmath36 , among which six sources ( ngc1399-eo12 , m104-eo01 , m60-ucd1 , ngc5128-eo02 , ngc5128-eo05 and ngc5128-eo01 ) are in common with our xeos . the remaining 15 sources are not included in our primary sample . a close examination indicates that some of these sources , considered ucds by pandya et al . ( 2016 ) , have an effective radius of 3 - 8 pc ( i.e. , more typical of gcs ) , and thus would not have appeared in the bk12 catalog . on the other hand , the 11 xeos that are identified by us but not included in pandya et al . ( 2016 ) are essentially sources not in their parent sample . in figure [ fig : hardness ] , we show the 0.5 - 8 kev intrinsic luminosity ( @xmath37 ) against hardness ratio of the 17 xeos . the hardness ratio , defined as @xmath38 and listed in table 2 , is calculated from the observed counts in the @xmath18 ( 0.5 - 2 kev ) and @xmath19 ( 2 - 8 kev ) bands , using a bayesian approach @xcite . for comparison , we also plot in figure [ fig : hardness ] the 360 xgcs from four host galaxies ( section 3.1 ; table 2 ) , which are presumably lmxbs . none of the xeos exhibits @xmath39 , i.e. , the regime of ultra - luminous x - ray sources , where black hole binary systems may be relevant . the majority of xgcs also fall short of this threshold ; only four xgcs in ngc1399 have @xmath39 . the xeos and xgcs are also similar in the distribution of their hardness ratios . notably , the ultra - massive m59-ucd3 is the softest among all xeos . this source is detected in the @xmath18-band but not in the @xmath19-band , thus having a hardness ratio of -1 . visual inspection of the _ chandra _ image indicates that all 7 photons from m59-ucd3 have an energy below 1.8 kev . several xeos have sufficient net counts for spectral analysis . for such sources , we extract their spectra from a 2@xmath40-radius circle , and the corresponding background spectra from a concentric ring with inner - to - outer radii of @xmath41-@xmath42 . spectra extracted from multiple exposures of the same source are co - added . all the spectra appear virtually featureless , and thus we fit them with an absorbed power - law model , requiring that the equivalent hydrogen column density is no less than the galactic foreground value ( kalberla et al . 2005 ) . we obtain meaningful constraints on the photon - index for m104-eo01 ( i.e. , sucd1 ) , m60-ucd1 , ngc1399-eo12 , ngc4365-eo117 , ngc5128-eo1 and ngc5128-eo5 , with best - fit values of @xmath43 , @xmath44 , @xmath45 , @xmath46 , @xmath47 and @xmath48 , respectively . these values are consistent with the typical range of lmxbs . we also examine the long - term flux variability of xeos , based on the source count rates measured from individual observations . following @xcite , we define source variability @xmath49 , where @xmath50 is the highest count rate among individual detections and @xmath51 the statistical upper limit of the lowest detected count rate . the values of @xmath52 are listed in table 2 . a value of @xmath53 is given for the two xeos with only one observation , while ngc4365-eo039 has an ill - defined @xmath52 because it is only detected in the combined image . the remaining xeos show moderate ( @xmath54 ) to strong ( @xmath55 ) variability , with the strongest variability found in ngc5128-eo01 ( @xmath56 ) . we note that one xeo , ngc4365-eo006 , has been identified in only one of the six observations available for ngc4365 ; its flux was apparently too low to be detected in the other five exposures as well as in the combined image . the prevalence of flux variability in the xeos , just as in the xgcs ( @xcite ) , suggest that the bulk of the detected x - ray emission arises from a single source rather than superposition of multiple sources . figure [ fig : reffmv ] shows the effective radius ( @xmath15 ) versus absolute v - band magnitude ( @xmath16 ) for the 511 ucds and escs studied in this work . by definition , all ucds and escs considered here have a size lower limit of 10 pc . the apparent paucity of objects around @xmath57 might have arisen from selection effect due to the heterogeneous nature of the bk12 catalog , but otherwise can be viewed as a technical division between ucds and the less luminous escs ( see discussion in forbes et al . bk12 noticed that the majority of eos in late - type galaxies have @xmath58 . we highlight the xeos with red diamonds in figure [ fig : reffmv ] , which distribute rather evenly across the entire range of @xmath16 . ten of the 17 xeos have @xmath59 , whereas six of the 9 most luminous eos ( with @xmath60 ) remain undetected in x - rays . for comparison , we show in figure [ fig : reffmv ] the gcs of m104 , among which those with an x - ray counterpart are further highlighted by blue squares . gcs in the other three galaxies are not shown due to the lack of available v - band magnitudes , but we expect that the m104 gcs are representative . we note that the great majority ( 94% ) of the gcs have @xmath61 4 pc . the median v - band magnitude of the gcs is -8.20 mag , and 52 out of the 65 xgcs ( i.e. , 80% ) are found in the brighter half . this clearly indicates that more massive gcs are more likely to host an lmxb , a familiar trend already noted by many previous work @xcite . the great majority of eos in our sample remain individually undetected in x - rays , partly owing to the relatively high limiting luminosity for most galaxies , which , at face value , is a good fraction of the eddington luminosity of neutron star binaries . we employ a stacking analysis for the undetected objects to shed light on their average x - ray properties . to do so , we collect the 0.5 - 8 kev counts registered within a @xmath62 box around each source of interest . these can be a subset of our total sample , e.g. , eos in a single galaxy . an eo is excluded if it is located within @xmath63 from an already detected x - ray source , to minimize contamination from psf - scattered photons . next , we measure signals from the stacked count image . after several tests we choose to accumulate the total on - source counts within a @xmath28-radius circle , and estimate the background counts within a ring with inner - to - outer radii of @xmath41-@xmath42 , after scaling the enclosed area . the signal - to - noise radio ( s / n ) is calculated accordingly . we have also measured the cumulative exposure time in a similar fashion . we examine the stacked signals of individual galaxies with at least 10 eos ( table 1 ) . however , none of these galaxies alone gives a s / n @xmath13 3 . we further stack all undetected eos from ngc1399 , ngc4365 , m104 and ngc5128 to enhance the s / n , which results in an average count rate of @xmath64 . defining a v - band luminosity - weighted mean distance , @xmath65^{-\frac{1}{2}}$ ] , where @xmath66 and @xmath67 are the v - band luminosity and distance of the @xmath68th eo , the above count rate corresponds to an equivalent 0.5 - 8 kev luminosity of @xmath69 for a distance of @xmath70 mpc . similarly , stacking all undetected gcs from the four galaxies , we obtain an average count rate of @xmath71 , or an equivalent 0.5 - 8 kev luminosity of @xmath72 per gc . this suggests that the individually undetected eos and gcs have on - average comparable x - ray emission . the similarity in the x - ray properties ( luminosity , spectra and variability ) of the xeos and xgcs ( section 3.2 ) strongly suggests that lmxbs also dominate the x - ray emission from eos , if the presumption that the xgcs are essentially lmxbs holds . pandya et al . ( 2016 ) drew a similar conclusion . the presence of lmxbs in the dense , predominately old stellar systems of eos has been naively expected . however , unlike the statistical behavior of the xgcs , the most luminous xeos do not show a clear tendency of hosting an x - ray source ( figure [ fig : reffmv ] ) . the overall incidence rate of xeos is also substantially low than that of the xgcs ( @xmath03% vs. 7% ; section 3.1 ) . these findings can be understood as follows . it has long been recognized that the abundance ( i.e. , number per unit stellar mass ) of luminous lmxbs in gcs is about two orders of magnitude higher than that of the galactic field ( @xcite ; @xcite ) . this over - abundance is widely accepted as the result of stellar dynamical interactions in the dense core of gcs , where an isolated neutron star ( ns ) can be captured by a main sequence star through tidal force ( @xcite ) , by a giant star through collision ( @xcite ) , or by a primordial binary through exchange scattering ( @xcite ) . all these processes are governed by the so - called stellar encounter rate , @xmath73 , where @xmath74 is the core stellar density , @xmath75 is the core radius and @xmath11 the velocity dispersion . the core stellar density of gcs can be as high as @xmath76-@xmath77 , compared to the typical mass density of 0.1 - 1@xmath78 in the field , and it is this crucial factor that leads to the high incidence rate of lmxbs in gcs . the ucds and escs are also dense stellar systems , but are less so than the gcs . this is demonstrated by the dashed lines in figure [ fig : reffmv ] , which mark equal values of the effective luminosity density , defined as @xmath79/@xmath80 . approximately , one may take the effective luminosity density as a proxy of the stellar mass density . most gcs show @xmath81 , incidentally a threshold above which no ucd / esc exists . notably , all but three xgcs have @xmath81 , consistent with the scenario of lmxbs having formed from dynamical interaction in gcs . on the other hand , the xeos can now be divided into two groups : those with @xmath82 and those with @xmath83 . members of the latter group all lie at @xmath84 mag , i.e. , they are practically ucds . in particular , m59-ucd3 and m60-ucd1 belong to this group , and indeed they have the highest @xmath85 of all ucds , implying that the x - ray counterparts of these two ultra - massive ucds are also lmxbs . we find that the incidence rate of the xeos with @xmath83 is @xmath86% , coming much closer to that of the xgcs . the likely reason is that the square dependence of stellar encounter rate on @xmath85 is partially compensated by the larger size of the ucds ( a cubic dependence on @xmath87 ) . in the same regard , the presence of xeos with @xmath82 is rather surprising , because the stellar encounter rate in these objects would be a factor of @xmath88 further lower . recall that some of the 6 xeos found in ngc4365 , typically with @xmath82 , might be interlopers rather than true associations ( section 3.1 ) . to investigate this issue further , we repeat the above stacking analysis ( section 3.3 ) for two subgroups of the individually undetected eos , one with @xmath82 and the other with @xmath89 . both subgroups show statistically significant signals , with average count rates of @xmath90 and @xmath91 @xmath92 per eo , and equivalent 0.5 - 8 kev luminosities of @xmath93 and @xmath94 , respectively . that the less dense subgroup has a lower average luminosity matches our anticipation , and also suggests that even the eos of the lower stellar densities can have a sizable population of dynamically - formed lmxbs . indeed , if dynamical effects are irrelevant , as is the case in the field , the expected number of ( field ) lmxbs is @xmath95 , empirically derived from the lmxb populations in the milky way and nearby galaxies ( @xcite ) . this relation predicts a negligibly small number of @xmath96 xeos , if we sum up the v - band light from all eos and assume a v - band mass - to - light ratio of 3.3 , appropriate for a simple stellar population with age of 10 gyr and half - solar metallicity . in the above discussion we have neglected the role of stellar - mass black holes ( bhs ) , which are usually thought to be absent in gcs due to their early segregation and subsequent mutual scattering at the core , although growing evidence now suggest that stellar - mass bhs do exist in some gcs ( @xcite ; @xcite ) . the eos , in particular the massive ucds which might be the remnant of stripped galaxies ( e.g. , @xcite ) , can harbor stellar - mass bhs . we note that the x - ray luminosities of all the xeos are compatible with ns binaries ( section 3.2 ) and do not seemingly require the presence of bh binaries . potentially also relevant is the stellar velocity dispersion . compared to gcs , the larger velocity dispersion of eos not only affects their stellar encounter rate , but also helps retain some of the otherwise escaping nss and bhs . a more quantitative treatment of all these affects is premature at this stage . the presence of lmxbs in eos implies that the underlying stellar population is an evolved one . the dynamical formation timescale of an ns binary via tidal capture , following hut & verbunt ( 1983 ) , is @xmath97^{-1}{\rm gyr}$ ] , where @xmath98 and @xmath99 are the characteristic stellar mass and radius , and @xmath100 is the mass of ns . this can be regarded as an independent evidence of eos being predominantly old stellar systems , consistent with existing optical spectroscopic studies . one caveat of this conclusion is that our _ chandra _ sample is biased against eos found in disk , typically late - type galaxies ( section 3.1 ) . such eos as possible descendants of recently aggregated massive star clusters ( e.g. , @xcite ) might not have sufficient time to form lmxbs , although they are also unlikely to harbor high - mass x - ray binaries unless with very recent star formation . this latter case can be tested using optical observations . while we have shown that the observed properties of the xeos can be reasonably understood as them being lmxbs , the alternative possibility that some of the xeos trace the x - ray emission from an embedded massive black hole ( mbh ) should not be easily dismissed . m60-ucd1 has been shown to harbor a mbh of @xmath10 ( seth et al . 2014 ) , and we find it to be an x - ray source with @xmath101 ( table 2 ) . while the current x - ray data does not unambiguously relate the detected x - ray emission to the mbh ( see also pandya et al . 2016 ) , we can infer an upper limit for its eddington ratio , @xmath102 , assuming that the x - ray band typically accounts for 10% of the mbh s bolometric luminosity . likewise , the ultra - massive m59-ucd3 is detected with a rather moderate @xmath103 , but has an atypical soft x - ray spectrum , whose nature remains to be understood with enhanced s / n . finally , we note that another ultra - massive ucd , m59co ( chilingarian & mamon 2008 ) , is undetected in the same _ chandra _ data of m59-ucd3 . future high - sensitivity x - ray and optical observations should continue to provide important clues to the existence of mbhs as well as dynamical structure in the ucds and escs . we have presented a systematic study of x - ray emission from ucds and escs based on archival chandra observations . the main results in this paper are as follows : * a total of 17 x - ray counterparts are identified with 0.5 - 8 kev luminosities above @xmath0@xmath1 , which are distributed in eight early - type host galaxies . in the meantime , 360 x - ray counterparts of gcs are identified in four of the eight host galaxies . the incidence rate of x - ray sources in ucds / escs and gcs are @xmath2% and @xmath34% respectively . * the spectral and temporal properties of the x - ray - detected ucds / escs are broadly similar to the x - ray - detected gcs , and are typical of lmxbs . * a stacking analysis further shows that there is on - average substantial x - ray emission from the individually non - detected ucds and escs , which is again comparable to that from the individually undetected gcs . * the x - ray properties of ucds / escs strongly suggest that they harbor a sizable population of lmxbs that have been formed from stellar dynamical interactions , consistent with the stellar populations in these dense systems being predominantly old . * for the most massive ucds , there remains the possibility that a central massive black hole produces the detected x - ray emission . future high - resolution , high - sensitivity x - ray and optical observations of a carefully selected sample of ucds and escs hold promise to solving their internal structure and dynamics . this work is supported by the national natural science foundation of china under grant 11133001 . the authors wish to thank eric peng and yanmei chen for helpful comments . m.h . is grateful to the hospitality of kiaa / pku during her summer visit . z.l . acknowledges support from the recruitment program of global youth experts . cccccccccc ngc247 & 11.785625 & -20.760389 & 3.6 & 10.0 ( 2 ) & 5.8@xmath104 & 2 + ngc891 & 35.639224 & 42.349146 & 10.0 & 171.6 ( 3 ) & 5.2@xmath104 & 6 + ngc1023 & 40.1000421 & 39.0632850 & 11.0 & 200.9 ( 5 ) & 5.5@xmath104 & 60 & 1 & + ngc1316 & 50.673750 & -37.208056 & 19.9 & 20.0 ( 1 ) & 6.3@xmath105 & 45 + ngc1380 & 54.113750 & -34.976028 & 18.3 & 41.6 ( 1 ) & 4.5@xmath105 & 13 + ngc1399 & 54.620941 & -35.450657 & 18.2 & 489.7 ( 14 ) & 3.7@xmath105 & 14 & 1 & 401 & 101 + m81 & 148.888221 & 69.065295 & 3.7 & 383.5 ( 25 ) & 5.1@xmath106 & 44 + ngc3115 & 151.308250 & -7.718583 & 9.8 & 1138.6 ( 11 ) & 1.4@xmath104 & 5 + ngc3311 & 159.175000 & -27.527500 & 53.7 & 31.9 ( 1 ) & 7.7@xmath107 & 19 + ngc3923 & 177.757059 & -28.806017 & 21.0 & 102.1 ( 2 ) & 3.0@xmath105 & 3 + ngc4278 & 185.028434 & 29.2807561 & 16.1 & 580.1 ( 9 ) & 5.8@xmath104 & 1 + ngc4365 & 186.117852 & 7.317673 & 21.4 & 195.8 ( 6 ) & 2.0@xmath105 & 216 & 6 & 3922 & 184 + m84 & 186.265597 & 12.886983 & 16.7 & 117.2 ( 4 ) & 2.0@xmath105 & 1 + ngc4382 & 186.350451 & 18.191487 & 15.2 & 49.9 ( 3 ) & 2.8@xmath105 & 4 + ngc4406 & 186.548928 & 12.946222 & 16.1 & 39.8 ( 3 ) & 9.5@xmath105 & 2 + ngc4449 & 187.046261 & 44.093630 & 3.8 & 100.9 ( 3 ) & 9.7@xmath106 & 7 + ngc4472 & 187.444841 & 8.000476 & 15.8 & 462.0 ( 10 ) & 1.4@xmath105 & 1 + m89 & 188.915864 & 12.5563414 & 16.0 & 201.4 ( 4 ) & 1.1@xmath105 & 2 & 1 & & + m104 & 189.997633 & -11.623054 & 11.1 & 194.2 ( 4 ) & 9.0@xmath104 & 10 & 2 & 659 & 65 + ic3652 & 190.243750 & 11.184556 & 15.2 & 5.1 ( 1 ) & 1.8@xmath107 & 1 + m59 & 190.509348 & 11.647027 & 15.5 & 30.1 ( 2 ) & 3.8@xmath105 & 1 & 1 & + m60 & 190.916564 & 11.552706 & 16.6 & 307.9 ( 6 ) & 1.3@xmath105 & 1 & 1 & + ngc4660 & 191.132917 & 11.190306 & 16.4 & 5.1 ( 1 ) & 2.1@xmath107 & 1 + ngc4696 & 192.205208 & -41.310833 & 37.6 & 779.3 ( 15 ) & 1.2@xmath107 & 2 + ngc5128 & 201.365063 & -43.019113 & 3.8 & 843.2 ( 24 ) & 6.6@xmath106 & 16 & 4 & 158 & 10 + m51 & 202.484200 & 47.230600 & 8.0 & 856.6 ( 14 ) & 1.3@xmath104 & 21 + ngc5846 & 226.622017 & 1.605625 & 26.9 & 149.9 ( 3 ) & 6.7@xmath105 & 13 + cccccccccccc ngc1023-eo1 & 40.1191 & 39.0608 & 40.1192 & 39.0610 & @xmath108 & @xmath109 & 2.1 & @xmath110 & 10.0 & -8.92 & 2.0 + ngc1399-eo12 & 54.82383 & -35.42506 & 54.82373 & -35.42503 & @xmath111 & @xmath112 & 33.0 & @xmath113 & 10.0 & -11.10 & 0.40 + ngc4365-eo019 & 186.06212 & 07.32031 & 186.06209 & 07.32052&@xmath114 & @xmath115 & 1.4 & @xmath116 & 17.4 & -8.23 & 0.3 + ngc4365-eo039 & 186.10850 & 07.36425 & 186.10835 & 07.36376&@xmath117 & @xmath118 & - & @xmath119 & 15.2 & -7.65 & 0.15 + ngc4365-eo006 & 186.11362 & 07.30964 & 186.11357 & 07.30957 & @xmath120 & @xmath121 & 10.9 & @xmath122 & 20.7 & -9.73 & 0.15 + ngc4365-eo117 & 186.11893 & 07.30381 & 186.11858 & 07.30391 & @xmath123 & @xmath124 & 8.1 & @xmath125 & 11.1 & -6.64 & 0.15 + ngc4365-eo206 & 186.15944 & 07.35225 & 186.15929 & 07.35184 & @xmath126 & @xmath127 & 10.9 & @xmath128 & 17.4&-5.53 & 0.3 + ngc4365-eo015 & 186.16258 & 07.28169 & 186.16245 & 07.28119 & @xmath129 & @xmath130 & 1.0 & @xmath131 & 21.2 & -8.43 & 0.02 + m89-eo1 & 188.90877 & 12.55032 & 188.90878 & 12.55081 & @xmath132 & @xmath133 & 1.4 & @xmath134 & 26.6 & -11.02 & 0.2 + m104-eo08 & 189.99429 & -11.63936 & 189.99474 & -11.63904 & @xmath135 & @xmath136 & 5.0 & @xmath137 & 15.6 & -6.15 & 0.1 + m104-eo01 & 190.01304 & -11.66786 & 190.01315 & -11.66781 & @xmath138 & @xmath139 & 9.3 & @xmath140 & 14.7 & -12.30 & 0.83 + m59-ucd3 & 190.54605 & 11.64479 & 190.54615 & 11.644546 & @xmath141 & @xmath142 & 1.0 & @xmath143 & 20 & -14.6 & 0.98 + m60-ucd1 & 190.89987 & 11.53464 & 190.89977 & 11.534675 & @xmath144 & @xmath145 & 5.6 & @xmath146 & 24 & -14.20 & 0.95 + ngc5128-eo03 & 201.24246 & -42.93619 & 201.24250 & -42.93621 & @xmath147 & @xmath148 & 1.8 & @xmath149 & 10.7 & -10.23 & 0.39 + ngc5128-eo02 & 201.27383 & -43.17519 & 201.27388 & -43.17508 & @xmath150 & @xmath151 & 4.6 & @xmath152 & 10.6 & -10.31 & - + ngc5128-eo05 & 201.37621 & -42.99300 & 201.37629 & -42.99297 & @xmath153 & @xmath154 & 9.3 & @xmath155 & 11.9 & -9.87 & 0.013 + ngc5128-eo01 & 201.38167 & -43.00078 & 201.38176 & -43.00079 & @xmath156 & @xmath157 & 49.1 & @xmath158 & 13.5 & -11.17 & 0.1 + bekki , k. , couch , w. j. , & drinkwater , m. j. , 2001 , , 522 , l105 blakeslee , j. p. , cho , h. , peng , e. w. , ferrarese , l. , et al . 2012 , , 746 , 88 blom , c. , spitler , l. r. , & forbes , d. a. , 2012 , , 420 , 37 brodie , j. p. , & larsen , s. s. , 2002 , , 124 , 1410 brns , r. c. , kroupa , p. , fellhauer , m. , metz , m. , & assmann , p. , 2011 , a&a , 529 , 138 brns , r. c. , & kroupa , p. , 2012 , , 547 , a65 ( bk12 ) caso , j. p. , bassino , l. p. , richtler , t. , smith castelli , a. v. , & faifer , f. r. , 2013 , , 430 , 1088 chattopadhyay , a. k. , chattopadhyay , t. , davoust , e. , et al . 2009 , , 705 , 1533 chiboucas , k. , tully , r. b. , marzke , r. o. , et al . 2011 , , 737 , 86 chilingarian , i. v. , & mamon , g. a. , 2008 , mnras , 385 , l83 clark , g. w. , 1975 , , 199 , l143 da rocha , c. , mieske , s. , georgiev , i. y. , hilker , m. , et al . 2011 , , 525 , 86 drinkwater , m. j. , jones , j. b. , gregg , m. d. , & phillipps , s. , 2000 , , 17 , 227 drinkwater , m. j. , gregg , m. d. , hilker , m. , et al . 2003 , , 423 , 519 drinkwater , m. j. , gregg , m. d. , couch , w. j. , et al . 2004 , , 21 , 375 fabbiano , g. , 2006 , , 44 , 323 fabian , a. c. , 1975 , , 173 , 161 fellhauer , m. , & kroupa , p. , 2002 , , 330 , 642 forbes , d.a . , pota , v. , usher , c. , et al . 2013 , mnras , 435 , l6 forbes , d. a. , almeida , a. , spitler , l. r. , & pota , v. , 2014 , , 442 , 1049 gilfanov , m. , 2004 , , 349 , 146 gregg , m. d. , drinkwater , m. j. , evstigneeva , e. , et al . 2009 , , 137 , 498 harris , g. l. h. , gmez , m. , harris , w. e. , johnston , k. , et al . 2012 , , 143 , 84 hasegan , m. , jordan , a. , cote , p. , et al . 2005 , , 627 , 203 hau , g. k. t. , spitler , l. r. , forbes , d. a. , et al . 2009 , , 394 , l97 hilker , m. , infante , l. , vieira , g. , kissler - 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we have conducted a systematic study of x - ray emission from ultra - compact dwarf ( ucd ) galaxies and extended star clusters ( escs ) , based on archival _ chandra _ observations . among a sample of 511 ucds and escs complied from the literature ,
17 x - ray counterparts with 0.5 - 8 kev luminosities above @xmath0@xmath1 are identified , which are distributed in eight early - type host galaxies . to facilitate comparison
, we also identify x - ray counterparts of 360 globular clusters ( gcs ) distributed in four of the eight galaxies .
the x - ray properties of the ucds and escs are found to be broadly similar to those of the gcs .
the incidence rate of x - ray - detected ucds and escs , @xmath2% , while lower than that of the x - ray - detected gcs [ ( @xmath3% ] , is substantially higher than expected from the field populations of external galaxies . a stacking analysis of the individually undetected ucds / escs further reveals significant x - ray signals , which corresponds to an equivalent 0.5 - 8 kev luminosity of @xmath0@xmath4 per source . taken together ,
these provide strong evidence that the x - ray emission from ucds and escs is dominated by low - mass x - ray binaries having formed from stellar dynamical interactions , consistent with the stellar populations in these dense systems being predominantly old . for the most massive ucds
, there remains the possibility that a putative central massive black hole gives rise to the observed x - ray emission .
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as is well known , a so - called excess current @xmath5 appears at large voltages @xmath1 in josephson junctions ( jj ) with a direct conductance,@xcite that is , the current @xmath5 arises in jjs of the s / n / s or s / c / s types , where n denotes a normal metal ( a wire or a film ) and c a constriction . this means that the current - voltage ( @xmath0-@xmath1 ) characteristics at large @xmath1 ( @xmath9 , where @xmath10 is the energy gap in the superconductors s ) has the form @xmath11 where @xmath12 is the resistance of the jj in the normal state and the constant @xmath5 is the excess current which can be written in the form @xmath13 here , @xmath14 is a numerical factor equal to @xmath15 in the diffusive limit,@xcite and @xmath16 in ballistic jjs with ideal ( fully transparent ) interfaces.@xcite eq . ( [ 1 ] ) also describes the asymptotic behavior ( @xmath17 ) of the @xmath0-@xmath1 characteristics of s / n / n contacts,@xcite where n is a normal metal reservoir . in the latter case , the excess current is twice smaller than in the s / n / s jjs . the excess current @xmath5 is an essential characteristics of s / n / n or s / n / s contacts which distinguishes them from the tunnel junctions s / i / n or s / i / s where this current does not arise . if the s / n or n / n interfaces are not ideal ( the transmission coefficient differs from 1 ) , the coefficient @xmath14 in eq . ( [ 1 ] ) can be either positive or negative . that is , an excess @xmath5 or deficit @xmath18 currents arise in this case . their values depend on the interface transparencies of both interfaces.@xcite the appearance of the excess current at large @xmath1 as well as the non - zero subgap conductance @xmath19 of the s / n / n contacts at @xmath20 and @xmath21 is explained@xcite in terms of andreev reflections ( ar).@xcite it has been shown in refs . that the zero bias conductance @xmath22 coincides with the conductance in the normal state and has a non - monotonous dependence on the applied voltage @xmath1 or temperature @xmath23 . similar behavior of the conductance takes place in the so - called andreev interferometers ( see experimental observations in refs . and theoretical explanations in refs . ) . the andreev reflection implies that an electron moving in the normal metal towards the superconductor is converted at the s / n interface into a hole with opposite spin which moves back along the same trajectory . physically , this process means that an electron with momentum @xmath24 and spin @xmath25 moving from the n - metal penetrates the superconductor s and forms there a cooper pair , i.e. , it pulls another electron with opposite momentum @xmath26 and spin @xmath27 . the absence of this electron in the n - metal is nothing else as the creation of a hole with momentum @xmath26 and spin @xmath27 . in the superconductor / ferromagnet ( s / f ) contacts , the ar is suppressed since the exchange field @xmath3 acting on spins breaks the symmetry of spin directions . de jong and beenakker@xcite have shown that the conductance @xmath28 in ballistic s / f systems is reduced with increasing @xmath3 and turns to zero at @xmath29 , where @xmath30 is the fermi energy . at high exchange energy , electrons with only one spin direction exist in the ferromagnet f so that the ar at s / f interfaces is not possible . one can expect a similar behavior of the conductance in s@xmath2/n / n contacts , where a `` magnetic '' superconductor with a spin filter s@xmath2 ( see below ) supplies only fully polarized triplet cooper pairs penetrating the n - metal . it consists of an s / f bilayer and a spin filter fl which passes electrons with only one spin direction , so that one deals with the s@xmath2 superconductor constructed as a multylayer structure of the type s / f / fl . in this case , the conventional ar at the s@xmath2/n interface is forbidden and , therefore , the subgap conductance at low temperatures as well as the excess current may disappear . as will be shown in this work , the subgap conductance as well as the excess current @xmath5 remain finite in s@xmath2/n / n contacts . the magnitude of the current @xmath5 and its sign depend on the value of the exchange field in the ferromagnet f. in the considered case of s@xmath2/n / n contacts , the subgap conductance and the excess current occur due to an unconventional ar in which two electrons with parallel spins in the n - film form a triplet cooper pair with the same direction of the total spin . therefore , the ar at the s@xmath2/n interface is not accompanied by spin - flip ( the hole in the n - wire has the same spin direction as the incident electron ) . note that , nowadays , the interest in studies of the excess current is revived in the light of recent measurements on s / sm / s jjs with unconventional semiconductor sm ( topological insulator ) in which the josephson effect can occur due to majorana modes ( see recent experimental papers refs . , and references therein ) . in these junctions , the excess current also has been observed . on the other hand , properties of high-@xmath31 superconductors including the iron - based pnictides have been also studied with the aid of point - contact spectroscopy in which the differential conductance of n / s point contacts has been measured.@xcite a theory of differential conductance of n / s point contacts composed by a two band superconductor with energy gaps of different signs [ @xmath32 has been presented in ref . . in this paper , we calculate the @xmath0-@xmath1 characteristics of diffusive superconductor / normal metal systems of two types . in the first type of contacts , s@xmath33/n / n , the `` magnetic '' superconductor s@xmath33 is a singlet superconductor s covered by a thin ferromagnetic layer [ see fig . [ fig : system1a ] ( a ) ] . in this case , both the singlet and the triplet cooper pairs penetrate into the n - wire . in the second type of contacts , s@xmath2/n / n , the magnetic superconductor s@xmath2 consists again of an s / f bilayer which is separated from the n - wire by a spin filter fl [ see fig . [ fig : system1a ] ( b ) ] . the spin filter fl is assumed to pass only electrons with spins oriented along the @xmath34 axis ( @xmath35 ) . using the quasiclassical theory , we show that in both types of contacts , s@xmath33/n / n and s@xmath2/n / n , the conductance @xmath36 is affected by the proximity effect and the excess ( deficit ) current @xmath5 ( @xmath18 ) as well as the subgap conductance are finite . we consider an s@xmath2/n / n contact , in which the `` magnetic '' superconductors are formed by a bcs superconductor s ( s - wave , singlet ) covered by a thin ferromagnetic layer f with an exchange field @xmath37 [ see fig . [ fig : system1a ] ( a ) ] . due to proximity effect , the singlet component penetrates from the superconductor into the f film , and also a triplet component arises under the action of the exchange field @xmath37 . as is well known ( see reviews refs . ) , in the case of homogeneous magnetization @xmath38 ( @xmath39 ) in the ferromagnet , the vector of the total spin of triplet cooper pairs @xmath40 lies in the plane perpendicular to @xmath38 . thus , the s / f bilayer with a sufficiently transparent interface can be considered as a `` magnetic '' superconductor with a built - in effective exchange field @xmath41 which has a nonzero projection onto the @xmath34 axis and an effective energy gap @xmath42 ( to be more exact , the condensate wave functions in the f film are analogous to those in a `` magnetic '' superconductor ) . the magnitudes of @xmath43 and @xmath42 are determined by certain conditions . for example , in case of thin f and s layers ( @xmath44 , @xmath45 , where @xmath46 are the thicknesses of the f(s ) layers , @xmath47 and @xmath48 ) and a low f / s interface resistance , one has@xcite @xmath49 in case of a high s / f interface resistance , we obtain ( see the appendix [ app : appendix_a ] ) @xmath50 where @xmath51 is the diffusion coefficient in the f film , @xmath52 , @xmath53 is the conductivity of the f film and @xmath54 is the s / f interface resistance per unit area [ see below eqs . ( [ 4 ] ) and ( [ 4 ] ) ] . the quantity @xmath55 determined by the interface resistance is the so - called subgap or minigap.@xcite in both cases , the effective exchange field @xmath56 may exceed the effective gap @xmath57 without causing a non - uniform state of the larkin ovchinnikov fulde ferrel type@xcite because the thicker s film is only weakly affected by the f film . for example in the famous experiment,@xcite where a long - range triplet component has been observed in a multilayered s / f/f / f/s josephson junction , the curie temperature in a weak ferromagnet f ( pd@xmath58ni@xmath59 ) was about @xmath60 , that is , much larger than the critical temperature of the superconducting transition in the superconductor s ( nb ) with a transition temperature @xmath61 . in principle , a similar f/s bilayer can be employed as a prototype of the presented s@xmath33 superconductor . the f layer in s@xmath2/n / n contacts is separated from the n - wire ( or film ) by a filter that passes electrons only with a certain spin direction , say , parallel or antiparallel to the @xmath34 axis [ see fig . [ fig : system1a ] ( b ) ] . as a filter , thin layers of strongly polarized magnetic insulator@xcite and dyn or gdn films@xcite can be used . /n / n contact the superconductor s@xmath33 consists of a bcs superconductor s and a thin ferromagnetic layer ( denoted by f@xmath62 ) , and is connected to a normal metal reservoir n on the right hand side via a normal metal wire n. ( b ) s@xmath2/n / n contact in addition to the case ( a ) , the s@xmath33 superconductor on the left hand side is covered by a spin filter fl that passes electrons only with a certain spin direction , say , parallel or antiparallel to the @xmath34 axis ( indicated by the thick blue arrow ) . the superconducting phase on the left hand side is @xmath63 . ( c ) sketch ( not to scale ) of a possible experimental realization of the case ( b ) . ] the convenient method to study the system under consideration is the theory of quasiclassical green s functions.@xcite this technique is generalized for the case of ferromagnet - superconductor structures where a non - trivial dependence of the quasiclassical green s functions @xmath64 on spin indices must be taken into account.@xcite in the considered non - equilibrium case , the green s function @xmath64 is a matrix with diagonal matrix elements ( @xmath65 and @xmath66 ) and non - diagonal element ( @xmath67 ) , where the matrices @xmath68 and @xmath67 are the retarded ( advanced ) and keldysh functions , respectively . all these functions are @xmath69 matrices in the gorkov - nambu and spin spaces . in the n - wire the matrix @xmath64 obeys an equation which looks similar to the usadel equation@xcite ( see also eq . ( 5 ) in ref . ) @xmath70 = 0 \ , , \label{2}\ ] ] where @xmath71 with the diffusion coefficient @xmath51 . the matrix @xmath72 is a tensor product of the pauli matrices @xmath73 ( @xmath74 ) and the @xmath75 unit matrix @xmath76 , which operate in the particle - hole and spin space , respectively . the matrix quasiclassical green s function @xmath64 obeys the normalization condition @xmath77 equation ( [ 2 ] ) is complemented by boundary conditions at the interfaces s@xmath33/n and n / n . they have the form [ see eq . ( 4.7 ) in ref . , refs . , and also the recent work ref . ] @xmath78|_{0 } \ , , \label{4 } \\ 2 \bar{r}_{\text{n } } l \check{g } \partial_{x } \check{g } & = [ \check{g}\ , , \hat{g}_{\text{n}}]|_{l } \ , . \label{4'}\end{aligned}\ ] ] here , the sub - indices @xmath79 and @xmath80 relate to the n / s@xmath33 and n / n interfaces , respectively , while @xmath81 , where @xmath53 is the conductivity of the n - wire , and @xmath82 denote the s@xmath33/n ( respectively , n / n ) interface resistance per unit area . the matrix @xmath83 describes the electron transmission with a spin - dependent probability @xmath84 . if the filters let to pass only electrons with spins parallel to the @xmath34 axis , then @xmath85 so that the probability for an electron with spin up ( down ) to pass into the n - wire is @xmath86 . we assume that @xmath87 with @xmath88 , and the coefficients @xmath89 and @xmath90 are normalized , i.e. , @xmath91 . note that coefficients @xmath92 are inverted with respect to the coefficients @xmath93 used in refs . . consider first eq . ( [ 2 ] ) for the keldysh green s function @xmath67 . in the considered one - dimensional case it has the form @xmath94 = 0 \ , . \label{5k}\ ] ] the keldysh function @xmath67 can be expressed in terms of the retarded and advanced green s functions @xmath68 , and the matrix distribution function @xmath95 , @xmath96 the distribution function @xmath97 determines the superconducting order parameter @xmath10 , whereas the function @xmath98 describes the dissipative current.@xcite we need to know only the distribution function @xmath98 . multiplying eq . ( [ 5k ] ) by @xmath99 and taking trace we obtain ( employing the normalization condition eq . ( [ 3 ] ) , in particular , the relations @xmath100 ) @xmath101 \partial_{x } n = j \ , , \label{6f}\]]where @xmath102 are , respectively , the diagonal and off - diagonal elements of @xmath68 matrices in the particle - hole space , and we introduced the notation @xmath103 . the quantity @xmath104 is independent of @xmath105 . integrating eq . ( [ 6f ] ) we obtain @xmath106 where @xmath107 . using the boundary conditions , eqs . ( [ 4 ] ) and ( [ 4 ] ) , we find @xmath108where @xmath109 - \tanh [ ( \epsilon - ev)/2 t ] \big]}$ ] is the distribution function in the normal metal reservoir ( we set the voltage in the s reservoir equal to zero ) , @xmath110 , @xmath111 , and @xmath112 . the current @xmath0 is expressed via the `` partial '' current @xmath113 as @xmath114 formula eq . ( [ 8j ] ) generalizes eq . ( 13 ) of ref . for the considered case of a spin - dependent interaction and can be applied to the description of contacts with a condensate consisting of singlet and triplet cooper pairs . in the normal state , above the critical temperature of the superconductor s , one has @xmath115 . thus , we obtain a standard expression for the current per unit area in an n / n / n contact @xmath116 the denominator is the sum of interface resistances and the resistance of the normal n - wire . the normalized differential conductance of the contacts under consideration @xmath117 at @xmath21 is @xmath118 where @xmath119 is the normalized voltage . the normalized current @xmath120 is given by the relation @xmath121 and , at large voltage , can be written in the form @xmath122 where @xmath123 is the normalized current through the contact in the normal state . the normalized excess ( @xmath124 ) or deficit current ( @xmath125 ) is determined by the expression @xmath126 \ , \mathrm{d } v \ , . \label{13exc}\ ] ] it is valid at arbitrary temperatures because for the function @xmath127 in eq . ( [ 8j ] ) we have @xmath128 for @xmath129 . the current @xmath130 can be presented as @xmath131 , where @xmath132 is a subgap current and @xmath133 is the contribution from quasiparticles with energies above the gap ; the normalized current in the normal state is @xmath134 . we see that the excess current is determined by the retarded ( advanced ) green s functions @xmath68 that obey an usadel - like equation . this equation can be solved in limiting cases . we consider a contact with a short n - wire ( @xmath135 ) in which the interface resistances dominate ( @xmath136 ) , i.e. , the interface resistances are much larger than the resistance of the n - wire , @xmath137 . in the case of a short contact , the last term in the denominator of eq . ( [ 11dif ] ) and the second term in eq . ( [ 5k ] ) can be neglected so that the usadel equation for the green s functions @xmath68 acquires the form @xmath138 provided that @xmath139 . we integrate eq . ( [ 14usadel ] ) once over @xmath105 and obtain @xmath140 from the boundary conditions eqs . ( [ 4 ] ) and ( [ 4 ] ) for the retarded ( advanced ) green s functions , we have @xmath141^{r(a ) } \ , , \label{15j } \\ 2 \hat{j}^{r(a)}l & = - \bar{r}_{\text{n}}^{-1 } [ \hat{g}_{\text{n } } \ , , \hat{g}(l)]^{r(a ) } \ , . \label{15j'}\end{aligned}\ ] ] subtracting the first equation from the second we arrive at @xmath142^{r(a ) } = 0 \ , , \label{15usadel}\ ] ] where the matrix @xmath143 is a sum of contributions of the n / n and s@xmath33/n interfaces , @xmath144 and @xmath145}$ ] . the form of matrices @xmath146 and @xmath147 depends on the type of a superconductor . [ [ magnetic - superconductors_textm . ] ] `` magnetic '' superconductor s@xmath33 . + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + that is , the superconductor s@xmath148 is represented by an s / f bilayer with a thin ferromagnetic layer f. we assume that the exchange field @xmath37 is aligned parallel to the z axis , @xmath149 . in this case , @xmath150 \ , , \label{eq : lambda_a}\ ] ] with@xcite @xmath151^{-1 } |\epsilon + h| \pm [ \zeta^{r(a)}(\epsilon - h)]^{-1 } |\epsilon - h| } { 2 } \ , , \label{g_s_mag}\\ f_{\text{s } \pm}^{r(a ) } & = \frac{\delta \big [ [ \zeta^{r(a)}(\epsilon + h)]^{-1 } \pm [ \zeta^{r(a)}(\epsilon - h)]^{-1 } \big]}{2 } \ , . \label{f_s_mag}\end{aligned}\ ] ] the terms @xmath152 and @xmath153 in eq . ( [ eq : lambda_a ] ) describe the singlet component and , respectively , the short - range triplet component with the total spin of triplet cooper pairs @xmath40 normal to the @xmath37 vector . note that the energy gap @xmath10 and the exchange field @xmath3 in eqs . ( [ g_s_mag ] ) and ( [ f_s_mag ] ) mean the effective @xmath42 and @xmath56 defined in eqs . ( [ 1a])([1b ] ) . in the following , for brevity , we drop the subindex `` eff '' . [ [ triplet - superconductor - s_textt . ] ] `` triplet '' superconductor s@xmath2 . + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + this case can be realized with the help of an s / f bilayer with the @xmath37 vector aligned , for instance , along the @xmath105 axis . the s / f bilayer is assumed to be separated from the n - wire by a spin filter oriented parallel to the @xmath34 axis . then , @xmath154 \,.\ ] ] the last term describes fully polarized triplet cooper pairs with the @xmath40 vector oriented along the @xmath34 axis . [ [ bcs - superconductor . ] ] bcs - superconductor . + + + + + + + + + + + + + + + + + + + for completeness , we consider also the case of the bcs superconductor which is obtained from the case of a `` magnetic '' superconductor s@xmath33 setting @xmath155 . here , @xmath156 \,,\ ] ] with @xmath157^{-1 } \ f_{\text{s}}^{r(a ) } & = \delta \big [ \zeta^{r(a ) } \big]^{-1 } \,,\end{aligned}\ ] ] and @xmath158 . in order to make the results more transparent , we assume that the parameter @xmath160 is small and both parameters @xmath161 are large ( @xmath162 ) . these conditions correspond to experimental systems and mean that the s / n interface resistance is much larger than the resistance of the n / n interface and both interface resistances are larger than the resistance of the short n - wire . then , the solution for a small correction @xmath163 [ where @xmath164 are the quasiclassical retarded ( advanced ) green s functions in the separated n - wire ] is @xmath165 we see that in the lowest approximation in the parameter @xmath160 only the condensate wave function , off - diagonal in the gorkov - nambu space , is changed due to proximity effect . the correction @xmath166 is small if the parameter @xmath167 is small or , in the case of the s@xmath2/n / n contact , if the parameter @xmath168 is small . using the known function @xmath169 and eq . ( [ 11dif ] ) , we can readily calculate the normalized conductance @xmath170 at @xmath21 . thus , we obtain @xmath171^{-1}}|_{\epsilon = v } \ , , \label{17con}\ ] ] with the functions @xmath172^{-1 } |\epsilon + h| + [ \zeta^{r}(\epsilon - h)]^{-1 } |\epsilon - h| \big\ } } { 2 } \ , , & \text{s$_{\text{m}}$/n / n } \ , , \\ \frac { \re\big\ { [ \zeta^{r}(\epsilon + h)]^{-1 } |\epsilon + h| + [ \zeta^{r}(\epsilon - h)]^{-1 } |\epsilon - h| \big\ } } { 2 } \ , , & \text{\text{s$_{\text{t}}$/n / n } } \ , , \\ \re\big\ { \frac{|\epsilon|}{\zeta^{r}_{+}(\epsilon ) } \big\ } \ , , & \text{s / n / n } \ , , \end{cases}\end{aligned}\ ] ] and @xmath173 ^ 2 + \big [ \re \big\ { \frac{\delta}{\zeta^{r}(\epsilon + h ) } - \frac{\delta}{\zeta^{r}(\epsilon - h ) } \big\ } \big]^2}{2 } \ , , & \text{s$_{\text{m}}$/n / n } \ , , \\ \frac{\big [ \re \big\ { \frac{\delta}{\zeta^{r}(\epsilon + h ) } - \frac{\delta}{\zeta^{r}(\epsilon - h ) } \big\ } \big]^2}{2 } \ , , & \text{\text{s$_{\text{t}}$/n / n } } \ , , \\ \big [ \re\big\ { \frac{\delta}{\zeta^{r}_{-}(\epsilon ) } \big\ } \big]^2 \ , , & \text{s / n / n } \ , , \end{cases } \notag\end{aligned}\ ] ] where @xmath174}}$ ] . equation ( [ 17con ] ) determines the dependence of the normalized differential conductance on the normalized voltage @xmath175 . ) as a function of normalized voltage for the ( a ) s@xmath33/n / n contact , ( b ) s@xmath2/n / n ( in both cases , the parameters are @xmath176 , @xmath177 for the black solid line and @xmath178 for the red dashed line ) , and ( c ) s / n / n contact , where s is a bcs superconductor ( the parameter is @xmath176 ) . note that the quantities @xmath10 and @xmath3 are not the true energy gap and the magnetic field , respectively , but the in eqs . ( [ 1a])([1b ] ) defined effective values ( see also appendix [ app : appendix_a ] ) . ] the first term in the denominator , @xmath179 determines the resistance of the n / n interface , while the second term is proportional to the resistance of the interface between the n - wire and the corresponding superconductor . the first term in the square brackets , @xmath180 , determines the conductance of this interface due to quasiparticles with energies above the gap , whereas the second term , @xmath181 , is related to the subgap conductance . we analyze the differential conductance @xmath170 and the @xmath0-@xmath1 characteristics @xmath182 , @xmath183 for contacts of different types . equations ( [ 17con])([17i_v ] ) allow one to calculate the conductance and the @xmath0-@xmath1 characteristics of contacts under consideration . in fig . [ fig : difcond2a ] , we show the dependence of the normalized differential conductance @xmath170 on the normalized voltage @xmath184 for the three types of contacts , i.e. , the s@xmath33/n / n contact [ fig . [ fig : difcond2a ] ( a ) ] , the s@xmath2/n / n contact [ fig . [ fig : difcond2a ] ( b ) ] , and the s / n / n contact , where s is a usual bcs superconductor [ fig . [ fig : difcond2a ] ( c ) ] . note that the dependence @xmath170 for the case of the bcs superconductor coincides with that for the case of a `` magnetic '' superconductor if one sets @xmath155 . although the function @xmath170 in fig . [ fig : difcond2a ] ( c ) looks like the voltage dependence of the differential conductance of an s / i / n junction ( where i stands for an insulating thin layer ) , it differs from the latter one because this dependence leads to an excess current @xmath5 . this current is given by the value of @xmath185 in fig [ fig : exc_current_on_h ] ( a ) at @xmath155 ( blue dashed line ) . the appearance of the excess current is a direct consequence of the fact that the integral @xmath186 $ ] is not zero as it takes place in tunnel s / i / n junctions . it is seen from fig . [ fig : difcond2a ] ( c ) that there is a nonzero subgap conductance in the s / n / n contact . it is caused by a subgap contribution related to the andreev reflection . this mechanism is also responsible for a zero - bias peak in the conductance that has been observed in early experiments on s / n / sm contacts ( here , sm is a n - doped semiconductor).@xcite theoretical explanations for the observed subgap conductance is given in refs . . in figs . [ fig : difcond2a ] ( a ) and [ fig : difcond2a ] ( b ) , we plot the voltage dependence of the normalized conductance of the contacts of s@xmath33/n / n and s@xmath2/n / n types for different values of @xmath3 . in both cases , the subgap conductance is not zero , but it is small in contacts of s@xmath33/n / n type if the exchange field @xmath3 is small compared to @xmath10 . the latter property is due to a negligible contribution to the conductance in the subgap region because this contribution is provided by fully polarized triplet cooper pairs the density of which , @xmath187 , decreases with decreasing @xmath3 since @xmath188 . note that similar results ( nonzero subgap conductance ) were obtained in ref . , where differential conductance of an f/f / s structure has been studied . however , the case of fully polarized triplet component has not been considered there . the subgap conductance in another , although similar , systems has been calculated in ref . on the basis of the scattering matrix approach . the authors considered a half - metal / ferromagnet / superconductor contact in the ballistic regime assuming that the magnetizations in half - metal and ferromagnet are not collinear . they assumed also that only a single conducting channel exists in the system so that the quasiclassical theory can not be applied to the system . to some extent , the results obtained in our paper and in ref . differ . although the subgap conductance @xmath170 calculated in ref . differs from zero , it turns to zero at @xmath189 whereas @xmath190 obtained by us in the present work is finite . a similar system consisting of a half - metallic ferromagnet and a superconductor has been studied in refs . the authors assumed that these materials are separated by a spin - active interface . they also obtained the vanishing zero - bias conductance for @xmath191 . in our case , the finite @xmath190 is caused by unconventional andreev reflection of triplet cooper pairs induced in the n - wire due to proximity effect . this ar make the s@xmath33/n interface partially transparent as it occurs in s / n contacts,@xcite . the zero - bias conductance @xmath190 as a function of @xmath3 is depicted in fig . [ fig : difcond3 ] for the s@xmath33/n / n and s@xmath2/n / n contacts . it is equal to zero at @xmath155 in the s@xmath2/n / n contact , where only triplet cooper pairs are present , and has a maximum at @xmath192 . as mentioned above , at @xmath155 the amplitude of the triplet component turns to zero , and hence the zero - bias conductance vanishes . for the contacts for the s@xmath2/n / n ( black solid line ) , respectively , s / n / n contact ( red dashed line)in both cases , the parameter @xmath193 . note that the quantities @xmath10 and @xmath3 are not the true energy gap and the magnetic field , respectively , but the in eqs . ( [ 1a])([1b ] ) defined effective values ( see also appendix [ app : appendix_a ] ) . ] we investigate the @xmath0-@xmath1 characteristics of the contacts of the types s@xmath33/n / n and s@xmath2/n / n . [ [ s_textmnn - contact . ] ] s@xmath33/n / n contact . + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + in the considered case of small but finite @xmath194 , the @xmath0-@xmath1 characteristics shows an excess current . in particular , for @xmath155 we obtain @xmath195 [ or , with dimension , @xmath196 with @xmath179 ] . the excess current increases with increasing the exchange field @xmath3 [ see fig . [ fig : exc_current_on_h ] ( a ) ] . the @xmath0-@xmath1 curve has a simple form for the case @xmath155 ( bcs superconductor ) . for small @xmath194 and @xmath198 , we obtain @xmath199 in this case , there is an excess current in the @xmath0-@xmath1 curve ( see fig . [ fig : exc5 ] ) . for the ( a ) s@xmath33/n / n and ( b ) s@xmath2/n / n contacts . noticeably is the nonmonotonic behavior of the @xmath0-@xmath1 curve in the s@xmath33/n / n contact . the excess current in the s@xmath2/n / n contact turns to deficit current at low @xmath7 ( see text ) . the parameter @xmath194 has the values @xmath200 ( black solid lines ) , @xmath176 ( blue dashed lines ) , @xmath201 ( red dash - dotted lines ) , and @xmath202 ( green dotted lines ) . the current is normalized to the value of the ohm s law current at the voltage @xmath203 , i.e. , @xmath204 , where @xmath205 with the resistance of the contact in the normal state @xmath12 . note that the quantities @xmath10 and @xmath3 are not the true energy gap and the magnetic field , respectively , but the in eqs . ( [ 1a])([1b ] ) defined effective values ( see also appendix [ app : appendix_a ] ) . ] ( black solid line ) and @xmath201 ( blue dashed line ) . the current is normalized to the value of the ohm s law current at the voltage @xmath203 , i.e. , @xmath204 . the black dotted line indicates the ohm s law , @xmath205 with the resistance of the contact in the normal state @xmath12 . note that the quantities @xmath10 and @xmath3 are not the true energy gap and the magnetic field , respectively , but the in eqs . ( [ 1a])([1b ] ) defined effective values ( see also appendix [ app : appendix_a ] ) . ] [ [ s_texttnn - contact . ] ] s@xmath2/n / n contact . + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + using eq . ( [ 13exc ] ) we find the excess or deficit current for small @xmath194 and @xmath3 , @xmath206 where @xmath207 . one can see that at @xmath208^{3/4}}$ ] , there is an excess current and at @xmath209 the excess current is converted into a deficit current , cf . [ fig : exc_current_on_h ] ( b ) . as is seen from fig . [ fig : exc_current_on_h ] , the magnitude of the excess current @xmath5 in the case of the s@xmath2/n / n junction is comparable with the excess current in an s / n / n junction with the same interface resistances . this means that it can be measured experimentally on existing experimental junctions . we studied transport properties of `` magnetic '' superconductor / normal metal point contacts of different types , in which both the singlet and triplet cooper pairs are present . it is shown that , as it takes place in point s / n / n contacts with bcs superconductor , the subgap conductance @xmath4 and the excess current @xmath5 are not zero even if only fully polarized triplet component exists in the n - wire . in this case , the @xmath4 and @xmath5 are caused by an unconventional andreev reflection without spin flip ; the hole moving back along the trajectory of an incident electron with a spin @xmath40 has the same spin direction as @xmath40 . a similar ar , equal - spin andreev reflection , has been studied in a recent paper,@xcite where a contact between a ferromagnet and topological superconductor with majorana modes has been considered . we considered two types of contacts , namely the s@xmath33/n / n contact , where both the singlet and triplet component exist , and the s@xmath2/n / n contact , in which only fully polarized triplet cooper pairs penetrate into the n - wire . in both types of contacts , the subgap conductance and the excess current are present . in the second type of contacts , in s@xmath2/n / n , these are caused by an equal - spin ar . with decreasing the magnitude of the exchange field @xmath3 the excess current in the s@xmath2/n / n contact is transformed into a deficit current @xmath18 . the systems considered by us can be realized experimentally taking into account a rapid progress in preparing s / f nanostructures of different kinds.@xcite the obtained results can be used for identifying the long - range triplet component and in future applications in spintronics.@xcite we appreciate the financial support from the dfg via the projekt ef 11/8 - 2 ; k. b. e. gratefully acknowledges the financial support of the ministry of education and science of the russian federation in the framework of increase competitiveness program of nust `` misis '' ( nr . k2 - 2014 - 015 ) . we consider an f / s bilayer and show that , under certain conditions , the matrix green s function @xmath210 coincides with that in a superconductor with a built - in exchange field @xmath3 . we assume that the thickness of the f layer @xmath211 is small so that the condition , @xmath212 , is fulfilled . then , the usadel - like equation ( [ 2 ] ) in the f region can be integrated over the thickness and we come to eq . ( [ 15usadel ] ) with @xmath213 + { \hat{\tau}_{2 } } f_{\text{s}}}$ ] , where @xmath214 is the green s function in s , @xmath215 . the exchange field vector @xmath37 is set along the @xmath34 axis . the subgap energy @xmath55 is defined in eq . ( [ 1b ] ) and we use the matsubara representation . the matrix @xmath210 is diagonal in the spin - space with elements @xmath216 . the retarded green s function @xmath65 can be directly obtained from @xmath210 using the relation @xmath217 . the solution for @xmath210 can be easily found as in sec . [ sec:2a ] and has the form @xmath218 where @xmath219 . if the resistance of the f / s interface is large enough ( this corresponds to real experiments ) so that the subgap energy @xmath220 is small in comparison with @xmath10 , the solution eq . ( [ a1 ] ) can be written as @xmath221 + { \hat{\tau}_{2 } } \epsilon_{\text{sg}}}{\sqrt{(\omega \pm ih)^{2 } + \epsilon_{\text{sg}}^{2 } } } \ , . \label{a2}\ ] ] equation ( [ a2 ] ) shows that the green s function in the f layer has the same form as in a superconductor with the energy gap @xmath55 and built - in exchange field @xmath3 . this equation is valid if the thickness @xmath211 satisfies the condition @xmath222 as follows from this condition , the exchange field @xmath3 can be much larger than the effective energy gap @xmath55 . 65ifxundefined [ 1 ] ifx#1 ifnum [ 1 ] # 1firstoftwo secondoftwo ifx [ 1 ] # 1firstoftwo secondoftwo `` `` # 1''''@noop [ 0]secondoftwosanitize@url [ 0 ] + 12$12 & 12#1212_12%12@startlink[1]@endlink[0]@bib@innerbibempty link:\doibase 10.1103/revmodphys.51.101 [ * * , ( ) ] `` , '' in link:\doibase 10.1002/352760278x.ch1 [ _ _ ] ( , , ) pp . @noop * * , ( ) @noop * * , ( ) @noop * * , ( ) link:\doibase 10.1103/physrevb.25.4515 [ * * , ( ) ] \doibase http://dx.doi.org/10.1016/0038-1098(79)90044-9 [ * * , ( ) ] \doibase http://dx.doi.org/10.1016/0921-4534(93)90005-b [ * * , ( ) ] @noop * * , ( ) link:\doibase 10.1103/physrevlett.73.2488 [ * * , ( ) ] link:\doibase 10.1103/physrevlett.73.1416 [ * * , ( ) ] link:\doibase 10.1103/physrevlett.74.602 [ * * , ( ) ] link:\doibase 10.1103/physrevlett.74.5268 [ * * , ( ) ] http://stacks.iop.org/0953-8984/8/i=4/a=001 [ * * , ( ) ] link:\doibase 10.1103/physrevlett.76.823 [ * * , ( ) ] link:\doibase 10.1103/physrevlett.74.1657 [ * * , ( ) ] @noop ( ) , @noop ( ) , link:\doibase 10.1038/nature07081 [ * * , ( ) ] link:\doibase 10.1126/science.1187399 [ * * , ( ) ] http://stacks.iop.org/0953-8984/22/i=46/a=465701 [ * * , ( ) ] link:\doibase 10.1103/physrevlett.105.167003 [ * * , ( ) ] link:\doibase 10.1103/physrevb.83.060510 [ * * , ( ) ] link:\doibase 10.1103/physrevb.84.214521 [ * * , ( ) ] link:\doibase 10.1103/revmodphys.77.935 [ * * , ( ) ] link:\doibase 10.1103/revmodphys.77.1321 [ * * , ( ) ] link:\doibase 10.1063/1.3541944 [ * * , ( ) ] http://stacks.iop.org/0034-4885/78/i=10/a=104501 [ * * , ( ) ] link:\doibase 10.1103/physrevlett.86.3140 [ * * , ( ) ] link:\doibase 10.1103/physrev.175.537 [ * * , ( ) ] @noop * * , ( ) link:\doibase 10.1103/physrev.135.a550 [ * * , ( ) ] link:\doibase 10.1103/physrevlett.104.137002 [ * * , ( ) ] http://stacks.iop.org/0034-4885/74/i=3/a=036501 [ * * , ( ) ] link:\doibase 10.1103/physrevlett.70.853 [ * * , ( ) ] \doibase http://dx.doi.org/10.1063/1.2172647 [ * * , ( ) , http://dx.doi.org/10.1063/1.2172647 ] \doibase http://dx.doi.org/10.1063/1.2787880 [ * * , ( ) , http://dx.doi.org/10.1063/1.2787880 ] @noop ( ) , link:\doibase 10.1103/revmodphys.58.323 [ * * , ( ) ] `` , '' ( , , ) link:\doibase 10.1006/spmi.1999.0710 [ * * , ( ) ] @noop _ _ , the international series of monographs on physics ( , , ) link:\doibase 10.1103/physrevlett.25.507 [ * * , ( ) ] link:\doibase 10.1103/physrevb.86.060506 [ * * , ( ) ] link:\doibase 10.1103/physrevb.86.214516 [ * * , ( ) ] link:\doibase 10.1103/physrevlett.110.047002 [ * * , ( ) ] http://stacks.iop.org/1367-2630/16/i=7/a=073002 [ * * , ( ) ] http://stacks.iop.org/1367-2630/17/i=8/a=083037 [ * * , ( ) ] @noop ( ) , link:\doibase 10.1103/physrevb.92.180506 [ * * , ( ) ] link:\doibase 10.1103/physrevb.92.214510 [ * * , ( ) ] link:\doibase 10.1007/bf00115264 [ * * , ] http://stacks.iop.org/0038-5670/22/i=5/a=r01 [ * * , ( ) ] link:\doibase 10.1103/physrevlett.67.3026 [ * * , ( ) ] @noop * * , ( ) link:\doibase 10.1103/physrevlett.71.1625 [ * * , ( ) ] link:\doibase 10.1103/physrevb.59.12264 [ * * , ( ) ] link:\doibase 10.1103/physrevb.79.024517 [ * * , ( ) ] link:\doibase 10.1103/physrevb.81.094508 [ * * , ( ) ] link:\doibase 10.1103/physrevlett.105.207001 [ * * , ( ) ] http://stacks.iop.org/1367-2630/18/i=2/a=023031 [ * * , ( ) ] @noop ( ) , link:\doibase 10.1103/physrevlett.108.127002 [ * * , ( ) ] link:\doibase 10.1103/physrevb.84.024517 [ * * , ( ) ] link:\doibase 10.1038/nphys3242 [ * * , ( ) ]
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we study the @xmath0-@xmath1 characteristics of s@xmath2/n / n contacts , where s@xmath2 is a bcs superconductor s with a built - in exchange field @xmath3 , n represents a normal metal wire , and n a normal metal reservoir .
the superconductor s@xmath2 is separated from the n - wire by a spin filter which allows the passage of electrons with a certain spin direction so that only fully polarized triplet cooper pairs penetrate into the n - wire .
we show that both the subgap conductance @xmath4 and the excess current @xmath5 , which occur in conventional s / n / n contacts due to andreev reflection ( ar ) , exist also in the considered system . in our case , they are caused by unconventional ar that is not accompanied by spin flip .
the excess current @xmath5 exists only if @xmath3 exceeds a certain magnitude @xmath6 . at @xmath7
the excess current is converted into a deficit current @xmath8 .
the dependencies of the differential conductance and the current @xmath5 are presented as a function of voltage and @xmath3 .
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the recent isolation of graphene layers a single atom thick@xcite has lead to a great deal of activity , because of their novel electronic properties and potential applications . the lattice structure of graphene is determined by the @xmath4 coordination between neighboring carbon atoms . each carbon atom has three nearest neighbors , leading to planar honeycomb lattice . with small modifications , the same structure describes other carbon allotropes , such as the fullerenes and the carbon nanotubes . in this paper we study the simplest systems which combine two of these allotropes : the junctions between a single graphene layer and carbon nanotubes . a regular array of closely spaced armchair nanotubes attached to a graphene layer has already been studied@xcite , and related systems are being considered for their potential applications@xcite . we undertake here the investigation of junctions made of nanotubes with different chiralities , which have in common a transition from the planar to the tubular geometry mediated by the presence of six heptagonal carbon rings . these induce the negative curvature needed to bend the honeycomb carbon lattice at the junction , playing a kind of dual role to that of the pentagonal carbon rings in the fullerene cages@xcite . we analyze first the electronic properties of a single junction between a carbon nanotube and a graphene layer . we discuss the possible structures of this type , concentrating on geometries where the heptagonal rings are evenly spaced around the junction . the nanotubes can be then either armchair @xmath0 or zig - zag with @xmath5 geometry ( that is , with @xmath6 hexagonal rings around the tube ) . we calculate their electronic structure , using the tight - binding model based on the @xmath7 orbitals of the carbon atoms widely applied to carbon allotropes with @xmath4 coordination . paying attention to the local density of states , we find that the junctions fall into two different classes , depending on the behavior in the low - energy regime . one of the classes , comprising the junctions made of armchair and @xmath5 nanotubes when @xmath2 is a multiple of 3 , is characterized by the presence of a peak in the density of states close to the fermi level . the peak is absent in the other class , formed by the junctions made with the rest of zig - zag geometries . in general , the density of states tends to be depleted in the junction at low energies , with peaks above and below the fermi level marking the threshold for the propagation of new states across the junction . we present next a continuum description , based on the formulation of dirac fermion fields in the curved geometry , which allows us to characterize the general properties of the junction , and which is consistent with the previous discrete analysis . thus , we see that the peak at the fermi level in the local density of states is in general a reflection of the existence of quasi - bound states ( zero modes ) for the dirac equation in the curved space of the junction . it is known that the topological defects of the honeycomb lattice ( pentagonal and heptagonal rings ) induce an effective gauge field in the space of the two dirac points of the planar graphene lattice@xcite . it turns out that the effective magnetic flux is enough to localize two states at the junctions made of armchair or @xmath5 nanotubes when @xmath2 is a multiple of 3 . at low energies , however , the generic behavior is given by evanescent states , which arise from the matching of modes with nonvanishing angular momentum and have exponential decay in the nanotube . we finally apply our computational framework to the analysis of the band structure of the arrays of nanotube - graphene junctions . considering the behavior of the low - energy bands close to the fermi level , we find that the arrays also fall into two different classes . the arrays made of armchair nanotubes or @xmath5 nanotubes with @xmath2 equal to a multiple of 3 tend to have a series of flat bands close to the fermi level , while the arrays made with the rest of zig - zag nanotubes have all the bands dispersing at low energies . such a different behavior has its origin in the existence of states confined in the nanotube side of the junction . we find that this feature can also be explained in the context of the continuum model . the armchair and the @xmath5 geometries with @xmath2 equal to a multiple of 3 allow for the formation of standing waves between the junction and the other end of the tube . this is the mechanism responsible for the confinement of the states in the nanotubes and the consequent development of the flat bands , whose number grows at low energies with the length of the nanotube , in agreement with the predictions of the continuum theory . our first aim is to analyze the density of states of a semi - infinite nanotube attached to a graphene layer in the tight - binding approximation . the possible setups that we will consider , keeping the threefold coordination of the carbon atoms , are sketched in fig . the structures can be wrapped by the graphene hexagonal lattice , with the exception of the six points where the sides of the hexagonal prism ( which describes the nanotube ) intersect the plane . the threefold coordination of the carbon atoms requires the existence of sevenfold rings at those positions . + ( a ) ( b ) ( c ) we describe the electronic states in the structures shown in fig . [ one ] by means of a nearest - neighbor tight - binding model . in general the relaxation of elastic energy will modify the bond lengths at the junction , depending on the nanotube radius . we will assume that this relaxation does not change significantly the electronic behavior . in this respect , a tight - binding model based on the @xmath8 carbon orbitals is well - suited for the purpose of discerning the extended or localized character of the different electronic states . our main achievement will be to assign the different features in the local density of states to the behavior of the electronic states near the nanotube - graphene junctions . to this aim , we have actually checked that slight modulations of the transfer integral @xmath9 near the junctions do not produce significant changes in the results shown in what follows . we concentrate on the analysis of geometries where the six heptagonal carbon rings are evenly spaced around the junction as in fig . this constrains the possible chiralities of the nanotubes , that can be then either armchair @xmath0 or zig - zag @xmath1 , with the number @xmath2 running over all the integers . nanotubes in which the carbon sheet is wrapped with helicity can be also attached at the expense of introducing an irregular distribution of the heptagonal rings . anyhow , we expect that the rules explaining the different features in the density of states are universal enough to hold even in these more general cases . we have obtained the spectra of different types of hybrid structures by diagonalization of the tight - binding hamiltonian for very large lattices , with up to @xmath10 carbon atoms in the graphene part and @xmath11 in the nanotube side . given that the whole geometry has @xmath3 symmetry , we have classified the energy eigenstates into six groups according to the eigenvalue @xmath12 under a rotation of @xmath13 . the nature of each electronic state is given in general by its behavior at the nanotube - graphene junction . for this reason , we have characterized the hybrid structures in terms of the local density of states averaged over a circular ring of atoms at the end of the nanotube close to the junction . our computations have covered a number of structures including armchair and zig - zag nanotubes with different radii . after inspection of all the spectra , it becomes clear that there are several generic features in the density of states . we have represented in fig . [ two ] the behavior near the junction between a graphene layer and a ( 54,0 ) zig - zag nanotube . we observe that , apart from the peak close to zero energy in the sectors corresponding to @xmath14 , for @xmath15 there is always a depletion in the density of states at low energies , delimited by two abrupt upturns . it is remarkable that the pattern in the sectors corresponding to @xmath16 reproduces the same observed for @xmath14 , but with a scale that is approximately twice larger . the density of states in the sector with @xmath17 displays in turn a wider depletion , with the position of the peaks scaled by an approximate factor of 3 with respect to those in the @xmath14 sectors . + ( a ) ( b ) + + + ( c ) ( d ) in the above behavior of the density of states , the appearance of the peak close to zero in the sectors with @xmath14 is the only feature not generic for all kinds of nanotubes . in the junctions made with zig - zag nanotubes , the peak actually appears for nanotube geometries of the type @xmath5 when @xmath2 is a multiple of 3 . in this series of hybrid structures , the patterns in the density of states for each value of @xmath12 are quite similar , with the position of the corresponding peaks scaled in proportion to the radius of the nanotube . on the other hand , the rest of junctions , for which @xmath2 is not a multiple of 3 , display a different behavior . we have represented in fig . [ three ] the local density of states averaged over a ring of atoms at the end of a @xmath18 nanotube close to the junction . it can be observed the depletion of the density of states at low energies in all but one of the @xmath12-sectors , and the absence of a peak at zero energy in any of the sectors . the local density of states of the @xmath18 nanotube is dominated at the junction by contributions from states with @xmath16 , and this has to do with the fact that the lowest - energy subbands of a @xmath5 zig - zag nanotube have a nonvanishing angular momentum equal to @xmath19 for the motion around the tubule . this corresponds to a quantum number @xmath20 in the case of the @xmath18 nanotube , while the low - energy states have @xmath21 in the @xmath22 nanotube . the present picture becomes then consistent with the fact that states in higher subbands may propagate across the junction only above ( or below ) some threshold energy . this feature will be established more precisely in the continuum approach derived below in terms of the dirac equation . + ( a ) ( b ) + + + ( c ) ( d ) at this point , the hybrid structures can be classified into two different groups , depending on whether there is a peak or not close to zero energy in the local density of states around the nanotube - graphene junction . the peak comes actually from the contribution of a doubly degenerated level with states having @xmath14 , and whose probability distribution decays exponentially in the nanotube . the character of these states will be established in the next section , after developing the continuum limit in terms of dirac fermion fields . the evidence for the two different classes of hybrid structures is reinforced by the fact that the junctions made with armchair nanotubes behave in a quite similar way to that shown by the @xmath5 nanotubes when @xmath2 is a multiple of 3 . the local density of states averaged over a circular ring of atoms around the junction between a @xmath23 nanotube and a graphene layer has been represented in fig . [ we observe the presence of the peak close to zero energy in the sectors with @xmath14 . there is a clear depletion in the local density of states at low energies except in the sector with @xmath21 , which is consistent with the fact that the lowest - energy subbands in the armchair nanotube correspond to zero angular momentum around the tubule . + ( a ) ( b ) + + + ( c ) ( d ) the existence of the two different classes of nanotube - graphene junctions is illustrated in fig . [ five ] , which shows the results for the local density of states around the junction ( after summing over the sectors with different values of @xmath12 ) for the different types of nanotube considered above . it is remarkable the similarity between the density of states for the @xmath24 and @xmath23 nanotube geometries , which have a very close value of the radius . this suggests that there must be a universal way of understanding the low - energy electronic properties of the two different classes of junctions , independent of the details of the lattice building the junction within each class . + ( a ) ( b ) + + + ( c ) ( d ) we observe that the density of states summed over all values of @xmath12 tends to have an approximate linear behavior away from the very low energy regime . this motivates the analysis of the junction in terms of the dirac equation in the hybrid geometry , as the dirac fermions provide an appropriate description of the electronic properties in the 1d carbon nanotube as well as in the 2d graphene layer . we will then assume that the radius @xmath25 of the nanotube is much larger than the graphene lattice constant , in order to obtain the continuum limit of the tight - binding model . within this approximation , the analysis of the electronic structure is reduced to the study of the dirac equation in a space with an abrupt change from a planar to a cylindrical structure . the transition from a geometry to the other takes place due to the presence of the six heptagonal rings at the junction . these defects are the source of negative curvature , playing a role opposite to that of the pentagonal rings in a fullerene cage@xcite . the heptagons , as well as the pentagons , also induce frustration in the honeycomb lattice , and lead to the exchange of the two dirac valleys of the planar geometry@xcite . this latter effect has to be accounted for by means of an effective non - abelian gauge field operating in the space of the two independent dirac points of graphene . it can be shown that the effective flux associated to an individual heptagonal ring is equal to @xmath26 @xcite ( in units such that @xmath27 ) . the flux provided by the six heptagons at the junction can reach therefore a maximum of 3/2 times the flux quantum . in general , however , the count of the total flux may not follow an additive rule , so that it can be lower than the maximum value , depending on the relative position of the heptagonal rings@xcite . we will see that this is actually the origin of the two different classes of junctions . note that the angular momentum around the axis of the nanotube is conserved and quantized in integer units , in a continuum description of the geometry analyzed here . on the other hand , the angular momentum in the plane is quantized and shifted by @xmath28 plus the number of flux quanta of the effective gauge field induced by the heptagonal rings . the existence of topological defects which induce an effective flux at the junction allows us to match wavefunctions with different angular momenta at either side of the junction , provided that the effective flux corresponds to a half - integer number of quanta . by looking at the effect of pairs of heptagonal rings , the non - abelian gauge field operating in the space of the two dirac points becomes anyhow proportional to a sigma matrix @xmath29 . we may consider then that the effect of the six heptagonal rings at the junction is described by an effective abelian field , standing for either of the eigenvalues of @xmath29 @xcite . we end up therefore with two different dirac equations , with effective magnetic fluxes of opposite sign . we will denote the dirac spinors satisfying the two dirac equations as @xmath30 and @xmath31 , respectively . given that the region of the space away from the junction has no curvature , we can use radial coordinates @xmath32 to write the set of two dirac equations in the plane for @xmath33 : @xmath34 @xmath35 is the fermi velocity and the components @xmath36 and @xmath37 denote the respective amplitudes of the electron in the two sublattices of the graphene lattice . the parameter @xmath38 corresponds to the quanta of effective flux felt by the electrons when making a complete tour around the junction . in the nanotube side , we use cylindrical coordinates @xmath39 , with @xmath40 . the dirac equation for the nanotube is @xmath41 we note that ( [ dp1])-([dp2 ] ) as well as ( [ dn1])-([dn2 ] ) are expressions of the dirac equation in flat space . this is consistent with the fact that , in the continuum limit , the curvature is localized at the circle connecting graphene and the nanotube . as an alternative to the coordinate @xmath42 , we could make for instance the change of variables @xmath43 , allowing us to map the nanotube into the region of the plane with @xmath44 . using a common radial coordinate @xmath45 to describe both graphene and the nanotube , the metric of the space turns out to be multiplied by the conformal factor @xmath46 . the first derivative of the metric becomes then discontinuous and the curvature scalar , computed in terms of second derivatives of the metric , is @xmath47 . the integral of this expression corresponds actually to the total curvature provided by the heptagonal rings . this makes clear that the effects of the curvature are implicit in the operation of matching the solutions of ( [ dp1])-([dp2 ] ) and ( [ dn1])-([dn2 ] ) at the circle @xmath48 . the solutions of the eqs . ( [ dp1])-([dp2 ] ) are of the form @xmath49 where @xmath50 and @xmath51 are constants , and @xmath52 and @xmath53 are bessel functions . the energy is @xmath54 . on the other hand , the resolution of eqs . ( [ dn1])-([dn2 ] ) shows that there are propagating and evanescent waves in the nanotube , which can be written as : @xmath55 where @xmath56 and @xmath57 are constants , the energy @xmath58 is given by : @xmath59 and the phase factor in eq . ( [ pe ] ) is : @xmath60 we note that the evanescent states with longitudinal decay @xmath61 arise for nonvanishing angular momentum @xmath2 . then there is an energy threshold @xmath62 for the appearance of propagating states in the nanotube . this is perfectly consistent with the behavior of the local density of states obtained for the different values of @xmath12 in the tight - binding approach . the depletion found in different @xmath12-sectors for the local density of states at the end of the nanotube ( close to the junction ) corresponds actually to the range of evanescent states given by ( [ e ] ) . as already mentioned , the position of the peaks delimiting the depletion in the tight - binding approach scales in proportion to the value of the angular momentum , which corresponds in the lattice to the different values of @xmath12 . moreover , we have also seen that such a position is inversely proportional to the nanotube radius @xmath25 , with values in the plots that can be approximately matched with the estimate @xmath63 ( after using the expression of the fermi velocity @xmath64 , in terms of the transfer integral @xmath9 and the c - c distance @xmath65 ) . we find therefore that the generic features found for the local density of states in the tight - binding approach are well captured by the continuum limit based on the dirac equation . the presence of a peak in the local density of states close to zero energy ( in the sectors @xmath66 ) is the only feature not generic for all types of nanotubes , and that can be also explained within our continuum approach . the rotation caused by each heptagonal ring in the space of the two dirac points corresponds to an effective magnetic flux of @xmath26 @xcite , but the way this flux is combined in the case of pairs of heptagons depends on their relative position . this has been studied in the case of pentagon pairs in ref . , arriving at a conclusion that can be readily generalized to the case of heptagonal defects . the result is that , when the distance between the heptagons is given by a vector @xmath67 ( using the same notation to classify the nanotubes ) such that @xmath68 is not a multiple of 3 , the effective flux of a pair of heptagons does not add to @xmath8 , but to the lower amount @xmath13 . the number @xmath68 for the distance between heptagons is a multiple of 3 only in the case of junctions with armchair nanotubes , or with @xmath1 nanotubes when @xmath2 is a multiple of 3 . in these instances , the total flux felt around the junction is equal to the sum of the fluxes provided by the individual heptagons , giving a value of @xmath69 . in the rest of the cases , the total flux corresponds instead to @xmath70 . the number @xmath38 of flux quanta has a direct correspondence with the number of zero modes of the dirac equation . their existence rests on the possibility of having localized states at the junction , with suitable decay in both the graphene part and the nanotube side . if we take for instance the maximum effective flux and @xmath69 , we have an equation for a zero - energy eigenstate in the region @xmath71 @xmath72 for a wavefunction with angular momentum @xmath2 , we obtain the behavior @xmath73 this gives rise to modes decaying from the junction for values @xmath74 . the wavefunction has to be matched at @xmath48 with the appropriate dependence in the nanotube , that is @xmath75 recalling that @xmath40 , we see that only the value @xmath76 provides a localized state at the junction . on the graphene side , the state is not strictly normalizable , in a similar way to other half - bound states induced by defects@xcite . anyhow , such a localized state has a reflection in the peak observed close to zero energy in the @xmath14 sectors of the tight - binding density of states . by inverting the direction of the flux and taking @xmath77 , it can be seen that the solutions have then a nonvanishing component @xmath78 similar to ( [ zm ] ) , but with angular momentum @xmath79 instead of @xmath2 . another localized state is found therefore with opposite chirality and @xmath80 . in the case of the junctions with @xmath1 nanotubes such that @xmath2 is not a multiple of 3 , the flux corresponding to @xmath70 is not enough to localize states at the junction . it can be seen that there are no zero - energy solutions of the dirac equation decaying simultaneously in the graphene plane and in the nanotube . this explains why in this type of junctions there is no low - energy peak within the depleted region of the local density of states . we complete in this way the correspondence between the tight - binding approach and the continuum limit based on the dirac equation , accounting for the main electronic features and unveiling also the origin of the different low - energy behavior in the two classes of junctions . ( [ e0 ] ) and ( [ pe ] ) allow us to analyze the scattering of a wave in the plane off the nanotube . the coefficients @xmath81 and @xmath82 define the transmission and reflection by the nanotube . for a wave coming from the plane , the coefficient @xmath82 is zero , and @xmath83 is proportional to the transmission coefficient . using the theory of scattering of two dimensional dirac electrons by an impurity@xcite , the transmission coefficient is given by : @xmath84 at high energies , @xmath85 or , alternatively , @xmath86 , we can use the asymptotic expansion : @xmath87 and @xmath88 . from these expansions , we obtain : @xmath89 so that @xmath90 . this estimate is valid for angular momenta @xmath2 such that @xmath91 . we can also obtain the reflection coefficient in this limit , @xmath92 , which is also independent of @xmath2 for @xmath91 . the angular dependence of the scattering cross section @xmath93 is : @xmath94 ^ 2\ ] ] and the total cross section , @xmath95 , is proportional to @xmath96 . the total flux of particles propagating inside the nanotube , normalized to the total incoming flux , is also proportional to @xmath96 . in the low energy limit , @xmath97 , we obtain : @xmath98 in this limit , most of the electrons reaching the junction are scattered back into the plane . we can make use of the continuum equations to analyze systems of very large sizes . we calculate the electronic green s functions numerically . the dirac equation in the plane , in radial coordinates , can be discretized@xcite . each radial equation can be approximated by a nearest neighbor tight binding model with two inequivalent hoppings : @xmath99 where @xmath100 and @xmath101 give the values of @xmath102 and @xmath103 at position @xmath104 , @xmath65 being a length scale which defines the discretization . ( [ discrete ] ) also include an energy scale , @xmath9 , which plays the role of an upper cutoff . the fermi velocity is equal to @xmath105 . the dirac equation is obtained for @xmath106 . we can write eqs . ( [ discrete ] ) in a more compact form using a single index , @xmath107 , such that @xmath108 and @xmath109 , so that : @xmath110 where , using as new length scale @xmath111 , we have : @xmath112\ ] ] where we start at position @xmath113 . the diagonal green s function at site @xmath114 can be written as : @xmath115 and : @xmath116 with boundary conditions at @xmath117 and @xmath118 : @xmath119 and : @xmath120 and : @xmath121 where the radius of the nanotube is @xmath122 . the boundary conditions in eq . ( [ bc ] ) describe a semi - infinite nanotube attached at position @xmath113 , and also approximate the boundary of a plane at position @xmath123 . the green s function deep inside the nanotube can be calculated analytically : @xmath124 the density of states in the plane has been calculated numerically , with @xmath125 and summing angular momenta from @xmath126 to @xmath127 . the calculation is equivalent to analyzing a cluster with @xmath128 sites . results for the green s function in the nanotube , and at the position @xmath129 , are shown in fig . [ dos_0 ] . the energy scale is set by @xmath130 . . the density of states is shown at position @xmath131 from the junction ( red , diamonds ) , and @xmath127 ( blue , hexagons).,width=302 ] the green s function for the system built up by the nanotube and the graphene sheet is shown in fig . the parameters are the same as the ones used for the calculation shown in fig . [ dos_0 ] , and the radius of the nanotube is @xmath132 . the density of states at distances from the nanotube @xmath133 are similar to those in the unperturbed sheet . near the juncture with the nanotube , there is a depletion of states at low energies , compensated by the existence of a localized state at @xmath134 . we have not analyzed so far the effect on the electronic structure of strains which may be induced near the junction . these strains deform the bonds , and induce an additional , intravalley gauge field acting on the electrons@xcite . the large in - plane stiffness of graphene implies that the bonds will tend to their equilibrium lengths throughout the system . the bending at the junction will be localized within a length scale @xmath135 , where @xmath136 ev is the bending rigidity of graphene , and @xmath137 ev @xmath138 is an average of the lam coefficients of graphene . this length is comparable to the lattice spacing . the mismatch between the diameter of the nanotube and the lattice constant of the graphene layer induces additional strains , with a long range decay into the bulk of the graphene plane and the nanotube , which can be calculated using the continuum theory of elasticity@xcite . we expect , however , the interatomic distance in graphene to be very close to the distance between carbon atoms along the radial direction of the nanotube , so that the strains induced by this effect will be small . the strains will decay as @xmath139 or @xmath140 as function of the distance to the junction . using dimensional analysis , the strain near the junction is of order @xmath141 , where @xmath142 is the change in the equilibrium radius of the nanotube induced by the plane or , alternatively , of order @xmath143 , where @xmath65 is the interatomic distance . we expect the value of @xmath143 to be similar on the plane side of the junction . the associated gauge field is @xmath144 , where @xmath145 gives the change of the tight - binding hopping @xmath9 with @xmath65 . thus , we expect that the elastic strains will induce changes on the electronic structure on energy scales of order @xmath146 near the junction . our computational framework allows us also to address the electronic properties of arrays of nanotube - graphene junctions . we consider the case in which the unit cell of the array has a hexagonal shape in the base , of the type shown in fig . the periodic arrangement of junctions is formed then by translating the unit cell by two independent vectors of the triangular array , in such a way that the 2d base is fully covered with the hexagonal patches . the brillouin zone of the superlattice is a hexagon , and the main electronic properties are encoded in the form of the bands from the center to the @xmath147 and @xmath148 points at the boundary of the zone . as long as the states in momentum space have well - defined transformation properties under translations by the lattice vectors of the triangular array , we can obtain the bands of the array of junctions by solving a tight - binding model in the unit cell , with appropriate momentum - dependent boundary conditions between opposite sides of its hexagonal base . the band structure of the array of junctions depends on the geometry of the nanotubes , as well as on their length and the distance between them . for simplicity , we are going to consider arrays where all the nanotubes have the same chirality . then , it can be checked that the arrays fall into two main classes , regarding the behavior of the bands close to the fermi level . the distinctive feature of one class with respect to the other is the presence of flat bands in the low - energy part of the spectrum . the arrays of junctions made of armchair nanotubes , for instance , always have a number of these flat bands , as illustrated by the representative in fig . [ eight](a ) . the appearance of flat bands in a particular array of junctions was noticed in ref . . we have found that the flat bands are actually generic in arrays made of armchair nanotubes , which display a series of them as one moves from the fermi level to higher ( or lower ) energies . the spacing in energy between the flat bands becomes inversely proportional to the length of the nanotubes . the bands dispersing at low energies in fig . [ eight](a ) are not affected however by variations of that variable , while they move instead closer to the fermi level as the distance between the nanotubes in the array is increased . + ( a ) ( b ) + + + ( c ) ( d ) on the other hand , the presence of flat bands at low energies is not generic in arrays made of zig - zag nanotubes . in general , we may expect a number of bands dispersing above and below the fermi level , as shown in figs . [ eight](b ) and ( d ) , which represent the low - energy bands of arrays made respectively of @xmath149 and @xmath150 nanotubes . in the case of the zig - zag nanotubes , flat bands only appear close to the fermi level when the junctions are formed with @xmath1 geometries such that @xmath2 is a multiple of 3 . this distinctive behavior can be appreciated in fig . [ eight](c ) , which displays the low - energy bands in the case of an array made of @xmath24 nanotubes . the shape of the bands resembles there the typical appearance of the spectra of arrays made of armchair nanotubes , as shown in fig . [ eight](a ) . the mentioned flat bands have their origin in the existence of localized states in the arrays of junctions . we have checked that the junctions made of armchair nanotubes and @xmath1 nanotubes with @xmath2 equal to a multiple of 3 have in common the formation of electron states confined mostly in the nanotube side . these are the states responsible for the development of the flat bands shown in fig . [ eight ] , as the wave functions with most of their weight in the nanotubes show little overlap in the graphene part of the lattice . this also explains in a natural way the proliferation of flat bands at low energies as the nanotube length is increased , by thinking of the confined modes as standing waves in the tube . on the other hand , the states that are preferentially localized at the junctions ( corresponding to the quasi - bound states of the dirac equation ) may be identified here as the pairs of branches degenerated at the @xmath151 point . these states may have in general a significant overlap between nearest junctions , which is reflected in the appreciable dispersion of the corresponding bands . we can reach in the continuum limit a qualitative understanding of the similar behavior of the arrays made of armchair and @xmath1 nanotubes when @xmath2 is a multiple of 3 , by noticing that these are the only geometries that support low - energy standing waves between the junction and the other end of the tube . this requires the superposition of two modes with opposite momenta along the tube , which is possible at low energies in the armchair nanotubes as the modes at opposite dirac points have then vanishing angular momentum . in general , this is not the case for the @xmath1 geometries , since in the zig - zag nanotubes the dirac points correspond to large momenta in the transverse direction . yet the formation of standing waves is possible when @xmath2 is a multiple of 3 , as the low - energy states about the two dirac points fall then in the same sector with quantum number @xmath21 regarding the @xmath3 symmetry . thus , it is possible to form a state confined in the nanotube by superposition of two modes with opposite longitudinal momenta and the same quantum number @xmath12 . this is consistent with the fact that the confined states are actually found in the @xmath152 sector in the diagonalization of very large lattices of individual nanotube - graphene junctions . in the real lattice of the array , the confinement of the electrons in the nanotubes is only approximate , but the decay of the wave functions in the graphene part away from the junctions is strong enough to account for the development of the flat bands shown above . in this paper we have studied the electronic structure of the hybrid material made of carbon nanotubes attached to a graphene sheet . by analyzing individual nanotube - graphene junctions , we have found the following features : \i ) low - energy electrons in the graphene layer , with @xmath153 , are scattered by the nanotube , and the probability of propagating into the tube is small . \ii ) high - energy electrons reaching the nanotube junction , with @xmath154 , are mostly transmitted into the nanotube . \iii ) at low energies , @xmath153 , and in the vicinity of the junction , @xmath155 , there is in general a depletion of the density of states . \iv ) in certain nanotube geometries ( armchair and @xmath1 with @xmath2 equal to a multiple of 3 ) , there are quasi - bound states near @xmath156 , partially localized at the junction . we have shown that these features can be accounted for in a continuum model of the hybrid geometry . this is based on the dirac fermion fields describing the electronic excitations , interacting with the curvature and the effective gauge field arising from the six heptagonal carbon rings at the junction . thus , properties i ) , ii ) and iii ) are intrinsic to the continuum dirac equation and universal for all nanotube geometries , while iv ) depends on the relative position of the six heptagonal rings and the consequent effective magnetic flux at the junction . while we have focused on the case of armchair and zig - zag nanotubes , it becomes clear that the continuum theory may account as well for the properties of junctions with other geometries . in this respect , it is likely that , by allowing for less regular distributions of the heptagonal rings , nanotubes with nontrivial helicity can also be attached to the graphene sheet . in a more general theoretical perspective , it may be interesting to analyze other discrete realizations of the 2d dirac equation , like the geometry of a square lattice with one half magnetic flux per plaquette . we have also shown that the arrays of nanotube - graphene junctions fall into two main classes , depending on whether their spectra exhibit or not flat bands close to the fermi level . the flat bands only appear in arrays made of armchair nanotubes or @xmath1 nanotubes when @xmath2 is a multiple of 3 . on the other hand , the semiconducting behavior seems to be a constant in the class characterized by the presence of the flat bands , as no dispersive bands cross then the fermi level . metallic behavior of the array of junctions is possible in the other class , as shown in fig . [ eight](b ) , though that behavior does not appear to be a generic trend , as illustrated by the absence of low - energy bands crossing the fermi level in the other representative of the class shown in fig . [ eight](d ) . in real experimental samples , it is quite likely that the arrays may be formed by junctions with nanotubes of different helicities . in this case , we can expect that the electronic structure of these arrays will be a mixture of the features already present in figs . [ eight](a)-(d ) . in particular , part of the electronic states will be still confined in some of the nanotubes , and other states will be partially localized at some of the junctions . the feasibility of using the arrays of nanotube - graphene junctions may depend on the possibility to tailor these hybrid structures to get specific functions . at this point , more input from experimental measurements on these arrays would be required , while the remarkable behavior predicted for these systems ( localization and confinement of states , flat bands ) opens good perspectives in the investigation of novel electronic devices . we acknowledge many helpful discussions with a. h. castro neto , who also mentioned to us the existence of ref . we acknowledge financial support from mec ( spain ) through grants fis2005 - 05478-c02 - 01 , fis2005 - 05478-c02 - 02 and consolider csd2007 - 00010 , by the comunidad de madrid , through citecnomik , cm2006-s-0505-esp-0337 . s. v. morozov , k. s. novoselov , m. i. katsnelson , f. schedin , l. a. ponomarenko , d. jiang and a. k. geim , phys . lett . * 97 * , 016801 ( 2006 ) . j. l. maes , phys . rev . b * 76 * , 045430 ( 2007 ) . f. guinea , m. i. katsnelson , and m. a. h. vozmediano , phys . rev . b * 77 * , 075422 ( 2008 ) . f. guinea , b. horowitz , and p. le doussal , phys . rev . b * 77 * , 205421 ( 2008 ) . l. d. landau and e. m. lifschitz , _ theory of elasticity _ , pergamon press , oxford ( 1959 ) .
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we study the electronic structure of the junctions between a single graphene layer and carbon nanotubes , using a tight - binding model and the continuum theory based on dirac fermion fields .
the latter provides a unified description of different lattice structures with curvature , which is always localized at six heptagonal carbon rings around each junction .
when these are evenly spaced , we find that it is possible to curve the planar lattice into armchair @xmath0 as well as zig - zag @xmath1 nanotubes . we show that the junctions fall into two different classes , regarding the low - energy electronic behavior .
one of them , constituted by the junctions made of the armchair nanotubes and the zig - zag @xmath1 geometries when @xmath2 is a multiple of 3 , is characterized by the presence of two quasi - bound states at the fermi level , which are absent for the rest of the zig - zag nanotubes .
these states , localized at the junction , are shown to arise from the effective gauge flux induced by the heptagonal carbon rings , which has a direct reflection in the local density of states around the junction .
furthermore , we also analyze the band structure of the arrays of junctions , finding out that they can also be classified into two different groups according to the low - energy behavior . in this regard ,
the arrays made of armchair and @xmath1 nanotubes with @xmath2 equal to a multiple of 3 are characterized by the presence of a series of flat bands , whose number grows with the length of the nanotubes .
we show that such flat bands have their origin in the formation of states confined to the nanotubes , with little overlap in the region between the junctions .
this is explained in the continuum theory from the possibility of forming standing waves in the mentioned nanotube geometries , as a superposition of modes with opposite momenta and the same quantum numbers under the @xmath3 symmetry of the junction .
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dans tout cet article , @xmath0 est un nombre premier , @xmath2 est un corps parfait de caractristique @xmath0 , et @xmath3 est une extension finie du corps des fractions @xmath4 de lanneau des vecteurs de witt sur @xmath2 . on se fixe une clture algbrique @xmath5 de @xmath3 , et on pose @xmath6 . utilisant lanneau de priodes @xmath0-adiques @xmath7 , fontaine a dfini la notion de reprsentation de _ de rham _ de @xmath8 . si @xmath9 est une telle reprsentation et si @xmath10 est une extension finie de @xmath3 , on dfinit @xmath11 comme tant lensemble des classes de cohomologie qui dterminent une extension @xmath12 de reprsentations de @xmath13 telle que @xmath14 est une reprsentation de de rham de @xmath13 . on crira @xmath15 pour dsigner lensemble des racines @xmath16-mes de lunit , et on dfinit @xmath17 ainsi que @xmath18 . on pose @xmath19 et @xmath20 . lalgbre diwasawa est lalgbre de groupe complte @xmath21 $ ] . la cohomologie diwasawa de @xmath9 est dfinie par @xmath22 o @xmath23 est un rseau de @xmath9 stable par @xmath8 et @xmath24 est la limite projective pour les applications corestriction ce qui fait de @xmath25 un @xmath26-module . par construction , on a des applications @xmath27 et lobjet de cet article est ltude du module @xmath28 des normes universelles , lensemble des @xmath29 tels que pour tout @xmath30 on ait @xmath31 . si @xmath32 est la plus grande sous - reprsentation de @xmath9 do nt tous les poids de hodge - tate sont @xmath33 , alors @xmath32 est de de rham et on peut montrer que @xmath34 . le rsultat principal de cet article est le suivant : [ main_theo ] si @xmath9 est une reprsentation de de rham , alors 1 . si @xmath9 na pas de sous - quotient stable par @xmath35 , alors @xmath36 ; 2 . en gnral , @xmath37 et le quotient est un @xmath26-module de torsion . la dmonstration est trs similaire celle quen a donne perrin - riou dans @xcite pour les reprsentations cristallines , et dans @xcite pour les reprsentations semi - stables , du groupe de galois dun corps non - ramifi sont entires , voir le dbut des paragraphes ( * ? ? ? * , 4.1 ) ] mais au lieu dutiliser son exponentielle largie , on utilise les constructions de @xcite ( qui redonnent lexponentielle largie dailleurs , comme on le montre dans @xcite ) et au lieu dutiliser des considrations de cran de la filtration et dordre , on utilise la thorie des @xmath38-modules qui encodent toutes ces informations . cela simplifie la dmonstration de perrin - riou et nous permet de plus de ltendre au cas des reprsentations de de rham du groupe de galois dun corps ventuellement ramifi . remarquons que lon nutilise pas le fait que les reprsentations de de rham sont potentiellement semi - stables . indiquons brivement do provient cette conjecture . on renvoie larticle @xcite de perrin - riou pour plus dinformations . si @xmath14 est une courbe elliptique dfinie sur @xmath3 , on sintresse au module des normes universelles @xmath39 . le calcul de ce module a t fait ( pour une courbe elliptique , une varit ablienne ou mme un groupe formel , et pour une @xmath40-extension quelconque ) dans certains cas par mazur @xcite , puis par hazewinkel @xcite , schneider @xcite et perrin - riou @xcite entre autres . dans @xcite , coates et greenberg ont formul une conjecture assez gnrale dcrivant le module des normes universelles . du rsultat que nous dmontrons est due nekov et concerne les normes universelles dans lextension cyclotomique pour une reprsentation @xmath0-adique de de rham . si on lapplique au module de tate dune courbe elliptique @xmath14 , on retrouve , via la thorie de kummer , certains rsultats des auteurs cits prcdemment : si @xmath14 est ordinaire , alors @xmath41 est un @xmath42-module de rang @xmath43 et si @xmath14 est supersingulire , alors il est nul . * remerciements * : je remercie pierre colmez pour ses nombreuses suggestions , de la dmonstration du thorme principal la rdaction finale de cet article . je remercie aussi jan nekov pour ses encouragements et ses commentaires . lobjet de ce premier chapitre est de rappeler et complter certains points de la thorie des @xmath38-modules . ensuite , on rsout un problme dalgbre diffrentielle . dans le deuxime chapitre , on verra comment cela sapplique aux reprsentations @xmath0-adiques . dans tout cet article , @xmath2 dsigne un corps parfait de caractristique @xmath0 , et @xmath3 est une extension finie de @xmath4 , le corps des fractions de lanneau des vecteurs de witt sur @xmath2 . on crit @xmath15 pour dsigner lensemble des racines @xmath16-mes de lunit , et on dfinit @xmath17 ainsi que @xmath18 . soit @xmath6 et @xmath19 le noyau du caractre cyclotomique @xmath44 et @xmath45 le groupe de galois de @xmath46 , qui sidentifie via le caractre cyclotomique un sous - groupe ouvert de @xmath47 . enfin , soit @xmath48 lextension maximale non - ramifie de @xmath4 contenue dans @xmath49 et @xmath50 le corps rsiduel de @xmath48 . on note @xmath51 le frobenius absolu ( qui relve @xmath52 sur @xmath50 ) . dfinissons ici quelques anneaux de sries formelles ( ces constructions sont faites en dtail dans @xcite ) : si @xmath53 est un rel positif , soit @xmath54 lanneau des sries formelles @xmath55 o @xmath56 est une suite borne telle que @xmath57 converge sur la couronne @xmath58 . cet anneau est muni dune action de @xmath59 , qui est triviale sur les coefficients et donne par @xmath60 et on peut dfinir un frobenius @xmath61 qui est @xmath51-semi - linaire sur les coefficients et tel que @xmath62 . le thorme de prparation de weierstrass montre que @xmath63 est un corps . ce corps nest pas complet pour la norme de gauss et on appelle @xmath64 son complt qui est un corps local de dimension @xmath65 do nt le corps rsiduel sidentifie @xmath66 . lextension @xmath67 est une extension finie de degr de ramification @xmath68 $ ] et par la thorie du corps de normes de @xcite il lui correspond une extension sparable @xmath69 de degr @xmath70 $ ] qui nous permet de dfinir des extensions non - ramifies @xmath71 et @xmath72 de degr @xmath70 $ ] . on peut montrer que @xmath73 o @xmath74 est un @xmath54-module libre de rang @xmath70 $ ] qui sidentifie un anneau de sries formelles @xmath75 o @xmath76 est une suite borne telle que @xmath77 converge sur la couronne @xmath78 . llment @xmath79 vrifie une quation deisenstein sur @xmath80 quon peut relever en une quation sur @xmath81 ; laction de @xmath82 stend naturellement @xmath74 de mme que le frobenius @xmath83 . un @xmath38-module est un @xmath84-espace vectoriel @xmath85 de dimension finie , muni dun frobenius @xmath86 et dune action de @xmath82 qui sont semi - linaires par rapport ceux de @xmath84 . on dit que @xmath85 est _ tale _ si @xmath87 possde un rseau @xmath88 stable par @xmath89 sur lanneau des entiers @xmath90 de @xmath91 , tel que @xmath92 engendre @xmath88 sur @xmath90 . dfinissons loprateur @xmath93 qui nous servira dans la suite . on peut montrer que tout lment @xmath94 scrit de manire unique @xmath95 . [ psi ] si @xmath95 , alors on pose @xmath96 . ceci fait de @xmath97 un inverse gauche de @xmath89 qui commute laction de @xmath82 et qui vrifie @xmath98 si @xmath99 et @xmath100 . il existe @xmath101 tel que si @xmath102 , alors on a une application injective @xmath103 $ ] ( cest lapplication @xmath104 de @xcite ) . par exemple si @xmath105 , alors @xmath106 o @xmath107 est une racine primitive @xmath16-me de @xmath43 et @xmath108 agit par @xmath109 sur les coefficients . on peut montrer ( voir @xcite ) que lensemble des sous @xmath74-modules de type fini @xmath110 de @xmath85 tels que @xmath111 admet un plus grand lment @xmath112 et quil existe @xmath113 que lon peut supposer @xmath114 tel que si @xmath115 , alors @xmath116 . on utilise @xmath108 pour dfinir @xmath117 \otimes^{\iota_n}_{{\mathbf{b}^{\dagger , r}}_k } d^{\dag , r}$ ] et @xmath118 . le lemme suivant sera utile par la suite : [ gam_tor_pg ] si @xmath119 , alors il existe @xmath120 $ ] tel que @xmath121 si et seulement si @xmath122 . un petit calcul montre que @xmath123 est un @xmath124-espace vectoriel de dimension @xmath125 ( il suffit de remarquer que des lments de @xmath123 qui sont lis sur @xmath84 le sont sur @xmath126 ) et donc quil existe @xmath127 $ ] en fait tel que @xmath128 annule @xmath123 . montrons donc la rciproque . supposons que @xmath129 est une relation de longueur minimale avec @xmath130 . on peut supposer que lun des @xmath131 est gal @xmath43 . en appliquant @xmath97 et en utilisant le fait que dune part @xmath132 et que dautre part @xmath97 agit par @xmath133 sur @xmath48 , on voit que @xmath134 pour tout @xmath135 et on suppose donc partir de maintenant que @xmath127 $ ] . nous utiliserons ci - dessous le rsultat suivant : si @xmath127 $ ] , alors il existe une constante @xmath136 telle que pour tout @xmath110 qui est un @xmath137$]-module libre de rang @xmath138 , muni dune action semi - linaire de @xmath82 par automorphismes , le @xmath48-espace vectoriel @xmath139 est de dimension @xmath140 . fixons @xmath115 et considrons , pour @xmath141 tel que @xmath142 , le @xmath117$]-module libre de rang @xmath138 dfini ci - dessus : @xmath117 \otimes^{\iota_n}_{{\mathbf{b}^{\dagger , r}}_k } d^{\dag , r}$ ] . on voit que lon a une injection : @xmath143 \otimes_{k_n[[t ] ] } k_n[[t ] ] \otimes^{\iota_n}_{{\mathbf{b}^{\dagger , r}}_k } d^{\dag , r } \right)^{p(\gamma)=0},\ ] ] ce qui fait que @xmath144 est un @xmath48-espace vectoriel de dimension @xmath140 et donc que @xmath145 est un @xmath48-espace vectoriel de dimension @xmath140 . comme @xmath89 commute @xmath128 , @xmath146 est un @xmath48-espace vectoriel de dimension finie et stable par @xmath89 ce qui fait que @xmath147 est bijectif . si @xmath148 , alors @xmath149 pour un @xmath150 et donc @xmath151 ce qui fait que @xmath152 et donc que @xmath153 ( ceci gnralise un rsultat de @xcite ) . pour conclure , il suffit de remarquer que si @xmath132 et @xmath154 , alors @xmath155 et donc @xmath156 . dans ce paragraphe , on dfinit les @xmath38-modules de de rham et on rappelle certains des rsultats de @xcite leur sujet . [ pg_dr ] on dit quun @xmath38-module @xmath85 est de _ de rham _ , si et seulement sil existe @xmath157 , @xmath158 avec @xmath159 , tels que le @xmath3-espace vectoriel @xmath160 est de dimension @xmath161 . si cest le cas , alors @xmath162 est de dimension @xmath161 pour tous les @xmath157 , @xmath158 tels que @xmath159 . nous allons rappeler les rsultats de @xcite qui permettent de donner une autre caractrisation des @xmath38-modules de de rham . ces rsultats sont aussi expliqus dans le sminaire bourbaki @xcite . lanneau @xmath74 sidentifiant un anneau de sries formelles convergeant sur une couronne , il est naturellement muni dune topologie de frchet , la topologie de la convergence compacte , et son complt @xmath163 pour cette topologie sidentifie lanneau de sries formelles @xmath75 o @xmath164 est une suite non ncessairement borne telle que @xmath77 converge sur la couronne @xmath78 . par exemple , si on pose @xmath165 , alors @xmath166 pour tout @xmath167 . lanneau @xmath168 est lanneau de robba . ces anneaux ont t tudis dans @xcite et nous allons rappeler quelques - uns des rsultats qui nous seront utiles dans la suite . lapplication @xmath108 se prolonge en une application injective @xmath169 $ ] . laction de @xmath82 sur @xmath163 stend en une action de lalgbre de lie de @xmath82 donne par @xmath170 pour @xmath171 assez proche de @xmath43 . si @xmath172 alors @xmath173 . si @xmath174 alors on pose @xmath175 ce qui fait que si @xmath172 alors @xmath176 et que si @xmath177 vrifie une quation algbrique @xmath178 sur @xmath179 telle que @xmath180 , alors on peut aussi calculer @xmath181 par la formule @xmath182 . en particulier @xmath183 si et seulement si @xmath184 . si @xmath85 est un @xmath38-module , on dfinit @xmath185 par @xmath186 . lalgbre de lie de @xmath82 agit sur @xmath185 par la formule @xmath187 pour @xmath171 assez proche de @xmath43 et on a donc aussi une application @xmath188 . la proposition suivante se dmontre exactement de la mme manire que ( * ? ? ? * thorme 5.10 ) . [ ndr_pg ] si @xmath85 est de de rham , si @xmath115 , et si @xmath189 est lensemble des @xmath190 $ ] tels que pour tout @xmath141 tel que @xmath142 , on ait @xmath191 \otimes_k ( k_n((t ) ) \otimes^{\iota_n}_{{\mathbf{b}^{\dagger , r}}_k } d^{\dag , r})^{\gamma_k},\ ] ] et si on pose @xmath192 , alors @xmath193 est un @xmath194-module libre de rang @xmath138 stable par les actions induites de @xmath89 et de @xmath82 , tel que @xmath195 et tel que @xmath196={d^\dag_{\mathrm{rig}}}[1/t]$ ] . nous tablirons quelques proprits de @xmath193 dans le paragraphe [ ii1 ] . donnons nous une variable @xmath197 sur laquelle on fait agir @xmath198 par @xmath199 . si @xmath161 , alors on dit que le @xmath38-module @xmath85 est 1 . _ cristallin _ , si @xmath200)^{\gamma_k}$ ] est un @xmath4-espace vectoriel de dimension @xmath138 ; 2 . _ semi - stable _ , si @xmath201[1/t])^{\gamma_k}$ ] est un @xmath4-espace vectoriel de dimension @xmath138 . il nest pas difficile de voir que cristallin implique semi - stable et que semi - stable implique de de rham ; on a dailleurs dans ce cas @xmath202 \otimes_f ( { d^\dag_{\mathrm{rig}}}[\ell_y][1/t])^{\gamma_k } \right)^{d / d\ell_y=0}.\ ] ] de plus , le thorme de monodromie @xmath0-adique pour les quations diffrentielles ( dmontr indpendamment dans @xcite ) montre que @xmath85 est de de rham si et seulement sil existe une extension finie @xmath10 de @xmath3 telle que @xmath203 est semi - stable . nous nutiliserons pas cela par la suite . on aura besoin dans la suite dun argument qui est une variante du dterminant wronskien et qui est lobjet de ce chapitre . soient @xmath204 un corps diffrentiel , do nt on notera @xmath205 la drivation , et @xmath2 , @xmath206 et @xmath207 trois entiers @xmath33 . on tend naturellement @xmath205 @xmath208 . on se donne @xmath209 vecteurs @xmath210 qui satisfont les deux conditions ci - dessous : 1 . dune part , les @xmath206 vecteurs @xmath211 pour @xmath212 sont linairement indpendants sur @xmath204 ; 2 . dautre part , les @xmath209 vecteurs @xmath213 pour @xmath214 sont linairement dpendants sur @xmath204 . [ wronski ] sous les hypothses ci - dessus , les @xmath209 vecteurs @xmath215 pour @xmath214 sont lis sur @xmath216 . comme on suppose que les @xmath217 sont lis sur @xmath204 , il existe des lments @xmath218 tels que @xmath219 , ce qui se traduit par le fait que pour tout @xmath220 , on a la relation @xmath221 : @xmath222 comme on a suppos que les @xmath223 sont libres , on voit que @xmath224 et on peut donc supposer que @xmath225 ce que lon fait partir de maintenant . si on drive @xmath221 , on trouve que @xmath226 si @xmath227 , alors le premier terme de la relation ci - dessus est nul par @xmath228 et on trouve que @xmath229 pour @xmath227 et donc que @xmath230 puisque @xmath225 . ceci nous donne une relation @xmath231 entre les @xmath223 et comme on a suppos que ceux - ci sont libres , cest que @xmath232 pour tout @xmath214 . on a donc montr que si les @xmath233 sont lis sur @xmath204 , alors ils sont lis sur @xmath216 . en considrant la premire composante de @xmath217 , on trouve : [ wronski_coro ] sous les hypothses ci - dessus , il existe des constantes @xmath234 pour @xmath214 , non toutes nulles , telles que @xmath235 dans ce paragraphe , nous allons montrer la proposition suivante : [ pg_diff ] soit @xmath85 un @xmath38-module de de rham qui vrifie @xmath236 et @xmath237 . si @xmath100 et @xmath238 sont tels que @xmath239 , alors il existe @xmath120 $ ] tel que @xmath121 . si @xmath240 , alors le fait que @xmath237 implique immdiatement que @xmath241 . on suppose dsormais que @xmath242 , et on se fixe @xmath198 dordre infini . comme @xmath85 est un @xmath84-espace vectoriel de dimension finie , il existe un entier @xmath243 tel que @xmath244 sont libres sur @xmath84 , mais @xmath245 sont lis sur @xmath84 , ce qui fait que pour tout @xmath246 , il existe des lments @xmath247 de @xmath84 tels que @xmath248 si lon drive @xmath2 fois la relation ci - dessus , on trouve que : @xmath249 supposons que lon se donne un entier @xmath250 et des lments @xmath251 de @xmath84 tels que pour tout @xmath252 et pour tout @xmath253 , on ait @xmath254 . bien sr , de tels lments ( non tous nuls ) existent si @xmath255 . on voit alors que @xmath256 montrons que cela implique que @xmath254 pour @xmath257 et pour @xmath253 , ce qui est quivalent au fait que les deux termes de lquation ( [ eqn01 ] ) ci - dessus sont nuls . on a @xmath258 et le terme de gauche de lquation ( [ eqn01 ] ) est donc dans @xmath259 tandis que le terme de droite est dans @xmath85 ce qui fait que les deux termes sont nuls puisquon a suppos que @xmath237 . pour @xmath260 , posons @xmath261 et @xmath262 ainsi que @xmath263 . si @xmath250 est le plus petit entier tel que les @xmath264 pour @xmath212 sont libres et les @xmath264 pour @xmath214 sont lis , alors les calculs prcdents montrent _ quen plus _ , les @xmath217 pour @xmath214 sont _ eux aussi _ lis . on peut alors appliquer le corollaire [ wronski_coro ] pour en dduire lexistence de @xmath265 pour @xmath214 tels que @xmath235 si lon combine cela avec le fait que par dfinition , on a @xmath266 , on trouve que @xmath267 ceci montre bien quil existe @xmath268 $ ] tel que @xmath154 . lobjet de ce chapitre est de montrer comment le rsultat dmontr au chapitre prcdent ( la proposition [ pg_diff ] ) implique le thorme principal de cet article . pour cela , on rappelle la correspondance entre reprsentations @xmath0-adiques et @xmath38-modules , puis on utilise la formule rciprocit de cherbonnier - colmez pour se ramener la situation du chapitre prcdent . on conclut par un argument de dvissage . une reprsentation @xmath0-adique est un @xmath124-espace vectoriel @xmath9 de dimension finie @xmath269 , muni dune action linaire et continue de @xmath8 . afin dtudier les reprsentations @xmath0-adiques , fontaine a construit ( voir @xcite par exemple ) un certains nombre danneaux de priodes @xmath0-adiques , ce qui conduit la dfinition des reprsentations _ cristallines _ , _ semi - stables _ ou de _ de rham_. dautre part , en combinant les constructions de fontaine ( voir @xcite ) et un thorme de cherbonnier - colmez ( voir @xcite ) , on dfinit un foncteur @xmath270 qui induit une quivalence de catgories entre la catgorie des reprsentations @xmath0-adiques et la catgorie des @xmath38-modules tales . si par exemple @xmath35 agit trivialement sur @xmath9 , alors @xmath271 et on rcupre @xmath9 partir de @xmath272 par @xmath273 . en gnral , la situation est plus complique mais on peut quand mme montrer que @xmath274 ce do nt nous aurons besoin par la suite . rappelons que pour un @xmath38-module @xmath85 , on a dfini au paragraphe [ i1 ] un rel @xmath113 tel que si @xmath275 , alors @xmath276 et on posera dans la suite @xmath277 . les rsultats de @xcite montrent que si @xmath9 est une reprsentation @xmath0-adique , et si @xmath278 , alors @xmath279 et donc que @xmath9 est de de rham si et seulement si @xmath272 est de de rham au sens de la dfinition [ pg_dr ] . on dispose alors dune part des applications @xmath280 et dautre part de lquation diffrentielle @xmath0-adique @xmath281 qui est le module @xmath193 do nt on a rappel la construction dans la proposition [ ndr_pg ] . les rsultats de @xcite montrent par ailleurs quune reprsentation @xmath9 est cristalline ( ou semi - stable ) si et seulement si son @xmath38-module @xmath272 est cristallin ( ou semi - stable ) . on retrouve alors les invariants associs @xmath9 par la thorie de hodge @xmath0-adique de la manire suivante : @xmath282)^{\gamma_k}$ ] et @xmath283[1/t])^{\gamma_k}$ ] . nous allons avoir besoin de quelques rsultats concernant @xmath281 , et on suppose partir de maintenant que @xmath9 est de de rham . [ ht_ndr ] si @xmath9 est une reprsentation de de rham , do nt les poids de hodge - tate sont @xmath284 , alors @xmath285 et si @xmath286 , alors les poids de hodge - tate de @xmath9 sont @xmath33 . tant donne la construction de @xmath287 que lon a donne dans la proposition [ ndr_pg ] , on voit quil suffit pour montrer le premier point de montrer que si @xmath288 et @xmath289 et si @xmath142 , alors @xmath290 \otimes_k { \mathbf{d}_{\mathrm{dr}}}(v)$ ] . on sait par les constructions de @xcite que @xmath291 et le rsultat suit du fait que si les poids de hodge - tate de @xmath9 sont @xmath284 , alors @xmath292 . dautre part , la dmonstration de ( * ? ? ? * proposition 5.15 ) montre que le @xmath293-module engendr par @xmath294 est @xmath295 . de mme , le @xmath293-module engendr par @xmath296 est @xmath297 et la deuxime assertion suit du fait que si @xmath298 , alors les poids de hodge - tate de @xmath9 sont @xmath33 . [ ndr_sub ] si @xmath9 est une reprsentation de de rham do nt les poids de hodge - tate sont @xmath284 et si @xmath299 est une sous - reprsentation de @xmath9 , alors @xmath300 . on voit que @xmath301 est un sous @xmath194-module de @xmath302 qui est libre de rang @xmath303 ( parce que @xmath304={\mathbf{d}^{\dagger } _ { \mathrm{rig}}}(v)[1/t]$ ] ) et qui est stable par @xmath305 ( qui concide avec loprateur induit par @xmath306 ) , ce qui fait que @xmath307 par dfinition de @xmath308 ( voir ( * ? ? ? * thorme 5.10 ) ) . [ no_divis ] si @xmath9 est une reprsentation de de rham do nt les poids de hodge - tate sont @xmath284 et qui nadmet pas de sous - reprsentation do nt les poids de hodge - tate sont @xmath33 , alors @xmath309 . si @xmath310 , alors on voit que lintersection @xmath311 des sous @xmath84-espaces vectoriels de @xmath272 stables par @xmath89 et @xmath82 et contenant @xmath312 vrifie @xmath313 . la proposition @xcite implique dautre part que @xmath311 est un sous @xmath38-module tale de @xmath272 et donc quil existe une reprsentation @xmath0-adique @xmath314 telle que @xmath315 . on voit alors que @xmath316 par le lemme [ ndr_sub ] et donc que les poids de hodge - tate de @xmath317 sont tous @xmath33 , par la proposition [ ht_ndr ] , et donc que @xmath318 ce qui fait que @xmath152 . rappelons que lon a une application @xmath319 et on en dduit une application @xmath320 o @xmath321 est lapplication coefficient de @xmath322 . [ divis_t ] si @xmath323 est tel que pour tout @xmath324 , on ait @xmath325 \otimes_k { \mathbf{d}_{\mathrm{dr}}}(v)$ ] , alors @xmath326 . en particulier , si @xmath327 pour tout @xmath324 , alors @xmath326 . fixons @xmath328 tel que @xmath329 . la dmonstration de la proposition @xcite montre est surjective - elle est surjective modulo @xmath330 pour tout @xmath331 . ] quil existe une base @xmath332 de @xmath189 do nt les images par @xmath108 sont une base de @xmath117 \otimes_k { \mathbf{d}_{\mathrm{dr}}}(v)$ ] pour tout @xmath324 . si lon crit que @xmath333 , et si @xmath325 \otimes_k { \mathbf{d}_{\mathrm{dr}}}(v)$ ] , alors @xmath334 pour tout @xmath324 et donc @xmath335 divise @xmath336 dans @xmath194 ce qui fait que @xmath326 . rappelons que herr a montr dans @xcite comment construire les groupes de cohomologie galoisienne @xmath337 partir du @xmath38-module associ @xmath9 . nous utiliserons la version quen ont donne cherbonnier et colmez dans @xcite et que nous rappelons brivement . rappelons que lon a dfini ( voir dfinition [ psi ] ) un oprateur @xmath338 . dans @xcite , une application @xmath339 est construite pour tout @xmath30 . ces applications @xmath340 donnent lieu ( par un thorme non publi de fontaine do nt on trouvera une dmonstration dans @xcite ) un isomorphisme entre @xmath341 et la cohomologie diwasawa @xmath25 par passage la limite sur @xmath141 . on suppose prsent que @xmath9 est de de rham , et on va voir quelle condition @xmath342 si @xmath343 . loprateur @xmath344 que lon a dfini au paragraphe [ secpgdr ] vrifie @xmath345 o @xmath346 si lon combine le thorme ( * ? ? ? * iv.2.1 ) avec des proprits bien connues de lexponentielle duale de bloch - kato , et le fait que @xmath347 pour @xmath238 , alors on trouve que : [ ch_co ] il existe @xmath348 que lon peut supposer @xmath349 tel que si @xmath350 , alors @xmath351 et si @xmath343 , @xmath352 et @xmath238 , alors on a @xmath353 si et seulement si @xmath354 . enfin pour terminer , on calcule la @xmath355$]-torsion de @xmath341 . [ gam_tor ] si @xmath343 , alors il existe @xmath120 $ ] tel que @xmath121 si et seulement si @xmath356 . tant donn que @xmath274 comme on la rappel au paragraphe [ ii1 ] , cest une consquence immdiate du lemme [ gam_tor_pg ] . perrin - riou a en fait dtermin la structure du @xmath357-module @xmath25 . son sous - module de torsion sidentifie @xmath358 et @xmath359 est ( au moins si @xmath360 est finie ) un @xmath357-module libre de rang @xmath361\dim(v)$ ] . dans ce paragraphe , on montre le thorme [ main_theo ] . on commence par un rsultat un peu plus faible ( la proposition [ univ_irred ] ci - dessous ) puis on montre comment en dduire le cas gnral en dvissant un peu . si @xmath299 est une reprsentation @xmath0-adique , soit @xmath362 lensemble des @xmath363 tels que pour tout @xmath30 , on ait @xmath364 . [ univ_irred ] si @xmath238 et si @xmath9 est une reprsentation de de rham do nt les poids de hodge - tate sont @xmath365 et qui nadmet pas de sous - reprsentation do nt les poids de hodge - tate sont @xmath366 , alors @xmath367 . dautre part , on a @xmath368 si @xmath369 et @xmath370 si @xmath242 . posons @xmath371 et @xmath372 . comme les poids de hodge - tate de @xmath373 sont @xmath284 , le lemme [ ht_ndr ] montre que @xmath374 . dautre part , le fait que @xmath9 nadmet pas de sous - reprsentation do nt les poids de hodge - tate sont @xmath366 et le lemme [ no_divis ] montrent que @xmath375 . enfin , la proposition [ ch_co ] montre que si @xmath376 pour tout @xmath30 , alors @xmath377 pour tout @xmath324 et donc que @xmath378 pour tout @xmath324 et le lemme [ divis_t ] ( appliqu @xmath373 ) montre que @xmath379 . on est donc en mesure dappliquer la proposition [ pg_diff ] , qui montre quil existe @xmath268 $ ] tel que @xmath121 . le lemme [ gam_tor ] montre enfin que @xmath380 , ce qui dmontre le premier point . ensuite , si @xmath240 et @xmath381 , alors @xmath382 et donc @xmath368 . enfin si @xmath242 et @xmath380 , alors @xmath383 pour tout @xmath30 et donc @xmath370 . pour terminer , on donne les arguments de dvissage permettant de dduire le thorme principal de la proposition [ univ_irred ] . [ g_ext ] si @xmath384 est une suite exacte de reprsentations @xmath0-adiques , alors on a une suite exacte : @xmath385 la thorie des @xmath38-modules nous donne une suite exacte : @xmath386 et le lemme du serpent implique que lon a : @xmath387 le lemme rsulte alors du fait ( voir ( * ? ? ? * , ( iii ) ) ) que les sous - quotients des reprsentations de rham sont de de rham . si @xmath299 est une reprsentation de hodge - tate et @xmath388 , soit @xmath389 la plus grande sous - reprsentation de @xmath299 do nt tous les poids sont @xmath390 . [ no_fil_fil ] on a @xmath391 . soit @xmath392 la projection naturelle . si @xmath393 a tous ses poids @xmath390 , alors on a une suite exacte @xmath394 et donc @xmath395 est une reprsentation de hodge - tate , extension de deux reprsentations de hodge - tate poids @xmath390 , et donc elle - mme poids @xmath390 , ce qui fait que @xmath396 et donc que @xmath397 . nous pouvons enfin montrer le rsultat principal de cet article . pour cela , rappelons que lon a un isomorphisme @xmath398 . [ main_in ] si @xmath9 est une reprsentation de de rham , alors 1 . si @xmath9 na pas de sous - quotient stable par @xmath35 , alors @xmath399 ; 2 . en gnral , @xmath400 et le quotient est un @xmath26-module de torsion . le fait que @xmath401 suit du fait ( dmontr en ( * ? ? ? * lemme 6.5 ) par exemple ) que si @xmath299 est une reprsentation de de rham do nt tous les poids de hodge - tate sont @xmath33 , alors toute extension de @xmath124 par @xmath299 est elle - mme de de rham . nous allons maintenant montrer le ( 2 ) , cest - - dire que @xmath402 est un @xmath26-module de torsion . par un argument de dvissage , il suffit de montrer que pour tout @xmath238 , @xmath403 est un @xmath357-module de torsion ce que nous allons maintenant faire . on a une suite exacte : @xmath404 et la reprsentation @xmath405 a des poids @xmath365 et nadmet pas de sous - reprsentation do nt les poids sont @xmath366 par le lemme [ no_fil_fil ] . la proposition [ univ_irred ] montre que @xmath406 et donc que @xmath407 est un @xmath26-module de torsion . ceci montre le ( 2 ) . les calculs ci - dessus montrent de plus que si @xmath9 na pas de sous - quotient stable par @xmath35 , alors en fait @xmath408 car dans ce cas , @xmath409 pour tout @xmath238 et donc @xmath410 pour tout @xmath238 . ceci montre le ( 1 ) . reprenons la dmonstration du thorme [ main_in ] ci - dessus . limage de @xmath411 dans @xmath412 sidentifie une sous - reprsentation @xmath413 de @xmath414 ; cest donc une reprsentation de @xmath8 fixe par @xmath35 et do nt le seul poids de hodge - tate est @xmath415 ( notons aussi que @xmath416 ) . comme il ny a pas dextensions non - triviales entre de tels objets qui soit encore fixe par @xmath35 ( ou , ce qui revient au mme , il ny a pas dextensions non - triviales entre de tels objets chez les @xmath26-modules ) , on en dduit que si lon splice les suites exactes ( [ eqn2 ] ) pour @xmath238 , on trouve : @xmath417 ce qui permet de prciser ( au moins en principe ) le thorme [ main_theo ] . voici une liste des principales notations du texte , dans lordre o elles apparaissent : [ i1 ] : @xmath2 , @xmath3 , @xmath4 , @xmath418 , @xmath419 , @xmath49 , @xmath8 , @xmath35 , @xmath420 , @xmath82 , @xmath48 , @xmath51 , @xmath54 , @xmath89 , @xmath421 , @xmath64 , @xmath422 , @xmath84 , @xmath91 , @xmath74 , @xmath85 , @xmath97 , @xmath101 , @xmath108 , @xmath112 , @xmath113 .
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on calcule le module des normes universelles pour une reprsentation @xmath0-adique de de rham .
le calcul utilise la thorie des @xmath1-modules ( la formule de rciprocit de cherbonnier - colmez ) et lquation diffrentielle associe une reprsentation de de rham .
we compute the module of universal norms for a de rham @xmath0-adic representation .
the computation uses the theory of @xmath1-modules ( cherbonnier - colmez s reciprocity formula ) and the differential equation attached to a de rham representation .
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recently , a lot of attention has been directed to the subject of excitons in bulk crystals due to experimental observation of the so - called yellow exciton series in cu@xmath0o up to a large principal quantum number of @xmath2 @xcite-@xcite . such excitons in copper oxide , in analogy to atomic physics , have been named rydberg excitons . by virtue of their special properties rydberg excitons are of fascination in solid state and optical physics . these objects , whose size scales as the square of the rydberg principal quantum number @xmath3 , are ideally suited for fundamental quantum interrogations , as well as detailed classical analysis . one could expect that rydberg excitons would have been described , in analogy to rydberg atoms , by rydberg series of hydrogen atoms , but it turned out that this generic method of description should has been revised because diameter of such exciton is much larger then wavelength of light needed to create it @xcite . the observation and detailed description of rydberg excitons have opened a new field in condensed matter spectroscopy . in analogy to medium of rydberg atoms , where it has been possible to obtain a large optical nonlinearity at the single photon level and to realize a lot of quantum optics sophisticated experiments such as optical kerr effect or correlated states @xcite , it is expected that the medium of rydberg excitons is also fertile area . the unique combination of their huge size , long radiative lifetimes , possible strong dipole - dipole interaction and , what is the most important advantage for future technological applications , miniaturization of media / samples they are realized in , can be exploited to perform robust light - exciton quantum interfaces for quantum information processing purposes . solid bulk media are systems well worth considering for storing quantum information because they have a number advantages over gases , where a lot of experiments have been done [ for recent review see ref.@xcite ] : they are easy to prepare , diffusion processes are not so fast , much higher densities of interacting particles can be achieved @xcite . a common class of solids used within a quantum information context are rare - earth - metal - doped crystals , where a long time over one minute of storing information has been achieved @xcite , @xcite , and nitrogen - vacancy centres in diamond @xcite which have relatively long spin coherence . however , the size of solid samples used in such experiments is of order of several millimetres . rydberg excitonic samples are much smaller : observation of dipolar blockade in bulk cu@xmath0o @xcite and quantum coherence @xcite were performed in samples as small as several tens of micrometers . realization of these experiments have unlocked the plethora of dynamical effect which might be observed in rydberg excitons media ; one of examples is the electromagnetically induced transparency , the performing of which in cu@xmath0o bulk crystal will be the next step toward potential implementation this medium for quantum information processing . electromagnetically induced transparency ( eit ) @xcite , is one of the important effect in quantum optics as it allows for the coherent control of materials optical properties . the generic eit bases on extraordinary dispersive properties of an atomic medium with three active states in the @xmath4 configuration . this phenomenon leads to the significant reduction of absorption of a resonant probe , weak laser field by irradiating the medium with a strong control field making an otherwise opaque medium transparent . it leads to dramatic changes of dispersion properties of the system : absorption forms a dip called the transparency window and approaches zero while dispersion in the vicinity of this region becomes normal with a slope , which increases for a decreasing control field . the resonant probe beam is now transmitted almost without losses . eit has been explained by destructive quantum interference between different excitation pathways of the excited state or alternatively in terms of a dark superposition of states . since at least 20 years there has been a considered level of activity devoted to eit @xcite , which has been motivated by recognition of a number of its applications among which slowing and storing the light , see i.e. , @xcite-@xcite and references therein , which allows for realization of delay lines and buffers in optical circuits are well - known examples . a remarkable quenching of absorption due to eit in an undoped bulk of cu@xmath0o in a @xmath4-type configuration involving lower levels was examining in @xcite . demonstration of eit in rydberg atoms involving the ladder levels scheme by mohapatra _ et al _ @xcite has taken advantage of their unique properties . in rydberg systems the ladder configuration enables to couple long - living metastable , initially empty upper level with the levels coupled by the probe field . rydberg excitons in cu@xmath0o offer a great variety of accessible states for creating the ladder configuration what enables such a choice of coupling which could be realized by accessible lasers or eventually to be suitable for desirable coherent interaction implementations . rydberg eit has also attracted attention with demonstration of interaction - enhanced absorption imaging of rydberg excitations @xcite . the non - linear response of rydberg medium is proportional to the group index and to the strength of the dipole - dipole interaction and both of them can be extremely large . eit provides a possibility of dissipation - free sensing and probing of highly excited rydberg states and coherent coupling of rydberg states via eit could be used for cross - phase modulation and photon entanglement . rydberg excitons offer an unprecedented potential to study above mentioned phenomena in solids . it seems that , similar to rydberg atoms , strong dipolar interaction between rydberg excitons could appear and leads to so - called photon - blockade , which offers promising means to tailor deterministic single - photon sources @xcite , and to realize photonic phase gates @xcite . up till now the researches involving in rydberg excitons , both theoretical and experimental , have concentrated on their static properties ( excitonic states , electro- and magneto - optical properties ) . the first step toward study the photon blockade , quantum non - linear optics , and many - body physics with rydberg excitons is the realization of rydberg eit . here we focuse our interest on dynamical aspects and properties of rydberg excitons in cu@xmath0o and show , that the rydberg excitonic states can be used to perform the eit . we indicate excitonic states which guarantee of the most efficient realization of the experiment taking into account our previous results concerning excitonic resonaces as well as damping parameters and the matrix elements of momentum operator . our paper is organized as follows . in sec . ii we present the assumptions of the considered model and solve the time evolution equations , obtaining an analytical expression for the susceptibility and the group index . we use the obtained expression to compute the real and imaginary part of the susceptibility and the group index ( sec . iii ) , for a cu@xmath0o crystal slab . we examine in details the changes of both real and imaginary part caused by changes of driving parameters ( for example , the rabi frequencies ) . in sec . iv we draw conclusions of the model studied in this paper and indicate the optimal choice of rydberg states to realize the eit and light slowing . the possible practical importance is also indicated . the phenomenon of eit , described qualitatively above , has been studied theoretically for various configurations of the transitions and probe and control beams . below we propose a theoretical description for the case when the atomic transitions are replaced by intra excitonic transitions in rydberg excitons media . condensed matter exhibits quite a variety of three - level systems where induced transparency could be achieved in much the same way as done with atoms . yet dephasings , which can easily break the coherence of the population trapping state , are typically much faster in solids than in atomic vapors ; it has caused difficulty to observe a large electromagnetic induced transparency effect in solids . this difficulty can be overcame by using the rydberg exciton states . higher states have much larger lifetimes and the dephasing can reach the value which enables observation of the eit effect . below we will consider a cu@xmath0o crystal as a medium where the eit phenomenon can be realized . we use the ladder configuration ( fig . [ fig_schemat ] ) , consisting of three levels @xmath5 , and @xmath6 . as in previous works on rydberg excitons , we focus our attention on the so - called yellow series associated with the lowest inter - band transition between the @xmath7 valence band and the @xmath8 conduction band . because both band - edge states are of even parity , the lowest @xmath9 exciton state is dipole - forbidden , whereas all the _ p _ states are dipole - allowed ; the @xmath9 to @xmath10 transition is also allowed . we have chosen the valence band as the @xmath11 state . the @xmath12 and @xmath13 excitonic states are the @xmath14 and @xmath6 states , respectively . to obtain the @xmath15 state we assume that a constant electric field @xmath16 is applied to the considered system . as it was recently shown , the applied field splits the @xmath17 levels into @xmath15 , @xmath17 , and @xmath18 states@xcite . let the probe / signal field of frequency @xmath19 and amplitude @xmath20 couples the ground state @xmath11 of energy @xmath21 with an excited state @xmath14 of energy @xmath22 . the control field of frequency @xmath23 and amplitude @xmath24 couples a state @xmath6 of energy @xmath25 with the state @xmath14 , as it is illustrated in fig . [ fig_schemat ] . the hamiltonian of such three level system interacting with an electromagnetic wave , in the rotating wave approximation reads @xmath26\vert b\vert\right.\\ & & \left.-\hbar\omega_{2}(z , t)\exp[-{\rm i}(\omega_{2}t - k_{2}z)]\vert a\rangle\langle c\vert+h.c.\right\},\nonumber\end{aligned}\ ] ] + where @xmath27 , @xmath28 , @xmath29 , @xmath30 are the wave vectors and rabi frequencies corresponding to the particular couplings and @xmath31 being the dipole transition moments related to the specific transitions . the time evolution of the system is governed by the von neumann equation with a phenomenological relaxation contribution @xmath32+r\rho,\ ] ] where @xmath33 denotes the density matrix for an exciton at the position @xmath34 , and @xmath35 is the relaxation operator accounting for all relaxation processes in the medium . setting @xmath36,\nonumber\\ \rho_{ac}&=&\sigma_{ac}\exp[-{\rm i}(\omega_{2}t - k_2 z)],\nonumber\\ \rho_{bc}&=&\sigma_{bc}\exp[-{\rm i}(\omega_1-\omega_2)t-(k_1-k_2)z)],\nonumber\\ \rho_{aa}&=&\sigma_{aa},\nonumber\\ \rho_{bb}&=&\sigma_{bb},\nonumber\\ \rho_{cc}&=&\sigma_{cc},\nonumber\end{aligned}\ ] ] one can get rid of time - dependent factors except for slowly varying factors accompanying probe field . we denote by @xmath37 and @xmath38 , respectively the probe and the control beam detunings . while propagation effects for the control field are neglected the evolution of the system is described by the following set of bloch equations @xmath39 parameters @xmath40 , @xmath41 , describe damping of exciton states and are determined by exciton - damping mechanisms comprising , temperature - dependent homogeneous broadening due to phonons and broadening due to structural imperfections and eventual impurities . the relaxation damping rates for the coherence are denoted by @xmath42 , @xmath41 @xcite . it should be noted that in the above equations only the relaxations inside the three level system are considered , so the total probability for the populations of the three levels is conserved : @xmath43 . in order to study propagation of the signal field inside the medium the bloch equations are accompanied by the maxwell propagation equation for the rabi frequency @xmath44 of the probe pulse , which reads in the slowly varying envelope approximation @xmath45 where @xmath46 @xmath47 being the density of excitons , @xmath48 is the transition dipole matrix element , @xmath19 is the electromagnetic wave frequency . in the first order perturbation with respect to the probe field , the evolution of our system reduces to the set of the following equations for the density matrix @xmath49 taking into account that for a weak probe pulse polarization of the medium for a given frequency is proportional to the signal field @xmath50 and to the susceptibility @xmath51 it has a form @xmath52 the steady state complex exciton susceptibility exhibited to the probe field has the form @xmath53 where @xmath54 is the vacuum dielectric constant . the susceptibility is a complex , rapidly varying function of @xmath55 , its real part describe the dispersion and imaginary show the absorption of the medium . due to the dependence of the refractive index @xmath56 on electric susceptibility we define the group index @xmath57 which account on time delay of a pulse propagating in a medium with the group velocity @xmath58 so the slope of dispersion inside the transparency window determines the propagation of the pulse inside the medium . note that derivative of the real part of the susceptibility may be positive ( normal dispersion ) or negative ( anomalous dispersion ) ; in the latter case the group velocity may even become negative . + another way of explaining electromagnetically induced transparency applies the notion of dressed states . consider the subspace spanned by the states a and c ( the energy @xmath25 of the latter being moved by the photon energy @xmath59 ) , coupled by the interaction @xmath60 . the dressed states are eigenvectors of the hamiltonian restricted to this subspace . the eigenenergies are shifted from their bare values ; if the control field is at resonance the shift is equal to @xmath61 . if the probe photon is tuned right in the middle between the dressed eigenenergies , the transition amplitudes from the state b interfere destructively . + in the case of spectrally not too wide pulses one can approximate the susceptibility to the lowest term of its taylor expansion at the line center and in such a case the probe pulse can has the form @xmath62 this means that the pulse moves with the velocity @xmath63 with its shape essentially unchanged apart from an exponential modification of its height and an overall phase shift . the group velocity is approximately the velocity of the pulse maximum ( exactly if there is no damping ) . we have performed numerical calculations for a cu@xmath0o crystal slab with a thickness 30 @xmath64 . a constant electric field @xmath16 is applied in the @xmath34-direction . using the formulas ( [ chiladder ] ) and ( [ group_index ] ) we have calculated the real and imaginary part of the susceptibility and the group index . the values of certain energies , dipole moments and damping parameters characteristic for the excitonic states of fig . [ fig_schemat ] the results from our previous paper @xcite have been used . the detailed calculations of @xmath65 are presented in the appendix . for @xmath47 being the density of excitons we have used the value @xmath66 . applying external electric field @xmath67 one obtains the following values of parameters + @xmath68=3266.576 thz ( = 2150.3 mev ) + @xmath69 thz ( = 20.6714 mev ) + @xmath70 ghz ( = 30 @xmath71ev/@xmath72 ) + @xmath73 ghz ( = 5 @xmath71ev/@xmath72 ) + @xmath74 @xmath75 + @xmath76 as can be seen in fig . [ fig_chi ] a ) for non - zero rabi frequency of the control field the imaginary part of the system s susceptibility reveals a dip in the lorentzian absorption profile called a transparency window . this means that a resonant probe beam which otherwise would be strongly absorbed , is now transmitted almost without losses . the width of the transparency window is proportional to the square of control field amplitude and therefore increasing the control field strength it is possible to open it out . the real part of susceptibility is shown in fig . [ fig_chi ] b ) . the dispersion inside transparency window becomes normal with the slope which increases from a decreasing control field . the normal dispersion inside the window is responsible for reduction of the group velocity the absorption at the resonance does not reaches zero due to finite value of the relaxation rate @xmath77 , but it is very small . this means that the medium has become transparent for a probe pulse which travels with a reduced velocity . it is well known that eit allows us to obtain a steeper slope of the refractive index and thereby a large group index , see fig . [ fig_chi ] c ) . the negative values of @xmath78 , corresponding to the regions of anomalous dispersion and high absorption , have been omitted for clarity . [ fig_chi ] d ) shows how the group index and absorption inside the transparency window depend on the rabi frequency @xmath60 . the plots represent a cross - sections of fig . [ fig_chi ] a ) and fig . [ fig_chi ] c ) at @xmath79 . for @xmath80 , the susceptibility given by eq . ( [ chiladder ] ) has a single resonance , so that the imaginary part of susceptibility is significant . as the rabi frequency @xmath60 increases , a transparency window is formed . for considered excitonic transitions optimal slowing down of order @xmath81 appears for transparency window of width of tens ghz ; as shown on the fig . [ fig_chi ] d ) , there is some optimum @xmath82 where the group index has a maximum value , corresponding to a narrow , but fully formed transparency window with significant dispersion @xmath83 . by increasing the @xmath60 further , one obtains a wider window , characterized by higher group velocity but also smaller absorption . it should be stress that the frequency of the signal to be slowed down or even processed determines the choice of the exciton level involved @xmath14 . the influence of the damping rate @xmath84 on the group index and on the group velocity is not crucial . its increase causes the widening of transparency window which is accompanied by decrease of the dispersion . a ) group velocity index @xmath85 . b ) real part of susceptibility @xmath86 . c ) imaginary part of susceptibility @xmath87 . d ) group index @xmath88 and @xmath89 , in the center of the transparency window.,title="fig : " ] b ) . a ) group velocity index @xmath85 . b ) real part of susceptibility @xmath86 . c ) imaginary part of susceptibility @xmath87 . d ) group index @xmath88 and @xmath89 , in the center of the transparency window.,title="fig : " ] c ) . a ) group velocity index @xmath85 . b ) real part of susceptibility @xmath86 . c ) imaginary part of susceptibility @xmath87 . d ) group index @xmath88 and @xmath89 , in the center of the transparency window.,title="fig : " ] d ) . a ) group velocity index @xmath85 . b ) real part of susceptibility @xmath86 . c ) imaginary part of susceptibility @xmath87 . d ) group index @xmath88 and @xmath89 , in the center of the transparency window.,title="fig : " ] the universal rydberg nature can be exploited in systems other than atomic gases . the impressive observation of a rydberg blockade shift on a very different platform offers a new approach for studying semiconductor systems and also provides entirely new long - term perspectives for developing novel devices , which are more robust and compact than atomic systems . we have indicated the optimal states and well justified parameters to attempt the observation eit in rydberg excitons cu@xmath0o media which allow to obtain considerable value of group index . due to coherence properties of rydberg excitons the manipulations of the medium transparency is possible ; the width of the window and slowing down the group velocity of the pulse travelled inside the sample might be changed in controlled way by the strength of control field . the ability to control on - demand group index enables storing and retrieving light pulses , which is a basis to quantum memory implementation . the way of a precise dynamical control of the optical properties of the medium by optical means reveals new aspects of excitons quantum optics and is supposed to lead to constructing efficient tools for photonics , e.g. , delay lines , quantum switches or multiplexers . so far experimental demonstration of eit in rydberg excitons media is probably difficult to realize , but it seems safe to expect such experiments in the future . performing eit in excitonic rydberg media will be the step toward realization of controlled interaction of rydberg excitons in integrated and scalable solid state devices . for the 2@xmath17 exciton we take the 2@xmath90 state , i.e. @xmath91 state , which couples with the @xmath34- component of the electromagnetic wave . so all the waves propagating in the considered medium must have the @xmath34- component . according to the notation of ref . @xcite the energy of 2@xmath90 exciton follows from the relations + @xmath92 with @xmath93 where for @xmath94 and in units @xmath95 , one obtains @xmath96 the energy eigenvalues have the form @xmath97 where @xmath98 , with @xmath99 defined by with @xmath100 @xmath101 being the exciton effective masses anisotropy parameter , @xmath102 are the laguerre polynomials , and @xmath103 being the spherical harmonics . the energy for the @xmath13 exciton results from eq . ( [ energy2p ] ) where we take the larger value from the two solutions @xmath104 . + the energy for @xmath12 exciton could be calculated from the equation @xmath105 where we take the smaller value from the solutions . the dipole matrix element @xmath106 follows from the relation @xmath107 @xmath108 being the longitudinal - transversal splitting , and @xmath109 the so - called coherence radius . positions of our resonances were in perfect agreement with the experimental data ( see fig . [ fig_comp ] a ) the damping parameters were determined by fitting our results regarding to the linewidths of resonances . we were able to estimate the dissipation rates of the p excitons ( fig . [ fig_comp ] a ) and s excitons ( fig . [ fig_comp ] b ) which we later used in our numerical calculations . the relaxation rates are comparable to the values found in prior literature @xcite . t. kazimierczuk , d. frhlich , s. scheel , h. stolz , and m. bayer , nature * 514 * , 344 ( 2014 ) . s. hfling and a. kavokin , nature * 514 * , 313 ( 2014 ) . j. thewes , j. hecktter , t. kazimierczuk , m. amann , d. frhlich , m. bayer , m. a. semina , and m. m. glazov , phys . * 115 * , 027402 ( 2015 ) . j. hecktter , _ stark - effect measurements on rydberg excitons in cu@xmath0o_. thesis . technical university dortmund , 2015 . s. zieliska - raczyska , g. czajkowski , and d. ziemkiewicz , phys . b * 93 * , 075206 ( 2016 ) . f. schweiner , j. main , and g. wunner , phys . b * 93 * , 085203 ( 2016 ) f. schne , s .- o . krger , p. grnwald , h. stolz , m. amann , j. hecktter , j. thewes , d. frhlich , and m. bayer , phys . b * 93 * , 075203 ( 2016 ) . m. feldmaier , j. main , f. schweiner , h. cartarius , and g. wunner , j. phys . b : at . mol . * 49 * , 144002 ( 2016 ) . f. schweiner , j. main , m. feldmaier , g. wunner , and ch . uihlein , arxiv:1604.00492v1 [ cond-mat.mtrl-sci ] 2 apr 2016 ; phys . b , accepted ( 2016 ) . m. amann , j. thewes , d. frlich , and m. bayer , nature materials , * 15 * , 741 ( 2016 ) . p. grnwald , m. amann , j. hecktter , d. frhlich , m. bayer , h. stolz , and s. scheel , phys . lett . * 117 * , 133003 ( 2016 ) . s. zieliska - raczyska , d. ziemkiewicz , and g. czajkowski , phys . b * 94 * , 045205 ( 2016 ) . o. firstenberg , c. s. adams , and s. hofferberth , j. phys , b : at . mol , opt . phys . * 49 * , 152003 ( 2016 ) . m. johnsson and k. molmer , phys . a * 70 * , 032320 ( 2004 ) . g. heinze , ch . hubrich , and t. halfmann , phys . . lett . * 111 * , 033601 ( 2013 ) . g. d. fuchs , g. burkard , p. v. klimov , and d. d. awschalom , nature physics * 7 * , 789 - 793 ( 2011 ) . d. schraft , m. hain , n. lorenz , and t. halfmann , phys . lett . * 116 * , 073602 , ( 2016 ) . s. e. harris , phys . today * 50 * , 36 ( 1997 ) . m. fleischhauer , a. immamoglu , and j. p. marangos , rev . 77 * , 633 ( 2005 ) . a. raczyski , j. zaremba , and s. zieliska - kaniasty , phys . a * 69 * , 043801 - 5 ( 2004 ) . m. perdian , a. raczyski , j. zaremba , and s. zieliska - kaniasty , optics comm . * 266 * , 552 - 557 ( 2006 ) . a. raczyski , j. zaremba , and s. zieliska - kaniasty , phys . a * 74 * , 031810 - 7 ( 2007 ) . m. artoni , g. c. la rocca , and f. bassani , europhys . lett . * 49 * , 445 , 2000 . a. k. mohapatra , t. r. jackson , and c. s. adams , phys . lett . * 98 * , 113030 ( 2007 ) . g. gnter , m. robert - de - saint - vincent , h. schempp , c. s. hofmann , s. whitlock , and m. weidemller , phys . lett . * 108 * , 013002 ( 2012 ) . t. peyronel , o. firstenberg , qi - yu liang , s. hofferberth , a. v. gorshkov , t. pohl , m. d. lukin , and v. vuleti , nature * 488 * , 57 ( 2012 ) . d. paredes - barato and c. s. adams , phys . lett . * 112 * , 040501 ( 2014 ) . j. hecktter , private correspondence .
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we show that the electromagnetically induced transparency ( eit ) is possible in a medium exhibiting rydberg excitons and indicate the realistic parameters to perform the experiment .
the calculations for a cu@xmath0o crystal are given which show that in this medium due to large group index one could expect slowing down a light pulse by a factor about @xmath1 .
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You are an expert at summarizing long articles. Proceed to summarize the following text:
in previous two papers @xcite , @xcite we defined and analyzed random processes with continuous time parameter describing the evolution of special trees consisting of _ living and dead nodes _ connected by _ lines_. it seems to be appropriate to repeat briefly the characteristic features of the evolution process . the initial state @xmath3 of the tree corresponds to a single living node called _ root _ which at the end of its life is capable of producing @xmath4 new living nodes , and after that it becomes immediately dead . if @xmath5 then the new nodes are promptly connected to the dead node and each of them _ independently of the others _ can evolve further like a root . the random evolution of trees with continuous time parameter has not been investigated intensively recently . the main interest since the late 1990s has been focussed on the study of non - equilibrium networks @xcite occurring in common real world . the evolution mechanism of trees with living and dead nodes may be useful in some of biological systems where the branching processes are dominant . in what follows , we will use notations applied in @xcite and @xcite . it seems to be useful to cite the basic definitions . the probability to find the number @xmath0 of living nodes produced by one dying precursor equal to @xmath6 was denoted by @xmath7 where @xmath8 . is the set of non - negative integers . ] for the generating function as well as the expectation value and the variance of @xmath0 we used the following notations : @xmath9 where @xmath10_{z=1 } , \;\;\;\;\;\ ; j = 1 , 2 , \ldots\ ] ] are _ factorial moments _ of @xmath0 . it was shown in @xcite that the time dependence of the random evolution is determined almost completely by the expectation value @xmath11 . in accordance to this the evolution was called subcritical if @xmath12 , critical if @xmath13 and supercritical if @xmath14 . in the further considerations we are going to use four distributions for the random variable @xmath0 . as shown in @xcite the equations derived for the first and the second moments of the number of nodes are _ independent of the detailed structure of the distribution of @xmath0 _ provided that the moments @xmath11 and @xmath15 are finite . we called distributions of this type for @xmath0 _ arbitrary _ and used the symbol @xmath16 for its notation . in many cases it seems to be enough to apply the truncated distribution of @xmath0 . if the possible values of the random variable @xmath0 are @xmath17 and @xmath18 with probabilities @xmath19 and @xmath20 , respectively , then in the previous paper @xcite the distributions of this type were denoted by @xmath21 . many times it is expedient to assume distributions to be completely determined by one parameter . as known the geometric and poisson distributions are such distributions . in paper @xcite we used the symbols @xmath22 and @xmath23 to identify these distributions . the distribution function of the lifetime @xmath24 of a living node will be supposed to be exponential , i.e. @xmath25 . in order to characterize the tree evolution two non - negative integer valued random functions @xmath26 and @xmath27 are introduced : @xmath26 is the number of living nodes , while @xmath27 is that of dead nodes at @xmath28 . the total number of nodes at @xmath29 is denoted by @xmath30 . clearly , the nodes can be sorted into groups according to _ the number of outgoing lines_. following the notation in @xcite the number of nodes with @xmath31 outgoing lines at time instant @xmath29 is denoted by @xmath32 . a node not having outgoing line is called _ end - node_. it is obvious that an end - node is either live or dead . therefore , the number of end - nodes @xmath33 can be written as a sum of numbers of living and dead end - nodes , i.e. @xmath34 . since all living nodes are end - nodes @xmath35 can be replaced by @xmath26 . the total number of dead nodes @xmath27 is given by @xmath36 . in this paper we are dealing with properties of @xmath37 and @xmath38 when @xmath39 . we will call the random trees arising from a single root after elapsing infinite time _ stationary_. in section @xmath18 the basic properties of probability distributions of the number of nodes , living and end - nodes are investigated when @xmath39 . special attention is paid in section @xmath40 to the effect of distribution law of the number of outgoing lines . three different distributions of @xmath0 are investigated . in order to simplify the notation , indices referring to different distributions of @xmath0 are usually omitted in formulas . finally , the characteristic properties of stationary random trees are summarized in section @xmath41 . let us introduce the notion of _ tree size _ which is nothing else but the total number of nodes @xmath30 at time moment @xmath28 . we want to analyze the asymptotic behavior of the tree size , i.e. the behavior of the random function @xmath30 when @xmath42 . we say the _ limit random variable _ @xmath43 exists in the sense that the relation : @xmath44 is true for all positive integers @xmath2 , where @xmath3 denotes the initial state of the tree . a randomly evolving tree is called _ `` very old '' _ when @xmath45 , and a very old tree , as mentioned already , will be named _ stationary random tree_. it is elementary to prove that if the limit probability @xmath46 exists , then the generating function @xmath47 is determined by one of the fixed points of the equation @xmath48.\ ] ] it can be shown that if @xmath49 , then the fixed point to be chosen has to satisfy the limit relation @xmath50 while if @xmath51 , then it should have the property @xmath52 and independently of @xmath11 the equation @xmath53 must hold . the relation ( [ 4 ] ) means that the probability to find stationary tree of finite size is evidently @xmath54 , if @xmath55 , but if @xmath51 , then @xmath56 i.e. the probability to find a stationary tree of infinite size is equal to @xmath57 the proof of the proposition is simple . let us assume that @xmath58 is a _ probability generating function _ , i.e. @xmath59 and @xmath60 . we need the following theorem : [ th1 ] if @xmath61 , then @xmath62 ; while if @xmath63 , then there is a point @xmath64 such that @xmath65 , i.e. @xmath66 and @xmath67 . let us introduce the function @xmath68 . since @xmath69 is convex , i.e. all derivatives are positive in the interval @xmath70 $ ] it is evident that @xmath71 is a nondecreasing function of @xmath72 $ ] . if @xmath73 , then @xmath74 , i.e. @xmath75 is nondecreasing , negative valued function of @xmath76 in @xmath77 . since @xmath78 it is obvious that @xmath79 , if @xmath80 , i.e. @xmath81 , and this is the first statement of theorem [ th1 ] . if @xmath82 , then @xmath83 , and since @xmath78 the inequality @xmath79 has to be true for all @xmath84 lying near @xmath54 . on the other side @xmath85 , hence there should exist one is excluded because @xmath69 is convex.]@xmath86 in @xmath87 which satisfies the equation @xmath88 and that implies the second statement of theorem [ th1 ] . it seems to be important to investigate _ how the living nodes behave _ in very old , i.e. stationary trees . intuitively one can say that stationary trees arising in subcritical evolution do not contain living nodes and they have finite average size . at the same time , it seems to be quite obvious that stationary trees originating in supercritical evolution could have living nodes with non - zero probability , i.e. they are entities of _ `` eternal life''_. in order to give more precise answer the generating function of the random variable @xmath89 should be derived . it is easy to show that @xmath90 is one of the fixed points of the equation @xmath91 = g^{(\ell)}(z).\ ] ] by theorem @xmath54 , equation ( [ 6 ] ) has one fixed point if @xmath92 , namely @xmath93 $ ] , and from this it follows that @xmath94 i.e. the probability that a subcritical very old tree does not have living node , is exactly @xmath54 . if @xmath51 , then besides @xmath95 equation ( [ 6 ] ) has another fixed point given by @xmath96 $ ] , hence the probability @xmath97 should be larger than zero . in other words , a supercritical stationary tree may evolve infinitely with certain non - zero probability . of subcritical stationary trees the measure of subset containing trees with living nodes is zero , while in that of supercritical trees the measure of subset consisting of trees with living nodes is larger than zero . ] the expectation value and the standard deviation of the total number of nodes can be used to characterize the size of a stationary random tree . from ( [ 3 ] ) , one obtains @xmath98_{z \uparrow 1 } = \frac{1}{1-q_{1 } } , \;\;\;\;\;\ ; \mbox{if } \;\;\;\;\ ; q_{1 } < 1.\ ] ] in a supercritical evolution , i.e. when @xmath99 , the expectation value @xmath100 does not exist , but the limit relation @xmath101 can be simply proved . we note here that in the critical case , i.e. when @xmath102 , the average tree size becomes infinite linearly , i.e. we have the relation @xmath103 . the standard deviation of the total number of nodes in stationary random trees is given by @xmath104 in a supercritical state the standard deviation does not exist , but it can be readily shown that @xmath105 finally , we would like to deal briefly with some of the _ properties of end - nodes _ in stationary random trees . by using equation ( @xmath106 ) of @xcite , it can be shown that the generating function @xmath107 is nothing else than one of fixed points of the following simple equation : @xmath108 = g_{0}(z ) + ( 1-z)\;f_{0}.\ ] ] substituting @xmath109 we obtain @xmath110^{n}\right\ } = 0,\ ] ] i.e. @xmath111 , hence @xmath112 . in the sequel we will investigate the properties of stationary trees when @xmath69 is known . in this case we can obtain exact expressions for probabilities @xmath113 from the corresponding generating function . the generating function of @xmath0 is given by @xmath114 with the restriction for @xmath11 and @xmath15 determined by the equality @xmath115 . ( see fig . 1 in @xcite . ) by using eq . ( [ 3 ] ) and applying ( [ 8 ] ) we have @xmath116 = g.\ ] ] the fixed point of this equation is @xmath117,\ ] ] is also the generating function of @xmath118 . it is an elementary task to show that @xmath119 [ ht ! ] since @xmath120 performing expansion of generating function ( [ 9 ] ) into power series of @xmath121 , we can determine the probabilities @xmath122 easily . after elementary calculations we obtain @xmath123,\ ] ] where @xmath124 @xmath125 and @xmath126 [ ht ] the probabilities @xmath127 can be seen in fig . these are the probabilities to find @xmath128 nodes in a `` very old '' tree developed according to the distribution of @xmath129 . the dark and light bars correspond to subcritical @xmath130 and slightly supercritical @xmath131 tree evolutions , respectively . in the last case @xmath132 and it is not surprising that the probabilities @xmath133 for finite @xmath2 are larger in sub- than in supercritical trees . [ fig2 ] shows the dependence of @xmath133 on @xmath2 in sub- and supercritical evolutions , respectively . if the difference @xmath134 is small enough , let us say @xmath135 , then one observes a special phenomenon , namely , an _ `` oscillation '' _ in the dependence of @xmath133 versus @xmath2 , what is clearly seen in fig . [ fig2 ] . [ ht ! ] in order to explain the origin of `` oscillation '' we calculated the dependence of probabilities @xmath136 and @xmath137 on @xmath11 at @xmath138 . in fig . [ fig3 ] , we can see that @xmath139 in the interval @xmath140 . for the sake of simplicity the upper limit @xmath141 will be called `` critical @xmath11 '' . the reason of oscillation is trivial . clearly , if @xmath142 , then there are no random trees having nodes of even number , i.e. @xmath143 , therefore , it should be an interval @xmath144 in which @xmath145 we calculated the critical values of @xmath11 corresponding to @xmath146 at @xmath147 . the results are shown in table 1 . as expected @xmath148 is slightly decreasing with @xmath6 when @xmath15 is fixed . table @xmath54 : critical values of @xmath148 at @xmath149 . [ cols="^,^,^,^",options="header " , ] [ tab4 ] [ ht ! ] it follows from eq . ( [ 22 ] immediately that the probability @xmath133 vs. @xmath11 has a maximum at @xmath150 the appearance of maxima is well seen in fig . [ fig13 ] in the cases of @xmath151 . finally , we would like to discuss the _ properties of end - nodes _ in stationary random trees evolved according to poisson distribution of @xmath0 . in order to obtain the probabilities @xmath152 we have to solve the equation @xmath153 = g_{0}(z ) + ( 1-z)\exp(-q_{1}),\ ] ] where @xmath154 by using a special expansion procedure we can get the probabilities @xmath155 @xmath156 step - by - step . introducing the notations @xmath157 @xmath158 @xmath159 @xmath160 @xmath161 the first six probabilities are given by the following eqs . : @xmath162^{3 } , \;\;\;\ ; p_{3}^{(0 ) } = \frac{1}{6 } q_{1}^{3 } \ell_{3}(q_{1 } ) [ u(q_{1})]^{5},\ ] ] @xmath163^{7 } , \;\;\;\ ; p_{5}^{(0 ) } = \frac{1}{120 } q_{1}^{5 } \ell_{5}(q_{1 } ) [ u(q_{1})]^{9},\ ] ] @xmath164^{11}.\ ] ] [ ht ! ] [ ht ! ] by using these formulas we are able to show the dependence of @xmath165 on @xmath2 and @xmath11 , respectively . we see in fig . [ fig14 ] that the evolution process prefers the one end - node structures , i.e. stationary random trees in which the probability of finding @xmath166 end - nodes is very small . the curves @xmath167 versus @xmath11 plotted in fig . [ fig15 ] are clearly demonstrating that the sites of maxima of probabilities are shifted to higher values of @xmath11 as the number of nodes @xmath2 is increasing . we have investigated the main properties of evolution of special trees when the continuous time parameter tends to infinity . a tree of infinite age is called _ stationary _ or simply `` very old '' . we have stated that if the limit relations @xmath168 are true , then the random function @xmath30 giving the number of nodes in a tree at @xmath29 _ converges in distribution _ to a _ random variable @xmath118 _ which counts the number of nodes in a stationary tree . for the generating function @xmath169 a simple equation has been derived , namely @xmath170 , \;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\ ; \mbox{{\bf ( a)}}\ ] ] where @xmath69 is nothing else than the generating function of the number of new nodes @xmath0 produced by one dying node . it has been proved that if @xmath0 has finite first and second factorial moments , i.e. if @xmath171 and @xmath172 are finite , then @xmath173 , if @xmath49 , and @xmath174 , if @xmath51 . it means , if @xmath49 , then the probability @xmath175 of finding infinite number of nodes in a stationary random tree is zero , while if @xmath51 , then that is larger than zero , namely , @xmath176 . here , one has to note that the expectation value @xmath177 exists only if the tree evolution is subcritical , i.e. if @xmath178 . in the case of @xmath51 , i.e. in supercritical evolution it has been shown that @xmath179\ } = q_{1}/(q_{1}-1),\ ] ] while if @xmath102 , then @xmath180 . it has been shown also that the generating function @xmath181 of the number of end - nodes @xmath182 in a stationary random tree has to satisfy the equation @xmath183 - ( 1-z)f_{0 } , \;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\ ; \mbox{{\bf ( b)}}\ ] ] which can be used to study end - node properties . when @xmath184 is known then by using appropriate method for comparison of coefficients of @xmath185 in both sides of equations * ( a ) * and * ( b ) * we could calculate step - by - step the probabilities @xmath186 to find @xmath187 nodes and end - nodes , respectively , in a stationary random tree . for the calculations , we have chosen three different distributions of @xmath0 . the generating functions of these distributions are given by the following formulas : @xmath188 , & \mbox{if $ \nu \in { \bf g}$ , } \\ \mbox { } & \mbox { } \\ e^{q_{1}(z-1 ) } , & \mbox{if $ \nu \in { \bf p}$. } \end{array } \right.\ ] ] analyzing the results of numerical calculations the first impression is that the qualitative properties of stationary random trees depend hardly on the character of distribution of @xmath0 . we have seen that in all cases the probability to find @xmath189 node ( or end - node ) in a stationary tree is significantly larger then to find @xmath190 nodes . one can conclude that in the evolution process the formation of a rod - like stationary random tree is much more probable than that with many branches . we have found special behavior in the dependence of @xmath133 on @xmath2 only in the case of @xmath21 when @xmath191 is smaller than a critical value . the appearance of the oscillation of @xmath133 versus @xmath2 is consequence of the following trivial statement : when @xmath192 then @xmath193 . it has been demonstrated that the probabilities @xmath133 and @xmath165 versus @xmath11 show a maximum the location of which is increasing with @xmath2 but remains always smaller than @xmath54 . this property is best seen in the case of poisson distribution of @xmath0 . 99 l. pl , randomly evolving trees i , arxiv : cond - mat/0205650 l. pl , randomly evolving trees ii , arxiv : cond - mat/0211092 s.n . dorogovtsev and j.f.f . mendes , _ evolutions of networks _ ( oxford university press , 2003 )
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the properties of randomly evolving special trees having defined and analyzed already in two earlier papers ( arxiv : cond - mat/0205650 and arxiv : cond - mat/0211092 ) have been investigated in the case when the continuous time parameter converges to infinity .
equations for generating functions of the number of nodes and end - nodes in a stationary ( i.e. infinitely old ) tree have been derived . in order to solve exactly these equations
we have chosen three different distributions for the number of new nodes @xmath0 produced by one dying node . by using appropriate method
we have calculated step - by - step the probabilities of finding @xmath1 nodes as well as end - nodes in a stationary random tree .
analyzing the results of numerical calculations we have observed that the qualitative properties of stationary random trees depend hardly on the character of distribution of @xmath0 . the conclusion to be correct that in the evolution process the formation of a rod - like stationary tree is much more probable than the formation of a tree with many branches .
we have established that the probability of finding @xmath2 nodes in a stationary tree depends sensitively on the average value of @xmath0 and has a maximum the location of which is increasing with @xmath2 but remains always smaller than unity .
this is also true for the end - nodes . *
pacs : 02.50.-r , 02.50.ey , 05.40.-a *
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You are an expert at summarizing long articles. Proceed to summarize the following text:
comets are cold icy bodies in the solar system that were formed in the solar nebula and are considered to be the signature bodies to understand the formation of the solar system . as the comet nucleus makes its journey towards its parent star , the ices start sublimating giving rise to a mixture of gas and dust which form the coma . for comets at heliocentric distances less than 3 au , the visible band spectrum shows strong molecular emission bands riding on the continuum radiation scattered by the cometary dust . studying these molecular emission bands has been an important part of the cometary study . + comet c/2014 q2 ( lovejoy ) , an oort cloud comet , was discovered by terry lovejoy in august 2014 using an 8 inch telescope when the comet had a visual magnitude of 15 . the comet brightened to a magnitude of 4 in the visual band in january 2015 being closest to earth at 0.469 au on january 7 . the comet s perihelion was on the 30th of january 2015 at a heliocentric distance of 1.29 au . according to the data from jpl horizons database , the comet orbit has an eccentricity of 0.9976 , orbital inclination of 80.3@xmath3 , semi - major axis of 576.34 au and orbital period of 13700 years . + @xcite have carried out a detailed survey of 85 comets during the years 1976 - 1992 . they have studied the variation of gas production rates and dust to gas ratio with heliocentric distances . they have defined certain limits in the gas ratios in order to classify the comets as to whether they are depleted in carbon - chain molecules or not . according to their study , most of the carbon depleted comets are from the jupiter family . they also say that cn is produced from the nucleus as well as from the dust in the comet s coma . several spectroscopic surveys have been carried out , e.g. @xcite , @xcite , @xcite . @xcite have spectroscopically studied five comets . they found a linear correlation between production rates of c@xmath4 , c@xmath5 and c@xmath1 with respect to cn . no correlation was found between the production rate ratios and heliocentric distance . @xcite have given new statistical methods in cometary spectroscopy for the extraction and subtraction of the sky using the comet frames itself . they have given a dynamical classification of comets by studying 26 different comets . more recently , @xcite have compiled and reported the spectroscopic results for 130 comets observed from mcdonald observatory . they have found remarkable similarity in the composition of most of the comets . they quote that the carbon chain depleted comets can be from any dynamical class . however they have not found any of them from the halley type comets ( tisserand parameter @xmath6 2 and period @xmath6 200 years ) due to significantly low number of these types in their sample . + in our study of comet c/2014 q2 , we have determined the production rates for various gas species , @xmath2 and dust to gas ratios at different heliocentric distances . the observations were carried out with lisa spectrograph mounted on the 0.5 m(f/6.8 ) telescope ( planewave instruments cdk20 ) at the mount abu infra - red observatory ( miro ) , mount abu , india . @xcite have given a brief overview of the telescope at the observatory . in the following we describe , briefly , the spectrograph and comet observations . lisais a low resolution high luminosity spectrograph which is designed for the spectroscopic study of faint and extended objects . the light from the object that passes through the slit falls on the grating . the rest of the light is reflected towards the guiding ccd camera . two detector cameras are placed on either side of the spectrograph : atik 314l is used for acquiring the object spectrum . this has a chip size of 1392 x 1040 pixels where each is a square pixel of size 6.45 @xmath7 . the light that passes through the slit is reflected by the mirror and is transformed into a parallel beam by a collimator . this is then diffracted by the grating and then passes through the objective lens and gets converged onto the detector . as the focal length of the collimator is 130 mm and that of the camera lens is 88 mm , the image on the slit plane is multiplied by a factor of 88/130=0.68 onto the plane of the ccd . the second detector is atik titan guiding camera . this has a chip size of 659 x 494 pixels where each is a square pixel of size 7.4 @xmath7 . all the reflected light that does not pass through the slit is directed to the guiding mirror . this is then sent to the guiding optics which refocuses the light on to the guiding camera . there are 8 different slits that are provided with the instrument . we have used a slit width of 1.37 arcsec for january and february , and a slit width of 2.08 arcsec for march and may observations . grating with 300 grooves / mm was used covering a total wavelength range from @xmath8 3800 @xmath9 to @xmath8 7400 @xmath9 with a wavelength scale of 2.6 @xmath9 per pixel . the plate scale on the ccd plane is 0.57 arcsec per pixel . the length of the slit imaged onto the ccd plane is 130 arcsec . we carried out spectroscopic observations from mount abu infra - red observatory ( miro ) using lisa , a low resolution spectrograph mounted on the 0.5 m telescope during january to may 2015 . the sky conditions were photometric during the observing period . bias images were taken at regular intervals . the slit was assumed to be uniformly illuminated . with this assumption , lamp flats were taken after every round of observation . this was used for flat field corrections . an argon - neon lamp was used for the wavelength calibration . details of the comet observations are given in table [ comet_obs ] . the exposure times mentioned in the table are for each individual frame . the slit was placed at the photo - center of the comet and was manually tracked through the guiding ccd throughout the exposure time . the observations were made using the scheme , sky - object - sky , for the proper background sky subtraction . the standard stars ( along with their spectral types ) from iraf s catalogue of spectroscopic standardsobserved for flux calibration are as follows : hd15318(b9 ) , hd19445 ( g2v ) , hr1544(a1 ) , hd126991(g2v ) , hd129357(g2v ) , hd140514(g2v ) , hr7596(a0 ) , hd192281(o4.5 ) , hd217086(o7 ) . these were preferably observed at a similar airmass as that of the comet . .observational log . columns : date , mid - ut , heliocentric distance in au , geocentric distance in au , phase angle in degree , exp.(exposure time ) in second , airmass [ cols="^,^,^,^,^,^,^",options="header " , ] [ afrho_table ] @xmath10 the gas production rate as well as @xmath2 increase after perihelion and show a decreasing trend only after february 2015 . it is interesting to note the simultaneous increase in gas and dust which indicates an increase in the overall activity of the comet after its perihelion passage . this kind of asymmetry has been seen in many comets . in fact @xcite have tabulated the difference in the pre and post perihelion values for different class of comets with positive values indicating post perihelion activity . although we do not have data points at exactly the same distance for pre and post perihelion passages , we can perhaps say that this comet may have a large positive asymmetry . this indicates that there might be volatile material present beneath the surface of the comet or the surface of the nucleus consists of layers of ice that have different vaporization rates . we have carried out spectral study of the comet c/2014 q2 ( lovejoy ) which shows strong emission lines of c@xmath0 , c@xmath1 and cn . the molecular production rates and the quantity af@xmath11 are estimated . we arrive at the following conclusions : * the comet shows a large positive asymmetry , which indicates increase in the post - perihelion activity . * considering the limits set by @xcite for the carbon abundance , c/2014 q2 is found to be in the typical class of comets . * the dust to gas ratio remains fairly constant within the observed heliocentric range which is in agreement with @xcite * the @xmath2 values indicate that the dust distribution favours smaller dust particles near perihelion which depletes as the comet moves away . * new haser model scale lengths were obtained which best fit the observed data . these differ significantly from the previously quoted values . this might be due to the comet s intrinsic nature itself or affected by the solar activity at the time of observation . a more exhaustive study is required to confirm some of the conclusions drawn here . this work is supported by the dept of space , govt of india . we acknowledge the local staff at the mount abu infra - red observatory for their help . we would also like to thank prof . david schleicher for his invaluable inputs and suggestions . we acknowledge with thanks prof jeremy tatum for useful discussions . we would like to specially thank ms . navpreet kaur and mr . sameer for their help in the observations . we also thank our colleagues in the astronomy & astrophysics division at prl for their comments and suggestions . 99 ahearn m.f . , schleicher d.g . , feldman p. , mills r. , thompson d. , 1984 , aj , 89 , 579 . ahearn m.f.,millis r.,schleicher d.g.,osip d.j , birch p. , 1995 , icarus , 118 , 223 cochran a.l . , barker e.s . , gray c.l . , 2012 icarus , 218 , 144 - 168 cochran a.l . , 1985 , aj , 90 , 12 farnham t l. , schleicher d g. , and ahearn m f. , 2000 , icarus 147,180 - 204 . michael r. , and disanti michael a. , 1991 , aj , 383 , 356 - 371 fink u. , hicks m.d . , 1996 , apj , 459 , 729 - 743 fink u.,2009 , icarus , 201 , 311 - 314 gilbert a.m. , wiegert p.a . , unda - sanzana e. , vaduvescu o. , 2010 , mnras , 85 , 220 ganesh , s. , baliyan , k. s. , chandra , s. , joshi , u. c. , kalyaan , a. , mathur , s. n. , 2013 , astronomical society of india conference series , 99 , 9 krishna swamy k.s . , `` the physics of comets '' , 3@xmath12 edition , world scientific series langland - shula l e. , smith g h. , 2011 , icarus , 213 , 280 - 322 newburn r.,spinard h.,1989 , aj , 97 , 552 schleicher d g. , 2010 , aj , 140 , 973 tatum j. , 1984 , aa , 135 , 183 - 187
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spectra of comet c/2014 q2 ( lovejoy ) were taken with a low resolution spectrograph mounted on the 0.5 m telescope at the mount abu infrared observatory ( miro ) , india during january to may 2015 covering the perihelion and post - perihelion periods .
the spectra showed strong molecular emission bands ( c@xmath0 , c@xmath1 and cn ) in january , close to perihelion .
we have obtained the scale lengths for these molecules by fitting the haser model to the observed column densities .
the variation of gas production rates and production rate ratios with heliocentric distance were studied .
the extent of the dust continuum using the @xmath2 parameter and its variation with the heliocentric distance were also investigated .
the comet is seen to become more active in the post - perihelion phase , thereby showing an asymmetric behaviour about the perihelion .
[ firstpage ] comet : general - comet : individual : c/2014 q2 - methods : observational - techniques : spectroscopic - telescopes - oort cloud
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the rare charmless b meson decays arouse more and more interest , since it is a good place for testing the standard model ( sm ) , studying cp violation and looking for possible new physics beyond the sm . since 1999 , the b factories in kek and slac collect more and more data sample of rare b decays . in the future cern large hadron collider beauty experiments ( lhc - b ) , the heavier @xmath6 and @xmath7 mesons can also be produced . with the bright hope in lhc - b experiments and btev experiments at fermilab , following a previous study of @xmath8 decay @xcite , we continue to investigate other @xmath6 rare decays . the most difficult problem in theoretical calculation of non - leptonic @xmath9 decays is the calculation of hadronic matrix element . the widely used method is the factorization approach ( fa ) @xcite . it is a great success in explaining the branching ratio of many decays @xcite , although it is a very simple method . in order to improve the theoretical precision , qcd factorization @xcite and perturbative qcd approach ( pqcd ) @xcite are developed . perturbative qcd factorization theorem for exclusive heavy - meson decays has been proved some time ago , and applied to semi - leptonic @xmath10 decays @xcite , the non - leptonic @xmath11 @xcite , @xmath12 @xcite decays . pqcd is a method to factorize hard components from a qcd process , which can be treated by perturbation theory . non - perturbative parts are organized in the form of universal hadron light cone wave functions , which can be extracted from experiments or constrained by lattice calculations and qcd sum rules . more information about pqcd approach can be found in @xcite . in this paper , we would like to study the @xmath13 and @xmath1 decays in the perturbative qcd approach . in our calculation , we ignore the soft final state interaction because there are not many resonances near the energy region of @xmath6 mass . our theoretical formulas for the decay @xmath14 in pqcd framework are given in the next section . in section [ sc : neval ] , we give the numerical results of the branching ratio of @xmath14 and discussions for cp asymmetries and the form factor of @xmath15 etc . at last , we give a short summary in section [ summ ] . decay . ] for decay @xmath16 , the related effective hamiltonian is given by @xcite @xmath17-v_{tb}^*v_{td}\sum_{i=3}^{10}c_i(\mu ) o_i(\mu)\right\},\label{hami}\ ] ] where @xmath18 are wilson coefficients at the renormalization scale @xmath19 and @xmath20 are the four quark operators @xmath21 here @xmath22 and @xmath23 are @xmath24 color indices ; the sum over @xmath25 runs over the quark fields that are active at the scale @xmath26 , i.e. , @xmath27 . operators @xmath28 come from tree level interaction , while @xmath29 are qcd - penguins operators and @xmath30 come from electroweak - penguins . working at the rest frame of @xmath6 meson , we take kaon and pion masses @xmath31 , which are much smaller than @xmath32 . in the light - cone coordinates , the momenta of the @xmath6 , @xmath33 and @xmath34 can be written as : @xmath35 denoting the light ( anti-)quark momenta in @xmath9 , @xmath33 and @xmath34 as @xmath36 , @xmath37 and @xmath38 , respectively , we can choose : @xmath39 in the following , we start to compute the decay amplitudes of @xmath40 . according to effective hamiltonian ( [ hami ] ) , we draw the lowest order diagrams of @xmath40 in fig . [ figure : fig1 ] . let us first look at the usual factorizable diagrams ( a ) and ( b ) . they can give the @xmath41 form factor if take away the wilson coefficients . the operators @xmath42 and @xmath43 are @xmath44 currents , and the sum of their contributions is given by @xmath45 = 16\pi c_f m_b^2 \int_0 ^ 1\!\!\ ! dx_1 dx_2 \int_0^\infty\!\!\!\!\ ! b_1 db_1\ , b_2 db_2\ \phi_b(x_1,b_1 ) \\ \times \bigl\ { [ ( 2-x_{2 } ) \phi_{k}^a(x_2)-r_k(1 - 2x_{2})\phi_{k}^p(x_2)\\ + r_k(1 - 2x_{2})\phi_{k}^t(x_2 ) ] \alpha_{s}(t_{a}^{1})h_{a}(x_{1},1-x_2,b_1,b_2 ) \exp[-s_b(t_{a}^{1})-s_k(t_{a}^{1})]c(t_{a}^{1})\\ + 2r_k\phi_{k}^p(x_2)\alpha_{s}(t_{a}^{2 } ) h_{a}(1-x_{2},x_1,b_2,b_1)\exp[-s_b(t_{a}^{2})-s_k(t_{a}^{2})]c(t_{a}^{2 } ) \bigr\},\label{fe}\end{gathered}\ ] ] where @xmath46 $ ] , @xmath47 $ ] . @xmath48 is the group factor of the @xmath49 gauge group . the expressions of the meson distribution amplitudes @xmath50 , the sudakov factor @xmath51 , and the functions @xmath52 are given in the appendix . in above formula , the wilson coefficients @xmath53 of the corresponding operators are process dependent . the operator @xmath54 have the structure of @xmath55 , their amplitude is @xmath56 = 32\pi c_f m_b^2 r_{\pi}\int_0 ^ 1\!\!\ ! dx_1 dx_2 \int_0^\infty\!\!\!\!\ ! b_1 db_1\ , b_2 db_2\ \phi_b(x_1,b_1 ) \\ \times \bigl\ { [ \phi_{k}^a(x_2)-r_k(x_{2}-3)\phi_{k}^p(x_2)\\ + r_k(1-x_{2 } ) \phi_{k}^t(x_2 ) ] \alpha_{s}(t_{a}^{1})h_{a}(x_{1},1-x_2,b_1,b_2)\exp[-s_b(t_{a}^{1 } ) -s_k(t_{a}^{1})]c(t_{a}^{1})\\ + 2r_k\phi_{k}^p(x_2 ) \alpha_{s}(t_{a}^{2})h_{a}(1-x_{2},x_1,b_2,b_1)\exp[-s_b(t_{a}^{2 } ) -s_k(t_{a}^{2})]c(t_{a}^{2})\bigr\}.\label{fep}\end{gathered}\ ] ] for the non - factorizable diagrams ( c ) and ( d ) , all three meson wave functions are involved . using @xmath57 function @xmath58 , the integration of @xmath59 can be preformed easily . for the @xmath44 operators the result is : @xmath60 = -\frac{32}{3}\pi c_f\sqrt{2n_{c } } m_b^2 \int_0 ^ 1\!\!\ ! dx_1 dx_2dx_3 \int_0^\infty\!\!\!\!\ ! b_3 db_3\ \phi_b(x_1,b_3 ) \\ \times \bigl\ { [ ( x_3 - 1)\phi_{\pi}^a(x_3)\phi_{k}^a(x_2)+r_k(1-x_{2})\phi_{\pi}^a(x_3 ) \phi_{k}^p(x_2 ) + r_k(1-x_{2})\phi_{\pi}^a(x_3)\phi_{k}^t(x_2)]c(t_c^1)\\ \alpha_{s}(t_{c}^{1})h_{c}^{(1)}(x_{1},x_2,x_3,b_2,b_3)\exp[-s_b(t_{c}^{1 } ) -s_{\pi}(t_{c}^{1})-s_k(t_{c}^{1 } ) ] -[(x_2-x_3 - 1)\phi_{\pi}^a(x_3)\phi_{k}^a(x_2)\\ + r_k(1-x_{2})\phi_{\pi}^a(x_3)\phi_{k}^p(x_2 ) -r_k(1-x_{2})\phi_{\pi}^a(x_3)\phi_{k}^t(x_2)]c(t_c^2)\\ \alpha_{s}(t_{c}^{2})h_{c}^{(2)}(x_{1},x_2,x_3,b_2,b_3)\exp[-s_b(t_{c}^{2 } ) -s_{\pi}(t_{c}^{2})-s_k(t_{c}^{2 } ) ] \bigr\}.\label{me}\end{gathered}\ ] ] for the@xmath55 operators , the formula is : @xmath61 = -\frac{32}{3}\pi c_f\sqrt{2n_{c } } m_b^2 r_{\pi}\int_0 ^ 1\!\!\ ! dx_1 dx_2dx_3 \int_0^\infty\!\!\!\!\ ! b_3 db_3\ \phi_b(x_1,b_3 ) \\ \times \bigl\{[r_k(x_{2}+x_3 - 2)\phi_{\pi}^p(x_3)\phi_{k}^p(x_2)-r_k ( x_2-x_3)\phi_{\pi}^p(x_3)\phi_{k}^t(x_2)-r_k(x_2-x_3)\phi_{\pi}^t(x_3 ) \phi_{k}^p(x_2)\\ -r_k(2-x_{2}-x_3)\phi_{\pi}^t(x_3)\phi_{k}^t(x_2)- ( 1-x_3)\phi_{\pi}^p(x_3)\phi_{k}^a(x_2)-(1-x_3)\phi_{\pi}^t(x_3 ) \phi_{k}^a(x_2)]c(t_c^1)\\ \alpha_{s}(t_{c}^{1})h_{c}^{(1)}(x_{1},x_2,x_3,b_2,b_3)\exp[-s_b(t_{c}^{1 } ) -s_{\pi}(t_{c}^{1})-s_k(t_{c}^{1 } ) ] + [ r_k(1-x_{2}+x_3)\phi_{\pi}^p(x_3)\phi_{k}^p(x_2)\\ + r_k(x_2+x_3 - 1)\phi_{\pi}^p(x_3)\phi_{k}^t(x_2 ) -r_k(x_2+x_3 - 1)\phi_{\pi}^t(x_3)\phi_{k}^p(x_2 ) -r_k(1-x_{2}+x_3)\phi_{\pi}^t(x_3)\phi_{k}^t(x_2)\\ + x_3\phi_{\pi}^p(x_3)\phi_{k}^a(x_2)-x_3\phi_{\pi}^t(x_3)\phi_{k}^a(x_2 ) ] c(t_c^2)\\ \alpha_{s}(t_{c}^{2})h_{c}^{(2)}(x_{1},x_2,x_3,b_2,b_3)\exp[-s_b(t_{c}^{2 } ) -s_{\pi}(t_{c}^{2})-s_k(t_{c}^{2 } ) ] \bigr\}.\label{mep}\end{gathered}\ ] ] similar to ( c),(d ) , the annihilation diagrams ( e ) and ( f ) also involve all three meson wave functions . here we have two kinds of amplitudes , @xmath62 is the contribution containing the operator of type @xmath44 , and @xmath63 is the contribution containing the operator of type @xmath55 . @xmath64 = -\frac{32}{3}\pi c_f\sqrt{2n_{c } } m_b^2 \int_0 ^ 1\!\!\ ! dx_1 dx_2dx_3 \int_0^\infty\!\!\!\!\ ! b_1 db_1\ b_2 db_2\ \phi_b(x_1,b_1 ) \\ \times \bigl\{[x_3\phi_{\pi}^a(x_3)\phi_{k}^a(x_2)+r_{\pi}r_k ( 2+x_2+x_3)\phi_{\pi}^p(x_3)\phi_{k}^p(x_2)-r_{\pi}r_k(x_2-x_3)\phi_{\pi}^p(x_3 ) \phi_{k}^t(x_2)\\ -r_{\pi}r_k(x_2-x_3)\phi_{\pi}^t(x_3)\phi_{k}^p(x_2)-r_{\pi}r_k ( 2-x_2-x_3)\phi_{\pi}^t(x_3)\phi_{k}^t(x_2)]c(t_e^1)\\ \alpha_{s}(t_{e}^{1})h_{e}^{(1)}(x_{1},x_2,x_3,b_1,b_2)\exp[-s_b(t_{e}^{1 } ) -s_{\pi}(t_{e}^{1})-s_k(t_{e}^{1 } ) ] -[x_2\phi_{\pi}^a(x_3)\phi_{k}^a(x_2)\\ + r_{\pi}r_k(x_2+x_3)\phi_{\pi}^p(x_3)\phi_{k}^p(x_2 ) + r_{\pi}r_k(x_2-x_3)\phi_{\pi}^p(x_3)\phi_{k}^t(x_2 ) + r_{\pi}r_k(x_2-x_3)\phi_{\pi}^t(x_3)\phi_{k}^p(x_2)\\ + r_{\pi}r_k(x_2+x_3)\phi_{\pi}^t(x_3)\phi_{k}^t(x_2)]c(t_e^2)\\ \alpha_{s}(t_{e}^{2})h_{e}^{(2)}(x_{1},x_2,x_3,b_1,b_2)\exp[-s_b(t_{e}^{2 } ) -s_{\pi}(t_{e}^{2})-s_k(t_{e}^{2 } ) ] \bigr\},\label{ma}\end{gathered}\ ] ] @xmath65 = -\frac{32}{3}\pi c_f\sqrt{2n_{c } } m_b^2 \int_0 ^ 1\!\!\ ! dx_1 dx_2dx_3 \int_0^\infty\!\!\!\!\ ! b_1 db_1\ b_2 db_2\ \phi_b(x_1,b_1 ) \\ \times \bigl\{[r_k(2-x_2)\phi_{\pi}^a(x_3)\phi_{k}^p(x_2)+r_k ( 2-x_2)\phi_{\pi}^a(x_3)\phi_{k}^t(x_2)-r_{\pi}(2-x_3)\phi_{\pi}^p(x_3 ) \phi_{k}^a(x_2)\\ -r_{\pi}(2-x_3)\phi_{\pi}^t(x_3)\phi_{k}^a(x_2)]c(t_e^1)\\ \alpha_{s}(t_{e}^{1})h_{e}^{(1)}(x_{1},x_2,x_3,b_1,b_2)\exp[-s_b(t_{e}^{1 } ) -s_{\pi } ( t_{e}^{1})-s_k(t_{e}^{1 } ) ] + [ r_kx_2\phi_{\pi}^a(x_3)\phi_{k}^p(x_2)\\ + r_kx_2\phi_{\pi}^a(x_3)\phi_{k}^t(x_2 ) -r_{\pi}x_3\phi_{\pi}^p(x_3)\phi_{k}^a(x_2 ) -r_{\pi}x_3\phi_{\pi}^t(x_3)\phi_{k}^a(x_2)]c(t_e^2)\\ \alpha_{s}(t_{e}^{2})h_{e}^{(2)}(x_{1},x_2,x_3,b_1,b_2)\exp[-s_b(t_{e}^{2 } ) -s_{\pi}(t_{e}^{2})-s_k(t_{e}^{2 } ) ] \bigr\}.\label{map}\end{gathered}\ ] ] the factorizable annihilation diagrams ( g ) and ( h ) involve only two light mesons wave functions . @xmath66 is for @xmath44 type operators , and @xmath67 is for @xmath55 type operators : @xmath68 = 16\pi c_f m_b^2 \int_0 ^ 1\!\!\ ! dx_2 dx_3 \int_0^\infty\!\!\!\!\ ! b_2 db_2\ , b_3 db_3\ \\ \times \bigl\ { [ -x_{2}\phi_{\pi}^a(x_3)\phi_{k}^a(x_2)-2r_{\pi}r_k(1+x_{2})\phi_{\pi}^p(x_3 ) \phi_{k}^p(x_2)+ 2r_{\pi}r_k(1-x_{2})\phi_{\pi}^p(x_3)\phi_{k}^t(x_2)]\\ \alpha_{s}(t_{g}^{1})h_{g}(x_2,x_3,b_2,b_3)\exp[-s_{\pi}(t_{g}^{1})-s_k(t_{g}^{1 } ) ] c(t_g^1)\\ + [ x_3\phi_{\pi}^a(x_3)\phi_{k}^a(x_2)+2r_{\pi}r_k(1+x_3)\phi_{\pi}^p(x_3 ) \phi_{k}^p(x_2 ) -2r_{\pi}r_k(1-x_3)\phi_{\pi}^t(x_3)\phi_{k}^p(x_2)]\\ c(t_g^2)\alpha_{s}(t_{g}^{2})h_{g}(x_{3},x_2,b_3,b_2 ) \exp[-s_{\pi}(t_{g}^{2})-s_k(t_{g}^{2})]\bigr\},\label{fa}\end{gathered}\ ] ] @xmath69 = 32\pi c_f m_b^2 \int_0 ^ 1\!\!\ ! dx_2 dx_3 \int_0^\infty\!\!\!\!\ ! b_2 db_2\ , b_3 db_3\ \\ \times \bigl\ { [ r_kx_{2}\phi_{\pi}^a(x_3)\phi_{k}^p(x_2)-r_kx_{2}\phi_{\pi}^a(x_3 ) \phi_{k}^t(x_2)+ 2r_{\pi}\phi_{\pi}^p(x_3)\phi_{k}^a(x_2)]\\ \alpha_{s}(t_{g}^{1})h_{g}(x_2,x_3,b_2,b_3)\exp[-s_{\pi}(t_{g}^{1})-s_k(t_{g}^{1 } ) ] c(t_g^1)\\ + [ 2r_k\phi_{\pi}^a(x_3)\phi_{k}^p(x_2)+r_{\pi}x_3\phi_{\pi}^p(x_3)\phi_{k}^a(x_2 ) -r_{\pi}x_3\phi_{\pi}^t(x_3)\phi_{k}^a(x_2)]\\ c(t_g^2 ) \alpha_{s}(t_{g}^{2})h_{g}(x_{3},x_2,b_3,b_2 ) \exp[-s_{\pi}(t_{g}^{2})-s_k(t_{g}^{2})]\bigr\}.\label{fap}\end{gathered}\ ] ] from equation ( [ fe])-([fap ] ) , the total decay amplitude for @xmath70 can be written as @xmath71\hspace*{3cm}\nonumber\\ -f_{\pi}v_{tb}^{*}v_{td}f_{e}^{p}\left[\frac{1}{3}c_{5}+c_{6 } + \frac{1}{3}c_{7}+c_{8}\right ] + m_{e}\left[v_{ud}v_{ub}^{*}c_{1}-v_{tb}^{*}v_{td}(c_{3}+c_{9})\right ] \hspace*{1cm}\nonumber\\ -v_{tb}^{*}v_{td}m_e^{p}\left(c_{5}+c_{7}\right)-v_{tb}^{*}v_{td}m_a\left(c_{3 } -\frac{1}{2}c_{9}\right ) -v_{tb}^{*}v_{td}m_a^p\left(c_{5}-\frac{1}{2}c_{7}\right)\hspace*{2cm}\nonumber\\ -f_{b } v_{tb}^{*}v_{td}f_a\left[\frac{1}{3}c_{3}+c_{4}-\frac{1}{6}c_{9 } -\frac{1}{2}c_{10}\right ] -f_{b}v_{tb}^{*}v_{td}f_a^{p}\left[\frac{1}{3}c_{5}+c_{6}-\frac{1}{6}c_{7 } -\frac{1}{2}c_{8}\right ] , \label{eq : width}\end{aligned}\ ] ] and the decay width is expressed as @xmath72 should be calculated at the appropriate scale t which can be found in the appendix of ref . the decay amplitude of the charge conjugate channel @xmath73 can be obtained by replacing @xmath74 to @xmath75 and @xmath76 to @xmath77 in eq.([eq : width ] ) . for the decay @xmath78 , its amplitude can be written as @xmath79 \hspace*{3.6cm}\nonumber\\ -f_{\pi}v_{tb}^{*}v_{td}f_{e}^{p}\left[-\frac{1}{3}c_{5}-c_{6}+\frac{1}{6}c_{7 } + \frac{1}{2}c_{8}\right ] + m_{e}\left[v_{ud}v_{ub}^{*}c_{2}-v_{tb}^{*}v_{td}(-c_{3}+\frac{1}{2}c_{9 } ) \right]\hspace*{1cm}\nonumber\\ -v_{tb}^{*}v_{td}m_{e}^{p}\left(\frac{1}{2}c_{7}-c_{5}\right)-v_{tb}^{*}v_{td}m_{a } \left(\frac{1}{2}c_{9}-c_{3}\right ) -v_{tb}^{*}v_{td}m_{a}^{p}\left(\frac{1}{2}c_{7}-c_{5}\right)\hspace*{2 cm } \nonumber\\ -f_{b}v_{tb}^{*}v_{td}f_{a}\left[-\frac{1}{3}c_{3}-c_{4}+\frac{1}{6}c_{9 } + \frac{1}{2}c_{10}\right ] -f_{b}v_{tb}^{*}v_{td}f_{a}^{p}\left[-\frac{1}{3}c_{5}-c_{6 } + \frac{1}{6}c_{7}+\frac{1}{2}c_{8}\right].\end{aligned}\ ] ] and the decay width is then expressed as @xmath80 the following parameters have been used in our numerical calculation @xcite : @xmath81 we leave the ckm phase angle @xmath82 as a free parameter , whose definition is @xmath83.\ ] ] in this language , the decay amplitude of @xmath84 in eq.([eq : width ] ) can be parameterized as @xmath85,\label{a1}\ ] ] where @xmath86 , and @xmath57 is the relative strong phase between tree diagrams @xmath87 and penguin diagrams @xmath88 . @xmath89 and @xmath57 can be calculated from pqcd . using the above parameters in ( [ eq : shapewv ] ) , we get @xmath90 and @xmath91 from pqcd calculation , which shows the dominance of the tree contribution in this decay and a large strong phase calculated from pqcd . similarly , the decay amplitude for @xmath92 can be parameterized as @xmath93.\label{a2}\ ] ] therefore the averaged decay width for @xmath94 is @xmath95 . \label{eq : width3}\end{aligned}\ ] ] it is a function of @xmath96 . in fig . [ figure : fig2 ] , we plot the averaged branching ratio of the decay @xmath97 with respect to the parameter @xmath98 . since the latest experiment constraint upon the ckm angle @xmath98 from belle and babar is @xmath98 around @xmath99 @xcite , we can arrive from fig . [ figure : fig2 ] : @xmath100 previous naive and generalized factorization approach gives a similar branching ratios at @xmath101 with the form factor @xmath102 @xcite . in paper @xcite , beneke _ _ also calculate this decay mode using qcd improved factorization approach ( bbns ) . it is based on naive factorization approach . the dominant contribution is still proportional to @xmath15 form factor , which is introduced as an input parameter . in principal , the decay amplitude expand as series of @xmath103 and @xmath104 . but in practice , only the first order of @xmath103 corrections is calculated , including the so called non - factorizable contributions . the annihilation type contribution is power ( @xmath104 ) suppressed in bbns approach . therefore , the branching ratio predicted in qcd factorization and pqcd should not differ too much ; but the cp violation in these two approaches will be different , since it depends on many non - leading order contributions ( see below for discussion ) . in ref.@xcite , the branching ratio is about @xmath105 , which is larger than our pqcd result and previous fa method @xcite , because their form factor @xmath106 @xcite is larger than the previous factorization approach and our calculation below . the diagrams ( a ) and ( b ) in fig . [ figure : fig1 ] correspond to the @xmath107 transition form factor @xmath108 , where @xmath109 is the momentum transfer . the sum of their amplitudes have been given by eq . ( [ fe ] ) , so we can use pqcd approach to compute this form factor . our result is @xmath110 , if @xmath111 ; and @xmath112 , if @xmath113 . in our approach , this form factor is sensitive to the decay constant and wave function of @xmath6 meson , where there is large uncertainty ; but not sensitive to the @xmath33 meson wave function . eventually this form factor can be extracted from semi - leptonic experiments @xmath114 in the future . in our calculation , the only input parameters are wave functions , which stand for the non - perturbative contributions . up to now , no exact solution is made for them . so the main uncertainty in pqcd approach comes from @xmath115 wave functions . in this paper , we choose the light cone wave functions which are obtained from qcd sum rules @xcite . for @xmath34 meson , the distribution amplitude of light cone wave function should take asymptotic form if the energy scale @xmath116 . but in our case , the scale is not more than @xmath117gev , so we choose the corrected asymptotic form for twist 2 distribution amplitude @xmath118 , and other twist 3 distribution amplitudes derived using equation of motion by neglecting three particle wave functions @xcite . these functions are listed in the appendix , which are also used in decay mode @xmath119 @xcite and @xmath120 @xcite etc . we also try to use the asymptotic form for @xmath34 meson , for all the three distribution amplitudes @xmath118 , @xmath121 and @xmath122 , since we have very poor knowledge about twist 3 distribution amplitudes @xcite . the branching ratio of @xmath123 is nearly unchanged ( only @xmath124 ) , because the branching ratio of @xmath125 is mainly determined by the form factor @xmath126 ( see fig.1(a ) and ( b ) ) which is not dependent on @xmath34 wave function . however , the cp asymmetry changes from @xmath127 to @xmath128 by @xmath129 , when @xmath130 . this is because the direct cp asymmetry depend on the strong phase ( see discussion below ) , which comes from non - factorizable and annihilation diagrams , where all three meson wave functions are involved . the cp asymmetry predicted here should be used with great care , since it depends on two much uncertainties . for heavy @xmath9 and @xmath6 meson , its wave function is still under discussion using different approaches @xcite . in this paper , we find the branching ratio of @xmath131 is sensitive to the wave function parameter @xmath132 . for @xmath133 , the resulted branching ratio will decrease from about @xmath134 to about @xmath135 . when we set @xmath113 , our result is more closer to that of qcd factorization @xcite . this sensitive dependence should be fixed by the @xmath15 form factors from the semi - leptonic @xmath6 decays . other uncertainties in our calculation include the next - to - leading order @xmath103 qcd corrections and higher twist contributions , which need more complicated calculations . from our calculation , we find that the dominant contribution comes from tree level diagrams ( see fig.1 ( a ) and ( b ) ) in this decay . if su(3 ) symmetry is good , the branching ratio of @xmath136 should be equal to that of @xmath137 . the experimental result of @xmath137 is @xmath138 @xcite . the predicted branching ratio of @xmath16 is about 1.7 times that of @xmath139 , where the difference comes mainly from su(3 ) symmetry breaking : the decay constant @xmath140 larger than @xmath141 and @xmath142 larger than @xmath143 . in the calculation , we also find that the electroweak - penguins contribution is negligibly small as @xmath144 in branching ratio . for the experimental side , there is recent upper limit on the decay @xmath145 @xcite , @xmath146 at 90% c.l . our predicted result is consistent with this upper limit . for the decays of @xmath147 , the tree level contribution is suppressed due to the small wilson coefficients @xmath148 . thus the penguin diagram contribution is comparable with the tree contribution . we study the averaged branching ratio of the decay @xmath149 as a function of @xmath98 in fig . [ figure : fig3 ] . it is similar with fig.[figure : fig2 ] . we find that the branching ratio of @xmath149 is about @xmath150 when @xmath98 is near @xmath99 , it is a little smaller than the result of ref . @xcite . in sm , the ckm phase angle is the origin of cp violation . using eqs.([a1 ] ) and ( [ a2 ] ) , the direct cp violation parameter can be derived as @xmath151 it is approximately proportional to ckm angle @xmath152 , strong phase @xmath153 and the relative size @xmath89 between penguin contribution and tree contribution . we show the direct cp violation parameters as a function of ckm angle @xmath98 in fig . [ figure : fig4 ] . from this figure one can see that the direct cp asymmetry parameter of @xmath154 and @xmath155 can be as large as @xmath156 and @xmath157 when @xmath98 is near @xmath158 . the larger direct cp asymmetry of @xmath159 decay is mainly due to a larger @xmath89 in @xmath159 than in @xmath154 . the direct cp asymmetry predicted in qcd factorization approach is quite different from our result , due to the different source of strong phases . in qcd factorization approach , the strong phase mainly comes from the perturbative charm quark loop diagram , which is @xmath103 suppressed @xcite . while the strong phase in pqcd comes mainly from non - factorizable and annihilation type diagrams . the sign of the direct cp asymmetry is different for these two approaches in @xmath160 decay , and the magnitude of cp asymmetry in qcd factorization ( about 5% ) is also smaller than pqcd . the future lhc - b experiments can make a test for the two methods . for the decays of @xmath147 , the final @xmath161 mesons can not be detected directly . what the experiments measured are their mixtures @xmath162 and @xmath163 , thus a mixing induced cp violation is involved . following notations in the previous literature @xcite , we define the mixing induced cp violation parameter as @xmath164 where @xmath165 using unitarity condition of the ckm matrix @xmath166 , and eqs.([a1],[a2 ] ) , we can get @xmath167 where @xmath168 . combining eq.([lam ] ) and ( [ mcp ] ) , we can get @xmath169 if @xmath170 is a very small number , the mixing induced cp asymmetry is proportional to @xmath171 , which will be a good place for the ckm angle @xmath172 measurement . however as we already mentioned , the tree contribution in this channel is suppressed , @xmath173 is a large number , so that the @xmath174 behavior is dominant in the eq . ( [ cpf ] ) . the result of mixing induced cp violation is shown in fig . [ cp ] , which is indeed a roughly @xmath174 behavior . the tail near @xmath175 also shows the contribution from @xmath171 in eq.([cpf ] ) . in this work , we study the branching ratio and cp asymmetry of the decays @xmath97 and @xmath149 in pqcd approach . from our calculation , we find that the branching ratio of @xmath97 is about @xmath176 ; @xmath4 around @xmath177 and there are large cp violation in the processes , which may be measured in the future lhc - b experiments and btev experiments at fermilab . the authors thank m - z yang for helpful discussions , they also thank professor dong - sheng du for reading the manuscript . this work is partly supported by national science foundation of china under grant no . 90103013 , 10475085 and 10135060 . [ appendix ] in the appendix we present the explicit expressions of the formulas used in section ii . first , we give the expressions of the meson distribution amplitudes @xmath50 . for @xmath178 meson wave function , we use the similar wave function as @xmath9 meson @xcite : @xmath179.\label{waveb}\ ] ] we set the central value of parameter @xmath180 in our numerical calculation , and @xmath181 is the normalization constant using @xmath182 . we use the distribution amplitude @xmath186 of the k meson from ref . @xcite : @xmath187 , \\ \nonumber \phi_{k}^p(x ) & = & \frac{f_{k}}{2\sqrt{2 n_c } } [ 1 + 0.106(3t^2 - 1)-0.148(3 - 30t^2 + 35t^4)/8 ] , \\ \phi_{k}^t(x ) & = & \frac{f_{k}}{2\sqrt{2 n_c}}t[1 + 0.1581(5t^2 - 3)],\end{aligned}\ ] ] whose coefficients correspond to @xmath188 . in our numerical analysis , we use the one loop expression for the strong running coupling constant , @xmath189 where @xmath190 and @xmath191 is the number of active quark flavor at the appropriate scale @xmath19 . @xmath192 is the qcd scale , which we take @xmath193mev at @xmath194 . @xmath195 , @xmath196 , @xmath197 used in the decay amplitudes are defined as @xmath198 where the so called sudakov factor @xmath199 resulting from the resummation of double logarithms is given as @xcite @xmath200 \label{su1}\ ] ] with @xmath201 \left(\frac{\alpha_s}{\pi}\right)^2 , \\ b=\frac{2}{3}\frac{\alpha_s}{\pi}\ln\left(\frac{e^{2\gamma_{e}-1}}{2}\right).\hspace{6cm}\end{gathered}\ ] ] here @xmath202 is the euler constant , @xmath191 is the active quark flavor number . for the detailed derivation of the sudakov factors , @xcite . the functions @xmath203 come from the fourier transformation of propagators of virtual quark and gluon in the hard part calculations . they are given as @xmath204 , \label{eq : propagator1}\end{aligned}\ ] ] we adopt the parametrization for @xmath213 contributing to the factorizable diagrams @xcite , @xmath214^{c } , \hspace{0.5cm}c=0.3.\end{aligned}\ ] ] the hard scale @xmath215 in eq.([fe])-([fap ] ) are chosen as @xmath216 they are given as the maximum energy scale appearing in each diagram to kill the large logarithmic radiative corrections . m. wirbel , b , stech , and m. bauer , z. phys . c29 , 637 ( 1985 ) ; + m. bauer , b , stech , and m. wirbel , _ ibid._34 , 103 ( 1987 ) ; + l .- l . chau , h .- y . cheng , w.k . sze , h. yao , and b. tseng , phys . d43 , 2176 ( 1991 ) ; 58 , 019902(e ) ( 1998 ) . a. ali , g. kramer and c.d . l , phys . rev . d 58 , 094009 ( 1998 ) ; + _ ibid . _ 59 , 014005 ( 1999 ) ; c.d . l , nucl . phys . b ( proc . suppl . ) 74 , 227 ( 1999 ) . y .- h . chen , h .- y . cheng , b. tseng , and k .- c . yang , phys . d60 , 094014 ( 1999 ) ; + h .- y . cheng and k .- c . yang , _ ibid . _ 62 , 054029 ( 2000 ) . m. beneke , g. buchalla , m. neubert , and c.t . sachrajda , phys . 83 , 1914 ( 1999 ) ; nucl . b591 , 313 ( 2000 ) . a.j . schwartz(for the belle collaboration ) , hep - ex/0411075 ; + a. bevan ( for the babar collaboration ) , hep - ex/0411090 . xing , phys . rev . d48 , 3400 ( 1993 ) ; d.s . yang , phys . b358 , 123 ( 1995 ) ; y.h . chen , h.y , cheng , b. tseng , phys . d59 , 074003 ( 1999 ) . m. beneke and m. neubert , nucl . b 675 , 333 ( 2003 ) ; + j .- f.sun , g.h . zhu , d .- s . du , phys . d68 , 054003 ( 2003 ) . a. khodjamirian , t. mannel and m. melcher , phys . d70 , 094002 ( 2004 ) ; v.m . braun and a. lenz , phys . d70 , 074020 ( 2004 ) . t. huang , x .- h . zhou , phys . d70 , 014013 ( 2004 ) ; + t. huang , x .- wu , x .- h . wu , phys . d70 , 053007 ( 2004 ) ; + t. huang , x .- wu , phys . d70 , 093013 ( 2004 ) ; + t. huang , m .- z . zhou , x .- h . wu , hep - ph/0501032 . h.kawamura , j.kodaira , c - f qiao and k.tanaka , nucl.phys.proc.suppl . 116 269(2003 ) ; + h .- liao , phys . d70 , 074030(2004 ) ; + tao huang , xing - gang wu and ming - zhen zhou , phys . b611 , 260(2005 ) ; + bodo geyer and oliver witzel , hep - ph/0502239 . cleo collaboration , d. cronin - hennessy et al.,hep - ex/0001010 . li and k. ukai , phys . b555 , 197 ( 2003 ) . n . li and b. melic , eur . j. c11 , 695 ( 1999 ) . n . li , phys . d52 , 3958 ( 1995 ) . t. kurimoto , h .- li , and a.i . sanda , phys . d65 , 014007 ( 2002 ) .
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in the framework of perturbative qcd approach , we calculate the branching ratio and cp asymmetry for @xmath0 and @xmath1 decays . besides the usual factorizable diagrams ,
both non - factorizable and annihilation type contributions are taken into account .
we find that ( a ) the branching ratio of @xmath2 is about @xmath3 ; @xmath4 about @xmath5 ; and ( b ) there are large cp asymmetries in the two processes , which can be tested in the near future lhc - b experiments at cern and btev experiments at fermilab .
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You are an expert at summarizing long articles. Proceed to summarize the following text:
in this paper , we present a formal analysis of electromagnetic ( em ) radiative corrections to @xmath0 transitions .. see however ref . @xcite ] only em corrections to the dominant octet nonleptonic hamiltonian are considered . such corrections modify not only the original @xmath3 amplitude but also induce @xmath4 contributions as well . by the standards of particle physics , this subject is very old @xcite . yet , there exists in the literature no satisfactory theoretical treatment . this is due largely to complications of the strong interactions at low energy . fortunately , the modern machinary of the standard model , especially the method of chiral lagrangians , provides the means to perform an analysis which is both correct and structurally complete . that doing so requires no fewer than _ eight _ distinct chiral langrangians is an indication of the complexity of the undertaking . there is , however , a problem with the usual chiral lagrangian methodology . the cost of implementing its calculational scheme is the introduction of many unknown constants , the finite counterterms associated with the regularization of divergent contributions . as regards em corrections to nonleptonic kaon decay , it is impractical to presume that these many unknowns will be inferred phenomenologically in the reasonably near future , or perhaps ever . as a consequence , in order to obtain an acceptable phenomenological description , it will be necessary to proceed beyond the confines of strict chiral perturbation theory . in a previous publication @xcite , we succeeded in accomplishing this task in a limited context , @xmath5 decay in the chiral limit . we shall extend this work to a full phenomenological treatment of the @xmath0 decays in the next paper @xcite of this series . the proper formal analysis , which is the subject of this paper , begins in sect . 2 where we briefly describe the construction of @xmath6 decay amplitudes in the presence of electromagnetic corrections . in section 3 , we begin to implement the chiral program by specifying the collection of strong and electroweak chiral lagrangians which bear on our analysis . the calculation of @xmath6 decay amplitudes is covered in section 4 and our concluding remarks appear in section 5 . .1 cm 0.2 cm there are three physical @xmath6 decay amplitudes , is defined via @xmath7 . ] _ k^0 ^+ ^- _ + - , _ k^0 ^0 ^0 _ 00 , _ k^+ ^+ ^0 _ + 0 . [ a0 ] we consider first these amplitudes in the limit of exact isospin symmetry and then identify which modifications must occur in the presence of electromagnetism . in the @xmath8 two - pion isospin basis , it follows from the unitarity constraint that _ + - & = & a_0 e^i _ 0 + a_2 e^i _ 2 , + a_00 & = & a_0 e^i _ 0 - a_2 e^i _ 2 , [ a1 ] + a_+0 & = & 3 2 a_2 e^i _ 2 . the phases @xmath9 and @xmath10 are just the @xmath8 pion - pion scattering phase shifts ( watson s theorem ) , and in a cp - invariant world the moduli @xmath11 and @xmath12 are real - valued . the large ratio @xmath13 is associated with the @xmath3 rule . when electromagnetism is turned on , several new features appear : 1 . charged external legs experience mass shifts ( _ cf _ fig . [ fig : f1](a ) ) . photon emission ( _ cf _ fig . [ fig : f1](b ) ) occurs off charged external legs . this effect is crucial to the cancelation of infrared singularities . final state coulomb rescattering ( _ cf _ fig . [ fig : f1](c ) ) occurs in @xmath14 . 4 . there are structure - dependent hadronic effects , hidden in fig . 1 within the large dark vertices . in this paper , we consider the leading contributions ( see fig . [ fig : f2 ] ) which arise from corrections to the @xmath3 hamiltonian . there will be modifications of the isospin symmetric unitarity relations and thus extensions of watson s theorem . any successful explanation of em corrections to @xmath6 decays must account for all these items . an analysis @xcite of the unitarity constraint which allows for the presence of electromagnetism yields _ + - & = & ( a_0 + a_0^em ) e^i(_0 + _ 0 ) + 1 ( a_2 + a_2^em ) e^i(_2 + _ 2 ) , + a_00 & = & ( a_0 + a_0^em ) e^i(_0 + _ 0 ) - ( a_2 + a_2^em ) e^i(_2 + _ 2 ) , [ a6 ] + a_+0 & = & 3 2 ( a_2 + a_2^+em ) e^i ( _ 2 + _ 2 ) , to be compared with the isospin invariant expressions in eq . ( [ a1 ] ) . this parameterization holds for the ir - finite amplitudes , whose proper definition is discussed later in sect . 4.3 . observe that the shifts @xmath15 and @xmath16 in @xmath17 are distinct from the corresponding shifts in @xmath18 and @xmath19 . this is a consequence of a @xmath20 component induced by electromagnetism . in particular , the @xmath20 signal can be recovered via _ 5/2 = 5 . [ a6f ] .1 cm 2.8 cm the preceding section has dealt with aspects of the @xmath6 decays which are free of hadronic complexities . in this section and the next , we use chiral methods to address these structure - dependent contributions . the implementation of chiral symmetry via the use of chiral lagrangians provides a logically consistent framework for carrying out a perturbative analysis . in chiral perturbation theory , the perturbative quantities of smallness are the momentum scale @xmath21 and the mass scale @xmath22 , where @xmath23 is the quark mass matrix . in addition , we work to first order in the electromagnetic fine structure constant @xmath24 , _ i = a_i^(0 ) + _ i^(1 ) + . [ c1 ] our goal is to determine the @xmath25 components @xmath26 . the fine structure constant thus represents a second perturbative parameter , and we consider contributions of chiral orders @xmath27 and @xmath28 , _ i^(1 ) a_i^(e^2 p^0 ) + a_i^(e^2 p^2 ) . [ c1a ] we shall restrict our attention to just the leading electromagnetic corrections to the @xmath6 amplitudes . since the weak @xmath29 amplitude is very much larger than the @xmath30 amplitude , our approach is to consider only electromagnetic corrections to @xmath29 amplitudes . as a class these arise via processes contained in fig . [ fig : f2 ] , where @xmath31 is the octet weak coupling defined below in eq . ( [ c6 ] ) . we adopt standard usage in our chiral analysis , taking the matrix @xmath32 of light pseudoscalar fields and its covariant derivative @xmath33 as u ( i _ k _ k /f _ ) ( k = 1, ,8 ) , d_u _ u + i e [ q , u ] a _ , [ c2 ] where @xmath34 is the quark charge matrix and @xmath35 is the photon field . the remainder of this section summarizes the eight distinct effective lagrangians ( strong , electromagnetic , weak and electroweak ) needed in the analysis . in the @xmath36 sector , we shall employ the strong / electromagnetic lagrangian ^(2)_str = f^2_04 ( d_u d^u^ ) + f^2_04 ( u^+ u^ ) , [ c3 ] where @xmath37 is the pseudoscalar meson decay constant in lowest order . @xmath38 will be used to produce @xmath39 and @xmath40 vertices in our calculation . the lagrangian @xmath38 will generate ( via tadpole diagrams ) strong self - energy effects on the external legs in the @xmath0 transitions . in order to regularize these divergent contributions , one employs the lagrangian @xcite @xmath41 . it is not necessary to write out this well - known set of operators , but simply to point out that the resulting wave function renormalization factors @xmath42 and @xmath43 obey = z _ f^3 , [ c3a ] up to logarithms . this explains the presence of @xmath44 in formulae such as eqs . ( [ d7]),([d8 ] ) in section 4 . two other nonweak effective lagrangians enter the calculation . the first is associated with electromagnetic effects at chiral order @xmath27 , _ ems^(0 ) = g_ems ( q u q u^ ) , [ c4 ] where the coupling @xmath45 is fixed ( in lowest chiral order ) from the pion electromagnetic mass splitting , g_ems = f_^2 2 m_^2 . [ c5 ] the second extends the description to chiral order @xmath28 . we need only the following subset of the lagrangian given in ref . @xcite , & & l_ems^(2 ) = f^2 e^2 . although the finite parts of the coefficients @xmath46 remain unconstrained , see however refs . @xcite for model determinations . the @xmath47 octet lagrangian begins at chiral order @xmath21 , _ 8^(2 ) = g_8 ( _ 6 d_u d^u^ ) , [ c6 ] with @xmath48 fit @xcite from @xmath0 decay rates . we use this to generate @xmath39 , @xmath40 and @xmath27 vertices . two chiral lagrangians will serve to provide counterterms for removing divergent contributions . the first @xcite is the octet @xmath47 lagrangian at chiral order @xmath49 , _ 8^(4 ) & = & n_5 _ 6 + & + & n_6 _ 6 u _ u^ ( ^^u - ^u^ ) + & + & n_7 _ 6 ( u ^+ u^ ) _ u ^u^ + & + & n_8 _ 6 _ u ^u^ ( u^+ ^u ) + & + & n_9 _ 6 + & + & n_10 _ 6 ( u ^u ^ + u^u^+ u ^u^ ) + & + & n_11 _ 6 ( u ^+ u^ ) ( u^+ ^u ) + & + & n_12 _ 6 ( u ^u ^ + u^u^- u ^u^ ) + & + & n_13 _ 6 ( u ^- u^ ) ( u ^- u^ ) . [ c7 ] at present , little is known of the finite parts of the couplings @xmath50 . the @xmath47 lagrangian at chiral order @xmath27 is _ emw^(0 ) = g_emw ( _ 6 u q u^ ) , [ c8 ] where @xmath51 is an _ a priori _ unknown coupling constant . it has been calculated recently in ref . @xcite , g_emw = ( -0.62 0.19 ) g_8 m_^2 . [ c9 ] we note in passing that despite the presence of just one charge matrix @xmath52 the lagrangian of eq . ( [ c8 ] ) indeed describes @xmath53 effects . a second factor of @xmath52 could be decomposed into a combination of the unit matrix and the @xmath54 matrix @xmath55 . the contribution from @xmath56 would vanish , leaving the form of eq . ( [ c8 ] ) . the second operator that we use to provide counterterm contributions is the @xmath47 lagrangian at chiral order @xmath28 . in terms of the notation @xmath57 , we have & & l_emw^(2 ) = e^2 g_8 . the first six operators in the above list appear in ref . @xcite . the remaining three are also required for our analysis . to our knowledge , none of the divergent or finite parts of the @xmath58 are yet known . the leading em corrections arise from the processes of fig . [ fig : f1 ] and fig . [ fig : f2 ] . contributions to fig . [ fig : f2 ] occur in two distinct classes , those explicitly containing virtual photons ( fig . [ fig : f3 ] ) and those with no explicit virtual photons ( fig . [ fig : f4 ] ) . the latter are induced by em mass corrections and by insertions of @xmath51 . in figs . [ fig : f3],[fig : f4 ] , the larger bold - face vertices are where the weak interaction occurs . the integrals which occur in our chiral analysis are standard and already appear in the literature ( _ e.g. _ see ref . @xcite or ref . it suffices here to point out that all divergent parts of the one - loop integrals are ultimately expressible in terms of the @xmath59-dimensional integral a ( m^2 ) dk 1 k^2 - m^2 = ^d-4 , [ bpp1 ] where @xmath60 is the integration measure , @xmath61 is the scale associated with dimensional regularization and @xmath62 is the singular quantity . [ bpp2 ] each amplitude in the discussion to follow will be expressed as a sum of a finite contribution and a singular term containing @xmath62 . .1 cm 0.2 cm we begin with the @xmath27 amplitudes , _ + -^(e^2 p^0 ) = - f_k f_^2 ( g_8 m_^2 + g_emw ) , _ 00^(e^2 p^0 ) = 0 , _ + 0^(e^2 p^0 ) = a_+-^(e^2 p^0 ) . [ d1 ] although these have already been determined in ref . @xcite , we include them here for the sake of completeness . they are finite - valued and require no regularization procedure . next come the amplitudes of order @xmath63 , expressed as _ p^2 ) = a_i^(expl ) + a_i^(impl ) + a_i^(ct ) . [ d2 ] the superscript ` expl ' refers to figs . [ fig : f1](a),(c ) and fig . [ fig : f3 ] where virtual photons are _ explicitly _ present , whereas superscript ` impl ' refers to fig . [ fig : f4 ] where em effects are _ implicitly _ present via em mass splittings and @xmath51 insertions . the final term @xmath64 is the counterterm amplitude . we turn first to the class @xmath65 of explicit photonic diagrams . for these contributions , it is consistent to take meson masses in the isospin limit . we find & & f_k f_^2 g_8 a_+-^(expl ) = ( m_k^2 - m_^2 ) _ + - ( m _ ) + & & + 4 - 6 ^d-4 e^2 m_k^2 , + & & f_k f_^2 g_8 a_00^(expl ) = 0 , [ d7 ] + & & f_k f_^2 g_8 a_+0^(expl ) = 4 m_^2 - 6 ^d-4 e^2 m_^2 . the quantity @xmath66 , which appears in the above expression for @xmath67 , is associated with the processes of figs . [ fig : f1](a),(c ) . due to such processes , the weak decay amplitudes @xmath68 will develop infrared ( ir ) singularities in the presence of electromagnetism . to tame such behavior , an ir regulator is introduced and appears as a parameter in the amplitudes . for our work , this takes the form of a photon squared - mass @xmath69 . @xmath66 is given by _ + - ( m_^2 ) & = & 1 4 , [ a50 ] where = ( 1 - 4 m_^2 /m_k^2 ) ^1/2 [ a50a ] and a ( ) & = & 1 + 1 + ^2 2 , + h ( ) & = & ^2 + + 2 f ( 1 + 2 ) - 2 f ( - 1 2 ) , [ a51 ] + f(x ) & = & - _ 0^x dt 1 t |1 - t| . notice that the function @xmath66 is complex , and both its real and imaginary parts have a logarithmic singularity as @xmath70 . the solution to this problem is well known ; in order to get an infrared - finite decay rate , one has to consider the process with emission of soft _ real _ photons , whose singularity will cancel the one coming from soft _ virtual _ photons . we shall be more explicit on this point in sect . .1 cm 0.5 cm the amplitudes @xmath67 and @xmath71 each contain an additive divergent term ( proportional to @xmath62 ) and also depend on the arbitrary scale @xmath61 introduced in dimensional regularization of loop integrals . both these features will require the introduction of counterterms . next comes the class @xmath72 of diagrams in fig . [ fig : f4 ] not containing explicit photons . for such contributions , one must be sure to include all possible effects of chiral order @xmath27 and @xmath28 and treat the various terms in a consistent manner . thus for the contributions to fig . [ fig : f4 ] , isospin - invariant meson masses are used in amplitudes involving @xmath73 and @xmath74 , whereas electromagnetic mass splittings appear in amplitudes involving @xmath75 . we write the results as sums of complex - valued finite amplitudes @xmath76 and divergent parts , essentially the amplitudes @xmath77 , _ i^(impl ) = re f_i ( ) + i i m f_i ( ) + ^d-4 d_i , ( i = + - , 00 , + 0 ) . [ d8 ] the scale - dependence in @xmath76 comes entirely from its real part @xmath78 . we express the @xmath79 in terms of dimensionless amplitudes @xmath80 , e f_i ( ) = _ i g_8 m_k^2 f_^2 f_k a_i^(impl ) ( ) , [ d8p ] with @xmath81 , @xmath82 . since the @xmath83 coefficients have rather cumbersome analytic forms , we find it most convenient to express them in the compact form a_i^(impl ) ( ) = b_i^(m ) m_^2 f^2 + b_i^(g ) g f^2 + , [ d8q ] where g g_emw / g_8 . [ d8r ] the coefficients appearing in eq . ( [ d8q ] ) are given in table 1 . l|cccc + & @xmath84 & @xmath85 & @xmath86 & @xmath87 + @xmath88 & @xmath89 & @xmath90 & @xmath91 & @xmath92 + @xmath93 & @xmath94 & @xmath95 & @xmath96 & @xmath97 + @xmath98 & @xmath99 & @xmath100 & @xmath101 & @xmath102 + the finite functions also have imaginary parts @xmath103 which arise entirely from the processes in fig . [ fig : f4](c ) . from direct calculation we find & & f_k f_^2 f^2 g_8 i m f_+- = - 16 , + & & f_k f_^2 f^2 g_8 i m f_00 = - 16 ( m_k^2 - m_^2 ) , [ d8s ] + & & f_k f_^2 f^2 g_8 i m f_+0 = 32 ( m_k^2 - 2 m_^2 ) ( m_^2 + g ) , where @xmath104 is defined in eq . ( [ a50a ] ) . as a check on our calculation , we have verified that the above results are identical to those obtained from unitarity . the singular parts of @xmath105 are embodied by the @xmath106-functions , & & f^2 f_k f_^2 g_8 d_+- = m_k^2 + m_^2 , + & & f^2 f_k f_^2 g_8 d_00 = ( m_k^2 - m_^2 ) , [ d8a ] + & & f^2 f_k f_^2 g_8 d_+0 = m_k^2 + m_^2 . to arrive at the above , we have used both the correspondence between @xmath107 and @xmath45 given in eq . ( [ c5 ] ) and also the relation m_^^2 - m_^0 ^ 2 = m_k^+^2 - m_k^0 ^ 2 , [ d8b ] in the evaluation of loop integrals . the latter follows from dashen s theorem @xcite and is justified since terms violating dashen s theorem would begin to contribute at the higher chiral order @xmath108 . .1 cm 0.2 cm in order to cancel the singular @xmath62-dependence in the @xmath0 amplitudes , it is necessary to calculate all possible counterterm amplitudes which can contribute . these enter in a variety of ways , as shown in fig . [ fig : f5 ] where the small bold - face square denotes the counterterm vertex . for figs . [ fig : f5](a),(b ) the counterterm vertex has @xmath47 whereas in fig . [ fig : f5](c ) it has @xmath36 . using the lagrangians @xmath109 , @xmath110 and @xmath111 we determine the counterterm amplitudes to be & & f^2 f_k f_^2 g_8 a_+-^(ct ) = + & & m_k^2 ( e^2 f^2 ( x_1 - 4 u_1 - 83 u_2 ) + m_^2 ( 8 n_7 - 4 n_8 - 4 n_9 ) ) + & & + m_^2 ( e^2 f^2 ( x_2 + 4 u_1 + 83 u_2 ) - m_^2 ( 4 n_5 + 8 n_7 + 2 n_8 ) ) , + & & f^2 f_k f_^2 g_8 a_00^(ct ) = ( m_k^2 - m_^2 ) e^2 f^2 , [ d5 ] + & & f^2 f_k f_^2 g_8 a_+0^(ct ) = m_k^2 ( e^2 f^2 x_3 - m_^2 ( 4 n_5 + 4 n_8 ) ) + & & + m_^2 ( e^2 f^2 x_4 - m_^2 ( 2 n_8 + 4 n_9 ) ) , where the @xmath112 are coefficients in the @xmath113 lagrangian @xmath109 of eq . ( [ c7 ] ) , the @xmath114 are combinations of coefficients in the @xmath115 lagrangian @xmath111 of eq . ( [ c5a ] ) , u_1 = _ 1 + _ 2 , u_2 = _ 5 + _ 6 , u_3 = -2 _ 3 + _ 4 , [ d6 ] and the @xmath116 are combinations of coefficients in the @xmath113 lagrangian @xmath110 of eq . ( [ c5a ] ) , x_1 & = & - 4 9 s_1 - 1 9 s_2 + 2 9 s_3 + 2 3 s_5 - 4 s_6 + 2 3 s_7 + s_8 + s_9 , + x_2 & = & 4 9 s_1 - 2 9 s_2 + 4 9 s_3 + 4 3 s_5 + 4 s_6 - 2 3 s_7 - s_8 - s_9 , + x_3 & = & - 2 3 s_1 - 1 3 s_2 + 4 3 s_4 + 2 3 s_5 + 2 3 s_7 , [ d6a ] + x_4 & = & 2 3 s_1 + 2 3 s_3 - 4 3 s_4 + 4 3 s_5 - 2 3 s_7 , + x_00 & = & 2 9 ( s_1 + s_2 + s_3 ) + 2 3 s_4 + s_8 + s_9 , the counterterms themselves have finite and singular parts , n_i & = & n_i ^d-4 + n_i^(r ) ( ) , + u_i & = & u_i ^d-4 + u_i^(r ) ( ) , [ d6b ] + x_i & = & x_i ^d-4 + x_i^(r ) ( ) . the coefficients @xmath117 of the divergent parts of @xmath118 have already been specified in the literature @xcite and hence the @xmath61-dependences of @xmath119 , @xmath120 are known from the renormalization group equations . we infer the @xmath121 coefficients in this paper by canceling divergences in the @xmath122 amplitudes . upon combining results obtained thus far , we find the new results x_00 & = & - 13 m_^2 e^2 f^2 - 3 g e^2 f^2 , + x_1 & = & 3 + 272 m_^2 e^2 f^2 - 13 2 g e^2 f^2 , + x_2 & = & 3 - 18 m_^2 e^2 f^2 - 7 g e^2 f^2 , [ d9c ] + x_3 & = & - 73 m_^2 e^2 f^2 - 89 18 g e^2 f^2 , + x_4 & = & 6 - 2 m_^2 e^2 f^2 - 86 9 g e^2 f^2 , where we recall @xmath123 . removal of the infrared divergence from the expression for the decay rate is achieved by taking into account the process @xmath124 . for soft photons , whose energy is below the detector resolution @xmath125 , this process can not be experimentally distinguished from @xmath126 , so the observable quantity involves the inclusive sum over the @xmath127 and @xmath128 final states . at the order we are working , it is sufficient to consider just the emission of a single photon . the amplitude for the radiative decay is given in lowest order by @xmath129 where @xmath130 and @xmath131 are the polarization and momentum of the emitted photon . + the infrared - finite observable decay rate is @xmath132 where @xmath133 and @xmath134 is the differential phase space factor for each process . the infrared divergent ( ird ) part of @xmath135 is seen to be @xmath136^{2 } \int \ , d \phi_{+ - } \ \ 2 \alpha \ , { \cal r}e { b}_{+ - } ( m_{\gamma } ) \right . \ \ . \label{v3}\ ] ] equation ( [ v3 ] ) displays explicitly the singularity and shows that the imaginary part of @xmath137 has no observable effect at this order . this result has been shown to be true to all orders in @xmath24 @xcite . for @xmath138 we get the following expression , up to terms of order @xmath139 , _ + - ( ) = ^2 d _ + - i_+-(m _ , ) , [ x1 ] where _ + - ( m _ , ) = , [ x2 ] with & & f ( ) = 1 + 1 + ^2 2 . [ x3 ] from these explicit expressions of @xmath140 and @xmath141 it is easy to see that the combination @xmath142 does not depend on the infrared regulator @xmath143 . however , this combination has a dependence on the experimental resolution @xmath125 . to obtain a meaningful prediction therefore requires knowledge of the experimental treatment of soft photons . a careful discussion of this point will appear in ref . @xcite . a generalization of the above considerations beyond the order @xmath144 in chpt leads to the following parameterization , _ + - ( ) = d _ + - g_+- ( ) | a_+ -^(0 ) + a_+ -^(1 ) |^2 , [ v4 ] where to first order in @xmath24 , g_+- ( ) = 1 + 2 re b_+ - ( m _ ) + i_+- ( m _ , ) . [ v4a ] with the prescription of dropping the term proportional to @xmath145 in the photonic loop contribution , the electromagnetic amplitude @xmath146 can be read from eqs . ( [ d1]),([d7]),([d8]),([d5 ] ) . the physical amplitudes will be complex - valued functions , as dictated by unitarity . the real parts are obtained by combining the finite loop amplitudes ( eq . ( [ d7 ] ) for @xmath147 and eqs . ( [ d8p]),([d8q ] ) along with table 1 for @xmath105 ) with the counterterm amplitudes of eq . ( [ d5 ] ) , e a_i^(e^2 p^2 ) = _ i g_8 m_k^2 f_^2 f_k . [ d15 ] in order to make the scale - dependence of @xmath148 explicit , we write e a_i^(loop ) = b_i + c_i . [ d16 ] numerical determination of the above quantities will depend on @xmath31 ( obtained from ref . @xcite ) , @xmath107 and @xmath51 ( given in eq . ( [ c9 ] ) ) . we obtain the central values l b_+- = 11.8 10 ^ -3 , + b_00 = - 0.5 10 ^ -3 , + b_+0 = - 1.3 10 ^ -3 , l c_+- = 7.1 10 ^ -3 , + c_00 = -3.9 10 ^ -3 , + c_+0 = -2.7 10 ^ -3 . [ d17 ] the imaginary parts of the physical amplitudes can be either determined from unitarity or read off from eqs . ( [ d8]),([d8s ] ) . of most interest is the em shift in @xmath149 , as only it receives the @xmath150 ( @xmath3 ) enhancement , ( i m a_2^em ) & = & 32 , [ d17a ] where @xmath151 and @xmath152 are pion - pion t - matrix elements in the isospin basis . the above three contributions have physically distinct origins ; the first involves the direct effect of electromagnetism on the @xmath153 decay amplitude , the second arises from final state scattering in which electromagnetism induces leakage from @xmath154 to @xmath155 , and the third is due to the shift in two - pion phase space produced by the electromagnetic mass shift @xcite . despite the presence of many unknown finite counterterms , it is possible to apply the numerical results of eq . ( [ d17 ] ) and obtain rough estimates of the em corrections . the reasoning is that since the physical amplitudes are independent of the scale @xmath61 , there must be compensating @xmath61-dependence between the chiral logarithms of eq . ( [ d16 ] ) and the counterterms . therefore the counterterms must be at least of the same order - of - magnitude as the chiral logs or even larger . we have adopted the operational procedure of assuming the counterterm contribution @xmath156 vanishes at the scale @xmath157 , and we assign an uncertainty given by @xmath158 . this leads to the numerical values ( a_0^em ) & = & ( 0.024 0.026 ) 10 ^ -7 m_k^0 , + ( a_2^em ) & = & ( 0.015 0.022 ) 10 ^ -7 m_k^0 , + ( a_2^+em ) & = & ( - 0.005 0.005 ) 10 ^ -7 m_k^0 , [ e1 ] + a_5/2 & = & ( 0.012 0.016 ) 10 ^ -7 m_k^0 , with @xmath159 and @xmath160 . specifically , for the em shift @xmath161 calculated in ref . @xcite , we now have the extended result = - ( 2.0 2.2 ) % . [ e2 ] if one allows for the uncertainty in @xmath51 in addition to those in the counterterm values , we find = - ( 2.0^+4.0_-2.2 ) % . [ e3 ] in the numerical findings of eqs . ( [ e1])-([e3 ] ) , the error bars are seen to be almost as large or larger than the signal . in our opinion , this is the best that one can do within a strict chiral perturbation theory approach . 1 . since the central values of the amplitudes have @xmath162 , the electromagnetic loop corrections are seen to produce @xmath20 effects , although the uncertainties of the counterterm values overwhelms the numerical result . a phenomenological analysis @xcite based on @xmath163-wave pion - pion scattering lengths and forward dispersion relations gives @xmath164 . yet an isospin analysis of @xmath6 decays yields @xmath165 . presumably this difference of nearly @xmath166 can be reconciled by subtracting em effects from the @xmath6 decays . the main em shift should be in @xmath10 as only this angle experiences a @xmath3 enhancement . using eq . ( [ d17a ] ) to calculate the angle @xmath167 of eq . ( [ a6 ] ) , we find _ 2 = a_0^(e^0p^2 ) _ 2^(e^0p^2 ) 4.5^o . [ e4 ] this evaluation , valid at order @xmath168 , is seen to worsen the discrepancy between the two determinations . to reveal the explanation behind this puzzle requires more work . @xcite 3 . finally , the most important implication of these estimates is that the electromagnetic shifts in @xmath12 are not large , being only a few percent . naive estimates allow the possibility that this shift could be much larger , perhaps even being a major portion of @xmath12 . our previous work at the leading order in the chiral expansion yielded a small effect . one motivation of the present calculation was to see if the next order effects upset this conclusion . our estimates show that the natural size of the shift in @xmath12 remains at the few percent level . this has been a complicated calculation with many different lagrangians , describing different aspects of electromagnetic physics , required to obtain the full effect . these include explicit photon loops , mass shifts in the mesons propagating in loops and the short - distance electroweak interaction . the chiral power counting was crucial in sorting out which effects must be included for a consistent calculation . the resulting structure is universal and model independent . however , it is a prelude to more fully predictive applications , as there remain unknown low energy constants which are not predicted by chiral symmetry alone . different models can be used to estimate the renormalized constants which appear in the chiral lagrangians , and these model predictions can then be readily translated into the physical amplitudes through the use of our calculation . in a following publication , we attempt to describe the extent that this may be accomplished using dispersive techniques to match long and short distance physics @xcite . 99 # 1 # 2 # 3 nucl . # 1 * , ( # 2 ) # 3 # 1 # 2 # 3 phys . # 1 * , ( # 2 ) # 3 # 1 # 2 # 3 phys . * # 1 * , ( # 2 ) # 3 # 1 # 2 # 3 phys . rev . * # 1 * , ( # 2 ) # 3 # 1 # 2 # 3 phys . rev . * # 1 * , ( # 2 ) # 3 # 1 # 2 # 3 rev . # 1 * , ( # 2 ) # 3 # 1 # 2 # 3 z. phys . * # 1 * , ( # 2 ) # 3 , c. wolfe , ph.d.thesis , univ . of toronto ( 1999 ) unpublished . for example , see f. abbud , b.w . lee and c.n . yang , phys . rev . lett . * 18 * ( 1967 ) 980 ; a.a . belavin and i.m . narodetskii , sov . * 8 * ( 1968 ) 568 ; a. neveu and j. scherk , phys b27 * ( 1968 ) 384 ; a.a . belkov and v.v . kostyuhkin , sov . * 51 * ( 1989 ) 326 . v. cirigliano , j.f . donoghue and e. golowich , phys . b450 * ( 1999 ) 241 . , v. cirigliano , j.f . donoghue and e. golowich , hep - ph/9909374 . j. gasser and h. leutwyler , nucl . * b250 * ( 1985 ) 465 . r. urech , nucl . phys . * b433 * ( 1995 ) 234 . j. bijnens and j. prades , nucl . b490 * ( 1997 ) 239 . r. baur and r. urech , nucl . b499 * ( 1997 ) 319 . b. moussallam , nucl . phys . * b504 * ( 1997 ) 381 . g. ecker , j. kambor , and d. wyler , nucl . b394 * ( 1993 ) 101 . e. de rafael , nucl . phys . * 7a * ( proc . suppl . ) ( 1989 ) 1 . j. gasser and h. leutwyler , ann . * 158 * ( 1984 ) 142 . e. golowich and j. kambor , nucl . b447 * , ( 1995 ) 373 . r. dashen , phys . rev . * 183 * , ( 1969 ) 1245 . j. kambor , j. missimer and d. wyler , phys . lett . * b261 * ( 1991 ) 496 . s. weinberg , phys . * b140 * ( 1965 ) 516 . yennie , s.c . frautschi and h. suura , ann . * 13 * ( 1961 ) 379 . e. chell and m.g . olsson , phys . * d48 * ( 1993 ) 4076 .
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an analysis of electromagnetic corrections to the ( dominant ) octet @xmath0 hamiltonian using chiral perturbation theory is carried out .
relative shifts in amplitudes at the several per cent level are found .
+ 2.0 cm 1.5 cm vincenzo cirigliano@xmath1 , john f. donoghue@xmath2 and eugene golowich@xmath2 + .15 cm @xmath1 dipartimento di fisica delluniversit and i.n.f.n .
+ via buonarroti,2 56100 pisa ( italy ) + [email protected] + .15 cm @xmath2 department of physics and astronomy + university of massachusetts + amherst ma 01003 usa + [email protected] + [email protected] + .3 cm 1.5 cm
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You are an expert at summarizing long articles. Proceed to summarize the following text:
on a daily basis , people undergo numerous interactions with objects that barely register on a conscious level . for instance , imagine a person shopping at a grocery store as shown in figure [ fig : main ] . suppose she picks up a can of juice to load it in her shopping cart . the distance of the can is maintained fixed due to the constant length of her arm . when she checks the expiration date on the can , the distance and orientation towards the can is adjusted with respect to her eyes so that she can read the label easily . in the next aisle , she may look at a lcd screen at a certain distance to check the discount list in the store . thus , this example shows that spatial arrangement between objects and humans is subconsciously established in 3d . in other words , even though people do not consciously plan to maintain a particular distance and orientation when interacting with various objects , these interactions usually have some consistent pattern . this suggests the existence of an egocentric object prior in the person s field of view , which implies that a 3d salient object should appear at a predictable location , orientation , depth , size and shape when mapped to an egocentric rgbd image . our main conjecture stems from the recent work on human visual perception @xcite , which shows that _ humans possess a fixed size prior for salient objects_. this finding suggests that a salient object in 3d undergoes a transformation such that people s visual system perceives it with an approximately fixed size . even though , each person s interactions with the objects are biased by a variety of factors such as hand dominance or visual acuity , common trends for interacting with objects certainly exist . in this work , we investigate whether one can discover such consistent patterns by exploiting egocentric object prior from the first - person view in rgbd frames . our problem can be viewed as an inverse object affordance task @xcite . while the goal of a traditional object affordance task is to predict human behavior based on the object locations , we are interested in predicting potential salient object locations based on the human behavior captured by an egocentric rgbd camera . the core challenge here is designing a representation that would encode generic characteristics of visual saliency without explicitly relying on object class templates @xcite or hand skin detection @xcite . specifically , we want to design a representation that captures how a salient object in the 3d world , maps to an egocentric rgbd image . assuming the existence of an egocentric object prior in the first - person view , we hypothesize that a 3d salient object would map to an egocentric rgbd image with a predictable shape , location , size and depth pattern . thus , we propose an egoobject representation that represents each region of interest in an egocentric rgbd video frame by its _ shape _ , _ location _ , _ size _ , and _ depth_. note that using egocentric camera in this context is important because it approximates the person s gaze direction and allows us to see objects from a first - person view , which is an important cue for saliency detection . additionally , depth information is also beneficial because it provides an accurate measure of object s distance to a person . we often interact with objects using our hands ( which have a fixed length ) , which suggests that depth defines an important cue for saliency detection as well . thus assuming the existence of an egocentric object prior , our egoobject representation should allow us to accurately predict pixelwise saliency maps in egocentric rgbd frames . to achieve our goals , we create a new egocentric rgbd saliency dataset . our dataset captures people s interactions with objects during various activities such as shopping , cooking , dining . additionally , due to the use of egocentric - stereo cameras , we can accurately capture depth information of each scene . finally we note that our dataset is annotated for the following three tasks : saliency detection , future saliency prediction , and interaction classification . we show that we can successfully apply our proposed egocentric representation on this dataset and achieve solid results for these three tasks . these results demonstrate that by using our egoobject representation , we can accurately characterize an egocentric object prior in the first - person view rgbd images , which implies that salient objects from the 3d world map to an egocentric rgbd image with predictable characteristics of shape , location , size and depth . we demonstrate that we can learn this egocentric object prior from our dataset and then exploit it for 3d saliency detection in egocentric rgbd images . region proposals . for each of the regions @xmath1 we then generate a feature vector @xmath2 that captures shape , location , size and depth cues and use these features to predict the 3d saliency of region @xmath1 . ] * saliency detection in images . * in the past , there has been much research on the task of saliency detection in 2d images . some of the earlier work employs bottom - up cues , such as color , brightness , and contrast to predict saliency in images @xcite . additionally , several methods demonstrate the importance of shape cues for saliency detection task @xcite . finally , some of the more recent work employ object - proposal methods to aid this task @xcite . unlike the above listed methods that try to predict saliency based on contrast , brightness or color cues , we are more interested in expressing an egocentric object prior based on shape , location , size and depth cues in an egocentric rgbd image . our goal is then to use such prior for 3d saliency detection in the egocentric rgbd images . * egocentric visual data analysis . * in the recent work , several methods employed egocentric ( first - person view ) cameras for the tasks such as video summarization @xcite , video stabilization @xcite , object recognition @xcite , and action and activity recognition @xcite . in comparison to the prior egocentric approaches we propose a novel problem , which can be formulated as an inverse object affordance problem : our goal is to detect 3d saliency in egocentric rgbd images based on human behavior that is captured by egocentric - stereo cameras . additionally , unlike prior approaches , we use * egocentric - stereo * cameras to capture egocentric rgbd data . in the context of saliency detection , the depth information is important because it allows us to accurately capture object s distance to a person . since people often use hands ( which have fixed length ) to interact with objects , depth information defines an important cue for saliency detection in egocentric rgbd environment . unlike other methods , which rely on object detectors @xcite , or hand and skin segmentation @xcite , we propose egoobject representation that is based solely on shape , location , size and depth cues in an egocentric rgbd images . we demonstrate that we can use our representation successfully to predict 3d saliency in egocentric rgbd images . based on our earlier hypothesis , we conjecture that objects from the 3d world map to an egocentric rgbd image with some predictable _ shape _ , _ location _ , _ size _ and _ depth_. we encode such characteristics in a region of interest @xmath3 using an egoobject map , @xmath4 \in \mathds{r}^{n_s\times n_l \times n_b \times n_d \times n_c}$ ] where @xmath5 , @xmath6 , @xmath7 , @xmath8 , and @xmath9 are the number of the feature dimension for shape @xmath10 , location @xmath11 , size @xmath12 , depth @xmath13 , and context @xmath14 , respectively . a shape feature , @xmath15^\mathsf{t}$ ] captures a geometric properties such as area , perimeter , edges , and orientation of @xmath3 . * @xmath16 : perimeter divided by the squared root of the area , the area of a region divided by the area of the bounding box , major and minor axes lengths . * @xmath17 : we employ boundary cues @xcite , which include , sum and average contour strength of boundaries in region @xmath3 and also minimum and maximum ultrametric - contour values that lead to appearance and disappearance of the smaller regions inside @xmath3 @xcite . * @xmath18 : eccentricity and orientation of @xmath3 and also the diameter of a circle with the same area as region @xmath3 . a location feature @xmath19^\mathsf{t}$ ] encode spatial prior of objects imaged in an egocentric view : * @xmath20 : normalized bounding box coordinates and the centroid of a region @xmath3 . * @xmath21 : we also compute horizontal and vertical distances from the centroid of @xmath1 to the center of an image , and also to the mid - points of each border in the image . a size feature @xmath22^\mathsf{t}$ ] encodes the size of the bounding box and area of the region . * @xmath23 : area and perimeter of region @xmath3 . * @xmath24 : area and aspect ratio of the bounding box corresponding to the region @xmath3 . @xmath25^\mathsf{t}$ ] encodes a spatial distribution depth within @xmath3 . * @xmath26 : minimum , average , maximum , depth and also standard deviation of depth in a region @xmath3 . * @xmath27 : @xmath28 spatial depth histograms over the region @xmath3 . * @xmath29 : @xmath30 depth histograms over the region @xmath3 aligned to its major axis . * @xmath31 : @xmath28 spatial _ normalized _ depth histograms over the region @xmath3 . * @xmath32 : @xmath30 _ normalized _ depth histograms over the region @xmath3 aligned to its major axis . in addition , we include a context feature @xmath14 that encodes a spatial relationship between near regions in the egocentric image . given two regions , @xmath33 computes a distance between two features , i.e. , @xmath34^\mathsf{t}.\nonumber\end{aligned}\ ] ] given a target region , @xmath3 , the context feature @xmath35^\mathsf{t}$ ] computes the relationship with @xmath36 neighboring regions , @xmath37 : * @xmath38 : @xmath39 * @xmath40 : @xmath41 * @xmath42 : @xmath43 * @xmath44 : @xmath45 where @xmath46 and @xmath47 are the feature vector constructued by the min - pooling and max - pooling of neighboring regions for each dimension . @xmath48 takes average of neighboring features and @xmath49 is the feature of the top @xmath50 nearest neighbor . * summary . * for every region of interest @xmath3 in an egocentric rgbd frame , we produce a @xmath51 dimensional feature vector denoted by @xmath52 . we note that some of these features have been successfully used previously in tasks other than 3d saliency detection @xcite . additionally , observe that we do not use any object - level feature or hand or skin detectors as is done @xcite . this is because , in this work , we are primarily interested in studying the idea that salient objects from the 3d world are mapped to an egocentric rgbd frame with a consistent shape , location , size and depth patterns . we encode these cues with our egoobject representation and show its effectiveness on egocentric rgbd data in the later sections of the paper . given an rgbd frame as an input to our problem , we first feed rgb channels to an mcg @xcite method , which generates @xmath53 proposal regions . then , for each of these regions @xmath3 , we generate our proposed features @xmath52 and use it as an input to the random forest classifier ( rf ) . using a rf , we aim to learn the function that takes the feature vector @xmath52 corresponding to a particular region @xmath3 as an input , and produces an output for one our proposed tasks for region @xmath3 ( i.e. saliency value or interaction classification ) . we can formally write this function as @xmath54 . we apply the following pipeline for the following three tasks : 3d saliency detection , future saliency prediction , and interaction classification . however , for each of these tasks we define a different output objective @xmath55 and train rf classifier according to that objective separately for each task . below we describe this procedure for each task in more detail . * 3d saliency detection . * we train a random forest _ regressor _ to predict region s @xmath3 intersection over union ( iou ) with a ground truth salient object . to train the rf regressor we sample @xmath56 regions from our dataset , and extract our features from each of these regions . we then assign a corresponding ground truth iou value to each of them and train a rf regressor using @xmath57 trees . our rf learns the mapping @xmath58 $ ] where @xmath55 denotes the ground truth iou value between the @xmath3 and the ground truth salient object . to deal with the imbalance issue , we sample an equal number of examples corresponding to the iou values of @xmath59,[0.25,0.5],[0.5,0.75]$ ] , and @xmath60 $ ] . at testing time , we use mcg @xcite to generate @xmath53 regions of interest . we then apply our trained rf for every region @xmath3 and predict @xmath61 , which denotes the saliency of a region @xmath3 . we note that mcg produces the set of regions that overlap with each other . thus , for the pixels belonging to multiple overlapping regions @xmath62 , we assign a saliency value that corresponds to the maximum predicted value across the overlapping regions ( i.e. @xmath63 ) . we illustrate the basic pipeline of our approach in fig . [ fig : method ] . * future saliency prediction . * for future saliency prediction , given a video frame , we want to predict , which object will be salient ( i.e. used by a person ) after @xmath64 seconds . we hypothesize that the gaze direction is one of the most informative cues that are indicative of person s future behavior . however , gaze signal may be noisy if we consider only a single frame in the video . for instance , this may happen due to the person s attention being focused somewhere else for a split second or due to the shift in the camera . to make our approach more robust to the fluctuations of person s gaze , we incorporate simple temporal features into our system . our goal is to use these temporal cues to normalize the gaze direction captured by an egocentric camera and make it more robust to the small camera shifts . thus , given a frame @xmath65 which encodes time @xmath66 , we also consider frames @xmath67 . we pair up each of these frames @xmath68 with @xmath65 and compute their respective homography matrix @xmath69 . we then use each @xmath69 to recompute the image center @xmath70 in the current frame @xmath65 . for every region @xmath1 we then recompute its distance @xmath71 to the new center @xmath70 for all @xmath72}$ ] and concatenate these new distances to the original features @xmath52 . such gaze normalization scheme ensures robustness to our system in the case of gaze fluctuations . seconds , where the same object is salient . our goal here is to predict an object that will be salient after @xmath66 seconds . ] seconds , where the same object is salient . our goal here is to predict an object that will be salient after @xmath66 seconds . ] seconds , where the same object is salient . our goal here is to predict an object that will be salient after @xmath66 seconds . ] seconds , where the same object is salient . our goal here is to predict an object that will be salient after @xmath66 seconds . ] seconds , where the same object is salient . our goal here is to predict an object that will be salient after @xmath66 seconds . ] seconds , where the same object is salient . our goal here is to predict an object that will be salient after @xmath66 seconds . ] * interaction classification . * most of the current computer vision systems classify objects by specific object class templates ( cup , phone , etc ) . however , these templates are not very informative and can not be used effectively beyond the tasks of object detection . adding object s function , and the type of interaction related to that object would allow researchers to tackle a wider array of problems overlapping vision and psychology . to predict an interaction type at a given frame , for each frame we select top @xmath73 highest ranked regions @xmath74 according to their predicted saliency score . we then classify each of these regions either as sight or touch . finally , we take the majority label from these @xmath73 classification predictions , and use it to classify an entire frame as sight or touch . | c | c | c | c | c | c | c | c | c | c | c | c | c | c | c | c | c ? c & mf & ap & mf & ap & mf & ap & mf & ap & mf & ap & mf & ap & mf & ap & mf & ap & mf & ap + fts @xcite & 7.3 & 0.6 & 8.9 & 1.3 & 10.6 & 1.2 & 5.8 & 0.5 & 4.5 & 1.1 & 17.2 & 2.0 & 20.0 & 2.5 & 5.0 & 0.7 & 9.9 & 1.2 + mcg @xcite & 10.4 & 4.7 & 13.8 & 7.0 & 21.1 & 12.7 & 7.1 & 2.9 & 12.5 & 5.7 & 23.6 & 12.2 & 31.2 & 14.9 & 11.1 & 5.1 & 16.4 & 8.1 + gbmr @xcite & 8.0 & 3.0 & 15.6 & 6.8 & 14.7 & 7.0 & 6.8 & 3.0 & 4.3 & 1.3 & 32.7 & 18.3 & 46.0 & 30.2 & 12.9 & 5.7 & 17.6 & 9.4 + salobj @xcite & 7.2 & 2.7 & 19.9 & 7.4 & 21.3 & 10.0 & 15.4 & 5.1 & 5.8 & 2.2 & 24.1 & 9.2 & 49.3 & 28.3 & 9.0 & 3.4 & 19.0 & 8.5 + gbvs @xcite & 7.2 & 3.0 & 21.3 & 11.4 & 20.0 & 10.6 & 16.1 & 8.8 & 4.3 & 1.4 & 23.1 & 13.8 & 50.9 & * 50.2 & 11.6 & 5.7 & 19.3 & 13.1 + objectness @xcite & 11.5 & 5.6 & * 35.1 & * 24.3 & 39.2 & 29.4 & 11.7 & 6.9 & 4.7 & 1.9 & 27.1 & 17.1 & 47.4 & 42.2 & 13.0 & 6.4 & 23.7 & 16.7 + * ours ( rgb ) & 25.7 & 16.2 & 34.9 & 21.8 & 37.0 & 23.0 & 23.3 & 14.4 & * 28.9 & * 18.5 & 32.0 & 18.7 & * 56.0 & 39.6 & 30.3 & 21.8 & 33.5 & 21.7 + * ours ( rgb - d ) & * 36.9 & * 26.6 & 30.6 & 18.2 & * 55.3 & * 45.4 & * 26.8 & * 19.3 & 18.8 & 10.5 & * 37.9 & * 25.4 & 50.6 & 38.4 & * 40.2 & * 28.5 & * 37.1 & * 26.5 + * * * * * * * * * * * * * * * * * * * * we now present our egocentric rgbd saliency dataset . our dataset records people s interactions with their environment from a first - person view in a variety of settings such as shopping , cooking , dining , etc . we use egocentric - stereo cameras to capture the depth of a scene as well . we note that in the context of our problem , the depth information is particularly useful because it provides an accurate distance from an object to a person . since we hypothesize that a salient object from the 3d world maps to an egocentric rgbd frame with a predictable depth characteristic , we can use depth information as an informative cue for 3d saliency detection task . our dataset has annotations for three different tasks : saliency detection , future saliency prediction , and interaction classification . these annotations enables us to train our models in a supervised fashion and quantitatively evaluate our results . we now describe particular characteristics of our dataset in more detail . * data collection . * we use two stereo gopro hero 3 ( black edition ) cameras with @xmath75 baseline to capture our dataset . all videos are recorded at @xmath76 with @xmath77 . the stereo cameras are calibrated prior to the data collection and synchronized manually with a synchronization token at the beginning of each sequence . * depth computation . * we compute disparity between the stereo pair after stereo rectification . a cost space of stereo matching is generated for each scan line and match each pixel by exploiting dynamic programming in a coarse - to- fine manner . * sequences . * we record @xmath78 video sequences that capture people s interactions with object in a variety of different environments . these sequences include : cooking , supermarket , eating , hotel @xmath79 , hotel @xmath80 , dishwashing , foodmart , and kitchen sequences . * saliency annotations . * we use grabcut software @xcite to annotate salient regions in our dataset . we generate @xmath81 annotated frames for kitchen , cooking , and eating sequences , @xmath82 and @xmath83 annotated frames for supermarket and foodmart sequences respectively , @xmath84 and @xmath85 annotated frames for hotel @xmath79 and hotel @xmath80 sequences respectively , and @xmath86 annotated frames for dishwashing sequence ( for a total of @xmath87 frames with per - pixel salient object annotations ) . in fig . [ fig : data_po ] , we illustrate a few images from our dataset and the depth channels corresponding to these images . to illustrate ground truth labels , we overlay these images with saliency annotations . additionally , in fig . [ fig : data_stats ] , we provide statistics that capture different properties of our dataset such as the location , depth , and size of annotated salient regions from all sequences . each video sequence from our dataset is marked by a different color in this figure . we observe that these statistics suggest that different video sequences in our dataset exhibit different characteristics , and captures a variety of diverse interactions between people and objects . * annotations for future saliency prediction . * in addition , we also label our dataset to predict future saliency in egocentric rgbd image after @xmath64 frames . specifically , we first find the frame pairs that are @xmath64 frames apart , such that the same object is present in both of the frames . we then check that this object is non - salient in the earlier frame and that it is salient in the later frame . finally , we generate per - pixel annotations for these objects in both frames . we do this for the cases where the pair of frames are @xmath88 , and @xmath89 seconds apart . we produce @xmath90 annotated frames for kitchen , @xmath91 for cooking , @xmath92 for eating , @xmath93 for supermarket , @xmath90 for hotel @xmath79 , @xmath94 for hotel @xmath80 , @xmath95 for foodmart , and @xmath90 frames dishwashing sequences . we present some examples of these annotations in fig . [ fig : data_fut ] . * annotations for interaction classification . * to better understand the nature of people s interactions with their environment we also annotate each interaction either as _ sight _ or as _ touch_. in this section , we present the results on our egocentric rgbd saliency dataset for three different tasks , which include 3d saliency detection , future saliency prediction and interaction classification . we show that using our egoobject feature representation , we achieve solid quantitative and qualitative results for each of these tasks . to evaluate our results , we use the following procedure for all three tasks . we first train random forest ( rf ) using the training data from @xmath96 sequences . we then use this trained rf to test it on the sequence that was not used in the training data . such a setup ensures that our classifier is learning a meaningful pattern in the data , and thus , can generalize well on new data instances . we perform this procedure for each of the @xmath78 sequences separately and then use the resulting rf model to test on its corresponding sequence . for the saliency detection and future saliency prediction tasks , our method predicts pixelwise saliency for each frame in the sequence . to evaluate our results we use two different measures : a maximum f - score ( mf ) along the precision - recall curve , and average precision ( ap ) . for the task of interaction classification , we simply classify each interaction either as sight or as touch . thus , to evaluate our performance we use the fraction of correctly classified predictions . we now present the results for each of these tasks in more detail . detecting 3d saliency in an egocentric rgbd setting is a novel and relatively unexplored problem . thus , we compare our method with the most successful saliency detection systems for 2d images . sequences . these visualizations demonstrate that in each of these sequences , our method captures an egocentric object prior that has a distinct shape , location , and size pattern . ] these visualizations demonstrate that in each of these sequences , our method captures an egocentric object prior that has a distinct shape , location , and size pattern . ] these visualizations demonstrate that in each of these sequences , our method captures an egocentric object prior that has a distinct shape , location , and size pattern . ] these visualizations demonstrate that in each of these sequences , our method captures an egocentric object prior that has a distinct shape , location , and size pattern . ] these visualizations demonstrate that in each of these sequences , our method captures an egocentric object prior that has a distinct shape , location , and size pattern . ] these visualizations demonstrate that in each of these sequences , our method captures an egocentric object prior that has a distinct shape , location , and size pattern . ] in table [ po_table ] , we present quantitative results for the saliency detection task on our dataset . we observe that our approach outperforms all the other methods by @xmath97 and @xmath98 in mf and ap evaluation metrics respectively . these results indicate that saliency detection methods designed for _ non - egocentric _ images do not generalize well to the _ egocentric _ images . this can be explained by the fact that in most _ non - egocentric _ saliency detection datasets , images are displayed at a pretty standard scale , with little occlusions , and also close to the center of an image . however , in the egocentric environment , salient objects are often occluded , they appear at a small scale and around many other objects , which makes this task more challenging . furthermore , we note that none of these baseline methods use depth information . based on the results , in table [ po_table ] , we observe that adding depth features to our framework provides accuracy gains of @xmath99 and @xmath100 according to mf and ap metrics respectively . finally , we observe that the results of different methods vary quite a bit from sequence to sequence . this confirms that our egocentric rgbd saliency dataset captures various aspects of people s interactions with their environment , which makes it challenging to design a method that would perform equally well in each of these sequences . based on the results , we see that our method achieves best results in @xmath96 and @xmath89 sequences ( out of @xmath78 ) according to mf and ap evaluation metrics respectively , which suggests that exploiting egocentric object prior via shape , location , size , and depth features allows us to predict visual saliency robustly across all sequences . additionally , we present our qualitative results in fig . [ fig : po_preds ] . our saliency heatmaps in this figure suggest that we can accurately capture different types of salient interactions with objects . furthermore , to provide a more interesting visualization of our learned egocentric object priors , we average our predicted saliency heatmaps for each of the @xmath89 selected sequences and visualize them in fig . [ fig : avg_preds ] . we note that these averaged heatmaps have a certain shape , location , and size characteristics , which suggests the existence of an egocentric object prior in egocentric rgbd images . in fig . [ fig : feats ] , we also analyze , which features contribute the most for the saliency detection task . the feature importance is quantified by the mean squared error reduction when splitting the node by that feature in a random forest . in this case , we manually assign each of our @xmath51 features to one of @xmath78 groups . these groups include shape , location , size , depth , shape context , location context , size context and depth context features ( as shown in fig . [ fig : feats ] ) . for each group , we average the importance value of all the features belonging to that group and present it in figure [ fig : feats ] . based on this figure , we observe that shape features contribute the most for saliency detection . additionally , since location features capture an approximate gaze of a person , they are deemed informative as well . furthermore , we observe that size and depth features also provide informative cues for capturing the saliency in an egocentric rgbd image . as expected , the context feature are least important . in this section , we present our results for the task of future saliency prediction . we test our trained rf model under three scenarios : predicting a salient object that will be used after @xmath88 , and @xmath89 seconds respectively . as one would expect , predicting the event further away from the present frame is more challenging , which is reflected by the results in table [ fut_table ] . for this task , we aim to use our egoobject representation to learn the cues captured by egocentric - stereo cameras that are indicative of person s future behavior . we compare our future saliency detector ( fsd ) to the saliency detector ( sd ) from the previous section and show that we can achieve superior results , which implies the existence and consistency of the cues that are indicative of person s future behavior . such cues may include person s gaze direction ( captured by an egocentric camera ) , or person s distance to an object ( captured by the depth channel ) , which are both pretty indicative of what the person may do next . in fig . [ fig : fut_preds ] , we visualize some of our future saliency predictions . based on these results , we observe , that even in a difficult environment such as supermarket , our method can make meaningful predictions . .future saliency results according to max f - score ( mf ) and average precision ( ap ) evaluation metrics . given a frame at time @xmath66 , our future saliency detector ( fsd ) predicts saliency for times @xmath101 , and @xmath102 ( denoted by seconds ) . as our baseline we use a saliency detector ( sd ) from section [ tech_approach ] of this paper . we show that in every case we outperform this baseline according to both metrics . this suggests that using our representation , we can consistently learn some of the egocentric cues such as gaze , or person s distance to an object that are indicative of people s future behavior . [ cols="^,^,^,^,^",options="header " , ] in this section , we report our results on the task of interaction classification . in this case , we only have two labels ( sight and touch ) and so we evaluate the performance as a fraction of correctly classified predictions . we compare our approach with a depth - based baseline , for which we learn an optimal depth threshold for each sequence , then for a given input frame , if a predicted salient region is further than this threshold , our baseline classifies that interaction as _ sight _ , otherwise the baseline classifies it as _ touch_. due to lack of space , we do not present the full results . however , we note that our approach outperforms depth - based baseline in @xmath89 out of @xmath78 categories and achieves @xmath103 higher accuracy on average in comparison to this baseline . we also illustrate some of the qualitative results in fig . [ fig : po_preds ] . these results indicate that we can use our representation to successfully classify people s interactions with objects by sight or touch . in this paper , we introduced a new psychologically inspired approach to a novel 3d saliency detection problem in egocentric rgbd images . we demonstrated that using our psychologically inspired egoobject representation we can achieve good results for the three following tasks : 3d saliency detection , future saliency prediction , and interaction classification . these results suggest that an egocentric object prior exists and that using our representation , we can capture and exploit it for accurate 3d saliency detection on our egocentric rgbd saliency dataset .
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on a minute - to - minute basis people undergo numerous fluid interactions with objects that barely register on a conscious level .
recent neuroscientific research demonstrates that humans have a fixed size prior for salient objects .
this suggests that a salient object in 3d undergoes a consistent transformation such that people s visual system perceives it with an approximately fixed size .
this finding indicates that there exists a consistent egocentric object prior that can be characterized by shape , size , depth , and location in the first person view . in this paper
, we develop an egoobject representation , which encodes these characteristics by incorporating shape , location , size and depth features from an egocentric rgbd image .
we empirically show that this representation can accurately characterize the egocentric object prior by testing it on an egocentric rgbd dataset for three tasks : the 3d saliency detection , future saliency prediction , and interaction classification .
this representation is evaluated on our new egocentric rgbd saliency dataset that includes various activities such as cooking , dining , and shopping . by using our egoobject representation ,
we outperform previously proposed models for saliency detection ( relative @xmath0 improvement for 3d saliency detection task ) on our dataset .
additionally , we demonstrate that this representation allows us to predict future salient objects based on the gaze cue and classify people s interactions with objects .
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completion of the knowledge of the generalized nuclear force , which includes not only the nucleon - nucleon ( nn ) interaction but also hyperon - nucleon ( yn ) and hyperon - hyperon ( yy ) interactions , brought the deeper understanding of atomic nuclei , structure of neutron stars and supernova explosions . however it is hard to know the properties of the yn and yy interactions because their scattering data in free - space are scarce . recently a method to extract the @xmath8 potential through the nbs wave function from lattice qcd simulations has been proposed in @xcite . the obtained potential is found to have desirable features , such as attractive well at long and medium distances , and the central repulsive core at short distance @xcite . further applications have been done in refs . @xcite . in this work , we focus on the @xmath0 , @xmath1 b - b system to seek the @xmath9 interaction and to see the su(3)@xmath10 breaking effects of b - b interaction from lattice qcd simulation . the @xmath11 baryon - baryon state consists of the @xmath9 , @xmath12 and @xmath13 components in terms of low - lying baryons . mass differences of these components are quite small , and it causes the contamination of nbs wave function from excited states . in sucn situation the source operator should be optimized to extract the energy eigen states through the variational method @xcite . the equal - time nbs wave function @xmath14 for an energy eigen state with @xmath15 is extracted from the four point function , @xmath16 where @xmath17 is diagonalized wall - source operator . the transition potential matrix of 3-states coupled channel equation can be acquired in a particle basis or a su(3 ) irreducible representation ( ir ) basis . they are connected by unitary trandformation ( see in appendix b in ref . the non - diagonal part of potential matrix in ir basis is a good measure of the su(3 ) breaking effect . .hadron masses in unit of [ mev ] are listed . [ cols="^,^,^,^,^,^,^,^",options="header " , ] in this calculation we employ the 2 + 1-flavor full qcd gauge configurations of japan lattice data grid(jldg)/international lattice data grid(ildg ) . they are generated by the cp - pacs and jlqcd collaborations with a renormalization - group improved gauge action and a non - perturbatively @xmath18 improved clover quark action at @xmath19 , corresponding to lattice spacings of @xmath20 @xcite . we choose three ensembles of the @xmath21 lattice which means the spatial volume of about @xmath22 . quark propagators are calculated from the spatial wall source at @xmath23 with the dirichlet boundary condition in temporal direction at @xmath24 . the numerical computation is carried out at kek supercomputer system , blue gene / l . the hadron masses are shown in table [ tab : gconf ] . in figure [ fig : potall ] we compare the results of potential matrix in the ir basis calculated in different configuration sets . we found the growth of repulstive core in the @xmath25 potential with decreasing the light quark mass . the @xmath26 and @xmath27 transition potential are consistent with zero within error bar . on the other hand , it is noteworthy that the @xmath28 transition potential which is not allowed in the su(3 ) symmetric world is strengthen as the su(3)@xmath10 breaking gets larger . we have investigated the @xmath29 bb state , which is known as the @xmath9 , @xmath12 and @xmath13 coupled state , from lattice qcd . we have found a small transition potential between the singlet and octet state in terms of the su(3 ) ir basis . such transition can not be allowed in the su(3 ) symmetric world . this method could greatly assist us to complete the knowledge of not only the generalized nuclear force but also the interaction of hadrons including mesons , baryons and quarks . * acknowledgements * : this work was supported by the large scale simulation program no.0923(fy2009 ) of high energy accelerator research organization ( kek ) , grant - in - aid of the ministry of education , science and technology , sports and culture ( nos . 20340047 , 22540268 , 19540261 ) and the grant - in - aid for scientific research on innovative areas ( no . 2004:20105001 , 20105003 ) . k. murano , n. ishii , s. aoki and t. hatsuda , pos * lattice2009 * ( 2009 ) 126 . y. ikeda et al . , arxiv:1002.2309 [ hep - lat ] . t. inoue et al . [ hal qcd collaboration ] , arxiv:1007.3559 [ hep - lat ] . c. michael , nucl . b * 259 * ( 1985 ) 58 . m. luscher and u. wolff , nucl . b * 339 * ( 1990 ) 222 .
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we investigate baryon - baryon interactions with strangeness @xmath0 and isospin @xmath1 system from lattice qcd . in order to solve this system ,
we prepare three types of baryon - baryon operators ( @xmath2 , @xmath3 and @xmath4 ) for the sink and construct three source operators diagonalizing the @xmath5 correlation matrix . combining of the prepared sink operators with the diagonalized source operators , we obtain nine effective nambu - bethe - salpeter ( nbs ) wave functions .
the @xmath5 potential matrix is calculated by solving the coupled - channel schrdinger equation .
the flavor @xmath6 breaking effects of the potential matrix are also discussed by comparing with the results of the @xmath6 limit calculation .
our numerical results are obtained from three sets of @xmath7 flavor qcd gauge configurations provided by the cp - pacs / jlqcd collaborations .
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