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arnold s cat map is a linear area - preserving map @xmath0 on a torus in phase space formed by identifying the boundaries of the interval @xmath1 $ ] in both the coordinate @xmath2 and the momentum @xmath3 directions @xcite . ( because of this the area of the torus is characterized by planck s constant which takes on the values @xmath4 , where @xmath5 is the number of sites in both the coordinate and the momentum directions in the phase space . ) from time step @xmath6 to @xmath7 it is given by @xmath8 where @xmath9 guarantees area preservation . the degree of chaos depends on the choice of @xmath0 . the eigenvalues of @xmath0 are either both real or both imaginary . in the latter case , @xmath0 is elliptic , the motion becomes periodic and no sensitive dependence on the initial condition is observed . when @xmath0 is hyperbolic , the motion is chaotic . quantized cat map is studied in detail by hannay and berry @xcite . the matrix has to assume a special form in order to yield nontrivial values of the progagator for the map . we choose @xmath10 for the elliptic and the hyperbolic cases respectively . for the special choice of the matrix elements @xmath11 made above the propagator takes on the simple forms , @xmath12 , ~~ u_2(j+1 , j ) = \sqrt{\frac{i}{\cal n}}\exp [ \frac{i}{\hbar } ( q_{j}^{2}- q_{j}q_{j+1}+q_{j+1}^{2})].\ ] ] since each iteration describes a permutation among sites , each site belongs to a periodic orbit . thus the quantum dynamics follows the classical way , resulting in the recurrence of the wave function ( or equivalently , the wigner function @xcite ) . we now couple the system linearly to a bath of @xmath13 harmonic oscillators with coordinates @xmath14 and momentum @xmath15 ( @xmath16 ) described by the hamiltonian @xmath17 and the interaction hamiltonian @xmath18 @xmath19 where @xmath2 is the coordinate of the system and @xmath20 is the coupling constant of @xmath2 to the @xmath21th oscillator in the bath . by integrating out the bath variables , we get the reduced density matrix , @xmath22.\ ] ] where @xmath23 is the classical action of the system which appears as the exponent of the propagator in ( 1.3 ) . @xmath24 , and @xmath25 are the actions for bath and interaction , respectively . the propagator @xmath26 for the reduced density matrix from time steps @xmath6 to @xmath7 is @xmath27,\ ] ] in a path - integral representation @xcite , where @xmath28\ ] ] is the influence action . here @xmath29 , and @xmath30 are the dissipation and noise kernels respectively @xcite . if we consider the simplest case of an ohmic bath at high temperature @xmath31 @xcite , and consider times shorter than the relaxation time , then we obtain a gaussian form for the influence functional , with @xmath32 where the noise kernel becomes local @xmath33 and @xmath34 is the damping coefficient . the unit - time propagator becomes @xmath35 \rangle.\ ] ] here @xmath36 is a gaussian white noise given by @xmath37\ ] ] where @xmath38 denotes statistical average over noise realization @xmath36 . for the elliptic map , we get @xmath39.\ ] ] and for the hyperbolic map , @xmath40\ ] ] the wigner function is defined as @xmath41 where @xmath42 is the momentum conjugate to @xmath43 . the propagator @xmath44 for the wigner function is @xmath45 this is reduced to the form of the classical cat map . for the elliptic case , @xmath46 for the hyperbolic case , @xmath47 without noise , quantum evolution follows classical permutation @xcite the phase space is divided by a finite number of different periodic orbits and the period is known to increase roughly proportional to @xmath5 with some irregular oscillation . when coupled to a bath , the cat map is exposed to a gaussian noise in each time step . the discretized noise induces transitions between different periodic orbits in an irregular way . interaction with an environment blurs the recurrence of physical quantities in the quantum map . fig.1 shows @xmath48 , the linearized entropy ( with the reversed sign ) for various cases . if there is no interaction with the environment , the entropy is constant for both regular and chaotic cases . quantum recurrence is evident even when the system is chaotic . when interaction sets in , @xmath48 decays exponentially , showing that the system rapidly decoheres . the rate of decoherence is much faster in chaotic systems than in regular systems @xcite . it suggests that recurrence would be less evident in a decohering chaotic system . in fig . 2 , we show the mean displacement of points in the phase space as a function of time steps . this is defined by @xmath49 , where @xmath50 and @xmath51 are the displacements from the initial phase space points , and @xmath52 denotes averaging over noise distributions . in the chaotic case , we see that recurrence disappears with just a small amount of noise ( fig . 2a ) whereas in the regular case , the same amount of noise does not alter the qualitative picture of recurrence ( fig . 2b ) . in both cases , the decohered quantum system behaves close to the classical picture in which the regular and chaotic dynamics are clearly distinguished . in spite of the discreteness of the points on the torus , the system behaves effectively classically due to the influence of the environment . the kicked rotor is one of the most intensively studied models from both the quantum and classical point of view @xcite . the hamiltonian of the kicked rotor is given by @xmath53 which describes a one - dimensional rotor subjected to a delta - functional periodic kick at @xmath54 . here @xmath55 is the angle of the rotor with period @xmath56 , @xmath57 is the moment of inertia , @xmath42 is the angular momentum , and @xmath44 is the strength of the kick which measures the nonlinearity . when @xmath58 , the system becomes chaotic over the entire phase space . the quantum dynamics of the kicked rotor is given by the corresponding schrdinger equation @xmath59 where @xmath60 is the wave function of the rotor . denoting @xmath61 as the wave function @xmath62 at each discrete time @xmath63 , and integrating ( 2.2 ) from @xmath6 to @xmath7 , we obtain @xmath64 \exp [ -i \frac{k\cos x}{\hbar } ] \psi_{j}(x ) \ ] ] the quantum kicked rotor ( qkr ) is known to exhibit dynamical localization . after some relaxation time scale , the wave function becomes exponentially localized in the momentum space @xcite . this may be interpreted as a particle moving in a lattice with a quasi - random potential . this heuristic picture seems to justify the analogy between the quantum kicked rotor to the tight binding model with an exponentially decaying hopping parameter which is known to show anderson localization @xcite . dynamical localization in this context arises from the suppression of classical diffusive behavior by the quantum dynamics . however , as shown by ott , et.al . @xcite , a small external noise can break the localization . sufficient amount of noise would induce the quantum system to exhibit classical diffusive behavior . dittrich and graham studied this problem @xcite by coupling the system to a zero temperature harmonic oscillator bath and analysed solutions to the master equation . cohen and fishman presented the most detailed study of this problem for an ohmic bath @xcite . here we want to approach these issues from an environment - induced decoherence point of view @xcite . we begin by calculating the density matrix for the kicked rotor coupled to an environment . we introduce a linear coupling of the system momentum @xmath42 with each oscillator coordinate @xmath65 in the bath in the form @xmath66 ( here @xmath67 without the subscript @xmath21 denote the system coordinate and momentum variables ) . as before , we assume an ohmic bath and examine the time period where dissipation is small . under these assumptions , the unit time propagator for the wave function @xmath68 is given by @xmath69 \exp [ -\frac{i}{\hbar } k\cos x ] \exp [ -\frac{i}{\hbar } \xi p ] \ ] ] where , as before , the noise term @xmath70 arises from using a gaussian identity in the integral transform of the term involving the noise kernel in the influence functional . summing over all noise realizations @xmath71 gives the desired reduced density matrix , @xmath72 where @xmath73 loss of quantum coherence is measured by the density matrix becoming approximately diagonal . @xmath48 can be expressed as @xmath74 where @xmath75 denotes the statistical average of all possible noise histories of two independent noises @xmath76 defined at each time interval from @xmath6 to @xmath7 . at high temperatures @xmath77 are reduced to two time - uncorrelated independent gaussian white noises defined at each time step . we see that there is a close relation between the breaking of dynamical localization and quantum decoherence . in fig.3 we plot the linearized entropy @xmath48 versus the energy @xmath78 . this shows that delocalization occurs as quantum coherence breaks down , suggesting that delocalization and decoherence occurs by the same mechanism . as the nonlinearity parameter @xmath44 increases , the system decoheres more rapidly . at the same time , the amount of delocalization measured by the diffusion constant increases . this may be explained in the following way : because the coupling is through the momentum , the noise term does not involve any nonlinearity . the time scale for the system to lose coherence is given by @xmath79 , where @xmath80 is the thermal de broglie wavelength , and @xmath81 is the relevant momentum scale . after this time , noise will destroy the quantum coherence between such momentum separations . in the kicked rotor case , localization will occur due to the coherence around @xmath82 , where @xmath83 is the localization length . since @xmath84 , this gives @xmath85 . this shows that nonlinearity increases the rate of decoherence . the relation between the diffusion constant @xmath86 and the noise strength is given in @xcite . for our case , @xmath87 and for weak noises , we can consider the particle as undergoing a random walk with hopping parameter @xmath88 . then @xmath89 . the wigner function is often used to examine the quantum to classical transition . the wigner function at time @xmath54 is defined as @xmath90 where @xmath91 . from ( 2.3 ) , the unit - time propagator for the wigner function of the qkr is found to be @xmath92 where @xmath93 measures the effect of the kick . we can see the effects of quantum corrections is seen more clearly if we expand @xmath94 in orders of @xmath95 : @xmath96 the first exponential contains the classical propagator and the second contains quantum corrections of even orders of @xmath95 . thus we get @xmath97 where the wigner function with the new arguments depicts classical evolution . this map alone is the source of streching and folding of volume in phase space which signify classical chaos . if the initial system wavefunction is described by a gaussian wave packet with width @xmath98 , we would expect to see a classical - like evolution of the packet at short times . when the width of the contracting wave packet gets so small as comparable to @xmath95 , the effect of quantum corrections from higher @xmath95 order terms in ( 3.4 ) set in . by comparing the classical and quantum terms , we see that quantum corrections will become important when @xmath99 . here @xmath100 , where the lyapunov exponent @xmath101 . thus we can deduce the ehrenfest time when the wigner function or the expectation value of any observable follow classical trajectories . ] for qkr to be @xmath102 note that in the continuum case , this definition gives us a different time scale for each term in the expansion @xcite . the major effect of the bath ( at times short compared with the relaxation time ) is the appearance of a diffusion term in ( 3.4 ) , @xmath103 competition amongst the three terms with different physical origins is apparent : the first term in ( 3.5 ) is the quantum diffusion term , the second is the quantum correction term , and the third is purely classical evolution . as discussed by zurek and paz @xcite , if d is sufficiently large , the effect of quantum corrections becomes inconspicuous . in this case , the diffusion term traces out a small scale oscillating behavior before quantum corrections have a chance to change classical evolution . then one may expect the time evolution of the wigner function to be like that of classical evolution with noise . the role of quantum diffusion is to add some gaussian averaging so that the contracting direction in phase space will be suppressed while it does not affect the stretching direction . as long as the width of the wave packet is large such that the first term is negligible , the evolution should be liouvillian ( time reversible if we assume infinite measurement precision ) . furthermore , we expect that after the width of the packet along the contracting direction becomes comparable to the diffusion generated width ( in the gaussian wave packet ) , the dynamics will start showing irreversible behavior arising from coarse graining ( as distict from irrreversibility from instability ) . consequently , entropy should increase in this regime . in fig . 4a , we plot the von neumann entropy for the dynamics of ( 3.5 ) . we can see three qualitatively different regimes : i. the liouville regime : the entropy is constant and the dynamics is time reversible . the decohering regime : the entropy keeps increasing due to coarse graining . the finite size regime : due to the bounded nature of the phase space , the entropy shows saturation . our result from quantitative analysis seems to confirm the qualitative description of zurek and paz @xcite who used the inverted harmonic oscillator potential as a generic source of instability . since the phase space in their model is not bounded they do not see regime iii . similar features appear in the quantum cap map ( fig . 4b ) in this case , the full quantum dynamics can be calculated in a simple way . resemblance with the result of a classical rotor with noise is obvious . however , in this case , the stable entropy is smaller than the maximum value which may be explained as a finite ( phase space ) size effect . + * acknowledgement * we thank drs . shmul fishman and juan pablo paz for explaining their work and drs . ed ott and richard prange for general discussions . research is supported in part by the national science foundation under grant phy91 - 19726 . blh gratefully acknowledges support from the general research board of the graduate school of the university of maryland and the dyson visiting professor fund at the institute for advanced study , princeton . + * figure captions * + * figure 1 * the linearized entropy ( with reversed sign ) @xmath104 is plotted here as a function of time . if there is no environment , the entropy is constant for both hyperbolic and elliptic cases , indicating the purity of the state . for the hyperbolic map , even though classically this system is strongly chaotic , the corresponding quantum system does not show chaotic behavior . this situation changes drastically when the system interacts with a thermal bath : entropy keeps increasing due to coarse graining . note that in the hyperbolic case ( solid line ) the rate of entropy increase is greater than in the elliptic case ( dotted line ) . n = 50 is used here ( also in fig.2 ) . + * figure 2 * the mean phase space point displacement is shown . when there is no environment ( dotted line ) , the system shows recurrence in both hyperbolic ( a ) and elliptic ( b ) cases . in the presence of an environment , the hyperbolic map loses the recurrence behavior ( solid line ) under a gaussian noise with @xmath105 and maintains a near - constant value , indicating the ergodicity of the classical map . on the other hand , the ellptic map still shows recurrence with the same amount of noise , suggesting classical periodicity . + * figure 3 * @xmath104 ( solid line , left scale ) and @xmath106 ( dashed line , right scale ) are plotted against time for @xmath107 and @xmath108 . the upper solid line and the lower dashed line correspond to the case when there is noise , with @xmath109 . as the noise strength increases to @xmath110 , the decoherence time shortens , and @xmath104 decays rapidly ( the lower solid line ) . this accompanies the increase of diffusive behavior in @xmath106 ( upper dashed line ) . + * figure 4 * the von neumann entropy is plotted versus time for ( a ) the quantum kicked rotor with an environment . here , @xmath108 , @xmath105 and @xmath111 . entropy stays at zero ( reversible dynamics ) until a transition regime , after which the dynamics becomes irreversible . ( b ) the quantum cat map , with the same parameters and the same amount of noise . we see the same qualitative feature as in the qkr case . b. l. hu , j. p. paz and y. zhang , phys . d45 , 2843 ( 1992 ) ; b. l. hu , j. p. paz and y. zhang , phys . rev . d47 , 1576 ( 1993 ) ; b. l. hu and yuhong zhang , in _ quantum dynamics of chaotic systems _ eds . j. m. yuan , d. h. feng and g. m. zaslavsky ( gordon and breach science publishers , langhorne , 1993 ) . a. tameshitit and j. e. sipe , in _ quantum dynamics of chaotic systems _ eds j. m. yuan , d. h. feng , g. m. zaslavsky ( gordon and breach science publishers , langhorne , 1993 ) . a45 , 8280 ( 1992 ) ; a47 , 1697 ( 1993 ) f. g. casati , b. v. chirikov , f. m. izrailev , and j. ford , in _ stochastic behavior in classical and quantum hamiltonian systems _ , lecture notes in physics vol.93 , eds f. g. casati , j. ford ( springer - verlag , berlin , 1979 ) . s. fishman , d. r. grempel , and r. e. prange , phys . * 49 * , 509 ( 1982 ) ; phys . rev . * a36 * , 289 ( 1987 ) ; d. r. grempel , r. e. prange , and s. fishman , phys . lett . * 49 * , 833 ( 1982 ) ; phys . rev . * a29 * , 1639 ( 1984 ) . w. h. zurek , phys . d24 , 1516 ( 1981 ) ; d26 , 1862 ( 1982 ) ; e. joos and h. d. zeh , z. phys . b59 , 223 ( 1985 ) ; a. o. caldeira and a. j. leggett , phys . a31 , 1059 ( 1985 ) ; w. g. unruh and w. h. zurek , phys . d40 , 1071 ( 1989 ) ; w. h. zurek , prog . 89 , 281 ( 1993 ) ; w. h. zurek and j. p. paz , phys . lett . * 72 * , 2508 ( 1994 ) .

decoherence in quantum systems which are classically chaotic is studied . it is well - known that a classically chaotic system when quantized loses many prominent chaotic traits . we show that interaction of the quantum system with an environment can under general circumstances quickly diminish quantum coherence and reenact some characteristic classical chaotic behavior . we use the feynman - vernon influence functional formalism to study the effect of an ohmic environment at high temperature on two classically - chaotic systems : the linear arnold cat map ( qcm ) and the nonlinear quantum kicked rotor ( qkr ) . features of quantum chaos such as recurrence in qcm and diffusion suppression leading to localization in qkr are destroyed in a short time due to environment - induced decoherence . decoherence also undermines localization and induces an apparent transition from reversible to irreversible dynamics in quantum chaotic systems . body of paper c i u plus 1000pt minus 1000pt # 1 # 1= to # 1= to = 8.5 in = 6.5 in = 0.in = 0.in = 0.in addtoresetequationsection

the observation and theoretical modeling of weak polarization signatures in spectral lines are opening a new window on the investigation of the weak magnetism of the solar atmosphere ( see , e.g. , the recent reviews by @xcite ) . to this aim , it is important to investigate carefully within the framework of the quantum theory of polarization ( e.g. , landi deglinnocenti 1983 ) the observable effects of the atomic polarization of the energy levels involved in the line transitions of interest , including their subtle modification by the presence of magnetic fields . in this respect , in a recent letter ( @xcite , hereafter paper i ) , we reported on an interesting property of the polarizability of the levels of the d@xmath1 line of : in spite of the fact that those levels can both be aligned , atomic _ alignment _ is a condition of population imbalances between the zeeman substates of a level , such that the total populations of substates with different values of @xmath6 are different . one speaks instead of atomic _ orientation _ when , for a given value of @xmath6 , the substates labeled by @xmath7 and @xmath8 have different populations . see , e.g. , @xcite , or the recent review by @xcite . ] when proper account is taken of the additional quantum numbers introduced by the hyperfine structure ( hfs ) of , the alignment is drastically reduced for fields larger than @xmath2 , and practically vanishes for @xmath9 , _ irrespective _ of the relative directions of the magnetic field and of the incident radiation . accordingly , any contribution to the linear polarization in the core of d@xmath1 that arises from atomic alignment is suppressed for magnetic fields sensibly larger than @xmath2 , so the only expected linear - polarization signal for such field strengths must be due to the transverse zeeman effect ( see fig . 2 of paper i ; the reader should note how the stokes-@xmath10 signature of single - scattering events taking place in the presence of a vertical magnetic field changes from antisymmetric for @xmath11 to symmetric for @xmath12 ) . in paper i , we were concerned mainly with a detailed calculation of the polarizability of the levels , and with the consequences it bears for our understanding of the magnetic - field distribution and topology in the solar atmosphere . in the present work , we focus instead on the investigation of the atomic physics that is behind the polarization properties of those lines . to this end , we follow our approach of paper i , and apply the quantum theory of line formation in the limit of complete frequency redistribution ( crd ) and in the collisionless regime , as developed by @xcite , to investigate the statistical equilibrium ( se ) of an ensemble of atoms illuminated by anisotropic radiation ( see also @xcite ) . the hypothesis of crd corresponds to the requirement that the incident radiation field coming from the underlying photosphere , and illuminating the scattering atoms , be spectrally flat over an interval much larger than the energy separation between atomic levels whose wavefunctions sensibly overlap ( leading to the phenomenon of quantum interferences ) . in the case of the d@xmath1 and d@xmath13 lines forming in the solar atmosphere , this is a good assumption only if we neglect the quantum interferences between the upper levels of d@xmath1 and d@xmath13 . more specifically , these are interferences between the levels p@xmath0 and p@xmath5 pertaining to the same atomic term . whereas the role of these so - called _ super - interferences _ is important for a correct interpretation of line polarization in the wings of d@xmath1 and d@xmath13 , the line - core polarization of those lines , which was the subject of the investigation of paper i , is expected to be largely unaffected by them . in 2 , we summarize our qualitative description of the polarization properties of the levels of ( see paper i ) , and introduce some useful new concepts and terminology . in [ sec : polar.quant ] , we put those concepts on a more quantitative basis , and provide an algebraic proof that the alignment of the levels of d@xmath1 is suppressed when a magnetic regime of complete decoupling of the angular momenta @xmath3 and @xmath4 is reached in the excited state p@xmath5 . finally , in the conclusive section , we provide further arguments to illuminate this interesting phenomenon . the stable isotope of sodium has a nuclear spin @xmath14 , therefore we must take into account the role of hfs in the solution of the se problem of . hfs was already indicated by @xcite as the only possible mechanism allowing for the existence of atomic alignment in the levels of the d@xmath1 line . in fact , levels with total angular momentum @xmath15 can not be aligned , whereas both hyperfine levels @xmath16 and @xmath17 , into which a level @xmath15 splits in the coupling process with a nuclear spin @xmath14 , can be aligned . for this reason , it is convenient to introduce the concept of _ intrinsic polarizability _ ( ip ) , for those levels whose values of @xmath18 allow the presence of atomic alignment , and of _ extrinsic polarizability _ ( ep ) , for those levels that can carry atomic alignment only through the `` internal '' @xmath19 quantum numbers , because of the presence of hfs . ( what distinguishes the roles of @xmath18 and @xmath19 as quantum numbers , in this context , is the assumption we made at the beginning , that quantum interferences can exist only between different @xmath19 levels , but not between different @xmath18 levels . ) in this sense , we can speak of ep only in the cases of @xmath20 and @xmath15 . therefore , both levels of the d@xmath1 line of are ep , whereas the upper level of d@xmath13 is ip , because @xmath21 . this nomenclature has a direct link with the physics of the interaction processes of the atom with the incident radiation field . we speak of ip of an atomic level when this level has the possibility of absorbing the multipole order @xmath22 of the polarization tensor of the incident radiation field ( @xcite ; see also @xcite ) , expressed in the irreducible spherical tensor representation , @xmath23 ( @xmath24 ) . in particular , if we assume that the incident radiation field is unpolarized , and has cylindrical symmetry around the local solar vertical through the scattering centre , only the multipole orders @xmath25 ( intensity ) and @xmath22 ( anisotropy ) are present in the radiation - field tensor . in this case , it is found that an ep level can only absorb the multipole order @xmath25 , so there is no atomic polarization directly induced by the incident radiation field . any atomic alignment ( @xmath22 , in the irreducible spherical representation of the density matrix ) that such level can show when proper account is taken of its sub - structure associated with hfs can only come from the transfer of atomic alignment from other atomic levels that are instead ip . in the case of , for instance , if the two levels of d@xmath1 were isolated ( i.e. , not radiatively connected with other levels in the atom ) , no atomic alignment could be created , even accounting for the presence of hfs . because of the presence of the upper level p@xmath5 of d@xmath13 in the se problem of , instead , transfer of atomic polarization from such ip level to the lower level of d@xmath1 can occur , via the radiative de - excitation associated with the formation of the d@xmath13 line . once ep has been created in the level s@xmath0 , this can be transferred via absorption processes to the upper level of d@xmath1 as well . in our case , the two levels of d@xmath1 manifest their ep because of the alignment induced onto the corresponding hfs levels , with @xmath26 ( see fig . 1 of paper i ; also fig . [ fig : alignment ] introduced below ) . on the other hand , the transfer of atomic alignment from an ip level to an ep level can be inhibited under particular conditions . for the three - level model of the atom considered here , and for the prescribed radiation field , we determined that the atomic polarization in the two ep levels vanishes when the ip level p@xmath5 reaches the regime of the complete paschen - back effect , in which the zeeman splittings of the @xmath19 levels due to the local magnetic field become much larger than the hfs separations between those levels . in this regime , the hfs coupling of the electronic and nuclear angular momenta , @xmath3 and @xmath4 , of the atoms in the excited state p@xmath5 , is `` relaxed '' by the presence of the strong magnetic field , through the electronic zeeman effect .. ] [ to understand the meaning of such decoupling process , we must observe that , in the regime of complete paschen - back effect , and assuming the direction of @xmath27 as the quantization axis , @xmath28 becomes a conserved quantity ( rigorously , an element of the complete set of commuting observables of the atomic system ) , along with @xmath29 . because @xmath30 must be conserved as well , both @xmath31 and @xmath32 become good quantum numbers , so the eigenvectors of the atomic system take the form @xmath33 . ] the inhibition of the transfer of atomic alignment from an ip level to an ep level for increasing magnetic strengths is clearly illustrated by the results presented in paper i. in figure [ fig : alignment ] , we reproduce similar results . we calculated the atomic alignment of the levels of d@xmath1 and d@xmath13 for magnetic strengths between @xmath34 and @xmath35 . a vertical field ( i.e. , aligned along the symmetry axis of the radiation cone from the photosphere illuminating the scattering atom ) was chosen , in order to clarify that the obtained trend of the alignment against the magnetic field strength is _ not _ due to hanle - effect depolarization . as we see , atomic alignment in the levels of d@xmath1 is drastically reduced for fields larger than @xmath2 , and practically vanishes for fields of the order of @xmath36 or larger . in figure [ fig : orientation ] , we show analogous results for the atomic orientation ( @xmath37 , in the irreducible spherical representation of the density matrix ; see note [ note : atompol ] for a description of atomic orientation ) , for magnetic strengths between @xmath38 and @xmath39 . also in this case , the orientation of the levels of the d@xmath1 line practically vanishes for @xmath9 . ( we note that , for @xmath40 , the level p@xmath5 approaches the regime of complete paschen - back effect . in fact , for @xmath41 , the typical zeeman splitting is already 10 times larger than the typical hfs separation for that level . ) on the other hand , as suggested by the work of @xcite concerning optical - pumping processes in cadmium , a sufficient condition for the vanishing of atomic alignment in the ep level is that the hfs frequency separation of the ip level be negligible with respect to the einstein @xmath42-coefficient of the atomic transition . this condition is very general , as it holds regardless of the magnetic - field strength ( in particular , it is valid also for zero magnetic fields ) . in the case of , the hfs frequency separation of the ip level p@xmath5 is comparable with the einstein @xmath42-coefficient of the d@xmath13 line . for this reason , transfer of atomic alignment from the ip upper level to the ep lower level is possible when @xmath43 , since @xmath3 and @xmath4 are still significantly coupled in the ip level p@xmath5 ( cf . also paper i , end of 3 ) . these results suggest that the inhibition of the transfer of atomic alignment from an ip level to an ep level should be regarded as an aspect of the so - called _ principle of spectroscopic stability _ applied to the ip level : whenever the hyperfine structure of the ip level becomes negligible , whether because a magnetic field is present which is strong enough to reach the complete paschen - back regime for that level , or because the hfs separation of that level is much smaller than its radiative width , the transfer of alignment from the ip level to the ep level is inhibited , so the ep level behaves as if the atomic hfs were not present at all . the reason for this is hidden in the complexity of the se problem , and it is addressed in the following section . we consider an ip level with total electronic angular momentum @xmath18 . we assume that this level can only interact with ep levels in the atom . beyond this restriction , the atomic system can be arbitrary , so the following formalism applies also for atoms other than . if a nuclear spin is present , of angular momentum @xmath44 , the density matrix for the ip level in the irreducible spherical tensor representation is ( cf . @xcite ) @xmath45 we write explicitly @xmath46 where @xmath47 in the previous equation , @xmath48 are clebsh - gordan coefficients , which can be expressed in terms of @xmath49 symbols as @xmath50 substitution of eq . ( [ eq : ket ] ) into eq . ( [ eq : rhoff ] ) , using eq . ( [ eq : cg ] ) , gives @xmath51 we now make the assumption that the electronic spin and the nuclear spin are decoupled ( or very weakly coupled ) when the atom is in the ip level . as anticipated in the previous section , this can be the case if the hfs separation is much smaller than the natural width of that level , or , in the presence of a magnetic field , if the level is in the regime of complete paschen - back effect . in either case , the atomic density matrix for the ip level can be factorized as @xmath52 we introduce at this point the formalism of the irreducible spherical tensors for both @xmath53 and @xmath54 , @xmath55 substitution of eq . ( [ eq : rho.fact ] ) into eq . ( [ eq : rhoff.1 ] ) , using eqs . ( [ eq : rhoj ] ) and ( [ eq : rhoi ] ) , gives @xmath56 finally , this equation must be substituted into eq . ( [ eq : rhokq ] ) . we then obtain an expression which involves the contraction over all magnetic quantum numbers of a product of five @xmath49 symbols . this contraction can be evaluated using , e.g. , eq . ( 14 ) , p. 456 , of @xcite , yielding the expression @xmath57 as a particular case , if nuclear polarization is absent ( @xmath58 ) , eq . ( [ eq : rhokq.ji ] ) reduces to @xmath59 in this case , the ( electronic ) atomic polarization of the @xmath18 level translates _ directly _ ( i.e. , with the same @xmath60 and @xmath10 ) into the atomic polarization of the @xmath61 pair . as an application of the former development , we consider a two - level atom @xmath62 endowed with hfs . neglecting stimulated emission for simplicity , the se equations for the two levels read ( @xcite ) @xmath63 and @xmath64 to understand how atomic polarization is created in an ep level , assuming that the other level is ip , we must consider explicitly the expressions of the transfer rates for absorption and spontaneous emission processes , respectively , @xmath65 and @xmath66 the relaxation rate due to spontaneous emission , @xmath67 , is completely diagonal , so it can only relate each of the elements @xmath68 to itself . the relaxation rate due to absorption , @xmath69 , is a necessary ingredient of this demonstration . however , the only fact we will rely upon is the presence in that rate of the @xmath70 symbol @xmath71 the rate @xmath72 , in both eqs . ( [ eq : seu ] ) and ( [ eq : sel ] ) , describes magnetic and hfs depolarization . the importance of this rate is that it accounts for the conversion mechanism of atomic alignment ( @xmath22 ) into atomic orientation ( @xmath37 ) discussed by @xcite . this is related to the fact that , in the algebraic expression of the rate ( not given here ) , @xmath73 and @xmath74 ( cf . [ [ eq : seu ] ] ) or @xmath75 and @xmath76 ( cf . [ [ eq : sel ] ] ) can have different parity . if the radiation illuminating the atom is not circularly polarized ( which is the case of the present discussion ) , this conversion mechanism is the only process capable of creating orientation in the atomic system ( see , e.g. , @xcite ) . on the other hand , this mechanism is only effective when quantum interferences between different @xmath19 levels are important , which corresponds to a regime of magnetic fields such that level crossing between @xmath19 levels can occur . therefore , for magnetic fields such that the upper level approaches the regime of complete paschen - back effect ( @xmath40 ) , the conversion of atomic alignment into atomic orientation is drastically reduced ( see fig . [ fig : orientation ] ) . for this reason , the role of the rate @xmath72 is not of immediate concern for the following arguments . we first consider the case in which @xmath77 is the ip level . when this level is in a regime of very weak coupling between @xmath3 and @xmath4 ( whether because the hfs separation is much smaller than @xmath78 , or because a magnetic field is present that is strong enough to establish a regime of complete paschen - back effect in that level ) , the irreducible components of the density matrix for that level , @xmath79 , can be written according to eq . ( [ eq : rhokq.ji ] ) . it is then found that the double summation over @xmath80 and @xmath81 in eq . ( [ eq : sel ] ) can be performed algebraically . this corresponds to a contracted product of a @xmath82 symbol with three @xmath70 symbols , which is evaluated using , e.g. , eq . ( 36 ) , p. 471 , of @xcite . the result is that the overall contribution of the transfer rate @xmath83 to eq . ( [ eq : sel ] ) is proportional to the product ( notice that @xmath84 ) @xmath85 since @xmath86 for the ep level , the former product vanishes unless @xmath87 . in particular , to create alignment in the ep lower level ( @xmath88 ) , either both electronic and nuclear orientations ( @xmath89 ) or only nuclear alignment ( @xmath90 ) must be present when the atom is in the excited state @xmath77 . to convince ourselves that these conditions can not be met , let us assume that initially ( i.e. , before irradiation ) atomic polarization is completely absent , in particular @xmath91 . since the level @xmath92 is ep , it is only sensitive ( through the relaxation rate @xmath69 ; cf . the @xmath70 symbol [ [ eq:6j.ra ] ] ) to the intensity of the incident radiation field , so lower - level polarization ( @xmath93 ) can not be created directly by irradiation . therefore , when irradiation begins , from eqs . ( [ eq : seu ] ) and ( [ eq : ta ] ) we see that the prescribed radiation field ( @xmath94 ) can only induce atomic alignment in the upper level ( besides populating it ) , because of the selection rule introduced by the @xmath49 symbol in eq . ( [ eq : ta ] ) . since the atom was initially unpolarized , and by assumption the electronic and nuclear systems are decoupled in the excited state , @xmath77 , the atomic alignment of the upper level can only be electronic . in fact , electric - dipole transitions can not affect the nuclear system , so the nuclear zeeman sublevels remain naturally populated in all cases of interest , even if strong @xmath18-@xmath44 coupling is present in the ep level . from this argument , we conclude that @xmath95 , and @xmath96 , as a result of the excitation process . as anticipated above , we can dismiss the alignment - to - orientation conversion mechanism as a possible source of upper - level orientation ( @xmath97 ) , because of the assumed regime of weak @xmath18-@xmath44 coupling . also , upper - level alignment ( @xmath98 ) can not be transferred in the de - excitation process , because the product ( [ eq : contr1 ] ) vanishes . therefore , nuclear polarization can never be created in this regime , and eq . ( [ eq : rhokq.ji.0 ] ) applies to the upper level . under these conditions , the product ( [ eq : contr1 ] ) vanishes identically for @xmath93 , so lower - level polarization can not be created . this is in agreement with the results of paper i , and of figures 1 and 2 in this paper . in summary , when the ip upper level is in a regime of very weak @xmath18-@xmath44 coupling , the creation of atomic alignment in the ep lower level ( @xmath88 ) by transfer of atomic alignment from the ip upper level ( @xmath98 ) is inhibited . in the case of , this implies that the ground level @xmath99 can not be aligned , and consequently also the upper level p@xmath0 of d@xmath1 must have zero alignment , as illustrated in paper i and by figure [ fig : alignment ] in this paper . lower - level orientation ( @xmath100 ) can in principle be created directly by irradiation , if @xmath101 , although it requires that the incident radiation be circularly polarized ( @xmath102 ; cf . the @xmath70 symbol [ [ eq:6j.ra ] ] ) . in our case , because of the prescribed radiation field , lower - level orientation can only be created by the transfer of atomic orientation from the upper level ( @xmath97 ) , which is not inhibited in principle . on the other hand , the alignment - to - orientation conversion mechanism in the upper level becomes very inefficient for very weak @xmath18-@xmath44 coupling ( see fig . [ fig : orientation ] ) , so also upper - level orientation can only be created if the incident radiation field is circularly polarized . we checked our conclusion that eq . ( [ eq : rhokq.ji.0 ] ) must apply to the ip upper level , in the regime of very weak @xmath18-@xmath44 coupling , against the numerical results of paper i ( cf . also fig . [ fig : alignment ] in this paper ) . in particular , we verified that the ratio of the quantities @xmath103 and @xmath104 for the upper level @xmath105 of ( @xmath14 ) in the strong - field limit ( @xmath106 ; see rightmost panels of fig . 1 in paper i ; also fig . [ fig : alignment ] in this paper ) is correctly reproduced by eq . this equation also accounts for the curious vanishing of the quantity @xmath107 in the same limit , which is due to the fact that @xmath108 vanishes identically because of the ( non - trivial ) nullity of the @xmath70 symbol in eq . ( [ eq : rhokq.ji.0 ] ) . within the same approximation of the two - level atom @xmath62 , we now assume that the decoupling of @xmath3 and @xmath4 is reached first in the lower level , while strong coupling is still present in the upper level . this time we assume that @xmath92 is the ip level , whereas @xmath77 is the ep level . since we assumed that the lower level is in a regime of very weak coupling between @xmath3 and @xmath4 , the irreducible components of the density matrix for that level , @xmath109 , can be written according to eq . ( [ eq : rhokq.ji ] ) . it is then found that the double summation over @xmath110 and @xmath111 in eq . ( [ eq : seu ] ) can be performed algebraically . this corresponds to a contracted product of two @xmath82 symbols with two @xmath70 symbols that is evaluated using , e.g. , eq . ( 37 ) , p. 471 , of @xcite . the result is that the overall contribution of the transfer rate @xmath112 to eq . ( [ eq : seu ] ) is proportional to the sum @xmath113 since @xmath114 for the ep level , this sum is limited to @xmath115 . again , we assume that the atomic polarization is absent before irradiation . because the lower level is ip , lower - level alignment can be created when irradiation begins . however , since @xmath3 and @xmath4 are decoupled in the lower level , nuclear polarization remains zero ( @xmath96 ) , so all the alignment of the lower level must be electronic ( @xmath116 ) . under these conditions , the sum ( [ eq : contr2 ] ) is restricted to @xmath117 only , because the first @xmath82 symbol in the sum ( [ eq : contr2 ] ) vanishes for @xmath118 unless @xmath119 is an odd integer . therefore atomic polarization in the upper level ( @xmath120 ) can never be created , because of nullity of the second @xmath82 symbol in the sum ( [ eq : contr2 ] ) . this shows , in particular , that the concept of ep is also valid for an upper level . in this case , the ep upper level is sensitive to the anisotropy of radiation ( @xmath121 ) through the transfer rate @xmath112 , but nonetheless creation of alignment in the upper level through the absorption of that anisotropy is not possible when the ip lower level is in a regime of very weak @xmath18-@xmath44 coupling , because of the selection rules implied by the sum ( [ eq : contr2 ] ) . upper - level orientation ( @xmath122 ) is not excluded in principle , if @xmath123 , although it can only be created by transfer of atomic orientation from the lower level ( @xmath124 ; see eq . [ [ eq : contr2 ] ] ) . however , when the lower level is in the regime of weak @xmath18-@xmath44 coupling , its orientation can only be due to the presence of circular polarization in the incident radiation field . in this paper we demonstrated analytically that the presence of @xmath18-@xmath44 coupling in the ip level p@xmath5 of is a necessary condition for the transfer of atomic alignment from that level to the ep ground level s@xmath0 . we based our demonstration on the quantum theory of line formation , as developed by @xcite , and assuming unpolarized incident radiation without spectral structure over the frequency intervals encompassing the hfs components of the atomic transitions of interest . under these conditions , we relied on the argument that nuclear polarization can not be created in an atom having only one ip level , if @xmath3 and @xmath4 are completely decoupled in that level , because the assumed incident radiation can not induce directly nuclear transitions in the atom . it follows , from the results of 3 , that atomic polarization can not be created in the ep levels , when @xmath3 and @xmath4 are completely decoupled in the ip level . we can further strengthen this argument by showing that the possibility of nuclear polarization actually resides in the presence of @xmath18-@xmath44 coupling in the ip level , whereas the presence of @xmath18-@xmath44 coupling in the ep level is not relevant . to this purpose , we repeated the calculation of figure [ fig : alignment ] after artificially zeroing the hfs separation in the level s@xmath0 of . the results of this calculation are shown in figure [ fig : small.hfs ] . since @xmath3 and @xmath4 are completely decoupled in the `` modified '' level s@xmath0 , the factorization ( [ eq : rho.fact ] ) always applies to this level . on the other hand , any atomic alignment in this modified ep level requires the presence of nuclear polarization ( cf . [ [ eq : rhokq.ji ] ] ) , since the electronic angular momentum of the level is @xmath15 . such nuclear polarization in the ep level s@xmath0 can only come from the atomic polarization of the ip level p@xmath5 ( which is transferred to the ep level via radiative de - excitation ) , since it is not possible for the prescribed radiation field to directly create atomic polarization in the ep level . from the results of figure [ fig : small.hfs ] , it is evident that the nuclear polarization in the modified level s@xmath0 vanishes when the regime of complete paschen - back effect is reached in the level p@xmath5 , and eq . ( [ eq : rho.fact ] ) also applies to that level . comparing the results of figures [ fig : alignment ] and [ fig : small.hfs ] , we see that the suppression of @xmath18-@xmath44 coupling in the level s@xmath0 does not alter substantially the se of the model atom . on the basis of these arguments , it seems safe to conclude that , even in the real case , nuclear polarization can not be created in the atom , when the regime of complete paschen - back effect is reached in the ( only ) ip level . finally , we must emphasize that the presence of atomic alignment in the upper level of the d@xmath1 line induces a characteristic _ antisymmetric _ signature in the core of the stokes-@xmath10 profile resulting from the scattering of the anisotropic radiation illuminating the atom ( see fig . 2 of paper i ) . this applies particularly to the optically thin `` prominence case '' considered in paper i , where the scattering polarization is solely due to the emission events following atomic excitation by the anisotropic radiation . currently we are investigating to what extent such antisymmetric signature can be modified through dichroism and radiative transfer effects , because of the presence of atomic alignment in the ground level of ( see , e.g. , @xcite ; for the observable effects of dichroism and ground - level polarization on the 10830 multiplet , see @xcite ) . in this respect , it is interesting to note that spectropolarimetric observations of the d - lines obtained with thmis in quiet regions close to the solar limb show an antisymmetric signature in the fractional linear polarization @xmath125 of the d@xmath1 line ( see fig . 1 of @xcite , which was adapted from @xcite ; see also @xcite ) . there seems to be an indication of a similar antisymmetric signature in the @xmath125 atlas of @xcite , which was obtained with the polarimeter zimpol - ii attached to the gregory coud telescope ( gct ) of irsol at locarno ( italy ) . on the contrary , analogous observations that @xcite had obtained previously with the polarimeter zimpol - i attached to the mcmath - pierce facility of the national solar observatory show almost symmetric profiles with a central positive peak ( see their fig . 3 ) . observations of quiet solar regions were obtained in march 1998 ( i.e. , two years earlier than the above mentioned thmis observations ) , when the sun had not yet reached the maximum of its magnetic activity cycle . ] as shown in paper i , for single - scattering events , one should expect a _ shape of the stokes-@xmath10 signature in the core of the d@xmath1 line for magnetic fields @xmath12 ( see fig . 2 of paper i ; note that such symmetric signature would change its sign if we considered , say , a horizontal canopy - like field instead of the vertical field assumed for the calculation of that figure ) . nevertheless , we think that the above mentioned linear - polarization observations of the d@xmath1 line in very quiet regions of the solar disk , with thmis and zimpol , both have the same physical origin , i.e. , atomic alignment in the levels of the d@xmath1 line . now that we understand how the ground level of becomes polarized , and how its polarization is modified by the presence of weak magnetic fields , it will be worthwhile to investigate the sodium polarization problem by means of full radiative transfer simulations , taking also into account the quantum interferences among the two upper levels of the `` enigmatic '' d - lines . the authors are grateful to philip judge and arturo lpez ariste ( both of hao ) , and to rafael manso sainz ( universit di firenze , italy ) , for reading the manuscript , and for helpful comments and suggestions . they also thank maurizio landi deglinnocenti ( italian council for national research ) for helpful discussions about the principle of spectroscopic stability during the early stages of this work . thanks are also due to jan stenflo and co - workers for some useful discussions and clarifications concerning their spectropolarimetric observations . one of the authors ( j.t.b . ) acknowledges the support of the spanish ministerio de ciencia y tecnologa through project aya2001 - 1649 . gandorfer , j. 2000 , _ the second solar spectrum : a high spectral resolution polarimetric survey of scattering polarization at the solar limb in graphical representation _ , vdf hochschulverlag ag an der eth zrich

in a recent letter ( @xcite ) , we showed the remarkable result that the atomic alignment of the levels p@xmath0 and s@xmath0 of the d@xmath1 line of is practically destroyed in the presence of magnetic fields sensibly larger than @xmath2 , irrespectively of the field direction . in this paper , we demonstrate analytically that this property is a consequence of the decoupling of the electronic and nuclear angular momenta , @xmath3 and @xmath4 , in the excited state p@xmath5 , which is achieved when the zeeman splitting from the local magnetic field becomes much larger than the typical hyperfine separation for that level .

the question of the origin of @xmath0 violation remains one of the outstanding puzzles in particle physics . although @xmath0 violation has now been seen in a number of processes in the kaon and @xmath6-meson systems @xcite , it is still far from clear whether its explanation lies exclusively within the picture provided by the standard model @xcite . to pin down the sources of @xmath0 violation , it is essential to observe it in many other processes . hyperon nonleptonic decays provide an environment where it is possible to make additional observations of @xmath0 violation @xcite . currently , there are @xmath0-violation searches in such processes being conducted by the hypercp ( e871 ) collaboration at fermilab . its main reactions of interest are the decay chain @xmath7 @xmath8 and its antiparticle counterpart @xcite . a different , but related , system also being studied by hypercp involves the spin-@xmath9 hyperon @xmath10 , namely the sequence @xmath11 @xmath8 and its antiparticle process @xcite . for each of these decays , the decay distribution in the rest frame of the parent hyperon with known polarization @xmath12 has the form @xmath13 where @xmath14 is the final - state solid angle , @xmath15 is the unit vector of the daughter - baryon momentum , and @xmath16 is the parameter relevant to the @xmath0 violation of interest . in the case of @xmath17 the hypercp experiment is sensitive to the _ sum _ of @xmath0 violation in the @xmath18 decay and @xmath0 violation in the @xmath19 decay , measuring @xcite @xmath20 where @xmath21 are the @xmath0-violating asymmetries in @xmath22 and @xmath23 respectively . similarly , the observable it measures in @xmath24 is @xmath25 @xcite . on the theoretical side , @xmath0 violation in @xmath26 and @xmath27 has been extensively studied @xcite . in contrast , the literature on @xmath0 violation in @xmath18 decays is minimal , perhaps the only study being ref . @xcite which deals with the partial - rate asymmetry in @xmath28 there is presently no data available or experiment being done on this rate asymmetry . in view of the upcoming measurement of @xmath29 by hypercp , it is important to have theoretical expectations of this observable . clearly , the information to be gained from @xmath29 will complement that from @xmath30 . since the estimates of @xmath2 and @xmath31 within and beyond the standard model ( sm ) have been updated very recently in refs . @xcite , in this paper we focus on @xmath1 . we begin in sec . [ observables ] by relating the observables of interest in @xmath22 to the strong and @xmath0-violating weak phases in the decay amplitudes . we discuss the role played by final - state interactions in this decay , which not only affect @xmath1 , but also cause its partial - rate asymmetry to be nonvanishing , thereby providing another @xmath0-violating observable . in sec . [ strong_phases ] , we employ heavy - baryon chiral perturbation theory ( @xmath32pt ) to calculate @xmath33- and @xmath34-wave amplitudes for baryon - meson scattering in channels with isospin @xmath35 and strangeness @xmath36 we use the derived amplitudes in a coupled - channel @xmath37-matrix formalism to determine the strong parameters needed in evaluating the @xmath0-violating asymmetries . in sec . [ a_sm ] , we estimate the asymmetries within the standard model . working in the framework of @xmath32pt , we calculate the weak phases by considering factorizable and nonfactorizable contributions to the matrix elements of the leading penguin operator . subsequently , we compare the resulting @xmath1 with @xmath2 , which was previously evaluated , as both asymmetries appear in @xmath29 . in sec . [ a_np ] , we address contributions to the @xmath0-violating asymmetries from possible new physics , taking into account constraints from @xmath0 violation in the kaon system . specifically , we consider contributions induced by chromomagnetic - penguin operators , which in certain models can be enhanced compared to the sm effects . sec . [ conclusion ] contains our conclusions . the amplitudes for @xmath38 and @xmath39 each contain parity - conserving @xmath33-wave and parity - violating @xmath34-wave components , with the former being empirically known to be dominant @xcite . they are related to the parameters @xmath40 and @xmath41 by @xmath42 where @xmath43 and @xmath44 @xmath45 and @xmath46 are the @xmath33- and @xmath34-wave components , respectively , for the @xmath10 @xmath47 decay . since both @xmath18 and @xmath19 have @xmath48 each of these decays is an exclusively @xmath49 transition . before writing down the amplitudes in terms their phases , we note that the strong phases in @xmath50 are not generated by the strong rescattering of @xmath51 alone . watson s theorem for elastic unitarity @xcite does not apply here , though it does in the cases of @xmath26 and @xmath52 final - state interactions also allow @xmath53 to contribute , yielding additional strong phases as well as weak ones , because the channel @xmath54 is open at the scattering energy @xmath55 since the @xmath56 rates overwhelmingly dominate the @xmath18 width @xcite , we expect other contributions via final - state rescattering to be negligible . the requirements of @xmath57 invariance and unitarity provide us with a relationship between the amplitudes for @xmath58 and its antiparticle counterpart . thus , with @xmath59 denoting the amplitude corresponding to @xmath60 being in a state with orbital angular momentum @xmath61 , we have @xmath62 where @xmath63 is the element of the strong @xmath64-matrix associated with the @xmath61 partial - wave of @xmath65 and only the @xmath35 component of the @xmath66 state is involved in the second term . assuming that the @xmath66 and @xmath51 channels are the only ones open , we can express the @xmath64-matrix as @xcite @xmath67 where @xmath68 is the inelasticity factor and @xmath69 denotes the phase shift in @xmath70 clearly @xmath71 is unitary , and each partial - wave has its own @xmath71 . now , since @xmath68 is expected to be close to and smaller than 1 , it is convenient to introduce a parameter @xmath72 defined by @xmath73 and so @xmath72 is positive and small . consequently , for @xmath74 and @xmath75 , to first order in @xmath76 we have @xcite @xmath77 where @xmath78 and @xmath79 are real , associated with @xmath80 and @xmath81 denote the corresponding weak phases in the @xmath49 amplitudes . putting together the results above , and keeping only the terms at lowest order in small quantities , we obtain @xmath82 where we have made use of the expectation that @xmath83 , @xmath84 , and @xmath85 are also small . unlike the strong phases in @xmath19 and @xmath86 decays , there are no data currently available for @xmath87 , and so we will calculate them here . to estimate the weak phases @xmath88 , we will consider contributions coming from the sm as well as from possible new physics . as for @xmath78 and @xmath79 , we will extract their approximate values from data shortly , under the assumption of no final - state interactions and no @xmath0 violation . now , the presence of the @xmath76 terms with additional weak and strong phases in the decay amplitudes in eq . ( [ pd ] ) implies that the rate of @xmath89 @xmath90 evaluated in the rest frame of @xmath18 , is no longer identical to that of @xmath91 hence these decays yield another @xmath0-violating observable , namely the partial - rate asymmetry @xmath92 it follows that to leading order @xmath93 we will also estimate this asymmetry below . was evaluated under the assumption that @xmath94 since @xmath95 results from the interference of @xmath33-wave amplitudes , a future measurement of it will probe @xmath0 violation in the underlying parity - conserving interactions . we note that the strong parameters entering eq . ( [ deltao ] ) , and the second and third terms in eq . ( [ ao ] ) , are not the strong phases , but @xmath96 . before ending this section , we determine the values of @xmath78 and @xmath79 which are needed in eqs . ( [ ao ] ) and ( [ deltao ] ) , and also in evaluating the weak phases . to do so , we apply the measured values of @xmath16 and @xmath97 , as well as of the masses involved , in the corresponding formulas , as those in eqs . ( [ alpha ] ) and ( [ width ] ) , assuming that the strong and weak phases are zero . the experimental values of @xmath97 for @xmath98 are well determined , but those of @xmath16 are not @xcite . hypercp is currently also measuring @xmath40 , in @xmath89 with much better precision , and has reported @xcite preliminary results of @xmath99 and @xmath100 applying the pdg averaging procedure @xcite to all the experimental results , including the preliminary ones from hypercp , yields the average @xmath101 which we adopt in the following . in the case of @xmath102 we use the data given by the pdg @xcite , and also @xmath103 to project out the @xmath49 amplitudes . thus we extract @xmath104 all in units of @xmath105 , with @xmath106 being the fermi coupling constant . to calculate the strong parameters needed in eq . ( [ ao ] ) , we take a @xmath37-matrix approach @xcite . furthermore , we include the contributions of other @xmath60 states with @xmath35 and @xmath107 namely @xmath108 and @xmath109 , which are coupled to @xmath51 and @xmath66 through unitarity constraints . although at @xmath110 the @xmath108 and @xmath109 channels are below their thresholds , it is important to incorporate their contributions to the open ones . such kinematically closed channels have been shown to have sizable influence on the open ones in some other cases @xcite . the @xmath37 matrix for the four coupled channels can be written as @xmath111 where the subscripts `` o '' and `` c '' refer to open and closed channels , respectively , at @xmath112 thus @xmath113 are all 2@xmath1142 matrices in this case and @xmath115 now , it is convenient to introduce the matrix @xmath116 where @xmath117 is the 2@xmath1142 unit matrix and @xmath118 with @xmath119 being the magnitude of the cm three - momentum in @xmath60 scattering , implying that @xmath120 and @xmath121 are purely imaginary at @xmath112 the elements of @xmath71 in eq . ( [ s ] ) can then be evaluated using @xcite @xmath122 where @xmath123 for the @xmath37-matrix elements , we make the simplest approximation by adopting the partial - wave amplitudes @xmath124 at leading order in chiral perturbation theory , namely @xmath125 before deriving them , we remark that time - reversal invariance of the strong interaction implies @xmath126 the chiral lagrangian that describes the interactions of the lowest - lying mesons and baryons is written down in terms of the lightest meson - octet , baryon - octet , and baryon - decuplet fields @xcite . the meson and baryon octets are collected into @xmath127 matrices @xmath128 and @xmath6 , respectively , and the decuplet fields are represented by the rarita - schwinger tensor @xmath129 , which is completely symmetric in its su(3 ) indices ( @xmath130 ) . the octet mesons enter through the exponential @xmath131 where @xmath132 is the pion - decay constant . in the heavy - baryon formalism @xcite , the baryons in the chiral lagrangian are described by velocity - dependent fields , @xmath133 and @xmath134 . for the strong interactions , the lagrangian at lowest order in the derivative and @xmath135 expansions is given by @xcite @xmath136 \right\rangle \nonumber \\ & & \!\ ! -\,\ , \bar{t}_v^\mu\ , { \rm i}v\cdot{\cal d } t_{v\mu}^ { } + \delta m\ , \bar{t}_v^\mu t_{v\mu}^ { } + { \cal c } \left ( \bar{t}_v^\mu { \cal a}_\mu^ { } b_v^ { } + \bar{b}_v^ { } { \cal a}_\mu^ { } t_v^\mu \right ) \nonumber \\ & & \!\ ! + \,\ , \frac{b_d^{}}{2 b_0^ { } } \left\langle \bar b_v^ { } \left\ { \chi_+^ { } , b_v^ { } \right\ } \right\rangle + \frac{b_f^{}}{2 b_0^ { } } \left\langle \bar b_v^ { } \left [ \chi_+^ { } , b_v^ { } \right ] \right\rangle + \frac{b_0^{}}{2 b_0^ { } } \left\langle \chi_+^ { } \right\rangle \left\langle \bar b_v^ { } b_v^ { } \right\rangle \nonumber \\ & & \!\ ! + \,\ , \frac{c}{2 b_0^{}}\ , \bar t_v^\mu \chi_+^ { } t_{v\mu}^ { } - \frac{c_0^{}}{2 b_0^ { } } \left\langle \chi_+^ { } \right\rangle \bar t_v^\mu t_{v\mu}^ { } \,\,+\,\ , \mbox{$\frac{1}{4}$ } f^2 \left\langle \chi_+^ { } \right\rangle \,\,+\,\ , \cdots \,\,,\end{aligned}\ ] ] where @xmath137 denotes @xmath138 in flavor - su(3 ) space , and we have shown only the relevant terms . in the first two lines , @xmath139 is the spin operator and @xmath140 with further details given in ref . the last two lines of @xmath141 contain @xmath142 with @xmath143 which explicitly breaks chiral symmetry . we will take the isospin limit @xmath144 and consequently @xmath145 the constants @xmath34 , @xmath146 , @xmath147 , @xmath148 , @xmath149 , @xmath150 , @xmath151 are free parameters which can be fixed from data . in the center - of - mass ( cm ) frame , the @xmath33-wave amplitude for @xmath152 with total angular - momentum @xmath153 has the form @xmath154 \hat{k}{}'\cdot\hat{k } + \left [ f_{b\phi\to b'\phi'}^{(p , j=\frac{1}{2 } ) } - f_{b\phi\to b'\phi'}^{(p , j=\frac{3}{2 } ) } \right ] { \rm i}\bm{\sigma}\cdot\hat{k}{}'\times\hat{k } \right\ } \chi_{b}^ { } \,\ , , \nonumber \\\end{aligned}\ ] ] where @xmath155 is the cm energy , @xmath156 and @xmath157 are the pauli spinors of the baryons , @xmath158 and @xmath159 denote the unit vectors of the momenta of @xmath6 and @xmath160 , respectively , and @xmath161 are the partial - wave amplitudes . at lowest order in @xmath32pt , the @xmath162 amplitude arises from the lagrangian in eq . ( [ ls ] ) , and the pertinent diagrams are displayed in fig . [ pwave ] . the amplitudes in the @xmath35 channels are then extracted using the @xmath35 states in eq . ( [ |xp > ] ) and @xmath163 which follow a phase convention consistent with the structure of the @xmath128 and @xmath133 matrices . we write the results as @xmath164 where the expressions for @xmath165 corresponding to the four channels have been collected in appendix [ pd ] . diagrams contributing to the @xmath33-wave @xmath162 amplitude for @xmath152 at leading order in @xmath32pt . in all figures , a dashed line denotes a meson field , a single ( double ) solid - line denotes an octet - baryon ( decuplet - baryon ) field , and each solid vertex is generated by @xmath141 in eq . ( [ ls ] ) . ] since a @xmath34-wave amplitude has to be at least of second order in momentum , @xmath166 , it can not arise from the lagrangian in eq . ( [ ls ] ) alone . also required is the lagrangian involving baryons at second order in the derivative expansion , namely @xmath167 b_v^ { } \,+\ , \frac{1}{2 m_0^{}}\ , \bar{t}{}_v^\mu\ , \bigl[{\cal d}^2-(v\cdot{\cal d})^2 \bigr ] t_{v\mu}^ { } \,\,+\,\ , \cdots \,\,,\end{aligned}\ ] ] where @xmath168 is the octet - baryon mass in the chiral limit , and we have shown only the relevant terms . these are two of the relativistic - correction terms in the @xmath166 lagrangian , and so their coefficients are fixed . in the cm frame , the @xmath34-wave amplitude for @xmath152 has the form @xmath169 \bigl [ \mbox{$\frac{3}{2}$ } \bigl ( \hat{k}{}'\cdot \hat{k}\bigr)^2 - \mbox{$\frac{1}{2}$ } \bigr ] \right . \nonumber\\ & & \hspace*{6em } + \left . \left [ f_{b\phi\to b'\phi'}^{(d , j=\frac{3}{2 } ) } - f_{b\phi\to b'\phi'}^{(d , j=\frac{5}{2 } ) } \right ] \bigl ( 3\hat{k}{}'\cdot\hat{k } \bigr)\ , { \rm i}\bm{\sigma}\cdot\hat{k}{}'\times\hat{k } \right\ } \chi_b^ { } \,\,.\end{aligned}\ ] ] the leading nonzero contribution to this amplitude for @xmath162 comes from diagrams shown in fig . [ dwave ] . the resulting @xmath35 partial - wave amplitudes are given by @xmath170 where the expressions for @xmath171 corresponding to the four channels have also been collected in appendix [ pd ] . diagrams for the leading nonzero contribution to the @xmath34-wave @xmath162 amplitude for @xmath172 each hollow vertex is generated by @xmath173 in eq . ( [ ls ] ) . ] numerically , we adopt the tree - level values @xmath174 and @xmath175 extracted from hyperon semileptonic decays @xcite , as well as @xmath176 from the strong decays @xmath177 after nonrelativistic quark models @xcite , which predict @xmath178 and @xmath179 both well satisfied by the adopted @xmath34 , @xmath146 , and @xmath147 values . ] we also employ @xmath180 @xmath181 value comes from simultaneously fitting the tree - level formulas for the octet - baryon masses and the sigma term , @xmath182 all derived from eq . ( [ ls ] ) , to the measured masses and the empirical value @xcite @xmath183 and the isospin - averaged masses @xmath184 all in units of mev . thus , putting together all the results above and setting @xmath185 from the @xmath33- and @xmath34-wave @xmath71-matrices we obtain @xmath186 which are pertinent to eqs . ( [ ao ] ) and ( [ deltao ] ) . the effects of the closed channels turn out to be significant on @xmath187 and @xmath188 . excluding the @xmath108 and @xmath109 channels would lead to @xmath189 and @xmath190 the closed channels have minor effects on the @xmath34-wave parameters . since the numbers in eq . ( [ deltapd ] ) proceed from the leading nonzero amplitudes in @xmath32pt , part of the uncertainties in these predictions comes from our lack of knowledge about the higher - order contributions , which are presently incalculable . to get an idea of how they might affect our results , we redo the calculation using the one - loop values @xmath191 @xmath192 and @xmath193 @xcite , finding @xmath194 @xmath195 @xmath196 and @xmath197 the differences between the two sets of results then provide an indication of the size of this part of the uncertainties . another part is due to our lack of knowledge about the reliability of our @xmath37-matrix approximation . a comparison of @xmath37-matrix results in @xmath198 scattering with experiment suggests that this approach gives results with the correct order - of - magnitude and sign @xcite . for these reasons , we may conclude that @xmath199 we will employ these numbers in evaluating the asymmetries . to calculate the @xmath0-violating phases , we will work in the framework of heavy - baryon @xmath32pt . the amplitude for the weak decay @xmath58 in the heavy - baryon approach has the general form @xmath201 where @xmath202 is the four - momentum of @xmath128 , and the superscripts refer to the @xmath33- and @xmath34-wave components of the amplitude . in the rest frame of @xmath18 , these components are related to the @xmath43 and @xmath44 amplitudes by @xmath203 we will follow the usual prescription for estimating a weak phase @xcite , namely , first calculating the imaginary part of the amplitude and then dividing it by the real part of the amplitude extracted from experiment under the assumption of no strong phases and no @xmath0 violation . within the sm , the weak interactions responsible for hyperon nonleptonic decays are described by the short - distance effective @xmath204 hamiltonian @xcite @xmath205 where @xmath206 are the elements of the cabibbo - kobayashi - maskawa ( ckm ) matrix @xcite , @xmath207 are the wilson coefficients , and @xmath208 are four - quark operators whose expressions can be found in ref . @xcite . in this case , the weak phases @xmath209 of eq . ( [ ao ] ) proceed from the @xmath0-violating phase residing in the ckm matrix , and its elements appearing in @xmath210 above can be expressed in the wolfenstein parametrization @xcite as @xmath211 at lowest order in @xmath212 . as is well known , @xmath213 transforms mainly as @xmath214 under su(3@xmath215@xmath114su(3@xmath216 rotations . it is also known from experiment that the octet term dominates the 27-plet term @xcite . we , therefore , assume in what follows that within the sm the decays of interest are completely characterized by the @xmath217 , @xmath49 interactions . the leading - order chiral lagrangian for such interactions is @xcite @xmath218 \right\rangle + h_c^{}\ , \bar t_v^\mu\ , \xi^\dagger h \xi\ , t_{v\mu}^ { } \,\,+\,\ , { \rm h.c . } \,\,,\end{aligned}\ ] ] where the 3@xmath1143-matrix @xmath219 selects out @xmath220 transitions , having elements @xmath221 and the parameters @xmath222 contain the weak phases of interest . these phases are induced primarily by the imaginary part of @xmath223 associated with the penguin operator @xmath224 , and this is due to its chiral structure and the relative size of @xmath225 . in order to relate the imaginary part of @xmath222 to @xmath225 , we use the results of ref . @xcite , obtained from factorizable and nonfactorizable contributions . accordingly , we have @xmath226 all in units of @xmath227 from @xmath228 together with @xmath141 , we can derive the diagrams displayed in fig . [ pwave_sm ] , which represent the leading - order contributions to the @xmath33-wave transitions in @xmath229 and yield the amplitudes @xmath230 diagrams representing standard - model contributions to the leading - order @xmath33-wave amplitude for @xmath231 each square represents a weak vertex generated by @xmath228 in eq . ( [ lw_sm ] ) . ] applying eq . ( [ imh ] ) in @xmath232 then leads to @xmath233 where @xmath234 are the central values of @xmath235 in eq . ( [ px , dx ] ) . the uncertainties in these predictions are due to our neglect of higher - order terms that are presently incalculable and to our lack of knowledge on the reliability of the matrix - element calculation . therefore , we assign an error of 100@xmath236 to these ratios , as was similarly done in ref . @xcite for the weak phases in @xmath26 and @xmath52 thus , using @xmath237 and @xmath238 as in ref . @xcite , we obtain @xmath239 the @xmath240 result is comparable in size to that estimated in ref . @xcite using the vacuum - saturation method . value in ref . @xcite . ] turning now to the @xmath34-wave phases , we note that the expression for the @xmath241 term in eq . ( [ i m ] ) implies that @xmath228 , in conjunction with @xmath141 and @xmath173 , can not solely give rise to diagrams for the @xmath34-wave components . rather , the weak lagrangian that can generate the leading nonzero contributions to this term must have the dirac structure @xmath242 which is of @xmath166 . the @xmath34-wave amplitude at @xmath166 can also receive contributions from so - called tadpole diagrams , each being a combination of a strong @xmath243 vertex , generated by a lagrangian having the structure @xmath244 and a @xmath245-vacuum vertex coming from a weak lagrangian of @xmath246 . unfortunately , at present the parameters of these strong and weak lagrangians of @xmath166 are incalculable . the best that we can do is to make a crude estimate based on naive dimensional analysis @xcite . thus , since the lowest - order chiral lagrangian yielding @xmath247 is of @xmath248 , whereas that yielding @xmath249 is of @xmath166 , and since @xmath250 in hyperon nonleptonic decays , we expect that @xmath251 where @xmath252 is the chiral - symmetry breaking scale . it is worth remarking here that for @xmath253 @xcite this naive expectation is compatible with the value of @xmath254 from eq . ( [ px , dx ] ) , in which the @xmath255 number is determined largely by the preliminary data from hypercp @xcite . for these reasons , we make the approximation @xmath256 for the magnitude of the phase , where @xmath257 comes from eq . ( [ phip_sm ] ) . since @xmath258 as quoted in eq . ( [ px , dx ] ) is poorly determined , we take the further approximation @xmath259 for its magnitude in order to estimate @xmath260 . all this leads to @xmath261 the errors that we quote in @xmath81 are obviously not gaussian and simply indicate the ranges resulting from our calculation . putting together the numbers from eqs . ( [ px , dx ] ) , ( [ deltapd ] ) , ( [ phip_sm ] ) , and ( [ phid_sm ] ) in eq . ( [ ao ] ) yields @xmath262 we note that the second term on the right - hand side of eq . ( [ ao ] ) , which would vanish if the @xmath54 rescattering were ignored , has turned out to be the largest one . this is due to @xmath240 and @xmath188 being much larger than @xmath263 and @xmath264 , respectively , as well as to @xmath83 being small . for the partial - rate asymmetry in eq . ( [ deltao ] ) , we find @xmath265 this is comparable to the corresponding asymmetry in @xmath266 @xcite , but larger than those in octet - hyperon decays @xcite . since the asymmetry measured by hypercp is the sum @xmath267 it is important to know how @xmath1 compares with @xmath2 . the sm contribution to @xmath2 has been evaluated most recently to be @xmath268 @xcite . thus within the standard model @xmath1 is smaller than @xmath2 , but not negligibly so , and the resulting @xmath29 has a value within the range @xmath269 for this observable , hypercp expects to have a statistical precision of @xmath270 @xcite , and so its measurement will unlikely be sensitive to the sm effects . here we evaluate @xmath1 and @xmath95 arising from possible physics beyond the standard model . in particular , we consider contributions generated by the chromomagnetic - penguin operators ( cmo ) , which in some new - physics models could be significantly larger that their sm counterparts @xcite . the relevant effective hamiltonian can be written as @xcite @xmath271 where @xmath272 and @xmath273 are the wilson coefficients , and @xmath274 are the cmo , with @xmath275 being the gluon field - strength tensor , @xmath276 the gluon coupling constant , and @xmath277 since various new - physics scenarios may contribute differently to the coefficients of the operators , we will not focus on specific models , but will instead adopt a model - independent approach , only assuming that the contributions are potentially sizable , in order to estimate bounds on the resulting asymmetries as allowed by constraints from kaon measurements . the chiral lagrangian proceeding from the cmo has to respect their symmetry properties . under @xmath278@xmath114@xmath279 rotations @xmath280 and @xmath281 transform as @xmath282 and @xmath283 respectively . moreover , under a @xmath284 transformation ( a @xmath0 operation followed by interchanging the @xmath285 and @xmath44 quarks ) @xmath280 and @xmath281 change into each other . these symmetry properties are also those of the quark densities @xmath286 of which the lowest - order chiral realization has been derived in ref . @xcite . from this realization , we can infer the leading - order chiral lagrangian induced by the cmo , namely @xmath287 \right\rangle + \beta_0^ { } \left\langle h\sigma^\dagger \right\rangle \left\langle \bar{b}{}_v^ { } b_v^ { } \right\rangle \nonumber \\ & & \!\ ! + \,\ , \tilde\beta_d^ { } \left\langle \bar{b}{}_v^ { } \left\ { \xi h\xi , b_v^ { } \right\ } \right\rangle + \tilde\beta_f^ { } \left\langle \bar{b}{}_v^ { } \left [ \xi h\xi , b_v^ { } \right ] \right\rangle + \tilde\beta_0^ { } \left\langle h\sigma \right\rangle \left\langle \bar{b}{}_v^ { } b_v^ { } \right\rangle \nonumber \\ & & \!\ ! + \,\ , \beta_c^{}\ , \bar{t}{}_v^\alpha\ , \xi^\dagger h\xi^\dagger\ , t_{v\alpha}^ { } - \beta_0 ' \left\langle h\sigma^\dagger \right\rangle \bar{t}{}_v^\alpha t_{v\alpha}^ { } + \tilde\beta_c^{}\ , \bar{t}{}_v^\alpha\ , \xi h\xi\ , t_{v\alpha}^ { } - \tilde\beta_0 ' \left\langle h\sigma \right\rangle \bar{t}{}_v^\alpha t_{v\alpha}^ { } \nonumber \\ & & \!\ ! + \,\ , \beta_\varphi^{}\ , f^2 b_0^ { } \left\langle h\sigma^\dagger \right\rangle \,\,+\,\ , \tilde\beta_\varphi^{}\ , f^2 b_0^ { } \left\langle h\sigma \right\rangle \,\,+\,\ , { \rm h.c . } \,\,,\end{aligned}\ ] ] where @xmath288 @xmath289 are parameters containing the coefficient @xmath290 @xmath291 . the part of this lagrangian without the decuplet - baryon fields was first written down in ref . @xcite . from @xmath292 along with @xmath293 , we derive the diagrams shown in fig . [ pwave_np ] , which represent the lowest - order contributions induced by the cmo to the @xmath33-wave transitions in @xmath294 we remark that each of the three diagrams in the figure is of @xmath248 in the @xmath135 expansion , and that fig . [ pwave_sm ] does not include the meson - pole diagram because within the sm it contributes only at next - to - leading order . the amplitudes following from fig . [ pwave_np ] are @xmath295 where @xmath296 and we have used @xmath297 derived from eq . ( [ ls ] ) . and @xmath27 cases @xcite , each of the two amplitudes in eq . ( [ ap_np ] ) vanishes if we set @xmath298 @xmath299 and @xmath300 with @xmath301 being a constant , take the limit @xmath302 and use the relations @xmath303 and @xmath304 both derived from eq . ( [ ls ] ) . this satisfies the requirement implied by the feinberg - kabir - weinberg theorem @xcite that the operator @xmath305 can not contribute to physical decay amplitudes @xcite , and thus serves as a check for the formulas in eq . ( [ ap_np ] ) . ] diagrams representing chromomagnetic - penguin contributions to the leading - order @xmath33-wave amplitude for @xmath231 each square represents a weak vertex generated by @xmath306 in eq . ( [ lw_np ] ) . ] in order to estimate the weak phases in @xmath1 , we need to determine the parameters @xmath307 in terms of the underlying coefficient @xmath308 which is the combination corresponding to parity - conserving transitions . from the effective hamiltonian in eq . ( [ hw_np ] ) and the chiral lagrangian in eq . ( [ lw_np ] ) , we can derive the one - particle matrix elements @xmath309 since there is presently no reliable way to determine these matrix elements from first principles , we employ the mit bag model to estimate them . the results for @xmath310 have already been derived in ref . @xcite using the bag - model calculations of ref . @xcite and are given by @xmath311 where @xmath312 , @xmath313 , and @xmath314 are bag parameters . for @xmath315 , extending the work of ref . @xcite we find @xmath316 numerically , we take @xmath317 for the octet baryons , @xmath318 for the decuplet baryons , and @xmath319 for the mesons , after refs . in addition , as in ref . @xcite , we have @xmath320 and @xmath321 for both the baryons and mesons . it follows that @xmath322 we note that @xmath323 here is the wilson coefficient at the low scale @xmath324 and hence already contains the qcd running from the new - physics scales . we also note that the bag - model numbers in eq . ( [ beta_i ] ) are comparable in magnitude to the natural values of the parameters as obtained from naive dimensional analysis @xcite , @xmath325 where we have chosen @xmath326 the differences between the two sets of numbers provide an indication of the level of uncertainty in estimating the matrix elements .. this will be taken into account in our results below . applying eq . ( [ beta_i ] ) in @xmath232 then leads to the cmo contributions @xmath327 where , as in the @xmath26 and @xmath27 cases @xcite , we have assigned an error of 200@xmath236 to each of these numbers to reflect the uncertainty due to our neglect of higher - order terms that are presently incalculable and the uncertainty in estimating the matrix elements above . for the @xmath34-wave phases , we have here the same problem in estimating them as in the standard - model case , and so we have to resort again to dimensional arguments . thus , since the @xmath34-wave amplitude is parity violating , we have @xmath328 where @xmath329 is the combination corresponding to parity - violating transitions . putting together the numbers from eqs . ( [ px , dx ] ) , ( [ deltapd ] ) , ( [ phip_np ] ) , and ( [ phid_np ] ) in eq . ( [ ao ] ) , we find @xmath330 as in the sm result , the second term in @xmath1 dominates these numbers . for the partial - rate asymmetry , we obtain @xmath331 we can now write down the contribution of the cmo to the sum of asymmetries @xmath332 being measured by hypercp . the most recent evaluation of their contribution to @xmath2 has been done in ref . @xcite , the result being @xmath333 evidently , @xmath334 is much smaller than , though still not negligible compared to , @xmath335 . summing the two asymmetries yields @xmath336 since the cmo also contribute to the @xmath0-violating parameters @xmath337 in kaon mixing and @xmath338 in kaon decay , which are now well measured , it is possible to obtain bounds on @xmath339 and @xmath340 using the @xmath337 and @xmath338 data . as discussed in ref . @xcite , the experimental values @xmath341 and @xmath342 @xcite imply that @xmath343 then , from eqs . ( [ delta_o^np ] ) and ( [ a_ol^np ] ) , it follows that @xmath344 the upper limits of these ranges well exceed those within the sm in eqs . ( [ delta_o^sm ] ) and ( [ a_ol^sm ] ) , but the largest size of @xmath339 is still an order of magnitude below the expected sensitivity of hypercp @xcite . this , nevertheless , implies that a nonzero measurement by hypercp would be an unmistakable signal of new physics . we have evaluated the sum of the @xmath0-violating asymmetries @xmath1 and @xmath2 occurring in the decay chain @xmath17 which is currently being studied by the hypercp experiment . the dominant contribution to @xmath1 has turned out to be due to final - state interactions via @xmath345 we have found that both within and beyond the standard model @xmath1 is smaller than @xmath2 , but not negligibly so . taking a model - independent approach , we have also found that contributions to @xmath346 from possible new - physics through the chromomagnetic - penguin operators are allowed by constraints from kaon data to exceed the sm effects by up to two orders of magnitude . in summary , @xmath347 since the sm contribution is well beyond the expected reach of hypercp , a finding of nonzero asymmetry would definitely indicate the presence of new physics . in any case , the upcoming data on @xmath29 will yield information which complements that to be gained from the measurement of @xmath30 in @xmath348 finally , we have shown that the contribution of @xmath349 also causes the partial - rate asymmetry @xmath95 in @xmath22 to be nonvanishing , thereby providing another means to observe @xmath0 violation in this decay . this asymmetry and that in @xmath266 tend to be larger than the corresponding asymmetries in octet - hyperon decays and hence are potentially useful probes of @xmath0 violation in future experiments . since @xmath95 results from the interference of @xmath33-wave amplitudes , a measurement of it will probe the underlying parity - conserving interactions . numerically , we have found @xmath350 where the bound on the contribution of the cmo arises from the constraint imposed by @xmath337 data . i would like to thank g. valencia for helpful discussions and comments . i am also grateful to e.c . dukes and l .- c . lu for experimental information . this work was supported in part by the lightner - sams foundation . for the four coupled channels , the @xmath359 factors are @xmath360 @xmath361 @xmath362 @xmath363 and the @xmath364 factors @xmath365 @xmath366 @xmath367 @xmath368 where @xmath369 is the energy of @xmath128 in the final state . we note that contributions to the propagators from the @xmath370 and quark - mass terms in eq . ( [ ls ] ) have been implicitly included in these results . christenson _ et al . _ , phys . * 13 * , 138 ( 1964 ) ; a. alavi - harati _ et al . _ [ ktev collaboration ] , _ ibid . _ * 83 * , 22 ( 1999 ) ; v. fanti _ et al . _ [ na48 collaboration ] , phys . b * 465 * , 335 ( 1999 ) ; b. aubert _ et al . _ [ babar collaboration ] , phys . lett . * 87 * , 091801 ( 2001 ) ; k. abe _ et al . _ [ belle collaboration ] , _ ibid . _ * 87 * , 091802 ( 2001 ) . see , e.g. , a. j. buras , arxiv : hep - ph/0402191 and references therein . s. okubo , phys . rev . * 109 * , 984 ( 1958 ) ; a. pais , phys . lett . * 3 * , 242 ( 1959 ) ; t.brown , s.f . tuan , and s. pakvasa , _ ibid . _ * 51 * , 1823 ( 1983 ) ; l.l . chau and h.y . cheng , phys . b * 131 * , 202 ( 1983 ) ; j.f . donoghue and s. pakvasa , phys . lett . * 55 * , 162 ( 1985 ) . luk , arxiv : hep - ex/9803002 . luk _ et al . _ [ e756 and hypercp collaborations ] , arxiv : hep - ex/0005004 . lu [ hypercp collaboration ] , aip conf . * 675 * ( 2003 ) 251 ; talk given at the meeting of the division of particles and fields of the american physical society , philadelphia , pennsylvania , 5 - 8 april 2003 . donoghue , x .- he , and s. pakvasa , phys . d * 34 * , 833 ( 1986 ) . iqbal and g.a . miller , phys . d * 41 * , 2817 ( 1990 ) ; x .- g . he , h. steger , and g. valencia , phys . b * 272 * , 411 ( 1991 ) ; n.g . deshpande , x .- he , and s. pakvasa , _ ibid . _ * 326 * , 307 ( 1994 ) . d. chang , x .- he , and s. pakvasa , phys . lett . * 74 * , 3927 ( 1995 ) ; x .- g . he and g. valencia , phys . d * 52 * , 5257 ( 1995 ) ; x .- g . he , h. murayama , s. pakvasa , and g. valencia , _ ibid . _ * 61 * , 071701 ( 2000 ) ; c .- h . chen , phys . b * 521 * , 315 ( 2001 ) ; j .- h . jiang and m .- l . yan , j. phys . g * 30 * , b1 ( 2004 ) . j. tandean and g. valencia , phys . d * 67 * , 056001 ( 2003 ) . j. tandean , phys . d * 69 * , 076008 ( 2004 ) . j. tandean and g. valencia , phys . b * 451 * , 382 ( 1999 ) . k. hagiwara _ et al . _ [ particle data group collaboration ] , phys . d * 66 * , 010001 ( 2002 ) . watson , phys . * 95 * , 228 ( 1954 ) . see , e.g. , h. pilkuhn , _ the interactions of hadrons _ ( wiley , new york , 1967 ) . l. wolfenstein , phys . d * 43 * , 151 ( 1991 ) . oller , e. oset , and a. ramos , prog . phys . * 45 * , 157 ( 2000 ) . j. tandean , a.w . thomas , and g. valencia , phys . d * 64 * , 014005 ( 2001 ) ; references therein . j. bijnens , h. sonoda , and m.b . wise , nucl . * b261 * , 185 ( 1985 ) . e. jenkins and a.v . manohar , phys . b * 255 * , 558 ( 1991 ) ; _ ibid . _ * 259 * , 353 ( 1991 ) . in _ effective field theories of the standard model _ , edited by u .- meissner ( world scientific , singapore , 1992 ) . e. jenkins , nucl . * b368 * , 190 ( 1992 ) ; n. kaiser , p.b . siegel , and w. weise , _ ibid . * a594 * , 325 ( 1995 ) ; n. kaiser , t. waas , and w. weise , _ ibid . _ * a612 * , 297 ( 1997 ) ; j. caro ramon , n. kaiser , s. wetzel , and w. weise , _ ibid . _ * a672 * , 249 ( 2000 ) ; c.h . lee , g.e . brown , d .- min , and m. rho , _ ibid . _ * a585 * , 401 ( 1995 ) ; j.w . bos _ et al . d * 51 * , 6308 ( 1995 ) ; _ ibid . _ * 57 * , 4101 ( 1998 ) ; g. mller and u .- g . meissner , nucl . phys . * b492 * , 379 ( 1997 ) . a. abd el - hady , j. tandean , and g. valencia , nucl . phys . a * 651 * , 71 ( 1999 ) . j. gasser , h. leutwyler , and m.e . sainio , phys . b * 253 * , 252 ( 1991 ) ; * 253 * , 260 ( 1991 ) . m.n . butler , m.j . savage , and r.p . springer , nucl . b * 399 * , 69 ( 1993 ) . c. dukes [ hypercp collaboration ] , talk given at da@xmath371ne 2004 : physics at meson factories , laboratori nazionali di frascati , italy , 7 - 11 june 2004 . n. cabibbo , phys . * 10 * , 531 ( 1963 ) ; m. kobayashi and t. maskawa , prog . phys . * 49 * , 652 ( 1973 ) . l. wolfenstein , phys . lett . * 51 * , 1945 ( 1983 ) . see , e.g. , j.f . donoghue , e. golowich , and b.r . holstein , _ dynamics of the standard model _ ( cambridge university press , cambridge , 1992 ) . e. jenkins , nucl . * b375 * , 561 ( 1992 ) . a. manohar and h. georgi , nucl . phys . * b234 * 189 ( 1984 ) ; h. georgi and l. randall , _ ibid . _ * b276 * 241 ( 1986 ) ; s. weinberg , phys . lett . * 63 * , 2333 ( 1989 ) . buras , arxiv : hep - ph/0307203 . f. gabbiani , e. gabrielli , a. masiero , and l. silvestrini , nucl . b * 477 * , 321 ( 1996 ) ; a. masiero and h. murayama , phys . lett . * 83 * , 907 ( 1999 ) ; x .- g . he and g. valencia , phys . d * 61 * , 075003 ( 2000 ) ; g. colangelo , g. isidori , and j. portoles , phys . b * 470 * , 134 ( 1999 ) ; j. tandean , phys . d * 61 * , 114022 ( 2000 ) ; j. tandean and g. valencia , _ ibid . _ * 62 * , 116007 ( 2000 ) . buras _ et al . _ , nucl . b * 566 * , 3 ( 2000 ) . g. feinberg , p. kabir , and s. weinberg , phys . * 3 * , 527 ( 1959 ) . donoghue and b.r . holstein , phys . d * 33 * , 2717 ( 1986 ) ; j.f . donoghue , e. golowich , and b.r . holstein , phys . * 131 * , 319 ( 1986 ) . donoghue , e. golowich , and b.r . holstein , phys . d * 15 * , 1341 ( 1977 ) ; j.f . donoghue , e. golowich , b.r . holstein , and w.a . ponce , _ ibid . _ * 23 * , 1213 ( 1981 ) . t. degrand , r.l . jaffe , k. johnson , and j.e . kiskis , phys . d * 12 * , 2060 ( 1975 ) .

the sum of the @xmath0-violating asymmetries @xmath1 and @xmath2 in the decay sequence @xmath3 @xmath4 is presently being measured by the e871 experiment . we evaluate contributions to @xmath1 from the standard model and from possible new physics , and find them to be smaller than the corresponding contributions to @xmath2 , although not negligibly so . we also show that the partial - rate asymmetry in @xmath5 is nonvanishing due to final - state interactions . taking into account constraints from kaon data , we discuss how the upcoming result of e871 and future measurements may probe the various contributions to the observables . smu - hep-04 - 06

the concept of elementary excitations and the diagrammatic perturbation - theoretic methods borrowed from quantum field theory have given us , over the past decades , many powerful insights into the behavior of materials . in a number of cases , however , these concepts and methods do nt seem to work . in previous papers@xcite , we presented results on a nonperturbative extension of the magnetic interaction model , which had until then been extensively used in the context of diagrammatic approaches . these latter applications were successful in many respects : in the eliashberg approximation , the magnetic interaction model correctly anticipated the pairing symmetry of the cooper state in the copper oxide superconductors@xcite and is consistent with spin - triplet p - wave pairing in superfluid @xmath0 [ for a recent review see , e.g. , ref . one also gets the correct order of magnitude of the superconducting and superfluid transition temperature @xmath1 when the model parameters are inferred from experiments in the normal state of the above systems . however , in ref.@xcite it was found that when the model was treated nonperturbatively and one approached the border of magnetic long - range order , the quasiparticle spectrum showed qualitative changes not captured by the eliashberg approximation . in ref.@xcite , we raised the possibility that these qualitative changes , namely the opening of a pseudogap in the quasiparticle spectrum , were intrinsically nonperturbative in nature . in this paper , we examine this possibility by comparing the nonperturbative results to various kinds of perturbation - theoretic approximations . the paper is organized as follows . in the next section we describe the model as well as the various perturbation - theoretic approximations to be compared to the monte carlo calculations . section iii contains the results of the nonperturbative and diagrammatic calculations . section iv contains a discussion of the results and finally we give a summary and outlook the model and its motivation have been extensively discussed in ref.@xcite . here we only give the definitions relevant to the present discussion . we consider particles on a two - dimensional square lattice whose hamiltonian in the absence of interactions is @xmath2 where @xmath3 is the tight - binding hopping matrix , @xmath4 the chemical potential and @xmath5 , @xmath6 respectively create and annihilate a fermion of spin orientation @xmath7 at site @xmath8 . we take @xmath9 if sites @xmath8 and @xmath10 are nearest neighbors and @xmath11 if sites @xmath8 and @xmath10 are next - nearest neighbors . to introduce interactions between the particles , we couple them to a dynamical molecular ( or hubbard - stratonovich ) field . it is instructive to consider two different types of molecular fields . in the first instance , we consider a vector hubbard - stratonovich field that couples locally to the fermion spin density . we also consider the case of a scalar field that couples locally to the fermion number density . this case corresponds to a coupling to charge - fluctuations or , within the approximation we are using here , `` ising''-like magnetic fluctuations where only longitudinal modes are present . the hamiltonians at imaginary time @xmath12 for particles coupled to the fluctuating exchange or scalar dynamical field are then @xmath13 where @xmath14 and @xmath15 are the real vector exchange and scalar hubbard - stratonovich fields respectively , and @xmath16 the coupling constant . the reason for the choice of an extra factor @xmath17 in eq . ( [ ham1 ] ) becomes clear later . since we ignore the self - interactions of the molecular fields , their distribution is gaussian and given by@xcite @xmath18 = { 1\over z } \exp\bigg(-\sum_{{\bf q},\nu_n } { { \bf m}({\bf q},i\nu_n)\cdot { \bf m}(-{\bf q},-i\nu_n ) \over 2\alpha({\bf q},i\nu_n)}\bigg ) \label{probm } \\ z = \int d{\bf m}\exp\bigg(-\sum_{{\bf q},\nu_n } { { \bf m}({\bf q},i\nu_n)\cdot { \bf m}(-{\bf q},-i\nu_n ) \over 2\alpha({\bf q},i\nu_n)}\bigg ) \label{normm}\end{aligned}\ ] ] in the case of a vector exchange molecular field and @xmath19 = { 1\over z}\exp\bigg(-\sum_{{\bf q},\nu_n } { \phi({\bf q},i\nu_n)\phi(-{\bf q},-i\nu_n ) \over 2\alpha({\bf q},i\nu_n)}\bigg ) \label{probphi } \\ z = \int d\phi\exp\bigg(-\sum_{{\bf q},\nu_n } { \phi({\bf q},i\nu_n)\phi(-{\bf q},-i\nu_n ) \over 2\alpha({\bf q},i\nu_n)}\bigg ) \label{normphi}\end{aligned}\ ] ] in the case of a scalar hubbard - stratonovich field . in both cases @xmath20 since the dynamical molecular fields are periodic functions in the interval @xmath21 $ ] . the fourier transforms of the molecular fields are defined as @xmath22\big ) \label{fourierm } \\ \phi_{\bf r}(\tau ) & = & \sum_{{\bf q},\nu_n } \phi({\bf q},i\nu_n ) \exp\big(-i[{\bf q}\cdot{\bf r}-\nu_n\tau]\big ) \label{fourierp}\end{aligned}\ ] ] we consider the case where there is no long - range magnetic or charge order . the average of the dynamical molecular fields must then vanish and their gaussian distributions eqs . ( [ probm],[probphi ] ) are completely determined by their variance @xmath23 , which we take to be @xmath24 where @xmath25 is the number of allowed wavevectors in the brillouin zone . then @xmath26 where @xmath27 denotes an average over the probability distributions eq . ( [ probm ] ) and eq . ( [ probphi ] ) for the vector and scalar cases respectively . in order to compare the scalar and vector molecular fields , we take the same form for their correlation function @xmath28 and parametrize it as in refs.@xcite . in what follows , we set the lattice spacing @xmath29 to unity . for real frequencies , we have @xmath30 where @xmath31 and @xmath32 are the correlation wavevectors or inverse correlation lengths in units of the lattice spacing , with and without strong correlations , respectively . let @xmath33 we consider commensurate charge fluctuations and antiferromagnetic spin fluctuations , in which case the parameters @xmath34 and @xmath35 in eq . ( [ chiml ] ) are defined as @xmath36 where @xmath37 is a characteristic temperature . we also consider the case of ferromagnetic spin - fluctuations , where the parameters @xmath34 and @xmath35 in eq . ( [ chiml ] ) are given by @xmath38 @xmath28 is related to the imaginary part of the response function @xmath39 , eq . ( [ chiml ] ) , via the spectral representation @xmath40 to get @xmath28 to decay as @xmath41 as @xmath42 , as it should , we introduce a cutoff @xmath43 and take @xmath44 for @xmath45 . a natural choice for the cutoff is @xmath46 . in our model , the single particle green s function is the average over the probability distributions @xmath47 $ ] ( eq . ( [ probm ] ) ) or @xmath48 $ ] ( eq . ( [ probphi ] ) ) of the fermion green s function in a dynamical vector or scalar field . @xmath49\ ; g(i\sigma\tau ; j\sigma'\tau'|[{\bf m } ] ) \label{gm } \\ { \cal g}(i\sigma\tau ; j\sigma'\tau ' ) & = & \int d\phi\ ; { \cal p}[\phi]\ ; g(i\sigma\tau ; j\sigma'\tau'|[\phi ] ) \label{gphi}\end{aligned}\ ] ] where @xmath50\;or \;[\phi ] ) = -\big < t_\tau\{\psi_{i\sigma}(\tau ) \psi^\dagger_{j\sigma'}(\tau')\}\big > \label{gfield}\ ] ] is the single particle green s function in a dynamical molecular field and is discussed at length in ref.@xcite . in evaluating expressions eqs . ( [ gm],[gphi ] ) one is summing over all feynman diagrams corresponding to spin or charge - fluctuation exchanges@xcite . the diagrammatic expansion of the green s function , eq . ( [ gm ] ) is shown pictorially in fig . 1 . since in our model no virtual fermion loops are present , there is no fermion sign problem@xcite . in this paper we compare the results of the monte carlo simulations to various diagrammatic approximations for the same model . we denote by @xmath51 and @xmath52 the bare and dressed quasiparticle propagators respectively . they are given by @xmath53 where @xmath54 is the quasiparticle self - energy and @xmath55 the tight - binding dispersion relation obtained from fourier transforming the hopping matrix @xmath3 in eq . ( [ ham0 ] ) and @xmath4 the chemical potential . we consider four approximations to the quasiparticle self - energy @xmath54 whose diagrammatic representations are shown in fig . 2 . in fig . 2a , the self - energy is approximated by first order perturbation theory in the exchange of magnetic or charge fluctuations and denoted @xmath56 . 2b shows the eliashberg approximation in which the self - energy denoted @xmath57 is given by the first order self - consistent ( or brillouin - wigner ) perturbation theory . the expressions for @xmath56 or @xmath57 in the case of quasiparticles coupled to magnetic or charge fluctuations are identical ( this is the reason for our choice of the factor @xmath17 in eq . ( [ ham1 ] ) ) and given by @xmath58 where @xmath59 and @xmath60 are the bare and dressed quasiparticle green s functions defined in eq . ( [ gbare ] ) and eq . ( [ gdressed ] ) respectively . 2c shows the diagrammatic expansion corresponding to second order perturbation theory and we denote the self - energy corresponding to that approximation @xmath61 . the second order self - consistent approximation to the quasiparticle self - energy , denoted @xmath62 , is shown diagrammatically in fig . 2d . the expressions for @xmath61 and @xmath62 now depend on whether the quasiparticles are coupled to the vector hubbard - stratonovich field ( magnetic fluctuations ) or scalar hubbard - stratonovich field ( charge fluctuations ) , because vertex corrections in the two cases do not have the same coefficient or even the same sign . the expressions for @xmath61 and @xmath62 for quasiparticles coupled to magnetic fluctuations are given by @xmath63 in the case of the scalar hubbard - stratonovich field , or coupling to charge fluctuations , the corresponding expressions are @xmath64 in eqs . ( [ 2ptm],[2scm],[2ptc],[2scc ] ) , @xmath59 and @xmath60 are the bare and dressed quasiparticle green s functions defined in eq . ( [ gbare ] ) and eq . ( [ gdressed ] ) respectively . the strength of the coupling to the magnetic or charge flucutations can be parametrized by a dimensionless mass renormalization parameter @xmath65 , which is defined as @xmath66 the fermi surface averages are given by @xmath67 in practice , we compute the fermi surface average with a discrete set of momenta and we replace the delta function by a finite temperature expression @xmath68 where @xmath69 is the fermi function . note that @xmath70 as @xmath71 . we have used @xmath72 and @xmath73 in all of our calculations . the finite temperature effectively means that van hove singularities will be smeared out . note that the fermi surface average that appears in @xmath65 , eq . ( [ lambda1 ] ) plays a role similar to that of @xmath74 in the case of phonon mediated superconductivity . one therefore expects @xmath75 to indicate the crossover between weak and strong coupling . the quasiparticle dispersion relation for the two - dimensional square lattice is obtained from eq . ( [ ham0 ] ) . we measure all energies and temperatures in units of the nearest - neighbor hopping parameter @xmath76 . we set the next - nearest - neighbor hopping parameter @xmath77 . the chemical potential is adjusted so that the electronic band filling is @xmath78 . the dimensionless parameters describing the molecular field correlations are @xmath79 , @xmath80 , @xmath32 and @xmath31 . we chose a representative value for @xmath81 , and set @xmath82 as in the earlier work@xcite . for an electronic bandwidth of @xmath83 , @xmath84k . we only consider one value of the coupling constant @xmath85 . in the random phase approximation , the magnetic instability would be obtained for a value of @xmath79 of the order of 10 . we consider what happens to the quasiparticle spectrum at a fixed temperature @xmath86 as the inverse correlation length @xmath31 changes , as in ref.@xcite . all the calculations were done on a 8 by 8 spatial lattice . in the monte carlo calculations we used 41 imaginary time slices , or equivalently 41 matsubara frequencies for the molecular fields , @xmath87 and @xmath88 ( @xmath20 , with @xmath89 ) . in the diagrammatic calculations , we used between 40 to 60 fermion matsubara frequencies . by analytic continuation of the single particle green s function @xmath90 one can obtain the quasiparticle spectral function @xmath91 and the tunneling density of states @xmath92 , where @xmath93 is the retarded single particle green s function . the imaginary time monte carlo data is analytically continued with the maximum entropy method@xcite , using the same methodology as in the earlier work@xcite . we used 10000 monte carlo samples grouped into 100 bins of 100 samples each . we always use a flat default model in the maximum entropy calculations . to provide a fair comparison between diagrammatic and nonperturbative calculations , one should use the same analytic continuation method ( with the same parameters ) in all cases . therefore , we generated 100 noisy measurements by adding gaussian random noise to the results of the diagrammatic calculations and analytically continued @xmath94 using the maximum entropy method as well , with the same default model as in the corresponding analytic continuation of the monte carlo data . the scheme is not perfect , however . while the variance of the gaussian noise added to the diagrammatic green s functions was chosen such that the statistical uncertainty of the average over the 100 noisy samples was identical to that in the corresponding monte carlo green s function , the correlations in the errors for different values of @xmath12 present in the monte carlo results can not be easily modeled . the gaussian random numbers added to the diagrammatic green s function were therefore taken to be independent of each other , and thus the noise in the diagrammatic and monte carlo green s functions did not have identical statistical properties . in spite of this , the present scheme is almost certainly better than the alternatives . 3,4 and 5 show the comparison , for different values of @xmath95 , between the nonperturbative calculations of the quasiparticle green s function @xmath90 , spectral function @xmath96 and tunneling density of states @xmath97 and those obtained from the approximations @xmath56 , @xmath61 , @xmath57 , and @xmath62 to the quasiparticle self - energy . 3 shows our results for @xmath98 . for this value of the inverse correlation length squared , the mass renormalization parameter @xmath99 . the coupling to the antiferromagnetic spin - fluctuations is therefore weak . not surprisingly , the quasiparticle green s function , spectral function and tunneling density of states obtained from the various diagrammatic approximation agree well with the monte carlo results . at @xmath100 , the difference between the nonperturbative green s function @xmath101 and its diagrammatic approximations is of the order of 0.001 t for all values of @xmath12 . there is virtually no difference between the straightforward perturbation - theoretic calculations of the spectral function and their self - consistent counterparts , in first and second order , which is expected for weak coupling . thus the small difference in the spectral functions @xmath96 at @xmath100 , seen in fig 3b , to the extent that they are not an artifact of the analytic continuation , must come from the vertex corrections . since the first order spectral functions are slightly sharper than the second order ones , the first order vertex corrections result in an increased spin - fluctuation interaction , as pointed out in refs.@xcite . the monte carlo spectral function is also somewhat broader than the diagrammatic calculations , and provided again that it is not an artifact of the analytic continuation , this suggests that the higher order diagrams lead to a further increase of the spin - fluctuation interaction . the results for @xmath102 are shown in fig . 4 . this value of @xmath95 gives a mass renormalization parameter @xmath103 . one is now in the intermediate coupling regime . the quasiparticle green s function @xmath90 and spectral function @xmath96 at @xmath100 as well as the tunneling density of states obtained from the various diagrammatic approximations agree qualitatively with the monte carlo results . there are , however , noticeable quantitative differences , not surprisingly much more so than for @xmath98 . the largest difference between the green s functions obtained from the diagrammatic approximations and the nonperturbative calculations is now bigger than the width of the lines and is roughly an order of magnitude ( 0.01 t ) larger than for @xmath98 , which is not unexpected since the mass renormalization parameter is also about an order of magnitude greater for @xmath102 than for @xmath98 . with the above caveat regarding the analytic continuation , one can make a few additional remarks . first of all , there is now a difference between the straightforward perturbation - theoretic results and the self - consistent calculations of the spectral function @xmath96 at @xmath100 , both at first and second order . in particular , the second order self - consistent spectral function is slightly broader than the first order self - consistent one , an indication that the first order vertex correction leads to an enhancement of the effective spin - fluctuation interaction , in agreement with refs.@xcite . the nonperturbative @xmath104 is broader than the second order self - consistent result , which would imply the higher order vertex corrections are further enhancing the magnetic interaction . note , that the second order perturbation - theoretic @xmath104 is slightly broader than its self - consistent counterpart ( the dressing of green s functions tends to reduce the effect of interactions ) and agrees very well with the monte carlo result . this may be due to a cancellation of errors ( or the analytic continuation procedure ) since the agreement between the nonperturbative tunneling density of states @xmath97 and the second order perturbation - theoretic @xmath97 is not as good . as @xmath105 , the quasiparticle mean free path becomes of the order of the magnetic correlation length for some wavevectors near the fermi surface , the quasiparticles then ca nt tell there is no long - range order , and this marks the onset of pseudogap behavior@xcite . for @xmath106 , the mass renormalization parameter is @xmath107 . one is therefore in the strong coupling regime . the results of our calculations for @xmath106 are shown in fig . 5 . the developing pseudogap in the spectral function @xmath96 at @xmath100 ( fig . 5b ) and in the tunneling density of states @xmath97 ( fig . 5e ) found in the nonperturbative monte carlo calculations is not seen in any of the diagrammatic approximations considered here , which therefore fail qualitatively . given that one is in the strong coupling regime @xmath108 , the breakdown of perturbation theory should not come as a surprise . the maximum difference in the quasiparticle green s function @xmath90 between the nonperturbative and diagrammatic calculations is now of the order 0.1 t , and hence an order of magnitude larger than for @xmath102 and a couple of orders of magnitude larger than in the weak coupling regime with @xmath98 . it is therefore not suprprising that the quasiparticle spectra that give rise to these rather different imaginary time green s functions turn out to show qualitative differences . note that for @xmath106 , there is nearly as much difference between the perturbation - theoretic and self - consistent approximations of the same order as there are between calculations of the same type at first and second order . figs . 6 - 9 show our results for the quasiparticle green s function @xmath90 , spectral function @xmath96 and tunneling density of states @xmath97 for several values of @xmath95 . we start with @xmath98 , for which the mass renormalization parameter @xmath99 for coupling to ferromagnetic spin - flucutations . in this weak coupling regime , fig . 6 shows that the results of the various diagrammatic calculations are in good agreement with the monte carlo results . as in the corresponding antiferromagnetic case , at @xmath100 , the difference between the nonperturbative green s function @xmath101 and its diagrammatic approximations is of the order of 0.001 t for all values of @xmath12 . moreover , there is virtually no difference between the straightforward perturbation - theoretic calculations of the spectral function and their self - consistent counterparts , in first and second order . thus the small difference in the spectral functions @xmath96 at @xmath100 , seen in fig . 6b must come from the vertex corrections . since the first order spectral functions are slightly sharper than the second order ones , the first order vertex corrections result in an increased spin - fluctuation interaction . given the smallness of the difference between the first and second order results and the ill - posed nature of the analytic continuation problem , one should take the above remark with some degree of caution . in ref.@xcite , however , it was shown that the increase in the effective interaction induced by the first order vertex correction is due to the spin dependence of the interaction , and thus should occur for quasiparticles coupled to either antiferromagnetic or ferromagnetic fluctuations . our analytically continued results are at least consistent with this . from fig . 6b , one also sees that the monte carlo spectral function is slightly broader than the first or second order results , as in the corresponding antiferromagnetic case . with the above caveat on the nature of the analytic continuation problem , this would suggest the higher order diagrams not included in our perturbation - theoretic approximations lead to a further enhancement of the magnetic interaction , as in the corresponding antiferromagnetic case . for @xmath102 , the mass renormalization parameter @xmath109 and one is therefore in an intermediate coupling regime . fig . 7 shows that for this value of @xmath95 , the diagrammatic approximations all qualitatively agree with the monte carlo results . the quantitative agreement is , not surprisingly , not as good as in the weak coupling limit with @xmath98 . one notes a number of similarities between the results of fig . 6 and the corresponding antiferromagnetic case , shown in fig . 4 : ( i ) the second order perturbation theory results for @xmath96 give the best agreement with the nonperturbative calculation , ( ii ) since the spectral function in either second order calculation , which include vertex corrections , is slightly broader in @xmath110 than the corresponding first order result , we conclude that first order vertex corrections lead to an enhancement of the effective quasiparticle interaction , which is what is expected on the basis of the arguments made in refs.@xcite ( iii ) the spectral function obtained by monte carlo sampling of the gaussian dynamical molecular fields is slightly broader than the second order results , which to the extent this is not an artifact of the maximum entropy analytic continuation is an indication that higher order spin - fluctuation exchanges not included in the diagrammatic approximations considered lead to a further enhancement of the effective quasiparticle interaction . the dynamical exponent @xmath111 is larger for ferromagnetic than antiferromagnetic spin fluctuations . hence the effective dimension @xmath112 in the ferromagnetic versus @xmath113 in the antiferromagnetic case and the standard theory of quantum critical phenomena@xcite leads one to expect weaker corrections for higher effective dimensions . the perturbative calculations qualitatively fail at @xmath105 in the antiferromagnetic case and on the basis of the above arguments one would expect that the breakdown of perturbation theory in the case of ferromagnetic fluctuations , if it happens , would occur for a smaller value of @xmath95 or larger values of the mass renormalization parameter @xmath65 . indeed , at @xmath106 , @xmath114 and therefore one is in the strong coupling regime . 8 shows that while for this value of @xmath95 the diagrammatic calculations still agree qualitatively with the monte carlo results , unsurprisingly there are larger quantitative differences than in the case @xmath102 shown in fig . 7 . our results for @xmath115 , for which the mass renormalization parameter @xmath116 are shown in fig . 9 . the spectral function @xmath96 obtained from the nonperturbative monte carlo calculations shows a double peak structure . this has been interpreted in ref.@xcite as an effective spin - splitting of the quasiparticle spectrum induced by the local ferromagnetic order . in looking at the evolution of the spectral function @xmath96 as @xmath95 is decreased , one first sees a broadening of @xmath96 and then , the broad quasiparticle peak splits into two . the monte carlo calculations show very little suppression of the quasiparticle spectral weight or density of states between the two split peaks . a look at figs . 7c , 8c , and 9e reveals that for @xmath117 , @xmath118 and depends very little on @xmath95 . this is is sharp contrast to the case of antiferromagnetic fluctuations discussed in the previous section . this difference is to be expected of course , since the antiferromagnetic state is gapped while the ferromagnetic state is not . it is clear that none of the diagrammatic approximations considered here reproduce this spin - splitting of the broad quasiparticle peak in @xmath96 and tunneling density of states @xmath97 well . in fact , the first order perturbation theoretic result shows a strong suppression of the tunelling density of states , which clearly does nt describe the precursor to the ferromagnetic state well , and therefore can be considered to fail qualitatively . we observe that the first order perturbation theoretic calculation failed to show a suppression of the tunneling density of states in the antiferromagnetic case where it is obtained in the nonperturbative calculations as expected ( see previous subsection ) but does show such a pseudogap in the ferromagnetic case where it is nt expected and does nt appear in the nonperturbative calculations . it therefore qualitatively fails in both cases . another clear sign that not all is well with the perturbation expansion is the large quantitative differences between the one - loop and two - loop results in fig.9 , something that could be expected at @xmath116 . in the view of the differences between the imaginary time green s function @xmath90 obtained from the monte carlo simulations and those of the various perturbation - theoretic approximations shown in figs . 9a and 9c which are of the order of 0.1 t , one would expect the spectral functions that produce these rather different imaginary time green s functions to be rather different themselves . the results of our calculations of the quasiparticle green s function @xmath90 , spectral function @xmath96 and tunneling density of states @xmath97 for several values of @xmath95 are shown in figs . 10 , 11 and 12 . for the model studied here , the mass renormalization parameter @xmath65 is the same for charge and antiferromagnetic fluctuations . therefore the results of the calculations for @xmath98 shown in fig . 10 correspond to @xmath119 , namely the coupling to the charge fluctuations is weak . the agreement between the monte carlo results and those of the various diagrammatic approximations is good . as seen in fig . 10a , the difference between the nonperturbative imaginary time green s function and its perturbative approximations at @xmath100 is less than the width of the line and of the order of 0.001 t for all imaginary times @xmath12 . if one compares the results of the perturbation - theoretic calculations at first and second order , one sees from fig . 10b that there is virtually no difference between the spectral functions @xmath96 obtained by straightforward perturbation theory or the self - consistent calculation at either first or second order . hence the slight difference bewteen the first and second order calculations , to the extent they are nt an artifact of the analytic continuation , must come from vertex corrections . in contrast to the case of coupling to antiferromagnetic fluctuations , the spectral functions at second order are slightly narrower than their first order counterpart . this suggests the first order vertex correction acts to reduce the effective charge fluctuation interaction , in agreement with the arguments presented in ref.@xcite . moreover , the nonperturbative @xmath96 at @xmath100 is slightly broader than the second order results , which would indicate that the higher order diagrams lead to an enhancement of the effective charge fluctuation interaction , as in the case of a coupling to antiferromagnetic fluctuations . while this observation is made on the basis of analytically continued results , it is consistent with the results for other values of @xmath95 presented below , where the enhancement of the effective charge fluctuation mediated interaction by higher than second order diagrams can be shown to occur on general grounds . 11 shows the the quasiparticle green s function @xmath90 , spectral function @xmath96 at @xmath100 and tunneling density of states @xmath97 for @xmath102 , for which @xmath103 . for this value of @xmath95 corresponding to an intermediate coupling regime , the reader will notice that the results of the second order perturbation theory ( self - consistent or not ) are not displayed in the figures . the reason is that both second order approximations , @xmath61 and @xmath62 , for the model parameters considered here , effectively violate causality requirements in that the eliashberg renormalization factor @xmath120 becomes less than one . in terms of the quasiparticle self - energy @xmath54 , @xmath121 where @xmath122 is the imaginary part of the retarded self - energy and we have made use of the spectral representation for the self - energy @xmath123 . causality demands that the retarded green s function be analytic in the upper - half complex frequency plane and therefore that the imaginary part of the retarded self - energy be always less than or equal to zero ( @xmath124 ) for all values of @xmath125 . this in turn means that @xmath126 for all values of @xmath125 . one can write the second order eliashberg renormalization factor @xmath127 , where @xmath128 is the change in @xmath120 coming from the @xmath129 order diagrams . @xmath130 is always greater than zero and therefore poses no problem as far as the condition @xmath126 is concerned . in the charge - fluctuation case , as was explained in ref.@xcite , the first order vertex correction has the opposite sign compared to the spin - fluctuation case , and leads to a suppression of the effective quasiparticle interaction . the enhancement of the quasiparticle spin - fluctuation vertex comes from the transverse magnetic fluctuations that manage to overcome the reduction of the effective coupling due to the longitudinal fluctuations . because of this cancellation effect , not only is the sign of the first order vertex correction different in the magnetic case , it is also smaller in magnitude than in the charge - fluctuation case , under otherwise similar conditions , as can be seen from the factor 1/3 in eqs . ( [ 2ptm],[2scm ] ) not present in the corresponding charge - fluctuation case in eqs . ( [ 2ptc],[2scc ] ) . the different sign of the vertex corrections in the charge and magnetic cases means that while in the magnetic case @xmath131 and at second order @xmath132 is always @xmath133 , in the charge fluctuation case @xmath134 . for @xmath117 , we find that the second order contribution to the eliashberg renormalization factor is greater in magnitude than the first order contribution , @xmath135 . note that the nonperturbative calculations always satisfy @xmath136 , and the problem only arises in the perturbative approximation and is a sign that , for @xmath117 , the perturbation expansion for the charge - fluctuation case is quite badly behaved , possibly even more so than for magnetic fluctuations . aso , the fact that the nonperturbative calculations always satisfy @xmath136 is a proof that the higher than second order diagrams contribute to an enhancement of the charge - fluctuation interaction for these values of @xmath95 . fig . 11b shows that the spectral function @xmath96 obtained from the nonperturbative calculations is noticeably sharper than those produced by the first order self - consistent calculations . this means that for @xmath85 and @xmath82 the first and higher order vertex corrections suppress the effective quasiparticle interaction . it is therefore not surprising that there are no qualitative differences between the nonperturbative and diagrammatic calculations . there are quantitative differences , however , and these are more pronounced than in the case of a coupling of quasiparticles to antiferromagnetic spin - fluctuations for the same value of @xmath65 shown in fig . 4 . finally , 12 shows our results for @xmath115 , for which @xmath137 , hence in the strong coupling regime . the difference between the nonperturbative imaginary time green s function and its first order diagrammatic approximations seen in figs . 12b and 12d is a clear indication of the breakdown of perturbation theory . but even in this strong coupling regime , a cdw - precursor pseudogap in the spectral function @xmath96 , which can be expected to occur on general grounds@xcite is not seen . the pseudogap effects in the charge fluctuation case thus require a stronger coupling still ( larger coupling constant @xmath138 or smaller value of @xmath95 ) . in ref.@xcite , we showed that the magnetic pseudogap induced by a coupling to antiferromagnetic spin - fluctuations and the spin - splitting of the quasiparticle peak induced by a coupling to ferromagnetic spin fluctuations were not captured by the first order self - consistent , or eliashberg , approximation . the main result of this paper , is that these phenomena also lie beyond the two magnetic - fluctuation exchange theories ( self - consistent or not ) , which contain first order vertex corrections . while this does obviously not constitute a proof , these results are consistent with the conjecture expressed in ref.@xcite that the pseudogap effects found in the monte carlo calculations are intrinsically nonperturbative in nature . since the calculations reported here show that the first order vertex corrections alone do not produce a magnetic pseudogap , the physics of that state must then mainly come from the higher order spin - fluctuation exchange processes . the results presented here and in ref.@xcite also indicate that a cdw pseudogap induced by coupling to the scalar dynamical molecular field ( eq . ( [ ham2 ] ) ) must also originate from high order charge - fluctuation exchange processes . close enough to a second order cdw transition , the diverging cdw correlation length is bound to exceed the characteristic length scale for quasiparticles and the calculations of ref.@xcite showed that when this happens a pseudogap opens in the quasiparticle spectrum . the first order vertex correction ca nt produce the pseudogap state , since as we have seen , in the case of charge fluctuations it leads to a suppression of the interaction . in fact we even found that for the range of model parameters considered here , the second order diagrams more than cancel the contribution from the first order terms leading to a second order eliashberg renormalization parameter @xmath139 , which is inconsistent with causality requirements . moreover , we expect this `` over - cancellation '' effect to get worse as @xmath95 gets smaller than the lowest value considered here , @xmath115 . since @xmath120 must be @xmath133 when all the diagrams are summed up , as in the monte carlo simulations , one can conclude that the higher than second order terms must give a contribution @xmath140 to @xmath141 which is positive . therefore , higher order charge - fluctuation exchange processes produce an enhancement of the effective quasiparticle interaction , as in the magnetic case , and it must be through this enhancement of the effective interaction that a pseudogap can appear in the quasiparticle spectrum on the border of long - range cdw order . these observations lead one to a unified picture of the pseudogap state found in our model of quasiparticles coupled to spin or charge fluctuations . when the dynamical molecular field correlation length exceeds the characteristic length scale for quasiparticles , either the thermal de broglie wavelength@xcite or mean free path@xcite , the quasiparticles effectively see long - range order and this marks the onset of the pseudogap state . this state must be produced by high order spin or charge - fluctuation exchanges which contain subtle quantum mechanical coherence effects . in the magnetic fluctuation case , the first order vertex correction favors the pseudogap state , while in the charge fluctuation case it suppresses it . this implies one has to be closer to the boundary of long - range charge order to observe a pseudogap than one has to be to the boundary of magnetic long - range order , under otherwise similar conditions . as the dynamical molecular field correlation length increases , the mass renormalization parameter @xmath65 gets larger , and therefore the many - body effects become stronger . our results show that the agreement between the results of the monte carlo simulations and the perturbation - theoretic results gets worse as @xmath65 increases , and that not surprisingly , the perturbation - theoretic calculations break down when one enters the strong coupling regime @xmath108 , where the pseudogap is found . a more rigourous analysis of the relevance of the effective quasiparticle interactions as @xmath95 increases or as the energy scales are decreased would require a renormalization group ( rg ) treatment@xcite . recent rg calculations@xcite on the border of the ferromagnetic state indicate that the quasiparticle interactions are indeed relevant in @xmath142 , and the rg flows to strong coupling as the energy cutoff is decreased . one would like to understand what property of the full vertex function @xmath143 is responsible for the appearance of the pseudogap and seems to be missing in the first order approximation to @xmath144 . our physical picture of the pseudogap state emerging from quantum mechanical coherence effects contained in high order feynman diagrams is to be contrasted with the results of refs.@xcite where a suppression of the quasiparticle tunneling density of states at the fermi level is obtained in the single spin or charge - fluctuation exchange approximation . this effect is typically obtained with relatively large magnetic or charge correlation lengths . in our model , the calculations reported here and in ref.@xcite show that as one approaches the border of magnetic long - range order , @xmath145 , the multiple spin - fluctuation exchange processes become important long before a suppression of the quasiparticle tunneling density of states at the fermi level is seen in the first order perturbation - theoretic and self - consistent calculations . indeed , pseudogap effects are only obtined in our calculations when the dimensionless mass renormalization parameter @xmath108 , i.e in the strong coupling regime where one does nt expect diagrammatic perturbation theory to give reliable approximations . the above finding is likely to be valid more generally , since the intuitive arguments for the physical origin of the pseudogap@xcite lead one to expect the breakdown of migdal s theorem to be a generic occurence near a spin or charge instability . there is also an important difference between a vertex correction induced pseudogap and a single - fluctuation exchange pseudogap . in the latter case , there is no essential distinction bewteen spin and charge fluctuations , in that at the single - fluctuation exchange level , for a given fluctuation spectrum the spin and charge - fluctuation theories of the quasiparticle spectral function can be made identical by an appropriate scaling of the coupling constant to the molecular field . this is no longer the case when vertex corrections are included , since these actually depend on the nature of the hubbard - stratonovich field , in our case vector versus scalar . the distinction could turn out to be essential , since we find , for a range of model parameters , that a pseudogap is observed for quasiparticles coupled to spin fluctuations but not in the corresponding charge - fluctuation case . moukouri et al.@xcite have developed a many - body theory of the precursor pseudogap to the mott transition in the half - filled hubbard model . their theory is inspired by the fluctuation exchange approximation ( flex)@xcite in which bare spin and charge susceptibilities are used to build up the effective quasiparticle interaction , corresponding to @xmath146 in our model . the key respect in which the theory of moukouri et al.@xcite differs from flex is that the coupling to spin and charge fluctuations are not given by the bare on - site coulomb repulsion , but by renormalized parameters determined self - consistently in such a way that an exact relationship between the single and two - particle green s functions is satisfied . this last step goes beyond perturbation theory and it is therefore plausible that the precursor pseudogap to the mott transition seen in the monte carlo simulations of the half - filled hubbard model@xcite is also nonperturbative in origin . the analog of their scheme for the present model would be the use of the first order perturbation theory approximation for the quasiparticle self - energy @xmath147 , eq . ( [ 1pt ] ) and a simultaneous renormalization of the coupling constant @xmath16 and the correlation wavevector @xmath95 . a renormalization of the coupling constant @xmath16 could account for all vertex corrections provided they are local in space and time . one can indeed get a pseudogap in the tunneling density of states with the first order perturbation theory approximation to @xmath148 ( eq . ( [ 1pt ] ) ) , as in refs.@xcite , provided @xmath95 is renormalized to lower values and @xmath16 renormalized to higher ones . one would thus have to renormalize the model to stronger coupling , roughly to values of @xmath149 . it should be clear that in this regime , first order perturbation theory is not controlled . one would also naively expect a sensible renormalization scheme that goes beyond the one - loop level to lead to renormalized values of @xmath95 larger than the bare value . the renormalized theory should be further away from the magnetic instability than the one - loop approximation rather than closer to it , since ideally one would like the improved theory to satisfy the mermin - wagner theorem in two dimensions . if @xmath95 were to be increased by the renormalization scheme , in order to obtain a pseudogap in @xmath97 one would likely need a large renormalization of the coupling @xmath16 and such a scheme for the present model does not look promising to the author . however , it is important to note that the model studied here , although similar in some respects , is actually different than the one considered in ref.@xcite . the renormalization scheme proposed by moukouri et al . which works well for the hubbard model need not necessarily apply to other theories . in ref.@xcite , we pointed out that in the case of quasiparticles coupled to ferromagnetic spin fluctuations , our results are at variance with expectations based on the standard theory of quantum critical phenomena@xcite . since the dynamical exponent @xmath150 , in @xmath151 spatial dimensions , the effective dimension is @xmath112 and is greater than the upper critical dimension @xmath152 above which one would expect the first order theory to be at least qualitatively correct . but our nonperturbative results show that at least for small enough @xmath95 , the first order theory qualitatively breaks down . al@xcite have shown that the quasiparticle interactions are indeed relevant in the rg sense for ferromagnetic fluctuations in @xmath142 , a result consistent with our findings . for antiferromagnetic and charge fluctuations , @xmath113 , the marginal case , and hence the qualitative breakdown of the first order approximation is nt necessarily inconsistent with the standard theory . however , the scaling relations derived in ref.@xcite rely on the applicability of perturbation theory . if the pseudogap effects are indeed intrinsically nonperturbative in nature , a conjecture that is consistent with the present work , it opens the possibility that the physics in the proximity of a quantum critical point is dominated by nonperturbative quantum mechanical effects and therefore even richer than anticipated in the earlier work@xcite . a number of new ideas in this field have recently been proposed@xcite and a discussion of some fundamental problems associated with quantum critical points can be found in ref.@xcite . we studied a nonperturbative formulation of the magnetic interaction model , in which quasiparticles are coupled to a gaussian distributed dynamical molecular exchange field . far from the magnetic boundary , the multiple magnetic fluctuation exchange processes do not bring about qualitative changes to the quasiparticle spectrum . but as one gets closer to the border of long - range magnetic order , we find , for a range of model parameters , that migdal s theorem does nt apply and the quasiparticle spectrum is qualitatively different from its eliashberg approximation . moreover , we find that going one step beyond the single spin - fluctuation exchange approximation and including first order vertex corrections , self - consistently or not , does nt help to reproduce the qualitative changes seen in the nonperturbative calculations . near the magnetic boundary , the simple perturbation expansion shows signs it is not well behaved , since the second order results differ greatly from their first order counterparts . the self - consistent , or renormalized perturbation expansion , which effectively consists in a reordering of the diagrammatic perturbation theory , is better behaved in that the differences between first and second order are much less pronounced . however , even if the renormalized perturbation expansion converges , our results show that it is quite likely to converge to the wrong answer , which could be explained if the original perturbation expansion is divergent . the intuitive argument for the onset of pseudogap behavior@xcite , namely that if the distance quasiparticles can travel during their lifetime becomes shorter than the molecular field correlation length , these quasiparticles effectively see long - range order , does not explain the failure of the single spin - fluctuation exchange approximation . as we pointed out in ref.@xcite , one can get in the regime where the mean - free path gets shorter than @xmath153 in the eliashberg approximation , but fail to observe a pseudogap in this regime . since we have not been able to produce a good fit to the monte carlo simulations by including either first order or vertex corrections that are local in space and time , i.e by a renormalization of the coupling constant @xmath16 to the molecular field , we conjecture that the physical origin of the pseudogap state found in the present calculations lies in non - local vertex corrections produced by high order spin - fluctuation exchanges . these vertex corrections effectively induce a quasiparticle coupling to the dynamical molecular field that is non - local in both space and time . the above conjecture raise the question of what essential property of the vertex function is not captured by its first order approximation . a study of the vertex function along the same lines as the work reported here for the single particle green s function should provide further insights into this problem . i would like to thank p. coleman , j.r . cooper , p.b . littlewood , g.g . lonzarich , j. loram , and d. pines for discussions on this and related topics . we acknowledge the support of the epsrc , the newton trust and the royal society . bickers , d.j . scalapino , and s.r . white , phys . rev . lett . * 62 * , 961 ( 1989 ) ; t. moriya , y. takahashi and k. ueda , j. phys . . jpn . * 52 * , 2905 ( 1990 ) ; p. monthoux , a.v . balatsky and d. pines , phys . * 67 * , 3448 ( 1991 ) .

we study a model of quasiparticles on a two - dimensional square lattice coupled to gaussian distributed dynamical molecular fields . we consider two types of such fields , a vector molecular field that couples to the quasiparticle spin - density and a scalar field coupled to the quasiparticle number density . the model describes quasiparticles coupled to spin or charge fluctuations , and is solved by a monte carlo sampling of the molecular field distributions . the nonperturbative solution is compared to various approximations based on diagrammatic perturbation theory . when the molecular field correlations are sufficiently weak , the diagrammatic calculations capture the qualitative aspects of the quasiparticle spectrum . for a range of model parameters near the magnetic boundary , we find that the quasiparticle spectrum is qualitatively different from that of a fermi liquid , in that it shows a double peak structure , and that the diagrammatic approximations we consider fail to reproduce , even qualitatively , the nonperturbative results of the monte carlo calculations . this suggests that the magnetic pseudogap induced by a coupling to antiferromagnetic spin - fluctuations and the spin - splitting of the quasiparticle peak induced by a coupling to ferromagnetic spin - fluctuations lie beyond diagrammatic perturbation theory . while a pseudogap opens when quasiparticles are coupled to antiferromagnetic fluctuations , such a pseudogap is not observed in the corresponding charge - fluctuation case for the range of parameters studied , where vertex corrections are found to effectively reduce the strength of the interaction . this suggests that one has to be closer to the border of long - range order to observe pseudogap effects in the charge - fluctuation case than for a spin - fluctuation induced interaction under otherwise similar conditions . the diagrammatic approximations that contain first order vertex corrections show the enhancement of the spin - fluctuation induced interaction and the suppression of the effective interaction in the charge - fluctuation case . however , for the range of model parameters considered here , the multiple spin or charge - fluctuation exchange processes not included in the diagrammatic approximations considered are found to be important , especially for quasiparticles coupled to charge fluctuations .

as widely discussed in many papers , the spectrum of the cosmic microwave background ( cmb ) carries unique informations on physical processes occurring during early cosmic epochs ( see e.g. danese & burigana 1993 and references therein ) . the comparison between models of cmb spectral distortions and cmb absolute temperature measures can constrain the physical parameters of the considered dissipation processes . we recently discussed ( salvaterra & burigana 2002 ) the implications of the current cmb spectrum data by jointly considering distortions generated in a wide range of early or intermediate cosmic epochs and at late cosmic epochs . various cmb spectrum experiments at long wavelengths , @xmath17 cm , are ongoing and planned for the future in order to improve the still quite poor accuracy of the data in this spectral region , where the maximum deviations from a pure blackbody spectrum are expected in the case of dissipation processes occurred at early and intermediate epochs . in this work we jointly consider the data from the firas instrument aboard the cobe satellite and simulated sets of cmb spectrum observations at wavelengths larger than 1 cm with the sensitivities expected from future experiments in order to discuss their impact for the recovery of the thermal history of the universe . in section 2 we briefly summarize the general properties of the cmb spectral distortions and the main physical informations that can be derived from the comparison with the observations . in section 3 we briefly discuss the performances of current and future cmb spectrum observations at long wavelengths and describe the generation of the simulated observations used in this work . the implications of observations with sensitivities typical of forthcoming and future ground and balloon experiments are presented in section 4 , while in section 5 we extensively discuss the implications of experiments at long wavelengths with a sensitivity comparable to that of cobe / firas , as foreseen for a space experiment , dimes , proposed to the nasa in the 1995 and designed to measure the cmb absolute temperature at @xmath18 cm with a sensitivity of @xmath3 mk ( kogut 1996 ) . in section 6 we present a detailed discussion of the capabilities of future cmb spectrum observations to discriminate between the firas calibration by the cobe / firas team ( referred as here `` standard '' calibration ; see fixsen et al . 1994 , 1996 , mather et al . 1999 , and references therein ) and that proposed by battistelli , fulcoli & macculi 2000 . the possibility to improve our knowledge of the free - free distortions is considered in section 7 , while section 8 is devoted to identify the experimental sensitivity requirements for an accurate baryon density evaluation through the detection of possible long wavelength distortions . finally , we draw our main conclusions in section 9 . the cmb spectrum emerges from the thermalization redshift , @xmath19 , with a shape very close to a planckian one , owing to the strict coupling between radiation and matter through compton scattering and photon production / absorption processes , radiative compton and bremsstrahlung , which were extremely efficient at early times and able to re - establish a blackbody ( bb ) spectrum from a perturbed one on timescales much shorter than the expansion time ( see e.g. danese & de zotti 1977 ) . the value of @xmath20 ( burigana , danese & de zotti 1991a ) depends on the baryon density ( in units of the critical density ) , @xmath21 , and the hubble constant , @xmath22 , through the product @xmath23 ( @xmath22 expressed in km / s / mpc ) . on the other hand , physical processes occurring at redshifts @xmath24 may lead imprints on the cmb spectrum . the timescale for the achievement of kinetic equilibrium between radiation and matter ( i.e. the relaxation time for the photon spectrum ) , @xmath25 , is @xmath26 where @xmath27 is the photon electron collision time , @xmath28 , @xmath29 being the electron temperature and @xmath30 ; @xmath31 is the mean fractional change of photon energy in a scattering of cool photons off hot electrons , i.e. @xmath32 ; @xmath33 is the present radiation temperature related to the present radiation energy density by @xmath34 ; a primordial helium abundance of 25% by mass is here assumed . it is useful to introduce the dimensionless time variable @xmath35 defined by @xmath36 where @xmath37 is the present time and @xmath38 is the expansion time given by @xmath39^{-1/2 } \sec \ , , \ ] ] @xmath40 being the redshift of equal non relativistic matter and photon energy densities ( @xmath41 is the density of non relativistic matter in units of critical density ) ; @xmath42 , @xmath43 being the number of relativistic , 2component , neutrino species ( for 3 species of massless neutrinos , @xmath44 ) , takes into account the contribution of relativistic neutrinos to the dynamics of the universe , is itself a function of the amount of energy dissipated . the effect , however , is never very important and is negligible for very small distortions . ] . burigana , de zotti & danese 1991b have reported on numerical solutions of the kompaneets equation ( kompaneets 1956 ) for a wide range of values of the relevant parameters and accurate analytical representations of these numerical solutions , suggested in part by the general properties of the kompaneets equation and by its well known asymptotic solutions , have been found ( burigana , de zotti & danese 1995 ) . the cmb distorted spectra depend on at least three main parameters : the fractional amount of energy exchanged between matter and radiation , @xmath45 , @xmath46 being the radiation energy density before the energy injection , the redshift @xmath47 at which the heating occurs , and the baryon density @xmath48 . the photon occupation number can be then expressed in the form @xmath49 where @xmath50 is the dimensionless frequency @xmath51 ( @xmath52 being the present frequency ) , and @xmath53 characterizes the epoch when the energy dissipation occurred , @xmath47 being the corresponding redshift ( we will refer to @xmath53 computed assuming @xmath54 , so that the epoch considered for the energy dissipation does not depend on the amount of released energy ) . the continuous behaviour of the distorted spectral shape with @xmath55 can be in principle used also to search for constraints on the epoch of the energy exchange . of course , by combining the approximations describing the distorted spectrum at early and intermediate epochs with the comptonization distortion expression describing late distortions , it is possible to jointly treat two heating processes ( see burigana et al . 1995 and salvaterra & burigana 2002 and references therein for a more exhaustive discussion ) . in this work the measures of the cmb absolute temperature are compared with the above models of distorted spectra for one or two heating processes by using a standard @xmath56 analysis . we determine the limits on the amount of energy possibly injected in the cosmic background at arbitrary primordial epochs corresponding to a redshift @xmath47 ( or equivalently to @xmath55 ) . this topic has been discussed in several works ( see e.g. burigana et al . 1991b , nordberg & smoot 1998 , salvaterra & burigana 2002 ) . as in salvaterra & burigana 2002 , we improve here the previous methods of analysis by investigating the possibility of properly combining firas data with longer wavelength measures with the sensitivities expected for forthcoming and future experiments and by refining the method of comparison with the theoretical models . we will consider the recent improvement in the calibration of the firas data , that sets the cmb scale temperature at @xmath57 k at 95 per cent confidence level ( cl ) ( mather et al . we do not consider the effect on the estimate of the amount of energy injected in the cmb at a given epoch introduced by the calibration uncertainty of firas scale temperature when firas data are treated jointly to longer wavelength measures , since the analysis of salvaterra & burigana 2002 shows that it introduces only minor effects . then , we study the combined effect of two different heating processes that may have distorted the cmb spectrum at different epochs . this hypothesis has been also taken into account by nordberg & smoot 1998 , who fit the observed data with a spectrum distorted by a single heating at @xmath58 , a second one at @xmath59 and by free - free emission , obtaining limits on the parameters that describe these processes . as in salvaterra & burigana 2002 , we extend their analysis by considering the full range of epochs for the early and intermediate energy injection process , by taking advantage of the analytical representation of spectral distortions at intermediate redshifts ( burigana et al . since the relationship between free - free distortion and comptonization distortion is highly model dependent , being related to the details of the thermal history at late epochs ( danese & burigana 1993 , burigana et al . 1995 ) , and can not be simply represented by integral parameters , we avoid a combined analysis of free - free distortions and other kinds of spectral distortions and separately discuss the implications of future , more accurate long wavelength measures on free - free distortions . it is also possible to extend the limits on @xmath5 for heatings occurred at @xmath60 , where @xmath61 is the redshift corresponding to @xmath62 , when the compton scattering was able to restore the kinetic equilibrium between matter and radiation on timescales much shorter than the expansion time and the evolution on the cmb spectrum can be easily studied by replacing the full kompaneets equation with the differential equations for the evolution of the electron temperature and the chemical potential . this study can be performed by using the simple analytical expressions by burigana et al . 1991b instead of numerical solutions . a recent analysis of the limits on the amount of the energy possibly injected in the cosmic background from the currently available data is reported in salvaterra & burigana 2002 . in particular , they found that the measures at @xmath17 cm do not significantly contribute to these constraints because of their poor sensitivity compared to that of firas . new and more accurate measurements are also needed in this range , which is particular sensitive to early energy injection processes . in fact , the current constraints on @xmath5 at @xmath63 are a factor @xmath64 less stringent than those at @xmath55 less than @xmath3 , because of the frequency coverage of firas , which mainly set the current constraints at the all cosmic epochs . thus , we are interested to investigate the role of future ground , balloon and space experiments at @xmath17 cm jointed to the firas measures at @xmath0 cm . to evaluate the scientific impact represented by the future experiment improvements , we create different data sets simulating the observation of a not distorted spectrum both from ground and balloon experiments and from a space experiment like dimes through the method described in section 3.1 . for a dimes - like experiment , we also explore the possibility of the observation of distorted spectra for different amounts of the energy injected in the radiation field and for different cosmic epochs . each data set will be then compared to models of distorted spectra by using the method described in salvaterra & burigana 2002 ( see also burigana & salvaterra 2000 for the details of the code ) to recover the value of @xmath5 or constraints on it , the heating epoch , @xmath55 , the free - free distortion parameter @xmath65 , and the combination , @xmath48 , of the baryon density and the hubble constant . for simplicity , we restrict to the case of a baryon density @xmath66 our analysis of the implications for the thermal history of the universe , but the method can be simply applied to different values of @xmath48 . in presence of an early distortion , @xmath48 could be in principle measured by cmb spectrum observations at long wavelengths provided that they have the required sensitivity about the minimum of the cmb absolute temperature ( see section 8) . the cmb spectrum experiments currently under study are dedicated to improve our knowledge at wavelengths longer than those covered by firas . at centimeter and decimeter wavelengths , the available measures typically show large error bars although some experiments are rather accurate ( i.e. , the measure of staggs et al . 1996 at @xmath67 cm ) . very accurate data at long wavelengths could give a significant improvement to our knowledge of physical processes in the primeval plasma , particularly at high redshifts . these projects regard measurements from ground , balloon and space . as representative cases , and without the ambition to cover the whole set of planned experiments , we briefly refer here to the ground experiment tris at very long wavelengths and to the dimes experiment from space ( kogut 1996 ) designed to reach an accuracy close to that of firas up to @xmath68 cm . tris is a set of total power radiometers designed to measure the absolute temperature of the cmb at three frequencies : 0.6 , 0.82 and 2.5 ghz . at these wavelengths ( @xmath69 cm ) the measurements are difficult because the cmb signal is comparable to other components of the antenna temperature : galactic background , unresolved extra - galactic sources , sidelobes pickup and atmospheric emission . to improve the experimental situation , tris will make absolute maps of large areas of the sky at the three frequencies , to disentangle the various components of the celestial signal ; all the lossy parts of the antenna front ends of the receivers will be cooled down liquid helium temperature , to reduce the thermal noise of these components ; the receiver temperatures will be very carefully stabilized to reduce drifts and gain variations . the tris expected sensitivity is of about 200 mk at the three frequencies . dimes ( diffuse microwave emission survey ) is a space mission submitted to the nasa in 1995 , designed to measure very accurately the cmb spectrum at wavelengths in the range @xmath18 cm ( kogut 1996 ) . dimes will compare the spectrum of each 10 degree pixel on the sky to a precisely known blackbody to precision of @xmath3 mk , close to that of firas ( @xmath70 mk ) . the set of receivers is given from cryogenic radiometers operating at six frequency bands about 2 , 4 , 6 , 10 e 90 ghz using a single external blackbody calibration target common to all channels . in each channel , a cryogenic radiometer switched for gain stability between an internal reference load and an antenna with 10 degree beam width , will measure the signal change as the antenna alternately views the sky and an external blackbody calibration target . the target temperature will be adjusted to match the sky signal in the lowest frequency band , allowing the absolute temperature to be read off from the target thermometry with minimal corrections for the instrumental signature . with its temperature held constant , the target will rapidly move over the higher - frequencies antenna apertures , effectively comparing the spectrum of diffuse emission from the sky to a precise blackbody . by comparing each channel to the same target , uncertainties in the target emission cancel so that deviations from a blackbody spectral shape may be determined much more precisely than the absolute temperature . the dimes design is driven by the need to reduce or eliminate systematic errors from instrumental artifacts . the instrument emission will be cooled to 2.7 k , whereas the calibration uncertainty will be minimized by using a single calibration target , common to all channels . the atmospheric emission will be observed from low earth orbit and the multiple channels measurements will minimize the foreground emission problems . the dimes sensitivity represents an improvement by a factor better than 300 with respect to previous measurements at cm wavelengths . we collect different data sets , simulating measurements of different cmb spectra , distorted or not , at the frequency ranges of the considered experiments . we add to these simulated data the firas data at higher frequencies according to the most recent calibration of the temperature scale at 2.725 k ( mather et al . 1999 ) . for the cases of distorted spectra we calculate the theoretical temperature of the cmb spectrum at the wavelengths of the new experiments as discussed in the previous section . of course , the thermodynamic temperature held obviously constant at all the frequencies for the case of a non distorted spectrum . the theoretical temperatures are then fouled to simulate real measurements affected by instrumental noise . the simulated temperature @xmath71 at the frequency @xmath52 is given by @xmath72 where @xmath73 is the theoretical temperature at the frequency @xmath52 and err(@xmath52 ) is the expected rms error ( at 1 @xmath74 ) of the experiment at this frequency . the numbers @xmath75 are a set of random numbers generated according to a gaussian distribution with null mean value and unit variance with the routine gasdev by press et al . 1992 ( 7 ) . we analyse here the impact of possible future observations from ground and balloon in the case of a not distorted spectrum at the temperature @xmath76 k. the results are thus comparable to those obtained with the firas data alone ( see e.g. salvaterra & burigana 2002 ) . to build the first simulated data set ( g&b1-bb ) , we split the region from 1 to 80 cm in three ranges and associate different values of sensitivity to each range according to the analysis of the main problems affecting the available observations in different spectral regions ( e.g. salvaterra & burigana 2000 ) . 1 . @xmath77 cm . in this range the measurements of staggs et al . 1996 show an uncertainty of @xmath78 mk . thus , quite accurate measures could be carried out in this range . we choose to associate to the future experiments at these wavelengths an improved typical sensitivity of 10 mk . 2 . @xmath79 cm . ground experiments in this range show error bars of about @xmath80 mk . progresses could be reached by improving the accuracy of the subtraction of the atmospheric contribution which dominates the final error at these wavelengths . thus , we choose to associate to the data in this region a typical sensitivity of 40 mk ; 3 . @xmath81 cm . observations in this range are still quite difficult , the typical sensitivities being between 200 mk for measures at 10 cm and 1.5 k for those at longer wavelengths . the expected sensitivity of the tris experiment ( see section 3 ) is of @xmath82 mk . thus , we choose to associate to future experiments in this range a typical sensitivity of 200 mk . the frequencies of the experiments at @xmath17 cm of the two last decades ( see e.g. table 1 of salvaterra & burigana 2002 ) , where suitable observation windows should exist and the presence of man made interferences should be not a concern , have been adopted in the generation of simulated observations . finally , we complete this data set by adding the firas measures calibrated at 2.725 k according to mather et al . 1999 to the above simulated data . a second data set ( g&b2-bb ) is built as before but by improving by a factor 10 the sensitivity associated to each of the above three frequency range in order to evaluate the impact of highly optimistic future progresses of ground and ballon experiments . the results of the fits to the simulated data g&b1-bb and g&b2-bb jointed to firas data are shown in fig . 1 [ for graphic purposes , we report in the plots the exact value of @xmath55 and the power - law approximation @xmath83 ( burigana et al . 1991b ) for the redshift ] . as evident , realistic improvements of future experiments from ground and balloon do not significantly change the firas limits . even under much more optimistic experimental conditions , able to decrease the errors by a factor 10 , the situation can not substantially improve , being the limits on @xmath84 obtained in this case only just more stringent than those based on firas data alone . we then conclude that , unfortunately , observations of the cmb absolute temperature with sensitivity levels typical of future ground and balloon experiments do not seem able to improve the limits on the amount of energy injected in the cosmic radiation field inferred on the basis of the currently available measures . we generate a set ( d - bb ) of simulated data in the case of a blackbody spectrum at a temperature of 2.725 k in order to evaluate the capability of an experiment with a sensitivity comparable to that expect for dimes to improve the constraints on the amount of the energy injected in the cosmic radiation field . the analysis of this case is in fact directly comparable with the results obtained from the fit to the firas data alone . then , we build up other data sets representing the observations of cmb spectra distorted by energy injections at different cosmic epochs in order to investigate the possibility of a dimes - like experiments to firmly determine the presence of spectral distortions . we consider processes occurring at a wide range of cosmic epochs , represented by the dimensionless time @xmath85 5 , 4 , 3 , 2 , 1 , 0.5 , 0.25 , 0.1 , 0.05 , 0.025 , 0.01 , and @xmath86 . we consider four representative values of fractional injected energy : @xmath87 , a value not much below the upper firas limits ; @xmath88 , well above the firas upper limit ( see section 6 ) ; @xmath89 and @xmath90 , two values well below the firas upper limit , to test the chances to detect very small distortions with a dimes - like experiment . as a further representative case , we simulate the observation of a spectrum distorted by two heating processes occurring at different epochs , the first at @xmath58 and the second at @xmath86 , both characterized by @xmath91 . all these distorted spectra are computed by setting @xmath92 and @xmath93 km / s / mpc . as a variance with respect the previous section , we choose here the frequencies of the simulated observations by adopting the five frequency channels of the dimes experiment . as in previous section , we complete these data set by adding the firas measures calibrated at 2.725 k. we fit the simulated data d - bb with a spectrum distorted by an energy injection at different values of @xmath55 in order to recover the value of @xmath5 , expected to be null , and the limits on it . the fit results are reported in fig . 2 . it is evident how future data at this sensitivity level will allow a strong improvement of the limits obtained with the firas data alone . the recovered best - fit value of @xmath5 is always compatible with the absence of distortions within the limits at 95 per cent cl . for heating processes at low @xmath94 ( @xmath95 ) the fit is substantially dominated by the firas data and the lower and the upper limits on @xmath5 are still @xmath96 . on the contrary , for early distortions ( @xmath6 ) the low frequency measures of a dimes - like experiment will allow to improve the firas constraints by a factor @xmath97 , the proper value increasing with the considered dissipation redshift . we conclude that measures from an instrument like dimes could represent a very good complement to the firas data . in the next sections we will analyse in detail the capability of a dimes - like experiment to determine the presence of spectral distortions possibly present in the cmb spectrum . the test reported in the previous section suggests that even small distortions could be determined provided that the dissipation would have occurred at relatively early epochs , @xmath6 . thus , we analyse the sensitivity of a dimes - like experiment in the recovery of the amount of energy possibly injected in the radiation field and explore also the possibility to determine the dissipation process epoch . firstly , we fit the data simulated as above under the hypothesis that the heating epoch is known ; more explicitly , we fit the data with a theoretical spectrum distorted by a process occurring at the considered @xmath55 by allowing to optimize @xmath5 ( and @xmath33 ) but by taking @xmath55 fixed . in this way we can see how accurately @xmath5 could be in principle recovered . on the other hand , unless we want to use the cmb spectrum data to constrain theoretical models with a well defined dissipation epoch , we are typically interested to set constraints on the value of @xmath5 possibly injected at a given unknown epoch occurring within a relatively wide cosmic period ; in addition , many classes of physical processes in the plasma epoch involve time parameters and it is important to understand how they can be possibly constrained by the comparison with cmb spectrum observations . thus , we focus on the cases of spectra distorted at high ( @xmath58 ) , medium ( @xmath98 ) and at low ( @xmath86 ) redshifts by fitting the simulated data by relaxing the a priori knowledge of the dissipation epoch . in this way we would be able to evaluate the possibility of determining also the epoch of the heating with a proper higher value of @xmath99 would give a distorted spectrum essentially indistinguishable by that generated in the case of a dissipation at @xmath62 with a lower value of @xmath99 , see section 5.4 . ] without a priori informations by jointly evaluating the impact of the unknowledge of the dissipation epoch on the recovery of injected energy . we will test also the possibility of deriving at the same time information on the baryon density . 5.2.2.1 @xmath100 energy injections at firas limits dissipation epoch : known _ 0.2 cm as a representative case we consider the simulated observation of a spectrum distorted at different values of @xmath55 by an energy injection with @xmath101 , a value below , but not much , the firas upper limit on @xmath5 . these data are then compared with the theoretical cmb spectrum distorted at the same @xmath55 ( assumed to be known ) by performing the fit only over @xmath5 and @xmath33 : this is appropriate to cases in which we have a quite well defined a priori information on the dissipation epoch but not on the amount of released energy . we find that for high redshift processes , @xmath102 , @xmath5 is precisely determined . for distortions at lower @xmath94 , @xmath103 , we obtain limits similar to those given from the currently available data , since the the fit result is mainly driven by firas data , more sensitive to these kinds of distortions , mainly located at high frequencies . 0.4 cm _ 5.2.2.2 @xmath100 energy injections at firas limits dissipation epoch : unknown _ 0.2 cm we relax here the assumption to know the dissipation epoch . we consider firstly the case of the fit to data simulated assuming a spectrum distorted at @xmath58 with @xmath101 with cmb theoretical spectra distorted at different values of @xmath55 . the best - fit to these data assuming @xmath58 gives a very accurate recovery of the input value of @xmath5 with a small quoted error ( we find an associated statistical error of @xmath104% at 95 per cent cl ) . the best - fit on @xmath5 assuming lower values of @xmath55 is far from the input value and the @xmath56 increases . thus , we search for a favourite value of @xmath55 by performing the fit over @xmath5 , @xmath33 , and @xmath55 . we obtain that the recovered best - fit value of @xmath55 is exactly the input one , 5.0 , and lower limit on @xmath55 at 95 per cent cl is 2.4 . by searching also for a favourite value of @xmath48 ( set to 0.05 in the data simulation ) , we obtain a 68 per cent cl range of @xmath105 . we repeated the same analysis in the case of a spectrum distorted at @xmath98 with @xmath101 . again , the recovered value of @xmath5 is close to the input one for fits with @xmath106 ( in this case we recover the input value of @xmath5 with an uncertainty of @xmath107% at 95 per cent cl ) and we are also able to determine a significative range ( @xmath6 at 95 per cent cl ) of favourite values of @xmath55 , although wider than in the previous case , while @xmath48 is found to be in the range @xmath108 at 68 per cent cl . similar results on @xmath55 and @xmath48 can not be obtained in the case of fit to the data simulating the observation of a spectrum distorted at @xmath109 . the fit result then is then compatible also with energy injections with smaller values of @xmath5 but at higher @xmath55 and with a non distorted spectrum . the @xmath56 value does not significantly change when @xmath55 varies . this is again the result of the main role of firas data for late dissipation processes . 5.2.2.3 @xmath100 energy injections below firas limits dissipation epoch : known _ 0.2 cm a dimes - like experiment should be able to detect also small spectral distortions . let consider here the case of a spectrum distorted from an energy injection with @xmath91 , about a factor 10 below the firas limits at 95 per cent cl . as shown in fig . 3 , if the dissipation epoch is known , we find that for processes at early and intermediate epochs the best - fit result is very close to the input value of the simulated data , although the limits on @xmath5 are not so stringent as in the case with a larger energy injection . for @xmath6 , a spectral distortion would be firmly detected at 95 per cent cl . 0.4 cm _ 5.2.2.4 @xmath100 energy injections below firas limits dissipation epoch : unknown _ 0.2 cm we relax here again the assumption to know the dissipation epoch . our results are summarized in fig . 4 : even for distortions well below the firas limits ( @xmath110 is assumed here ) an experiment like dimes would provide significative constraints both on the amount of dissipated energy and on the dissipation epoch in the case of and early processes . in this test the input dissipation epoch ( @xmath111 ) is again quite well recovered , the @xmath56 increasing of 4 when @xmath55 becomes close to unity . it would be also possible to provide an independent estimate of the baryon density : we find @xmath112 at 68 per cent cl . for a process occurring at intermediate epochs ( @xmath98 in this specific test ) we find that it is still possible to determine the amount of injected energy , but in this case of distortions significantly smaller than the firas limits the @xmath56 is no longer particularly sensitive to @xmath55 and the recovered range of dissipation epochs is wide . energy dissipations processes at intermediate epochs may then result still compatible with these simulated data and only energy injections at late epochs , could be excluded ( in this test we find that the @xmath56 increases of @xmath113 for @xmath114 ) . as already found for larger distortions , significant information on late processes can not be obtained from accurate long wavelength data because of the more relevant role of the firas data . 0.4 cm _ 5.2.2.5 @xmath100 very small energy injections _ 0.2 cm finally , we consider the possibility to detect very small energy injections , namely with @xmath115 . if the dissipation epoch is known , the result of the fit shows that is still possible to determine a no null distortion provided that the dissipation process occurs at high redshifts . for energy injections at epochs close to @xmath58 the recovered value of @xmath5 , @xmath116 in this test , is quite close to the input one and the associate statistical error gives a @xmath5 range of @xmath117 at 95 per cent cl . an indication of the dissipation epoch can be also derived ( in this case we find that the @xmath56 increases of @xmath113 for @xmath118 ) . unfortunately , no significant informations on @xmath48 can be obtained with the considered sensitivity and frequency coverage in the case of so small distortions . for dissipations at @xmath119 a significative distortion can be also determined , the recovered value of @xmath5 ranging between @xmath120 and @xmath121 at 95 per cent cl for @xmath122 where the @xmath123d.o.f . is about its minimum , but significative informations on the dissipation epoch can be no longer derived . finally , in the case of processes at late epochs , the fit results are compatible with an unperturbed spectrum , being these kind of distortions mainly detectable at firas frequencies . we discuss here the possibility to significantly improve the constraints on ( or to detect ) energy exchanges also in the more general case of a joint analysis of early / intermediate and late dissipation processes . by exploiting the data set d - bb presented in sect . 5.1 we consider the case of no significant deviations from a planckian spectrum . top panel of fig . 5 shows the limits on the energy exchange as function of @xmath55 by allowing for a later dissipation process possibly occurred at @xmath86 ; bottom panel of fig . 5 shows the constraints on the energy injected at low @xmath94 by allowing for a previous distortion occurred at any given @xmath55 . in fig . 5 we report also the comparison with the results based only on the firas data ( as shown in salvaterra & burigana 2002 , the current long wavelength measures do not change significantly these results ) . the conclusion is impressive : the constraints on @xmath5 for early and intermediate dissipation processes could be improved by a factor @xmath124 , depending on the considered dissipation epoch . in addition , the constraints on the energy dissipation at late epochs can be also improved , by a factor of about two , because of the reduction of the partial degeneracy introduced by the rough compensation ( salvaterra & burigana 2002 ) between the effect of early and late energy exchanges on the cmb spectrum when no accurate measures are available at long wavelengths . to complete the analysis of the impact of a possible dimes - like experiment , we consider the simulated observation of a spectrum distorted by a first energy dissipation at @xmath58 with @xmath91 and a second one at @xmath86 with @xmath91 . we then compare these data with theoretical spectra distorted by a process at @xmath86 and another at any given @xmath125 according to our grid of @xmath55 ( see section 5.1 ) . we find that a dimes - like experiment would allow to firmly determine the presence of the distortion at high @xmath94 ; in particular , at @xmath58 the recovered @xmath5 is very close to the input one , as already found for the tests described in sections 5.2.2.3 and 5.2.2.4 . on the contrary , for the dissipation process at @xmath86 the fit result is compatible with an unperturbed spectrum , since the firas data dominate the limits on the distortions at low redshifts . the limits on @xmath5 for processes at low redshifts are , however , again more stringent , by a factor 2 , than those obtained with the currently available data . we find that the @xmath56 significantly increases by assuming in the fit an earlier process at decreasing @xmath55 ( the @xmath56 increases of @xmath113 for @xmath126 ) : even in the case of a combination of an early and a late process a dimes - like experiment would be able to significantly constrain the epoch of the earlier energy exchange . we extend here at @xmath127 ( i.e. @xmath128 ) the constraints on @xmath5 that would be possible to derive at @xmath129 ( @xmath62 ) with a dimes - like experiment . we remember that at @xmath130 the compton scattering is able to restore , after an energy injection , the kinetic equilibrium between matter and radiation , yielding a bose - einstein ( be ) spectrum , and the combined effect of compton scattering and photon production processes tends to reduce the magnitude of spectral distortions , possibly leading to a blackbody spectrum . we firstly consider here the case of the simulated observation of a not distorted spectrum , that represents a good test of the possible improvements of an instrument like dimes , because the limits on @xmath5 at relevant redshifts can be directly compared to those obtained with firas data alone . for simplicity , we consider the case of a single energy injection possibly occurred in the cosmic thermal history . the comparison is shown in fig . 6 . as evident , the constraints on @xmath5 can be improved by a factor @xmath131 for processes possibly occurred in a wide range of cosmic epochs , corresponding to about a decade in redshift at @xmath94 about @xmath132 . of course , large energy injections are still possible at very early epochs close to the thermalization redshift , when primordial nucleosynthesis set the ultimately constraints on energy injections in the cosmic radiation field . for late dissipations , firas data mainly constrain @xmath5 . as a further example , we consider the constraints on the energy injections at @xmath133 ( @xmath134 ) in the case of a fit with a single energy injection to simulated observations of a spectrum distorted at @xmath62 with @xmath135 . as shown by the high redshift tails of the curves of fig . 3 , in this case the constraints on the thermal history of the universe would be completely different from those derived in the case in which distortions are not detected ( fig . . a firm detection of early energy injections would be clearly possible with the considered experimental performances and the constraints on the energy possibly injected at @xmath130 could be directly derived from such kind of future cmb spectrum data . in addition , fig . 6 shows that , contrariously to the case of the current observational status , the constraints on early energy exchanges based on future high accuracy long wavelength measures are no longer appreciably relaxed by assuming that a late process could be also occurred . in the previous sections , we considered simulated data with a planckian spectrum or with distortions compatible with the limits derived from firas data calibrated according to mather et al . 1999 . on the other hand , a recent analysis of firas calibration ( referred here as `` revised '' ) by battistelli et al . 2000 suggests a frequency dependence of the firas main calibrator emissivity . the firas data recalibrated according to their `` favourite '' calibration emissivity law ( r - firas data in what follows ) indicate the existence of deviations from a planckian shape or at least a significant relaxation of the constraints on them . salvaterra & burigana 2002 discussed the main implications of this analysis . although it seems quite difficult to fully explain from a physical point of view the r - firas data , two classes of phenomenological models may fit them : in the first one the main contribution derives from an intrinsic cmb spectral distortion with @xmath136 few @xmath137 occurring at early / intermediate epochs ; the second one involves a millimetric component possibly due to cold dust emission , described by a modified blackbody spectrum , added to a cmb blackbody spectrum at a temperature of @xmath113 mk below firas temperature scale of 2.725 k. fig . 7 shows as these models , very similar at millimeter wavelengths , predict significant differences at centimeter and decimeter wavelengths . in this section we carefully discuss the capabilities of forthcoming and future cmb spectrum measures at long wavelengths to discriminate between these two different firas calibrations and , in the case of the calibration by battistelli et al . 2000 , to distinguish between the two above scenarios . in the case in which the firas calibration by mather et al . 1999 is substantially correct , and therefore that revised by battistelli et al . 2000 is wrong , the cmb spectrum is expected to show an essentially planckian shape also at @xmath17 cm . to test the capabilities of forthcoming and future cmb spectrum experiments to rule out the `` revised '' calibration we fit simulated observations of a planckian spectrum at @xmath17 cm generated as described in section 3.1 added to the r - firas data set in terms of a single energy exchange process and of a combination of two processes at different cosmic times . we consider long wavelength data simulated assuming different values of @xmath33 , at steps of 0.1 mk within the firas temperature scale range , in order to reach the best compromise between them and the non - flat r - firas temperature data . the first five rows of table 1 summarize our results in the case of a dimes - like experiment assuming @xmath138 k , the case in which we find the best agreement . for comparison , we report in the last three rows of table 1 the results obtained by considering the r - firas data alone ( salvaterra & burigana 2002 ) . as evident from the significant increase of the @xmath123d.o.f . , long wavelength measures indicating a cmb planckian shape are not compatible with the firas data calibrated according to battistelli et al . 2000 . similar analyses carried out in the case of simulated long wavelength planckian data with the sensitivity of forthcoming , or improved , ground and balloon experiments ( data sets g&b1-bb and g&b2-bb ) show values of @xmath123d.o.f . close to @xmath139 ( similar to those obtained considering r - firas data alone ) , when the epoch ( or the epochs ) and the energy exchange ( exchanges ) of the dissipation process ( processes ) is ( are ) properly chosen . we then conclude that a cmb spectrum experiment at long wavelengths designed to rule out the firas calibration as revised by battistelli et al . 2002 should have a sensitivity comparable to that of a dimes - like experiment . .results of the fit to the r - firas data combined ( first five rows ) or not ( last three rows ) with long wavelength data with a sensitivity of a dimes - like experiment and simulated according to a planckian shape at @xmath140 k ( first 5 rows ) in terms of a single or two dissipation processes at different epochs ( fits to two or three parameters : @xmath33 and one or two values of @xmath5 ) . see also the text . [ cols= " < , < , < , < " , ] for distortions at relatively high redshifts ( @xmath6 ) , the value of @xmath48 can be simply determined by the knowledge of the frequency position of the minimum of the cmb absolute temperature : @xmath141 this opportunity is very powerful in principle , since the dependence of @xmath142 on @xmath48 is determined only by the well known physics of the radiation processes in an expanding universe during the radiation dominated era . for dissipations at @xmath143 , the amplitude of this temperature decrement is @xmath144 where @xmath145 is the chemical potential at the redshift @xmath61 corresponding to @xmath58 . ( 7 ) gives the range of wavelengths to observe for a firm evaluation of @xmath48 . as example , for @xmath146 we need to accurately measure the cmb absolute temperature up to wavelengths of about 50 cm , clearly out from the dimes range . ground and ballon experiments are currently planned to reach these wavelengths . moreover , the amplitude of the maximum dip of the brightness temperature for the energy dissipations at @xmath143 , see eq . ( 8) , turns to be at the mk level for @xmath147 and distortions within the firas limits ; burigana et al . 1991a shown that it is about 3 times smaller for energy injections at @xmath126 . experiments designed to estimate @xmath48 through the measure of @xmath142 should then have a sensitivity level of @xmath13 mk or better . for sake of illustration , we consider the simulated observation of a spectrum distorted at @xmath58 with @xmath101 ( or @xmath148 , as suggested by the `` revised '' firas data ) in a @xmath146 universe through a very precise experiment extended up to @xmath149 cm . more precisely , we consider the dimes channels combined to measures at 73.5 , 49.1 , 36.6 , 21.3 , 12 and 6.3 cm as those proposed for the space experiment lobo dedicated to measure the cmb spectrum at very low frequencies ( @xmath150 ghz ; see sironi et al . 1995 , pagana & villa 1996 ) , but we assume a much better sensitivity , @xmath151 mk , comparable to that of the dimes - like experiment , ( or @xmath152 mk ) . again , we generate the simulated data as described in section 2.3 . the fit to these simulated data by assuming to know the dissipation epoch shows that it would be possible to accurately determine both the amount of injected energy and the baryon density . we recover @xmath153 ( or @xmath154 ) and @xmath155 ( errors at 95 per cent cl ) . unfortunately , experiments at decimeter wavelengths with a sensitivity of @xmath13 mk or better , although very informative in principle , seem to be very far from current possibilities . we have studied the implications of possible future observations of the cmb absolute temperature at @xmath156 cm , where both ground , balloon and space experiments are currently under study to complement the accurate firas data at @xmath0 cm . our analysis shows that future measures from ground and balloon will not be able to significantly improve the constraints on energy exchanges in the primeval plasma already provided by the firas data . even observations with a sensitivity better by a factor 10 with respect to the realistic performances of the next experiments at different centimeter and decimeter wavelengths can not significantly improve this conclusion . thus , we have studied the impact of very high quality data , such those that could be in principle reached with a space experiment . for this analysis , we referred to the dimes experiment ( kogut 1996 ) , submitted to the nasa in 1995 , planned to measure the cmb absolute temperature at @xmath157 cm with a sensitivity of @xmath151 mk , close to that of firas . we have demonstrated that these data would represent a substantial improvement for our knowledge of energy dissipation processes at intermediate and high redshifts ( @xmath6 ) . dissipation processes at @xmath158 could be accurately constrained and possibly firmly detected even for very small amounts of the injected energy ( @xmath159 ) . for these early dissipation processes it would be possible to estimate also the energy injection epoch . distortions at intermediate redshifts ( @xmath119 ) could be also firmly detected , although in this case interesting information on the heating epoch can be derived only for energy injections , @xmath5 , larger than about @xmath160 . on the contrary , by considering the case of a single energy exchange in the thermal history of the universe , for late processes ( @xmath161 ) a such kind of experiment can not substantially improve the limits based on the firas data at @xmath0 cm , which would still set the constraints on @xmath5 at late epochs . by the jointed analysis of two dissipation processes occurring at different epochs , we demonstrated that the sensitivity and frequency coverage of a dimes - like experiment would allow to accurately recover the amount of energy exchanged in the primeval plasma at early and intermediate redshifts , and possibly the corresponding epoch , even in presence of a possible late distortion . even in this case , the constraints on @xmath5 can be improved by a factor @xmath162 for processes possibly occurred in a wide range of cosmic epochs , corresponding to about one two decades in redshift at @xmath94 about @xmath132 , while the constraints on the energy possibly dissipated at late epochs can be also improved by a factor @xmath7 , because the rough compensation between the distortion effects at millimetric wavelengths from an early and a late process with opposite signs becomes much less relevant in presence of very accurate long wavelength data . in addition , accurate long wavelength measures can provide an independent cross - check of the firas calibration : a dimes - like experiment could accurately distinguish between the firas calibrations by mather et al . 1999 and by battistelli et al . 2000 and in this second case could discriminate between different scenarios to account for it . interesting , although not fully exhaustive , indications on this aspect could be also obtained by improving the sensitivity of the next ground and balloon experiments by a factor @xmath1 . further , we have shown that a possible accurate observation of spectral distortions at @xmath163 cm compatible with relatively large energy injections , compared to the `` standard '' firas limits , can not be consistently reconciled with the firas data , at least for the class of distortion considered here . in this observational scenario , `` exotic '' models for spectral distortions should be carefully considered . we have shown that future long wavelength measures can significantly improve the current observational status of the free - free distortion : constraints on ( or detection of ) @xmath65 at the ( accuracy ) level of @xmath164 can be reached by forthcoming experiments , while improving the sensitivity up to that of a dimes - like experiment will allow to measure @xmath165 with an accuracy up to about @xmath166 ( errors at 95 per cent cl ) . of course , not only a very good sensitivity , but also an extreme control of the all systematical effects and , in particular , of the frequency calibration is crucial to reach these goals . finally , a dimes - like experiment will be able to provide indicative independent estimates of the baryon density : the product @xmath8 can be recovered within a factor @xmath9 even in the case of ( very small ) early distortions with @xmath10 . on the other hand , for @xmath11 , an independent baryon density determination with an accuracy at @xmath12 per cent level , comparable to that achievable with cmb anisotropy experiments , would require an accuracy of @xmath13 mk or better in the measure of possible early distortions but up to a wavelength from @xmath12 few @xmath14 dm to @xmath15 dm , according to the baryon density value . it is a pleasure to thank m. bersanelli , n. mandolesi , c. macculi , g. palumbo , and g. sironi for useful discussions on cmb spectrum observations . warmly thank l. danese and g. de zotti for numberless conversations on theoretical aspects of cmb spectral distortions . battistelli e.s . , fulcoli v. , macculi c. 2000 , new astronomy , 5 , 77 burigana c. , danese l. , de zotti g. 1991a , a&a , 246 , 59 burigana c. , de zotti g. , danese l. 1991b , apj , 379 , 1 burigana c. , de zotti g. , danese l. 1995 , a&a , 303 , 323 burigana c. & salvaterra r. 2000 , int . itesre / cnr 291/2000 , august danese l. & burigana c. 1993 , in : `` present and future of the cosmic microwave background '' , lecture in physics , vol . 429 , eds . sanz , e. martinez - gonzales , l. cayon , springer verlag , heidelberg ( frg ) , p. 28 danese l. & de zotti g. 1977 , riv . nuovo cimento , 7 , 277 fixsen d.j . 1994 , apj , 420 , 457 fixsen d.j . 1996 , apj , 473 , 576 kogut a. 1996 , `` diffuse microwave emission survey '' , in the proceedings from xvi moriond astrophysics meeting held march march 16 - 23 in les arcs , france , astro - ph/9607100 kompaneets a.s . 1956 , zh . , 31 , 876 [ sov . jept , 4 , 730 , ( 1957 ) ] mather j.c . , fixsen d.j . , shafer r.a . , mosier c. , wilkinson , d.t . 1999 , apj , 512 , 511 nordberg h.p . & smoot g.f . 1998 , astro - ph/9805123 pagana e. , villa f. , 1996 , `` the lobo satellite mission : feasibility study and preliminary cost evaluation '' , int . rep . c.i.f.s . - 1996 press w.h . , teukolsky s.a . , vetterling w.t . , flannery b.p . 1992 , `` numerical recipes in fortran '' , second edition , cambridge university press , usa salvaterra r. & burigana c. 2000 , int . itesre / cnr 270/2000 , march , astro - ph/0206350 salvaterra r. & burigana c. 2002 , mnras , 336 , 592 sironi g. , bonelli g. , dalloglio g. , pagana e. , de angeli s. , perelli m. , 1995 , astroph . comm . , 32 , 31 staggs s.t . , jarosik n.c . , meyer s.s . , wilkinson d.t . 1996 , apj , 473 , l1 [ lastpage ]

we analyse the implications of future observations of the cmb absolute temperature at centimeter and decimeter wavelengths , where both ground , balloon and space experiments are currently under study to complement the accurate cobe / firas data available at @xmath0 cm . our analysis shows that forthcoming ground and balloon measures will allow a better understanding of free - free distortions but will not be able to significantly improve the constraints already provided by the firas data on the possible energy exchanges in the primeval plasma . the same holds even for observations with sensitivities up to @xmath1 times better than those of forthcoming experiments . thus , we have studied the impact of very high quality data , such those in principle achievable with a space experiment like dimes planned to measure the cmb absolute temperature at @xmath2 cm with a sensitivity of @xmath3 mk , close to that of firas . we have demonstrated that such high quality data would improve by a factor @xmath4 the firas results on the fractional energy exchanges , @xmath5 , associated to dissipation processes possibly occurred in a wide range of cosmic epochs , at intermediate and high redshifts ( @xmath6 ) , and that the energy dissipation epoch could be also significantly constrained . by jointly considering two dissipation processes occurring at different epochs , we demonstrated that the sensitivity and frequency coverage of a dimes - like experiment would allow to accurately recover the epoch and the amount of energy possibly injected in the radiation field at early and intermediate epochs even in presence of a possible late distortion , while the constraints on the energy possibly dissipated at late epochs can be improved by a factor @xmath7 . in addition , such measures can provide an independent and very accurate cross - check of firas calibration . finally , a dimes - like experiment will be able to provide indicative independent estimates of the baryon density : the product @xmath8 can be recovered within a factor @xmath9 even in the case of ( very small ) early distortions with @xmath10 . on the other hand , for @xmath11 , an independent baryon density determination with an accuracy at @xmath12 per cent level , comparable to that achievable with cmb anisotropy experiments , would require an accuracy of @xmath13 mk or better in the measure of possible early distortions but up to a wavelength from @xmath12 few @xmath14 dm to @xmath15 dm , according to the baryon density value . # 1 # 1 2truept2truept 2truept2truept @xmath16 # 1#2 = cmbx12 et al.et al . [ firstpage ] cosmology : cosmic microwave background cosmological parameters - cosmology : theory

questions such as whether the universe will expand forever or eventually re - collapse and end with a big crunch , and what its shape and size may be , are among the most fundamental challenges in cosmology . regarding the former question , it is well known that the ultimate fate of the universe is intrinsically associated with the nature of its dominant components . in the friedmann - lematre - robertson - walker ( flrw ) class of models , for instance , a universe that is dominated by a pressureless fluid ( as , e.g. , baryons and/or dark matter ) or any kind of fluid with positive pressure ( as radiation , for example ) will expand forever if its spatial geometry is euclidean or hyperbolic , or will eventually re - collapse if it is spherical . this predictable destiny for the universe , however , may be completely modified if it is currently dominated by some sort of negative - pressure dark component , as indicated by a number of independent observational results ( see , e.g. , ref . @xcite ) . in this case , not only the dynamic but also the thermodynamic fate of the universe may be completely different , with the possibility of an eternally expanding closed model @xcite , an increasingly hot expanding universe @xcite or even a progressive rip - off of the large and small scale structure of matter ending with the occurrence of a curvature singularity , the so - called big smash @xcite . the remaining questions , concerning the shape and size of our @xmath2dimensional world , go in turn beyond the scope of general relativity ( gr ) , since they have an intrinsically topological nature . in this way , approaches or answers to these questions are ultimately associated with measurements of the _ global _ structure ( topology ) of the universe and , as a _ metric theory , gr can not say much about it , leaving the global topology of the universe undetermined . over the past few years , several aspects of the cosmic topology have become topical ( see , e.g. , the review articles ref . @xcite ) , given the wealth of increasingly accurate cosmological observations , especially the recent results from the wilkinson microwave anisotropy probe ( wmap ) experiment @xcite , which have heightened the interest in the possibility of a universe with a nontrivial spatial topology . a pertinent question the reader may ask at this point is whether the current values of cosmological density parameters , which help us to answer the above first question ( associated with the ultimate fate of the universe ) , can be constrained by a possible detection of the spatial topology of the universe . our primary objective here is to address this question by focusing our attention on possible topological constraints on the density parameters associated with the baryonic / dark matter ( @xmath0 ) and dark energy ( @xmath1 ) . motivated by the best fit value for the total energy density @xmath3 ( @xmath4 level ) reported by wmap team @xcite , which includes a positively curved universe as a realistic possibility , we shall consider globally homogeneous spherical manifolds , some of which account for the suppression of power at large scales observed by wmap @xcite , and also fits the wmap temperature two - point correlation function @xcite . to this end , in the next section we present our basic context and prerequisites , while in the last section we discuss our main results and present some concluding remarks . within the framework of standard cosmology , the universe is described by a space - time manifold @xmath5 with a locally homogeneous and isotropic robertson walker ( rw ) metric @xmath6 \;,\ ] ] where @xmath7 , @xmath8 , or @xmath9 depends on the sign of the constant spatial curvature ( @xmath10 , respectively ) . the @xmath2space @xmath11 is usually taken to be one of the following simply - connected spaces : euclidean @xmath12 , spherical @xmath13 , or hyperbolic @xmath14 . however , given that the simple - connectedness of our space @xmath11 has not been established , our @xmath2space may equally well be any one of the possible quotient manifolds @xmath15 , where @xmath16 is a fixed point - free group of isometries of the covering space @xmath17 . thus , for example , in a universe whose geometry of the spatial section is euclidean ( @xmath18 ) , besides @xmath19 there are 6 classes of topologically distinct compact orientable @xmath2spaces @xmath11 that admits this geometry , while for universes with either spherical ( @xmath20 ) and hyperbolic ( @xmath21 ) spatial geometries there is an infinite number of topologically non - homeomorphic ( inequivalent ) manifolds with nontrivial topology that can be endowed with these geometries . quotient manifolds are compact in three independent directions , or compact in two or at least one independent direction . in compact manifolds , any two given points may be joined by more than one geodesic . since the radiation emitted by cosmic sources follows geodesics , the immediate observational consequence of a nontrivial detectable spatial topology of @xmath11 is that the sky may show multiple images of radiating sources : cosmic objects or specific correlated spots of the cosmic microwave background radiation ( cmbr ) . at very large scales , the existence of these multiple images ( or pattern repetitions ) is a physical effect that can be used to probe the @xmath2-space topology . in this work , we use the so - called circles - in - the - sky " method ( for cosmic crystallographic methods see , e.g. , refs . @xcite ) , which relies on multiple copies of correlated circles in the cmbr maps @xcite , whose existence is clear from the following reasoning : in a space with a detectable nontrivial topology , the last scattering sphere ( lss ) intersects some of its topological images along pairs of circles of equal radii , centered at different points on the lss , with the same distribution of temperature fluctuations , @xmath22 . since the mapping from the lss to the night sky sphere preserves circles @xcite , these pairs of matching circles will be inprinted on the cmbr temperature fluctuations sky maps regardless of the background geometry and detectable topology . as a consequence , to observationally probe a nontrivial topology on the available largest scale , one should scrutinize the full - sky cmb maps in order to extract the correlated circles , whose angular radii and relative position of their centers can be used to determine the topology of the universe . thus , a nontrivial topology of the space section of the universe may be observed , and can be probed through the circles - in - the - sky for all locally homogeneous and isotropic universes with no assumption on the cosmological density parameters . let us now state our basic cosmological assumptions and fix some notation . in addition to the rw metric ( [ rwmetric ] ) , we assume that the current matter content of the universe is well approximated by cold dark matter ( cdm ) of density @xmath23 plus a cosmological constant @xmath24 . in this standard @xmath24cdm context , for nonflat spaces the scale factor @xmath25 can be identified with the curvature radius of the spatial section of the universe at time @xmath26 , which is given by @xmath27 where here and in what follows the subscript @xmath28 denotes evaluation at present time @xmath29 , @xmath30 is the hubble constant , and @xmath31 is the total density at @xmath26 . in this way , for nonflat spaces the distance @xmath32 of any point with coordinates @xmath33 to the origin ( in the covering space ) _ in units of the curvature radius _ , @xmath34 , reduces to @xmath35 where @xmath36 is an integration variable , and @xmath37 . throughout this paper we shall measure the lengths in unit of curvature radius @xmath38 . a typical characteristic length of nonflat manifolds @xmath11 , which we shall use in this paper , is the so - called injectivity radius @xmath39 , which is defined as the radius of the smallest sphere ` inscribable ' in @xmath11 . an important mathematical result is that @xmath39 , expressed in terms of the curvature radius , is a constant ( topological invariant ) for any given spherical and hyperbolic manifolds . in this work we shall focus our attention in globally homogeneous spherical manifolds , as presented in table [ singleaction ] ( see also its caption for more details ) . these manifolds satisfy a topological principle of homogeneity , in the sense that all points in @xmath11 are topologically equivalent . .the globally homogeneous spherical manifolds are of the form @xmath40 . the first column gives the name we use for the manifolds . the second column displays the covering groups @xmath16 . finally , the remaining columns present the order of the group @xmath16 and the injectivity radius @xmath39 . the cyclic and binary dihedral cases actually constitute families of manifolds , whose members are given by the different values of the integers @xmath41 and @xmath42 . the order of @xmath16 gives the number of fundamental polyhedra needed to fulfill the whole covering space @xmath13 . thus , for example , for the manifold @xmath43 which is the the well - known poincar dodecahedral space , the fundamental polyhedron is a regular spherical dodecahedron , @xmath44 of which tile the @xmath2sphere into identical cells that are copies of the fp . [ cols="^,^,^,^",options="header " , ] to investigate the extent to which a possible detection of a nontrival topology may place constraints on the cosmological density parameters , we consider here the globally homogeneous spherical manifolds . in these @xmath2spaces the number of pairs of matching circles depends on the ratio of the injectivity radius @xmath39 to the radius @xmath45 of lss , which in turn depends on the density parameters ( see ref . @xcite for examples of specific estimates of this number regarding @xmath46 , @xmath47 and @xmath43 ) . nevertheless , if the topology of a globally homogeneous spherical manifold is detectable the correlated pairs will be antipodal , i.e. the centers of correlated circles are separated by @xmath48 , as shown in figure [ cinthesky1 ] . clearly the distance between the centers of each pair of the _ first _ correlated circles is twice the injectivity radius @xmath39 . now , a straightforward use of known trigonometric rules to the right - angled spherical triangle shown in figure [ cinthesky1 ] yields a relation between the angular radius @xmath49 and the angular sides @xmath39 and radius @xmath45 of the last scattering sphere , namely @xmath50 where @xmath39 is a topological invariant , whose values are given in table [ singleaction ] , and the distance @xmath45 of the last scattering surface to the origin in units of the curvature radius is given by ( [ redshift - dist ] ) with @xmath51 @xcite . 0.1 in equations ( [ cosalpha ] ) along with ( [ redshift - dist ] ) give the relations between the angular radius @xmath49 and the cosmological density parameters @xmath1 and @xmath52 , and thus can be used to set bounds on these parameters . to quantify this we proceed in the following way . firstly , as an example , we assume the angular radius @xmath53 . secondly , since the measurements of the radius @xmath49 unavoidably involve observational uncertainties , in order to obtain very conservative results we take @xmath54 . and its uncertainty . ] in order to study the effect of the cosmic topology on the density parameters @xmath55 and @xmath56 , we consider the binary tetrahedral @xmath46 and the binary octahedral @xmath47 spatial topologies ( see table [ singleaction ] ) , to reanalyze with these two topological priors the constraints on these parameters that arise from the so - called _ gold _ sample of 157 sne ia , as compiled by riess _ et al . _ @xcite , along with the latest chandra measurements of the x - ray gas mass fraction in 26 x - ray luminous , dynamically relaxed galaxy clusters ( spanning the redshift range @xmath57 ) as provided by allen _ et al . _ @xcite ( see also @xcite for details on sne ia and x - ray statistics ) . the @xmath46 and @xmath47 spatial topology is added to the conventional sne ia plus clusters data analysis as a gaussian prior on the value of @xmath45 , which can be easily obtained from an elementary combination of ( [ cosalpha ] ) and ( [ redshift - dist ] ) . in other words , the contribution of the topology to @xmath58 is a term of the form @xmath59 . -0.2 cm -0.2 cm figures 2b and 2c ( central and right panels ) show the results of our statistical analysis . confidence regions 68.3% and 95.4% confidence limits ( c.l . ) in the parametric space @xmath52@xmath56 are displayed for the above described combination of observational data . for the sake of comparison , we also show in fig . 2a the @xmath52@xmath56 plane for the conventional sne ia plus galaxy clusters analysis , i.e. , the one without the above cosmic topology assumption . by comparing both analyses , it is clear that a nontrivial space topology reduces considerably the parametric space region allowed by the current observational data , and also breaks some degeneracies arising from the current sne ia and x - ray gas mass fraction measurements . at 95.4% c.l . our sne ia+x - ray+topology analysis provides @xmath60 and @xmath61 ( binary octahedral @xmath62 ) and @xmath63 and @xmath64 ( binary tetrahedral @xmath65 ) . concerning the above analysis it is worth emphasizing three important aspects . first , that the best - fit values depend weakly on the value used for radius @xmath49 of the circle . second , the uncertainty @xmath66 alters predominantly the area corresponding to the confidence regions , without having a significant effect on the best - fit values . third , we also notice that there is a topological degeneracy in that the same best fits and confidence regions for , e.g. , the @xmath46 topology , would equally arise from either @xmath67 or @xmath68 spatial topology . similarly , @xmath47 , @xmath69 and @xmath68 give rise to identical bounds on the density parameters . this kind of topological degeneracy passed unnoticed in refs . @xcite . finally , we emphasize that given the wealth of increasingly accurate cosmological observations , especially the recent results from the wmap , and the development of methods and strategies in the search for cosmic topology , it is reasonable to expect that we should be able to detect it . besides it importance as a major scientific achievement , we have shown through concrete examples that the knowledge of the spatial topology allows to place constraints on the density parameters associated to dark matter ( @xmath0 ) and dark energy ( @xmath1 ) . we thank cnpq for the grants under which this work was carried out . we also thank a.f.f . teixeira for the reading of the manuscript and indication of relevant misprints and omissions . v. sahni and a. starobinsky , int . j. mod d * 9 * , 373 ( 2000 ) ; j.e . peebles and b. ratra , rev . mod . phys . * 75 * , 559 ( 2003 ) ; t. padmanabhan , phys . rep . * 380 * , 235 ( 2003 ) ; j.a.s . lima , braz . j. phys . * 34 * , 194 ( 2004 ) . m. lachize - rey and j .- p . luminet , phys . rep . * 254 * , 135 ( 1995 ) ; g.d . starkman , class . quantum grav . * 15 * , 2529 ( 1998 ) ; j. levin , phys . rep . * 365 * , 251 ( 2002 ) ; m.j . rebouas and g.i . gomero , braz . j. phys . * 34 * , 1358 ( 2004 ) . astro - ph/0402324 ; m.j . rebouas , a brief introduction to cosmic topology , in _ proc . xith brazilian school of cosmology and gravitation _ , eds . m. novello and s. e. perez bergliaffa ( americal institute of physics , melville , new york , 2005 ) aip conference proceedings vol . * 782 * , p 188 ( 2005 ) . e. komatsu et al . , * 148 * , 119 ( 2003 ) ; h.k . eriksen , f.k . hansen , a.j . banday , k.m . gorski , and p.b . lilje , astrophys . j. * 605 * , 14 ( 2004 ) ; c.j . copi , d. huterer , and g.d . starkman , phys . d * 70 * , 043515 ( 2004 ) . spergel et al . , astrophys . j.suppl . * 148 * , 175 ( 2003 ) . m. tegmark , a. de oliveira - costa , and a.j.s . hamilton , phys . d * 68 * , 123523 ( 2003 ) ; a. de oliveira - costa , m. tegmark , m. zaldarriaga , and a. hamilton , phys . d * 69 * , 063516 ( 2004 ) ; j.r . weeks , astro - ph/0412231 ; p. bielewicz , h.k . eriksen , a.j . banday , and k.m . gorski , and p.b . lilje , astro - ph/0507186 ; k. land and j. magueijo , phys . * 95 * , 071301 ( 2005 ) . a. bernui , b. mota , m.j . rebouas , and r. tavakol , astro - ph/0511666 ; k. land and j. magueijo , mon . not . astron . 357 * , 994 ( 2005 ) . luminet , j. weeks , a. riazuelo , r. lehoucq and j .- p . uzan , nature * 425 * , 593 ( 2003 ) ; n.j . cornish , d.n . spergel , g.d . starkman , and e. komatsu , phys . * 92 * , 201302 ( 2004 ) ; j. gundermann , astro - ph/0503014 ; b.f . roukema , b. lew , m. cechowska , a. marecki , and s. bajtlik , astron . astrophys . * 423 * , 821 ( 2004 ) . r. lehoucq , m. lachize - rey , and j .- p luminet , astron . astrophys . * 313 * , 339 ( 1996 ) ; b.f . roukema and a. edge , _ mon . not . soc . _ * 292 * , 105 ( 1997 ) ; b.f . roukema , class . quantum grav . * 15 * , 2645 ( 1998 ) ; r. lehoucq , j .- p luminet , and j .- uzan , astron . astrophys . * 344 * , 735 ( 1999 ) ; h.v . fagundes and e. gausmann , phys . a * 238 * , 235 ( 1998 ) ; h.v . fagundes and e. gausmann , phys . a * 261 * , 235 ( 1999 ) ; j .- uzan , r. lehoucq and j .- p . luminet , astron . astrophys . * 351 * , 766 ( 1999 ) ; g.i . gomero , m.j . rebouas , and a.f.f . teixeira , int . d * 9 * , 687 ( 2000 ) ; r. lehoucq , j .- uzan , and j .- p luminet , astron . astrophys . * 363 * , 1 ( 2000 ) ; g.i . gomero , m.j . rebouas , and a.f.f . teixeira , phys . a * 275 * , 355 ( 2000 ) ; g.i . gomero , m.j . rebouas , and a.f.f . teixeira , class . quantum grav . * 18 * , 1885 ( 2001 ) ; g.i . gomero , a.f.f . teixeira , m.j . rebouas and a. bernui , int d * 11 * , 869 ( 2002 ) ; a. marecki , b. roukema , and s. bajtlik , astron . astrophys . * 435 * , 427 ( 2005 ) . gomero , m.j . rebouas and r. tavakol , class . quantum grav . * 18 * , 4461 ( 2001 ) ; g.i . gomero , m.j . rebouas , and r. tavakol , int . j. mod . a * 17 * , 4261 ( 2002 ) ; j.r . weeks , r. lehoucq , and j .- uzan , class . quantum grav . * 20 * , 1529 ( 2003 ) ; j.r . weeks , mod . phys . lett . a * 18 * , 2099 ( 2003 ) ; g.i . gomero and m.j . rebouas , phys . lett . a * 311 * , 319 ( 2003 ) ; b. mota , m.j . rebouas , and r. tavakol , class . quantum grav . * 20 * , 4837 ( 2003 ) ; b. mota , g.i . gomero , m. j. rebouas and r. tavakol , class . quantum grav . * 21 * , 3361 ( 2004 ) . s.w . allen , r.w . schmidt , h. ebeling , a.c . fabian , and l. van speybroeck , mon . not . soc . * 353 * , 457 ( 2004 ) . j.a.s . lima , j.v . cunha and j.s . alcaniz , phys . d * 68 * , 023510 ( 2003 ) ; j.s . alcaniz and z .- h . zhu , phys . d * 71 * , 083513 ( 2005 ) ; d. rapetti , s.w . allen and j. weller , mon . not . . soc . * 360 * , 546 ( 2005 ) .

given the wealth of increasingly accurate cosmological observations , especially the recent results from the wmap , and the development of methods and strategies in the search for cosmic topology , it is reasonable to expect that we should be able to detect the spatial topology of the universe in the near future . motivated by this , we examine to what extent a possible detection of a nontrivial topology of positively curved universe may be used to place constraints on the matter content of the universe . we show through concrete examples that the knowledge of the spatial topology allows to place constraints on the density parameters associated to dark matter ( @xmath0 ) and dark energy ( @xmath1 ) .

a series of rare elementary processes involving more than two particles in the final state are going to be measured with increasing precision . the multiplicity of the final state makes it difficult to extract predictions by the standard gauge theories even if semplifications arise when either partecipants are all massless or only some of the external particles are massive . however more accurate rate measurements of processes with heavy quark hadrons in the final state will soon be available as is the case of the chorus experiment where direct evidence for the associate charm production in charged current neutrino nucleon scattering has been shown @xcite . in the one loop calculation of such processes we encounter pentagon integrals with a massive line as skeched in figure 1 where massive particles are bold , massless ones thin and dashed ones can be either massive or not . in general the inclusion of masses makes things more involved , although the calculation simplifies when either external masses are equal to each other or they are equal to the internal masses or both eventualities occur as it is often the case in normal gauge theories . recently a lot of progress has been made in the technics for perturbative calculations with different approaches . a non - comprehensive list is given in @xcite and reference therein and in @xcite , @xcite , @xcite , @xcite and @xcite . in particular adopting the dimensional regularization approach for feynman parametrized integrand the authors of ref.@xcite derived simplifications and recursion formulas by the implementation of algebraic technic . using these methods the problem of the evaluation of a one loop @xmath0 points scalar integral is translated to the evaluation of a combination of @xmath1 points scalar integrals and the original @xmath0 points integral in @xmath2 dimensions ; moreover the original @xmath0 points one loop integral can be represented as the solution of a partial differential equation system . in the present paper we use this approach to perform the calculation of the pentagon integral represented in figure 1 . other massive pentagon integrals have been recently evaluated in next to leading order calculations of processes in which an higgs particle can be generated at hadron colliders . in particular two independent groups report the nlo corrections for the process in which an higgs particle is generated together with a @xmath3 pair , @xcite and @xcite . another nlo calculation involving massive pentagon integrals is given in @xcite in which the final state considered consists of an higgs particle plus two jets . the general methods employed here do not concern with the specific processes and the results must be analitically continued to describe a specific process . finally only the most simple tensor integral is given while we postpone other cases to a dedicated paper @xcite . the paper is organized as follow : in section ii relevant formulas from ref.@xcite are collected , in section iii they are applied to the scalar massive pentagon represented in figure 1 transforming it in a combination of four points integrals ; section iv is devoted to four point integrals evaluation and in section v more simple tensor integral ( vector ) is given with the conclusions . the initial condition for the differential equations originated in the four points evaluation are calculated in the appendix . the starting point is the integral in @xmath4 dimensions @xmath5 with the momenta @xmath6 taken to be outgoing , @xmath7 and @xmath8 applying feynman parametrization , wick rotating and integrating over loop momentum this integral can be cast in the form @xmath9 @xmath10 with @xmath11 and the matrix @xmath12 given by @xmath13 with @xmath14 and @xmath15 for @xmath16 . we will not repeat the derivations obtained in ref.@xcite but , to introduce notation and to be self - consistent , in the rest of this section we just collect relevant formulas that will be used in section iii and iv . performing a projective transformation @xcite with parameters @xmath17 in such a way that the denominator in eq.([aaa ] ) has no @xmath17 dependence the definition of a new matrix follows ( indices are not summed ) @xmath18 using the following definitions @xmath19 the authors of ref.@xcite find @xmath20\\ \frac{1}{n-4 + 2\varepsilon}\,\frac{\partial\hat{i}_n}{\partial\alpha_i}= \frac{1}{2n_n}\left[\sum_{j=1}^n\eta_{ij}\,\hat{i}_{n-1}^{(j)}+ ( n-5 + 2\varepsilon)\,\gamma_i\,\hat{i}_n^{d=6 - 2\varepsilon}\right]\end{aligned}\ ] ] where @xmath21 stands for the @xmath1 integral with the denominator obtained from an @xmath22 integral eliminating the propagator between legs @xmath23 and @xmath24 ; once feynman parameter has been introduced in the usual way for @xmath22 the denominator in @xmath21 is obtained putting @xmath25 . by the observation that @xmath26 and @xmath27 are finite in @xmath28 dimensions , performing one - loop calculation one can limit to evaluate @xmath29 taking only the divergent part from the @xmath30 integrals in eq.([bbb ] ) . to write down the integral in figure 1 we set @xmath31 , @xmath32 and @xmath33 giving @xmath34 and @xmath35 with @xmath36 given in eq.([ccc ] ) and the matrix @xmath12 given by @xmath37 with @xmath38 . we define @xmath39 and in the following we will assume @xmath40 . performing the projective transformation with @xmath41 we get for the @xmath42 matrix in eq.([rrhhoo ] ) @xmath43 with @xmath44 @xmath45 if @xmath46 we only have to take @xmath47 in eq.([eee ] ) . the coefficient relevant for the evaluation of the pentagon by eq.([ppp ] ) are given in the table 1 , keeping apart the case @xmath48 . due to the presence of masses we have not cyclic relations between the coefficients but only the relations @xmath49 table 1 . coefficients to be used in eq.([ppp ] ) [ cols="^,^,^ " , ] in terms of new kinematical variables @xmath17 , @xmath50 and @xmath51 the denominator in the @xmath27 integral represented in eq.([ccc ] ) is given by @xmath52 and the four points denominators in the @xmath53 integrals in eq.([ppp ] ) can be obtained putting @xmath54 to zero in the expression above . it is easy to verify the relations @xmath55 in the next section we proceed to the evaluation of @xmath56 , @xmath57 and @xmath58 using the set of partial differential eqs.([bbb ] ) . here we evaluate the integrals @xmath56 and @xmath57 , corrsponding to massive boxes with an internal massive line , in the variables defined in eqs.([c1 ] , [ c2 ] , and [ c3 ] ) and translate the integrals @xmath58 that are well known and correspond to massive boxes with massles internal lines . after putting @xmath59 in eq.([ggg ] ) we have for the denominator in @xmath56 @xmath60 before solving the integral we perform the following kinematic transformation : @xmath61 in terms of the new variables we get for the denominator : @xmath62 and @xmath63 the only divergent three points functions extracted by the expression above are @xmath64 and @xmath65 obtained putting @xmath66 and @xmath67 respectively ; these correspond to two two - mass triangles , while the other two obtained putting @xmath68 and @xmath69 respectively are three - mass triangles checked to be finite . at the @xmath70 we have @xmath71 the system of partial differential equations in eq.([bbb ] ) is then given by @xmath72 \nonumber \\ \frac{\partial\hat{i}_4^{(1)}}{\partial c_5}&= & \frac{2\,\gamma(1+\varepsilon)}{c_4-c_5 } \ , \log \left ( \frac{c_5}{c_4 } \right)\end{aligned}\ ] ] with the solution @xmath73.\end{aligned}\ ] ] the integration constant @xmath74 is evaluated in the appendix and its value is @xmath75 where @xmath76 is the dilogarithm function and @xmath77 . afetr some manipulation we have @xmath78 reintroducing the original variables inverting eq.([tr1 ] ) we get for @xmath56 : @xmath79 being @xmath56 independent from @xmath50 its value does not change in the limit @xmath80 . here and in the following subsection we proceed performing the same steps as in the derivation of @xmath56 . the limit @xmath80 now gives a different situation ; in fact in this limit there will be three divergent three - point integrals extracted by @xmath57 so as explained in @xcite the limit procedure is not smooth and the two case have to be taken separately . in this case @xmath57 is a three external mass box but , differently from @xmath56 , it has all external masses different from each other and so it needs evaluation . after putting @xmath66 in eq.([ggg ] ) we have for the denominator in @xmath57 @xmath82 rescaling the variables with @xmath83 we get @xmath84 giving @xmath85 and @xmath86 the only divergent three points functions extracted by the expression above are @xmath64 and @xmath65 obtained putting @xmath59 and @xmath67 respectively ; these correspond to two two - mass triangle , while the other two obtained putting @xmath68 and @xmath69 respectively are three - mass triangle checked to be finite . at the @xmath70 we have @xmath87 the system in eq.([bbb ] ) is then given by @xmath88 \nonumber \\ \frac{\partial\hat{i}_4^{(2)}}{\partial c_5}&= & \frac{2\,\gamma(1+\varepsilon)}{c_5-\delta \,c_4 } \log \left ( \frac{\delta \,c_4}{c_5 } \right)\end{aligned}\ ] ] with the solution @xmath89.\end{aligned}\ ] ] the integration constant @xmath90 is evaluated in the appendix and its value is @xmath91 after some maipulation we have @xmath92\ ] ] reintroducing the original variables inverting eq.([tr2 ] ) we get for @xmath57 : @xmath93 \\\end{aligned}\ ] ] in this case @xmath57 is a two external mass box . putting @xmath47 the denominator in eq.([den2 ] ) became @xmath95 rescaling the variables as in eq.([tr2 ] ) we get @xmath96 the divergent three point functions @xmath64 , @xmath65 and @xmath97 are obtained putting @xmath98 , @xmath99 and @xmath67 respectively . @xmath100 while @xmath101 is a three - mass triangle checked to be finite . the partial differential equation system is given by @xmath102 with the solution @xmath103\end{aligned}\ ] ] the integration constant @xmath104 is evaluated in the appendix and its value is @xmath105 after some manipulation we find @xmath106\ ] ] reintroducing the original variables we have @xmath107\ ] ] putting @xmath67 in eq.([ggg ] ) we eliminate the massive propagator and obtain the easy ( opposite ) two mass box @xcite or the one external massive box if we take respectively @xmath108 or @xmath94 . these integrals are well - known and are reported also in @xcite . here we just put these integrals in the kinematics specified in section 3 . after putting @xmath67 in eq.([ggg ] ) the denominator is given by @xmath109 using eq.(4.44 ) from the third paper in ref.@xcite the integral in the kinematics of section 3 reads @xmath110\end{aligned}\ ] ] putting @xmath47 in eq.([den3 ] ) the denominator of this integrals is given by @xmath111 using eqs.(4.27 , 4.40 ) from the third paper in ref.@xcite the integral in the kinematics of section 3 reads @xmath112\end{aligned}\ ] ] an expression for the scalar pentagon integral shown in figure 1 can be built via eqs.([ppp ] ) , ( [ rel ] ) , the four points integrals evaluated in the last section and the coefficients in table 1 . the expressions for @xmath27 are very long and are not reported . more familiar kinematics is realized by replacing the variables @xmath17 , @xmath50 and @xmath113 with their definitions in terms of @xmath114 , @xmath115 and @xmath116 . tensor integrals will be considered in a separate paper @xcite , however the simplest one of them , the vector integral , is related to the scalar integrals with one feynman parameter in the numerator by the following relation @xmath117\rightarrow i_n^d[\mathcal{p}^\mu]\ ] ] in which the arrow means integration over loop momentum @xmath118 , the integrand numerator is in the square brackets and @xmath119 with @xmath120 given in eqs.([pi ] ) . the integrals @xmath121 $ ] can be evaluated by @xcite @xmath122=\frac{1}{2n_5}\sum_{i=1}^5\eta_{ji}\hat{i}_{4}^{(i)}+ { \mathcal o}(\varepsilon)\ ] ] where @xmath123 defined in eq.([fff ] ) is deduced by @xmath124 given in eq.([eee ] ) @xmath125 higher tensor integrals can be evaluated considering that they are linked to scalar integrals with more powers of feynman parameters in the numerator @xcite . such a decompositioncan can also be organized in a way that drastically reduces numerical instabilities genarated by the presence of inverse powers of gram determinants @xcite . besides the deep inelastic case mentioned in the introduction , the results obtained in the present paper with @xmath81 can be useful in the evaluation at one loop of the decay amplitude of a real @xmath126 boson or a virtual photon in a heavy quark - antiquark pair and two light quarks . let us consider the case in which all massless particles and @xmath127 are gluons , then the pentagon studied with @xmath94 can be identified with one of the four pentagon in the perturbative evaluation of the one - loop associated production of heavy quark in the gluon - gluon - fusion with a gluon in the final state ( @xmath128 ) ; in this case , indeed , pentagons are found in which the propagators form chains with @xmath129 , @xmath130 , @xmath131 and @xmath132 equal mass fermions the first of which is calculated in the present paper while the other ones can be calculated analogously . the author gratefully acknowledges prof . p. strolin who supported the present research , profs . g. cosenza and a. della selva for discussions , dr . g. de lellis for suggesting the topic and for many comments on the manuscript , dr . d. falcone for suggestions on the manuscript , drs . r. mertig and f. orellana for help with @xmath133 @xcite , dr . f. di capua and dr . l. scotto lavina for a quick help with x - fig and dr . g. celentano for help with latex . in this appendix we report the calculation of the integration constants for the four points integral of section four systematically neglecting @xmath134 terms . instead of reporting all length passages , we give the steps that can be followed by programs of function manipulation like the used mathematica . the point chosen to evaluate @xmath56 is @xmath135 where the expression in eq.([sol1 ] ) gives @xmath136 the expression for the integral at the point selected deduced using eq.([quat ] ) and the first of eqs.([fff ] ) is @xmath137 the factor @xmath130 is given by @xmath138 . renaming @xmath99 with @xmath139 , @xmath140 with @xmath141 and @xmath142 with @xmath143 , and performing the transformation @xmath144 , @xmath145 and @xmath146 we arrive at the expression @xmath147 putting apart the gamma function for the moment , the @xmath143 integration gives @xmath148 the two integrals can be evaluated by shifting both in @xmath141 @xmath149 simplifying the @xmath139 integral @xmath150 and inverting the integration order @xcite . after some manipulation and expanding some hypergeometric and generalized hypergeometric functions the result is @xmath151 finally , making the substitution @xmath152 in eq.([n1 ] ) and taking into account eq.([ap1 ] ) we find @xmath153 in eq.([c01 ] ) . [ [ integration - constant - for - hati_42-integral - with - q2-neq-0 ] ] integration constant for @xmath57 integral with @xmath154 ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ to evaluate the integration constant we evaluate the integral in the point : @xmath155 with @xmath156 the integrations are trivial but the expression is very long . the @xmath157 chosen can not be simultaneously positive so we checked the result in the point @xmath158 where the expression in eq.([sol2 ] ) gives @xmath159 the expression for the integral at the point selected is @xmath160 renaming @xmath98 with @xmath139 , @xmath140 with @xmath141 and @xmath142 with @xmath143 , and performing the transformation @xmath144 , @xmath145 and @xmath146 we arrive at the expression @xmath161 putting apart the gamma function for the moment , the @xmath143 integration gives @xmath162 performing the @xmath139 integration before and adding and subtracting terms we find eq.([ss2 ] ) . [ [ integration - constant - for - hati_42-integral - with - q2 - 0 ] ] integration constant for @xmath57 integral with @xmath94 ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ the point chosen to evaluate @xmath57 is given by @xmath163 in which the expression in eq.([sol3 ] ) gives @xmath164 the expression for the integral at the point selected is @xmath165 renaming @xmath140 with @xmath139 , @xmath142 with @xmath141 and @xmath166 with @xmath143 , and performing the transformation @xmath144 , @xmath145 and @xmath146 we arrive at the expression @xmath167 putting apart the gamma function for the moment , the @xmath143 integration gives @xmath168 performing the shift @xmath149 gives @xmath169 finally performing the @xmath139 integration before and adding and subtracting terms before expanding in @xmath170 the result is @xmath171 substituting @xmath172 in eq.([n3 ] ) and taking into account eq.([ap3 ] ) we find @xmath104 in eq.([c03 ] ) .

* abstract * in this paper we present the calculation of a scalar pentagon integral with two consecutive massive external legs having an equal mass propagator embedded between them . we also deal with the two situations where the farest external leg is either massive or not . the relevance of the calculation comes from its application in many perturbative qcd calculations as well as in qcd corrections for weak precesses . = 0.5 cm = 0.0 cm = 0.75 cm dsf-22/2002 * pentagon integrals for heavy quark physics * dipartimento di scienze fisiche , universit di napoli , complesso di monte s. angelo , via cintia , napoli , italy

over 200 extrasolar giant planets have now been discovered by radial velocity surveys . fourteen of these planets have been observed transiting their parent star , allowing an accurate determination of their radius and mean density . for half of these objects , notably the first ever discovered transit hd209458b , the predicted theoretical radius lies several @xmath0 s ( up to @xmath1 ) below the observed mean value @xcite . the various scenarios proposed so far to solve this discrepancy either have been rejected on observational or theoretical arguments @xcite or lack an identified robust mechanism to convert surface kinetic energy into thermal energy at depth @xcite . a fair conclusion of these studies is that an important physical mechanism is probably missing in our present description of at least some short period planets , for which we have a radius determination , but possibly of all extrasolar or even solar giant planets . according to the conventional core - accretion model for planet formation @xcite , planets are believed to have a substantial enrichment in heavy elements compared with their parent star , with a total of @xmath2 for a jupiter - mass object @xcite . observational constraints of jupiter and saturn show that these planets do have a significantly enhanced z - abundance compared with the sun , with a global mean mass fraction @xmath3=@xmath4 10 - 20% @xcite . in all these calculations , the big planetesimals are generally supposed to drown to the core during the early phase of solid accretion , while the smaller ones are distributed _ uniformly _ throughout the envelope , leading to a uniform heavy element abundance . this is a rather simplistic description of the planet internal structure , which implies ( i ) well defined interfaces between the central core and the ( highly diffusive ) h / he rich envelope and ( ii ) very efficient large - scale thermal convection throughout the entire gaseous envelope . the observed atmospheric abundances of jupiter and saturn , however , seem to require the redistribution of a subsequent fraction of heavy elements in the interior of these planets @xcite . in this letter , we explore the consequences of the presence of an initial compositional gradient in the envelope , as a result of either early planetesimal accretion or subsequent core erosion , and of the resulting less efficient heat transport and compositional mixing on the fate of gaseous planets . we show that layered convection , if it occurs as a result of this compositional gradient , might be the lacking physical mechanism to explain the transiting planet abnormally large radii . the presence of a positive compositional gradient , i.e. a gradient of mean molecular weight @xmath5=@xmath6 , tends to stabilize the fluid against convective instability according to the ledoux stability condition : @xmath7 where @xmath8 and @xmath9 denote the usual temperature and adiabatic gradients , respectively , and @xmath10=@xmath11 , @xmath12=@xmath13 . in most of the giant planet interiors , superadiabaticity is extremely small , with @xmath14 , so a small molecular weight gradient over a typical mixing length size region can affect significantly and even damp out convection . in convective systems where buoyancy effects of ( destabilizing ) heat and ( stabilizing ) composition are opposed , the process leads generally to quasi - static uniformly mixed convective layers separated by small diffusive interfaces with steep gradients , @xmath15 . such stable layered convection is indeed observed in some areas of the earth s oceans , due to the presence of the stabilizing salt gradient ( thermohaline convection ) leading to a stratified step - like temperature profile with stable boundary layers @xcite . laboratory experiments have also confirmed this layering @xcite . once formed , the stratification is stable provided the compositional gradient remains large enough at the interfaces to satisfy eq.([ledoux ] ) . would such a stratification occur in planetary interiors , the layered part of the interior can be considered as a _ semiconvection _ zone with a reduced efficiency to transport the internal heat and composition flux compared with large - scale convection . one may argue that the conditions in planetary interiors differ from the ones in the oceans or in the experiments . characteristic thermal diffusivity in h / he planetary interiors , dominated by electronic transport in the central , ionized parts , and by molecular motions in the outer envelope , lies in the range @xmath16-@xmath17 @xmath18 , while the kinematic viscosity is @xmath19-@xmath17 @xmath18 @xcite . the characteristic prandtl number thus ranges from @xmath20 @xmath21 to 1 . these values do not differ by large factors from the ones characteristic in the oceans or in laboratory experiments , @xmath22 - 10 , in contrast to the ones characteristic of stellar conditions ( @xmath23 ) . therefore , a layering process in giant planet interiors can not be excluded and it is worth exploring the consequences on the planet evolution . such layered convection may occur near the discontinuity in composition at the boundary of the central rocky - icy core or in chemically inhomogeneous regions in the interior , reminiscent of the early planetesimal accretion episodes . there is presently no widely accepted description of semiconvection . water - salt experiments @xcite show that a series of quasi - static convective layers separated by diffusive interfaces develop when a balance is reached between the variation of potential energy ( i.e. of buoyancy ) due to mixing at the interface and the kinetic energy of the eddies available at the interface . this translates into a critical richardson number @xmath24=@xmath25 of order 1 to 10 , where @xmath26 is the density contrast between the diffusive and convective layers , @xmath27 is the gravity and @xmath28 is the kinetic energy of the convective flow , of characteristic average length scale @xmath29 and rms velocity @xmath30 . guided by experimental results @xcite and energetic arguments , stevenson ( 1979 ) , in a wave description of semiconvection , showed that , should layers form as a result of small - scale wave breaking whereupon the compositional gradient is redistributed , they would be stable if @xmath31 where the inverse lewis number @xmath32=@xmath33=@xmath34 is the ratio of the solute microscopic diffusivity to the thermal diffusivity . spruit s ( 1992 ) stability condition is less restrictive , since layered formation is supposed to always occur , and is given essentially by eq.(1 ) . under jovian planet conditions , typical values are @xmath35-@xmath36 @xmath18 @xcite , then @xmath37 , so that , according to this criterion , diffusive layers could be at least marginally stable in giant planet interiors . it is worth noting that the molecular to thermal diffusivity ratio is the same for a h / he mixture under jovian interior conditions as for salty water , @xmath38 , so that the extent of the solute versus thermal layer is about the same . since , according to experiments and condition ( 2 ) , this ratio is the relevant criterion for stability of the layers , this adds some support to the planetary case . in a layered convection stratification , heat is carried away from the interfaces by descending and ascending plumes in the overturning regions while transport across the interface occurs by diffusion . because of the boundary layers , only a part of the fluid transports heat efficiently . on average , one has `` convective - like '' motions having a much shorter length scale than for ordinary convection . the _ thermal _ thickness of the diffusive layers , @xmath39 , is determined by a balance between the thickening due to diffusion and the entrainment due to convective motions so the convective time , @xmath40 , in the mixed layer of size @xmath29 must be comparable to the thermal diffusion time , @xmath41 , across the boundary layer @xcite . this yields @xmath42 and @xmath43 for the thickness of the heat and compositional interfacial layers , respectively . the number @xmath44 of layers is of course very uncertain . it can be crudely estimated as follows . the heat flux @xmath45 transported by convection in each mixed layer is the mass flux carried by the plumes fed from the diffusive layers , and thus of width comparable to these layer thickness , times the energy variation across the convective layer , @xmath46 : @xmath47 for a semiconvective region extending over a planet - size region , the total number of layers is thus given by : @xmath48 using characteristic numbers for jovian planet conditions ( with @xmath49 ) , one gets @xmath50-@xmath51 as a rough estimate . if convection is inhibited , however , the smaller heat flux and larger superadiabaticity require less layers . we have conducted calculations following the evolution of a template jupiter - mass planet , representative of hd209458b and similar short - period planets , with a global metal content @xmath52 , including a 6 @xmath53 core , i.e. @xmath54 ( @xmath55 ) , in agreement with previously mentioned planet formation models and jupiter and saturn s observational constraints . the amount of heavy elements is distributed initially throughout the planet following a gradient , distributed within a certain number of boundary layers @xmath44 where condition ( 1 ) is fulfilled . the layers are located within the inner @xmath56 by mass ( 60% in radius ) of the planet , where h and he are fully ionized , to ensure high enough thermal conductivity . the present calculations have been done with @xmath44=50 and @xmath44=100 ; the width of each boundary layer corresponds to @xmath57 cm @xmath58 . a larger number of layers would be computationally too difficult to resolve correctly . these boundary layers are separated by larger convective , mixed layers , with a uniform composition ( @xmath59 ) , where the usual mixing - length formalism applies . the sizes of the boundary and mixed layers ( @xmath60 ) obey the aforementioned relationships . the heat flux @xmath61 and solute flux @xmath62 in the boundary layers are calculated with the appropriate diffusion equations : @xmath61=@xmath63 , where @xmath64=@xmath65 and @xmath66 denote the conductive @xcite and radiative @xcite mean opacities , respectively , and @xmath62=@xmath67 , where @xmath68=@xmath69 , with @xmath70 , is the mean concentration gradient across the layer . conduction remains efficient enough in the thin boundary layers to fulfill condition ( 1 ) . because diffusion limits the heat transport , the internal heat flow of the planet is significantly reduced compared with that of a fully convective object . the signatures of double - diffusive convection in a planetary interior are thus a reduced heat output and a larger radius compared with an object where heat is transported efficiently by large - scale convection . this is illustrated in fig . 1 , which compares the evolution of the radius and thermal intrinsic luminosity of the planet in both cases . the excellent agreement with the otherwise unexplained observed radii of hd209458b and similar irradiated planets suggests that diffusive convection might be taking place in the interior of at least certain giant planets . as seen , the expected luminosity at young ages is more than one order of magnitude fainter than that of a fully convective planet evolving from a comparable initial state . the observational confirmation of the present scenario would be either the determination of an exoplanet temperature or luminosity at young ages , is smaller than the absorbed and reflected contributions of the incident stellar flux , @xmath71 . for long - period planets or for planed telescopes like the lbt or the jwst , dedicated to infrared planet searches , however , the planet intrinsic luminosity can be determined . ] or the observation of an inflated radius for a transiting planet at large enough orbital distance , @xmath72 au for a solar - type parent star , for stellar irradiation not to affect the planet s internal structure . figure 1 illustrates also the dependence of the evolution upon the number of layers . less boundary layers implies larger convective layers and thus more efficient heat transport , as illustrated by the more rapidly decreasing radius in the 50-layer calculations . a key question is to know if diffusive interfaces can persist on time scales comparable to the characteristic time for the evolution of the planet . according to the aforementioned critical richardson number criterion , supported by experiments @xcite , a quantitative argument is that if the average kinetic energy in the convective layers is smaller than ( a fraction of ) the potential energy wall of the interface , convection can not penetrate deeply into this latter and significant entrainment across the interface can not occur . this implies that the molecular diffusion time scale be long enough . this latter can be estimated for the entire stack of layers , distributed over a region of size @xmath73 in the planet ( presently @xmath74 cm ) . the flux of element across an interface is @xmath75 , where @xmath76 is the jump in the element mass fraction at each interface while @xmath77 is the total variation over the entire semiconvective region . the time scale to redistribute the entire gradient over the entire region is then @xmath78 10 gyr . this admitedly crude estimate shows that the stable diffusive convection configuration might last long enough to affect substantially the evolution . with the typical value @xmath79 , about 10% of the initial gradient @xmath77 has been transported by diffusion over a gyr , as confirmed by our numerical calculations . in principle , the compositional gradient thus remains large enough during the evolution for the ledoux criterion to remain valid in a majority of layers . in other words , the temperature jump at interfaces is too small to offset the molecular weight stabilization of interfaces ( @xmath80 ) . the composition and temperature profiles in our calculations at 5 gyr are portrayed in fig . 2 , with @xmath81 and @xmath82 k at each diffusive interface . note that , if layers form in sequence through turbulent entrainment or from sporadically breaking internal waves generated by oscillatory instabilities , interfaces may be dynamically renewed with time , if some compositional gradient or stirring effects remain present . such a process occurs in laboratory systems and oceans . different reasons can be advocated for the cause of the initial compositional gradient . this latter can be inherited from the formation process . large incoming planetesimals could disseminate part of their constituents , iron , silicates , ices , by ablation and break - up as they penetrate the building gaseous envelope @xcite . note also that accretion will not proceed homogeneously as capture mechanisms differ for the gas ( h , he ) , ice ( essentially c , o , n ) and rock ( silicates and iron ) components . this will increase substantially the compositional gradients before the core is reached . as mentioned earlier , even modest gradients can easily offset superadiabatic excess over planet - size regions , preventing large - scale convective motions . a compositional gradient might also result from disruption and redistribution of the core due to a giant impact or erosion at the core - envelope interface because of metallic hydrogen high diffusivity , leading to a core diluted into a fraction of the planet @xcite . the redistribution of these elements might be partially inhibited by diffusive processes , forming diffusive interfaces because of the opposite buoyancy effects of heat and composition . furthermore , when accreting the envelope , part of the outermost regions of the protoplanet might be nearly isothermal @xcite , which favors the stability of a compositional gradient . interestingly , when distributing the layers in the _ outer _ 10% by mass ( @xmath83 by radius ) of the planet , where _ radiative _ thermal diffusivity starts to dominate over conductive diffusivity , we get an effect similar to the one portrayed in fig . the situation for the formation of diffusive convection is particularly favorable for short - period exoplanets for several reasons . first of all , a substantial fraction of the gaseous envelope has been eliminated by evaporation @xcite , leading to a larger metal fraction . second of all , for short - period exoplanets , the numerous collisions tend to eject the gas , leading to a larger enrichment in planetesimals than for the other planets . third of all , the higher internal temperatures for short - period , irradiated planets , favor ( i ) ionization of the various elements and thus the thermal conductivity , ( ii ) solubility of the core material into the envelope . at last , because of the stellar irradiation , the outer layers of short - period planets are isothermal and not adiabatic . the aim of the present letter is to suggest an alternative , possibly important energy transport mechanism in giant planet interiors and to explore its effects on the evolution . these calculations provide a consistent description of the evolution of giant planets , _ with a metal enrichment in agreement with observational constraints _ , in case heat is transported by layered convection . assuming an initial compositional stratification within a certain number of double - diffusive interfaces , diffusive and convective transport in the respective layers are calculated consistently during the evolution . only the outermost 40% in radius ( 70% in mass ) of the planet can convect freely . these calculations , however , can not be expected to give an accurate description of the onset and stability of layered convection . there is presently no accurate treatment of this mechanism under conditions characteristic of giant planets . the only attempt to study the onset of double - diffusive layer formation at low prandtl numbers ( @xmath84 ) @xcite remains inconclusive . indeed , insufficient numerical resolution and artificially enhanced viscous and molecular diffusivities in the simulations might suppress small - scale motions / instabilities and the formation of a statistically steady state of intermitent diffusive layers . simulations of vertical salinity in water , @xmath85=@xmath86 , on the other hand , well reproduce the experiments and confirm the formation of quasi - static convective layers separated by diffusive interfaces above a critical richardson number @xcite . even though the present calculations rely on some uncertain ground , the agreement with the puzzling and otherwise unexplained observed radius of hd209458b and other abnormally large exoplanets leads to the conclusion that layered convection might be taking place in at least some planets and could explain their particular properties . this transport mechanism yields a much reduced heat escaping rate compared with a homogeneous adiabatic structure . even if a stable layered configuration does not occur , overstable modes of convection ( fullfilling eq.(1 ) but not eq.(2 ) ) , due to the presence of opposite diffusive processes ( composition and heat ) of different efficiencies , can lead to the growth of small - scale fluid oscillations @xcite . overstability , however , is more similar to an enhanced diffusion process than to a convective mixing process , with a much smaller energy transport efficiency , as shown by experiments @xcite . the onset or persistence of layered or overstable ( oscillatory ) convection might require optimal conditions , inherited from particularly favorable formation or evolution histories ( e.g. late accretion of large unmixed planetesimals or giant impacts stirring up completely the planet interior ) . the present paper suggests that , under such appropriate conditions , the heat transport mechanism in giant planet interiors can be severely affected , decreasing the efficiency of or even inhibiting large - scale convection . this should motivate 3d investigations of convection in the presence of a stabilizing compositional gradient under conditions suitable to giant planets and the search for transits at larger orbital distances . alibert , y. , mordasini , c. , benz , w. & winisdoerffer , c. , 2005 , , 434 , 343 baraffe , i. et al . , 2005 , , 436 , 47 baraffe , i. , alibert , y. , chabrier , g. & benz , w. , 2006 , , 450 , 25 ferguson et al . , j. , 2005 , , 623 , 585 fernando , h. , 1989 , , 209 , 1 guillot , t. , stevenson , d. , hubbard , w.b . , saumon , d. , 2004 , jupiter : the planet , satellites and magnetosphere , cambridge university press iaroslavitz , e. & podolak , m. , 2007 , icarus , 187 , 600 levrard , b. et al . , 2007 , , 462 , l5 linden , p.f . & shirtcliffe , t. , 1978 , , 87 , 417 merryfield , w. , 1995 , , 444 , 318 mizuno , h. , 1980 , prog . physics , 64 , 544 molemaker , m. & dijkstra , h. , 1997 , , 331 , 199 pollack , j. et al . , 1996 , icarus , 124 , 62 potekhin , a. , 1999 , , 351 , 787 rogers , f. , swenson , f. & iglesias , c. , 1996 , , 456 , 902 saumon , d. & guillot , t. , 2004 , , 609 , 1170 showman , a. , & guillot , t. , 2002 , , 385 , 156 schmitt , r. , 1994 , ann . rev . fluid mech . , 26 , 255 spruit , h. , 1992 , , 253 , 131 stevenson , d. , 1979 , , 187 , 129 stevenson , d. , 1982 , planet . space sci . , 755 stevenson , d. , salpeter , e. , 1977a , , 35 , 221 stevenson , d. , salpeter , e. , 1977b , , 35 , 239

we explore the possibility that large - scale convection be inhibited over some regions of giant planet interiors , as a consequence of a gradient of composition inherited either from their formation history or from particular events like giant impacts or core erosion during their evolution . under appropriate circumstances , the redistribution of the gradient of molecular weight can lead to double diffusive layered or overstable convection . this leads to much less efficient heat transport and compositional mixing than large - scale adiabatic convection . we show that this process can explain the abnormally large radius of the transit planet hd209458b and similar objects , and may be at play in some giant planets , with short - period planets offering the most favorable conditions . observational signatures of this transport mechanism are a large radius and a reduced heat flux output compared with uniformly mixed objects . if our suggestion is correct , it bears major consequences on our understanding of giant planet formation , structure and evolution , including possibly our own jovian planets .

the mission has recently made public a catalog of all transiting planet candidates identified during the first four months of observation by the spacecraft @xcite . included in this list of 1235 objects are nearly 300 in the category of super - earths ( defined here as having radii in the range @xmath5 ) , and several dozen of earth size ( @xmath6 ) . the wealth of new information promises to revolutionize our knowledge of extrasolar planets . although strictly speaking these are still only _ candidates _ since confirmation by spectroscopic or other means is not yet in hand , expectations are high that the rate of false positives in this list is relatively small ( see * ? ? ? * ; * ? ? ? consequently , results from this sample concerning the general properties of exoplanets have already begun to emerge , including studies of the architecture and dynamics of multiple transiting systems @xcite , an investigation of the statistical distribution of eccentricities @xcite , and first estimates of the rate of occurrence of planets larger than 2@xmath7 with orbital periods up to 50 days @xcite , among others . for good reasons the confirmation or `` validation '' of small transiting planets ( earth - size or super - earth - size ) has attracted considerable attention , but has proven to be non - trivial in many cases because of the difficulty of detecting the tiny radial - velocity ( rv ) signatures that these objects cause on their parent stars , as exemplified by the cases of corot-7b @xcite , kepler-9d @xcite , and kepler-11 g @xcite . in fact , such spectroscopic signals are often too small to detect with current instrumentation , and the planetary nature of the candidate must be established statistically , as in the latter two cases . the smallest planet discovered to date , kepler-10b , was announced recently by @xcite , and is the mission s first rocky planet . it has a measured radius of @xmath8 and a mass of @xmath9 , leading to a mean density of @xmath10 that implies a significant iron mass fraction @xcite . its parent star , kepler-10 ( kic11904151 , 2mass119024305 + 5014286 ) , is relatively bright among the targets ( magnitude @xmath11 ) and displays _ two _ periodic signals with periods of 0.84 days and 45.3 days , and flux decrements ( ignoring limb darkening ) of @xmath12 ppm and @xmath1 ppm , respectively @xcite . the extensive observations that followed the detection of these signals are documented in detail by those authors , and include the difficult measurement of the reflex radial - velocity motion of the star with a semi - amplitude of only @xmath13 and a period that is consistent with the shorter signal . as is customary also in ground - based searches for transiting planets , the shapes of the spectral lines were examined carefully to rule out changes of similar amplitude correlating with orbital phase that might indicate a false positive , such as a background eclipsing binary ( eb ) blended with the target , or an eb physically associated with it . however the precision of the measurements ( bisector spans ) compared to the small rv amplitude did not allow such changes to be ruled out unambiguously . false positive scenarios were explored with the aid of , a technique that models the transit light curves to test a wide range of blend configurations @xcite , and it was found that the overwhelming majority of them can be rejected . this and other evidence presented by @xcite allowed the planetary nature of kepler-10b to be established with very high confidence . this was not the case , however , for the 45-day period signal referred to as koi-072.02 ( object of interest 72.02 ) , which is the subject of this paper . no significant rv signal was detected at this period , and only an upper limit on its amplitude could be placed . using , @xcite were able to rule out a large fraction of the blend scenarios involving circular orbits ( including hierarchical triples ) , but eccentric orbits were not explored because of the increased complexity of the problem and the much larger space of parameters for false positives . while circular orbits are a reasonable assumption for kepler-10b because of the strong effects of tidal forces at close range , this is not true for koi-072.02 on account of its much longer orbital period ( see , e.g. , * ? ? ? * ) ; eccentric orbits can not be ruled out . this provides the motivation for the present work , in which we set out to examine all viable astrophysical false positive scenarios for koi-072.02with the goal of validating it as a bona - fide planet . in addition to improvements in the modeling , we bring to bear new near - infrared observations obtained with the _ spitzer _ space telescope in which the transits are clearly detected , as well as the complete arsenal of follow - up observations gathered by the team , including high - resolution adaptive optics imaging and speckle interferometry , high - resolution spectroscopy , and an analysis based on the observations themselves of the difference images in and out of transit for positional displacements ( centroid motion ) . all of these observations combined with the strong constraints provided by significantly limit the kinds of blends that remain possible , and as we describe below they allow us to claim with very high confidence that koi-072.02 is indeed a planet . its estimated radius is approximately 60% of that of neptune . with this , kepler-10 becomes the mission s third confirmed multi - planet system ( after kepler-9 and kepler-11 ; * ? ? ? * ; * ? ? ? * ) containing a transiting super - earth - size planet and at least one larger planet that also transits . we begin with a brief recapitulation of the technique , including recent improvements . we then present the _ warm spitzer_observations at 4.5 that help rule out many blends , and we summarize additional constraints available from other observations . this is followed by the application of to koi-072.02 in order to identify all blends scenarios that can mimic the transit light curve . next we combine this information with the other constraints and carry out a statistical assessment of the false alarm rate for the planet hypothesis , leading to the validation of the candidate as kepler-10c . we conclude with a discussion of the possible constitution of the new planet in the light of current models , and the significance of this type of validation . the detailed morphology of a transit light curve ( length of ingress / egress , total duration ) contains important information that can be used to reject many false positive scenarios producing brightness variations that do not quite have the right shape , even though they may well match the observed transit depth ( see , e.g. , * ? ? ? @xcite takes advantage of this to explore a very large range of scenarios , including background or foreground eclipsing binaries blended with the target , as well as eclipsing binaries physically associated with the target in a hierarchical triple configuration . following the notation introduced by @xcite , the objects composing the binary are referred to as the `` secondary '' and `` tertiary '' , and the candidate is the `` primary '' . the tertiary can be either a star ( including a white dwarf ) or a planet , and the secondary can be a main - sequence star or a ( background ) giant . with the help of model isochrones to set the stellar properties , simulates blend light curves resulting from the flux of the eclipsing pair diluted by the brighter target ( and any additional stars that may fall within the photometric aperture ) . each simulated light curve is compared with the observations in a @xmath14 sense to identify which of them result in acceptable fits ( to be defined later ) . the parameters varied during the simulations are the mass of the secondary star ( @xmath15 ) , the mass of the tertiary ( @xmath16 , or its radius @xmath17 if a planet ) , the impact parameter ( @xmath18 ) , the relative linear distance ( @xmath19 ) between the eclipsing pair and the target , and the relative duration ( @xmath20 ) of the transit compared to the duration for a circular orbit ( see below ) . for convenience the relative linear distance is parametrized in terms of the difference in distance modulus , @xmath21 , where @xmath22 . in the case of hierarchical triple configurations the isochrone for the binary is assumed to be the same as for the primary ( metallicity of @xmath23 } = -0.15 $ ] and a nominal age of 11.9gyr ; see * ? ? ? * ) , whereas for background blends we have adopted for the binary a representative 3gyr isochrone of solar metallicity , although these parameters have a minimal impact on the results . for full details of the technique we refer the reader to the references above . three recent changes and improvements that are especially relevant to the application to koi-072.02 are described next : ( _ i _ ) the relatively long orbital period of koi-072.02 ( 45.3 days ) precludes us from assuming that the eccentricity ( @xmath24 ) is zero , as we were able to suppose in previous applications of to kepler-9d and kepler-10b , which have periods of 1.59 and 0.84 days , respectively . the reason this matters is that the duration of the transit is set , among other factors , by the size of the secondary star . eccentricity can alter the speed of the tertiary around the secondary , making it slower or faster than in the circular case depending on the orientation of the orbit ( longitude of periastron , @xmath25 ) . given a fixed ( measured ) duration , blends with smaller or larger secondary stars than in the circular case may still provide satisfactory fits to the light curve , effectively increasing the pool of potential false positives . now takes this into account , although rather than using as parameters @xmath24 and @xmath25 , which are the natural variables employed in the binary light - curve generating routine at the core of ( see * ? ? ? * ) , a more convenient variable that captures the effects of both is the duration relative to a circular orbit . following @xcite , this may be expressed as @xmath26 . operationally , then , we vary @xmath20 over wide ranges as we explore different blend scenarios , and for each value we infer the corresponding values of @xmath24 and @xmath25 . in practice , in order to solve for \{@xmath24 , @xmath25 } from @xmath20 it is only necessary to consider the limiting cases with @xmath27 and 270 , corresponding to transits occurring at periastron and apastron , respectively , since these are the orientations resulting in the minimum and maximum durations for a given eccentricity . other combinations of @xmath24 and @xmath25 will lead to intermediate relative durations that are already sampled in our @xmath20 grid . it is worth noting that use of only these two values of @xmath25 leads to predicted secondary eclipses in the simulated light curves that are always located at phase 0.5 , whereas secondary eclipses in the real data might be present at any phase . for our purposes this is of no consequence , as koi-072.02 has already had its light curve screened for secondary eclipses at any phase that might betray a false positive , as part of the vetting process . no such features are present down to the 100 ppm level . thus , any simulated light curves from that display a significant secondary eclipse will yield poor fits no matter where the secondary eclipse happens to be , and will lead to the rejection of that particular blend scenario . ( _ ii _ ) for each false positive configuration can predict the overall photometric color of the blend , for comparison with the measured color index of the candidate as reported in the input catalog ( kic ; * ? ? ? * ) . a color index such as @xmath28 , where @xmath29 is the magnitude and @xmath30 derives from the 2mass catalog , provides a reasonable compromise between wavelength leverage and the precision of the index . the latter varies typically between 0.015 and 0.030 mag , depending on the passband and the brightness of the star ( see * ? ? ? we consider a particular blend to be rejected when its predicted color deviates from the kic value by more than three times the error of the latter . as it turns out , color is a particularly effective way of rejecting blends that include secondary stars of a different spectral type than the primary , such as those that become possible when allowing for eccentric orbits . ( _ iii _ ) recent refinements in the resolution of the simulations to better explore parameter space , in addition to the inclusion of eccentricity ( or @xmath20 ) as an extra variable , have increased the complexity of the problem as well as the computing time ( by nearly two orders of magnitude ) compared to the relatively simple case of circular orbits . the number of different parameter combinations examined with ( and corresponding light - curve fits ) can approach @xmath31 in some cases . consequently the simulations are now performed on the pleiades cluster at the nasa advanced supercomputing division , located at the ames research center ( california ) , typically on 1024 processors running in parallel . for convenience hierarchical triple configurations ( 4 parameters ) and background / foreground blends ( 5 parameters ) are studied separately , each for the two separate cases of stellar and planetary tertiaries ( for a total of four grids ) . one additional fit is carried out using a true transiting planet model to provide a reference for the quality of the false positive fits in the other grids . the discriminating value of the shape information contained in the light curves , mentioned at the beginning of this section , is highlighted by our results for kepler-10b , as described by @xcite . in that study it was found that _ all _ background eclipsing binary configurations with stellar tertiaries yield very poor fits to the light curve , and are easily rejected . the underlying reason is that all such blend models predict obvious brightness changes out of eclipse ( ellipsoidal variations ) with an amplitude that is not seen in the data , and that are a consequence of the very short orbital period . ) artificially suppresses out - of - eclipse variations to some extent , typically by median filtering , so that the light curves for periods as short as that of kepler-10b ( 0.84 days ) are rendered essentially flat except for the transits themselves . in this sense the situation is similar to that mentioned earlier regarding the presence of secondary eclipses : obvious ellipsoidal variability in the raw data would normally trigger a false positive warning during the vetting process , preventing the target from becoming an object of interest . but if it reaches koi status , we assume that out - of - eclipse modulations are insignificant so that the comparison with any model in which those variations are present is meaningful and would yield a poor fit , sufficient in most cases to reject the blend . ] hierarchical triple scenarios were also excluded based on joint constraints from and other follow - up observations . the only configurations providing suitable alternatives to the true planet scenario involved stars in the foreground or background of the target that are orbited by a larger transiting planet . the considerable reduction in the blend frequency from the exclusion of all background eclipsing binaries led to a false alarm probability low enough to validate kepler-10b with a very high level of confidence , _ independently _ of any spectroscopic evidence . this remarkable result speaks to the power of when combined with all other observational constraints . it also assumes considerable significance for kepler-10b , given that it was not possible to provide separate proof of the planetary nature of this signal in the @xcite study from an examination of the bisector spans . the scatter of the bisector span measurements ( 10.5 ) was three times larger than the rv semi - amplitude ( 3.3 ) , rendering them inconclusive . the situation regarding the analysis of the koi-072.02 signal in the @xcite study was very different : the orbital period is much longer , and ellipsoidal variations are predicted to be negligible , so that background eclipsing binaries with stellar tertiaries remain viable blends . this , and the added complication from eccentric orbits , hindered the efforts of those authors to validate this candidate . with the benefit of the enhancements in described above , we are now in a better position to approach this problem anew . as follow - up observations provide important constraints that are complementary to those supplied by , and play an important role in determining the false alarm rate for the planetary nature of koi-072.02 ( sect . [ sec : statistics ] ) , we describe those first below , beginning with our new near - infrared _ spitzer _ observations . koi-072.02 was observed during two transits with the irac instrument on the _ spitzer _ space telescope @xcite at 4.5(program i d 60028 ) . the observations were obtained on ut 2010 august 30 and november 15 , with each visit lasting approximately 15hr 10min . the data were gathered in full - frame mode ( @xmath32 pixels ) with an exposure time of 6.0s per image , which resulted in approximately a 7.1s cadence and yielded 7700 images per visit . the method we used to produce photometric time series from the images is described by @xcite . it consists of finding the centroid position of the stellar point spread function ( psf ) and performing aperture photometry using a circular aperture on individual exposures . the images used are the basic calibrated data ( bcd ) delivered by the _ spitzer _ archive . these files are corrected for dark current , flat - fielding , and detector non - linearity , and are converted to flux units . we converted the pixel intensities to electrons using the information on the detector gain and exposure time provided in the fits headers . this facilitates the evaluation of the photometric errors . we extracted the utc - based julian date for each image from the fits header ( keyword date_obs ) and corrected to mid - exposure . we converted to tdb - based barycentric julian dates using the ` utc2bjd ` procedure developed by @xcite . this program uses the jpl horizons ephemeris to estimate the position of the _ spitzer _ spacecraft during the observations . we then corrected for transient pixels in each individual image using a 20-point sliding median filter of the pixel intensity versus time . to do so , we compared each pixel s intensity to the median of the 10 preceding and 10 following exposures at the same pixel position , and we replaced outliers greater than @xmath33 with their median value . the fraction of all pixels we corrected is 0.02% for the first visit and 0.06% for the second . -10pt the centroid position of the stellar psf was determined using the daophot - related procedures ` gcntrd ` , from the idl astronomy library . we applied the ` aper ` routine to perform aperture photometry with a circular aperture of variable radius , using a range of radii between 1.5 and 8 pixels in steps of 0.5 . the propagated uncertainties were derived as a function of the aperture radius , and we adopted the aperture providing the smallest errors . we found that the transit depths and errors varied only weakly with aperture radius for all light - curves analyzed in this project . the optimal aperture was found to have a radius of 4.0 pixels . we estimated the background by examining a histogram of counts from the full array . we fit a gaussian curve to the central region of this distribution ( ignoring bins with high counts , which correspond to pixels containing stars ) , and we adopted the center of this gaussian as the value of the residual background intensity . as seen already in previous _ warm spitzer _ observations @xcite , we found that the background varies by 20% between three distinct levels from image to image , and displays a ramp - like behavior as function of time . the contribution of the background to the total flux from the stars is low for both observations , from 0.1% to 0.55% depending on the image . therefore , photometric errors are not dominated by fluctuations in the background . we used a sliding median filter to select and trim outliers in flux and position greater than 5@xmath34 , representing 1.6% and 1.3% of the data for the first and second visits , respectively . we also discarded the first half - hour s worth of observations , which is affected by significant telescope jitter before stabilization . the final number of photometric measurements used is 7277 and 7362 . the raw time series are presented in the top panel of figure [ fig : spitzerlightcurves ] . we find that the point - to - point scatter in the photometry gives a typical signal - to - noise ratio ( s / n ) of 280 per image , which corresponds to 90% of the theoretical signal - to - noise . therefore , the noise is dominated by poisson statistics . in order to determine the transit parameters and associated uncertainties from the _ spitzer _ time series we used a transit light curve model multiplied by instrumental decorrelation functions , as described by @xcite the transit light curves were computed with the idl transit routine ` occultsmall ` from @xcite . for the present case we allowed for a single free parameter in the model , which is the planet - to - star radius ratio @xmath35 ( or equivalently , the depth , in the absence of limb darkening ) . the normalized orbital semi - major axis ( system scale ) @xmath36 , the impact parameter @xmath18 , the period @xmath37 , and the time of mid transit @xmath38 were held fixed at the values derived from the light curve , as reported by @xcite and summarized below in sect . [ sec : discussion ] . limb darkening is small at 4.5 , but was nevertheless included in our modeling using the 4-parameter law by @xcite and theoretical coefficients published by @xcite . spitzer_/irac photometry is known to be systematically affected by the so - called `` pixel - phase effect '' ( see , e.g. , * ? ? ? * ; * ? ? ? this effect is seen as oscillations in the measured fluxes with a period corresponding to that of the telescope pointing jitter . for the first visit this period was 70 min , and the amplitude of the oscillations was approximately 2% peak - to - peak ; for the second visit the period was 35 min , and the amplitude about 1% . we decorrelated our signal in each channel using a linear function of time for the baseline ( two parameters ) and a quadratic function of the psf position ( four parameters ) to correct the data for each channel . we performed a simultaneous levenberg - marquardt least - squares fit to the data @xcite to determine the transit and instrumental model parameters ( 7 in total ) . the errors on each photometric point were assumed to be identical , and were set to the rms residual of the initial best fit . to obtain an estimate of the correlated and systematic errors in our measurements @xcite we used the residual permutation bootstrap technique , or `` prayer bead '' method , as described by @xcite . in this method the residuals of the initial fit are shifted systematically and sequentially by one frame , and then added to the transit light curve model before fitting again . we considered asymmetric error bars spanning 34% of the points above and below the median of the distributions to derive the @xmath39 uncertainties for each parameter , as described by @xcite . the bottom panel of figure [ fig : spitzerlightcurves ] shows the best - fit model superimposed on the observations from the two visits combined , with the data binned in 36min bins for clarity ( 295 points per bin ) . the transit depths at 4.5 ( after removing limb - darkening effects ) are @xmath40 ppm for the first visit and @xmath41 for the second , which are in good agreement with each other . the weighted average depth of @xmath0 is consistent with the non - limb - darkened value of @xmath1 ppm derived from the light curve @xcite well within the 1@xmath34 errors , strongly suggesting the transit is achromatic , as expected for a planet . the above _ spitzer _ observations provide a useful constraint on the kinds of false positives ( blends ) that may be mimicking the koi-072.02signal . for example , if kepler-10 were blended with a faint unresolved background eclipsing binary of much later spectral type that manages to reproduce the transit depth in the passband , the predicted depth at 4.5 may be expected to be larger because of the higher flux of the contaminating binary at longer wavelengths compared to kepler-10 . since the transit depth we measure in the near infrared is about the same as in the optical , this argues against blends composed of stars of much later spectral type . based on model isochrones and the properties of the target star ( see below ) , we determine an upper limit to the secondary masses of 0.77@xmath42 . spitzer _ constraint is used in sect . [ sec : blender_app ] to eliminate many blends . further constraints of a different kind are provided by high - resolution imaging as described in more detail by @xcite . briefly , these consist of speckle observations obtained on ut 2010 june 18 with a two - color ( approximately @xmath43 and @xmath44 ) speckle camera on the wiyn 3.5 m telescope on kitt peak ( see * ? ? ? * ) , and near - infrared ( @xmath45-band ) adaptive optics ( ao ) observations conducted on ut 2009 september 8 with the pharo camera on the 5 m palomar telescope . no companions were detected around kepler-10 within 15 ( for speckle ) or 125 ( ao ) , and more generally these observations place strong limits on the presence of other stars as a function of angular separation ( down to 005 in the case of speckle ) and relative brightness ( companions as faint as @xmath46 for ao ) . these sensitivity curves are shown in fig . 9 of @xcite , and we make use of that information below . high - resolution spectra described also by @xcite and obtained with the hires instrument on the 10 m keck i telescope place additional limits on the presence of close companions falling within the spectrograph slit ( 087 ) , such that stars within about 2 magnitudes of the target would generally have been seen . a small chance remains that these companions could escape detection if their radial velocity happens to be within a few of that of the target ( which is a narrow - lined , slowly rotating star with @xmath47 ; * ? ? ? * ) , so that the spectral lines are completely blended . this would be extremely unlikely for a chance alignment with a background / foreground star , but not necessarily for physically associated companions in wide orbits , i.e. , with slow orbital motions . we explored this through monte carlo simulations . the results indicate that the probability of having a physical companion within a conservative range of @xmath4810 of the rv of the target that would also go unnoticed in our speckle observations , and that additionally would not induce a rv drift on the target large enough to have been detected in the high - precision measurements of @xcite , is only about 0.1% . finally , an analysis of the image centroids measured from the observations rules out background objects of any brightness beyond about 2 of the target . this exclusion limit ( equivalent to half a pixel ) is considerably more conservative than the 06 reported by @xcite , and accounts for saturation effects not considered earlier ( given that at @xmath11 the star is very bright by standards ) as well as quarter - to - quarter variations ( where `` quarters '' usually represent 3-month observing blocks interrupted by spacecraft rolls required to maintain the proper illumination of the solar panels ) . the photometry used here is the same as employed in the work of @xcite , and was collected between 2009 may 2 and 2010 january 9 . these dates correspond to quarter 0 ( first nine days of commissioning data ) through the first month of quarter 4 . for this study we used only the long - cadence observations ( 10,870 measurements ) obtained by the spacecraft at regular intervals of about 29.4 min . all blend models generated with were integrated over this time interval for comparison with the measurements . the original data have been de - trended for this work by removing a first - order polynomial , and then applying median filtering with a 2-day wide sliding window . observations that occur during transits were masked and did not contribute to the median calculation . because this sliding window is considerably shorter than the 45.3-day orbital period , any ellipsoidal variations present in the original data should be largely preserved , although in any case they are expected to be very small for binaries with periods as long as this . we adopted also the ephemeris of mid - transit for koi-072.02 as reported by @xcite , which is @xmath49 } = 2,\!454,\!971.6761 + n \times 45.29485 $ ] days , where @xmath50 is the number of cycles from the reference epoch . because it is relatively bright ( @xmath11 ) , kepler-10 was also observed by the mission with a shorter cadence of approximately 1 min for a period of several months to allow an asteroseismic characterization of the star . a total of 19 oscillation frequencies were detected , and enabled a very precise determination of the mean stellar density . when combined with stellar evolution models and a spectroscopic determination of the effective temperature and chemical composition , the resulting parameters for the star are very well determined . kepler-10 is relatively old ( @xmath51gyr ) but is otherwise quite similar to the sun , with a temperature of @xmath52k , a mass and radius of @xmath53@xmath42 and @xmath54@xmath55 , and a composition [ fe / h ] @xmath56 slightly below solar @xcite . as indicated earlier we considered four general scenarios for false positives : chance alignments ( a pair of background / foreground eclipsing objects ) and hierarchical triple systems , each with tertiaries that can be either stars or planets . the free parameters were varied over the following ranges : secondary mass @xmath15 between 0.10 and 1.40@xmath42 , in steps of 0.02@xmath42 ; tertiary mass @xmath16 between 0.10 and @xmath15 , also in steps of 0.02@xmath42 ; tertiary radius @xmath17 between 0.06 and 2.00@xmath57 in steps of 0.02@xmath57 ; impact parameter @xmath18 between 0.00 and 1.00 in steps of 0.05 ; relative duration @xmath20 between 0.2 and 4.6 in steps of 0.2 , corresponding to eccentricities up to 0.92 and values of @xmath25 of 90 and 270 ( see sect . [ sec : blender ] ) ; and relative distance @xmath21 ( distance modulus difference ) between @xmath58 and @xmath59 in steps of 0.5 mag , except for hierarchical triple configurations , for which @xmath60 . the goodness of the fit of each of the large number of synthetic light curves generated by is quantified here by computing the @xmath14 statistic and comparing it with that of the best planet model fit . the difference can be assigned a significance level ( or false alarm rate ) that depends on the number of free parameters of the problem . for example , for a blend scenario corresponding to a hierarchical triple system ( 4 degrees of freedom ) , a trial model giving a worse fit than the planet solution by @xmath61 4.72 is statistically different at the 1@xmath34 level , assuming gaussian errors ( see , e.g. , * ? ? ? a fit that is worse by @xmath61 16.3 is different at the 3@xmath34 level . hierarchical triple blends giving poorer fits than this are considered here to be ruled out by the photometry . for background / foreground scenarios ( 5 degrees of freedom ) the 3@xmath34 blend rejection level is @xmath61 18.2 . in this section we describe the simulations carried out for the four general blend configurations mentioned above . although the secondaries for the background scenarios can in principle also be evolved stars ( giants ) , as opposed to main - sequence stars , we consistently found that the transit light curves generated by such systems give a very poor match to the observations because they do not have the right shape ( the ingress / egress phases are too long ) . therefore , we restricted our exploration of parameter space to main - sequence stars only . an additional possibility for a false positive may stem from an error in the determination of the orbital period . if the true period were twice the nominal value , alternating transit events would correspond to primary and secondary eclipses , implicating a blended eclipsing binary . the primary and secondary eclipses would often ( but not always ) be of different depth . as part of the vetting process for each candidate , the team examines the even - numbered and odd - numbered events to look for differences in depth that may indicate a false positive of this kind . as described by @xcite , no significant differences were found for koi-072.02 beyond the 2@xmath34 level , where @xmath34 represents the uncertainty in the transit depth ( 9 ppm ) . nevertheless , as the possibility still exists that the components of the eclipsing binary are identical , experiments were run with to examine the transit shape produced by such scenarios , and it was found that the ingress and egress phases are always much too long compared to the observations , as expected for two equal - size stars eclipsing each other . thus , these scenarios are easily ruled out as well . the simulations with indicate that few background blend scenarios with stellar tertiaries are able to mimic the transit features in the light curve at an acceptable level , and they all correspond to somewhat eccentric orbits . in figure [ fig : backstar ] we show the goodness of fit of these scenarios , with the small closed 3@xmath34 contour representing the region of parameter space within which the fits are satisfactory , according to the criteria given above . only blends with secondary masses @xmath15 larger than about 1.3@xmath42 are allowed , and the eclipsing binary can only be within a small range of distances behind the target ( @xmath62 ) for the dilution effect to be just right , such that the corresponding apparent brightness difference @xmath63 is between 2.5 and 3.5 mag ( see figure ) . the best among these blend models ( located near the center of the contour ) provides a fit that is about 2.1@xmath34 worse than a planet model ( but still acceptable ) , and is shown in the top panel of figure [ fig : fits ] compared against the planet model . the tertiary stars in these blends are constrained to be very small , between 0.10 and 0.16@xmath42 . that most blends involving background eclipsing binaries can be ruled out may appear somewhat surprising , and is worth investigating . indeed , for a given measured transit depth @xmath64 , a blend can only reproduce the light curve if it contributes at least a fraction @xmath64 of the total flux collected in the aperture . thus , one would expect that binaries as faint as @xmath65 mag relative to the target should be able to match that amount of dimming if they were totally eclipsed ( see , e.g. , * ? ? ? * ) , and furthermore , that the measured duration could also be reproduced by a large range of secondary sizes with an appropriate combination of orbital eccentricity and @xmath25 . yet we find that no blends fainter than @xmath66 give tolerable fits to the light curve ( see figure [ fig : backstar ] ) . a visual understanding of the underlying reason for this may be seen in the bottom panel of figure [ fig : fits ] , in which we show a blend model that one would naively expect should be able to match the observations , according to the crude recipe described above . this particular blend scenario is marked with a cross in figure [ fig : backstar ] , and corresponds to @xmath67 and @xmath68@xmath42 , resulting in a magnitude difference of @xmath69 for the eb relative to the target . while this model does yield a good match to the measured depth , and even the total duration , it does nt perform nearly as well in the ingress / egress phases , which are too long when compared against the observations . the quality of this fit relative to the best planet fit , which can also be seen in the figure , corresponds to a 10.1@xmath34 difference , and therefore rejects it . thus , the reason most blends of this class can be ruled out is ultimately the high precision of the light curves , which provides a very strong constraint on the shape of the transit light curve , and in particular on the size ratio between the secondary and tertiary , which sets the duration of the ingress and egress phases . there is a very broad range of blends consisting of a background or foreground star transited by a planet ( as opposed to a star ) that are found by to give satisfactory fits to the data , as shown in figure [ fig : back_plan ] . these viable blends occupy the area below the 3@xmath34 contour represented with a thick white line . secondary stars of all spectral types ( masses ) are permitted , in principle , although in practice other constraints described below eliminate a substantial fraction of them . all of these blends involve secondary+tertiary pairs that are within 4 magnitudes of the target in the passband ( diagonal dashed line in the figure ) . the tertiary sizes in these blends range from 0.42@xmath57 to 1.84@xmath57 . our _ warm spitzer _ observations set a lower limit of about 0.77@xmath42 for the secondary masses of these blends , as described earlier ; scenarios involving redder stars would result in transits at 4.5 significantly deeper than we observe ( i.e. , deeper than the measured depth + 3@xmath34 ) . this exclusion region is indicated by the shaded area . additionally , blends that are much brighter than @xmath70 would most likely have been detected spectroscopically ( see * ? ? ? * ) , so we consider those to be ruled out as well . we indicate this with the green hatched region in the lower right - hand side of the figure finally , the colors of the background / foreground configurations simulated with provide a further constraint which is represented by the blue hatched area on the lower left of the figure . this swath of parameter space is excluded because the blends are significantly redder than the color index measured for kepler-10 ( @xmath71 ) , by more than three times the uncertainty in the observed index . as a result of these complementary constraints , the only section of parameter space remaining for viable blends involving star+planet pairs is the area under the 3@xmath34 contour and limited from below and on the left by the hatched areas ( color and brightness conditions ) and shaded area ( _ spitzer _ constraint ) , respectively . all of these blends have the eclipsing pair behind the target ( foreground scenarios are all ruled out ) . we note that in this star+planet blend scenario white dwarfs can also act as tertiaries , as long as they are cooler than the secondaries so that they do not lead to deep occultation events that would have been seen in the light curve of koi-072.02 . the above range of tertiary radii ( 0.42@xmath57 to 1.84@xmath57 ) excludes essentially all cool carbon - oxygen and oxygen - neon white dwarfs more massive than about 0.4@xmath42 , as these are smaller than the lower limit set by , which corresponds to 4.7@xmath7 ( see , e.g. , * ? ? ? low - mass helium - core or oxygen - core white dwarfs that are the product of common - envelope evolution in binary stars can be considerably larger in size , although they appear to be very rare . the mission itself has uncovered only three examples to date @xcite . however , all of them are very hot ( @xmath72k ) , and produce deep and unmistakable flat - bottomed occultation signals . model calculations such as those of @xcite show that as these helium - core white dwarfs cool , their radii quickly become earth - size or smaller . therefore , we do not consider white dwarfs to be a significant source of blends for koi-072.02 . eclipsing binaries composed of two stars physically associated with the target are clearly ruled out by , as they produce very poor fits to the light curves . for cases in which the tertiaries are planets , viable scenarios identified by span a range of secondary masses and tertiary radii within the 3@xmath34 contour shown in figure [ fig : htp_r3 ] . most of these configurations turn out to involve eccentric orbits , with transit durations longer than those corresponding to circular orbits along with secondary stars that are smaller than the primary ( see figure [ fig : htp_dur ] ) . once again other observational constraints are very complementary , and in this case they are sufficient to exclude all of these blends . for example , the shaded area of parameter space to the left of 0.77@xmath42 is eliminated by the _ spitzer _ observations , as described earlier . the constraint on the @xmath73 color ( hatched area on the left ) is partly redundant with the nir observations , but extends to slightly larger secondary masses . and finally , the spectroscopic constraint removes the remaining scenarios corresponding to higher - mass ( brighter ) secondaries . we conclude that of all the hierarchical triple blend scenarios that are capable of precisely reproducing the detailed shape of the transit light curve , _ none _ would have escaped detection by one or more of our follow - up efforts , including nir _ spitzer_observations , high - resolution spectroscopy , or absolute photometry ( colors ) . ) , not only here but in all previously discovered transiting planets . for the present purposes we do not consider this `` twin star '' scenario as a false positive in the strict sense ( see also * ? ? ? * ) , as the transiting object would still be a planet , only that it would be larger than we thought by about a factor of @xmath74 because of the extra dilution from the companion.[twin ] ] this highlights the importance of these types of constraints for validating candidates , given that blends involving physically associated stars would generally be spatially unresolved by our high - resolution imaging with adaptive optics or speckle interferometry , and they would typically also be below the sensitivity limits of our centroid motion analysis , so that they would not be detected by those means . therefore , the only blends we need to be concerned about for koi-072.02 are those consisting of stars in the background of the target that are orbited by other stars or by transiting planets . in order to estimate the frequency of the blend scenarios ( i.e. , background configurations ) that remain possible after applying and all other observational constraints , we follow a procedure similar to that described by @xcite for kepler-9d . we appeal to the besanon galactic structure models of @xcite to predict the number density of background stars of each spectral type ( mass ) and brightness around kepler-10 , in half - magnitude bins , and we make use of estimates of the frequencies of transiting planets and of eclipsing binaries from recent studies by the team to infer the number density of blends . using constraints from our high - resolution imaging ( specifically , the sensitivity curves presented by * ? ? ? * their fig . 9 ) we calculate the area around the target within which blends would go undetected , and with this the expected number of blends . ) , is significantly less constraining than the high - resolution imaging , so is not as useful here as it was for kepler-9d . ] the recent release by @xcite of a list of 1235 candidate transiting planets ( kois ) from provides a means to estimate planet frequencies needed for our calculations , with significant advantages over the calculations of @xcite for kepler-9d , which were based on the earlier list of candidates published by @xcite . not only is the sample now much larger , but the knowledge of the rate of false positives for is also much improved , and that rate is believed to be relatively small ( 2040% depending on the level of vetting of the candidate , according to @xcite ; less than 10% according to the recent study by @xcite ) . thus , our results will not be significantly affected by the assumption that all of the candidates are planets ( see also below ) . an additional assumption we make is that this census is largely complete . among these candidates we count a total of 267 having radii in the range allowed by for the tertiaries of viable blends ( i.e. , between 0.42 and 1.84@xmath57 ) . with the total number of targets being 156,453 @xcite , the relevant frequency of transiting planets for our blend calculation is @xmath75 . @xcite have recently published a catalog of the 2165 eclipsing binaries found in the field , from the first four months of observation . only the 1225 detached systems among these are considered here , since binaries in the category of semi - detached , over - contact , or ellipsoidal variables would not produce light curves with a shape consistent with a transit . the frequency of eclipsing binaries for our purposes is then @xmath76 . table [ tab : new_stats ] presents the results of our calculation of the frequency of blends , separately for background blends with stellar tertiaries ( eclipsing binaries ) and with planetary tertiaries . columns 1 and 2 give the @xmath29 magnitude range of each bin and the magnitude difference @xmath77 relative to the target , calculated at the upper edge of each bin . column 3 reports the mean number density of stars per square degree obtained from the besanon models , for stars in the mass range allowed by as shown in figure [ fig : backstar ] . in column 4 we list the maximum angular separation @xmath78 at which stars in the corresponding magnitude bin would go undetected in our imaging observations , taken from the information in the work of @xcite . the product of the area implied by this radius and the stellar densities in the previous column give the number of stars in the appropriate mass range , listed in column 6 in units of @xmath79 . multiplying these figures by the frequency of eclipsing binaries @xmath80 then gives the number of background star+star blends in column 7 . a similar calculation for the background star+planet blends , making use of @xmath81 , is presented in columns 710 . we sum up the contributions from each magnitude bin at the bottom of columns 6 and 10 . the total number of blends we expect _ a priori _ ( blend frequency ) is given in the last line of the table by adding these two values together , and is @xmath82 . the calculations show that background blends consisting of star+planet pairs contribute to this frequency about three times more than background eclipsing binaries . while we have assumed up to now that any companions to koi-072.02 within @xmath70 mag of the target would have been seen spectroscopically , we note that relaxing this condition to a much more conservative @xmath83 has no effect at all on the contribution from eclipsing binaries , and a negligible effect on the contribution of star+planet scenarios . to obtain a bayesian estimate of the probability that koi-072.02 is indeed a planet as opposed to a false positive ( or equivalently , the `` false alarm rate '' , far ) we follow the general methodology of @xcite and compare the _ a priori _ likelihoods of blends and of planets : far = bf / pf . if the _ a priori _ blend frequency is sufficiently small compared the planet frequency ( pf ) , we consider the planet validated . our _ a priori _ blend frequencies above correspond to false positive scenarios giving fits to the light curve that are within 3@xmath34 of the best planet fit . we use a similar criterion to estimate the _ a priori _ planet frequency by counting the kois in the @xcite sample that have radii within 3@xmath34 of the best fit from a planet model ( @xmath84 ; see table [ tab : systemparams ] below ) . we find that 157 among the 1235 kois are in this radius range ( 2.062.38@xmath7 ) , giving pf @xmath85 . this results in a false alarm rate for koi-072.02 of @xmath86 , which is so small that it allows us to validate the candidate with a very high level of confidence . the planet is designated kepler-10c . this result rests heavily on the _ a priori _ frequency of planets from the mission , derived from the assumption that all 1235 candidates reported by @xcite are indeed planets rather than false positives . if we were to be as pessimistic as to assume that as many as 90% of the small - size candidates are actually false positives ( a similar rate of false positives as is typically found in ground - based surveys for transiting planets ) , and at the same time that all of the larger - size candidates that come into the blend frequency calculation are real planets ( thereby maximizing bf and minimizing pf ) , the false alarm rate would be 10 times larger than before , or @xmath87 . this is still a very small number , and our conclusion regarding validation is unchanged . we note that a rate of false positives as high as 90% yields a planet frequency that is strongly inconsistent not only with the expectations of @xcite and @xcite , but also with the independent results of ground based doppler surveys as reported by @xcite . in the above calculations we have implicitly assumed similar period distributions for planets of all sizes and for eclipsing binaries . however , it is conceivable that the results could change if the period distribution of planets such as kepler-10c were significantly different from the one for larger planets that go into the blend frequency calculations , or from the one for ebs ( which have a smaller contribution to bf ; see table [ tab : new_stats ] ) . therefore , as a further test we considered the impact of restricting the periods to be within an arbitrary factor of two of the kepler-10c period of 45.3 days , both in our blend frequency calculations and for the _ a priori _ estimate of the planet frequency , pf . we find that the planet frequencies are reduced by a factor of 4.5 , and the eclipsing binary frequency by a factor of 10.4 , and as a result the false alarm rate for koi-072.02 is @xmath88 , which is about the same as before . thus , our conclusions are robust against assumptions about the period distributions . finally , our false alarm rate is conservative in the sense that we have not accounted for the flatness ( coplanarity ) of the kepler-10 system . only a small fraction of single transiting planets with periods as long as 45 days orbiting background stars ( i.e. , those acting as blends ) are likely to transit , _ a priori _ , whereas a planet of this period such as kepler-10c is much more likely to transit if it is coplanar with kepler-10b . taking this into account would boost the planet frequency ( pf ) and decrease the far by as much as an order of magnitude ( see , e.g. , * ? ? ? coplanarity in multiple systems is in fact supported by the large number of multiple transiting system candidates found by @xcite , and their mutual inclinations seem to be small ( 15 ; * ? ? ? therefore , we consider our estimate of the far for kepler-10c to be conservative . the stellar , orbital , and planetary parameters inferred for the system as determined by @xcite are summarized in table [ tab : systemparams ] , to which we add the transit duration . the small formal uncertainty in the planetary radius ( @xmath892.4% ) derives from the relatively high precision of the stellar radius , which is based on asteroseismic constraints on the mean density of the star . with its radius of about 2.2@xmath7 , kepler-10c is among the smallest exoplanets discovered to date . the mass is undetermined as the doppler signature has not been detected . nevertheless , @xcite placed a constraint on it based on the distribution of masses resulting from the markov chain monte carlo fitting procedure they applied to the existing radial - velocity measurements of kepler-10 . their conservative 3-@xmath34 upper limit for the mass is 20@xmath4 . the corresponding maximum mean density is 10 g @xmath90 . given a precise radius measurement and mass upper limit of 20@xmath4 , some minimal constraints can be placed on the composition of kepler-10c . using the models of @xcite , we find that an earth - like rock - iron composition is only possible at @xmath89 20 @xmath4 . lower masses would require a depletion in iron compared to rock , or more likely an enrichment in low - density volatiles such as water and/or h@xmath91/he gas . a 50/50 rock / water composition yields 2.23@xmath7 at 7@xmath4 . still lower masses are possible with a h@xmath91/he gas envelope . using models presented in @xcite , a planet with a rock / iron core and a 5% h@xmath91/he atmosphere ( by mass ) matches the measured radius of kepler-10c at only 3@xmath4 . a massive 20@xmath4 core should have attained a h@xmath91/he envelope , and it would appear to be stable at kepler-10c s relatively modest irradiation level , which would lead to a planetary radius dramatically larger than 2.23@xmath7 . this would tend to favor a scenario where kepler-10c is more akin to gj 1214b @xcite and kepler-11b and kepler-11f , which are all below 7@xmath4 and enriched in volatiles . the well measured inclinations of both kepler-10b and kepler-10c allow us to put a weak constraint on the true mutual inclination ( @xmath92 ) between the orbital planes of the two planets . although the relative orientation in the plane of the sky ( i.e. , the mutual nodal angle ) is unknown , the different impact parameters and resulting apparent inclinations place a lower limit on @xmath92 . as discussed by @xcite , the geometric limits to the mutual inclination are given by @xmath93 , where @xmath94 @xcite and @xmath95 ( table [ tab : systemparams ] ) are the usual inclinations with respect to the line of sight . assuming a random orientation of the lines of nodes ( which does not account for the _ a priori _ knowledge that both planets are transiting ) , the mutual inclination is constrained to be in the interval @xmath96 , with the most likely values being at the extremes of this distribution . making the reasonable supposition of non - retrograde orbits , a mutual inclination close to the lower limit of about 5 is most likely for these planets . a more detailed probabilistic argument requires making assumptions about the number of planets in the kepler-10 system . this mutual inclination is on the high end of the distribution inferred for other multiple candidate systems ( 15 ) by @xcite . if this mutual inclination is typical for planets in this system , then it is relatively likely ( depending on the orbital period ) that other planets , if present , are not transiting . when considering the set of candidates in multiple systems that have periods less than 125 days , the ratio of periods between kepler-10c and kepler-10b ( which is 54.1 ) is by far the highest of all period ratios of neighboring pairs of candidates ( the next highest being 23.4 ) , and is even higher than the period ratios between non - neighboring planets . clearly , there is room for multiple additional planets between kepler-10b and kepler-10c . the preponderance of tightly - packed multiple candidate systems suggests that additional planets may exist , and these may be revealed in the future with more detailed transit timing variation measurements . kepler-10c is the first target observed with _ warm spitzer_with the aim of testing the wavelength dependence of the transit depth . this is currently the only facility available that has the capability of detecting such shallow transits at wavelengths that are sufficiently separated from the passband to be helpful . in this case the observations were successful , and the transit at 4.5 is shown to have virtually the same depth as in the optical . this places a very strong constraint on the color of potential blends , which are restricted to have secondaries of similar spectral type as the primary star . the detailed analysis of the photometry with combined with constraints from other observations eliminates the vast majority of possible blend scenarios . this includes most background eclipsing binaries ( leaving only a small range of possible spectral types and relative fluxes for the secondaries ) , most of the scenarios involving chance alignments with a star transited by a larger planet , and all possible hierarchical triple configurations . the latter are among the most difficult to detect observationally since they are typically spatially unresolved . the key factors that have allowed this , and made possible the validation of the planet , are the high - precision of the photometry , the relatively short ingress and egress phases ( which places strong constraints on the size ratio between the secondary and tertiary ) , and the near equatorial orientation , resulting in a relatively flat transit that leaves less freedom for the parameters of the eclipsing binaries . we expect to be similarly effective for other candidates that show similar features in their light curves . kepler-10c along with kepler-9d and kepler-11 g are examples of transiting planets that have not received the usual confirmation by dynamical means that previous discoveries have enjoyed ( including essentially all ground - based discoveries ) , in which either the doppler signature is detected unambiguously ( and verified by the lack of bisector span variations ) , or transit timing variations in a multiple system are directly measured ( as in kepler-9b and c as well as the five inner planets of the kepler-11 system ) . instead , the planets in those three cases have been _ validated _ statistically , with a bayesian approach to estimate the probability that the transit signals are due to a planet rather than a false positive . this probability has been computed by first estimating the _ a priori _ likelihood of a false positive , and then comparing it with the _ a priori _ chance of having observed a true planet . in the three cases mentioned above the ratio of the false positive to planet likelihoods is small enough that the planetary nature of the signal is established with a very high degree of confidence . for kepler-10c the false alarm rate is @xmath2 . the recent work of @xcite has provided a means of assessing a rough false alarm rate for candidates as a function of the depth of the transit signal and the brightness of the object . as noted also by those authors , while these estimates are extremely valuable for statistical studies , the validation of candidates on an individual basis with a sufficiently high degree of confidence will usually require a much more detailed analysis of false positives , such as we have performed here . masses for these objects ( other than upper limits ) may of course be difficult or impractical to determine in many cases , but it is worth keeping in mind that some of the most exciting candidates to be discovered by will be in this category , namely , earth - size planets in the habitable zones of their parent stars . except for stars of late spectral type , the rv signals will generally be very challenging to detect with the sensitivity of current instrumentation . thus , statistical validation of planets is likely to play an important role for in the years to come . funding for this discovery mission is provided by nasa s science mission directorate . this research has made use of the facilities at the nasa advanced supercomputing division ( nasa ames research center ) , and is based also on observations made with the spitzer space telescope which is operated by the jet propulsion laboratory , california institute of technology under a contract with nasa . support for this work was provided by nasa through an award issued by jpl / caltech . we thank mukremin kilic and rosanne di stefano for helpful discussions about white dwarfs , and the anonymous referee for constructive comments . ccccccccccc 11.011.5 & 0.5 & & & & & & & & & + 11.512.0 & 1.0 & & & & & & & & & + 12.013.0 & 1.5 & & & & & & & & & + 12.513.0 & 2.0 & & & & & & & & & + 13.013.5 & 2.5 & & & & & & 139 & 0.12 & 0.485 & 0.0008 + 13.514.0 & 3.0 & 32 & 0.15 & 0.175 & 0.0014 & & 197 & 0.15 & 1.074 & 0.0018 + 14.014.5 & 3.5 & 44 & 0.18 & 0.346 & 0.0027 & & 278 & 0.18 & 2.183 & 0.0037 + 14.515.0 & 4.0 & & & & & & 351 & 0.20 & 3.403 & 0.0058 + 15.015.5 & 4.5 & & & & & & & & & + 15.516.0 & 5.0 & & & & & & & & & + 16.016.5 & 5.5 & & & & & & & & & + 16.517.0 & 6.0 & & & & & & & & & + 17.017.5 & 6.5 & & & & & & & & & + 17.518.0 & 7.0 & & & & & & & & & + 18.018.5 & 7.5 & & & & & & & & & + 18.519.0 & 8.0 & & & & & & & & & + & 76 & & 0.521 & * 0.0041 * & & 965 & & 7.145 & * 0.0121 * + + [ -1.5ex ] effective temperature , @xmath97 ( k ) & @xmath98 & a + surface gravity , @xmath99 ( cgs ) & @xmath100 & a + metallicity , [ fe / h ] & @xmath101 & a + projected rotation , @xmath102 ( ) & @xmath103 & a + mass , @xmath104 ( @xmath42 ) & @xmath105 & b + radius , @xmath106 ( @xmath55 ) & @xmath107 & b + surface gravity , @xmath108 ( cgs ) & @xmath109 & b + luminosity , @xmath110 ( @xmath111 ) & @xmath112 & b + absolute @xmath43 magnitude , @xmath113 ( mag ) & @xmath114 & b + age ( gyr ) & @xmath115 & b + distance ( pc ) & @xmath116 & b + orbital period , @xmath37 ( days ) & @xmath117 & c + mid - transit time , @xmath38 ( hjd ) & @xmath118 & c + scaled semimajor axis , @xmath119 & @xmath120 & c + scaled planet radius , @xmath121 & @xmath122 & c + impact parameter , @xmath18 & @xmath123 & c + orbital inclination , @xmath124 ( deg ) & @xmath125 & c + transit duration , @xmath126 ( hours ) & @xmath127 & c + radius , @xmath128 ( @xmath7 ) & @xmath129 & b , c + mass , @xmath130 ( @xmath4 ) & @xmath131 & d + mean density , @xmath132 ( g @xmath90 ) & @xmath133 & d + orbital semimajor axis , @xmath134 ( au ) & @xmath135 & e + equilibrium temperature , @xmath136 ( k ) & 485 & f + [ -1.5ex ]

the mission has recently announced the discovery of kepler-10b , the smallest exoplanet discovered to date and the first rocky planet found by the spacecraft . a second , 45-day period transit - like signal present in the photometry from the first eight months of data could not be confirmed as being caused by a planet at the time of that announcement . here we apply the light - curve modeling technique known as to explore the possibility that the signal might be due to an astrophysical false positive ( blend ) . to aid in this analysis we report the observation of two transits with the _ spitzer _ space telescope at 4.5 . when combined they yield a transit depth of @xmath0 ppm that is consistent with the depth in the passband ( @xmath1 ppm , ignoring limb darkening ) , which rules out blends with an eclipsing binary of a significantly different color than the target . using these observations along with other constraints from high - resolution imaging and spectroscopy we are able to exclude the vast majority of possible false positives . we assess the likelihood of the remaining blends , and arrive conservatively at a false alarm rate of @xmath2 that is small enough to validate the candidate as a planet ( designated kepler-10c ) with a very high level of confidence . the radius of this object is measured to be @xmath3 ( in which the error includes the uncertainty in the stellar properties ) , but currently available radial - velocity measurements only place an upper limit on its mass of about 20@xmath4 . kepler-10c represents another example ( with kepler-9d and kepler-11 g ) of statistical `` validation '' of a transiting exoplanet , as opposed to the usual `` confirmation '' that can take place when the doppler signal is detected or transit timing variations are measured . it is anticipated that many of s smaller candidates will receive a similar treatment since dynamical confirmation may be difficult or impractical with the sensitivity of current instrumentation .

the measurement of masses of galaxies has been , over a long period of time , an interesting and difficult problem , which has elicited the application of various and diverse techniques @xcite . since the determination of rotation curves for a large number of spiral galaxies @xcite and the suggestion that these rotation curves are flat because of the presence of an unseen amount of mass which has been called ` dark matter ' , the determination of the mass of all types of galaxies has become a pressing concern of modern astronomy . it is fair to say at this point that there is no direct evidence of the existence of dark matter and that there are other explanations which , although not as currently popular as dark matter , may explain the observations quite reasonably . the total mass of a galaxy is composed of two elements ; luminous matter and dark matter . if we assume that both luminous and dark matter respond to the newtonian gravitational law in the same way , then the difference between the dynamical mass and the luminous mass of a galaxy provides us with an estimation of the amount of dark matter present in the galactic system in question . from such a determination we would be able to study if a dependence of the amount of dark matter with dynamical mass and/or redshift exists . measuring the amount of radiation from a particular galaxy , combined with typical mass to light ratios ( @xmath3/@xmath4 ) that have been calibrated using different stellar samples in our own galaxy , allows us to estimate its stellar , gas and dust content . moreover , rotation curves for spiral galaxies permit the calculation of dynamical mass inside any radius for which a value of rotation velocity is known , allowing us , in principle , to calculate from these two determinations the amount of dark matter present in the galaxy under study . as is well known , rotation velocity curves are used for studying the kinematics of galaxies , determining the amount and distribution of mass interior to a given radius , to derive an insight into galactic evolutionary histories and the possible role that interactions with other systems may have played . since rotation curves may be obtained at different wavelengths they provide information as to the kinematics of different constituents of a galaxy . they may be observed in the infrared as well as in the optical , which may be used to trace ionised gas and the stellar motions , also in the radio and microwave regimes which trace the neutral and molecular gas components of a galaxy . recently , stellar population synthesis models have been used to calculate galactic masses . these models also give us an idea of the total stellar content of a galaxy as well as the distribution of stars of all the different spectral types and luminosity classes @xcite . dynamical theoretical models can also be used to calculate masses for early - type galaxies ( etgs ) , such as those which @xcite constructed for 37 bright elliptical galaxies . from these models he found an average ( m/@xmath5 . discrepancies of the observed velocities in the outer parts with those predicted by the models may be explained by the inclusion of massive dark haloes . @xcite performed dynamical studies of the shapes of line - profiles for 21 elliptical galaxies ; they used them to investigate the dark halo properties and dynamical family relations of these galaxies . they appear to have minimal haloes implied from the fact that the ratio m/@xmath6 turned out maximal . some of these galaxies showed no dark matter within @xmath7 . @xcite investigated the correlations between the mass - to - light m/@xmath4 ratios of 25 elliptical and lenticular galaxies . field and cluster galaxies presented no difference , and their dark matter content within an effective radius @xmath8 was @xmath9 of the total mass contained there . it appeared that the amount of dark matter correlates with galactic rotation velocity ; in the sense that more massive slow - rotating galaxies contain less dark matter that the fast - rotating galaxies . there have been many papers in which dynamical arguments are used to calculate the dynamical mass of galaxies , and hence , by comparison with the amount of luminous mass , they calculate the amount of dark matter present , see for example : @xcite , @xcite , @xcite , @xcite to mention a few . also check the detailed introduction to this topic published in @xcite . the gravitational lens phenomenon provides direct and precise measurements of masses of galaxies at different scales , and allows us to establish the nature and presence of dark matter in a galactic system . elliptical galaxies have been considered to have extended dark - matter massive haloes @xcite that follow the @xcite density profiles . @xcite and @xcite have studied the kinematics of different components in nearby elliptical systems and have concluded that dark matter haloes are required to explain the dynamics of massive elliptical galaxies , provided that newtonian gravity be valid at these scales . galactic mass determinations have also been made using weak and strong lensing observations @xcite . the fraction of total mass in the form of dark matter in etgs , @xmath10 , appears to increase with growing radius reaching values of @xmath11 at five effective radii @xcite . furthermore , @xmath10 within a fixed radius seems to grow with galaxy stellar mass and with velocity dispersion @xcite . @xmath10 varies from small values as in the case of bright giant elliptical galaxies @xcite to very large values , as has been found for dwarf spheroidal galaxies by @xcite . studies of the virgo giant elliptical galaxy ngc 4949 ( m60 ) by @xcite reveal that the kinematics of planetary nebulae in this object is consistent with the presence of a dark matter halo with @xmath12 for @xmath13 . @xcite presented three - integral axisymmetric models for ngc 4649 and ngc 7097 and concluded that the kinematic data for ngc 4649 only require a small amount of dark matter , however @xcite determine @xmath14 at @xmath15 for ngc 4649 . using gravitational lensing experiments , @xcite find a projected dark matter fraction of @xmath16 for 15 etgs , while @xcite studying sixteen early - type lens galaxies determine the lower limit for dark matter @xmath10 inside the effective radius . the median value for this fraction is @xmath17 with variations from almost 0 to up to @xmath18 . as mentioned above , direct detection of dark matter has not been achieved yet . its presence requires the validity of newtonian gravity . if we were to assume that at these very low acceleration regimes newtonian gravity is not valid or may be slightly modified @xcite then further developments have explained several phenomena without the need of dark matter e.g. spiral galaxies , flat - rotation curves @xcite , projected surface density profiles and observational parameters of the local dwarf spheroidal galaxies @xcite , the relative velocity of wide binaries in the solar neighbourhood @xcite , fully self - consistent equilibrium models for ngc 4649 @xcite and references within among others . in this paper we present a study of luminous and dynamical mass inside the effective radius of etgs considering newtonian dynamics . we search for differences between these masses and assume that any difference is due to dark matter or a non - universal imf or a combination of both . the structure of this study is as follows ; in 2 we present the sample of etgs used in this work , in 3 we discuss the calculation of the stellar and virial masses for the galaxies in the sample , in 4 we discuss the distribution of stellar mass as a function of virial mass , in 5 and 6 we outline the difference between virial and stellar mass as a function of mass and redshift , in 7 we discuss our results in the fundamental plane context and finally in 8 we present the conclusions . we use a sample of etgs from the ninth data release ( dr9 ) of the sloan digital sky survey ( sdss ) @xcite and two subsamples of it , all of them in the @xmath19 and @xmath20 filters . these samples were compiled by and described in great detail in @xcite . here we shall describe briefly the selection criteria used . \1 ) the brightness profile of the galaxy must be well adjusted by a de vaucouleurs profile , in both the @xmath19 and @xmath20 filters ( fracdevg = 1 and fracdevr = 1 according to the sdss nomenclature ) . \2 ) the de vaucouleurs magnitude of the galaxies must be contained in the interval @xmath21 and its equivalent in the g filter . \3 ) the quotient of the semi axes ( b / a ) for the galaxies must be larger than 0.6 in both filters @xmath19 and @xmath20 . \4 ) the galaxies must have a velocity dispersion of @xmath22 60 @xmath23 and a signal - to - noise ratio ( s / n ) @xmath24 10 . the main sample is called total - sdss - sample " . it contains approximately 98000 galaxies , is distributed in a redshift interval @xmath1 and within a magnitude range @xmath25 @xmath26 @xmath27 ( @xmath28 ) . the first subsample is named the - morphological - sample " . the main characteristic of the - morphological - sample is that the selection criteria for the morphological classification are more rigorous than in the total - sdss - sample , due to the fact that @xcite use the morphological classification from the galaxy zoo project ( see @xcite ) . with these added criteria they obtain approximately 27,000 etgs . the last subsample is named the - homogeneous - sdss - sample " . in this case @xcite consider a volume limited sample ( 0.04 @xmath29 0.08 ) with the objective of obtaining a complete sample in the bright end of the magnitude range . in this volume they obtain approximately 19 000 etgs . this subsample covers a magnitude range @xmath25 @xmath30 @xmath27 ( @xmath31 ) and is approximately complete for @xmath32 ( see nigoche - netro et al . 2015 for details ) . the photometry and spectroscopy of the samples of galaxies were corrected due to different biases . below we list these corrections : * seeing correction : the seeing - corrected parameters were obtained from the sdss pipeline . * extinction correction : the extinction correction values were obtained from the sdss pipeline . * k correction : the k correction was obtained from @xcite . * cosmological dimming correction : the cosmological dimming correction was obtained from @xcite . * evolution correction : the evolution correction was obtained from @xcite . * effective radius correction : the effective radius correction to the rest reference frame was obtained from @xcite . * aperture correction to the velocity dispersion : the velocity dispersion inside the radius subtended by the sdss fibre was corrected using the aperture correction from @xcite . we use the stellar and virial masses obtained in @xcite . here we shall describe briefly the procedure used to calculate those masses and some terms that are important for the present work . the total stellar mass was obtained by @xcite considering different stellar population synthesis models , using a universal imf ( salpeter or kroupa ) and different brightness profiles ( de vaucouleurs or srsic ) . the combination of these ingredients results in three mass estimations , as follows : \i ) de vaucouleurs salpeter - imf stellar mass . \ii ) srsic salpeter - imf stellar mass . \iii ) kroupa - imf stellar mass . according to @xcite , within a sphere of radius equal to @xmath0 , 42% of the total stellar mass is contained . the stellar masses described before assume a universal imf . however , some papers in the astronomical literature claim that the imf is not universal but rather it depends on the stellar mass @xcite . we do not correct the stellar mass for the behaviour of the imf as a function of mass because there is no accurate equation describing this effect . since our results have to take into account this effect , we will discussed them in the subsequent sections . the total virial mass was obtained by @xcite using an equation from @xcite . this method assumes newtonian mechanics and virial equilibrium for the galaxies in question . the equation is as follows : @xmath33 where the variables @xmath34 , @xmath0 and , @xmath35 represent respectively the total virial mass , the effective radius and the velocity dispersion inside @xmath0 . @xmath36 stands for the gravitational constant and @xmath37 is a scale factor . for the de vaucouleurs profile case @xmath38 @xcite . the amount of mass within an effective radius corresponds to 0.42 times the value calculated from equation ( 1 ) this mass may or may not be luminous . the errors calculated for the different parameters reported in this paper are obtained using the rules of error propagation and considering possible systematics on the photometric and spectroscopic parameters as discussed in detail by @xcite . in the following sections , and taking into consideration only the region internal to @xmath0 , we will carry out an analysis of the behaviour of the virial vs. stellar mass . in @xcite we have made a complete analysis of the distribution of the stellar mass with respect to the virial mass for etgs samples . in this section we present an extract of that analysis only with the relevant information for the goals of the present work . figure 1 is the most important part of the extract because it shows the comparison of viral and stellar mass for each galaxy in our samples . in this figure , column 1 represents the total sample , column 2 the morphologic sample and column 3 the homogeneous sample . the rows correspond to different profiles and imfs , being the first one associated with the de vaucouleurs salpeter - imf stellar mass , the second with the sersic salpeter - imf stellar mass and the third with the kroupa - imf stellar mass . the solid line is the one - to - one line . @xcite discuss different procedures to analyse the distribution of masses shown in figure 1 considering that the mentioned distribution may depend on observational biases , on physical properties of the galaxies , and on arbitrary cuts performed in the observed samples ( see also * ? ? ? * ; * ? ? ? * ; * ? ? ? * ; * ? ? ? . those procedures may be helpful in investigating whether there is dark matter inside etgs . particularly they found that the application of the weighted bisector fit ( @xmath39 fit ) to the mean value of the distribution at quasi - constant mass , results in a reduction of the possible biases which may creep in the process ( for details see section 7.3 of appendix a from * ? ? ? * ) . this method is only a first approximation to the study of dark matter inside etgs because the distribution of masses seen in figure 1 have a high intrinsic dispersion and the physical causes of the intrinsic dispersion are at present yet unknown . refinements to this method must be sought for in the physical causes of the intrinsic dispersion seen in the mass distributions . in the previous section we have mentioned that there are different procedures to analyse the properties of our etgs samples . in particular , we can study the difference between the virial and stellar mass using the mass distribution of the samples and the @xmath39 fit . to obtain the parameters of the linear regression ( @xmath39 fit ) for the different etgs samples we use the nigoche - netro et al . ( 2015 ) procedure as follows : * we calculate the mean value of the logarithm of stellar mass at quasi - constant logarithm of virial mass . * we calculate the mean value of the logarithm of virial mass at quasi - constant logarithm of stellar mass . * we perform a linear regression ( @xmath39 fit ) to the previously mentioned mean values . the term quasi - constant mass in this context means mass intervals in the logarithm of width equal to 0.1 . in figure 2 we show a mosaic of the behaviour of the stellar mass with respect to the virial mass for the etgs total ( column 1 ) , morphologic ( column 2 ) and homogeneous ( column 3 ) samples . the first row corresponds to the de vaucouleurs salpeter - imf stellar mass , the second one corresponds to the srsic salpeter - imf stellar mass and the third one corresponds to the kroupa - imf stellar mass . each graph shows the mean values of the luminous mass distribution at quasi - constant virial mass ( black dots ) , the mean values of the virial mass distribution at quasi - constant stellar mass ( blue squares ) and the @xmath39 fit ( dashed line ) to both point distributions . the solid line is the one - to - one line . in table 1 we show the results of the @xmath39 fit to the different samples of etgs . the difference between virial and stellar mass shown has been obtained considering the slope and the zero point of the fit for each sample . for each value of virial mass we calculated the stellar mass and the average , maximum and minimum differences among these masses . the mean error in the average , maximum and minimum values of the difference between log@xmath40 and log@xmath41 is approximately 0.12 . from table 1 we find the following : * the average difference between log@xmath40 and log@xmath41 considering the samples with salpeter - imf stellar mass is 0.380 , whereas if we consider kroupa - imf stellar mass profiles the average difference is 0.281 . this seems to indicate that the estimated difference between masses is affected by the imf used in the calculation of luminous mass . however the mentioned difference seems to be due to the zero point because the slopes of the samples are similar . * considering only those samples where the masses were obtained using de vaucouleurs profiles the average difference is 0.376 , whereas if we consider srsic profiles the average difference is 0.384 . this seems to indicate that the estimated difference between masses is not affected by the profile used in the calculation of luminous and virial mass . this result is confirmed by the slopes of the samples which are similar . * if we consider only samples in large intervals of redshift ( total and morphological samples ) the average difference is 0.348 , whereas if we consider only the sample restricted in redshift ( homogeneous sample ) the average difference is 0.344 . this seems to indicate that the average difference between masses is similar when we move from a wide to a narrow redshift interval . however the slopes of the samples seems to refute this result . the previous results are in agreement with the results found by @xcite . however , the comparison of the mean values of the difference in masses could mask the real behaviour of the samples because the midpoints of two straight lines could be similar even if the slopes of those straight lines are different . so it is necessary to compare the masses in a different way considering that the difference in mass could depend on other variables such as mass and/or redshift . from table 1 we can see that , if we consider all the samples , the average difference between the maximum and minimum values for log(@xmath42 - log@xmath43 ) is 0.401 . this relatively large difference as well as the equations corresponding to table 1 suggest that there is a dependence of the log(@xmath42 - log@xmath43 ) on virial and stellar mass . this behaviour can be easily seen in figure 3 where , for all samples , the difference between virial and stellar mass depends on virial mass . from figure 3 , it is interesting to note that the behaviour of the samples with different imf - stellar mass is similar , although with different zero points . a similar behaviour also occurs when we consider different profiles . also from figure 3 , it is important to note that the slope for the restricted in redshift sample ( homogeneous sample ) is steeper than the slope for the samples in large intervals of redshift . that is to say , the difference between virial and stellar mass seems to depend on redshift . the behaviour of the differences between virial and stellar mass as a function of mass and redshift requires a deeper analysis which we will address in the following section . in the previous section we have found that the difference between virial and stellar mass depends on mass and seems to depend on redshift . to investigate these dependences in a deeper way we can analyse the difference in masses considering quasi - constant mass ( virial and stellar ) and quasi - constant redshift . the term quasi - constant mass means mass interval in the logarithm of width equal to 0.1 . the term quasi - constant redshift means redshift interval equal to 0.01 . in figure 4 we can see the behaviour of the stellar mass as a function of virial mass considering quasi - constant redshift for the kroupa - imf stellar mass sample . each colour and symbol represents quasi - constant redshift . the redshift value goes from approximately zero ( lower - left part of the graph ) to approximately 0.3 ( upper - right part of the graph ) . in this figure we can see that for the same value of the virial mass , the stellar mass grows with redshift , that is to say , the difference between virial and stellar mass diminishes with redshfit . we can also see that for the high - redshift and high - mass regime the dispersion of the distribution is lower than in the low - mass regime of the samples . from this figure we can see that the redshift plays an important role in the intrinsic dispersion seen in figure 1 . in figure 5 we show the difference between virial and stellar mass as a function of virial mass considering quasi - constant redshift for the kroupa - imf stellar mass sample . the redshift value goes from approximately zero ( upper - left part of the graph ) to approximately 0.3 ( lower - right part of the graph ) . in this graph we can see that for the same value of the virial mass the difference between virial and stellar mass diminishes with redshift . we can also see that the more massive galaxies have a greater difference between virial and stellar mass and that for the high - redshfit and high - mass regime the dispersion of the distribution is lower than in the low - mass regime of the samples . the difference between virial and stellar mass at different redshift is directly related with the dispersion seen in figures 1 and 4 . in figure 6 we show the behaviour of the stellar mass as function of redshift considering quasi - constant virial mass for the kroupa - imf stellar mass sample . the logarithm of virial mass value goes from approximately 10 ( lower - left part of the graph ) to approximately 12 ( upper - right part of the graph ) . in this graph we can see that for a constant value of virial mass the stellar mass increases as function of redshift . we can also see that for the high - mass regime the behaviour of the stellar mass as a function of redshift is less steep than in the low - mass regime of the samples . the slope of the stellar mass as function of redshift at quasi - constant virial mass is related with the intrinsic dispersion seen in figures 1 and 4 . from the previous results we can conclude that the redshift plays an important role in the intrinsic dispersion of the distribution of log(@xmath42 vs. log@xmath43 ) . we also can conclude that the difference between virial and stellar mass , in the redshift range @xmath1 and in the dynamical mass range @xmath44 , depends on mass and redshift . the difference between dynamical and stellar mass increases as a function of dynamical mass and decreases as a function of redshift . this last result is in agreement with recent works from the literature where it is shown that the amount of dark matter could depend on mass @xcite and redshift @xcite . if we convert the data shown in figure 5 to percentages , we find that the difference between masses goes from almost zero to approximately 70% of the virial mass . this difference could be due to the dark matter and/or a non universal imf . therefore , the amount of dark matter , in the redshift range @xmath1 and in the dynamical mass range @xmath44 , goes from almost zero to 70% of the virial mass depending on mass and redshift and on the impact of the imf on the stellar mass estimation . it is important to note that we have found similar results for the de vaucouleurs salpeter - imf sample and srsic salpeter - imf sample . the results previously described can be analysed in the fundamental plane ( fp ) context . during the last 30 years a lot of scientific papers about the fp have been published @xcite . the fp is a relation among the variables ; effective radii ( @xmath45 ) , the effective mean surface brightness ( @xmath46 ) and the central velocity dispersion ( log @xmath47 ) , as follows : @xmath48 where @xmath49 , @xmath50 and @xmath51 represent scale factors . this relation seems to be due to the virial equilibrium of etgs , however the theoretical and observational results do not agree . the difference between the theoretical and observational results is known as the tilt of the fp . there are different explanations for the fp tilt , for example , it could be due to : the increase of m / l with l @xcite , the variation of the fp parameters with redshift @xcite , the variation in the homology of etgs ( non - constant k in the virial relation -see equation 1- ) @xcite , the variation of the shape of the light profile and the content and concentration of dark matter relative to luminous matter @xcite among others . some of these works have found that the tilt could be due to a combination of several of the mentioned effects @xcite which seems to be the most plausible explanation . given that the fp relates dynamic variables and stellar formation processes as log(@xmath42 vs. log@xmath43 ) does , we can extrapolate our findings to the fp . in this sense , the mass and redshift dependence of log(@xmath42 - log@xmath43 ) found in this work go in the same direction as the increase of the content of dark matter relative to luminous matter along fp and the variation of the fp parameters with redshift . that is to say , our findings go in the same direction of the ` hybrid ' explanation to the tilt of the fp . however we have to take into account that in this work we consider that the dark matter follows the light and that the scale factor k in the virial relation ( see equation 1 ) depends only on the light profile which , according to some authors @xcite is not necessarily appropriate . in a forthcoming paper we will analyse these variables and their relation with the difference between log(@xmath42 and log@xmath43 ) . the analysis of the distribution of stellar mass with respect to virial mass on several samples of etgs from the sdss dr9 in the redshift range @xmath1 and in the dynamical mass range @xmath44 has yielded the following results : * a significant part of the intrinsic dispersion of the distribution of log(@xmath42 vs. log@xmath43 ) is due to redshift ( see fig . the difference between dynamical and stellar mass depends on mass and redshift . * the difference between dynamical and stellar mass increases as a function of dynamical mass and decreases as a function of redshift . * the difference between dynamical and stellar mass goes from almost zero to approximately 70% of the virial mass depending on mass and redshift ( see figure 5 ) . this difference is due to dark matter or a non - universal imf or a combination of both . * the amount of dark matter inside etgs would be equal to or less than the difference between dynamical and stellar mass depending on the impact of the imf on the stellar mass estimation . the previous results have been analysed in the fp context and we have found that they go in the same direction as some fp results found in the literature in the sense that they could be interpreted as an increase of dark matter along the fp and a dependence of the fp on redshift . however in this work we have considered that the dark matter follows the same density profile as the stellar component and that the scale factor k in the virial relation ( see equation 1 ) depends only on the light profile which , according to some authors @xcite , is not appropriate for massive and compact galaxies respectively . in a forthcoming paper we will analyse these variables and their possible relation with the log(@xmath42-log@xmath43 ) difference . to the memory of mrs . eutiquia netro castillo , an extraordinary woman . we thank and acknowledge the comments made by an anonymous referee , they improved greatly the presentation of this paper . we also thank to instituto de astronoma y meteorologa ( udg , mxico ) and instituto de astronoma ( unam , mxico ) for all the facilities provided for the realisation of this project . a. nigoche - netro and g. ramos - larios acknowledge support from conacyt and prodep ( mxico ) . patricio lagos is supported by a postdoctoral grant sfrh / bpd/72308/2010 , funded by fct ( portugal ) and fundao para a cincia e a tecnologia ( fct ) under project fcomp-01 - 0124-feder-029170 ( reference fct ptdc / fis - ast/3214/2012 ) , funded by the feder program . a. ruelas - mayorga thanks direccin general de asuntos del personal acadmico , dgapa at unam for financial support under project number papiit in103813 . a. m. hidalgo - gmez thanks instituto politcnico nacional sip20161416 for financial support under project number sip20161416 . cappellari , m. , bacon , r. , bureau , r. , damen , m.c . , davies , r.l . , de zeeuw , p.t . , emsellem , e. , falcn - barroso , j. , krajnovi , d. , kuntscher , h. , mcdermid , r.m . , peletier , r.f . , sarzi , m. , van den bosch , r.c.e . & van de ven , g. 2006 , mnras , 366 , 1126 donofrio , m. , fasano , g. , moretti , a. , marziani , p. , bindoni , d. , j. fritz , j. , varela , j. , bettoni , d. , cava , a. , poggianti , b. gullieuszik , m. , kjrgaard , p. , moles , m. , vulcani , b. , omizzolo , a. , w.j . couch , w. j. & dressler , a 2013 , mnras , 435 , 45 peralta de arriba , l. , balcells , m. , trujillo , i. , falcon - barroso , j. , tapia , t. , cardiel , n. , gallego , j. , guzman , r. , hempel , a. , martn - navarro , i. , perez - gonzalez , p. g. & sanchez - blaquez , p. mnras , 453 , 704

we study the behaviour of the dynamical and stellar mass inside the effective radius ( @xmath0 ) of early - type galaxies ( etgs ) . we use several samples of etgs -ranging from 19 000 to 98 000 objects- from the ninth data release of the sloan digital sky survey . we consider newtonian dynamics , different light profiles and different initial mass functions ( imf ) to calculate the dynamical and stellar mass . we assume that any difference between these two masses is due to dark matter and/or a non universal imf . the main results for galaxies in the redshift range @xmath1 and in the dynamical mass range 9.5 @xmath2 log(m ) @xmath2 12.5 are : i ) a significant part of the intrinsic dispersion of the distribution of dynamical vs. stellar mass is due to redshift . ii ) the difference between dynamical and stellar mass increases as a function of dynamical mass and decreases as a function of redshift . iii ) the difference between dynamical and stellar mass goes from approximately 0% to 70% of the dynamical mass depending on mass and redshift . iv ) these differences could be due to dark matter or a non universal imf or a combination of both . v ) the amount of dark matter inside etgs would be equal to or less than the difference between dynamical and stellar mass depending on the impact of the imf on the stellar mass estimation . vi ) the previous results go in the same direction of some results of the fundamental plane ( fp ) found in the literature in the sense that they could be interpreted as an increase of dark matter along the fp and a dependence of the fp on redshift . [ firstpage ] galaxies : fundamental parameters , photometry , distances and redshifts . cosmology : dark matter .

the ua5 collaboration noticed for the first time that charged - particle multiplicity distributions measured in high energy proton-(anti)proton collisions in limited intervals of pseudo - rapidity have the negative binomial form @xcite . in the present paper this observation will be verified for the collisions at @xmath0 = 0.9 and 2.36 tev performed by ua5 @xcite and alice collaborations @xcite . only non - single diffractive ( nsd ) events will be considered because such a case was analyzed with this respect by both collaborations . in fact , the author investigated alice inelastic events also ( including the case of @xmath0 = 7 tev @xcite ) , but all fits were entirely unacceptable . the negative binomial distribution ( nbd ) is defined as @xmath1 where @xmath2 , @xmath3 and @xmath4 is a positive real number . in the application to high energy physics @xmath5 has the meaning of the number of charged particles detected in an event . the expected value @xmath6 and variance @xmath7 and the variance @xmath8 . [ przyp1 ] ] are expressed as : @xmath9 in this analysis the hypothesis that the charged - particle multiplicities measured in high energy @xmath10 collisions are distributed according to the nbd is verified with the use of the maximum likelihood method ( ml ) and the likelihood ratio test . more details of this approach can be found in refs . . there are two crucial reasons for this approach : the fitted quantity is a probability distribution function ( p.d.f . ) , so the most natural way is to use the ml method , where the likelihood function is constructed directly from the tested p.d.f .. but more important is that because of wilks s theorem ( see appendix b ) one can easily define a statistic , the distribution of which converges to a @xmath11 distribution as the number of measurements goes to infinity . thus for the large sample the goodness - of - fit can be expressed as a @xmath12-value computed with the corresponding @xmath11 distribution . the most commonly used method , the least - squares method ( ls ) ( called also @xmath11 minimization ) , has the disadvantage of providing only the qualitative measure of the significance of the fit , in general . only if observables are represented by gaussian random variables with known variances , the conclusion about the goodness - of - fit equivalent to that mentioned in the point ( i ) can be derived @xcite . it is worth noting that the ml method with binned data and poisson fluctuations within a bin was already applied to fitting multiplicity distributions to the nbd but at much lower energies ( e-802 collaboration @xcite ) . the number of charged particles @xmath13 is assumed to be a random variable with the p.d.f . given by eq . ( [ nbdist ] ) . each event is treated as an independent observation of @xmath13 and a set of a given class of events is a sample . for @xmath14 events in the class there are @xmath14 measurements of @xmath13 , say @xmath15 . some of these measurements can be equal , _ i.e. _ @xmath16 for @xmath17 can happen . the whole population consists of all possible events with the measurements of 0 , 1 , 2 , ... charged particles and by definition is infinite . ] for the class of events one can defined the likelihood function @xmath18 where @xmath19 is the nbd , eq . ( [ nbdist ] ) . the values @xmath20 and @xmath21 for which @xmath22 has its maximum are the maximum likelihood estimates of parameters @xmath12 and @xmath4 . this is equivalent to the maximization of the log - likelihood function @xmath23 thus the values @xmath20 and @xmath21 are the solutions of the equations : @xmath24 @xmath25 it can be proven that one of the necessary conditions for the existence of the maximum is ( see appendix a for details ) : @xmath26 _ i.e. _ the distribution average has to be equal to the experimental average . let divide the sample defined in sect . [ likmeth ] into @xmath27 bins characterized by @xmath28 - the number of measured charged particles for @xmath17 and @xmath29 . [ przyp3 ] ] and @xmath30 - the number of entries in the @xmath31th bin , @xmath32 ( details of the theoretical framework of this section can be found in refs . ) . then the expectation value of the number of events in the @xmath31th bin can be written as @xmath33 where @xmath34 is the expected number of all events in the sample , @xmath35 . this is because one can treat the number of events in the sample @xmath14 also as a random variable with its own distribution - poisson one . generally , the whole histogram can be treated as one measurement of @xmath27-dimensional random vector @xmath36 which has a multinomial distribution , so the joint p.d.f . for the measurement of @xmath14 and @xmath37 can be converted to the form @xcite : @xmath38 since now @xmath39 is the p.d.f . for one measurement , @xmath40 is also the likelihood function @xmath41 with the use of eq . ( [ neventi ] ) the corresponding likelihood function can be written as @xmath42 then the likelihood ratio is defined as @xmath43 where @xmath44 , @xmath20 and @xmath21 are the ml estimates of @xmath34 , @xmath12 and @xmath4 with the likelihood function given by eq . ( [ liksubset ] ) and @xmath45 , @xmath46 are the ml estimates of @xmath47 treated as free parameters . note that since the denominator in eq . ( [ likeliratio ] ) does not depend on parameters , the log - ratio defined as @xmath48 where @xmath47 are expressed by eq . ( [ neventi ] ) , can be used to find the ml estimates of @xmath34 , @xmath12 and @xmath4 . further , the statistic given by @xmath49 approaches the @xmath11 distribution asymptotically , _ i.e. _ as the number of measurements , here the number of events @xmath14 , goes to infinity ( the consequence of the wilks s theorem , see appendix b ) . the values @xmath50 are the estimates of @xmath47 given by @xmath51 and if one assumes that @xmath34 does not depend on @xmath12 and @xmath4 then @xmath52 . for such a case @xmath53 and eq . ( [ poissonchi ] ) becomes @xmath54 also then one can just put @xmath55 and eq . ( [ logratio ] ) can be rewritten as @xmath56 with the term depending on @xmath12 and @xmath4 the same as eq . ( [ loglikfubi ] ) and @xmath57 . therefore conclusions of appendix a holds here , _ i.e. _ the necessary conditions for the existence of the maximum is @xmath58 , eq . ( [ dloglikfk ] ) is the equation which determines @xmath21 and @xmath20 is obtained with the help of eq . ( [ oneoverp ] ) . note that the maximum of @xmath59 is the minimum of @xmath60 , so from eqs . ( [ multinomchi ] ) and ( [ logratfreq ] ) one arrives at @xmath61 in fact , the method just described assumes that the sum of @xmath62 over all bins equals 1 . but only the infinite sum of @xmath63 is 1 . however the measured values of @xmath64 are big enough ( of the order of 20 at least for all considered cases ) so in the vicinity of @xmath20 and @xmath21 the sum of @xmath63 up to @xmath65 equals 1 approximately ( see the seventh column in table [ table1 ] ) . nevertheless , to calculate @xmath66 , eq . ( [ finalchi ] ) , @xmath67 were normalized appropriately and these results are listed in the fifth column of tables [ table1]-[table3 ] . another way to treat this problem is to create arbitrary the @xmath68st bin for all possible @xmath69 and with @xmath70 . bins with @xmath71 ( @xmath72 equivalently ) do not contribute in eq . ( [ finalchi ] ) ( see ref . ) . in practice , it means that @xmath66 would be calculated also from eq . ( [ finalchi ] ) but without the normalization . it has turned out that that way leads to much greater values of @xmath66 . the method described in sections [ likmeth ] and [ liktest ] requires that all bins in a given data set have the width equal to 1 , so as the experimental probability @xmath73 to measure a signal in the @xmath31th bin was equivalent to the probability of the measurement of @xmath74 charged particles ( the first bin is the bin of 0 charged particles detected ) . this is fulfilled for all bins of the considered data sets except the ends of their tails . in these tails the measured values of @xmath73 have been uniformly distributed over the bin range so as the method could be applied directly . if the bin width is not significantly greater than 1 then this approximation should not change substantially the value of @xmath66 given by eq . ( [ finalchi ] ) because in the most cases @xmath73 at tails are two orders smaller than in the main part of distributions . also errors in tails are bigger , in the range @xmath75 , increasing with @xmath31 . since the test statistic @xmath76 has a @xmath77 distribution approximately in the large sample limit , it can be used as a test of the goodness - of - fit . the result of the test is given by the so - called @xmath12-value which is the probability of obtaining the value of the statistic , eq . ( [ poissonchi ] ) , equal to or greater then the value just obtained by the ml method for the present data set , when repeating the whole experiment many times : @xmath78 where @xmath79 is the @xmath77 p.d.f . and @xmath80 the number of degrees of freedom , @xmath81 here . @cccccccccc@ & & & & @xmath11/@xmath80 & & & + experiment & n & @xmath21 & @xmath20 & @xmath11 & p - value & @xmath82 & quadrature & sum & statistical + @xmath0 & & & & ( @xmath80 ) & [ % ] & & sum & & only + ua5 & 8550.0 & 1.5574 & 0.3012 & 0.339 & 99.97 & 0.99996 & 0.375 & na & na + @xmath83 tev & ( 80 % _ eff_. ) & @xmath84 & @xmath85 & 10.16 & & & & & + @xmath86 0.5 & & & & ( 30 ) & & & & & + & & & & & + ua5 & 10000.0 & 1.5574 & 0.3012 & 0.396 & 99.87 & 0.99996 & 0.375 & na & na + @xmath83 tev & ( 70 % _ eff_. ) & @xmath87 & @xmath88 & 11.88 & & & & & + @xmath86 0.5 & & & & ( 30 ) & & & & & + & & & & & + alice & 149663.16 & 1.3764 & 0.2767 & 14.155 & 0 & 0.99960 & 1.116 & 0.576 & 3.089 + @xmath83 tev & & @xmath89 & @xmath90 & 353.88 & & & & & + @xmath86 0.5 & & & & ( 25 ) & & & & & + & & & & & + alice & 128476.45 & 1.4316 & 0.1625 & 37.761 & 0 & 0.99865 & 1.886 & 1.034 & 11.51 + @xmath83 tev & & @xmath91 & @xmath92 & 1548.21 & & & & & + @xmath86 1.0 & & & & ( 41 ) & & & & & + & & & & & + alice & 60142.77 & 1.4955 & 0.1332 & 22.051 & 0 & 0.99876 & 2.993 & 1.671 & 15.31 + @xmath83 tev & & @xmath93 & @xmath94 & 1168.69 & & & & & + @xmath86 1.3 & & & & ( 53 ) & & & & & + & & & & & + ua5 & 8550.0 & 1.7987 & 0.1385 & 0.812 & 87.81 & 0.99991 & 0.487 & na & na + @xmath83 tev & ( 80 % _ eff_. ) & @xmath95 & @xmath96 & 60.12 & & & & & + @xmath86 1.5 & & & & ( 74 ) & & & & & + & & & & & + ua5 & 10000.0 & 1.7987 & 0.1385 & 0.950 & 59.99 & 0.99991 & 0.487 & na & na + @xmath83 tev & ( 70 % _ eff_. ) & @xmath97 & @xmath98 & 70.31 & & & & & + @xmath86 1.5 & & & & ( 74 ) & & & & & + & & & & & + alice & 38970.79 & 1.1778 & 0.2084 & 6.266 & 0 & 0.99930 & 0.888 & 0.501 & 3.592 + @xmath99 tev & & @xmath100 & @xmath101 & 194.26 & & & & & + @xmath86 0.5 & & & & ( 31 ) & & & & & + & & & & & + alice & 37883.99 & 1.2139 & 0.1180 & 17.416 & 0 & 0.99726 & 2.209 & 1.312 & 17.73 + @xmath99 tev & & @xmath102 & @xmath103 & 853.37 & & & & & + @xmath86 1.0 & & & & ( 49 ) & & & & & + & & & & & + alice & 22189.40 & 1.2123 & 0.0927 & 15.561 & 0 & 0.99644 & 4.0557 & 2.4537 & 34.40 + @xmath99 tev & & @xmath104 & @xmath103 & 949.22 & & & & & + @xmath86 1.3 & & & & ( 61 ) & & & & & + & & & & & + [ table1 ] @cccccccccc@ & & & & @xmath11/@xmath80 & & + experiment & n & @xmath21 & @xmath20 & @xmath11 & p - value & quadrature & sum & statistical & @xmath105 + @xmath0 & & & & ( @xmath80 ) & [ % ] & sum & & only & + ua5 & 8550.0 & 1.5574 & 0.3012 & 0.211 & 99.998 & 0.072 & na & na & 0.203 + @xmath83 tev & ( 80 % _ eff_. ) & @xmath84 & @xmath85 & 4.859 & & & & & + @xmath86 0.5 & & & & ( 23 ) & & & & & + & & & & & + ua5 & 10000.0 & 1.5574 & 0.3012 & 0.247 & 99.991 & 0.072 & na & na & 0.237 + @xmath83 tev & ( 70 % _ eff_. ) & @xmath87 & @xmath88 & 5.683 & & & & & + @xmath86 0.5 & & & & ( 23 ) & & & & & + & & & & & + alice & 149663.16 & 1.3764 & 0.2767 & 14.498 & 0 & 0.728 & 0.381 & 2.458 & 15.107 + @xmath83 tev & & @xmath89 & @xmath90 & 347.95 & & & & & + @xmath86 0.5 & & & & ( 24 ) & & & & & + & & & & & + alice & 128476.45 & 1.4316 & 0.1625 & 36.855 & 0 & 1.718 & 0.948 & 11.010 & 38.017 + @xmath83 tev & & @xmath91 & @xmath92 & 1547.91 & & & & & + @xmath86 1.0 & & & & ( 42 ) & & & & & + & & & & & + alice & 60142.77 & 1.4955 & 0.1332 & 24.323 & 0 & 2.213 & 1.276 & 15.201 & 25.771 + @xmath83 tev & & @xmath93 & @xmath94 & 1167.51 & & & & & + @xmath86 1.3 & & & & ( 48 ) & & & & & + & & & & & + ua5 & 8550.0 & 1.7987 & 0.1385 & 1.099 & 28.94 & 0.362 & na & na & 1.14 + @xmath83 tev & ( 80 % _ eff_. ) & @xmath95 & @xmath96 & 57.16 & & & & & + @xmath86 1.5 & & & & ( 52 ) & & & & & + & & & & & + ua5 & 10000.0 & 1.7987 & 0.1385 & 1.286 & 8.06 & 0.362 & na & na & 1.33 + @xmath83 tev & ( 70 % _ eff_. ) & @xmath97 & @xmath98 & 66.85 & & & & & + @xmath86 1.5 & & & & ( 52 ) & & & & & + & & & & & + alice & 38970.79 & 1.1778 & 0.2084 & 7.030 & 0 & 0.761 & 0.428 & 3.805 & 7.465 + @xmath99 tev & & @xmath100 & @xmath101 & 189.82 & & & & & + @xmath86 0.5 & & & & ( 27 ) & & & & & + & & & & & + alice & 37883.99 & 1.2139 & 0.1180 & 18.535 & 0 & 2.288 & 1.362 & 18.802 & 20.282 + @xmath99 tev & & @xmath102 & @xmath103 & 852.59 & & & & & + @xmath86 1.0 & & & & ( 46 ) & & & & & + & & & & & + alice & 22189.40 & 1.2123 & 0.0927 & 18.233 & 0 & 4.245 & 2.599 & 39.647 & 19.980 + @xmath99 tev & & @xmath104 & @xmath103 & 948.11 & & & & & + @xmath86 1.3 & & & & ( 52 ) & & & & & + & & & & & + [ table2 ] the results of the analysis are presented in table [ table1 ] . note that for ua5 cases two possibilities of the corrected number of events are listed . this is because only the measured number of events , @xmath106 , is given in ref . . however , the fits have been done to the corrected distributions , so also the corrected number of events should be put into eq . ( [ finalchi ] ) . the number have been estimated in the following way : in fig.4 of ref . the mean of the observed distribution versus the corrected ( true ) number of particles is plotted , the curve is a straight line roughly with the tangent equal to @xmath107 , so one can guess that the efficiency is also about @xmath108 . just to check how results are stable with respect to a change in the number of events , the case with @xmath109 efficiency has been also calculated . as one can see , for all alice cases the hypothesis in question should be rejected , whereas for the listed ua5 cases should be accepted . but it was claimed that charged - particle multiplicities measured in the limited pseudo - rapidity windows by the alice collaboration are distributed according to the nbd @xcite . however that conclusion was the result of the @xmath11 minimization ( the ls method ) . therefore it seems to be reasonable to check what are the values of the ls @xmath11 function at the ml estimators listed in the third and fourth columns of table [ table1 ] . for the sample described in sect . [ liktest ] one can define the ls @xmath77 function as : @xmath110 where @xmath111 is the uncertainty of the @xmath31th measurement . here this function * is not minimized * with respect to @xmath12 and @xmath4 as in the ls method but is calculated at ml estimates of @xmath12 and @xmath4 , _ i.e. _ at @xmath20 and @xmath21 . one can see from the eight and ninth columns of table [ table1 ] that @xmath112/@xmath80 values are significant for the alice narrowest pseudo - rapidity windows , what agrees with the results of ref . . since the determination of @xmath21 and @xmath20 has been done for the distributions modified in their tails , as it has been just explained , one should check what values of @xmath11 and @xmath112 are at @xmath21 and @xmath20 for the original data sets . it means that if the @xmath31th bin width is greater than 1 , instead of @xmath67 in eq . ( [ finalchi ] ) the appropriate sum @xmath113 over @xmath114 is taken . the results of the check are presented in table [ table2 ] . qualitatively the results are the same as in table [ table1 ] , only slight differences in numbers can be noticed except the ua5 cases ( for @xmath86 0.5 @xmath11 has decreased more than 2 times , but the change is in the good direction ) . this is because the maximal width of a tail bin is 2 for all alice cases , but is 8 and 17 for ua5 windows @xmath86 0.5 and @xmath86 1.5 , respectively . of course , the assumption of the uniform distribution inside a wider bin causes greater discrepancies . nevertheless , the results of the test for both ua5 cases are positive even if ( @xmath21 , @xmath20 ) is not the maximum of the exact likelihood function ( in fact , values of @xmath21 are the same as those obtained by ua5 collaboration in ref . ) . this is guaranteed by the wilks s theorem ( see appendix b ) , which allows for the test of a single point in the parameter space . then the tested point might not be the best estimate of the true value but the hypothesis in question becomes the hypothesis only about a particular distribution ( a _ simple _ hypothesis ) . this is also the reason why @xmath115 in table [ table2 ] . in terms of rigorous statistics single points are tested in there . in all alice cases @xmath11 values listed in the fifth column of table [ table2 ] are only slightly smaller than corresponding ones from table [ table1 ] . for @xmath86 0.5 the decrease is about @xmath116 , for other cases is less than @xmath117 . also @xmath11/@xmath80 values are much greater than 1 . therefore it is reasonable to recognize @xmath21 and @xmath20 determined for modified data sets as a good approximations of the ml estimators . thus the hypothesis about the nbd should be rejected on the basis of obtained values of @xmath11/@xmath80 and @xmath12-values . one can also compare @xmath11/@xmath80 with @xmath112/@xmath80 calculated for the original data sets and the same @xmath21 and @xmath20 . the results are listed in four last columns of table [ table2 ] for various treatment of errors . note that for ua5 conclusions from both statistics are exactly the same . in the alice both cases of the window @xmath86 0.5 , @xmath112/@xmath118 is acceptable for errors expressed as the quadrature sum of statistical and systematical components and is smaller than the corresponding values in table [ table1 ] . in other alice cases @xmath112/@xmath80 is substantially greater than 1 for the same treatment of errors . this is in the full agreement with the results of ref . . one can also check what @xmath112/@xmath80 is if only statistical errors are taken into account . the results are listed in the next to last column of table [ table2 ] . for all alice cases the values are much greater than 1 . this means that acceptable @xmath112/@xmath80 was obtained only because of significant systematic errors of alice measurements . significant is subjective , here means significant with respect to the sample size , not to the value of @xmath73 . the crucial question is now why the conclusions from @xmath11 and @xmath112 test statistics are the same for ua5 data but entirely opposite for alice measurements ? the main difference between both statistics is that @xmath11 depends explicitly on the number of events but @xmath112 does not . on opposite , @xmath11 does not depend on the actual errors but @xmath112 does . in fact , @xmath11 statistic implicitly assumes errors of the type @xmath119 , what is the straightforward result of the form of the likelihood function , eqs . ( [ jointpdf ] ) and ( [ ljointpdf ] ) , namely the product of poisson distributions . this is revealed when one compare @xmath11/@xmath80 and @xmath112/@xmath80 with errors @xmath120 ( the fifth and last column in table [ table2 ] ) . the values are practically the same . @cccccccccc@ & & & & @xmath11/@xmath80 & & + experiment & n & @xmath21 & @xmath20 & @xmath11 & p - value & quadrature & sum & statistical & @xmath105 + @xmath0 & & & & ( @xmath80 ) & [ % ] & sum & & only & + ua5 & 8550.0 & 1.5574 & 0.3012 & 0.211 & 99.998 & 0.072 & na & na & 0.203 + @xmath83 tev & ( 80 % _ eff_. ) & @xmath84 & @xmath85 & 4.859 & & & & & + @xmath86 0.5 & & & & ( 23 ) & & & & & + & & & & & + alice & 8550.0 & 1.3764 & 0.2767 & 0.828 & 70.37 & 0.728 & 0.381 & 2.458 & 0.863 + @xmath83 tev & & @xmath121 & @xmath122 & 19.88 & & & & & + @xmath86 0.5 & & & & ( 24 ) & & & & & + & & & & & + alice & 8550.0 & 1.4316 & 0.1625 & 2.453 & 5 @xmath123 & 1.718 & 0.948 & 11.010 & 2.530 + @xmath83 tev & & @xmath124 & @xmath125 & 103.01 & & & & & + @xmath86 1.0 & & & & ( 42 ) & & & & & + & & & & & + alice & 8550.0 & 1.4955 & 0.1332 & 3.458 & 7 @xmath126 & 2.213 & 1.276 & 15.201 & 3.664 + @xmath83 tev & & @xmath127 & @xmath128 & 165.97 & & & & & + @xmath86 1.3 & & & & ( 48 ) & & & & & + & & & & & + ua5 & 8550.0 & 1.7987 & 0.1385 & 1.099 & 28.94 & 0.362 & na & na & 1.14 + @xmath83 tev & ( 80 % _ eff_. ) & @xmath95 & @xmath96 & 57.16 & & & & & + @xmath86 1.5 & & & & ( 52 ) & & & & & + & & & & & + [ table3 ] to find out what is the reason for the above - mentioned disagreement the calculations of table [ table2 ] have been repeated for alice measurements at @xmath83 tev but with the arbitrary assumption that all cases have the same number of events as ua5 ones . the results are listed in table [ table3 ] . one can see that now there is full agreement between @xmath11 and @xmath112 test statistic results for all alice cases . this means that the accuracy with which experimental distributions approximate the nbd has not increased in alice data even though the sample sizes are one order greater . but the accuracy should increase with the sample size because if the hypothesis is true the postulated form of distribution is exact for the whole population . so with the growing number of events , the experimental distribution should be closer to the postulated one . this is also seen in the form of @xmath66 , eq . ( [ finalchi ] ) , where the linear dependence on @xmath14 is explicit . to keep @xmath66 at least constant when @xmath14 ( the sample size ) is growing the relative differences between @xmath129 and @xmath73 have to decrease . the main conclusion is that the hypothesis of the nbd of charged - particle multiplicities measured by the alice collaboration in proton - proton collisions at @xmath0 = 0.9 and 2.36 tev should be rejected for all pseudo - rapidity window classes . this is the result of likelihood ratio tests performed for the corresponding data samples . the significant systematic errors are the reasons for acceptable values of the least squares test statistic for the narrowest pseudo - rapidity window measurements . the second conclusion is that the size of proper errors ( _ i.e. _ not too big and not too small , both extremes cause the false inference from @xmath112/@xmath80 values ) is somehow related to the sample size . here , for instance , errors of the type @xmath119 could be a frame of reference as it has been revealed from the results gathered in tables [ table2 ] and [ table3 ] . this is connected with the meaning of the formulation of a hypothesis . if the hypothesis is true , it means that the form of a distribution postulated by this hypothesis is exact for the whole population . thus for the very large samples ( as in all alice cases ) the measured distribution should be very close to that postulated . the performed analysis has shown that the alice experimental errors are much bigger than the acceptable discrepancies ( acceptable for these sample sizes ) . therefore @xmath11 and @xmath112 test statistics give the opposite answers in the narrowest pseudo - rapidity windows of the alice measurements . for the ua5 sample sizes , which are much smaller than the alice ones , the experimental errors have turned out to be of the order of acceptable discrepancies , so both test statistics give the same answer . the author thanks jan fiete grosse - oetringhaus for providing him with the numbers of entries in the alice event classes . this work was supported in part by the polish ministry of science and higher education under contract no . n n202 231837 . [ [ section ] ] the sample defined in sect . [ likmeth ] can be divided into @xmath27 bins with the different value of measured @xmath13 in each bin . let @xmath30 be the number of events in the @xmath31th bin , _ i.e. _ events with the same measured value of @xmath13 , say @xmath28 . then the number of events in the sample equals @xmath130 dividing by @xmath14 one can obtain the condition for experimental probabilities ( frequencies ) @xmath73 : @xmath131 now the likelihood function , eq . ( [ likelfun ] ) , can be rewritten as @xmath132^{n } \cr \cr & & = \bigg [ \ ; \prod_{i=1}^{m}\ ; p(y_i ; p , k)^{p_i^{ex } } \bigg ] ^{n}\ ; , \label{likfunbin}\end{aligned}\ ] ] and the corresponding log - likelihood function reads @xmath133 since the logarithm of the nbd is given by @xmath134 the necessary conditions for the existence of the maximum , eqs . ( [ loglikeqp ] ) , have the following form : @xmath135 \cr \cr & & = n \bigg [ -\frac{1}{1-p } \sum_{i=1}^{m}\;p_i^{ex } y_i + \frac{k}{p } \sum_{i=1}^{m}\;p_i^{ex } \bigg ] \cr \cr & & = n \bigg [ -\frac{1}{1-p } \langle n_{ch } \rangle + \frac{k}{p } \bigg ] = 0 \ ; , \label{dloglikeqp}\end{aligned}\ ] ] @xmath136 \cr \cr & & = n \bigg [ \sum_{i=1}^{m}\;p_i^{ex}\ ; \sum_{j=1}^{y_i}\ ; \frac{1}{k+j-1 } + \ln{p } \bigg ] = 0 \ ; , \label{dloglikeqk}\end{aligned}\ ] ] where the sum over @xmath137 is 0 if @xmath138 . from eqs . ( [ dloglikeqp ] ) and ( [ parametpk ] ) one can obtain : @xmath139 expressing @xmath12 as a function of @xmath4 and @xmath140 @xmath141 and substituting it to eq . ( [ dloglikeqk ] ) the equation which determines @xmath21 is obtained : @xmath142 = 0 \;. \cr & & \label{dloglikfk}\end{aligned}\ ] ] the above equation can be solved numerically . having obtained @xmath21 and substituting it into eq . ( [ oneoverp ] ) @xmath20 is derived . let @xmath143 be a random variable with p.d.f @xmath144 , which depends on parameters @xmath145 , where a parameter space @xmath146 is an open set in @xmath147 . for the set of @xmath14 independent observations of @xmath143 , @xmath148 , one can defined the likelihood function this is a statistic because it does not depend on parameters @xmath153 no more , in the numerator and the denominator there are likelihood function values at the ml estimators of parameters @xmath153 with respect to sets @xmath150 and @xmath146 , respectively . the wilks s theorem says that under certain regularity conditions if the hypothesis @xmath150 is true ( _ i.e. _ it is true that @xmath154 ) , then the distribution of the statistic @xmath155 converges to a @xmath11 distribution with @xmath156 degrees of freedom as @xmath157 @xcite . the proof can be found in ref . . note that @xmath158 is possible , so one point in the parameter space ( one value of the parameter ) can be tested as well . 00 ua5 collab . ( g. j. alner _ _ ) , _ phys . lett . b _ * 160 * , 193 ( 1985 ) . ua5 collab . ( r. e. ansorge _ et al_. ) , _ z. phys . c _ * 43 * , 357 ( 1989 ) . alice collab . ( k. aamodt _ et al_. ) , _ eur . c _ * 68 * , 89 ( 2010 ) . alice collab . ( k. aamodt _ et al_. ) , _ eur . c _ * 68 * , 345 ( 2010 ) . g. cowan , _ statistical data analysis _ , ( oxford university press , oxford , 1998 ) f. james , _ statistical methods in experimental physics _ , ( world scientific , singapore , 2006 ) s. baker and r. d. cousins , _ nucl . meth . _ * 221 * , 437 ( 1984 ) . e-802 collab . ( t. abbott _ et al_. ) , _ phys . c _ * 52 * , 2663 ( 1995 ) . j. f. grosse - oetringhaus , private communication . r. m. dudley , _ 18.466 mathematical statistics , spring 2003 _ , ( massachusetts institute of technology : mit opencourseware ) , http://ocw.mit.edu/courses/ mathematics/18 - 466-mathematical - statistics - spring-2003/lecture - notes/

likelihood ratio tests are performed for the hypothesis that charged - particle multiplicities measured in proton-(anti)proton collisions at @xmath0 = 0.9 and 2.36 tev are distributed according to the negative binomial form . results indicate that the hypothesis should be rejected in the all cases of alice - lhc measurements in the limited pseudo - rapidity windows , whereas should be accepted in the corresponding cases of ua5 data . possible explanations of that and of the disagreement with the least - squares fitting method are given .

first , the problem of the influence of the weak microlensing effect on the pulsar timing observations was discussed in @xcite . it was considered as interstellar shapiro effect . the massive body that flies not far from the line pulsar - observer produces changes in the observing frequency of the pulsar similar to glitches . estimations were made for crab and vela pulsars , glitches in these pulsars can be partially explained by the influence of the effect . substantial contribution to the problem was made by @xcite ; they mainly investigated the case of microlensing ( i.e. the gravitational deflector flied very close to the line observer - pulsar ) . it was shown that the microlensing effect would cause short - term growth of the residuals and follow - up relaxation . whole interaction would take less then several years and the maximum amplitude of the residuals would be 20 - 30 ms . such remarkable events are very rare , but all the pulsars are affected by the weak microlensing effect to a greater or lesser extent . this effect was considered in @xcite , where timing of millisecond pulsars was proposed as detection method for machos . growth of number of observed pulsars and time span of observation would make such detection easier . numerical estimates were made in @xcite . they stated that even when the measurement accuracy reaches to 10 ns , probability of the remarkable influence would be in the order of @xmath0 for the pulsar of a few kpc distance from us observed over ten years . on the other hand there s well developed formalism for the effect that came from the optics . the weak microlensing effect causes distant sources like quasars from icrf to `` tremble '' on the level of tens of mas . it was shown in @xcite that these angular fluctuations range from a few up to hundreds of microarcseconds and this leads to a small rotation of the celestial reference frame . in @xcite influence of the effect on parallax measurements was considered - apparent parallax can be even negative due to the influence of the effect . also , the weak microlensing effect can affect vlbi observations @xcite and it should be taken into account with new generation of space - based vlbi . in @xcite some statistical studies with toy - models were made , that was applied later to real model of the galaxy . in fact , both weak microlensing effect and fly - by effect on timing are very similar and can be considered as manifestation of 4d ( four - dimensional ) astrometry @xcite in this work we tried to apply eikonal formalism that was developed earlier for investigation of weak microlensing effect for use in pulsar timing studies . the paper is organized as follows . in section 2 we give a short review of influence of a passing body on pulsar timing in eikonal approximation . in section 3 we apply a model of distribution of stars in our galaxy to numerical estimations of their influence on pulsar timing and conclude our consideration in section 4 . change of phase during the propagation of electromagnetic wave can be obtained as a solution of hamilton - jacobi equation for a massless particle : @xmath1 though @xmath2 formally is a function of action , we hereafter identify it as eikonal or wave phase along the trajectory of the ray of light . in weak field approximation the metric tensor of gravitational field can be written down in a following form @xmath3 here @xmath4 is flat minkowskian metric , @xmath5 small additions to the flat metric that describes gravitational field of spherically symmetric body ( star ) equation ( [ eik1 ] ) can be solved in the following form : we take an exact solution @xcite and then take its asymptotic when the impact parameter of the propagating ray is much larger then the schwarzschild radius @xmath6 ( @xmath7-mass of the deflector ) @xmath8 here @xmath9 is full change of the phase along the trajectory , @xmath10- change of the phase along the trajectory that corresponds to the propagation in the flat space and time , @xmath11 - schwarzschild radius of the deflector , @xmath12-frequency of the electromagnetic wave , r- some point on the trajectory , @xmath13 - impact parameter ( i.e. minimal distance between deflector @xmath14 and curve of photon propagation ) . -deflecting object . ] only the second term in ( [ eik3 ] ) is a matter of interest to us , though it s only a small addition to the usual change of phase during the propagation.the complete phase shift can be obtained as a sum of two solutions . the first is a phase shift during propagation from the source of the electromagnetic waves ( which is located in @xmath15 ) to the closest approach to the deflector ( we set the point of origin to the center of the deflector ) : @xmath16 the second - is a phase shift during propagation from the closest approach to the deflector to the observers ( at @xmath17 ): @xmath18 and the total phase shift is : @xmath19 we treated the deflector as a motionless body in this solution . in fact , all stars , including machos of our galaxy are moving . approximate solution of space - time metric in the case of moving deflector and trajectory of photon in such a variable gravitational field was calculated in @xcite . the metric which originates from a moving body and small perturbations of photon trajectory in gravitational field of this body , differ in @xmath20 terms from our solutions and we will omit this difference . to describe the motion we take @xmath13 ( impact parameter ) as function of @xmath21 only : @xmath22 \end{array}\ ] ] indices denotes values at different epochs @xmath23 and @xmath24 . the first two terms are negligibly small , so we can rewrite expression ( [ eik7 ] ) : @xmath25 also we can write out time dependence of @xmath26 : + @xmath27 , here @xmath28-minimal impact parameter , @xmath29- velocity of relative motion of pulsar and deflector , @xmath30 - epoch of the closest approach . we can rewrite the equation for the phase shift and obtain equation for time delays or residuals of time of arrival ( toa ) . it s worth noting , that these delays do nt depend on frequency of electromagnetic wave : @xmath31 and impact parameter @xmath32 is passing by near this position ] we can set the first epoch @xmath23 equal to zero and discard the second index , @xmath33 : @xmath34 here , @xmath21 is time span of observations ( we set the epoch of initial observations equal to 0 ) , @xmath30 - is the epoch of the closest approach of the deflector to the line of propagation . it s convenient to consider this problem on the `` plane of deflector '' . thus we convert all linear measures into angular ones : @xmath35 , @xmath36 , @xmath32 -angular distance of the closest approach of deflector to pulsar , @xmath37 - angular velocity of the relative motion ( mainly due to the proper motion of pulsar ) , @xmath38- distance between the deflector and the observer . hereafter phrases like `` deflector s close to pulsar '' mean we observe close angular coincidence of the bodies , not in 3d space . @xmath39 value @xmath32 depends on location of pulsar in galaxy and its proper motion . the higher is density of deflectors in the neighborhood of pulsar on the celestial sphere , the smaller that value would be . we take into consideration only deflectors between the pulsar and the observer , because they make the largest contribution on the effect . we chose two pulsars j1643 - 1224 and b1937 + 21 for further estimates , because they re quite distant and located in populated regions of our galaxy ( b1937 + 21 : @xmath40 ; j1643 - 1224 : @xmath41 ) @xcite , so probability that effect would have place is much higher than for other millisecond pulsars . it s essential to define values @xmath32 and @xmath42 - average duration of influence . they can be approximately found in such way @xcite : stars are nearly uniformly distributed in the neighborhood of the pulsar on the celestial sphere ; the angular distance to the nearest star , which would affect the pulsar timing depends on the location of pulsar . we calculated the density of the stars in the neighborhood , using accepted model of the disk of our galaxy @xcite . @xmath43 @xmath44 sought density in the direction of the pulsar , which is assigned by the angles @xmath45,@xmath46 . @xmath45- angle between the line of sight and the galactic plane , @xmath46- angle between the projection of the line observer - pulsar to the galactic plane and the line solar system - galactic center;@xmath47- distance from the observer . @xmath48 @xmath49 -density of the stars in sun s neighborhood , @xmath50 -distance from the axis of the galaxy , z - distance from the galactic plane , @xmath51 -distance between the solar system and the galactic center , @xmath52 and @xmath53 - radial and vertical scales of the model , accordingly . @xmath54 @xmath55 @xmath56 shows that the influence of higher - power order items should be taken into consideration . ] shows that the influence of higher - power order items should be taken into consideration . ] average angular distance @xmath57 between the pulsar and the closest deflector ( star ) can be found with taking into account @xmath44 . values @xmath32 and @xmath42 were calculated using monte - carlo simulation : a circle of @xmath57 were circumscribed around the pulsar on the celestial sphere , then a large amount ( 1000 ) of test deflectors with proper motion @xmath37 were started from this circle under random angles @xmath58 . as a result we found distributions for values @xmath32 and @xmath42 , and their averages , that were used in following estimates . only known distribution of stars in our galaxy was used in our estimates and if we take into account possible influence of dark matter , then sought values can be lower in 2 - 3 times , because mass of dm does nt exceed mass of ordinary matter more than 4 - 5 times . also , we set mass of deflectors equal to @xmath59 . values that are essential for further estimations ( j1643 - 1224 , b1937 + 21 ) are given in the table below . @xmath32 & @xmath42 + j1643 - 1224 & 7.3 & @xmath60 & 4.7 `` & 470 yr + b1937 + 21 & 2.5 '' & @xmath60 & 1.5 " & 150 yr + we can see the influence of the effect on the residuals , but only trends of cubic order and higher will survive during usual fitting procedure @xcite . linear and quadratic terms will redefine apparent period of pulsar @xmath61 and its first derivative @xmath62 and ca nt be found . residuals of toa due to the weak microlensing effect can be written as follows : @xmath63 @xmath64 are coefficients in taylor s series of function ( 5 ) where @xmath65 . plotted coefficient @xmath66 depending on @xmath30 is represented in fig . [ fig : c ] ( plotted for b1937 + 21 ) . one can see from the plot that the fastest increase of residuals takes place when the epoch of the initial observation are 50 - 150 years away from the epoch @xmath30 , because the third derivative have maximum in that interval maximal . if the initial observation coincides with the closest approach , then only fourth and higher orders term will affect timing and the residuals will increase much slowly . magnitude of the residuals after subtraction of linear and quadratic terms can be expressed as follows : @xmath67 , where @xmath68 , @xmath69 -linear and quadratic coefficients at @xmath65 . @xmath70 @xmath71 the plot in fig . [ fig : timing ] shows magnitude of the residuals at different @xmath30 ( 0 , 50 , 100 years ; blue , green and red graphs accordingly ) . module of that magnitude depends only on module @xmath30 . residuals of 10 ns magnitude due to the effect of weak microlensing will appear with probability of @xmath72 if time span of observations exceeds 20 years . we can also calculate allan variance ( avar ) for pulsar time scale with time residuals caused by the effect . toa residuals due to the effect can be significant , if @xmath57 ( angular distance between the pulsar and the nearest affecting body ) is much smaller than average . the plot fig.7 represents situation when @xmath73 . this situation has @xmath74 chance of probability in case of b1937 + 21 ; probability reduces like @xmath32 inverse squared . the magnitude of the residuals can be as a great as 800 - 1000 ns in the same 20 years span . however , if we used in fitting procedure terms of cubic and higher orders , then the magnitude of the effect can be effectively set to 0 . the magnitude can be much greater for pulsars in gc ( or pulsars behind gc ) ( @xmath75 , @xmath76 ( length of path of ray in gc)@xmath77 , @xmath38(distance to gc ) = @xmath78 ) ) . @xmath42 and @xmath32 can be much smaller because the density of stars in gc is large , the magnitude of the effect will be much greater ( the same 1 ms in 20 years span ) . time of one significant interaction will be quite small ( 20 - 30 years ) . complete investigation of the question can be found in @xcite . so , we can make several conclusions : average toa residuals due to a weak microlensing effect is about 10 ns ( b1937 + 21 ) in 20 years span . toa residuals can be effectively set to zero by using higher order terms in fitting procedure ( not for pulsars in globular clusters . residuals can be much greater if pulsar is located in a globular cluster , so the pulsars in globular clusters ca nt be recommended for using in pt scale . j. n. bahcall , annual review of astronomy and astrophysics , v. 24 , p. 577 , 1986 baker&hellings , annual review of astronomy and astrophysics , v. 24 , p. 537 hosokava , ohnishi , fukushima , astronomy and astrophysics , v. 351 , p. 393,1999 ilyasov et al . , iau symp . 141 , p. 213 , 1989 kalinina , pshirkov , astronomy reports , vol . 50 , issue 6,p . 427 - 431 , 2006 6 . kopeikin s.m.,schaffer g. , phys . d 60 , no.12 , p. 4002 , 1999 or arxiv : gr - qc 9902030 larchenkova , doroshenko , astronomy and astrophysics , v. 297 , p. 607 , 1995 larchenkova , kopeikin , astronomy letters , v. 32 , issue 1 , p. 18 , 2006 manchester , r. n. , hobbs , g. b. , teoh , a. & hobbs , m. , aj , 129 , 1993 - 2006 ( 2005 ) ohnishi et al . in asp conference series v. 105 , p. 1250 , 1996 sazhin m.v . , in proc . conference on general relativity and gravity , stockholm , p. 519 sazhin m.v . , astronomy letters , vol . 22 , issue 5 , p. 573 , 1996 sazhin m.v . , zharov a.f . , kalinina t.a . , mnras , v . 300 , p. 287 , 1998 sazhin m.v . , zharov a.f . , kalinina t.a . , mnras , v . 323 , p. 952 , 2001 sazhin , saphonova , astrophysics and space science , v. 208 , p. 93 , 1993 sazhin , pshirkov , http://zhurnal.ape.relarn.ru/articles/2005/119.pdf , ( in russian ) , 2005 gravitation and cosmology : principles and applications of the general theory of relativity , 1972

an influence of the weak microlensing effect on the pulsar timing is investigated for pulsar b1937 + 21 . average residuals of time of arrival ( toa ) due to the effect would be as large as 10 ns in 20 years observation span . these residuals can be much greater ( up to 1 ms in 20 years span ) if pulsar is located in globular cluster ( or behind it ) .

studies of galaxy evolution have revealed surprisingly recent changes in galaxy populations . comparisons of present day galaxies with those at moderate ( @xmath13 ) and high ( @xmath14 ) redshift have uncovered trends which are often dramatic , and may trace galaxies to the time at which they were first assembled into recognizable entities . these discoveries have shed new light on the formation of galaxies , and have provided clues as to the nature of their evolution . at @xmath15 , the picture that is emerging is one in which early type galaxies evolve slowly and passively , while late type galaxies become more numerous with increasing redshift ( e.g. , @xcite ) . at higher redshifts , deep surveys such as the hubble deep field ( @xcite ) indicate an increase in the cosmic star formation rate out to @xmath16 ( e.g. , madau , pozzetti , and dickinson 1998 ) . while considerable progress has been made in the observational description of galaxy evolution , important questions remain regarding the physical processes driving this evolution . mechanisms that have been postulated include galaxy - galaxy mergers , luminosity - dependent luminosity evolution , and the existence of a new population of galaxies that has faded by the present epoch ( see reviews by @xcite and @xcite ) . in this study , we will investigate the relative importance of mergers in the evolution of field galaxies . mergers transform the mass function of galaxies , marking a progression from small galaxies to larger ones . in addition , mergers can completely disrupt their constituent galaxies , changing gas - rich spiral galaxies into quiescent ellipticals ( e.g. , toomre and toomre 1972 ) . during a collision , a merging system may also go through a dramatic transition , with the possible onset of triggered star formation and/or accretion onto a central black hole ( see review by barnes & hernquist 1992 ) . it is clear that mergers do occur , even during the relatively quiet present epoch . however , the frequency of these events , and the distribution of masses involved , has yet to be accurately established . this is true at both low and high redshift . furthermore , while a number of attempts have been made , a secure measurement of evolution in the galaxy merger rate remains elusive , and a comparable measure of the accretion rate has yet to be attempted . in this study , we introduce a new approach for relating dynamically close galaxy pairs to merger and accretion rates . these new techniques yield robust measurements for disparate samples , thereby allowing meaningful comparisons of mergers at low and high redshift . in addition , these pair statistics can be adapted to a variety of redshift samples , and to studies of both major and minor mergers . we apply these techniques to a large sample of galaxies at low redshift ( ssrs2 ) , providing a much needed local benchmark for comparison with samples at higher redshift . in a forthcoming paper ( patton et al . 2000 ) , we will apply these techniques to a large sample of galaxies at moderate redshift ( cnoc2 ; @xmath17 ) , yielding a secure estimate for the rate of evolution in the galaxy merger and accretion rates . an overview of earlier pair studies , and a discussion of their limitations and shortcomings , are given in the next section . the ssrs2 data are described in [ ssrs2mr : data ] . section [ ssrs2mr : mrate ] discusses the connection between close pairs and the merger and accretion rates , while [ ssrs2mr : nclc ] introduces new statistics for relating these quantities . section [ ssrs2mr : flux ] describes how these statistics can be applied to flux - limited surveys in a robust manner . a pair classification experiment is presented in [ ssrs2mr : class ] , giving empirical justification for our close pair criteria . pair statistics are then computed for the ssrs2 survey in [ ssrs2mr : sample ] , and the implications are discussed in [ ssrs2mr : discuss ] . conclusions are given in the final section . throughout this paper , we use a hubble constant of @xmath18 km s@xmath19 mpc@xmath19 . we assume @xmath20=1 and @xmath21=0.1 , unless stated otherwise . every estimate of evolution in the merger and/or accretion rate begins with the definition of a merger statistic . ideally , this statistic should be independent of selection effects such as optical contamination due to unrelated foreground / background galaxies , redshift incompleteness , redshift - dependent changes in minimum luminosity resulting from flux limits , contamination due to non - merging systems , @xmath22-corrections , and luminosity evolution . in addition , it should be straightforward to relate the statistic to the global galaxy population , and to measurements on larger scales . the statistic should then be applied to large , well - defined samples from low to high redshift , yielding secure estimates of how the merger and/or accretion rates vary with redshift . within the past decade , there have been a number of attempts to estimate evolution in the galaxy merger rate using close pairs of galaxies ( e.g. , @xcite , @xcite , @xcite , @xcite , @xcite , @xcite , @xcite ) . the statistic that has been most commonly employed is the traditional pair fraction , which gives the fraction of galaxies with suitably close physical companions . this statistic is assumed to be proportional to the galaxy merger rate . the local ( low - redshift ) pair fraction was estimated by patton et al . ( 1997 ) , using a flux - limited ( @xmath23 ) sample of galaxies from the ugc catalog ( @xcite ) . using pairs with projected physical separations of less than 20 @xmath5 kpc , they estimated the local pair fraction to be @xmath24 . this result was shown to be consistent with the local pair fraction estimates of carlberg et al . ( 1994 ) and yee & ellingson ( 1995 ) , both of whom also used the ugc catalog . the pair fraction has been measured for samples of galaxies at moderate redshift ( @xmath13 ) , yielding published estimates ranging from approximately 0% ( @xcite ) to 34% @xmath25 9% ( @xcite ) . evolution in the galaxy merger rate is often parameterized as @xmath26 . close pair studies have yielded a wide variety of results , spanning the range @xmath27 . there are several reasons for the large spread in results . first , different methods have been used to relate the pair fraction to the merger rate . in addition , some estimates have been found to suffer from biases due to optical contamination or redshift completeness . after taking all of these effects into account , patton et al . ( 1997 ) demonstrated that most results are broadly consistent with their estimate of @xmath28 , made using the largest redshift sample ( 545 galaxies ) to date . while this convergence seems promising , all of these results have suffered from a number of very significant difficulties . the central ( and most serious ) problem has been the comparison between low and moderate redshift samples . low-@xmath4 samples have been poorly defined , due to a lack of suitable redshift surveys . in addition , the pair fraction depends on both the clustering and mean density of galaxies . the latter is very sensitive to the limiting absolute magnitude of galaxies , leading to severe redshift - dependent biases when using flux - limited galaxy samples . these biases have not been taken into account in the computation of pair fractions , or in the comparison between samples at different redshifts . while these problems are the most serious , there are several other areas of concern . a lack of redshift information has meant dealing with optical contamination due to unrelated foreground and background galaxies . moreover , while one can statistically correct for this contamination , it is still not possible to discern low velocity companions from those that are physically associated but unbound , unless additional redshift information is available . finally , there is no direct connection between the pair fraction and the galaxy correlation function ( cf ) and luminosity function ( lf ) , making the results more difficult to interpret . to address these issues , we have developed a novel approach to measuring pair statistics . we will introduce new statistics that overcome many of the afflictions of the traditional pair fraction . we will then apply these statistics to a large , well - defined sample of galaxies at low redshift . the second southern sky redshift survey ( @xcite ; hereafter ssrs2 ) consists of 5426 galaxies with @xmath29 , in two regions spanning a total of 1.69 steradians in the southern celestial hemisphere . the first region , denoted ssrs2 south , has boundaries @xmath30 and @xmath31 . the second region , ssrs2 north , is a more recent addition , and is bounded by @xmath32 and @xmath33 . galaxies were selected primarily from the list of non - stellar objects in the _ hubble space telescope _ guide star catalog , with positions accurate to @xmath10 1 and photometry with an rms scatter of @xmath10 0.3 magnitudes ( alonso et al . 1993 , alonso et al . 1994 ) . steps were taken to ensure that single galaxies were not mistakenly identified as close pairs , due to the presence of dust lanes , etc . ( @xcite ) . in addition , careful attention was paid to cases where a very close pair might be mistaken for a single galaxy . this was found to make a negligible contribution to the catalog as a whole ( @xmath34 of galaxies are affected ) . the effect on the pairs analysis in this paper is further reduced by imposing a minimum pair separation of 5 @xmath5 kpc ( see [ ssrs2mr : class ] ) . the sample now includes redshifts for all galaxies brighter than @xmath29 . we correct all velocities to the local group barycenter using equation 6 from courteau and van den bergh ( 1999 ) . we restrict our analysis to the redshift range @xmath35 . this eliminates nearby galaxies , for which recession velocities are dominated by peculiar velocities , giving poor distance estimates . we also avoid the sparsely sampled high redshift regime . this leaves us with a well - defined sample of 4852 galaxies . the primary goal of earlier close pair studies has been to determine how the galaxy merger rate evolves with redshift . the merger rate affects the mass function of galaxies , and may also be connected to the cosmic star formation rate . before attempting to measure the merger rate , it is important to begin with a clear definition of a merger and a merger rate . here , we refer to mergers between two galaxies which are both above some minimum mass or luminosity . if this minimum corresponds roughly to a typical bright galaxy ( @xmath36 ) , this criterion can be thought of as selecting so - called major mergers . we consider two merger rate definitions . first , it is of interest to determine the number of mergers that a typical galaxy will undergo per unit time . in this case , the relevant rate may be termed the galaxy merger rate ( hereafter @xmath37 ) . a related quantity is the total number of mergers taking place per unit time per unit co - moving volume . we will refer to this as the volume merger rate ( hereafter @xmath38 ) . clearly , @xmath39 , where @xmath40 is the co - moving number density of galaxies . while both of these merger rates provide useful measures of galaxy interactions , they have their limitations . as one probes to faint luminosities , one will find an increasing number of faint companions ; hence , the number of inferred mergers will increase in turn . for all realistic lfs , this statistic will become dominated by dwarf galaxies . in addition , it is of interest to determine how the mass of galaxies will change due to mergers . to address these issues , we will also investigate the rate at which mass is being accreted onto a typical galaxy . this quantity , the total mass accreted per galaxy per unit time , will be referred to as the galaxy accretion rate ( hereafter @xmath41 ) . this is related to the rate of mass accretion per unit co - moving volume ( @xmath42 ) by @xmath43 . the mass ( or luminosity ) dependence of the accretion rate means that it will be dominated by relatively massive ( or luminous ) galaxies , with dwarfs playing a very minor role unless the mass function is very steep . in order to determine @xmath37 observationally , one may begin by identifying systems which are destined to merge . by combining information about the number of these systems and the timescale on which they will undergo mergers , one can estimate an overall merger rate . specifically , if one identifies @xmath44 ongoing mergers per galaxy , and if the average merging timescale for these systems is @xmath45 , then @xmath46 . if the mass involved in these mergers ( per galaxy ) is @xmath47 , then @xmath48 . in practice , direct measurement of these quantities is a daunting task . it is difficult to determine if a given system will merge ; furthermore , estimating the merger timescale for individual systems is challenging with the limited information generally available . however , if one simply wishes to determine how the merger rate is _ changing _ with redshift , then the task is more manageable . if one has the same definition of a merger in all samples under consideration , then it is reasonable to assume that the merger timescale is the same for these samples . in this case , we are left with the task of measuring quantities which are directly proportional to the number or mass of mergers per galaxy or per unit co - moving volume . if one wishes to consider luminosity instead of mass , the relation between mass and luminosity must either be the same at all epochs , or understood well enough to correct for the differences . we have considered several quantities that fit this description . all involve the identification of close physical associations of galaxies . a `` close companion '' is defined as a neighbour which will merge within a relatively short period of time ( @xmath49 ) , which allows an estimate of the instantaneous merger / accretion rate . if a galaxy is destined to undergo a merger in the very near future , it must have a companion close at hand . one might attempt to estimate the number of mergers taking place within a sample of galaxies . for example , a close pair of galaxies would be considered one merger , while a close triple would lead to two mergers , etc . owing to the difficulty of determining with certainty which systems are undergoing mergers , we will not use this approach . one alternative is to estimate the number of galaxies with one or more close companions , otherwise known as the pair fraction . one drawback of this approach is that close triples or higher order n - tuples complicate the analysis , since they are related to higher orders of the correlation function . this also makes it difficult to correct for the flux - limited nature of most redshift surveys . as a result , we choose to steer clear of this method also . in this study , we choose instead to use the _ number and luminosity of close companions _ per galaxy . the number of close companions per galaxy , hereafter @xmath0 , is similar in nature to the pair fraction . in fact , they are identical in a volume - limited sample with no triples or higher order n - tuples . however , @xmath0 will prove to be much more robust and versatile . we assume that @xmath0 is directly proportional to the number of mergers per galaxy , such that @xmath50 ( @xmath22 is a constant ) . this pairwise statistic is preferable to the number of mergers per galaxy or the fraction of galaxies in merging systems , in that it is related , in a direct and straightforward manner , to the galaxy two - point cf and the lf ( see section [ ssrs2mr : nclc ] ) . we note that it is not necessary that there be a one - to - one correspondence between companions and mergers , as long as the correspondence is the same , on average , in all samples under consideration . using this approach to estimate the number of mergers per galaxy , the merger rate is then given by @xmath51 . the actual value of @xmath22 depends on the merging systems under consideration . if one identifies a pure set of galaxy pairs , each definitely undergoing a merger , then each pair , consisting of 2 companions , would lead to one merger , giving @xmath22=0.5 . for a pair sample which includes some triples and perhaps higher order n - tuples , if the merging sample under investigation contains some systems which are not truly merging ( for instance , close pairs with hyperbolic orbits ) , then @xmath22 will also be reduced . while @xmath22 clearly varies with the type of merging system used , the key is for @xmath22 to be the same for all samples under consideration . we take a similar approach with the accretion rate . we again use close companions , and in this case we simply add up the luminosity in companions , per galaxy ( @xmath1 ) . defining the mean companion mass - to - light ratio as @xmath53 , it follows that @xmath54 and @xmath55 . when comparing different samples , any significant differences in @xmath53 must be accounted for . in order to motivate further the need for merger rate measurements , and to set the stage for future work relating pair statistics to the mass and luminosity function , we develop a simple model which relates these important quantities . suppose the galaxy mass function is given by @xmath56 . this function gives the number density of galaxies of mass @xmath57 at time @xmath58 , per unit mass . the model that follows can also be expressed in terms of luminosity or absolute magnitude , rather than mass . we begin by assuming that all changes in the mass function are due to mergers . while this is clearly simplistic , this model will serve to demonstrate the effects that various merger rates can have on the mass function . in order to relate the mass function to the observable luminosity function , we further assume that mergers do not induce star formation . again , this is clearly an oversimplification ; however , this simple case will still provide a useful lower limit on the relative contribution of mergers to lf evolution . finally , we assume that merging is a binary process . following @xcite , we consider how @xmath56 evolves as the universe ages from time @xmath58 to time @xmath59 . each merger will remove two galaxies from the mass function , and produce one new galaxy . let @xmath60 represent the decrease in the mass function due to galaxies removed by mergers , while @xmath61 gives the increase due to the remnants produced by these mergers . evolution in the mass function can then be given by @xmath62 we model this function by considering all galaxy pairs , along with an expression for the merging likelihood of each . let @xmath63 denote the probability that a galaxy of mass @xmath57 will merge with a galaxy of mass @xmath64 in time interval @xmath65 . in order to estimate @xmath60 , we need to take all galaxies of mass @xmath57 , and integrate over all companions , yielding @xmath66 ^ 3dm^{\prime}.\end{aligned}\ ] ] we devise a comparable expression for @xmath61 by integrating over all pairs with end@xmath67products of mass @xmath57 . this is achieved by considering all pairs with component of mass @xmath68 and @xmath64 , such that @xmath69 ^ 3dm^{\prime}.\end{aligned}\ ] ] we can also express equation [ ssrs2mr : eqinout ] in terms of the pair statistics outlined in section [ ssrs2mr : observables ] . if one considers close companions of mass @xmath70 next to primary galaxies of mass @xmath57 , the volume merger rate can be expressed as @xmath71 , yielding @xmath72 similarly , if one defines a merger remnant statistic , @xmath73 , to be the co - moving number density of merger remnants per unit time corresponding to these same mergers , then @xmath74 therefore , it is possible , in principle , to use pair statistics to measure the evolution in the mass or luminosity function due to mergers . however , current pair samples are too small to permit useful pair statistics for different mass combinations . in addition , present day observations of close pairs are not of sufficient detail to determine the proportion of pairs that will result in mergers ( factor @xmath22 in previous section ) . moreover , timescale estimates for these mergers are not known with any degree of certainty . hence , useful observations of mass function evolution due to mergers will have to wait for improved pair samples and detailed estimates of merger timescales . in this section , we outline the procedure for measuring the mean number ( @xmath0 ) and luminosity ( @xmath1 ) of close companions for a sample of galaxies with measured redshifts . we begin by defining these statistics in real space , demonstrating how they are related to the galaxy lf and cf . we then show how these statistics can be applied in redshift space . in this study , we will measure pair statistics for a complete low - redshift sample of galaxies ( ssrs2 ) . however , we wish to make these statistics applicable to a wide variety of redshift samples . we would also like this method to be useful for studies of minor mergers , where one is interested in faint companions around bright galaxies . moreover , these techniques should be adaptable to redshift samples with varying degrees of completeness ( that is , with redshifts not necessarily available for every galaxy ) . therefore , in the following analysis , we treat host galaxies and companions differently . consider a primary sample of @xmath75 host galaxies with absolute magnitudes @xmath76 , lying in some volume @xmath77 . suppose this volume also contains a secondary sample of @xmath78 galaxies with @xmath79 . in the general case , the primary and secondary samples may have galaxies in common . this includes the special case in which the two samples are identical . if @xmath80 , this will tend to probe major mergers . if @xmath81 is chosen to be significantly fainter than @xmath82 , this will allow for the study of minor mergers . we assume here that both samples are complete to the given absolute magnitude limits ; in section [ ssrs2mr : flux ] , we extend the analysis from volume - limited samples to those that are flux - limited . we wish to determine the mean number and luminosity of companions ( in the secondary sample ) for galaxies in the primary sample . in real space , we define a close companion to be one that lies at a true physical separation of @xmath83 , where @xmath84 is some appropriate maximum physical separation . to compute the observed mean number ( @xmath0 ) and luminosity ( @xmath1 ) of companions , we simply add up the number ( @xmath85 ) and luminosity ( @xmath86 ) of companions for each of the @xmath75 galaxies in the primary sample , and then compute the mean . therefore , @xmath87 and @xmath88 we can also estimate what these statistics should be , given detailed knowledge of the galaxy two - point cf @xmath2 and the lf @xmath3 . this is necessary if one wishes to relate these pair statistics to measurements on larger scales . consider a galaxy in the primary sample with absolute magnitude @xmath89 at redshift @xmath90 . we would first like to estimate the number and luminosity of companions lying in a shell at physical distance ( _ proper _ co - ordinates ) @xmath91 from this primary galaxy . to make this estimate , we need to know the mean density of galaxies ( related to the lf ) , and the expected overdensity in the volume of interest ( given by the cf ) . the mean physical number density of galaxies at redshift @xmath90 in the secondary sample , with absolute magnitudes @xmath92 , is given by @xmath93 where @xmath94 is the differential galaxy lf , which specifies the _ co - moving _ number density of galaxies at redshift @xmath90 , in units of @xmath95 . the actual density of objects in the region of interest is determined by multiplying the mean density by ( 1+@xmath2 ) , where @xmath2 is the overdensity given by the two - point cf ( @xcite ) . in general , @xmath2 depends on the pair separation @xmath96 , the mean redshift @xmath90 , the absolute magnitude of each galaxy ( @xmath89 , @xmath97 ) , and the orbits involved , specified by components parallel ( @xmath98 ) and perpendicular ( @xmath99 ) to the line of sight . it follows that the mean number of companions with @xmath92 and @xmath91 is given by @xmath1004\pi r^2dr.\end{aligned}\ ] ] we must now integrate this expression for all companions with @xmath79 and @xmath83 . integration over the lf yields @xmath101 integration over the cf is non - trivial , because of the complex nature of @xmath102 . with redshift samples that are currently available , it is not possible to measure this dependence accurately for the systems of interest . hence , we must make three important assumptions at this stage . first , we assume that @xmath2 is independent of luminosity . later in the paper , we demonstrate empirically that this is a reasonable assumption , provided one selects a sample with appropriate ranges in absolute magnitude ( see section [ ssrs2mr : clust ] ) . secondly , we assume that the distribution of velocities is isotropic . if one averages over a reasonable number of pairs , this is bound to be true , and therefore @xmath2 is independent of @xmath99 and @xmath98 . finally , we assume that the form of the cf , as measured on large scales , can be extrapolated to the small scales of interest here . this assumption applies only to the method of relating pairs to large scale measures , and not to the actual measurement of pair statistics . it is now straightforward to integrate equation [ ssrs2mr : ncdiff ] . the mean number of companions with @xmath79 and @xmath83 for a primary galaxy at redshift @xmath90 is given by @xmath1034\pi r^2dr.\end{aligned}\ ] ] we derive an analogous expression for the mean luminosity in companions . the integrated luminosity density is given by @xmath104 where @xmath105 therefore , @xmath1064\pi r^2dr.\end{aligned}\ ] ] given measurements of the cf on large scales , it is then straightforward to integrate these equations to arrive at predicted values of @xmath85 and @xmath86 . it is important to note that these statistics are directly dependent on @xmath81 , which affects the mean density of galaxies in the secondary sample . this is different from statistics such as the correlation function , which are independent of density . hence , this serves as a reminder that we must exercise caution when choosing our samples , to ensure that differences in the pair statistics ( and hence in the merger and accretion rates ) are not simply due to apparent density differences resulting from selection effects . in addition , note that the choice of @xmath82 has no density - related effects on @xmath0 and @xmath1 . while it is preferable to identify companions based on their true physical pair separation , this is clearly not feasible when dealing with data from redshift surveys . in the absence of independent distance measurements for each galaxy , one must resort to identifying companions in redshift space . in this section , we outline a straightforward approach for measuring our new pair statistics in redshift space . we then attempt to relate these statistics to their counterparts in real space . for any given pair of galaxies in redshift space , one can compute two basic properties which describe the intrinsic pair separation : the projected physical separation ( hereafter @xmath107 ) and the rest - frame relative velocity along the line of sight ( hereafter @xmath108 ) . for a pair of galaxies with redshifts @xmath90 ( primary galaxy ) and @xmath109 ( secondary ) , with angular separation @xmath110 , these quantities are given by @xmath111 and @xmath112 , where @xmath113 is the angular diameter distance at redshift @xmath90 . note that @xmath107 gives the projected separation at the redshift of the primary galaxy . we define a close companion as one in which the separation ( both projected and line - of - sight ) is less than some appropriate separation , such that @xmath114 and @xmath115 . the line - of - sight criterion depends on both the physical line - of - sight separation and the line - of - sight peculiar velocity of the companion . it is of course not possible to determine the relative contributions of these components without distance information . however , for the small companion separations we will be concerned with , the peculiar velocity component is likely to be dominant in most cases , as we will be dealing with a field sample of galaxies ( the same would not be true in the high velocity environment of rich clusters ) . hence , this criterion serves primarily to identify companions with low peculiar velocities . while this is fundamentally different from the pure separation criterion used in real space , it too will serve to identify companions with the highest likelihood of undergoing imminent mergers . using this definition of a close companion , it is straightforward to compute @xmath0 and @xmath1 , using equations [ ssrs2mr : eqncbasic ] and [ ssrs2mr : eqlcbasic ] . thus , the complexities of redshift space do not greatly complicate the computation of these pair statistics . as in real space , we wish to relate these statistics to measurements on larger scales , given reasonable assumptions about the lf and cf . the situation is more complicated in redshift space , and therefore involves additional assumptions . we stress , however , that these assumptions apply only to the method of relating pair statistics to large scale measures , and not to the measured pair statistics themselves . to outline an algorithm for generating these predictions , we follow the approach of the previous section . we begin by modifying equations [ ssrs2mr : ncint ] and [ ssrs2mr : lcint ] , integrating over the new pair volume defined in redshift space . in order to do this , we use the two dimensional correlation function in redshift space , @xmath116 , giving @xmath1172\pi r_pdr_pdr_v\end{aligned}\ ] ] and @xmath1182\pi r_pdr_pdr_v.\end{aligned}\ ] ] the two dimensional correlation function is the convolution of the velocity distribution in the redshift direction , @xmath119 , with the spatial correlation function @xmath120 , given by @xmath121)dy.\ ] ] here , @xmath122 is the hubble constant at redshift @xmath4 , given by @xmath123 . we have ignored the effect of infall velocities , which must be taken into account at larger radii but is an acceptable approximation for small separations . if the form of the cf and lf are known , it is straightforward to integrate equations [ ssrs2mr : nc2int ] and [ ssrs2mr : lc2int ] , yielding predictions of @xmath0 and @xmath1 . it is not always possible to have precise redshifts for all galaxies of interest in a sample . a common scenario with redshift surveys is to have redshifts available for a subset of galaxies identified in a flux - limited photometric sample . the photometric sample used to select galaxies for follow - up spectroscopy probes to fainter apparent magnitudes than the spectroscopic sample . in addition , the spectroscopic sample may be incomplete , even at the bright end of the sample . in this section , we will describe the procedure for applying pair statistics to this class of samples . suppose the primary sample is defined as all galaxies in the spectroscopic sample with absolute magnitudes @xmath76 . the secondary sample consists of all galaxies lying in the photometric sample , regardless of whether or not they have measured redshifts . once again , there may be some overlap between the primary and secondary samples . we must now identify close pairs . for each primary - secondary pair , we can compute @xmath107 in precisely the same manner as before ( see previous section ) , since we need only the redshift of the primary galaxy and the angular separation of the pair . however , we are no longer able to compute the relative velocity along the line of sight , since this requires redshifts for both members of the pair . thus , we do not have enough information to identify close dynamical pairs . however , it is still possible to determine , in a statistical manner , how many physically associated companions are present . this is done by comparing the number ( or luminosity ) of observed companions with the number ( or luminosity ) expected in a random distribution . as stressed in section [ ssrs2mr : realspace ] , pair statistics depend on the minimum luminosity @xmath81 imposed on the secondary sample . while we are now unable to compute the actual luminosity for galaxies in the secondary sample , we must still impose @xmath81 if the ensuing pair statistics are to be meaningful . to do this , we make use of the fact that all physical companions must lie at approximately the same redshift as the primary galaxy under consideration . therefore , @xmath81 corresponds to a limiting _ apparent _ magnitude @xmath124 at redshift @xmath90 , such that @xmath125 where @xmath126 is the luminosity distance at redshift @xmath90 , and @xmath127 is the @xmath22-correction . to begin , one finds all observed close companions with @xmath128 , using only the @xmath107 criterion . this results in the quantities @xmath129 and @xmath130 , where the `` d '' superscript denote companions found in the data sample . one must then estimate the number ( @xmath131 ) and luminosity ( @xmath132 ) of companions expected at random . the final pair statistics for close physical companions are then given by @xmath133 and @xmath134 . we will now describe how to predict these statistics using the known lf and cf . this is relatively straightforward , since the excess @xmath2 given by the cf is determined by the relative proportions of real and random companions . the pair statistics are once again integrals over the two dimensional cf in redshift space , as specified by equations [ ssrs2mr : nc2int ] and [ ssrs2mr : lc2int ] . in the `` @xmath135 '' term , the first part gives the random contribution , while the second gives the excess over random . thus , these pair statistics give the true density of companions , rather than the `` excess '' density . this is intentional , since mergers will occur even in an uncorrelated , randomly distributed sample of galaxies . at the small separations of interest , usually less than 1% of the correlation length , the difference between the mean density and the mean overdensity is less than about 0.01% in real space . for practical measurements in redshift space , where @xmath136 is of order the correlation length , the background contribution is substantially larger than real space , but still amounts to less than 1% . thus , for the close pairs considered in this study , it is reasonable to ignore the contribution that random companions make to the sample of physical companions . that is , we take @xmath137 . this allows us to relate the predictions to the measured pair statistics set out above . in principle , equations [ ssrs2mr : nc2int ] and [ ssrs2mr : lc2int ] can be integrated over the range @xmath138 to obtain predictions of @xmath0 and @xmath1 . to illustrate the concepts introduced so far , and to emphasize how these statistics depend on @xmath81 , we apply these techniques to volume - limited monte carlo simulations , which mimic the global distribution of galaxies in the ssrs2 north and south catalogs . using @xmath21=0.5 , galaxies are distributed randomly within the co - moving volume enclosed by @xmath139 and the ssrs2 boundaries on the sky ( see section [ ssrs2mr : data ] ) . all peculiar velocities are set to zero . to create a volume - limited sample , we impose a minimum luminosity of @xmath140=@xmath141 , and assign luminosities using the ssrs2 lf ( @xcite ) , which has schechter function parameters @xmath142=@xmath143 , @xmath144=@xmath145 , and @xmath146 = @xmath147 . an arbitrarily large number of galaxies can be generated , which is of great assistance when looking for small systematic effects . we produce 16000 galaxies in the south , and 8070 in the north ; this gives the same density of galaxies in both regions . using these simulations , we compute @xmath0 and @xmath1 . as these galaxies are distributed randomly ( as opposed to real galaxies which are clustered ) , close pairs are relatively rare . to ensure a reasonable yield of pairs , we use a pair definition of @xmath148=1 @xmath5 mpc and @xmath149=1000 km / s . we note that there are no peculiar velocities in these simulations ; hence , the @xmath149 criterion provides upper and lower limits on the line - of - sight distance to companions . also , recall from the preceding section that the choice of @xmath82 has no effect on the pair statistics if clustering is independent of luminosity . hence , we choose @xmath82=@xmath81 , which maximizes the size of the primary sample , and therefore minimizes the measurement errors in @xmath0 and @xmath1 . with these assumptions , we compute pair statistics for a range of choices of @xmath81 . errors are computed using the jackknife technique . for this resampling method , partial standard deviations , @xmath150 , are computed for each object by taking the difference between the quantity being measuring , @xmath151 , and the same quantity with the @xmath152 galaxy removed from the sample , @xmath153 , such that @xmath154 . for a sample of @xmath155 galaxies , the variance is given by @xmath156^{1/2}$ ] ( efron 1981 ; efron & tibshirani 1986 ) . results are given in figure [ ssrs2mr : figm2vl ] . both statistics continue to increase as @xmath81 becomes fainter . @xmath0 diverges at faint magnitudes , while @xmath1 is seen to converge . this behaviour is a direct consequence of the shape of the lf ; @xmath0 converges for @xmath157 , while @xmath1 converges for @xmath158 . the existence and magnitude of these trends clearly demonstrate the need to specify @xmath81 when measuring pair statistics . the preceding section gives a straightforward prescription for computing pair statistics in volume - limited samples . however , redshift surveys are generally flux - limited . by defining a volume - limited sample within such a survey , one must discard a large proportion of the data . in this section , we will outline how these pair statistics can be applied to flux - limited surveys . pair statistics necessarily depend on both clustering and mean density , as shown by equations [ ssrs2mr : nc2int ] and [ ssrs2mr : lc2int ] . in a flux - limited sample , both clustering and mean density will vary throughout the sample . we will use these equations to account for redshift - dependent changes in mean density , and we will demonstrate how to minimize the effects of clustering differences . these techniques will then be tested with monte carlo simulations . by removing the fixed luminosity limit , the overall distribution of galaxy luminosities will vary with redshift within the sample , and the mean luminosity of the sample will differ from the volume - limited sample . however , galaxy clustering is known to be luminosity dependent . measures of the galaxy correlation function ( e.g. , loveday et al . 1995 , willmer et al . 1998 ) , power spectrum ( e.g. , vogeley 1993 ) , and counts in cells ( benoist et al . 1996 ) all find that luminous galaxies ( @xmath159 ) are more clustered than sub-@xmath36 galaxies , typically by a factor of @xmath160 . this increase in clustering may be particularly strong ( factor @xmath161 ) for very luminous galaxies ( @xmath162 ) . clearly , this effect should not be ignored when computing pair statistics . in principle , this could be incorporated into the measurement of these pair statistics . however , available pair samples are too small to measure this dependence . we choose instead to minimize these effects by restricting the analysis to a fixed range in absolute magnitude , within which luminosity - dependent clustering is small or negligible . this is done by imposing additional bright ( @xmath163 ) and faint ( @xmath164 ) absolute magnitude limits on the sample . having thereby reduced the effects of luminosity segregation , we then assume that the remaining differences will not have a significant effect on the measured pair statistics . in section [ ssrs2mr : sense ] , we use the ssrs2 sample to demonstrate empirically that this is in fact a reasonable assumption . in section [ ssrs2mr : nclc ] , we demonstrated that these pair statistics are meaningful only if one specifies the minimum luminosity of the primary and secondary samples . for a flux - limited sample , however , the minimum luminosity of the sample increases with redshift . one must therefore decide on a representative minimum luminosity , and account for differences between the desired minimum luminosity and the redshift - dependent minimum imposed by the apparent magnitude limit of the sample . if the lf is known , this can be achieved by weighting each galaxy appropriately . in this section , we outline a weighting scheme which makes this correction . consider a flux - limited sample in which host galaxies are located at a variety of redshifts . those at low redshift will have the greatest probability of having close companions that lie above the flux limit , since the flux limit corresponds to an intrinsic luminosity that is fainter than that for galaxies at higher redshift . if we wish to avoid an inherent bias in the pair statistics , we must correct for this effect . furthermore , we must account for any limits in absolute magnitude imposed on the sample to reduce the effects of luminosity - dependent clustering ( [ ssrs2mr : clust ] ) . finally , we have demonstrated the importance of specifying a limiting absolute magnitude for companions ( @xmath81 ) when computing pair statistics . therefore , we must attempt to correct the pair statistics to the values that would have been achieved for a volume - limited secondary sample with @xmath165 . qualitatively , this correction should assign greater importance ( or weight ) to the more rare companions found at the high redshift end of the flux - limited sample . to make this correction as rigorous as possible , we will use the galaxy lf . by integrating the lf over a given range in absolute magnitude , one can obtain an estimate of the mean number or luminosity density of galaxies in the sample . by performing this integration at any given redshift , accounting for the allowed ranges in absolute magnitude and the flux limit , it is possible to quantify how the mean density varies with redshift within the defined sample . this information can be used to remove this unwanted bias from the pair statistics . we assign a weight to each galaxy in the secondary sample , which renormalizes the sample to the density corresponding to @xmath165 . we must first determine @xmath166 , which gives the limiting absolute magnitude allowed at redshift @xmath90 . at most redshifts , this is imposed by the limiting apparent magnitude @xmath167 , such that @xmath168 . at the low redshift end of the sample , however , @xmath164 ( defined in [ ssrs2mr : clust ] ) will take over . that is , the limiting absolute magnitude used for identifying galaxies in the secondary sample is given by @xmath169.\ ] ] the selection function , denoted @xmath170 , is defined as the ratio of densities in flux - limited versus volume - limited samples . this function , given in terms of number density ( @xmath171 ) and luminosity density ( @xmath172 ) , is as follows : @xmath173 @xmath174 where @xmath175 is defined in equation [ ssrs2mr : eql ] . in order to recover the correct pair statistics , each companion must be assigned weights @xmath176 and @xmath177 . the total number and luminosity of close companions for the @xmath178 primary galaxy , computed by summing over the @xmath179 galaxies satisfying the `` close companion '' criteria , is given by @xmath180 and @xmath181 respectively . by applying this weighting scheme to all galaxies in the secondary sample , we will retrieve pair statistics that correspond to a volume - limited sample with @xmath165 . the above weighting scheme ensures that the number and luminosity of companions found around each primary galaxy is normalized to @xmath165 . however , these estimates are obviously better for galaxies at the low redshift end of the primary sample , since they will have the largest number of _ observed _ companions . recall that @xmath0 and @xmath1 are quantities that are averaged over a sample of primary galaxies . in order to minimize the errors in these statistics , we assign weights to the primary galaxies ( denoted @xmath182 and @xmath183 ) which are inversely proportional to the square of their uncertainty . if the observed number and luminosity of companions around the @xmath178 primary galaxy are given by @xmath184 and @xmath185 respectively , and if we assume that the uncertainties are determined by poisson counting statistics , then @xmath186 and @xmath187 . on average , these quantities will be related to expectation values @xmath188 and @xmath189 by @xmath190 and @xmath191 . combining these relations yields @xmath192 @xmath193 that is , the optimal weighting is the reciprocal of the weighting scheme used for companions . therefore , weights @xmath194 and @xmath195 should be assigned to primary galaxies . the pair statistics are then computed as follows : @xmath196 @xmath197 it is worth noting that , for a close pair , both galaxies will lie at roughly the same redshift , meaning that @xmath198 . we choose not to make this approximation , in order to keep these relations valid for pairs that are not close , and to allow for future application to pairs with additional selection weights . however , we stress that , with or without this approximation , the primary weights in the denominator provide an overall correction for the flux limit , unlike the traditional pair fraction . note also that , for a volume - limited sample , weights for all galaxies in the primary and secondary samples are equal , reducing these equations to @xmath199 and @xmath200 , as defined in section [ ssrs2mr : realspace ] . a small correction must be made to these weights if a primary galaxy lies close to a region of space that is not covered by the survey . this will happen if a galaxy lies close to the boundaries on the sky , or close to the minimum or maximum redshift allowed . if this is the case , it is possible that close companions will be missed , leading to an underestimate of the pair statistics . therefore , we must account for these effects . first , we consider galaxies lying close to the survey boundaries on the sky , as defined in section [ ssrs2mr : data ] . for each galaxy in the primary sample , we compute the fraction of sky with @xmath201 that lies within the survey boundaries . this fraction will be denoted @xmath202 . for ssrs2 , our usual choices of @xmath203 and @xmath148 ( see [ ssrs2mr : class ] ) make this a very small effect , with @xmath204=1 for 99.75% of galaxies in the primary sample . having measured @xmath202 for each galaxy in the primary sample , we must incorporate this into the measurement of the pair statistics . the first task is to ensure that we correct the number of companions to match what would be expected if coverage was complete . we do this by assigning each companion a boundary weight @xmath205 = @xmath206 , where @xmath202 is associated with its host galaxy from the primary sample . by multiplying each companion by its boundary weight , we will recover the correct number of companions . we must also adjust weights for the primary galaxies . following the method described in the previous section , we wish to give less weight to galaxies that are likely to have fewer observed companions . therefore , each primary galaxy is assigned a boundary weight @xmath207 = @xmath202 . we now consider galaxies which lie near the survey boundaries along the line of sight . if a primary galaxy lies close to the minimum or maximum redshift allowed , it is possible that we will miss companions because they lie just across this redshift boundary . in order to account correctly for this effect , one would need to model the velocity distribution of companions . as this requires several assumptions , we choose instead to exclude all companions that lie between a primary galaxy and its nearest redshift boundary , provided the boundary lies within @xmath149 of the primary galaxy . to account for this exclusion , we assume that the velocity distribution is symmetric along the line of sight . thus , as we will miss half of the companions for these galaxies , we assign a weight of @xmath208=2 to any companions found in the direction opposite to the boundary . we must also consider how to weight the primary galaxies themselves . clearly , primary galaxies close to the redshift boundaries will be expected to have half as many _ observed _ companions as other primary galaxies . to minimize the errors in computing the pair statistics , we assign these primary galaxies weights @xmath209=0.5 . to summarize , weights for companions in the secondary sample are given by @xmath210 @xmath211 while primary galaxies are assigned weights @xmath212 @xmath213 we will now perform a test to see if this weighting scheme achieves the desired effects . to do this , we will use flux - limited monte carlo simulations , for which the intrinsic density and clustering are fixed . therefore , the _ intrinsic _ pair statistics do not depend on redshift or luminosity . if the secondary sample weights are correct , the measured pair statistics will be the same everywhere ( within the measurement errors ) , regardless of redshift or luminosity . we will also check to see if the weights for the primary sample are correct . if they are , the errors on the pair statistics will be minimized , as desired . the flux - limited monte carlo simulations were generated in a similar manner to the simulations described in section [ ssrs2mr : mc ] ; however , a limiting apparent magnitude of @xmath29 was imposed . sample sizes of 8000 ( south ) and 4035 ( north ) were used , providing a good match to the overall density in ssrs2 . the resulting simulations are similar to ssrs2 in all respects , except for the absence of clustering . we have already established how the pair statistics depend on the choice of @xmath82 and @xmath81 . in the following analysis , we choose @xmath81=@xmath82=@xmath214 . in section [ ssrs2mr : clust ] , we outlined reasons for restricting the sample to a fixed range in absolute magnitude . here , we demonstrate how the chosen range affects @xmath0 and @xmath1 . for comparison , we also compute @xmath0 without normalizing to a specified range in absolute magnitude ( in this case , @xmath215=@xmath215=1 ) . this provides some insight into the behaviour of the traditional ( uncorrected ) pair fraction . these tests are most straightforward if the intrinsic pair statistics are the same everywhere in the enclosed volume . this is not quite true for these simulations , however . galaxies are distributed randomly within the enclosed _ co - moving _ volume . as a result , the physical density varies with redshift as @xmath216 . in addition , the volume element encompassed by the line - of - sight pair criterion @xmath108 varies with redshift as @xmath217 for @xmath21=0.5 . in order to have the simulations mimic a sample with universal pair statistics , we normalize the sample for these effects by weighting each galaxy by @xmath217 . we stress that this is done only for the monte carlo simulations . one should _ not _ apply either of these corrections to real redshift data . in figure [ ssrs2mr : figrmfaint ] , the pair statistics are computed for a range of @xmath164 . in addition , we compute @xmath0 without weighting by the luminosity function , to demonstrate the danger of ignoring this important correction . this statistic is directly analagous to the traditional ( uncorrected ) pair fraction used in the literature . it is clear that both @xmath0 and @xmath1 are independent of the choice of @xmath164 , within the errors . this verifies that we have correctly accounted for the biases introduced by the apparent magnitude limit . in contrast , the unweighted @xmath0 is seen to have a strong dependence on @xmath164 . as expected , it increases as @xmath164 becomes fainter , due to the increase in sample density . we stress that this does not happen with the normalized @xmath0 and @xmath1 statistics , because both are corrected to a fixed range in limiting absolute magnitude . finally , we demonstrate that the weighting scheme used for the primary sample ( [ ssrs2mr : w1 ] ) does in fact minimize errors in @xmath0 and @xmath1 . recall that the weighting used was the reciprocal of the weights for the secondary sample . here we will assume that @xmath218 and @xmath219 . in section [ ssrs2mr : w1 ] , justification was given for setting @xmath220=@xmath221 . here , we will allow @xmath220 to vary , in order to investigate empirically which value minimizes the errors . special cases of interest are @xmath220=0 ( no weighting ) and @xmath220=1 ( same weighting as _ secondary _ sample ) . the results are given in figure [ ssrs2mr : figw1 ] . the relative errors in @xmath0 and @xmath1 reach a minimum at @xmath222 , as expected . errors are @xmath10 40% larger if no weighting is used ( @xmath220=0 ) . for @xmath220=1 , errors are much larger , increasing by nearly a factor of 5 . while errors increase dramatically for @xmath223 , they change slowly around @xmath220=@xmath221 . clearly , @xmath220 = @xmath221 is an excellent choice . the first step in applying these techniques to a real survey of galaxies is to decide on a useful close pair definition . this involves imposing a maximum projected physical separation ( @xmath148 ) and , if possible , a maximum line - of - sight rest - frame velocity difference ( @xmath149 ) . the limits should be chosen so as to extract information on mergers in an optimal manner . this involves a compromise between the number and merging likelihood of pairs . while one should focus on companions which are most likely to be involved in mergers , a very stringent pair definition may yield a small and statistically insignificant sample . in previous close pair studies , the convention has been to set @xmath148 = 20 @xmath5 kpc . pairs with separations of @xmath224 20 @xmath5 kpc are expected to merge within 0.5 gyr ( e.g. , @xcite , @xcite ) . we note , however , that timescale estimates are approximate in nature , and have yet to be verified . in earlier work , it has not been possible to apply a velocity criterion , since redshift samples have been too small to yield useful pair statistics using only galaxies with measured redshifts . instead , all physical companions have been used , with statistical correction for optical contamination ( @xcite ) . with a complete redshift sample , we can improve on this . this can be seen by inspecting a plot of @xmath107 versus @xmath108 for the ssrs2 pairs , given in figure [ ssrs2mr : figrpdelv ] . by imposing a velocity criterion , we can eliminate optical contamination ; furthermore , we are able to concentrate on the physical pairs with the lowest relative velocities , and hence the greatest likelihood of merging . we can now use our large sample of low-@xmath4 pairs to shed new light on these issues . we will use images of these pairs in an attempt to determine how signs of interactions are related to pair separation . we begin by finding all 255 ssrs2 pairs with @xmath225 @xmath5 kpc , computing @xmath107 and @xmath108 for each . images for these pairs were extracted from the digitized sky survey . interactions were immediately apparent in some of these pairs , and the images were deemed to be of sufficient quality that a visual classification scheme would be useful . an interaction classification parameter ( i@xmath226 ) was devised , where i@xmath226=0 indicates that a given pair is `` definitely not interacting '' , and i@xmath226=10 indicates `` definitely interacting '' . in order to avoid a built - in bias , the classifier is not given the computed values of @xmath107 and @xmath108 . the classifier uses all visible information available ( tidal tails and bridges , distortions / asymmetries in member galaxies , apparent proximity , etc . ) . classifications were performed by three of us ( drp , rom , rgc ) , and the median classification was determined for each system . the results are presented in figure [ ssrs2mr : figpc ] . a clickable version of this plot , which allows the user to see the corresponding digitized sky survey image for each pair , is available at http://www.astro.utoronto.ca/@xmath10patton/ssrs2/ic . there are several important features in this plot . first , there is a clear correlation between @xmath227 and @xmath107 , with closer pairs exhibiting stronger signs of interactions . there are several interacting pairs with @xmath228 @xmath5 kpc . while these separations are fairly large , it is not surprising that there would be some early - stage mergers with these separations ( e.g. , barton , bromley , & geller 1998 ) . an excellent example of this phenomenon is the striking tail - bridge system arp 295a / b ( cf . @xcite ) , which has @xmath107 = 95 @xmath5 kpc . however , these systems clearly do not dominate ; almost all pairs with large separations have very low interaction classifications . the majority of pairs showing clear signs of interactions / mergers have @xmath229 @xmath5 kpc . there is also a clear connection with @xmath108 . pairs with @xmath230 km / s do not exhibit signs of interactions , with 61/63 ( 97% ) classified as @xmath231 . this indicates that interactions are most likely to be seen in low velocity pairs , as expected . we note , however , that there are very few optical pairs ( i.e. , small @xmath107 and large @xmath108 ) in this low redshift sample . at higher redshift , increased optical contamination may lead to difficulties in identifying interacting systems when the galaxies are close enough to have overlapping isophotes . clearly , it is necessary to have redshift information for both members of each pair if one is to exclude these close optical pairs . after close inspection of figure [ ssrs2mr : figpc ] , we decided on close pair criteria of @xmath148 = 20 @xmath5 kpc and @xmath149 = 500 km / s . a mosaic of some of these pairs is given in figure [ ssrs2mr : figim ] . in this regime , 31% ( 9/29 ) exhibit convincing evidence for interactions ( i@xmath226 @xmath232 ) , while 69% ( 20/29 ) show some indication of interactions ( i@xmath226 @xmath233 ) . furthermore , the vast majority ( 9/10 ) of pairs with clear signs of interactions ( i@xmath226 @xmath232 ) are found in this regime . these criteria appear to identify a sample of pairs which are likely to be undergoing mergers ; moreover , the resulting sample includes most of the systems classified as interacting . we also impose an inner boundary of @xmath107 = 5 @xmath5 kpc . this limit is chosen so as to avoid the confusion that is often present on the smallest scales . in this regime , it is often difficult to distinguish between small galaxies and sub - galactic units , particularly in merging systems . while we are omitting the most likely merger candidates , those at separations @xmath234 5 @xmath5 kpc are not expected to account for more than @xmath10 5% of the companions within 20 @xmath5 kpc . this expectation , which has yet to be verified , is based both on pair counts in _ hst _ imaging ( @xcite ) and on inward extrapolation of the correlation function ( @xcite ) . while this inner boundary will lead to a slight decrease in @xmath0 and @xmath1 , it should have no significant effect on estimates of merger / accretion rate evolution , provided the same restriction is applied to comparison samples at other redshifts . in the preceding sections , we have outlined techniques for measuring pair statistics in a wide variety of samples . we have demonstrated a robust method of applying this approach to flux - limited samples , accounting for redshift - dependent density changes and minimizing differences in clustering . we have also selected pair definitions that identify the most probable imminent mergers . we will now apply these techniques to the ssrs2 survey . as this is a complete redshift survey , redshifts are available for all close companions ; hence , for the first time , we will measure pair statistics using only close _ dynamical _ pairs . after limiting the analysis to a reasonable range in absolute magnitude , we compute @xmath0 and @xmath1 for the ssrs2 survey . in section [ ssrs2mr : clust ] , we emphasized the importance of restricting the sample in absolute magnitude , to minimize bias due to luminosity - dependent clustering . for ssrs2 , we first impose a bright limit of @xmath163 = @xmath235 . all galaxies brighter than this are hereafter excluded from the analysis . this allows us to avoid the most luminous galaxies , which are probably the most susceptible to luminosity - dependent clustering ; however , this reduces the size of the sample by only 0.5% . we also impose a faint absolute magnitude limit of @xmath164 = @xmath236 , which results in the exclusion of intrinsically faint galaxies at @xmath237 . this guards against the possibility that these intrinsically faint galaxies are clustered differently than the bulk of the galaxies in the sample . this pruning of the sample is illustrated in figure [ ssrs2mr : figmbz ] . these restrictions allow us to minimize concerns about luminosity - dependent clustering while retaining 90% of the sample . the final results are insensitive to these particular choices ( see section [ ssrs2mr : sense ] ) . the above limits in absolute magnitude , along with the flux limit , define the usable sample of galaxies . in order to compute pair statistics , we must also normalize the measurements to a given range in absolute magnitude , for both the primary and secondary samples . the mean limiting absolute magnitude of the primary sample , weighted according to section [ ssrs2mr : w1 ] , is @xmath140 = @xmath238 . for convenience , we set @xmath239 = @xmath240 ( we will compute pair statistics for @xmath241 in the following section ) . for reference , we note that this corresponds to @xmath242=@xmath243 at @xmath4=0.017 . as we are dealing with a complete redshift sample , we set @xmath244=@xmath239 in order to use all of the available information . finally , as we have limited the sample using @xmath163 = @xmath235 , this will be used in conjunction with @xmath239 to derive pair statistics for galaxies with @xmath11 . using these parameters , we identified all close companions in ssrs2 . the north sample yielded 27 companions , and 53 were found in the south , giving a total of 80 . we emphasize that it is _ companions _ that are counted , rather than pairs ; hence , if both members of a pair fall within the primary sample , the pair will usually yield 2 companions . a histogram of companion absolute magnitudes is given in figure [ ssrs2mr : figlh ] . this plot shows that 90% of the companions we observe in our flux - limited sample fall in the range @xmath11 . hence , galaxies with @xmath245 do not dominate the sample . tables [ ssrs2mr : tabcpn ] and [ ssrs2mr : tabcps ] give complete lists of close aggregates ( pairs and triples ) for ssrs2 north and south respectively . these systems contain all companions used in the computation of pair statistics . these tables list system i d , number of members , @xmath107 ( @xmath5 kpc ) , @xmath108 ( km / s ) , ra ( 1950.0 ) , dec ( 1950.0 ) , and recession velocity ( km / s ) . dss images for these systems were given earlier in figure [ ssrs2mr : figim ] . using this sample of companions , the pair statistics were computed . the results are given in table [ ssrs2mr : tabstats ] . errors were computed using the jackknife technique . results from the two subsamples were combined , weighting by jackknife errors , to give @xmath246 and @xmath247 at @xmath4 = 0.015 . results from the two subsamples agree within the quoted 1@xmath248 errors . to facilitate future comparison with other samples , we also generate pair statistics spanning the range @xmath241 ( see table [ ssrs2mr : tabm2 ] ) . we note , however , that while we account for changes in number and luminosity density over this luminosity range ( using lf weights described in section [ ssrs2mr : flux ] ) , there is no correction for changes in clustering . hence , our statistics should be considered most appropriate for @xmath239 = @xmath249 , and more approximate in nature at brighter and fainter levels . the results in table [ ssrs2mr : tabm2 ] indicate that @xmath0 increases by a factor of 5 between @xmath239 = @xmath214 and @xmath239 = @xmath236 , resulting solely from an increase in mean number density . the change in @xmath1 is less pronounced , with an increase by a factor of 2 over the same luminosity range . these substantial changes in both statistics emphasize the need to specify @xmath81 when computing pair statistics and comparing results from different samples . in addition , the smaller change in @xmath1 is indicative of the benefits of using a luminosity statistic such as @xmath1 , which is more likely to converge as one goes to fainter luminosities ( see section [ ssrs2mr : mc ] ) . @xmath1 will always converge faster than @xmath0 , thereby reducing the sensitivity to @xmath81 . furthermore , it is possible to retrieve most of the relevant luminosity information without probing to extremely faint levels . for example , for the ssrs2 lf , 70% of the total integrated luminosity density is sampled by probing down to @xmath239 = @xmath249 . to first order , the same will be true for @xmath1 . going 2 magnitudes fainter would increase the completeness to 95% . while we are currently unable to apply pair statistics down to these faint limits , this will be pursued when deeper surveys become available . in this section , we explore the effects of choosing different survey parameters . earlier in this study , we demonstrated that @xmath0 and @xmath1 are insensitive to the choice of survey limits in absolute magnitude , provided clustering is independent of luminosity and the pair statistics are normalized correctly . here , we test this hypothesis empirically . first , we compute the pair statistics for a range in @xmath164 , normalizing the statistics to @xmath11 in each case . figure [ ssrs2mr : figmfaint ] demonstrates a possible trend of decreasing pair statistics with fainter @xmath164 . this trend , however , is significant only for the brightest galaxies ( @xmath250 ) . this is consistent with the findings of willmer et al . ( 1998 ) , who measure an increase in clustering for bright galaxies in ssrs2 , on scales of @xmath251 @xmath5 mpc . for fainter @xmath164 , there is no significant dependence . the pair statistics vary by @xmath10 5% over the range @xmath252 , which is well within the error bars . therefore , we conclude that our choice of @xmath164 = @xmath236 has a negligible effect on @xmath0 and @xmath1 . this implies that , to first order , clustering is independent of luminosity within this sample . next , we investigate how the pair statistics depend on our particular choices of @xmath148 and @xmath149 , which comprise our definition of a close companion . first , we compute pair statistics for 10 @xmath5 kpc@xmath253@xmath5 kpc , with @xmath254 km / s . results are given in figure [ ssrs2mr : figrp ] . this plot indicates a smooth increase in both statistics with @xmath148 . this trend is expected from measurements of the galaxy cf . the cf is commonly expressed as a power law of the form @xmath255 , with @xmath256=1.8 ( @xcite ) . integration over this function yields pair statistics that vary as @xmath257 , which is in good agreement with the trend found in figure [ ssrs2mr : figrp ] . from this plot , it also appears likely that there are systematic differences between the two subsamples . this is hardly surprising , since there are known differences in density between the subsamples , and it is likely that there are non - negligible differences in clustering as well . this cosmic variance is not currently measurable on the smaller scales ( @xmath258 @xmath5 kpc ) relevant to our main pair statistics . hence , we choose to ignore these differences for now . however , these field - to - field variations are certain to add some systematic error to our quoted pair statistics . we also compute pair statistics for a range in @xmath149 . this is done first for @xmath259 @xmath5 kpc , showing the relative contributions at different velocities to the main pair statistics quoted in this paper . we also compute statistics using @xmath260@xmath5 kpc , in order to improve the statistics . results are given in figure [ ssrs2mr : figrl ] . several important conclusions may be drawn from this plot . first , at small velocities ( @xmath261 km / s ) , both pair statistics increase with @xmath149 , as expected . this simply indicates that one continues to find additional companions as the velocity threshold increases . secondly , it appears that our choice of @xmath149 was a good one . the @xmath258 @xmath5 kpc pair statistics increase very little beyond @xmath254 km / s , while the contamination due to non@xmath67merging pairs would continue to increase ( see figure [ ssrs2mr : figrpdelv ] ) . moreover , as both pair statistics flatten out at around @xmath254 km / s , small differences in the velocity distributions of different samples should not result in large differences in their pair statistics . finally , for @xmath262 @xmath5 kpc , the pair statistics continue to increase out to @xmath263 km / s . this indicates an increase in velocity dispersion at these larger separations . this provides additional confirmation that one is less likely to find low@xmath67velocity pairs at larger separations , thereby implying that mergers should also be less probable . all published estimates of the local pair fraction have been hindered by small sample sizes and a lack of redshifts . in addition , as demonstrated throughout this paper , the traditional pair fraction is not a robust statistic , particularly when applied to flux - limited surveys . the new statistics introduced in this paper , along with careful accounting for selection effects such as the flux limit , yield the first secure measures of pair statistics at low redshift . therefore , strictly speaking , the results in this paper can not be compared directly with earlier pair statistics . however , it is possible to check for general consistency in results , and we will attempt to do so . as discussed in section [ ssrs2mr : background ] , patton et al . ( 1997 ) estimated the local pair fraction to be @xmath264 , using the ugc catalog . the patton et al . ( 1997 ) estimate was based on a flux - limited sample with @xmath23 , and a mean redshift of @xmath4=0.0076 . this corresponds roughly to an average limiting absolute magnitude of @xmath140 = @xmath265 . loosely speaking , this is analogous to @xmath81 . the pair definition used in their estimate was @xmath258 @xmath5 kpc , with no @xmath108 criterion . @xmath0 may be interpreted as an approximation to the traditional pair fraction , provided the relative proportion of triples is small . we recompute the ssrs2 pair statistics , using @xmath266 km / s in an attempt to match the results that would be found using no @xmath108 criterion ( see figure [ ssrs2mr : figrl ] ) . we find @xmath267 . this implies a local pair fraction of @xmath268 . this value is somewhat smaller than the earlier result , with larger errors . we strongly emphasize that , while these results are broadly consistent , we would not expect excellent agreement , due to the improved techniques used in this study . the quantities @xmath0 and @xmath1 are practical measures of the average numbers and luminosities of companions with relatively high merging probabilities . however , the ambiguity of redshift space is such that some of these companions can be entirely safe from ever merging . that is , @xmath270 km / s can correspond either to two galaxies at a small physical separation with a large infall velocity , or , to two galaxies at a separation of 5@xmath5 mpc , with no relative peculiar velocity . in order to transform from @xmath271 ( and @xmath272 ) to an estimate of the incidence of mergers , we must determine what fraction of our close companions have true 3-dimensional physical separations of @xmath273 @xmath5 kpc . this quantity , which we refer to as @xmath274 , has been discussed previously by yee and ellingson ( 1995 ) . this quantity needs to be evaluated based on the small separation clustering and kinematics of the galaxy population , @xmath275 , @xmath256 , and @xmath276 , the parameters @xmath108 and @xmath148 , and the projected separation at which two galaxies `` optically overlap '' in the image . the quantity @xmath274 is then evaluated with a triple integral , first placing the correlation function into redshift space , then integrating over projected and velocity separation . for reasonable choices of pair selection parameters , the outcome is fairly stable at @xmath277 . the most important parameter is @xmath278 , which should lie in the range of 2 to 4 . if this parameter is too small , physical pairs will be missed ; if it is too large , too many distant companions will be incorporated . other reasonable choices include setting the ratio of the overlap separation to maximum pair separation to be at least three , and the ratio of the maximum pair separation to correlation length to be at least a factor of 30 . we will take @xmath274=0.5 to be the best estimate currently available . we can now estimate the merger fraction ( @xmath279 ) at the present epoch . in this study , we have found @xmath271=0.0226@xmath280 . as most companions are found in pairs , rather than triplets or higher order n - tuples , this is comparable to the fraction of galaxies in close pairs . from our estimate of @xmath274 , we infer that half of these galaxies are in merging systems , yielding @xmath281 . this implies that approximately 1.1% of @xmath11 galaxies are undergoing mergers at the present epoch . we stress here that this result applies only to galaxies within the specified absolute magnitude limits . probing to fainter luminosities would cause @xmath279 to increase substantially . in addition , this result applies only to the close companions defined in this analysis . clearly , by modifying this definition ( and therefore changing the typical merger timescale under consideration ) , the merger fraction would also be certain to change . we now have an idea of how prevalent ongoing mergers are at the present epoch . in order to relate this result to the overall importance of mergers , we must estimate the merger timescale ( @xmath45 ) . we will use the properties of our ssrs2 pairs to estimate the mean dynamical friction timescale for pairs in our sample . following binney and tremaine ( 1987 ) , we assume circular orbits and a dark matter density profile given by @xmath282 . the dynamical friction timescale @xmath283 ( in gyr ) is given by @xmath284 where @xmath96 is the initial physical pair separation in kpc , @xmath285 is the circular velocity in km / s , @xmath57 is the mass ( @xmath286 ) , and @xmath287 is the coulomb logarithm . we estimate @xmath96 and @xmath285 using the pairs in tables [ ssrs2mr : tabcpn ] and [ ssrs2mr : tabcps ] . the mean projected separation is @xmath288@xmath5 kpc . as our procedure already includes a correction from projected separation ( @xmath107 ) to 3-dimensional separation ( @xmath96 ) , we take @xmath289 . assuming @xmath20=0.7 , this leads to @xmath290 kpc . the mean line of sight velocity difference is @xmath291 km / s . we assume the velocity distribution is isotropic , which implies that @xmath292 km / s . the mean absolute magnitude of companions is @xmath293 ( see figure [ ssrs2mr : figlh ] ) . we assume a representative estimate of the galaxy mass - to - light ratio of @xmath294 , yielding a mean mass of @xmath295 . finally , dubinski , mihos , and hernquist ( 1999 ) estimate @xmath296 , which fits the orbital decay of equal mass mergers seen in simulations . using equation [ ssrs2mr : eqntfric ] , we find @xmath297 gyr . we caution that this estimate is an approximation , and is averaged over systems with a wide range in merger timescales . nevertheless , we will take @xmath45 = 0.5 gyr as being representative of the merger timescale for the pairs in our sample . now that we have estimated the present epoch merger fraction and the merger timescale , we will attempt to ascertain what fraction of present galaxies have undergone mergers in the past . these galaxies can be classified as merger remnants ; hence , we will refer to this fraction as the remnant fraction ( @xmath299 ) . we begin by imagining the state of affairs at a lookback time of @xmath300 . suppose the merger fraction at the corresponding redshift is given by @xmath301 . in the time interval between then and the present , a fraction @xmath301 of galaxies will undergo mergers , yielding @xmath302 merger remnants . therefore , the remnant fraction at the present epoch is given by @xmath303 similarly , if we extend this to a lookback time of @xmath304 , where @xmath155 is an integer , then the remnant fraction is given by @xmath305 where @xmath109 corresponds to a lookback time of @xmath306 . we now make the simple assumption that the merger rate does not change with time . in this case , our present epoch estimate of the merger fraction holds at all redshifts , giving @xmath301=0.011 . in order to convert between redshift and lookback time , we must specify a cosmological model . we assume a hubble constant of @xmath20=0.7 . for simplicity , we assume @xmath21=0.5 . therefore , @xmath307 . using our merger timescale estimate of @xmath45=0.5 gyr , we can now investigate the cumulative effect of mergers . with the chosen cosmology , @xmath4=1 corresponds to a lookback time of @xmath10 6 gyr , or 12@xmath45 ( @xmath155=12 ) . with this lookback time , equation [ ssrs2mr : eqnremz ] yields @xmath299=0.066 . this implies that @xmath10 6.6% of galaxies with @xmath11 have undergone mergers since @xmath308 . if the mergers taking place in our sample produce elliptical galaxies , it is worthwhile comparing the remnant fraction to the elliptical fraction ( cf . toomre 1977 ) . the elliptical fraction for bright field galaxies is generally found to be about 10% ( e.g. , dressler 1980 , postman and geller 1984 ) . this result is broadly consistent with the remnant fraction found in this study . while our estimate of the remnant fraction is based on our statistically secure measurement of @xmath271 , it also relies on fairly crude assumptions regarding the merger fraction and merger timescale . in particular , the merger rate has been assumed to be constant . there is no physical basis for this assumption ; in fact , a number of studies have predicted a rise in the merger rate with redshift . if this is true , we will have underestimated the remnant fraction , and the relative importance of mergers . in a future paper ( patton et al . 2000 ) , we will address this issue by investigating how the merger rate changes with redshift . we have introduced two new pair statistics , @xmath0 and @xmath1 , which are shown to be related to the galaxy merger and accretion rates respectively . using monte carlo simulations , these statistics are found to be robust to the redshift - dependent density changes inherent in flux - limited samples ; this represents a very significant improvement over all previous estimators . in addition , we provide a clear prescription for relating @xmath0 and @xmath1 to the galaxy cf and lf , enabling straightforward comparison with measurements on larger scales . these statistics are applied to the ssrs2 survey , providing the first statistically sound measurements of pair statistics at low redshift . for an effective range in absolute magnitude of @xmath11 , we find @xmath246 at @xmath4=0.015 , implying that @xmath10 2.3% of these galaxies have companions within a projected physical separation of 5 @xmath5 kpc @xmath6 20 @xmath5 kpc and 500 km / s along the line of sight . if this pair statistic remains fixed with redshift , simple assumptions imply that @xmath10 6.6% of present day galaxies with @xmath11 have undergone mergers since @xmath4=1 . for our luminosity statistic , we find @xmath309 . this statistic gives the mean luminosity in companions , per galaxy . both of these statistics will serve as local benchmarks in ongoing and future studies aimed at detecting redshift evolution in the galaxy merger and accretion rates . it is our hope that these techniques will be applied to a wide range of future redshift surveys . as we have demonstrated , one must carefully account for differences in sampling effects between pairs and field galaxies . this will be of increased importance when applying pair statistics at higher redshift , as @xmath22-corrections , luminosity evolution , band - shifting effects , and spectroscopic completeness have to be properly accounted for . the general approach outlined in this paper indicates the steps that must be taken to allow for a fair comparison between disparate surveys at low and high redshift . these techniques are applicable to redshift surveys with varying degrees of completeness , and are also adaptable to redshift surveys with additional photometric information , such as photometric redshifts , or even simply photometric identifications . finally , this approach can be used for detailed studies of both major and minor mergers . we wish to thank all members of the ssrs2 colloboration for their work in compiling the ssrs2 survey , and for making these data available in a timely manner . digitized sky survey images used in this research were obtained from the canadian astronomical data centre ( cadc ) , and are based on photographic data of the national geographic society palomar observatory sky survey ( ngs - poss ) . the digitized sky surveys were produced at the space telescope science institute under u.s . government grant nag w-2166 . this work was supported by the natural sciences and engineering research council of canada , through research grants to r.g.c . and c.j.p .. alonso , m. v. , da costa , l. n. , pellegrini , p. s. , & kurtz , m. j. 1993 , , 106 , 676 alonso , m. v. , da costa , l. n. , latham , d. w. , pellegrini , p. s. , & milone , a. e. 1994 , , 108 , 1987 bahcall , s. r. & tremaine , s. 1988 , , 326 , l1 barnes , j. e. 1988 , , 331 , 699 barnes , j. e. & hernquist , l. 1992 , , 30 , 705 barton , e. j. , bromley , b. c. , & geller , m. j. 1998 , , 511 , l25 benoist , c. , maurogordato , s. , da costa , l. n. , cappi , a. , & schaeffer , r. 1996 , , 472 , 452 binney , j. , and tremaine , s. 1987 , in galactic dynamics ( princeton : princeton university press ) burkey , j. m. , keel , w. c. , windhorst , r. a. , & franklin , b. e. 1994 , , 429 , l13 carlberg , r. g. , pritchet , c. j. , & infante , l. 1994 , , 435 , 540 courteau , s. , & van den bergh , s. 1999 , , 118 , 337 da costa , l. n. , willmer , c. n. a. , pellegrini , p. s. , chaves , o. l. , rit , c. , maia , m. a. g. , geller , m. j. , latham , d. w. , kurtz , m. j. , huchra , j. p. , ramella , m. , fairall , a. p. , smith , c. , & lpari , s. 1998 , , 116 , 1 davis , m. , & peebles , p. j. e. 1983 , , 267 , 465 dressler , a. 1980 , , 236 , 351 dubinski , j. , mihos , j. c. , and hernquist , l. 1999 , , submitted efron , b. 1981 , biometrika , 68 , 589 efron , b. , & tibshirani , r. 1986 , statistical science , 1 , 54 ellis , r. s. 1997 , , 35 , 389 hibbard , j. e. , & van gorkom , j. h. 1996 , , 111 , 655 koo , d. , & kron , r. 1992 , , 30 , 613 le fvre , o. , abraham , r. , lilly , s. j. , ellis , r. s. , brinchmann , j. , tresse , l. , colless , m. , crampton , d. , glazebrook , k. , hammer , f. , & broadhurst , t. 1999 , , in press lin , h. , yee , h. k. c. , carlberg , r. g. , morris , s. l. , sawicki , m. , patton , d. r. , wirth , g. d. , & shepherd , c. w. 1999 , , 518 , 533 loveday , j. , maddox , s. j. , efstathiou , g. , & peterson , b. a. 1995 , , 442 , 457 marzke , r. o. , geller , m. j. , da costa , l. n. , & huchra , j. p. 1995 , , 110 , 477 marzke , r. o. , da costa , l. n. , pellegrini , p. s. , willmer , c. n. a. , & geller , m. j. 1998 , , 503 , 617 madau , p. , pozzetti , l. , & dickinson , m. 1998 , , 498 , 106 nilson , p. 1973 , uppsala general catalog of galaxies ( uppsala : royal society of sciences of uppsala ) patton , d. r. , pritchet , c. j. , yee , h. k. c. , ellingson , e. , & carlberg , r. g. 1997 , , 475 , 29 patton , d. r. , pritchet , c. j. , carlberg , r. g. , marzke , r. o. , yee , h. k. c. , ellingson , e. , hall , p. b. , lin , h. , morris , s. l. , sawicki , m. , schade , d. , shepherd , c. w. , & wirth , g. d. 2000 , in preparation peebles , p. j. e. 1980 , _ the large - scale structure of the universe _ ( princeton university press : princeton ) postman , m. , & geller , m. j. 1984 , , 281 , 95 toomre , a. , & toomre , j. 1972 , , 179 , 623 toomre , a. 1977 , in evolution of galaxies and stellar populations , ed . b. m. tinsley , & r. b. larson ( new haven : yale obs . ) , p. 401 vogeley , m. s. 1993 , phd dissertation , harvard university , cambridge , ma usa williams , r. e. et al . 1996 , , 112 , 1335 willmer , c. n. a. , da costa , l. n. , pellegrini , p. s. 1998 , , 115 , 869 woods , d. , fahlman , g. g. , & richer , h. b. 1995 , , 454 , 32 yee , h. k. c. , & ellingson , e. 1995 , , 445 , 37 zepf , s. e. , & koo , d. c. 1989 , , 337 , 34

the galaxy merger and accretion rates , and their evolution with time , provide important tests for models of galaxy formation and evolution . close pairs of galaxies are the best available means of measuring redshift evolution in these quantities . in this study , we introduce two new pair statistics , which relate close pairs to the merger and accretion rates . we demonstrate the importance of correcting these ( and other ) pair statistics for selection effects related to sample depth and completeness . in particular , we highlight the severe bias that can result from the use of a flux - limited survey . the first statistic , denoted @xmath0 , gives the number of companions per galaxy , within a specified range in absolute magnitude . @xmath0 is directly related to the galaxy merger rate . the second statistic , called @xmath1 , gives the total luminosity in companions , per galaxy . this quantity can be used to investigate the mass accretion rate . both @xmath0 and @xmath1 are related to the galaxy correlation function @xmath2 and luminosity function @xmath3 in a straightforward manner . both statistics have been designed with selection effects in mind . we outline techniques which account for various selection effects , and demonstrate the success of this approach using monte carlo simulations . if one assumes that clustering is independent of luminosity ( which is appropriate for reasonable ranges in luminosity ) , then these statistics may be applied to flux - limited surveys . these techniques are applied to a sample of 5426 galaxies in the ssrs2 redshift survey . this is the first large , well - defined low-@xmath4 survey to be used for pair statistics . using close ( 5 @xmath5 kpc @xmath6 20 @xmath5 kpc ) dynamical ( @xmath7 km / s ) pairs , we find @xmath8 and @xmath9 at @xmath4=0.015 . these are the first secure estimates of low - redshift pair statistics , and they will provide local benchmarks for ongoing and future pair studies . if @xmath0 remains fixed with redshift , simple assumptions imply that @xmath10 6.6% of present day galaxies with @xmath11 have undergone mergers since @xmath4=1 . when applied to redshift surveys of more distant galaxies , these techniques will yield the first robust estimates of evolution in the galaxy merger and accretion rates . @#1 @@size@false # 1 @@size@false # 1 @@size@false # 1 @mathfonts @#1#2#3 # 1#2#1@xmath12roman#2 2#1#1@xmath12 = =

we study a kazhdan lusztig - like correspondence between a vertex - operator algebra and a quantum group in the case where the conformal field theory associated with the vertex - operator algebra is logarithmic . in its full extent , the kazhdan lusztig correspondence comprises the following claims : 1 . [ item : equiv - cat ] a suitable representation category of the vertex - operator algebra is equivalent to the category of finite - dimensional quantum group representations . [ item : grring ] the fusion algebra associated with the conformal field theory coincides with the quantum - group grothendieck ring . [ item : sliiz ] the modular group representation associated with conformal blocks on a torus is equivalent to the modular group representation on the center of the quantum group . such full - fledged claims of the kazhdan lusztig correspondence @xcite have been established for affine lie algebras at a negative integer level and for some other algebras `` in the negative zone . '' but in the positive zone , the correspondence holds for rational conformal field models @xcite ( such as @xmath6-minimal virasoro models and @xmath7 models with @xmath8 ) with certain `` corrections . '' notably , the semisimple fusion in rational models corresponds to a semisimple quasitensor category obtained as the quotient of the representation category of a quantum group by the tensor ideal of indecomposable tilting modules . taking the quotient ( `` neglecting the negligible '' in @xcite , cf . @xcite ) makes the correspondence somewhat indirect ; in principle , a given semisimple category can thus correspond to different quantum groups . remarkably , the situation is greatly improved for the class of logarithmic ( nonsemisimple ) models considered in this paper , where the quantum group itself ( not only a quasitensor category ) can be reconstructed from the conformal field theory data . in this paper , we are mostly interested in claims [ item : sliiz ] and [ item : grring ] . claim [ item : sliiz ] of the kazhdan lusztig correspondence involves the statement that the counterpart of the quantum group center on the vertex - operator algebra side is given by the endomorphisms of the identity functor in the category of vertex - operator algebra representations . this object morally , the `` center '' of the associated conformal field theory can be identified with the finite - dimensional space @xmath9 of conformal blocks on a torus . in the semisimple case , @xmath9 coincides with the space of conformal field theory characters , but in the nonsemisimple case , it is not exhausted by the characters , although we conveniently call it the ( space of ) extended characters ( all these are functions on the upper complex half - plane ) . the space @xmath9 carries a modular group representation , and the kazhdan lusztig correspondence suggests looking for its relation to the modular group representation on the quantum group center . we recall that an @xmath0-representation can be defined for a class of quantum groups ( in fact , for ribbon quasitriangular categories ) @xcite . remarkably , the two @xmath0-representations ( on @xmath9 and on the quantum group center @xmath10 ) are indeed equivalent for the logarithmic conformal field theory models studied here . the details of our study and the main results are as follows . on the vertex - operator algebra side , we consider the `` triplet '' w - algebra @xmath11 that was studied in @xcite in relation to the logarithmic @xmath4 models of conformal field theory with @xmath12 . the algebra @xmath11 has @xmath3 irreducible highest - weight representations @xmath13 , @xmath14 , which ( in contrast to the case of rational conformal field models ) admit nontrivial extensions among themselves ( @xmath15 is nondiagonalizable on some of extensions , which makes the theory logarithmic ) . the space @xmath9 in the @xmath4-model is @xmath16-dimensional ( cf . @xcite ) . on the quantum - group side , we consider the _ restricted _ ( `` baby '' in a different nomenclature ) quantum group @xmath2 at the primitive @xmath3th root of unity @xmath17 . we define it in * [ sec : cas ] * below , and here only note the key relations @xmath18 , @xmath19 ( with @xmath20 then being central ) . it has @xmath3 irreducible representations and a @xmath16-dimensional center ( prop . * [ prop - center ] * below ) . the center @xmath10 of @xmath2 is endowed with an @xmath0-representation constructed as in @xcite , even though @xmath2 is not quasitriangular @xcite ( the last fact may partly explain why @xmath2 is not as popular as the _ small _ quantum group ) . [ thm:1.1 ] the @xmath0-representations on @xmath9 and on @xmath10 are equivalent . thus , claim [ item : sliiz ] of the kazhdan lusztig correspondence is fully valid for @xmath11 and @xmath2 at @xmath21 . we let @xmath1 denote the @xmath0-representation in the theorem . regarding claim [ item : grring ] , we first note that , strictly speaking , the fusion for @xmath11 , understood in its `` primary '' sense of calculation of the coinvariants , has been derived only for @xmath22 @xcite . in rational conformal field theories , the verlinde formula @xcite allows recovering fusion from the modular group action on characters . in the @xmath4 logarithmic models , the procedure proposed in @xcite as a nonsemisimple generalization of the verlinde formula allows constructing a commutative associative algebra from the @xmath0-action on the @xmath11-characters . this algebra @xmath23 on @xmath3 elements @xmath24 ( @xmath25 , @xmath14 ) is given by @xmath26 where @xmath27 for @xmath22 , this algebra coincides with the fusion in @xcite , and we believe that it is indeed the fusion for all @xmath28 . our next result in this paper strongly supports this claim , setting it in the framework of the kazhdan lusztig correspondence between @xmath11 and @xmath2 at @xmath29 . [ thm:1.2 ] let @xmath29 . under the identification of @xmath24 , @xmath30 , @xmath31 , with the @xmath3 irreducible @xmath2-representations , the algebra @xmath23 in is the grothendieck ring of @xmath32 . we emphasize that the algebras are isomorphic as fusion algebras , i.e. , including the identification of the respective preferred bases given by the irreducible representations . the procedure in @xcite leading to fusion is based on the following structure of the @xmath0-representation @xmath1 on @xmath9 in the @xmath4 model : @xmath33 here , @xmath34 is a @xmath35-dimensional @xmath0-representation ( actually , on characters of a lattice vertex - operator algebra ) , @xmath36 is a @xmath37-dimensional @xmath0-representation ( actually , the representation on the unitary @xmath38-characters at the level @xmath39 ) , and @xmath40 is the standard two - dimensional @xmath0-representation . equivalently , is reformulated as follows . we have two @xmath0-representations @xmath41 and @xmath42 on @xmath9 in terms of which @xmath1 factors as @xmath43 @xmath44 and which commute with each other , @xmath45 ; moreover , @xmath41 restricts to the @xmath3-dimensional space of the @xmath11-characters . in view of theorem * [ thm:1.1 ] * , this structure of the @xmath0-representation is reproduced on the quantum - group side : there exist @xmath0-representations @xmath41 and @xmath42on the center @xmath10 of @xmath2 in terms of which the representation in @xcite factors . remarkably , these representations @xmath41 and @xmath42 on @xmath10 can be constructed in intrinsic quantum - group terms , by modifying the construction in @xcite . we recall that the @xmath46 generator of @xmath0 is essentially given by the ribbon element @xmath47 , and the @xmath48 generator is constructed as the composition of the radford and drinfeld mappings . that @xmath41 and @xmath42 exist is related to the multiplicative jordan decomposition of the ribbon element @xmath49 , where @xmath50 is the semisimple part and @xmath51 is the unipotent ( one - plus - nilpotent ) part . then @xmath50 and @xmath51 yield the respective `` @xmath52''-generators @xmath53 and @xmath54 . the corresponding `` @xmath55''-generators @xmath56 and @xmath57 are constructed by deforming the radford and drinfeld mappings _ respectively _ , as we describe in sec . * [ two - rep - on - z ] * below . we temporarily call the @xmath0-representations @xmath41 and @xmath42 the representations _ associated with _ @xmath50 and @xmath51 . [ thm : modular-2 ] let @xmath49 be the multiplicative jordan decomposition of the @xmath2 ribbon element ( with @xmath50 being the semisimple part ) and let @xmath41 and @xmath42 be the respective @xmath0-representations on @xmath10 associated with @xmath50 and @xmath51 . then 1 . @xmath58 for all @xmath59 , 2 . @xmath60 for all @xmath61 , and 3 . the representation @xmath41 restricts to the image of the grothendieck ring in the center . the image of the grothendieck ring in this theorem is under the drinfeld mapping . the construction showing how the representations @xmath41 and @xmath42 on the center are derived from the jordan decomposition of the ribbon element is developed in sec . * [ two - rep - on - z ] * only for @xmath2 , but we expect it to be valid in general . the multiplicative jordan decomposition of the ribbon element gives rise to @xmath0-representations @xmath41 and @xmath42 with the properties as in theorem * [ thm : modular-2 ] * for any factorizable ribbon quantum group . regarding claim [ item : equiv - cat ] of the kazhdan lusztig correspondence associated with the @xmath4 logarithmic models , we only formulate a conjecture ; we expect to address this issue in the future , beginning with @xcite , where , in particular , the representation category is studied in great detail . in a sense , the expected result is more natural than in the semisimple@xmath62rational case because ( as in theorem * [ thm:1.2 ] * ) it requires no `` semisimplification '' on the quantum - group side . the category of @xmath11-representations is equivalent to the category of finite - dimensional @xmath2-representations with @xmath21 . from the reformulation of fusion in quantum - group terms ( explicit evaluation of the product in the image of the grothendieck ring in the center under the drinfeld mapping ) , we obtain a combinatorial corollary of theorem * [ thm:1.2 ] * ( see for the notation regarding @xmath5-binomial coefficients ) : [ lemma : the - identity ] for @xmath63 , there is the @xmath5-binomial identity @xmath64}{0pt}{}{n - i}{j}$}}}{{\genfrac{[}{]}{0pt}{}{n - i}{j } } } { { \genfrac{[}{]}{0pt}{}{n - i}{j}}}{{\genfrac{[}{]}{0pt}{}{n - i}{j } } } } { \mathchoice { { \mbox{\footnotesize$\displaystyle \genfrac{[}{]}{0pt}{}{i}{m - j}$}}}{{\genfrac{[}{]}{0pt}{}{i}{m - j } } } { { \genfrac{[}{]}{0pt}{}{i}{m - j}}}{{\genfrac{[}{]}{0pt}{}{i}{m - j } } } } { \mathchoice { { \mbox{\footnotesize$\displaystyle \genfrac{[}{]}{0pt}{}{i + j + s - n}{j}$}}}{{\genfrac{[}{]}{0pt}{}{i + j + s - n}{j } } } { { \genfrac{[}{]}{0pt}{}{i + j + s - n}{j}}}{{\genfrac{[}{]}{0pt}{}{i + j + s - n}{j } } } } { \mathchoice { { \mbox{\footnotesize$\displaystyle \genfrac{[}{]}{0pt}{}{m - i - j + s'}{m - j}$}}}{{\genfrac{[}{]}{0pt}{}{m - i - j + s'}{m - j } } } { { \genfrac{[}{]}{0pt}{}{m - i - j + s'}{m - j}}}{{\genfrac{[}{]}{0pt}{}{m - i - j + s'}{m - j}}}}={}\\ { } = q^{2 m n } \sum_{\ell=0}^{\min(s , s ' ) } { \mathchoice { { \mbox{\footnotesize$\displaystyle \genfrac{[}{]}{0pt}{}{n - \ell}{m}$}}}{{\genfrac{[}{]}{0pt}{}{n - \ell}{m } } } { { \genfrac{[}{]}{0pt}{}{n - \ell}{m}}}{{\genfrac{[}{]}{0pt}{}{n - \ell}{m } } } } { \mathchoice { { \mbox{\footnotesize$\displaystyle \genfrac{[}{]}{0pt}{}{m + s + s ' - \ell - n}{m}$}}}{{\genfrac{[}{]}{0pt}{}{m + s + s ' - \ell - n}{m } } } { { \genfrac{[}{]}{0pt}{}{m + s + s ' - \ell - n}{m}}}{{\genfrac{[}{]}{0pt}{}{m + s + s ' - \ell - n}{m}}}}. \end{gathered}\ ] ] the multiplication in algebra , which underlies this identity , is alternatively characterized in terms of chebyshev polynomials , see * [ prop : quotient ] * below . there are numerous relations to the previous work . the fundamental results in @xcite regarding the modular group action on the center of a drinfeld double can be `` pushed forward '' to @xmath2 , which is a ribbon quantum group . we note that in the standard setting @xcite , a ribbon hopf algebra is assumed to be quasitriangular . this is not the case with @xmath2 , but we keep the term `` ribbon '' with the understanding that @xmath2 is a subalgebra in a quasitriangular hopf algebra from which it inherits the ribbon structure , as is detailed in what follows . the structure , already implicit in @xcite , is parallel to the property conjectured in @xcite for the @xmath0-representation on the center of the _ small _ quantum group @xmath65 . albeit for a different quantum group , we extend the argument in @xcite by choosing the bases in the center that lead to a simple proof and by giving the underlying jordan decomposition of the ribbon element and the corresponding deformations of the radford and drinfeld mappings . the @xmath16-dimensional center of @xmath2 at @xmath17 the primitive @xmath3th root of unity is twice as big as the center of @xmath65 for @xmath17 the primitive @xmath28th root of unity ( for odd @xmath28 ) @xcite . we actually find the center of @xmath2 by studying the bimodule decomposition of the regular representation ( the decomposition of @xmath65 under the _ adjoint _ action has been the subject of some interest ; see @xcite and the references therein ) . there naturally occur indecomposable @xmath3-dimensional @xmath2-representations ( projective modules ) , which have also appeared in @xcite . on the conformal field theory side , the @xmath11 algebra was originally studied in @xcite , also see @xcite . this paper can be considered a continuation ( or a quantum - group counterpart ) of @xcite and is partly motivated by remarks already made there . that the quantum dimensions of the irreducible @xmath11-representations are dimensions of quantum - group representations was noted in @xcite as an indication of a quantum group underlying the fusion algebra derived there . for the convenience of the reader , we give most of the necessary reference to @xcite in sec . [ sec : cftetc ] and recall the crucial conformal field theory formulas there . in sec . [ sec : all - usl2 ] , we define the restricted quantum group @xmath2 , describe some classes of its representations ( most importantly , irreducible ) , and find its grothendieck ring . in sec . [ sec : new ] , we collect the facts pertaining to the ribbon structure and the structure of a factorizable hopf algebra on @xmath2 . there , we also find the center of @xmath2 in rather explicit terms . in sec . [ sec : sliiz - restr ] , we study @xmath0-representations on the center of @xmath2 and establish the equivalence to the representation in sec . [ sec : cftetc ] and the factorization associated with the jordan decomposition of the ribbon element . the appendices contain auxiliary or bulky material . in appendix [ app : hopf ] , we collect a number of standard facts about hopf algebras that we use in the paper . in appendix [ sec : double ] , we construct a drinfeld double that we use to derive the @xmath66-matrix and the ribbon element for @xmath2 . in appendix [ verma - proj - mod - base ] , we give the necessary details about indecomposable @xmath2-modules . the `` canonical '' basis in the center of @xmath2 is explicitly constructed in appendix [ app : center ] . as an elegant corollary of the description of the grothendieck ring in terms of chebyshev polynomials , we reproduce the formulas for the eigenmatrix in @xcite . appendix [ app : derivation ] is just a calculation leading to identity . we use the standard notation @xmath67 = { \mbox{\footnotesize$\displaystyle\frac{q^n - q^{-n}}{q - q^{-1}}$}},\quad n\in{\mathbb{z}},\quad [ n ] ! = [ 1][2]\dots[n],\quad n\in{\mathbb{n}},\quad[0]!=1\ ] ] ( without indicating the `` base '' @xmath5 explicitly ) and set @xmath68}{0pt}{}{m}{n}$}}}{{\genfrac{[}{]}{0pt}{}{m}{n } } } { { \genfrac{[}{]}{0pt}{}{m}{n}}}{{\genfrac{[}{]}{0pt}{}{m}{n}}}}= \begin{cases } 0 , & n<0\quad\text{or}\quad m - n<0,\\ { \mbox{\footnotesize$\displaystyle\frac{[m]!}{[n]!\,[m - n]!}$}}&\text{otherwise}. \end{cases}\ ] ] in referring to the root - of - unity case , we set @xmath69 for an integer @xmath70 . the @xmath28 parameter is as in sec . * [ sec : cftetc]*. for hopf algebras in general ( in the appendices ) and for @xmath2 specifically , we write @xmath71 , @xmath72 , and @xmath55 for the comultiplication , counit , and antipode respectively . some other conventions are as follows : * the quantum group center , * the space of @xmath5-characters ( see * [ sec : q - chars ] * ) , * the integral ( see * [ app : int ] * ) , * the cointegral ( see * [ app : int ] * ) , * the balancing element ( see * [ app : int ] * ) , * the ribbon element ( see * [ sec : ribbon ] * ) , * the @xmath66-matrix ( see * [ app : m ] * ; @xmath73 is used for @xmath2 and @xmath66 in general ) , * the drinfeld mapping @xmath74 ( see * [ sec : drpdef ] * ) , * the image of the irreducible @xmath2-representation @xmath13 in the center under the drinfeld mapping ( see * [ fusion - center ] * ) , * the radford mapping @xmath74 ( see * [ sec : radford - all ] * ) , * the image of the irreducible @xmath2-representation @xmath13 in the center under the radford mapping ( see * [ sec : radford - sl2 ] * ) , * irreducible @xmath2-representations ( see * [ subsec : irrep ] * ) ; in * [ sec : voa ] * , irreducible @xmath11-rerpesentations . * verma modules ( see * [ subsec : verma - mod ] * and * [ verma - mod - base ] * ) , * contragredient verma modules ( see * [ verma - mod - base ] * ) , * projective @xmath2-modules ( see * [ subsec : proj - mod ] * and * [ proj - mod - base ] * ) , * the @xmath5-character of a @xmath2-representation @xmath75 ( see * [ app : qch ] * ) , * the @xmath2 grothendieck ring ; @xmath76 is the grothendieck ring of a hopf algebra @xmath77 , * the grothendieck ring image in the center under the drinfeld mapping , * the grothendieck ring image in the center under the radford mapping . we write @xmath78 , @xmath79 , @xmath80 , etc . ( sweedler s notation ) in constructions like @xmath81 for a linear function @xmath82 , we use the notation @xmath83 , where @xmath84 indicates the position of its argument in more complicated constructions . we choose two elements generating @xmath0 as @xmath85 and @xmath86 and use the notation of the type @xmath48 , @xmath57 , @xmath56 , and @xmath46 , @xmath54 , @xmath53 , for these elements in various representations . logarithmic models of conformal field theory , of which the @xmath4-models are an example , were introduced in @xcite and were considered , in particular , in @xcite ( also see the references therein ) . such models are typically defined as kernels of certain screening operators . the actual symmetry of the theory is the maximal local algebra in this kernel . in the @xmath4-model , which is the kernel of the `` short '' screening operator , see @xcite , this is the w - algebra @xmath11 studied in @xcite . we briefly recall it in * [ sec : voa]*. in * [ mod - on - char ] * , we give the modular transformation properties of the @xmath11-characters and identify the @xmath16-dimensional @xmath0-representation on @xmath9 ( the space of extended characters ) . in * [ thm : r - decomp ] * , we describe the structure of this representation . following @xcite , we consider the vertex - operator algebra @xmath11the w - algebra studied in @xcite , which can be described in terms of a single free field @xmath87 with the operator product expansion @xmath88 . for this , we introduce the energy - momentum tensor @xmath89 with central charge @xmath90 , and the set of vertex operators @xmath91 with @xmath92 . let @xmath93 be the sum of fock spaces corresponding to @xmath94 for @xmath95 and @xmath96 ( see the details in @xcite ) . there exist two screening operators @xmath97 satisfying @xmath98\,{=}\,0 $ ] . we define @xmath11 as a maximal local subalgebra in the kernel of the `` short '' screening @xmath99 . the algebra @xmath11 is generated by the currents @xmath100,\quad\ ; w^+(z)=[s_+,w^0(z)]\ ] ] ( which are primary fields of dimension @xmath101 with respect to energy - momentum tensor ) . the algebra @xmath11 has @xmath3 irreducible highest - weight representations , denoted as @xmath102 and @xmath103 , @xmath96 ( the respective representations @xmath104 and @xmath105 in @xcite ) . the highest - weight vectors in @xmath102 and @xmath103 can be chosen as @xmath106 and @xmath107 respectively . it turns out that @xmath108 we now recall @xcite the modular transformation properties of the @xmath11-characters @xmath109 ( the respective characters @xmath110 and @xmath111 in @xcite ) , where @xmath15 is a virasoro generator , the zero mode of energy - momentum tensor . under the @xmath48-transformation of @xmath112 , these characters transform as @xmath113 - \sum_{s'=1}^{p-1}(-1)^{p+s+s'}{\mathfrak{q}}^{ss'}_{- } \varphi_{s'{\relax}}(\tau ) \biggr)\end{gathered}\ ] ] and @xmath114 + \smash[b]{\sum_{s'=1}^{p-1}}(-1)^{s+1 } { \mathfrak{q}}^{s ' s}_{- } \varphi_{s'{\relax}}(\tau)\biggr),\end{gathered}\ ] ] where @xmath115 , @xmath116 , and we introduce the notation @xmath117 the @xmath11-characters are in fact combinations of modular forms of different weights , and hence their modular transformations involve explicit occurrences of @xmath112 ; in the formulas above , @xmath112 enters only linearly , but much more complicated functions of @xmath112 ( and other arguments of the characters ) can be involved in nonrational theories , cf . @xcite . in the present case , because of the explicit occurrences of @xmath112 , the @xmath0-representation space turns out to be @xmath16-dimensional , spanned by @xmath118 , @xmath96 , and @xmath119 , @xmath120 . indeed , we have @xmath121 where for the future convenience we introduce a special notation for certain linear combinations of the characters : @xmath122 under the @xmath46-transformation of @xmath112 , the @xmath11-characters transform as @xmath123 where @xmath124 and hence @xmath125 we let @xmath9 denote this @xmath16-dimensional space spanned by @xmath118 , @xmath96 , and @xmath119 , @xmath120 . as noted in the introduction , @xmath9 is the space of conformal blocks on the torus , which is in turn isomorphic to the endomorphisms of the identity functor . let @xmath1 be the @xmath0-representation on @xmath9 defined by the above formulas . [ thm : r - decomp ] the @xmath0-representation on @xmath9 has the structure @xmath126 where @xmath34 and @xmath36 are @xmath0-representations of the respective dimensions @xmath127 and @xmath128 , and @xmath40 is the two - dimensional representation . this implies that there exist @xmath0-representations @xmath41 and @xmath42 on @xmath9 such that @xmath129 let @xmath34 be spanned by @xmath130 ( these are the characters of _ verma _ modules over @xmath11 ) . the formulas in * [ mod - on - char ] * show that @xmath34 is an @xmath0-representation ; namely , it follows that @xmath131 and @xmath132 where @xmath133 is another basis in @xmath34 . next , let @xmath134 be the space spanned by @xmath119 in ; another basis in @xmath134 is @xmath135 finally , let another @xmath136-dimensional space @xmath137 be spanned by @xmath138 in ; another basis in @xmath137 is given by @xmath139 equations then imply that @xmath140 and the @xmath46-transformations in eqs . are expressed as @xmath141 therefore , the representation @xmath1 has the structure @xmath142 , where @xmath143 is spanned by @xmath144 , @xmath120 . we now let @xmath145 and @xmath146 act on @xmath147 as @xmath148 and let @xmath149 and @xmath150 act as @xmath151 and @xmath152 it follows that under @xmath42 , we have the decomposition @xmath153 ( where @xmath154 is the trivial representation ) and under @xmath41 , the decomposition @xmath155 it is now straightforward to verify that @xmath41 and @xmath42 satisfy the required relations . 1 . up to some simple multipliers , @xmath42 is just the inverse matrix automorphy factor in @xcite and the restriction of @xmath41 to @xmath156 is the @xmath0-representation in @xcite that leads to the fusion algebra via a nonsemisimple generalization of the verlinde formula . @xmath36 is the @xmath0-representation realized in the @xmath157 minimal model @xcite . in sec . [ sec : sliiz - restr ] , the structure described in * [ thm : r - decomp ] * is established for the @xmath0-representation on the quantum group center . the version of the quantum @xmath158 that is kazhdan lusztig - dual to the @xmath4 conformal field theory model is the restricted quantum group @xmath2 at @xmath17 the primitive @xmath3th root of unity . we introduce it in * [ sec : cas ] * , consider its representations in * [ sec : repr ] * , and find its grothendieck ring in * [ sec : grring]*. the hopf algebra @xmath2 ( henceforth , at @xmath29 ) is generated by @xmath159 , @xmath160 , and @xmath161 with the relations @xmath162 and the hopf - algebra structure given by @xmath163={\mbox{\footnotesize$\displaystyle\frac{k - k^{-1}}{{\mathfrak{q}}-{\mathfrak{q}}^{-1}}$}},\\ \delta(e)={\boldsymbol{1}}\otimes e+e\otimes k,\quad \delta(f)=k^{-1}\otimes f+f\otimes{\boldsymbol{1}},\quad \delta(k)=k\otimes k,\\ \epsilon(e)=\epsilon(f)=0,\quad\epsilon(k)=1,\\ s(e)=-ek^{-1},\quad s(f)=-kf,\quad s(k)=k^{-1}.\end{gathered}\ ] ] the elements of the pbw - basis of @xmath2 are enumerated as @xmath164 with @xmath165 , @xmath166 , @xmath167 , and its dimension is therefore @xmath168 . [ [ section ] ] it follows ( e.g. , by induction ) that @xmath169}{0pt}{}{m}{r}$}}}{{\genfrac{[}{]}{0pt}{}{m}{r } } } { { \genfrac{[}{]}{0pt}{}{m}{r}}}{{\genfrac{[}{]}{0pt}{}{m}{r}}}}{\mathchoice { { \mbox{\footnotesize$\displaystyle \genfrac{[}{]}{0pt}{}{n}{s}$}}}{{\genfrac{[}{]}{0pt}{}{n}{s } } } { { \genfrac{[}{]}{0pt}{}{n}{s}}}{{\genfrac{[}{]}{0pt}{}{n}{s}}}}\\ * { } \times f^r e^{n - s } k^{r - m+j}{\otimes}f^{m - r } e^s k^{n - s+j}.\end{gathered}\ ] ] for @xmath2 , the right integral and the left right cointegral ( see the definitions in * [ app : int ] * ) are given by @xmath170 and @xmath171 where we choose the normalization as @xmath172!)^2}$}}\ ] ] for future convenience . next , simple calculation shows that the comodulus for @xmath2 ( see * [ app : int ] * ) is @xmath173 . this allows us to find the balancing element using . there are _ four _ possibilities for the square root of @xmath174 , two of which are group - like , and we choose @xmath175 this choice determines a ribbon element for @xmath2 , and hence a particular version of the @xmath0-action on the quantum group studied below . the balancing element allows constructing the `` canonical '' @xmath5-characters of @xmath2-representations ( see * [ app : qch ] * ) . let @xmath10 denote the center of @xmath2 . it contains the element @xmath176 called the casimir element . it satisfies the minimal polynomial relation @xmath177 where @xmath178 a proof of is given in * [ fusion - center ] * below as a spin - off of the technology developed for the grothendieck ring ( we do not need before that ) . it follows from the definition of @xmath2 that @xmath179 . in fact , @xmath20 is in the @xmath3-dimensional subalgebra in @xmath10 generated by @xmath180 because of the identity @xmath181 where we set @xmath182 the @xmath2-representation theory at @xmath29 is not difficult to describe ( also see @xcite ) . there turn out to be just @xmath3 irreducible representations . in what follows , we also need verma modules ( all of which except two are extensions of a pair of irreducible representations ) and projective modules ( which are further extensions ) . the category of all finite - dimensional @xmath2-representations at the primitive @xmath3th root of unity is fully described in @xcite . the irreducible @xmath2-representations @xmath183 are labeled by @xmath30 and @xmath96 . the module @xmath13 is linearly spanned by elements @xmath184 , @xmath185 , where @xmath186 is the highest - weight vector and the @xmath2-action is given by @xmath187[s - n]{| s , n - 1\rangle}^{\pm},\\ f { | s , n\rangle}^{\pm } & = { | s , n + 1\rangle}^{\pm},\end{aligned}\ ] ] where we set @xmath188 . @xmath189 is the trivial module . for later use , we list the weights occurring in the module @xmath190 , i.e. , the eigenvalues that @xmath161 has on vectors in @xmath190 , @xmath191 and in the module @xmath192 , @xmath193 we also note the dimensions and quantum dimensions ( see * [ app : qch ] * ) @xmath194 and @xmath195 $ ] . it follows that @xmath196 and @xmath197 . there are @xmath3 verma modules @xmath198 , @xmath96 . first , these are the two steinberg modules @xmath199 next , for each @xmath200 and @xmath30 , the verma module @xmath201 is explicitly described in * [ verma - mod - base ] * as an extension @xmath202 ; for consistency with more complicated extensions considered below , we represent it as @xmath203 with the convention that the arrow is directed to a _ _ sub__module . we note that @xmath204 and @xmath205 ( negligible modules @xcite ) . for @xmath200 , there are nontrivial extensions yielding the projective modules @xmath206 and @xmath207 , @xmath208 their structure can be schematically depicted as @xmath209 \ar@/_/[dr ] & \\ & \stackrel{{\mathscr{x}}^{-\alpha}(p{-}s)}{\bullet}\ar@/^/[dr ] & & \stackrel{{\mathscr{x}}^{-\alpha}(p{-}s)}{\bullet}\ar@/_/[dl ] \\ & & \stackrel{{\mathscr{x}}^{\alpha}(s)}{\bullet } & } \ ] ] it follows that @xmath210 and @xmath211 . the bases and the action of @xmath2 in @xmath206 and @xmath207 are described in * [ module - l ] * and * [ module - p]*. we next find the grothendieck ring of @xmath2 . [ thm : gr - ring ] multiplication in the @xmath2 grothendieck ring @xmath23 is given by @xmath212{\sum_{\substack{s''=|s - s'| + 1\\ \mathrm{step}=2}}^{s + s ' - 1 } } \widetilde{\mathscr{x}}^{\alpha\alpha'}(s''),\ ] ] where @xmath213 to prove this , we use ( i ) a property of the tensor products of any representation with a verma module , ( ii ) an explicit evaluation of the tensor product of any irreducible representation with a two - dimensional one , and ( iii ) the observation that the information gained in ( i ) and ( ii ) suffices for finding the entire grothendieck ring . we first of all note that the trivial representation @xmath214 is the unit in the grothendieck ring and , obviously , @xmath215 for all @xmath216 and @xmath30 . moreover , @xmath217 and it therefore suffices to find all the products @xmath218 and , furthermore , just the products @xmath219 . in the grothendieck ring , the verma module @xmath201 ( with @xmath120 ) is indistinguishable from @xmath220 , and we choose to consider only the @xmath221 verma modules @xmath222 , @xmath223 , given by @xmath224 their highest weights @xmath225 coincide with the respective highest weights of @xmath226 , @xmath227 , @xmath228 . taking the tensor product of a verma module @xmath222 and an irreducible representation gives a module that is filtered by verma modules . in the grothendieck ring , this tensor product therefore evaluates as a sum of verma modules ; moreover , the verma modules that occur in this sum are known , their highest weights being given by @xmath229 , where @xmath230 is the highest weight of @xmath222 and @xmath231 are the weights of vectors in the irreducible representation . with , this readily gives the grothendieck - ring multiplication @xmath232 where we set @xmath233 for @xmath234 and @xmath235 for @xmath236 . for @xmath237 , we have @xmath238 let @xmath239 for @xmath240 and @xmath241 be the respective bases in @xmath183 and in @xmath242 . under the action of @xmath160 , the highest - weight vector @xmath243 with the weight @xmath244 generates the module @xmath245 . the vector @xmath246e_{0}\otimes f_{1}$ ] satisfies the relations @xmath247 under the action of @xmath160 , it generates the module @xmath248 . as regards the product @xmath249 , we already know it from because @xmath250 is a verma module : with the two relevant verma modules replaced by the sum of the corresponding irreducible representations , the resulting four terms can be written as @xmath251 as we have noted , the products @xmath252 are given by the above formulas with the reversed `` @xmath253 '' signs in the right - hand sides . [ [ section-1 ] ] we next evaluate the products @xmath254 as @xmath255 where the products with @xmath242 are already known . by induction on @xmath256 , this allows finding all the products @xmath218 as @xmath257{\sum_{\substack{s''=|s - s'| + 1\\s''\neq p,\ ; \mathrm{step}=2}}^{p - 1 - |p - s - s'| } } { \mathscr{x}}^{\alpha}(s '' ) + \delta_{p , s , s'}{\mathscr{x}}^{\alpha}(p)\\ * { } + \sum_{\substack{s''= 2p - s - s ' + 1\\ \mathrm{step}=2}}^{p - 1 } ( 2{\mathscr{x}}^{\alpha}(s '' ) + 2{\mathscr{x}}^{-\alpha}(p - s'')),\end{gathered}\ ] ] where @xmath258 is equal to @xmath259 if @xmath260 and @xmath261 , and is @xmath262 otherwise . the statement in * [ thm : gr - ring ] * is a mere rewriting of , taken together with the relations @xmath263 . it shows that the @xmath2 grothendieck ring is the @xmath4-model fusion algebra derived in @xcite . this concludes the proof of * [ thm : gr - ring]*. [ cor : quotient ] the @xmath2 grothendieck ring contains the ideal @xmath264 of verma modules generated by @xmath265 the quotient @xmath266 is a _ fusion _ algebra with the basis @xmath267 , @xmath120 the canonical images of the corresponding @xmath102 and multiplication @xmath268 this is a _ semisimple _ fusion algebra , which coincides with the fusion of the unitary @xmath269 representations of level @xmath270 . [ cor : generated ] the @xmath2 grothendieck ring @xmath23 is generated by @xmath271 . this easily follows from theorem [ thm : gr - ring ] ; therefore , @xmath23 can be identified with a quotient of the polynomial ring @xmath272 $ ] . let @xmath273 denote the chebyshev polynomials of the second kind @xmath274 the lower such polynomials are @xmath275 , @xmath276 , @xmath277 , and @xmath278 . [ prop : quotient ] the @xmath2 grothendieck ring is the quotient of the polynomial ring @xmath272 $ ] over the ideal generated by the polynomial @xmath279 moreover , let @xmath280 under the quotient mapping , the image of each polynomial @xmath281 coincides with @xmath282 for @xmath96 and with @xmath283 for @xmath284 . it follows from * [ thm : gr - ring ] * that @xmath285 we recall that the chebyshev polynomials of the second kind satisfy ( and are determined by ) the recursive relation @xmath286 with the initial data @xmath276 , @xmath277 . from , we then obtain that polynomials satisfy relations after the identifications @xmath287 for @xmath96 and @xmath288 for @xmath284 . then , for eq . to be satisfied , we must impose the relation @xmath289 ; this shows that the grothendieck ring is the quotient of @xmath272 $ ] over the ideal generated by polynomial . [ prop : factor ] the polynomial @xmath290 can be factored as @xmath291 this is verified by direct calculation using the representation @xmath292 which follows from . we note that @xmath293 for @xmath294 . the restricted quantum group @xmath2 is not quasitriangular @xcite ; however , it admits a drinfeld mapping , and hence there exists a homomorphic image @xmath295 of the grothendieck ring in the center . in * [ sec : from ] * , we first identify @xmath2 as a subalgebra in a quotient of a drinfeld double . we then obtain the @xmath66-matrix in * [ sec : m - matrix ] * , characterize the subalgebra @xmath296 in * [ fusion - center ] * , and find the center @xmath10 of @xmath2 at @xmath29 in * [ the - center]*. furthermore , we give some explicit results for the radford mapping for @xmath2 in * [ sec : radford - sl2 ] * and we find a ribbon element for @xmath2 in * [ sl2-ribbon]*. the hopf algebra @xmath2 is not quasitriangular , but it can be realized as a hopf subalgebra of a quasitriangular hopf algebra @xmath297 ( which is in turn a quotient of a drinfeld double ) . the @xmath66-matrix ( see * [ app : m ] * ) for @xmath298 is in fact an element of @xmath299 , and hence @xmath2 can be thought of as a factorizable hopf algebra , even though relation required of an @xmath66-matrix is satisfied not in @xmath2 but in @xmath298 ( but on the other hand , holds only with @xmath300 and @xmath301 being bases in @xmath2 ) . the hopf algebra @xmath298 is generated by @xmath302 , @xmath303 , and @xmath304 with the relations @xmath305={\mbox{\footnotesize$\displaystyle\frac{{k}^2-{k}^{-2}}{{\mathfrak{q}}-{\mathfrak{q}}^{-1}}$ } } , \\ { e}^p=0,\quad{\phi}^p=0,\quad{k}^{4p}={\boldsymbol{1 } } , \\ \epsilon({e})=0,\quad\epsilon({\phi})=0,\quad \epsilon({k})=1 , \\ \delta({e})={\boldsymbol{1}}\otimes{e}+{e}\otimes{k}^2,\quad \delta({\phi})={k}^{-2}{\otimes}{\phi}+{\phi}{\otimes}{\boldsymbol{1}},\quad \delta({k})={k}\otimes{k } , \\ s({e})=-{e}{k}^{-2},\quad s({\phi})=-{k}^{2}{\phi},\quad s({k})={k}^{-1}.\ ] ] a hopf algebra embedding @xmath306 is given by @xmath307 in what follows , we often do not distinguish between @xmath159 and @xmath302 , @xmath160 and @xmath303 , and @xmath161 and @xmath308 . [ thm - bar - one ] @xmath298 is a ribbon quasitriangular hopf algebra , with the universal @xmath309-matrix @xmath310!}$}}\,{\mathfrak{q}}^{m(m-1)/2+m(n - j)-nj/2 } { e}^m{k}^{n}\otimes{\phi}^m{k}^{j}\ ] ] and the ribbon element @xmath311!}$}}\ , { \mathfrak{q}}^{-\frac{m}{2}+mj+ { \mathchoice{{\mbox{\footnotesize$\displaystyle\frac{1}{2}$}}}{\frac{1}{2}}{\frac{1}{2}}{\frac{1}{2}}}(j+p+1)^2}{\phi}^m{e}^m{k}^{2j}.\ ] ] equation follows from the realization of @xmath298 as a quotient of the drinfeld double @xmath312 in * [ thm : double]*. the quotient is over the hopf ideal generated by the central element @xmath313 . it follows that @xmath298 inherits a quasitriangular hopf algebra structure from @xmath312 and @xmath309-matrix is the image of under the quotient mapping . using @xmath309-matrix , we calculate the canonical element @xmath314 ( see ) as @xmath315!}$}}\ , { \mathfrak{q}}^{-m(m+3)/2-rn/2}{\phi}^m{k}^{-r}{e}^m{k}^n.\ ] ] we note that actually @xmath316 . indeed , @xmath317!}$}}\ , { \mathfrak{q}}^{-m(m+3)/2-rm - rn/2}{\phi}^m{e}^m{k}^{n - r}={}\\ = { \mbox{\footnotesize$\displaystyle\frac{1}{4p}$}}\sum_{m=0}^{p-1}\sum_{j=0}^{2p-1 } \bigl(\sum_{r=0}^{4p-1}e^{-i\pi\frac{1}{2p}r(r+2m+2j)}\bigr ) ( -1)^m{\mbox{\footnotesize$\displaystyle\frac{({\mathfrak{q}}-{\mathfrak{q}}^{-1})^m}{[m]!}$}}\ , { \mathfrak{q}}^{- { \mathchoice{{\mbox{\footnotesize$\displaystyle\frac{1}{2}$}}}{\frac{1}{2}}{\frac{1}{2}}{\frac{1}{2}}}m(m+3)}{\phi}^m{e}^m{k}^{2j}\\ + { \mbox{\footnotesize$\displaystyle\frac{1}{4p}$}}\sum_{m=0}^{p-1}\sum_{j=0}^{2p-1 } \bigl(\sum_{r=0}^{4p-1}e^{-i\pi\frac{1}{2p}r(r+2m+2j+1)}\bigr ) ( -1)^m{\mbox{\footnotesize$\displaystyle\frac{({\mathfrak{q}}-{\mathfrak{q}}^{-1})^m}{[m]!}$}}\ , { \mathfrak{q}}^{- { \mathchoice{{\mbox{\footnotesize$\displaystyle\frac{1}{2}$}}}{\frac{1}{2}}{\frac{1}{2}}{\frac{1}{2}}}m(m+3)}{\phi}^m{e}^m{k}^{2j+1}. \end{gathered}\ ] ] the second gaussian sum vanishes , @xmath318 to evaluate the first gaussian sum , we make the substitution @xmath319 : @xmath320!}$}}\ , { \mathfrak{q}}^{- { \mathchoice{{\mbox{\footnotesize$\displaystyle\frac{1}{2}$}}}{\frac{1}{2}}{\frac{1}{2}}{\frac{1}{2}}}m(m+3)+ { \mathchoice{{\mbox{\footnotesize$\displaystyle\frac{1}{2}$}}}{\frac{1}{2}}{\frac{1}{2}}{\frac{1}{2}}}(j+m)^2}{\phi}^m{e}^m{k}^{2j}\\ = { \mbox{\footnotesize$\displaystyle\frac{1}{4p}$}}\sum_{m=0}^{p-1}\sum_{j=0}^{2p-1 } \bigl(\sum_{r=0}^{4p-1}e^{-i\pi\frac{1}{2p}r^2}\bigr ) { \mbox{\footnotesize$\displaystyle\frac{({\mathfrak{q}}-{\mathfrak{q}}^{-1})^m}{[m]!}$}}\ , { \mathfrak{q}}^{- { \mathchoice{{\mbox{\footnotesize$\displaystyle\frac{1}{2}$}}}{\frac{1}{2}}{\frac{1}{2}}{\frac{1}{2}}}m+m(j - p-1)+ { \mathchoice{{\mbox{\footnotesize$\displaystyle\frac{1}{2}$}}}{\frac{1}{2}}{\frac{1}{2}}{\frac{1}{2}}}j^2}{\phi}^m{e}^m{k}^{2j}. \end{gathered}\ ] ] then evaluating @xmath321 we obtain @xmath322!}$}}\ , { \mathfrak{q}}^{- { \mathchoice{{\mbox{\footnotesize$\displaystyle\frac{1}{2}$}}}{\frac{1}{2}}{\frac{1}{2}}{\frac{1}{2}}}m+mj+ { \mathchoice{{\mbox{\footnotesize$\displaystyle\frac{1}{2}$}}}{\frac{1}{2}}{\frac{1}{2}}{\frac{1}{2}}}(j+p+1)^2}{\phi}^m{e}^m{k}^{2j+2p+2}.\ ] ] we then find the ribbon element from relation using the balancing element @xmath323[page : balancing ] from , which gives . we next obtain the @xmath66-matrix ( see * [ app : m ] * ) for @xmath2 from the universal @xmath309-matrix for @xmath298 in . because @xmath324 , it follows from that the @xmath66-matrix for @xmath298 , @xmath325 , actually lies in @xmath299 , and does not therefore satisfy condition in @xmath298 ( and hence @xmath298 is not factorizable ) . but this _ is _ an @xmath66-matrix for @xmath326 . a simple calculation shows that @xmath327 is explicitly rewritten in terms of the @xmath2-generators as @xmath328 ! [ n]!}$}}\ , { \mathfrak{q}}^{m(m - 1)/2 + n(n - 1)/2}\\ * \times { \mathfrak{q}}^{- m^2 - m j + 2n j - 2n i - i j + m i } f^{m } e^{n } k^{j}{\otimes}e^{m } f^{n } k^{i}.\end{gathered}\ ] ] given the @xmath66-matrix , we can identify the @xmath2 grothendieck ring with its image in the center using the homomorphism in * [ lemma : dr - hom]*. we evaluate this homomorphism on the preferred basis elements in the grothendieck ring , i.e. , on the irreducible representations . with the balancing element for @xmath2 in and the @xmath66-matrix in , the mapping in * [ lemma : dr - hom ] * is @xmath330 clearly , @xmath331 . we let @xmath296 denote the image of the grothendieck ring under this mapping . [ prop - eval ] for @xmath216 and @xmath30 , @xmath332}{0pt}{}{s - n+m-1}{m}$}}}{{\genfrac{[}{]}{0pt}{}{s - n+m-1}{m } } } { { \genfrac{[}{]}{0pt}{}{s - n+m-1}{m}}}{{\genfrac{[}{]}{0pt}{}{s - n+m-1}{m } } } } { \mathchoice { { \mbox{\footnotesize$\displaystyle \genfrac{[}{]}{0pt}{}{n}{m}$}}}{{\genfrac{[}{]}{0pt}{}{n}{m } } } { { \genfrac{[}{]}{0pt}{}{n}{m}}}{{\genfrac{[}{]}{0pt}{}{n}{m } } } } e^m f^m k^{s-1+\beta p - 2n + m } , \end{gathered}\ ] ] where we set @xmath333 if @xmath334 and @xmath335 if @xmath336 . in particular , it follows that @xmath337 the proof of is a straightforward calculation based on the well - known identity ( see , e.g. , @xcite ) @xmath338 which readily implies that @xmath339!)^2\sum_{n=0}^{s-1}{\mathfrak{q}}^{a(s-1 - 2n ) } { \mathchoice { { \mbox{\footnotesize$\displaystyle \genfrac{[}{]}{0pt}{}{s - n+m-1}{m}$}}}{{\genfrac{[}{]}{0pt}{}{s - n+m-1}{m } } } { { \genfrac{[}{]}{0pt}{}{s - n+m-1}{m}}}{{\genfrac{[}{]}{0pt}{}{s - n+m-1}{m}}}}{\mathchoice { { \mbox{\footnotesize$\displaystyle \genfrac{[}{]}{0pt}{}{n}{m}$}}}{{\genfrac{[}{]}{0pt}{}{n}{m } } } { { \genfrac{[}{]}{0pt}{}{n}{m}}}{{\genfrac{[}{]}{0pt}{}{n}{m}}}}.\ ] ] using this in gives . for @xmath340 , we then have @xmath341}{0pt}{}{1-n+m}{m}$}}}{{\genfrac{[}{]}{0pt}{}{1-n+m}{m } } } { { \genfrac{[}{]}{0pt}{}{1-n+m}{m}}}{{\genfrac{[}{]}{0pt}{}{1-n+m}{m}}}}{\mathchoice { { \mbox{\footnotesize$\displaystyle \genfrac{[}{]}{0pt}{}{n}{m}$}}}{{\genfrac{[}{]}{0pt}{}{n}{m } } } { { \genfrac{[}{]}{0pt}{}{n}{m}}}{{\genfrac{[}{]}{0pt}{}{n}{m } } } } e^m f^m k^{1 - 2n + m}=\\ = -{\mathfrak{q}}^{-1}k - { \mathfrak{q}}k^{-1 } - ( { \mathfrak{q}}- { \mathfrak{q}}^{-1})^2 e f. \end{gathered}\ ] ] combining * [ prop - eval ] * and * [ cor : generated ] * , we obtain [ dr - alg - cas ] @xmath295 coincides with the algebra generated by the casimir element . the following corollary is now immediate in view of * [ prop : quotient ] * and * [ prop : factor]*. relation holds for the casimir element . identity holds . the derivation of from the algebra of the @xmath342 is given in appendix * [ app : derivation ] * in some detail . we note that although the left - hand side of is not manifestly symmetric in @xmath343 and @xmath256 , the identity shows that it is . [ [ verma - in - center ] ] in what follows , we keep the notation @xmath264 for the verma - module ideal ( more precisely , for its image in the center ) generated by @xmath344 this ideal is the socle ( annihilator of the radical ) of @xmath295 . we now find the center of @xmath2 at the primitive @xmath3th root of unity . for this , we use the isomorphism between the center and the algebra of _ bimodule _ endomorphisms of the regular representation . the results are in * [ prop - center ] * and * [ prop - center - explicit]*. the @xmath168-dimensional regular representation of @xmath2 , viewed as a free left module , decomposes into indecomposable projective modules , each of which enters with the multiplicity given by the dimension of its simple quotient : @xmath345 we now study the regular representation as a @xmath2-bimodule . in what follows , @xmath346 denotes the external tensor product . as a @xmath2-bimodule , the regular representation decomposes as @xmath347{\bigoplus_{s=0}^{p}{\mathscr{q}}(s)},\ ] ] where 1 . the bimodules @xmath348 are simple , 2 . the bimodules @xmath349 , @xmath120 , are indecomposable and admit the filtration @xmath350 where the structure of subquotients is given by @xmath351 and @xmath352 and where @xmath353 is isomorphic to the quotient @xmath354 . the proof given below shows that @xmath355 is in fact the jacobson radical of @xmath349 and @xmath356 , with @xmath357 , and hence @xmath353 is the socle of @xmath349 . for @xmath200 , the left @xmath2-action on @xmath349 and the structure of subquotients can be visualized with the aid of the diagram @xmath358[l]{${\boxtimes}{{\mathscr{x}}^{+}}(s)$ } } \ar[1,-1 ] \ar[1,1 ] & * { } & * { } & * { } * { } & { { { \mathscr{x}}^{-}}(p{-}s)\makebox[0pt][l]{${\boxtimes}{{\mathscr{x}}^{-}}(p{-}s)$ } } \ar[1,-1 ] \ar[1,1 ] \\ { { { \mathscr{x}}^{-}}(p{-}s)\makebox[0pt][l]{${\boxtimes}{{\mathscr{x}}^{+}}(s)$ } } \ar[1,1 ] & * { } & { { { \mathscr{x}}^{-}}(p{-}s)\makebox[0pt][l]{${\boxtimes}{{\mathscr{x}}^{+}}(s)$ } } \ar[1,-1 ] & * { \quad } & { { { \mathscr{x}}^{+}}(s)\makebox[0pt][l]{${\boxtimes}{{\mathscr{x}}^{-}}(p{-}s)$ } } \ar[1,1 ] & * { } & { { { \mathscr{x}}^{+}}(s)\makebox[0pt][l]{${\boxtimes}{{\mathscr{x}}^{-}}(p{-}s)$ } } \ar[1,-1 ] \\ * { } & { { { \mathscr{x}}^{+}}(s)\makebox[0pt][l]{${\boxtimes}{{\mathscr{x}}^{+}}(s)$ } } & * { } & * { } & * { } * { } & { { { \mathscr{x}}^{-}}(p{-}s)\makebox[0pt][l]{${\boxtimes}{{\mathscr{x}}^{-}}(p{-}s)$ } } } \ ] ] and the right action with the reader may find it convenient to look at these diagrams in reading the proof below . first , the category @xmath359 of finite - dimensional left @xmath2-modules has the decomposition @xcite @xmath360 where each @xmath361 is a full subcategory . the full subcategories @xmath362 and @xmath363 are semisimple and contain precisely one irreducible module each , @xmath228 and @xmath226 respectively . each @xmath361 , @xmath364 , contains precisely two irreducible modules @xmath190 and @xmath365 , and we have the vector - space isomorphisms @xcite @xmath366 where a basis in each @xmath40 can be chosen as the extensions corresponding to the verma module @xmath198 and to the contragredient verma module @xmath367 ( see * [ verma - mod - base ] * ) . in view of , the regular representation viewed as a @xmath2-bimodule has the decomposition @xmath368{\bigoplus_{s=0}^{p}{\mathscr{q}}(s)}\ ] ] into a direct sum of indecomposable two - sided ideals @xmath349 . we now study the structure of subquotients of @xmath349 . let @xmath355 denote the jacobson radical of @xmath349 . by the wedderburn artin theorem , the quotient @xmath354 is a semisimple matrix algebra over @xmath154 , @xmath369 ( where we note that @xmath370 ) . as a bimodule , @xmath354 has the decomposition @xmath371 for @xmath120 , we now consider the quotient @xmath372 , where we set @xmath356 . for brevity , we write @xmath373 , @xmath374 , @xmath375 and @xmath376 , @xmath377 , @xmath378 , and similarly for the contragredient verma modules @xmath379 in view of , there are the natural bimodule homomorphisms @xmath380 the image of @xmath381 has the structure of the lower - triangular matrix @xmath382 clearly , the radical of @xmath383 is the bimodule @xmath384 . it follows that @xmath385 and the bimodule @xmath384 is a subquotient of @xmath386 . in a similar way , we obtain that @xmath387 and @xmath388 . therefore , we have the inclusion @xmath389 next , the radford mapping @xmath390 ( see * [ sec : radford - all ] * ) establishes a bimodule isomorphism between @xmath391 and @xmath392 , and therefore the socle of @xmath393 is isomorphic to @xmath394 . this suffices for finishing the proof : by counting the dimensions of the subquotients given in and , and the dimension of the socle of @xmath393 , we obtain the statement of the proposition . to find the center of @xmath2 , we consider bimodule endomorphisms of the regular representation ; such endomorphisms are in a @xmath395 correspondence with elements in the center . clearly , @xmath396 for each @xmath349 , @xmath397 , there is a bimodule endomorphism @xmath398 that acts as identity on @xmath349 and is zero on @xmath399 with @xmath400 . these endomorphisms give rise to @xmath221 primitive idempotents in the center of @xmath2 . next , for each @xmath349 with @xmath120 , there is a homomorphism @xmath401 ( defined up to a nonzero factor ) whose kernel , as a linear space , is given by @xmath402 ( see ) ; in other words , @xmath403 sends the quotient @xmath404 into the subbimodule @xmath404 at the bottom of @xmath349 and is zero on @xmath399 with @xmath400 . similarly , for each @xmath200 , there is a central element associated with the homomorphism @xmath405 with the kernel @xmath406 , i.e. , the homomorphism sending the quotient @xmath407 into the subbimodule @xmath408 ( and acting by zero on @xmath399 with @xmath400 ) . in total , there are @xmath409 elements @xmath410 , @xmath120 , which are obviously in the radical of the center . by construction , the @xmath411 and @xmath412 have the properties summarized in the following proposition . [ prop - center ] the center @xmath10 of @xmath2 at @xmath29 is @xmath16-dimensional . its associative commutative algebra structure is described as follows : there are two `` special '' primitive idempotents @xmath413 and @xmath414 , @xmath128 other primitive idempotents @xmath411 , @xmath120 , and @xmath415 elements @xmath410 @xmath416 in the radical such that @xmath417 we call @xmath411 , @xmath410 the canonical basis elements in the center , or simply the _ canonical central elements_. they are constructed somewhat more explicitly in * [ prop - center - explicit]*. we note that the choice of a bimodule isomorphism @xmath418 fixes the normalization of the @xmath410 . [ [ rem : coeffs ] ] for any central element @xmath77 and its decomposition @xmath419 with respect to the canonical central elements , _ the coefficient @xmath420 is the eigenvalue of @xmath77 in the irreducible representation @xmath190_. to determine the @xmath421 and @xmath422 coefficients similarly , we fix the normalization of the basis vectors as in * [ proj - mod - base ] * , i.e. , such that @xmath403 and @xmath423 act as @xmath424 in terms of the respective bases in the projective modules @xmath206 and @xmath425 defined in * [ module - l ] * and * [ module - p]*. then _ the coefficient @xmath421 is read off from the relation @xmath426 in @xmath206 , and @xmath422 , similarly , from the relation @xmath427 in @xmath425_. for a hopf algebra @xmath77 with a given cointegral , we recall the radford mapping @xmath428 , see * [ sec : radford - all ] * ( we use the hat for notational consistency in what follows ) . for @xmath429 , with the cointegral @xmath430 in , we are interested in the restriction of the radford mapping to the space of @xmath5-characters @xmath431 and , more specifically , to the image of the grothendieck ring in @xmath431 via the mapping @xmath432 ( see ) . we thus consider the mapping @xmath433 let @xmath434 be the linear span of the @xmath435 ( the image of the grothendieck ring in the center under the radford mapping ) . as we see momentarily , @xmath434 is @xmath3-dimensional and coincides with the algebra generated by the @xmath436 . it follows that @xmath437 in accordance with the fact that @xmath430 furnishes an embedding of the trivial representation @xmath438 into @xmath2 . a general argument based on the properties of the radford mapping ( cf . @xcite ) and on the definition of the canonical nilpotents @xmath412 above implies that for @xmath439 , @xmath440 coincides with @xmath403 up to a factor and @xmath441 coincides with @xmath442 up to a factor . we now give a purely computational proof of this fact , which at the same time fixes the factors ; we describe this in some detail because similar calculations are used in what follows . [ lemma : phi - idem ] for @xmath120 , @xmath443 ^ 2}$}}.\ ] ] also , @xmath444 therefore , the image of the grothendieck ring under the radford mapping is the socle ( annihilator of the radical ) of @xmath10 . first , we recall and use and to evaluate @xmath445!)^2 { \mathfrak{q}}^{j(s - 1 - 2n)}{\mathchoice { { \mbox{\footnotesize$\displaystyle \genfrac{[}{]}{0pt}{}{s - n + i - 1}{i}$}}}{{\genfrac{[}{]}{0pt}{}{s - n + i - 1}{i } } } { { \genfrac{[}{]}{0pt}{}{s - n + i - 1}{i}}}{{\genfrac{[}{]}{0pt}{}{s - n + i - 1}{i } } } } { \mathchoice { { \mbox{\footnotesize$\displaystyle \genfrac{[}{]}{0pt}{}{n}{i}$}}}{{\genfrac{[}{]}{0pt}{}{n}{i } } } { { \genfrac{[}{]}{0pt}{}{n}{i}}}{{\genfrac{[}{]}{0pt}{}{n}{i } } } } f^{p - 1 - i } e^{p - 1 - i } k^j\ ] ] ( the calculation is very similar to the one in * [ prop - eval ] * ) . next , we decompose @xmath436 with respect to the canonical basis following the strategy in * [ rem : coeffs]*. that is , we use to calculate the action of @xmath440 on the module @xmath446 ( @xmath447 ) . this action is nonzero only on the vectors @xmath448 ( see * [ module - l ] * ) ; because @xmath440 is central , it suffices to evaluate it on any single vector , which we choose as @xmath449 . for @xmath120 , using and , we then have @xmath450!)^2 { \mathfrak{q}}^{j(s + s ' - 2 - 2n ) } { \mathchoice { { \mbox{\footnotesize$\displaystyle \genfrac{[}{]}{0pt}{}{s - n + i - 1}{i}$}}}{{\genfrac{[}{]}{0pt}{}{s - n + i - 1}{i } } } { { \genfrac{[}{]}{0pt}{}{s - n + i - 1}{i}}}{{\genfrac{[}{]}{0pt}{}{s - n + i - 1}{i } } } } { \mathchoice { { \mbox{\footnotesize$\displaystyle \genfrac{[}{]}{0pt}{}{n}{i}$}}}{{\genfrac{[}{]}{0pt}{}{n}{i } } } { { \genfrac{[}{]}{0pt}{}{n}{i}}}{{\genfrac{[}{]}{0pt}{}{n}{i}}}}\\ * \shoveright{{}\times \prod_{r=0}^{p-2-i } \bigl({\boldsymbol{c}}-{\mbox{\small$\displaystyle\frac{{\mathfrak{q}}^{2r+1}k+{\mathfrak{q}}^{-2r-1}k^{-1}}{({\mathfrak{q}}-{\mathfrak{q}}^{-1})^2}$ } } \bigr ) { \mathsf{b}}^{(+,s')}_0}\\ { } = \zeta \sum_{n=0}^{s - 1}\ ! \sum_{i=0}^{n}\ ! \sum_{j=0}^{2p - 1 } ( -1)^{p+i } ( [ i]!)^2 { \mathfrak{q}}^{j(s + s ' - 2 - 2n ) } { \mathchoice { { \mbox{\footnotesize$\displaystyle \genfrac{[}{]}{0pt}{}{s - n + i - 1}{i}$}}}{{\genfrac{[}{]}{0pt}{}{s - n + i - 1}{i } } } { { \genfrac{[}{]}{0pt}{}{s - n + i - 1}{i}}}{{\genfrac{[}{]}{0pt}{}{s - n + i - 1}{i } } } } { \mathchoice { { \mbox{\footnotesize$\displaystyle \genfrac{[}{]}{0pt}{}{n}{i}$}}}{{\genfrac{[}{]}{0pt}{}{n}{i } } } { { \genfrac{[}{]}{0pt}{}{n}{i}}}{{\genfrac{[}{]}{0pt}{}{n}{i } } } } \prod_{r=1}^{p-2-i}\ ! [ s'+r][r]\ , { \mathsf{a}}^{(+,s')}_0 , \end{gathered}\ ] ] with the convention that whenever @xmath451 , the product over @xmath452 evaluates as @xmath259 . we simultaneously see that the diagonal part of the action of @xmath440 on @xmath446 vanishes . analyzing the cases where the product over @xmath452 in involves @xmath453=0 $ ] , it is immediate to see that a necessary condition for the right - hand side to be nonzero is @xmath454 . let therefore @xmath455 , where @xmath456 . it is then readily seen that vanishes for odd @xmath457 ; we thus set @xmath458 , which allows us to evaluate @xmath459!)^2 { \mathchoice { { \mbox{\footnotesize$\displaystyle \genfrac{[}{]}{0pt}{}{m+i}{i}$}}}{{\genfrac{[}{]}{0pt}{}{m+i}{i } } } { { \genfrac{[}{]}{0pt}{}{m+i}{i}}}{{\genfrac{[}{]}{0pt}{}{m+i}{i}}}}{\mathchoice { { \mbox{\footnotesize$\displaystyle \genfrac{[}{]}{0pt}{}{m+s'-1}{i}$}}}{{\genfrac{[}{]}{0pt}{}{m+s'-1}{i } } } { { \genfrac{[}{]}{0pt}{}{m+s'-1}{i}}}{{\genfrac{[}{]}{0pt}{}{m+s'-1}{i}}}}{\mbox{\footnotesize$\displaystyle\frac{[p-2-i+s']!}{[s']!}$}}\ , [ p-2-i]!\,{\mathsf{a}}^{(+,s')}_0 . \end{gathered}\ ] ] but this vanishes for all @xmath460 in view of the identity @xmath461\dots[j+s'+m-1]}{[j]![m - j]!}$ } } = { \mbox{\footnotesize$\displaystyle\frac{1}{[m]}$}}\sum_{j\in{\mathbb{z}}}(-1)^j{\mathchoice { { \mbox{\footnotesize$\displaystyle \genfrac{[}{]}{0pt}{}{m}{j}$}}}{{\genfrac{[}{]}{0pt}{}{m}{j } } } { { \genfrac{[}{]}{0pt}{}{m}{j}}}{{\genfrac{[}{]}{0pt}{}{m}{j}}}}{\mathchoice { { \mbox{\footnotesize$\displaystyle \genfrac{[}{]}{0pt}{}{m+s'-1+j}{m-1}$}}}{{\genfrac{[}{]}{0pt}{}{m+s'-1+j}{m-1 } } } { { \genfrac{[}{]}{0pt}{}{m+s'-1+j}{m-1}}}{{\genfrac{[}{]}{0pt}{}{m+s'-1+j}{m-1 } } } } = 0 , \quad m{\,{\geqslant}\,}1 . \end{gathered}\ ] ] thus , @xmath440 acts by zero on @xmath446 for all @xmath462 ; it follows similarly that @xmath440 acts by zero on @xmath463 for all @xmath256 and on both steinberg modules @xmath464 . therefore , @xmath440 is necessarily proportional to @xmath403 , with the proportionality coefficient to be found from the action on @xmath206 . but for @xmath465 , the sum over @xmath466 in the right - hand side of is zero unless @xmath467 , and we have @xmath468}$ } } \sum_{i=0}^{s-1 } ( -1)^{p+i } [ i]!{\mbox{\footnotesize$\displaystyle\frac{[p-2-i]![s+p-2-i]!}{[s-1-i]!}$}}\ , { \mathsf{a}}^{(+,s)}_0 , \intertext{where the terms in the sum are readily seen to vanish unless $ i = s-1 $ , and therefore } & = 2p\,\zeta\,(-1)^{p+s+1}{\mbox{\footnotesize$\displaystyle\frac{[p-1]!\,[s-1]!\,[p-1-s]!}{[s]}$}}\ , { \mathsf{a}}^{(+,s)}_0 , \end{aligned}\ ] ] which gives @xmath469 as claimed . the results for @xmath441 ( @xmath120 ) and @xmath470 are established similarly . it follows ( from the expression in terms of the canonical central elements ; cf . @xcite for the small quantum group ) that the two images of the grothendieck ring in the center , @xmath295 and @xmath434 , span the entire center : @xmath471 we next describe the intersection of the two grothendieck ring images in the center ( cf . @xcite for the small quantum group ) . this turns out to be the verma - module ideal ( see * [ verma - in - center ] * ) . [ prop : phi+phi ] @xmath472 . proceeding similarly to the proof of * [ lemma : phi - idem ] * , we establish the formulas @xmath473!)^2\!}{p}$}}\\ * { } \times\bigl((-1)^{p - s}{\pmb{\boldsymbol{\varkappa}}}(0 ) + \sum_{s'=1}^{p-1 } ( -1)^{p+s+s'}\bigl({\mathfrak{q}}^{s s ' } + { \mathfrak{q}}^{-s s'}\bigr ) { \pmb{\boldsymbol{\varkappa}}}(s ' ) + { \pmb{\boldsymbol{\varkappa}}}(p)\bigr ) \end{gathered}\ ] ] for @xmath200 , and @xmath474 which imply the proposition . the derivation may in fact be simplified by noting that as a consequence of and * [ remarks - d]*([item : w ] ) , @xmath475 belongs to the subalgebra generated by the casimir element , which allows using . we finally recall ( see * [ sec : ribbon ] * and @xcite ) that a ribbon element @xmath476 in a hopf algebra @xmath77 is an invertible central element satisfying . for @xmath2 , the ribbon element is actually given in , rewritten as @xmath477!}$}}\ , { \mathfrak{q}}^{-\frac{m}{2}+mj+ { \mathchoice{{\mbox{\footnotesize$\displaystyle\frac{1}{2}$}}}{\frac{1}{2}}{\frac{1}{2}}{\frac{1}{2}}}(j+p+1)^2 } f^m e^m k^{j}\ ] ] in terms of the @xmath2 generators . a calculation similar to the one in the proof of * [ lemma : phi - idem ] * shows the following proposition . [ ribbon - basis ] the @xmath2 ribbon element is decomposed in terms of the canonical central elements as @xmath478\,{\mbox{\footnotesize$\displaystyle\frac{{\mathfrak{q}}- { \mathfrak{q}}^{-1}}{\sqrt{2 p}}$}}\,{{\widehat}{\pmb{\varphi}}}(s ) , \end{aligned}\ ] ] where @xmath479 strictly speaking , expressing @xmath47 through the canonical central elements requires using * [ lemma : phi - idem ] * , but below we need @xmath47 expressed just through @xmath435 . in this section , we first recall the standard @xmath0-action @xcite reformulated for the center @xmath10 of @xmath2 . its definition involves the ribbon element and the drinfeld and radford mappings . from the multiplicative jordan decomposition for the ribbon element , we derive a factorization of the standard @xmath0-representation @xmath1 , @xmath480 , where @xmath41 and @xmath42 are also @xmath0-representations on @xmath10 . we then establish the equivalence to the @xmath0-representation on @xmath9 in * [ mod - on - char]*. let @xmath1 denote the @xmath0-representation on the center @xmath10 of @xmath2 constructed , as a slight modification of the representation in @xcite , as follows . we let @xmath481 and @xmath482 be defined as @xmath483 where @xmath47 is the ribbon element , @xmath484 is the drinfeld mapping , @xmath485 is the radford mapping , and @xmath486 is the normalization factor @xmath487 we call it the _ standard @xmath0-representation _ , to distinguish it from other representations introduced in what follows . we recall that @xmath488 acts via the antipode on the center of the quantum group , and hence acts identically on the center of @xmath2 , @xmath489 [ thm : equiv ] the standard @xmath0-representation on the center @xmath10 of @xmath490 at @xmath116 is equivalent to the @xmath16-dimensional @xmath0-representation on @xmath9 the extended characters of the @xmath4 conformal field theory model in * [ mod - on - char]*. we therefore abuse the notation by letting @xmath1 denote both representations . we introduce a basis in @xmath10 as @xmath491 where @xmath492 @xmath493 are defined in , and @xmath494 ( with @xmath495 defined in ) . that this is a basis in the center follows , e.g. , from the decomposition into the canonical central elements . the mapping @xmath496 between the bases in @xmath147 and in @xmath10 establishes the equivalence . showing this amounts to the following checks . first , we evaluate @xmath497 as @xmath498 and hence , in view of , @xmath499 we also need this formula rewritten in terms of @xmath500 that is , @xmath501 further , we use and to evaluate @xmath502 as @xmath503 where we set @xmath504 . this shows that @xmath48 acts on @xmath505 , @xmath493 , and @xmath506 as on the respective basis elements @xmath507 , @xmath508 , and @xmath509 in @xmath9 . next , it follows from * [ ribbon - basis ] * that @xmath47 acts on @xmath510 as @xmath511 as an immediate consequence , in view of @xmath512 , we have @xmath513 where @xmath514 is defined in . it follows that @xmath46 acts on @xmath505 and @xmath493 as on the respective basis elements @xmath507 and @xmath508 in @xmath9 . finally , we evaluate @xmath515 . recalling * [ ribbon - basis ] * to rewrite @xmath47 as @xmath516 we use and , with the result @xmath517 but ( a simple rewriting of the formulas in * [ eigenp ] * ) @xmath518 ^ 2}$}}\,{\boldsymbol{w}}_s\bigr),\ ] ] and therefore ( also recalling the projector properties to see that only one term survives in the sum over @xmath519 ) @xmath520 ^ 2}$}}\,{\boldsymbol{w}}_s\bigr)={}\\ = -b{\mbox{\footnotesize$\displaystyle\frac{\sqrt{2p}}{{\mathfrak{q}}^{s } - { \mathfrak{q}}^{-s}}$}}\,{\mathscr{s}}\ , ( -1)^{p } { \mathfrak{q}}^ { { \mathchoice{{\mbox{\footnotesize$\displaystyle\frac{1}{2}$}}}{\frac{1}{2}}{\frac{1}{2}}{\frac{1}{2}}}(s^2 - 1)}{\boldsymbol{e}}_s \bigl({\boldsymbol{e}}_s - { \mbox{\footnotesize$\displaystyle\frac{{\mathfrak{q}}^{s } + { \mathfrak{q}}^{-s}}{[s]^2}$}}\,{\boldsymbol{w}}_s - { \boldsymbol{\varphi}}(1){\boldsymbol{e}}_s\bigr)\\ = b(-1)^{s+1}{\mathfrak{q}}^ { { \mathchoice{{\mbox{\footnotesize$\displaystyle\frac{1}{2}$}}}{\frac{1}{2}}{\frac{1}{2}}{\frac{1}{2}}}(s^2 - 1)}\ , { \mathscr{s}}\,{{\widehat}{\boldsymbol{\rho}}}(r ) + b{\mbox{\footnotesize$\displaystyle\frac{(-1)^{p}\sqrt{2p}\,{\mathfrak{q}}^ { { \mathchoice{{\mbox{\footnotesize$\displaystyle\frac{1}{2}$}}}{\frac{1}{2}}{\frac{1}{2}}{\frac{1}{2}}}(s^2 - 1)}}{{\mathfrak{q}}^{s } - { \mathfrak{q}}^{-s}}$}}\ , { \mathscr{s}}\ , { \boldsymbol{\varphi}}(1){\boldsymbol{e}}_s . \end{gathered}\ ] ] here , @xmath521 and @xmath522 , and hence @xmath523 this completes the proof . in view of the equivalence of representations , the @xmath0-representation @xmath1 on the center admits the factorization established in * [ thm : r - decomp]*. remarkably , this factorization can be described in `` intrinsic '' quantum - group terms , as we now show . that is , we construct two more @xmath0-representations on @xmath10 with the properties described in * [ thm : modular-2]*. [ [ section-2 ] ] for the ribbon element @xmath47 , we consider its multiplicative jordan decomposition @xmath524 into the semisimple part @xmath525 and the unipotent part @xmath526 with , we now let @xmath527 and @xmath528 be defined by the corresponding parts of the ribbon element , similarly to : @xmath529 then , evidently , @xmath530 [ [ section-3 ] ] we next define a mapping @xmath531 as @xmath532 where @xmath533 it intertwines the coadjoint and adjoint actions of @xmath2 , and we therefore have the mapping @xmath534 , which is moreover an isomorphism of vector spaces . we set @xmath535 this gives the decomposition @xmath536 [ thm : factorization ] the action of @xmath537 and @xmath538 on the center generates the @xmath0-representation @xmath42 , and the action of @xmath56 and @xmath53 on the center generates the @xmath0-representation @xmath41 , such that 1 . @xmath539 for all @xmath59 , 2 . the representation @xmath41 restricts to the grothendieck ring ( i.e. , to its isomorphic image in the center ) , and 3 . @xmath480 for all @xmath61 , and @xmath1 and @xmath41 are isomorphic to the respective @xmath0-representations on @xmath9 in * [ thm : r - decomp]*. the verification is similar to the proof of * [ thm : equiv ] * , with @xmath540 and @xmath541 ( and similarly for @xmath56 ) , based on the formula @xmath542 [ [ section-4 ] ] the three mappings involved in @xmath485 defined in , @xmath484 defined in , and @xmath543 in can be described in a unified way as follows . let @xmath77 be a ribbon hopf algebra endowed with the standard @xmath0-representation . for @xmath544 , we define @xmath545\]]as @xmath546 where @xmath48 is the standard action of @xmath85 . taking @xmath547 to be the three elements @xmath548 , @xmath47 , and @xmath51 , we have @xmath549 we have shown that the kazhdan lusztig correspondence , understood in a broad sense as a correspondence between conformal field theories and quantum groups , extends into the nonsemisimple realm such that a number of structures on the conformal field theory side and on the quantum group side are actually isomorphic , which signifies an `` improvement '' over the case of rational@xmath62semisimple conformal field theories . although much of the argument in this paper is somewhat too `` calculational , '' and hence apparently `` accidental , '' we hope that a more systematic derivation can be given . in fact , the task to place the structures encountered in the study of nonsemisimple verlinde algebras into the categorical context @xcite was already formulated in @xcite . with the quantum - group counterpart of nonsemisimple verlinde algebras and of the @xmath0-representations on the conformal blocks studied in this paper in the @xmath4 example , this task becomes even more compelling . we plan to address claim [ item : equiv - cat ] of the kazhdan lusztig correspondence ( see page ) between the representation categories of the @xmath11 algebra and of @xmath2 @xcite . this requires constructing vertex - operator analogues of extensions among the irreducible representations ( generalizing the @xmath550 case studied in @xcite ) . another direction where development is welcome is to go over from @xmath4 to @xmath6 models of logarithmic conformal field theories , starting with the simplest such model , @xmath551 , whose content as a minimal theory is trivial , but whose logarithmic version may be quite interesting . we are grateful to a. belavin , e. feigin , m. finkelberg , k. hori , b. khesin , s. loktev , s. parkhomenko , y. soibelman , m.a . soloviev , and b.l . voronov for useful discussions . this paper was supported in part by the rfbr grants 04 - 01 - 00303 ( blf , amg , ams , and iyt ) , lss-1578.2003.2 ( ams and iyt ) , 02 - 01 - 01015 and lss-2044.2003.2 ( blf ) , intas grant 03 - 51 - 3350 ( blf ) . ams is grateful to the fields institute , where a part of this paper was written , for hospitality . we let @xmath77 denote a hopf algebra with comultiplication @xmath71 , counit @xmath72 , and antipode @xmath55 . the general facts summarized here can be found in @xcite . for a hopf algebra @xmath77 , the adjoint and coadjoint actions @xmath552 and @xmath553 ( @xmath554 ) are defined as @xmath555 the center @xmath556 of @xmath77 can be characterized as the set @xmath557 by definition , the space @xmath558 of @xmath5-characters is @xmath559 given an invertible element @xmath560 satisfying @xmath561 for all @xmath544 , we define the linear mapping @xmath562 for any @xmath77-module @xmath75 as @xmath563 [ lemma : qch ] for any @xmath77-module @xmath75 and an element @xmath519 such that @xmath561 , we have 1 . @xmath564 2 . if in addition @xmath519 is group - like , i.e. , @xmath565 , then @xmath566 is a homomorphism of the grothendieck ring to the ring of @xmath5-characters . for a hopf algebra @xmath77 , a _ right integral _ @xmath567 is a linear functional on @xmath77 satisfying @xmath568 for all @xmath544 . whenever such a functional exists , it is unique up to multiplication with a nonzero constant . a _ comodulus _ @xmath174 is an element in @xmath77 such that @xmath569 the left cointegral _ @xmath430 is an element in @xmath77 such that @xmath570 if it exists , this element is unique up to multiplication with a nonzero constant . we also note that the cointegral gives an embedding of the trivial representation of @xmath77 in the bimodule @xmath77 . we use the normalization @xmath571 . whenever a square root of the comodulus @xmath174 can be calculated in a hopf algebra @xmath77 , the algebra admits the _ balancing element _ @xmath572 that satisfies @xmath573 in fact , we have the following lemma . @xmath574 let @xmath77 be a hopf algebra with the right integral @xmath567 and the left right cointegral @xmath430 . the radford mapping @xmath428 and its inverse @xmath575 are given by @xmath576 [ lemma : rad - map ] @xmath485 and @xmath577 are inverse to each other , @xmath578 , @xmath579 , and intertwine the left actions of @xmath77 on @xmath77 and @xmath580 , and similarly for the right actions . here , the left-@xmath77-module structure on @xmath580 is given by @xmath581 ( and on @xmath77 , by the regular action ) . a quasitriangular hopf algebra @xmath77 has an invertible element @xmath582 satisfying @xmath583 for a quasitriangular hopf algebra @xmath77 , the @xmath66-matrix is defined as @xmath584 it satisfies the relations @xmath585 indeed , using , we find @xmath586 and then using , we obtain . next , from , which we write as @xmath587 , it follows that @xmath588 , that is , . if in addition @xmath66 can be represented as @xmath589 where @xmath300 and @xmath301 are two _ bases _ in @xmath77 , the hopf algebra @xmath77 is called _ factorizable_. in any quasitriangular hopf algebra , the square of the antipode is represented by a similarity transformation @xmath590 where the _ canonical element _ @xmath314 is given by @xmath591 ( where @xmath592 ) and satisfies the property @xmath593 any invertible element @xmath519 such that @xmath561 for all @xmath594 can be expressed as @xmath595 , where @xmath596 is an invertible central element . given an @xmath66-matrix ( see * [ app : m ] * ) , we define the drinfeld mapping @xmath597 as @xmath598 [ lemma : dr - map ] in a factorizable hopf algebra @xmath77 , the drinfeld mapping @xmath597 intertwines the adjoint and coadjoint actions of @xmath77 and its restriction to the space @xmath431 of @xmath5-characters gives an isomorphism of associative algebras @xmath599 a _ ribbon hopf algebra _ @xcite is a quasitriangular hopf algebra equipped with an invertible central element @xmath47 , called the _ ribbon element _ , such that @xmath600 in a ribbon hopf algebra , @xmath601 where @xmath572 is the balancing element ( see * [ app : int ] * ) . [ [ app : qch ] ] let @xmath77 be a ribbon hopf algebra and @xmath75 an @xmath77-module . the balancing element @xmath572 allows constructing the `` canonical '' @xmath5-character of @xmath75 : @xmath602 we also define the quantum dimension of a module @xmath75 as @xmath603 it satisfies the relation @xmath604 for any two modules @xmath605 and @xmath606 . let now @xmath77 be a factorizable ribbon hopf algebra and let @xmath76 be its grothendieck ring . we combine the mapping @xmath607 given by @xmath432 and the drinfeld mapping @xmath597 . [ lemma : dr - hom ] in a factorizable ribbon hopf algebra @xmath77 , the mapping @xmath608 is a homomorphism of associative commutative algebras . in this appendix , we construct a double of the hopf algebra @xmath609 associated with the short screening in the logarithmic conformal field theory outlined in * [ sec : voa]*. the main structure resulting from the double is the @xmath309-matrix , which is then used to construct the @xmath66-matrix @xmath73 for @xmath2 . for @xmath21 , we let @xmath609 denote the hopf algebra generated by @xmath302 and @xmath304 with the relations @xmath610 the pbw - basis in @xmath609 is @xmath611 the space @xmath612 of linear functions on @xmath609 is a hopf algebra with the multiplication , comultiplication , unit , counit , and antipode given by @xmath613 for any @xmath614 and @xmath615 . the quantum double @xmath312 is a hopf algebra with the underlying vector space @xmath616 and with the multiplication , comultiplication , unit , counit , and antipode given by eqs . and and by @xmath617 [ thm : double ] @xmath312 is the hopf algebra generated by @xmath302 , @xmath303 , @xmath304 , and @xmath618 with the relations @xmath619={\mbox{\footnotesize$\displaystyle\frac{{k}^2-{\kappa}^2}{{\mathfrak{q}}-{\mathfrak{q}}^{-1}}$}},\label{rel - bb*}\\ \delta({e})={\boldsymbol{1}}\otimes{e}+{e}\otimes{k}^2,\quad \delta({k})={k}\otimes{k},\quad \epsilon({e})=0,\quad \epsilon({k})=1,\label{rel - b-2}\\ \delta({\phi})={\kappa}^2{\otimes}{\phi}+{\phi}{\otimes}{\boldsymbol{1}},\quad \delta({\kappa})={\kappa}{\otimes}{\kappa},\quad \epsilon({\phi})=0,\quad \epsilon({\kappa})=1,\label{rel - b*-2}\\ s({e})=-{e}{k}^{-2},\quad s({k})={k}^{-1},\label{rel - b-3}\\ s({\phi})=-{\kappa}^{-2}{\phi},\quad s({\kappa})={\kappa}^{-1}.\label{rel - b*-3 } \end{gathered}\ ] ] equations , , and are relations in @xmath609 . the unit in @xmath612 is given by the function @xmath548 such that @xmath620 the elements @xmath621 are uniquely defined by @xmath622 for elements of the pbw - basis of @xmath609 , the first relation in becomes @xmath623 where we use the notation @xmath624,\quad { \langlen\rangle } ! = { \langle1\rangle}{\langle2\rangle}\dots { \langlen\rangle},\quad { \mathchoice { { \mbox{\footnotesize$\displaystyle \genfrac{\langle}{\rangle}{0pt}{}{m}{n}$}}}{{\genfrac{\langle}{\rangle}{0pt}{}{m}{n } } } { { \genfrac{\langle}{\rangle}{0pt}{}{m}{n}}}{{\genfrac{\langle}{\rangle}{0pt}{}{m}{n}}}}={\mbox{\footnotesize$\displaystyle\frac{{\langlem\rangle}!}{{\langlen\rangle}!\,{\langlem - n\rangle}!}$}}.\ ] ] we then check that the elements @xmath625 with @xmath165 and @xmath626 constitute a basis in @xmath612 and evaluate on the basis elements of @xmath609 as @xmath627 the easiest way to see that holds is to use to calculate @xmath628 and @xmath629 by induction on @xmath466 and then calculate @xmath630 using again , with @xmath631 and @xmath632 . next , we must show that @xmath625 are linearly independent for @xmath165 and @xmath633@xmath634 . possible linear dependences are @xmath635 with some @xmath636 , that is , @xmath637 for all @xmath638 and @xmath639 . using , we obtain the system of @xmath640 linear equations @xmath641{\sum_{i=0}^{p-1}\sum_{j=0}^{4p-1 } \delta_{mi}{\mbox{\footnotesize$\displaystyle\frac{{\langlei\rangle}!}{({\mathfrak{q}}-{\mathfrak{q}}^{-1})^i}$}}}\,{\mathfrak{q}}^{-(j+2i)n/2-ij -i(i-1 ) } \lambda_{ij}={}\\ { } = { \mbox{\footnotesize$\displaystyle\frac{{\langlem\rangle } ! } { ( { \mathfrak{q}}-{\mathfrak{q}}^{-1})^m}$}}\ , { \mathfrak{q}}^{-mn -m(m-1)}\sum_{j=0}^{4p-1 } { \mathfrak{q}}^{-\frac{1}{2 } j(n+2 m ) } \lambda_{mj}=0 \end{gathered}\ ] ] for the @xmath640 variables @xmath642 . the system decomposes into @xmath28 independent systems of @xmath643 linear equations @xmath644 for @xmath643 variables @xmath645 , @xmath626 ( with @xmath646 fixed ) , where @xmath647 . the determinant of the matrix @xmath648 is the vandermonde determinant , which is nonzero because no two numbers among @xmath649 coincide . with established , we verify , , and . next , to verify , we write for @xmath650 and @xmath651 as the respective relations @xmath652 valid for all @xmath653 . the following formulas are obtained by direct calculation using : @xmath654 these relations and imply , which finishes the proof . as any drinfeld double , @xmath312 is a quasitriangular hopf algebra , with the universal @xmath309-matrix given by @xmath655 where @xmath656 are elements of a basis in @xmath609 and @xmath657 are elements of the dual basis , @xmath658 [ lemma : r ] for @xmath312 constructed in * [ subsec : double ] * , the dual basis is expressed in terms of the generators @xmath303 and @xmath618 as @xmath659!}$}}\,{\mathfrak{q}}^{i(i-1)/2 } { \mbox{\footnotesize$\displaystyle\frac{1}{4p}$}}\sum_{r=0}^{4p-1 } { \mathfrak{q}}^{i(j+r)+rj/2}{\phi}^i{\kappa}^r,\ ] ] and therefore the @xmath309-matrix is given by @xmath660!}$}}\,{\mathfrak{q}}^{m(m-1)/2+m(i - j)-ij/2 } { e}^m{k}^{i}\otimes{\phi}^m{\kappa}^{-j}.\ ] ] by a direct calculation using , we verify that eqs . are satisfied with @xmath661 given by . let @xmath343 be an integer @xmath120 and @xmath30 . the verma module @xmath201 has the basis @xmath662 where @xmath663 correspond to the submodule @xmath664 and @xmath665 correspond to the quotient module @xmath183 in @xmath666 with the @xmath2-action given by @xmath667[s - k]{\mathsf{x}}_{k-1 } , & \quad & 0{\,{\leqslant}\,}k{\,{\leqslant}\,}s-1 \quad(\text{with}\quad{\mathsf{x}}_{-1}\equiv0 ) , \kern-60pt\end{aligned}\ ] ] @xmath668[p - s - n]{\mathsf{a}}_{n-1 } , \quad 0{\,{\leqslant}\,}n{\,{\leqslant}\,}p - s-1\quad(\text{with}\quad{\mathsf{a}}_{-1}\equiv0)\notag\\ \intertext{and } f{\mathsf{x}}_k&= \begin{cases } { \mathsf{x}}_{k+1 } , & 0{\,{\leqslant}\,}k{\,{\leqslant}\,}s-2,\\ { \mathsf{a}}_0 , & k = s-1,\\ \end{cases } \\ f{\mathsf{a}}_n&={\mathsf{a}}_{n+1 } , \quad 0{\,{\leqslant}\,}n{\,{\leqslant}\,}p - s-1 \quad(\text{with}\quad{\mathsf{a}}_{p - s}\equiv0).\notag\end{aligned}\ ] ] in addition , there are verma modules @xmath669 . the contragredient verma module @xmath670 is defined in the basis by the same formulas except and , replaced by the respective formulas @xmath671[s - k]{\mathsf{x}}_{k-1 } , & 1{\,{\leqslant}\,}k{\,{\leqslant}\,}s-1,\\ \end{cases } \\ f{\mathsf{x}}_k&={\mathsf{x}}_{k+1 } , \quad 0{\,{\leqslant}\,}k{\,{\leqslant}\,}s-1 \quad(\text{with}\quad{\mathsf{x}}_{s}\equiv0).\end{aligned}\ ] ] the module @xmath672 , @xmath120 , is the projective module whose irreducible quotient is given by @xmath13 . the modules @xmath672 appeared in the literature several times , see @xcite . in explicitly describing their structure , we follow @xcite most closely . let @xmath343 be an integer @xmath120 . the projective module @xmath206 has the basis @xmath674 where @xmath675 is the basis corresponding to the top module in , + @xmath676 to the bottom , @xmath677 to the left , and @xmath678 to the right module , with the @xmath2-action given by @xmath679[p - s - k]{\mathsf{x}}^{(+,s)}_{k-1 } , & \quad 0{\,{\leqslant}\,}k&{\,{\leqslant}\,}p - s-1 \quad(\text{with}\quad{\mathsf{x}}^{(+,s)}_{-1}\equiv0 ) , \kern-60pt\end{aligned}\ ] ] @xmath680[p - s - k]{\mathsf{y}}^{(+,s)}_{k-1 } , & 1{\,{\leqslant}\,}k{\,{\leqslant}\,}p - s-1,\\ { \mathsf{a}}^{(+,s)}_{s-1 } , & k=0,\\ \end{cases } \\ e{\mathsf{a}}^{(+,s)}_n&=[n][s - n]{\mathsf{a}}^{(+,s)}_{n-1 } , \quad 0{\,{\leqslant}\,}n{\,{\leqslant}\,}s-1\quad(\text{with}\quad{\mathsf{a}}^{(+,s)}_{-1}\equiv0),\\ e{\mathsf{b}}^{(+,s)}_n&= \begin{cases } [ n][s - n]{\mathsf{b}}^{(+,s)}_{n-1}+{\mathsf{a}}^{(+,s)}_{n-1 } , & 1{\,{\leqslant}\,}n{\,{\leqslant}\,}s-1,\\ { \mathsf{x}}^{(+,s)}_{p - s-1 } , & n=0,\\ \end{cases } \\ \intertext{and } f{\mathsf{x}}^{(+,s)}_k&= \begin{cases } { \mathsf{x}}^{(+,s)}_{k+1 } , & 0{\,{\leqslant}\,}k{\,{\leqslant}\,}p - s-2,\\ { \mathsf{a}}^{(+,s)}_0 , & k = p - s-1,\\ \end{cases } \\ f{\mathsf{y}}^{(+,s)}_k&={\mathsf{y}}^{(+,s)}_{k+1 } , \quad 0{\,{\leqslant}\,}k{\,{\leqslant}\,}p - s-1 \quad(\text{with}\quad{\mathsf{y}}^{(+,s)}_{p - s}\equiv0),\\ f{\mathsf{a}}^{(+,s)}_n&={\mathsf{a}}^{(+,s)}_{n+1 } , \quad 0{\,{\leqslant}\,}n{\,{\leqslant}\,}s-1 \quad(\text{with}\quad{\mathsf{a}}^{(+,s)}_s\equiv0),\\ f{\mathsf{b}}^{(+,s)}_n&= \begin{cases } { \mathsf{b}}^{(+,s)}_{n+1 } , & 0{\,{\leqslant}\,}n{\,{\leqslant}\,}s-2,\\ { \mathsf{y}}^{(+,s)}_0 , & n = s-1 . \end{cases}\end{aligned}\ ] ] let @xmath343 be an integer @xmath120 . the projective module @xmath425 has the basis @xmath682 where @xmath683 is the basis corresponding to the top module in , + @xmath684 to the bottom , @xmath685 to the left , and @xmath686 to the right module , with the @xmath2-action given by @xmath687[p - s - k]{\mathsf{x}}^{(-,s)}_{k-1 } , & \quad 0{\,{\leqslant}\,}k&{\,{\leqslant}\,}p - s-1\quad(\text{with}\quad{\mathsf{x}}^{(-,s)}_{-1}\equiv0 ) , \kern-60pt\end{aligned}\ ] ] @xmath688[p - s - k]{\mathsf{y}}^{(-,s)}_{k-1}+{\mathsf{x}}^{(-,s)}_{k-1 } , & 1{\,{\leqslant}\,}k{\,{\leqslant}\,}p - s-1,\\ { \mathsf{a}}^{(-,s)}_{s-1 } , & k=0,\\ \end{cases } \\ e{\mathsf{a}}^{(-,s)}_n&=[n][s - n]{\mathsf{a}}^{(-,s)}_{n-1 } , \quad 0{\,{\leqslant}\,}n{\,{\leqslant}\,}s-1\quad(\text{with}\quad { \mathsf{a}}^{(-,s)}_{-1}\equiv0),\\ e{\mathsf{b}}^{(-,s)}_n&= \begin{cases } [ n][s - n]{\mathsf{b}}^{(-,s)}_{n-1 } , & 1{\,{\leqslant}\,}n{\,{\leqslant}\,}s-1,\\ { \mathsf{x}}^{(-,s)}_{p - s-1 } , & n=0,\\ \end{cases } \\ \intertext{and } f{\mathsf{x}}^{(-,s)}_k&={\mathsf{x}}^{(-,s)}_{k+1 } , \quad 0{\,{\leqslant}\,}k{\,{\leqslant}\,}p - s-1 \quad(\text{with}\quad{\mathsf{x}}^{(-,s)}_{p - s}\equiv0),\\ f{\mathsf{y}}^{(-,s)}_k&= \begin{cases } { \mathsf{y}}^{(-,s)}_{k+1 } , & 0{\,{\leqslant}\,}k{\,{\leqslant}\,}p - s-2,\\ { \mathsf{b}}^{(-,s)}_0 , & k = p - s-1,\\ \end{cases } \\ f{\mathsf{a}}^{(-,s)}_n&= \begin{cases } { \mathsf{a}}^{(-,s)}_{n+1 } , & 0{\,{\leqslant}\,}n{\,{\leqslant}\,}s-2,\\ { \mathsf{x}}^{(-,s)}_0 , & n = s-1 , \end{cases } \\ f{\mathsf{b}}^{(-,s)}_n&={\mathsf{b}}^{(-,s)}_{n+1 } , \quad 0{\,{\leqslant}\,}n{\,{\leqslant}\,}s-1 \quad(\text{with}\quad{\mathsf{b}}^{(-,s)}_s\equiv0).\end{aligned}\ ] ] to explicitly construct the canonical central elements in * [ prop - center ] * in terms of the @xmath2 generators , we use the standard formulas in ( * ? ? v.2 ) ( also cf . @xcite ; we are somewhat more explicit about the representation - theory side , based on the analysis in * [ the - center ] * ) . we first introduce projectors @xmath689 and @xmath690 on the direct sums of the eigenspaces of @xmath161 appearing in the respective representations @xmath190 and @xmath192 for @xmath120 , eqs . and . these projectors are @xmath691 it follows that @xmath692 second , we recall polynomial relation for the casimir element and define the polynomials @xmath693 where we recall that @xmath694 , with @xmath695 for @xmath294 . [ prop - center - explicit ] the canonical central elements @xmath411 , @xmath397 , and @xmath696 , @xmath120 , are explicitly given as follows . the elements in the radical of @xmath10 are @xmath697 where @xmath698 the canonical central idempotents are given by @xmath699 where we formally set @xmath700 . first , @xmath701 acts by zero on @xmath702 and @xmath703 . we next consider its action on @xmath349 for @xmath120 . it follows from * [ proj - mod - base ] * that the casimir element acts on the basis of @xmath206 as @xmath704 for all @xmath185 . clearly , @xmath705 annihilates the entire @xmath206 , and therefore @xmath701 acts by zero on each @xmath349 with @xmath706 . on the other hand , for @xmath707 , we have @xmath708 similar formulas describe the action of the casimir element on the module @xmath425 . it thus follows that @xmath709 sends the quotient of the bimodule @xmath710 in , i.e. , @xmath711 , into the subbimodule @xmath711 at the bottom of @xmath710 . therefore , @xmath712 . to obtain @xmath713 and @xmath714 , we multiply @xmath709 with the respective operators projecting on the direct sums of the eigenspaces of @xmath161 occurring in @xmath190 and @xmath192 . this gives ( the reader may verify independently that although the projectors @xmath715 are not central , their products with @xmath709 are ) . the normalization in is chosen such that we have @xmath716 . to obtain the idempotents @xmath717 , we note that @xmath718 annihilates all @xmath349 for @xmath706 , while on @xmath710 , we have @xmath719 , @xmath720 , @xmath721 , and furthermore , by taylor expanding the polynomial , @xmath722 with higher - order terms in @xmath723 annihilating @xmath724 . similar formulas hold for the action on @xmath425 . therefore , @xmath710 is the root space of @xmath725 with eigenvalue @xmath259 , and the second term in is precisely the subtraction of the nondiagonal part . 1 . [ item : w ] we note that @xmath726 . this follows because @xmath727 . 2 . for any polynomial @xmath728 , decomposition takes the form @xmath729 for example , implies that for @xmath730 defined in * [ sec : casimir ] * , we have @xmath731 using and expressions through the chebyshev polynomials in * [ prop : quotient ] * , we recover the eigenmatrix @xmath732 of the fusion algebra . this eigenmatrix was obtained in @xcite by different means , from the matrix of the modular @xmath55-transformation on @xmath11-characters . the eigenmatrix relates the preferred basis ( the basis of irreducible representations ) and the basis of idempotents and nilpotents in the fusion algebra . specifically , if we order the irreducible representations as @xmath733 and the idempotents and nilpotents that form a basis of @xmath734 as @xmath735 then the eigenmatrix @xmath736 is defined as @xmath737 the calculation of the entries of @xmath736 via is remarkably simple : for example , with @xmath738 taken as @xmath739 ( see * [ prop : quotient ] * ) , we have @xmath740 in accordance with . evaluating the other case in similarly and taking the derivatives , we obtain the eigenmatrix @xmath741 with the @xmath742 blocks @xcite corrects a misprint in @xcite , where @xmath743 occurred in a wrong matrix entry . ] @xmath744 0\;&-{\mbox{\footnotesize$\displaystyle\frac{2\lambda_j}{p}$}}\sin{\mbox{\footnotesize$\displaystyle\frac{j\pi}{p}$ } } \end{pmatrix}\ ! , \\ p_{s,0}&= \begin{pmatrix } s\ ; & ( -1)^{s+1}s\\[2pt ] p{-}s\ ; & ( -1)^{s+1}(p{-}s ) \end{pmatrix}\ ! , \end{alignedat}\\ p_{s , j}= ( -1)^s \begin{pmatrix } -{\mbox{\footnotesize$\displaystyle\frac{\sin\frac{sj\pi}{p}}{\sin\frac{j\pi}{p}}$ } } & { \mbox{\footnotesize$\displaystyle\frac{2\lambda_j}{p^2}$ } } \bigl(-s\cos{\mbox{\footnotesize$\displaystyle\frac{sj\pi}{p}$}}\sin{\mbox{\footnotesize$\displaystyle\frac{j\pi}{p}$ } } + \sin{\mbox{\footnotesize$\displaystyle\frac{sj\pi}{p}$}}\cos{\mbox{\footnotesize$\displaystyle\frac{j\pi}{p}$}}\bigr)\\[12pt ] { \mbox{\footnotesize$\displaystyle\frac{\sin\frac{sj\pi}{p}}{\sin\frac{j\pi}{p}}$ } } & { \mbox{\footnotesize$\displaystyle\frac{2\lambda_j}{p^2}$}}\bigl ( -(p{-}s)\cos{\mbox{\footnotesize$\displaystyle\frac{sj\pi}{p}$}}\sin{\mbox{\footnotesize$\displaystyle\frac{j\pi}{p}$ } } -\sin{\mbox{\footnotesize$\displaystyle\frac{sj\pi}{p}$}}\cos{\mbox{\footnotesize$\displaystyle\frac{j\pi}{p}$}}\bigr ) \end{pmatrix}\end{gathered}\ ] ] for @xmath745 , where , for the sake of comparison , we isolated the factor @xmath746 ^ 3\sin\frac{\pi}{p}}$ } } = { \mbox{\small$\displaystyle\frac{p^2\,\bigl(\sin\frac{\pi}{p}\bigr)^2 } { \bigl(\sin\frac{j\pi}{p}\bigr)^3}$}}\ ] ] whereby the normalization of each nilpotent element , and hence of each even column of @xmath732 starting with the fourth , differs from the normalization chosen in @xcite ( both are arbitrary because the nilpotents can not be canonically normalized ) . we derive identity from the fusion algebra realized on the central elements @xmath747 . in view of * [ lemma : dr - hom ] * , the central elements @xmath748 in with @xmath30 , @xmath216 satisfy the algebra@xmath749 where @xmath750 we now equate the coefficients at the respective pbw - basis elements in both sides of . because of , it suffices to do this for the algebra relation for @xmath751 . writing it as in , we have @xmath752{\sum_{\substack{s''= 2p - s - s ' + 1\\ \mathrm{step}=2}}^{p - 1 } } ( 2{\boldsymbol{\chi}}^{+}(s '' ) + 2{\boldsymbol{\chi}}^{-}(p - s'')).\end{gathered}\ ] ] we first calculate the right - hand side . simple manipulations with @xmath5-binomial coefficients show that @xmath753{\sum_{n=0}^{p-1}\sum_{m=0}^{p-1 } } ( { \mathfrak{q}}-{\mathfrak{q}}^{-1})^{2 m } { \mathfrak{q}}^{-(m+1)(m+s-1 - 2n)}\\ * { } \times{{{\mathchoice { { \mbox{\footnotesize$\displaystyle \genfrac{[}{]}{0pt}{}{s+m - n-1}{m}$}}}{{\genfrac{[}{]}{0pt}{}{s+m - n-1}{m } } } { { \genfrac{[}{]}{0pt}{}{s+m - n-1}{m}}}{{\genfrac{[}{]}{0pt}{}{s+m - n-1}{m}}}}}_*}{\mathchoice { { \mbox{\footnotesize$\displaystyle \genfrac{[}{]}{0pt}{}{n}{m}$}}}{{\genfrac{[}{]}{0pt}{}{n}{m } } } { { \genfrac{[}{]}{0pt}{}{n}{m}}}{{\genfrac{[}{]}{0pt}{}{n}{m } } } } e^m f^m k^{s-1 - 2n + m},\end{gathered}\ ] ] where @xmath754}{0pt}{}{m}{n}$}}}{{\genfrac{[}{]}{0pt}{}{m}{n } } } { { \genfrac{[}{]}{0pt}{}{m}{n}}}{{\genfrac{[}{]}{0pt}{}{m}{n}}}}}_*}= \begin{cases } 0 , & n<0,\\ { \mbox{\footnotesize$\displaystyle\frac{[m - n+1]\dots[m]}{[n]!}$}}&\text{otherwise } , \end{cases}\ ] ] which leads to @xmath755}{0pt}{}{s + s ' - 2 - \ell - n + m}{m}$}}}{{\genfrac{[}{]}{0pt}{}{s + s ' - 2 - \ell - n + m}{m } } } { { \genfrac{[}{]}{0pt}{}{s + s ' - 2 - \ell - n + m}{m}}}{{\genfrac{[}{]}{0pt}{}{s + s ' - 2 - \ell - n + m}{m } } } } { \mathchoice { { \mbox{\footnotesize$\displaystyle \genfrac{[}{]}{0pt}{}{n - \ell}{m}$}}}{{\genfrac{[}{]}{0pt}{}{n - \ell}{m } } } { { \genfrac{[}{]}{0pt}{}{n - \ell}{m}}}{{\genfrac{[}{]}{0pt}{}{n - \ell}{m } } } } e^m f^m k^{s + s ' - 2 - 2n + m}.\end{gathered}\ ] ] changing the order of summations , using that the @xmath5-binomial coefficients vanish in the cases specified in , and summing over even and odd @xmath646 separately , we have @xmath756}{0pt}{}{s + s ' - 2 - \ell - n + \frac{m}{2}}{m}$}}}{{\genfrac{[}{]}{0pt}{}{s + s ' - 2 - \ell - n + \frac{m}{2}}{m } } } { { \genfrac{[}{]}{0pt}{}{s + s ' - 2 - \ell - n + \frac{m}{2}}{m}}}{{\genfrac{[}{]}{0pt}{}{s + s ' - 2 - \ell - n + \frac{m}{2}}{m } } } } { \mathchoice { { \mbox{\footnotesize$\displaystyle \genfrac{[}{]}{0pt}{}{n + \frac{m}{2 } - \ell}{m}$}}}{{\genfrac{[}{]}{0pt}{}{n + \frac{m}{2 } - \ell}{m } } } { { \genfrac{[}{]}{0pt}{}{n + \frac{m}{2 } - \ell}{m}}}{{\genfrac{[}{]}{0pt}{}{n + \frac{m}{2 } - \ell}{m } } } } e^m f^m k^{s + s ' - 2 - 2n}+{}}\\ { } + \smash[b ] { ( -1)^{s+s'}\sum_{\substack{m=1\\ \text{odd}}}^{p - 1 } } \sum_{n=0}^{2p - 1 } \sum_{\ell=0}^{\min(n+\frac{m-1}{2 } , s - 1 , s ' - 1 ) } ( { \mathfrak{q}}- { \mathfrak{q}}^{-1})^{2m}{\mathfrak{q}}^{-(m + 1)(s + s ' - 2n - 1)}\\ * { } \times { \mathchoice { { \mbox{\footnotesize$\displaystyle \genfrac{[}{]}{0pt}{}{s + s ' - 2 - \ell - n + \frac{m+1}{2}}{m}$}}}{{\genfrac{[}{]}{0pt}{}{s + s ' - 2 - \ell - n + \frac{m+1}{2}}{m } } } { { \genfrac{[}{]}{0pt}{}{s + s ' - 2 - \ell - n + \frac{m+1}{2}}{m}}}{{\genfrac{[}{]}{0pt}{}{s + s ' - 2 - \ell - n + \frac{m+1}{2}}{m } } } } { \mathchoice { { \mbox{\footnotesize$\displaystyle \genfrac{[}{]}{0pt}{}{n + \frac{m-1}{2 } - \ell}{m}$}}}{{\genfrac{[}{]}{0pt}{}{n + \frac{m-1}{2 } - \ell}{m } } } { { \genfrac{[}{]}{0pt}{}{n + \frac{m-1}{2 } - \ell}{m}}}{{\genfrac{[}{]}{0pt}{}{n + \frac{m-1}{2 } - \ell}{m } } } } e^m f^m k^{s + s ' - 2n - 1}.\end{gathered}\ ] ] next , in the left - hand side of , we use that @xmath757 are central and readily calculate @xmath758{(-1)^{s+1}\sum_{n=0}^{s-1 } \sum_{m=0}^{n } } ( { \mathfrak{q}}-{\mathfrak{q}}^{-1})^{2 m } { \mathfrak{q}}^{-(m+1)(m+s-1 - 2n)}\\ * \shoveright{{}\times{\mathchoice { { \mbox{\footnotesize$\displaystyle \genfrac{[}{]}{0pt}{}{s - n+m-1}{m}$}}}{{\genfrac{[}{]}{0pt}{}{s - n+m-1}{m } } } { { \genfrac{[}{]}{0pt}{}{s - n+m-1}{m}}}{{\genfrac{[}{]}{0pt}{}{s - n+m-1}{m}}}}{\mathchoice { { \mbox{\footnotesize$\displaystyle \genfrac{[}{]}{0pt}{}{n}{m}$}}}{{\genfrac{[}{]}{0pt}{}{n}{m } } } { { \genfrac{[}{]}{0pt}{}{n}{m}}}{{\genfrac{[}{]}{0pt}{}{n}{m } } } } e^m { \boldsymbol{\chi}}^{+}(s ' ) f^m k^{s - 1 - 2n + m}={}}\\ \shoveleft { { } = { ( -1)^{s+s'}\sum_{m=0}^{p - 1 } \sum_{n'=0}^{s ' - 1 } \sum_{n = n'}^{s + n ' - 1 } \sum_{j=0}^{p-1 } } ( { \mathfrak{q}}- { \mathfrak{q}}^{-1})^{2 m } { \mathfrak{q}}^{-m(m + s ' - 2n ' ) } { \mathfrak{q}}^{-(j + 1)(s + s ' - 2 - 2n)}}\\ * { } \times { \mathchoice { { \mbox{\footnotesize$\displaystyle \genfrac{[}{]}{0pt}{}{s { - } n { + } n ' { + } j { - } 1}{j}$}}}{{\genfrac{[}{]}{0pt}{}{s { - } n { + } n ' { + } j { - } 1}{j } } } { { \genfrac{[}{]}{0pt}{}{s { - } n { + } n ' { + } j { - } 1}{j}}}{{\genfrac{[}{]}{0pt}{}{s { - } n { + } n ' { + } j { - } 1}{j } } } } { \mathchoice { { \mbox{\footnotesize$\displaystyle \genfrac{[}{]}{0pt}{}{n { - } n'}{j}$}}}{{\genfrac{[}{]}{0pt}{}{n { - } n'}{j } } } { { \genfrac{[}{]}{0pt}{}{n { - } n'}{j}}}{{\genfrac{[}{]}{0pt}{}{n { - } n'}{j } } } } { \mathchoice { { \mbox{\footnotesize$\displaystyle \genfrac{[}{]}{0pt}{}{s ' { - } n ' { + } m { - } j { - } 1}{m { - } j}$}}}{{\genfrac{[}{]}{0pt}{}{s ' { - } n ' { + } m { - } j { - } 1}{m { - } j } } } { { \genfrac{[}{]}{0pt}{}{s ' { - } n ' { + } m { - } j { - } 1}{m { - } j}}}{{\genfrac{[}{]}{0pt}{}{s ' { - } n ' { + } m { - } j { - } 1}{m { - } j}}}}{\mathchoice { { \mbox{\footnotesize$\displaystyle \genfrac{[}{]}{0pt}{}{n'}{m { - } j}$}}}{{\genfrac{[}{]}{0pt}{}{n'}{m { - } j } } } { { \genfrac{[}{]}{0pt}{}{n'}{m { - } j}}}{{\genfrac{[}{]}{0pt}{}{n'}{m { - } j } } } } e^m f^m k^{s + s ' - 2 - 2n + m}.\end{gathered}\ ] ] changing the order of summations , using that the @xmath5-binomial coefficients vanish in the cases specified in , and summing over even and odd @xmath646 separately , we have @xmath759}{0pt}{}{s { - } n { - } \frac{m}{2}+ n ' + j { - } 1}{j}$}}}{{\genfrac{[}{]}{0pt}{}{s { - } n { - } \frac{m}{2}+ n ' + j { - } 1}{j } } } { { \genfrac{[}{]}{0pt}{}{s { - } n { - } \frac{m}{2}+ n ' + j { - } 1}{j}}}{{\genfrac{[}{]}{0pt}{}{s { - } n { - } \frac{m}{2}+ n ' + j { - } 1}{j } } } } { \mathchoice { { \mbox{\footnotesize$\displaystyle \genfrac{[}{]}{0pt}{}{n { + } \frac{m}{2 } { - } n'}{j}$}}}{{\genfrac{[}{]}{0pt}{}{n { + } \frac{m}{2 } { - } n'}{j } } } { { \genfrac{[}{]}{0pt}{}{n { + } \frac{m}{2 } { - } n'}{j}}}{{\genfrac{[}{]}{0pt}{}{n { + } \frac{m}{2 } { - } n'}{j } } } } { \mathchoice { { \mbox{\footnotesize$\displaystyle \genfrac{[}{]}{0pt}{}{s ' { - } n ' { + } m { - } j { - } 1}{m { - } j}$}}}{{\genfrac{[}{]}{0pt}{}{s ' { - } n ' { + } m { - } j { - } 1}{m { - } j } } } { { \genfrac{[}{]}{0pt}{}{s ' { - } n ' { + } m { - } j { - } 1}{m { - } j}}}{{\genfrac{[}{]}{0pt}{}{s ' { - } n ' { + } m { - } j { - } 1}{m { - } j } } } } { \mathchoice { { \mbox{\footnotesize$\displaystyle \genfrac{[}{]}{0pt}{}{n'}{m { - } j}$}}}{{\genfrac{[}{]}{0pt}{}{n'}{m { - } j } } } { { \genfrac{[}{]}{0pt}{}{n'}{m { - } j}}}{{\genfrac{[}{]}{0pt}{}{n'}{m { - } j } } } } e^m f^m k^{s + s ' - 2 - 2n}}\\ * \shoveleft{{}+(-1)^{s+s'}\sum_{\substack{m=1\\ \text{odd}}}^{p - 1 } \sum_{j=0}^{p - 1 } \sum_{n=0}^{2p - 1 } \sum_{n'=0}^{s ' - 1 } ( { \mathfrak{q}}- { \mathfrak{q}}^{-1})^{2 m } { \mathfrak{q}}^{-m(m + s ' - 2n ' ) } { \mathfrak{q}}^{-(j + 1)(s + s ' - 2 n - m - 1)}}\\ * { } \times { \mathchoice { { \mbox{\footnotesize$\displaystyle \genfrac{[}{]}{0pt}{}{s { - } n { - } \frac{m-1}{2 } { + } n ' { + } j { - } 1}{j}$}}}{{\genfrac{[}{]}{0pt}{}{s { - } n { - } \frac{m-1}{2 } { + } n ' { + } j { - } 1}{j } } } { { \genfrac{[}{]}{0pt}{}{s { - } n { - } \frac{m-1}{2 } { + } n ' { + } j { - } 1}{j}}}{{\genfrac{[}{]}{0pt}{}{s { - } n { - } \frac{m-1}{2 } { + } n ' { + } j { - } 1}{j } } } } { \mathchoice { { \mbox{\footnotesize$\displaystyle \genfrac{[}{]}{0pt}{}{n { + } \frac{m-1}{2 } { - } n'}{j}$}}}{{\genfrac{[}{]}{0pt}{}{n { + } \frac{m-1}{2 } { - } n'}{j } } } { { \genfrac{[}{]}{0pt}{}{n { + } \frac{m-1}{2 } { - } n'}{j}}}{{\genfrac{[}{]}{0pt}{}{n { + } \frac{m-1}{2 } { - } n'}{j } } } } { \mathchoice { { \mbox{\footnotesize$\displaystyle \genfrac{[}{]}{0pt}{}{s ' { - } n ' { + } m { - } j { - } 1}{m { - } j}$}}}{{\genfrac{[}{]}{0pt}{}{s ' { - } n ' { + } m { - } j { - } 1}{m { - } j } } } { { \genfrac{[}{]}{0pt}{}{s ' { - } n ' { + } m { - } j { - } 1}{m { - } j}}}{{\genfrac{[}{]}{0pt}{}{s ' { - } n ' { + } m { - } j { - } 1}{m { - } j } } } } { \mathchoice { { \mbox{\footnotesize$\displaystyle \genfrac{[}{]}{0pt}{}{n'}{m { - } j}$}}}{{\genfrac{[}{]}{0pt}{}{n'}{m { - } j } } } { { \genfrac{[}{]}{0pt}{}{n'}{m { - } j}}}{{\genfrac{[}{]}{0pt}{}{n'}{m { - } j } } } } e^m f^m k^{s + s ' - 2n - 1}.\end{gathered}\ ] ] equating the respective coefficients at the pbw - basis elements in and , we obtain @xmath760}{0pt}{}{n - i}{j}$}}}{{\genfrac{[}{]}{0pt}{}{n - i}{j } } } { { \genfrac{[}{]}{0pt}{}{n - i}{j}}}{{\genfrac{[}{]}{0pt}{}{n - i}{j } } } } { \mathchoice { { \mbox{\footnotesize$\displaystyle \genfrac{[}{]}{0pt}{}{i}{m - j}$}}}{{\genfrac{[}{]}{0pt}{}{i}{m - j } } } { { \genfrac{[}{]}{0pt}{}{i}{m - j}}}{{\genfrac{[}{]}{0pt}{}{i}{m - j } } } } { \mathchoice { { \mbox{\footnotesize$\displaystyle \genfrac{[}{]}{0pt}{}{i + j + s - 1 - n}{j}$}}}{{\genfrac{[}{]}{0pt}{}{i + j + s - 1 - n}{j } } } { { \genfrac{[}{]}{0pt}{}{i + j + s - 1 - n}{j}}}{{\genfrac{[}{]}{0pt}{}{i + j + s - 1 - n}{j } } } } { \mathchoice { { \mbox{\footnotesize$\displaystyle \genfrac{[}{]}{0pt}{}{m - i - j - 1 + s'}{m - j}$}}}{{\genfrac{[}{]}{0pt}{}{m - i - j - 1 + s'}{m - j } } } { { \genfrac{[}{]}{0pt}{}{m - i - j - 1 + s'}{m - j}}}{{\genfrac{[}{]}{0pt}{}{m - i - j - 1 + s'}{m - j}}}}={}\\ { } = { \mathfrak{q}}^{m(2 n + 1 - s ) } \sum_{\ell=0}^{\min(s - 1 , s ' - 1 ) } { \mathchoice { { \mbox{\footnotesize$\displaystyle \genfrac{[}{]}{0pt}{}{n - \ell}{m}$}}}{{\genfrac{[}{]}{0pt}{}{n - \ell}{m } } } { { \genfrac{[}{]}{0pt}{}{n - \ell}{m}}}{{\genfrac{[}{]}{0pt}{}{n - \ell}{m}}}}{\mathchoice { { \mbox{\footnotesize$\displaystyle \genfrac{[}{]}{0pt}{}{m + s + s ' - 2 - \ell - n}{m}$}}}{{\genfrac{[}{]}{0pt}{}{m + s + s ' - 2 - \ell - n}{m } } } { { \genfrac{[}{]}{0pt}{}{m + s + s ' - 2 - \ell - n}{m}}}{{\genfrac{[}{]}{0pt}{}{m + s + s ' - 2 - \ell - n}{m}}}},\end{gathered}\ ] ] where @xmath761 , @xmath762 , @xmath763 . because of the vanishing of @xmath5-binomial coefficients ( see ) , the summations over @xmath466 and @xmath764 in the left - hand side can be extended to @xmath765 , which gives after the shifts @xmath766 , @xmath767 . in the above derivation , @xmath17 was the @xmath3th primitive root of unity , but because @xmath28 does not explicitly enter the resultant identity and because @xmath5-binomial coefficients are ( laurent ) polynomials in @xmath5 , we conclude that is valid for all @xmath5 . 99 d. kazhdan and g. lusztig , _ tensor structures arising from affine lie algebras , _ i , j. amer . soc . 6 ( 1993 ) 905947 ; ii , j. amer . math . soc . 6 ( 1993 ) 9491011 ; iii , j. amer . 7 ( 1994 ) 335381 ; iv , j. amer . soc . 7 ( 1994 ) 383453 . v. lyubashenko , _ invariants of @xmath768-manifolds and projective representations of mapping class groups via quantum groups at roots of unity _ , commun . ( 1995 ) 467516 [ hep - th/9405167 ] ; _ modular properties of ribbon abelian categories _ , symposia gaussiana , proc . of the 2nd gauss symposium , munich , 1993 , conf . a ( berlin , new york ) , walter de gruyter , ( 1995 ) 529579 [ hep - th/9405168 ] ; _ modular transformations for tensor categories _ , j. pure applied algebra 98 ( 1995 ) 279327 . feigin , a.m. gainutdinov , a.m. semikhatov , and i.yu . tipunin , _ kazhdan lusztig correspondence for the representation category of the triplet @xmath770-algebra in logarithmic cft _ , math.qa/0512621 . reshetikhin and v.g . turaev , _ ribbon graphs and their invariants derived from quantum groups _ , comm . phys . , 127 ( 1990 ) 126 . a. lachowska , _ on the center of the small quantum group _ , math.qa/0107098 .

the @xmath0-representation @xmath1 on the center of the restricted quantum group @xmath2 at the primitive @xmath3th root of unity is shown to be equivalent to the @xmath0-representation on the _ extended _ characters of the logarithmic @xmath4 conformal field theory model . the multiplicative jordan decomposition of the @xmath2 ribbon element determines the decomposition of @xmath1 into a `` pointwise '' product of two commuting @xmath0-representations , one of which restricts to the grothendieck ring ; this restriction is equivalent to the @xmath0-representation on the @xmath4-characters , related to the fusion algebra via a nonsemisimple verlinde formula . the grothendieck ring of @xmath2 at the primitive @xmath3th root of unity is shown to coincide with the fusion algebra of the @xmath4 logarithmic conformal field theory model . as a by - product , we derive @xmath5-binomial identities implied by the fusion algebra realized in the center of @xmath2 .

in recent years , there have been a considerable number of important developments in the extension of ( classical ) information - theoretic concepts to a quantum - mechanical setting . bennett and shor @xcite have surveyed this progress in the outstanding commemorative issue 19481998 of the _ ieee transactions on information theory_. in particular , they pointed out in strict analogy to the classical case , successfully studied some fifty years ago by shannon in famous landmark work @xcite that quantum data compression allows signals from a redundant quantum source to be compressed into a bulk approaching the source s ( quantum ) entropy . bennett and shor did not , however , discuss the intriguing case which arises when the specific nature of the quantum source is _ unknown_. this , of course , corresponds to the classical question of _ universal _ coding or data compression ( see @xcite , ( * ? ? ? ii.e ) ) . we do address this interesting issue here , by investigating whether or not it is possible to extend to the quantum domain , recent ( classical ) seminal results of clarke and barron @xcite . they , in fact , derived various forms of asymptotic redundancy of universal data compression for parameterized families of probability distributions . their analyses provide a rigorous basis for the reference prior method in bayesian statistical analysis . for an extensive commentary on the results of clarke and barron , see @xcite . also see @xcite , for some recent related research , as well as a discussion of various rationales that have been employed for using the ( classical ) jeffreys prior a possible quantum counterpart of which will be of interest here for bayesian purposes , cf . let us also bring to the attention of the reader that in a brief review @xcite of @xcite , the noted statistician , i. j. good , commented that clarke and barron `` have presumably overlooked the reviewer s work '' and cited , in this regard @xcite . let us briefly recall the basic setup and the results of clarke and barron that are relevant to the analyses of our paper . clarke and barron work in a noninformative bayesian framework , in which we are given a parametric family of probability densities @xmath7 on a space @xmath8 . these probability densities generate independent identically distributed random variables @xmath9 , which , for a fixed @xmath10 , we consider as producing strings of length @xmath4 according to the probability density @xmath11 of the @xmath4-fold product of probability distributions . now suppose that nature picks a @xmath10 from @xmath12 , that is a joint density @xmath11 on the product space @xmath13 , the space of strings of length @xmath4 . on the other hand , a statistician chooses a distribution @xmath14 on @xmath15 as his best guess of @xmath11 . of course , there is a loss of information , which is measured by the total relative entropy @xmath16 , where @xmath17 is the _ kullback leibler divergence _ of @xmath18 and @xmath19 ( the _ relative entropy _ of @xmath18 with respect to @xmath19 ) . for finite @xmath4 , and for a given _ prior _ @xmath20 on @xmath12 , by a result of aitchison @xcite , the best strategy @xmath14 to minimize the average risk @xmath21 is to choose for @xmath14 the mixture density @xmath22 . this is called a _ bayes procedure _ or a _ bayes strategy_. the quantities corresponding to such a procedure that must be investigated are the _ risk _ ( _ redundancy _ ) _ of the bayes strategy _ @xmath23 and the _ bayes risk _ , the average of risks , @xmath24 . the bayes risk equals shannon s mutual information @xmath25 ( see @xcite ) . moreover , the bayes risk is bounded above by the _ minimax redundancy _ @xmath26 . in fact , by a result of gallager @xcite and davisson and leon garcia @xcite ( see @xcite for a generalization ) , for each fixed @xmath4 there is a prior @xmath27 which realizes this upper bound , i.e. , the _ maximin redundancy _ @xmath28 and the minimax redundancy are the same . such a prior @xmath27 is called _ capacity achieving _ or _ least favorable_. clarke and barron investigate the above - mentioned quantities _ asymptotically _ , that is , for @xmath4 tending to infinity . first of all , in ( * ? ? ? * ( 1.4 ) ) , ( * ? ? ? * ( 2.1b ) ) , they show that the redundancy @xmath29 of the bayes strategy is asymptotically @xmath30 as @xmath4 tends to infinity . here , @xmath31 is the @xmath32 fisher information matrix the negative of the expected value of the hessian of the logarithm of the density function . ( although the binary logarithm is usually used in the quantum coding literature , we employ the natural logarithm throughout this paper , chiefly to facilitate comparisons of our results with those of clarke and barron @xcite . ) for priors supported on a compact subset @xmath33 in the interior of the domain @xmath12 of parameters , the asymptotic minimax redundancy @xmath26 was shown to be ( * ? ? ? * ( 2.4 ) ) , @xcite , @xmath34 moreover ( * ? ? ? * ( 2.6 ) ) , it is _ jeffreys prior _ @xmath35 ( with @xmath36 a normalizing constant ; see also @xcite ) which is the unique continuous and positive prior on @xmath33 which is asymptotically least favorable , i.e. , for which the asymptotic maximin redundancy achieves the value ( [ eq:3 ] ) . in particular , asymptotically the maximin and minimax redundancies are the same . in obvious contrast to classical information theory , quantum information theory directly relies upon the fundamental principles of quantum mechanics . this is due to the fact that the basic unit of quantum computing , the quantum bit " or `` qubit , '' is typically a ( two - state ) microscopic system , possibly an atom or nuclear spin or polarized photon , the behavior of which ( e.g. entanglement , interference , superposition , stochasticity , ) can only be accurately explained using the rules of quantum theory @xcite . we refer the reader to @xcite for a comprehensive introduction to these matters ( including the subjects of quantum error - correcting codes and quantum cryptography ) . here , we shall restrict ourselves to describing , in mathematical terms , the basic notions of quantum information theory , how they pertain to data compression , and in what manner they parallel the corresponding notions from classical information theory . in quantum information theory , the role of probability densities is played by _ density matrices _ , which are , by definition , nonnegative definite hermitian matrices of unit trace , and which can be considered as operators acting on a ( finite - dimensional ) hilbert space . any probability density on a ( finite ) set @xmath37 , where the probability of @xmath38 equals @xmath39 , is representable in this framework by a diagonal matrix @xmath40 ( which is quite clearly itself , a nonnegative definite hermitian matrix with unit trace ) . given two density matrices @xmath41 and @xmath42 , the quantum counterpart of the relative entropy , that is , the _ relative entropy _ of @xmath43 with respect to @xmath44 , is @xcite ( cf . @xcite ) , @xmath45 where the logarithm of a matrix @xmath46 is defined as @xmath47 , with @xmath48 the appropriate identity matrix . ( alternatively , if @xmath49 acts diagonally on a basis @xmath50 of the hilbert space by @xmath51 , then @xmath52 acts by @xmath53 , @xmath54 . ) clearly , if @xmath41 and @xmath42 are diagonal matrices , corresponding to classical probability densities , then ( [ eq:5 ] ) reduces to the usual kullback leibler divergence . as we said earlier , our goal is to examine the possibility of extending the results of clarke and barron to quantum theory . that is , first of all we have to replace the ( classical ) probability densities @xmath55 by density matrices . we are not able to proceed in complete generality , but rather we will restrict ourselves to considering the first nontrivial case , that is , we will replace @xmath55 by @xmath56 density matrices . such matrices can be written in the form , @xmath57 where , in order to guarantee nonnegative definiteness , the points @xmath58 must lie within the unit ball ( `` bloch sphere '' @xcite ) , @xmath59 . ( the points on the bounding spherical surface , @xmath60 , corresponding to the _ pure states _ , will be shown to exhibit nongeneric behavior , see ( [ a5 ] ) and the respective comments in sec . [ s3 ] ( cf . @xcite ) . ) such @xmath56 density matrices correspond , in a one - to - one fashion , to the standard ( complex ) two - level quantum systems notably , those of spin-@xmath61 ( electrons , protons , ) and massless spin-@xmath62 particles ( photons ) . these systems carry the basic units of quantum computing , the _ quantum bits_. ( if we set @xmath63 in ( [ eq:6 ] ) , we recover a classical binomial distribution , with the probability of `` success '' , say , being @xmath64 and of `` failure '' , @xmath65 . setting either @xmath66 or @xmath67 to zero , puts us in the framework of real as opposed to complex quantum mechanics . ) the quantum analogue of the product of ( classical ) probability distributions is the _ tensor product _ of density matrices . ( again , it is easily seen that , for diagonal matrices , this reduces to the classical product . ) hence , we will replace @xmath11 by the tensor products @xmath68 , where @xmath49 is a @xmath56 density matrix ( [ eq:6 ] ) . these tensor products are @xmath69 matrices , and can be used to compute ( _ via _ the fundamental rule that the expected value of an observable is the trace of the matrix product of the observable and the density matrix ; see @xcite ) the probability of strings of quantum bits of length @xmath4 . in @xcite it was argued that the quantum fisher information matrix ( requiring due to noncommutativity the computation of symmetric logarithmic derivatives @xcite ) , one must find the symmetric logarithmic derivatives ( @xmath70 ) satisfying @xmath71 and then compute the entries of ( [ eq:8 ] ) in the form ( * ? ? ? ( 2 ) , ( 3 ) ) @xmath72 , \quad \beta , \gamma = x , y , z.\ ] ] for a well - motivated discussion of these formulas and the manner in which classical and quantum fisher information are related , see @xcite . ] ) for the density matrices ( [ eq:6 ] ) should be taken to be of the form @xmath73 the quantum counterpart of the jeffreys prior was , then , taken to be the normalized form ( dividing by @xmath74 ) of the square root of the determinant of ( [ eq:8 ] ) , that is , @xmath75 on the basis of the above - mentioned result of clarke and barron that the jeffreys prior yields the asymptotic common minimax and maximin redundancy , it was conjectured @xcite that its assumed quantum counterpart ( [ eq:9 ] ) would have similar properties , as well . to examine this possibility , ( [ eq:9 ] ) was embedded as a specific member ( @xmath76 ) of a one - parameter family of spherically - symmetric / unitarily - invariant probability densities ( i.e. , under unitary transformations of @xmath49 , the assigned probability is invariant ) , @xmath77 ( embeddings of ( [ eq:9 ] ) in other ( possibly , multiparameter ) families are , of course , possible and may be pursued in further research . in this regard , see theorem [ t15 ] in sec . [ s3 ] . ) for @xmath78 , we obtain a uniform distribution over the unit ball . ( this has been used as a prior over the two - level quantum systems , at least , in one study @xcite . ) for @xmath79 , the uniform distribution over the spherical boundary ( the locus of the pure states ) is approached . ( this is often employed as a prior , for example @xcite . ) for @xmath80 , a dirac distribution concentrated at the origin ( corresponding to the fully mixed state ) is approached . for a treatment in our setting that is analogous to that of clarke and barron , we average @xmath81 with respect to @xmath0 . doing so yields a one - parameter family of @xmath2 _ bayesian density matrices _ @xcite , @xmath82 @xmath1 , which are the analogues of the mixtures @xmath83 , and which exhibit highly interesting properties . now , still following clarke and barron , we have to compute the analogue of the risk @xmath29 , i.e. , the relative entropy @xmath84 . keeping the definition ( [ eq:5 ] ) in mind , this requires us to explicitly find the eigenvalues and eigenvectors of the matrices @xmath6 , which we do in sec . subsequently , in sec . [ s2.3 ] , we determine explicitly the relative entropy of @xmath81 with respect to @xmath6 . we do this by using identities for hypergeometric series and some combinatorics . ( it is also possible to obtain some of our results by making use of representation theory of @xmath85 . an even more general result was derived by combining these two approaches . we comment on this issue at the end of sec . [ s3 ] . ) on the basis of these results , we then address the question of finding asymptotic estimations in sec . [ s2.4 ] and [ s2.5 ] . these , in turn , form the basis of examining to what degree the results of clarke and barron are capable of extension to the quantum domain . let us ( naively ) attempt to apply the formulas of clarke and barron @xcite ( [ eq:4 ] ) and ( [ eq:3 ] ) above to the quantum context under investigation here . we do this by setting @xmath86 to 3 ( the dimensionality of the unit ball which we take as @xmath33 ) , @xmath87 to @xmath88 ( the determinant of the quantum fisher information matrix ( [ eq:8 ] ) ) , so that @xmath89 is @xmath74 , and @xmath90 to @xmath91 . then , we obtain from the expression for the asymptotic redundancy ( [ eq:4 ] ) , @xmath92 where @xmath93 , and from the expression for the asymptotic minimax redundancy ( [ eq:3 ] ) , @xmath94 we shall ( in sec . [ s3 ] ) compare these two formulas , ( [ eq:12 ] ) and ( [ eq:11 ] ) , with the results of sec . [ s2 ] and find some striking similarities and coincidences , particularly associated with the fully mixed state ( @xmath95 ) . these findings will help to support the working hypothesis of this study that there are meaningful extensions to the quantum domain of the ( commutative probabilistic ) theorems of clarke and barron . however , we find that the minimax and maximin properties of the jeffreys prior do not strictly carry over , but transfer only in an approximate sense , which is , nevertheless , still quite remarkable . in any case , we can not formally rule out the possibility that the actual global ( perhaps common ) minimax and maximin are achieved for probability distributions not belonging to the one - parameter family @xmath0 . in analogy to ( * ? ? ? 5.2 ) , the matrices @xmath6 should prove useful for the _ universal _ version of schumacher data compression @xcite . schumacher s result @xcite must be considered as the quantum analogue of shannon s noiseless coding theorem ( see e.g. ( * ? ? ? roughly , _ quantum data compression _ , as proposed by schumacher @xcite , works as follows : a ( quantum ) signal source ( sender " ) generates signal states of a quantum system @xmath96 , the ensemble of possible signals being described by a density operator @xmath97 . the signals are projected down to a dominant " subspace of @xmath96 , the rest is discarded . the information in this dominant subspace is transmitted through a ( quantum ) channel . the receiver tries to reconstruct the original signal by replacing the discarded information by some typical " state . the quality ( or _ faithfulness _ ) of a coding scheme is measured by the _ fidelity _ , which is by definition the overall probability that a signal from the signal ensemble @xmath96 that is transmitted to the receiver passes a validation test comparing it to its original ( see ( * ? ? ? what schumacher shows is that , for each @xmath98 and @xmath99 , under the above coding scheme a compression rate of @xmath100 qubits per signal is possible , where @xmath101 is the _ von neumann entropy _ of @xmath97 , @xmath102 at a fidelity of at least @xmath103 . ( thus , the von neumann entropy is the quantum analogue of the shannon entropy , which features in shannon s classical noiseless coding theorem . indeed , as is easy to see , for diagonal matrices , corresponding to classical probability densities , the right - hand side of ( [ eq:1 ] ) reduces to the shannon entropy . ) this is achieved by choosing as the dominant subspace that subspace of the quantum system @xmath96 which is the span of the eigenvectors of @xmath97 corresponding to the largest eigenvalues , with the property that the eigenvalues add up to at least @xmath104 . consequently , in a universal compression scheme , we propose to project blocks of @xmath4 signals ( qubits ) onto those `` typical '' subspaces of @xmath105-dimensional hilbert space corresponding to as many of the dominant eigenvalues of @xmath3 as it takes to exceed a sum @xmath106 . for all @xmath107 , the leading one of the @xmath108 distinct eigenvalues has multiplicity @xmath109 , and belongs to the ( @xmath109)-dimensional ( bose einstein ) symmetric subspace @xcite . ( projection onto the symmetric subspace has been proposed as a method for stabilizing quantum computations , including quantum state storage @xcite . ) for @xmath110 , the leading eigenvalue can be obtained by dividing the @xmath111-st catalan number that is , @xmath112 by @xmath113 . ( the catalan numbers `` are probably the most frequently occurring combinatorial numbers after the binomial coefficients '' @xcite . ) let us point out to the reader the quite recent important work of petz and sudr @xcite . they demonstrated that in the quantum case in contrast to the classical situation in which there is , as originally shown by chentsov @xcite , essentially only one monotone metric and , therefore , essentially only one form of the fisher information there exists an infinitude of such metrics . `` the monotonicity of the riemannian metric @xmath114 is crucial when one likes to imitate the geometrical approach of [ chentsov ] . an infinitesimal statistical distance has to be monotone under stochastic mappings . we note that the monotonicity of @xmath114 is a strengthening of the concavity of the von neumann entropy . indeed , positive definiteness of @xmath114 is equivalent to the strict concavity of the von neumann entropy and monotonicity is much more than positivity '' @xcite . the monotone metrics on the space of density matrices are given @xcite by the operator monotone functions @xmath115 , such that @xmath116 and @xmath117 . for the choice @xmath118 , one obtains the minimal metric ( of the symmetric logarithmic derivative ) , which serves as the basis of our analysis here . `` in accordance with the work of braunstein and caves , this seems to be the canonical metric of parameter estimation theory . however , expectation values of certain relevant observables are known to lead to statistical inference theory provided by the maximum entropy principle or the minimum relative entropy principle when _ a priori _ information on the state is available . the best prediction is a kind of generalized gibbs state . on the manifold of those states , the differentiation of the entropy functional yields the kubo - mori / bogoliubov metric , which is different from the metric of the symmetric logarithmic derivative . therefore , more than one privileged metric shows up in quantum mechanics . the exact clarification of this point requires and is worth further studies '' @xcite . it remains a possibility , then , that a monotone metric other than the minimal one ( which corresponds to @xmath119 , that is ( [ eq:9 ] ) ) may yield a common global asymptotic minimax and maximin redundancy , thus , fully paralleling the classical / nonquantum results of clarke and barron @xcite . we intend to investigate such a possibility , in particular , for the kubo - mori / bogoliubov metric @xcite . in this section , we implement the analytical approach described in the introduction to extending the work of clarke and barron @xcite to the realm of quantum mechanics , specifically , the two - level systems . such systems are representable by density matrices @xmath49 of the form ( [ eq:6 ] ) . a composite system of @xmath4 independent ( unentangled ) and identical two - level quantum systems is , then , represented by the @xmath4-fold tensor product @xmath81 . in theorem [ t1 ] of sec . [ s2.1 ] , we average @xmath120 with respect to the one - parameter family of probability densities @xmath0 defined in ( [ eq:10 ] ) , obtaining the bayesian density matrices @xmath6 and formulas for their @xmath121 entries . then , in theorem [ t2 ] of sec . [ s2.2 ] , we are able to explicitly determine the @xmath105 eigenvalues and eigenvectors of @xmath6 . using these results , in sec . [ s2.3 ] , we compute the relative entropy of @xmath122 with respect to @xmath6 . then , in sec . [ s2.4 ] , we obtain the asymptotics of this relative entropy for @xmath123 . in sec . [ s2.5 ] , we compute the asymptotics of the von neumann entropy ( see ( [ eq:1 ] ) ) of @xmath6 . all these results will enable us , in sec . [ s3 ] , to ascertain to what extent the results of clarke and barron could be said to carry over to the quantum domain . the @xmath4-fold tensor product @xmath81 is a @xmath69 matrix . to refer to specific rows and columns of @xmath81 , we index them by subsets of the @xmath4-element set @xmath124 . we choose to employ this notation instead of the more familiar use of binary strings , in order to have a more succinct way of writing our formulas . for convenience , we will subsequently write @xmath125 $ ] for @xmath124 . thus , @xmath81 can be written in the form @xmath126},\ ] ] where @xmath127 with @xmath128 denoting the number of elements of @xmath125 $ ] contained in both @xmath48 and @xmath129 , @xmath130 denoting the number of elements _ not _ in both @xmath48 and @xmath129 , @xmath131 denoting the number of elements not in @xmath48 but in @xmath129 , and @xmath132 denoting the number of elements in @xmath48 but not in @xmath129 . in symbols , @xmath133 \backslash ( i\cup j)},\\ n_{\notin\in}&=\v{j\backslash i},\\ n_{\in\notin}&=\v{i\backslash j}.\end{aligned}\ ] ] we consider the average @xmath6 of @xmath81 with respect to the probability density @xmath134 defined in ( [ eq:10 ] ) taken over the unit sphere @xmath135 . this average can be described explicitly as follows . [ t1 ] the average @xmath6 , @xmath136 equals the matrix @xmath137}$ ] , where @xmath138 here , @xmath139 denotes the kronecker delta , @xmath140 if @xmath141 and @xmath142 otherwise . _ it is important for later considerations to observe that because of the term @xmath143 in ( [ e4 ] ) the entry @xmath144 is nonzero if and only if the sets @xmath48 and @xmath129 have the same cardinality . if @xmath48 and @xmath129 have the same cardinality , @xmath145 say , then @xmath144 only depends on @xmath128 , the number of common elements of @xmath48 and @xmath129 , since in this case @xmath130 is expressible as @xmath146 . _ . to compute @xmath144 , we have to compute the integral @xmath147 for convenience , we treat the case that @xmath148 and @xmath149 . the other four cases are treated similarly . first , we rewrite the matrix entries @xmath150 , @xmath151 of course , in order to compute the integral ( [ e5 ] ) , we transform the cartesian coordinates into polar coordinates , @xmath152 thus , using ( [ e6 ] ) , the integral ( [ e5 ] ) is transformed into @xmath153 to evaluate this triple integral we use the following standard formulas : [ e8 ] @xmath154 for any nonnegative integers @xmath96 and @xmath155 . furthermore , we need the beta integral @xmath156 now we consider the integral over @xmath157 in ( [ e7 ] ) . using ( [ e8b ] ) and ( [ e8c ] ) , we see that each summand in ( [ e7 ] ) vanishes if @xmath131 has a parity different from @xmath132 . on the other hand , if @xmath131 has the same parity as @xmath132 , then we can evaluate the integrals over @xmath157 using ( [ e8a ] ) and ( [ e8d ] ) . discarding for a moment the terms independent of @xmath157 and @xmath158 , we have @xmath159 the last line being due to the binomial theorem . these considerations reduce ( [ e7 ] ) to @xmath160 using ( [ e8b ] ) , ( [ e8d ] ) and ( [ e9 ] ) this can be further simplified to @xmath161 next we interchange sums over @xmath162 and @xmath163 and write the sum over @xmath163 in terms of the standard hypergeometric notation @xmath164=\sum _ { k=0 } ^{\infty}\frac { \po{a_1}{k}\cdots\po{a_r}{k } } { k!\,\po{b_1}{k}\cdots\po{b_s}{k } } z^k\ , \ ] ] where the shifted factorial @xmath165 is given by @xmath166 , @xmath167 , @xmath168 . thus we can write ( [ e10 ] ) in the form @xmath169.\end{gathered}\ ] ] the @xmath170 series can be summed by means of gau @xmath170 summation ( see e.g. ( * ? ? ? * ( 1.7.6 ) ; appendix ( iii.3 ) ) ) @xmath171 = \frac { \ga(c)\,\ga(c - a - b ) } { \ga(c - a)\,\ga(c - b ) } , \ ] ] provided the series terminates or @xmath172 . applying ( [ e12 ] ) to the @xmath170 in ( [ e11 ] ) ( observe that it is terminating ) and writing the sum over @xmath163 as a hypergeometric series , the expression ( [ e11 ] ) becomes @xmath173.\end{gathered}\ ] ] another application of ( [ e12 ] ) gives @xmath174 trivially , we have @xmath175 . since ( [ e13 ] ) vanishes unless @xmath176 , we can substitute @xmath177 for @xmath131 in the arguments of the gamma functions . thus , we see that ( [ e13 ] ) equals ( [ e4 ] ) . this completes the proof of the theorem . with the explicit description of the result @xmath6 of averaging @xmath120 with respect to @xmath0 at our disposal , we now proceed to describe the eigenvalues and eigenspaces of @xmath6 . the eigenvalues are given in theorem [ t2 ] . lemma [ l4 ] gives a complete set of eigenvectors of @xmath6 . the reader should note that , though complete , this is simply a set of linearly independent eigenvectors and not a fully orthogonal set . [ t2 ] the eigenvalues of the @xmath178 matrix @xmath6 , the entries of which are given by _ ( [ e4 ] ) _ , are @xmath179 with respective multiplicities @xmath180 the theorem will follow from a sequence of lemmas . we state the lemmas first , then prove theorem [ t2 ] assuming the truth of the lemmas , and after that provide proofs of the lemmas . in the first lemma some eigenvectors of the matrix @xmath6 are described . clearly , since @xmath6 is a @xmath178 matrix , the eigenvectors are in @xmath105-dimensional space . as we did previously , we index coordinates by subsets of @xmath125 $ ] , so that a generic vector is @xmath181}$ ] . in particular , given a subset @xmath182 of @xmath125 $ ] , the symbol @xmath183 denotes the standard unit vector with a 1 in the @xmath182-th coordinate and 0 elsewhere , i.e. , @xmath184}$ ] . now let @xmath185 be integers with @xmath186 and let @xmath187 and @xmath188 be two disjoint @xmath189-element subsets @xmath187 and @xmath188 of @xmath125 $ ] . then we define the vector @xmath190 by @xmath191\backslash ( a\cup b),\ \v{y}=s - h } { \sum _ { x\subseteq a } ^{}}(-1)^{\v{x}}\ , e_{x\cup x'\cup y},\ ] ] where @xmath192 is the _ complement of @xmath8 in @xmath188 _ " by which we mean that if @xmath8 consists of the @xmath193- , @xmath194- , -largest elements of @xmath187 , @xmath195 , then @xmath192 consists of all elements of @xmath188 _ except for _ the @xmath193- , @xmath194- , -largest elements of @xmath188 . for example , let @xmath196 . then the vector @xmath197 is given by @xmath198 ( in this special case , the possible subsets @xmath8 of @xmath199 in the sum in ( [ e16 ] ) are @xmath200 , @xmath201 , @xmath202 , @xmath203 , with corresponding complements in @xmath204 being @xmath205 , @xmath206 , @xmath207 , @xmath200 , respectively , and the possible sets @xmath208 are @xmath209 , @xmath210 , @xmath211 . ) observe that all sets @xmath212 which occur as indices in ( [ e16 ] ) have the same cardinality @xmath213 . [ l3 ] let @xmath185 be integers with @xmath186 and let @xmath187 and @xmath188 be disjoint @xmath189-element subsets of @xmath125 $ ] . then @xmath190 as defined in _ ( [ e16 ] ) _ is an eigenvector of the matrix @xmath6 , the entries of which are given by _ ( [ e4 ] ) _ , for the eigenvalue @xmath214 , where @xmath214 is given by _ ( [ e14])_. we want to show that the multiplicity of @xmath214 equals the expression in ( [ e15 ] ) . of course , lemma [ l3 ] gives many more eigenvectors for @xmath214 . therefore , in order to describe a basis for the corresponding eigenspace , we have to restrict the collection of vectors in lemma [ l3 ] . we do this in the following way . fix @xmath189 , @xmath215 . let @xmath18 be a lattice path in the plane integer lattice @xmath216 , starting in @xmath217 , consisting of @xmath218 up - steps @xmath219 and @xmath189 down - steps @xmath220 , which never goes below the @xmath66-axis . figure 1 displays an example with @xmath196 and @xmath221 . clearly , the end point of @xmath18 is @xmath222 . we call a lattice path which starts in @xmath217 and never goes below the @xmath66-axes a _ ballot path_. ( this terminology is motivated by its relation to the ( two - candidate ) _ ballot problem _ , see e.g. ( * ? ? ? * ch . 1 , sec . 1 ) . an alternative term for ballot path which is often used is `` dyck path '' , see e.g. @xcite . ) we will use the abbreviation `` b.p . '' for `` ballot path '' in displayed formulas . given such a lattice path @xmath18 , label the steps from @xmath62 to @xmath4 , as is indicated in figure 1 . then define @xmath223 to be set of all labels corresponding to the first @xmath189 up - steps of @xmath18 and @xmath224 to be set of all labels corresponding to the @xmath189 down - steps of @xmath18 . in the example of figure 1 we have for the choice @xmath221 that @xmath225 and @xmath226 . thus , to each @xmath189 and @xmath213 , @xmath227 , and @xmath18 as above we can associate the vector @xmath228 . in our running example of figure 1 the vector @xmath229 would hence be @xmath197 , the vector in ( [ e17 ] ) . to have a more concise form of notation , we will write @xmath230 for @xmath228 from now on . [ l4 ] the set of vectors @xmath231 is linearly independent . the final lemma tells us how many such vectors @xmath230 there are . [ l5 ] the number of ballot paths from @xmath217 to @xmath222 is @xmath232 . the total number of all vectors in the set _ ( [ e18 ] ) _ is @xmath105 . now , let us for a moment assume that lemmas [ l3][l5 ] are already proved . then , theorem [ t2 ] follows immediately , as it turns out . proof of theorem [ t2 ] . consider the set of vectors in ( [ e18 ] ) . by lemma [ l3 ] we know that it consists of eigenvectors for the matrix @xmath6 . in addition , lemma [ l4 ] tells us that this set of vectors is linearly independent . furthermore , by lemma [ l5 ] the number of vectors in this set is exactly @xmath105 , which is the dimension of the space where all these vectors are contained . therefore , they must form a basis of the space . lemma [ l3 ] says more precisely that @xmath230 is an eigenvector for the eigenvalue @xmath214 . from what we already know , this implies that for fixed @xmath189 the set @xmath233 forms a basis for the eigenspace corresponding to @xmath214 . therefore , the dimension of the eigenspace corresponding to @xmath214 equals the number of possible numbers @xmath213 times the number of possible lattice paths @xmath18 . this is exactly @xmath234 the number of possible lattice paths @xmath18 being given by the first statement of lemma [ l5 ] . this expression equals exactly the expression ( [ e15 ] ) . thus , theorem [ t2 ] is proved . now we turn to the proofs of the lemmas . proof of lemma [ l3 ] . let @xmath185 and @xmath235 be fixed , satisfying the restrictions in the statement of the lemma . we have to show that @xmath236 restricting our attention to the @xmath48-th component , we see from the definition ( [ e16 ] ) of @xmath190 that we need to establish @xmath237\backslash ( a\cup b),\ \v{y}=s - h } { \sum _ { x\subseteq a } ^{}}\kern-1 cm z_{i , x\cup x'\cup y}\,(-1)^{\v{x}}= \begin{cases } \la_h ( -1)^{\v{u}}&\text { if $ i$ is of the form $ u\cup u'\cup v$}\\ & \text { for some $ u$ and $ v$ , $ u\subseteq a$,}\\ & \text { $ v\subseteq [ n]\backslash ( a\cup b)$ , $ \v{v}=s - h$}\\ 0&\text { otherwise.}\end{cases}\ ] ] we prove ( [ e19 ] ) by a case by case analysis . the first two cases cover the case otherwise " in ( [ e19 ] ) , the third case treats the first alternative in ( [ e19 ] ) . . as we observed earlier , the cardinality of any set @xmath212 which occurs as index at the left - hand side of ( [ e19 ] ) equals @xmath213 . the cardinality of @xmath48 however is different from @xmath213 . as we observed in the remark after theorem [ t1 ] , this implies that any coefficient @xmath238 on the left - hand side vanishes . thus , ( [ e19 ] ) is proved in this case . . now the sum on the left - hand side of ( [ e19 ] ) contains nonzero contributions . we have to show that they cancel each other . we do this by grouping summands in pairs , the sum of each pair being 0 . consider a set @xmath212 which occurs as index at the left - hand side of ( [ e19 ] ) . let @xmath239 be minimal such that 1 . either : the @xmath239-th largest element of @xmath187 and the @xmath239-th largest element of @xmath188 are both in @xmath48 , 2 . or : the @xmath239-th largest element of @xmath187 and the @xmath239-th largest element of @xmath188 are both not in @xmath48 . that such an @xmath239 must exist is guaranteed by our assumptions about @xmath48 . now consider @xmath8 and @xmath192 . if the @xmath239-th largest element of @xmath187 is contained in @xmath8 then the @xmath239-th largest element of @xmath188 is not contained in @xmath192 , and vice versa . define a new set @xmath240 by adding to @xmath8 the @xmath239-th largest element of @xmath187 if it is not already contained in @xmath8 , respectively by removing it from @xmath8 if it is contained in @xmath8 . then , it is easily checked that @xmath241 on the other hand , we have @xmath242 since the cardinalities of @xmath8 and @xmath240 differ by @xmath243 . both facts combined give @xmath244 hence , we have found two summands on the left - hand side of ( [ e19 ] ) which cancel each other . summarizing , this construction finds for any @xmath245 sets @xmath246 such that the corresponding summands on the left - hand side of ( [ e19 ] ) cancel each other . moreover , this construction applied to @xmath246 gives back @xmath245 . hence , what the construction does is exactly what we claimed , namely it groups the summands into pairs which contribute 0 to the whole sum . therefore the sum is 0 , which establishes ( [ e19 ] ) in this case also . . this assumption implies in particular that the cardinality of @xmath48 is @xmath213 . from the remark after the statement of theorem [ t1 ] we know that in our situation @xmath238 depends only on the number of common elements in @xmath48 and @xmath212 . thus , the left - hand side in ( [ e19 ] ) reduces to @xmath247 where @xmath248 is the number of sets @xmath212 , for some @xmath8 and @xmath208 , @xmath249 , @xmath250\backslash ( a\cup b)$ ] , @xmath251 , which have @xmath252 elements in common with @xmath48 , and which have @xmath253 elements in common with @xmath254 . clearly , we used expression ( [ e4 ] ) with @xmath255 and @xmath256 . to determine @xmath248 , note first that there are @xmath257 possible sets @xmath258 which intersect @xmath259 in exactly @xmath253 elements . next , let us assume that we already made a choice for @xmath258 . in order to determine the number of possible sets @xmath208 such that @xmath212 has @xmath252 elements in common with @xmath48 , we have to choose @xmath260 elements from @xmath261 , for which we have @xmath262 possibilities , and we have to choose @xmath263 elements from @xmath125\backslash ( i\cup a\cup b)$ ] to obtain a total number of @xmath213 elements , for which we have @xmath264 possibilities . hence , @xmath265 so it remains to evaluate the double sum ( [ e20 ] ) , using the expression ( [ e21 ] ) for @xmath248 . we start by writing the sum over @xmath162 in ( [ e20 ] ) in hypergeometric notation , @xmath266 } .\end{gathered}\ ] ] to the @xmath267 series we apply a transformation formula of thomae ( see e.g. ( * ? ? ? * ( 3.1.1 ) ) ) , @xmath268 = { \frac { ( { \textstyle -b + e } ) _ { m } } { ( { \textstyle e } ) _ { m } } } { } _ { 3 } f _ { 2 } \!\left [ \begin{matrix } { -m , b , -a + d}\\ { d , 1 + b - e - m}\end{matrix } ; { \displaystyle 1}\right ] \ ] ] where @xmath269 is a nonnegative integer . we write the resulting @xmath267 again as a sum over @xmath162 , then interchange sums over @xmath163 and @xmath162 , and write the ( now ) inner sum over @xmath163 in hypergeometric notation . thus we obtain @xmath270.\end{gathered}\ ] ] the @xmath170 series in this expression is terminating because @xmath271 is a nonpositive integer . hence , it can be summed by means of gau sum ( [ e12 ] ) . writing the remaining sum over @xmath162 in hypergeometric notation , the above expression becomes @xmath272.\ ] ] again , the @xmath170 series is terminating and so is summable by means of ( [ e12 ] ) . thus , we get @xmath273 which is exactly the expression ( [ e14 ] ) for @xmath214 times @xmath274 . this proves ( [ e19 ] ) in this case . the proof of lemma [ l3 ] is now complete . . we know from lemma [ l3 ] that @xmath230 lies in the eigenspace for the eigenvalue @xmath214 , with @xmath214 being given in ( [ e14 ] ) . the @xmath214 s , @xmath275 , are all distinct , so the corresponding eigenspaces are linearly independent . therefore it suffices to show that for any _ fixed _ @xmath189 the set of vectors @xmath233 is linearly independent . on the other hand , a vector @xmath190 lies in the space spanned by the standard unit vectors @xmath183 with @xmath276 . clearly , as @xmath213 varies , these spaces are linearly independent . therefore , it suffices to show that for any _ fixed _ @xmath189 _ and _ @xmath213 the set of vectors @xmath277 is linearly independent . so , let us fix integers @xmath189 and @xmath213 with @xmath227 , and let us suppose that there is some vanishing linear combination @xmath278 we have to establish that @xmath279 for all ballot paths @xmath18 from @xmath217 to @xmath222 . we prove this fact by induction on the set of ballot paths from @xmath217 to @xmath222 . in order to make this more precise , we need to impose a certain order on the ballot paths . given a ballot path @xmath18 from @xmath217 to @xmath222 , we define its _ front portion _ @xmath280 to be the portion of @xmath18 from the beginning up to and including @xmath18 s @xmath189-th up - step . for example , choosing @xmath221 , the front portion of the ballot path in figure 1 is the subpath from @xmath217 to @xmath281 . note that @xmath280 can be any ballot path starting in @xmath217 with @xmath189 up - steps and less than @xmath189 down - steps . we order such front portions lexicographically , in the sense that @xmath282 is before @xmath283 if and only if @xmath282 and @xmath283 agree up to some point and then @xmath282 continues with an up - step while @xmath283 continues with a down - step . now , here is what we are going to prove : fix any possible front portion @xmath284 . we shall show that @xmath279 for all @xmath18 with front portion @xmath280 equal to @xmath284 , _ given that it is already known that @xmath285 for all @xmath286 with a front portion @xmath287 that is before @xmath284 . _ clearly , by induction , this would prove @xmath279 for _ all _ ballot paths @xmath18 from @xmath217 to @xmath222 . let @xmath284 be a possible front portion , i.e. , a ballot path starting in @xmath217 with exactly @xmath189 up - steps and less than @xmath189 down - steps . as we did earlier , label the steps of @xmath284 by @xmath288 , and denote the set of labels corresponding to the down - steps of @xmath284 by @xmath289 . we write @xmath290 for @xmath291 , the number of all down - steps of @xmath284 . observe that then the total number of steps of @xmath284 is @xmath292 . now , let @xmath182 be a fixed @xmath293-element subset of @xmath294 . furthermore , let @xmath295 be a set of the form @xmath296 , where @xmath297 and @xmath298 , and such that @xmath299 . we consider the coefficient of @xmath300 in the left - hand side of ( [ e23 ] ) . to determine this coefficient , we have to determine the coefficient of @xmath300 in @xmath230 , for all @xmath18 . we may concentrate on those @xmath18 whose front portion @xmath280 is equal to or later than @xmath284 , since our induction hypothesis says that @xmath279 for all @xmath18 with @xmath280 before @xmath284 . so , let @xmath18 be a ballot path from @xmath217 to @xmath222 with front portion equal to or later than @xmath284 . we claim that the coefficient of @xmath300 in @xmath230 is zero unless the set @xmath224 of down - steps of @xmath18 is contained in @xmath295 . let the coefficient of @xmath300 in @xmath230 be nonzero . to establish the claim , we first prove that the front portion @xmath280 of @xmath18 has to equal @xmath284 . suppose that this is not the case . then the front portion of @xmath18 runs in parallel with @xmath284 for some time , say for the first @xmath301 steps , with some @xmath302 , and then @xmath284 continues with an up - step and @xmath280 continues with a down - step ( recall that @xmath280 is equal to or later than @xmath284 ) . by ( [ e16 ] ) we have @xmath303\backslash ( a_p\cup b_p),\ \v{y}=s - h } { \sum _ { x\subseteq a_p } ^{}}(-1)^{\v{x}}\ , e_{x\cup x'\cup y}.\ ] ] we are assuming that the coefficient of @xmath300 in @xmath230 is nonzero , therefore @xmath295 must be of the form @xmath304 , with @xmath245 as described in ( [ e24 ] ) . we are considering the case that the @xmath269-th step of @xmath280 is a down - step , whence @xmath305 , while the @xmath269-th step of @xmath284 is an up - step , whence @xmath306 . by definition of @xmath295 , we have @xmath307 , whence @xmath308 . summarizing so far , we have @xmath305 , @xmath308 , for some @xmath309 , and @xmath304 , for some @xmath245 as described in ( [ e24 ] ) . in particular we have @xmath310 . now recall that @xmath192 is the `` complement of @xmath8 in @xmath224 '' . this says in particular that , if @xmath269 is the @xmath311-th largest element in @xmath224 , then the @xmath311-th largest element of @xmath223 , @xmath312 say , is an element of @xmath8 , and so of @xmath295 . by construction of @xmath223 and @xmath224 , @xmath312 is smaller than @xmath269 , so in particular @xmath313 . as we already observed , there holds @xmath314 , so we have @xmath315 , i.e. , the @xmath312-th step of @xmath284 is a down - step . on the other hand , we assumed that @xmath18 and @xmath284 run in parallel for the first @xmath301 steps . since @xmath316 , the set of up - steps of @xmath18 , the @xmath312-th step of @xmath18 is an up - step . we have @xmath317 , therefore the @xmath312-th step of @xmath284 must be an up - step also . this is absurd . therefore , given that the coefficient of @xmath300 in @xmath230 is nonzero , the front portion @xmath280 of @xmath18 has to equal @xmath284 . now , let @xmath18 be a ballot path from @xmath217 to @xmath222 with front portion equal to @xmath284 , and suppose that @xmath295 has the form @xmath318 , for some @xmath245 as described in ( [ e24 ] ) . by definition of the front portion , the set @xmath223 of up - steps of @xmath18 has the property @xmath319 . since @xmath320 , these are the labels of exactly @xmath189 up - steps . since the cardinality of @xmath223 is exactly @xmath189 by definition , we must have @xmath321 . because of @xmath314 , which we already used a number of times , @xmath223 and @xmath295 are disjoint , which in particular implies that @xmath223 and @xmath8 are disjoint . however , @xmath8 is a subset of @xmath223 by definition , so @xmath8 must be empty . this in turn implies that @xmath322 . this says nothing else but that the set @xmath224 of down - steps of @xmath18 equals @xmath192 and so is contained in @xmath295 . this establishes our claim . in fact , we proved more . we saw that @xmath295 has the form @xmath323 , with @xmath324 . this implies that the coefficient of @xmath300 in @xmath230 , as given by ( [ e24 ] ) , is actually @xmath325 . comparison of coefficients of @xmath300 in ( [ e23 ] ) then gives @xmath326 for any @xmath296 , where @xmath297 and @xmath298 , and such that @xmath299 . now , we sum both sides of ( [ e25 ] ) over all such sets @xmath295 , keeping the cardinality of @xmath327 and @xmath328 fixed , say @xmath329 , enforcing @xmath330 , for a fixed @xmath162 , @xmath331 . for a fixed ballot path @xmath18 from @xmath217 to @xmath222 , with front portion @xmath284 , with @xmath332 down - steps in @xmath182 , and hence with @xmath163 down - steps in @xmath333 , there are @xmath334 such sets @xmath297 containing all the @xmath332 down - steps of @xmath18 in @xmath182 , and there are @xmath335 such sets @xmath298 containing all the @xmath163 down - steps of @xmath18 in @xmath333 . therefore , summing up ( [ e25 ] ) gives @xmath336 denoting the inner sum in ( [ e26 ] ) by @xmath337 , we see that ( [ e26 ] ) represents a non - degenerate triangular system of linear equations for @xmath338 . therefore , all the quantities @xmath338 have to equal 0 . in particular , we have @xmath339 . now , @xmath340 consists of just a single term @xmath341 , with @xmath18 being the ballot path from @xmath217 to @xmath222 , with front portion @xmath284 , and the labels of the @xmath342 down - steps besides those of @xmath284 being exactly the elements of @xmath182 . therefore , we have @xmath279 for this ballot path . the set @xmath182 was an arbitrary @xmath293-subset of @xmath294 . thus , we have proved @xmath279 for any ballot path @xmath18 from @xmath217 to @xmath222 with front portion @xmath284 . this completes our induction proof . . that the number of ballot paths from @xmath217 to @xmath222 equals @xmath232 is a classical combinatorial result ( see e.g. ( * ? ? ? * theorem 1 with @xmath343 ) ) . from this it follows that the total number of vectors in the set ( [ e18 ] ) is @xmath344 to evaluate this sum , note that the summand is invariant under the substitution @xmath345 . therefore , extending the range of summation in ( [ e27 ] ) to @xmath346 and dividing the result by @xmath347 gives the same value . so , the cardinality of the set ( [ e18 ] ) is also given by @xmath348 the reader will not have any difficulty in splitting this sum into three parts so that each part can be summed by means of the binomial theorem . ( computer algebra systems like _ maple _ or _ mathematica _ do this automatically . ) the result is exactly @xmath105 , as was claimed . in fact , theorem [ t2 ] can be generalized to a wider class of matrices . [ t6 ] let @xmath349}$ ] be the @xmath350 matrix defined by @xmath351 where @xmath128 , etc . , have the same meaning as earlier , and where @xmath352 is a function of @xmath66 which is symmetric , i.e. , @xmath353 . then , the eigenvalues of @xmath354 are @xmath355 with respective multiplicities @xmath356 independent of @xmath213 . the above proof of theorem [ t2 ] has to be adjusted only insignificantly to yield a proof of theorem [ t6 ] . in particular , the vector @xmath190 as defined in ( [ e16 ] ) is an eigenvector for @xmath357 , for any two disjoint @xmath189-element subsets @xmath187 and @xmath188 of @xmath125 $ ] , and the set ( [ e18 ] ) is a basis of eigenvectors for @xmath354 . we now apply the preceding results to compute the relative entropy @xmath359 of @xmath81 with respect to @xmath6 . utilizing the definition ( [ eq:5 ] ) of relative entropy and employing the property @xcite that @xmath360 , it is given by @xmath361 for the first term , for the entropy @xmath362 of @xmath49 , @xmath49 being given by ( [ eq:6 ] ) , we have , using spherical coordinates @xmath363 , so that @xmath364 , @xmath365 concerning the second term in ( [ e31 ] ) , we have the following theorem . [ t7 ] let @xmath366}$ ] be the matrix with entries @xmath144 given in _ ( [ e4])_. then , we have @xmath367 with @xmath214 as given in _ ( [ e14 ] ) _ , and with @xmath93 . before we move on to the proof , we note that theorem [ t7 ] gives us the following expression for the relative entropy of @xmath81 with respect to @xmath6 [ c8 ] the relative entropy @xmath359 of @xmath81 with respect to @xmath6 equals @xmath368 with @xmath214 as given in _ ( [ e14 ] ) _ , and with @xmath93 . . one way of determining the trace of a linear operator @xmath369 is to choose a basis of the vector space , @xmath370\}$ ] say , write the action of @xmath369 on the basis elements in the form @xmath371 and then form the sum @xmath372 of the `` diagonal '' coefficients , which gives exactly the trace of @xmath369 . clearly , we choose as a basis our set ( [ e18 ] ) of eigenvectors for @xmath6 . to determine the action of @xmath373 we need only to find the action of @xmath122 on the vectors in the set ( [ e18 ] ) . we claim that this action can be described as @xmath374 for any basis vector @xmath230 in ( [ e18 ] ) . to see this , consider the @xmath48-th component of @xmath375 , i.e. , the coefficient of @xmath376 in @xmath375 , @xmath377 $ ] . by the definition ( [ e16 ] ) of @xmath230 it equals @xmath378\backslash ( a_p\cup b_p),\ \v{y}=s - h } { \sum _ { x\subseteq a_p } ^{}}\kern-1 cm r_{i , x\cup x'\cup y}\,(-1)^{\v{x}},\ ] ] where @xmath150 denotes the @xmath379-entry of @xmath122 . ( recall that @xmath150 is given explicitly in ( [ e2 ] ) . ) now , it should be observed that we did a similar calculation already , namely in the proof of lemma [ l3 ] . in fact , the expression ( [ e34 ] ) is almost identical with the left - hand side of ( [ e19 ] ) . the essential difference is that @xmath144 is replaced by @xmath150 for all @xmath129 ( the nonessential difference is that @xmath235 are replaced by @xmath380 , respectively ) . therefore , we can partially rely upon what was done in the proof of lemma [ l3 ] . we distinguish between the same cases as in the proof of lemma [ l3 ] . . we do not have to worry about this case , since @xmath376 then lies in the span of vectors @xmath381 with @xmath382 , which is taken care of in ( [ e33 ] ) . . essentially the same arguments as those in case 2 in the proof of lemma [ l3 ] show that the term ( [ e34 ] ) vanishes for this choice of @xmath48 . of course , one has to use the explicit expression ( [ e2 ] ) for @xmath150 . . in case 3 in the proof of lemma [ l3 ] we observed that there are @xmath248 sets @xmath212 , for some @xmath8 and @xmath208 , @xmath383 , @xmath250\backslash ( a_p\cup b_p)$ ] , @xmath251 , which have @xmath252 elements in common with @xmath48 , and which have @xmath253 elements in common with @xmath384 , where @xmath248 is given by ( [ e21 ] ) . then , using the explicit expression ( [ e2 ] ) for @xmath150 , it is straightforward to see that the expression ( [ e34 ] ) equals @xmath385 in this case . this establishes ( [ e33 ] ) . now we are in the position to write down an expression for the trace of @xmath386 . by theorem [ t2 ] and by ( [ e33 ] ) we have @xmath387 from what was said at the beginning of this proof , in order to obtain the trace of @xmath386 , we have to form the sum of all the `` diagonal '' coefficients in ( [ e35 ] ) . using the first statement of lemma [ l5 ] and replacing @xmath388 by @xmath389 , we see that it is @xmath390 in order to see that this expression equals ( [ e30 ] ) , we have to prove @xmath391 we start with the left - hand side of ( [ e37 ] ) and write the inner sum in hypergeometric notation , thus obtaining @xmath392.\ ] ] to the @xmath170 series we apply the transformation formula ( ( * ? ? ? * ( 1.8.10 ) , terminating form ) @xmath393 = { \frac { ( { \textstyle c - a } ) _ { m } } { ( { \textstyle c } ) _ { m } } } { } _ { 2 } f _ { 1 } \!\left [ \begin{matrix } { -m , a}\\ { 1 + a - c - m}\end{matrix } ; { \displaystyle 1 - z}\right ] , \ ] ] where @xmath269 is a nonnegative integer . we write the resulting @xmath170 series again as a sum over @xmath163 . in the resulting expression we exchange sums so that the sum over @xmath162 becomes the innermost sum . thus , we obtain @xmath394 clearly , the innermost sum can be evaluated by the binomial theorem . then , we interchange sums over @xmath213 and @xmath163 . the expression that results is @xmath395 again , we can apply the binomial theorem . thus , we reduce our expression on the left - hand side of ( [ e37 ] ) to @xmath396 now , we replace @xmath397 by its binomial expansion @xmath398 , interchange sums over @xmath163 and @xmath158 , and write the ( now ) inner sum over @xmath163 in hypergeometric notation . this gives @xmath399\bigg ) .\end{gathered}\ ] ] finally , this @xmath170 series can be summed by means of gau summation ( [ e12 ] ) . simplifying , we have @xmath400 which is easily seen to equal the right - hand side in ( [ e37 ] ) . this completes the proof of the theorem . in the preceding subsection , we obtained in corollary [ c8 ] the general formula ( [ e32 ] ) for the relative entropy of @xmath122 with respect to the bayesian density matrix @xmath6 . we , now , proceed to find its asymptotics for @xmath123 . we prove the following theorem . [ t9 ] the asymptotics of the relative entropy @xmath359 of @xmath81 with respect to @xmath6 for a fixed @xmath93 with @xmath402 is given by @xmath403 in the case @xmath95 , this means that the asymptotics is given by the expression _ ( [ a3 ] ) _ in the limit @xmath404 , i.e. , by @xmath405 for any fixed @xmath98 , the @xmath406 term in _ ( [ a3 ] ) _ is uniform in @xmath107 and @xmath407 as long as @xmath408 . for @xmath409 the asymptotics is given by @xmath410 also here , the @xmath406 term is uniform in @xmath107 . _ it is instructive to observe that , although a comparison of ( [ a3 ] ) and ( [ a5 ] ) seems to suggest that the asymptotics of the relative entropy of @xmath81 with respect to @xmath6 behaves completely differently for @xmath402 and @xmath409 , the two cases are really quite compatible . in fact , letting @xmath407 tend to @xmath62 in ( [ a3 ] ) shows that ( ignoring the error term ) the asymptotic expression approaches @xmath411 for @xmath412 , @xmath413 for @xmath414 , and it approaches @xmath415 for @xmath110 . this indicates that , for @xmath409 , the order of magnitude of the relative entropy of @xmath81 with respect to @xmath6 should be larger than @xmath416 if @xmath412 , smaller than @xmath417 if @xmath414 , and exactly @xmath416 if @xmath110 . how much larger or smaller is precisely what formula ( [ a5 ] ) tells us : the order of magnitude is @xmath418 , and in the case @xmath110 the asymptotics is , in fact , @xmath419 . _ sketch of proof of theorem [ t9 ] . we have to estimate the expression ( [ e32 ] ) for large @xmath4 . clearly , it suffices to concentrate on the sum in ( [ e32 ] ) . because of @xmath420 , this sum can be also expressed as @xmath421 for @xmath409 this sum reduces to @xmath422 , @xmath423 being given by ( [ e14 ] ) . a straightforward application of stirling s formula then leads to ( [ a5 ] ) . from now on let @xmath402 . we recall that @xmath214 is given by ( [ e14 ] ) . consequently , we expand the logarithm in ( [ a6 ] ) according to the addition rule , and split the sum ( [ a6 ] ) into the corresponding parts . the individual parts can be summed by means of the binomial theorem , except for the parts which involve @xmath424 . ( to be precise , they have to be split appropriately before the binomial theorem can be applied . computer algebra systems like _ maple _ or _ mathematica _ do this automatically . ) in order to handle the terms which contain @xmath424 , we use stirling s formula @xmath425 again , after splitting , all the resulting sums can be evaluated by means of the binomial theorem , except for @xmath426 the asymptotics of this sum can now easily ( if though tediously ) be determined by making use of a taylor expansion of @xmath427 about @xmath428 ( i.e. , at @xmath429 ) with sufficiently many terms . if everything is put together , the result is ( [ a3 ] ) . the main result of this section describes the asymptotics of the von neumann entropy ( [ eq:1 ] ) of @xmath6 . in view of the explicit description of the eigenvalues of @xmath6 and their multiplicities in theorem [ t2 ] , this entropy equals @xmath430 with @xmath214 being given by ( [ e14 ] ) . [ t11 ] the asymptotics of the von neumann entropy @xmath431 of @xmath6 is given by @xmath432 where @xmath433 is the digamma function , @xmath434 sketch of proof . we have to estimate the expression ( [ eq : vonn ] ) for large @xmath4 . we proceed as in the proof of theorem [ t9 ] . first we use the property @xmath420 to rewrite the sum ( [ eq : vonn ] ) as @xmath435 next , while recalling that @xmath214 is given by ( [ e14 ] ) , we expand the logarithm in ( [ b11 ] ) according to the addition rule , and split the sum ( [ b11 ] ) into the corresponding parts . here , the individual parts can be summed by means of gau @xmath170 summation ( [ e12 ] ) , except for the parts which involve @xmath424 . ( again , to be precise , they have to be split appropriately before the gau summation can be applied , which is done automatically by computer algebra systems like _ maple _ or _ mathematica_. ) to handle the terms which contain @xmath424 , we invoke again stirling s formula ( [ eq : stirling ] ) . after splitting , all the resulting sums can be evaluated by means of gau @xmath170 summation ( [ e12 ] ) , except for @xmath436 now , to get an asymptotic estimate for this sum , as @xmath4 tends to infinity , is not as obvious as it was for ( [ a7 ] ) . the essential `` trick '' needed was kindly indicated to us by peter grabner : an asymptotic estimate ( in fact , an exact result ) for ( [ b15 ] ) with @xmath427 replaced by @xmath437 can be obtained without difficulty ( but with some amount of tedious calculation ) by starting with the sum @xmath438 evaluating it by applying gau @xmath170 summation ( [ e12 ] ) , differentiating both sides of the resulting equation with respect to @xmath439 , and by finally setting @xmath440 . finally one relates the result to ( [ b15 ] ) by using the asymptotic expansion @xmath441 . if everything is put together , the right - hand side of ( [ b2 ] ) is obtained . let us , first , compare the formula ( [ eq:4 ] ) for the asymptotic redundancy of clarke and barron to that derived here ( [ a3 ] ) for the two - level quantum systems , in terms of the one - parameter family of probability densities @xmath0 , @xmath442 , given in ( [ eq:10 ] ) . since the unit ball or bloch sphere of such systems is three - dimensional in nature , we are led to set the dimension @xmath86 of the parameter space in ( [ eq:4 ] ) to 3 . the quantum fisher information matrix @xmath31 for that case was taken to be ( [ eq:8 ] ) , while the role of the probability function @xmath90 is played by @xmath0 . under these substitutions , it was seen in the introduction that formula ( [ eq:4 ] ) reduces to ( [ eq:12 ] ) . then , we see that for @xmath443 , formulas ( [ a3 ] ) and ( [ eq:12 ] ) coincide except for the presence of the monotonically increasing ( nonclassical / quantum ) term @xmath444 ( see figure 2 for a plot of this term @xmath445 `` nats '' of information equalling one `` bit '' ) in ( [ a3 ] ) . ( this term would have to be replaced by @xmath446 that is , its limit for @xmath447 to give ( [ eq:12 ] ) . ) in particular , the order of magnitude , @xmath448 , is precisely the same in both formulas . for the particular case @xmath95 , the asymptotic formula ( [ a3 ] ) ( see ( [ a4 ] ) ) precisely coincides with ( [ eq:12 ] ) . in the case @xmath409 , however , i.e. , when we consider the boundary of the parameter space ( represented by the unit sphere ) , the situation is slightly tricky . due to the fact that the formula of clarke and barron holds only for interior points of the parameter space , we can not expect that , in general , our formula will resemble that of clarke and barron . however , if the probability density , @xmath0 , is concentrated on the boundary of the sphere , then we may disregard the interior of the sphere , and consider the boundary of the sphere as the _ true _ parameter space . this parameter space is _ two - dimensional _ and consists of interior points throughout . indeed , the probability density @xmath0 is concentrated on the boundary of the sphere if we choose @xmath449 since , as we remarked in the introduction , in the limit @xmath450 , the distribution determined by @xmath0 tends to the uniform distribution over the boundary of the sphere . let us , again , ( naively ) attempt to apply clarke and barron s formula ( [ eq:4 ] ) to that case . we parameterize the boundary of the sphere by polar coordinates @xmath451 , @xmath452 the probability density induced by @xmath0 in the limit @xmath450 then is @xmath453 , the density of the uniform distribution . using ( * ? ? ? * eq . ( 8) ) ( see footnote 2 ) , the quantum ( symmetric logarithmic derivative ) fisher information matrix turns out to be @xmath454 its determinant equalling , therefore , @xmath455 . so , setting @xmath456 and substituting @xmath453 for @xmath457 and @xmath455 for @xmath458 in ( [ eq:4 ] ) gives @xmath459 . on the other hand , our formula ( [ a5 ] ) , for @xmath449 , gives @xmath460 . so , again , the terms differ only by a constant . in particular , the order of magnitude is again the same . let us now focus our attention on the asymptotic minimax redundancy ( [ eq:3 ] ) of clarke and barron . if in ( [ eq:3 ] ) we again set @xmath86 to 3 , we obtain ( [ eq:11 ] ) , which , numerically , is @xmath461 . clarke and barron prove that this minimax expression is only attained by the ( classical ) jeffreys prior . in order to derive its quantum counterpart at least , a restricted ( to the family @xmath0 ) version we have to determine the behavior of @xmath462 for @xmath463 . by theorem [ t9 ] we know that for large @xmath4 the relative entropy @xmath359 equals @xmath464 up to an error of the order @xmath465 , which is uniform in @xmath107 and @xmath407 as long as @xmath408 for any fixed @xmath98 . let us for the moment ignore the error term . then what we have to do is to determine the minimax of the expression ( [ d6a ] ) , that is @xmath466 where @xmath467 this is an easy task . first of all , if @xmath468 then the function @xmath469 is unbounded at @xmath409 . hence , to determine the minimax , we can ignore that range of @xmath107 . if @xmath76 , then @xmath469 is maximal at @xmath409 , at which it attains the value @xmath470 . on the other hand , if @xmath471 then @xmath469 attains a maximum in the interior of the interval @xmath472 . to determine this maximum , we differentiate @xmath469 with respect to @xmath407 , to obtain @xmath473 equating this to 0 gives @xmath474 now we have to express @xmath407 in terms of @xmath107 , @xmath475 say , substitute in @xmath469 , and determine @xmath476 . however , equivalently , we can express @xmath107 in terms of @xmath407 , @xmath477 say ( as was previously done in ( [ d6d ] ) ) , substitute in @xmath469 , and determine @xmath478 . in order to do so , we differentiate @xmath479 with respect to @xmath407 , equate the result to 0 , and solve for @xmath407 . numerically , the result is @xmath480 . substituting this back into ( [ d6d ] ) , we obtain @xmath481 . the value of @xmath469 at these values of @xmath407 and @xmath107 is @xmath482 . this is smaller than that previously found ( @xmath483 ) for @xmath76 , so that particular value of @xmath107 is not of concern for the minimax , as well . in the beginning , we did ignore the error term . in fact , as is not very difficult to see , since the error term is uniform in @xmath107 and @xmath407 as long as @xmath408 for any fixed @xmath98 , it is legitimate to ignore the error term . to be precise , the asymptotic minimax is the result above , subject to an error of @xmath484 , that is , the value of ( [ d6a ] ) for @xmath480 and @xmath481 . this is @xmath485 . for @xmath76 , on the other hand , asymptotically , the maximum of the redundancy ( [ e32 ] ) ( which , by the considerations above , is ( [ d6a ] ) for @xmath409 ) equals @xmath486 . we must , therefore , conclude that in contrast to the classical case @xcite our trial candidate ( @xmath119 ) for the quantum counterpart of jeffreys prior does not exactly achieve the minimax redundancy , although the prior @xmath487 is remarkably close to @xmath119 , the hypothesized `` quantum jeffreys prior '' from @xcite . we now concern ourselves with the asymptotic _ maximin _ redundancy . clarke and barron @xcite prove that the maximin redundancy is attained asymptotically , again , by the jeffreys prior . to derive the quantum counterpart of the maximin redundancy within our analytical framework , we would have to calculate @xmath488 where @xmath489 varies over the @xmath490-dimensional convex set of @xmath2 density matrices and @xmath491 varies over all probability densities over the unit ball . as we already mentioned in the introduction , in the classical case , due to a result of aitchison @xcite , the minimum is achieved by setting @xmath489 to be the bayes estimator , i.e. , the average of all possible probability densities in the family that is considered with respect to the given probability distribution . in the quantum domain the same assertion is true . for the sake of completeness , we include the proof in the appendix . we can , thus , take the quantum analog of the bayes estimator to be the bayesian density matrix @xmath3 . that is , we set @xmath492 in ( [ d7 ] ) . let us , for the moment , restrict the possible @xmath491 s over which the maximum is to be taken to the family @xmath0 , @xmath1 . thus , we consider @xmath493 by the definition ( [ eq:5 ] ) of relative entropy , we have @xmath494 the second line being due to ( [ eq:7 ] ) . therefore , we get @xmath495 from theorem [ t11 ] , we know the asymptotics of the von neumann entropy @xmath431 . hence , we find that the expression ( [ d7b ] ) is asymptotically equal to @xmath496 we have to , first , perform the maximization required in ( [ d7a ] ) , and then determine the asymptotics of the result . due to the form of the asymptotics in ( [ d8 ] ) , we can , in fact , derive the proper result by proceeding in the reverse order . that is , we first determine the asymptotics of @xmath497 , which we did in ( [ d8 ] ) , and then we maximize the @xmath107-dependent part in ( [ d8 ] ) with respect to @xmath107 ( ignoring the error term ) . ( in figure 3 we display this @xmath107-dependent part over the range @xmath498 $ ] . ) of course , we do the latter step by equating the first derivative of the @xmath107-dependent part in ( [ d8 ] ) with respect to @xmath107 to zero and solving for @xmath107 . it turns out that this equation takes the appealingly simple form @xmath499 numerically , we find this equation to have the solution @xmath500 , at which the asymptotic maximin redundancy assumes the value @xmath501 . for @xmath76 , on the other hand , we have for the asymptotic redundancy ( [ d8 ] ) , @xmath502 . again , we must , therefore , conclude that in contrast to the classical case @xcite our trial candidate ( @xmath119 ) for the quantum counterpart of jeffreys prior can not serve as a `` reference prior , '' in the sense introduced by bernardo @xcite . moreover , again in contrast to the classical situation @xcite we find that the minimax and the maximin are _ not _ identical ( although remarkably close ) . the two distinct priors yielding these values ( @xmath487 , respectively @xmath503 ) are themselves remarkably close , as well . since they are mixtures of product states , the matrices @xmath3 are classically as opposed to epr ( einstein podolsky rosen ) correlated @xcite . therefore , @xmath504 must not be less than the sum of the von neumann entropies of any set of reduced density matrices obtained from it , through computation of partial traces . for positive integers , @xmath505 , the corresponding reduced density matrices are simply @xmath506 , due to the mixing ( * ? ? ? * exercise 7.10 ) . using these reduced density matrices , one can compute _ conditional _ density matrices and quantum entropies @xcite . clarke and barron @xcite have an alternative expression for the redundancy in terms of conditional entropies , and it would be of interest to ascertain whether a quantum analogue of this expression exists . let us note that the theorem of clarke and barron utilized the uniform convergence property of the asymptotic expansion of the relative entropy ( kullback leibler divergence ) . condition 2 in their paper @xcite is , therefore , crucial . it assumes as is typically the case classically that the matrix of second derivatives , @xmath507 , of the relative entropy is identical to the fisher information matrix @xmath31 . in the quantum domain , however , in general , @xmath508 , where @xmath507 is the matrix of second derivatives of the quantum relative entropy ( [ eq:5 ] ) and @xmath31 is the symmetric logarithmic derivative fisher information matrix @xcite . the equality holds only for special cases . for instance , @xmath509 does hold if @xmath510 for the situation considered in this paper . the volume element of the kubo - mori / bogoliubov ( monotone ) metric @xcite is given by @xmath511 . this can be normalized for the two - level quantum systems to be a member ( @xmath110 ) of a one - parameter family of probability densities @xmath512 and similarly studied , it is presumed , in the manner of the family @xmath0 ( cf . ( [ eq:10 ] ) and ( [ e7 ] ) ) analyzed here . these two families can be seen to differ up to the normalization factor by the replacement of @xmath513 in ( [ eq : kubo ] ) by , simply , @xmath407 . ( these two last expressions are , of course , equal for @xmath95 . ) in general , the volume element of a monotone metric over the two - level quantum systems is of the form ( * ? ? ? * eq . 3.17 ) @xmath514 where @xmath515 is an operator monotone function such that @xmath516 and @xmath517 . for @xmath518 , one recovers the volume element ( @xmath519 ) of the metric of the symmetric logarithmic derivative , and for @xmath520 , that ( @xmath521 ) of the kubo - mori / bogoliubov metric @xcite . ( it would appear , then , that the only member of the family @xmath0 proportional to a monotone metric is @xmath119 , that is ( [ eq:9 ] ) . the maximin result we have obtained above corresponding to @xmath500 the solution of ( [ maximin ] ) would appear unlikely , then , to extend globally beyond the family . of course , a similar remark could be made in regard to to the minimax , corresponding to @xmath522 , as shown above . ) while @xmath507 can be generated from the relative entropy ( [ eq:5 ] ) ( which is a limiting case of the @xmath523-entropies @xcite ) , @xmath31 is similarly obtained from ( * ? ? ? * eq . 3.16 ) @xmath524 it might prove of interest to repeat the general line of analysis carried out in this paper , but with the use of ( [ jan ] ) rather than ( [ eq:5 ] ) . also of importance might be an analysis in which the relative entropy ( [ eq:5 ] ) is retained , but the family ( [ eq : kubo ] ) based on the kubo - mori / bogoliubov metric is used instead of @xmath0 . let us also indicate that if one equates the asymptotic redundancy formula of clarke and barron ( [ eq:4 ] ) ( using @xmath525 ) to that derived here ( [ a3 ] ) , neglecting the residual terms , solves for @xmath526 , and takes the square root of the result , one obtains a prior of the form ( [ monotone ] ) based on the monotone function @xmath527 . ( let us note that the reciprocal of the related `` morozova - chentsov '' function @xcite , @xmath528 , in this case , is the _ exponential _ mean @xcite of @xmath66 and @xmath67 , while for the minimal monotone metric , the reciprocal of the morozova - chentsov function is the _ arithmetic _ mean . it is , therefore , quite interesting from an information - theoretic point of view that these are , in fact , the only two means which furnish additive quasiarithmetic average codeword lengths @xcite . also , it appears to be a quite important , challenging question bearing upon the relationship between classical and quantum probability to determine whether or not a family of probability distributions over the bloch sphere exists , which yields as its volume element for the corresponding fisher information matrix , a prior of the form ( [ monotone ] ) with the noted @xmath529 . ) as we said in the introduction , ideally we would like to start with a ( suitably well - behaved ) _ arbitrary _ probability density on the unit ball , determine the relative entropy of @xmath81 with respect to the average of @xmath81 over the probability density , then find its asymptotics , and finally , among all such probability densities , find the one(s ) for which the minimax and maximin are attained . in this regard , we wish to mention that a suitable combination of results and computations from sec . [ s2 ] with basic facts from representation theory of @xmath85 ( cf . @xcite for more information on that topic ) yields the following result . [ t15 ] let @xmath491 be a spherically symmetric probability density on the unit ball , i.e. , @xmath530 depends only on @xmath93 . furthermore , let @xmath531 be the average @xmath532 . then the eigenvalues of @xmath533 are @xmath534 with respective multiplicities @xmath535 and corresponding eigenspaces @xmath536 a ballot path from @xmath217 to @xmath537 , which were described in sec . [ s2.2 ] . the relative entropy of @xmath538 with respect to @xmath533 is given by _ ( [ e32 ] ) _ , with @xmath214 as given in _ ( [ e50])_. we hope that this theorem enables us to determine the asymptotics of the relative entropy and , eventually , to find , at least within the family of spherically symmetric ( that is , unitarily - invariant ) probability densities on the unit ball , the corresponding minimax and maximin redundancies . doing so , would resolve the outstanding question of whether these two redundancies , in fact , coincide , as classical results would suggest @xcite . clarke and barron @xcite ( cf . @xcite ) have derived several forms of asymptotic redundancy for arbitrarily parameterized families of probability distributions . we have been motivated to undertake this study by the possibility that their results may generalize , in some yet not fully understood fashion , to the quantum domain of noncommutative probability . ( thus , rather than probability densities , we have been concerned here with density matrices . ) we have only , so far , been able to examine this possibility in a somewhat restricted manner . by this , we mean that we have limited our consideration to two - level quantum systems ( rather than @xmath4-level ones , @xmath539 ) , and for the case @xmath540 , we have studied ( what has proven to be ) an analytically tractable one - parameter family of possible prior probability densities , @xmath0 , @xmath541 ( rather than the totality of arbitrary probability densities ) . consequently , our results can not be as definitive in nature as those of clarke and barron . nevertheless , the analyses presented here reveal that our trial candidate ( @xmath119 , that is ( [ eq:9 ] ) ) for the quantum counterpart of the jeffreys prior closely approximates those probability distributions which we have , in fact , found to yield the minimax ( @xmath487 ) and maximin ( @xmath503 ) for our one - parameter family ( @xmath542 ) . future research might be devoted to expanding the family of probability distributions used to generate the bayesian density matrices for @xmath540 , as well as similarly studying the @xmath4-level quantum systems ( @xmath543 ) . ( in this regard , we have examined the situation in which @xmath544 , and the only @xmath545 density matrices considered are simply the tensor products of @xmath269 identical @xmath5 density matrices . surprisingly , for @xmath546 , the associated trivariate candidate quantum jeffreys prior , taken , as throughout this study , to be proportional to the volume elements of the metrics of the symmetric logarithmic derivative ( cf . @xcite ) , have been found to be _ improper _ ( nonnormalizable ) over the bloch sphere . the minimality of such metrics is guaranteed , however , only if `` the whole state space of a spin is parameterized '' @xcite . ) in all such cases , it will be of interest to evaluate the characteristics of the relevant candidate quantum jeffreys prior _ vis - - vis _ all other members of the family of probability distributions employed over the @xmath547-dimensional convex set of @xmath545 density matrices . we have also conducted analyses parallel to those reported above , but having , _ ab initio _ , set either @xmath66 or @xmath67 to zero in the @xmath5 density matrices ( [ eq:6 ] ) . this , then , places us in the realm of real as opposed to complex ( standard or conventional ) quantum mechanics . ( of course , setting _ both _ @xmath66 and @xmath67 to zero would return us to a strictly classical situation , in which the results of clarke and barron @xcite , as applied to binomial distributions , would be directly applicable . ) though we have on the basis of detailed computations developed strong conjectures as to the nature of the associated results , we have not , at this stage of our investigation , yet succeeded in formally demonstrating their validity . in conclusion , again in analogy to classical results , we would like to raise the possibility that the quantum asymptotic redundancies derived here might prove of value in deriving formulas for the _ stochastic complexity _ @xcite ( cf . @xcite ) the shortest description length of a string of @xmath4 _ quantum _ bits . the competing possible models for the data string might be taken to be the @xmath5 density matrices ( @xmath49 ) corresponding to different values of @xmath407 , or equivalently , different values of the von neumann entropy , @xmath362 . let @xmath55 , @xmath548 , be a family of density matrices , and let @xmath457 , @xmath548 , be a probability density on @xmath12 . the minimum @xmath549 taken over all density matrices @xmath19 , is achieved by @xmath550 . proof . we look at the difference @xmath551 and show that it is nonnegative . indeed , @xmath552 since relative entropies of density matrices are nonnegative ( * ? ? ? * bottom of p. 17 ) . christian krattenthaler did part of this research at the mathematical sciences research institute , berkeley , during the combinatorics program 1996/97 . paul slater would like to express appreciation to the institute for theoretical physics for computational support . this research was undertaken , in part , to respond to concerns ( regarding the rationale for the presumed quantum jeffreys prior ) conveyed to him by walter kohn and members of the informal seminar group he leads . the co - authors are grateful to : ira gessel for bringing them into initial contact _ via _ the internet ; to helmut prodinger and peter grabner for their hints regarding the asymptotic computations ; to a. r. bishop and an anonymous referee of @xcite ; and to the two anonymous referees of this paper itself , whose comments helped to considerably improve the presentation . j. aczl and z. darczy , _ on measures of information and their characterizations_. academic prss : new york , 1975 . j. aitchison , `` goodness of prediction fit , '' _ biometrika _ , vol . 3 , pp.547554 , 1975 . a. bach and a. srivastav , `` a characterization of the classical states of the quantum harmonic oscillator by means of de finetti s theorem '' _ comm . math . phys . _ , vol . 3 , pp . 453462 , 1989 . a. barenco , a. berthiaume , d. deutsch , a. ekert , r. jozsa , and c. macchiavello , `` stabilisation of quantum computations by symmetrisation , '' _ siam j. comput . 5 , pp . 15411547 , 1997 . h. barnum , c. a. fuchs , r. jozsa , and b. schumacher , `` general fidelity limit for quantum channels , '' _ phys . a _ , vol . 54 , no . 6 , pp . 47074711 , dec 1996 . e. g. beltrametti and g. cassinelli , _ the logic of quantum mechanics _ , addison - wesley : reading , 1981 . c. h. bennett , `` quantum information and computation , '' _ physics today _ , 2430 , oct . 1995 . c. h. bennett and p. w. shor , `` quantum information theory , '' _ ieee trans . 44 , no . 6 , pp . 2724 - 2742 ( 1998 ) . j. m. bernardo , `` reference posterior distributions for bayesian inference , '' , _ b _ , vol . 41 , pp . 113147 , 1979 . j. m. bernardo and a. f. m. smith , _ bayesian theory_. wiley : new york , 1994 . l. c. biedenharn and j. d. louck , _ angular momentum in quantum physics _ , addison wesley : massachusetts , 1981 . s. l. braunstein and g. j. milburn , `` dynamics of statistical distance : quantum limits of two - level clocks , '' _ phys . a _ , vol . 3 , pp . 18201826 , mar . n. j. cerf and c. adami , `` information theory of quantum entanglement and measurement , '' _ physica d _ 1 , pp . 6281 , 1998 . n. n. chentsov , _ statistical decision rules and optimal inference_. amer . soc . : providence , 1982 . b. s. clarke , `` implications of reference priors for prior information and for sample size , '' _ 173184 , march 1996 . b. s. clarke and a. r. barron , information - theoretic asymptotics of bayes methods , " _ ieee trans . inform . theory _ 3 , pp . 453471 , may , 1990 . b. s. clarke and a. r. barron , jeffreys prior is asymptotically least favorable under entropy risk , " _ j. statist . planning and inference _ , vol . 1 , pp . 3761 , aug . 1994 . b. s. clarke and a. r. barron , `` jeffreys prior yields the asymptotic minimax redundancy , '' in _ ieee - ims workshop on information theory and statistics _ , piscataway , nj : ieee , 1995 , p. 14 . r. cleve and d. p. divincenzo , `` schumacher s quantum data compression as a quantum computation , '' _ phys . 4 , pp . 26362650 , oct . 1996 . l. d. davisson , universal noiseless coding , " _ ieee trans . inform . theory _ it-19 , pp . 783795 , 1980 . l. davisson and a. leon garcia , a source matching approach to finding minimax codes , " _ ieee trans . inform . theory _ it-26 , pp . 166174 , 1980 . a. fujiwara and h. nagaoka , quantum fisher metric and estimation for pure state models , _ phys . a _ , vol . 119124 , 1995 . r. gallager , source coding with side information and universal coding , " technical report lids - p-937 , m.i.t . laboratory for information and decision systems , 1979 . g. gasper and m. rahman , _ basic hypergeometric series _ , encyclopedia of mathematics and its applications 35 , cambridge university press , cambridge , 1990 . i. j. good , _ math . _ , 95k:62011 , nov . 1995 . i. j. good , `` utility of a distribution , '' _ nature _ , vol . 219 , no . 5161 , p. 1392 , 28 sept . 1968 . i. j. good , `` what is the use of a distribution , '' in _ multivariate analysis - ii _ ( p. r. krishnaiah , ed . ) . new york : academic press , 1969 , pp . d. haussler , a general minimax result for relative entropy " , _ ieee trans . inform . theory _ 4 , pp . 12761280 , 1997 . k. r. w. jones , `` principles of quantum inference , '' _ ann . 1 , pp . 140170 , 1991 . r. jozsa and b. schumacher , a new proof of the quantum noiseless coding theorem , " _ 41 , no . 12 , pp . 23432349 , 1994 . r. e. kass and l. wasserman , `` the selection of prior distributions by formal rules , '' _ 435 , pp . 13431370 , sept . e. g. larson and p. r. dukes , `` the evolution of our probability image for the spin orientation of a spin-1/2ensemble connection with information theory and bayesian statistics , '' in _ maximum entropy and bayesian methods _ ( w. t. grandy , jr . and l. h. schick , eds . ) . dordrecht : kluwer , 1991 , pp . lo , quantum coding theorem for mixed states , " _ opt . 119 , pp . 552556 , j. d. malley and j. hornstein , `` quantum statistical inference , '' _ statist . _ , vol . 8 , no . 433 - 457 ( 1993 ) . s. massar and s. popescu , `` optimal extraction of information from finite quantum ensembles , '' _ phys . 74 , no . 8 , pp . 12591263 , feb . t. matsushima , h. inazumi , and s. hirasawa , `` a class of distortionless codes designed by bayes decision theory , '' _ ieee trans . 5 , pp . 12881293 , sept . s. g. mohanty , _ lattice path counting and applications _ , academic press , new york , 1979 . m. ohya and d. petz , _ quantum entropy and its use_. berlin : springer - verlag , 1993 . a. peres , _ quantum theory : concepts and methods_. dordrecht : kluwer , 1993 . d. petz , `` geometry of canonical correlation on the state space of a quantum system , '' _ j. math . 780795 , feb . d. petz and h. hasegawa , `` on the riemannian metric of @xmath523-entropies of density matrices , '' _ lett . 38 , pp . 221225 , 1996 . d. petz and c. sudr , geometries of quantum states , " _ j. math . 26622673 , june 1996 . d. petz and g. toth , `` the bogoliubov inner product in quantum statistics , '' _ lett . 27 , pp . 205216 , 1993 . f. qi , `` generalized weighted mean values with two parameters , '' _ proc . a _ , vol . 1978 , pp . 27232732 , 1998 . j. rissanen , fisher information and stochastic complexity , " _ ieee trans . inform . theory _ 1 , pp . 4047 , jan . j. rissanen , _ stochastic complexity in statistical inquiry_. world scientific : singapore , 1989 . b. schumacher , quantum coding , `` _ phys . a _ , vol . 4 , pp . 27382747 , april 1995 . c. e. shannon , ' ' a mathematical theory of communication , " _ bell . 27 , pp . 379423 , 623656 , july oct . 1948 . l. j. slater , _ generalized hypergeometric functions _ , cambridge university press , cambridge , 1966 . p. b. slater , applications of quantum and classical fisher information to two - level complex and quaternionic and three - level complex systems , " _ j. math 6 , pp . 26822693 , june 1996 . p. b. slater , quantum fisher - bures information of two - level systems and a three - level extension , " _ j. phys . a _ , vol . l271l275 , 21 may 1996 . p. b. slater , `` the quantum jeffreys prior / bures metric volume element for squeezed thermal states and a universal coding conjecture , '' _ j. phys . a math l601l605 , 1996 . p. b. slater , _ universal coding of multiple copies of two - level quantum systems _ , march 1996 . n. j. a. sloane and s. plouffe , _ the encyclopaedia of integer sequences _ , academic press , san diego , 1995 . k. svozil , _ quantum algorithmic information theory _ , los alamos preprint archive , quant - ph/9510005 , 5 oct . s. verd , fifty years of shannon theory , " _ ieee trans . inform . theory _ , vol . 44 , no . 6 , pp . 20572078 , 1998 . x. viennot , _ une thorie combinatoire des polynmes orthogonaux generaux _ , uqam : montreal , quebec , 1983 . n. j. vilenkin and a. u. klimyk , _ representation of lie groups and special functions _ , vol . 1 , kluwer : dordrecht , boston , london , 1991 . a. wehrl , `` general properties of entropy , '' _ rev . 2 , pp . 221260 , apr . 1978 . d. welsh , _ codes and cryptography _ , clarendon press , oxford , 1989 . r. f. werner , `` quantum states with einstein - podolsky - rosen correlations admitting a hidden - variable model , '' _ phys . 40 , no . 8 , pp . 42774281 , 15 oct . 1989 .

clarke and barron have recently shown that the jeffreys invariant prior of bayesian theory yields the common asymptotic ( minimax and maximin ) redundancy of universal data compression in a parametric setting . we seek a possible analogue of this result for the two - level _ quantum _ systems . we restrict our considerations to prior probability distributions belonging to a certain one - parameter family , @xmath0 , @xmath1 . within this setting , we are able to compute exact redundancy formulas , for which we find the asymptotic limits . we compare our quantum asymptotic redundancy formulas to those derived by naively applying the classical counterparts of clarke and barron , and find certain common features . our results are based on formulas we obtain for the eigenvalues and eigenvectors of @xmath2 ( bayesian density ) matrices , @xmath3 . these matrices are the weighted averages ( with respect to @xmath0 ) of all possible tensor products of @xmath4 identical @xmath5 density matrices , representing the two - level quantum systems . we propose a form of _ universal _ coding for the situation in which the density matrix describing an ensemble of quantum signal states is unknown . a sequence of @xmath4 signals would be projected onto the dominant eigenspaces of @xmath6 . _ index terms _ quantum information theory , two - level quantum systems , universal data compression , asymptotic redundancy , jeffreys prior , bayes redundancy , schumacher compression , ballot paths , dyck paths , relative entropy , bayesian density matrices , quantum coding , bayes codes , monotone metric , symmetric logarithmic derivative , kubo - mori / bogoliubov metric = 5.9pt plus2pt minus 4pt = 5.9pt plus2pt minus 4pt research supported in part by the msri , berkeley ]

anharmonic oscillators are useful examples of nonlinear phenomena . many vibrating systems found in the real world are nonlinear whether they be macroscopic mechanical oscillators @xcite or microscopic atomic oscillators@xcite . the pendulum at high angles is a classic example of an anharmonic oscillator@xcite . a particular feature is that the restoring force is equivalent to a spring that softens at large amplitudes i.e. @xmath1@xcite in this work we do a detailed analysis of an oscillator materialized by the bobbing cone ; we assume that the movement is a vertical translation and that the top of the solid is always emerged and the base is always submerged in an ideal fluid for which the restoring force is due to archimedes principle . the restoring force being of the type @xmath2 means that the oscillator does not move symmetrically about the origin notwithstanding being periodic . figure([fig : figum ] ) schematically represents a floating cone of radius @xmath3 , height @xmath4 and specific mass @xmath5 , partially immersed in an ideal fluid of specific mass @xmath6 ( @xmath7 ) . the position of the solid is chosen to be the coordinate of the point @xmath8 which coincides with the origin of the reference axis when the buoy is at equilibrium . this origin ( point @xmath9 ) is the intersection of the axis of the cone at equilibrium with the plane of the free liquid . the resultant of the forces acting on the solid is @xmath10 where @xmath11 is the volume of the immersed cone in static equilibrium ( @xmath12 i.e. @xmath13 ) , @xmath14 is the volume of the cone , @xmath15 is the difference between the immersed volume @xmath16 and @xmath11 and @xmath17 is the gravity acceleration . therefore : @xmath18\end{aligned}\ ] ] where @xmath19 then , @xmath20\ ] ] where @xmath21 . introducing the reduced variables @xmath22 and @xmath23 , the reduced restoring force is : @xmath24=-p_3(\overline{y},\alpha)\ ] ] where @xmath25 is a third degree polynomial with coefficients that depend on the parameter @xmath26 . the reduced coordinates of the vertex and of the center of the base are , at equilibrium , respectively : @xmath27{1-\alpha}$ ] and @xmath28{1-\alpha}$ ] . therefore , due to the restrictions of the movement @xmath29 . however , another condition should be imposed on the value of the initial position @xmath30 ( or its reduced form @xmath31 ) as a consequence of the potential energy barrier , i.e. @xmath32 and @xmath33 . the potential energy associated with the restoring force @xmath34 , assuming that @xmath35 at the equilibrium position ( @xmath13 ) is @xmath36dy\\ & = & \rho_{\mbox{\scriptsize f}}\ , g v_{\mbox{\scriptsize c}}\int_0^{\overline{y } } \left[\overline{y}^3 - 3(1-\alpha)^{1/3}\,\overline{y}^2 + 3(1-\alpha)^{2/3}\,\overline{y}\right]d\overline{y}.\end{aligned}\ ] ] in reduced form , @xmath37 , with @xmath38 the choice for the initial values ( @xmath39 and zero velocity ) is subjected to the relations @xmath40{1-\alpha}\leq \overline{y}_0\leq \sqrt[3]{1-\alpha}\\\vspace{-5 mm } \\ p_4(\overline{y}_0)= \min\,\left\{\overline{p}_4(-\overline{y}_{\mbox{\scriptsize c } } ) , \overline{p}_4(-\overline{y}_{\mbox{\scriptsize v}})\right\ } \end{array}\right.\ ] ] these equations express the condition that the oscillation has the maximum energy compatible with the fact that the cone is neither completely immersed nor completely emerged . it is easy to verify that @xmath41{1-\alpha})=\frac{3}{4}\,(1-\alpha)^{4/3}+\alpha -\frac{3}{4 } \quad \mbox{and}\\ p_4(-\overline{y}_{\mbox{\scriptsize v}})=p_4(\sqrt[3]{1-\alpha})=\frac{3}{4}\,(1-\alpha)^{4/3}.\end{aligned}\ ] ] then @xmath42 for @xmath43 @xmath44 @xmath45 . so , if the initial velocity is zero then the initial amplitude @xmath31 should be chosen along with @xmath46{1-\alpha}~ \mbox{and}~ \overline{e}_{total}=\frac{3}{4}\,(1-\alpha)^{4/3}+\alpha -\frac{3}{4}\\\vspace{-5 mm } \\ \alpha > \frac{3}{4 } \longrightarrow \overline{y}_0= -\overline{y}_{\mbox{\scriptsize v}}= \sqrt[3]{1-\alpha}~ \mbox{and}~ \overline{e}_{total}= \frac{3}{4}\,(1-\alpha)^{4/3}. \end{array}\right.\ ] ] the other limit @xmath47 of the interval of @xmath48 can be obtained from the roots of the polynomial @xmath49 . there are four solutions ( two real and two complex conjugate ) ; the real are the relevant solutions : @xmath31 [ eqs.([eqyzero ] ) ] and @xmath47 [ eqs.([eqyum ] ) ] @xmath50^{1/3}}{3}\,\raisebox{-0.1mm}{\large -}\\\vspace{-6mm}\vspace{2.5 mm } \\ & \displaystyle -\frac{2}{3\left[-44 + 54\alpha+6\sqrt{-27 + 81\,(1-\alpha)^2 + 30\,\alpha}\right]^{1/3 } } + ( 1-\alpha)^{1/3 } + { \displaystyle\frac{1}{3}}\\\vspace{-5 mm } \\ \alpha > \frac{3}{4 } \longrightarrow \overline{y}_1&= -(\sqrt[3]{4}-1)\,\sqrt[3]{1-\alpha}. \end{array}\right.\ ] ] for each @xmath26 value there is a definite maximum of total energy and a corresponding interval of amplitude @xmath51\equiv [ -\overline{y}_{\mbox{\scriptsize c } } , \overline{y}_1]$ ] for @xmath52 and @xmath53 $ ] for @xmath54 given by eqs.([eqyzero ] ) and ( [ eqyum ] ) . it means that the procedure to initiate the movement should be as follows : for @xmath55 the base of the cone should be raised near the free surface of the liquid ; for @xmath54 the vertex of the cone should be lowered until near complete immersion ; for @xmath56 the choice of the base or vertex to initiate the movement is irrelevant since the amplitude of the oscillation , that has a maximum ( equal to @xmath4 ) corresponds to a displacement between the base and the vertex i.e. @xmath57 $ ] . this can be seen in fig.([fig : figuredoissemcor ] ) . , and the reduced interval , @xmath58 , of the oscillations as a function of @xmath26 . the maximum of the energy @xmath59 occurs at the crossover @xmath43.,width=4 ] the restoring force ( reduced value ) @xmath60 and the potential energy @xmath61 can be represented in the interval of oscillation with @xmath26 as parameter [ fig.([fig : figuretressemcor ] ) and fig.([fig : figurequatrosemcor ] ) ] . , within the proper interval of oscillation taking @xmath26 as a parameter.,width=4 ] , within the proper interval of oscillation taking @xmath26 as a parameter.,width=4 ] we use the newton equation to find the position of the point @xmath8 of the cone as a function of time . the adoption of reduced values continues to offer simplification in the final equation . let us define the unit of time @xmath62 . therefore @xmath63 . the unit for @xmath64 is @xmath65 which points to @xmath66 . on the other hand @xmath67 , i.e. the equation of motion in terms of the reduced variables is @xmath68 finally @xmath69 the system does not oscillate symmetrically about the origin since the force ( polynomial @xmath70 ) shows odd and even powers . for small displacements @xmath71 , the equation is approximately linear and so the movement is quasi harmonic with period @xmath72{1/(1-\alpha)}$ ] . in other physical situations the oscillator exhibits anharmonic displacements and the period will depend on the amplitude ( or total energy ) . the solutions @xmath73 and @xmath74 were obtained by computational methods and are represented in figs . ( [ fig : figuracinco ] ) , ( [ fig : figuraseis ] ) , ( [ fig : figurasete ] ) and ( [ fig : figuraoito ] ) . .,width=4 ] .,width=4 ] .,width=4 ] .,width=4 ] the conservation of the energy of the oscillator allows the determination of the velocity as a function of displacement : @xmath75 . @xmath76 the phase space representation is shown in fig . ( [ fig : novafiguranove ] ) . from the symmetry of these curves relatively to the horizontal axis , we can conclude that the interval of time between two zeros of the velocity or two extremes of the displacements is equivalent to half of the period of the movement . since @xmath77 , @xmath78 . then , as a function of @xmath48 ) for some values of @xmath26.,width=4 ] @xmath79 the limits of integration and the integrand function have been determined above . the result is only a function of @xmath26 . it is possible to get a close solution for the period in terms of complete elliptic integrals of the first kind@xcite . for @xmath80 , the analytical solution is far too complicated to be considered and so numerical methods were used . however , the solution for @xmath81 was easily obtained : @xmath82{4}-1)\sqrt[3]{1-\alpha}}^{\sqrt[3]{1-\alpha}}\,\frac{d\overline{y}}{\sqrt{3(1-\alpha)^{4/3}-\overline{y}^4 + 4(1-\alpha)^{1/3}\overline{y}^3 - 6(1-\alpha)^{2/3}\overline{y}^2}}\\\vspace{-4 mm } \\\nonumber \overline{t}=&\frac{\sqrt{\alpha}}{\sqrt[3]{1-\alpha}}\,\frac{2^{4/3}}{3^{1/4}}~k\hspace{-1mm}\left(\frac{\sqrt{2}}{2(1+\sqrt{3})}\right)=c\frac{\sqrt{\alpha}}{\sqrt[3]{1-\alpha}},\end{aligned}\ ] ] where @xmath83 is a complete elliptic integral of the first kind of argument @xmath84 . the numerical value of c is : @xmath85 . an interesting result that should be outlined is the fact that the normalized value of the period for @xmath81 , @xmath86 ( @xmath87 , period of small oscillations ) is a constant , i.e. does not depend on @xmath26 : @xmath88{1-\alpha}}\,\frac{\sqrt{6}\sqrt[3]{1-\alpha}}{2\pi\,\sqrt{\alpha}}=\frac{\sqrt{6}}{2\pi}c= 1.192900269.\ ] ] the period of the cone oscillations at the highest energy as a function of @xmath26 is represented in fig . ( [ fig : figuradezpretobranco ] ) together with the normalized values . , calculated for the highest value of its total energy . the derivative @xmath89 is also shown in the inset.,width=4 ] as a parameter . the energy varies from @xmath90 to its maximum value ( eq.([eqyzero])).,width=4 ] so far , we have represented the oscillation with the highest energy or maximum value of the initial amplitude satisfying conditions ( [ conditons ] ) . it is interesting to examine closely the dependence of the period with the energy of the oscillator from zero to the highest value given by eqs.([eqyzero ] ) . this is shown in fig.([fig : figureonzesemcor ] ) taking , as usual @xmath26 as a parameter . it is interesting to observe that the numerically obtained values ( @xmath91 ) fit @xmath92 a second degree polynomial . the fourier analysis is a very useful tool to investigate the harmonic components of a periodic function of time that derives from a non linear equation , as it happens in our study . from figs.([fig : figuracinco ] ) and ( [ fig : figuraseis ] ) one can conclude that the displacement @xmath73 is a periodic symmetric even function with a non null mean value . the fourier decomposition gives for the general case : @xmath93 ; in this case all @xmath94 since @xmath95 is an even function . the coefficients @xmath96 were calculated by numerical methods once the fundamental frequency is known . only the first four coefficients were significant . the fourier components for the particular case @xmath97 that corresponds to the maximum of the highest value of energy are represented in fig.([fig : figuredozesemcor ] ) . it also exhibits the highest coefficients which indicate the highest degree of anharmonicity . the reduced values of all calculated coefficients ( @xmath98 ) are also shown in table[table ] . llllllll & @xmath99&@xmath100&@xmath101&@xmath102&@xmath103&@xmath104&@xmath105 + @xmath106&@xmath107&@xmath108&@xmath109&@xmath110&@xmath111&@xmath112&@xmath113 + @xmath114&@xmath115&@xmath116&@xmath117&@xmath118&@xmath119&@xmath120&@xmath121 + @xmath122&@xmath123&@xmath124&@xmath125&@xmath126&@xmath127&@xmath128&@xmath129 + @xmath130&@xmath131&@xmath132&@xmath133&@xmath134&@xmath135&@xmath136&@xmath137 + @xmath138&@xmath139&@xmath140&@xmath141&@xmath142&@xmath143&@xmath144&@xmath145 + @xmath146&@xmath147&@xmath148&@xmath149&@xmath150&@xmath151&@xmath152&@xmath153 + @xmath154&@xmath155&@xmath156&@xmath157&@xmath158&@xmath159&@xmath160&@xmath161 + @xmath162&@xmath163&@xmath164&@xmath165&@xmath166&@xmath167&@xmath168&@xmath169 + @xmath170&@xmath171&@xmath172&@xmath173&@xmath174&@xmath175&@xmath176&@xmath177 + most real oscillators contain anharmonic components . the present study of a floating cone movement is by all means an interesting case of an anharmonic oscillator . the fact that the restoring force is a polynomial of the third degree with non null coefficients ( except the independent term ) imply an asymmetry of this force and the related potential curve . comparing this to well known based mass - spring systems results that the equivalent spring stiffness is no longer constant . it varies as a sum of two contributions that have opposite sign from a specific position . in solid state physics we encounter these kind of forces such as the cohesive force containing a short range repulsion ( hard sphere interaction ) and a long range attraction . the former varies with the displacement much faster than the latter . 10 filipponi a and cavicchia d r 2011 _ am . j. phys . _ * 79 * 730 arnold t w and case w 1982 _ am . j. phys . _ * 50 * 220 whineray s 1991 _ _ * 12 * 90 pecori b , torzo g and sconza a 1999 _ am . j. phys . _ * 67 * 228 ashcroft n w and mermin n d 1976 _ solid state physics _ ( ny : holt rinehart and winston ) lewowski t and wozniak k 2002 _ _ * 23 * 461 lima f m s and arun p 2006 _ am . j. phys . _ * 74 * 892 boyd j n 1991 _ virginia journal of science _ * 42 * lebedev n n 1972 _ special functions and their applications _ ( ny : dover )

a study of the floating of a circular cone shaped buoy in an ideal fluid has revealed some new interesting results . using reduced variables it is shown , that at a crossover value @xmath0 of the ratio of the specific masses of the fluid and of the buoy , the anharmonicity of the oscillation is the highest and that , unexpectedly , above this crossover value the normalized period is constant .

active galactic nuclei ( agn ) are the most luminous objects in the universe . they emit large amounts of radiation over a wide range of wave - bands ( from radio to @xmath6-rays ) , and sometimes produce relativistic jets . it is a goal of modern astronomy to understand the origin of the extreme activities in agn . one of the key observational clues in the x - ray band to understand agn is the iron line profile in the energy spectrum @xcite . a broad and skewed line feature around 57 kev was clearly detected by asca for the first time in an energy spectrum of the seyfert galaxy mcg6 - 30 - 15 @xcite . since then , a similar feature was detected by asca from several seyfert galaxies ( e.g. @xcite ) . the broad and skewed feature has been interpreted as an iron fluorescent line originating from the innermost region of an accretion disk ( @xcite , so - called `` disk - line '' model ) . in this model , a broad and skewed feature is explained by the combination of doppler broadening due to the relativistic motion of the line emitting matter and gravitational redshift due to the central massive black hole . if the broad and skewed feature in the energy spectra is really a `` disk - line '' , we can determine the inclination angle of the accretion disk through an analysis of the line profile @xcite . however , the thus - obtained inclination angle is sometimes inconsistent with that estimated with other methods ( e.g. @xcite ) . it may also be inconsistent with an expectation from the unified scheme , in which the accretion disk in seyfert 2 galaxies should have an edge - on geometry . asca observations have revealed the presence of a broad and skewed line feature in several seyfert 2 galaxies . the inclination angles have been deduced for these seyfert galaxies from an analysis of the `` disk - line '' , and always found to be around @xmath7 @xcite . this inclination angle is unexpectedly small . this problem may be partially solved if we assume that the line is a composite and consists of a narrow line centered at 6.4 kev and a disk line with intermediate inclination angle @xcite . however , we still need an observational confirmation on such a composite model . ngc 4151 is a bright , nearby seyfert galaxy ( type 1.5 ; @xcite ; for a review see @xcite ) . the presence of the broad and skewed line feature in 4.57.5 kev has been known through asca observations @xcite . thus , ngc 4151 is one of the best targets to precisely study the line feature . a `` disk - line '' analysis of ngc 4151 has resulted in a face - on geometry of the accretion disk @xcite , while observations of the [ o iii ] @xmath85007 image with the hubble space telescope indicate an inclination angle of @xmath9 @xcite . this inconsistency might be partially resolved if we assume two disk lines and an additional narrow peak centered at 6.4 kev to reproduce the profile @xcite . however , it is not clear how the two disk lines with different inclination angles can be produced simultaneously . in this paper , we analyze the broad and skewed feature observed from ngc 4151 using a model - independent method as much as possible , and try to identify the origin of the feature . for this purpose , we analyze the long observations of ngc 4151 made with asca ( in 2000 ) and rxte ( in 1999 ) , while focusing on the spectral variations on various time scales . the results from another set of long asca observations of ngc 4151 have been presented by @xcite . the primary asca data set used for the present analysis was acquired from 2000 may 13 through 25 . concerning the instrumentation on board asca , see the following references : @xcite for a general description of asca , @xcite for the x - ray telescope , @xcite and @xcite for the gas imaging spectrometer ( gis ) , and @xcite for the solid - state imaging spectrometer ( sis ) . in the observations , sis was operated in the 1-ccd faint mode and gis in the standard ph mode . a lower level discriminator was applied to sis at 0.48 kev to avoid telemetry saturation due to flickering pixels . the data selection criteria for sis and gis are summarized in table [ tbl : crit ] . when the elevation angle , i.e. the angle between the field of view and the earth limb , was less than @xmath10 , the data were discarded because they were affected by scattered x - rays from the day earth or by the absorption due to the atmosphere . some of the optical light leakage is known in sis , and the sis data obtained when the elevation from the day earth was less than @xmath11 were discarded . the data with telemetry saturation are also discarded to avoid any deterioration of the detection efficiency . the net exposure of sis was about 350 ks , which spans @xmath12 s , and an average count rate of sis was 0.8 count s@xmath13 after data screening and reduction . an x - ray source was detected with both sis and gis at a position consistent with the optical position of ngc 4151 within the positional accuracy of asca . a bl lac object ( 1207 + 39w4 ; @xcite ) was detected in the field of view of asca about north of ngc 4151 . contamination by this source was found to be negligible above 2 kev , which was used for the current data analysis . according to the chandra observations , a faint source was also present at from ngc 4151 to the southwest @xcite . this source was not resolved with asca . contamination by this source was completely negligible . the energy spectra and light curves of each detector were accumulated from a circular region centered on the source . the extraction radius of sis was set to , which is the maximum radius fit in both chips , on the whole . the background spectra of sis are accumulated from the source free region on the chip . the extraction radius of gis was set to 6 arcmin , because various calibrations of gis had been done with this standard extraction radius . the background data of gis were accumulated from a circular region opposite to the target position against the bore - sight axis , excluding the source region . we usually obtained 4 sets of energy spectra from the asca data , two from sis and the other two from gis , for a single set of observations . to make the analysis simple , we summed up the two energy spectra from sis and the two corresponding response matrices , respectively , which produced a single sis spectrum with the corresponding response . similarly , we summed up the two gis spectra and the corresponding responses , respectively . the summed sis and gis spectra were fitted with a model spectrum simultaneously . the simultaneous fitting made a full utilization of the asca data in the sense that the higher detection efficiency below 5 kev and better energy resolution of sis , and the higher detection efficiency above 5 kev of gis are all reflected in the best - fit spectral parameters . because there is 3% uncertainty of flux normalization between sis and gis , a systematic error of 3% was conservatively added to the sis energy spectra in the simultaneous fitting . although a decrement of sis detection efficiency below 2 kev was reported by the instrument team ( t. yaqoob , private communications ) , we did not include the effect in the analysis because the spectral fittings were carried out only for the energy range above 2 kev . data obtained by the proportional counter array ( pca ) on board the rossi x - ray timing experiment ( rxte ) were also used in the present analysis . details of rxte and pca can be found in @xcite . rxte observations of ngc 4151 were carried out every @xmath45 days from the beginning of 1999 to study the long - term variations of the x - ray flux . we used the data obtained from 1999 january 1 through july 24 , which were publicly available from the archive in heasarc at the time of data analysis . reduction of the pca data was carried out using the standard procedure @xcite . in the analysis , we discarded data obtained when the elevation from the earth limb was less than @xmath14 , data obtained within 30 minute after the satellite s passage of saa , data obtained when the pointing offset was greater than , and data when the count rate of electrons was greater than 0.1 . the background spectra were calculated using a faint - source model according to a method described in the rxte cook book . figure [ fig : ascalc ] shows the x - ray light curve at 0.710 kev from ngc 4151 obtained with asca gis on 2000 may 1325 . here , the data are shown in 5.6 ks binning . this 5.6 ks bin - time corresponds to the orbital period of asca around the earth . we chose this bin time because : ( 1 ) time variations shorter than the orbital period were not significant , ( 2 ) data gaps are usually much shorter than the orbital period and we can get continuous light curve , ( 3 ) possible background variations , which are correlated with the orbital period , could be largely reduced . note that data gaps were produced mainly by the earth occultation of the source , high background regions on the earth , and down - link to the ground stations . x - ray flux variations of a factor of 23 are clearly seen in the figure . we next studied the energy dependence of the flux variations . for this purpose , we calculated the ratios of the time - resolved energy spectra to the average energy spectrum . considering the statistics of the data and the amplitude of the time variations , the time - resolved energy spectra were calculated every @xmath15 s and six spectra were obtained in total . we also calculated the average spectra for gis and sis , each of which correspond to the average of 6 time - resolved spectra , as shown in figure [ fig : avespe ] . the ratios of the six time - resolved spectra to the average spectrum are presented in figure [ fig : pharatio ] . large time variations in both the flux and the shape can be seen at 25 kev , while the ratios are almost flat at 0.71.5 kev and 710 kev . the flat spectral ratios mean that the shape of the energy spectra does not vary at 0.71.5 kev and 710 kev . furthermore , those at 0.71.5 kev are always close to unity this indicates that the energy spectra change neither their shapes nor the fluxes in this energy range . it should be noted that a local structure is seen just around 6.4 kev and the ratios are closer to unity only in this narrow energy band . at least three spectral components may be required to reproduce these time variations of the spectral ratios : a stable soft component below @xmath42 kev , a variable hard component above @xmath42 kev , and a relatively stable component around 67 kev . because we are mostly interested in the feature around 67 kev , we do not discuss the soft stable component hereafter by limiting the energy range in the spectral analyses to above @xmath42 kev . in order to study the nature of the relatively stable component around 67 kev , we need to separate the component from the underlying variable hard component . we first determined the shape of the variable hard component by performing a model fit to the average spectrum at 24 kev and 810 kev after masking the 48 kev range . the results show that a simple model of a power law with a single absorption can not reproduce the spectrum , and that a `` dual absorber '' model is necessary . the need for a dual absorber for the asca spectrum was already pointed out by @xcite . the `` dual absorber '' model is expressed by the following formula : @xmath16 + ( 1 - c_{\rm f } ) \exp[-\sigma(e ) n^{2}_{\rm h } ] \right\ } e^{-\gamma } , \label{eq : dualabs}\ ] ] where @xmath17 is a normalization factor , @xmath18 the photon index , @xmath19 the column density , @xmath20 the photo - electric absorption cross - section , and @xmath21 the source covering fraction ( @xmath22 ) . the superscript to @xmath19 indicates two different column densities . in the following analysis , the photo - electric absorption cross - sections compiled by @xcite are considered . the thus - determined continuum spectrum was subtracted from the average spectrum ; the residual feature at 48 kev is shown in figure [ fig : fe ] . the presence of a narrow peak at 6.4 kev and a broad feature skewed toward the lower energy can be clearly seen . we then studied the time variations of the residual feature at around 6.4 kev . for this purpose , we used the 6 sets of sis spectra , which have a better energy resolution than the gis spectra . the continuum shape was determined separately for the 6 sets of spectra using the same method for the average spectrum . we set @xmath23 in equation ( [ eq : dualabs ] ) to 0 to simplify the model fitting , considering the statistics of the data . we confirmed that setting @xmath23 to zero did not change the continuum shape at 48 kev more than 3% while taking the 4th data set ( those with the largest flux ) as an example . the model was found to be acceptable for all 6 sets of the spectra with the largest value of @xmath24 . when the continuum model was interpolated to the 48 kev band , an excess feature could be clearly seen in all 6 sets of the energy spectra . we show them in figure [ fig : feindiv ] after subtracting the continuum model from the observed spectra . a prominent narrow peak at around 6.4 kev , and broad features at 4.56.0 kev ( hereafter referred to as a red wing ) and at 6.87.5 kev ( a blue wing ) are clearly seen in all 6 data sets . in order to study any profile change of the broad and skewed feature , we made the ratios of each of the 6 excess features to their average . the ratios were calculated by dividing the individual excess feature ( after subtracting the interpolated continuum ) by the excess feature in the average spectrum ( figure [ fig : fe ] ) . the results are shown in figure [ fig : excesspha ] . by performing a @xmath25 test for the hypothesis that the ratios have no energy dependence , we found that the 6 sets of ratios are consistent with being constant in terms of the energy , and that the time variation of the profile of the broad and skewed feature is not significant . at the same time , we calculated the flux ratio of the red wing ( 4.56.0 kev ) to the narrow core ( 6.06.8 kev ) and that of the blue wing ( 6.87.5 kev ) to the narrow core for the 6 sets of sis spectra . no significant time variation was found at the 90% confidence limit in the two sets of ratios ( @xmath26 at most ) . the standard deviation to the mean for the flux ratio of the red wing to the narrow core was calculated to be 20% at most . although the time variation is not significantly found , a variation of less than 20% in amplitude is not rejected . we analyzed the rxte pca data in order to study the spectral variations on a time scale longer than the @xmath12 s covered by asca . we show in figure [ fig : rxtelc ] the light curve calculated from the pca data obtained from 1999 january 1 through july 24 . significant flux variations can be clearly seen . the pca spectrum averaged over the rxte observations is shown in figure [ fig : rxteave ] . as was done for the asca spectra , we fit a power law modified by `` dual absorption '' ( @xmath27 ) to the average spectrum in 2.810 kev after masking the 48 kev energy band . we also checked how it affected the continuum shape to set @xmath23 to zero , and found that there was no noticeable impact if the fit range was restricted to 2.84 and 810 kev . the ratio of the observed spectrum to the continuum model is shown in the lower panel of figure [ fig : rxteave ] . we can clearly see the presence of a broad and skewed feature at 48 kev again . next , in order to search for the time variations of the broad and skewed feature on time scales longer than @xmath12 s , we divided the data into 11 subsets on a time bin of @xmath28 s and calculated an energy spectrum for each subset of data . we then fit a power law modified by `` dual absorption '' to the 11 spectra in 2.810 kev while excluding the 48 kev band again . the fit was acceptable for all of the spectra . the residuals of the 11 spectra after subtracting the best - fit continuum model are shown in figure [ fig : rxteindiv ] . a significant excess flux over the continuum model is always seen at 4.57.5 kev . the excess has a peak at around 6.06.5 kev as well as a significant tail feature on the lower energy side . then , we again investigated the time variations of the spectral shape of the excess by comparing the 11 excess spectra with their average . the spectral ratios of the 11 spectra to their average are shown in figure [ fig : rxteratio ] . we performed @xmath25 tests to check whether or not each of the 11 ratio - spectra is consistent with being flat . a flat model was found to be acceptable for all 11 ratios with @xmath29 , although the flux level of the excess shows a significant variation . this indicates that , although the excess flux at 4.57.5 kev is variable on a time scale longer than @xmath28 s , the profile does not change significantly . in the previous section we discussed the time variability of the flux and the profile of the broad and skewed feature consisting of a narrow peak at 6.4 kev and red / blue wings in a way fairly independent of any spectral models . in this section , we introduce two different model functions to reproduce the profile : one is the so - called `` disk - line '' model , and the other is a `` reflection '' model . it is widely accepted that the broad and skewed feature at around 57 kev often seen from agn is composed of gravitationally redshifted iron - fluorescent lines from matter orbiting in an accretion disk very close to the central black hole @xcite . hence , we first tried to fit the broad line - like feature observed from ngc 4151 with the disk - line model . we adopted a spectrum described by equation ( [ eq : dualabs ] ) as the continuum , where @xmath30 was set to 0 to simplify the model . the outer and inner radii of the accretion disk were set to 1000 @xmath31 and 10 @xmath31 , respectively , where @xmath31 is the schwarzschild radius . the emissivity of the line emission is assumed to have a power - law dependence on the radius with the index of @xmath322 @xcite . the model fitting was carried out in the energy range of 2.210 kev simultaneously to the sis and gis data for each of the 6 sets of the asca energy spectra . the photon index was fixed to 1.55 , which was the average value when we fit a power law with a partial covering absorber to the spectra after masking 48 kev . the model was found to be acceptable for all of the spectra at 2.210.0 kev with @xmath33 . the best - fit parameters are listed in table [ tbl : ascapar ] . the inclination angles and the line center energies obtained from the spectral fits to the 6 spectra are not significantly different from one another . the reduced @xmath25 values for the hypothesis that the 6 values are constant were 1.35 and 0.89 for 5 d.o.f . for the inclination angle and the line center energy , respectively . this means that these two parameters are consistent with being constant . the average inclination angle and the line center energy are @xmath34 and @xmath35 kev , respectively . we applied the same model to the 11 energy spectra of pca at 2.824 kev . however , the fit was not acceptable with @xmath36 . the deviation of the model from the observed spectra became noticeable above 16 kev . hence , we replaced the power law in equation ( [ eq : dualabs ] ) with a broken power law as described below : @xmath37 where @xmath17 is a normalization , @xmath38 and @xmath39 are photon indices , and @xmath40 is a break energy at which the spectral index changes . note that the broken power - law is required due to the apparent change in the spectral slope above @xmath416 kev , not because we set @xmath30 to zero . the 11 energy spectra of pca were all reproduced by the new model with @xmath41 . the best - fit parameters are listed in table [ tbl : rxtepar ] . we checked the time variation of the 11 values of each parameter by performing a @xmath25 test . the photon indices , @xmath38 and @xmath39 , and the break energy , @xmath40 , are found to have not changed significantly during the observational period . we cross - checked the spectral slope above the break energy , which was found to be @xmath42 , using the data from another detector hexte ( high energy x - ray timing experiment ) on board rxte . hexte consists of two clusters of 4 nai / csi phoswich scintillation detectors sensitive to x - rays from 15 to 250 kev . the photon indices obtained from a simultaneous fit of a power - law model to the 11 sets of the pca spectrum ( 1824 kev ) and hexte spectrum ( 18100 kev ) are found to fall around @xmath43 . this is consistent with @xmath44 . it is noteworthy that , although the disk line model can always reproduce the excess feature around 57 kev , the continuum spectrum needs a break at around 17 kev to reproduce the energy spectrum up to 100 kev . in the disk - line model , the peak at 6.4 kev is considered to be a blue part of fluorescent lines coming from matter orbiting in an accretion disk very close to the central black hole . however , the peak energy is just the k@xmath45 line of a neutral , or a lowly ionized iron , in the rest frame ; furthermore , the profile around the peak can be reproduced by a single narrow line . hence , it would be more natural to consider that the peak at 6.4 kev is a single fluorescent iron line coming from a region free from relativistic effects . if the line is emitted through the fluorescent process as a result of x - ray illumination on relatively cold matter far outside the central x - ray source , x - ray reflection at the surface of the matter should take place simultaneously with the fluorescent line emission . the x - rays reflected by the cold matter emitting the fluorescent lines could be a possible origin of the broad red and blue wings accompanied by the narrow line feature on both sides . because x - ray reflection is due to thomson scattering , the reflected x - rays should have the same spectrum as the illuminating x - rays in a typical x - ray band , but suffer from absorption when they pass through the cold mater . thus , the reflected component may be approximated by applying cold - matter absorption to the continuum model . in order to test this possibility , we introduced the following model spectrum : @xmath46 + 1-c_{\rm f}^1\ } \exp[-\sigma(e ) n_{\rm h}^2 ] + i(e_{\rm line } ) + c_{\rm f}^2 p(e,\gamma ) \exp[-\sigma(e ) n_{\rm h}^3 ] . \label{eq : ref}\ ] ] here , the first term is the same as equation ( [ eq : dualabs ] ) , representing continuum x - rays directly coming from the central x - ray source . the second term represents the fluorescent iron line at 6.4 kev and @xmath47 is a gaussian function . the line width is assumed to be zero . the third term is for the reflection component . the efficiency of the x - ray reflection is represented by @xmath48 , which corresponds to a covering fraction of the x - ray reflector optically thick for thomson scattering . the function , @xmath49 , is a power law with a photon index of @xmath18 . the photon index is optimized in the course of the fitting , because it may be affected by the introduction of a highly absorbed component . the column densities and covering fractions are also optimized . model fitting was firstly carried out to the 11 pca spectra in the energy range of 2.824 kev . in this energy range , because the effect of absorption by @xmath30 was negligible , we omitted this term . the fit was acceptable for all energy spectra with @xmath50 . an example of the results of the model fitting is shown in figure [ fig : rxteref ] and the best - fit parameters are listed in table [ tbl : rxteref ] . the results show that 4050% of the direct flux is reflected . we checked the constancy of the 11 values of each spectral parameter with a @xmath25 test . we found that the photon index and the absorption column of the reflected component , @xmath51 , are both consistent with being constant ( @xmath52 , and 1.25/10 , respectively ) . we confirmed that the profile of the broad and skewed feature did not change significantly during the rxte observations . the weighted mean of the 11 photon indices is @xmath53 . this is comparable to @xmath54 , which we obtained from a simultaneous fitting to the 11 sets of the pca energy spectra at 1824 kev and the hexte energy spectra at 18100 kev . if we extrapolate the best - fit model at 2.824 kev to 100 kev , no significant discrepancy is recognized between the model and the observed energy spectra with hexte . this indicates that the intrinsic spectrum emitted from the central source should be a single power law with an index of @xmath41.9 , and should extend up to 100 kev without a break , if the `` reflection '' model explains the broad and skewed feature at around 57 kev . although the best - fit line energies are found to be consistent with being constant , the mean line center energy is @xmath55 kev . this is slightly lower than the center energy of the iron fluorescent line from neutral matter ( 6.4 kev ) . however , it is known that there is a systematic uncertainty of about 23% in the energy scale of the rxte instrument . thus , the line energy can still be interpreted as the iron fluorescent line . the time variation of the line flux is shown together with that of the continuum flux at 810 kev in figure [ fig : rxtefe ] . the time variation of the line flux is clearly seen in a time scale of @xmath56@xmath57 s. next , we also applied the model function ( equation [ eq : ref ] ) to the asca data . because the energy range of the asca detectors is limited below 10 kev , some of the model parameters were not well constrained . hence , we fixed the following parameters in the fitting : @xmath18 ( @xmath58 ) , @xmath51 ( @xmath59 atom @xmath60 ) and @xmath48 ( @xmath61 % ) . other parameters were optimized during the course of fitting . the model fitting was carried out to each of the 6 sets of the asca data , simultaneously to the sis and gis spectra . fittings of this model were all acceptable with @xmath62 . an example of the results of the model fittings to the asca sis spectra is shown in figure [ fig : ascaref ] . the best - fit parameters are listed in table [ tbl : ascaref ] . the time histories of the line flux and the continuum flux at 810 kev for the asca data are plotted in figure [ fig : ascalcfe ] . no significant change of the line flux can be seen on time scales of @xmath63@xmath56 s. as can be seen from the table , the values of @xmath64 , @xmath65 and @xmath30 obtained from the spectral fits to the 6 sets of asca spectra significantly change on time scales of @xmath63@xmath56 s. the relative amplitude of the variations of @xmath64 and @xmath30 are @xmath420% and @xmath412% , respectively . finally , we investigated the parameters of the narrow line component . we optimized the center energy and the intrinsic width of the narrow line component in the `` reflection '' model by fitting it to the average spectra of the asca observations in 2000 . the line center energy and the intrinsic width were found to be @xmath66 kev and less than 92 ev ( gaussian @xmath67 ) , respectively . an analysis of the time - resolved energy spectra showed that the iron line flux was consistent with being constant on time scales of @xmath63@xmath56 s , while significant variations were seen in its flux on time scales of @xmath56@xmath57 s. in order to study the origin of the line emission , the relation between the continuum flux and the line flux was studied . here , the continuum flux was calculated in the energy band of 810 kev , which is higher than the k edge energy of neutral iron . in this energy range , the flux variation should reflect the variation of the intrinsic , power - law component . because the line emission probably results from a reprocessing of the continuum x - rays by matter ambient to the continuum source , the light curve of the line flux might suffer from some amount of smearing and/or time delay to the continuum flux light curve . in fact , if we compare the light curves of the continuum flux and the line flux ( figure [ fig : rxtefe ] ) , the line flux variation seems to follow the smeared variations of the continuum flux . in order to see the effect of smearing , we applied smearing to the continuum light curve using a following simple method . we redistributed the continuum flux of @xmath68-th time bin into @xmath69 bins starting from @xmath68-th bin . thus , the smeared light curve is expressed as @xmath70 where @xmath71 is the flux of @xmath68-th bin in the original light curve , @xmath72 the flux of @xmath68-th bin in the smeared light curve , and @xmath69 the number of bins to smear . we then fitted the smeared light curve to the line flux light curve . we increased @xmath69 from 1 until we obtain an acceptable fit . note that @xmath73 corresponds to the case without smearing . the relation between @xmath69 and @xmath74 is plotted in figure [ fig : rxtesmear ] . we could obtain an acceptable agreement with @xmath75 between the smeared light curve of the continuum flux and the line flux light curve . the smeared light curve in the case of @xmath75 is compared with the light curve of the line flux in figure [ fig : rxtesmcomp ] . the above - mentioned smearing algorithm of the continuum flux also introduces a delay of @xmath76 bins together with smearing of @xmath76 bins to both sides . thus the typical delay and smearing time scale may be regarded as @xmath77 s. we then checked whether or not smearing of @xmath78 s can explain the absence of a correlation between the continuum flux and the line flux on shorter time scales for the asca data . since the asca data is shorter than @xmath78 s , we could not smear the continuum flux light curve directly . hence , we assumed that the observed light curve of the continuum flux in the asca observations repeats periodically . we then smeared the assumed light curve using the method described above . the fractional variation ( i.e. standard deviation divided by the average ) of the continuum flux was originally @xmath40.25 , but was reduced to @xmath40.06 by smearing . the reduced value is consistent with the observed upper limit , 7% , of the fractional variation of the line flux in the asca observations . we analyzed the asca data obtained in 2000 may which covered time intervals of @xmath79 s. the data were divided into 6 sets with an integration time of @xmath80 s each to study the time variabilities of the spectrum . in order to analyze the excess component separately from the continuum , we masked a range of 48 kev in spectral fitting and determined the continuum model . we found that a power law with a photon index of @xmath41.55 modified by two absorbers with different column densities and covering fractions can smoothly connect the two spectral parts at 24 kev and 810 kev . above this continuum , an excess is clearly detected in 4.57.5 kev , which has a broad and skewed feature . the feature has a prominent narrow peak at 6.4 kev , but accompanies a broad red wing at 4.56.0 kev and a blue wing at 6.87.5 kev . the flux and shape of the excess feature obtained from each of the 6 spectra were compared with one another . it is found that the excess flux is consistent with being stable , concerning both the flux and the spectral shape , over the observations of @xmath79 s. we also analyzed data obtained by rxte from january through july in 1999 in order to search for any time variations of the iron line on a time scale of @xmath81 s. we obtained 11 sets of time - resolved energy spectra , each of which covers a time interval of @xmath82 s. if we selected only the energy bands free from the iron structures , i.e. the 2.84.0 kev and 810 kev bands , each energy spectrum could be reproduced again by a power law with a photon index of @xmath41.5 modified by the partial covering absorption . this result is consistent with that of asca . in 4.57.5 kev , an excess flux over the continuum is clearly seen as a broad feature , in which significant flux variations are detected . however , no clear change was noticed in its profile . the absence of a profile change in spite of the significant flux variations in the broad and skewed feature strongly indicates that the feature is produced through a single mechanism . we studied the time scale of the flux variations of the narrow line . it was found that the flux variations are not significant on time scales of @xmath63@xmath56 s , whereas they become significant on time scales of @xmath56@xmath57 s. we also studied whether or not an introduction of a smearing effect can improve the correlation between the line flux and the continuum flux . we redistributed the continuum flux light curve with a simple method approximating a smear and delay . we found that the agreement between the line flux light curve and the smeared light curve of the continuum flux becomes acceptable when we introduce a smear and a delay on a time scale of @xmath77 s. these results concerning the line flux variations and the effect of a smear strongly suggest that the line emitting region should have a size extent of 10@xmath83 cm . we further studied the time scale of the variation in the absorption column density . a column density of the order of @xmath84 atom @xmath60 is necessary to reproduce the continuum spectrum , and is found to vary significantly on a time scale of @xmath63@xmath56 s with a relative amplitude of @xmath420% . taking account of the relative amplitude of the variation , the size of the absorber is indicated to be no larger than @xmath1 cm , which is just the size of the line emitting region . the absorber should be around , or inside , the line emitting region . we tried a continuum plus a disk - line model to reproduce the energy spectra , and found that this model can reproduce the energy spectra at 210 kev for both asca and rxte data . as a result of the previous discussion , the size of the line emitting region should be as large as @xmath1 cm . in the disk - line model , the line is assumed to be emitted from a region with a size of several to ten times the schwarzschild radius . if the size of @xmath1 cm deduced from observational results corresponds to several to ten - times the schwarzschild radius , we need to assume a central mass close to @xmath2 @xmath3 . this central mass is not consistent with @xmath57 @xmath3 estimated for ngc 4151 from various methods , e.g. x - ray variability time scale @xcite and the analysis of the high ionization lines in the broad line region @xcite . this mass is rather close to the typical mass of a galaxy , and would be too large as the mass of the central black hole . the presence of the absorber around or inside the line emitting region would also be difficult to explain in the context of the `` disk - line '' model . if the line emitting region has a size of several to ten - times the schwarzschild radius , the absorber should necessarily be located at a region very close to the central black hole . this may not be consistent with the unified scheme in which the heavy absorption as seen in ngc 4151 is considered to be due to a dusty torus around the central active region . we obtain an inclination angle of the disk to be @xmath85 . this inclination angle is smaller than the generally accepted value @xmath86 , which is supported by the chandra observation @xcite and the optical observations . this is also a disadvantage of the `` disk - line '' model . as discussed above , there exist some serious difficulties in the disk - line model , which are not consistent with a reasonable , common picture of agn , although we can not completely exclude its possibility . we consider here an alternative explanation for the origin of the broad and skewed profile , in which the narrow peak at 6.4 kev should be a narrow line and the red and blue wings should be a part of the continuum . however , according to the constant profile of the broad feature around 57 kev , the narrow line and the continuum should have a strong physical connection . the line energy at 6.4 kev strongly suggests that the line is the fluorescent k - line from neutral or low ionization iron . if the line is really emitted through a fluorescent process , it implies that some amount of matter exists in the vicinity of the x - ray source . this matter should be irradiated by x - rays from the x - ray source and the fluorescent lines should be emitted there . if this is the case , continuum x - rays should also be emitted from the matter through electron scattering of irradiating x - rays . this reflected component should be observed together with the fluorescent line , and could be the red and blue wings . when an x - ray from the source is absorbed by an iron atom in the x - ray reflector through photo - ionization of a k - electron , a fluorescent k - line is emitted with a certain probability called a fluorescent yield . on the other hand , when an x - ray from the source hits an electron in the reflector , it could be re - emitted towards us through thomson ( compton ) scattering . the cross section of thomson scattering is about @xmath87 @xmath88 . hence , if the column density of the x - ray reflector is sufficiently large , an x - ray penetrated into the reflector would , on average , experiences photo - electric absorption by matter with a column density of about @xmath89 @xmath60 before being scattered by an electron . as a result , x - rays reflected by sufficiently thick matter should have a spectrum with a continuum shape being the same as the intrinsic x - rays , but suffering from photo - electric absorption by matter with a column density of about @xmath89 @xmath60 . in fact , the broad and skewed feature can be reproduced by a model consisting of a narrow line at 6.4 kev , and a power law with the same slope as the remaining continuum and with a photo - electric absorption by a column of @xmath89 @xmath60 . the equivalent width of the 6.4 kev line with respect to the heavily absorbed component is about 2 kev . this is roughly consistent with a value expected from a case when the heavily absorbed component is just the reflected component ( see e.g. @xcite ) . if we adopt the above model for the broad and skewed feature around 4.58 kev , the total spectrum becomes to have three differently absorbed components with the same continuum shape . the absorption column densities are about @xmath90 @xmath60 , ( 12)@xmath91 @xmath60 , and about @xmath89 @xmath60 . among the three components , two components absorbed by smaller columns are found to be variable on a time scale of @xmath63@xmath56 s , whereas the component with the absorption by @xmath92 @xmath60 was steady on that time scale . this strongly suggests that the heavily absorbed component has a different origin from the other two components . the time variability of the heavily absorbed component is rather similar to that of the fluorescent iron line . this indicates a strong physical coupling between the fluorescent iron line and the heavily absorbed component , and strongly supports the idea that the heavily absorbed component is a reflected emission by cold matter which is also responsible for the fluorescent iron line . the above model fits both the asca spectra at 210 kev and the rxte spectra at 224 kev well . the model can also fit the data in a higher energy range . actually , if we extrapolate the best - fit model in the energy range of 2.824 kev to 100 kev , no significant discrepancy is recognized between the model and the observed energy spectra . therefore , this model can explain wider range of energy spectra without the spectral break than the model based on the disk - line . if the reflection is really at work , the covering fraction of the absorber may be related to the solid angle of the reflector , @xmath93 , subtending to the x - ray source . the covering fraction that we obtained for an absorber of @xmath89 @xmath60 is @xmath94 , which corresponds to @xmath95 . the reflection structure was detected in the energy spectra of ngc 4151 with the ginga / osse data @xcite and the bepposax data @xcite . the bepposax data indicated that the relative contribution of the reflection changed with time . the covering fraction of the reflector so far reported ranges over @xmath96 @xcite . the covering fraction which we obtained is slightly larger than this range , but is not very different , if we consider the different modeling of the reflection component . based on the reflection model , we may be able to constrain the mass of the central black hole using the iron line parameters . from the model fitting , we found that the iron line is consistent with having no intrinsic width , and its upper limit is 92 ev . we also found that the iron line emitting region has a size of @xmath1 cm . it may be natural to assume that the line emission region is located around the central black hole and is rotating at the kepler velocity around the black hole . then , the line width may be determined by the doppler effect ; the upper limit of the line width can be related to the upper limit of the kepler velocity as @xmath97 . this may be converted to the mass of the black hole using the relation @xmath98 , which yields the upper limit of the black hole mass , @xmath99 @xmath100 . here , we assume @xmath101 cm . this is consistent with the estimated mass of @xmath57 @xmath100 . if the mass of the central black hole is @xmath57 @xmath100 , a few times 10@xmath102 cm corresponds to @xmath103 @xmath31 . at this distance , the x - ray reflector should exist . since the reflected component shows evidence of absorption by a column of @xmath89 @xmath60 , the reflector should be thomson thick . the solid angle of the reflector , as viewed from the central x - ray source , should be about @xmath104 . this reflector would be a so - called dust torus , which is generally believed to exist around the central engine in seyfert galaxies . it has been shown that the continuum spectrum of ngc 4151 needs a partial covering by x - ray absorbers with a column density of ( 12)@xmath91 @xmath60 . the absorption column and the covering factor have been found to vary on a time scale of @xmath63@xmath56 s. as already discussed above , this leaky absorber should exist around or inside the line emitting region . if a dusty torus emits the line , a geometrical relation between a dusty torus and a broad line region seems to be consistent with the observed constraint on the geometrical relation between the line emitting region and the leaky absorber . the absorbers could be relatively cold clouds in the broad line region . if so , the typical time scale of a change of the leaky absorption should be roughly estimated by @xmath105 , where @xmath106 is the size of the x - ray emitting region and @xmath107 the velocity of the x - ray absorbing clouds . since the typical velocity of the broad line clouds is several times @xmath108 cm s@xmath13 and the typical time scale of the absorption change is a few times @xmath63 s , the size of the emission region may be about @xmath109 cm . this size corresponds to a few ten - times @xmath31 of a black hole with a mass of @xmath57 @xmath100 , and is consistent with a natural expectation that a region close to the central black hole should be emitting the continuum x - rays . arp , h. 1997 , , 319 , 33 burke , b. e. , mountain , r. w. , daniels , p. j. , & dolat , v. s. 1994 , ieee trans . 41 , 375 clavel , j. , altamore , a. , boksenberg , a. , bromage , g. e. , elvius , a. , pelat , d. , penston , m. v. , perola , g. c. , snijders , m. a. , & ulrich , m. h. 1987 , , 321 , 251 edelson , r , & nandra , k. 1999 , , 514 , 682 evans , i. n. , tsvetanov , z. , kriss , g. a. , ford , h. c. , caganoff , s. , & koratkar , a. p. 1993 , , 417 , 82 fabian , a. c. , rees , m. j. , stella , l. , & white , n. e. 1989 , , 238 , 729 fabian , a. c. , nandra , k. , reynolds , c. s. , brandt , w. n. , otani , c. , tanaka , y. , inoue , h. , & iwasawa , k. 1995 , , 277 , l11 fabian , a. c. , iwasawa , k. , reynolds , c. s. , & young , a. j. 2000 , , 112 , 1145 hayashida , k. , miyamoto , s. , kitamoto , s. , negoro , h. , & inoue , h. 1998 , , 500 , 642 inoue , h. 1985 , , 40 , 317 jahoda , k. , swank , j. h. , giles , a. b. , stark , m. j. , strohmayer , t. , zhang , w. , & morgan , e. h. 1996 , spie , 2808 , 59 makishima , k. , tashiro , m. , ebisawa , k. , ezawa , h. , fukazawa , y. , gunji , s. , hirayama , m. , idesawa , e. et al . 1996 , , 48 , 171 morrison , r. , & mccammon , d. 1983 , , 270 , 119 nandra , k. , george , i. m. , mushotzky , r. f. , turner , t. j. , & yaqoob , t. 1997 , , 477 , 602 nishiura , s. , murayama , t. , & taniguchi , y. 1998 , , 50 , 31 ogle , p. m. , marshall , h. m. , lee , j. c. , & canizares , c. r. 2000 , , 545 , l81 ohashi , t. , ebisawa , k. , fukazawa , y. , hiyoshi , k. , horii , m. , ikebe , y. , ikeda , h. , inoue , h. et al . 1996 , , 48 , 157 osterbrock , d. e. & koski , a. t. 1976 , , 176 , 61 piro , l. , nicastro , f. , feroci , m. , grandi , p. , parmar , a. , oosterbroek , t. , mineo , t. , piraino , s. et al . 1998 , b ( proc . suppl . ) , 69 , 481 serlemitsos , p. j. , jalota , l. , soong , y. , kunieda , h. , tawara , y. , tsusaka , y. , suzuki , h. , sakima , y. et al . 1995 , , 47 , 105 tanaka , y. , inoue , h. , & holt , s. s. 1994 , , 46 , l37 tanaka , y. , nandra , k. , fabian , a. c. , inoue , h. , otani , c. , dotani , t. , hayashida , k. , iwasawa , k. et al . 1995 , , 375 , 659 turner , t. j. , george , i. m. , nandra , k. , & mushotzky , r. f. 1998 , , 493 , 91 ulrich , m .- h . , & horne , k. 1996 , , 283 , 748 ulrich , m .- h . 2000 , , 10 , 135 wang , j .- x . , zhou , y .- y . , & wang , t .- 1999 , , 523 , l129 wang , j .- x . , wang , t .- , & zhou , y .- y . 2001 , , 549 , 891 weaver , k. a. , mushotzky , r. f. , arnaud , k. a. , serlemitos , p. j. , marshall , f. e. , petre , r. , jahoda , k. m. , smale , a. p. , & netzer , m. 1994 , , 423 , 621 weaver , k. a. , & reynolds , c. s. 1998 , , 503 , l39 yaqoob , t. , edelson , r. , weaver , k. a. , warwick , r. s. , mushotzky , r. f. , serlemitsos , p. j. , & holt , s. s. , 1995 , , 453 , l81 yang , y. , wilson , a. s. , & ferruit , p. 2001 , , 563 , 124 zdziarski , a. a. , johnson , w. n. , & magdziarz , p. 1996 , , 283 , 193 lccccccsequence number & 1 & 2 & 3 & 4 & 5 & 6 + @xmath64 [ @xmath110 atom @xmath60 ] & @xmath112 & @xmath113 & @xmath114 & @xmath115 & @xmath116 & @xmath116 + @xmath65 [ % ] & @xmath117 & @xmath118 & @xmath118 & @xmath119 & @xmath120 & @xmath121 + inclination angle [ deg ] & @xmath122 & @xmath123 & @xmath122 & @xmath124 & @xmath125 & @xmath126 + line flux [ @xmath127 photon @xmath60 s@xmath13 ] & @xmath129 & @xmath129 & @xmath129 & @xmath130 & @xmath131 & @xmath129 + @xmath132 & 1.10/140 & 1.02/140 & 0.99/140 & 1.03/140 & 1.02/140 & 1.03/140 + + lccccccsequence number & 1 & 2 & 3 & 4 & 5 & 6 + photon index ( @xmath38 ) & @xmath133 & @xmath134 & @xmath135 & @xmath136 & 1.55@xmath137 & 1.45@xmath138 + break energy [ kev ] & @xmath139 & @xmath140 & @xmath139 & @xmath140 & @xmath141 & @xmath142 + photon index ( @xmath39 ) & @xmath143 & @xmath144 & @xmath145 & @xmath146 & 1.89@xmath147 & 1.87@xmath148 + @xmath149 [ 10@xmath150 atom @xmath60 ] & @xmath151 & @xmath152 & @xmath153 & @xmath154 & @xmath155 & @xmath156 + @xmath65 [ % ] & @xmath157 & @xmath158 & @xmath159 & @xmath160 & @xmath161 & @xmath162 + line flux [ 10@xmath163 photon @xmath60 s@xmath13 ] & @xmath164 & @xmath165 & @xmath166 & @xmath167 & @xmath168 & @xmath169 + @xmath132 & 1.07/46 & 1.40/46 & 1.12/46 & 1.48/46 & 1.14/46 & 1.27/46 + sequence number & 7 & 8 & 9 & 10 & 11 + photon index ( @xmath38 ) & @xmath170 & @xmath171 & @xmath172 & @xmath134 & @xmath173 + break energy [ kev ] & @xmath139 & @xmath174 & @xmath175 & @xmath176 & @xmath174 + photon index ( @xmath39 ) & @xmath177 & @xmath178 & @xmath179 & @xmath180 & @xmath181 + @xmath149 [ 10@xmath150 atom @xmath60 ] & @xmath182 & @xmath183 & 9.7@xmath184 & @xmath185 & @xmath186 + @xmath65 [ % ] & @xmath187 & @xmath188 & @xmath189 & @xmath190 & @xmath191 + line flux [ 10@xmath163 photon @xmath60 s@xmath13 ] & @xmath192 & @xmath193 & @xmath194 & @xmath195 & @xmath196 + @xmath132 & 1.12/46 & 0.99/46 & 1.39/46 & 1.28/46 & 1.20/46 + + lccccccsequence number & 1 & 2 & 3 & 4 & 5 & 6 + photon index & @xmath197 & @xmath198 & @xmath199 & @xmath200 & @xmath201 & @xmath202 + @xmath64 [ 10@xmath150 atom @xmath60 ] & @xmath203 & @xmath112 & @xmath204 & @xmath205 & @xmath204 & @xmath203 + @xmath206 [ % ] & @xmath207 & @xmath208 & @xmath209 & @xmath210 & @xmath211 & @xmath212 + @xmath51 [ 10@xmath150 atom @xmath60 ] & @xmath213 & @xmath214 & @xmath215 & @xmath216 & @xmath217 & @xmath218 + @xmath48 [ % ] & @xmath219 & @xmath220 & @xmath221 & @xmath222 & @xmath223 & @xmath224 + line flux [ @xmath127 photon @xmath60 s@xmath13 ] & @xmath225 & @xmath226 & @xmath227 & @xmath228 & @xmath229 & @xmath230 + @xmath132 & 0.99/46 & 0.94/46 & 0.91/46 & 0.88/46 & 0.97/46 & 0.87/46 + sequence number & 7 & 8 & 9 & 10 & 11 + photon index & @xmath231 & @xmath232 & @xmath233 & @xmath234 & @xmath235 + @xmath64 [ 10@xmath150 atom @xmath60 ] & @xmath236 & @xmath205 & @xmath237 & @xmath140 & @xmath236 + @xmath206 [ % ] & @xmath238 & @xmath239 & @xmath240 & @xmath241 & @xmath242 + @xmath51 [ 10@xmath150 atom @xmath60 ] & @xmath243 & @xmath244 & @xmath245 & @xmath246 & @xmath247 + @xmath48 [ % ] & @xmath248 & @xmath249 & @xmath250 & @xmath251 & @xmath252 + line flux [ 10@xmath163 photon @xmath60 s@xmath13 ] & @xmath253 & @xmath254 & @xmath254 & @xmath255 & @xmath256 + @xmath132 & 0.98/46 & 0.91/46 & 0.99/46 & 0.91/46 & 0.90/46 + + lccccccsequence number & 1 & 2 & 3 & 4 & 5 & 6 + @xmath64 [ @xmath110 atom @xmath60 ] & @xmath139 & @xmath257 & @xmath141 & @xmath258 & @xmath259 & @xmath142 + @xmath65 [ % ] & @xmath260 & @xmath261 & @xmath262 & @xmath263 & @xmath264 & @xmath265 + @xmath30 [ @xmath110 atom @xmath60 ] & @xmath266 & @xmath267 & @xmath268 & @xmath225 & @xmath269 & @xmath270 + line flux [ @xmath127 photon @xmath60 s@xmath13 ] & @xmath271 & @xmath272 & @xmath273 & @xmath272 & @xmath273 & @xmath274 + @xmath132 & 1.04/140 & 0.97/140 & 0.94/140 & 0.99/140 & 0.99/140 & 1.02/140 +

we have studied the origin of the broad and skewed feature at 4.57.5 kev in the energy spectra of ngc 4151 using the asca and rxte data . the feature consists of a narrow peak at 6.4 kev and a broad wing extended between 4.57.5 kev . an analysis of the long - term variations revealed that the feature became variable only on a time scale longer than @xmath0 s. through a comparison with the continuum variabilities , we found that the emission region of the excess flux at 4.57.5 kev has an extent of @xmath1 cm . the broad and skewed feature at 4.57.5 kev may be explained by the so - called `` disk - line '' model . if so , the size of the line - emitting region , @xmath1 cm , should be equal to several or ten - times the schwarzschild radius of the central black hole . this results in a black hole mass of @xmath2 @xmath3 , which may be too large for ngc 4151 . we propose an alternative explanation for the broad and skewed feature , i.e. a `` reflection '' model , which can also reproduce the overall energy spectra very well . in this model , cold matter with a sufficiently large column density is irradiated by x - rays to produce a reflected continuum , which constitutes the broad wing of the feature , and narrow fluorescent lines . the equivalent width of the iron fluorescent line ( @xmath42 kev ) and the upper limit of its width ( @xmath5 ev ) are also consistent with this model . from these results and considerations , we conclude that the `` disk - line '' model has difficulty to explain the spectral variations of ngc 4151 , and the reflection model is more plausible .

transient lunar phenomena ( tlps or ltps ) are defined for the purposes of this investigation as localized ( smaller than a few hundred km across ) , transient ( up to a few hours duration , and probably longer than typical impact events - less than 1s to a few seconds ) , and presumably confined to processes near the lunar surface . how such events are manifest is summarized by cameron ( 1972 ) . in paper i ( crotts 2008 ; see also crotts 2009 ) we study the systematic behavior ( especially the spatial distribution ) of tlp observations - particularly their significant correlations with tracers of lunar surface outgassing , and we are thereby motivated to understand if this correlation is directly causal . numerous works have offered hypotheses for the physical cause of tlps ( mills 1970 , garlick et al . 1972a , b , geake & mills 1977 , cameron 1977 , middlehurst 1977 , hughes 1980 , robinson 1986 , zito 1989 , carbognani 2004 , davis 2009 ) , but we present a methodical examination of the influence of outgassing , exploring quantitatively how outgassing might produce tlps . furthermore , it seems likely that outgassing activity is concentrated in several areas , which leads one to ask how outgassing might interact with and alter the regolith presumably overlying the source of gas . reviews of similar processes exist but few integrate apollo - era data e.g. , stern ( 1999 ) , mukherjee ( 1975 ) , friesen ( 1975 ) . as the final version of this paper approached completion , several papers were published regarding the confirmed discovery of hydration of the lunar regolith . fortunately , we deal here with the special effects of water on lunar regolith and find that many of our predictions are borne out in the recently announced data . we will deal with this explicitly in 5 . several experiments from apollo indicate that gas is produced in the vicinity of the moon , even though these experiments disagree on the total rate : 1 ) lace ( lunar atmosphere composition experiment on _ apollo 17 _ ) , @xmath10.1 g s@xmath2 over the entire lunar surface ( hodges et al . 1973 , 1974 ) ; 2 ) side ( suprathermal ion detector experiment on _ apollo 12 , 14 , 15 _ ) , @xmath17 g s@xmath2 ( vondrak et al . 1974 ) ; 3 ) ccge ( cold cathode gauge experiment on _ apollo 12 , 14 , 15 _ ) , @xmath360 g s@xmath2 ( hodges et al . 1972 ) . these measurements not only vary by more than two orders of magnitude but also in assayed species and detection methods . lace results here applies only to neutral @xmath4ar , @xmath5ar and @xmath6ne . by mass @xmath4ar predominates . side results all relate to ions , and perhaps include a large contribution from molecular species ( vondrak et al . 1974 ) . ccge measures only neutral species , not easily distinguishing between them . the lace data indicate @xmath4ar episodic outgassing on timescales of a few months or less ( hodges & hoffman 1975 ) , but resolving this into faster timescales is more ambiguous . in this discussion we adopt the intermediate rate ( side ) , about 200 tonne y@xmath2 for the total production of gas , of all species , ionized or neutral . the lace is the only instrument to provide compositional ratios , which also include additional , rarer components in detail . we will use these ratios and in some cases normalize them against the side total . much of the following discussion is only marginally sensitive to the actual composition of the gas . for many components of molecular gas at the lunar surface , however , there is a significant possible contribution from cometary or meteoritic impacts , and a lesser amount from solar wind / regolith interactions . the influx of molecular gas from comets and meteorites are variously estimated , usually in the range of tonnes or tens of tonnes per year over the lunar surface ( see anders et al . 1973 , morgan & shemansky 1991 ) . cometary contributions may be sporadically greater ( thomas 1974 ) . except for h@xmath7 , solar wind interactions ( mukhergee 1975 ) provide only a small fraction of the molecular concentration seen at the surface ( which are only marginally detected : hoffman & hodges 1975 ) . there is still uncertainty as to what fraction of this gas is endogenous . current data do not succeed in resolving these questions , but we will return to consider them later in the context of gas seepage / regolith interactions . in this paper we consider various effects of outgassing through the regolith , and find the most interesting simple effect occurs when the flow is high enough to cause disruption of the regolith by an explosion to relieve pressure ( 2 ) , which we compare to fluidization . another interesting effect occurs when the gas undergoes a phase change while passing through the regolith ( 3 ) , which seems to apply only to water vapor . this leads primarily to the prediction of the likely production of subsurface ice , particularly in the vicinity of the lunar poles . these effects suggest a variety of observational / experimental approaches , which we summarize in 4 . in 5 we discuss the general implications of these findings , with specific suggestions as to how these might guide further exploration , particularly in respect to contamination by anthropogenic volatiles . we also discuss the relevance of the predictions in 4 to very recent discoveries regarding lunar regolith hydration . first , let us make a few basic points about outgassing and the regolith . one can easily picture several modes in which outgassing volatiles might interact with regolith on the way to the surface . these modes will come into play with increasing gas flow rate and/or decreasing regolith depth , and we simply list them with mneumonic labels along with descriptions : \1 ) choke : complete blockage below the regolith , meaning that any chemistry or phase changes occur within the bedrock / megaregolith ; \2 ) seep : gas is introduced slowly into the regolith , essentially molecule by molecule ; \3 ) bubble : gas is introduced in macroscopic packets which stir or otherwise rearrange the regolith ( such as `` fluidization '' e.g. , mills 1969 ) ; \4 ) gulp : gas is introduced in packets whose adiabatic expansion deposits kinetic energy into regolith and cools the gas , which therefore might even undergo a phase change ; \5 ) explode : gas is deposited in packets at base of the regolith leading to an explosion ; and \6 ) jet : gas simply flows into the vacuum at nearly sound speed with little entrained material . while the intermediate processes might prove interesting , the extreme cases are probably more likely to be in effect and will receive more of our attention . in fact choking behavior might lead to explosions or geysers , when the pressure blockage is released . since these latter two processes involve primarily simple hydrodynamics ( and eventually , newtonian ballistics ) , we will consider them first , and how they might relate to tlps . if outgassing occurs at a rate faster than simple percolation can sustain , and where regolith obstructs its path to the surface , the accumulation of the gas will disrupt and cause bulk motion of the intervening regolith . the outgassing can lift the regolith into a cloud in the temporary atmosphere caused by the event . the presence of such a cloud has the potential to increase the local albedo from the perspective of an outside observer due to increased reflectivity and possible mie scattering of underlying regolith . additionally , volatiles buried in the regolith layer could become entrained in this gas further changing the reflective properties of such a cloud . garlick et al . ( 1972b ) describe fluidization of lunar regolith , in which dust is displaced only temporarily and/or over small distances compared to ballistic trajectories , but we will assume that we are dealing with more rapid changes . let us construct a simple model of explosive outgassing through the lunar surface . for such an event to occur , we assume a pocket of pressurized gas builds at the base of the regolith , where it is delivered by transport through the crust / megaregolith presumably via channels or cracks , or at least faster diffusion from below . given a sufficient flow rate ( which we consider below ) , gas will accumulate at this depth until its internal pressure is sufficient to displace the overlying regolith mass , or some event releases downward pressure e.g. , impact , moonquake , incipient fluidization , puncturing a seal , etc . we can estimate the minimal amount of gas alone required to cause explosive outgassing by assuming that the internal energy of the buried gas is equal to the total energy necessary to raise the overlying cone of regolith to the surface . this `` minimal tlp '' is the smallest outgassing event likely to produce potentially observable disruption at a new site , although re - eruption through thinned regolith will require less gas . we consider the outgassing event occurring in two parts illustrated in figure 1 . initially , the gas bubble explodes upward propelling regolith with it until it reaches the level of the surface ; we assume that the plug consists of a cone of regolith within 45@xmath8 of the axis passing from the gas reservoir to the surface , normal to the surface . through this process the gas and regolith become mixed , and we assume they now populate a uniform hemispherical distribution of radius @xmath9 m on the surface . at this point , the gas expands into the vacuum and drags the entrained regolith outward until the dust cloud reaches a sufficiently small density to allow the gas to escape freely into the vacuum and the regolith to fall eventually to the surface . we consider this to be the `` minimal tlp '' for explosive outgassing , as there is no additional reservoir that is liberated by the event beyond the minimum to puncture the regolith . one could also imagine triggering the event by other means , many of which might release larger amounts of gas other than that poised at hydrodynamical instability . for the initial conditions of the first phase of our model , we assume the gas builds up at the base of the regolith layer at a depth of 15 m ( for more discussion of this depth , see 3 ) . we set the bulk density of the regolith at @xmath10 g @xmath11 ( mckay et al . 1991 ) , thereby setting the pressure at this depth at 0.45 atm . because of the violent nature of an explosive outgassing , we assume that the cone of dust displaced will be 45@xmath8 from vertical ( comparable to the angle of repose for a disturbed slope of this depth : carrier , olhoeft & mendell 1991 ) . the mass of overlying regolith defined by this cone is @xmath12 kg . in order to determine the mass of gas required to displace this regolith cone , we equate the internal energy of this gas bubble with the potential energy ( @xmath13 , where @xmath14 m s@xmath15 ) required to lift the cone of regolith a height @xmath16 m to the surface , requiring 47,000 moles of gas . much of the gas found in outgassing events consists of @xmath17he , @xmath5ar and @xmath4ar ( see 1 ) , so we assume a mean molar mass for the model gas of @xmath18 g mol@xmath2 , hence 940 kg of gas is necessary to create an explosive outgassing event . the temperature at this depth is @xmath19c ( see 3 ) , consequently implying an overall volume of gas of 2400 m@xmath20 or a sphere 8.3 m in radius . what flow rate is needed to support this ? using fick s diffusion law , @xmath21 , where the gas number density @xmath22 @xmath11 is taken from above , and drops to zero through 15 m of regolith in @xmath23 . the diffusivity @xmath24 is 7.7 and 2.3 cm@xmath0 s@xmath2 for he and ar , respectively , in the knudsen flow regime for basaltic lunar soil simulant ( martin et al . 1973 , where @xmath26 is the absolute temperature . the sticking time of the gas molecules , or heat of absorption , becomes significant if the gas is more reactive or the temperature is reduced . unfortunately we find no such numbers for real regolith , although we discuss realistic diffusivities for other gases below . ] ) , so we adopt @xmath27 cm@xmath0 s@xmath2 for our assumed he / ar mixture . ( for any other gas mixture of this molecular weight , @xmath24 would likely be smaller ; below we also show that @xmath24 tends to be lower for real regolith . ) over the area of the gas reservoir , this implies a mass leakage rate of 2.8 g s@xmath2 , or @xmath140% of the total side rate . with the particular approximations made about the regolith diffusivity , this is probably near the upper limit on the leakage rate . at the surface of the regolith , this flow is spread to a particle flux of only @xmath28 cm@xmath15 , which presumably causes no directly observable optical effects . the characteristic time to drain ( or presumably fill ) the reservoir is 4 d. the second phase of the simulation models the evolution of dust shells and expanding gas with a spherically symmetric 1d simulation centered on the explosion point . the steps of the model include : 1 ) the regolith is divided into over 600 bins of different mean particle size . these bins are logarithmically spaced over the range @xmath30 mm to @xmath31 m according to the regolith particle size distribution from sample 72141,1 ( from mckay et al . the published distribution for sample 72141,1 only goes to 2@xmath32 m , but other sources ( basu & molinaroli 2001 ) indicate a component extending below 2@xmath32 m , so we extend our size distribution linearly from 2@xmath32 m to 0@xmath32 m . furthermore , we assume the regolith particles are spherical in shape and do not change in shape or size during the explosion ; 2 ) to represent the volume of regolith uniformly entrained in the gas , we create a series of 1000 concentric hemispherical shells for each of the different particle size bins ( i.e. , roughly 600,000 shells ) . each of these shells is now independent of each other and totally dependent on the gas - pressure and gravity for motion ; 3 ) we further assume that each regolith shell remains hemispherical throughout the simulation . explicitly , we trace the dynamics of each shell with a point particle , located initially 45 degrees up the side of the shell ; 4 ) we calculate the outward pressure of the gas exerted on the dust shells . the force from this pressure is distributed among different shells of regolith particle size weighted by the total surface area of the grains in each shell . we calculate each shell s outward acceleration and consequently integrate their equations of motion using a timestep @xmath33 s ; 5 ) we calculate the diffusivity of each radial shell ( in terms of the ability of the gas to move through it ) by dividing the total surface area of all dust grains in a shell by the surface area of the shell itself ( assuming grains surfaces to be spherical ) ; 6 ) starting with the largest radius shell , we sum the opacities of each shell until we reach a gas diffusive opacity of unity . gas interior to this radius can not `` see '' out of the external regolith shells and therefore remains trapped . gas outside of this unit opacity shell is assumed to escape and is dropped from the force expansion calculation . dust shells outside the unit opacity radius are now assumed to be ballistic ; 7 ) we monitor the trajectory of each dust shell ( represented by its initially 45@xmath8 particle ) until it drops to an elevation angle of 30@xmath8 ( when most of the gas is expanding above the particle ) , at which time this particle shell is no longer supported by the gas , and is dropped from the gas - opacity calculation ; 8) an optical opacity calculation is made to determine the ability of an observer to see the lunar surface when looking down on the cloud . we calculate the downward optical opacity ( such as from earth ) by dividing the total surface area of the dust grains in a shell by the surface area of the shell as seen from above ( @xmath34 ) . starting with the outmost dust shell , we sum downward - view optical opacities until we reach optical depth @xmath35 and @xmath36 to keep track the evolution of this cloud s appearance as seen from a distance ; 9 ) we return to step # 4 above and iterate another timestep @xmath37 , integrating again the equations of motion . we continue this algorithm until all gas is lost and all regolith has fallen to the ground . finally , when all dust has fallen out , we calculate where the regolith ejecta have been deposited . because we re representing each shell as a single point for the purposes of the equations of motion calculations , we want to do more than simply plot the location of each shell - particle on the ground to determine the deposition profile of ejected regolith . thus we create a template function for the deposition of a ballistic explosion of a single spherical shell of material . by applying this template to each shell - particle s final resting location , we better approximate the total deposition of material from that shell . we then sum all of the material from the @xmath38 shells to determine the overall dust ejecta deposition profile . there are obvious caveats to this calculation . undoubtably the release mechanism is more complex than that adopted here , but this release mode is sufficiently simple to be modeled . secondly , the diffusion constant , and therefore the minimal flow rate , might be overestimated due to the significant ( but still largely unknown ) decrease in regolith porosity with increasing depth on the scale of meters ( carrier et al . 1991 ) , plus the liklihood that simulants used have larger particles and greater porosity than typical regolith . lastly , the regolith depth of 15 meters might be an overestimate for some of these regions , which are among the volcanically youngest and/or freshest impacts on the lunar surface . this exception does not apply to plato and its active highland vicinity , however , and aristarchus is thickly covered with apparent pyroclastic deposits which likely have different but unknown depths and diffusion characteristics . we find results for this `` minimal tlp '' numerical model of explosive outgassing through the lunar regolith interesting in terms of the reported properties of tlps . figure 2 shows the evolution of the model explosion with time , as might be seen from an observer above , in terms of the optical depth @xmath39 and @xmath40 profiles of the model event , where @xmath39 is a rough measure of order unity changes in the appearance of the surface features , whereas @xmath40 is close to the threshold of the human eye for changes in contrast , which is how many tlps are detected ( especially the many without noticable color change ) . in both cases the cloud at the particular @xmath41 threshold value expands rapidly to a nearly fixed physical extent , and maintains this size until sufficient dust has fallen out so as to prevent any part of the cloud from obscuring the surface to this degree . easily - seen effects on features ( @xmath39 ) lasts for 50 s and extends over a radius of 2 km , corresponding to 2 arcsec in diameter , resolvable by a typical optical telescope but often only marginally so . in contrast , the marginally detectable @xmath40 feature extends over 14 km diameter ( 7.5 arcsec ) , lasting for 90 s , but is easily resolved . this model `` minimal tlp '' is an interesting match to the reported behavior of non - instantaneous ( not @xmath42 s ) tlps : about 7% of duration 90 s or less , and half lasting under about 1300 s. certainly there should be selection biases suppressing reports of shorter events . most tlp reports land in an envelope between about two minutes and one hour duration , and this model event lands at the lower edge of this envelope . furthermore , most tlps , particularly shorter ones , are marginally resolved spatially , as would be the easily - detectable component of the model event . this correspondence also seems interesting , given the simplicity of our model and the state of ignorance regarding relevant parameters . how might this dust cloud actually affect the appearance of the lunar surface ? first , the cloud should cast a shadow that will be even more observable than simple surface obscuration , blocking the solar flux from an area comparable to the @xmath39 region and visible in many orientations . experiments with agitation of lunar regolith ( garlick et al . 1972b ) show that the reflectance of dust is nearly always increased under fluidization , typically by about 20% and often by about 50% depending on the particular orientation of the observer versus the light source and the cloud . similar results should be expected here for our simulated regolith cloud . these increases in lunar surface brightness would be easily observable spread over the many square kilometers indicated by our model . furthermore , because the sub - micron particle sizes dominate the outer regions of the cloud , it seems reasonable to expect mie - scattering effects in these regions with both blue and red clouds expected from different sun - earth - moon orientations . figure 3 shows the typical fall - out time of dust particles as a function of size . particles larger than @xmath130 @xmath32 m all fall out within the first few seconds , whereas after a few tens of seconds , particles are differentiated for radii capable of contributing to wavelength - dedendent scattering . later in the event we should expect significant color shifts ( albeit not order - unity changes in flux ratios ) . the larger dynamical effects in the explosion cloud change rapidly over the event . half of the initially entrained gas is lost from the cloud in the first 3 s , and 99% is lost in the first 15 s. throughout the observable event , the remaining gas stays in good thermal contact with the dust , which acts as an isothermal reservoir . gas escaping the outer portions of the dust cloud does so at nearly the sound speed ( @xmath43 m s@xmath2 ) , and the outer shells of dust also contain particles accelerated to similar velocities . gas escaping after about 3 s does so from the interior of the cloud in parcels of gas with velocities decreasing roughly inversely with time . one observable consequence of this is the expectation that much of the gas and significant dust will be launched to altitudes up to about 50 km , where it may be observed and might affect spacecraft in lunar orbit . the longterm effects of the explosion are largely contained in the initial explosion crater ( nominally 14 m in radius ) , although exactly how the ejecta ultimately settle in the crater is not handled by the model . at larger radii the model is likely to be more reliable ; figure 4 shows how much dust ejecta is deposited by the explosion as a function of radius . beyond the initial crater , the surface density of deposited material varies roughly as @xmath44 , so it converges rapidly with distance . inside a radius of @xmath45 m , the covering factor of ejecta is greater than unity ; beyond this one expects coverage to be patchy . this assumes that the crater explosion is symmetric and produces few `` rays . '' the explosion can change the reflectivity by excavating fresh material . this would be evidenced by a @xmath110% drop in reflectance at wavelength @xmath46 nm caused by surface fe@xmath47 states in pyroxene and similar minerals ( adams 1974 , charette et al . likewise there is an increase in reflectivity in bluer optical bands ( buratti et al . 2000 ) over hundreds of nm . even though these photometric effects are compositionally dependent , we are interested only in differential effects : gradients over small distances and rapid changes in time . the lifetime of even these effects at 300 m radius is short , however , due to impact `` gardening '' turnover . the half - life of the ejecta layer at 300 m radius is only of order 1000 y ( from gault et al . 1974 ) , and shorter at large radius ( unless multiple explosions accumulate multiple layers of ejecta ) . at 30 m radius the half - life is of order 10@xmath48 y. from maturation studies of the 950 nm feature ( lucey et al . 2000 , 1998 ) , even at 30 m , overturn predominates over optical maturation rates ( over hundreds of my ) . the scale of outgassing in this model event , both in terms of gas release ( @xmath49 1 tonne ) and timescale ( @xmath49 4 d ) , are consistent with the total gas output and temporal granularity of outgassing seen in @xmath4ar , a dominant lunar atmospheric component . the fact that this model also recovers the scale of many features actually reported for tlps lends credence to the idea that outgassing and tlps might be related to each other causally in this way , as well as circumstantially via the rn@xmath50 episodes and tlp geographical correlation ( paper i ) . how often such an explosive puncturing of the regolith layer by outgassing should occur is unknown , due to the uncertainty in the magnitude and distribution of endogenous gas flow to the surface , and to some degree how the regolith reacts in detail to large gas flows propagating to the surface . also , a new crater caused by explosive outgassing will change the regolith depth , its temperature structure , and eventually its diffusivity . we will not attempt here to follow the next steps in the evolution of an outgassing `` fumerole '' in this way , but are inspired to understand how regolith , its temperature profile , and gas interact , as in the next section . furthermore , such outgassing might happen on much larger scales , or might over time affect a larger area . indeed , such a hypothesis is offered for the scoured region of depleted regolith forming the ina d feature and may extend to other regions around imbrium ( schultz et al . 2006 ) . our results here and in paper i bear directly on the argument of vondrak ( 1977 ) that tlps as outgassing events are inconsistent with side episodic outgassing results . the detection limits from alsep sites _ apollo 12 , 14 _ and _ 15 _ correspond to 16 - 71 tonne of gas per event at common tlp sites , particularly aristarchus . ( vondrak states that given the uncertainties in gas transportation , these levels are uncertain at the level of an order of magnitude . ) our `` minimal tlp event '' described above is 20 - 80 times less massive than this , however , and still visible from earth . it seems implausible that a spectrum of such events would never exceed the side limit , but it is not so obvious such a large event would occur in the seven - year alsep operations interval . also , this side limit interpretation rests crucially on alphonsus ( and ross d ) as prime tlp sites , both features which are rejected by our robust geographical tlp sieve in paper i. dust elutriation or particle segregation in a cloud agitated by a low - density gas , occurring in this model , could potentially generate large electrostatic voltages , perhaps relating to tlps ( mills 1970 ) . luminous discharges are generated in terrestrial volcanic dust clouds ( anderson et al . 1965 , thomas et al . 2007 ) . above we see dust particles remain suspended in a gas of number density @xmath51 to @xmath52 @xmath11 on scales of several tenths of a km to several km , a plausible venue for large voltages . in the heterogeneous lunar regolith , several predominant minerals with differing particle size may segregate under gas flow suspension and acceleration . assuming a typical particle size of @xmath53 m , and typical work function differences for particles of even well - defined compositions is problematic due to surface effects such as solar - wind / micrometeoritic weathering and exposed surface fe@xmath47 states . the following analysis suffices for two particles of different conducting composition ; a similar result arises via triboelectric interaction of two different dielectrics although the details are less understood . disturbed dust is readily charged for long periods in the lunar surface environment ( stubbs , vondrak & farell 2005 ) . ] of @xmath54 ev , two particles exchange charge upon contact until the equivalent of @xmath550.25v is maintained , amounting to @xmath56 coul = 1700 e@xmath57 . when these particles separate to distance @xmath58 , their mutual capacitance becomes @xmath59 . for @xmath60 m , if particles retain @xmath61 , voltages increase by @xmath62 times ! such voltages can not be maintained . paschen s coronal discharge curve reaches minimum potential at 137v for ar , 156v for he , for column densities @xmath63 of @xmath64 and @xmath65 cm@xmath15 , respectively , and rises steeply for lesser column densities ( and roughly proportional to @xmath63 for larger @xmath63 ) . similar optimal @xmath63 are found for molecules , with minimum voltages a few times higher e.g. , 420v at @xmath66 cm@xmath15 for co@xmath7 , 414v for h@xmath7s , 410v for ch@xmath67 , and @xmath63 for other molecules @xmath68 cm@xmath15 . the visual appearance of atomic emission in high voltage discharge tubes is well know , with he glowing pink - orange ( primarily at 4471.5 and 5875.7 : reader & corliss 1980 , pearse & gaydon 1963 ) , and ar glowing violet ( from lines 4159 - 4880 ) . if this applies to tlps , the incidence of intense red emission in some tlp reports ( cameron 1978 ) argues for another gas . to 13 ( mag arcsec@xmath15 ) in v , compared to 3.4 at full moon , so visible sources could be faint . ] ne is not an endogenous gas . common candidate molecules appear white or violet - white ( co@xmath7 , so@xmath7 ) or red ( water vapor - primarily h@xmath69 , which is produced in many hydrogen compounds ; ch@xmath67 - balmer lines plus ch bands at 390 and 431 nm ) . the initial gas density at the surface from a minimal tlp is @xmath70 @xmath11 , so initially the optimal @xmath63 for coronal discharge is on cm scales ( versus the initial outburst over tens of meters ) . as the tlp expands to 1 km radius , @xmath71 drops to @xmath72 @xmath11 , so the optimal @xmath63 holds over the scale of the entire cloud , likely the most favorable condition for coronal discharge . if gas kinetic energy converts to luminescence with , for instance , 2% efficiency , at this density this amounts to @xmath73 j m@xmath74 , or 100 j m@xmath15 , compared to the reflected solar flux of 100 w m@xmath15 , capable of a visible color shift for several seconds . perhaps a minimal tlp could sustain a visible coronal discharge over much of its @xmath75 min lifetime . these should also be observable on the nightside surface , too , since solar photoionization is seemingly unimportant in initiating the discharge , and there are additional factors to consider . referring back to scenarios ( 2 ) and ( 3 ) in 1 , the onset of fluidization ( mills 1969 ) marks the division between these two regimes of seepage and `` bubbling '' and has been studied ( siegal & gold 1973 , schumm 1970 ) . although laboratory test are made with coarser sieve particulates and much thinner dust layers in 1 @xmath76 gravity , we can scale the gas pressure needed for incipient fluidization by @xmath77 and thickness @xmath78 to find the threshold @xmath79 atm ( siegal & gold 1973 ) . correcting for less diffusive regolith , this pressure estimate is likely a lower limit . below this pressure simple gas percolation likely predominates . what processes occur during `` simple '' percolation ? were it not for phase changes of venting gas within the regolith , the composition of the gas might be a weak consideration in this paper ( except for perhaps the molecular / atomic mass ) , and temperature would likely only affect seepage as @xmath80 in the diffusivity . water plays a special role in this study ( separate from concerns regarding resource exploitation or astrobiology ) , in that it is the only common substance encountering its triple point temperature in passing through the regolith , at least in many locations . in this case water might not contribute to overpressure underneath the regolith leading to explosive outgassing . this would also imply that even relatively small volatile flows containing water would tend to freeze in place and remain until after the flow stops . for water this occurs at 0.01@xmath8c , corresponding to 0.006 atm in pressure ( the pressure dropping by a factor of 10 every @xmath125@xmath8 . ) effectively , water is the only relevant substance to behave in this fashion . the next most common substances may be large hydrocarbons such as nonane or benzene , obviously not likely abundant endogenous effluents from the interior . also h@xmath7so@xmath67 reaches its triple point , but changes radically with even modest concentrations of water . a similar statement can be made about hno@xmath81 , not a likely outgassing constituent . these will not behave as their pure state , either ; this leaves only h@xmath7o . water ( and sulfur ) has been found in significant concentration in volcanic glasses from the deep lunar interior ( saal et al . 2008 , friedman et al . 2009 ) , and has been liberated in large quantities in past volcanic eruptions . the measured quantities of tens of ppm imply juvenile concentrations of hundreds of ppm . from the heat flow measurements at the _ apollo 15 _ and _ 17 _ lunar surface experiment ( alsep ) sites ( langseth & keihm 1977 ) , we know that just below the surface , the stable regolith temperature is in the range of 247 - 253k ( dependent on latitude , of course ) , with gradients ( below 1 - 2 m ) of 1.2 - 1.8 deg m@xmath2 , which extrapolates to @xmath82c at @xmath83 m depths subsurface . with the exception of the outermost few centimeters , the entire regolith is below the triple point temperature and is too deep to be affected significantly by variations in heating over monthly timescales . this is an interesting depth , since in many areas the regolith is not quite this deep , as small as under a few meters near lichtenberg ( schultz & spudis 1983 ) and at the surveyor 1 site near flamsteed ( shoemaker & morris 1970 ) to depths at apollo sites ( summarized by mckay et al . 1991 ) near the @xmath82c depths calculated above , up to probably 20 m or more in the highlands , and 40 m deep north of the south pole - aitken basin ( bart & melosh 2005 ) . presumably , the fractured megaregolith supporting the regolith likely does not contain as many small particles useful for retaining water ice , as we detail below , but it may accumulate ice temporarily . recent heat flow analyses ( saito et al . 2007 ) account for longer timescale fluctuations placing the @xmath82c depth twice as far subsurface , increasing the lifetime of retained volatiles against sublimation accordingly ; for now we proceed with a more conventional , shorter - lived analysis . the escape of water and other volatiles into the vacuum is regulated by the state of the regolith and is presumably largely diffusive . we assume the knudsen flow regime ( low - density , non - collisional gas ) . of special importance is the measured abundance of small dust grains in the upper levels of the regolith , which perhaps pertains to depths @xmath84 m ( where bulk density is probably higher : carrier et al . assuming that particle distributions are self - similar in size distribution ( constant porosity ) , for random - walk diffusion out of a volume element @xmath85 , the diffusion time step presumably scales with the particle size @xmath86 , so the diffusion time @xmath87 . for particles of the same density , therefore , one should compute the diffusion time by taking a @xmath88-weighted average of particle sizes counted by mass , @xmath89 . this same moment of the distribution is relevant in 2 . published size distributions measured to sufficiently small sizes include again mckay et al . ( 1974 ) with @xmath90 @xmath32 m , and supplemented on smaller sizes with _ apollo 11 _ sample 10018 ( basu & molinaroli 2001 ) , which reduces the average to about 20 @xmath32 m . this is an overestimate because a large fraction ( 34 - 63% ) are agglutinates , which are groupings of much smaller particles . many agglutinates have large effective areas e.g. , @xmath91 , with values of a few up to 8 . ( here @xmath92 is a mean radius from the center of mass to a surface element . ) to a gas particle , the sub - particle size is more relevant than the agglutinate size , so the effective particle size of the entire sample might be much smaller , conceivably by a factor of a few . we compare this to experimental simulations , a reasonably close analogy being the sublimation of a slab of ice buried up to 0.2 m below a medium of simulant jsc mars-1 ( allen et al . 1998 ) operating at @xmath93k and 7 mbar ( chevrier et al . 2007 ) , close to lunar regolith conditions . this corresponds to the lifetime of 800 y for a 1 m thick ice layer covered by 1 m of regolith . the porosity of jsc mars-1 is 44 - 54% , depending on compactification whereas lunar soil has @xmath149% at the surface , perhaps 40% at a depth of 60 cm , and slightly lower at large depths ( carrier et al . lunar soil is somewhat less diffusive by solely this measure . the mean size @xmath89 of jsc mars-1 is 93 @xmath32 m , @xmath4910 times larger than that for _ apollo 17 _ and _ 11 _ regolith , accounting for agglutinates , so the sublimation timescale for regolith material is , very approximately , @xmath4910 ky ( perhaps up to @xmath130 ky ) . other simulants are more analogous to lunar regolith , so future experiments might be more closely relevant . converting a loss rate for 1 m below the surface to 15 m involves the depth ratio @xmath94 . farmer ( 1976 ) predicts an evaporation rate scaling as @xmath95 ( as opposed to the no - overburden analysis : ingersoll 1970 ) . experiments with varying depths of simulated regolith ( chevrier et al . 2007 ) show that the variation in lifetime indeed goes roughly as @xmath96 , implying a 1 m ice slab lifetime at 15 m on the order of @xmath97 to @xmath98 y. the vapor pressure for water ice drops a factor of 10 in passing from @xmath82c to current temperatures of about @xmath99c just below the surface ( also the naked - ice sublimation rate : andreas 2007 ) , which would indicate that @xmath190% of water vapor tends to stick in overlying layers ( without affecting the lifetime of the original layer , coincidentally ) . this begs the question of the preferred depth for an ice layer to form . the regolith porosity decreases significantly between zero and 1 m depth ( carrier et al . 1991 ) which argues weakly for preferred formation at greater depth . at 30 m depth or more , the force of overburden tends to close off porosity . the current best limit on water abundance is from the sunrise terminator abundances from lace , which produces a number ratio of h@xmath7o/@xmath4ar with a central value of 0.014 ( with @xmath100 limits of 0 - 0.04 ) . this potentially indicates an actual h@xmath7o/@xmath4ar outgassing rate ratio up to 5 times higher ( hoffman & hodges 1975 ) . adopting the side rate of 7 g s@xmath2 in the @xmath120 - 44 amu mass range , and assuming most of this is @xmath4ar ( vondrak , freeman & lindeman 1974 : given the much lower solar wind contributions of other species in this range ) , this translates to 0.1 g s@xmath2 of water ( perhaps up to 0.5 g s@xmath2 or 15 tonne y@xmath2 ) , in which case most of the gas must be ionized . the disagreement between side and lace is a major source of uncertainty ( perhaps due to the neutral / ionized component ambiguity ) . we discuss below that at earlier times the subsurface temperature was likely lower , but let us consider now the situation in which a source arises into pre - established regolith in recent times . we assume a planar diffusion geometry , again . in this case , we take spatial gradients over 15 m and scale the jsc mars-1 diffusivity of 1.7 cm@xmath0 s@xmath2 to 0.17 cm@xmath0 s@xmath2 for lunar regolith . since the triple - point pressure corresponds to number density @xmath101 @xmath11 , the areal particle flux density is @xmath102 s@xmath2 cm@xmath15 . for a large outgassing site , with the same water fraction of water indicated by lace e.g. , total outgassing of 7 g s@xmath2 including 0.1 g s@xmath2 of water , this rate can maintain a total area of 0.012 km@xmath0 at the triple - point pressure i.e. , a 125 m diameter patch . this is much larger than the 15 m regolith depth , bearing out our assumed geometry . if this ice patch were 1 m thick , for example , the ice would need to be replenished every 4000 y. of course , this is a simple model and many complications could enter . we consider briefly the effects of latitude , change in lunar surface temperature over geological time , and the effects of aqueous chemistry on the regolith . the temperature just below the surface is legislated by the time - averaged energy flux in sunlight , so it scales according to the stefan - boltzmann law from the temperature at the equator ( @xmath103 ) according to @xmath104 . this predicts a 6k temperature drop from the equator ( at about 252k ) to the latitude of the aristarchus plateau ( @xmath105 ) or the most polar subsurface temperature measurement by _ apollo 15 _ , a drop to 224k at plato ( @xmath106 ) , @xmath107200k for the coldest 10% of the lunar surface ( @xmath108 ) and @xmath107150k for the coldest 1% ( @xmath109 ) . , but there are flooded craters much higher . ] these translate into a regolith depth at the water triple - point of @xmath14 , 18 , 33 or 65 m deeper than at the equator , respectively , probably deeper in the latter cases than the actual regolith layer . permanently shadowed cold traps , covering perhaps 0.1% of the surface , have temperatures @xmath10760k ( e.g. , adorjan 1970 , hodges 1980 ) . ( note that the lunar south pole is a minor tlp site responsible for about 1% of robust report counts . ) since even at the equator the h@xmath7o triple point temperature occurs @xmath113 m below the surface , at increasing latitude this zone quickly moves into the megaregolith where the diffusivity is largely unknown but presumably higher ( neglecting the decrease in porosity due to compression by overburden ) . to study this , we assume a low diffusivity regolith layer 15 m deep overlying a high diffusivity layer which may contain channels directing gas quickly upward ( although perhaps not so easily horizontally ) . the diffusivity of the regolith near 0@xmath8c is dominated by elastic reflection from mineral surfaces , without sticking , whereas at lower temperatures h@xmath7o molecules stick during most collisions ( haynes , tro & george 1992 ) . this is especially the case if the surfaces are coated with at least a few molecular layers of h@xmath7o molecules , of negligible mass . the sticking behavior of h@xmath7o molecules on water ice has been studied over most of the temperature range relevant here ( washburn et al . 2003 ) ; but does depend somewhat on whether the ice is crystalline or amorphous ( speedy et al . in contrast the sticking behavior of h@xmath7o molecules on lunar minerals is much less well known . the lunar simulant diffusivity value above corresponds to a mean free path time of @xmath11 @xmath32s for h@xmath7o molecules near 0@xmath8c . in contrast the timescale for h@xmath7o molecules sticking on ice is ( from schorghofer & taylor [ 2007 ] and references therein ) : @xmath110 where @xmath111 is the areal density of h@xmath7o molecules on ice @xmath112 cm@xmath15 for density @xmath113 , and molecular mass @xmath32 . the sticking fraction @xmath69 varies from about 70% to 100% for @xmath114k to 120k . the equilibrium vapor pressure is given by @xmath115 $ ] where @xmath116 and @xmath117 are the triple point pressure and temperature , respectively , and sublimation enthalpy @xmath118 kj / mole . this expression and laboratory measurements imply a sticking timescale @xmath119s at @xmath120k , 1 ms at 200k , 1 s at 165k , 1 hr at 134k , and 1 yr at 113k . the sticking timescale quickly and drastically overwhelms the kinetic timescale at lower temperatures . this molecular behavior has a strong effect on the size of the ice patch maintained by the example source considered above . simply scaling by the time between molecular collisions , corresponding to a 125 m diameter ice patch at @xmath103 , we find at the base of the regolith a 160 m patch at @xmath121 ( aristarchus plateau ) , 580 m at @xmath122 ( plato ) , 2.3 km at @xmath123 ( 10% polar cap ) , and an essentially divergent value , 522 km at @xmath124 ( 1% polar cap ) . if in fact the regolith layer is much deeper than suspected , the added depth of low diffusivity dust significantly increases the patch area : 170 m at @xmath125 , 830 m at @xmath106 , and 4 km at @xmath123 . figure 5 presents graphically how the growth of the ice patch varies with latitude , plus also the effects of flow rate and the assumed regolith depth . most portions of the lunar surface have been been largely geologically inactive during the past 3 gy or more ( with some of the notable exceptions listed above ) . during this time several important modifications of the scenario above are relevant . the current heat flow from the lunar interior , @xmath126 w m@xmath15 ( langseth et al . 1972 , 1973 ) , is only a @xmath127 part of the solar constant , so it affects the temperature near the lunar surface at the level of only @xmath128 millidegree . there were times in the past , however , when interior heating likely pushed the temperature near the surface over @xmath82c . a zero - degree zone near the maria presumably could not form until @xmath13 gy ago , probably sufficient for the moon globally ( see spohn et al . after this the @xmath82c depth receded into the regolith , while the regolith layer was also growing . simultaneously , the average surface temperature was cooler by @xmath129 degree due to standard solar evolution ( gough 1981 perhaps @xmath130 lower in the highlands at 4 gy ago ) . since the the thickness of regolith after 3 gy ago grows at only about 1 m per gy ( quaide & oberbeck 1975 ) , within the maria the @xmath82c depth sinks into bedrock / fractured zone . whatever interaction and modification might be involved between the regolith and volatiles will proceed inwards , leaving previous epochs effects between the surface vacuum and the @xmath82c layer now at @xmath131 m. another issue to consider is possible regolithic chemical reactions with outgassing volatiles , especially over prolonged geological timescales . the key issue is the possible presence of water vapor , and perhaps so@xmath7 . there is little experimental work on the aqueous chemistry of lunar regolith ( which will vary due to spatial inhomogeneity ) . dissolution of lunar fines by water vapor is greatly accelerated in the absence of other gases such as o@xmath7 and n@xmath7 ( gammage & holmes 1975 ) and appears to proceed by etching the numerous damage tracks from solar - wind particles . this process acts in a way to spread material from existing grains without reducing their size ( which would otherwise tend to increase porosity ) . liquid water is more effective than vapor , not surprisingly , and ice tends to establish a pseudo - liquid layer on its surface . this is separate from any discussion of water retention on hydrated minerals surfaces robust to temperatures above 500@xmath8c ( cocks et al . 2002 & op cit . ) . this is a complex chemical system that will probably not be understood without simulation experiments . the major constituents are presumably silicates , which will migrate in solution only over geologic time . ( on earth , consider relative timescales of order 30 my typical migration times for quartz , 700 ky for orthoclase feldspar , kalsi@xmath81o@xmath132 and 80 ky for anorthite , caal@xmath7si@xmath7o@xmath132 : brantley 2004 . ) one might also expect the production of ca(oh)@xmath7 , plus perhaps mg(oh)@xmath7 and fe(oh)@xmath7 . it is not clear that fe(oh)@xmath7 would oxidize to more insoluble feo(oh ) , but any free electrons would tend to encourage this . it seems that the result would be generally alkaline . since feldspar appears to be a major component in some outgassing regions e.g. , aristarchus ( mcewen et al . 1994 ) , one should also anticipate the production of clays . this is not accounting for water reactions with other volatiles e.g. , ammonia , which has been observed as a trace gas ( hoffman & hodges 1975 ) perhaps in part endogenous to the moon , and which near @xmath82c can dissolve in water at nearly unit mass ratio ( also to make an alkaline solution ) . carbon dioxide is a possible volatile constituent , and along with water can metamorphose olivine / pyroxene into mg@xmath81si@xmath67o@xmath133(oh)@xmath7 i.e. talc , albeit slowly under these conditions ; in general the presence of co@xmath7 and thereby h@xmath7co@xmath81 opens a wide range of possible reactions into carbonates . likewise the presence of sulfur ( or so@xmath7 ) opens many possibilities e.g. , caso@xmath1342h@xmath7o ( gypsum ) , etc . since we do not know the composition of outgassing volatiles in detail , we will probably need to inform simulation experiments with further remote sensing or in situ measurements . the mechanical properties of this processed regolith are difficult to predict . some possible products have very low hardness and not high ductility . some of these products expand but will likely fill the interstitial volume with material , which will raise its density and make it more homogeneous . regolith is already ideal in having a nearly power - law particle distribution with many small particles . it seems likely that any such void - filling will sharply reduce diffusivity . the volatiles actually discovered in volcanic glasses from the deep interior ( saal et al . 2008 , friedman et al . 2009 ) include primarily h@xmath7o and so@xmath7 but not co@xmath7 or co. with the addition of water , regolithic mineral combinations tend to be cement - like , and experiments with anorthositic lunar chemical simulants have produced high quality cement without addition of other substances , except sio@xmath7 ( horiguchi et al . 1996 , 1998 ) . whether this happens @xmath135 @xmath136 depends upon whether over geological time ( cao)@xmath81sio@xmath7 or other ca can act as a binder without heating to sintering temperatures . the possible production of gypsum due to the high concentrations of sulfur would add to this cement - like quality . the extent to which ordinary mixes such as portland cement lose water into the vacuum depend on their content of expansive admixture ( kanamori 1995 ) . portland cement mixes show little evidence of loss of compressional strength in a vacuum ( cullingford & keller 1992 ) . we need to think in terms of possibly cemented slabs in some vicinities , and need to consider the effects of cracks or impacts into this concrete medium . this is probably not a dominant process , since the overturn timescale to depths even as shallow as 1 m is more than 1 gy ( gault et al . 1974 , quaide & oberbeck 1975 ) , whereas we discuss processes at @xmath115 m or more . craters 75 m in diameter will permanently excavate to a 15 m depth ( e.g. , collins 2001 , and ignoring the effects of fractures and breccia formation ) , and are formed at a rate of about 1 gy@xmath2 km@xmath15 ( extending neukum et al . [ 2001 ] with a shoemaker number / size power - law index 2.9 ) . this will affect some of the areal scales discussed above , but not all . we speculate that vapor or solution flow might tend to deliver ice and/or solute to these areas and eventually act to isolate the system from the vacuum . finally , we note above that over geological timescales this ice layer will tend to sink slowly into the regolith , at a rate of order 1 m gy@xmath2 , setting up a situation where any relatively impermeable concrete zone will tend to isolate volatiles from the vacuum . in this case volatile leakage will tend to be reduced to a peripheral region around the ice patch . assuming that volatiles leak out through the entire 15 m thickness of regolith at the patch boundary , the 125 m diameter patch area for @xmath137 from above corresponds to a peripheral zone expected from a 520 m diameter patch . thus any such concrete overburden will encourage growth of small patches , and will do so even more for larger ones ( assuming @xmath10715 m regolith depth ) . how much water might reasonably be expected to outgas at these sites ? the earliest analyses of apollo samples argued for extreme scarcity of water and other volatiles ( anders 1970 , charles , hewitt & wones 1971 , epstein & taylor 1972 ) . on the earth , water is the predominant juvenile outgassing component ( gerlach & graeber 1985 , rubey 1964 ) , whereas even the highest water concentrations discussed below ( saal et al . 2008 ) imply values an order of magnitude smaller . on the moon , water content is drastically smaller , with a current atmospheric water content much less than what would affect hydration in lunar minerals ( mukherkjee & siscoe 1973 ) , although some lunar minerals seem to involve water in their formation environments ( agrell et al . 1972 , williams & gibson 1972 , gibson & moore 1973 , and perhaps akhmanova et al . 1978 ) . the origin of the water in volcanic glasses ( saal et al . 2008 , friedman et al . 2009 ) is still poorly understood but implies internal concentrations that at first look seems in contradiction with earlier limits e.g. , anders 1970 . for much different lunar minerals , high concentration is implied for water ( mccubbin et al . 2007 ) as well as other volatiles ( krhenbhl et al . 1973 ) . it is not a goal of this paper to explain detected water in lunar samples , but its origin at great depth is salient here . as a point of reference , hodges and hoffman ( 1975 ) show that the @xmath4ar in the lunar atmosphere derives from deep in the interior , of order 100 km or more . they hypothesize that the gas could just as easily derive from the asthenosphere , 1000 km deep or more ( see also hodges 1977 ) . the picritic glasses analyzed by saal , friedman , et al . derive from depths of @xmath1300 - 400 km or greater ( elkins - tanton et al . 2003 , shearer , layne & papike 1994 ) . o@xmath7 fugacity measurements e.g. , sato 1979 , are based on glasses from equal or lesser depths . water originating from below the magma ocean might provide one explanation ( saal et al . 2008 ) , as might inhomogeneity over the lunar surface . differentiation might not have cleared volatiles from the deep interior despite its depletion partially into the mantle . one might also consider that geographical variation between terranes e.g. , kreep ( k - rare earth element - p ) or not , might be important . in the moon s formation temperatures of proto - earth and progenitor impactor material in simulations grow to thousands of kelvins , sufficient to drive off the great majority of all volatiles , but these are not necessarily the only masses in the system . either body might have been orbited by satellites containing appreciable volatiles , which would likely not be heated to a great degree and which would have had a significant probability of being incorporated into the final moon . furthermore , there is recent discussion of significant water being delivered to earth / moon distances from the sun in the minerals themselves ( lunine et al . 2007 , drake & stimpfl 2007 ) , and these remaining mineral - bound even at high temperatures up to 1000k ( stimpfl et al . the volume of surface water on earth is at least @xmath138 km@xmath20 , so even if the specific abundance of lunar water is depleted to @xmath139 terrestrial , one should still expect over @xmath140 tonnes endogenous to the moon , and it is unclear that later differentiation would eliminate this . this residual quantity of water would be more than sufficient to concern us with the regolith seepage processes outlined above . for carbon compounds , models of the gas filling basaltic vesicles ( sato 1976 ; also ohara 2000 , wilson & head 2003 , taylor 1975 ) predict co , cos , and perhaps co@xmath7 as major components . negligible co@xmath7 is found in fire - fountain glasses originating from the deep interior ( saal et al . 2008 ) ; this should be considered in light of co on the moon ( and co@xmath7 on earth ) forming the likely predominant gas driving the eruption ( rutherford & papale 2009 ) . we suspect that water outgassing was likely higher in the past than it is now . furthermore , no site of activity traced by @xmath50rn or by robust tlp counts ( paper i ) has been sampled . ( the sample return closest to aristarchus , _ apollo 12 _ , is 1100 km away . ) from our discussion above the behavior of outgassing sites near the poles versus near the equator might differ greatly , with volatile retention near the poles being long - term and perhaps making the processing of volatiles much more subterranean and covert . ( note that the lunar south pole is a minor tlp site responsible for about 1% of robust report counts , as per paper i. ) with these uncertainties we feel unable to predict exactly how or where particular evidence of lunar surface outgassing might be found , although the results from above offer specific and varied signals that might be targeted at the lunar surface . for this reason we turn attention to how such effects might be detected realistically from the earth , lunar orbit , and near the moon s surface , and we suggest strategies not only for how these might be tested but also how targeted observations might economically provide vital information about the nature of lunar outgassing . we consider the impact of recent hydration detections in 5 . we appreciate the controversial nature of suggesting small but significant patches of subsurface water ice , given the history of the topic . we take care to avoid `` cargo cult science '' - selection of data and interpretion to produce dramatic but subjectively biased conclusions that do not withstand further objective scrutiny ( feynman 1974 ) . despite the advances made primarily by apollo - era research , we are still skirting the frontiers of ignorance . we are operating in many cases in a regime where interesting observations have been made but the parameters e.g. , the endogenous lunar molecular production ( water vapor or otherwise ) , required to evaluate alternative models and interpretations are sufficiently uncertain to frustrate immediate progress . below we offer several straightforward and prompt tests of our conclusions and hypotheses which offer prospects of settling many of these issues . tlps are rare and short - lived , which hampers their study . we advance supplanting the current anecdotal catalog with data with _ a priori _ explicit , calculable selection effects . this might seem daunting ; paper i used in essence all known reports from lunar visual observers since the telescope s invention ! with modern imaging and computing , it is tractable . another problem clear above is the variety of ways in which outgassing can interact with the regolith . in cases of slow seepage , gases may long delay their escape from the regolith . if the gases are volcanic , they might interact along the way , and water vapor might trap it and other gases in the regolith . these factors bear on designing future investigations . we can make significant headway exploiting more modern technology . table 1 lists the many methods detailed in this section . there has been no areal - encompassing , digital image monitoring of the near side with appreciable time coverage using modern software techniques to isolate transients . numerous particle detection methods are promising . the relevant experiments on apollo were limited in duration , a week or less , or 5 - 8 years in the case of alsep . furthermore _ clementine _ and the relevant portion of _ lunar prospector _ were also relatively short . these limitations serve as background to the following discussions . in this section we provide a potential roadmap to detailed study of outgassing . -0.5 in optical imaging advances several goals . transient monitoring recreates how tlps were originally reported . not yet knowing tlp emission spectra , our bandpass should span the visual , 400 - 700 nm . after an event , surface morphology / photometry changes might persist , betrayed by 0.95 and 1.9 @xmath32 m surface fe@xmath47 bands and increased blue reflectivity ( 2.2 ) . hydration is manifest in the infrared . asteroidal regolith 3 @xmath32 m hydration signals are common ( lebofsky et al . 1981 , rivkin et al . 1995 , 2002 , volquardsen et al . 2004 ) , and stronger than those at 700 nm ( vilas et al . 1999 ) seen in lunar polar regions . absorption near 3 @xmath32 m appears in lunar samples exposed to terrestrial atmosphere for a few years ( markov et al . 1980 , pieters et al . 2005 ) but not immediately ( akhmanova et al . 1972 ) , disappearing within a few days in a dry environment . further sample experiments are needed . -0.5 in earth - based monitoring favors the near side , as do tlp - correlated effects : @xmath50rn outgassing ( all four events on nearside , plus most @xmath141po residual ) and mare edges . the best , consistent resolution comes from the @xmath142 @xmath143 @xmath144 with @xmath145 arcsec fwhm ( @xmath1150 m ) but with large overhead times . competing high - resolution imaging from `` lucky exposures '' ( le , also `` lucky imaging '' ) exploits occasionally superlative imaging within a series of rapid exposures ( fried 1978 , tubbs 2003 ) . amateur setups achieve excellent le results , and the cambridge group ( law , mackay & baldwin 2006 ) attains diffraction - limited imaging on a 2.5-meter telescope , @xmath1200 - 300 m fwhm . only @xmath1461% of observing time survives image selection , but for the moon this requires little time . le resolution is limited to a seeing isoplanatic patch , @xmath11000 arcsec@xmath0 , 3000 times smaller than the moon . likewise , @xmath147 s wide field camera 3 , covering 3000 arcsec@xmath0 , can not practically survey the near side . -0.1 in high resolution imaging can monitor small areas over time or in one - shot applications compared with other sources i.e. , lunar imaging missions . le or @xmath147 match the resolution of global maps from _ lunar reconnaissance orbiter _ camera s ( lroc ) wide - angle camera ( robinson et al . 2005 ) , and @xmath148/uvis , over 0.3 - 1 @xmath32 m . @xmath149 s multiband imager ( ohtake et al . 2007 ) has 40-m 2-pixel resolution . lroc s narrow angle camera has 2 m resolution in one band , targeted . _ change-1_/ccd ( yue et al . 2007 ) might also aid `` before / after '' sequences . lunar orbiter images , resolving to @xmath11 m , form excellent `` before '' data for many sites , for morphological changes e.g. , cores of explosive events over 40 years . the prime technique for detecting changes between epochs of similar images is image subtraction , standard in studying supernovae , microlensing and variable stars . this produces photon poisson noise - limited performance ( tomaney & crotts 1996 ) and is well - matched to ccd or cmos imagers , which at 1 - 2 arcsec fwhm resolution cover the moon with 10 - 20 mpixels , readily available . one needs @xmath492 pixels fwhm , otherwise non - poisson residuals dominate . our group has automated tlp monitors on the summit of cerro tololo , chile and at rutherfurd observatory in new york that produce regular lunar imaging ( crotts et al . 2009 ) , often simultaneously . each cover the near side at 0.6 arcsec / pixel with images processed in 10s . this is sufficient to time - sample nearly all reported tlps ( see paper i ) and produce residual images free of systematic errors at poisson levels ( figure 6 ) . imaging monitors open several possibilities for tlp studies , with extensive , objective records of changes in lunar appearance , at sensitivity levels @xmath110 times better than the human eye . an automated system can distinguish contrast changes of 1% or better , whereas the human eye is limited to @xmath4910% . we will measure the frequency of tlps soon enough ; paper i indicates perhaps one tlp per month visible to a human observing at full duty cycle . tlp monitors open new potential to alert other observers , triggering le imaging of an active area , or spectroscopy of non - thermal processes and the gas associated with tlps . spatially resolved spectroscopy can 1 ) elucidate tlp physics , including identification of gas released , or 2 ) probe quasi - permanent changes in tlp sites . we must find changes in a four - dimensional dataset : two spatial dimensions , wavelength , and time , too much to monitor for transients . fortunately , tlp monitoring can alert to an event in under 1000s , and a larger telescope with a spectrograph can observe the target ( within @xmath1300s ) . whereas `` hyperspectral '' imaging usually refers to resolving power @xmath150 , where @xmath151 is the fwhm resolution , tlp emission might be much narrower , thereby diluted at low resolution . for line emission , rejecting photons beyond the line profile yields contrasts up to 10@xmath17 times better than the human eye using a telescope . ir hydration band near 3.4 @xmath32 m have substructure over @xmath120 nm , requiring @xmath152 , compared to the ir spex on the nasa infrared telescope facility with @xmath153 . the 950 nm and 1.9 @xmath32 m pyroxene bands show compositional shifts ( hazen , bell & mao 1978 ) seen at @xmath154 . differentiating pyroxenes from fe - bearing glass ( farr et al . 1980 ) requires @xmath155 . observations involve scanning across the lunar face with a long slit spectrograph ( figure 7a ) . since lunar surface spectral reflectance is homogenized by impact mixing , @xmath15699% of the light in such a spectrum is `` subtracted away '' by imposing this average spectrum and looking for deviations ( figure 7b ) . the data cube can be sliced in any wavelength to construct maps of lunar features in various bands . figure 8 shows that surface features are reconstructed in detail and fidelity . what narrow lines might we search for ? the emission measure of gas in our model excited by solar radiation is undetectable except for the first few seconds . coronal discharge offers a caveat . reddish discharge may indicate h@xmath69 from dissociation of many possible molecules . emission from aristarchus ( absent balmer lines ) , and transient c@xmath7 swan bands ( kozyrev 1958 ) . we do not advance a model to explain these observations . ] rather than relying on h@xmath69 plus faint optical lines / bands to distinguish molecules , note that near - ir vibrational / rotational bands are brighter and more discriminatory . as in 3 , internal water vapor might have produced ice in the regolith @xmath10715 m subsurface , a venue for ground - penetrating radar , from lunar orbit . while epithermal neutrons and gamma radiation can detect hydrogen , they can not penetrate @xmath491 m. near the poles or subject to chemical modification ( 3.3 ) , ice might range closer to the surface . past and current lunar radar include _ apollo 17 _ s lunar sounder experiment ( lse ) ( brown 1972 , porcello 1974 ) at 5 , 16 and 260 mhz , @xmath149 s lunar radar sounder ( ono & oya 2000 ) at 5 mhz ( optionally , 1 mhz or 15 mhz ) , @xmath157 s mini - rf ( mini radio - frequency technology demonstration ) at 3 and @xmath110 ghz , and mini - sar on _ chandrayaan-1 _ at 3 ghz ( bussey et al . shorter wavelength radar could map possible changes in surface features in explosive outgassing , over tens of meters , in before / after radar sequences meshed with optical monitoring e.g. , with _ mini - rf ( chin et al . 2007 ) . for regolith and shallow bedrock , we need @xmath1100 - 300 mhz ; lse operated only a few orbits and near the equator . near side maps at @xmath11 km resolution at 430 ghz ( campbell et al . 2007 ) could improve with intensive ground - based programs , or from lunar orbit , penetrating @xmath158 m. orbital missions can combine different frequencies and/or reception angles to improve spatial resolution and ground clutter , and reduce interference speckle noise . earth - based radar maps exist at 40 , 430 and 800 mhz ( thompson & campbell 2005 ) , also 2.3 ghz ( stacy 1993 , campbell et al . 2006a , b ) . angles of incidence from earth are large e.g. , @xmath159 , with echoes dominated by diffuse scattering not easily modulated . circular polarization return can probe for surface water ice ( nozette 1996 , 2001 ) but is questioned ( simpson 1998 , campbell et al . 2006a ) . applying these to subsurface ice is at least as problematic , especially at @xmath1300 mhz to penetrate @xmath115 m. finding subsurface ice is challenging . the dielectric constant is @xmath160 for regolith , water ice ( slightly higher ) , and many relevant powders of comparable specific gravity e.g. , anorthosite and various basalts . these have attenuation lengths similar to ice , as well . using net radar return alone , it will be difficult to distinguish ice from regolith . in terrestrial situations massive ice bodies reflect little internally ( moorman , robinson & burgess 2003 ) . ice - bearing regions should be relatively dark in radar images , if lunar ice - infused volumes homogenize or `` anneal , '' either forming a uniform slab or by binding together regolith with ice in a uniform @xmath24 bulk . on the other hand , hydrated regolith has @xmath24 much higher than unhydrated ( up to 10 times ) , and attenuation lengths over 10 times shorter ( chung 1972 ) . hydration effects are largest at lower frequencies , even below 100 mhz . if water ice perturbs regolith chemistry , increasing charge mobility as in a solution , @xmath24 and conductivity increase , raising the loss tangent ( conductivity divided by @xmath24 and frequency ) . this high-@xmath24 zone should cause reflections , depending strongly on the suddenness of the transition interface . the 430 mhz radar map ( ghent et al . 2004 ) of aristarchus and vicinity , site of @xmath150% of tlp and radon reports shows the 43-km diameter crater surrounded by low radar - reflectivity some 150 km across , especially downhill from the aristarchus plateau , which is dark to radar , except bright craters and vallis schrteri . the darker radar halo centered on aristarchus itself is uniquely smooth , indicating that it was probably formed or modified by the impact , a few hundred my ago . this darkness might be interpreted as high loss tangent , as above , or simply fewer scatterers ( ghent et al . 2004 ) i.e. , rocks of @xmath11 m size ; it is undemonstrated why the latter applies in the ejecta blanket within the bright radar halo within 70 km of the aristarchus center . other craters , some as large as aristarchus , have dark radar haloes , but none so extended ( ghent et al . the aristarchus region matches subsurface ice redistributed by impact melt : dark , smooth radar - return centered on the impact ( although tending downslope ) . one should search for dark radar areas around likely outgassing sites . _ alpha - particle spectrometry : _ a @xmath50rn atom random walks only @xmath1200 km before decaying ( or sticking to a cold surface ) . in under a day , @xmath50rn dispersal makes superfluous placing detectors @xmath146100 km above the surface ( excepting @xmath161 sensitivity considerations ) . alpha - particle spectrometers observed the moon successfully for short times . the latitude coverage was limited on _ apollo 15 _ ( @xmath162 for 145 hours ) and _ apollo 16 _ ( @xmath163 , 128 h ) . _ lunar prospector s _ alpha particle spectrometer ( covering the entire moon over 229 days spanning 16 months ) was partially damaged and suffered sensitivity drops due to solar activity ( binder 1998 ) . @xmath149 s alpha ray detector ( ard ) promised 25 times more sensitivity than apollo ( nishimura et al . 2006 ) , but sharing a failed power supply it has yet to produce results . _ apollo 15 _ observed outgassing events from aristarchus and grimaldi , _ apollo 16 _ none , and _ lunar prospector _ aristarchus and kepler integrated over the mission . apollo and _ lunar prospector _ detected decay product @xmath141po at mare / highlands boundaries from @xmath50rn leakage over the past @xmath1100 y. an expected detection rate might be grossly estimated , consistent with an event 1 - 2 times per month detectable by _ apollo 15 _ , and by _ lunar prospector _ over the mission , with aristarchus responsible for @xmath150% . a polar orbiting alpha - particle spectrometer with a lifetime of a year or more and instantaneous sensitivity equal to apollo s could produce a detailed map of outgassing on the lunar surface separate from optical manifestation . two in polar orbit could cover the lunar surface every 1.8 half - lives of @xmath50rn , nearly doubling sensitivity . sensitivity can be increased if detectors incorporate solar wind vetos , or operate during solar minimum , and if detectors orient towards the lunar surface . _ on - orbit mass spectrometry : _ unlike @xmath50rn and its long surface residence , other outgassing events call for several instruments for efficient localization e.g. , by mass spectrometry . with outgassing of hundreds of tons and tens of events per year , particle mass fluence from one outburst seen 1000 km away approaches @xmath164 amu cm@xmath15 . a burst that is seen by a few detectors could be well constrained . gas scale heights @xmath1100 km imply detectors near the ground . conversely , an instantaneous outburst seen 100 km away will disperse less than one minute in arrival ; detectors must operate rapidly . this was a problem e.g. , the _ apollo 15 _ orbital mass spectrometer experiment ( hoffman & hodges 1972 ) requiring 62s to scan through a factor of 2.3 in mass . clearly there are two separate modes of gas propagation above the surface , neutral and ionized ( vondrak , freeman & lindeman 1974 , hodges et al . 1972 ) , at rates of one to hundreds of tonne y@xmath2 for each . operational strategies of these detectors are paramount . consider an event 1000 km away , which will spread @xmath165s in arrival time . a simple gas pressure gauge is too insensitive ; with an ambient atmosphere not atypical e.g. , number density @xmath166 @xmath11 ( varying day / night e.g. , hodges , hoffman & johnson 2000 ) , the collisonal background rate in 500 s amounts to 10@xmath167 or more than the fluence for a typical outburst ( assuming @xmath158 amu particles ) . since interplanetary solar proton densities can vary by order unity in an hour or less ( e.g. , mcguire 2006 ) , pressure alone is insufficient . mass spectrometry subdivides incoming flux in mass , but also in direction , decreasing effective background rates . one satellite particle detector can not distinguish episodic behavior of outgassing versus spacecraft motion at @xmath168 km s@xmath2 . localizing such signals between two platforms is ideal , at least for neutral species , if they constrain temporal / spatial location of specific outbursts using timing and signal strength differences . a timing difference indicates the distance difference to the source , with the source confined to a hyperboloid locus . location on this hyperboloid is fixed by signal strengths , plus left / right ambiguity from detector directionality . a mean nearest satellite distance of 1000 km from arbitrary sources requires @xmath4910 low orbital platforms . mass spectrometers on the surface can maintain such density over smaller areas efficiently once we know roughly where sources may be . a mass spectrometer planned for _ lunar atmospheric and dust environment explorer _ ( @xmath169 ) sits on one platform in equatorial orbit ; geographical resolution of outgassing events will be poor.27 , 50 , 76 or @xmath170 ( e.g. , ramanan & adimurthy 2005 ) . if we want to maintain a position over the terminator ( sun - synchronous orbit ) , we requires a precession rate @xmath171 rad s@xmath2 . alternatively , ladee achieves this by a precessing , highly eccentric orbit , but spends a small fracton of its time near the lunar surface . precession is fixed by coefficient @xmath172 ( konopliv et al . 1998 ) according to @xmath173 , where @xmath86 is lunar radius , @xmath174 orbital angular speed , @xmath175 lunar mass and @xmath92 orbital radius . precession due to the sun and earth are much smaller . one can not effectively institute both sun synchronicity in a polar orbit , however , since the maximum inclination orbit with @xmath176 s@xmath2 occurs at @xmath177 ( or else below the surface ) . to force sun - synchronicity at @xmath178 , @xmath179 s@xmath2 , requires only @xmath180 mm s@xmath15 which could even be accomplished by a hall - effect ion engine or even a solar sail ( with 330 cm@xmath0 g@xmath2 ) . ] surveying _ in situ _ approaches to studying volatiles is beyond the scope of this paper ; we emphasize a few key points . the key effort is to focus from wide - ranging reconnaissance down to scales where lunar volatiles can be sampled near their source . primary global strategies are optical transient monitoring ( near side , resolution @xmath11 km ) and orbital @xmath50rn and 3@xmath32 m detection ( both hemispheres , @xmath1100 km and @xmath1461 km , respectively ) . even trusting that tlps trace volatiles and centroiding tlps to 10% of a resolution element , localization error ( @xmath1100 m ) could preclude easy _ in situ _ followup . ( in appendix i we outline improving this to @xmath181 m. ) two simple _ in situ _ technologies could isolate outgassing sources below 100 km scales . first , three alpha particle detectors on the surface can triangulate nearby @xmath50rn outgassing sources , using strength and time delay in arrival of random walking @xmath50rn . secondly , a mass spectrometer that can reconstruct the ballistic trajectory of neutrals from the source ( austin et al . 2008 , daly , radebaugh & austin 2009 ) can construct an `` image '' of transient outgassing sources over regions up to 1000 km across . this spectrometer is not overwhelmed by pulsed sources while measuring masses over a wide range . further technologies could pinpoint subsurface structure at 10 m scales from information at 1 km . , or tuned to one of several species vibrational - rotational states . on smaller scales ( 1 - 100 m ) several varieties of mass spectroscopy might prove effective , including downward - sniffing spectrometers , triangulating outburst detectors arrays , and pyrolysis mass spectrometers ( ten kate et al . 2009 ) which heat regolith samples in search of absorbed species from previous outgassing . ] by lace s deployment with the final apollo landing , the outgassing environment was contaminated by anthropogenic gas ( freeman & hills 1991 ) especially near landing sites ; each mission of human exploration will deliver tens of tonnes gases to the surface , with species relevant to endogenous volcanic gas , approaching or exceeding the annual endogenous output of such gases.o@xmath67 ( nitrogen teroxide ) and ch@xmath81n@xmath7h@xmath81 ( mono - methyl hydrazine ) , with altair propelled by liquid oxygen and hydrogen . future missions might use liquid o@xmath7 and ch@xmath67 . earth departure stages might deliver residual @xmath182 and @xmath183 in lunar impact . altair ( and eds ) produce water , and orion exhausts h@xmath7o , co@xmath7 and n@xmath7 . n@xmath7 was the prime candidate constituent in an outburst seen by the _ apollo 15 _ over mare orientale : hoffman & hodges 1972 , perhaps anthropogenically - hodges 1991 . ] depending on spacecraft orientations and trajectories when thrusting , they may deliver @xmath120 tonnes of mostly water to the surface , which will remain up to about one lunation , making suspect measurements of these and other species for years . the origin of tlps has been mysterious , and their correlation to outgassing , while strong , was only circumstantial . the plausible generation of tlp - like events as simple consequence of outgassing from the interior lends credence to a possible causal link . we present a model tied to outgassing from deep below the regolith that reproduces the time and spatial scale of reported tlps , suggesting a causal link to outgassing . radiogenic gas evolved from the regolith can not provide the concentration to produce a noticeable explosive event . apollo and later data were insufficiently sensitive to establish the level of outgassing beyond @xmath50rn , and isotopes of ar , plus he , presumably , and did detect marginally molecular gas , but of uncertain origin , particularly ch@xmath67 , reviewing the evidence and available techniques , there are several gases that should be highlighted as crucial outgassing tracers . @xmath50rn ( and its products e.g. , @xmath141po ) can be detected remotely of course and are unique in terms of their mapping potential , while being a minor fraction of escaping gas , presumably . @xmath4ar is a major mass constituent of the atmosphere and unlike @xmath17he is not confused with the solar wind . both @xmath50rn and @xmath4ar will favor kreep terrane in the western maria , presumably . if outgassing arises in the deep interior , one can not neglect indications that at one time this was dominated by volcanic , molecular gas . particular among these is water vapor , passing its triple point temperature in rising through the regolith . given a high enough concentration , therefore , one should expect the production of water ice . the conditions under the regolith , particularly near the lunar poles , are favorable for such ice to persist even over geological time interval . it is possible that ice generated there when outgassing was more active still remains . we further point out that the plausible chemical interaction of such molecular gases with the regolith is the production of cement - like compounds that might radically alter the diffusivity of the regolith . given the temperature evolution of the regolith , this non - diffusive layer would isolate the volatile outflow from the vacuum . the question remains how we will detect such molecular outgassing effects , given their largely covert nature ; this is greatly complicated by possible anthropogenic contamination in the future . among molecular gases , sulfur e.g. , so@xmath7 is the predominate volatile detected in deep - interior fire fountain glasses , a key factor in deciding what to pursue as a volcanic tracer . furthermore , essential no liquid or hybrid rocket propellant candidate contain sulfur , and the only sulfuric solid propellants are fairly outdated e.g. , black powder and zn - s . nasa and hopefully other space agencies have no plans to use these on lunar missions . despite the hypotheses and methods outlined above , there is great doubt regarding the nature of lunar outgassing . water is of obvious and diverse interest , and co@xmath7 and co , while missing as apparent constituents , are interesting as drivers for fire fountain eruption . plausibly the only way to study these components reliably is before the new introduction of large spacecraft into the lunar environment . given uncertainty of how these gases and so@xmath7 might interact with the regolith , this early study appears paramount . significantly , many years to come monitoring for optical transients will be best done from earth s surface , even considering the important contributions that will be made by lunar spacecraft probes in the near future . however , these spacecraft will be very useful in evaluating the nature of transient events in synergy with ground - based monitoring . given the likely behavior of outgassing events , it is unclear that in - situ efforts alone will necessarily isolate their sources within significant winnowing of the field by remote sensing . early placement of capable mass spectrometers of the lunar surface , however , might prove very useful in refining our knowledge of outgassing composition , in particular a dominant component that could be used as a tracer to monitor outgassing activity with more simple detectors . this should take place before significant atmospheric pollution by large spacecraft , which will produce many candidate tracer gases in their exhaust . finally , as we edit this paper s final version , several works have become available indicating confirmed lunar regolith hydration signals ( pieters et al . 2009 , clark 2009 , sunshine et al . 2009 ) in the 3@xmath32 m band , and we comments about these here . these show a strong increase in hydration signal towards the poles , as predicted in 4 . to our knowledge our model is uniquely consistent with this and the general hydration signal strength , in places @xmath49700 ppm by mass ( also with vilas et al . 1999 , 2008 ) . unfortunately the moon mineralogy mapper ( m@xmath20 : pieters et al . 2006 ) , in finding this signal , but not completely mapping it , provided tentative indication of its large scale distribution varying over a lunation . this variation can be studied from earth with a simple near / mid ir camera ( insb or red - extended hgcdte ) with on- and off - band filters for ir hydration bands ( or 0.7@xmath32 m : vilas et al . 1999 ) . to complete this valuable work , along with other instruments needing a lunar polar orbiter ( alpha - particle spectrometer , ground - penetrating radar , mass spectrometers , etc . ) , an instrument similar to m@xmath20 should probably fly again before human lunar missions . note that the same type and level of signal was detected by _ luna 24 _ ( akhmanova et al . 1978 ) , and these authors believed it not due to terrestrial contamination . they detected increasing hydration with depth into the regolith , a likely circumstance in our model . this core sample reached 2 meters depth , several times deeper than epithermal neutrons e.g. , seen on _ lunar prospector _ or _ lro _ , and corresponding to an impact gardening over @xmath12 g.y . such a gradient arises naturally from seepage of water vapor , but water and/or hydroxyl from solar wind proton implantation may not explain the concentration of 3@xmath32 m signal to the poles and the _ luna 24 _ hydration depth profile . this offers a challenge for this model , or for water delivered by comets and/or meteoroids . in a separate paper we will review further evidence supporting endogenous origin of lunar hydration . we would much like to thank alan binder and james applegate , as well as daniel savin , daniel austin , ed spiegel and the other members of aeolus ( `` atmosphere as seen from , earth , orbit and lunar orbit '' ) for helpful discussion . this research was supported in part by nasa ( 07-past07 - 0028 and 07-laser07 - 0005 ) , the national geographic society ( cre grant 8304 - 07 ) , and columbia university . given constraints on imaging from earth , we consider imaging monitors closer to the moon . we propose no special - purpose missions , but detectors that could ride on other platforms e.g. , does lunar exploration require communications with line - of - sight access to all points on the moon s surface ( except within deep craters , etc . ) ? this might also serve for comprehensive imaging monitoring . a minimal full network has a tetrahedral geometry with points @xmath160000 km above the surface : a single platform at earth - moon lagrange point l1 , covering most of the near side , and three points in wide halo orbits around l2 for the far side plus limb seen from earth . proposals exist for an l1 orbital transfer facility ( lo 2004 , ross 2006 ) . no single satellite sees the entire far side , especially since farside radio astronomy might restrict low - frequency transmission i.e. , lasers only . one l2 satellite covers @xmath10797% of the far side ( subtending @xmath184 , selenocentrically ) ; full coverage ( plus some redundancy ) requires three satellites ( plus l1 ) . with this configuration , the farthest point from a satellite will be typically @xmath185 ( selenocentrically ) , foreshortened by @xmath186 times . such an imaging monitor might be ambitious ; to achieve 100 m fwhm at the lunar sub - satellite point requires @xmath14 gpixels , aperture @xmath187 m , and field - of - view 3@xmath188 . each such monitor on an existing platform will cost perhaps $ 100 m . in the meantime , we should accomplish what we can from the ground . martin , r.t . , winkler , j.l . , johnson s.w . & carrier , iii , w.d . 1973 , `` measurement of conductance of apollo 12 lunar simulant taken in the molecular flow range for helium , argon , and krypton gases . '' unpublished report quoted in carrier et al . ( 1991 ) . map of tlp activity & imaging monitor , entire nearside , & optical & comprehensive schedulability ; more & limited resolution + & @xmath12 km resolution . & & sensitive than human eye & + & & & & + polarimetric study of & compare reflectivity in two & optical & easy to schedule ; further constrains & requires use of two monitors + dust & monitors with perpendicular & & dust behavior & + & polarizers & & & + & & & & + changes in small , & adaptive optic imaging , @xmath1100 m&0.95@xmath32 m , etc.&``on demand '' given good conditions&undemonstrated , depends on + active areas & resolution & & & seeing ; covers @xmath150 km + & & & & diameter maximum + & & & & + & `` lucky imaging , '' @xmath1200 m & 0.95@xmath32 m , etc.&on demand given good conditions & low duty cycle , depends on + & resolution & & & seeing + & & & & + & _ hubble space telescope _ , @xmath1100 m&0.95@xmath32 m , etc.&on demand given advanced notice & limited availability ; low + & resolution & & & efficiency + & & & & + & _ clementine / lro / chandrayaan-1 _ & 0.95@xmath32 m , etc.&existing or planned survey & limited epochs ; low flexibility + & imaging , @xmath1100 m resolution & & & + & & & & + & _ lro / kaguya / chang-1 _ imaging , & 0.95@xmath32 m , etc.&existing or planned survey & limited epochs ; low flexibility + & higher resolution & & & + & & & & + tlp spectrum & scanning spectrometer map , plus & nir , & may be best method to find & requires alert from tlp image + & spectra taken during tlp event & optical & composition & tlp mechanism & monitor ; limited to long events + & & & & + & & & & + regolith hydration & nir hydration bands seen before vs.&2.9 , 3.4@xmath32m&directly probe regolith / water & requires alert from monitor + measurement & after tlp in nir imaging & & chemistry ; may detect water & and flexible scheduling + & & & & + & scanning spectrometer map , then & 2.9 , 3.4@xmath32m&directly probe regolith / water & requires alert from monitor + & spectra taken soon after tlp & & chemistry ; may detect water & and flexible scheduling + & & & & + relationship between & simultaneous monitoring : @xmath50rn @xmath69&@xmath50rn @xmath69 & & refute / confirm tlp / outgassing&optical monitor only covers + tlps & outgassing & particles by @xmath149 & optical tlps & optical & correlation ; find outgassing loci & nearside ; more monitors better + & & & & + subsurface water ice & penetrating radar from earth & @xmath1430 mhz&directly find subsurface ice with&ice signal is easily confused + & & & existing technique & with others + & & & & + & penetrating radar from lunar orbit&@xmath1300 mhz&better resolution ; deeper than&ice signal is easily confused ; + & & & neutron or gamma probes & more expensive + & & & & + & surface radar from lunar orbit & @xmath1561 ghz & better resolution ; study tlp site & redundant with high resolution + & & & surface changes & imaging ? + & & & & + high resolution tlp & imagers at / near l1 , l2 points & optical & map tlps with greater resolution & & expensive , but could piggyback + activity map & covering entire moon , at 100 m & & sensitivity , entire moon & on communications network + & resolution & & & + & & & & + comprehensive @xmath50rn @xmath69 & two @xmath50rn @xmath69 detectors in polar&@xmath50rn @xmath69&map outgassing events at full&expensive ; even better response + particle map & orbits 90@xmath8 apart in longitude & & sensitivity & with 4 detectors + & & & & + comprehensive map of & two mass spectrometers in adjacent & ions & & map outgassing events & find & expensive ; even better with more + outgas components & polar orbits & neutrals & composition & detectors +

we follow paper i with predictions of how gas leaking through the lunar surface could influence the regolith , as might be observed via optical transient lunar phenomena ( tlps ) and related effects . we touch on several processes , but concentrate on low and high flow rate extremes , perhaps the most likely . we model explosive outgassing for the smallest gas overpressure at the regolith base that releases the regolith plug above it . this disturbance s timescale and affected area are consistent with observed tlps ; we also discuss other effects . for slow flow , escape through the regolith is prolonged by low diffusivity . water , found recently in deep magma samples , is unique among candidate volatiles , capable of freezing between the regolith base and surface , especially near the lunar poles . for major outgassing sites , we consider the possible accumulation of water ice . over geological time ice accumulation can evolve downward through the regolith . depending on gases additional to water , regolith diffusivity might be suppressed chemically , blocking seepage and forcing the ice zone to expand to larger areas , up to km@xmath0 scales , again , particularly at high latitudes . we propose an empirical path forward , wherein current and forthcoming technologies provide controlled , sensitive probes of outgassing . the optical transient / outgassing connection , addressed via earth - based remote sensing , suggests imaging and/or spectroscopy , but aspects of lunar outgassing might be more covert , as indicated above . tlps betray some outgassing , but does outgassing necessarily produces tlps ? we also suggest more intrusive techniques from radar to in - situ probes . understanding lunar volatiles seems promising in terms of resource exploitation for human exploration of the moon and beyond , and offers interesting scientific goals in its own right . many of these approaches should be practiced in a pristine lunar atmosphere , before significant confusing signals likely to be produced upon humans returning to the moon . 6.5 in 8.5 in 0.0 in 0.0 in

the past few years have been a watershed in our ability to directly observe galaxy evolution . deep field surveys such as the canada - france redshift survey ( cfrs lilly 1995 ) and color - selected field samples such as that of steidel ( 1996 , 1999 ) have provided critical information on the evolution of field galaxies . madau ( 1996 ) integrated the results at @xmath8 into a coherent picture of the star formation history of the universe , suggesting that the global star formation rate peaked between @xmath9 and @xmath10 . since then , recognition of the importance of both dust and cosmic variance has changed the steep decline in the cosmic star formation rate inferred at @xmath11 into a flat plateau for @xmath12 ( steidel 1999 ) . cowie ( 1999 ) also show a more gradual rise at @xmath13 than initially inferred by the cfrs . there remain four substantial caveats regarding these findings . first , the number of spectroscopically measured redshifts between @xmath14 and @xmath10 is small . second , since the uv dropout technique used to identify the @xmath15 population requires them to be uv bright , it is possible that a substantial amount of star - forming activity in dusty systems has been overlooked . third , redshift surveys from which cosmic star formation rates are measured must be of sufficient depth and wavelength coverage that star formation indicators ( , @xmath16(2800 ) ) can be measured with limited extrapolation over wide redshift intervals . finally , small area surveys , such as the hdf , are vulnerable to perturbations from large scale structure . infrared - selected surveys provide a powerful tool for addressing these issues ( see dickinson , these proceedings ) . among the benefits , infrared @xmath17-corrections are small and relatively independent of galaxy type , age , and redshift . since the long - wavelength light of galaxies is dominated by lower mass stars rather than short - lived high - mass stars , infrared luminosities track galaxy mass , thereby providing a more direct comparison to theories of galaxy formation without relying on the poorly - understood physics of star formation . infrared light is also less vulnerable to dust absorption . on the negative side , since spectroscopy is primarily performed at optical wavelengths , infrared - selected samples are challenging to follow - up . also , since evolved stars become important contributors to the long - wavelength flux of a galaxy , poorly - understood phases of stellar evolution can make interpretation of broad - band colors ambiguous ( spinrad 1997 ) . finally , since infrared - surveys do not select for young stars , they are suboptimal for studying the cosmic star - formation history , though they provide a natural basis for studying the mass - aggregation history . we present the spices survey ( eisenhardt 2001 , in prep . ) , a deep @xmath2 imaging and spectroscopic survey covering over 100 arcmin@xmath1 spread over four fields . table [ tab1 ] lists the vega magnitude 3@xmath18 depths in 3 diameter apertures for the imaging . the relatively large area mitigates the effects of large - scale structure while the @xmath19-band depth is more than sufficient to detect @xmath20 galaxies to @xmath21 . the area and depth are a significant improvement over several recent surveys ( cowie 1996 ) , but are modest compared to several programs currently in production mode ( cimatti , these proceedings ; mccarthy , these proceedings ) . an important strength of spices is the spectroscopic program : we currently have 626 spectroscopic redshifts of @xmath22 sources selected from the sample , approximately one - third of the complete @xmath22 sample ( see figure 1 ) . these spectroscopic redshifts are being used to directly construct an eigenbasis of galaxy spectral energy distributions with which to determine photometric redshifts for the complete sample ( see budavari 2000 ) . wu ( these proceedings ) discusses _ hst _ imaging of one of the spices fields . here we discuss two initial results from the survey . .depth of spices imaging ( vega magnitudes ) [ cols="<,^,^,^,^,^,^ " , ] [ tab1 ] extremely red objects ( ero s ) are an intriguing class of extragalactic object , likely associated with @xmath23 galaxies ( cimatti , these proceedings ) . we find that the surface density of these sources is elevated in the spices fields relative to some of the surface densities reported previously in the literature . for @xmath24 , we find a surface density of 1.4 ero s arcmin@xmath25 , with a range in this value of 1.3 to 1.9 across the four fields . for the same magnitude range and color criterion , barger ( 2000 ) find a surface density of @xmath26 ero s arcmin@xmath25 over a field of view of 61.8 arcmin@xmath1 while mccracken ( 2000 ) find a surface density of ero s in the herschel deep field of @xmath27 arcmin@xmath25 over a 47.2 arcmin@xmath1 field . similarly , if we consider ero s defined as @xmath28 sources with @xmath29 , the spices fields have 0.13 ero s arcmin@xmath25 with a range of @xmath30 ero s arcmin@xmath25 across the four fields . using the same definition , the cadis survey finds @xmath31 ero s arcmin@xmath25 across a 154 arcmin@xmath1 field ( thompson 1999 ) while daddi ( 2000 ) find 0.07 ero s arcmin@xmath25 across a 447.5 arcmin@xmath1 field with strong clustering reported . what is the source of this discrepancy ? one possibility is that the depth and area of the spices imaging are significantly improved over many of the surveys mentioned above : @xmath32 is a 10@xmath18 detection in the spices survey . another possibility is large scale structure . though the spices fields cover @xmath33 arcmin@xmath1 , larger than several of the above surveys , fluctuations in the ero surface density on these scales have been reported by more recent larger area deep infrared surveys ( daddi 2000 ; cimatti , these proceedings ; mccarthy , these proceedings ) . indeed , one of the spices fields ( the lynx field : @xmath34 ) has a higher surface density of red objects than the other three fields . keck / lris spectroscopy and has subsequently identified many of these red sources with galaxies in two x - ray emitting clusters at @xmath35 ( stanford 1997 ; rosati 1998 ) . the @xmath19-band luminosity function ( klf ) at @xmath9 offers a powerful constraint on theories of galaxy formation . since the @xmath19-band light tracks mass better than ultraviolet / optical light , the klf is more directly comparable to theories of the collapse and merging of galaxies . kauffmann & charlot ( kc98 ; 1998 ) show that pure luminosity evolution ( ple ) models , models in which galaxies collapse monolithically at high redshift with little subsequent merging activity , predict that many massive galaxies exist at @xmath9 : @xmath36% of an infrared - selected field galaxy sample with @xmath37 should be at @xmath7 . alternatively , their hierarchical model predicts only @xmath38% of @xmath37 field galaxies should be at @xmath7 . ignoring the spices field with the @xmath35 clusters and another field with very limited spectroscopy , we conservatively find that @xmath39% of @xmath37 spices sources are at @xmath7 . this assumes that @xmath40% of @xmath28 , @xmath41 ( red ) sources are at @xmath7 , as our spectroscopic program shows thus far , and we only count those @xmath37 , @xmath42 ( blue ) sources already spectroscopically confirmed to be at @xmath7 . early photometric redshift analysis on these fields suggests a value @xmath3% of the @xmath37 being at @xmath7 . these numbers show that neither ple nor the kc98 hierarchical model correctly predicts the @xmath43 klf , implying that substantial merging occurs at @xmath7 . the identification of two clusters at @xmath35 and one cluster at @xmath45 in the lynx spices fields has led to a deep , 190 ksec _ chandra _ map of the field . analysis of the diffuse high- and low - redshift cluster x - ray emission are discussed in stanford ( 2001 , submitted ) and holden ( 2001 , in prep . ) , respectively . stern ( 2001 , in prep . ) discusses x - ray background ( xrb ) results from this data set . we confirm results of recently published _ chandra _ studies ( giacconi 2001 ) : most of the @xmath46 kev xrb is resolved into discrete sources ; the fainter soft - band sources have harder x - ray spectra , providing a coherent solution to the long - standing ` spectral paradox ' ; and @xmath47% of the sources have optical / near - infrared identifications in deep ground - based imaging . a preliminary spectroscopic program shows a mix of obvious agn , apparently normal galaxies , and , perhaps surprisingly , several x - ray emitting stars , some with hard x - ray spectra . we are also targeting the spices fields with very deep imaging in @xmath48 to identify high - redshift sources using the lyman break technique . this work has led to the discovery of a faint quasar at @xmath49 ( stern 2000 ) and several high - redshift galaxies out to @xmath50 . strong emission - line galaxies have also been identified serendipitously during the spices spectroscopic campaign , the highest redshift source being a likely @xmath51 ly@xmath52 emitter with @xmath53 erg cm@xmath25 s@xmath55 . we present first results from the spices survey , an infrared - selected photometric and spectroscopic survey . we find an elevated surface density of ero s compared to several recent deep , infrared surveys , likely due to fluctuations in that quantity from large scale structure at moderate redshifts . perhaps relatedly , we also find a large fraction of infrared - bright ( @xmath28 ) galaxies residing at @xmath7 . a good measure of this quantity provides a powerful constraint on models of galaxy formation . budavri , t. ( 2000 ) , , 120 , 1588 . barger , a. ( 1999 ) , , 117 , 102 . cowie , l. ( 1996 ) , , 112 , 839 . cowie , l. , songaila , a. , & barger , a. ( 1999 ) , , 118 , 603 . daddi , e. ( 2000 ) , , 361 , 535 . giacconi , r. ( 2001 ) , , in press . kauffmann , g. & charlot , s. ( 1998 ) , , 297 , 23 . lilly , s. ( 1995 ) , , 455 , 108 . madau , p. ( 1996 ) , , 283 , 1388 . mccracken , h. j. ( 2000 ) , , 311 , 707 . rosati , p. ( 1999 ) , , 118 , 76 . spinrad , h. ( 1997 ) , , 484 , 581 . stanford , s.a . ( 1997 ) , , 114 , 2332 . steidel , c.c . ( 1996 ) , , 112 , 352 . steidel , c.c . ( 1999 ) , , 519 , 1 . stern , d. et al . ( 2000 ) , , 533 , l75 thompson , d. et al . ( 2000 ) , , 523 , 100

we present first results from spices , the spectroscopic , photometric , infrared - chosen extragalactic survey . spices is comprised of four @xmath0 arcmin@xmath1 high galactic latitude fields with deep @xmath2 imaging reaching depths of @xmath3 mag ( ab ) in the optical and @xmath4 mag ( ab ) in the near - infrared . to date we have 626 spectroscopic redshifts for infrared - selected spices sources with @xmath5 ( vega ) . the project is poised to address galaxy formation and evolution to redshift @xmath6 . we discuss initial results from the survey , including the surface density of extremely red objects and the fraction of infrared sources at @xmath7 . one of the spices fields has been the target of a deep 190 ksec _ chandra _ exposure ; we discuss initial results from analysis of that data set . finally , we briefly discuss a successful campaign to identify high - redshift sources in the spices fields .

as one of the most compelling evidences for new physics beyond the standard model ( sm ) , the cosmic dark matter ( dm ) has been widely studied in particle physics @xcite . recently , the cdms - ii collaboration observed three events which can be explained by a light dm with mass about 8.6 gev and a spin - independent dm - nucleon scattering cross section of about @xmath1 pb @xcite . the existence of such a light dm seems to be corroborated by other direct detections such as the cogent @xcite , cresst @xcite and dama / libra @xcite . moreover , a light dm is also hinted by fermi - lat , a satellite - based dm indirect detection experiment @xcite . recent analysis of the fermi - lat data exhibits peaks in the gamma - ray spectrum at energies around 1 - 10 gev , which could be interpreted in terms of the annihilation of a dm with mass low than about 60 gev into leptons or bottom quarks @xcite . about these experimental results , it should be noted that they are not completely consistent with each other , and more seriously , they conflict with the xenon data @xcite and the latest lux result @xcite . so the issue of light dm leaves unresolved and will be a focal point both experimentally and theoretically . on the experimental side , many experiments like lux , xenon , cdms and cdex @xcite will continue their searches , while on the theoretical side we need to examine if such a light dark matter can naturally be predicted in popular new physics theories such as low energy supersymmetry ( susy ) . previous studies @xcite showed that , in the framework of the next - to - minimal supersymmetric standard model ( nmssm ) @xcite , a light neutralino dm around 10 gev is allowed by the collider constraints and dm relic density ( in contrast such a light dm is not easy to obtain in the mssm @xcite or cmssm @xcite ) . in nmssm , due to the presence of a singlet superfield @xmath2 , we have five neutralinos , three cp - even higgs bosons ( @xmath3 ) and two cp - odd higgs bosons ( @xmath4 ) @xcite . the mass eigenstates of neutralinos are the mixture of the neutral singlino ( @xmath5 ) , bino ( @xmath6 ) , wino ( @xmath7 ) and higgsinos ( @xmath8 , @xmath9 ) ; while the cp - even ( odd ) higgs mass eigenstates are the mixture of the real ( imaginary ) part of the singlet scalar @xmath10 and the cp - even ( odd ) mssm doublet higgs fields . an important feature of the nmssm is that the lightest cp - even ( odd ) higgs boson @xmath11 can be singlet - like and very light , and the lightest neutralino ( @xmath12 ) can be singlino - like and also very light . as a result , the spin - independent neutralino - nucleon scattering cross section can be enhanced to reach the cdms - ii value by the @xmath13channel mediation of a light @xmath14 @xcite . meanwhile , the dm relic density can be consistent with the measured value either through the @xmath15channel resonance effect of @xmath11 in dm annihilation or through the annihilation into a pair of light @xmath14 or @xmath16 @xcite . note that such a light dm in the nmssm should be re - examined because the latest lhc data may give severe constraints . due to the presence of a light dm and concurrently a light @xmath16 or @xmath14 , the sm - like higgs boson ( @xmath17 ) can have new decays @xmath18 and @xmath19 @xcite . as analyzed in @xcite , such decays may be subject to stringent constraints from the current lhc higgs data @xcite . besides , since a certain amount of higgsino component in @xmath12 is needed to strengthen the coupling of @xmath20 ( or @xmath21 ) which is necessary for the dm annihilation , the higgsino - dominated neutralinos and the chargino @xmath22 are generally not very heavy and will be constrained by the searches for events with three leptons and missing transverse momentum ( @xmath23 + /@xmath24 ) at 8 tev lhc @xcite . in this work , we consider these latest lhc data and examine the status of a light dm in the nmssm . we note that a recent study @xcite tried to explain the cdms - ii results in terms of a light dm in the nmssm . compared to @xcite which only studied three representative benchmark points , we perform a numerical scan under various experimental constraints and display the allowed parameter space in comparison with the the direct detection results of cogent , cdms - ii and lux . we also perform a global fit of the higgs data using the package @xcite , in which we further consider the latest lhc results of higgs invisible decay from the channel @xmath25 @xcite . moreover , we consider the constraints from the searches for events with @xmath23 + /@xmath24 signal at 8 tev lhc @xcite . the paper is organized as follows . in sec.ii we list the experimental constraints and describe our scan . in sect.iii we present our results and perform detailed analysis . finally , we draw our conclusions in sec.iv . in order to reduce the number of free parameters in our scan over the nmssm parameter space , we make some assumptions on the parameters that do not influence dm properties significantly . explicitly speaking , we fix gluino mass and all the soft mass parameters in squark sector at 2 tev , and those in slepton sector at 300 gev . we also assume the soft trilinear couplings @xmath26 and let them vary to tune the higgs mass . moreover , in order to predict a bino - like light dm and also to avoid the constraints from @xmath27 invisible decay @xcite , we abandon the gut relation between @xmath28 and @xmath29 . the free parameters are then @xmath30 in the higgs sector , the gaugino and higgsino mass parameters @xmath31 and @xmath32 , and the soft trilinear couplings of the third generation squarks @xmath33 . in this work , we define all these parameters at @xmath34 scale and adopt the markov chain monte carlo ( mcmc ) method to scan the following parameter ranges using @xmath35 @xcite : @xmath36 note here that the ranges of @xmath37 and @xmath38 are motivated to avoid landau pole , generally corresponding to the requirement of @xmath39 . this has been encoded in @xmath35 including the consideration of the interplay between @xmath37 and @xmath38 in the renormalization group running . a relatively small @xmath32 is chosen to avoid strong cancelation in getting the @xmath27 boson mass @xcite , and as we will see below , the upper bound of @xmath40 for @xmath32 here suffices our study and does not affect our main conclusions . also note that we artificially impose a lower bound of 320 gev for @xmath29 . this is motivated by the fact that @xmath29 in our study is not an important parameter , and that as required by the @xmath23 + /@xmath24 constraint @xmath29 should be larger than about 320 gev in the simplified model discussed in @xcite ( also see the constraint ( viii ) discussed below ) . the relevant @xmath41 function for the mcmc scan is build to guarantee the dm relic density and the sm - like higgs boson mass around their measured values . in our discussion , we consider the samples surviving the following constraints : * @xmath42 and @xmath43 . * _ the constraints from b - physics_. the light cp - even / odd higgs bosons can significantly affect the b - physics observables . especially , the precise measurements of radiative decays @xmath44 @xcite , @xmath45 @xcite and @xmath46 @xcite can give stringent constraints . so we require the samples to satisfy these b - physics bounds at 2@xmath47 level . * _ dm relic density_. as the sole dark matter candidate , the lightest neutralino @xmath12 is required to produce the correct thermal relic density . we require the neutralino relic density to be in the @xmath48 range of the planck and wmap 9-year data , @xmath49 , where a 10% theoretical uncertainty is included @xcite . * _ muon g-2_. we require nmssm to explain the muon anomalous magnetic moment data @xmath50 @xcite at @xmath48 level . * _ the absence of landau pole_. we impose this constraint using @xmath35 @xcite , where the interplay of @xmath37 and @xmath38 in the renormalization group running has been considered . * _ lep searches for susy_. for the lep experiments , the strongest constraints come from the chargino mass and the invisible @xmath27 decay . we require @xmath51 and the non - sm invisible decay width of @xmath52 to be smaller than 1.71 mev , which is consistent with the precision electroweak measurement result @xmath53 mev at @xmath54 confidence level @xcite . * _ higgs data_. firstly , we consider the exclusion limits of the lep , tevatron and lhc in higgs searches with the package @xcite . this package also takes into account the results of the lhc searches for non - sm higgs bosons , such as @xmath55 and @xmath56 @xcite . secondly , noticing that a light @xmath14 ( or @xmath16 ) may induce the distinguished signal @xmath57 , we consider the limitation of the @xmath58 signal on the parameter space using the latest cms results @xcite . finally , since a large invisible branching ratio of the higgs may be predicted in the light dm case , we perform a global fit of the higgs data using the package @xcite , where the systematics and correlations for the signal rate predictions , luminosity and higgs mass predictions are taken into account . in our fit , we further consider the latest lhc results of higgs invisible decay from the channel @xmath25 @xcite . we require our samples to be consistent with the higgs data at @xmath48 level , which corresponds to @xmath59 with @xmath41 obtained with the and @xmath60 denoting the minimum value of @xmath41 for the surviving samples in our scan . * _ lhc searches for susy_. based on the 20 fb@xmath61 data collected at the 8 tev run , the atlas and cms collaborations performed a search for the @xmath62 production with @xmath23 + /@xmath24 signal in a simplified model , where both @xmath63 and @xmath64 are assumed to be wino - like with @xmath65 , and a 95% c.l . upper limit on @xmath66 was obtained on the @xmath67 plane @xcite . + in this work , in order to implement this constraint we perform an analysis similar to @xcite with the code checkmate @xcite for each sample surviving the constraints ( i ) - ( vii ) . we consider the contributions from all @xmath68 ( @xmath69 and @xmath70 ) associated production processes to the signal , and calculate the production rates and the branching ratios with the code prospino2 @xcite and nmsdecay @xcite , respectively . our analysis indicates that this constraint can exclude effectively those samples with small values of @xmath32 below 115 gev , and also some samples with moderate @xmath32 in the range from 115 gev to 200 gev . nevertheless , compared to the results without considering this constraint , our conclusions do not change much such as the upper bounds on @xmath71 presented below . in order to study the implication of the dm direct detection experiments on the nmssm , we also calculate the dm spin - independent elastic scattering cross section off nucleon with the formulae used in our previous work @xcite . in getting the cross section , we set the parameter of the strange quark content in the nucleon as @xmath72 . in the rest of this work , we categorize the dm by its component , i.e. either bino - like or singlino - like , in presenting our results . since the interactions of the neutralinos with the higgs bosons come from the following lagrangian @xmath73 where the fields @xmath74 , @xmath75 and @xmath76 denote the neutral scalar parts of the higgs superfields @xmath2 , @xmath77 and @xmath78 , respectively , one can infer that if the dm is bino - like , the coupling strength of the @xmath79 interaction is mainly determined by the higgsino - component in @xmath80 , or more basically by the value of @xmath32 . to be more specific , if @xmath81 is sm - like , the coupling strength is mainly determined by the first two terms in the second row of eq.[interaction ] , while if @xmath81 is singlet - dominated , the coupling of @xmath79 is mainly determined by the first term of eq.[interaction ] . however , if the dm is singlino - like , the coupling strength is fundamentally determined by the parameters @xmath37 and @xmath38 and a low @xmath32 value may be helpful to enhance the coupling . in this work , we are also interested in the couplings of the sm - like higgs to light singlet - like scalars @xmath14 and @xmath16 . these couplings are mainly determined by the following terms in the higgs potential @xcite @xmath82 this equation indicates that , if @xmath37 and @xmath38 approach zero , the couplings @xmath83 can not be very large ; while if both of them have a moderate value , accidental cancelation is very essential to suppress the couplings . in fig.[fig1 ] we project the samples surviving the above constraints on the plane of neutralino dark matter mass versus spin - independent neutralino - nucleon scattering cross section . about this figure , we want to emphasize two points . the first one is that some of the experimental constraints , such as the dm relic density and the higgs data , play an important role in limiting the parameter space of the nmssm . so in the following , we pay special attention to investigate how the samples in fig.[fig1 ] survive these constraints . the other one is that the various experimental constraints will cut into the parameter space and the interplay among them is very complicated . as a result , the sample distributions on the @xmath84 plane might be very wired . the strategy of analyzing this figure is to get a general picture of the current status of light dm confronting the direct detection results and then focus on some interesting regions . as we will discuss later , we will mainly focus on those samples that either can explain the cdms - ii results or can survive the first lux exclusion . we will not consider the up - right region ( @xmath85 ) in fig.[fig1 ] since it is not experimentally hinted . after carefully analyzing our results , we have the following observations from fig.[fig1 ] : 1 . in the nmssm , dm as light as @xmath86 is still allowed by the current higgs data . both the bino and singlino - like dm are capable of explaining the results of cdms - ii and cogent , or surviving the current lux results and future lux exclusion limits . 2 . as pointed out in @xcite , light dm in the nmssm may annihilate in the early universe through @xmath74-channel resonance effect of some mediators or into light higgs scalar pair to get a correct relic density . we checked that , for @xmath87 , singlino - like dm annihilated in the early universe mainly through the @xmath74-channel resonance effect of @xmath88 for the most case ; while bino - like dm might annihilate either through the resonance effect or into @xmath14 ( @xmath16 ) pair . we will discuss this issue in more detail later . + in fact , the long thick band of grey samples ( for bino - like dm ) around @xmath89 exactly corresponds to the resonance case , and samples along this band are characterized by @xmath90 with @xmath91 denoting the mediator mass . for @xmath92 and @xmath93 , the mediator is @xmath27 boson and the sm - like higgs boson , respectively , while in other cases the mediator is either @xmath14 or @xmath16 . these conclusions can also apply to the singlino - like dm ( see fig.[fig2 ] ) . 3 . for samples with @xmath94 , generally @xmath14 needs to be lighter than about @xmath95 to push up the scattering rate . for the bino - like dm with mass varying from @xmath96 to @xmath97 , such a light @xmath14 is difficult to obtain after considering the constraint from the relic density ( see fig.[fig2 ] ) . while in the @xmath27 ( @xmath17 ) resonance region , the relic density has rather weak limitation on @xmath14 properties . in this case , @xmath14 may be as light as several gev so that the scattering rate is rather large , or the coupling @xmath98 may be greatly reduced to result in a relatively small @xmath99 . 4 . for bino - like dm , generally it is not easy to obtain samples with @xmath100 . this is because the @xmath101 interaction is still sizable even after considering the various constraints ( see discussions on fig.[fig3 ] ) , and in this case , the @xmath17-mediated contribution to the dm - nucleon scattering is important . however , in the extreme case when the bino - like dm is close to about @xmath102 , due to the lower bound of @xmath32 , the higgsino component in the dm will get further reduced and result in an even smaller @xmath103 . 5 . when focusing on the xenon and lux experiments , the bino - like and singlino - like dm exhibit quite different behaviors . the first lux-300 kg result can exclude a large part of the allowed parameter space , but still leaves both the bino - like and singlino - like light dm viable . the future xenon-1 t and lux-7.2ton results can cut further deeply into the parameter space . especially , they limit tightly the bino - like dm case and constrain most of the bino - like dm mass to be lower than about 17 gev and 12 gev , respectively , while the singlino - like dm can still survive leisurely . 6 . for bino - like dm samples there is a gap in the right half part of the cdms - ii @xmath48 region . this is due to the tension between the lhc higgs data and the constraint from @xmath104 . as discussed in @xcite ( and see table i ) , the cdms - ii favored samples in bino - like dm scenario usually require a moderate @xmath37 along with a moderate @xmath38 to achieve the accidental cancelation in @xmath105 so that the sm - like higgs decay to @xmath14 or @xmath16 pair is suppressed . while on the other hand , this may increase the effective coupling of @xmath14 to down - type fermions which is proportional to @xmath106 with @xmath107 $ ] @xcite , and receive constraint from the measurement of @xmath104 . we checked that most of the excluded bino - like dm samples in the gap have a relatively large @xmath37 , while the singlino - like dm samples usually correspond to a small @xmath37 ( see following discussion on table i ) and thus receive less constraint . compared to bino - like dm which is restricted in certain areas on the @xmath84 plane , singlino - like dm can spread nearly to the whole region of the plane . this reflects the fact that singlino - like dm is more adaptable in light dm physics . .the ranges of relevant nmssm input parameters corresponding to part of the samples in fig.1 , which predict a dm lighter than 35 gev and meanwhile can explain the cdms - ii at 2@xmath47 level or survive the lux-300 kg exclusion limit . parameters with the mass dimension are in the unit of gev . [ cols="^ , < , < , < , < " , ] in the following , we concentrate on the samples in fig.1 that can either explain the cdms - ii experiment at @xmath48 level or survive the lux-300 kg exclusion limit . since the results of the cdms - ii and lux experiments are so incompatible , it would be interesting to investigate the difference of these two types of samples . to simply our analysis , we mainly consider the samples predicting a dm lighter than about 35 gev . these samples are not easy to obtain with traditional random scan method when exploring the susy parameter space due to its rather specific particle spectrum , but as we will see below , the underlying physics of these samples are clear and easy to understood . in table [ table1 ] , we list the ranges of relevant nmssm input parameters corresponding to these samples , which are classified by the component of the dm ( i.e. bino - like or singlino - like ) and meanwhile by its scattering cross section off the nucleon ( i.e. can explain the cdms - ii results at @xmath48 level or survive the lux-300 kg exclusion limit ) . from table i , one can learn the following facts : * the survived parameter ranges for lux - safe samples are generally wider than those of cdms - ii preferred samples . this is totally expectable from the experimental data of lux and cdms - ii . on the @xmath84 plane , cdms - ii @xmath48 region is constrained in a relatively narrow range @xmath108 and @xmath109 . to survive the first lux exclusion , however , a properly large @xmath110 for a certain @xmath111 will be enough . @xmath111 can cover the whole range @xmath112 and @xmath99 can vary from @xmath113 to @xmath114 . therefore , compared to cdms - ii region , there will be more freedom for the parameter space to satisfy the lux exclusion . * to obtain a dm lighter than 35 gev , one needs to have @xmath115 for the bino - like dm and @xmath116 for the singlino - like dm . this can be easily understood from the neutralino mass matrix @xcite . @xmath117 where @xmath118 and @xmath119 are gauge couplings , and @xmath120 and @xmath121 are higgs vacuum expectation values . in fact , a simple estimation can be made for singlino - like dm mass . table [ table1 ] shows that @xmath122 is usually at least one order smaller than @xmath37 . assuming @xmath123 and @xmath124 , we will have @xmath125 . * the cdms - ii samples usually have @xmath126 for both bino - like and singlino - like dm . the underlying reason is that a small value of @xmath32 and consequently a sufficient amount of higgsino component in the dm is helpful to increase the coupling strength of the dm with the light higgs bosons . this will in return push up the rate of the dm - nucleon scattering which is required by the cdms - ii results . * more interestingly , we find that for samples in the whole range of @xmath127 , the value of @xmath32 is upper bounded by about @xmath128 and @xmath129 for bino - like and singlino - like dm , respectively . two reasons can account for this . the first one is that in our scan , we required the nmssm to explain the muon anomalous magnetic moment . the parameter @xmath32 influences the contribution of the nmssm to the moment through chargino and neutralino mass , and a large value of @xmath32 will reduce the contribution significantly . another important reason is that , as mentioned above and also discussed below , in order to get a correct dm relic density , a light @xmath14 or @xmath16 must be present . noting that @xmath32 enters explicitly the squared mass of the singlet scalar @xcite , one can infer that too large values of @xmath32 can not be favored to get the desired light scalar masses . + we also want to emphasize that , for the bino - like dm , an upper bound of @xmath32 will result in a lower limit of the higgsino component in the dm and thus a lower bound of the invisible branching ratio for @xmath130 . this can be explicitly seen in the left panel of fig.[fig3 ] below . * for singlino - like dm case , both @xmath37 and @xmath38 are small and especially , @xmath122 is very close to 0 . as indicated by eqs.([interaction],[higgs_potential],[neutralino_mass_matrix ] ) , the couplings of sm - like higgs boson to dm and also to the light higgs scalars @xmath131 will usually be suppressed . this can result in a @xmath99 as low as @xmath132 ( see fig.[fig1 ] ) and also a relatively small rate for the decays @xmath133 ( see fig.[fig3 ] and fig.[fig4 ] ) . while for the bino - like dm case with a moderate value of @xmath37 and @xmath38 , accidental cancelation is very essential to suppress the couplings of @xmath17 to @xmath134 and obtain an allowed higgs signal . as discussed in fig.[fig1 ] , given @xmath135 , at least one light scalar is needed to accelerate the annihilation . in order to illustrate this feature , in fig.[fig2 ] we project the @xmath135 samples of fig.1 which can explain the cdms - ii results at 2@xmath47 level or survive the lux-300 kg exclusion limits on the plane of dm mass versus @xmath136 . red codes represent samples suggested by the cdms - ii experiment and meanwhile satisfying @xmath137 , while cyan ( blue ) codes correspond to samples surviving the lux-300 kg exclusion limits and also satisfying @xmath137 ( @xmath138 ) . note that due to the large scattering cross section favored by the cdms - ii results , a light @xmath14 is needed ( as the t - channel propagator ) and the case @xmath139 is absent . from fig.[fig2 ] we have the following observations : samples in fig.[fig1 ] which can explain the cdms - ii results at 2@xmath47 level or survive the lux-300 kg exclusion limits , projected on the plane of dm mass versus @xmath136 . red codes represent samples suggested by the cdms - ii experiment and meanwhile satisfying @xmath137 , while cyan(blue ) codes correspond to samples surviving the lux-300 kg exclusion limits and also satisfying @xmath137 ( @xmath138 ) . note that due to the large scattering cross section favored by the cdms - ii results , a light @xmath14 is needed ( as the t - channel propagator ) and the case @xmath139 is absent.,width=604 ] 1 . in both bino - like and singlino - like dm scenario , the straight line @xmath140 is very obvious , which corresponds to the s - channel resonance effect of @xmath14 or @xmath16 . however , in the singlino - like scenario with @xmath141 , there are some small regions where the line seems to be not continuous . in fact , this is not the case . we checked that there still exits a scalar ( either @xmath14 or @xmath16 ) with mass around @xmath142 . it is just that this scalar does not correspond to the lightest higgs boson . moreover , for the scalars shown in fig.[fig2 ] , we checked that they are highly singlet - dominated , which agree with previous study in @xcite . 2 . since @xmath14 contributes to the spin - independent dm - nucleon scattering as the t - channel propagator @xcite , a very light @xmath14 is needed to explain the cdms - ii result . for the bino - like dm , the cdms - ii samples are mainly distributed in low @xmath143 region with @xmath110 upper bounded by about 4 gev , while for the singlino - like dm , the corresponding samples spread a larger region in @xmath144 plane . moreover , when focusing on the cdms - ii samples , we checked that if the dm is bino - like , the channel @xmath145 plays the dominant role in contributing to the dm annihilation , while if the dm is singlino - like , the s - channel resonance effect is the main contribution . since the constraint from the lux-300 kg data on the scattering rate is rather weak in the very light dm region , @xmath14 as light as 1 gev is still allowed for @xmath146 . with the increase of dm mass , the constraint becomes much stronger and @xmath14 generally needs to be heavier than about 10 gev for @xmath147 in both scenarios . , but projected on the plane of the invisible branching fractions of the sm - like higgs boson versus dm mass , and extended the dm mass to about 60 gev.,width=604 ] for the sm - like higgs boson , since the decay channel @xmath148 is opened when @xmath149 , one can expect that the higgs data will impose rather tight constraints on this decay rate . in fig.[fig3 ] , we show the samples of fig.[fig2 ] on the plane of @xmath150 versus dm mass and extend the dm mass to about 60 gev . we have the following observations : 1 . the current higgs data still allow for an invisible decay branching ratio as large as @xmath0 at @xmath48 level . the tolerance of such a large invisible branching ratio is owe to the large uncertainties of the current data , especially the fact that atlas and cms data point to two opposite directions in the di - photon rate . obviously , an invisible decay branching ratio reaching @xmath0 may be easily tested at the 14 tev lhc with @xmath151 fb@xmath61 , where a 95% c.l . upper limit on the invisible decay , i.e. @xmath152 , can be imposed @xcite . 2 . in the bino - like dm scenario , due to the necessary higgsino component in the dm required by an efficient dm annihilation rate , the interaction between dm and @xmath17 can be relatively large . as a result , @xmath150 as large as @xmath0 is possible . note that for the cdms - ii samples , @xmath150 is always larger than about @xmath153 . the underlying reason is that , as we mentioned earlier , the channel @xmath145 plays an important role in contributing to the dm annihilation . this requires the strength of the @xmath154 interaction to be sufficiently large , and so is the @xmath155 interaction . also note that since @xmath32 is upper bounded for @xmath156 ( see table [ table1 ] ) , generally there is a lower bound of @xmath157 . 3 . in the singlino - like scenario , since the @xmath158 coupling is determined by @xmath37 and @xmath38 and table [ table1 ] indicates that these two parameters are generally small , @xmath150 is usually suppressed and can reach about @xmath159 in the optimal case . , but showing the branching fraction of decays @xmath160 versus dm mass.,width=604 ] due to the existence of light scalars in light dm scenario , the sm - like higgs may also decay into the lighter scalars , @xmath161 . unlike the @xmath162 coupling , the coupling strengthes of @xmath17 to these scalars are mainly determined by @xmath37 and @xmath38 ( see eq.([higgs_potential ] ) and also note that both @xmath14 and @xmath16 are highly singlet - dominated @xcite ) . consequently , according to table [ table1 ] , the maximum decay rate in the bino - like dm scenario should in principle be larger than that in the singlino - like case . similar to fig.[fig3 ] , we show the total branching fractions of these two decays versus dm mass in fig.[fig4 ] . one can learn that this branching ratio can reach @xmath0 in the bino - like dm scenario , while in the singlino - like case the maximum can only reach about @xmath159 . under current experimental constraints including the latest lhc higgs data and the dark matter relic density , we examined the status of a light nmssm dark matter and confronted it with the direct detection results of cogent , cdms - ii and lux . we have the following observations : ( i ) a dark matter as light as 8 gev is still allowed and its scattering cross section off the nucleon can be large enough to explain the cogent / cdms - ii favored region ; ( ii ) the lux data can exclude a sizable part of the allowed parameter space , but still leaves a light dark matter viable ; ( iii ) the sm - like higgs boson can decay into the light dark matter pair and its branching ratio can reach @xmath0 at @xmath48 level under the current lhc higgs data , which may be covered largely at the 14 tev lhc experiment . we thank nima arkani - hamed , archil kobakhidze , yang zhang and jie ren for helpful discussions . this work was supported by the arc center of excellence for particle physics at the tera - scale , by the national natural science foundation of china ( nnsfc ) under grant no . 10821504 , 11222548 , 11305049 and 11135003 , and also by program for new century excellent talents in university . s. schael _ et al . _ [ aleph and delphi and l3 and opal and sld and lep electroweak working group and sld electroweak group and sld heavy flavour group collaborations ] , phys . rept . * 427 * , 257 ( 2006 ) [ hep - ex/0509008 ] .

in susy , a light dark matter is usually accompanied by light scalars to achieve the correct relic density , which opens new decay channels of the sm - like higgs boson . under current experimental constraints including the latest lhc higgs data and the dark matter relic density , we examine the status of a light neutralino dark matter in the framework of nmssm and confront it with the direct detection results of cogent , cdms - ii and lux . we have the following observations : ( i ) a dark matter as light as 8 gev is still allowed and its scattering cross section off the nucleon can be large enough to explain the cogent / cdms - ii favored region ; ( ii ) the lux data can exclude a sizable part of the allowed parameter space , but still leaves a light dark matter viable ; ( iii ) the sm - like higgs boson can decay into the light dark matter pair with an invisible branching ratio reaching @xmath0 under the current lhc higgs data , which may be tested at the 14 tev lhc experiment .

understanding the formation mechanisms and evolution with cosmic time of galaxies is one of the major goals of observational cosmology . in the current picture of structure formation , dark matter halos build up in a hierarchical fashion controlled by the nature of the dark matter , the power spectrum of density fluctuations , and the parameters of the cosmological model . the assembly of the stellar content of galaxies is governed by much more complicated physics , such as the mechanisms of star formation , gaseous dissipation , the feedback of stellar and central supermassive black hole energetic output on the baryonic material of the galaxies , and mergers . the mean space density of galaxies per unit luminosity , or luminosity function ( lf ) , is one of the most fundamental of all cosmological observables , and it is one of the most basic descriptors of a galaxy population . the shape of the lf retains the imprint of galaxy formation and evolution processes ; the evolution of the lf as a function of cosmic time , galaxy type and environment provides insights into the physical processes that govern the assembly and the following evolution of galaxies . therefore , the lf represents one of the fundamental observational tools to constrain the free parameters of theoretical models . the local ( @xmath6 ) lf has been very well determined from several wide - area , multi - wave band surveys with follow - up spectroscopy ( @xcite ; @xcite ; @xcite ; @xcite ) . at intermediate redshifts ( @xmath16 ) , spectroscopic surveys found a steepening of the faint - end lf with increasing redshift in the global lf , mainly due to the contribution by later type galaxies ( @xcite ; @xcite ) . from the combo-17 survey , @xcite measured the rest - frame optical lf up to @xmath17 , finding that early - type galaxies show a decrease of a factor of @xmath18 in the characteristic density @xmath8 of the lf . the latest type galaxies show a brightening of @xmath19 mag in @xmath20 ( the characteristic magnitude ) and an increase of @xmath21 in @xmath8 in their highest redshift bin in the blue band . further progress in the measurement of the lf at @xmath22 was obtained with the vimos vlt deep survey ( vvds ; @xcite ) and the deep-2 galaxy redshift survey @xcite . from the vvds data , @xcite measured the rest - frame optical lf from @xmath23 to @xmath24 . from the same data set , @xcite performed a similar analysis for different spectral galaxy types , finding a significant steepening of the lf going from early to late types . their results indicate a strong type - dependent evolution of the lf , and identify the latest spectral types as responsible for most of the evolution of the uv - optical lf out to @xmath25 . contrary to low - redshift studies , the selection of high - redshift ( @xmath5 ) galaxies still largely relies on their colors . one of the most efficient ways to select high - redshift galaxies is the lyman drop - out technique , which enabled steidel and collaborators to build large samples of @xmath26 star - forming galaxies ( @xcite , 1999 ) . extensive studies of these optically ( rest - frame ultraviolet ) selected galaxies at @xmath26 ( lyman break galaxies [ lbgs ] ) and at @xmath27 ( bm / bx galaxies ; @xcite ; @xcite ) have shown that they are typically characterized by low extinction , modest ages , stellar masses @xmath28 m@xmath29 , and star formation rates of 10100 m@xmath29yr@xmath30 ( @xcite ; @xcite ; @xcite ) . @xcite recovered the rest - frame @xmath1-band lf of lbgs at @xmath26 from the rest - frame uv lf ( @xcite ; but see also @xcite ) , finding that the lbg lf is characterized by a very steep faint end . lbgs dominate the uv luminosity density at @xmath31 , as well as possibly the global star formation rate density at these redshifts @xcite . however , since the lyman break selection technique requires galaxies to be very bright in the rest - frame uv in order to be selected , it might miss galaxies that are heavily obscured by dust or whose light is dominated by evolved stellar populations . these objects can be selected in the near - infrared ( nir ) , which corresponds to the rest - frame optical out to @xmath26 . using the nir selection criterion @xmath32 ( also suggested by @xcite ) , @xcite and @xcite discovered a new population of high - redshift galaxies ( distant red galaxies [ drgs ] ) that would be largely missed by optically selected surveys . follow - up studies have shown that drgs constitute a heterogeneous population . they are mostly actively forming stars at @xmath33 ( @xcite ; @xcite ; @xcite ; @xcite ; @xcite ; @xcite ) . however , some show no signs of active star formation and appear to be passively evolving ( @xcite ; @xcite ; @xcite ) , while others seem to host powerful active galactic nuclei ( @xcite ; @xcite ) . compared to lbgs , drgs have systematically older ages and larger masses @xcite , although some overlap between the two exists ( @xcite ; @xcite ) . recently , @xcite have demonstrated that in a mass - selected sample ( @xmath34 @xmath35 ) at @xmath36 , drgs make up 77% in mass , compared to only 17% from lbgs ( see also @xcite ) , implying that the rest - frame optical lf determined by @xcite is incomplete . the global ( i.e. , including all galaxy types ) rest - frame optical lf at @xmath5 can be studied by combining multiwavelength catalogs with photometric redshift information . @xcite studied the @xmath0-band lfs of red and blue galaxies . they find that the @xmath0-band number densities of red and blue galaxies have different evolution , with a strong decrease of the red population at @xmath37 compared to @xmath38 and a corresponding increase of the blue population , in broad agreement with the predictions from their hierarchical cold dark matter models . as all previous works at @xmath5 are based on either very deep photometry but small total survey area ( @xcite ; @xcite ) or larger but still single field surveys ( @xcite ) , their results are strongly affected by field - to - field variations and by low number statistics , especially at the bright end . moreover , @xcite used an i - band selected data set from the fors deep field . the @xmath39 band corresponds to the rest - frame uv at @xmath27 , which means that significant extrapolation is required . in this paper we take advantage of the deep nir musyc survey to measure the rest - frame optical ( @xmath0 , @xmath1 , and @xmath2 band ) lfs of galaxies at @xmath3 . its unique combination of surveyed area and depth allows us to ( 1 ) minimize the effects of field - to - field variations , ( 2 ) better probe the bright end of the lf with good statistics , and ( 3 ) sample the lf down to luminosities @xmath40 mag fainter than the characteristic magnitude . to constrain the faint - end slope of the lf and to increase the statistics , we also made use of the fires and the goods - cdfs surveys , by constructing a composite sample . the large number of galaxies in our composite sample also allows us to measure the lfs of several subsamples of galaxies , such as drgs and non - drgs ( defined based on their observed @xmath15 color ) , and of intrinsically red and blue galaxies ( defined based on their rest - frame @xmath12 color ) . this paper is structured as follows . in [ sec - cs ] we present the composite sample used to measure the lf of galaxies at @xmath3 ; in [ sec - lf ] we describe the methods applied to measure the lf and discuss the uncertainties in the measured lf due to field - to - field variations and errors in the photometric redshift estimates ; the results ( of all galaxies and of the individual subsamples considered in this work ) are presented in [ sec - results ] , while the estimates of the number and luminosity densities and the contribution of drgs ( red galaxies ) to the global stellar mass density are given in [ sec - densities ] . our results are summarized in [ sec - concl ] . we assume @xmath41 , @xmath42 , and @xmath43 km s@xmath30 mpc@xmath30 throughout the paper . all magnitudes and colors are on the vega system , unless identified as `` ab '' . throughout the paper , the @xmath15 color is in the observed frame , while the @xmath12 color refers to the rest frame . the data set we have used to estimate the lf consists of a composite sample of galaxies built from three deep multiwavelength surveys , all having high - quality optical to nir photometry : the `` ultradeep '' faint infrared extragalactic survey ( fires ; @xcite ) , the great observatories origins deep survey ( goods ; @xcite ; chandra deep field south [ cdf - s ] ) , and the multi - wavelength survey by yale - chile ( musyc ; @xcite ; @xcite ) . photometric catalogs were created for all fields in the same way , following the procedures of @xcite . fires consists of two fields , namely , the hubble deep field south proper ( hdf - s ) and the field around ms 105403 , a foreground cluster at @xmath44 . a complete description of the fires observations , reduction procedures , and the construction of photometric catalogs is presented in detail in @xcite and @xcite for hdf - s and ms 105403 , respectively . briefly , the fires hdf - s and ms 105403 ( hereafter fh and fms , respectively ) are @xmath45 band limited multicolor source catalogs down to @xmath46 and @xmath47 , for a total of 833 and 1858 sources over fields of @xmath48 and @xmath49 , respectively . the fh and fms catalogs have 90% completeness level at @xmath50 and @xmath51 , respectively . the final fh ( fms ) catalogs used in the construction of the composite sample has 358 ( 1427 ) objects over an effective area of 4.74 ( 21.2 ) arcmin@xmath52 , with @xmath53 ( 22.54 ) , which for point sources corresponds to a 10 ( 8) @xmath54 signal - to - noise ratio ( @xmath55 ) in the custom isophotal aperture . lccccrr[!t ] fires - hdfs & u@xmath56b@xmath57v@xmath58i@xmath59j@xmath60hk@xmath60 & 23.80 & 23.14 & 4.74 & 358 & 68 + fires - ms1054 & ubvv@xmath58i@xmath59j@xmath60hk@xmath60 & 22.85 & 22.54 & 21.2 & 1427 & 297 + goods - cdfs & b@xmath61v@xmath58i@xmath62z@xmath63jhk@xmath60 & 21.94 & 21.34 & 65.6 & 1588 & 215 + musyc & ubvrizjhk@xmath60 & 21.33 & 21.09 & 286.1 & 5507 & 116 + from the goods / eis observations of the cdf - s ( data release version 1.0 ) a @xmath45 band limited multicolor source catalog ( hereafter cdfs ) was constructed , described in s. wuyts et al . ( 2007 , in preparation ) . goods zero points were adopted for @xmath64 and @xmath45 . the @xmath65-band zero point was obtained by matching the stellar locus on a @xmath15 versus @xmath66 color - color diagram to the stellar locus in fires hdf - s and ms 105403 . the difference with the official goods @xmath65-band zero point varies across the field , but on average our @xmath65-band zero points are @xmath67 mag brighter . a total effective area of 65.6 arcmin@xmath52 is well exposed in all bands . the final catalog contains 1588 objects with @xmath68 in this area . at @xmath69 the median @xmath55 in the @xmath45 isophotal aperture is @xmath70 . musyc consists of optical and nir imaging of four independent @xmath71 fields with extensive spectroscopic follow - up @xcite . deeper nir @xmath72 imaging was obtained over four @xmath73 subfields with the ispi camera at the cerro tololo inter - american observatory ( ctio ) blanco 4 m telescope . a complete description of the deep nir musyc observations , reduction procedures , and the construction of photometric catalogs will be presented in @xcite . the 5 @xmath54 point - source limiting depths are @xmath74 , @xmath75 , and @xmath76 . the optical @xmath77 data are described in @xcite . the present work is restricted to three of the four deep fields : the two adjacent fields centered around hdf - s proper ( hereafter mh1 and mh2 ) and the field centered around the quasar sdss 1030 + 05 ( m1030 ) . the final musyc @xmath45-selected catalog used in the construction of the composite sample has 5507 objects over an effective area of 286.1 arcmin@xmath52 , with @xmath78 , which for point sources corresponds to a @xmath1410 @xmath54 @xmath55 in the isophotal aperture . table [ tab - ref1 ] summarizes the specifications of each field , including wave band coverage , @xmath79 band total magnitude 90% completeness limit ( @xmath80 ) , effective area , the @xmath79 band total magnitude limit used to construct the composite sample ( @xmath81 ) , the number of objects , and the number of sources with spectroscopic redshifts . only a few percent of the sources in the considered catalogs have spectroscopic redshift measurements . consequently , we must rely primarily on photometric redshift estimates . photometric redshifts @xmath82 for all galaxies are derived using an identical code to that presented in @xcite , but with a slightly modified template set . this code models the observed spectral energy distribution ( sed ) using nonnegative linear combinations of a set of eight galaxy templates . as in @xcite , we use the e , sbc , scd , and i m seds from @xcite , the two least reddened starburst templates from @xcite , and a synthetic template corresponding to a 10 myr old simple stellar population ( ssp ) with a @xcite stellar initial mass function ( imf ) . we also added a 1 gyr old ssp with a salpeter imf , generated with the @xcite evolutionary synthesis code . the empirical templates have been extended into the uv and the nir using models . comparing the photometric redshifts with 696 spectroscopic redshifts ( 63 at @xmath83 ) collected from the literature and from our own observations gives a scatter in @xmath84 of @xmath85 . restricting the analysis to galaxies at @xmath83 in the musyc fields gives @xmath86 , corresponding to @xmath87 at @xmath88 . approximately 5% of galaxies in this sample are `` catastrophic '' outliers . a full discussion of the quality of the photometric redshifts is given elsewhere @xcite . the effects of photometric redshift errors on the derived lfs are modeled in [ sub - zphot ] . rest - frame luminosities are computed from the observed seds and redshift information using the method extensively described in the appendix of @xcite . this method does not depend directly on template fits to the data but rather interpolates directly between the observed fluxes , using the best - fit templates as a guide . we computed rest - frame luminosities in the @xmath89 , @xmath0 , @xmath1 , and @xmath2 filters of @xcite . for these filters we use @xmath90 , @xmath91 , @xmath92 , and @xmath93 . in all cases where a spectroscopic redshift is available we computed the rest - frame luminosities fixed at @xmath94 . stars in all @xmath45-selected catalogs were identified by spectroscopy , by fitting the object seds with stellar templates from @xcite and/or inspecting their morphologies , as in @xcite . on average , approximately 10% of all the objects were classified as stars . we constructed a composite sample of high - redshift ( @xmath95 ) galaxies to be used in the estimate of the lf in [ sec - lf ] . a large composite sample with a wide range of luminosities is required to sample both the faint and the bright end of the lf well ; moreover , a large surveyed area is necessary to account for sample variance . the very deep fires allows us to constrain the faint end of the lf , while the large area of musyc allows us to sample the bright end of the lf very well . the cdfs catalog bridges the two slightly overlapping regimes and improves the number statistics . the final composite sample includes 442 , 405 , and 547 @xmath45-selected galaxies in the three targeted redshift intervals @xmath96 , @xmath97 , and @xmath98 , respectively , for a total of 989 galaxies with @xmath99 at @xmath95 . of these , @xmath144% have spectroscopic redshifts . in figure [ fig2_ref ] we show the rest - frame @xmath0-band absolute magnitude versus the redshift for the composite sample in the studied redshift range @xmath3 . to estimate the observed lf in the case of a composite sample , we have applied an extended version of the @xmath100 algorithm @xcite as defined in @xcite so that several samples can be combined in one calculation . for a given redshift interval [ @xmath101,@xmath102 , we computed the galaxy number density @xmath103 in each magnitude bin @xmath104 in the following way : @xmath105 where @xmath106 is the number of objects in the chosen bin and @xmath107 is : @xmath108 where @xmath109 is the area in units of steradians corresponding to the @xmath110th field , @xmath111 is the number of samples combined together , @xmath112 is the comoving volume element per steradian , and @xmath113 is the minimum of @xmath114 and the maximum redshift at which the @xmath115th object could have been observed within the magnitude limit of the @xmath110th sample . the poisson error in each magnitude bin was computed adopting the recipe of @xcite valid also for small numbers . the @xmath100 estimator has the advantages of simplicity and no a priori assumption of a functional form for the luminosity distribution ; it also yields a fully normalized solution . however , it can be affected by the presence of clustering in the sample , leading to a poor estimate of the faint - end slope of the lf . although field - to - field variation represents a significant source of uncertainty in deep surveys ( since they are characterized by very small areas and hence small sampled volumes ) , the majority of published cosmological number densities and related quantities do not properly account for sample variance in their quoted error budgets . our composite sample is made of several independent fields with a large total effective area of @xmath116 arcmin@xmath52 ( about a factor of 3 larger than the nominal area of the @xmath45-selected cdfs - goods catalog used in @xcite ) , which significantly reduces the uncertainties due to sample variance . also , the large number of fields considered in this work with their large individual areas allows us to empirically measure the field - to - field variations from one field to the other in the estimate of the lf with the @xmath100 method , especially at the bright end , and to properly account for it in the error budget . in order to quantify the uncertainties due to field - to - field variations in the determination of the lf , we proceeded as follows . first , for each magnitude bin @xmath104 , we measured @xmath117 for each individual @xmath110th field using equation ( [ eq-1vmax ] ) . for each magnitude bin with @xmath118 , we estimated the contribution to the error budget of @xmath119 from sample variance using : @xmath120 with @xmath111 the number of individual fields used . for the magnitude bins with @xmath121 ( usually the brightest bin and the 3 - 4 faintest ones ) , we adopted the mean of the @xmath122 with @xmath118 . the final 1 @xmath54 error associated to @xmath119 is then @xmath123 , with @xmath124 the poisson error in each magnitude bin . we also measured the observed lf using the sty method @xcite , which is a parametric maximum likelihood estimator . the sty method has been shown to be unbiased with respect to density inhomogeneities ( e.g. , @xcite ) , it has well - defined asymptotic error properties ( e.g. @xcite ) , and does not require binning of the data . the sty method assumes that @xmath119 has a universal form , i.e. , the number density of galaxies is separable into a function of luminosity times a function of position : @xmath125 . therefore , the shape of @xmath119 is determined independently of its normalization . we have assumed that @xmath119 is described by a @xcite function , @xmath126 { } \nonumber\\ \times \exp{\big [ -10^{0.4(m^{\star}-m ) } \big]},\end{aligned}\ ] ] where @xmath127 is the faint - end slope parameter , @xmath20 is the characteristic absolute magnitude at which the lf exhibits a rapid change in the slope , and @xmath8 is the normalization . the probability of seeing a galaxy of absolute magnitude @xmath128 at redshift @xmath129 in a magnitude - limited catalog is given by @xmath130 where @xmath131 and @xmath132 are the faintest and brightest absolute magnitudes observable at the redshift @xmath129 in a magnitude - limited sample . the likelihood @xmath133 ( where the product extends over all galaxies in the sample ) is maximized with respect to the parameters @xmath127 and @xmath20 describing the lf @xmath103 . the best - fit solution is obtained by minimizing @xmath134 . a simple and accurate method of estimating errors is to determine the ellipsoid of parameter values defined by @xmath135 where @xmath136 is the @xmath137-point of the @xmath138 distribution with @xmath139 degrees of freedom . parameter @xmath136 is chosen in the standard way depending on the desired confidence level in the estimate ( as described , e.g. , by @xcite ; @xcite ) : @xmath140 , 6.2 , and 11.8 to estimate @xmath141 error contours with 68% , 95% , and 99% confidence level ( 1 , 2 , and 3 @xmath54 , respectively ) . the value of @xmath8 is then obtained by imposing a normalization on the best - fit lf such that the total number of observed galaxies in the composite sample is reproduced . the 1 , 2 , and 3 @xmath54 errors on @xmath8 are estimated from the minimum and maximum values of @xmath8 allowed by the 1 , 2 , and 3 @xmath54 confidence contours in the @xmath141 parameter space , respectively . studies of high - redshift galaxies still largely rely on photometric redshift estimates . it is therefore important to understand how the photometric redshift uncertainties affect the derived lf and to quantify the systematic effects on the lf best - fit parameters . @xcite have shown that at lower redshifts ( @xmath142 ) the measurement of the lf is strongly affected by errors associated with @xmath143 . specifically , large redshift errors together with the steep slope at the bright end of the galaxy lf tend to flatten the observed lf and result in measured @xmath20 systematically brighter than the intrinsic value , since there are more intrinsically faint galaxies scattered into the bright end of the lf than intrinsically bright galaxies scattered into the faint end . using monte carlo simulations , @xcite obtained a best - fit @xmath20 that was 0.8 mag brighter than the intrinsic value in the redshift range @xmath144 . in order to quantify the systematic effect on the lf parameters @xmath127 and @xmath20 in our redshift range of interest ( @xmath3 ) , we performed a series of monte carlo simulations . the details of these simulations and the results are presented in appendix [ app-1 ] . briefly , we generated several model catalogs of galaxies of different brightness according to an input schechter lf , extracted the redshifts of the objects from a probability distribution proportional to the comoving volume per unit redshift ( @xmath112 ) , and obtained the final mock catalogs after applying a limit in the observed apparent magnitude . to simulate the errors in the redshifts , we assumed a redshift error function parametrized as a gaussian distribution function of 1 @xmath54 width @xmath145 , with @xmath146 being the scatter in @xmath147 , and we formed the observed redshift catalog by perturbing the input galaxy redshift within the redshift error function . finally , we determined the lf for the galaxies at @xmath148 using the @xmath100 and maximum likelihood methods described in [ sec-1vmaxmeth ] and [ sec - stymeth ] , respectively . as shown in appendix [ app-1 ] , the systematic effects on the measured @xmath127 and @xmath20 in the redshift interval @xmath3 are negligible with respect to the other uncertainties in the lf estimate if the errors on the photometric redshifts are characterized by a scatter in @xmath147 of @xmath149 , which is the appropriate value for the @xmath150 sample considered in this work . this is not true at @xmath151 , where we find large systematic effects on both @xmath20 and @xmath127 , consistent with @xcite . as explained in detail in appendix [ app-1 ] , the large systematic effects found at @xmath151 arise from the strong redshift dependency of both @xmath112 and @xmath152 at low-@xmath153 ; at @xmath5 these dependencies are much less steep , and this results in smaller systematic effects on the measured lf . from the monte carlo simulations we also quantified that the effects of photometric redshift errors on the estimated luminosity density are typically a few percent ( always @xmath154% ) . we conclude that the parameters of the lf and the luminosity density estimates presented in this work are not significantly affected by the uncertainties in the photometric redshift estimates we also investigated the effects of non - gaussian redshift error probability distributions . systematic outliers in the photometric redshift distribution can potentially cause systematic errors in the lf measurements , although these are much smaller than the random uncertainties in the lf estimates ( if the outliers are randomly distributed ) . ] . in order to include this contribution in the error budget , we conservatively assume a 10% error contribution to the luminosity density error budget due to uncertainties in the photometric redshift estimates . in this section we present the results of the measurement of the lf of galaxies at @xmath150 . we have measured the global lf in the rest - frame @xmath2 and @xmath1 band at redshift @xmath155 and @xmath156 , respectively . as shown in figure [ fig - filters ] , at these redshifts , the rest - frame @xmath2 and @xmath1 bands correspond approximately to the observed @xmath45 band , which is the selection band of the composite sample . we also measured the global lf in the rest - frame @xmath0 band in the redshift interval @xmath155 , to compare it with the rest - frame @xmath2-band lf , and at redshift @xmath157 , to compare it with previous studies . for each redshift interval and rest - frame band we also split the sample based on the _ observed _ @xmath15 color ( @xmath158 , drgs ; @xmath159 , non - drgs ) and the _ rest - frame _ @xmath12 color ( @xmath160 , red galaxies ; @xmath161 , blue galaxies ) . in [ subsec - lf ] we present the global lf of all galaxies , and in [ subsec - sublf ] we present the lfs of the considered subsamples ( drgs , non - drgs , red and blue galaxies ) in the rest - frame @xmath2 band . the results for the rest - frame @xmath1 and @xmath0 bands are shown in appendix [ app-2 ] ; in appendix [ app-3 ] we compare our results with those in the literature . figure [ fig - lf_bvr_all ] shows the global rest - frame @xmath2- and @xmath0-band lfs for galaxies at @xmath155 , the rest - frame @xmath1-band lf at @xmath156 , and the rest - frame @xmath0-band lf at @xmath162 . the large surveyed area of the composite sample allows the determination of the bright end of the optical lf at @xmath150 with unprecedented accuracy , while the depth of fires allows us to constrain also the faint - end slope . this is particularly important because of the well - known correlation between the two parameters @xmath127 and @xmath20 . the best - fit parameters with their 1 , 2 and 3 @xmath54 errors ( from the maximum likelihood analysis ) are listed in table [ tab-3 ] , together with the schechter parameters of the local rest - frame @xmath2-band ( from @xcite ) and @xmath0-band ( from @xcite ) lfs . lcccc[!t ] @xmath163 & @xmath2 & @xmath164 & @xmath165 & @xmath166 + @xmath38 & @xmath2 & @xmath167 & @xmath168 & @xmath169 + @xmath156 & @xmath1 & @xmath170 & @xmath171 & @xmath172 + @xmath163 & @xmath0 & @xmath173 & @xmath174 & @xmath175 + @xmath157 & @xmath0 & @xmath176 & @xmath177 & @xmath178 + @xmath38 & @xmath0 & @xmath179 & @xmath180 & @xmath181 + at redshift @xmath155 , the faint - end slope of the rest - frame @xmath2-band lf is slightly flatter than in the rest - frame @xmath0-band , although the difference is within the errors . in the two higher redshift bins , the faint - end slope of the rest - frame @xmath1-band lf is flatter ( by @xmath182 ) than in the rest - frame @xmath0-band , although the difference is only at the 1 @xmath54 level . similarly , the faint - end slopes of the rest - frame @xmath0-band global lf in the low- and high - redshift bins are statistically identical . the characteristic magnitude @xmath183 in the low-@xmath153 interval is about 0.5 mag fainter with respect to the high - redshift one , although the difference is significant only at the @xmath141.5 @xmath54 level . we therefore conclude that the rest - frame @xmath0-band global lfs in the low- and high - redshift bins are consistent with no evolution within their errors ( @xmath184 @xmath54 ) . in figure [ fig - lf_bvr_all ] we have also plotted the local ( @xmath185 ) rest - frame @xmath2-band ( from @xcite ) and @xmath0-band ( from @xcite ) lfs . the faint - end slope of the @xmath2-band lf at @xmath155 is very similar to the faint - end slope of the local lf ; the characteristic magnitude is instead significantly ( @xmath186 @xmath54 ) brighter than the local value ( by @xmath7 mag ) , and the characteristic density is a factor of @xmath187 smaller than the local value . the rest - frame @xmath0-band lf at @xmath162 is characterized by a faint - end slope consistent with the local @xmath0-band lf ; the characteristic magnitude is significantly brighter ( @xmath186 @xmath54 ) than the local value by @xmath188 mag , while the characteristic density is a factor of @xmath189 smaller with respect to the local value . in this section we present the results of the lfs for different subsamples , by splitting the composite sample based on the observed @xmath15 color ( @xmath158 , drgs ; @xcite ) and on the rest - frame @xmath12 color ( by defining the red galaxies as those having @xmath160 , which is the median value of @xmath12 of the composite sample at @xmath155 ) . in figure [ fig3_ref ] , we show the rest - frame @xmath12 color versus the observed @xmath15 color for the composite sample at @xmath3 . in figure [ lf_r_lowz.ps ] we show the rest - frame @xmath2-band lf at @xmath155 of drgs versus non - drgs and red versus blue galaxies , together with the 1 , 2 , and 3 @xmath54 contour levels in the @xmath190 parameter space from the sty analysis . the lfs of the different subsamples in the rest - frame @xmath1 band at @xmath156 and in the rest - frame @xmath0 band at @xmath155 and @xmath162 are shown in appendix [ app-2 ] in figures . [ lf_v_highz.ps ] , [ lf_b_lowz.ps ] , and [ lf_b_highz.ps ] , respectively . in table [ tab-4 ] the best - fit parameters and their 1 , 2 , and 3 @xmath54 errors from the sty method are listed for all the considered rest - frame bands and redshift intervals . cccccc[!t ] @xmath163 & @xmath2 & @xmath158 & @xmath191 & @xmath192 & @xmath193 + & & @xmath159 & @xmath194 & @xmath195 & @xmath196 + & & @xmath160 & @xmath197 & @xmath198 & @xmath199 + & & @xmath161 & @xmath200 & @xmath201 & @xmath202 + @xmath156 & @xmath1 & @xmath158 & @xmath203 & @xmath204 & @xmath205 + & & @xmath159 & @xmath206 & @xmath207 & @xmath208 + & & @xmath160 & @xmath209 & @xmath210 & @xmath211 + & & @xmath161 & @xmath212 & @xmath213 & @xmath214 + @xmath163 & @xmath0 & @xmath158 & @xmath215 & @xmath216 & @xmath217 + & & @xmath159 & @xmath218 & @xmath219 & @xmath220 + & & @xmath160 & @xmath221 & @xmath222 & @xmath223 + & & @xmath161 & @xmath224 & @xmath225 & @xmath226 + @xmath162 & @xmath0 & @xmath158 & @xmath227 & @xmath228 & @xmath229 + & & @xmath159 & @xmath230 & @xmath231 & @xmath232 + & & @xmath160 & @xmath233 & @xmath234 & @xmath235 + & & @xmath161 & @xmath236 & @xmath237 & @xmath238 + as shown in figure [ lf_r_lowz.ps ] , the rest - frame @xmath2-band lf at @xmath155 of drgs is significantly ( @xmath186 @xmath54 ) different from that of non - drgs . the faint - end slope of the non - drg lf is much steeper , indicating that the contribution of drgs to the global luminosity and number density at faint luminosities is very small compared to that of non - drgs . the bright end of the drg lf is instead very similar to that of non - drgs , with the two subsamples contributing equally to the global lf . splitting the composite sample based on the rest - frame @xmath12 color , we find a qualitatively similar result , with the faint - end slope of the blue galaxy lf being much steeper than that of red galaxies ( although the red galaxies clearly dominate the bright end of the lf ) . the difference between the lfs of drgs ( red galaxies ) and non - drgs ( blue galaxies ) is mainly driven by the different faint - end slopes . a similar result holds in the rest - frame @xmath1 band at @xmath156 , although it is slightly less significant ( at the 2 - 3 @xmath54 level ) : the non - drg ( blue galaxy ) lf is very similar to that of drgs ( red galaxies ) at the bright end , while at the faint end , the lf of non - drgs ( blue galaxies ) is steeper than that of drgs ( red galaxies ) . in the rest - frame @xmath0 band , the differences between the lfs of drgs / red galaxies and non - drgs / blue galaxies become even less significant . although drgs / red galaxies are always characterized by lfs with flatter faint - end slopes , the significance of this result is only marginal ( @xmath184 @xmath54 ) , especially in the higher redshift interval . within our sample , there is marginal evidence for evolution with redshift : the rest - frame @xmath0-band non - drg / blue galaxy lfs in the two targeted redshift bins are characterized by similar ( within the errors ) faint - end slopes , while the characteristic magnitude is brighter by @xmath239 mag in the higher redshift bin . the lf of drgs / red galaxies tends to get steeper from low to high redshifts and @xmath20 gets brighter by @xmath240 mag . however , because of the large uncertainties ( especially for drgs and red galaxies ) on the measured schechter parameters , the differences in the rest - frame @xmath0 band between the high- and the low - redshift bins are at most at the 2 @xmath54 significance level . we note that the uncertainties on the estimated schechter parameters mainly arise from the small number statistics at the very faint end , which is probed only by fires . very deep ( down to the deepest fires ) nir imaging over large spatially disjoint fields is required for further progress in our understanding of the lowest luminosity galaxies at @xmath5 . @xcite computed the rest - frame optical ( @xmath1 band ) lf of @xmath26 lbgs using the distribution of optical @xmath241 magnitudes ( i.e. , the rest - frame uv lf ) and the distribution of @xmath241-@xmath45 colors as a function of @xmath241 magnitude . the rest - frame uv lf of lbgs was taken from @xcite with best - fit schechter parameters @xmath242 , @xmath243 mag , and @xmath244 mpc@xmath245 in our adopted cosmology . @xcite detected a correlation with 98% confidence between @xmath241-@xmath45 color and @xmath241 magnitude , such that fainter galaxies have redder @xmath241-@xmath45 colors . this trend was included in their lf analysis by using the relationship implied by the best - fit regression slope to the correlation , @xmath246 ( the scatter around this regression is very large ) . the schechter function was then fitted to the average lf values , obtaining best - fit schechter parameters @xmath247 , @xmath248 mag , and @xmath249 mpc@xmath245 . the overall shape of the rest - frame optical lf of lbgs is determined by the way in which the @xmath241-@xmath45 distribution as a function of @xmath241 magnitude redistributes @xmath241 magnitudes into @xmath45 magnitudes . therefore , as a result of the detected positive correlation between @xmath241 and @xmath241-@xmath45 , the faint - end slope of the lbg rest - frame optical lf is steeper than that of the uv lf @xcite . in figure [ v_drg_lbg.ps ] we compare the rest - frame @xmath1-band lf of blue galaxies at @xmath156 and the lbg lf from @xcite in the same rest - frame band and redshift interval . the blue galaxy lf estimated with the @xmath100 method appears consistent within the errors with the average lf values of lbgs ( shown as stars in figure [ v_drg_lbg.ps ] ) . however , the best - fit schechter parameters from the maximum likelihood analysis are only marginally consistent , with the faint - end slope of the lbg lf being significantly steeper than the one of blue galaxies , as shown in the inset of figure [ v_drg_lbg.ps ] . the same result is obtained if the rest - frame @xmath1-band lf of non - drgs ( rather than rest - frame blue galaxies ) is compared to that of lbgs . in appendix [ app-3 ] we compare our results with previously published lfs . specifically , we have compared our rest - frame @xmath0-band lf with that derived by @xcite , @xcite , and @xcite in the redshift intervals @xmath155 and @xmath157 , and our rest - frame @xmath2-band lf with the rest - frame @xmath250-band lf derived by @xcite at @xmath155 . we also compared our rest - frame @xmath0-band lfs of red and blue galaxies at @xmath157 with those measured by @xcite . the estimates of the number density @xmath251 obtained by integrating the best - fit schechter function to the faintest observed rest - frame luminosity are listed in table [ tab - ndall ] . for completeness , we also list @xmath252 , calculated by integrating the best - fit schechter lf to the rest - frame magnitude limits of the nir musyc , and @xmath253 , calculated by integrating the best - fit schechter lf to 2 mag fainter than the faintest observed luminosities . we find that the contribution of drgs ( red galaxies ) to the total number density down to the faintest probed rest - frame luminosities is 13%-25% ( 18%-29% ) depending on the redshift interval . by integrating the rest - frame @xmath0-band lf down to a fixed rest - frame magnitude limit ( @xmath254 ) , we find a hint of an increase of the contribution of blue galaxies from the low redshift bin ( 62% ) to the higher bin ( 74% ) , but the differences are not significant . if only the bright end of the lf is considered ( integrating the lf down to the fixed nir musyc limit , @xmath255 ) , the increase of the contribution of the blue galaxy population becomes significant at the 2 @xmath54 level , going from 42% in the low-@xmath153 bin to 66% in the high-@xmath153 bin . lccccc @xmath163 & @xmath2 & all & @xmath256 & @xmath257 & @xmath258 + & & @xmath158 & @xmath259 & @xmath260 & @xmath261 + & & @xmath159 & @xmath262 & @xmath263 & @xmath264 + & & @xmath160 & @xmath265 & @xmath266 & @xmath267 + & & @xmath161 & @xmath268 & @xmath269 & @xmath270 + @xmath156 & @xmath1 & all & @xmath271 & @xmath272 & @xmath273 + & & @xmath158 & @xmath274 & @xmath275 & @xmath276 + & & @xmath159 & @xmath277 & @xmath278 & @xmath279 + & & @xmath160 & @xmath280 & @xmath281 & @xmath282 + & & @xmath161 & @xmath283 & @xmath284 & @xmath285 + @xmath163 & @xmath0 & all & @xmath286 & @xmath287 & @xmath288 + & & @xmath158 & @xmath289 & @xmath290 & @xmath291 + & & @xmath159 & @xmath292 & @xmath293 & @xmath294 + & & @xmath160 & @xmath295 & @xmath296 & @xmath297 + & & @xmath161 & @xmath298 & @xmath299 & @xmath300 + @xmath162 & @xmath0 & all & @xmath301 & @xmath302 & @xmath303 + & & @xmath158 & @xmath304 & @xmath305 & @xmath306 + & & @xmath159 & @xmath307 & @xmath308 & @xmath309 + & & @xmath160 & @xmath310 & @xmath311 & @xmath312 + & & @xmath161 & @xmath313 & @xmath314 & @xmath315 + we determine the field - to - field variance in the density by fixing the parameters @xmath127 and @xmath20 to the best - fit values measured using the composite sample , and estimating @xmath316 for each @xmath110th field separately by imposing a normalization on the lf such that the total number of observed galaxies in each field is reproduced . in table [ tab - ndcomp ] , the derived @xmath316 of drgs and non - drgs in each field are listed for the three targeted redshift intervals and compared to @xmath8 measured from the composite sample . the results in the redshift range @xmath162 are plotted in figure [ fig - phistar ] . we find an overdensity of drgs in the m1030 field at all redshifts , with the excess ( as compared to the characteristic density of the composite sample ) varying from a factor of @xmath7 in the lowest redshift bin up to a factor of @xmath188 in the redshift interval @xmath162 . we also find an underdensity of drgs ( a factor of 0.82 - 0.86 ) in the goods - cdfs field , although only at @xmath317 . the value of @xmath8 for drgs in m1030 is a factor of @xmath212.4 larger than that in the goods - cdfs field at @xmath317 , although they are similar at @xmath155 . cccccccc + @xmath163 & @xmath318 & @xmath319 & @xmath320 & @xmath321 & @xmath322 & @xmath323 & @xmath324 + @xmath156 & @xmath325 & @xmath326 & @xmath327 & @xmath328 & @xmath329 & @xmath330 & @xmath331 + @xmath157 & @xmath332 & @xmath333 & @xmath334 & @xmath335 & @xmath336 & @xmath337 & @xmath338 + @xmath163 & @xmath339 & @xmath340 & @xmath341 & @xmath342 & @xmath343 & @xmath344 & @xmath345 + @xmath156 & @xmath346 & @xmath347 & @xmath348 & @xmath349 & @xmath350 & @xmath351 & @xmath352 + @xmath157 & @xmath353 & @xmath354 & @xmath355 & @xmath356 & @xmath357 & @xmath358 & @xmath359 + these results are qualitatively consistent with @xcite , who showed that the goods - cdfs field is underdense in massive ( @xmath360 @xmath35 ) galaxies at @xmath36 , with a surface density that is about 60% of the mean and a factor of 3 lower than that of their highest density field ( m1030 ) . however , our results seem to show systematically smaller underdensities for the goods - cdfs field compared to their work . in order to understand the origin of the smaller underdensity of drgs found for the goods - cdfs field in our work compared to that of massive galaxies in @xcite , we have estimated the surface density of drgs in the redshift range @xmath36 down to @xmath361 . we find that the surface density of drgs in the goods - cdfs field is @xmath362% of the mean and a factor of @xmath363 lower than that of the m1030 field , in good agreement with the values in @xcite . therefore , the smaller underdensities of drgs found for the goods - cdfs field in our work appear to arise mainly from the different targeted redshift ranges . the approach adopted in this work to quantify field - to - field variance by comparing the @xmath316 of the individual fields might also mitigate field - to - field differences , especially at the bright end . we note that there are significant differences in the observed characteristic densities even within the musyc fields , although they have areas of @xmath364 arcmin@xmath52 . for example , the observed @xmath8 of drgs in the mh1 field is consistent with the one derived from the composite sample , but it is 0.61 - 0.76 times the value in the m1030 field . these results demonstrate that densities inferred from individual @xmath364 arcmin@xmath52 fields should be treated with caution . in this section we present estimates of the luminosity density . because of the coupling between the two parameters @xmath127 and @xmath20 , the luminosity density ( obtained by integrating the lf over all magnitudes ) is a robust way to characterize the contribution to the total lf from the different subpopulations and to characterize the evolution of the lf with redshift . the luminosity density @xmath365 is calculated using : @xmath366 which assumes that the schechter parametrization of the observed lf is a good approximation and valid also at luminosities fainter than probed by our composite sample . table [ tab-5 ] lists @xmath365 with the corresponding 1 , 2 , and 3 @xmath54 errors errors of the luminosity densities were calculated by deriving the distribution of all the values of @xmath365 allowed within the 1 , 2 , and 3 @xmath54 solutions , respectively , of the schechter lf parameters from the maximum likelihood analysis . the contribution from the uncertainties in the photometric redshift estimates derived in appendix [ app-1 ] was added in quadrature . ] for all of the considered samples . we also list @xmath367 , the luminosity density calculated to the faintest probed rest - frame luminosity , and @xmath368 , the luminosity density calculated to the rest - frame magnitude limits of the deep nir musyc . while the difference between @xmath365 and @xmath367 is very small ( negligible for drgs and red galaxies , and @xmath67 dex on average for non - drgs and blue galaxies ) , the difference between @xmath365 and @xmath368 is significant , especially for non - drgs and blue galaxies ( @xmath369 dex on average ) . cccccc[!t ] @xmath163 & @xmath2 & all & @xmath370 & @xmath371 & @xmath372 + & & @xmath158 & @xmath373 & @xmath373 & @xmath374 + & & @xmath159 & @xmath375 & @xmath376 & @xmath377 + & & @xmath160 & @xmath378 & @xmath378 & @xmath379 + & & @xmath161 & @xmath380 & @xmath381 & @xmath382 + @xmath156 & @xmath1 & all & @xmath383 & @xmath384 & @xmath385 + & & @xmath158 & @xmath386 & @xmath387 & @xmath388 + & & @xmath159 & @xmath389 & @xmath390 & @xmath391 + & & @xmath160 & @xmath392 & @xmath393 & @xmath394 + & & @xmath161 & @xmath395 & @xmath396 & @xmath397 + @xmath163 & @xmath0 & all & @xmath398 & @xmath399 & @xmath400 + & & @xmath158 & @xmath401 & @xmath402 & @xmath403 + & & @xmath159 & @xmath404 & @xmath405 & @xmath406 + & & @xmath160 & @xmath407 & @xmath408 & @xmath409 + & & @xmath161 & @xmath410 & @xmath411 & @xmath412 + @xmath162 & @xmath0 & all & @xmath413 & @xmath414 & @xmath415 + & & @xmath158 & @xmath416 & @xmath417 & @xmath418 + & & @xmath159 & @xmath419 & @xmath420 & @xmath421 + & & @xmath160 & @xmath422 & @xmath423 & @xmath424 + & & @xmath161 & @xmath425 & @xmath426 & @xmath427 + in the top panel of figure [ lumdensbr_z.ps ] we have plotted the total rest - frame @xmath0-band luminosity density @xmath365 versus the redshift , including a compilation of results from the literature . only the results from the literature which are not significantly affected by field - to - field variations , or that have taken these into account , are plotted . our measurement of the total rest - frame @xmath0-band luminosity density is the only one at @xmath5 that is not significantly affected by field - to - field variance . from figure [ lumdensbr_z.ps ] , there is an indication of a possible increase of the total luminosity density in the highest redshift bin , significant at the @xmath428 @xmath54 level . the measurement at @xmath155 is consistent with the one at @xmath429 from @xcite . in figure[lumdensbr_z.ps ] we have also plotted the computed @xmath0-band rest - frame luminosity density as a function of @xmath153 predicted from large - scale @xmath430cdm hydrodynamical simulations from @xcite and from a semianalytical model taken from @xcite . while the predicted luminosity densities match the measurements at @xmath431 well , they clearly overpredict them at larger redshifts . only the prediction at @xmath26 from @xcite is consistent with our measurement , although their model still overpredicts significantly the luminosity densities in the range @xmath432 . the bottom panel of figure [ lumdensbr_z.ps ] shows the total rest - frame @xmath2-band luminosity density @xmath365 versus the redshift , including a compilation of results from the literature as in the top panel . as for the rest - frame @xmath0 band , our measurement of the rest - frame @xmath2-band @xmath365 is the first one at @xmath5 for which sample variance does not significantly contribute to the error budget . our point at @xmath5 is consistent with the trend observed at @xmath22 of decreasing luminosity densities with increasing redshifts , although the point at @xmath433 from the lf analysis of the goods - cdfs survey by @xcite is only @xmath434% of our measurement at @xmath435 . from table [ tab - ndcomp ] we see that in the redshift range @xmath155 , the cdfs field is underdense in non - drgs . the value of @xmath8 for all galaxies in the cdfs field is @xmath436 mpc@xmath437 mag@xmath30 ( estimated as in [ sec - numdens ] ) , which is a factor of @xmath438 the value of @xmath8 from the composite sample ( see table [ tab-3 ] ) . therefore , the lower value of the rest - frame @xmath2-band @xmath365 from @xcite at @xmath439 could be due to an underdensity of galaxies at @xmath27 . in figure [ lumdensbr_z.ps ] we have also plotted the computed @xmath2-band rest - frame luminosity density as function of @xmath153 predicted from large - scale @xmath430cdm hydrodynamical simulations from @xcite . as for the rest - frame @xmath0 band , the predictions match the observations well enough at @xmath440 , but at larger redshifts they significantly overpredict them . we can also compare our estimated luminosity densities in the rest - frame @xmath0 and @xmath1 bands with those from @xcite , who presented the evolution of the rest - frame optical luminosity and stellar mass densities at @xmath441 . the luminosity density in @xcite was computed by simply adding up the luminosities of all galaxies in the targeted redshift bins with rest - frame @xmath1-band luminosities @xmath442 l@xmath29 . if we integrate our measured lf down to the same limit adopted in @xcite , we obtain @xmath443 and @xmath444 for the @xmath1 and @xmath0 band , respectively , at @xmath26 , and @xmath445 for the @xmath0 band at @xmath435 ( units in erg s@xmath30 hz@xmath30 mpc@xmath245 ) , in excellent agreement with their estimates ( @xmath446 , @xmath447 , and @xmath448 , respectively ) . in [ subsec - sublf ] we showed that the contribution of drgs and red galaxies to the global lf is comparable to ( or larger than ) that of non - drgs and blue galaxies at the bright end , but it becomes significant smaller at the faint end , where non - drgs and blue galaxies dominate the global lf . in [ sec - numdens ] , the contribution of drgs to the global number density has been shown to be 13%-25% down to the faintest probed rest - frame luminosities . the contribution of drgs to the global luminosity density is 19%-29% depending on the considered rest - frame band and redshift interval ( see table [ tab-50 ] ) . their contribution increases up to 31%-44% if we cut the composite sample to the rest - frame absolute brightness limit of musyc ( @xmath449 , @xmath450 , @xmath451 of the global lf in the rest - frame @xmath2 , @xmath1 , and @xmath0 band , respectively ) , which reflects the increasing importance of drgs at the bright end . a similar result holds if we consider the red galaxy subsample . their contribution to the global luminosity density is 29%-52% down to the faintest observed luminosities and increases up to 36%-69% if we limit the analysis to the musyc brightness limits . from table [ tab-50 ] we conclude that the total luminosity density is dominated by non - drgs / blue galaxies , especially in the bluer rest - frame optical bands , although drgs / red galaxies contribute about 50% at the bright end . ccccc[!t ] @xmath163 & @xmath2 & @xmath158 & @xmath452 @xmath453 @xmath454 & @xmath455 @xmath456 @xmath457 + & & @xmath160 & @xmath458 @xmath459 @xmath460 & @xmath461 @xmath462 @xmath463 + @xmath156 & @xmath1 & @xmath158 & @xmath464 @xmath465 @xmath466 & @xmath467 @xmath468 @xmath469 + & & @xmath160 & @xmath470 @xmath471 @xmath466 & @xmath472 @xmath473 @xmath474 + @xmath163 & @xmath0 & @xmath158 & @xmath475 @xmath476 @xmath477 & @xmath478 @xmath479 @xmath480 + & & @xmath160 & @xmath481 @xmath482 @xmath483 & @xmath484 @xmath485 @xmath486 + @xmath162 & @xmath0 & @xmath158 & @xmath487 @xmath488 @xmath489 & @xmath490 @xmath491 @xmath492 + & & @xmath160 & @xmath493 @xmath494 @xmath495 & @xmath496 @xmath497 @xmath498 + although non - drgs and blue galaxies represent the major contribution to the total luminosity and number densities in the rest - frame optical bands , it has been shown that drgs usually have larger mass - to - light ratios than non - drgs ( e.g. , @xcite ; @xcite ; @xcite ) , as is generally true for red versus blue galaxies ( e.g. , @xcite ; @xcite ) . it is therefore interesting to quantify the contribution of drgs ( red galaxies ) to the total stellar mass density . following the method described in @xcite , we estimated the stellar mass density from the measured global luminosity density modulo the mass - to - light ratio @xmath13 . for each subsample we have measured the median rest - frame @xmath12 color color of the sample , @xcite computed the global @xmath12 color from the relation : @xmath499 , where @xmath500 and @xmath501 are computed by adding the luminosities of the individual galaxies . the two methods return very similar values for the @xmath12 colors . ] , estimated the corresponding @xmath13 ratio from the relation between @xmath12 color and @xmath13 ratio obtained from stellar population synthesis models , and multiplied the estimated @xmath13 ratio by the measured luminosity density to obtain the stellar mass density . to convert between the measured rest - frame @xmath12 color and the mass - to - light ratio @xmath13 , we have generated stellar population synthesis models with the evolutionary synthesis code developed by g. bruzual and s. charlot @xcite . we selected the `` padova 1994 '' evolutionary tracks , which are preferred by bruzual & charlot over the more recent `` padova 2000 '' tracks because the latter may be less reliable and predict a hotter red giant branch leading to worse agreement with observed galaxy colors . we used the solar metallicity set of tracks . the metallicities of the drgs are poorly known , with evidence for solar and supersolar metallicities for luminous drgs @xcite . these drgs appear more metal - rich than the five lbgs at @xmath26 studied by @xcite and similar to the seven uv - selected star - forming `` bx / md '' objects at @xmath27 for which @xcite inferred solar , and possibly supersolar , metallicities . in all cases , however , the determinations rely on limited samples and suffer from large uncertainties . as shown by @xcite adopting subsolar metallicity ( @xmath502 ) , the estimated @xmath13 ratios are systematically lower by a factor of @xmath503 on average . therefore , if non - drgs ( or blue galaxies ) are characterized by lower metallicities with respect to drgs ( red galaxies ) , the differences in @xmath13 ( and stellar mass densities ) would be even larger than what is estimated assuming solar metallicities for both subsamples . for the star formation history ( sfh ) we used three different prescriptions : a constant star formation history ( csf model ) , an exponentially declining in time sfh characterized by the parameter @xmath504 ( tau - model ) , and an instantaneous burst model ( ssp model ) . several values of @xmath504 were used , from @xmath505 gyr ( the resulting model being similar to the ssp model ) to 6 gyr ( closer to the csf model ) . we adopted the @xcite imf with lower and upper imf mass cutoffs @xmath506 @xmath35 and @xmath507 @xmath35 , respectively . adopting a different imf would result in different derived mass - to - light ratios , which strongly depend on the shape and cutoff of the low - mass imf ( for example , assuming a salpeter [ 1955 ] imf , the estimated @xmath13 ratio would be systematically larger by a factor of @xmath508 ) . however , since we are interested only in the relative contribution of drgs ( red galaxies ) to the global stellar mass density , the results do not depend on the adopted imf as long as all galaxies are characterized by the same imf . we assumed that the interstellar extinction by dust within the objects followed the attenuation law of @xcite derived empirically from observations of local uv - bright starburst galaxies under the formalism of a foreground screen of obscuring dust . we plot in figure [ fig - mlrr ] the relation between the rest - frame @xmath12 color and the mass - to - light ratio in the rest - frame @xmath2 band , @xmath509 , for the generated model tracks in the two cases with no extinction and with @xmath510 . it is seen that dust extinction moves the tracks roughly parallel to the model tracks . as emphasized by @xcite , dust is a second - order effect for estimating stellar @xmath13 ratios . dust extinguishes light from the stellar population , making it dimmer ; however , dust also reddens the stellar population , making it appear to have a somewhat larger stellar @xmath13 ratio . to first order , these effects cancel out , leaving a dust - reddened galaxy on the same color stellar @xmath13 ratio correlation . using the relation between color and @xmath13 , we convert the estimated median rest - frame @xmath12 color and the measurements of the luminosity densities @xmath365 to stellar mass density estimates @xmath511 . specifically , we adopted the median value of @xmath13 within the family of considered model tracks with @xmath510 ( blue dashed line plotted in figure [ fig - mlrr ] for the rest - frame @xmath2 band ) ; the error on the @xmath13 was chosen as half of the difference between the upper and lower envelope of the model tracks . the median rest - frame @xmath12 color of each subsample and the corresponding @xmath13 in the rest - frame @xmath0 , @xmath1 , and @xmath2 bands are listed in table [ tab-60 ] . cclcc @xmath163 & @xmath2 & @xmath158 & @xmath512 ( @xmath513 ) & @xmath514 ( @xmath515 ) + & & @xmath159 & @xmath516 ( @xmath517 ) & @xmath518 ( @xmath519 ) + & & @xmath520 & @xmath521 ( @xmath522 ) & @xmath523 ( @xmath524 ) + & & @xmath161 & @xmath525 ( @xmath526 ) & @xmath527 ( @xmath528 ) + @xmath156 & @xmath1 & @xmath158 & @xmath529 ( @xmath529 ) & @xmath530 ( @xmath530 ) + & & @xmath159 & @xmath531 ( @xmath532 ) & @xmath533 ( @xmath534 ) + & & @xmath520 & @xmath535 ( @xmath529 ) & @xmath536 ( @xmath530 ) + & & @xmath161 & @xmath537 ( @xmath532 ) & @xmath534 ( @xmath534 ) + @xmath163 & @xmath0 & @xmath158 & @xmath512 ( @xmath513 ) & @xmath538 ( @xmath538 ) + & & @xmath159 & @xmath516 ( @xmath517 ) & @xmath539 ( @xmath540 ) + & & @xmath520 & @xmath521 ( @xmath522 ) & @xmath541 ( @xmath542 ) + & & @xmath161 & @xmath525 ( @xmath526 ) & @xmath543 ( @xmath539 ) + @xmath162 & @xmath0 & @xmath158 & @xmath544 ( @xmath544 ) & @xmath545 ( @xmath545 ) + & & @xmath159 & @xmath546 ( @xmath547 ) & @xmath543 ( @xmath548 ) + & & @xmath520 & @xmath529 ( @xmath529 ) & @xmath545 ( @xmath545 ) + & & @xmath161 & @xmath549 ( @xmath547 ) & @xmath543 ( @xmath548 ) + although the values of the stellar mass densities of the individual subsamples might be affected by very large uncertainties , the relative contribution to the global stellar mass density of drgs ( red galaxies ) and non - drgs ( blue galaxies ) should be more robust . the contribution @xmath550 of the drgs ( red galaxies ) to the global stellar mass density is listed in table [ tab-50 ] . adopting the same assumptions for the stellar population synthesis models of drgs ( red galaxies ) and non - drgs ( blue galaxies ) ( i.e. , the median value of the considered model track ) , we see from table [ tab-60 ] that drgs ( red galaxies ) have @xmath13 ratios systematically higher than non - drgs ( blue galaxies ) by a factor of @xmath9 - 11 depending on the rest - frame band ( higher in the bluer bands ) . the differences in @xmath13 are smaller when the brighter samples ( down to the musyc limit ) are considered , with the @xmath13 of drgs ( red galaxies ) being a factor of @xmath551 - 9 ( 4 - 9 ) larger with respect to non - drgs ( blue galaxies ) . for comparison , from the analysis of _ spitzer_-irac imaging on hdf - s , @xcite found that the average mass - to - light ratio of drgs in the rest - frame @xmath4 band is about a factor of @xmath551 larger than lbgs , finding a correlation between @xmath552 and rest - frame @xmath12 color . consistently , from sed modeling of the @xmath3 fires galaxies , @xcite found that the median rest - frame @xmath1-band @xmath13 of drgs is @xmath7 - 2.3 m@xmath29 l@xmath553 ( @xmath554 for lbgs ) . the higher values of @xmath13 for drgs agree very well also with the results from sed fitting of individual galaxies in @xcite , who found for drgs a median value @xmath555 @xmath556 . finally , our estimated @xmath557 are in excellent agreement with those estimated in @xcite . because of the systematically larger mass - to - light ratios , drgs ( red galaxies ) dominate the global stellar mass density , with contributions in the range 66%-82% ( 69%-89% ) down to the faintest probed rest - frame luminosities . the contribution of drgs ( red galaxies ) increases up to 67%-85% ( 79%-92% ) if the brightest sample is considered ( down to the rest - frame magnitude limits of the deep nir musyc ) . these numbers are consistent with the results of @xcite . from a complete mass - selected sample ( @xmath558 @xmath35 ) constructed with musyc , fires , and goods - cdfs , they estimated that drgs in the redshift interval @xmath36 make up 77% of the total stellar mass , in very good agreement with our results . our results are in qualitative good agreement also with the results from @xcite , who found that the drgs contribute 64% of the stellar mass density at @xmath559 and 30%-50% at @xmath27 . we stress that the estimated contributions of drgs and red galaxies to the global stellar mass density are very uncertain and need confirmation from detailed sed analysis of mass - limited ( rather than luminosity - limited ) samples . in this paper we have measured the rest - frame optical ( @xmath2 , @xmath1 , and @xmath0 band ) luminosity functions of galaxies at redshifts @xmath3 from a composite sample constructed with the deep nir musyc , the ultra - deep fires , and the goods - cdfs . the large surveyed area ( @xmath560 arcmin@xmath52 , 76% of which comes from the deep nir musyc ) of the composite sample and the large range of luminosities spanned allows us to measure the bright end of the lf and to constrain the faint - end slope . moreover , the several independent fields and their large area enabled us to largely reduce uncertainties due to sample variance , especially at the bright end . we have used monte carlo simulations to show that the uncertainties in the photometric redshift estimates do not significantly affect the measured parameters of the lf in the studied redshift regimes . there is a hint for a steepening in the faint - end slope of the lf from the rest - frame @xmath2 band to the @xmath0 band , although the differences are not significant . the measured lf faint - end slopes at @xmath5 are consistent , within the errors , with those in the local lfs . the characteristic magnitudes are significantly brighter than the local ones ( e.g. , @xmath7 mag in the rest - frame @xmath2 band ) , while the measured values for @xmath8 are typically a factor of @xmath9 smaller with respect to the local values . the large number of objects in the composite sample allowed the first measurement of the lf of drgs ( defined based on their observed @xmath15 color ) , which we compared to that of non - drgs in the same redshift range . the drg population is characterized by a very different lf than that of non - drgs , especially at the faint end . while at the bright end the lf of drgs is similar to that non - drgs , at the faint end the latter one has a significantly steeper faint - end slope , especially in the rest - frame @xmath2 band . the significance of the difference between the lfs of drgs and non - drgs decreases going to bluer rest - frame bands and to higher redshifts , although this is mainly caused by decreasing constraints on the faint end of the lf of drgs . qualitatively similar results are found if we compare the lfs of red ( rest - frame @xmath160 ) and blue ( rest - frame @xmath561 ) galaxies in the same redshift intervals , with the former equally contributing ( or even dominating ) at the bright end and the latter dominating the faint end . in the rest - frame @xmath1 band we have also compared the lfs of blue galaxies ( non - drgs ) with those of lbgs in the same redshift range . although the two lfs agree very well at the bright end , the faint - end slope estimated by @xcite is much steeper than the one measured in this paper . as the rest - frame optical lf of lbgs was estimated in @xcite from the rest - frame uv lf and the observed distribution of @xmath241-@xmath45 colors as a function of @xmath241 magnitude , their steeper slope could be a result of an overestimate of the regression slope of the correlation between @xmath241-@xmath45 and @xmath241 and/or of the faint - end slope of the adopted rest - frame uv lf of lbgs . supporting the former possibility is the work of @xcite , who do not find any positive correlation between @xmath2 magnitudes and @xmath562 colors in their deeper sample . support for the latter comes from the very recent work of @xcite , who measured the rest - frame uv lf for @xmath563 lbgs from the keck deep fields ( kdf ; @xcite ) and find a faint - end slope @xmath564 , significantly shallower than the one adopted in @xcite . alternatively , our blue @xmath45-selected ( i.e. , rest - frame optical selected ) galaxies might simply constitute a different population than the @xmath26 lbgs ( rest - frame uv selected ) , with different characterizations of the lf at the faint end . we also caution that our measurements of the faint - end slopes still have significant uncertainties due to small number statistics . we generally find good agreement between our measured rest - frame @xmath0-band lfs at @xmath157 and those previously published by @xcite , @xcite , and @xcite . in the redshift range @xmath155 , the agreement between our rest - frame @xmath0- and @xmath2-band lfs and those measured by @xcite from the fdf survey is less good , especially for the @xmath2 band . their schechter parameters @xmath127 and @xmath20 are consistent with ours only at the 2 @xmath54 level in the rest - frame @xmath2 band , while their estimated @xmath8 is larger than ours by a factor of @xmath141.3 - 1.6 in the rest - frame @xmath0 and @xmath2 band , respectively . we have shown that this disagreement may be due to the spectroscopically confirmed existence of a ( proto)cluster at @xmath565 in the single field fdf survey . from the measured lfs we have estimated the number and luminosity densities of the global population of high-@xmath153 galaxies and of various subsamples . the contribution of drgs ( red galaxies ) to the global number density is only @xmath566%-25% ( 18%-29% ) down to the faintest probed rest - frame luminosities . however , we have shown that field - to - field variations can be very significant ( up to a factor of @xmath551 ) , especially for relatively bright samples , in accord with the highly clustered nature of high luminosity and red galaxies ( @xcite ; @xcite ; @xcite ) . the contribution of drgs ( red galaxies ) to the global luminosity density is @xmath567%-30% ( 30%-50% ) , higher in the redder rest - frame bands ( which are less affected by extinction and better tracers of the underlying stellar mass ) and at lower redshifts . with respect to the lower @xmath153 luminosity density estimates from the literature , we confirm the trend of slowly decreasing rest - frame @xmath2-band luminosity densities beyond @xmath568 , with @xmath569 at @xmath570 being a factor of @xmath11 smaller than the local one . in the rest - frame @xmath0 band , the measured global luminosity density at @xmath570 is similar to the local value . at @xmath571 , the estimated global luminosity density may be a factor of @xmath11 higher , similar to the values around @xmath568 . finally , using stellar population synthesis models , we have derived the mass - to - light ratios of the considered subsamples by converting the estimated median rest - frame @xmath12 color into @xmath13 . in the rest - frame @xmath2 and @xmath1 bands , the mass - to - light ratios of drgs ( red galaxies ) are a factor of @xmath9 larger than non - drgs ( blue galaxies ) , consistent with previous works . in the rest - frame @xmath0 band the difference in @xmath13 is higher , up to a factor of @xmath572 . using the estimated @xmath13 , we have quantified the contribution of drgs and red galaxies to the global stellar mass density , finding that the total stellar mass budget is dominated by drgs ( red galaxies ) , whose contribution is of order @xmath573%-80% of the global value . we caution that our @xmath13 ratios estimates are very rough and characterized by very large uncertainties and need confirmation from detailed sed analysis . the main limitation of this work is the small number statistics at the very faint end of the lf , which is probed only by the ultra - deep fires . the faint - end slopes of the drg and red galaxy subsamples are especially uncertain . to make further progress in the determination of the lf of different galaxy populations at @xmath5 and to better constrain the global lfs , it is crucial to better probe the faint end of the lfs . this can only be achieved with ultradeep nir imaging with high - quality optical data over many spatially disjoint fields , in order to improve the statistics at the faint end and to mitigate the effect of field - to - field variations . although we have shown that well - behaved photometric redshift errors do not affect significantly the measurement of the lf , the heavy reliance on photometric redshifts is another limitation of this work , since `` catastrophic '' failures and systematic errors could potentially affect the lf measurements . obtaining large numbers of spectroscopic redshifts for @xmath4-selected high-@xmath153 sources has proven difficult and extremely time consuming . even though the success rate for measuring spectroscopic redshift for bright galaxies is high with nir spectroscopy @xcite , only the use of multiobject nir spectrographs will make it possible to construct a large sample of high-@xmath153 @xmath4-selected galaxies with spectroscopic redshift measurements . further advances can be expected from further extension of the wavelength range into the red . scheduled _ spitzer _ irac observations on the deep nir musyc fields will allow us to ( 1 ) separate old and passively evolving galaxies from heavily obscured and strongly active star - forming galaxies ( see @xcite ) , making it possible to study the lf of physically different types of galaxies ; ( 2 ) extend the robust measurement of the lf at redshift @xmath36 into the rest - frame nir , which is much closer to a selection by stellar mass ; ( 3 ) convert the measured rest - frame nir luminosity function into a mass function and study the evolution of the stellar mass density ; and ( 4 ) extend the study of the rest - frame optical lfs to even higher redshifts . we thank all the members of the musyc collaboration for their contribution to this research . musyc has greatly benefited from the support of fundacin andes and the yale astronomy department . d.m . is supported by nasa ltsa nng04ge12 g . the authors acknowledge support from nsf carrer ast-0449678 . e.g. is supported by nsf fellowship ast-0201667 . is supported by fondecyt grant # 1040719 . we thank the anonymous referee for comments and suggestions which helped improve the paper . in order to quantify the systematic effects on the lf parameters @xmath127 and @xmath20 due to the uncertainties in the photometric redshift estimates , we performed a series of monte carlo simulations . first , we generated several model catalogs of 25,000 galaxies with redshifts between @xmath574 and @xmath575 and with luminosities drawn from an input schechter lf . while in the monte carlo simulations of @xcite the redshifts of the objects in the mock catalogs were extracted from a random uniform distribution , we took into account the fact that , under the assumption of no evolution in the number density , the probability of a galaxy existing at the redshift @xmath153 is proportional to the volume : @xmath576 where @xmath577 is the luminosity distance . since in a real survey galaxies are selected down to a limiting apparent magnitude , the final mock catalogs are obtained after applying a limit in the observed apparent magnitude . the effect of a limiting apparent magnitude is that , at a fixed observed magnitude , intrinsically fainter sources are systematically excluded from the catalog at higher redshift . next , as done by @xcite , we assumed a redshift error function parameterized as a gaussian distribution function of 1 @xmath54 width @xmath145 , with @xmath146 the scatter in @xmath147 , and we formed an observed redshift catalog by perturbing the input galaxy redshift within the redshift error function . finally , we determined the lf for the galaxies at @xmath148 using the @xmath100 and maximum likelihood methods described in [ sec - lf ] . note that we ignore @xmath4-correction in our monte carlo simulations . we first studied the effects of the photometric redshift uncertainties at @xmath151 , by using @xmath578 , @xmath579 , and assuming an input schechter lf with parameters @xmath580 and @xmath581 ( as in @xcite ) . in order to compare our results with those from the monte carlo simulations in @xcite , we used @xmath582 and we measured the lf in the same redshift range @xmath144 . we find that the median measured @xmath20 is brighter than the intrinsic value by @xmath554 mag and that the measured @xmath127 is steeper on average than the intrinsic value by @xmath67 . this result is shown in figure [ fig - chen4v ] : in the left panel , the input schechter lf is compared to the median monte carlo realization ; in the right panel , the 100 monte carlo realizations are plotted in the @xmath141 plane and compared to the best - fit values and the corresponding 1 , 2 , and 3 @xmath54 contour levels of the lf measured on the redshift - unperturbed mock catalog . as shown in figure [ fig - chen4v ] , the measured systematic effects on @xmath127 and @xmath20 caused by the redshift uncertainties arise from an excess of sources at both the faint and the bright end of the lf . a careful analysis of the monte carlo simulations reveals the origin of these excesses . because of the uncertainties in the redshifts , sources can scatter from high to low redshifts , and vice versa . as the probability of a galaxy existing at redshift @xmath153 is proportional to @xmath112 ( which peaks at @xmath583 ) , the number of sources that scatter from higher redshifts to lower ones is much larger than vice versa . this is evident in figure [ fig - mc_mvsz ] , where the rest - frame absolute magnitudes of the mock catalog are plotted versus redshift : at a fixed magnitude , the number of sources is larger at higher redshifts . since an object in the mock catalog is characterized by a fixed apparent magnitude , a new redshift estimate translates into a new rest - frame absolute magnitude ; e.g. , when a source `` scatters '' from high to low redshift , the estimated absolute brightness is fainter than the intrinsic value , and the object moves , in figure [ fig - mc_mvsz ] , from right to left along a line parallel to the black solid line . moreover , at any redshift , there are more sources at faint magnitudes than at bright ones because of the shape of the lf . therefore , the sources scattering from high redshifts into the considered redshift bin ( plotted in fig . [ fig - mc_mvsz ] in cyan ) preferentially end up at fainter magnitudes ( plotted in red ) , producing the excess at the faint end of the lf with respect to the input lf . the excess at the bright end is instead mainly produced by those sources that scatter into the considered redshift range coming from lower redshifts ( represented in fig . [ fig - mc_mvsz ] with cyan and green filled circles ) ; since at the bright end the number of objects is very small , even a handful of new sources can significantly increase the measured density with respect to the intrinsic one . because at low redshifts the dependency of @xmath152 with redshift is strong , even small uncertainties in the redshift estimate have large effects on the rest - frame absolute magnitude . for example , at @xmath584 , a @xmath585 results in a @xmath586 mag ( only 0.1 - 0.2 mag at @xmath587 ) . the systematic effect on @xmath20 that we derive ( @xmath588 mag ) is about half the effect found by @xcite . also almost no systematic effect in @xmath127 was found in their work . it seems very likely that these differences might be due to the fact that the redshifts are drawn from a random uniform distribution in @xcite while in our monte carlo simulations they are extracted from the probability function specified in eq . [ eq - pz ] . in the former case , there would be a much larger number of low-@xmath153 sources that can scatter into the considered redshift bin , resulting in a larger excess of sources at the bright end and therefore a larger systematic effect in @xmath20 . if we repeat our monte carlo simulations extracting the redshifts of the sources from a random uniform distribution , we obtain @xmath589 mag and @xmath590 , consistent with the result of @xcite as shown in the right panel of figure [ fig - chen4v ] . next , we repeated our monte carlo simulations at higher redshift by generating model catalogs with galaxies at redshifts between @xmath591 and @xmath592 , since the goal of this paper is to measure the lf of galaxies in the redshift intervals @xmath155 and @xmath157 . in figure [ fig - mock7wv ] we plot the results of our monte carlo simulations at @xmath155 , assuming an input schechter lf with @xmath593 and @xmath594 and for @xmath595 ( which corresponds to the photometric redshift errors in the deep nir musyc for @xmath596 objects ) . the systematic effects on the measured @xmath127 and @xmath20 are now very small , @xmath597 mag and @xmath598 , and negligible with respect to the other uncertainties on the estimated best - fit parameters . similar results are obtained in the redshift bin @xmath157 . at @xmath3 , the effect due to @xmath599 is much smaller than at @xmath151 , since @xmath600 peaks at @xmath601 and then decreases very slowly , so that the number of high- and low-@xmath153 sources scattering into the considered redshift bin is similar to the number of sources scattering out . also , at @xmath5 , the error on the rest - frame absolute magnitude corresponding to a redshift error is significantly smaller than at @xmath151 ; e.g. , @xmath602 mag for @xmath603 at @xmath587 . therefore , the measured lf is similar to the input one and the systematic effects on @xmath127 and @xmath20 are negligible compared to the uncertainties in the lf estimates for reasonable values of @xmath146 ( @xmath604 ) . we repeated the above monte carlo simulations assuming different input @xmath127 ( @xmath605 , @xmath606 , and @xmath607 ) to study the behavior of the systematic effects as function of the faint - end slope . no significant differences are found : @xmath608 mag for @xmath609 and @xmath607 ( depending on the considered redshift bin ) ; for @xmath610 , the systematic effect is slightly larger ( @xmath611 mag for @xmath155 and @xmath162 , respectively ) , but also the uncertainties on the best - fit @xmath20 increase with steeper faint - end slopes ( since the observed lf appears more like a power law ) so that the systematic effects on the measured best - fit schechter parameters remain very small with respect to the uncertainties on the best - fit values . finally , we investigated the effects of non - gaussian redshift error probability distributions . first , using a model catalog with galaxies at redshifts between @xmath612 and @xmath592 , we simulated the effect of a 5% `` catastrophic '' outliers by assigning random redshifts to 5% of the mock catalog . adopting the input lf with @xmath593 and @xmath613 and assuming @xmath595 , we find larger systematic effect in both @xmath127 and @xmath20 by a factor of almost 2 . next , we built the mean redshift probability distribution of the deep nir musyc by averaging the individual redshift probability distribution for each galaxy calculated by the used photometric redshift code ( for details see @xcite ) . the average musyc redshift probability distribution is well modeled by a lorentzian function , rather than a gaussian function . we find a larger systematic effect in @xmath127 , twice as much as the corresponding effect assuming a gaussian parametrization for the redshift probability distribution , but similar systematic effect in @xmath20 , although in the opposite direction . to summarize , although the systematic effects in @xmath127 and @xmath20 expectedly get larger when we simulate `` catastrophic '' outliers or we adopt a redshift error function with broader wings compared to the gaussian model , they remain much smaller than the random uncertainties in the lf estimates . we also quantified the systematic effect on the luminosity density estimates . we find that the effect is of the order of a few percent ( always @xmath6146% ) depending on the considered redshift interval and on the input schechter lf . in order to include this contribution in the error budget , we conservatively assume a 10% error contribution to the luminosity density error budget due to uncertainties in the photometric redshift estimates . in the body of the paper we showed the rest - frame @xmath2 band lfs of drgs , non - drgs , red and blue galaxies ( see fig . [ lf_r_lowz.ps ] ) . for completeness , we show here the comparison of the lfs of drgs ( red galaxies ) and non - drgs ( blue galaxies ) discussed in [ subsec - sublf ] in the rest - frame @xmath1 band at @xmath97 ( fig . [ lf_v_highz.ps ] ) and in the rest - frame @xmath0 band at @xmath96 ( fig . [ lf_b_lowz.ps ] ) and at @xmath98 ( fig . [ lf_b_highz.ps ] ) . the corresponding best - fit schechter parameters are listed in table [ tab-4 ] . here we compare our results to previous rest - frame optical lf studies at @xmath5 , which were based on smaller samples . we note that these studies are affected by significant uncertainties due to field - to - field variance ( as they are based on a single field or on a very small total surveyed area ) and by small number statistics at the bright end . @xcite analyzed a sample of 138 @xmath4-selected galaxies down to @xmath615 to construct the rest - frame @xmath0-band lf in the redshift range @xmath616 . the total area of their composite sample is @xmath617 arcmin@xmath52 , a factor of @xmath618 smaller than the area sampled in this work . @xcite repeated the analysis in @xcite with an improved composite sample ( although with the same area ) and allowing the schechter parameters @xmath8 and @xmath20 to vary with the redshift , while @xmath127 is kept constant at the low - redshift value ( @xmath151 ) . a direct comparison between the lfs of @xcite and @xcite and the lf measured in this work is shown in figure [ fig - poli ] ( _ left panel _ ) for the redshift range @xmath619 . first , we note in figure [ fig - poli ] how much better the bright end of the lf is constrained from our work : the large area of the composite sample ( @xmath620% of which comes from the deep nir musyc alone ) allows us to sample the lf up to @xmath0-band magnitudes @xmath621 mag brighter than done in @xcite and @xcite . our measurements of the lf using the @xmath100 method are consistent within the errors with those in both @xcite and @xcite . the best - fit lf estimated with the maximum likelihood analysis in @xcite is consistent , within the errors , with our best - fit solution . however , the best - fit lf estimated with the maximum likelihood analysis in @xcite is significantly different from ours , as clearly shown in the inset of figure [ fig - poli ] ( _ left panel _ ) . while their faint - end slope is consistent with our best - fit @xmath127 , their lack of constraints on the bright end of the lf results in a much fainter @xmath183 ( by @xmath240 mag ) . @xcite analyzed a sample of 5558 @xmath39-selected galaxies down to @xmath622 ( 50% compleness limit ) from the fdf survey @xcite to study the evolution of the rest - frame @xmath0- and @xmath250-band lf over the redshift range @xmath623 . the total area of their sample is @xmath624 arcmin@xmath52 over a single field , a factor of @xmath625 smaller than the total area sampled in this work . a direct comparison between the @xmath0-band lf from @xcite and ours is possible in the redshift range @xmath98 , and it is shown in the left panel of figure [ fig - poli ] ( _ red symbols _ ) . the best - fit schechter parameters @xmath127 , @xmath20 , and @xmath8 are consistent with ours within the errors . in the redshift range @xmath96 we compared our estimated @xmath0-band lf with the one defined by their best - fit schechter parameters ( @xmath127 , @xmath20 , and @xmath8 ) estimated at @xmath626 . while their best - fit schechter parameters @xmath127 and @xmath20 are consistent with ours , their best - fit @xmath8 is a factor of @xmath627 larger . this difference can be entirely accounted for by field - to - field variations ( see [ sec - numdens ] ) . in fact , the fdf survey consists of a single pointing of only @xmath624 arcmin@xmath52 and thus it is potentially strongly affected by sample variance . moreover , @xcite spectroscopically identified an overdensity of galaxies at @xmath565 ( possibly a [ proto]cluster , with more than 10 identical redshifts ) , which can potentially strongly bias the estimate of @xmath8 in this redshift bin . in the right panel of figure [ fig - poli ] we compare our measured @xmath2-band lf with that measured by @xcite in the redshift range @xmath628 . the agreement between the two lfs is much worse than for the rest - frame @xmath0-band . their best - fit schechter parameters @xmath127 and @xmath20 are now consistent with ours only at the @xmath11 @xmath54 level , with their @xmath20 about 0.3 mag brighter than ours ; their best - fit @xmath8 is a factor of @xmath21 larger than our best - fit value . as mentioned above , an overdensity of galaxies was spectroscopically found in this redshift interval in the fdf . although it is hard to quantify the effect of the presence of a ( proto)cluster at this redshift on the measured lf , it is interesting to make a connection with the work of @xcite , who spectroscopically identified a protocluster at @xmath629 in the hs 1700 + 643 field . in the spectroscopically identified galaxy sample , @xcite found 19 ( out of 55 ) objects at @xmath630 in the redshift range @xmath631 . within the @xmath632 field of view ( similar to the fdf field of view ) over which the protocluster objects are distributed , the protocluster and `` field '' galaxy sky distributions are the same . assuming that the distribution of the spectroscopically identified galaxies is representative of the whole sample , the estimated density would be a factor of @xmath141.3 - 1.6 that of `` field '' galaxies only , consistent with the difference found in @xmath8 between our composite sample and the fdf in the redshift range @xmath631 . @xcite measured the rest - frame @xmath0-band lf of red and blue galaxies at redshift @xmath633 using a sample of 138 @xmath4-selected galaxies in the redshift range @xmath616 down to @xmath615 . their red and blue populations were defined on the basis of an `` s0 color track '' . from their figure 1 , the average rest - frame @xmath12 color of their model at redshift @xmath633 is @xmath634 , very similar to our definition of blue and red galaxies ( @xmath161 and @xmath160 , respectively ) . in figure [ fig - bgiallongo ] we have compared the lfs of red and blue galaxies from this work to those presented in @xcite . as for the lf of all galaxies , we are able to constrain the bright end of the lf much better by sampling the lf to luminosities @xmath621 mag brighter . for the red galaxy population , their measurements of the lf with the @xmath100 method are consistent within the errors with ours ; their best - fit faint - end slope is also consistent , within the errors , with our estimate , although @xmath127 was fixed at the local value in @xcite ; our @xmath20 is @xmath554 mag brighter than their best - fit value , but still consistent within the errors at the 1 @xmath54 level . the best - fit normalization of @xcite is a factor of @xmath621 smaller than ours , only marginally consistent at the 2 @xmath54 level . for the blue population , the @xmath100 measurements of the lf from @xcite are consistent with ours . however , the agreement gets worse when we compare our lf estimated using the sty method with theirs , especially at the bright end , where they lack information . while their assumed faint - end slope is consistent with our best - fit solution , the characteristic magnitude from @xcite is @xmath635 mag fainter than ours . their best - fit solution is , however , very uncertain , characterized by very large error bars and still consistent with ours at the @xmath19 @xmath54 level . adelberger , k. l. , erb , d. k. , steidel , c. c. , reddy , n. a. , pettini , m. , & shapley , a. e. 2005 , , 620 , l75 adelberger , k. l. , & steidel , c. c. 2000 , , 544 , 218 adelberger , k. l. , steidel , c. c. , shapley , a. e. , hunt , m. p. , erb , d. k. , reddy , n. a. , & pettini , m. 2004 , , 607 , 226 avni , y. 1976 , , 210 , 642 avni , y. , & bahcall , j. n. 1980 , , 235 , 694 bell , e. f. , de jong , r. s. 2001 , , 550 , 212 bessell , m. s. 1990 , , 102 , 1181 blanton , m. r. , et al . 2001 , , 121 , 2358 blanton , m. r. , et al . 2003 , , 592 , 819 brown , w. r. , geller , m. j. , fabricant , d. g. , kurtz , m. j. 2001 , , 2001 , 122 , 714 bruzual , g. , & charlot , s. 2003 , , 344 , 1000 calzetti , d. , armus , l. , bohlin , r. c. , kinney , a. l. , koornneef , j. , & storchi - bergmann , t. 2000 , , 533 , 682 chabrier , g. 2003 , , 115 , 763 chen , h .- w , et al . 2003 , , 586 , 745 cole , s. , et al . 2001 , , 326 , 255 coleman , g. d. , wu , c .- c . , weedman , d. w. 1980 , , 43 , 393 daddi , e. , et al . 2003 , , 588 , 50 dahlen , t. , mobasher , b. , somerville , r. s. , moustakas , l. a. , dickinson , m. , ferguson , h. c. , giavalisco , m. 2005 , , 631 , 126 davis , m. , et al . 2003 , proc . spie , 4834 , 161 efstathiou , g. , ellis r. s. , & peterson , b. a. 1988 , , 232 , 431 ellis r. s. , colless , m. , broadhurst , t. , heyl , j. , glazebrook , k. 1996 , , 280 , 235 frster schreiber , n. m. , et al . 2004 , , 616 , 40 frster schreiber , n. m. , et al . 2006 , , 131 , 1891 franx , m. , et al . 2003 , , 587 , l79 gabasch , a. , et al . 2004 , , 421 , 41 gabasch , a. , et al . 2006 , , 448 , 101 gawiser , e. , et al . 2006 , , 162 , 1 gehrels , n. 1986 , , 303 , 336 giallongo , e. , salimbeni , s. , menci , n. , zamorani , g. , fontana , a. , dickinson , m. , cristiani , s. , pozzetti , l. 2005 , , 622 , 116 giavalisco , m. , et al . 2004 , , 600 , l93 hauschildt , p. h. , allard , f. , & baron , e. 1999 , , 512 , 377 heidt , j. , et al . 2003 , , 398 , 49 ilbert , o. , et al . 2005 , , 439 , 86 kauffmann , g. , et al . 2003 , , 341 , 33 kendall , m. g. , & stuart , a. 1961 , the advanced theory of statistics , vol . 2 ( london : griffin & griffin ) kinney , a. l. , calzetti , d. , bohlin , r. c. , mcquade , k. , storchi - bergmann , t. , & schmitt , h. r. 1996 , , 467 , 38 knudsen , k. k. , et al . 2005 , , 632 , l9 kochanek , c. s. , et al . 2001 , , 560 , 566 kriek , m. et al . 2006 , , 649 , l71 labb , i. , et al . 2003 , , 125 , 1107 labb , i. , et al . 2005 , , 624 , l81 labb , i. , et al . 2006 , , submitted lampton , m. , margon , b. , & bowyer , s. 1976 , , 208 , 177 le fvre , et al . 2004 , , 417 , 839 lilly , s.j . , tresse , l. , hammer , f. , crampton , d. , le fvre , o. 1995 , , 455 , 108 lin , h. , yee , h. k. c. , carlberg , r. g. , ellingson , e. 1997 , , 475 , 494 nagamine , k. , cen , r. , & ostriker , j. p. 2000 , , 541 , 25 nagamine , k. , fukugita , m. , cen , r. , & ostriker , j. p. 2001 , , 327 , l10 norberg , p. , et al . 2002 , , 336 , 907 papovich , c. , et al . 2006 , , 640 , 92 pettini , m. , shapley , a. e. , steidel , c. c. , cuby , j .- g . , dickinson , m. , moorwood , a. f. m. , adelberger , k. l. , & giavalisco , m. 2001 , , 554 , 981 poli , f. , et al . 2003 , , 593 , l1 quadri , r. , et al . 2007a , , 654 , 138 quadri , r. , et al . 2007b , , submitted ( astro - ph/0612612 ) reddy , n. a. , erb , d. k. , steidel , c. c. , shapley , a. e. , adelberger , k. l. , pettini , m. 2005 , , 633 , 748 reddy , n. a. , steidel , c. c. , fadda , d. , yan , l. , pettini , m. , shapley , a. e. , erb , d. k. , adelberger , k. l. 2006 , , 644 , 792 rubin , k. h. r. , van dokkum , p. g. , coppi , p. , johnson , o. , frster schreiber , n. m. , franx , m. , & van der werf , p. 2004 , , 613 , l5 rudnick , g. , et al . 2001 , , 122 , 2205 rudnick , g. , et al . 2003 , , 599 , 847 rudnick , g. , et al . 2006 , , 650 , 624 salpeter , e. e. 1955 , , 121 , 161 sandage , a. , tammann , g. a. , & yahil , a. 1979 , , 232 , 352 saracco , p. , giallongo , e. , cristiani , s. , dodorico , s. , fontana , a. , iovino , a. , poli , f. , & vanzella , e. 2001 , , 375 , 1 sawicki , m. , & thompson , d. 2005 , , 635 , 100 sawicki , m. , & thompson , d. 2006 , , 642 , 653 schechter , p. 1976 , , 203 , 297 schmidt , m. 1968 , , 151 , 393 shapley , a. e. , erb , d. k. , pettini , m. , steidel , c. c. , & adelberger , k. l. 2004 , , 612 , 108 shapley , a. e. , steidel , c. c. , adelberger , k. l. , dickinson , m. , giavalisco , m. , & pettini , m. 2001 , , 562 , 95 shapley , a. e. , steidel , c. c. , erb , d. k. , reddy , n. a. , adelberger , k. l. , pettini , m. , barmby , p. , & huang , j. 2005 , , 626 , 698 steidel , c. c. , adelberger , k. l. , giavalisco , m. , dickinson , m. , & pettini , m. 1999 , , 519 , 1 steidel , c. c. , adelberger , k. l. , shapley , a. e. , erb , d. k. , reddy , n. a. , & pettini , m. 2005 , , 626 , l44 steidel , c. c. , adelberger , k. l. , shapley , a. e. , pettini , m. , dickinson , m. , & giavalisco , m. 2003 , , 592 , 728 steidel , c. c. , giavalisco , m. , pettini , m. , dickinson , m. , & adelberger , k. l. 1996 , , 462 , l17 steidel , c. c. , shapley , a. e. , pettini , m. , adelberger , k. l. , erb , d. k. , reddy , n. a. , & hunt , m. p. 2004 , , 604 , 534 van dokkum , p. g. , et al . 2003 , , 587 , l83 van dokkum , p. g. , et al . 2004 , , 611 , 703 van dokkum , p. g. , et al . 2006 , , 638 , 59 wolf , c. , meisenheimer , k. , rix , h .- w . , borch , a. , dye , s. , kleinheinrich , m. 2003 , , 401 , 73 zucca , e. , et al . 2006 , , 445 , 879

we present the rest - frame optical ( @xmath0 , @xmath1 , and @xmath2 band ) luminosity functions ( lfs ) of galaxies at @xmath3 , measured from a @xmath4-selected sample constructed from the deep nir musyc , the ultradeep fires , and the goods - cdfs . this sample is unique for its combination of area and range of luminosities . the faint - end slopes of the lfs at @xmath5 are consistent with those at @xmath6 . the characteristic magnitudes are significantly brighter than the local values ( e.g. , @xmath7 mag in the @xmath2 band ) , while the measured values for @xmath8 are typically @xmath9 times smaller . the @xmath0-band luminosity density at @xmath10 is similar to the local value , and in the @xmath2 band it is @xmath11 times smaller than the local value . we present the lf of distant red galaxies ( drgs ) , which we compare to that of non - drgs . while drgs and non - drgs are characterized by similar lfs at the bright end , the faint - end slope of the non - drg lf is much steeper than that of drgs . the contribution of drgs to the global densities down to the faintest probed luminosities is 14%-25% in number and 22%-33% in luminosity . from the derived rest - frame @xmath12 colors and stellar population synthesis models , we estimate the mass - to - light ratios ( @xmath13 ) of the different subsamples . the @xmath13 ratios of drgs are @xmath9 times higher ( in the @xmath2 and @xmath1 bands ) than those of non - drgs . the global stellar mass density at @xmath3 appears to be dominated by drgs , whose contribution is of order @xmath1460%-80% of the global value . qualitatively similar results are obtained when the population is split by rest - frame @xmath12 color instead of observed @xmath15 color .

the jaynes cummings ( jc ) model is a fundamental building block in quantum optics ; it describes the interaction of a qubit with a quantum electromagnetic field under long wave and rotating wave approximations . it is exactly solvable @xcite and has proven useful to describe phenomena as rabi oscillations @xcite and collapse and revivals of the atomic inversion @xcite , among others ; see @xcite for a review on the model . if the number of qubits increases , the model , known as the dicke or tavis cummings model , shows many - body phenomena in the form of a superradiant phase @xcite . the dicke model is also exactly solvable @xcite and is known to show super - fluorescence and amplified spontaneous emission ; see @xcite for a recent review . in recent years , a general dicke hamiltonian , including quadratic self - interactions on both the field and qubit ensemble was introduced to study the effect of the nonlinearities and their relation to the stark shift , in units of @xmath0 , @xmath1 in this model the frequencies for the field and two - level system transitions are given by @xmath2 and @xmath3 , the quadratic interactions are assumed to be equal and given by @xmath4 , while the coupling between field and qubit is given by the parameter @xmath5 . an exact solution to this system was found by quantum inverse methods involving bethe anzats @xcite . the importance of the dicke model and its generalizations lies in its ability to describe more than atoms interacting with the quantized field of a cavity ; i.e. lasers . for example , it may describe open dynamical cavity - qed systems @xcite , ion trap systems @xcite , circuit - qed systems @xcite , and bose - einstein condensates interacting with classical or quantized electromagnetic fields @xcite . in this contribution , we present an exact solution , up to the roots of a polynomial , to a more general dicke hamiltonian by considering non - identical nonlinear interactions in ( [ eq : hbogoliubov ] ) . in the following , we will discuss our general dicke hamiltonian and the physical systems it can describe . we then show how a novel right unitary transform involving susskind glogower operators helps us diagonalize the hamiltonian in the field basis . with this at hand , it is simple to diagonalize the resulting tridiagonal hamiltonian in the dicke basis via orthogonal polynomials satisfying a three - term recurrence relation . in order to verify the validity of our exact solution , we recover the time evolution for a system involving just the single qubit . finally , we study the time evolution of different ensemble sizes to illustrate the simplicity of our approach and the results it yields ; we focus on the population inversion dynamics of the qubit ensemble as well as the evolution of the entropy and q - function of the field . let us consider a system composed by an ensemble of @xmath6 identical two - level systems ( ` qubits ' ) that interact with each other . these qubitas are in the presence of a quantized field and a kerr medium . for the sake of simplicity , we move into the frame defined by the transformation @xmath7 , where the excitation number operator is given by @xmath8 , and work with the hamiltonian in units of @xmath0 , @xmath9 the qubits ensemble is described by collective dicke operators satisfying the @xmath10 algebra , @xmath11 = 2 \hat{j}_{z}$ ] , @xmath12 = \pm \hat{j}_{\pm}$ ] , while the annihilation and creation operators for a single mode field satisfy @xmath13=1 $ ] . the transition frequency of each qubit , @xmath14 , and the frequency of the field , @xmath15 , are summarized by the detuning @xmath16 . the kerr medium is described by the parameter @xmath17 , while the qubit - qubit and ensemble - field couplings are given by @xmath4 and @xmath18 , in that order . the hamiltonian ( [ eq : hamiltonian ] ) describes the @xmath6-atom maser in general . in the special case of equal self - interactions , @xmath19 , it can be transformed into the @xmath6-atom maser including , kerr nonlinearity and stark shift as discussed in @xcite . different parameter sets describe particular physical models ; e.g.,@xmath20 delivers the kerr model @xcite , @xmath21 yield the dicke or tavis cummings model @xcite and @xmath22 gives the micromaser with kerr nonlinearity @xcite . furthermore , the general hamiltonian ( [ eq : hamiltonian ] ) and its reductions are experimentally feasible in cavity- and circuit - qed as well as trapped ions . it may also be possible to realize some of these models with two - mode bose - einstein condensates coupled to radiation fields @xcite . the case of equal - self interactions , @xmath19 , has been solved by inverse quantum methods in the past @xcite . this solution involves the bethe ansatz method . the general hamiltonian ( [ eq : hamiltonian ] ) can also be solved by extending our right unitary approach to the quantum landau zener problem for a single two - level system presented in @xcite , which delivers an evolution operator with the form @xmath23 where the auxiliary hamiltonians are given by @xmath24 where the ket @xmath25 is a dicke state , the operator @xmath26 is the density matrix for the pure state of the field with @xmath27 photons , the operator @xmath28 is the photon number operator and the symbol @xmath29 represents kronecker delta . these auxiliary hamiltonians are diagonal in the field basis ; i.e. they are given in terms of the photon number functions @xmath30^{1/2 } \left [ \hat{n } + 1 + \frac{n}{2 } - j \right]^{1/2}.\end{aligned}\ ] ] there is , however , a simpler approach to solve this general radiation - matter interaction model . in order to present a simpler approach to solve hamiltonian ( [ eq : hamiltonian ] ) , let us define the right unitary transformation @xmath31 where we have used the susskind glogower operators , @xmath32 which act as lowering and raising ladder operators on the fock state basis , @xmath33 and @xmath34 in that order , and are right - unitary , @xmath35 and @xmath36 , where @xmath26 is the density matrix for the pure state of the field with @xmath27 photons . again , the ket @xmath25 is a dicke or angular momentum state . then , it is possible to write the general hamiltonian ( [ eq : hamiltonian ] ) as : @xmath37 where the auxiliary hamiltonian is given by , @xmath38 we have used the notation @xmath39 to bring forward that this hamiltonian is _ semi - classical_-like because it is only expressed in terms of the number operator , @xmath40^{1/2 } \left [ \hat{n } + 1 - \frac{n}{2 } -j \right]^{1/2}. \end{aligned}\ ] ] it is possible to express the dynamics of this model as the evolution operator @xmath41 where powers of the form @xmath42 are needed . these powers can be obtained by realizing from ( [ eq : rightunitary ] ) and ( [ eq : scham ] ) that @xmath43 leads to @xmath44 by means of @xmath45 and @xmath46 for @xmath47 and @xmath48 . thus , the evolution operator in the reduced form is given by the expression @xmath49 the hamiltonian @xmath39 is diagonal in the field basis and is symmetric tridiagonal in the dicke basis ; i.e. it is diagonalizable in the angular momentum basis . the eigenvalues of this hamiltonian can be found by the method of minors and are given by the roots of the characteristic polynomial @xmath50 p_{n}\left(\nu\right ) - g^2\left(-\frac{n}{2}+1,\hat{n}\right ) p_{n-1}\left(\nu\right)\end{aligned}\ ] ] with @xmath51 p_{j-1}\left(\nu\right ) + \nonumber \\ & & - g^2\left(\frac{n}{2 } + 2 -j , \hat{n}\right ) p_{j-2}\left(\nu\right ) , \quad j \ge 2 \nonumber \\\end{aligned}\ ] ] the eigenvectors are calculated from the eigenvalue equation for the hamiltonian and give @xmath52 where the amplitudes answer to the following recurrence relations , @xmath53 c_{\frac{n}{2}}^{\left(j\right ) } + g\left ( \frac{n}{2 } , \hat{n } \right ) c_{\frac{n}{2 } - 1}^{\left(j\right)}&= & 0 , \\ \left [ f\left(j , \hat{n } \right ) - \nu_{j } \right ] c_{k}^{\left(j\right ) } + g \left(j , \hat{n } \right ) c_{k- 1}^{\left(j\right ) } + g \left(j+1 , \hat{n } \right ) c_{k + 1}^{\left(j\right ) } & = & 0 , \\ \left [ f\left(-\frac{n}{2 } , \hat{n } \right ) - \nu_{j } \right ] c_{-\frac{n}{2}}^{\left(j\right ) } + g\left(-\frac{n}{2}+1 , \hat{n } \right ) c_{\frac{n}{2 } + 1}^{\left(j\right)}&= & 0 . \end{aligned}\ ] ] the time evolution given in the previous section accounts for the full dynamics of the system and helps calculating any given quantity of interest . as an example , we will focus on the time evolution of the reduced density matrix for the field where the initial state is given by a pure state @xmath54 , @xmath55 the notation @xmath56 is used to describe the components of the _ semi - classical _ time evolution operator . this allows us to calculate the mean photon number evolution , @xmath57 and in consequence the population inversion @xmath58 . other interesting quantities are the purity of the field , @xmath59 and von neumann entropy , @xmath60 , \end{aligned}\ ] ] which are a good measure of the degree of mixedness of the reduced system . let us consider a system with just the single qubit , @xmath61 the _ semi - classical _ hamiltonian is given by @xmath62 and it is possible to give a closed form time evolution operator as @xmath63 } \left\ { \cos \frac{\omega(\hat{n } ) t}{2 } - \frac{i \left[\beta(\hat{n } ) \hat{\sigma}_{z } + 2 \lambda \sqrt{n } \hat{\sigma}_{x } \right]}{\omega(\hat{n } ) } \sin \frac{\omega(\hat{n})t}{2 } \right\},\\ \beta(\hat{n } ) & = & \delta + \kappa \left(1 - 2 \hat{n } \right ) , \\ \omega(\hat{n } ) & = & \sqrt { \left [ \beta(\hat{n } ) \right]^2 + 4 \hat{n } \lambda^2 } \end{aligned}\ ] ] it is trivial to apply the operator @xmath64 ( @xmath65 ) to any given initial state ket ( bra ) and then apply the _ semi - classical _ exponential . figure [ fig : fig1 ] shows the time evolution of the mean population inversion ( first row ) , entropy of the reduced field ( second row ) and husimi s q function of the field ( third row ) for a single qubit as given by a jaynes cummings model ( left column ) and a jaynes cummings kerr model ( right column ) . our results are in accordance with those in the literature @xcite and we can proceed to sample the dynamics of ensembles . , under the jaynes - cummings model , left column ( a , c , e , f ) , and under a jaynes - cummings - kerr model , right column ( b , d , g , h ) . the initial state for both cases is @xmath66 with @xmath67 . ] for an ensemble of qubits , the task of finding a closed form expression for the time evolution becomes cumbersome but it is possible to numerically diagonalize the _ semi - classical _ hamiltonian and implement the time evolution of any given initial state . as an example , we consider the evolution of ensembles of three , fig . [ fig : fig2 ] , and twenty five , fig . [ fig : fig3 ] , qubits . the information about the particular initial conditions and parameter values can be found in the figures and their captions . at the time , it is not our goal to report and in - depth analysis of the dynamics of generalized dicke models but just to present our diagonalization scheme to obtain an exact solution via susskind - glogower operators . for this reason , we will just briefly comment some basic characteristics of the dynamics . by considering an initial state given by the separable state consisting of a coherent field and the ensemble in its ground state , @xmath68 , it is possible to see that the dicke model presents strong collapse and revivals of the population inversion as long as the mean photon number is larger than the number of qubits in the system . a clear collapse of the population inversion is seen in any case studied here , up to @xmath69 . the strength of the oscillations in the population inversion diminishes as the number of qubits in the system gets close to the mean photon number of the coherent state but they become ever - present at smaller times as we get larger ensemble sizes for a fixed value of the coherent state parameter . meanwhile , the purity and entropy of such a dicke model signals an ever - present entangled state between the field and the ensemble as the number of qubits gets close to or equal to the mean number of photons ; i.e. the plots change from strong , well - defined , unmodulated dips in the functions to a strongly modulated flat - liner close to the value of a mixed reduced density matrix @xcite . the q - function for the reduced field behaves as expected . for @xmath70 , @xmath71 well - defined phase blobs appear and evolve half of them clock - wise and the other half counter - clock - wise as time goes by . the revivals in the population inversion are associated to the overlapping of these phase blobs ; a stronger revival corresponding to a better overlapping . however , when an interacting ensemble of qubits is considered under dicke kerr dynamics , the collapse and revivals of the population inversion are always weak but well defined and periodical . purity and entropy functions point a return to a quasi - separable state on the first revival for the cases analyzed with the number of qubits less or equal to the mean photon number of the field . the mean value of these functions gradually increases with time and some dips appear periodically due to the constructive interference of the wavefunction components , leading to revivals in the population inversion . under dicke kerr dynamics the phase blobs seem heavily defined by the kerr process and for @xmath67 four phase blobs appear and two of them evolve clockwise while the other two do it counter - clockwise . this process produces an overlap of two and two of the phase blobs leading to a weak local minimum in the purity / entropy but does not register in the population inversion . it is only when the four phase blobs overlap that a pronounced local minimum and a revival of the population inversion appears . , with three qubits under the dicke model , left column ( a , c , e , f ) , and with three interacting qubits under a dicke - kerr model , right column ( b , d , g , h ) . the initial state for both cases is @xmath72 with @xmath67 . ] , with twenty five qubits under the dicke model , left column ( a , c , e , f ) , and with twenty five interacting qubits under a dicke - kerr model , right column ( b , d , g , h ) . the initial state for both cases is @xmath73 with @xmath67 . ] we have considered the general @xmath6-atom maser model which can be described by the dicke model plus dipople dipole interactions and kerr nonlinearity . as a side result , we extend a previous result based on susskind glogower operators that gives the exact dynamics of a jaynes cummings model as the product of two time evolution operators . our main result is a different and simpler approach involving susskind glogower operators and right unitary transformations that allow us to represent our generalized dicke model as a transformed _ semi - classical_-like hamiltonian which is diagonal in the field basis and tridiagonal in the dicke basis ; thus , the diagonalization of this _ semi - classical _ hamiltonian is known up to the roots of its characteristic polynomial . the transformed _ semi - classical_-like hamiltonian gives the time evolution of the system and provides access to the dynamics of any quantity of interest . we use our result to derive a closed analytical form for the time evolution operator of a single qubit interacting with a quantized field in the presence of a kerr medium , a jaynes - cummings - kerr model . also , we present the time evolution of the population inversion , reduced field entropy and husimi s q - function of the field for ensembles consisting of three and twenty - five interacting two - level systems under a dicke - kerr model where the interaction and kerr nonlinearity are equal . this is done to show how simple it is to deal with many atoms with our partial diagonalization approach . it is possible that one could follow the dynamics of hundreds and maybe a few thousands of qubits with our approach in a simple workstation with efficient programming ; e.g. , this is of importance in the description of realistic micromasers and may be relevant to the study of fields interacting with bose - einstein condensates in the two - mode approximation . some systems , e.g. circuit - qed and open - dynamical systems , may deliver a strong coupled version of the general dicke hamiltonian in ( [ eq : hamiltonian ] ) , @xmath74 notice that the @xmath75 term @xcite has been kept for the sake of generality . the presence of the strong interaction term deters the use of the approach presented above . here , we want to show two things . the first is that we can get rid of the second order nonlinearity , @xmath76 , if it is weak compared to the frequency of the field . this allows us to use a squeezed states basis for the field , described by the transformation , @xmath77 that helps us get rid of the @xmath76 term . the second thing we want to show is that a _ small rotation _ @xcite , @xmath78 has an effect similar to that of the rotating wave approximation . this _ small rotation _ leads to just a dicke hamiltonian including a kerr medium and dipole - dipole interactions between the qubit ensemble components , @xmath79 after we have moved to a frame defined by the total excitation number @xmath80 rotating at the frequency of the field and defined the parameters @xmath81 , @xmath82 and @xmath83 . note that we have taken the self - interaction nonlinearities @xmath17 and @xmath84 a couple orders of magnitude smaller than the transition frequency @xmath14 in order to neglect products of couplings and nonlinearities . we want to emphasize that , while we can not deal with the strong - coupling regime , this _ small rotation _ method may be valid in the regime where phase transitions appear @xmath85 @xcite . 10 url # 1#1urlprefix[2][]#2 jaynes e t and cummings f w 1963 _ proc . ieee _ * 51 * 89 109 rodrguez - lara b m and lee r k 2012 classical dynamics of a two - species bose - einstein condensate in the presence of nonlinear maser processes _ spontaneous symmetry breaking , self - trapping , and josephson oscillations in nonlinear systems _ progress in optical science and photonics ( springer berlin heidelberg )

we show a right unitary transformation approach based on susskind glogower operators that diagonalizes a generalized dicke hamiltonian in the field basis and delivers a tridiagonal hamiltonian in the dicke basis . this tridiagonal hamiltonian is diagonalized by a set of orthogonal polynomials satisfying a three - term recurrence relation . our result is used to deliver a closed form , analytic time evolution for the case of a jaynes cummings kerr model and to study the time evolution of the population inversion , reduced field entropy , and husimi s q - function of the field for ensembles of interacting two - level systems under a dicke kerr model .

preprocessing and the analysis of preprocessed data are ubiquitous components of statistical inference , but their treatment has often been informal . we aim to develop a theory that provides a set of formal statistical principles for such problems under the banner of multiphase inference . the term `` multiphase '' refers to settings in which inferences are obtained through the application of multiple procedures in sequence , with each procedure taking the output of the previous phase as its input . this encompasses settings such as multiple imputation ( mi , @xcite ) and extends to other situations . in a multiphase setting , information can be passed between phases in an arbitrary form ; it need not consist of ( independent ) draws from a posterior predictive distribution , as is typical with multiple imputation . moreover , the analysis procedure for subsequent phases is not constrained to a particular recipe , such as rubin s mi combining rules ( @xcite ) . the practice of multiphase inference is currently widespread in applied statistics . it is widely used as an analysis technique within many publications any paper that uses a `` pipeline '' to obtain its final inputs or clusters estimates from a previous analysis provides an example . furthermore , projects in astronomy , biology , ecology , and social sciences ( to name a small sampling ) increasingly focus on building databases for future analyses as a primary objective . these projects must decide what levels of preprocessing to apply to their data and what additional information to provide to their users . providing all of the original data clearly allows the most flexibility in subsequent analyses . in practice , the journey from raw data to a complete model is typically too intricate and problematic for the majority of users , who instead choose to use preprocessed output . unfortunately , decisions made at this stage can be quite treacherous . preprocessing is typically irreversible , necessitating assumptions about both the observation mechanisms and future analyses . these assumptions constrain all subsequent analyses . consequently , improper processing can cause a disproportionate amount of damage to a whole body of statistical results . however , preprocessing can be a powerful tool . it alleviates complexity for downstream researchers , allowing them to deal with smaller inputs and ( hopefully ) less intricate models . this can provide large mental and computational savings . two examples of such trade - offs come from nasa and high - throughput biology . when nasa satellites collect readings , the raw data are usually massive . these raw data are referred to as the `` level 0 '' data ( @xcite ) . the level 0 data are rarely used directly for scientific analyses . instead , they are processed to levels 1 , 2 , and 3 , each of which involves a greater degree of reduction and adjustment . level 2 is typically the point at which the processing becomes irreversible . @xcite provide an excellent illustration of this process for the atmospheric infrared sounder ( airs ) experiment . this processing can be quite controversial within the astronomical community . several upcoming projects , such as the advanced technology solar telescope ( atst ) will not be able to retain the level 0 or level 1 data ( @xcite ) . this inability to obtain raw data and increased dependence on preprocessing has transformed low - level technical issues of calibration and reduction into a pressing concern . high - throughput biology faces similar challenges . whereas reproducibility is much needed ( e.g. , @xcite ) , sharing raw datasets is difficult because of their sizes . the situation within each analysis is similar . confronted with an overwhelming onslaught of raw data , extensive preprocessing has become crucial and ubiquitous . complex models for genomic , proteomic , and transcriptomic data are usually built upon these heavily - processed inputs . this has made the intricate details of observation models and the corresponding preprocessing steps the groundwork for entire fields . to many statisticians , this setting presents something of a conundrum . after all , the ideal inference and prediction will generally use a complete correctly - specified model encompassing the underlying process of interest and all observation processes . then , why are we interested in multiphase ? we focus on settings where there is a natural separation of knowledge between analysts , which translates into a separation of effort . the first analyst(s ) involved in preprocessing often have better knowledge of the observation model than those performing subsequent analyses . for example , the first analyst may have detailed knowledge of the structure of experimental errors , the equipment used , or the particulars of various protocols . this knowledge may not be easy to encapsulate for later analysts the relevant information may be too large or complex , or the methods required to exploit this information in subsequent analyses may be prohibitively intricate . hence , the practical objective in such settings is to enable the best possible inference given the constraints imposed and provide an account of the trade - offs and dangers involved . to borrow the phrasing of @xcite and @xcite , we aim for achievable practical efficiency rather than theoretical efficiency that is practically unattainable . multiphase inference currently represents a serious gap between statistical theory and practice . we typically delineate between the informal work of preprocessing and feature engineering and formal , theoretically - motivated work of estimation , testing , and so forth . however , the former fundamentally constrains what the latter can accomplish . as a result , we believe that it represents a great challenge and opportunity to build new statistical foundations to inform statistical practice . we present two examples that show both the impetus for and perils of undertaking multiphase analyses in place of inference with a complete , joint model . the first concerns microarrays , which allow the analysis of thousands of genes in parallel . we focus on expression microarrays , which measure the level of gene expression in populations of cells based upon the concentration of rna from different genes . these are typically used to study changes in gene expression between different experimental conditions . in such studies , the estimand of interest is typically the log - fold change in gene expression between conditions . however , the raw data consist only of intensity measurements for each probe on the array , which are grouped by gene along with some form of controls . these intensities are subject to several forms of observation noise , including additive background variation and additional forms of interprobe and interchip variation ( typically modeled as multiplicative noise ) . to deal with these forms of observation noise , a wide range of background correction and normalization strategies have been developed ( for a sampling , see @xcite , @xcite , @xcite , @xcite , @xcite , @xcite , @xcite ) . later analyses then focus on the scientific question of interest without , for the most part , addressing the underlying details of the observation mechanisms . background correction is a particularly crucial step in this process , as it is typically the point at which the analysis moves from the original intensity scale to the log - transformed scale . as a result , it can have a large effect on subsequent inferences about log - fold changes , especially for genes with low expression levels in one condition ( @xcite , @xcite ) . one common method ( mas5 ) , provided by one microarray manufacturer , uses a combination of background subtraction and truncation at a fixed lower threshold for this task ( @xcite ) . other more sophisticated techniques use explicit probability models for this de - convolution . a model with normally - distributed background variation and exponentially distributed expression levels has proven to be the most popular in this field ( @xcite , @xcite ) . unfortunately , even the most sophisticated available techniques pass only point estimates onto downstream analyses . this necessitates ad - hoc screening and corrections in subsequent analyses , especially when searching for significant changes in expression ( e.g. , @xcite ) . retaining more information from the preprocessing phases of these analyses would allow for better , simpler inference techniques with greater power and fewer hacks . the motivation behind the current approach is quite understandable : scientific investigators want to focus on their processes of interest without becoming entangled in the low - level details of observation mechanisms . nevertheless , this separation can clearly compromise the validity of their results . the role of preprocessing in microarray studies extends well beyond background correction . normalization of expression levels across arrays , screening for data corruption , and other transformations preceding formal analysis are standard . each technique can dramatically affect downstream analyses . for instance , quantile normalization equates quantiles of expression distributions between arrays , removing a considerable amount of information . this mutes systematic errors ( @xcite ) , but it can seriously compromise analyses in certain contexts ( e.g. , mirna studies ) . another example of multiphase inference can be found in the estimation of correlations based upon indirect measurements . this appears in many fields , but astrophysics provides one recent and striking case . the relationships between the dust s density , spectral properties , and temperature are of interest in studies of star - forming dust clouds . these characteristics shed light on the mechanisms underlying star formation and other astronomical processes . several studies ( e.g. , @xcite , @xcite , @xcite , @xcite ) have investigated these relationships , finding negative correlations between the dust s temperature and spectral index . this finding is counter to previous astrophysical theory , but it has generated many alternative explanations . such investigations may , however , be chasing a phantasm . these correlations have been estimated by simply correlating point estimates of the relevant quantities ( temperature @xmath0 and spectral index @xmath1 ) based on a single set of underlying observations . as a result , they may conflate properties of this estimation procedure with the underlying physical mechanisms of interest . this has been noted in the field by @xcite , but the scientific debate on this topic continues . @xcite provide a particularly strong argument , using a cohesive hierarchical bayesian approach , that improper multiphase analyses have been a pervasive issue in this setting . improper preprocessing led to incorrect , negative estimates of the correlation between temperature and spectral index , according to @xcite . these incorrect estimates even appeared statistically significant with narrow confidence intervals based on standard methods . on a broader level , this case again demonstrates some of the dangers of multiphase analyses when they are not carried out properly . those analyzing this data followed an intuitive strategy : estimate what we want to work with ( @xmath0 and @xmath1 ) , then use it to estimate the relationship of interest . unfortunately , such intuition is not a recipe for valid statistical inference . multiphase inference has wide - ranging connections to both the theoretical and applied literatures . it is intimately related to previous work on multiple imputation and missing data ( @xcite ( @xcite ) , @xcite , @xcite , @xcite ) . in general , the problem of multiphase inference can be formulated as one of missing data . however , in the multiphase setting , missingness arises from the preprocessing choices made , not a probabilistic response mechanism . thus , we can leverage the mathematical and computational methods of this literature , but many of its conceptual tools need to be modified . multiple imputation addresses many of the same issues as multiphase inference and is indeed a special case of the latter . concepts such as congeniality between imputation and analysis models and self - efficiency ( @xcite ) have natural analogues and roles to play in the analysis of multiphase inference problems . multiphase inference is also tightly connected to work on the comparison of experiments and approximate sufficiency , going back to @xcite ( @xcite ) and continuing through @xcite and @xcite , among others . this literature has addressed the relationship between decision properties and the probabilistic structure of experiments , the relationship between different notions of statistical information , and notions of approximate sufficiency all of these are quite relevant for the study of multiphase inference . we view the multiphase setting as an extension of this work to address a broader range of real - world problems , as we will discuss in section [ sec : riskmonotone ] . the literature on bayesian combinations of experts also informs our thinking on multiphase procedures . @xcite provides an excellent review of the field , while @xcite provides the core formalisms of interest for the multiphase setting . overall , this literature has focused on obtaining coherent ( or otherwise favorable ) decision rules when combining information from multiple bayesian agents , in the form of multiple posterior distributions . we view this as a best - case scenario , focusing our theoretical development towards the mechanics of passing information between phases . we also focus on the sequential nature of multiphase settings and the challenges this brings for both preprocessors and downstream analysts , in contrast to the more `` parallel '' or simultaneous focus of the literature mentioned above . there are also fascinating links between multiphase inference and the signal processing literature . there has been extensive research on the design of quantizers and other compression systems ; see for example @xcite . such work is often focused on practical questions , but it has also yielded some remarkable theory . in particular , the work of @xcite on the relationship between surrogate loss functions in quantizer design and @xmath2-divergences suggests possible ways to develop and analyze a wide class of multiphase procedures , as we shall discuss in section [ sec : future ] . to formalize the notion of multiphase inference , we begin with a formal model for two - phase settings . the first phase consists of the data generation , collection , and preprocessing , while the second phase consists of inference using the output from the first phase . we will call the first - phase agent the `` preprocessor '' and the second - phase agent the `` downstream analyst '' . the preprocessor observes the raw data @xmath3 . this is a noisy realization of @xmath4 , variables of interest that are not directly obtainable from a given experiment , e.g. , gene expression from sequencing data , or stellar intensity from telescopic observations . we assume that the joint density of @xmath4 and @xmath3 with respect to product measure @xmath5 can be factored as @xmath6 here , @xmath7 encapsulates the underlying process of interest and @xmath8 encapsulates the observation process . we assume that @xmath9 is of fixed dimension in all asymptotic settings . in practice , the preprocessor should be able to postulate a reasonable `` observation model '' @xmath10 , but will not always know the true `` scientific model '' @xmath11 . this is analogous to the mi setting , where the imputer does not know the form of the final analysis . from the original data generating process and outputs @xmath0 , with @xmath4 as missing data . the downstream analyst observes the preprocessor s output @xmath0 and has both @xmath4 and @xmath3 missing . ] using this model , the preprocessor provides the downstream analyst with some output @xmath12 , where @xmath13 is a ( possibly stochastic ) additional input . when @xmath14 is stochastic ( e.g. , an mcmc output ) , the conditional distribution @xmath15 is its theoretical description instead of its functional form . however , for simplicity , we will present our results when @xmath0 is a deterministic function of @xmath3 only , but many results generalize easily . given such @xmath0 , downstream analysts can carry out their inference procedures . figure [ fig : models ] depicts our general model setup . this model incorporates several restrictions . first , it is markovian with respect to @xmath3 , @xmath4 , and @xmath9 ; @xmath3 is conditionally independent of @xmath9 given @xmath4 ( and @xmath16 ) . second , the parameters governing the observation process ( @xmath16 ) and those governing the scientific process ( @xmath9 ) are distinct . in bayesian settings , we further assume that @xmath16 and @xmath9 are independent _ a priori_. the parameters @xmath16 are nuisance from the perspective of all involved ; the downstream analyst wants to draw inferences about @xmath4 and @xmath9 , and the preprocessor wants to pass forward information that will be useful for said inferences . if downstream inferences are bayesian with respect to @xmath16 , then @xmath17 ( which holds under ( [ e : model ] ) ) is sufficient for all inference under the given model and prior . hence , this conditional density is frequently of interest in our theoretical development , as is the corresponding marginalized model @xmath18 . we will compare results obtained with a fixed prior to those obtained in a more general setting to better understand the effects of nuisance parameters in multiphase inference . these restrictions are somewhat similar to those underlying rubin s ( @xcite ) definition of `` missing at random '' ; however , we do not have missing data mechanism ( mdm ) in this setting _ per se_. the distinction between missing and observed data ( @xmath4 and @xmath3 ) is fixed by the structure of our model . in place of mdm , we have two imposed patterns of missingness : one for the data - generating process , and one for the inference process . the first is @xmath10 , which creates a noisy version of the desired scientific variables . here , @xmath4 can be considered the missing data and @xmath3 the observed . for the inference process , the downstream analyst observes @xmath0 in place of @xmath3 but desires inference for @xmath9 based upon @xmath19 . hence , @xmath3 and @xmath4 are both missing for the downstream analyst . neither pattern is entirely intrinsic to the problem both are fixed by choice . the selection of scientific variables @xmath4 for a given marginal likelihood @xmath20 is a modeling decision . the selection of preprocessing @xmath21 is a design decision . this contrasts with the typical missing data setting , where mdm is forced upon the analyst by nature . with multiphase problems , we seek to design and evaluate engineered missingness . thus the investigation of multiphase inference requires tools and ideas from design , inference , and computation in addition to the established theory of missing data . with this model in place , we turn to formally defining multiphase procedures . this is more subtle than it initially appears . in the mi setting , we focus on complete - data procedures for the downstream analyst s estimation and do not restrict the dependence structure between missing data and observations . in contrast , we restrict the dependence structure as in ( [ e : model ] ) , but place far fewer constraints on the analysts procedures . here , we focus our definitions and discussion on the two - phase case of a single preprocessor and downstream analyst . this provides the formal structure to describe the interface between any two phases in a chain of multiphase analyses . in our multiphase setting , downstream analysts need not have any complete - data procedure in the sense of one for inferring @xmath9 from @xmath4 and @xmath3 ; indeed , they need not formally have one based only upon @xmath4 for inferring @xmath9 . we require only that they have a set of procedures for their desired inference using the quantities provided from earlier phases as inputs ( @xmath0 ) , not necessarily using direct observations of @xmath4 or @xmath3 . such situations are common in practice , as methods are often built around properties of preprocessed data such as smoothness or sparsity that need not hold for the actual values of @xmath4 . for the preprocessor , the input is @xmath3 and the output is @xmath0 . here @xmath0 could consist of a vector of means with corresponding standard errors , or , for discrete @xmath3 , @xmath0 could consist of carefully selected cross - tabulations . in general , @xmath0 clearly needs to be related to @xmath4 to capture inferential information , but its actual form is influenced by practical constraints ( e.g. , aggregation to lower than desired resolutions due to data storage capacity ) . for the downstream analyst , the input is @xmath0 and the output is an inference for @xmath22 . this analyst can obviously adapt . for example , suppose @xmath23 for each entry @xmath24 of @xmath4 . if the preprocessor provides @xmath25 , the analyst may simply use an unweighted mean to estimate @xmath9 . if the preprocessor instead gives the analyst @xmath26 , where @xmath27 contains standard errors , the latter could instead use a weighted mean to estimate @xmath9 . this adaptation extends to an arbitrary number of possible inputs @xmath28 , each of which corresponds to a set of constraints facing the preprocessor . to formalize this notion of adaptation , we first define an index set @xmath29 with one entry for each such set of constraints . this maps between forms of input provided by the preprocessor and estimators selected by the downstream analyst . in this way , @xmath29 captures the downstream analyst s knowledge of previous processing and the underlying probability model . thus , this index set plays an central role in the definition of multiphase inference problems , far beyond that of a mere mathematical formality ; it regulates the amount of mutual knowledge shared between the preprocessor and the downstream analyst . now , we turn to the estimators themselves . we start with point estimation as a foundation for a broader class of problems . testing begins with estimating rejection regions , interval estimation with estimating coverage , classification with estimating class membership , and prediction with estimating future observations and , frequently , intermediate parameters . the framework we present therefore provides tools that can be adapted for more than estimation theory . we define multiphase estimation procedures as follows : a _ multiphase estimation procedure _ @xmath30 is a set of estimators @xmath31 indexed by the set @xmath29 , where @xmath28 corresponds to the output of the @xmath32th first - phase method ; that is , @xmath30 is a family of estimators with different inputs . when clear , we will drop the subscripts @xmath32 and index the estimators in @xmath30 by their inputs . this definition provides enough flexibility to capture many practical issues with multiphase inference , and it can be iterated to define procedures for analyses involving a longer sequence of preprocessors and analysts . it also encompasses the definition of a missing data procedure used by @xcite . such procedures can not , of course , be arbitrarily constructed if they are to deliver results with general validity . hence , having defined these procedures , we will cull many of them from consideration in section [ sec : riskmonotone ] . the obvious choice of our estimand , suggested by our notation thus far , is the parameter for the scientific model , @xmath9 . this is very amenable to mathematical analysis and relevant to many investigations . hence , it forms the basis for our results in section [ sec : theory ] . however , for multiphase analyses , other classes of estimands may prove more useful in practice . in particular , functions of @xmath4 , future scientific variables @xmath33 , or future observations @xmath34 may be of interest . prediction of such quantities is a natural focus in the multiphase setting because such statements are meaningful to both the preprocessor and downstream analyst . such estimands naturally encompass a broad range of statistical problems including prediction , classification , and clustering . however , there is often a lack of mutual knowledge about @xmath35 , so the preprocessor can not expect to `` target '' estimation of @xmath9 in general , as we shall discuss in section [ sec : remarks ] . it is not automatic for multiphase estimation procedures to produce better results as the first phase provides more information . to obtain a sensible context for theoretical development , we must regulate the way that the downstream analyst adapts to different inputs . for instance , they should obtain better results ( in some sense ) when provided with higher - resolution information . this carries over from the mi setting ( @xcite , @xcite , @xcite , @xcite ) , where notions such as self - efficiency are useful for regulating the downstream analyst s procedures . we define a similar property for multiphase estimation procedures , but without restricting ourselves to the missing data setting . specifically , let @xmath36 indicate @xmath37 is a deterministic function of @xmath38 . in practice , @xmath37 could be a subvector , aggregation , or other summary of @xmath38 . a multiphase estimation procedure @xmath30 is _ risk monotone _ with respect to a loss function @xmath39 if , for all pairs of outputs @xmath40 , @xmath36 implies @xmath41 . an asymptotic analogue of risk monotonicity is defined as would be expected , scaling the relevant risks at an appropriate rate to obtain nontrivial limits . this is a natural starting point for regulating multiphase estimation procedures ; stronger notions may be required for certain theoretical results . note that this definition does not require that `` higher - quality '' inputs necessarily lead to lower risk estimators . risk monotonicity requires only that estimators based upon a larger set of inputs perform no worse than those with strictly less information ( in a deterministic sense ) . however , risk monotonicity is actually quite tight in another sense . it requires that additional information can not be misused by the downstream analyst , imposing a strong constraint on mutual knowledge . for an example , consider the case of unweighted and weighted means . to obtain better results when presented with standard errors , the downstream analyst must know that they are being given ( the correct ) standard errors and to weight by inverse variances . this definition is related to the comparison of experiments , as explored by @xcite ( @xcite ) , but diverges on a fundamental level . our ordering of experiments , based on deterministic functions , is more stringent than that of @xcite , but they are related . indeed , our @xmath42 relation implies that of @xcite . in the latter work , an experiment @xmath43 is defined as more informative than experiment @xmath1 , denoted @xmath44 , if all losses attainable from @xmath1 are also attainable from @xmath43 . this relation is also implied when @xmath43 is sufficient for @xmath1 . our stringency stems from our broader objectives in the multiphase setting . from a decision - theoretic perspective , the partial ordering of experiments investigated by blackwell and others deal with which risks are attainable given pairs of experiments , allowing for arbitrary decision procedures . in contrast , our criterion restricts procedures based on whether such risks are actually attained , with respect to a particular loss function . this is because , in the multiphase setting , it is not generally realistic to expect downstream analysts to be capable of obtaining optimal estimators for all forms of preprocessing . the conceptually - simplest way to generate such a procedure is to begin with a complete probability model for @xmath45 . under traditional asymptotic regimes , all procedures consisting of bayes estimators based upon such a model will ( with full knowledge of the transformations involved in each @xmath28 and a fixed prior ) be risk monotone . the same is true asymptotically under the same regimes ( for squared - error loss ) for procedures consisting of mles under a fixed model . under some other asymptotic regimes , however , these principles of estimation do not guarantee risk - monotonicity ; we explore this further in section [ sec : missinfo ] . but such techniques are not the only way to generate risk monotone procedures from probability models . this is analogous to self - efficiency , which can be achieved by procedures that are neither bayesian nor mle ( @xcite , @xcite ) . and @xmath38 form the basis set of statistics . each of these has three descendants ( @xmath46 from @xmath37 and @xmath47 from @xmath38 ) . these descendants are deterministic functions of their parent , but they are not deterministic functions of any other basis statistics . given correctly - specified models for @xmath37 and @xmath38 , a risk monotone procedure can be constructed for all statistics ( @xmath48 ) shown here as described in the text . ] a risk monotone procedure can be generated from any set of probability models for distinct inputs that `` span '' the space of possible inputs . suppose that an analyst has a set of probability models , all correctly specified , for @xmath49 , where @xmath50 ranges over a subset @xmath51 of the relevant index set @xmath29 . we also assume that this analyst has a prior distribution @xmath52 for each such basis models . these priors need not agree between models ; the analyst can build a risk - monotone procedure from an inconsistent set of prior beliefs . suppose that the inputs @xmath53 are not deterministic functions of each other and all other inputs can be generated as nontrivial deterministic transformations of one of these inputs . formally , we require @xmath54 for all distinct @xmath55 and , for each @xmath56 there exists a unique @xmath57 such that @xmath58 ( each output is uniquely descended from a single @xmath59 ) , as illustrated in figure [ fig : risk - monotone ] . this set can form a basis , in a sense , for the given procedure . using the given probability models with a single loss function and set of priors ( potentially different for each model ) , the analyst can derive a bayes rule under each model . for each @xmath57 , we require @xmath60 to be an appropriate bayes rule on said model . as @xmath61 for some function @xmath62 , we then have the implied @xmath63 , yielding the bayes rule for estimating @xmath9 based on @xmath28 , which is no less risky than @xmath60 . the requirement that each output @xmath28 derives from a unique @xmath59 means that each basis component @xmath59 has a unique line of descendants . within each line , each descendant is comparable to only a single @xmath59 in the sense of deterministic dependence . between these lines , such comparisons are not possible . this ensures the overall risk - monotonicity . biology provides an illustration of such bases . a wide array of methodological approaches have been used to analyze high - throughput gene expression data . one approach , builds upon order and rank statistics ( @xcite , @xcite , @xcite ) . another common approach uses differences in gene expression between conditions or experiments , often aggregating over pathways , replicates , and so forth . each class of methods is based upon a different form of preprocessing : ranks transformations for the former , normalization and aggregation for the latter . taking procedures based on rank statistics and aggregate differences in expression as a basis , we can consider constructing a risk - monotone procedure as above . thus , the given formulation can bring together apparently disparate methods as a first step in analyzing their multiphase properties . such constructions are , unfortunately , not sufficient to generate all possible risk monotone procedures . obtaining more general conditions and constructions for risk monotone procedures is a topic for further work . by casting the examples in section [ sec : examples ] into the formal structure just established , we can clarify the practical role of each mathematical component and see how to map theoretical results into applied guidance . we also provide an example that illustrates the boundaries of the framework s utility , and another that demonstrates its formal limits . these provide perspective on the trade - offs made in formalizing the multiphase inference problem . the case of microarray preprocessing presented previously fits quite nicely into the model of section [ sec : model ] . there , @xmath3 corresponds to the observed probe - level intensities , @xmath4 corresponds to the true expression level for each gene under each condition , and @xmath9 corresponds to the parameters governing the organism s patterns of gene expression . in the microarray setting , @xmath8 would characterize the relationship between expression levels and observed intensities , governed by @xmath16 . these nuisance parameters could include chip - level offsets , properties of any additive background , and the magnitudes of other sources of variation . the assumptions of a markovian dependence structure and distinct parameters for each part of the model appear quite reasonable in this case , as ( 1 ) the observation @xmath3 can only ( physically ) depend upon the sample preparation , experimental protocol , and rna concentrations in the sample and ( 2 ) the distributions @xmath7 and @xmath8 capture physically distinct portions of the experiment . background correction , normalization , and the reduction of observations to log - fold changes are common examples of preprocessing @xmath21 . as discussed previously , estimands based upon @xmath4 may be of greater scientific interest than those based upon @xmath9 . for instance , we may want to know whether gene expression changed between two treatments in a particular experiment ( a statement about @xmath4 ) than whether a parameter regulating the overall patterns of gene expression takes on a particular value . for the astrophysical example , the fit is similarly tidy . the raw astronomical observations correspond to @xmath3 , the true temperature , density , and spectral properties of each part of the dust cloud become @xmath4 , and the parameters governing the relationship between these quantities ( e.g. , their correlation ) form @xmath9 . the @xmath8 distribution governs the physical observation process , controlled by @xmath16 . this process typically includes the instruments response to astronomical signals , atmospheric distortions , and other earthbound phenomena . as before , the conditional independence of @xmath9 and @xmath3 given @xmath4 and @xmath16 is sensible based upon the problem structure , as is the separation of @xmath9 and @xmath16 . here @xmath4 corresponds to signals emitted billions or trillions of miles from earth , whereas the observation process occurs within ground- or space - based telescopes . hence , any non - markovian effects are quite implausible . preprocessing @xmath21 corresponds to the ( point ) estimates of temperature , density , and spectral properties from simple models of @xmath3 given @xmath4 and @xmath16 . the multiphase framework encompasses a broad range of settings , but it does not shed additional light on all of them . if @xmath0 is a many - to - one transformation of @xmath3 , then our framework implies that the preprocessor and downstream analyst face structurally different inference ( and missing data ) problems . this is the essence of multiphase inference , in our view . settings where @xmath64 is degenerate or @xmath0 is a one - to - one function of @xmath3 are boundary cases where our multiphase interpretation and framework add little . for a concrete example of these cases , consider a time - to - failure experiment , with the times of failure @xmath65 , @xmath66 . now , suppose that the experimenters actually ran the experiment in @xmath67 equally - sized batches . they observe each batch only until its first failure ; that is , they observe and report @xmath68 for each batch @xmath50 . subsequent analysts have access only to @xmath69 . this seems to be a case of preprocessing , but it actually resides at the very edge of our framework . we could take the complete observations to be @xmath4 and the batch minima to be @xmath3 . this would satisfy our markov constraint , with a singular , and hence deterministic , observation process @xmath70 simply selecting a particular order statistic within each batch . however , @xmath21 is one - to - one ; the preprocessor observes only the order statistics , as does the downstream analyst . there is no separation of inference between phases ; the same quantities are observed and missing to both the preprocessor and the downstream analyst . squeezing this case into the multiphase framework is technically valid but unproductive . the framework we present is not , however , completely generic . consider a chemical experiment involving a set of reactions . the underlying parameters @xmath9 describe the chemical properties driving the reactions , @xmath4 are the actual states of the reaction , and @xmath3 are the ( indirectly ) measured outputs of the reactions . the measurement process for these experiments , as described by @xmath64 , could easily violate the structure of our model in this case . for instance , the same chemical parameters could affect both the measurement and reaction processes , violating the assumed separation of @xmath9 and @xmath16 . even careful preprocessing in such a setting can create a fundamental incoherence . suppose the downstream analysis will be bayesian , so the preprocessor provides the conditional density of @xmath3 as a function of @xmath4 , latexmath:[${p_y}(y @xmath16 share components , and the preprocessor uses their prior on @xmath16 to create @xmath70 , the conditional density need not be sufficient for @xmath9 under the downstream analyst s model . because the downstream analyst s prior on @xmath9 need not be compatible with the preprocessor s prior on @xmath16 , inferences based on the preprocessor s @xmath70 can be seriously flawed in this setting . hence , we exclude such cases from our investigation for the time being . thinking bayesianly , our model ( [ e : model ] ) obviously does not exclude the possibility that the downstream analyst has more knowledge about @xmath9 than the preprocessor in the form of a prior on @xmath22 . however , _ prior _ information means that it is based on studies that do not overlap with the current one . probabilistically speaking , this means that our model permits the downstream analyst to formally incorporate another data set @xmath72 , as long as @xmath72 is conditionally independent of the scientific variables @xmath4 and observations @xmath3 given @xmath73 or @xmath9 . for example , the downstream analyst could observe completely separate experiments pertaining to the same underlying process governed by @xmath9 or the outcomes of separate calibration pertaining to @xmath16 , but not additional replicates governed by the same realization of @xmath4 . in a biological setting , this means that the downstream analyst could have access to results from samples not available to the preprocessor ( e.g. , biological replicates ) , possibly using the same equipment ; however , they could not have access to additional analyses of the same biological sample ( e.g. , technical replicates ) , as a single biological sample would typically correspond to a single realization of @xmath4 . these examples remind us that our multiphase setting does not encompass all of statistical inference . this is quite a relief to us . our work aims to open new directions for statistical research , but it can not possibly address every problem under the sun ! multiphase theory hinges on procedural constraints . consider , for example , finding the optimal multiphase estimation procedure in terms of the final estimator s bayes risk . without stringent procedural constraints , the result is trivial : compute the appropriate bayes estimator using the distribution of @xmath0 given @xmath9 . similarly , the optimal preprocessing @xmath0 will , without tight constraints , simply compute an optimal estimator using @xmath3 and pass it forward . note that both of these cases respect risk - monotonicity to the letter ; it is not sufficiently tight to enable interesting , relevant theory . more constraints , based upon careful consideration of applied problems , are clearly required . this is not altogether bad news . we need only look to the history of multiple imputation to see how rich theory can arise from stringent , pragmatic constraints . multiple imputation forms a narrow subset of multiphase procedures : @xmath4 corresponds to the complete data ( @xmath74 , in mi notation ) , @xmath3 corresponds to the observed data @xmath75 and missing data indicator @xmath76 , and @xmath0 usually consists of posterior predictive draws of the missing data together with the observed data . the markovian property depicted in figure [ fig : models ] holds when the parameter ( @xmath77 ) for the missing data mechanism @xmath78 ) is distinct from the parameter of interest ( @xmath22 ) in @xmath79 , which is a common assumption in practice . the second - phase procedure is then restricted to repeatedly applying a complete - data procedure and combining the results . these constraints were originally imposed for practical reasons in particular , to make the resulting procedure feasible with existing software . however , they have opened the door to deep theoretical investigations . in that spirit , we consider two types of practically - motivated constraints for multiphase inference : restrictions on the downstream analyst s procedure and restrictions on the preprocessor s methods . these constraints are intended to work in concert with coherence conditions ( e.g. , risk monotonicity ) , not in isolation , to enable meaningful theory . constraints on the downstream analyst are intended to reflect practical limitations of their analytic capacity . examples include restricting the downstream analyst to narrow classes of estimators ( e.g. , linear functions of preprocessed inputs ) , to specific principles of estimation ( e.g. , mles ) , or to special cases of a method we can reasonably assume the downstream analyst could handle , such as a complete - data estimator @xmath80 , available from software with appropriate inputs . estimators derived from nested families of models are often suitable for this purpose . for example , whereas @xmath80 may involve only an ordinary regression , the computation of @xmath81 may require a weighted least - squares regression.=1 another constraint on the downstream analyst pertains to nuisance parameters . such constraints are of great practical and theoretical interest , as we believe that the preprocessor will typically have better knowledge and statistical resources available to address nuisance parameters than the downstream analyst . an extreme but realistic case of this is to assume that the downstream analyst can not address nuisance parameters at all . as we shall discuss in section [ sec : theory ] , this would force the preprocessor to either marginalize over the nuisance parameters , find a pivot with respect to them , or trust the downstream analyst to use a method robust to the problematic parameters . turning to the preprocessor , we consider restricting either the form of the preprocessor s output or the mechanics of their methods . in the simplest case of the former , we could require that @xmath0 consist of the posterior mean ( @xmath82 ) and posterior covariance ( @xmath83 ) of the unknown @xmath4 under the preprocessor s model . a richer , but still realistic , class of output would be finite - dimensional real or integer vectors . restricting output to such a class would prevent the preprocessor from passing arbitrary functions onto the downstream analyst . this leads naturally to the investigation of ( finite - dimensional ) approximations to the preprocessor s conditional density , aggregation , and other such techniques . on the mechanical side , we can restrict either the particulars of the preprocessor s methods or their broader properties . examples of the former include particular computational approximations to the likelihood function or restrictions to particular principles of inference ( e.g. , summaries of the likelihood or posterior distribution of @xmath4 given @xmath84 ) . such can focus our inquiries to specific , feasible methods of interest or reflect the core statistical principles we believe the preprocessor should take into account . in a different vein , we can require that preprocessor s procedures be distributable across multiple researchers , each with their own experiments and scientific variables of interest . such settings are of interest for both the accumulation of scientific results for later use and for the development of distributed statistical computation . this leads to preprocessing based upon factored `` working '' models for @xmath4 , as we explore further in section [ sec : sufficiency ] . nuisance parameters play an important role in these constraints , narrowing the class of feasible methods ( e.g. , marginalization over such parameters may be exceedingly difficult ) and largely determining the extent to which preprocessing can be distributed . we explore these issues in more detail throughout section [ sec : theory ] . we now present a few steps towards a theory of multiphase inference . in this , we endeavor to address three basic questions : ( 1 ) how can we determine what to retain , ( 2 ) what limits the performance of multiphase procedures , and ( 3 ) what are some minimal requirements for being an ideal preprocessor ? we find insight into the first question from the language of classical sufficiency . we leverage and specialize results from the missing - data literature to address the second . for the third question , we turn to the tools of decision theory . suppose we have a group of researchers , each with their own experiments . they want to preprocess their data to reduce storage requirements , ease subsequent analyses , and ( potentially ) provide robustness to measurement errors . this group is keenly aware of the perils of preprocessing and want to ensure that the output they provide will be maximally useful for later analyses . their question is , `` which statistics should we retain ? '' if each of these researchers was conducting the final analysis themselves , using only their own data , they would be in a single - phase setting . the optimal strategy then is to keep a minimal sufficient statistic for each researcher s model . similarly , if the final analysis were planned and agreed upon among all researchers , we would again have a single - phase setting , and it is optimal to retain the sufficient statistics for the agreed - upon model . we use the term _ optimal _ here because it achieves maximal data reduction without losing information about the parameters of interest . such lossless compression in the general sense of avoiding statistical redundancy is often impractical , but it provides a useful theoretical gold standard . in the multiphase setting , especially with multiple researchers in the first phase , achieving optimal preprocessing is far more complicated even in theory . if @xmath21 is the output of the _ entire _ preprocessing phase , then in order to retain all information we must require @xmath21 to be a sufficient statistics for @xmath85 under model ( [ e : model ] ) ; that is , @xmath86 where @xmath39 denotes a likelihood function ; or at least in the ( marginal ) bayesian sense , @xmath87 where @xmath88 is the posterior of @xmath22 given data @xmath89 with the likelihood given by ( [ e : model ] ) . note that ( [ e : con1 ] ) implies ( [ e : con2 ] ) , and ( [ e : con2 ] ) is useful when the downstream analyst wants only a bayesian inference of @xmath22 . in either case the construction of the sufficient statistic generally depends on the joint model for @xmath3 as implied by ( [ e : model ] ) , requiring more knowledge than individual researchers typically possess . often , however , it is reasonable to assume the following conditional independence . let @xmath90 be the specification of @xmath91 for researcher @xmath92 , where @xmath93 forms a _ partition _ of @xmath3 . we then assume that @xmath94 note in the above definition implicitly we also assume the baseline measure @xmath95 is a product measure @xmath96 , such as lebesgue measure . the assumption ( [ e : obsm ] ) holds , for example , in microarray applications , when different labs provide conditionally - independent observations of probe - level intensities . the preceding discussion suggests that this assumption is necessary for ensuring ( [ e : con1 ] ) or even ( [ e : con2 ] ) , but obviously it is far from sufficient because it says nothing about the model on @xmath4 . it is reasonable or at least more logical than not to assume each researcher has the best knowledge to specify his / her own observation model @xmath97 ( @xmath98 . but , for the scientific model @xmath99 used by the downstream analyst , the best we can hope is that each researcher has _ a working model _ @xmath100 that is in some way related to @xmath99 . the notation @xmath101 reflects our hope to construct a common working parameter @xmath102 that can ultimately be _ linked _ to the scientific parameter @xmath22 . given this working model , the @xmath24th researcher can obtain the corresponding ( minimal ) sufficient statistic @xmath103 for @xmath104 with respect to @xmath105 when one has a prior @xmath106 for @xmath107 , one could alternately decide to retain the ( bayesian ) sufficient statistic @xmath108 with respect to the model @xmath109 our central interest here is to determine when the collection @xmath110 will satisfy ( [ e : con1 ] ) and when @xmath111 will satisfy ( [ e : con2 ] ) . this turns out to be an exceedingly difficult problem if we seek a necessary and sufficient condition for _ when _ this occurs . however , it is not difficult to identify sufficient conditions that can provide useful practical guidelines . we proceed by first considering cases where @xmath112 forms a partition of @xmath4 . compared to the assumption on partitioning @xmath3 , this assumption is less likely to hold in practice because different researchers can share common parts of @xmath4 s or even the entire scientific variable @xmath4 . however , as we shall demonstrate shortly , we can extend our results formally to all models for @xmath4 , as long as we are willing to put tight restrictions on the allowed class of working models . specifically , the following condition describes a class of working models that are ideal because they permit separate preprocessing yet retain joint information . note again that an implicit assumption here is that the baseline measure @xmath113 is a product measure @xmath114 . [ ( distributed separability condition ( dsc ) ) ] a set of working models @xmath115 is said to satisfy the _ distributed separability condition _ with respect to @xmath116 if there exists a probability measure @xmath117 such that @xmath118\,{\mathrm{d}}p_{\eta}({\eta}| { \theta } ) . \label{eq : dsc}\ ] ] [ thm : dsc ] under the assumptions ( [ e : obsm ] ) and ( [ eq : dsc ] ) , we have the collection of individual sufficient statistics from ( [ e : prob ] ) , that is , @xmath119 , is jointly sufficient for @xmath85 in the sense that ( [ e : con1 ] ) holds . under the additional assumption that @xmath120 forms a partition of @xmath77 and @xmath121 , both @xmath21 corresponding to ( [ e : prob ] ) and @xmath122 corresponding to ( [ e : prom ] ) are bayesianly sufficient for @xmath22 in the sense that ( [ e : con2 ] ) holds . by the sufficiency of @xmath123 for @xmath124 , we can write @xmath125 this implies that , @xmath126 } & = & \int _ { x } \biggl [ \prod_{i=1}^{r } { p_y}(y_i | x_i , { \xi}_i ) \biggr ] \\ & & { } \times\biggl [ \int_{\eta } \biggl [ \prod _ { i=1}^{r } { \tilde{p}_x}\bigl(x_i | g_i({\eta})\bigr ) \biggr]\,{\mathrm{d}}p_{\eta}({\eta}| { \theta } ) \biggr]\,{\mathrm{d}}\mu _ x(x ) , \\ { [ \mbox{by factorization of } \mu_x ] } & = & \int _ { \eta } \prod_{i=1}^{r } \biggl [ \int_{x_i } { p_y}(y_i | x_i , { \xi}_i ) { \tilde{p}_x}\bigl(x_i | g_i ( { \eta})\bigr)\,{\mathrm{d}}\mu_{x_i}(x_i ) \biggr]\,{\mathrm{d}}p_{\eta}({\eta}| { \theta } ) , \\ { \bigl[\mbox{by } ( \ref{eq : a1})\bigr ] } & = & \biggl [ \prod _ { i=1}^{r } h_i(y_i ) \biggr ] \biggl [ \int_{\eta } \prod_{i=1}^{r } f_i\bigl(t_i;g_i({\eta } ) , { \xi}_i\bigr)\,{\mathrm{d}}p_{\eta}({\eta}| { \theta } ) \biggr ] .\end{aligned}\ ] ] this establishes ( 1 ) by the factorization theorem . assertion ( 2 ) is easily established via an analogous argument , by integrating all the expressions above with respect to @xmath127 . we emphasize that dsc does not require individual researchers to model their parts of @xmath4 in the same way as the downstream analyst would , which would make it an essentially tautological condition . rather , it requires that individual researchers understand their own problems and how they can fit into the broader analysis hierarchically . this means that the working model for each @xmath128 @xmath129 can be more saturated than the downstream analyst s model for the same part of @xmath4 . consider a simple case with @xmath130 , where the preprocessor correctly assumes the multivariate normality for @xmath4 but is unaware that its covariance actually has a block structure or is unwilling to impose such a restriction to allow for more flexible downstream analyses . clearly any sufficient statistic under the unstructured multivariate model is also sufficient for any ( nested ) structured ones . the price paid here is failing to achieve the greatest possible sufficient reduction of the data , but this sacrifice may be necessary to ensure the broader validity of downstream analyses . for example , even if downstream analysts adopt a block - structured covariance , they may still want to perform a model checking , which would not be possible if all they are given is a _ minimal _ sufficient statistic for the model to be checked . knowledge suitable for specifying a saturated model is more attainable than complete knowledge of @xmath116 , although ensuring common knowledge of its ( potential ) hierarchical structure still requires some coordination among the researchers . each of them could independently determine for which classes of scientific models their working model satisfies the dsc . however , without knowledge of the partition of @xmath4 across researchers and the overarching model(s ) of interest , their evaluations need not provide any useful consensus . this suggests the necessity of some general communications and a practical guideline for distributed preprocessing , even when we have chosen a wise division of labors that permits dsc to hold . formally , dsc is similar in flavor to de finetti s theorem , but it does not require the components of the factorized working model to be exchangeable . dsc , however , is by no means necessary ( even under ( [ e : con1 ] ) ) , as an example in section [ sec : counterexamples ] will demonstrate . its limits stem from `` unparameterized '' dependence dependence between @xmath128 s that is not controlled by @xmath9 . when such dependence is present , statistics can exist that are sufficient for both @xmath131 and @xmath9 without the working model satisfying dsc . however , a simple necessary condition for distributed sufficiency is available . unsurprisingly , it links the joint sufficiency of @xmath132 under @xmath133 to the joint sufficiency of @xmath134 under the scientific model @xmath99 , where @xmath135 is any sufficient statistic for the working model @xmath136 . [ thm : necessary ] if , for all observation models satisfying ( [ e : obsm ] ) , the collections of individual sufficient statistics from ( [ e : prob ] ) @xmath119 are jointly sufficient for @xmath137 in the sense that ( [ e : con1 ] ) holds , then any collection of individual sufficient statistics under @xmath138 , that is , @xmath139 , must be sufficient for @xmath22 under @xmath140 . the proof of this condition emerges easily by considering the trivial observation model @xmath141 , where @xmath142 is the indicator function of set @xmath143 . theorem [ thm : necessary ] holds even if we require the observation model to be nontrivial , as the case of @xmath144\}}$ ] for arbitrary @xmath145-neighborhoods of @xmath128 demonstrates . the result says that if we want distributed preprocessing to provide a lossless compression regardless of the actual form of the observation model , then even under the conditional independence assumption ( [ e : obsm ] ) , we must require the individual working models to _ collectively _ preserve sufficiency under the scientific model . note that preserving sufficiency for a model is a much weaker requirement than preserving the model itself . indeed , two models can have very different model spaces yet share the same _ form _ of sufficient statistics , as seen with i.i.d . @xmath146 and @xmath147 models , both yielding the sample average as a complete sufficient statistic . although we find this sufficiency - preserving condition quite informative about the limits of lossless distributed preprocessing , it is not a sufficient condition . as a counterexample , consider @xmath148 independent for @xmath149 , @xmath150 , where @xmath151 . for the true model , we assume @xmath152 as follows : @xmath153 , @xmath154 , and all variables are mutually independent . for the working model , we take @xmath155 as follows : @xmath156 independently , and @xmath157 with probability 1 for all @xmath158 . obviously @xmath159 is a sufficient statistic for both @xmath155 and @xmath99 because of their normality . because @xmath27 is _ minimally _ sufficient for @xmath102 , this implies that any sufficient statistic for @xmath155 must be sufficient for @xmath99 , therefore the sufficiency preserving condition holds . however , the collection of the complete sufficient statistics @xmath160 for @xmath102 under @xmath161 is not sufficient for @xmath22 under @xmath162 because the latter is no longer an exponential family . the trouble is caused by the failure of the working models to capture additional flexibility in the scientific model that is not controlled by its parameter @xmath22 . therefore , obtaining a condition that is both necessary and sufficient for lossless compression via distributed preprocessing is a challenging task . such a condition appears substantially more intricate than those presented in theorems [ thm : dsc ] and [ thm : necessary ] and may therefore be less useful as an applied guideline . below we discuss a few further subtleties . although theorem [ thm : dsc ] covers both likelihood and bayesian cases , it is important to note a subtle distinction between their general implications . in the likelihood setting ( [ e : con1 ] ) , we achieve lossless compression for all downstream analyses targeting @xmath73 . this allows the downstream analyst to obtain inferences that are robust to the preprocessor s beliefs about @xmath16 , and they are free to revise their inferences if new information about @xmath16 becomes available . but , the downstream analyst must address the nuisance parameter @xmath77 from the preprocessing step , a task a downstream analyst may not be able or willing to handle . in contrast , the downstream analyst need not worry about @xmath77 in the bayesian setting ( [ e : con2 ] ) . however , this is achieved at the cost of robustness . all downstream analyses are potentially affected by the preprocessors beliefs about @xmath77 . furthermore , because @xmath122 is required only to be sufficient for @xmath22 , it may not carry any information for a downstream analyst to check the preprocessor s assumptions about @xmath16 . fortunately , as it is generally logical to expect the preprocessor to have better knowledge addressing @xmath77 than the downstream analyst , such robustness may not be a serious concern from a practical perspective . theoretically , the trade - off between robustness and convenience is not clear - cut ; they can coincide for other types of preprocessing , as seen in section [ sec : missinfo ] below . as discussed earlier , ( conditional ) dependencies among the observation variables @xmath163 across different @xmath24 s will generally rule out the possibility of achieving lossless compression by collecting individual sufficient statistics . this points to the importance of appropriate separation of labors when designing distributed preprocessing . in contrast , dependencies among @xmath128 s are permitted , at the expense of redundancy in sufficient statistics . we first consider deterministic dependencies , and for simplicity , take @xmath164 and constrain attention to the case of sufficiency for @xmath9 . suppose we have @xmath165 and @xmath166 forming a partition of @xmath4 , with a working model @xmath167 that satisfied the dsc for some @xmath168 . imagine we need to add a common variable @xmath72 to both @xmath165 and @xmath166 that is conditionally independent of @xmath169 given @xmath22 and has density @xmath170 , with the remaining model unchanged . however , the two researchers are unaware of the sharing of @xmath72 , so they set up @xmath171 and @xmath172 , with @xmath173 does not correspond to the scientific variable @xmath174 of interest . however , we notice that if we can force @xmath175 in @xmath176 , then we can recover @xmath4 . this forcing is not a mere mathematical trick . rather , it reflects an extreme yet practical strategy when researchers are unsure whether they share some components of their @xmath177 with others . the strategy is simply to retain statistics sufficient for the entire part that they may _ suspect _ to be common , which in this case means that both researchers will retain statistics sufficient for the @xmath178s @xmath179 in their entirety . mathematically , this corresponds to letting @xmath180 , where @xmath181 . it is then easy to verify that dsc holds , if we take @xmath182 , where @xmath183 . this is because when @xmath184 , both sides of ( [ eq : dsc ] ) are zero . when @xmath175 , we have ( adopting integration over @xmath185 functions ) @xmath186\,{\mathrm{d}}p_{\eta } ' \bigl ( { \eta } ' | { \theta}\bigr ) \\ & & \quad= \int_\eta \int_{\zeta _ 1 } \biggl [ \prod_{i=1}^2 \tilde p_{x_i}(x_i |\eta_i ) \delta_{\ { z=\zeta_i\ } } \biggr]\,{\mathrm{d}}p_{\eta } ( { \eta}| { \theta } ) \delta_{\{\zeta_1=\zeta_2\}}\,{\mathrm{d}}p_{z}(\zeta_1|\theta ) \\ & & \quad = \biggl [ \int_\eta\prod_{i=1}^2 \tilde p_{x_i}(x_i |\eta_i)\,{\mathrm{d}}p_{\eta } ( { \eta}| { \theta } ) \biggr ] \int_{\zeta_1 } \delta_{\{\zeta_1=z\}}\,{\mathrm{d}}p_{z}(\zeta_1|\theta ) \\ & & \quad = p_{x_{}}(x_1 , x_2|\theta)p_{z}(z| \theta)=p_{x}(x|\theta).\end{aligned}\ ] ] this technique of expanding @xmath102 to include shared parts of the @xmath4 allows the dsc and theorem [ thm : dsc ] to be applied to all models @xmath140 , not only those with with distinct @xmath128 s . however , this construction also restricts working models to those with deterministic relationships between parts of @xmath102 and each @xmath128 . the derivation above demonstrates both the broader applications of dsc as a theoretical condition and its restrictive nature as a practical guideline . retaining sufficient statistics for both @xmath187 and @xmath188 can create redundancy . if each preprocessor observes @xmath72 without noise , then only one of them actually needs to retain and report their observation of @xmath72 . however , if each observes @xmath72 with independent noise , then both of their observations are required to obtain a sufficient statistic for @xmath22 . the noise - free case also provides a straightforward counterexample to the necessity of dsc . assuming both preprocessors observe @xmath72 directly , as long as one of the copies of @xmath72 is retained via the use of the saturated @xmath185 density , the other copy can be modeled in any way and hence can be made to violate dsc without affecting their joint sufficiency for @xmath22 . regardless of the dependencies among the @xmath128 s , there is always a safe option open to the preprocessors for data reduction : retain @xmath123 sufficient for @xmath189 under @xmath190 . this will preserve sufficiency for @xmath9 under any scientific model @xmath116 : [ thm : safe ] if @xmath191 is correctly specified and satisfies ( [ e : obsm ] ) , then any collection of individual sufficient statistics @xmath192 with each @xmath123 sufficient for @xmath189 is jointly sufficient for @xmath73 in the sense of ( [ e : con1 ] ) for all @xmath116 . by the factorization theorem , we have @xmath193 for any @xmath24 . hence , by ( [ e : obsm ] ) , @xmath194 \int_x [ \prod_{i=1}^r p_t(t_i | x_i , { \xi}_i ) ] { p_x}(x @xmath195 is sufficient for @xmath22 , by the factorization theorem for sufficiency . theorem [ thm : safe ] provides a universal , safe strategy for sufficient preprocessing and a lower bound on the compression attainable from distributed sufficient preprocessing . as all minimal sufficient statistics for @xmath22 are functions of any sufficient statistic for @xmath196 , retaining minimal sufficient statistics for each @xmath189 results in less compression than any approach properly using knowledge of @xmath140 . however , the compression achieved relative to retaining @xmath3 itself may still be significant . minimal sufficient statistics for @xmath22 provide an upper bound on the attainable degree of compression by the same argument . achieving this compression generally requires that each preprocessor knows the true scientific model @xmath140 . between these bounds , the dsc ( [ eq : dsc ] ) shows a trade - off between the generality of preprocessing ( with respect to different scientific models ) and the compression achieved : the smaller the set of scientific models for which a given working model satisfies ( [ eq : dsc ] ) , the greater the potential compression from its sufficient statistics . more generally , stochastic dependence among @xmath128 s reduces compression and increases redundancy in distributed preprocessing . these costs are particularly acute when elements of @xmath9 control dependence among @xmath128 s , as seen in the following example where @xmath197 here @xmath198 is a column vector with @xmath199 @xmath200 s as its components , and @xmath201 is the usual kronecker product . if @xmath202 is known , then each researcher can reduce their observations @xmath163 to a scalar statistic @xmath203 and preserve sufficiency for @xmath204 . if @xmath202 is unknown , then each researcher must retain all of @xmath205 ( but not @xmath206 for @xmath207 ) in addition to these sums to ensure sufficiency for @xmath208 , because the minimal sufficient statistic for @xmath209 requires the computation of @xmath210 . thus , the cost of dependence here is @xmath89 additional pieces of information per preprocessor . dependence among the @xmath128 s forces the preprocessors to retain enough information to properly combine their individual contributions in the final analysis , downweighting redundant information . this is true even if they are interested only in efficient estimation of @xmath204 , leading to less reduction of their raw data and less compression from preprocessing than the independent case . from this investigation , we see that it is generally not enough for each researcher involved in preprocessing to reduce data based on even a correctly - specified model for their problem at hand . we instead need to look to other models that include each experimenter s data hierarchically , explicitly considering higher - level structure and relationships . however , significant reductions of the data are still possible despite these limitations . each @xmath123 need not be sufficient for each @xmath128 , nor must @xmath0 be sufficient for @xmath4 overall . this often implies that much less data need to be retained and shared than retaining sufficient statistics for each @xmath128 would demand . for instance , if a working model with @xmath211 satisfies the dsc for a given model @xmath116 and @xmath212 , then only means and covariance matrices of @xmath206 within each experiment @xmath24 need to be retained . the discussions above demonstrate the importance of involving downstream analysts in the design of preprocessing techniques . their knowledge of @xmath116 is extremely useful in determining what compression is appropriate , even if said knowledge is imperfect . constraining the scientific model to a broad class may be enough to guarantee effective preprocessing . for example , suppose we fix a working model and consider all scientific models that can be expressed as ( [ eq : dsc ] ) by varying the choices of @xmath117 . this yields a very broad class of hierarchical scientific models for downstream analysts to evaluate , while permitting effective distributed preprocessing based on the given working model.=1 practically , we see two paths to distributed preprocessing : coordination and caution . coordination refers to the downstream analyst evaluating and guiding the design of preprocessing as needed . such guidance can guarantee that preprocessed outputs will be as compact and useful as possible . however , it is not always feasible . it may be possible to specify preprocessing in detail in some industrial and purely computational settings . accomplishing the same in academic research or for any research conducted over time is an impractical goal . without such overall coordination , caution is needed . it is not generally possible to maintain sufficiency for @xmath9 without knowledge of the possible models @xmath116 unless the retained summaries are sufficient for @xmath4 itself . preprocessors should therefore proceed cautiously , carefully considering which scientific models they effectively exclude through their preprocessing choices . this is analogous to the oft - repeated guidance to include as many covariates and interactions as possible in imputation models ( @xcite , @xcite ) . having considered the lossless preprocessing , we now turn to more realistic but less clear - cut situations . we consider a less careful preprocessor and a sophisticated downstream analyst . the preprocessor selects an output @xmath0 , which may discard much information in @xmath3 but nevertheless preserves the identifiability of @xmath22 , and the downstream analyst knows enough to make the best of whatever output they are given . that is , the index set @xmath29 completely and accurately captures all relevant preprocessing methods @xmath213 . this does not completely capture all the practical constraints discussed in section [ sec : concepts ] . however , it is important to establish an upper bound on the performance of multiphase procedures before incorporating such issues . this upper bound is on the fisher information , and hence a lower bound on the asymptotic variances of estimators @xmath214 of @xmath9 . as we will see , nuisance parameters ( @xmath16 ) play a crucial role in these investigations . when using a lossy compression , an obvious question is how much information is lost compared to a lossless compression . this question has a standard asymptotic answer when the downstream analyst adopts an mle or bayes estimator , so long as nuisance parameters behave appropriately ( as will be discussed shortly ) . if the downstream analyst adopts some other procedures , such as an estimating equation , then there is no guarantee that the procedure based on @xmath3 is more efficient than the one based on @xmath0 . that is , one can actually obtain a more efficient estimator with less data when one is not using _ probabilistically principled _ methods , as discussed in detail in @xcite . therefore , as a first step in our theoretical investigations , we will focus on mles ; the results also apply to bayesian estimators under the usual regularity conditions to guarantee the asymptotic equivalence between mles and bayesian estimators . specifically , let @xmath215 and @xmath216 be the mles of @xmath217 based respectively on @xmath3 and @xmath0 under model ( [ e : model ] ) . we place standard regularity conditions for the joint likelihood of @xmath73 , assuming bounded third derivatives of the log - likelihood , common supports of the observation distributions with respect to @xmath73 , full rank for all information matrices at the true parameter value @xmath218 , and the existence of an open subset of the parameter space that contains @xmath218 . these conditions imply the first and second bartlett identities . however , the most crucial assumption here is a sufficient accumulation of information , indexed by an _ information size _ @xmath219 , to constrain the behavior of remainder terms in quadratic approximations of the relevant score functions . independent identically distributed observations and fixed - dimensional parameters would satisfy this requirement , in which case @xmath219 is simply the data size of @xmath3 , but weaker conditions can suffice ( for an overview , see @xcite ) . in general , this assumption requires that the dimension of both @xmath220 and @xmath77 are bounded as we accumulate more data , preventing the type of phenomenon revealed in @xcite . for multiphase inferences , cases where these dimensions are unbounded are common ( at least in theory ) and represent interesting settings where preprocessing can actually improve asymptotic efficiency , as we discuss shortly . to eliminate the nuisance parameter @xmath77 , we work with the observed fisher information matrices based on the profile likelihoods for @xmath9 , denoted by @xmath221 and @xmath222 respectively . let @xmath223 be the limit of @xmath224 , the so - called _ fraction of missing information _ ( see @xcite ) , as @xmath225 . the proof of the following result follows the standard asymptotic arguments for mles , with the small twist of applying them to profile likelihoods instead of full likelihoods . ( we can also invoke the more general arguments based on decomposing estimating equations , as given in @xcite . ) [ thm : missinfo ] under the conditions given above , we have asymptotically as @xmath225 , @xmath226^{-1 } \rightarrow f\ ] ] and @xmath227^{-1 } \rightarrow i - f.\ ] ] this establishes the central role of the fraction of missing information @xmath223 in determining the asymptotic efficiency of multiphase procedures under the usual asymptotic regime . as mentioned above , this is an ideal - case bound on the performance of multiphase procedures , and it is based on the usual squared - error loss ; both the asymptotic regime and amount of knowledge held by the downstream analyst are optimistic . we explore these issues below , focusing on ( 1 ) mutual knowledge and alternative definitions of efficiency , ( 2 ) the role of reparameterization , ( 3 ) asymptotic regimes and multiphase efficiency , and ( 4 ) the issue of robustness in multiphase inference . in practice , downstream analysts are unlikely to have complete knowledge of @xmath8 . therefore , even if they were given the entire @xmath3 , they would not be able to produce the optimal estimator @xmath228 , making the @xmath223 value given by theorem [ thm : missinfo ] an unrealistic yardstick . nevertheless , theorem [ thm : missinfo ] suggests a direction for a more realistic standard . the classical theory of estimation focuses on losses of the form @xmath229 , where @xmath230 denotes the truth . risk based on this type of loss , given by @xmath231 $ ] , is a raw measure of performance , using the truth as a baseline . an alternative is regret , the difference between the risk of a given estimator and an ideal estimator @xmath232 ; that is , @xmath233 . regret is popular in the learning theory community and forms the basis for oracle inequalities . it provides a more adaptive baseline for comparison than raw risk , but we can push further . consider evaluating loss with respect to an estimator rather than the truth . for mean - squared error , this yields @xmath234 .\ ] ] can this provide a better baseline , and what are its properties ? for mles , @xmath235 behaves the same ( asymptotically ) as additive regret because theorem [ thm : missinfo ] implies that , as @xmath225 under the classical asymptotic regime , @xmath236 \\[-8pt ] \nonumber & = & r\bigl({\hat{{\theta}}}(t ) , { \theta}_0\bigr)-r\bigl({\hat{{\theta}}}(y ) , { \theta}_0\bigr ) .\end{aligned}\ ] ] for inefficient estimators , ( [ eq : same ] ) does not hold in general because @xmath237 is no longer guaranteed to be asymptotically uncorrelated with @xmath238 . in such cases , this is precisely the reason @xmath81 can be more efficient than @xmath238 or , more generally , there exists a constant @xmath239 such that @xmath240 is ( asymptotically ) more efficient than @xmath238 . in the terminology of @xcite , the estimation procedure @xmath241 is not _ self - efficient _ if ( [ eq : same ] ) does not hold , viewing @xmath3 as the complete data @xmath242 and @xmath0 as the observed data @xmath243 . indeed , if @xmath244 , @xmath245 may actually be _ larger _ for a _ better _ @xmath81 because of the inappropriate baseline @xmath238 ; it is a measure of difference , not dominance , in such cases . hence , some care is needed in interpreting this measure . therefore , we can view ( [ eq : risk ] ) as a generalization of the usual notion of regret , or the relative regret if we divide it by @xmath246 . this generalization is appealing for the study of preprocessing : we are evaluating the estimator based on preprocessed data directly against what could be done with the complete raw data , sample by sample , and we no longer need to impose the restriction that the downstream analysts must carry out the most efficient estimation under a model that captures the actual preprocessing . this direction is closely related to the idea of strong efficiency from @xcite and @xcite , which generalizes the idea of asymptotic decorrelation beyond the simple ( but instructive ) setting covered here . such ideas from the theory of missing data provide a strong underpinning for the study of multiphase inference and preprocessing . theorem [ thm : missinfo ] also emphasizes the range of effects that preprocessing can have , even in ideal cases . consider the role that @xmath223 plays under different transformations of @xmath9 . although the eigenvalues of @xmath223 are invariant under one - to - one transformations of the parameters , submatrices of @xmath223 can change substantially . formally , if @xmath208 is transformed to @xmath247 , then the fraction of missing information for @xmath248 can be very different from that for @xmath204 . these changes mean that changes in parameterization can reallocate the fractions of missing information among resulting subparameters in unexpected and sometimes very unpleasant ways . this is true even for linear transformations ; a given preprocessing technique can preserve efficiency for @xmath204 and @xmath202 individually while performing poorly for @xmath249 . such issues have arisen in , for instance , the work of @xcite when attempting to characterize the behavior of multiple imputation estimators under uncongeniality . on a fundamental level , theorem [ thm : missinfo ] is a negative result for preprocessing , at least for mles . reducing the data from @xmath3 to @xmath0 can only hinder the downstream analyst . formally , this means that @xmath250 ( asymptotically ) in the sense that @xmath251 is positive semi - definite . as a result , @xmath238 will dominate @xmath81 in asymptotic variance for any preprocessing @xmath0 . thus , the only justification for preprocessing appears to be pragmatic ; if the downstream analyst could not make use of @xmath8 for efficient inference or such knowledge could not be effectively transmitted , preprocessing provides a feasible way to obtain the inferences of interest . however , this conclusion depends crucially on the assumed behavior of the nuisance parameter @xmath16 . the usual asymptotic regime is not realistic for many multiphase settings , particularly with regards to @xmath16 . in many problems of interest , @xmath252 does not tend to zero as @xmath219 increases , preventing sufficient accumulation of information on the nuisance parameter @xmath77 . a typical regime of this type would accumulate observations @xmath163 from individual experiments @xmath24 , each of which brings its own nuisance parameter @xmath253 . such a process could describe the accumulation of data from microarrays , for instance , with each experiment corresponding to a chip with its own observation parameters , or the growth of astronomical datasets with time - varying calibration . in such a regime , preprocessing can have much more dramatic effects on asymptotic efficiency . in the presence of nuisance parameters , inference based on @xmath0 can be more robust and even more efficient than inference based on @xmath3 . it is well - known that the mle can be inefficient and even inconsistent in regimes where @xmath254 ( going back to at least @xcite ) . bayesian methods provide no panacea either . marginalization over the nuisance parameter @xmath16 is appealing , but resulting inferences are typically sensitive to the prior on @xmath16 , even asymptotically . in many cases ( such as the canonical neyman scott problem ) , only a minimal set of priors provide even consistent bayes estimators . careful preprocessing can , however , enable principled inference in such regimes . such phenomena stand in stark contrast to the theory of multiple imputation . in that theory , complete data inferences are typically assumed to be valid . thus , under traditional missing data mechanisms , the observed data ( corresponding to @xmath0 ) can not provide better inferences than @xmath3 . this is not necessarily true in multiphase settings . if the downstream analyst is constrained to particular principles of inference ( e.g. , mle or bayes ) , then estimators based on @xmath0 can provide lower asymptotic variance than those based on @xmath3 . this occurs , in part , because the mechanisms generating @xmath3 and @xmath0 from @xmath4 are less restricted in the multiphase setting compared to the traditional missing - data framework . principled inferences based on @xmath4 would , in the multiphase setting , generally dominate those based on either @xmath3 or @xmath0 . however , such a relationship need not hold between @xmath3 and @xmath0 without restrictions on the behavior of @xmath16 . we emphasize that this does not contradict the general call in @xcite to follow the probabilistically - principled methods ( such as mle and bayes recipes ) to prevent violations of self - efficiency , precisely because the well - established principles of single - phase inference may need to be `` re - principled '' before they can be equally effective in the far more complicated multiphase setting . in the simplest case , if a @xmath0 can be found such that it is a pivot with respect to @xmath16 and remains dependent upon @xmath9 , then sensitivity to the behavior of @xmath77 can be eliminated by preprocessing . in such cases , an mle or bayes rule based on @xmath0 can dominate that based on @xmath3 even asymptotically . one such example would be providing @xmath255-statistics from each of a set of experiments to the downstream analyst . this clearly limits the range of feasible downstream inferences . with these @xmath255-statistics , detection of signals via multiple testing ( e.g. , @xcite ) would be straightforward , but efficient combination of information across experiments could be difficult . this is a ubiquitous trade - off of preprocessing : reductions that remove nuisance parameters and improve robustness necessarily reduce the amount of information available from the data . these trade - offs must be considered carefully when designing preprocessing techniques universal utility is unattainable without the original data . a more subtle case involves the selection of @xmath0 as a `` partial pivot '' . in some settings , there exists a decomposition of @xmath16 as @xmath256 such that @xmath257 for some fixed @xmath89 and all @xmath219 , and the distribution of @xmath0 is free of @xmath258 for all values of @xmath259 . many normalization techniques used in the microarray application of section [ sec : examples ] can be interpreted in this light . these methods attempt to reduce the unbounded set of experiment - specific nuisance parameters affecting @xmath0 to a bounded , manageable size . for example , suppose each processor @xmath24 observes @xmath260 , @xmath261 . the downstream analyst wants to estimate @xmath262 , considering @xmath263 and @xmath264 as nuisance parameters . in our previous notation , we have @xmath265 and @xmath266 . suppose each preprocessor reduces her data to @xmath267 , where @xmath268 is the ols estimator of @xmath269 based on @xmath270 . the distribution of each @xmath123 depends on @xmath264 but is free of @xmath269 . hence , @xmath271 is a partial pivot as defined above , with @xmath272 and @xmath273 . such pivoting techniques can allow @xmath81 to possess favorable properties even when @xmath238 is inconsistent or grossly inefficient . as mentioned before , this kind of careful preprocessing can dominate bayesian procedures in the presence of nuisance parameters when @xmath274 can grow with @xmath275 . in these regimes , informative priors on @xmath16 can affect inferences even asymptotically . however , reducing @xmath3 to @xmath0 so only the @xmath276-part of @xmath16 is relevant for @xmath0 s distribution allows information to accumulate on @xmath259 , making inferences far more robust to the preprocessor s beliefs about @xmath16 . these techniques share a common conceptual framework : invariance . invariance has a rich history in the bayesian literature , primarily as a motivation for the construction of noninformative or reference priors ( e.g. , @xcite , @xcite , @xcite , @xcite , @xcite ) . it is fundamental to the pivotal methods discussed above and arises in the theory of partial likelihood ( @xcite ) . we see invariance as a core principle of preprocessing , although its application is somewhat different from most bayesian settings . we are interested in finding functions of the data whose distributions are invariant to subsets of the parameter , not priors invariant to reparameterization . for instance , the rank statistics that form the basis for cox s proportional hazards regression in the absence of censoring ( @xcite ) can be obtained by requiring a statistic invariant to monotone transformations of time . indeed , cox s regression based on rank statistics can be viewed as an excellent example of eliminating an infinite dimensional nuisance parameter , i.e. , the baseline hazard , via preprocessing , which retains only the rank statistics . the relationship between invariance in preprocessing , modeling , and prior formulation is a rich direction for further investigation . an interesting practical question arises from this discussion of robustness : how realistic is it to assume efficient inference with preprocessed data ? this may seem unrealistic as preprocessing is frequently used to simplify problems so common methods can be applied . however , preprocessing can make many assumptions more appropriate . for example , aggregation can make normality assumptions more realistic , normalization can eliminate nuisance parameters , and discretization greatly reduces reliance on parametric distributional assumptions altogether . it may therefore be more appropriate to assume that efficient estimators are generally used with preprocessed data than with raw data . the results and examples explored here show that preprocessing is a complex topic in even large - sample settings . it appears formally futile ( but practically useful ) in standard asymptotic regimes . under other realistic asymptotic regimes , preprocessing emerges as a powerful tool for addressing nuisance parameters and improving the robustness of inferences . having established some of the formal motivation and trade - offs for preprocessing , we discuss further extensions of these ideas into more difficult settings in section [ sec : future ] . in some cases , effective preprocessing techniques are quite apparent . if @xmath10 forms an exponential family with parameter @xmath4 or @xmath196 , then we have a straightforward procedure : retain a minimal sufficient statistic . to be precise , we mean that one of the following factorizations holds for a sufficient statistic @xmath21 of bounded dimension : @xmath277 retaining this sufficient statistic will lead to a lossless compression , assuming that the first - phase model is correct . unfortunately , such nice cases are rare . even the bayesian approach offers little reprieve . integrating @xmath10 with respect to a prior @xmath278 typically removes the observation model from the exponential family consider , for instance , a normal model with unknown variance becoming a @xmath279 distribution . if @xmath280 is approximately quadratic as a function of @xmath4 , then retaining its mode and curvature would seem to provide much of the information available from the data to downstream analysts . however , such intuition can be treacherous . if a downstream analyst is combining inferences from a set of experiments , each of which yielded an approximately quadratic likelihood , the individual approximations may not be enough to provide efficient inferences . approximations that hold near the mode of each experiment s likelihood need not hold away from these modes including at the mode of the joint likelihood from all experiments . thus , remainder terms can accumulate in the combination of such approximations , degrading the final inference on @xmath9 . furthermore , the requirement that @xmath280 be approximately quadratic in @xmath4 is quite stringent . to justify such approximations , we must either appeal to asymptotic results from likelihood theory or confine our attention to a narrow class of observation models @xmath281 . unfortunately , asymptotic theory is often an inappropriate justification in multiphase settings , because @xmath4 grows in dimension with @xmath3 in many asymptotic regimes of interest , so there is no general reason to expect information to accumulate on @xmath4 . these issues are of particular concern as such quadratic approximations are a standard implicit justification for passing point estimates with standard errors onto downstream analysts . moving away from these cases , solutions become less apparent . no processing ( short of passing the entire likelihood function ) will preserve all information from the sample when sufficient statistics of bounded dimension do not exist . however , multiphase approaches can still possess favorable properties in such settings . we begin by considering a stubborn downstream analyst she has her method and will not consider anything else . for example , this analyst could be dead set on using linear discriminant analysis or anova . the preprocessor has only one way to affect her results : carefully designing a particular @xmath0 given to the downstream analyst . such a setting is extreme . we are saying that the downstream analyst will charge ahead with a given estimator regardless of her input with neither reflection nor judgment . we investigate this setting because it maximizes the preprocessor s burden in terms of her contribution to the final estimate s quality . formally , we consider a fixed second - stage estimator @xmath81 ; that is , the form of its input @xmath0 and the function producing @xmath214 are fixed , but the mechanism actually used to generate @xmath0 is not . @xmath0 could be , for example , a vector of fixed dimension . as we discuss below , admissible designs for the first - phase with a fixed second - phase method are given by a ( generalized ) bayes rule . this uses the known portion of the model @xmath10 to construct inputs for the second stage and assumes that any prior the preprocessor has on @xmath16 is equivalent to what a downstream analyst would have used in the preprocessor s position . formally , this describes all rules that are admissible among the class of procedures using a given second - stage method , following from previous complete class results in statistical decision theory ( e.g. , @xcite , @xcite ) . assume that the second - stage procedure @xmath81 is fixed as discussed above and we are operating under the model ( [ e : model ] ) . further assume that the preprocessor s prior on @xmath16 is the only such prior used in all bayes rule constructions . for @xmath282 , consider a smooth , strictly convex loss function @xmath39 . then , under appropriate regularity conditions ( e.g. , @xcite , @xcite ) , if @xmath81 is a smooth function of @xmath0 , then all admissible procedures for generating @xmath0 are bayes or generalized bayes rules with respect to the risk @xmath283 . the same holds when @xmath0 is restricted to a finite set . this guideline follows directly from conventional complete class results in decision theory . we omit technical details here , focusing instead on the guideline s implications . however , a sketch of its proof proceeds along the following lines . there are two ways to approach this argument : intermediate loss and geometry . the intermediate loss approach uses an intermediate loss function @xmath284 . this @xmath285 is the loss facing the preprocessor given a fixed downstream procedure @xmath81 . if @xmath285 is well - behaved , in the sense of satisfying standard conditions ( strict convexity , or a finite parameter space , and so on ) , then the proof is complete from previous results for real @xmath0 . similarly , if @xmath0 is restricted to a finite discrete set , then we face a classical multiple decision problem and can apply previous results to @xmath286 . these straightforward arguments cover a wide range of realistic cases , as @xcite has shown . otherwise , we must turn to a more intricate geometric argument . broadly , this construction uses a convex hull of risks generated by attainable rules . this guideline has direct bearing upon the development of inputs for machine learning algorithms , typically known as _ feature engineering_. given an algorithm that uses a fixed set of inputs , it implies that using a correctly - specified observation model to design these inputs is necessary to obtain admissible inferences . thus , it is conceptually similar to `` rao - blackwellization '' over part of a probability model . however , several major caveats apply to this result . first , on a practical level , deriving such bayes rules is quite difficult for most settings of interest . second , and more worryingly , this result s scope is actually quite limited . as we discussed in section [ sec : missinfo ] , even bayesian estimators can be inconsistent in realistic multiphase regimes . however , these estimators are still admissible , as they can not be dominated in risk for particular values of the nuisance parameters @xmath77 . admissibility therefore is a minimal requirement ; without it , the procedure can be improved uniformly , but with it , it can still behave badly in many ways . finally , there is the problem of robustness . an optimal input for one downstream estimator @xmath287 may be a terrible input for another estimator @xmath288 , even if @xmath289 and @xmath290 take the same form of inputs . such considerations are central to many real - world applications of preprocessing , as researchers aim to construct databases for a broad array of later analyses . however , this result does show that engineering inputs for downstream analyses using bayesian observation models can improve overall inferences . how to best go about this in practice is a rich area for further work . as befits first steps , we are left with a few loose ends and puzzles . starting with the dsc condition ( [ eq : dsc ] ) of section [ sec : sufficiency ] , we provide a simple counterexample to its necessity . suppose we have @xmath291 . let @xmath292 independent of each other . now , let @xmath293 , @xmath294 , @xmath295 , where @xmath296 , @xmath297 , @xmath298 is a vector of signs @xmath299 , or @xmath300 for @xmath165 , @xmath301 denotes the element - wise absolute value , and @xmath302 denotes the hadamard product . we fix @xmath303 . as our working model , we posit that @xmath304 independently . then , we clearly have @xmath305 as a sufficient statistic for both @xmath131 and @xmath9 . however , the dsc does not hold for this working model . we can not write the actual joint distribution of @xmath4 as a marginalization of @xmath306 with respect to some distribution over @xmath131 in such a way that @xmath307 is sufficient for @xmath131 . to enforce @xmath308 under the working model , any such model must use @xmath131 to share this information . for this example , we can obtain a stronger result : no factored working model @xmath306 exists such that ( 1 ) @xmath309 is sufficient for @xmath310 under @xmath311 and ( 2 ) the dsc holds . for contradiction , assume such a working model exists . under this working model , @xmath163 is conditionally independent of @xmath101 given @xmath309 , so we can write @xmath312 . as the dsc holds for this working model , we have @xmath313 \int_{\eta } \biggl [ \prod _ { i=1}^2 h_i\bigl(y_i^\top y_i ; g_i(\eta)\bigr ) \biggr ] p_{\eta}({\mathrm{d}}{\eta}| { \theta } ) .\ ] ] hence , we must have @xmath314 conditionally independent of @xmath315 given @xmath316 . however , this conditional independence does not hold under the true model . hence , the given working model can not both satisfy the dsc and have @xmath309 sufficient for each @xmath101 . the issue here is unparameterized dependence , as mentioned in section [ sec : sufficiency ] . the @xmath4 s have a dependence structure that is not captured by @xmath9 . thus , requiring that a working model preserves sufficiency for @xmath9 does not ensure that it has enough flexibility to capture the true distribution of @xmath3 . a weaker condition than the dsc ( [ eq : dsc ] ) that is necessary and sufficient to ensure that all sufficient statistics for @xmath131 are sufficient for @xmath9 may be possible . from sections [ sec : missinfo ] and [ sec : completeclass ] , we are left with puzzles rather than counterexamples . as mentioned previously , many optimality results are trivial without sufficient constraints . for instance , minimizing risk or maximizing fisher information naively yield uninteresting ( and impractical ) multiphase strategies : have the preprocessor compute optimal estimators , then pass them downstream . overly tight constraints bring their own issues . restricting downstream procedures to excessively narrow classes ( e.g. , point estimates with standard errors ) limits the applied utility of resulting theory and yields little insight on the overall landscape of multiphase inference . striking the correct balance with these constraints is a core challenge for the theory of multiphase inference and will require a combination of computational , engineering , and statistical insights . as we discussed in sections [ sec : concepts ] and [ sec : theory ] , we have a deep well of questions that motivate further research on multiphase inference . these range from the extremely applied ( e.g. , enhancing preprocessing in astrophysical systems ) to the deeply theoretical ( e.g. , bounding the performance of multiphase procedures in the presence of nuisance parameters and computational constraints ) . we outline a few directions for this research below . but , before we look forward , we take a moment to look back and place multiphase inference within the context of broader historical debates . such `` navel gazing '' helps us to understand the connections and implications of the theory of multiphase inference . on a historical note , the study of multiphase inference touches the long - running debate over the role of decision theory in statistics . one side of this debate , championed by wald and lehmann ( among others ) , has argued that decision theory lies at the core of statistical inference . risk - minimizing estimators and , more generally , optimal decision rules play a central role in their narrative . even subjectivists such as savage and de finetti have embraced the decision theoretic formulation to a large extent . other eminent statisticians have objected to such a focus on decisions . as noted by @xcite , fisher in particular vehemently rejected the decision theoretic formulation of statistical inference . one interpretation of fisher s objections is that he considered decision theory useful for eventual economic decision - making , but not for the growth of scientific knowledge . we believe that the study of multiphase inference brings a unifying perspective to this debate . fisher s distinction between intermediate processing and final decisions is fundamental to the problem of multiphase inference . however , we also view decision theory as a vital theoretical tool for the study of multiphase inference . passing only risk - minimizing point estimators to later analysts is clearly not a recipe for valid inference . the key is to consider the use of previously generated results explicitly in the final decision problem . in the study of multiphase inference , we do so by focusing on the separation of knowledge and objectives between agents . such separation between preprocessing and downstream inference maps nicely to fisher s distinction between building scientific knowledge and reaching actionable decisions . thus , we interpret fisher s line of objections to decision - theoretic statistics as , in part , a rejection of adopting a myopic single - phase perspective in multiphase settings . we certainly do not believe that our work will bring closure to such an intense historical debate . however , we do see multiphase inference as an important bridge between these competing schools of thought . we see a wide range of open questions in multiphase inference . can more systematic ways to leverage the potential of preprocessing be developed ? is it possible to create a mathematical `` warning system , '' alerting practitioners when their inferences from preprocessed data are subject to severe degradation and showing where additional forms of preprocessing are required ? and , can multiphase inference inform developments in distributed statistical computation and massive - data inference ( as outlined below in section [ sec : computation ] ) ? all of these problems call for a shared collection of statistical principles , theory , and methods . below , we outline a few directions for the development of these tools for multiphase inference . the mechanics of passing information between phases constitute a major direction for further research . one approach leverages the fact that the likelihood function itself is always a minimal sufficient statistic . thus , a set of ( computationally ) efficient approximations to the likelihood function @xmath317 for @xmath318 could provide the foundation for a wide range of multiphase methods . many probabilistic inference techniques for the downstream model ( e.g. , mcmc samplers ) would be quite straightforward to use given such an approximation . the study of such multiphase approximations also offers great dividends for distributed statistical computation , as discussed below . we believe these approximations are promising direction for general - purpose preprocessing . however , there are stumbling blocks . first , nuisance parameters remain an issue . we want to harness and understand the robustness benefits offered by preprocessing , but likelihood techniques themselves offer little guidance in this direction . even the work of @xcite on partial likelihood focuses on the details of estimation once the likelihood has been partitioned . we would like to identify the set of formal principles underlying techniques such as partial pivoting ( to mute the effect of infinite - dimensional nuisance parameters ) , building a more rigorous understanding of the role of preprocessing in providing robust inferences . as discussed in section [ sec : missinfo ] , invariance relationships may be a useful focus for such investigations , guiding both bayesian and algorithmic developments . second , we must consider the burden placed on downstream analysts by our choice of approximation . probabilistic , model - based techniques can integrate such information with little additional development . however , it would be difficult for a downstream analyst accustomed to , say , standard regression methods to make use of a complex emulator for the likelihood function . the burden may be substantial for even sophisticated analysts . for instance , it could require a significant amount of effort and computational sophistication to obtain estimates of @xmath4 from such an approximation , and estimates of @xmath4 are often of interest to downstream analysts in addition to estimates of @xmath9 . with these trade - offs in mind and through the formal analysis of widely - applicable multiphase techniques , we can begin to establish bounds on the error properties of such techniques in a broad range of problems under realistic constraints ( in both technical and human terms ) . more general constraints , for instance , can take the form of upper bounds on the regret attainable with a fixed amount of information passed from preprocessor to downstream analyst for fixed classes of scientific models . extensions to nonparametric downstream methods would have both practical and theoretical implications . in cases where the observation model is well - specified but the scientific model is less clearly defined , multiphase techniques can provide a useful alternative to computationally - expensive semi - parametric techniques . fusing principled preprocessing with flexible downstream inference may provide an interesting way to incorporate model - based subject - matter knowledge while effectively managing the bias - variance trade - off . the directions discussed above share a conceptual , if not technical , history with the development of congeniality ( @xcite ) . both the study of congeniality in mi and our study of multiphase inference seek to bound and measure the amount of degradation in inferences that can occur when agents attempt ( imperfectly ) to combine information . despite these similarities , the treatment of nuisance parameters are rather different . nuisance parameters lie at the very heart of multiphase inference , defining many of its core issues and techniques . for mi , the typical approaches have been to integrate them out in a bayesian analysis ( e.g. , @xcite ) or assume that the final analyst will handle them ( e.g. , @xcite ) . recent work by @xcite has shed new light on the role of nuisance parameters in mi , but the results are largely negative , demonstrating that nuisance parameters are often a stumbling block for practical mi inference . understanding the role of preprocessing in addressing nuisance parameters , providing robust analyses , and effectively distributing statistical inference represent further challenges beyond those pursued with mi . therefore , much remains to be done in the study of multiphase inference , both theoretical and methodological . we also see multiphase inference as a source for computational techniques , drawing inspiration from the history of mi . mi was initially developed as a strategy for handling missing data in public data releases . however , because mi separates the task of dealing with incomplete data from the task of making inferences , its use spread . it has frequently been used as a practical tool for dealing with missing - data problems where the joint inference of missing data and model parameters would impose excessive modeling or computational burdens . that is , increasingly the mi inference is carried out from imputation through analysis by a single analyst or research group . this is feasible as a computational strategy only because the error properties and conditions necessary for the validity of mi are relatively well - understood ( e.g. , @xcite , @xcite).=1 multiphase methods can similarly guide the development of efficient , statistically - valid computational strategies . once we have a theory showing the trade - offs and pitfalls of multiphase methods , we will be equipped to develop them into general computational techniques . in particular , our experience suggests that models with a high degree of conditional independence ( e.g. , exchangeable distributions for @xmath4 ) can often provide useful inputs for multiphase inferences , even when the true overall model has a greater degree of stochastic structure . the conditional independence structure of such models allows for highly parallel computation with first - phase procedures , providing huge computational gains on modern distributed systems compared to methods based on the joint model.=1 for example , in @xcite , a factored model was used to preprocess a massive collection of irregularly - sampled astronomical time series . the model was sophisticated enough to account for complex observation noise , yet its independence structure allowed for efficient parallelization of the necessary computation . its output was then combined and used for population - level analyses . just as markov chain monte - carlo ( mcmc ) has produced a windfall of tools for approximate high - dimensional integration ( see @xcite for many examples ) , we believe that this type of principled preprocessing , with further theoretical underpinnings , has the potential to become a core tool for the statistical analysis of massive datasets.=1 we would like to acknowledge support from the arthur p. dempster award and partial financial support from the nsf . we would also like to thank arthur p. dempster and stephen blyth for their generous feedback . this work developed from the inaugural winning submission for said award . we also thank david van dyk , brandon kelly , nathan stein , alex damour , and edo airoldi for valuable discussions and feedback , and steven finch for proofreading . finally , we would like to thank our reviewers for their thorough and thoughtful comments , which have significantly enhanced this paper.=1

preprocessing forms an oft - neglected foundation for a wide range of statistical and scientific analyses . however , it is rife with subtleties and pitfalls . decisions made in preprocessing constrain all later analyses and are typically irreversible . hence , data analysis becomes a collaborative endeavor by all parties involved in data collection , preprocessing and curation , and downstream inference . even if each party has done its best given the information and resources available to them , the final result may still fall short of the best possible in the traditional single - phase inference framework . this is particularly relevant as we enter the era of `` big data '' . the technologies driving this data explosion are subject to complex new forms of measurement error . simultaneously , we are accumulating increasingly massive databases of scientific analyses . as a result , preprocessing has become more vital ( and potentially more dangerous ) than ever before . we propose a theoretical framework for the analysis of preprocessing under the banner of multiphase inference . we provide some initial theoretical foundations for this area , including distributed preprocessing , building upon previous work in multiple imputation . we motivate this foundation with two problems from biology and astrophysics , illustrating multiphase pitfalls and potential solutions . these examples also emphasize the motivations behind multiphase analyses both practical and theoretical . we demonstrate that multiphase inferences can , in some cases , even surpass standard single - phase estimators in efficiency and robustness . our work suggests several rich paths for further research into the statistical principles underlying preprocessing . to tackle our increasingly complex and massive data , we must ensure that our inferences are built upon solid inputs and sound principles . principled investigation of preprocessing is thus a vital direction for statistical research .

electronic structure calculations have played an important role in understanding the properties of a wide range of materials systems @xcite . in particular , the kohn - sham formalism of density functional theory @xcite has been the workhorse of ground - state electronic structure calculations . however , the kohn - sham approach requires the computation of single - electron wavefunctions to compute the kinetic energy of non - interacting electrons , whose computational complexity typically scales as @xmath1 for an @xmath2-electron system , thus , limiting standard calculations to materials systems containing few hundreds of atoms . while there has been progress in developing close to linear - scaling algorithms for the kohn - sham approach @xcite , these are still limited to a few thousands of atoms , especially for metallic systems @xcite . the orbital - free approach to dft @xcite , on the other hand , models the kinetic energy of non - interacting electrons as an explicit functional of the electron density , thus circumventing the computationally intensive step of computing the single - electron wavefunctions . further , the computational complexity of orbital - free dft scales linearly with the system size as the ground - state dft problem reduces to a minimization problem in a single field the electron density . the past two decades has seen considerable progress in the development of accurate models for orbital - free kinetic energy functionals @xcite , and , in particular , for systems whose electronic - structure is close to a free electron gas ( for e.g. al , mg ) . also , orbital - free dft calculations are being increasingly used in the simulations of warm dense matter where the electronic structure is close to that of a free electron gas at very high temperatures @xcite . as the reduced computational complexity of orbital - free dft enables consideration of larger computational domains , recent studies have also focused on studying extended defects in al and mg , and have provided important insights into the energetics of these defects @xcite . the widely used numerical implementation of orbital - free dft is based on a fourier space formalism using a plane - wave discretization @xcite . a fourier space formulation provides an efficient computation of the extended interactions arising in orbital - free dft electrostatics and kinetic energy functionals through fourier transforms . further , the plane - wave basis is a complete basis and provides variational convergence in ground - state energy with exponential convergence rates . however , the fourier space formulations are restricted to periodic geometries and boundary conditions that are suitable for perfect bulk materials , but not for materials systems containing extended defects . also , the extended spatial nature of the plane - wave basis affects the parallel scalability of the numerical implementation and is also not suitable for multi - scale methods that rely on coarse - graining . in order to address the aforementioned limitations of fourier space techniques , recent efforts have focussed on developing real - space formulations for orbital - free dft and numerical implementations based on finite - element @xcite and finite difference discretizations @xcite . in the present work , we build on these prior efforts to develop an efficient real - space formulation of orbital - free dft employing the widely used non - local wang - govind - carter ( wgc ) @xcite kinetic energy functional . as in prior efforts @xcite , we reformulate the extended interactions in electrostatics and the non - local terms in the wgc kinetic energy functionals as local variational problems in auxiliary potential fields . however , the proposed reformulation of electrostatic interactions is notably different from previous works , and enables the evaluation of variational configurational forces corresponding to both internal atomic relaxations as well as external cell relaxation under a single framework . further , the proposed formulation naturally extends to all - electron orbital - free dft calculations of warm dense matter @xcite . in the proposed real - space formulation , the ground - state orbital - free dft problem is reformulated as an equivalent saddle point problem of a local functional in electron density , electrostatic potential and the auxiliary potential fields ( kernel potentials ) accounting for the extended interactions in the kinetic energy functional . we employ a higher - order finite - element basis to discretize the formulation , and demonstrate the optimal numerical convergence of both the ground - state energy and configurational forces with respect to the discretization . further , we propose an efficient numerical approach to compute the saddle point problem in electron density , electrostatic potential and kernel potentials by expressing the saddle point problem as a fixed point iteration problem , and using a self - consistent field approach to solve the fixed point iteration problem . we subsequently investigate the accuracy and transferability of the proposed real - space formulation of orbital - free dft for al and mg materials systems . to this end , we compute the bulk properties of al , mg and al - mg intermetallics , and compare it with kohn - sham dft . as orbital - free dft only admits local pseudopotentials , the kohn - sham dft calculations are conducted using both local and non - local psedupotentials . our studies indicates that the bulk properties computed using orbital - free dft for al , mg and al - mg intermetallics are in good agreement with kohn - sham dft . we further investigate the accuracy of orbital - free dft by computing the interatomic forces in al and mg , which are also in good agreement with kohn - sham dft calculations . our studies demonstrate that orbital - free dft is accurate and transferable across a wide range of properties for al , mg and al - mg intermetallics , and can be used to study properties of these materials systems that require computational domains that are not accessible using kohn - sham dft . for instance , in the present study we computed the formation energy of @xmath3 al - mg alloy containing @xmath4 atoms in a unit cell employing the proposed real - space formulation of orbital - free dft , but the same system was found to be prohibitively expensive using kohn - sham dft . we finally investigate the cell - size effects in the electronic structure of point defects , in particular a mono - vacancy in al . prior studies using fourier - based formulations of orbital - free dft have suggested that the formation energy of a mono - vacancy in al is well converged by 108 - 256 atom cell - sizes @xcite . however , coarse - grained real - space calculations have suggested that much larger cell - sizes of the order of 1,000 atoms are required for convergence of vacancy formation energies @xcite , which was also supported by asymptotic estimates @xcite . in order to understand the underpinnings of this discrepancy , we use the finite - element discretized real - space formulation of orbital - free dft and compute the vacancy formation energy using two boundary conditions : ( i ) periodic boundary conditions , equivalent to fourier - space based formulations ; ( ii ) bulk dirichlet boundary conditions , where the perturbations in the electronic structure arising due to the vacancy vanishes on the boundary of the computational domain . our study suggests that while the vacancy formation energy is well converged by 108 atom cell - size using periodic boundary conditions , the electronic fields are not well - converged by this cell - size . on the other hand the bulk dirichlet boundary conditions show well converged formation energy as well as electronic fields by cell sizes of @xmath51,000 atoms , which is consistent with prior real - space calculations . this study reveals that while periodic boundary conditions show a superior convergence in formation energies due to the variational nature of the formalism , the true cell - size effects which also measure convergence of electronic fields are provided by the bulk dirichlet boundary conditions . we note that the proposed real - space formulation with finite - element discretization are crucial to employing bulk dirichlet boundary conditions , which enable the study of isolated defects in bulk . the remainder of the paper is organized as follows . section ii provides a description of the orbital - free dft problem . section iii presents the proposed real - space formulation of the orbital - free dft problem , the configurational forces associated with structural relaxations , and the finite - element discretization of the formulation . section iv discusses the numerical implementation of the formulation and presents an efficient numerical approach for the solution of the saddle point real - space variational problem . section v presents the numerical convergence results of the finite - element discretization of the real - space formulation , the accuracy and transferability of the real - space orbital - free dft formalism for al - mg materials system , and the study of the role of boundary conditions on the cell - size effects in electronic structure calculations of point defects . we finally conclude with a summary and outlook in section vi . the ground - state energy of a charge neutral materials system containing @xmath6 nuclei and @xmath2 valence electrons in density functional theory is given by @xcite @xmath7 where @xmath8 denotes the electron - density and @xmath9 denotes the vector containing the positions of @xmath6 nuclei . in the above , @xmath10 denotes the kinetic energy of non - interacting electrons , @xmath11 is the exchange - correlation energy , @xmath12 is the hartree energy or classical electrostatic interaction energy between electrons , @xmath13 is the classical electrostatic interaction energy between electrons and nuclei , and @xmath14 denotes the electrostatic repulsion energy between nuclei . we now discuss the various contributions to the ground - state energy , beginning with the exchange - correlation energy . the exchange - correlation energy , denoted by @xmath11 , incorporates all the quantum - mechanical interactions in the ground - state energy of a materials system . while the existence of a universal exchange - correlation energy as a functional of electron - density has been established by hohenberg , kohn and sham @xcite , its exact functional form has been elusive to date , and various models have been proposed over the past decades . for solid - state calculations , the local density approximation ( lda ) @xcite and the generalized gradient approximation @xcite have been widely adopted across a range of materials systems . in particular , the lda exchange - correlation energy , which is adopted in the present work , has the following functional form : @xmath15 where @xmath16 , and @xmath17 @xmath18 and @xmath19 . in the present work , we use the ceperley and alder constants @xcite in equation . the last three terms in equation represent electrostatic interactions between electrons and nuclei . the hartree energy , or the electrostatic interaction energy between electrons , is given by @xmath20 the interaction energy between electrons and nuclei , in the case of local pseudopotentials that are adopted in the present work , is given by @xmath21 where @xmath22 denotes the pseudopotential corresponding to the @xmath23 nucleus , which , beyond a core radius is the coulomb potential corresponding to the effective nuclear charge on the @xmath23 nucleus . the nuclear repulsive energy is given by @xmath24 where @xmath25 denotes the effective nuclear charge on the @xmath26 nucleus . the above expression assumes that the core radius of the pseudopotential is smaller than internuclear distances , which is often the case in most solid - state materials systems . we note that in a non - periodic setting , representing a finite atomic system , all the integrals in equations - are over @xmath27 and the summations in equations - include all the atoms . in the case of an infinite periodic crystal , all the integrals over @xmath28 in equations - are over the unit cell whereas the integrals over @xmath29 are over @xmath27 . similarly , in equations - , the summation over @xmath30 is on the atoms in the unit cell , and the summation over @xmath31 extends over all lattice sites . henceforth , we will adopt these notions for the domain of integration and summation . the remainder of the contribution to the ground - state energy is the kinetic energy of non - interacting electrons , denoted by @xmath10 , which is computed exactly in the kohn - sham formalism by computing the single - electron wavefunctions ( eigenfunctions ) in the mean - field @xcite . the conventional solution of the kohn - sham eigenvalue problem , which entails the computation of the lowest @xmath2 eigenfunctions and eigenvalues of the kohn - sham hamiltonian , scales as @xmath32 that becomes prohibitively expensive for materials systems containing a few thousand atoms . while efforts have been focused towards reducing the computational complexity of the kohn - sham eigenvalue problem @xcite , this remains a significant challenge especially in the case of metallic systems . in order to avoid the computational complexity of solving for the wavefunctions to compute @xmath10 , the orbital - free approach to dft models the kinetic energy of non - interacting electrons as an explicit functional of electron density @xcite . these models are based on theoretically known properties of @xmath33 for a uniform electron gas , perturbations of uniform electron gas , and the linear response of uniform electron gas @xcite . as the orbital - free models for the kinetic energy functional are based on properties of uniform electron gas , their validity is often limited to materials systems whose electronic structure is close to a free electron gas , in particular , the alkali and alkali earth metals . further , as the orbital - free approach describes the ground - state energy as an explicit functional of electron - density , it limits the pseudopotentials calculations to local pseudopotentials . while these restrictions constrain the applicability of the orbital - free approach , numerical investigations @xcite indicate that recently developed orbital - free kinetic energy functionals and local pseudopotentials can provide good accuracy for al and mg , which comprise of technologically important materials systems . further , there are ongoing efforts in developing orbital - free kinetic energy models for covalently bonded systems and transition metals @xcite . in the present work , we restrict our focus to the wang - goving - carter ( wgc ) density - dependent orbital - free kinetic energy functional @xcite , which is a widely used kinetic energy functional for ground - state calculations of materials systems with an electronic structure close to a free electron gas . in particular , the functional form of the wgc orbital - free kinetic energy functional is given by @xmath34 where @xmath35 in equation , the first term denotes the thomas - fermi energy with @xmath36 , and the second term denotes the von - weizs@xmath37cker correction @xcite . the last term denotes the density dependent kernel energy , @xmath38 , where the kernel @xmath39 is chosen such that the linear response of a uniform electron gas is given by the lindhard response @xcite . in the wgc functional @xcite , the parameters are chosen to be @xmath40 and @xmath41 . for materials systems whose electronic structure is close to a free - electron gas , the taylor expansion of the density dependent kernel about a reference electron density ( @xmath42 ) , often considered to be the average electron density of the bulk crystal , is employed and is given by @xmath43 in the above equation , @xmath44 and the density independent kernels resulting from the taylor expansion are given by @xmath45 numerical investigations have suggested that the taylor expansion to second order provides a good approximation of the density dependent kernel for materials systems with electronic structure close to a free electron gas @xcite . in particular , in the second order taylor expansion , the contribution from @xmath46 has been found to dominate contributions from @xmath47 . thus , in practical implementations , often , only contributions from @xmath46 in the second order terms are retained for computational efficiency . in this section , we present the local variational real - space reformulation of orbital - free dft , the configurational forces associated with internal ionic relaxations and cell relaxation , and the finite - element discretization of the formulation . [ sec : rs - formulation ] we recall that the various components of the ground - state energy of a materials system ( cf . section [ sec : ofdft ] ) are local in real - space , except the electrostatic interaction energy and the kernel energy component of the wgc orbital - free kinetic energy functional that are extended in real - space . conventionally , these extended interactions are computed in fourier space to take advantage of the efficient evaluation of convolution integrals using fourier transforms . for this reason , fourier space formulations have been the most popular and widely used in orbital - free dft calculations @xcite . however , fourier space formulations employing the plane - wave basis result in some significant limitations . foremost of these is the severe restriction of periodic geometries and boundary conditions . while this is not a limitation in the study of bulk properties of materials , this is a significant limitation in the study of defects in materials . for instance , the geometry of a single isolated dislocation in bulk is not compatible with periodic geometries , and , thus , prior electronic structure studies have mostly been limited to artificial dipole and quadrapole arrangements of dislocations . further , numerical implementations of fourier - space formulations also suffer from limited scalability on parallel computing platforms . moreover , the plane - wave discretization employed in a fourier space formulation provides a uniform spatial resolution , which is not suitable for the development of coarse - graining techniques such as the quasi - continuum method @xcite that rely on an adaptive spatial resolution of the basis . we now propose a real - space formulation that is devoid of the aforementioned limitations of a fourier space formulation . the proposed approach , in spirit , follows along similar lines as recent efforts @xcite , but the proposed formulation differs importantly in the way the extended electrostatic interactions are treated . in particular , the proposed formulation provides a unified framework to compute the configurational forces associated with both internal ionic and cell relaxations discussed in [ sec : configurationalforces ] . we begin by considering the electrostatic interactions that are extended in the real - space . we denote by @xmath48 a regularized dirac distribution located at @xmath49 , and the @xmath26 nuclear charge is given by the charge distribution @xmath50 . defining @xmath51 and @xmath52 , the repulsive energy @xmath14 can subsequently be reformulated as @xmath53 where @xmath54 denotes the self energy of the nuclear charges and is given by @xmath55 we denote the electrostatic potential corresponding to the @xmath26 nuclear charge ( @xmath56 ) as @xmath57 , and is given by @xmath58 the self energy , thus , can be expressed as @xmath59 noting that the kernel corresponding to the extended electrostatic interactions in equations - is the green s function of the laplace operator , the electrostatic potential and the electrostatic energy can be computed by taking recourse to the solution of a poisson equation , or , equivalently , the following local variational problem : [ eq : selfenergylocal ] @xmath60 @xmath61 in the above , @xmath62 denotes the hilbert space of functions such that the functions and their first - order derivatives are square integrable on @xmath27 . we next consider the electrostatic interaction energy corresponding to both electron and nuclear charge distribution . we denote this by @xmath63 , which is given by @xmath64 we denote the electrostatic potential corresponding to the total charge distribution ( electron and nuclear charge distribution ) as @xmath65 , which is given by @xmath66 the electrostatic interaction energy of the total charge distribution , in terms of @xmath65 , is given by @xmath67 as before , the electrostatic interaction energy as well as the potential of the total charge distribution can be reformulated as the following local variational problem : [ eq : totelecenergylocal ] @xmath68 @xmath69 in the above , @xmath70 is a suitable function space corresponding to the boundary conditions of the problem . in particular , for non - periodic problems such as isolated cluster of atoms @xmath71 . for periodic problems , @xmath72 where @xmath73 denotes the unit cell and @xmath74 denotes the space of periodic functions on @xmath73 such that the functions and their first - order derivatives are square integrable . the electrostatic interaction energy in dft , comprising of @xmath75 , @xmath13 and @xmath14 ( cf . equations - ) , can be rewritten in terms of @xmath63 and @xmath54 as @xmath76 for the sake of convenience of representation , we will denote by @xmath77 the vector containing the electrostatic potentials corresponding to all nuclear charges in the simulation domain . using the local reformulation of @xmath63 and @xmath54 ( cf . equations and ) , the electrostatic interaction energy in dft can now be expressed as the following local variational problem : [ eq : elecrsreformulation ] @xmath78 @xmath79 in the above , the minimization over @xmath80 represents a simultaneous minimization over all electrostatic potentials corresponding to @xmath81 . we note that , while the above reformulation of electrostatic interactions has been developed for pseudopotential calculations , this can also be extended to all - electron calculations in a straightforward manner by using @xmath82 and @xmath25 to be the total nuclear charge in the above expressions . thus , this local reformulation provides a unified framework for both pseudopotential as well as all - electron dft calculations . we now consider the local reformulation of the extended interactions in the kernel energy component of the wgc orbital - free kinetic energy functional ( cf . ) . here we adopt the recently developed local real - space reformulation of the kernel energy @xcite , and recall the key ideas and local reformulation for the sake of completeness . we present the local reformulation of @xmath83 and the local reformulations for other kernels ( @xmath84 , @xmath47 , @xmath46 ) follows along similar lines . consider the kernel energy corresponding to @xmath83 given by @xmath85 we define potentials @xmath86 and @xmath87 given by @xmath88 taking the fourier transform of the above expressions we obtain @xmath89 following the ideas developed by choly & kaxiras @xcite , @xmath90 can be approximated to very good accuracy by using a sum of partial fractions of the following form @xmath91 where @xmath92 , @xmath93 , @xmath94 are constants , possibly complex , that are determined using a best fit approximation . using this approximation and taking the inverse fourier transform of equation , the potentials in equation ( [ eq : kerpotential_v ] ) reduce to @xmath95\,,\notag \\ v^0_{\beta}({\boldsymbol{\textbf{x}}})=\sum\limits_{j=1}^m\,[\omega^0_{\beta_j}({\boldsymbol{\textbf{x}}})+a_j \rho^{\beta}({\boldsymbol{\textbf{x}}})]\,.\end{aligned}\ ] ] where @xmath96 and @xmath97 for @xmath94 are given by the following helmholtz equations : @xmath98 we refer to these auxiliary potentials , @xmath99 and @xmath100 introduced in the local reformulation of the kernel energy as _ kernel potentials_. expressing the helmholtz equations in a variational form , we reformulate @xmath101 in ( [ eq : ker0_energy ] ) as the following local variational problem in kernel potentials : [ eq : kernel_variational ] @xmath102 @xmath103d{\boldsymbol{\textbf{x}}}\big\}\ , . \end{split}\ ] ] the variational problem in equation represents a simultaneous saddle point problem on kernel potentials @xmath104 and @xmath105 for @xmath106 . following a similar procedure , we construct the local variational reformulations for the kernel energies @xmath107 , @xmath108 and @xmath109 corresponding to kernels @xmath110 , @xmath47 and @xmath46 , respectively . we denote by @xmath111 , @xmath112 and @xmath113 the lagrangians with respective kernel potentials corresponding to kernel energies of @xmath110 , @xmath47 and @xmath46 , respectively . we refer to the supplemental material for the numerical details of the approximations for each of the kernels used in the present work . finally , using the local variational reformulations of the extended electrostatic and kernel energies , the problem of computing the ground - state energy for a given positions of atoms is given by the following local variational problem in electron - density , electrostatic potentials , and kernel potentials : @xmath114 in the above , @xmath115 denotes the index corresponding to a kernel , and @xmath116 and @xmath70 are suitable function spaces corresponding to the boundary conditions of the problem . in particular , for periodic problems , @xmath72 and @xmath117 . it is convenient to use the substitution @xmath118 , and enforce the integral constraint in @xmath116 using a lagrange multiplier . also , for the sake of notational simplicity , we will denote by @xmath119 and @xmath120 the array of kernel potentials @xmath121 and @xmath122 , respectively . subsequently , the variational problem in equation can be expressed as @xmath123 [ sec : configurationalforces ] we now turn our attention to the configurational forces corresponding to geometry optimization . to this end , we employ the approach of inner variations , where we evaluate the generalized forces corresponding to perturbations of underlying space , which provides a unified expression for the generalized force corresponding to the geometry of the simulation cell internal atomic positions , as well as , the external cell domain . we consider infinitesimal perturbations of the underlying space @xmath124 corresponding to a generator @xmath125 given by @xmath126 such that @xmath127 . we constrain the generator @xmath128 such that it only admits rigid body deformations in the compact support of the regularized nuclear charge distribution @xmath129 in order to preserve the integral constraint @xmath130 . let @xmath28 denote a point in @xmath73 , whose image in @xmath131 is @xmath132 . the ground - state energy on @xmath133 is given by @xmath134 where @xmath135 , @xmath136 , @xmath137 and @xmath138 are solutions of the saddle point variational problem given by equation evaluated over the function space @xmath139 . the subscript @xmath140 on @xmath141 is used to denote that the variational problem is solved on @xmath131 . for the sake of convenience , we will represent the integrand of the lagrangian @xmath141 in equation by @xmath142 and @xmath143 , where @xmath144 denotes the integrand whose integrals are over @xmath73 and @xmath145 denotes the integrand whose integrals are over @xmath27 . the ground - state energy on @xmath133 in terms of @xmath144 and @xmath145 can be expressed as @xmath146 transforming the above integral to domain @xmath73 , we obtain @xmath147 we now evaluate the configurational force given by the gteaux derivative of @xmath148 : @xmath149 in the above , we denote by ` @xmath150 ' the outer product between two vector , by ` @xmath151 ' the dot product between two vectors and by ` @xmath152 ' the dot product between two tensors . we note that in the above expression there are no terms involving the explicit derivatives of @xmath144 and @xmath145 with respect to @xmath153 as @xmath154 , which follows from the restriction that @xmath155 corresponds to rigid body deformations in the compact support of @xmath129 . we further note that terms arising from the inner variations of @xmath148 with respect to @xmath135 , @xmath136 , @xmath137 , @xmath138 and @xmath156 vanish as @xmath157 @xmath158 , @xmath159 , @xmath160 and @xmath161 are the solutions of the saddle point variational problem corresponding to @xmath162 . we now note the following identities @xmath163 @xmath164 using these identities in equation , and rearranging terms , we arrive at @xmath165 where @xmath166 and @xmath167 denote eshelby tensors corresponding to @xmath144 and @xmath145 , respectively . the expressions for the eshelby tensors @xmath166 and @xmath168 explicitly in terms of @xmath169 , @xmath170 , @xmath119 , @xmath120 , @xmath171 and @xmath172 are given by @xmath173 in the above , for the sake of brevity , we represented by @xmath174 the integrand corresponding to @xmath175 . we also note that the terms @xmath176 and @xmath177 do not appear in the expressions for @xmath166 and @xmath178 , respectively , as @xmath179 on the compact support of @xmath129 owing to the restriction that @xmath128 corresponds to rigid body deformations in these regions . it may appear that evaluation of the second term in equation is not tractable as it involves an integral over @xmath27 . to this end , we split this integral on a bounded domain @xmath180 containing the compact support of @xmath48 , and its complement . the integral on @xmath181 can be computed as a surface integral . thus , @xmath182 where @xmath183 denotes the outward normal to the surface @xmath184 . the last equality follows from the fact that @xmath185 on @xmath181 . the configurational force in equation provides the generalized variational force with respect to both the internal positions of atoms as well as the external cell domain . in order to compute the force on any given atom , we restrict the compact support of @xmath128 to only include the atom of interest . in order to compute the stresses associated with cell relaxation ( keeping the fractional coordinates of atoms fixed ) , we restrict @xmath128 to affine deformations . thus , this provides a unified expression for geometry optimization corresponding to both internal ionic relaxations as well as cell relaxation . we further note that , while we derived the configurational force for the case of pseudopotential calculations , the derived expression is equally applicable for all - electron calculations by using @xmath186 . [ sec : fe - discretization ] among numerical discretization techniques , the plane - wave discretization has been the most popular and widely used in orbital - free dft @xcite as it naturally lends itself to the evaluation of the extended interactions in electrostatic energy and kernel kinetic energy functionals using fourier transforms . further , the plane wave basis offers systematic convergence with exponential convergence in the number of basis functions . however , as noted previously , the plane - wave basis also suffers from notable drawbacks . importantly , plane - wave discretization is restricted to periodic geometries and boundary conditions which introduces a significant limitation , especially in the study of defects in bulk materials @xcite . further , the plane - wave basis has a uniform spatial resolution , and thus is not amenable to adaptive coarse - graining . moreover , the use of plane - wave discretization involves the numerical evaluation of fourier transforms whose scalability is limited on parallel computing platforms . in order to circumvent these limitations of the plane - wave basis , there is an increasing focus on developing real - space discretization techniques for orbital - free dft based on finite - difference discretization @xcite and finite - element discretization @xcite . in particular , the finite - element basis @xcite , which is a piecewise continuous polynomial basis , has many features of a desirable basis in electronic structure calculations . while being a complete basis , the finite - element basis naturally allows for the consideration of complex geometries and boundary conditions , is amenable to unstructured coarse - graining , and exhibits good scalability on massively parallel computing platforms . moreover , the adaptive nature of the finite - element discretization also enables the consideration of all - electron orbital - free dft calculations that are widely used in studies of warm dense matter @xcite . further , recent numerical studies have shown that by using a higher - order finite - element discretization significant computational savings can be realized for both orbital - free dft @xcite and kohn - sham dft calculations @xcite , effectively overcoming the degree of freedom disadvantage of the finite - element basis in comparison to the plane - wave basis . let @xmath187 denote the finite - element subspace of @xmath70 , where @xmath188 represents the finite - element mesh size . the discrete problem of computing the ground - state energy for a given positions of atoms , corresponding to equation , is given by the constrained variational problem : @xmath189 in the above , @xmath190 , @xmath191 , @xmath192 and @xmath193 denote the finite - element discretized fields corresponding to square - root electron - density , electrostatic potential , and kernel potentials , respectively . we restrict our finite - element discretization such that atoms are located on the nodes of the finite - element mesh . in order to compute the finite - element discretized solution of @xmath194 , we represent @xmath195 as a point charge on the finite - element node located at @xmath196 , and the finite - element discretization provides a regularization for @xmath194 . previous investigations have suggested that such an approach provides optimal rates of convergence of the ground - state energy ( cf . @xcite for a discussion ) . the finite - element basis functions also provide the generator of the deformations of the underlying space in the isoparametric formulation , where the same finite - element shape functions are used to discretize both the spatial domain as well as the fields prescribed over the domain . thus , the configurational force associated with the location of any node in the finite - element mesh can be computed by substituting for @xmath128 , in equation , the finite - element shape function associated with the node . thus , the configurational force on any finite - element node located at an atom location corresponds to the variational ionic force , which are used to drive the internal atomic relaxation . the forces on the finite - element nodes that do not correspond to an atom location represent the generalized force of the energy with respect to the location of the finite - element nodes , and these can be used to obtain the optimal location of the finite - element nodes a basis adaptation technique . we note that the local real - space variational formulation in section [ sec : rs - formulation ] , where the extended interactions in the electrostatic energy and kernel functionals are reformulated as local variational problems , is essential for the finite - element discretization of the formulation . in this section , we present the details of the numerical implementation of the finite - element discretization of the real - space formulation of orbital - free dft discussed in section [ sec : rs - ofdft ] . subsequently , we discuss the solution procedure for the resulting discrete coupled equations in square - root electron - density , electrostatic potential and kernel potentials . [ sec : fe - basis ] a finite - element discretization using linear tetrahedral finite - elements has been the most widely used discretization technique for a wide range of partial differential equations . linear tetrahedral elements are well suited for problems involving complex geometries and moderate levels of accuracy . however in electronic structure calculations , where the desired accuracy is commensurate with chemical accuracy , linear finite elements are computationally inefficient requiring of the order of hundred thousand basis functions per atom to achieve chemical accuracy . a recent study @xcite has demonstrated the significant computational savings of the order of 1000-fold compared to linear finite - elements that can be realized by using higher - order finite - element discretizations . thus , in the present work we use higher - order hexahedral finite elements , where the basis functions are constructed as a tensor product of basis functions in one - dimension @xcite . [ sec : numersoln ] the discrete variational problem in equation involves the computation of the following fields square - root electron - density , electrostatic potential and kernel potentials . two solution procedures , suggested in prior efforts @xcite , for solving this discrete variational problem include : ( i ) a simultaneous solution of all the discrete fields in the problem ; ( ii ) a nested solution procedure , where for every trial square - root electron - density the discrete electrostatic and kernel potential fields are computed . given the non - linear nature of the problem , the simultaneous approach is very sensitive to the starting guess and often suffers from lack of robust convergence , especially for large - scale problems . the nested solution approach , on the other hand , while constituting a robust solution procedure , is computationally inefficient due to the huge computational costs incurred in computing the kernel potentials which involves the solution of a series of helmholtz equations ( cf . equation ) . thus , in the present work , we will recast the local variational problem in equation as the following fixed point iteration problem : [ eq : fixedpoint ] @xmath197 @xmath198 we solve this fixed point iteration problem using a mixing scheme , and , in particular , we employ the anderson mixing scheme @xcite with full history in this work . our numerical investigations suggest that the fixed point iteration converges , typically , in less than ten self - consistent iterations even for large - scale problems , thus , providing a numerically efficient and robust solution procedure for the solution of the local variational orbital - free dft problem . we note that this idea of fixed point iteration has independently and simultaneously been investigated by another group in the context of finite difference discretization @xcite , and have resulted in similar findings . in the fixed point iteration problem , we employ a simultaneous solution procedure to solve the non - linear saddle point variational problem in @xmath190 and @xmath191 ( equation ) . we employ an inexact newton solver provided by the petsc package @xcite with field split preconditioning and generalized - minimal residual method ( gmres ) @xcite as the linear solver . the discrete helmholtz equations in equation are solved by employing block jacobi preconditioning and using gmres as the linear solver . an efficient and scalable parallel implementation of the solution procedure has been developed to take advantage of the parallel computing resources for conducting the large - scale simulations reported in this work . in this section , we discuss the numerical studies on al , mg and al - mg intermetallics to investigate the accuracy and transferability of the real - space formulation of orbital - free dft ( rs - ofdft ) proposed in section [ sec : rs - ofdft ] . wherever applicable , we benchmark the real - space orbital - free dft calculations with plane - wave based orbital - free dft calculations conducted using profess @xcite , and compare with kohn - sham dft ( ks - dft ) calculations conducted using the plane - wave based abinit code @xcite . further , we demonstrate the usefulness of the proposed real - space formulation in studying the electronic structure of isolated defects . [ sec : calc ] , scaledwidth=46.0% ] , scaledwidth=46.0% ] in all the real - space orbital - free dft calculations reported in this section , we use the local reformulation of the density - dependent wgc @xcite kinetic energy functional proposed in section [ sec : rs - formulation ] , the local density approximation ( lda ) @xcite for the exchange - correlation energy , and bulk derived local pseudopotentials ( blps ) @xcite for al and mg . cell stresses and ionic forces are calculated using the unified variational formulation of configurational forces developed in section [ sec : configurationalforces ] . in the second order taylor expansion of the density - dependent wgc functional about the bulk electron density ( cf . section [ sec : ofdft ] ) , we only retain the @xmath46 term for the computation of bulk properties as the contributions from @xmath46 dominate those of @xmath47 for bulk materials systems . however , in the calculations involving mono - vacancies , where significant spatial perturbations in the electronic structure are present , we use the full second order taylor expansion of the density dependent wgc functional . we recall from section [ sec : rs - formulation ] that in order to obtain a local real - space reformulation of the extended interactions in the kinetic energy functionals , the kernels ( @xmath199 , @xmath110 , @xmath47 , @xmath46 ) are approximated using a sum of @xmath200 partial fractions where the coefficients of the partial fractions are computed using a best fit approximation ( cf . equation ) . these best fit approximations for @xmath201 that are employed in the present work are given in the supplemental material . it has been shown in recent studies that @xmath202 suffices for al @xcite . however , we find that @xmath203 is required to obtain the desired accuracy in the bulk properties of mg , and table [ tab : bulktrf2 ] shows the comparison between the kernel approximation with @xmath203 and plane - wave based orbital - free dft calculations conducted using profess @xcite for mg . thus , we use the best fit approximation of the kernels with @xmath202 for al , and employ the approximation with @xmath203 for mg and al - mg intermetallics . henceforth , we will refer by rs - ofdft - fe the real - space orbital - dft calculations conducted by employing the local formulation and finite element discretization proposed in section [ sec : rs - ofdft ] . the ks - dft calculations used to assess the accuracy and transferability of the proposed real - space orbital - free dft formalism are performed using the lda exchange correlation functional @xcite . the ks - dft calculations are conducted using both local blps as well as the non - local troullier - martins pseudopotential ( tm - nlps ) @xcite in order to assess the accuracy and transferability of both the model kinetic energy functionals in orbital - free dft as well as the local pseudopotentials to which the orbital - free dft formalism is restricted to . the tm - nlps for al and mg are generated using the fhi98pp code @xcite . within the fhi98pp code , we use the following inputs : @xmath204 angular momentum channel as the local pseudopotential component for both al and mg , default core cutoff radii for the @xmath205 , @xmath206 , and @xmath204 angular momentum channels , which are @xmath207 bohr and @xmath208 bohr for al and mg respectively , and the lda @xcite exchange - correlation . for brevity , henceforth , we refer to the ks - dft calculations with blps and tm - nlps as ks - blps and ks - nlps , respectively . in all the rs - ofdft - fe calculations reported in this work , the finite - element discretization , order of the finite - elements , numerical quadrature rules and stopping tolerances are chosen such that we obtain 1 mev/ atom accuracy in energies , @xmath209 accuracy in cell stresses and @xmath210 accuracy in ionic forces . similar accuracies in energies , stresses and ionic forces are achieved for ks - dft calculations by choosing the appropriate k - point mesh , plane - wave energy cutoff , and stopping tolerances within abinit s framework . all calculations involving geometry optimization are conducted until cell stresses and ionic forces are below threshold values of @xmath211 and @xmath212 , respectively . [ sec : convergence ] we now study the convergence of energy and stresses with respect to the finite - element discretization of the proposed real - space orbital - free dft formulation . in a prior study on the computational efficiency afforded by higher - order finite - element discretization in orbital - free dft @xcite , it was shown that second and third - order finite - elements offer an optimal choice between accuracy and computational efficiency . thus , in the present study , we limit our convergence studies to hex27 and hex64 finite - elements , which correspond to second- and third - order finite - elements . as a benchmark system , we consider a stressed fcc al unit cell with a lattice constant @xmath213 bohr . we first construct a coarse finite - element mesh and subsequently perform a uniform subdivision to obtain a sequence of increasingly refined meshes . we denote by @xmath188 the measure of the size of the finite - element . for these sequence of meshes , we hold the cell geometry fixed and compute the discrete ground - state energy , @xmath214 , and hydrostatic stress , @xmath215 . the extrapolation procedure proposed in motamarri et . al @xcite allows us to estimate the ground - state energy and hydrostatic stress in the limit as @xmath216 , denoted by @xmath217 and @xmath218 . to this end , the energy and hydrostatic stress computed from the sequence of meshes using hex64 finite - elements are fitted to expressions of the form @xmath219 to determine @xmath220 . in the above expression , @xmath221 denotes the number of elements in a finite - element mesh . we subsequently use @xmath217 and @xmath218 as the exact values of the ground - state energy and hydrostatic stress , respectively , for the benchmark system . figures [ fig : energyconv ] and [ fig : stressconv ] show the relative errors in energy and hydrostatic stress plotted against @xmath222 , which represents a measure of @xmath188 . we note that the slopes of these curves provide the rates of convergence of the finite - element approximation for energy and stresses . these results show that we obtain close to optimal rates of convergence in energy of @xmath223 , where @xmath224 is polynomial interpolation order ( @xmath225 for hex27 and @xmath226 for hex64 ) . further , we obtain close to @xmath227 convergence in the stresses , which represents optimal convergence for stresses . the results also suggest that higher accuracies in energy and stress are obtained with hex64 in comparison to hex27 . thus , we will employ hex64 finite - elements for the remainder of our study . .[tab : bulktrf1 ] [ cols="^,^,^,^,^,^",options="header " , ] in order to understand this boundary condition dependence of the cell - size effects , we compute the perturbations in the electronic fields due to the presence of the mono - vacancy by subtracting from the electronic fields corresponding to the mono - vacancy the electronic fields of a perfect crystal . to this end , we define the normalized perturbations in the electronic fields computed on the finite - element mesh to be @xmath228 in the above , @xmath229 and @xmath230 denote the electronic fields in the computational domain with the vacancy and those without the vacancy ( perfect crystal ) , respectively . @xmath231 denotes the volume average of an electronic field over the computational cell . as a representative metric , in the definition of @xmath232 and @xmath233 we only consider the kernel potentials corresponding to @xmath83 . figures [ fig : monovaccf1 ] and [ fig : monovaccf2 ] shows the normalized corrector fields for the mono - vacancy , computed using periodic boundary conditions , along the face - diagonal of the periodic boundary . it is interesting to note from these results that the perturbations in the electronic structure due to the vacancy are significant up to @xmath234 computational cells . thus , although the vacancy formation energy appears converged by @xmath235 computational cell while using periodic boundary conditions , the electronic fields are not converged till a cell - size of @xmath234 computational cell . on the other hand , the cell - size convergence in mono - vacancy formation energy suggested by the bulk dirichlet boundary conditions is inline with the convergence of electronic fields . these results unambiguously demonstrate that the cell - size effects in the electronic structure of defects are larger than those suggested by a cell - size study of defect formation energies employing periodic boundary conditions . using bulk dirichlet boundary conditions for the cell - size study of defect formation energies provides a more accurate estimate of the cell - size effects in the electronic structure of defects , and the extent of electronic structure perturbations due to a defect . further , while periodic boundary conditions are limited to the study of point defects , bulk dirichlet boundary conditions can be used to also study defects like isolated dislocations @xcite , whose geometry does not admit periodic boundary conditions . we have developed a local real - space formulation of orbital - free dft with wgc kinetic energy functionals by reformulating the extended interactions in electrostatic and kinetic energy functionals as local variational problems in auxiliary potentials . the proposed real - space formulation readily extends to all - electron orbital - free dft calculations that are commonly employed in warm dense matter calculations . building on the proposed real - space formulation we have developed a unified variational framework for computing configurational forces associated with both ionic and cell relaxations . further , we also proposed a numerically efficient approach for the solution of ground - state orbital - free dft problem , by recasting the local saddle point problem in the electronic fields electron density and auxiliary potential fields as a fixed point iteration problem and employing a self - consistent iteration procedure . we have employed a finite - element basis for the numerical discretization of the proposed real - space formulation of orbital - free dft . our numerical convergence studies indicate that we obtain close to optimal rates of convergence in both ground - state energy and configurational forces with respect to the finite - element discretization . we subsequently investigated the accuracy and transferability of the proposed real - space formulation of orbital - free dft for al - mg materials system . to this end , we conducted a wide range of studies on al , mg and al - mg intermetallics , including computation of bulk properties for these systems , formation energies of al - mg intermetallics , and ionic forces in bulk and in the presence of point defects . our studies indicate that orbital - free dft and the proposed real - space formulation is in good agreement with kohn - sham dft calculations using both local pseudopotentials as well as non - local pseudpotentials , thus providing an alternate linear - scaling approach for electronic structure studies in al - mg materials system . we finally investigated the cell - size effects in the electronic structure of a mono - vacancy in al , and demonstrated that the cell - size convergence in the vacancy formation energy computed by employing periodic boundary conditions is not commensurate with the convergence of the electronic fields . on the other hand , the true cell - size effects in the electronic structure are revealed by employing the bulk dirichlet boundary conditions , where the perturbations in the electronic fields due to the defect vanish on the boundary of the computational domain . our studies indicate that the true cell - size effects are much larger than those suggested by periodic calculations even for simple defects like point defects . we note that the proposed real - space formulation and the finite - element basis are crucial to employing the bulk dirichlet boundary conditions that are otherwise inaccessible using fourier based formulations . the proposed formulation , besides being amenable to complex geometries , boundary conditions , and providing excellent scalability on parallel computing platforms , also enables coarse - graining techniques like the quasi - continuum reduction @xcite to conduct large - scale electronic structure calculations on the energetics of extended defects in al - mg materials system , and is an important direction for future studies . we gratefully acknowledge the support from the u.s . department of energy , office of basic energy sciences , division of materials science and engineering under award no . de - sc0008637 that funds the predictive integrated structural materials science ( prisms ) center at university of michigan , under the auspices of which this work was performed . v.g . also gratefully acknowledges the hospitality of the division of engineering and applied sciences at the california institute of technology while completing this work . we also acknowledge advanced research computing at university of michigan for providing the computing resources through the flux computing platform . 999 r. m. martin , _ electronic structure : basic theory and practical methods _ ( cambridge university press , cambridge , 2011 ) . p. hohenberg and w. kohn , phys . rev . * 136 * , b864 ( 1964 ) . w. kohn and l. j. sham , phys . rev . * 140 * , a1133 ( 1965 ) . s. goedecker , rev . 71 * , 1085 ( 1999 ) . d. r. bowler and t. miyazaki , rep . prog . phys . * 75 * , 036503 ( 2012 ) . p. motamarri and v. gavini , phys . b * 90 * , 115127 ( 2014 ) . r. parr and w. yang , _ density - functional theory of atoms and molecules _ ( oxford university press , 1989 ) . wang and m. p. teter , phys . b * 45 * , 13196 ( 1992 ) . e. smargiassi and p. a. madden , phys . b * 49 * , 5220 ( 1994 ) . y. a. wang , n. govind , and e. a. carter , phys . b * 58 * , 13465 ( 1998 ) . y. a. wang , n. govind , and e. a. carter , phys . b * 60 * , 16350 ( 1999 ) . karasiev , travis sjostrom , and s.b . trickey , phys . b * 86 * , 115101 ( 2012 ) . y. ke , f. libisch , j. xia , l .- w . wang , and e. a. carter , phys . lett . , * 111 * , 066402 ( 2013 ) . karasiev , d.chakraborty , and s.b . trickey , in _ many - electron approaches in physics , chemistry , and mathematics _ , l. delle site and v. bach eds . ( springer , heidelberg , 2014 ) , 113 - 134 . y. ke , f. libisch , j. xia , and e. a. carter , phys . b , * 89 * , 155112 ( 2014 ) . f. lambert , j. clrouin , and g. zrah , phys . e * 73 * , 016403 ( 2006 ) . f. lambert , j. clrouin , j .- f . danel , l. kazandjian , and g. zrah , phys . e * 77 * , 026402 ( 2008 ) . d. a. horner , f. lambert , j. d. kress , and l. a. collins , phys . b * 80 * , 024305 ( 2009 ) . l. burakovsky , c. ticknor , j. d. kress , l. a. collins , and f. lambert , phys . e * 87 * , 023104 ( 2013 ) . d. sheppard , j. d. kress , s. crockett , l. a. collins , and m. p. desjarlais , phys . e * 90 * , 063314 ( 2014 ) . v. gavini , k. bhattacharya , and m. ortiz , phys . b * 76 * , 180101 ( 2007 ) . g. ho , m. t. ong , k. j. caspersen , and e. a. carter , phys . * 9 * , 4951 ( 2007 ) . q. peng , x. zhang , l. hung , e. a. carter , and g. lu , phys . b * 78 * , 054118 ( 2008 ) . i. shin , a. ramasubramaniam , c. huang , l. hung , and e. a. carter , philos . mag . * 89 * , 3195 ( 2009 ) . i. shin and e. a. carter , phys . b * 88 * , 064106 ( 2013 ) . m. iyer , b. radhakrishnan , and v. gavini , j. mech . solids * 76 * , 260 ( 2015 ) . l. hung , c. huang , i.shin , g. ho , v. l. ligneres , and e. a. carter , comput . comm . , * 181 * , 2208 ( 2010 ) . m. chen , j. xia , c. huang , j. m. dieterich , l. hung , i. shin , and e. a. carter , comp . * 190 * , 228 ( 2015 ) . v. gavini , j. knap , k. bhattacharya , and m. ortiz , j. mech . solids * 55 * , 669 ( 2007 ) . b. g. radhakrishnan and v. gavini , phys . b * 82 * , 094117 ( 2010 ) . p. motamarri , m. iyer , j. knap , and v. gavini , j. comput . phys . * 231 * , 6596 ( 2012 ) . garcia - cervera , comm . comp . phys . * 2 * , 334 ( 2007 ) . p. suryanarayana and d. phanish , j. comp . phys . * 275 * , 524 ( 2014 ) . s. ghosh and p. suryanarayana , arxiv preprint arxiv:1412.8250 ( 2014 ) . v. gavini , l. liu , j. mech . solids * 59 * , 1536 ( 2011 ) . d. m. ceperley and b. j. alder , phys . rev . lett . * 45 * 566 ( 1980 ) . j. p. perdew and a. zunger , phys . b * 23 * 5048 ( 1981 ) . d. c. langreth and m. j. mehl , phys . b * 28 * , 1809 ( 1983 ) . j. p. perdew , j. a. chevary , s. h. vosko , k. a. jackson , m. r. pederson , d. j. singh , and c. fiolhais , phys . b * 46 * , 6671 ( 1992 ) . c. huang and e. a. carter , phys . chem . phys.*10 * , 7109 ( 2008 ) . j. xia and e. a. carter , phys . b * 86 * , 235109 ( 2012 ) . c. huang and e. a. carter , phys . rev . b * 85 * , 045126 ( 2012 ) . m. finnis , _ interatomic forces in condensed matter _ ( oxford university press , 2003 ) . n. choly and e. kaxiras , solid state communications * 121 * , 281 ( 2002 ) . v. gavini , k. bhattacharya , and m. ortiz , j. mech . solids * 55 * , 697 ( 2007 ) . brenner and l.r . scott , _ the mathematical theory of finite - element methods _ ( springer , new york , 2002 ) . p. motamarri , m. nowak , k. leiter , j. knap , and v. gavini , j. comput . phys . * 253 * , 308 ( 2013 ) . d. g. anderson , j. assoc . * 12 * , 547 ( 1965 ) . s. balay , j. brown , k. buschelman , v. eijkhout , w. d. gropp , d. kaushik , m. g. knepley , l. c. mcinnes , b. f. smith , h. zhang , _ petsc 3.4 users manual _ , argonne national laboratory , 2013 . y. saad and m. h. schultz , siam j. sci . * 7 * , 856 ( 1986 ) . x. gonze , j .- m . beuken , r. caracas , f. detraux , m. fuchs , g .- rignanese , l. sindic , m. verstraete , g. zerah , f. jollet , m. torrent , a. roy , m. mikami , p. ghosez , j .- y . raty and d. c. allan , comput . sci . * 25 * , 478 ( 2002 ) . x. gonze , b. amadon , p.m. anglade , j .- beuken , f. bottin , p. boulanger , f. bruneval , d. caliste , r. caracas , m. cote , t. deutsch , l. genovese , ph . ghosez , m. giantomassi , s. goedecker , d. hamann , p. hermet , f. jollet , g. jomard , s. leroux , m. mancini , s. mazevet , m.j.t . oliveira , g. onida , y. pouillon , t. rangel , g .- rignanese , d. sangalli , r. shaltaf , m. torrent , m.j . verstraete , g. zrah , j.w . zwanziger , comput . . commun . * 180 * , 2582 ( 2009 ) . n. troullier and j. l. martins , phys . rev . b * 43 * , 1993 , ( 1991 ) . m. fuchs and m. scheffler , comput . * 119 * , 67 ( 1999 ) . m. feuerbacher , c. thomas , j.p.a . makongo , s. hoffmann , w. carrillo - cabrera , r. cardoso , y. grin , g. kreiner , j .- joubert , t. schenk , j. gastaldi , h. nguyen - thi , n. mangelinck - noel , b. billia , p. donnadieu , a. czyrska - filemonowicz , a. zielinska - lipiec , b. dubiel , t. weber , p. schaub , g.r krauss , v. gramlich , j. christensen , s. lidin , d. fredrickson , m. mihalkovic , w. sikora , j. malinowski , s. brhne , t. proffen , w. assmus , m. de boissieu , f. bley , j .- l . chemin , j. schreuer , z. kristallographie * 222 * , 259 ( 2007 ) . n. chetty , m. weinert , t. s. rahman and j. w. davenport , phys . b * 52 * , 6313 ( 1995 ) . m. j. gillan , j. phys . : condens . matter * 1 * , 689 ( 1989 ) . iyer , m. , gavini , v. , j. mech . solids * 59 * , 1506 ( 2011 ) .

we propose a local real - space formulation for orbital - free dft with density dependent kinetic energy functionals and a unified variational framework for computing the configurational forces associated with geometry optimization of both internal atomic positions as well as the cell geometry . the proposed real - space formulation , which involves a reformulation of the extended interactions in electrostatic and kinetic energy functionals as local variational problems in auxiliary potential fields , also readily extends to all - electron orbital - free dft calculations that are employed in warm dense matter calculations . we use the local real - space formulation in conjunction with higher - order finite - element discretization to demonstrate the accuracy of orbital - free dft and the proposed formalism for the al - mg materials system , where we obtain good agreement with kohn - sham dft calculations on a wide range of properties and benchmark calculations . finally , we investigate the cell - size effects in the electronic structure of point defects , in particular a mono - vacancy in al . we unambiguously demonstrate that the cell - size effects observed from vacancy formation energies computed using periodic boundary conditions underestimate the extent of the electronic structure perturbations created by the defect . on the contrary , the bulk dirichlet boundary conditions , accessible only through the proposed real - space formulation , which correspond to an isolated defect embedded in the bulk , show cell - size effects in the defect formation energy that are commensurate with the perturbations in the electronic structure . our studies suggest that even for a simple defect like a vacancy in al , we require cell - sizes of @xmath0 atoms for convergence in the electronic structure .

thermal dilepton production in the mass region below 1 gev / c@xmath1 is largely mediated by the light vector mesons @xmath0 , @xmath2 and @xmath3 . among these , the @xmath0(770 mev / c@xmath1 ) is the most important , due to its strong coupling to the @xmath4 channel and its short lifetime of only 1.3 fm / c . changes both in width and in mass were originally suggested as precursor signatures of the chiral transition @xcite and subsequent models have tied these changes directly @xcite or indirectly @xcite to chiral symmetry restoration . the first na60 results focused on the space - time averaged spectral function of the @xmath0 @xcite ; more details were added recently @xcite . the present paper concentrates mainly on new developments , in particular first results on acceptance - corrected @xmath5 and @xmath6 spectra for different mass windows details of the na60 apparatus can be found in @xcite , while the different analysis steps ( including the critical assessment of the combinatorial background from @xmath7 and @xmath8 decays through event mixing ) are described in @xcite . the results reported here were obtained from the analysis of data taken in 2003 for in - in at 158 agev . the left part of fig . [ fig1 ] shows the opposite - sign , background and signal dimuon mass spectra , integrated over all collision centralities . after subtracting the combinatorial background and the signal fake matches , the resulting net spectrum contains about 360000 muon pairs in the mass range 0 - 2 gev / c@xmath9 , roughly 50% of the total available statistics . the associated average charged - particle multiplicity density measured by the vertex tracker is @xmath10 = 120 . vector mesons @xmath2 and @xmath3 are completely resolved with a mass resolution at the @xmath2 of 20 mev / c@xmath1 . most of the analysis is done in four classes of collision centrality ( defined through the charged - particle multiplicity density ) : peripheral ( 4@xmath1130 ) , semiperipheral ( 30@xmath11110 ) , semicentral ( 110@xmath11170 ) and central ( 170@xmath11240 ) . the peripheral data can essentially be described by the expected electromagnetic decays of the neutral mesons @xcite . this is not the case in the more central bins , due to the presence of a strong excess . to isolate this excess without any fits , a novel procedure has been devised @xcite . the resulting spectrum for all centralities and all @xmath12 shows a a peaked structure residing on a broad continuum . the same feature can be seen for each centrality bin , with a yield strongly increasing with centrality , but remaining essentially centered around the position of the nominal @xmath0 pole . a more quantitative analysis of the shape of the excess mass spectra _ vs. _ centrality has been performed , using a finer subdivision of the data into 12 centrality bins @xcite . on the basis of this analysis it is possible to rule out that the excess shape can be accounted for by the cocktail @xmath0 residing on a broad continuum , independent of centrality . for different mass windows . right : rapidity distribution of the excess data for the mass window @xmath13 gev/@xmath14 and for three selected @xmath5 bins . the measured data ( full markers ) are reflected around midrapidity ( open markers).,title="fig:",scaledwidth=40.0% ] for different mass windows . right : rapidity distribution of the excess data for the mass window @xmath13 gev/@xmath14 and for three selected @xmath5 bins . the measured data ( full markers ) are reflected around midrapidity ( open markers).,title="fig:",scaledwidth=40.0% ] a differential analysis in @xmath12 was originaly performed and mass spectra associated with three different @xmath12 windows , without acceptance correction , were obtained @xcite . the na60 acceptance relative to @xmath15 as a function of @xmath12 as shown in fig . [ fig3 ] ( left ) , implies that one should in principle perform the acceptance correction using a 3-dimensional grid in ( @xmath16 ) space . this can lead , however , to large errors once the correction is applied . instead , the correction is performed in 2-dimensional ( @xmath17 ) space , using the measured rapidity distribution as an input . the latter was determined with an acceptance correction found , in an iterative way , from monte carlo simulations matched to the data in @xmath18 and @xmath12 . on the basis of this rapidity distribution , 0.1 gev/@xmath19 bins in m and 0.2 gev/@xmath20 bins in @xmath12 were used to determine the remaining 2-dimensional correction . after correction the results were integrated over the three extended mass windows @xmath21 , @xmath13 and @xmath22 gev / c@xmath1 . in fig . [ fig3 ] ( right ) the rapidity distribution of the central mass window is shown for three different @xmath12 windows , exhibiting a close resemblance to the distribution of inclusive pion production , as measured by na49 for pb - pb and na60 for in - in . the results for the acceptance - corrected @xmath5 spectra are summarized in fig . the errors shown are purely statistical . systematic errors arise from the acceptance corrections including the rapidity distribution used , the subtraction of the cocktail , and the subtraction of the combinatorial background plus fake matches . for @xmath23 gev / c , the combinatorial background contributes most , ranging from 10 to 25% for semiperipheral up to central . for @xmath24 gev / c , the statistical errors dominate . the data show a significant dependence on mass , but hardly any on centrality . to bear out the differences in mass more clearly , the data were summed over the three more central bins and plotted in fig . [ fig5 ] ( left ) as a function of @xmath25 . the inverse slope parameter @xmath26 as determined from differential fits of the @xmath6 spectra with @xmath27 , using a sliding window in @xmath5 , is plotted on the right . instead of flattening at very low @xmath6 as should be expected from radial flow , strikingly all spectra _ steepen _ , which is equivalent to very small values of @xmath26 . for comparison , the @xmath3 resonance , placed just in between the upper two mass windows , does flatten as expected . moreover , depending on the fit region , @xmath26 covers an unusually large dynamic range . finally , the largest masses have the steepest @xmath6 spectrum , _ i.e. _ the smallest value of @xmath26 everywhere , again contrary to radial flow and to what is usually observed for hadrons . all this suggests that different mass regions are coupled to basically different emission sources . summarizing , the previously measured excess mass spectra and the new acceptance - corrected @xmath5 and @xmath6 spectra present an unexpected behaviour . beyond the @xmath0 spectral function this may lead to a better understanding of the continuum part of the spectra for @xmath28 gev / c@xmath9 , possibly disentangling parton - hadron duality . 00 r. d. pisarski , phys . * 110b * , 155 ( 1982 ) g. e. brown , m. rho , phys . rept . * 363 * , 85 ( 2002 ) r. rapp and j. wambach , adv . . phys . * 25 * , 1 ( 2000 ) r. arnaldi _ et al . _ ( na60 collaboration ) , phys . * 96 * ( 2006 ) 162302 s. damjanovic _ et al . _ ( na60 collaboration ) , nucl - ex/0609026 ; s. damjanovic _ et al . _ ( na60 collaboration ) , nucl - ex/0701015 g. usai _ et al . _ ( na60 collaboration ) , eur . j. * c43 * , 415 ( 2005 ) ; g. usai _ et al . _ ( na60 collaboration ) , these proceedings . m. keil _ et al . _ , nucl . instrum . meth . * a539 * , 137 ( 2005 ) and * a546 * , 448 ( 2005 ) r. shahoyan _ et al . _ ( na60 collaboration ) , nucl . * a774 * , 677 ( 2006 ) r. shahoyan _ et al . _ ( na60 collaboration ) , these proceedings

the na60 experiment at the cern sps has studied low - mass muon pairs in 158 agev in - in collisions . a strong excess of pairs is observed above the yield expected from neutral meson decays . the unprecedented sample size of close to 400k events and the good mass resolution of about 2% have made it possible to isolate the excess by subtraction of the decay sources ( keeping the @xmath0 ) . the shape of the resulting mass spectrum exhibits considerable broadening , but essentially no shift in mass . the acceptance - corrected transverse - momentum spectra have a shape atypical for radial flow and show a significant mass dependence , pointing to different sources in different mass regions .

galaxy cluster mergers are ideal probes of gravitational collapse and the hierarchical structure formation in the universe . observations of the evolving cluster mass function provide a sensitive cosmological test that is both independent of , and complementary to , other methods ( e.g. , bao , sn , cmb ) @xcite . the use of galaxy clusters as cosmological probes relies on the accuracy of scaling relations between the total mass and observable quantities . galaxy cluster mergers will disrupt the intracluster gas and cause departures from these scaling relations ( e.g. , * ? ? ? * ; * ? ? ? given that these mass scaling relations are a necessary ingredient for the interpretation of on - going cosmological surveys , a detailed understanding of the intracluster medium ( icm ) gas physics in mergers has become increasingly important . llccclcc + satellite & pointing & obsid & r.a . & dec & date obs & exposure & pi + & & & & & & xis0/xis1/xis3 & + & & & & & & ( ks ) + + _ suzaku _ & north & 806096010 & 13 31 15.53 & - 01 39 13.3 & 2011 jul 2 & 74.7/74.7/74.7 & s. randall + _ suzaku _ & center & 806095010 & 13 30 46.63 & - 01 53 14.3 & 2011 jul 24 & 38.0/38.0/38.0 & s. randall + _ suzaku _ & south & 806097010 & 13 30 13.15 & - 02 06 22.7 & 2011 jul 9 & 70.2/70.2/70.2 & s. randall + _ suzaku _ & southeast & 806098010 & 13 31 27.19 & - 02 04 19.9 & 2011 jul 6 & 55.9/55.9/56.0 & s. randall + _ suzaku _ & southeast & 806098020 & 13 31 28.58 & - 02 02 29.4 & 2011 dec 23 & 11.3/11.3/11.3 & s. randall + _ chandra _ & north & 11878 & 13 31 10.83 & - 01 43 21.0 & 2010 may 11 & 19.4@xmath3 & s. murray + _ chandra _ & center & 11879 & 13 30 50.30 & - 01 52 28.0 & 2010 may 9 & 19.7@xmath3 & s. murray + _ chandra _ & south & 12914 & 13 30 15.80 & - 02 02 28.7 & 2011 mar 16 & 36.8@xmath3 & s. murray + + + [ table : obs ] the properties of the icm in the cores of merging clusters have been studied in detail , since the high density and surface brightness of the gas in this region is well - suited to high angular resolution observations with _ chandra _ and _ xmm - newton _ ( see * ? ? ? * for a review ) . with the launch of the low particle - background _ suzaku _ mission , it has become possible to probe the low gas density and faint surface brightness regions at the virial radii of nearby galaxy clusters ( e.g. * ? ? ? * ; * ? ? ? * ; * ? ? ? * ; * ? ? ? * ; * ? ? ? observational studies at these radii have mostly focused on relatively relaxed , massive , cool - core systems . due to the limited number of observations , the dynamical evolution of the icm in strong merger events out to the viral radius is not clearly understood . strongly merging , bimodal clusters are where we expect to find the large - scale filaments and accretion shocks . comparing results from observations of mergers and relaxed clusters at the virial radius will provide an important confirmation of our current picture of large - scale structure formation . the double clusters identified from _ einstein _ observations ( a1750 , a98 , a115 , a3395 ) are ideal targets for studying the virial radii of strongly merging clusters @xcite . these canonical binary galaxy clusters have two separated peaks of x - ray emission , and distortions in their x - ray surface brightness distributions suggest ongoing merger events ( e.g. , * ? ? ? most of these systems are in fact triple clusters , with all sub - clusters lying roughly along the same line , suggesting the presence of large - scale structure filaments . a1750 is a triple merger system at a redshift of 0.085 , with an average temperature of 4.5 kev @xcite . it contains three main sub - clusters with x - ray centroids : a1750n ( j2000 , ra : 202.79@xmath4 , dec : @xmath51.73@xmath4 ) , a1750c ( j2000 , ra : 202.71@xmath4 , dec : @xmath51.86@xmath4 ) , and a1750s ( j2000 , ra : 202.54@xmath4 , dec : @xmath52.105@xmath4 ) . mmt data provided redshifts for the brightest cluster galaxies of 0.0836 , 0.0878 , and 0.0865 ( see section [ sec : resultsopt ] for details ) . a1750 was identified as a strongly merging double " cluster due to the presence of two bright x - ray subcluster peaks , which are clearly visible in the _ einstein _ image @xcite . the centers of a1750n and a1750c are separated by 9.7@xmath6 ( 930 kpc ; see figure [ fig : image ] ) . and observations indicate possible shock heated gas with an elevated temperature of 5.5 kev between these sub - clusters , suggesting that they are in an early stage merger @xcite . more recent _ xmm - newton _ observations confirm this region of elevated temperature , and also indicate that a1750c may itself be undergoing a merger @xcite . a1750s was identified with observations . its center is located 17.5@xmath6 ( 1.68 mpc ) to the southwest of a1750c , along the same line connecting a1750c and a1750n , presumably tracing a large - scale filament . the 0.2 @xmath5 10 kev luminosities of the two brighter sub - clusters are 1.3 @xmath7 ergs s@xmath8 for a1750n and 2.2 @xmath7 ergs s@xmath8 for a1750c @xcite . the x - ray luminosity of the fainter , southern sub - cluster a1750s is 6.4 @xmath9 ergs s@xmath8 , estimated from pspc observations . here , we present results from mosaic _ suzaku _ observations of a1750 out to the virial radius . these new observations , together with archival _ chandra _ and _ xmm - newton _ observations , probe the icm properties from the subcluster cores out to their viral radii . previous studies of other ( non - merging ) systems have found entropy profiles that flatten at large radii , in contradiction with theoretical predictions , possibly due to the presence of unresolved cool gas clumps @xcite . this behavior shows some variation with azimuth , suggesting a connection with large - scale structure and gas accretion @xcite . we use our observations , which extend both along and perpendicular to the putative large - scale structure filament , to look for correlations between the icm properties , the surrounding large - scale environment , and to examine the merger dynamics . this paper is organized as follows : in section [ sec : obs ] , we describe the _ suzaku _ , _ chandra _ , and _ mmt _ data used in our analysis . in section [ sec : analysis ] , the analysis of the x - ray and optical observations is described in detail . in section [ sec : syst ] , we discuss systematic errors that are relevant to the _ suzaku _ x - ray measurements at large radii . in sections [ sec : resultsopt],[sec : resultsxray ] , and [ sec : thermo ] we discuss our results and present our conclusions in section [ sec : conc ] . throughout the paper , a standard @xmath10cdm cosmology with h@xmath11 = 70 km s@xmath8 mpc@xmath8 , @xmath12 = 0.7 , and @xmath13 = 0.3 is assumed . in this cosmology , 1@xmath14 at the redshift of the cluster corresponds to @xmath15 96.9 kpc . unless otherwise stated , reported errors correspond to 90% confidence intervals . r vs r ) plot of galaxies included in the spectroscopic catalog , with selected passive cluster members plotted in red . * lower panel : * color - color ( g @xmath5 r vs r @xmath5 i ) plot of galaxies included in the spectroscopic catalog , with selected passive cluster members plotted in red.[fig : colormag ] , width=340 ] the majority of the galaxy spectroscopic redshifts used in this analysis are new observations obtained using the hectospec instrument @xcite at the _ mmt _ observatory 6.5 m telescope on mt . hopkins , az . a single hectoscpec configuration places up to 300 fibers in a region of the sky approximately one degree in diameter . we use data from two such configurations , which resulted in 517 individual spectroscopic redshift measurements . to supplement our hectospec spectroscopy , we include data from the literature , when available . specifically , we use 12 spectroscopic redshift measurements from @xcite , 68 from @xcite , seven from @xcite , 19 from the 6df galaxy survey @xcite , and 200 from the sloan digital sky survey ( sdss ; * ? ? ? the sdss selection includes all objects within a 0.5 degree radius of the centroids of the x - ray emission of a1750n , a1750c , and a1750s , and with a spectroscopic redshift falling in the interval 0.03 @xmath16 0.15 , which easily captures the range of recessional velocities of galaxies associated with a1750 . we then check for duplicate entries across the different input redshift catalogs , resulting in 24 removals and a final data set of 799 spectroscopic redshifts . in addition to optical spectroscopy , we also use optical photometry from the sdss catalogs . we perform a query of all objects classified as galaxies within a 0.5 degree radius of the centroid of the x - ray surface brightness of each subcluster and download all of the available optical photometry in the _ ugriz _ bands for those sources . [ fig : image ] a1750 was observed with _ suzaku _ with five pointings during july 2011 and december 2011 ( see table [ table : obs ] ) . we process the unfiltered _ suzaku _ data with _ heasoft _ version 6.13 , and the latest calibration database caldb as of may 2014 . the raw event files are filtered using the ftool _ aepipeline_. in addition to the standard filtering performed by _ aepipeline _ , we require an earth elevation angle @xmath17 5@xmath18 , a geomagnetic cut - off rigidity of @xmath17 6 gv / c , and exclude data collected during passages through the south atlantic anomaly as described in @xcite . the data taken with 3@xmath193 and 5@xmath195 clocking modes are merged and the corners of the chips illuminated by the fe - calibration sources are excluded from further analysis . we carefully examine each light curve after the initial screening to ensure that the data are free from background flaring events . due to the increase in charge injection in data taken with xis1 after 2011 june 1 , the two rows adjacent to the standard charge - injected rows are removed . the region lost due to a putative micrometeorite hit on xis0 is also excluded from our analysis . the net exposure times of each xis0 , xis1 , and xis3 pointing after filtering are given in table [ table : obs ] . due to our strict filtering , 30 ks of the total exposure time was lost . the total filtered _ suzaku _ xis0/xis1/xis3 exposure time is 250.1/250.1/250.2 ks . the _ chandra _ observations that were used in the analysis are summarized in table [ table : obs ] . for each observation , the aimpoint was on the front - side illuminated acis - i ccd . all data were reprocessed from the level 1 event files using ciao 4.6 and caldb 4.4.7 . cti and time - dependent gain corrections were applied . lc_clean was used to check for periods of background flares . the mean event rate was calculated from a source free region using time bins within 3@xmath20 of the overall mean , and bins outside a factor of 1.2 of this mean were discarded . there were no periods of strong background flares . to model the background we used the caldb blank sky background files appropriate for this observation , normalized to match the 10 - 12 kev count rate in our observations to account for variations in the particle background . the total filtered acis - i exposure time is 75.9 ks . sdss _ ugriz _ photometry samples the full optical spectral energy distribution ( sed ) for galaxies in a1750 , including the 4000 break that is located in the @xmath21-band at the redshift of a1750 . the 4000 break is a strong feature , characteristic of the passive red sequence galaxies that dominate the galaxy populations of evolved galaxy clusters @xcite . we identify candidate cluster member galaxies of a1750 using the red sequence in the _ gri _ bands , which span the break . the red sequence selection involves two steps . the initial selection is made in color - magnitude space ( @xmath22 vs. @xmath23 ; figure [ fig : colormag ] top panel ) with a manual identification of the over - density of galaxies with approximately the same @xmath22 color . we then perform a linear fit in color - magnitude space to define the red sequence in a1750 , and flag all galaxies within @xmath240.125 in @xmath22 magnitudes as candidate red sequence galaxies . the second step occurs in color - color space ( @xmath22 vs @xmath25 ; figure [ fig : colormag ] bottom panel ) , where we identify an over - density of candidate red sequence galaxies with similar @xmath25 colors . galaxies that satisfy the initial color - magnitude selection while also falling within @xmath240.125 magnitudes of the mean @xmath25 color of the over density in color - color space are flagged as red sequence galaxies . the range of color values that we use accounts for both the observed intrinsic scatter in the red sequence of massive galaxy clusters ( @xmath260.05 - 0.1 mags ; * ? ? ? * ; * ? ? ? * ; * ? ? ? * ) and the typical sdss photometric uncertainties of @xmath150.025 magnitudes . we extract an image of a1750 in the 0.5 @xmath5 7 kev energy band and mosaic the pointings in sky coordinates . the non - x - ray background ( nxb ) images are generated using the ` night - earth ' data ( nte ) using the ftool _ xisnxbgen _ @xcite . the nxb images are then subtracted from the mosaicked image prior to exposure correction . to generate the exposure maps , we first simulate a monochromatic photon list assuming a 20@xmath6 uniform extended source for each observation with the _ xrt _ ray - tracing simulator _ xissim _ @xcite . these vignetting - corrected photon lists are then used with _ xisexpmapgen _ to generate exposure maps of each pointing , as described in detail in @xcite . regions with @xmath27% of the maximum exposure time are removed . the resulting exposure maps for each pointing are merged . the particle background subtracted , vignetting - corrected image is shown in figure [ fig : image ] ( left panel ) . 1.11.5 lccccccccccc + region & n1 & n2 & n3 & n4 & se1 & se2 & se3 & s1 & s2 + + + n1 & 40.88 & 15.64 & 1.41 & 0.45 & - & - & - & - & - + n2 & 7.82 & 56.23 & 15.95 & 1.11 & - & - & - & - & - + n3 & 1.18 & 15.95 & 58.37 & 13.58 & - & - & - & - & - + n4 & 0.07 & 0.57 & 8.54 & 55.84 & - & - & - & - & - + se1 & - & - & - & - & 56.64 & 7.86 & 0.05 & - & - + se2 & - & - & - & - & 10.81 & 61.54 & 2.20 & - & - + se3 & - & - & - & - & 0.45 & 4.65 & 57.16 & - & - + s1 & - & - & - & - & - & - & - & 60.78 & 14.35 + s2 & - & - & - & - & - & - & - & 9.05 & 69.38 + + + + [ table : psf ] to detect x - ray point sources unresolved by _ , we use the three _ chandra _ pointings of the cluster , which overlap with the northern , central , and southern _ suzaku _ pointings . the locations of the point sources in the field - of - view ( fov ) are detected using ciao s _ wavdetect _ tool and are shown in the right panel of figure [ fig : image ] . since the point spread function ( psf ) sizes of _ suzaku _ and _ chandra _ are different , the extents of the point sources reported by _ wavdetect _ can not be used directly to exclude point source in the _ suzaku _ fov . we use the following procedure to determine a reliable and conservative radius for point source exclusion . the brightest point source within both the _ chandra _ and _ suzaku _ fov ( j2000 ; ra : 202.603@xmath4 , dec : @xmath51.808@xmath4 ) is selected as a test case ( shown with green circle in figure [ fig : image ] left panel ) . the source is located in a fairly faint region ( 9@xmath14 away from the center of a1750 to northeast ) . spectrum of the point - source is extracted using ciao s _ specextract _ tool and is fitted with an absorbed power - law model with an index fixed to 1.4 ( the slope associated with the x - ray background spectrum at 0.5@xmath58 kev ; e.g. @xcite ) , while the normalization is left free . based on the best - fit power - law index and normalization ( 5.22 @xmath28 photons kev@xmath8 @xmath29 s@xmath8 ) obtained from the _ chandra _ fits , a 120 ks long _ suzaku _ xis observation is simulated using the _ xissim _ tool . to assess the impact of the source flux on the measured parameters of the diffuse emission , we add the simulated source spectrum to a typical diffuse emission spectrum with 1000 net counts . we then incrementally increase the source exclusion radius ( thereby decreasing the contribution of the point source to the total emission ) and examine the effect on the best - fitting parameters to the total ( source plus diffuse emission ) spectrum . we find that for all exclusion radii @xmath30 the best - fitting parameters ( kt , abundance , and normalization ) are not significantly affected by the point source contribution . since this estimate is based on the analysis of the brightest brightest point source in a faint region , the exclusion radius for fainter point sources would be smaller . we note that since all our spectral extraction regions include at least 2000 total counts , this radius represents a conservative estimate . we therefore exclude regions with radii of 35@xmath31 around point sources detected by _ chandra _ from our _ suzaku _ analysis . the southeast _ suzaku _ pointing does not have an overlapping _ chandra _ observation . therefore , the point sources in this region are detected from the suzaku data using ciao s _ wavdetect _ tool . the detection is performed using _ suzaku _ s half - power radius of 1@xmath6 as the wavelet radius , as done in @xcite . the point sources detected with _ suzaku _ are shown as green regions in the left panel in figure [ fig : image ] . spectra are extracted from the filtered event files in _ xselect_. corresponding detector redistribution function ( rmf ) files are constructed using the _ xisrmfgen _ tool , while the ancillary response function ( arf ) files are constructed using the _ xisarfgen _ tool assuming a uniform surface brightness in a 20@xmath6 radius . cutoff - rigidity - weighted particle - induced background spectra are extracted from the nte data for each detector using the _ xisnxbgen _ tool . the particle induced background spectrum is subtracted from each source spectrum prior to fitting . spectral fitting is performed in the 0.5 @xmath5 7 kev energy band where the _ suzaku _ xis is the most sensitive . the cluster emission is modeled with an absorbed single temperature thermal plasma model with atomdb version 2.0.2 @xcite . _ xspec _ v12.8.2 is used to perform the spectral fits @xcite with the extended c - statistic as an estimator of the goodness of fits . we co - add front illuminated ( fi ) xis0 and xis3 data to increase the signal - to - noise , while the back illuminated ( bi ) xis1 data are modeled simultaneously with the front illuminated observations due to the difference in energy responses . we adopt the solar abundance table from @xcite . the galactic column density is frozen at the leiden / argentine / bonn ( lab ) galactic hi survey value @xcite of 2.37 @xmath19 10@xmath32 @xmath29 in our fits . we examine the local x - ray background emission using the rosat all sky survey ( _ rass _ ) data extracted from a 1@xmath52 degree annulus surrounding the central sub - cluster s centroid . a region 19@xmath6 @xmath5 21@xmath6 away from the central sub - cluster a1750c in the southeast pointing is used to extract the local background ( see figure [ fig : image ] ) . the rass spectrum is simultaneously fit with the local background xis fi and bi spectra using two gaussian models for solar wind charge exchange at 0.56 and 0.65 kev , an unabsorbed _ apec _ model for local hot bubble ( lhb ) emission , and an absorbed _ apec _ model for galactic halo ( gh ) emission @xcite . the abundances of these models are set to solar , while the redshifts are fixed at zero . an absorbed power - law component with a photon index of 1.4 is added to the model to include emission from unresolved extragalactic sources ( primarily agn ) . we note that statistical uncertainties in the observed local background parameters given in this section are 1@xmath20 . the best - fit temperature of the lhb component is 0.14@xmath33 kev , with a normalization of 3.22@xmath34 @xmath35 arcmin@xmath36 . the best - fit temperature and normalization of the gh component is 0.69@xmath37 kev and 1.79@xmath38 @xmath39 @xmath35 arcmin@xmath36 . the normalization for the cxb power - law component is 5.84@xmath40 photons kev@xmath8 @xmath29 s@xmath8 arcmin@xmath41 at 1 kev , corresponding to a cxb flux of ( 1.15 @xmath42 ergs s@xmath8 @xmath29 deg@xmath41 . the flux ( 6.22 @xmath24 0.16 ) @xmath43 ergs s@xmath8 @xmath29 deg@xmath41 in the 0.5@xmath52 kev band is in agreement with the value ( 7.7@xmath240.4 @xmath43 ergs s@xmath8 @xmath29 deg@xmath41 ) reported by @xcite . in studies of low surface brightness emission , it is crucial to estimate the contribution of various systematic uncertainties , particularly those related to background modeling , to the total error budget . we consider the following potential sources of systematic error in our analysis ; i ) uncertainties due to stray light contamination and the large size of the psf of _ suzaku _ s mirrors ; ii ) uncertainties due to intrinsic spatial variations in the local soft background ; iii ) systematics associated with the nxb ; vi ) uncertainties due to the intrinsic spatial variation of unresolved point sources . due to _ suzaku s relatively large psf , some x - ray photons that originate from one particular region on the sky may be detected elsewhere on the detector . the psf spreading in each direction is calculated by generating simulated event files using the ray - tracing simulator _ xissim _ @xcite . _ chandra _ x - ray images of each annular sector ( shown in figure [ fig : image ] left panel ) and the best - fit spectral models obtained from the _ suzaku _ observations are used to simulate event files with 1@xmath44 photons . the fraction of photons that are spread into the surrounding annuli is calculated for each xis detector and annulus sector . relative contributions are weighted by the effective area at 1.5 kev of each detector to calculate the overall percentage contribution ( given in table [ table : psf ] ) . we find that the majority of photons originating in an annulus on the sky are detected in the same annulus ( except region n1 ) on the detector . up to 15% of the photons may be detected in surrounding annuli . however , the percentage fraction of photons that scatter into the outermost annuli at @xmath2 is defined as the radius within which the average matter density of the cluster is 200 times the critical density of the universe at the cluster redshift . ] to the north and southeast from the bright cores is small ( @xmath11% ) . these results are consistent with the photon fractions reported in @xcite and @xcite . considering the shallow temperature distribution of a1750 measured by _ observations , the psf is expected to have a minimal effect on the measurements of temperature in the outermost regions . to estimate the effect of psf spreading on our temperature and normalization measurements , we jointly fit the spectra of sectors with models , with the normalizations scaled according to the fractions listed in table [ table : psf ] . in all cases , the change in best - fit parameter values due to scattered flux from other annuli is significantly less than the statistical errors on the measured observables ( see table [ table:1tfits ] ) . 1.11.7 lcccc pointing & region 1 & region 2 & region 3 & region 4 + north & 1 & 4 & 8 & 19 + southeast & 0.7 & 3.4 & 25 & - + south & 4 & 20 & - & - + + + + [ table : bkgfluc ] to model the soft x - ray foreground and cosmic x - ray background , we jointly fit rass data with local xis background spectra , as described in detail in section [ sec : suzanalysis ] . we find that the local x - ray background is consistent with the rass data . however , spatial variations in the background level can introduce additional systematic uncertainties on x - ray observables . to estimate the effect of these uncertainties , we perform 10,000 monte carlo realizations of the background model . the model parameters are allowed to vary simultaneously within their 1@xmath20 uncertainty ranges obtained from the joint rass - local background fit . a variation of up to @xmath153.6% of the nxb level is also taken into account @xcite . the percent systematic uncertainty contributions due to the variance in cosmic , local , and particle background on the temperature estimates are given in table [ table : bkgfluc ] . we find that the effect on temperature and normalization is negligible ( @xmath15 1% ) and smaller than the statistical uncertainties in the inner regions ( shown in table [ table:1tfits ] ) , while it can be as large as 25% in the outskirts near @xmath2 . these uncertainties are included in the total error budget in our analysis by adding them in quadrature . 1.11.7 .estimated 1@xmath20 fluctuations in the cxb level due to unresolved point sources in the _ suzaku _ fov in units of @xmath45 ergs @xmath29 s@xmath8 deg@xmath41 . [ cols="<,^,^,^,^",options="header " , ] [ table : cxbfluc ] the intrinsic variations in the unresolved cxb component can be an important source of uncertainty in the analyses of cluster outskirts with _ suzaku_. to estimate the magnitude of this component , we follow a similar approach to that described in @xcite . the _ suzaku _ data alone allow us to detect point sources to a limiting flux of 1.3 @xmath46 ergs @xmath29 s@xmath8 deg@xmath41 in our observations . the contribution of unresolved point sources to the total flux in ergs @xmath29 s@xmath8 deg@xmath41 can be estimated as @xcite : @xmath47 the source flux distribution in the 2@xmath510 kev band is described by the analytical function @xmath48 \rm{erg\ cm^{-2}\ s^{-1 } } , \label{eqn : psflux}\ ] ] where @xmath49 , @xmath50 , @xmath51 , and @xmath52 . we then integrate equation [ eqn : cxbflux ] from a lower limit of @xmath53 ergs @xmath29 s@xmath8 ( the flux of the faintest source in our fov ) up to the upper limit of @xmath54 ergs @xmath29 s@xmath8 @xcite . the integration gives an unresolved 2 @xmath5 10 kev flux of ( 1.20 @xmath55 ergs @xmath29 s@xmath8 deg@xmath41 . the expected deviation in the cxb level due to unresolved point sources is @xmath56 where @xmath57 is the solid angle @xcite . using the power - law relation ( given in equation [ eqn : psflux ] ) in equation [ eqn : fluc ] , we calculate the 1@xmath58 rms fluctuations in the cxb ( given in table [ table : cxbfluc ] ) . we find that the variation is 4.3 @xmath43 erg @xmath29 s@xmath8 deg@xmath41 in the faintest outermost se3 region , which extends to @xmath2 . these estimates are consistent with the values reported by @xcite and @xcite . the 1@xmath20 uncertainty on the measured cxb ( from joint rass and local background fits ) is comparable to the expectation value of fluctuations on the cxb brightness calculated here . we include this variation in our monte - carlo markov realizations of the x - ray background to account for the cxb variation ( as described in detail in section [ sec : systcxb ] ) . the final systematic errors on the observed quantities were added in quadrature . [ fig : rsmap ] in this section , we describe our search for substructure in the optical data and use them to further constrain the dynamical state and merger history of a1750 and determine whether the subclusters are bound to each other . in figure [ fig : rsmap ] we overplot the spatial density of red sequence ( rs ) galaxies on the sky on the _ suzaku _ x - ray image . this map is created from a cloud - in - cell interpolation of the spatial distribution of red sequence selected galaxies on the sky , where galaxies are weighted by their r - band magnitudes ( brighter galaxies weighted more heavily ) . the resulting map of the surface density of red sequence light traces the collisionless galaxy component of the system . we then applied a broad gaussian smoothing kernel of 54@xmath31 to generate the contours shown in the image . the peaks observed in the red light distribution roughly align with the peaks in the x - ray emission . we note that we do not find strong evidence for extended filaments along the axis of the aligned clumps in rs light , thus no large - size groups are detected along the filament direction to the north . the lack of evidence indicates that the filaments do not contain significant large group - like structures with a detectable red sequence population . the large sample of spectroscopic redshifts available in the a1750 field provides an opportunity to investigate the dynamical state of cluster member galaxies . we first characterize the cluster member dynamics for the entire system by making an initial selection of cluster members that are within a projected physical radius of 1 mpc of the centroid of the x - ray emission of each subcluster ( this is approximately equal to the region covered by our hectospec observations ) , and which have redshifts in the interval 0.07 @xmath59 0.1 . we then use the bi - weight location and scale estimators @xcite as the starting guess for the median and dispersion of cluster member velocities . we then iterate this process , rejecting galaxies with redshifts more than 3-@xmath20 away from the median until the redshift sample converges . this results in an estimate of the velocity dispersion for the entire system of @xmath60 km s@xmath8 , and a median redshift of @xmath61 ( a recession velocity , @xmath62 km s@xmath8 ) based on 243 cluster member redshifts . . , width=302 ] [ fig : vdist ] 1.11.5 lcccc + region & @xmath63 & @xmath64 & @xmath65 & @xmath66 + & & & ( km s@xmath8 ) & ( km s@xmath8 ) + total system & 243 & 0.0861@xmath240.0028 & 780@xmath2430 & 0 + a1750n & 25 & 0.0832@xmath240.0006 & 750@xmath24160 & -810 + a1750c & 40 & 0.0864@xmath240.0004 & 835@xmath24120 & 90 + a1750s & 33 & 0.0868@xmath240.0019 & 532@xmath2460 & 200 + + + + [ table : memberdynamics ] given that we have hundreds of spectroscopic cluster members , we can also test for line - of - sight velocity differences between the three individual x - ray sub - clusters . we define subsets of spectra that originate from galaxies located in three non - overlapping 3@xmath14 radius circular regions on the sky that are centered on the x - ray peaks of each of the three distinct sub - clusters . for each of these regions , we use all of the cluster member galaxies that satisfy the @xmath243-@xmath20 velocity range for the total cluster system from above , and compute the bi - weight location and scale estimates of the median and dispersion in the galaxy velocities . these regions extend out radially @xmath15300 kpc from each x - ray centroid , and therefore only include a relatively small fraction of the full sample of 243 cluster member spectra ( between 25@xmath540 cluster members per subregion ) . the resulting kinematics estimates are given in table [ table : memberdynamics ] ; the central and southern x - ray clumps have redshifts that are similar to the median for the total system , but the northern clump is blue - shifted , with a peculiar velocity of @xmath67 km s@xmath8 ( see figure [ fig : vdist ] ) . the observed peculiar velocities imply that any relative motion between the central and southern clumps is in the plane of the sky , while the northern clump is moving at least partly along a vector that is normal to the sky . the velocity dispersion of galaxies within the total structure is not larger than the velocity dispersion of the individual clumps , indicating that the system is unrelaxed . the individual subclusters havent begun to virialize into the final larger cluster , i.e. the total mass of all three clumps is @xmath68 , and 840 km s@xmath8 is well below the velocity dispersion of a virialized structure of that mass . the shape of the velocity distributions within the different subcluster regions ( plotted in figure [ fig : vdist ] ) , suggest that northern and central subclusters are less well - structured ( with asymmetric velocity dispersion profiles ) than that of the southern subcluster . this could be due to some degree of interaction between the central and northern subclusters , while the southern subcluster may still be infalling ( i.e. , has not started tidally interacting with the other systems ) . we further calculated the implied virial masses of individual subclusters based on the velocity dispersions . using the @xcite scaling relations , the virial masses of a1750n , a1750c , and a1750s are 4.6@xmath69 m@xmath70 , 6.4@xmath71 m@xmath70 , and 1.7@xmath72 m@xmath70 , respectively . these masses are consistent with total masses of each subcluster obtained from x - ray observations ( see section [ sec : mass ] for detailed calculations ) . 1.11.5 cccccc + @xmath73&@xmath74 & @xmath75 & @xmath76 & @xmath77 & @xmath78 + ( rad ) & ( degrees ) & ( kpc ) & ( kpc ) & ( @xmath79 ) & ( @xmath80 ) + 4.542 & 71.73 & 2966.7 & 5072.2 & 931.2 & 22 + 5.319 & 23.86 & 1016.9 & 4727.4 & 2186.2 & 78 + + + + we apply a dynamical model introduced by @xcite and @xcite to evaluate the dynamical state of the subclusters a1750n and a1750c . this model allows us to estimate the most likely angle between the merger axis and the plane of the sky . the equations of motion take two different forms , depending on whether the subclusters are gravitationally bound or not . for the case where they are gravitationally bound , we parameterize the equations of motion in the following form : @xmath81 where @xmath76 is the subclusters separation at the moment of maximum expansion , @xmath82 is the system s total mass , and @xmath73 is the variable used to parametrize friedmann s equation , also know as development angle . for the case of not gravitationally bound subclusters , the equations are parametrized as : @xmath83 where @xmath84 is the velocity of expansion at the asymptotic limit . @xmath85 , the radial velocity difference , and @xmath86 , the projected distance , are related to the parameters of the equations by @xmath87 where @xmath74 is the projection angle of the system with respect to the plane of the sky . the virial mass of this subclusters is @xmath88 ( sum of the masses of both subclusters within @xmath2 uncertainties are quoted here as 68% confidence intervals ; see section [ sec : mass ] for detailed calculations ) derived from @xcite icm models . we assume that the subclusters velocities are the median velocities of their galaxies . the projected distance on the plane of the sky between the x - ray center of each subcluster is @xmath89 = 0.93 mpc . the difference of the median redshifts of these subclusters yields a radial velocity difference of @xmath90 . by setting @xmath91 = 12.4 gyr , the age of the universe at the mean redshift of these subclusters ( @xmath92 ) , we close the system of equation . the parametric equations are then solved via an iterative procedure , which computes the radial velocity difference @xmath93 for each projection angle @xmath74 . using simple energy considerations , we determine the limits of the bound solutions : @xmath94 figure [ fig : alpha_vr ] presents the projection angle ( @xmath74 ) as a function of the radial velocity difference ( @xmath93 ) between the subclusters . the uncertainties in the measured radial velocity and mass of the subclusters lead to a range in the solutions for the projection angles ( @xmath95 and @xmath96 ) . we compute the relative probabilities of these solutions by : @xmath97 where each solution is represented by the index @xmath98 . we then normalize the probabilities by @xmath99 . ) as a function of the radial velocity difference ( @xmath93 ) between the subclusters . uo , bi , and bo stand for unbound outgoing , bound incoming , and bound outgoing solutions . solid red and blue lines correspond to unbound and bound solutions , respectively . the vertical solid line corresponds to the radial velocity difference between the median velocities of the galaxies in each subcluster . dashed lines correspond to 68@xmath80 confidence ranges . [ fig : alpha_vr ] ] 1.11.5 cccccc + @xmath73&@xmath74 & @xmath75 & @xmath76 & @xmath77 & @xmath78 + ( rad ) & ( degrees ) & ( kpc ) & ( kpc ) & ( @xmath79 ) & ( @xmath80 ) + 1.692 & 86.45 & 15025.5 & 886.0 & 610.4 & 0.02 + + + + 1.11.5 ccccccc + @xmath73&@xmath74 & @xmath75 & @xmath76 & @xmath77 & @xmath78 & relative + ( rad ) & ( degrees ) & ( kpc ) & ( kpc ) & ( @xmath79 ) & ( @xmath80 ) & motion + 2.871 & 77.68 & 7871.0 & 8017.1 & 113.0 & 10 & outgoing + 3.385 & 74.76 & 6390.7 & 6486.5 & 114.4 & 15 & incoming + 4.987 & 4.39 & 1684.9 & 4620.9 & 1442.4 & 75 & incoming + + + + solving the parametric equations we obtain two bound solutions and one unbound solution . for the case of the bound solutions , the subclusters are either approaching each other at 931 km s@xmath8 ( 22@xmath80 probability ) or at 2186 km s@xmath8 ( 78@xmath80 probability ) . the former solution corresponds to a collision in less than 3.1 gyr , given their separation of @xmath15 2.97 mpc . the latter corresponds to a collision in less than 460 myr , given their separation of @xmath15 1017 kpc . the unbound solution ( 0.02@xmath80 probability ) corresponds to a separation of @xmath15 15 mpc . the parameters of these solutions are presented in tables [ tab : bound ] and [ tab : unbound ] . given its very low probability , the unbound solution can be neglected , while the bound solution in which the separation between the clusters is @xmath15 1017 kpc is highly favored ( @xmath100 probability ) . as mentioned in @xcite , the method to determine the dynamical state of a system of clusters from @xcite assumes a purely radial infall . also , the way the probabilities are computed , by integrating over the angles determined by the uncertainties on the mass of the system , favors small angle solutions . therefore , the probabilities for the solutions should be treated with caution , as we have no information about the angular momentum of this subclusters . now , we apply the same procedure to determine the dynamical state of the pair a1750c - a1750s . using the virial mass estimated from the velocity dispersion of the galaxies in the southern subcluster , the total mass of this system is @xmath101 ( uncertainties are quoted here at the 68% confidence level ) . the difference between the median redshifts of these subclusters yields a radial velocity difference of @xmath102 . solving the system of parametric equations ( equations ( [ eq : r_bound ] ) ( [ eq : v_r_alpha ] ) ) yields the results presented in table [ tab : bound2 ] , with a1750c - a1750s being bound in all solutions . the most likely solution ( 75% probability ) indicates that the merger is happening very close to the plane of the sky ( @xmath103 degrees ) , also supporting the scenario in which all three subclusters are merging along a cosmic filament . we extract spectra in concentric annular sectors along the north ( filament ) , south ( filament ) , and southeast ( off - filament ) directions from the regions shown in figure [ fig : image ] . each spectral extraction region is selected to include at least 2000 net source counts . the lhb+cxb+gh components are fixed to the values determined from fits to local background and rass data as described in section [ sec : suzanalysis ] . we stress that the systematic errors are included as explained in section [ sec : syst ] . the _ suzaku _ spectra are fitted using an absorbed single temperature ( 1 t ) model with free temperature , abundance , and normalization . we first examine the _ suzaku _ spectra extracted along the north direction starting from the center of a1750n . the spectra are extracted from four consecutive annular sectors ; 0@xmath6@xmath52.5@xmath6 , 2.5@xmath6@xmath55@xmath6 , 5.0@xmath6@xmath57.5@xmath6 , and 7.5@xmath6@xmath512.5@xmath6 . the total source counts in the co - added fi observations in regions n1 , n2 , n3 , and n4 are 3300 , 4400 , 2700 , and 3300 , respectively . the bi spectra in the same regions have total source counts of 2400 , 3300 , 2100 , and 2600 . both fi and bi spectra of the outermost 7.5@xmath6 @xmath5 12.5@xmath6 region are dominated by the nxb background at @xmath176 kev , thus this band is excluded from further analysis . 1.11.4 lcccc + region & @xmath104 & abund & @xmath105 & c - stat + & ( kev ) & ( a@xmath106 ) & ( @xmath107 @xmath35 ) & ( dof ) + + + n@xmath108 & 3.33 @xmath109 & 0.28 @xmath24 0.5 & 110.18 @xmath24 2.95 & 178.10 ( 177 ) + n@xmath110 & 2.80 @xmath111 & 0.15 @xmath24 0.4 & 48.87 @xmath24 1.70 & 183.31 ( 244 ) + n@xmath112 & 1.98 @xmath24 0.18 & 0.2@xmath3 & 17.55 @xmath24 0.77 & 272.86 ( 165 ) + n@xmath113 & 1.61 @xmath24 0.30 & 0.2@xmath3 & 6.08 @xmath24 1.09 & 322.35 ( 241 ) + + s@xmath108 & 2.61 @xmath24 0.21 & 0.19 @xmath24 0.08 & 22.28 @xmath24 1.79 & 148.68 ( 144 ) + s@xmath110 & 2.04 @xmath114 & 0.20 @xmath115 & 2.76 @xmath24 1.15 & 201.25 ( 187 ) + + se@xmath108 & 4.72 @xmath116 & 0.31 @xmath24 0.01 & 154.79 @xmath24 5.33 & 712.29 ( 746 ) + se@xmath110 & 4.83 @xmath24 0.40 & 0.2@xmath3 & 31.93 @xmath24 1.44 & 776.43 ( 747 ) + se@xmath112 & 2.47 @xmath117 & 0.2@xmath3 & 4.26 @xmath24 1.10 & 495.63 ( 473 ) + + + + + [ table:1tfits ] to investigate the nature of the gas along the filament , we first fit the fi and bi spectra simultaneously with a 1 t model . the parameters of the fi and bi spectral models are tied to each other . the abundances are only constrained by the observations in regions n1 and n2 . the best - fit temperatures are 3.33@xmath109 kev and 2.80@xmath111 kev , respectively . a 1 t model produces an acceptable fit to the spectra of the innermost two regions . adding an additional model does not significantly improve the fits for these regions . the model parameters are given in table [ table:1tfits ] . 1.11.7 lccccc + region & @xmath118 & abund & @xmath119 & @xmath120 & @xmath121 + & ( kev ) & a@xmath106 & ( @xmath107 @xmath35 ) & ( kev ) & ( @xmath107 @xmath35 ) + + + n@xmath112 & 3.24@xmath122 & 0.1@xmath123 & 10.32@xmath242.05 & 1.01@xmath124 & 9.19@xmath125 + n@xmath112 & 2.93@xmath126 & 0.2@xmath123 & 11.76@xmath241.66 & 0.99@xmath240.07 & 4.87@xmath127 + n@xmath112 & 2.93@xmath128 & 0.3@xmath123 & 11.75@xmath241.53 & 0.95@xmath240.08 & 3.29@xmath129 + n@xmath113 & 1.95@xmath130 & 0.1@xmath123 & 4.81@xmath131 & 0.79@xmath132 & 2.98@xmath133 + n@xmath113 & 2.12@xmath134 & 0.2@xmath123 & 4.53@xmath135 & 0.81@xmath240.12 & 1.74@xmath136 + n@xmath113 & 2.29@xmath137 & 0.3@xmath123 & 4.21@xmath24 1.33 & 0.80@xmath138 & 1.28@xmath24 0.25 + + [ table:2tfits ] error bars of _ xmm - newton _ and _ suzaku _ temperatures include systematic and statistical uncertainties . temperatures reported by three satellites are in a good agreement . we are able to extend the gas temperature measurements out to 0.9 @xmath2 of a1750n ( r@xmath139 ) and @xmath2 of a1750c ( r@xmath140 ) clusters.,width=325 ] a 1 t model produces best - fit temperatures of 1.98@xmath240.18 kev and 1.61@xmath240.30 kev in regions n3 and n4 , respectively . abundances are not constrained ; we therefore assume an abundance of 0.2a@xmath106 , as observed in the outskirts of low mass clusters @xcite . the projected temperature profile to the north is shown in figure [ fig : projtprof ] . we compare the _ suzaku _ results with those from _ chandra _ ( this work ) and _ xmm - newton _ @xcite . we note that the _ chandra _ results shown in figure [ fig : projtprof ] do not include the systematic uncertainties , and are shown here for a rough check on the _ suzaku _ temperature estimates . we find good agreement between measurements from each satellite . we note that both _ suzaku _ and _ chandra _ observations cover the radial range out to 0.9 @xmath2 ( @xmath15 14@xmath14 , see section [ sec : mass ] ) , and the best - fit temperatures measured by _ suzaku _ and _ chandra _ are in agreement at the 1@xmath20 confidence level . however , since _ suzaku _ has a lower background at large radii ( @xmath15@xmath2 ) and more precise temperature measurements ( i.e. , smaller systematic+statistical uncertainties ) , we will use _ suzaku _ temperature and density measurements hereafter . the residuals in the spectrum after a model fit in the softer 0.5 @xmath5 2.0 kev band ( shown in the left panel of figure [ fig : nn3 ] ) suggest the possible presence of a second , cooler thermal component in the regions n3 and n4 . to investigate this , we add another absorbed component to the model ( 2 t ) and re - do the fit . both the temperature and the normalization of the second component are left free , while the abundances are tied to each other between the two models . the best - fit parameters of the 2 t model and the improvement in the fits are given in table [ table:2tfits ] . figure [ fig : nn3 ] ( right panel ) shows the improvements in the fits of both region n3 and n4 . the temperature of the primary component increases from 1.98 @xmath24 0.09 kev to 2.93@xmath126 kev , while the temperature of the secondary component is estimated to be 0.99 @xmath24 0.07 kev in region n3 . the change in the goodness of the fit statistics is significant , with a @xmath141c - statistic of 64.5 for an additional two d.o.f . the c - statistic value does not provide a statistical test to quantify the significance of the improvement in the fit from adding the second component , thus we calculate the corresponding @xmath142 values before and after addition of the secondary model . we find that adding two d.o.f . ( additional temperature and its normalization ) improves the @xmath142 by 28.5 . in region n4 , the best - fit temperature of the primary _ apec _ becomes 2.12@xmath134 kev in the 2 t fits , while the temperature of the secondary component is 0.81 @xmath24 0.12 kev . the @xmath143 value of 24.4 with an additional two d.o.f . , corresponding to a null hypothesis probability of @xmath144 , suggests that the detection is significant . the best - fit parameters of these 2 t models are summarized in table [ table:2tfits ] . the derived _ xspec _ normalizations , i.e. emission measures , and temperatures depend on the assumed metallicity . we provide the measurements of these observables for various solar abundance fractions . we note that the assumed metallically does not have a significant impact on temperature or emission measure of the hotter component in our fits . the discussion of the nature of this gas is provided in section [ sec : fila ] . considering that the calibration of xis below 0.7 kev is uncertain , we re - perform the 1 t and 2 t model fits in the n3 and n4 regions to investigate the effect of this uncertainty on the temperature and normalization ( i.e. density ) . fixing the abundance at 0.2 @xmath145 , we find that the temperatures and normalizations of both models are consistent with results from the 0.57 kev band fits within the total ( statistical plus systematic ) uncertainties . the results from the 1 t and 2 t model fits in the 0.77 kev band are given in table [ table : fits-0p7 ] . we conclude that the detection of the cooler @xmath151 kev gas is not significantly affected by the effective area uncertainties below 0.7 kev . 1.11.7 lccccc + region & @xmath118 & @xmath119 & @xmath120 & @xmath121 + & ( kev ) & ( @xmath107 @xmath35 ) & ( kev ) & ( @xmath107 @xmath146 ) + + + n@xmath112 & 1.96@xmath240.18 & 16.45@xmath147 & @xmath5 & @xmath5 + n@xmath112 & 2.90@xmath148 & 10.82@xmath242.19 & 0.99@xmath149 & 4.75 @xmath24 1.21 + n@xmath112 & 1.59@xmath240.29 & 5.73@xmath241.12 & @xmath5 & @xmath5 + n@xmath113 & 2.09@xmath150 & 4.22@xmath151 & 0.79@xmath152 & 1.73@xmath153 + + [ table : fits-0p7 ] to investigate the x - ray emission along the filament to the south , we extract spectra from two annular sectors ( regions s1 and s2 ) extending south from the center of a1750s . these regions are shown in figure [ fig : image ] . region s1 extends from the cluster core to 4@xmath6 , and region s2 extends from 4@xmath6 to 9.7@xmath6 . the source counts in the combined fi and bi observations are 2600 and 1700 in region s1 , and 2200 and 2000 in region s2 . we first fit the spectra with a 1 t model . the best - fit temperatures of 2.61 @xmath24 0.21 kev and 2.04@xmath154 kev , and abundances of 0.19 @xmath24 0.08 a@xmath106 and 0.20@xmath115 a@xmath106 are measured in regions s1 and s2 , respectively . the results are shown in table [ table:1tfits ] with the goodness of the fits . abundance measurements of 0.2a@xmath106 are consistent with the abundances measured in low mass systems @xcite . the possible presence of the cool @xmath15 1 kev gas is tested by performing 2 t fits . the additional secondary model does not significantly improve the fits . unlike the detection in the north , we find no evidence for such a component in the south . the x - ray emission to the southeast , perpendicular to the putative large - scale filament , is examined using spectra extracted in annular sectors ( se1 , se2 , and se3 shown in figure [ fig : image ] ) with radii of 0@xmath6@xmath54@xmath6 , 4.0@xmath6@xmath58.0@xmath6 , and 8@xmath14 extending out to @xmath2 ( @xmath1516@xmath6 ) of the central sub - cluster . the total source counts in the fi and bi observations are 7800 and 5000 in region se1 , and 4000 and 2600 in region se2 , and 3000 and 2500 in region se3 . to study the nature of the gas along the off - filament direction we followed a similar approach to that outlined in section [ sec : fila ] . the fi and bi spectra of each region are first fit with a 1 t model . the best - fit parameters and the goodness of these fits are given in table [ table:1tfits ] . the temperature and abundance in the innermost region are 4.72@xmath116 kev and 0.31 @xmath24 0.01 a@xmath106 . the best - fit temperature of the se2 region is 4.83 @xmath24 0.40 kev . unlike in region se1 , we are not able to constrain the abundance in region se2 , thus the abundance parameter is fixed at 0.2a@xmath106 . to test if the best - fit temperature is sensitive to the assumed metallicity , we perform the fit with abundances of 0.1a@xmath106 and 0.3a@xmath106.the best - fit temperature declines to 4.73 @xmath24 0.39 kev for an assumed abundance of 0.1a@xmath106 , while it increases to 4.95 @xmath24 0.38 kev for an abundance of 0.3a@xmath106 . however , the change in the measured temperature is not statistically significant . the spectrum from region se3 are dominated by the nxb above 5 kev . therefore , we perform our fits in the 0.5@xmath55 kev energy band in this region . the best - fit temperature is 2.47@xmath117 kev for an assumed abundance of 0.2a@xmath106 . the temperature is 2.56@xmath155 kev and 2.85@xmath156 kev for fixed abundances of 0.1a@xmath106 and 0.3a@xmath106 , respectively . the temperatures for our assumed abundances are all consistent within the 1@xmath20 level . in all cases , we observe a significant sharp decline in the projected temperature at @xmath157 ( 10.6@xmath14 ; see section [ sec : mass ] ) to the southeast . taking a similar approach as in section [ sec : fila ] , we fit the spectra of the outermost regions se2 and se3 with a 2 t model . the temperature of the secondary component is not constrained , and this addition does not improve the fit significantly . thus , we find no evidence for a softer thermal component in the off - filament direction . to further test if the @xmath151 kev gas detected along the filament to the north is observable along the off - filament southeast direction , we scale the normalization of the softer component detected in region n4 ( see table [ table:2tfits ] ) by the ratio of the area of regions se3 and n4 . freezing the normalization to the scaled value of 1.5@xmath158 @xmath35 and the observed temperature to 0.99 kev , we refit the fi and bi spectra of the se3 region . the temperature and normalizations of the primary component are unconstrained after the fit is performed . the sharp decline in the goodness - of the fit ( c - statistics value of 4931.15 for 471 d.o.f . ) suggests that if the @xmath151 kev gas detected along the filament direction existed in this region with the same surface brightness , it would be detected . thus , this component is clearly absent in the off - filament direction . we investigate the distribution of the gas temperature between a1750n @xmath5 a1750c and between a1750c @xmath5 a1750s along the merger axis . we define rectangular regions along the line connecting the centroids of the three sub - clusters ( figure [ fig : bridge ] , right ) , which are marked with dashed lines in figure [ fig : bridge ] , left . we fit the spectra of the selected regions using a 1 t model . figure [ fig : bridge ] ( left panel ) displays the projected temperature as a function of distance . we find that , starting from the northernmost region , the temperature keeps rising towards the center of a1750n , and reaches a peak temperature of 3.37 @xmath24 0.10 kev . due to the large psf of _ suzaku _ , we can not rule out or confirm the suggestion that a1750n is a cool core cluster @xcite . continuing past a1750n , the temperature rises up to 5.49 @xmath24 0.59 kev with a sharp increase at @xmath156@xmath6 ( @xmath15 0.5 mpc ) . this increase in the temperature is significant at a level of 2.7@xmath20 . hot , presumably shock - heated gas between a1750n and a1750c , coinciding with the location where we detect hot gas with _ suzaku _ , has previously been observed in _ chandra _ and _ xmm - newton _ data @xcite . the presence of hot gas in this region is an indication of an interaction between the a1750n and a1750c sub - clusters . a1750c shows a relatively uniform temperature around the centroid , with a peak temperature of 4.25 @xmath24 0.16 kev . we detected another temperature peak located 7@xmath6 away from a1750c , in the southwest direction , with a temperature of 4.74 @xmath24 0.70 kev . southwest of this peak , the temperature declines to 3.19 @xmath240.42 kev . this sharp decrease is significant at a 4@xmath20 level , suggesting an interaction between the sub - clusters a1750s and a1750c . a hot region , where the peak detected by _ suzaku _ observations , was previously detected in the vicinity of a1750c @xcite . due to large error bars on the temperature ( 5.7@xmath159 kev ) , the authors were unable to determine the true nature of the structure and claimed that it could due to a point source . similarly a hot region was observed in _ data @xcite coinciding with the reported location of the peak . here we confirm the extended nature of the emission and suggest a potential interaction between a1750c and a1750s . although , we note that the optical data do nt show any evidence of interaction between these clusters ( see section [ sec : optspec ] for discussion ) . the projected temperature continues to decline towards the center of the southern sub - cluster a1750s . the central temperature of a1750s is 2.93 @xmath24 0.21 kev . the radial temperature profile shows that the temperature decreases smoothly moving across the center of a1750s towards the southwest . to examine the radial profiles of cluster masses and thermodynamical quantities such as entropy and pressure , we determine the deprojected density and temperature . the electron density is obtained from the best - fit normalization @xmath160 of the model in _ xspec _ using the relation , @xmath161 where @xmath162 is the angular size distance to the source in units of cm , and @xmath163 and @xmath164 are the electron and hydrogen number densities in units of @xmath165 . we note that the arfs generated by _ assume a uniform source occupying an area of 400@xmath166 square arcminutes . we therefore apply a correction factor to each region and normalization prior to deprojection . an ` onion - peeling ' method is used to deproject the temperature and density profiles @xcite . the resulting deprojected density and temperature profiles to the north , southeast , and south directions are shown in figure [ fig : deprojprof ] . we extend the temperature and density profiles out to 0.9@xmath2 for a1750n to the north and @xmath2 for a1750c to the southeast with the new _ suzaku _ observations ( see in figure [ fig : image ] ) . the temperature profiles to the north and southeast decline with radius and reach half of the peak value at @xmath2 . similar temperature declines have been reported for other clusters ( e.g. , * ? ? ? * ; * ? ? ? we observe a rather gradual decline in temperature to the north and south . however , the profile to the southeast indicates a uniform temperature within 8@xmath6 and falls relatively rapidly beyond @xmath167 . based on the average deprojected density and temperature , we estimated the mass of each sub - cluster within @xmath168 using the @xmath169 scaling relation ( * ? ? ? * v09 hereafter ) . the spectra between 0.15 @xmath5 1@xmath168 are extracted to determine the global properties for each cluster . to avoid flux contamination , adjacent sub - clusters were excluded . a1750n has a best - fit global temperature of 3.14@xmath170 kev , and an abundance of 0.15 @xmath24 0.03 a@xmath106 . our measurement is consistent with the temperature of 3.17 @xmath24 0.1 kev reported in @xcite . the scaling relation predicts a total mass of 1.98 @xmath171 at @xmath168 ( 9.3@xmath14 ) . the best - fit temperature of a1750c is 4.15@xmath172 kev , with an abundance of 0.21@xmath173 a@xmath106 . the global temperature reported in @xcite is slightly lower ( @xmath174 kev ) . their extraction region excludes the hotter plasma between a1750n and a1750c , which may account for the difference observed in temperature . the v09 scaling relation predicts a total mass of 3.03 @xmath171 enclosed within @xmath175 ) . the spectral fit to a1750s gives a best - fit temperature of 3.59@xmath176 kev and an abundance of 0.20@xmath177 . the estimated total mass within @xmath168 ( 9.9@xmath14 ) is 2.43 @xmath171 . lcccccc + & north & southeast + + @xmath178 ( @xmath179 @xmath165 ) & 1.78 @xmath180 & 2.19 @xmath181 + @xmath182 ( kev ) & 3.89 @xmath24 0.22 & 5.49 @xmath24 0.27 + @xmath183 & 4.49 @xmath184 & 6.01 @xmath185 + @xmath186 ( arcmin ) & 300@xmath3 & 480@xmath3 + @xmath187 & 2.0@xmath3 & 2.0@xmath3 + @xmath142 ( dof ) & 5.35 ( 5 ) & 3.21 ( 3 ) + + [ table : b10params ] [ fig : fits ] to investigate the radial behavior of the gas mass , the total mass , and the gas mass fraction , we employ a physically motivated icm model described in ( * ? ? ? * ; * ? ? ? * b10 , hereafter ) . the b10 model is based on the assumption that the icm is a polytropic gas in hydrostatic equilibrium in the cluster s gravitational potential . the deprojected density and temperature profiles are fit simultaneously using the b10 model . the fitting was performed using a markov chain monte - carlo ( mcmc ) approach , with metropolis - hastings sampling , to determine posterior distributions for the best - fit model parameters . the temperature profile is @xmath188 , \label{eqn_polytropic_temperature}\ ] ] where the normalization constant @xmath189 is @xmath190 using the relation between temperature and gas density provided by the polytropic relation , the gas density is @xmath191^{n},\ ] ] where @xmath187 + 1 is the slope of the total density distribution , @xmath183 is the polytropic index , @xmath186 is the scale radius , and @xmath189 and @xmath178 are the central temperature and density of the polytropic function . this model has sufficient fitting flexibility to describe x - ray data , while making simple physical assumptions @xcite . we note that the core taper function in the b10 model is omitted in the fits performed in this work , since observations are not able to resolve the cluster cores . figure [ fig : fits ] shows the best - fit models to the density ( left panel ) and temperature ( right panel ) in the off - filament and filament directions . @lcccccccc cluster & @xmath168&@xmath192 & @xmath193 & @xmath194 & @xmath2 & @xmath195 & @xmath196 & @xmath197 + & ( arcmin ) & ( @xmath198 ) & ( @xmath199 ) & & ( arcmin ) & ( @xmath198 ) & ( @xmath199 ) + + a1750n & 9.3@xmath14 & 1.86 @xmath24 0.38 & 1.54 @xmath200 & 0.12 @xmath201 & 14.1@xmath14 & 3.41 @xmath202 & 2.32 @xmath203 & 0.15 @xmath204 + + a1750c & 10.6@xmath14 & 3.15 @xmath205 & 3.04 @xmath206 & 0.10 @xmath201 & 16.2@xmath14 & 5.46 @xmath24 0.16 & 4.85 @xmath207 & 0.11 @xmath208 + + [ table : mass ] due to the limited number of data points compared to the number of free model parameters of the b10 model ( five in this case ) , we were not able to constrain all of the free parameters of the model . the @xmath187 parameter is fixed to the slope of the navarro - frenk - white profile @xcite , while the scale radius r@xmath209 ( fixed in our fits ) , the radius beyond which the temperature starts declining , is estimated from the temperature profiles ( see figure [ fig : fits ] ) . the rest of the model parameters ( @xmath183 , @xmath178 , and @xmath182 ) are allowed to vary independently . the best - fit parameters of the model are given in table [ table : b10params ] , along with the goodness of the fits . the best - fit models for the density and temperature profiles are displayed in figure [ fig : fits ] , with 90% confidence intervals . given the limited number of data points , the profiles to the south are not constrained . the total mass enclosed within radius @xmath23 is @xmath210 . \end{aligned } \label{eqn : totalmass}\ ] ] the normalization factor for the total matter density is @xmath211 $ ] . the gas mass @xmath212 is computed by integrating the gas density profile within the volume , @xmath213 where @xmath214 and @xmath215 are the mean molecular weight per electron and the proton mass . the gas mass fraction is @xmath216 the gas mass , total mass , and @xmath217 are measured at @xmath168 , determined using the v09 scaling relations , and are given in table [ table : mass ] . following @xcite , we assume @xmath168 = 0.659@xmath2 . the total mass , gas mass , and gas mass fraction profiles are plotted in figure [ fig : mass ] . we find that the total masses enclosed within @xmath168 are well within agreement with the total masses estimated using the v09 scaling relations . the gas mass fractions of a1750n and a1750c are consistent with the gas mass fraction expected for clusters in this mass range based on the v09 scaling relations ( @xmath218 0.11 ) at @xmath168 . the b10 model was then used to calculate the masses and mass fraction at @xmath2 . we found that the gas mass fraction of a1750c and a1750n at @xmath2 is 0.11@xmath208 and 0.15@xmath204 . the virial masses of the a1750n and a1750c subclusters are in agreement with the mass estimates from the optical observations at a 2.7@xmath20 level ( see section [ sec : optspec ] ) . however , we note that , the cluster mass inferred from x - ray analysis depends on the geometry of the merger , hydrostatic equilibrium , and other model parameters ( e.g. scale radius ) of the merging clusters . the gas fractions derived in the filament and off - filament directions are consistent with the cosmic baryon fraction derived from wmap seven - year data of 0.166 @xcite . similarly , gas mass fractions consistent with the cosmic value , were observed in rx j1159 + 5531 @xcite , a1689 @xcite , and a1246 @xcite . however , we note that the total mass estimates are based on a few assumptions on the distribution of the gas properties . spherical symmetry and isotropy are assumed when calculating these masses . such assumptions may bias our results , particularly in a merger system at large radii . [ fig : mass ] the entropy ( @xmath219 ) and pressure ( @xmath220 ) profiles are calculated using the electron density ( @xmath163 ) and deprojected temperature ( @xmath104 ) . the profiles along the filament and off - filament directions are shown in figure [ fig : deprojentropress ] . in the absence of non - gravitational processes , such as radiative cooling and feedback , cluster entropy profiles are expected to follow the simple power - law relation @xmath221 where we assume a cosmic baryon fraction of @xmath222 = 0.15 , with a characteristic entropy of @xmath223 @xcite . we used an @xmath224 ( the total mass within @xmath168 of a1750c ) of 3 @xmath171 , as determined in [ sec : mass ] . the resulting expected self - similar entropy profile for a1750c is shown as the dashed lines in figure [ fig : deprojentropress ] ( left ) . we find that the entropy along the filament directions ( to the north and south ) and off - filament direction derived from _ suzaku _ data alone are in good agreement with each other within @xmath225 . profiles obtained from _ xmm - newton _ observations are consistent with those from _ suzaku _ data within @xmath226 . the observed entropy exceeds the self - similar model prediction within @xmath227 , which we attribute to the influence of non - gravitational processes ( e.g. , agn feedback , infalling substructures due to violent merging events ) in the subcluster cores . such an influence on the entropy profiles of a sample of low redshift clusters ( @xmath228 0.25 ) was reported by @xcite . the entropy profiles follow a flatter profile beyond a radius of @xmath229 , and become consistent with the self - similar model , both along the northern filament and off - filament directions . we find that the entropy profile towards the northern filament reaches the self - similar level at smaller radii ( @xmath150.4r@xmath230 ) as compared with the off - filament direction . this may be due to the lower temperature gas ( @xmath151 kev ) observed to the north , which biases the average temperature low , and depresses the measured value of the entropy . the entropy profile along the off - filament direction stays above the self - similar expectation to @xmath231 . beyond this radius it remains consistent with the self - similar prediction . if the entropy contribution from the cool gas detected to the north is removed , the entropy rises to 1245.6 @xmath24 486.5 kev @xmath232 ( shown in figure [ fig : deprojentropress ] with the dashed data point in red ) and becomes more consistent with the entropy to the southeast . this provides evidence that the cool gas does indeed lead to a slight decrease in the entropy , although not at the level seen in other systems where it is likely arises from gas clumping @xcite . [ fig : deprojentropress ] unlike the rising , self - similar entropy observed in a1750 , a flattening of entropy profiles near @xmath2 appears to be a common feature in other relaxed and disturbed clusters ( for a review , see * ? ? ? * ) . a few detailed studies of nearby bright merging systems have probed the physical properties of the icm at large radii , e.g. the coma cluster @xcite and the virgo cluster @xcite . _ xmm - newton _ observations of the dynamically young cool virgo cluster revealed a suppressed entropy profile beyond 450 kpc by a factor of 22.5 below the expectation from pure gravitational collapse models . authors attributed this flattening to gas clumping at large radii . in the merging coma cluster , @xcite find no evidence for entropy flattening along the relatively relaxed directions , although due to large uncertainties they are unable to exclude entropy flattening at the level of what is observed in some relaxed clusters . there has been great effort in the literature to explain the seeming ubiquity of flattened entropy profiles at large radii . in the hierarchical model of structure formation , clusters form by accreting material from their surrounding large - scale structure . accretion of infalling subhalos can cause gas motions and clumpiness " around @xmath2 . these subhalos tend to have lower temperature and higher density than the surrounding icm , leading to a bias towards lower temperatures and higher densities in the emission measure - weighted spectra , if the subhalos are unresolved . the level of gas inhomogeneities is characterized through the clumping factor ( @xmath233 ) . as a result of overestimation of density , the gas mass , and subsequently the gas mass fraction , are biased high ( i.e. above the cosmic baryon fraction ) . the observed excess in the gas mass fraction ( @xmath234 ) in the _ suzaku _ observations of the perseus cluster was explained with a very large clumping factor of 3 @xmath5 4 around @xmath2 @xcite . @xcite reported that the expected clumpiness factor at @xmath2 can be as large as 2 and confirmed the flattened entropy profiles beyond @xmath235 in their non - radiative and cooling+star - formation simulations . however , @xcite examined entropy profiles for a sample of relaxed clusters at @xmath236 0.25 out to @xmath2 and concluded that the gas clumping calculated in the numerical simulations is insufficient to reproduce the observed flattening of the entropy . an alternative explanation to the flattening was proposed by @xcite and @xcite , and is based on the electron - ion non - equilibrium in the cluster outskirts . if the energy is not transferred to the electrons through electron - ion collisions sufficiently rapidly , the electron temperature remains low compared to that of ions , leading to an apparent entropy suppression at @xmath2 . @xcite and @xcite proposed that the flattening in the entropy is a result of a weakened accretion shock as it expands . the bulk energy carried along with the shock increases the turbulence and non - thermal pressure support in the outskirts , but the shock is not energetic enough to raise the intra - cluster entropy . the decreasing thermalization in low - density regions results in a tapered entropy around @xmath15@xmath2 . this claim supports the observed azimuthal variations in entropy in cool - core clusters @xcite and in the non - cool core coma cluster @xcite . other proposed explanations of entropy flattening include a rapid radial fall of the gas temperature caused by non - gravitational effects @xcite and cosmic - rays consuming as a significant sink for the kinetic energy in the outskirts @xcite . on the other hand , @xcite have performed a joint planck sz and rosat x - ray analysis of 18 galaxy clusters and concluded that entropy profiles are consistent with a self - similar power - law increase expected from pure gravitational infall . the discrepancy between the @xcite and the @xcite results is due to the differing dependence on sz and x - ray signals to the electron pressure used to derive entropy profiles @xcite . self - similar entropy profiles at @xmath2 have been previously observed in low mass relaxed fossil groups , e.g. rx j1159 + 5531 @xcite . on the other hand , the entropy of morphologically relaxed groups has been found to be significantly higher than self - similar at @xmath237 @xcite . however , massive mergers ( @xmath238 ) are expected to have a higher level of gas clumping , since they have a larger fraction of lower - temperature gas that is not detectable in the x - ray band @xcite . although a1750 is a dynamically young , massive system , we do not find evidence for gas clumping in this merger system . entropy profile measurements along the off - filament and filament directions are in agreement with each other and with the universal expectation with a power - law relation @xmath239 . remarkably in a1750 , the entropy profiles within @xmath2 do not seem to have been influenced by the apparent filamentary structure of the system . our results suggest that gravitational collapse is the main driver of the temperature and density profiles in the outskirts . we also examine the pressure profiles along the off - filament and filament directions . pressure profiles are calculated assuming an ideal gas law with @xmath240 , and compared to the universal pressure profiles of @xcite ( a10 , hereafter ) and @xcite ( planck13 , hereafter ) for clusters with mean redshifts of 0.11 and 0.17 , respectively . the a10 universal pressure profile is @xmath241^{2/3+\kappa}\\ & \times \mathbb{p}(r / r_{500})\ h_{70}^{2 } \ \ \rm{kev}\ { cm^{-3 } } , \end{aligned } \label{eqn : arnaud}\ ] ] where the scaled pressure profile is characterized based on the generalized navarro - frenk - white profile @xcite @xmath242^{(\beta-\gamma)/\alpha}},\ ] ] and @xmath243 with best - fit parameters of @xmath244 = 8.403@xmath245 , @xmath246 = 1.177 , @xmath247 = 0.3081 , @xmath74 = 1.0510 , and @xmath187 = 5.4905 . the first term in equation [ eqn : kappa ] , @xmath248 , is an approximation which depends on the departures from the standard scaling relations , while the second term , @xmath249 , represents a break from self - similarity . since non - gravitational processes become less dominant at large radii , the latter term is negligible at @xmath250 . the a10 universal pressure profile primarily samples the inner regions , while the planck13 profile samples the cluster outskirts . the pressure profiles derived from the planck observations for a sample of 62 galaxy clusters found slightly higher pressure than that predicted by a10 in the outskirts of clusters . these profiles were obtained by averaging pressure profiles from all azimuths for a large sample of clusters with different dynamical states . the dispersion over the universal profiles can be as large as 100 % at @xmath15@xmath168 ( see figure 8 in a10 ) . we compare the pressure profiles of a1750 with the universal profiles of a10 and planck13 in the right panel of figure [ fig : deprojentropress ] . while the pressure profile along the filament direction to the north agrees with the universal profile , the profile along the off - filament direction is higher and the profile to the south is lower than the expectation within @xmath1 0.2@xmath2 . on the other hand , the profile in the filament direction to the south and to the north is consistent with the a10 and planck13 universal profiles at large radii ( @xmath15@xmath2 ) at the 2.7@xmath20 level . the pressure to the southeast exceeds the universal models at all radii . pressure excesses at large radii have been previously reported in other relaxed clusters , e.g. pks 0745 - 191 @xcite , the centaurus cluster @xcite , and the fossil group rx j1159 + 5531 @xcite , and were attributed to gas clumping . figure [ fig : deprojprof ] indicates that the excess in the pressure along the southeast direction compared to the north or south directions in a1750 is due to high temperature ( not high density ) . on the contrary , clumping ( if it existed in this system ) would bias the density measurements high , leading to an excess in pressure and a decrement in entropy in the outskirts . therefore , the deviation from the universal profile in a1750 is unlikely to be due to clumpy gas , since other evidence for clumping , e.g. entropy flattening and an excess in gas mass fraction ( see section [ sec : mass ] ) , are not observed in this system . we note that @xcite reported the detection of a weak @xmath251 , shock resulting from a merger event intrinsic to a1750c along the southeast direction . this merger event may elevate the temperature and cause deviations from the universal profile . in any event , given the large dispersion among pressure profiles of clusters in the a10 and planck13 samples , we do not expect the pressure profiles derived in a1750 in perfect agreement with their results . the cool @xmath15 1 kev gas detected in regions n@xmath112 and n@xmath113 ( see figure [ fig : nn3 ] ) may be 1 ) the hot dense whim connecting a1750n to the large - scale filament , 2 ) stripped icm gas formed as a result of infalling groups , or 3 ) gas stripped from a1750n itself , as it interacts with filament gas or with a1750c . the feature is relatively extended with an observed radial range of @xmath17 0.62 mpc . assuming a geometry for the merger system , the mass of the feature can be calculated ( see section [ sec : dynamics ] for the detailed calculation ) . assuming that the density of the feature is constant within each region ( 5.56 @xmath252 @xmath165 ) , and can be described as a cylinder that extends to 1.2 mpc with a line - of - sight depth of the structure ( _ l _ ) , we obtain a gas mass of 4.13 @xmath253 . the observed flux of the feature ( 1 @xmath254 erg @xmath29 s@xmath8 arcmin@xmath41 ) , density , and temperature ( @xmath150.8 kev ) are consistent with the expected surface brightness and temperature of the dense portion of the whim , where the large - scale structure interacts with the cluster s icm @xcite . such a filamentary feature also may be due to an additional small subcluster infalling into a1750n which is being disrupted as it interacts with the main cluster . the bulk of the halo gas lags behind the infalling groups , and is stripped by the ram pressure of the ambient icm . such halos are expected to have an average temperature of @xmath151 kev with a typical halo mass of 3@xmath255 @xcite . bright , large - scale ( @xmath15 700 kpc ) stripped tails have been observed in the outskirts of galaxy clusters , e.g. the virgo cluster @xcite , a85 @xcite , and a2142 @xcite . the stripped gas from infalling halos may seed gas inhomogeneities ( i.e. clumping ) , which suppress the average entropy inferred at large radii . in such systems , a flattening of the entropy profile , as well as an excess in the gas mass fraction as compared with the cosmic value , have often been observed in cluster outskirts . in the case of a1750 , the entropy profile remains consistent with the self - similar prediction out to @xmath2 , and the gas mass fraction is consistent with the cosmic value ( see section [ sec : mass ] ) , implying that the observed cool gas could indeed be the densest and hottest parts of the warm - hot intergalactic medium . in addition , dense and cool clumps in the outer cluster regions are expected to lead to more entropy flattening since they will lower the average temperature , and , more importantly raise the average density @xcite . along the north direction of a1750n , there is sufficient cool gas to be detected , but it does not cause the dramatic entropy flattening seen in some other clusters , suggesting that its density can not be too high . completely ruling out the ram pressure - stripping scenario for this cool gas requires deeper _ chandra _ observations with good angular resolution . the whim interpretation of this feature can not be firmly established based on the _ suzaku _ data . we present an analysis of the strongly merging cluster a1750 using _ suzaku _ and _ chandra _ x - ray observations , and _ mmt _ optical observations out to the cluster s virial radius . the deep _ suzaku _ observations allow us to constrain the entropy , pressure , and mass profiles at the outskirts , both along and perpendicular to the large - scale filament . we use optical observations to constrain the dynamical state of the cluster . our major results are : 1 . a1750n and a1750c have a 78% chance of being bound . there is an apparent hot region with a temperature of 5.49 @xmath24 0.59 kev in between these subclusters implying an interaction . the red galaxy distribution and the velocity dispersion data prefer a pre - merger scenario . in an early pre - merger scenario , one expects the outer icm atmospheres of the subclusters to interact subsonically , driving shocks , and ultimately creating a heated icm region between the subclusters , e.g. , n7619 and n7626 @xcite . we find overall a good agreement between the measured entropy profiles and the self - similar expectation predicted by gravitational collapse near @xmath2 both along and perpendicular to the putative large - scale structure filament . unlike some other clusters , the entropy profiles at large radii , both perpendicular and along the filamentary directions , are consistent with each other . agreement of the entropy with the self - similar expectation at @xmath2 in this massive and dynamically young system suggests that a1750 exhibits little gas clumping at large radii . the gas mass fractions in both the filament and off - filament directions are consistent with the cosmic baryon fraction at @xmath2 . this may indicate that gas clumping may be less common in such smaller , lower temperature ( @xmath256 4 kev ) systems ( with a few exceptions , e.g. the virgo cluster , * ? ? ? cluster mass may therefore play a more important role in gas clumping than dynamical state . 4 . an extended gas ( @xmath17 0.62 mpc ) is observed to the north of the a1750n subcluster along the large scale structure , where one would expect to detect the densest part of the whim in a filament , near a massive cluster . the measured temperature ( 0.8 @xmath51 kev ) , density , and radial extent of this cool gas is consistent with the whim emission . the thermodynamical state of the gas at that radius ( i.e. self - similar like entropy profile , and gas mass fraction consistent with the cosmic value ) favors the whim emission interpretation . however , a deeper observation with _ resolution is required to distinguish this diffuse filamentary gas from an infalling substructure , or gas from ram pressure - stripping . we thank gabriel pratt for kindly providing temperature , density , and entropy profiles from _ xmm - newton _ data . we also thank mike mcdonald and john zuhone for useful comments and suggestions . eb was supported in part by nasa grants nnx13ae83 g and nnx10ar29 g . swr was supported by the _ chandra _ x - ray center through nasa contract nas8 - 03060 and by the smithsonian institution . mbb acknowledges support from the nsf through grant ast-1009012 . elb and rpm was partially supported by the national science foundation through grant ast-1309032 . cls was funded in part by _ chandra _ grants go4 - 15123x and go5 - 16131x and nasa xmm grant nnx15ag26 g . aedm acknowledges partial support by _ chandra _ grants go2 - 13152x and go3 - 14132x . authors thank prof . nihal e. ercan for providing the support for ce .

we present results from recent _ suzaku _ and _ chandra _ x - ray , and _ mmt _ optical observations of the strongly merging double cluster " a1750 out to its virial radius , both along and perpendicular to a putative large - scale structure filament . some previous studies of individual clusters have found evidence for icm entropy profiles that flatten at large cluster radii , as compared with the self - similar prediction based on purely gravitational models of hierarchical cluster formation , and gas fractions that rise above the mean cosmic value . weakening accretion shocks and the presence of unresolved cool gas clumps , both of which are expected to correlate with large scale structure filaments , have been invoked to explain these results . in the outskirts of a1750 , we find entropy profiles that are consistent with self - similar expectations , and gas fractions that are consistent with the mean cosmic value , both along and perpendicular to the putative large scale filament . thus , we find no evidence for gas clumping in the outskirts of a1750 , in either direction . this may indicate that gas clumping is less common in lower temperature ( @xmath0 kev ) , less massive systems , consistent with some ( but not all ) previous studies of low mass clusters and groups . cluster mass may therefore play a more important role in gas clumping than dynamical state . finally , we find evidence for diffuse , cool ( @xmath1 1 kev ) gas at large cluster radii ( @xmath2 ) along the filament , which is consistent with the expected properties of the denser , hotter phase of the whim .

@xmath2li is destroyed in stellar interiors where temperatures exceed @xmath3 k , and li - depleted material can in principle reach the stellar surfaces where it can be observed . thus , if one is to infer pre - stellar @xmath2li abundances from current - epoch observations , it is important to understand the stellar processing of this species . it has widely , though not universally , been supposed that warm ( @xmath4 k ) , metal - poor ( [ fe / h ] @xmath5 ) stars retain their pre - stellar abundances ( spite & spite 1982 ; bonifacio & molaro et al . 1997 ; but see also deliyannis 1995 ; ryan et al . 1996 ) . although claims had been made of an intrinsic spread in the li abundances by 0.04 0.1 dex ( deliyannis , pinsonneault , & duncan 1993 ; thorburn 1994 ) , ryan , norris & beers ( 1999 ) attributed these to an embedded @xmath6(li ) vs [ fe / h ] dependence , and underestimated errors , respectively . ryan et al . ( 1999 ) set tight limits on the intrinsic spread of @xmath2li in metal - poor field stars as essentially zero , stated conservatively as @xmath7 dex . however , the subset of ultra - li - deficient stars identified by spite , maillard , & spite ( 1984 ) , hobbs & mathieu ( 1991 ) , hobbs , welty & thorburn ( 1991 ) , thorburn ( 1992 ) , and spite et al . ( 1993 ) stands out as a particular exceptional counter - example to the general result . these stars have only upper limits on their @xmath2li abundances , typically 0.5 dex or more below otherwise similar stars of the same @xmath8 and metallicity . detailed studies of other elements in these objects have revealed some chemical anomalies , but none common to all , or which might explain _ why _ their li abundances differ so clearly from those of otherwise similar stars ( norris et al . 1997a ; ryan , norris & beers 1998 ) . in contrast to the situation for population ii stars , a wider range of li behaviors is seen in population i. in addition to a stronger increase with metallicity , thought to be due to the greater production of li in later phases of galactic chemical evolution ( ryan et al . 2001 ) , there is also substantial evidence of li depletion in certain temperature ranges . open cluster observations , for example , show steep dependences on temperature for @xmath9 k ( e.g. , hobbs & pilachowski 1988 ) and in the region of the f - star li gap ( 6400 k @xmath10 7000 k ; boesgaard & tripicco 1986 ) . more problematic , for the young cluster @xmath11 per ( age 50 myr ) and the pleiades ( age 100 myr ) , is the presence of a large apparent li spread even at a given mass . various explanations have been proposed involving mixing in addition to that due to convection . extra mixing processes include rotationally - induced mixing ( e.g. , chaboyer , demarque & pinsonneault 1995 ) , structural changes associated with rapid rotation ( martn & claret 1996 ) , and different degrees of suppression of mixing by dynamo - induced magnetic fields ( ventura et al . gravity waves have been proposed as yet another different mixing mechanism ( schatzman 1993 ; montalbn & schatzman 1996 ) . consensus has not yet emerged concerning the range of possible mechanisms , or the relative importance of each . jeffries ( 1999 ) even questions the reality of a li abundance spread in low mass pleiades stars , due to a similar spread being seen in the resonance line . amongst older open clusters , the spread at a given effective temperature is generally much less , though m67 ( jones , fischer , & soderblom 1999 ) is an exception . a class of stars with higher lithium abundances than otherwise similar stars is short - period tidally - locked binaries ( deliyannis et al . 1994 ; ryan & deliyannis 1995 ) which give credence to the view that physics related to stellar rotation can and does influence the evolution of li in approximately solar - mass stars . the fraction of warm , metal - poor stars that fall in the ultra - li - deficient category has previously been estimated at approximately 5% ( thorburn 1994 ) . however , recent measurements of li in a sample of 18 warm ( @xmath12 k ) , metal - poor ( @xmath13[fe / h ] @xmath14 ) stars yielded four ultra - li - deficient objects , i.e. more than 20% of the sample ( ryan et al . the poisson probability of a 5% population yielding 4 or more objects in a sample of this size is just 0.013 . clearly , the selection criteria for this sample have opened up a regime rich in ultra - li - poor stars . we now examine those criteria , and discuss the implications for the origin of such systems and for our understanding of li - poor and li - normal stars . we note some similarities between li - deficient halo stars and blue stragglers . although these two groups have traditionally been separated due to the different circumstances of their _ discovery _ , we question whether there is a reliable _ astrophysical _ basis for this separation . one must ask whether the process(es ) that gives rise to blue stragglers is capable only of producing stars whose mass is greater than that of the main sequence turnoff of a @xmath1513 gyr old population . if , as we think is reasonable , the answer is `` no '' , then one may ask what the sub - turnoff mass products of this process(es ) would be . our proposal is that they would be li - deficient , but otherwise difficult to distinguish from the general population , and in this regard very similar to the ultra - li - deficient halo stars . the ultra - li - poor halo stars we consider were identified serendipitously in a study of predominantly high proper - motion halo stars having @xmath16 k and @xmath17 [ fe / h ] @xmath18 , and are listed in table 1(a ) . details of the sample selection and abundance analysis are given by ryan et al . ( 2001 ) ; the key points are that high resolving power ( @xmath19 ) chelle spectra were obtained , equivalent widths were measured , and abundances were computed using a model atmosphere spectrum - synthesis approach . two of the li - poor stars were subsequently found to have previous li measurements ; wolf 550 was identified as g66 - 30 , and g202 - 65 had been observed by hobbs & mathieu ( 1991 ) in a study targeted at blue stragglers . the new spectra of the four stars , plus one with normal li for comparison , are shown in figure 1 . the comparison star , cd@xmath20305 , has @xmath21 k , [ fe / h ] = @xmath22 , and @xmath6(li ) = 2.24 ( ryan et al . 2001 ) . for convenience , previously known li - depleted halo stars are listed in table 1(b ) . the full sample of ryan et al . ( 2001 ) is plotted in figure 2 , along with additional stars from the literature . it is immediately apparent that three of the four ultra - li - deficient stars are amongst the hottest in our sample , though not _ the _ hottest in the figure . it seems likely that high temperature is one biasing characteristic of these objects . the stars with @xmath23 k and _ normal _ li abundances are listed in table 1(c ) . these have had comparatively high values of de - reddening applied , and it is possible that they are in reality cooler than table 1 shows . an indication that high temperature is not the _ only _ biasing characteristic of ultra - li - poor stars is that the ryan et al . ( 1999 ) study of 23 very metal - poor ( @xmath24 [ fe / h ] @xmath25 ) stars in the same temperature range included only one ultra - li - deficient star , g186 - 26 . this rate , 1 in 23 , is consistent with previous estimates for population ii stars as a whole . however , very few relatively metal - rich ( @xmath26 [ fe / h ] @xmath18 ) halo stars in this temperature range had been studied previously , so earlier works may have been biased against discovering ultra - li - poor objects . it appears , then , that the fraction of ultra - li - deficient stars is higher as metallicity increases . this may explain why our study , which targeted stars in the higher metallicity range _ and _ with @xmath27 k , was so successful at yielding ultra - li - deficient stars . figure 3 shows the distribution of objects in the @xmath28 , [ fe / h ] plane . whereas previously no region of parameter space stood out as `` preferred '' by li - deficient stars , the objects are now more conspicuous as a result of their high temperatures and relatively high metallicities . also shown in figure 3 are the @xmath28 of the main - sequence turnoff as a function of metallicity , for 14 and 18 gyr isochrones . the isochrones shown are the oxygen - enhanced curves of bergbusch & vandenberg ( 1992 ; solid curves ; y=0.235 ) , and , for comparison , the revised yale isochrones of green , demarque & king ( 1987 ; dotted curves ; y=0.24 ) . clearly there is disagreement of @xmath294 gyr between the two sets as to the ages that would be assigned to these stars , and there are uncertainties in the color-@xmath28 transformations that have been applied to the observed data . however , these difficulties are not the issue here . rather , we use the isochrones to indicate the _ shape _ of the turnoff locus in the @xmath28 vs [ fe / h ] plane , and on that point the four isochrones are in overall agreement . they emphasise that even though hd 97916 is cooler than five other li - depleted stars in the study , it is nevertheless close to the turnoff . that is , a star with @xmath28 = 6100 k would appear below the turnoff if [ fe / h ] = @xmath30 , but will be close to the turnoff if [ fe / h ] = @xmath31 . even excluding the definite blue straggler bd+25@xmath321981 , there are four li - depleted stars amongst the eight whose symbols lie above or touch the 14 gyr revised yale isochrone . clearly , all of these are very close to the turnoff once their metallicities are taken into account . besides these li - depleted stars close to the turnoff , four are 100200 k cooler than the turnoff . we discuss later in this paper whether the these two groupings might have different origins . blue stragglers are recognised observationally as stars that are considerably bluer than the main - sequence turnoff of the population to which they belong , but having a luminosity consistent with main - sequence membership . such objects were originally identified in globular clusters ( e.g. , m3 ; sandage 1953 ) , but are also known in the field ( e.g. , carney & peterson 1981 ) , and in population i as well as population ii ( e.g. , leonard 1989 ; stryker 1993 ) . their origin is not known with certainty , and it is possible that more than one mechanism is responsible for their presence . a range of explanations was examined by leonard ( 1989 ) , but the discovery of li destruction in blue stragglers in the halo field and the open cluster m67 led hobbs & mathieu ( 1991 ) and pritchet & glaspey ( 1991 ) to conclude that `` virtually all mechanisms for the production of blue stragglers _ other than _ mixing , binary mass transfer , or binary coalescence appear to be ruled out ... . '' as hobbs & mathieu emphasized , internal mixing alone is also ruled out ; mixing out to the surface is required . recent advances in high - resolution imaging have verified that the blue straggler fractions in at least some globular clusters are higher in their cores , strongly supporting the view that some blue stragglers are formed through stellar collisions , probably involving the coalescence of binary stars formed and/or hardened through exchanges , in these dense stellar environments ( e.g. , ferraro et al . however , it is neither established nor required that a single mechanism will explain all blue stragglers , and it is unclear how the field examples and those in the tenuous dwarf galaxy ursa minor ( feltzing 2000 , priv . comm . ) relate to those in the dense cores of globular clusters . probably even the halo field and dwarf spheroidal stars formed in clusters of some description ( since the formation of stars in isolation is unlikely ) , but one should not be too quick to link the properties of surviving globular clusters to diffuse populations . this view is supported by preston & sneden s ( 2000 ) conclusion that at more than half ( 62% 100% ) of their field blue metal - poor binaries are blue stragglers formed by mass transfer rather than mergers , due to the long orbital periods and low eccentricities of the field systems they observed . their conclusion is entirely consistent with the views of ferraro et al . ( 1995 ) , who ascribed blue straggler formation to interactions _ between _ systems in high - density environments , but _ within _ systems ( primordial binaries ) in lower - density clusters . in contrast to but not contradicting preston & sneden s result for field systems , mateo et al . ( 1990 ) argue that all of the blue stragglers in the globular cluster ngc 5466 are the result of close binary mergers . the mechanism for li destruction in field blue stragglers is not known . it is unclear what degree of mixing will occur as a result of coalescence . early work by webbink ( 1976 ) suggested substantial mixing would occur , whereas more recent simulations of head - on collisions by sills et al . ( 1997 ) , and grazing collisions and binary mergers by sandquist , bolte , & hernquist ( 1997 ) , have suggested otherwise . sills , bailyn & demarque ( 1995 ) argue , however , that to account for the blue stragglers observed in ngc 6397 , mixing is nevertheless required ( unless the collision products have more than twice the turnoff mass ) , and may occur after the initial coalescence . this is perhaps consistent with the result of lombardi , rasio , & shapiro ( 1996 ) that some mixing could occur as a merger remnant re - contracts to the main sequence . due to the fragility of li , if some mixing of surface material does occur during the coalescence it will at least dilute , and possibly also destroy , any lithium remaining in the stars thin convective surface zones up to that time . one might suppose that mass transfer in a detached system also destroys li , though one could also imagine gentle mass - transfer processes where the rate is slow enough that the original envelope is not subjected to additional mixing , and where the transferred matter itself does not undergo additional li - destruction . of course , mass transfer via roche lobe overflow in a detached system , or wind accretion from a more distant companion , involve mass from an evolved star which may have _ already _ depleted its surface li due to single - star evolutionary processes . consequently , the mass transferred may be already devoid of li , as in the scenario quantified by norris et al . ( 1997a ) . we also note the possibility that the accretor in a mass - transfer system , or the progenitors of a coalescence , was ( were ) devoid of li prior to that event . li is ( normally ) preserved in halo stars only over the temperature range from the turnoff ( @xmath1 k ) to about @xmath33 k , corresponding to a mass range from 0.80 down to 0.70 @xmath34 . therefore it is likely that any mass accretor , and certain that any merger remnant , now seen in this mass range began life as one ( or two ) stars with initial mass(es ) less than 0.70 @xmath34 and had already destroyed li normally , as lower - mass stars are known to do , prior to mass exchange . in such a scenario , it is not _ necessary _ for any li to have been destroyed as a result of the blue - straggler formation process itself , though this could occur as well . in view of the distributions of the ultra - li - deficient stars in the @xmath28 , [ fe / h ] plane , with four at the turnoff and four 100200 k cooler , we consider whether all represent the same phenomenon , or the possibility that two distinct processes have been in operation . it is not a trivial matter to answer this question , because we do not know with certainty what mechanism(s ) has affected any of the stars . however , we explore a number of possibilities in the discussion that follows . ignoring again the obvious blue straggler bd+25@xmath321981 , of the 111 stars shown in figure 3 , 8 are ultra - li - deficient . if all ultra - li - poor stars have the same origin , then we should begin by restating the frequency of such li - weak objects as @xmath297% of plateau stars rather than @xmath295% as estimated previously when the parameter space was incompletely sampled , and with strong metallicity and temperature dependences in that fraction . historically , blue stragglers and ultra - li - deficient stars have been regarded as separate phenomena . however , we have been driven to consider whether there is any astrophysical basis for this separation . one must ask whether the process(es ) that gives rise to blue stragglers is capable only of producing stars whose mass is greater than that of the main sequence turnoff of a @xmath1513 gyr old population . if , as we think is reasonable , the answer is `` no '' , then one may ask what the sub - turnoff mass products of this process(es ) would be . our proposal is that they would be li - deficient , but otherwise difficult to distinguish from the general population . for ultra - li - poor stars redder than the main sequence turnoff , hipparcos parallaxes have established that g186 - 26 is on the main sequence rather than on the subgiant branch . of those _ at _ the turnoff , wolf 550 , g202 - 65 , and bd+51@xmath321817 also have hipparcos parallaxes ; two are almost certainly dwarfs , while g202 - 65 is subject to larger uncertainties and may be more evolved ( see ryan et al . 2001 , table 2 ) . the argument that the evolutionary rate of subgiants is too rapid to explain the high frequency of observed li - deficient objects , which persuaded norris et al . ( 1997a ) to reject the proposition that they might be the _ redward_-evolving ( post - turnoff ) progeny of blue - stragglers , is therefore redundant . however , the detection of several li - weak stars at the bluest edge of the colour distribution has prompted us to re - examine their possible association with blue stragglers . we would describe g202 - 65 as `` at '' the turnoff rather than classify it as a blue straggler in the conventional sense , as it is only marginally hotter ( bluer ) than the main sequence turnoff for its metallicity ( see figure 3 ) . hobbs & mathieu , on the other hand , classified it as a blue straggler , based presumably on the photometry of laird , carney & latham ( 1988 ) which they referenced . ( indeed , carney et al ( 1994 ) declare it as a `` blue straggler candidate '' , and carney et al . ( 2000 ) treat it as one , though acknowledging at the same time that some normal stars may be included in this classification . ) our purpose is _ not _ to debate how this star should be classified , but rather to underline the main suggestion of our work , that the blue straggler and halo ultra - li - deficient stars may have a common origin . although blue stragglers have historically been recognised because they are bluer than the main - sequence turnoff , it is essential to remember that stars that have accreted mass from a companion , or that result from a coalescence can have a mass less than the current turnoff . such stars would be expected to share many of the properties of blue stragglers , but would not _ yet _ appear bluer than the turnoff . however , at some future time , once the main - sequence turnoff reaches lower masses , these non - standard objects would lag the evolution of normal stars and hence appear bluer , showing canonical straggling behaviour . therefore , such stars might , for the present , be regarded as `` blue - stragglers - to - be , '' and our speculation is that the ultra - li - deficient halo stars in are in fact members of such a population . note that this proposition is distinct from that of _ redward_-evolving systems considered and rejected by norris et al . . if ultra - li - deficient stars and blue stragglers are manifestations of the same process , then li deficiency may be the only way of distinguishing sub - turnoff - mass blue - stragglers - to - be from normal main - sequence stars , prior to their becoming classical blue stragglers . mass transfer during their formation may also help clarify some of the unusual element abundances found by norris et al . ( 1997a ; see also ryan et al . 1998 ) . whereas an appeal to extra mixing ( in a single - star framework ) to explain the li depletion would not necessarily affect other elements , mass transfer in a binary with an agb donor may be capable of altering s - process abundances as well . in this regard , we recall that two of the ultra - li - deficient stars studied by norris et al . ( 1997a ; also ryan et al . 1998 ) had non - standard sr and ba abundances . mass transfer from an rgb donor would presumably leave a different chemical signature . @xmath36 , as many as 25% have c overabundances ( e.g. norris , ryan , & beers 1997b ) . at least some but not all of these ( norris , ryan , & beers 1997c ) have s - process anomalies . detailed studies have yet to be completed , so it is unclear what fraction of stars are formed from anomalous material and what fraction became modified later in their life . whilst we can not presently rule out the possibility that the s - process anomalies seen in some ultra - li - deficient stars were inherited at birth , our expectation is that mass transfer from a companion star will be a more common mechanism . ] some constraints on the progenitors of the li - deficient stars may be obtained from their rotation rates and radial velocity variations . webbink s ( 1976 ) calculations of a coalesced star ( @xmath37 ) show that a high rotation rate is maintained at least until it reaches the giant branch . in contrast , previously known blue stragglers appear not to have uncommonly high rotation rates ( e.g. , carney & peterson 1981 ; pritchet & glaspey 1991 ) . this tends to argue against the blue stragglers as having originated from coalesced main - sequence contact binaries , and points towards one of the other binary mass - transfer scenarios , unless mass loss ( e.g. , via webbink s excretion disk ) and magnetic breaking can dissipate envelope angular momentum during the main sequence lifetime of a coalesced star . to spin down , stars must have a way of losing surface angular momentum . in single stars , most of this is believed to occur during the pre- and early - main - sequence phase when magnetic coupling of the stellar surface to surrounding dust creates a decelerating torque on the star . it is not clear that two mature stars which merge will still have this coupling , because of the much lower mass loss rates beyond the early stages of evolution ( unless they produce an excretion disk ) and lower magnetic field strengths . ( see also discussion by sills et al . 1997 , 5.5 . ) leonard & livio ( 1995 ) have proposed that the merger product acquires the distended form of a pre - main - sequence - like star which then spins down as it again approaches the main sequence , losing angular momentum in much the same way as conventional pre - main - sequence stars . for the four stars observed in this work , three had previous radial velocity measurements accurate to @xmath29 1 km s@xmath38 ( carney et al . 1994 ) . the new measurements ( ryan et al . 2001 ; table 2 ) showed residuals of + 1.0 ( bd+51@xmath321817 ) , @xmath393.3 ( g202 - 65 ) , and @xmath396.9 km s@xmath38 ( wolf 550 ) ; compared with the expected radial velocity accuracy of @xmath40 = 0.30.7 km s@xmath38 , these are consistent with significant motion . carney et al . ( 2000 ) indicate periods of 168 to 694 days for these systems , and low eccentricities , except for wolf 550 ( @xmath41 = 0.3 ) . similarly , the metal - poor field blue straggler cs 22966 - 043 has an orbital period of 319 days ( preston & landolt 1999 ) . if the brighter component has a mass of 0.8 m@xmath42 and its companion has a mass between 0.4 and 1.4 m@xmath42 ( appropriate to a white dwarf ) then the _ current _ semi - major axis of the system will be in the range @xmath43 = 200260 r@xmath42 ( from kepler s third law ) . companions having a canonical white - dwarf mass . ] their second system , cs 29499 - 057 , may have an even longer period of 2750 days , implying @xmath43 = 9001100 r@xmath42 . the periods of these and carney et al s systems , and hence their large current separations , are more compatible with mass loss from an evolved companion rather than being short - period systems in contact on the main sequence . the evidence presented to date has argued against internal mixing alone as an adequate explanation for the ultra - li - deficient stars whose neutron - capture elements show abundance anomalies . note , though , that certainly not all ultra - li - deficient stars and blue stragglers exhibit neutron - capture element anomalies ( carney & peterson 1981 ; norris et al . 1997a ; ryan et al . if mass transfer has occurred , systems in which s - process elements are abnormal would presumably indicate material originating with an agb companion , whereas s - process - normal remnants would indicate mass transfer during an earlier stage of evolution ( rgb ) or from a pre - thermal - pulsing agb mass donor . ( we have no data on the n abundance , and the ch band in these stars is too weak to hope to measure the @xmath44c/@xmath45c ratio . ) likewise , the rotation rates of both blue stragglers and ultra - li - deficient stars are apparently normal , arguing against coalescences having already occurred on the main sequence . of the three mechanisms found to be viable by pritchet & glaspey ( 1991 ) and hobbs & mathieu ( 1991 ) , this leaves mass transfer from a companion as the only one remaining , _ if _ we are correct in speculating that the ultra - li - deficient and blue straggler phenomena are manifestations of the same process . in the absence of an adequate theory for why eight otherwise - normal halo stars ( excluding the traditional blue straggler bd+21@xmath321981 ) should have low ( zero ? ) li abundances , it may be useful to consider the hot subsample ( 6200 k @xmath46 @xmath28 @xmath46 6300 k ) as a distinct group . several possibilities then arise that might account for the observed li deficiency , including diffusion ( the sinking of li to below the photosphere ) , the f - star li dip , and an unknown process that may be responsible for depletion in some ( but not all ) disk stars . we consider each of these in turn . we note that the three li - deficient stars with @xmath1 k are confirmed binaries , whereas most cooler ones show no evidence of binary motion . the binary / single distinction between warmer / cooler li - depleted stars is pronounced ; see table 1 , where the binary status ( carney et al . 1994 , 2000 ; latham 2000 , priv.comm . ) is given in the final column . if such a dichotomy is maintained as more li - poor systems are discovered , it may indicate a genuine difference in the origin of the turnoff and sub - turnoff systems . deliyannis , demarque & kawaler ( 1990 ) and proffitt & michaud ( 1991 ) have computed the predicted effects of diffusion on the surface li abundances of warm halo stars . diffusion is more significant in hotter stars because their surface convective zone is thinner . the degree of depletion expected at @xmath0 k is a function of effective temperature , changing by @xmath29 0.2 dex per 100 k in the former ( for @xmath11 = 1.1 ) , and @xmath29 0.2 and @xmath47 dex per 100 k in the latter ( for @xmath11 = 1.7 and 1.5 respectively ) . this does not match the behavior observed ( see figure 2 ) . for comparison , our ultra - li - poor stars are depleted by @xmath48 dex . this alone appears to rule out diffusion as the explanation , except possibly for the lower-@xmath11 model of proffitt & michaud . however , li diffusion appears to have been inhibited in all other metal - poor samples ( e.g. , ryan et al . 1996 ) , so it would be unusual to see it suddenly present and with such effect only in isolated stars in our new sample . boesgaard & tripicco ( 1986 ) and hobbs & pilachowski ( 1988 ) showed that li is severely depleted in population i open cluster stars over the interval 6400 k @xmath49 k. various explanations have been proposed , including mass loss ( e.g. , schramm , steigman , & dearborn 1990 ) , diffusion ( e.g. , turcotte , richer & michaud 1998 ) , and slow mixing of various forms ( e.g. , deliyannis & pinsonneault 1997 ) , but none has been convincingly established as responsible , and several mechanisms may be acting in concert ( e.g. , turcotte et al . ) . whatever the correct explanation(s ) , is it possible that the hottest ultra - li - deficient stars are encroaching on this regime and are affected by this phenomenon ? although this can not be ruled out completely for the hot subset , especially since we have questioned the reliability of the @xmath50 ( and hence @xmath28 ) values of the hottest li - preserving stars in figure 2 , the onset of destruction in the f - star dip seems too gradual with @xmath28 to explain the new data . the hyades observations ( boesgaard & tripicco 1986 ) show a decrease of only 0.3 dex from 6200 to 6400 k , substantially less than the @xmath51 dex deficit in the ultra - metal - poor objects around 6300 k. as noted above , hipparcos parallaxes are available for five of the eight known ultra - li - deficient stars and , with the possible exception of g202 - 65 , rule out the possibility that these stars are redward - evolving _ descendants _ of the li - dip . lambert , heath & edvardsson ( 1991 ) found that , in almost all cases , the low li abundances in their population i sample could be ascribed to their being evolved descendants of li - dip stars , or else being dwarfs exhibiting the li - depletion that increases towards _ lower _ temperature , as is normally associated with pre - main - sequence and/or main - sequence burning . anomalously high li depletions were found in only 13 cases out of some 26 old - disk stars , and for a similar fraction of young - disk stars . based on this fraction , lambert et al . proposed that a new class of highly li - depleted stars , comprising less than about 10% of the population , might exist . it is interesting to note that this proposal pre - dated the discovery of ultra - li - deficient halo dwarfs . the uncertain number of cases stated above arises because lambert et al . recognised that uncertainties in the stellar luminosities , and hence mass , could drive stars into or out of the region of importance . we now have the benefit of accurate hipparcos parallaxes . these indicate that two of the seven stars highlighted by their study , hd 219476 and hr 4285 , are indeed considerably more massive than reported in lambert et al.s tables and hence are probably descendants of the li gap , thus reducing the number of _ genuine _ cases to 2 out of 26 old - disk stars , and 3 out of a similar number of young - disk stars . that is , the fraction of anomalously li - depleted stars appears to be around 8 - 10% , albeit sensitive to small - number statistics . ultra - li - depleted population i stars are also seen in young open clusters . they can be recognised , for example , in fig . 1 of ryan & deliyannis ( 1995 ) , where @xmath296% of the hyades stars cooler than the f - star dip appear to be ultra - li - deficient . is it possible that the li - depleted halo stars are of the same type ? the lack of examples in the two pop i and pop ii classes to compare with precludes a detailed analysis , but we note that we see li deficiency in about 7% of halo objects , which is comparable to the ratio for the pop i objects . that is , the pop i and pop ii examples could arise due to the same process , even though it remains unclear what that process is . we note , for completeness , that ryan et al . ( 2001 ) showed that the kinematics of the new ultra - li - depleted stars are clearly those of halo objects , and thus they genuinely belong to the halo population , despite their metallicities being close to those of the most metal - poor thick - disk stars . the stars remaining on lambert et al s list of unusually li - deficient objects are : hr 3648 , hr 4657 , hr 5968 , hr 6541 , and hd 30649 . upon searching the literature for evidence of binarity or abundance anomalies in these systems , we found that not only was hr 4657 a 850 day period binary , but fuhrmann & bernkopf ( 1999 ) had also been driven to consider this star as a blue straggler . it has an unexpectedly high rotational velocity ( in contrast to the blue stragglers studied by carney & peterson 1981 ) . there is no evidence of s - process anomalies , but other unusual characteristics of the system include an observable soft x - ray flux and the very likely association of this object with grb 930131 . hr 3648 (= 16 uma = hd 79028 ) is a 16.2 day period chromospherically - active single - lined spectroscopic binary ( basri , laurent , & walter 1985 ) . hd 30649 (= g81 - 38 ) and hr 6541 ( = hd 159332 ) , in contrast , show no significant evidence of binarity ( carney et al . hr 5968 (= @xmath52 crb ) does not appear to have a stellar companion , though it has a planetary companion ( noyes et al . 1997 ) , but ryan ( 2000 ) argues that li in this star is _ not _ anomalous . hr 3648 and hr 4657 have ba abundance measurements from the study by chen et al . the latter also has been observed by fuhrmann & bernkopf ( 1999 ) , but neither star appears abnormal in this element . ryan et al . ( 1999 ) have argued that the ultra - li - deficient halo stars are distinct from the majority of halo stars that occupy the spite plateau , and , in particular , that they do _ not _ merely represent the most extreme examples of a _ continuum _ of li depletion . if the association with blue stragglers ( or , for that matter , any distinct evolutionary phenomenon ) is correct , then the mechanism for their unusual abundances will at last be understood and they will be able to be neglected with certainty from future discussion of the spite plateau . in the present work , we have proposed and discussed the possibility that ultra - li - depleted halo stars and blue stragglers are manifestations of the same phenomenon , and described the former as `` blue - stragglers - to - be . '' we proposed that their li was destroyed either during the formation process of blue stragglers or during the _ normal _ single - star evolutionary processes of their precursors , namely during pre - main - sequence and/or main - sequence phases of low - mass stars , or during post - main - sequence evolution of mass donors , as in the scenario quantified by norris et al . ( 1997a ) . we note that in a study carried out separately but over the same time period as ours , carney et al . ( 2000 ) have examined the orbital characteristics of blue stragglers , and have been driven towards similar considerations as we have . there are clearly still details to be clarified , but our two groups appear to be converging on a view unifying blue stragglers and ultra - li - deficient systems . because there are numerous observational and theoretical issues surrounding this unified view , we seek to clarify the main arguments and possibilities using an itemised summary . observations : @xmath53 in a study of 18 halo stars with @xmath17 [ fe / h ] @xmath18 and 6000 k @xmath54 6400 k , we have found four ultra - li - deficient objects , i.e. a 22% detection rate . @xmath53 the fraction of ultra - li - deficient stars is very much higher amongst the hottest and most metal - rich halo main - sequence stars ( @xmath2920% ) than amongst cooler and more metal - poor ones ( @xmath295% ) . @xmath53 ultra li - deficient stars exist both at the turnoff , and cooler than the turnoff , and with well - determined main - sequence luminosities from hipparcos . @xmath53 all of the turnoff ultra - li - deficient halo stars , but none of the sub - turnoff ultra - li - deficient halo stars , appear to be binaries . this may indicate that two different mechanisms are causing the halo ultra - li - deficient phenomenon . theoretical framework : @xmath53 blue stragglers may form from _ several _ mechanisms , but seem to require at least one of either complete mixing , binary mass transfer , or coalescence ( hobbs & mathieu 1991 ; pritchet & glaspey 1991 ) . origins : @xmath53 we speculate that ultra - li - deficient stars and blue stragglers are manifestations of the same process , and that sub - turnoff - mass ultra - li - deficient stars may be regarded as `` blue - stragglers - to - be . '' @xmath53 li could be destroyed at several stages : ( i ) in a mass - transfer event which induces extensive mixing ; ( ii ) by single - star evolutionary processes ( convective mixing ) in a post - main - sequence mass donor ; ( iii ) by single - star evolutionary processes ( mixing ) in pre - main - sequence ( or possibly main - sequence ) low - mass stars prior to their gaining mass . @xmath53 mass - transfer scenarios from an agb star seem better able to explain the unusual neutron - capture element ratios _ sometimes _ seen in ultra - li - depleted stars ( norris et al . 1997a ) than internal mixing , since @xmath29 0.8 m@xmath42 core - hydrogen - burning stars are not expected to process neutron - capture elements . this argues against internal mixing as the sole explanation for the existence of ultra - li - depleted stars with unusual neutron - capture abundances . ( mass transfer from pre - agb ( most likely rgb ) donors would produce the stars with normal neutron - capture abundances . ) @xmath53 coalesced binaries are expected to maintain high rotation rates until they reach the giant branch , but neither blue stragglers nor ultra - li - depleted halo stars have high rotation rates . this argues against coalescence of a binary as the explanation for these objects unless they have spun down . @xmath53 the orbital periods of metal - poor field blue stragglers ( preston & landolt 1999 ; carney et al . 2000 ) suggest current semi - major axes in the range 2001100 r@xmath42 , arguing against these being coalescing stars ( unless they began their lives as triple systems ) . @xmath53 the arguments against solely internal mixing , and against coalescence of main - sequence contact binaries , leaves mass transfer as the most viable mechanism for field binaries . this is _ not _ to say that li was destroyed during the transfer ; it may have been destroyed by single - star mechanisms already . @xmath53 the observed d@xmath6(li)/d@xmath28 is too steep compared with models of diffusion to be due to that process . @xmath53 the observed d@xmath6(li)/d@xmath28 is too steep compared with the hyades data to be due to the f - star li dip . @xmath53 the halo ultra - li - deficient stars could be related to the pop i anomalously - li - depleted stars identified in the field by lambert et al . ( 1991 ) and also seen in open clusters . @xmath53 hipparcos parallaxes rule out the possibility that the ultra - li - deficient stars are redward - evolving post - turnoff stars . they have not descended from the f - star li dip . implications : @xmath53 severe li depletion may be the ( only ? ) signature of sub - turnoff - mass blue stragglers . the halo population fraction comprising ultra - li - poor stars is 7% . @xmath53 understanding the ultra - li - depleted stars as resulting from a distinct process ( not normally affecting single stars ) would eliminate the need to include them in discussions of processes affecting the evolution of normal spite plateau stars , and would explain why they appear so radically different from the vast majority of halo stars ( ryan et al . the authors gratefully acknowledge the support for this project given by the australian time assignment committee ( atac ) and panel for the allocation of telescope time ( patt ) of the aat and wht respectively , and for practical support given by the staff of these facilities . they also express gratitude to d. a. latham and b. w. carney for conveying the results of their program in advance of publication , and to an anonymous referee for his / her comments that helped us clarify our arguments . s.g.r . sends a special thanks to colleagues at the university of victoria : to c. j. pritchet for a most memorable snow - shoeing expedition on 1991 february 10 during which li deficiency in blue stragglers was discussed , to d. a. vandenberg for discussing and supplying isochrones , and to f. d. a. hartwick for once asking whether there were blue stragglers in the halo field . t.c.b acknowledges partial support from grant ast 95 - 29454 from the national science foundation . basri , g. , laurent , r. , & walter , f. m. 1985 , apj , 298 , 761 bergbusch , p. a. , & vandenberg , d. a. 1992 , apjs , 81 , 163 boesgaard , a. m. & tripicco , m. j. 1986 , apj , 302 , l49 bonifacio , p. & molaro , p. 1997 , mnras , 285 , 847 carney , b. w. , latham , d. w. , laird , j. b. , grant , c. e , & morse , j. a. 2000 , preprint carney , b. w. , latham , d. w. , laird , j. b. , & aguilar , l. a. 1994 , aj , 107 , 2240 carney , b. w. , & peterson , r. c. 1981 , apj , 251 , 190 chaboyer , b. 2000 , `` the galactic halo ; from globular clusters to field stars '' a. noels , p. magain , d. caro , e. jehin , g. parmentier , & a. thoul ( eds ) ( u. lige , lige ) in press chaboyer , b. , demarque , p. , & pinsonneault , m. h. 1995 , apj , 441 , 876 chen , y. q. , nissen , p. e. , zhao , g. , zhang , h.w . , & benoni , t. 2000 , a&as , 141 , 491 deliyannis , c. p. 1995 , the light element abundances , ed . p. crane , ( berlin : springer - verlag ) , 395 deliyannis , c. p. , demarque , p. , & kawaler , s. d. 1990 , apjs , 73 , 21 deliyannis , c. p. , king , j. r. , boesgaard , a. m. , & ryan , s. g. 1994 , apj , 434 , l71 deliyannis , c. p. & pinsonneault , m. h. 1997 , apj , 488 , 836 deliyannis , c. p. , pinsonneault , m. h. & duncan , d. k. 1993 , apj , 414 , 740 ferraro , f. r. , fusi pecci , f. , & bellazini , m. 1995 , a&a , 294 , 80 ferraro , f. r. , palterinieri , b. , rood , r. t. , & dorman , b. 1999 , apj , 522 , 983 fuhrmann , k. , & bernkopf , j. 1999 , a&a , 247 , 897 to field stars , ed . a noels & p. magain , 35@xmath55 lige int . astroph . , in press green , e. m. , demarque , p. , & king , c. r. 1987 , the revised yale isochrones and luminosity functions ( new haven : yale univ . press ) hobbs , l. m. , & mathieu , r. d. 1991 , pasp , 103 , 431 hobbs , l. m. & pilachowski , c. 1988 , apj , 334 , 734 hobbs , l. m. , welty , d. e. , & thorburn , j. a. 1991 , apj , 373 , l47 jeffries , r. d. 1999 , mnras , 309 , 189 jones , b. f. , fischer , d. , & soderblom , d. r. 1999 , aj , 117 , 330 laird , j. b. , carney , b. w. , & latham , d. w. 1988 , aj , 95 , 1843 lambert , d. l. , heath , j. e. , and edvardsson , b. 1991 , mnras , 253 , 610 leonard , p. j. t. 1989 , aj , 98 , 217 leonard , p. j. t. , & livio , m. 1995 , apj , 447 , l121 lombardi , j. c. jr , rasio , f. a. , & shapiro , s. l. 1996 , apj , 468 , 797 martn , e. l. , & claret , a. 1996 , a&a , 306 , 408 mateo , m. , harris , h. c. , nemec , j. , & olszewski , e. w. 1990 , aj , 100 , 469 montalbn , j. , & schatzman , e. 1996 , a&a , 305 , 513 norris , j. e. , ryan , s. g. , & beers , t. c. 1997b , apj , 488 , 350 norris , j. e. , ryan , s. g. , & beers , t. c. 1997c , apj , 489 , l169 norris , j. e. , ryan , s. g. , beers , t. c. , & deliyannis , c. p. 1997a , apj , 485 , 370 noyes , r. w. , jha , s. , korzennik , s. g. , krockenberger , m. , nisenson , p. , brown , t. m. , kennelly , e. j. , & horner , s. d. 1997 , apj , 483 , l111 piotto , g. 2000 , `` the galactic halo ; from globular clusters to field stars '' a. noels , p. magain , d. caro , e. jehin , g. parmentier , & a. thoul ( eds ) ( u. lige , lige ) in press portegies zwart , s. 2000 , proc . . the influence of binaries on stellar population studies , brussels 2000 , d. vanbeveren ( ed ) , in press preston , g. w. , & landolt , a. u. 1999 , aj , 118 , 3006 preston , g. p. & sneden , c. 2000 , in press pritchet , c. j. , & glaspey , j. w. 1991 , apj , 373 , 105 proffitt , c. r. , & michaud , g. 1991 , apj , 371 , 584 rebolo , r. , molaro , p. & beckman , j. e. 1988 , a&a , 192 , 192 ryan , s. g. 2000 , mnras , 316 , l35 ryan , s. g. , beers , t. c. , deliyannis , c. p. , & thorburn , j. a. 1996 , apj , 458 , 543 ryan , s. g. & deliyannis , c. p. 1995 , apj , 453 , 819 ryan , s. g. , kajino , t. , beers , t. c. , suzuki , t. , romano , d. , matteucci , f. , & rosolankova , k. 2001 , apj , 548 ( 20 feb ) ryan , s. g. , norris , j. e. , & beers , t. c. 1998 , apj , 506 , 892 ryan , s. g. , norris , j. e. , & beers , t. c. 1999 , apj , 523 , 654 sandage , a. r. 1953 , aj , 58 , 61 sandquist , e. l. , bolte , m. , & hernquist , l. 1997 , apj , 477 , 335 schatzman , e. , 1993 , a&a , 279 , 431 schramm , d. n. , steigman , g. , and dearborn , d. s. p. 1990 , apj , 359 , l55 sills , a. p. , bailyn , c. d. , & demarque , p. 1995 , apj , 455 , l163 sills , a. , lombardi , j. c. jr , bailyn , c. d. , demarque , p. , rasio , f. a. , & shapiro , s. l. 1997 , apj , 487 , 290 spite , f. & spite , m. 1982 , a&a , 115 , 357 spite , m. , maillard , j. p. , & spite , f. 1984 , a&a , 141 , 56 spite , m. , molaro , p. , franois , p. , & spite , f. 1993 , a&a , 271 , l1 spite , m. , spite , f. , cayrel , r. , hill , v. , depagne , e. , nordstrom , b. , beers , t. , & nissen , p. e. 2000 , `` the evolution of the light elements : iau symp . 198 '' l. da silva , r. de medeiros , & m. spite ( eds ) , ( asp conf ser . ) , in press stryker , l. l. 1993 , pasp , 105 , 1081 thorburn , j. a. 1992 , apj , 399 , l83 thorburn , j. a. 1994 , apj , 421 , 318 thorburn , j. a. & beers , t. c. 1993 , apj , 404 , l13 turcotte , s. , richer , j. , & michaud , g. 1998 , apj , 504 , 559 ventura , p. , zeppieri , mazzitelli , i. , & dantona , f. 1998 , a&a , 331 , 1011 webbink , r. f. 1976 , apj , 209 , 829

we present data for four ultra - li - deficient , warm , halo stars . the li deficiency of two of these is a new discovery . three of the four stars have effective temperatures @xmath0 k , in contrast to previously known li - deficient halo stars which spanned the temperature range of the spite plateau . in this paper we propose that these , and previously known ultra - li - deficient halo stars , may have had their surface lithium abundances reduced by the same mechanism as produces halo field blue stragglers . even though these stars have yet to reveal themselves as blue stragglers , they might be regarded as `` blue - stragglers - to - be . '' in our proposed scenario , the surface abundance of li in these stars could be destroyed ( a ) during the normal pre - main - sequence single star evolution of their low mass precursors , ( b ) during the post - main - sequence evolution of a evolved mass donor , and/or ( c ) via mixing during a mass - transfer event or stellar merger . the warmest li - deficient stars at the turnoff would be regarded as emerging `` canonical '' blue stragglers , whereas cooler ones represent sub - turnoff - mass `` blue - stragglers - to - be . '' the latter are presently hidden on the main sequence , li depletion being possibly the clearest signature of their past history and future significance . eventually , the main sequence turnoff will reach down to their mass , exposing those li - depleted stars as canonical blue stragglers when normal stars of that mass evolve away . arguing _ against _ this unified view is the observation that the three li - depleted stars at @xmath1 k are _ all _ binaries , whereas very few of the cooler systems show evidence for binarity ; it is thus possible that two separate mechanisms are responsible for the production of li - deficient main - sequence halo stars .

expanding maps of the unit interval have been widely studied in the last decades and the associated transfer operators have proven to be of vital importance in solving problems concerning the statistical behaviour of the underlying interval maps @xcite . in recent years an increasing amount of interest has developed in maps which are expanding everywhere except on an unstable fixed point ( that is , an indifferent fixed point ) at which trajectories are considerably slowed down . this leads to an interplay of chaotic and regular dynamics , a characteristic of intermittent systems @xcite . from an ergodic theory viewpoint , this phenomenon leads to an absolutely continuous invariant measure having infinite mass . therefore , standard methods of ergodic theory can not be applied in this setting ; indeed it is wellknown that birkhoff s ergodic theorem does not hold under these circumstances , see for instance @xcite . in this paper we will be concerned with @xmath0-farey maps , see figure [ fig1 ] . these maps are of great interest since they are piecewise linear and expanding everywhere except for at the indifferent fixed point where they have ( right ) derivative one . this makes the @xmath0-farey maps a simple model for studying the physical phenomenon of intermittency @xcite . moreover , an induced version of the @xmath0-farey maps are given by the @xmath0-lroth maps introduced in @xcite , which have significant meaning in number theory , see for instance @xcite . thaler @xcite was the first to discern the asymptotics of the transfer operator of a class of interval maps preserving an infinite measure . this class of maps , to which the @xmath0-farey maps do not belong , have become to be known as thaler maps . in an effort to generalise this work , by combining renewal theoretical arguments and functional analytic techniques , a new approach to estimate the decay of correlation of a dynamical system was achieved by sarig @xcite . subsequently , gouzel @xcite generalised these methods . using these ideas and employing the methods of garsia and lamperti @xcite , erickson @xcite and doney @xcite , recently melbourne and terhesiu ( * * theorem 2.1 to 2.3 ) proved a landmark result on the asymptotic rate of convergence of iterates of the induced transfer operator and showed that these result can be applied to gibbs - markov maps , thaler maps , afn maps , and pomeau - manneville maps . thus , the question which naturally arises is , whether this asymptotic rate can be related to the asymptotic rate of convergence of iterates of the actual transfer operator . the results of this paper give some positive answers to this question for @xmath1-expansive @xmath0-farey maps . as mentioned above , in this paper , we will consider the @xmath0-farey map @xmath2 \to [ 0 , 1]$ ] , which is given for a countable infinite partition @xmath3 of @xmath4 by non - empty intervals @xmath5 . it is assumed throughout that the atoms of @xmath0 are ordered from right to left , starting with @xmath6 , and that these atoms only accumulate at zero . further , we assume that @xmath5 is right - open and left - closed , for all natural numbers @xmath7 . we define the @xmath0-_farey map _ \to [ 0 , 1]$ ] by @xmath8 where @xmath9 is equal to the lebesgue measure @xmath10 of the atom @xmath11 and @xmath12 denotes the lebesgue measure of the @xmath7-th tail @xmath13 of @xmath0 , see figure [ fig1 ] . throughout , we will assume that the partition @xmath0 satisfies the condition that the sequence @xmath14 is not summable . for @xmath15 $ ] , an @xmath0-farey map @xmath16 is said to be _ @xmath1-expansive _ if the sequence @xmath17 is regularly varying of order @xmath18 , that is , if there exists a slowly varying function @xmath19 such that @xmath20 , for all @xmath21 . ( recall that @xmath22 is called a _ slowly varying function _ , if it is measurable , locally riemann integrable and @xmath23 , for each @xmath24 and for some @xmath25 , see @xcite for further details . ) in this situation , ( * ? ? ? * theorem 1.5.10 ) implies that @xmath26 therefore , the lebesgue measure of the @xmath7-th tail of @xmath0 is asymptotic to a regularly varying function of order @xmath27 . thus , @xmath1-expansive implies expansive of order @xmath1 in the sense of @xcite . however , an expansive @xmath0-farey map of order @xmath1 is not necessarily @xmath1-expansive . throughout , let @xmath28 denote the @xmath16-invariant measure which is determined by @xmath29 and let @xmath30 denote the borel @xmath31-algebra of @xmath32 $ ] . here and in the sequel , for a given borel set @xmath33 , we let @xmath34 denote the indicator function on @xmath35 . it is verified in @xcite that , since the sequence @xmath17 is regularly varying of order @xmath36 , the map @xmath16 is conservative , ergodic and measure preserving on the infinite and @xmath31-finite measure space @xmath37 , \mathscr{b } , \mu_{\alpha})$ ] . the dynamical system @xmath38 , \mathscr{b } , \mu_{\alpha } , f_{\alpha})$ ] will be referred to as a _ @xmath0-farey system_. following the definitions and notations of @xcite , throughout , we let @xmath39)$ ] ( respectively @xmath40)$ ] ) denote the class of measurable functions @xmath41 with domain @xmath32 $ ] for which @xmath42 is @xmath28-integrable ( respectively @xmath43-integrable ) , and for @xmath44)$ ] ( respectively @xmath40)$ ] ) , define @xmath45 ( respectively @xmath46 ) by @xmath47 further , for a given measurable function @xmath48 , we set @xmath49 } \rvert w(x ) \lvert$ ] . the _ @xmath0-farey transfer operator _ @xmath50 ) \to \mathcal{l}^{1}_{\mu_{\alpha}}([0,1])$ ] is the positive linear operator given by @xmath51 where @xmath52})^{-1}$ ] and @xmath53})^{-1}$ ] refer to the inverse branches of @xmath16 . in particular , for all @xmath54)$ ] and all measurable functions @xmath48 with @xmath55 , @xmath56 ( the above equality is a direct consequence of ( * ? ? ? * lemma 2.5 ) . ) note that the equality given in is the usual defining relation for the _ transfer operator _ of @xmath16 . however , the relation in only determines values of the transfer operator of @xmath16 applied to an observable @xmath28-almost everywhere . thus the @xmath0-farey transfer operator is a version of the transfer operator of @xmath16 . in order to state our main theorems , we will also require the following function spaces . we let @xmath57 denote the _ first return time _ , given by @xmath58 , and we write @xmath59 . let @xmath60 denote the countable - infinite partition @xmath61 of @xmath6 and let @xmath62 denote the set of functions with domain @xmath32 $ ] that are supported on a subset of @xmath63 and which have finite @xmath64-norm , where @xmath65 and where @xmath66 in particular , if @xmath67 , then @xmath41 is lipschitz continuous on each atom of @xmath60 , zero outside of @xmath63 and bounded ( everywhere ) . we then define @xmath68 ) : \lvert v \rvert_{\infty } < \infty\ ; \text{and } \ ; \widehat{f}^{n-1}_{\alpha}(v \cdot \mathds{1}_{a_{n } } ) \in \mathcal{b}_{\alpha } \ ; \text{for all } \ ; n \in \mathbb{n } \right\}. \ ] ] for examples of observables belonging to @xmath69 , we refer the reader to example [ ex : ex1 ] and the discussion succeeding our main results , theorems [ thm : main2 ] and [ thm : main1 ] . let us also recall from @xcite that the _ wandering rate _ of @xmath16 is given by @xmath70 further , as we will see in , if @xmath71 and if the given @xmath0-farey system is @xmath1-expansive , then the wandering rate is regularly varying of order @xmath72 . also , in the case that @xmath73 , if @xmath74 is a bounded sequence , then we say that the wandering rate @xmath75 is _ moderately increasing_. here and in the sequel for @xmath76 we let @xmath77 denote the smallest integer greater than or equal to @xmath78 . with the above preparations , we are now in a position to state the main results , theorems [ thm : main2 ] and [ thm : main1 ] . theorem [ thm : main2 ] provides mild conditions under which the asymptotic behavior of the iterates of an @xmath0-farey transfer operator _ restricted to _ @xmath6 can be extended to all of @xmath79 $ ] and is used in our proof of theorem [ thm : main1 ] . ( note that , by , any @xmath1-expansive @xmath0-farey system satisfies the requirements of theorem [ thm : main2 ] . ) one of the facets of theorem [ thm : main1 ] is that it gives sufficient conditions on observables which guarantee that iterates of an @xmath0-farey transfer operator applied to such an observable is asymptotic to a constant times the wandering rate . these results complement ( * ? ? ? * theorem 10.5 ) and show that additional assumptions are required in ( * ? ? ? * theorem 10.4 ) . namely , in the case that @xmath73 , we show that the statement of ( * ? ? ? * theorem 10.4 ) holds true , with the additional assumption that the wandering rate is moderately increasing ( theorem [ thm : main1][i ] ) ; for @xmath80 , we provide an example which demonstrates that additional requirements are necessary for the expected convergence ( theorem [ thm : main1][iii ] ) and provide sufficient conditions ( theorem [ thm : main1][ii ] ) . [ thm : main2 ] for an @xmath0-farey system @xmath37 , \mathscr{b } , \mu_{\alpha } , f_{\alpha})$ ] for which the wandering rate satisfies the condition @xmath81 , we have that , if @xmath82)$ ] satisfies @xmath83 uniformly on @xmath63 , then the same holds on any compact subset of @xmath79 $ ] . the same statement holds when replacing uniform convergence by almost everywhere uniform convergence . for @xmath71 , a @xmath1-expansive @xmath0-farey system has wandering rate satisfying @xmath81 . [ thm : main1 ] let @xmath37 , \mathscr{b } , \mu_{\alpha } , f_{\alpha})$ ] be a @xmath1-expansive @xmath0-farey system . 1 . [ i ] let @xmath84 and assume that the wandering rate is moderately increasing.if@xmath85andif @xmath86 then uniformly on compact subsets of @xmath79 $ ] , @xmath87 2 . [ ii ] for @xmath88 $ ] , if @xmath89)$ ] with @xmath90 bounded and if 1 . the sequence @xmath91 is bounded and 2 . there exist constants @xmath92 and @xmath93 with @xmath94 , for all @xmath21 . + then uniformly on compact subsets of @xmath79 $ ] , @xmath95 here , @xmath96 and @xmath97 denotes the gamma function . [ iii ] for @xmath80 , there exists a positive , locally constant , riemann integrable function @xmath98 of bounded variation satisfying the inequality in , such that , for all @xmath99 , @xmath100 it is immediate that if @xmath101 , then @xmath102 and @xmath103 , and that these parameters give rise to an example of an @xmath0-farey system which satisfies the conditions of theorem [ thm : main1][i ] . indeed there exist many examples of @xmath0-farey systems for which the conditions of theorem [ thm : main1][i ] are satisfied , but where the wandering rate behaves very differently to the function @xmath104 . letting @xmath73 , as we will see in lemma [ lem : powerslowly][sv(vi ) ] , the sequence @xmath105 is slowly varying and @xmath106 . we also have that @xmath107 using this we deduce the following . 1 . if @xmath108 , then @xmath109 and @xmath110 . if @xmath111 , then @xmath112 and @xmath113 , + where @xmath114 . indeed the above two sets of parameters give rise to examples of @xmath115-expansive @xmath0-farey systems whose wandering rate is moderately increasing . moreover , @xmath116 demonstrating that two moderately increasing wandering rates , although they are all slowly varying , do not have to be asymptotic to each other nor to the function @xmath104 . in the case that @xmath16 is a @xmath115-expansive @xmath0-farey map , we have that the wandering rate @xmath75 is a slowly varying function . we remark here that it is not the case that every slowly varying function is moderately increasing , namely , it is not the case that if @xmath117 \to \mathbb{r}$ ] is a slowly varying function , then the sequence @xmath118 is bounded . for instance consider the following . let @xmath119 be a decreasing sequence of positive real numbers which converge to zero and , for @xmath120 , set @xmath121 where @xmath122 . we define @xmath123 by @xmath124 for @xmath125 $ ] . the function @xmath126 \to \mathbb{r}$ ] defined by @xmath127 is , by construction , slowly varying . however , the sequence given in is unbounded . ( we are grateful to fredrik ekstrm for providing this example ) . if in the definition of the norm @xmath64 , one replaces the norm @xmath128 by the _ essential supremum norm _ @xmath129 , then by appropriately adapting the proofs given in the sequel , one can obtain a proof of theorem [ thm : main1 ] where the uniform convergence on compact subsets of @xmath79 $ ] is replaced by uniform convergence almost everywhere on compact subsets of @xmath4 . the first part of the proof of theorem [ thm : main1 ] [ i ] and [ ii ] are inspired by the first paragraph in the proof of ( * ? ? ? * theorem 10.4 ) . the structure of this paper is as follows . in section [ sec : pre ] we collect basic properties of @xmath0-farey maps and their corresponding transfer operators . in section [ j_diploma_thesis ] we provide a proof of theorem [ thm : main2 ] . this proof is inspired by arguments originally presented in @xcite . then in section [ sec : main1 ] we present the proof of theorem [ thm : main1 ] , breaking the proof into three constituent parts . in section [ section : counterexamples2.2 ] we obtain part [ i ] and give explicit examples of observables satisfying the given properties . in section [ mre_alpha_delta_in_051 ] we prove part [ ii ] , for explicit examples of observables which satisfy the pre - requests of theorem [ thm : main1 ] [ ii ] we refer the reader to remark [ rmk : rmk2 ] . finally we conclude with section [ section : counterexamples ] where part [ iii ] is proven using a constructive argument . before we conclude this section with a series of remarks , remarks [ rmk : rmk1 ] to [ rmk : rmk3 ] , in which we comment on how theorem [ thm : main1 ] , and hence theorem [ thm : main2 ] , complement the results obtained in @xcite , we introduce the _ perron - frobenius operator _ @xmath130 ) \to \mathcal{l}_{\lambda}^{1}([0 , 1])$ ] which is defined by @xmath131 where @xmath132 denotes the right derivative of @xmath16 and where @xmath133 . ( note , by construction , if @xmath16 is @xmath1-expansive , then the right derivative of @xmath16 at zero is equal to one . ) a useful relation between the operators @xmath134 and @xmath135 is that @xmath136 we refer the reader to @xcite for a proof of the equality in . [ rmk : rmk1 ] for certain interval maps @xmath137 \to [ 0,1]$ ] with two monotonically increasing , differentiable branches whose invariant measure has infinite mass and whose tail probabilities are regularly varying with exponent @xmath138 , thaler @xcite discerned the precise asymptotic behaviour of iterates of the associated perron - frobenius operator @xmath139 , namely , that for all riemann integrable functions @xmath140 with domain @xmath32 $ ] , one has that @xmath141 uniformly almost everywhere on compact subsets of @xmath142 $ ] . here , @xmath143 denotes the associated invariant density and @xmath144 denotes the wandering rate of @xmath145 . however , @xmath0-farey maps do not fall into this class of interval maps . using the relationship between the transfer and the perron - frobenius operator , theorem [ thm : main1 ] [ ii ] together with the assumption that the banach space of functions of bounded variation[page : bv - banach ] with the norm @xmath146 satisfies certain functional analytic conditions ( namely , conditions ( h1 ) and ( h2 ) given in section [ sec : pre ] ) , show that thaler s result can be extended to @xmath1-expansive @xmath0-farey maps . results of this form have also been obtained in @xcite for afn maps . ( note , an @xmath0-farey map is also not an afn map . ) [ rmk : rmk2 ] kessebhmer and slassi @xcite showed that for the classical farey map the convergence given in holds uniformly almost everywhere on @xmath147 $ ] for convex . likewise , for a @xmath1-expansive @xmath0-farey map , theorems [ thm : main1 ] [ ii ] implies that if @xmath140 is a convex @xmath148-observable , then the convergence in holds uniformly on compact subsets of @xmath142 $ ] . to see that a convex @xmath148-observable satisfies the requirements of theorem [ thm : main1 ] [ ii ] , one employs arguments similar to those used in example [ ex : ex1 ] together with and . [ rmk : rmk3 ] the consequences of theorem [ thm : main1 ] go even further , in that for a map , we are able to obtain that the convergence given in holds uniformly on compact subsets of @xmath79 $ ] , for certain non - riemann integrable observables which are not necessarily bounded . for instance , if @xmath149 is an observable such that @xmath150 , @xmath151 and @xmath152 , for some @xmath93 , then , as we will see in lemma [ lem : partiiclaim3 ] , since @xmath153 , this observable fulfils the conditions of theorem [ thm : main1 ] [ ii ] and it is neither riemann integrable nor is it bounded . we use the symbol @xmath154 between the elements of two sequences of real numbers @xmath155 and @xmath156 to mean that the sequences are asymptotically equivalent , namely that @xmath157 . we use the landau notation @xmath158 , if @xmath159 . the same notation is used between two real - valued function @xmath41 and @xmath160 , defined on the set of real numbers @xmath161 , positive real numbers @xmath162 , natural numbers @xmath163 or non - negative integers @xmath164 . specifically , if @xmath165 , then we will write @xmath166 , and if @xmath167 , then we will write @xmath168 . the map @xmath169 defined by @xmath170 is called the _ first return map _ and it is well known that @xmath171 is conservative , ergodic and measure preserving on @xmath172 , see for instance ( * ? ? ? * propositions 1.4.8 and 1.5.3 ) . from this point on , we write @xmath28 for both @xmath28 and @xmath173 and @xmath30 for both @xmath30 and @xmath174 . also , throughout , unless otherwise stated , we assume that @xmath16 is @xmath1-expansive . we denote the open unit disk in @xmath175 by @xmath176 , its closure by @xmath177 and its boundary by @xmath178 . given @xmath179 , define @xmath180 ) \to \mathcal{l}_{\mu_{\alpha}}^{1}([0,1])$ ] by @xmath181 it is an easy exercise to show that @xmath182 is a version of the transfer operator of the map @xmath171 . namely , for all @xmath183)$ ] and all measurable functions @xmath48 with @xmath184 finite , we have that @xmath185 we will see in proposition [ prop : conditions_h1_h2 ] that @xmath186 is a banach space , that the operators @xmath187 and @xmath182 map @xmath62 into itself and that the following properties are fulfilled . ( h1 ) : : there exists a constant @xmath188 such that the operator @xmath189 is a bounded linear operator with @xmath190 , for all @xmath21 . ( here , the operator norm @xmath191 is taken with respect to the banach space @xmath186 . ) ( h2 ) : : a result that will be crucial in the proof of theorem [ thm : main1 ] is ( * ? ? ? * theorem 2.1 ) . in order to see how this result reads in our situation , note , for a @xmath1-expansive @xmath0-farey map , that @xmath192 , which is essential in the proof of ( * ? ? ? * theorem 2.1 ) given in @xcite . further , since @xmath193 and @xmath194 , for all @xmath21 , karamata s tauberian theorem for power series ( * ? ? ? * corollary 1.7.3 ) implies that , for @xmath71 , @xmath195 here , @xmath196 . [ thm : mt2011:thm2.1 ] assuming the above setting , in particular that conditions ( h1 ) and ( h2 ) are satisfied , we have that @xmath197 in the sequel , we will also use of the following auxiliary results , where we set @xmath198 and for each @xmath21 and for each word @xmath199 we let @xmath200 \to [ 0 , 1]$ ] denote the function @xmath201 . if @xmath202 is equal to the empty word , then we set @xmath203 to be equal to the identity map . [ lamma : partiiclaim1 ] let @xmath2 \to [ 0 , 1]$ ] denote an arbitrary @xmath0-farey map . for each @xmath120 , we have that @xmath204 where the constants @xmath205 are given recursively by @xmath206 in particular , letting @xmath207 , we have that @xmath208 , for each @xmath120 . we proceed by induction on @xmath209 . the start of the induction is an immediate consequence of . suppose that the statement is true for some @xmath120 . we then have that @xmath210 this completes the proof of . the remaining assertion is proven by a straight forward inductive argument , using the defining relations given in . [ lem : partiiclaim3 ] for each @xmath21 , we have that @xmath211 and hence , by the definition of the norm , @xmath212 . for @xmath213 the result is immediate . for @xmath214 , we have , by lemma [ lamma : partiiclaim1 ] , that , on @xmath215 , @xmath216 to complete the proof , we need to evaluate the function @xmath217 at the point @xmath115 for @xmath218 . by lemma [ lamma : partiiclaim1 ] , we have that @xmath219 this completes the proof . we will now show that conditions ( h1 ) and ( h2 ) are satisfied for every @xmath1-expansive @xmath0-farey system and for the banach space @xmath186 . [ prop : conditions_h1_h2 ] the pair @xmath186 forms a banach space and for a @xmath1-expansive @xmath0-farey system , the operators @xmath187 and @xmath182 map @xmath62 into itself . moreover , ( h1 ) and ( h2 ) are satisfied . in the proof of the above proposition we will make use of the following lemma . [ lemma : c_10 ] for any @xmath0-farey map @xmath16 , we have that @xmath220 , where @xmath221 , for each @xmath21 . by construction of the @xmath0-farey map @xmath16 , we have that @xmath222 $ ] and that @xmath223 $ ] , for all integers @xmath224 . thus , @xmath225 we will now show by induction on @xmath7 that , for each @xmath120 , @xmath226 from , we have that @xmath227 , for each @xmath120 . suppose that the statement in is true for some @xmath21 . from , we have that @xmath228 for each @xmath120 , which gives @xmath229 this completes the proof of the statement in . setting @xmath230 in , we obtain that @xmath231 , for all @xmath21 . combining this with , completes the proof . it is shown in ( * ? ? ? * section 1 ) that the pair @xmath186 forms a banach space . we now prove that condition ( h1 ) holds and the invariance of @xmath62 . for this , let @xmath232 and fix @xmath120 . applying lemmas [ lamma : partiiclaim1 ] and [ lemma : c_10 ] we have that @xmath233 hence , by definition of the partition @xmath60 , we have that @xmath234 , and so , the operator @xmath235 maps @xmath62 into itself . further , by definition of @xmath182 , this gives that @xmath236 and so , the operator @xmath182 maps @xmath62 into itself . linearity of @xmath235 and @xmath182 follows from the linearity of @xmath135 . for the proof of property ( h2)(i ) , observe that @xmath171 is a piecewise linear expansive map with the following properties . 1 . on the set @xmath237 , the absolute value of the derivative of @xmath171 is equal to @xmath238 . moreover , since @xmath14 is a positive monotonically decreasing sequence which is bounded above by @xmath115 , it follows that there exists a constant @xmath239 with @xmath240 , for all @xmath21 . 2 . the partition @xmath60 is a countable - infinite partition of @xmath6 and @xmath241 and @xmath242 if @xmath218 , and hence , @xmath243 , for all @xmath21 . moreover , the @xmath31-algebra generated by @xmath244 is equal to the borel @xmath31-algebra on @xmath63 . 3 . for each @xmath21 and @xmath245 , we have that @xmath246 ) = \overline{\ { \phi = n \ } } \quad \text{and } \quad \frac{\mathrm{d } \mu_{\alpha } \circ f_{\alpha , \psi}}{\mathrm{d } \mu_{\alpha } } = t_{n } - t_{n+1 } = a_{n}. \ ] ] given these properties , ( h2)(i ) is a consequence of ( * ? ? ? * theorem 1.6 ) : the proof of which is based on the _ theorem on the difference of two norms _ by ionescu - tulcea and marinescu @xcite . for the proof of property ( h2)(ii ) , we distinguish between the cases @xmath247 and @xmath248 . _ case 1_. ( @xmath247 ) : : sarig showed in ( * ? ? ? * section 3 ) that @xmath249 where the operators @xmath250 ) \to \mathcal{l}_{\mu_{\alpha}}^{1}([0 , 1])$ ] are defined by @xmath251 by way of contradiction , suppose that @xmath115 is an eigenvalue of @xmath252 restricted to @xmath62 . then there exists a non - zero measurable function @xmath253 such that @xmath254 . substituting this into shows that @xmath48 is equal to zero @xmath28-almost everywhere , which gives a contradiction . _ case 2_. ( @xmath248 ) : : we will now show that @xmath115 is not an eigenvalue of @xmath252 . ( this part of the proof is based on the proof of ( * ? ? ? * lemma 6.7 ) . ) since @xmath248 , there exists a @xmath255 such that @xmath256 . suppose that @xmath257 , for some non - zero @xmath67 . let @xmath258 denote the class of complex - valued measurable functions @xmath41 with domain @xmath63 for which @xmath259 is @xmath28-integrable , and let it be equipped with the standard @xmath260-inner product , @xmath261 for each @xmath262 set @xmath263 . further , set @xmath264 and define @xmath265 by @xmath266 . noting that @xmath267 and using , we have that , for all @xmath268 and @xmath269 , @xmath270 further , @xmath271 and , as @xmath171 preserves the measure @xmath28 restricted to @xmath63 , we have that @xmath272 combining and , it follows that @xmath273 vanishes @xmath28-almost everywhere on @xmath63 . thus , by taking the modulus , the ergodicity of @xmath171 implies that @xmath42 is equal to a constant , everywhere on @xmath63 . as @xmath41 does not vanish @xmath28-almost everywhere , this constant is non - zero , and so , we obtain that @xmath274 almost everywhere on @xmath63 . now , for each @xmath21 , let @xmath275 be the interval of positive measure , such that @xmath276 and let @xmath277 since @xmath278 almost everywhere , and since the map @xmath171 is linear and expanding , we have that @xmath279 . in particular , the set @xmath280 is non - empty . we claim that there exists @xmath281 such that @xmath282 , for each @xmath283 . by way of contradiction , suppose that @xmath284 , for all @xmath285 . since @xmath41 is constant almost everywhere on @xmath63 , we have that @xmath41 is constant almost everywhere on @xmath286 , which gives an immediate contradiction to the assumption that @xmath284 , for all @xmath285 . therefore , we have that there exists @xmath287 , such that @xmath288 , @xmath282 and @xmath289 , for each @xmath283 . hence , we have that @xmath290 , for all @xmath291 , contradicting the initial choice of @xmath292 . finally , in preparation for the proof of theorem [ thm : main1 ] , let us make note of the following well known properties of slowly varying functions . [ lem : powerslowly ] let @xmath293 \to \mathbb{r}$ ] be a positive slowly varying function , for some @xmath294 . 1 . [ sv(i ) ] for a compact interval @xmath295 we have that @xmath296 holds uniformly with respect to @xmath297 , and hence , for a fixed @xmath298 , @xmath299 2 . [ sv(ii ) ] for a fixed @xmath298 we have that @xmath300 3 . [ sv(v ) ] if @xmath301 is continuous , strictly increasing and @xmath302 then , for a fixed @xmath303 , @xmath304 4 . [ sv(vi ) ] if @xmath305 is defined to be the linear interpolation of the function @xmath306 then @xmath307 is a slowly varying function and @xmath308 let us first recall that , for @xmath309 $ ] and @xmath291 , @xmath310 which gives @xmath311 we proceed by induction as follows . the start of the induction is given by the assumption in the theorem . for the inductive step , assume that the statement holds for @xmath312 , for some @xmath313 . then consider some arbitrary @xmath314 , and let @xmath315 denote the unique element in @xmath316 such that @xmath317 . using , the fact that @xmath318 and the inductive hypothesis in tandem with the assumption that @xmath319 , we obtain that @xmath320 where the last equality is a consequence of the eigenequation @xmath321 . using an analogous proof to that given above , one can obtain that the result of theorem [ thm : main2 ] holds for other interval maps , such as gibbs - markov maps , thaler maps and pomeau - manneville maps . throughout this section , we let @xmath37 , \mathscr{b } , \mu_{\alpha } , f_{\alpha})$ ] be a @xmath115-expansive @xmath0-farey system . in order to prove theorem [ thm : main1 ] [ i ] , we will use the following auxiliary results ( lemmas [ lem : convergnecelemma ] and [ lem : jsigman ] ) . before which we require the following notation . define the function @xmath322 by @xmath323,\\ t_{n+1 } ( x -n ) + w_{n } & \text{if } \ ; x \in [ n , n+1 ] , \ ; \text{for } \ ; n \in \mathbb{n}. \end{cases } \ ] ] note that @xmath324 is the linear interpolation of the function @xmath325 defined on @xmath164 , where @xmath326 . further , for @xmath327 , define @xmath328 for all @xmath329 . [ lem : convergnecelemma ] for a given @xmath327 , we have that @xmath330 . for @xmath327 , we have that @xmath331 where the last equality follows from the fact that @xmath324 is a positive , strictly monotonically increasing function and lemma [ lem : powerslowly ] [ sv(v ) ] . here and in the sequel we will use the following notation . for @xmath76 we let @xmath332 denote the largest integer not exceeding @xmath78 . [ lem : jsigman ] let @xmath333 denote a sequence of positive real numbers such that @xmath334 . if the wandering rate is moderately increasing , then @xmath335 without loss of generality , assume that @xmath336 and let @xmath327 be fixed . by definition of @xmath324 , we have , for @xmath337 , that @xmath338 by definition of @xmath339 , we have that @xmath340 . further , by lemma [ lem : powerslowly ] [ sv(vi ) ] and since @xmath341 is regularly varying sequence of order @xmath342 , we have that , @xmath343 if @xmath344 , then this completes the proof . otherwise , note that , by lemma [ lem : powerslowly ] [ sv(ii ) ] , we have that @xmath324 is a slowly varying function . also , since @xmath324 is an unbounded monotonically increasing function we have that @xmath345 and , by lemma [ lem : convergnecelemma ] , we have that @xmath346 . the above three statements in tandem with the assumptions that @xmath334 and that the wandering rate is moderately increasing , yield the following : @xmath347 this completes the proof in the case @xmath348 . by theorem [ thm : mt2011:thm2.1 ] and proposition [ prop : conditions_h1_h2 ] , we have for each @xmath21 that there exists @xmath349 such that @xmath350 and @xmath351 set @xmath352 , for @xmath21 and @xmath353 . by , we have on @xmath63 that @xmath354 since @xmath355)$ ] , it follows that in the final line of the third term converges to zero . to see that the first and the second term in the final line of converge to to zero , observe that 1 . since @xmath98 , we have that @xmath82)$ ] and , moreover , @xmath356 2 . since @xmath98 we have that @xmath357 is finite , and so the sequence @xmath358 is a bounded sequence ; 3 . using lemma [ lem : partiiclaim3 ] together with the fact that @xmath135 is positive and linear and the fact that if @xmath98 , then @xmath357 is finite , we have that @xmath359 ; 4 . given @xmath360 , there exists @xmath361 such that @xmath362 , for all @xmath363 . combining these observations with lemma [ lem : jsigman ] and , we have that the first and the second term in the final line of converge to to zero . since the arguments given above are independent of a given point in @xmath63 , an application of theorem [ thm : main2 ] now finishes the proof . in the proof of theorem [ thm : main1 ] [ i ] we have not used the specific structure of @xmath62 . we only used that @xmath62 is a banach space which satisfies conditions ( h1 ) and ( h2 ) . thus , we may replace @xmath62 by an arbitrary banach space which satisfies conditions ( h1 ) and ( h2 ) . for such alternative banach spaces see remark [ rmk : rmk1 ] . in doing such a substitution one may change the uniform convergence to almost everywhere uniform convergence . to conclude , we give examples of @xmath115-expansive @xmath0-farey systems and of observables which belong to the set @xmath69 and which satisfy the summability condition given in . [ ex : ex1 ] let @xmath37 , \mathscr{b } , \mu_{\alpha } , f_{\alpha})$ ] denote a @xmath115-expansive @xmath0-farey system with moderately increasing wandering rate . set @xmath364 , where @xmath365 , for @xmath366 ) \ ; \text{and } \ ; f \in c^{2}((0,1 ) ) \ ; \text{with } \ ; f ' > 0 \ ; \text{and } \ ; f''\leq 0\}.\ ] ] we claim that @xmath367 and moreover , that @xmath140 satisfies the summability condition given in . we first verify that @xmath367 . for this , we are required to show that @xmath368)$ ] , that @xmath369 and that @xmath370 , for all @xmath371 . by definition , any function belonging to @xmath372 is convex and continuous on @xmath4 , twice differentiable and @xmath28-integrable . thus , @xmath373)$ ] and @xmath369 . combining this with the fact that @xmath374 is @xmath28 integrable , non - negative and bounded , we have that @xmath368)$ ] and @xmath369 . let us now turn to the second assertion , namely that @xmath375 , for all @xmath21 . we immediately have that @xmath376 . for @xmath218 , note that , if @xmath160 is a differentiable lipschitz function on @xmath63 , then @xmath377 . thus , by lemma [ lem : partiiclaim3 ] and the chain rule , we have that , for each integer @xmath218 , @xmath378 since @xmath379 , we have that @xmath380 and that @xmath381 , for all @xmath382 . therefore , since @xmath383 , it follows that there exists @xmath188 , such that @xmath384 combining this with and using the facts that the sequence @xmath385 is summable and that @xmath386 and @xmath387 are finite , the summability condition in follows . hence it follows that @xmath367 . recall that @xmath37 , \mathscr{b } , \mu_{\alpha } , f_{\alpha})$ ] is a @xmath1-expansive @xmath0-farey system and that @xmath389 . from and , we have that , for all @xmath21 , @xmath390 combining this with the assumptions of the theorem , there exists a constant @xmath391 such that @xmath392 , for all @xmath21 . as in the proof of theorem [ thm : main1 ] [ i ] we have , by theorem [ thm : mt2011:thm2.1 ] and proposition [ prop : conditions_h1_h2 ] , that there exists @xmath393 \to \mathbb{r}$ ] such that @xmath394 and , for each @xmath395 , @xmath396 set @xmath397 , for each @xmath21 and @xmath398 . by a calculation similar as in , we have on @xmath63 that @xmath399 as @xmath400)$ ] , the third summand on the rhs of tends to zero . to see that the second term on the rhs of converges to zero , recall that the sequence @xmath401 is positive , monotonically decreasing and bounded above by one and that by assumption @xmath20 . further , by , given @xmath402 , there exist constants @xmath403 and @xmath404 such that , for all @xmath405 with @xmath406 , @xmath407 this implies that given @xmath408 , we have for all @xmath21 such that @xmath409 , @xmath410 ( recall that @xmath411 is the value given in condition ( b ) of theorem [ thm : main1 ] [ ii ] ) . a simple calculation shows that the rhs of the latter inequality converges to zero as @xmath7 tends to infinity . further , since all of the summand are positive , we have that the limit as @xmath7 tends to infinity exists and equals zero . this together with lemma [ lem : powerslowly ] [ sv(i ) ] implies that , for each @xmath412 , @xmath413 furthermore , @xmath414 since @xmath415 are arbitrary , it follows that @xmath416 and hence , the second term on the rhs of converges to zero as @xmath7 tends to infinity . we now show that the first term on the rhs of converges to zero . by and the fact that @xmath394 , given @xmath402 there exists @xmath417 such that , for all @xmath418 , @xmath419 moreover , there exists a constants @xmath420 such that , for all @xmath395 , @xmath421 furthermore , since @xmath16 is @xmath1-expansive , by condition ( b ) in theorem [ thm : main1 ] [ ii ] , we have that the sequence @xmath422 is summable . these properties together with lemma [ lem : partiiclaim3 ] and an argument similar to that presented in , imply the existence of a constant @xmath423 such that @xmath424 since @xmath402 was chosen arbitrarily , the result follows and , since the arguments given above are independent of a given point in @xmath425 , an application of theorem [ thm : main2 ] finishes the proof . in this section we provide a constructive proof of theorem [ thm : main1 ] [ iii ] . the proof is divided into several parts . first , we define a class of observables @xmath427 . second , in proposition [ prop : l1andlinfty ] we will show that if @xmath428 , then @xmath149 is bounded , of bounded variation , riemann integrable and belongs to @xmath39)$ ] . third , in proposition [ prop : balpha ] we will show that if @xmath428 , then it belongs to the space @xmath69 , and in proposition [ prop : summand ] we will show that the summability condition given in is satisfied for all @xmath428 . finally , in proposition [ prop : liminfandlimsup ] we will show that , if @xmath428 , then @xmath429 combing these results will then yield a proof of theorem [ thm : main1 ] [ iii ] let us now begin by defining the set @xmath427 . we let @xmath427 denote the class of observable @xmath430 \to \mathbb{r}$ ] which are of the following form : @xmath431 where @xmath432 and where @xmath433 , @xmath434 and @xmath435 denote three positive constants , depending on @xmath1 , such that [ ex : exc1c2c31 ] for @xmath80 , choose @xmath440 and @xmath441 . then it is clear that @xmath442 and @xmath443 satisfy the conditions ( c1 ) and ( c2 ) . with these choices one immediately verifies that ( c3 ) is equivalent to @xmath444 , for @xmath445 . hence , by choosing @xmath446 sufficiently small , it follows that the conditions ( c1 ) , ( c2 ) and ( c3 ) can be fulfilled simultaneously . the main reason why we require the sequence @xmath451 is to ensure that @xmath149 is of bounded variation . further , condition ( c3 ) is only required in the proof of the second statement of proposition [ prop : liminfandlimsup ] , specifically when lemma [ lemma : keylemma ] is used . by ( c1 ) and ( c2 ) , we have that @xmath460 and @xmath461 . this implies that , for all @xmath120 , @xmath462 using ( c1 ) and ( c2 ) once more immediately verifies that the latter term is strictly greater than one . clearly the observable @xmath149 is riemann integrable . moreover , @xmath149 is measurable , as each of the atoms of @xmath0 is measurable and @xmath149 is the sum of indicator functions of atoms of @xmath0 . further , the range of @xmath149 is equal to @xmath463 , and thus , @xmath464 . by lemma [ lem : nknk-1 ] , we have that @xmath459 , and so the variation of @xmath149 is equal to @xmath465 , which is finite , as @xmath466 and as @xmath435 is positive . this shows that @xmath149 is of bounded variation . it remains to show that @xmath149 is @xmath28-integrable . for this recall that @xmath467 , for each @xmath120 . choose a positive constant @xmath468 and recall that @xmath193 . by lemma [ lem : powerslowly ] [ sv(ii ) ] , there exists a constant @xmath188 such that @xmath469 , for each @xmath21 . therefore , by lemma [ lemmamvt ] and lemma [ lem : nknk-1 ] , we have that @xmath470 the latter series converges , since @xmath468 , @xmath471 and @xmath472 , for all @xmath120 . by proposition [ prop : l1andlinfty ] , we have that @xmath54)$ ] and that @xmath473 . moreover , by lemma [ lem : partiiclaim3 ] , we have on @xmath32 $ ] , that , for each @xmath371 , @xmath474 therefore , @xmath475 , for all @xmath371 , and hence , it follows that @xmath98 . 1 . using the facts that @xmath426 and @xmath360 , that the sequence @xmath479 is not bounded above and is strictly monotonically increasing , that @xmath480 and that @xmath481 is a fixed natural number , we have that @xmath482 2 . for each @xmath120 , we have that @xmath483 using condition ( c3 ) with the facts that @xmath426 , @xmath360 and that @xmath484 and @xmath435 are positive , it follows that @xmath485 3 . there exist constants @xmath486 such that , for all @xmath120 sufficiently large , @xmath487 here , the first inequality follows from the facts that @xmath488 is a slowly varying function and that @xmath489 together with lemma [ lem : powerslowly ] [ sv(i ) ] . the second inequality follows from lemma [ lem : powerslowly ] [ sv(ii ) ] , which guarantees the existence of the constant @xmath490 such that @xmath491 , for all @xmath21 . by theorem [ thm : mt2011:thm2.1 ] and proposition [ prop : conditions_h1_h2 ] , we have uniformly on @xmath63 that @xmath493 thus , given @xmath402 , there exists @xmath404 such that , for all @xmath406 on @xmath63 , @xmath494 we will first show the second statement in . for this , observe that by it is sufficient to show that , on @xmath6 , @xmath495 in order to see this , let @xmath496 be fixed and let @xmath497 denote the smallest integer for which @xmath498 . since @xmath135 is a positive linear operator , we have , for all @xmath499 , that @xmath500 now , lemma [ lem : powerslowly ] [ sv(i ) ] implies that @xmath501 . as the sequence @xmath502 is positive and since @xmath503 the value @xmath504 is finite and strictly greater than zero . hence , by , and and the fact that @xmath193 , we have on @xmath63 that , for each @xmath120 sufficiently large , @xmath505 by lemma [ lemma : keylemma ] , the latter term diverges . all that remains to show is that the first statement of holds . for this , observe that , by positivity and linearity of @xmath506 , theorem [ thm : mt2011:thm2.1 ] , proposition [ prop : conditions_h1_h2 ] and , we have on @xmath63 that , for each @xmath120 , @xmath507 since @xmath120 was arbitrary , the above inequalities imply that on @xmath63 , @xmath508 suppose that the latter inequality is strict , namely , suppose that there exists a constant @xmath188 such that on @xmath63 , @xmath509 this assumption together with implies that , given @xmath402 , there exists @xmath417 such that , for all @xmath510 and @xmath99 , @xmath511 thus , by karamata s tauberian theorem for power series ( * ? ? ? * corollary 1.7.3 ) , it follows that , for all @xmath510 and @xmath99 , @xmath512 hence , @xmath513 this is a contradiction , since by and by combining theorem [ thm : mt2011:thm2.1 ] with karamata s tauberian theorem for power series ( * ? ? ? * corollary 1.7.3 ) , we have that the set @xmath63 is a set and therefore , by ( * ? ? ? * proposition 3.7.5 ) , the @xmath0-farey system is pointwise dual ergodic , meaning that , for @xmath28-almost every @xmath514 $ ] , we have that @xmath515

we study the asymptotics of iterates of the transfer operator for non - uniformly hyperbolic @xmath0-farey maps . we provide a family of observables which are riemann integrable , locally constant and of bounded variation , and for which the iterates of the transfer operator , when applied to one of these observables , is not asymptotic to a constant times the wandering rate on the first element of the partition @xmath0 . subsequently , sufficient conditions on observables are given under which this expected asymptotic holds . in particular , we obtain an extension theorem which establishes that , if the asymptotic behaviour of iterates of the transfer operator is known on the first element of the partition @xmath0 , then the same asymptotic holds on any compact set bounded away from the indifferent fixed point .

without doubt , most particles can be regarded as composite particles , such as molecules composed of atoms , atoms composed of electrons and nuclei , nuclei composed of nucleons , so on , it is important to recognize that physics must be invariant for the composite particles and their constituent particles , this requirement is called particle invariance in this paper . but difficulties arise immediately because for fermion we use dirac equation , for meson we use klein - gordon equation and for classical particle we use newtonian mechanics , while the connections between these equations are quite indirect . thus if the particle invariance is held in physics , i.e. , only one physical formalism exists for any particle , we can expect to find out the differences between these equations by employing the particle invariance . using this approach is one of the goals of this paper , consequently , several new relationships between them are found , the most important result is that the obstacles that cluttered the path from classical mechanics to quantum mechanics are found , it becomes possible to derive the quantum wave equations from relativistic mechanics after the obstacles are removed . another goal is just to discuss interactions between particles under the particle invariance , several new formulae of interactions are derived and discussed . the new results provide an insight into improving quark model . fermions satisfy fermi - dirac statistics , bosons satisfy bose - einstein statistics , there is a connection between the spin of a particle and the statistics . it is clear that the spin is a key concept for particle physics . in this section we shall show that the spin of a particle is one of the consequences of the particle invariance . according to newtonian mechanics , in a hydrogen atom , the single electron revolves in an orbit about the nucleus , its motion can be described by its position in an inertial cartesian coordinate system @xmath0 . as the time elapses , the electron draws a spiral path ( or orbit ) , as shown in fig.[dfig1](a ) in imagination . if the reference frame @xmath1 rotates through an angle about the @xmath2-axis in fig.[dfig1](a ) , becomes a new reference frame @xmath3 ( there will be a lorentz transformation linking the frames @xmath1 and @xmath3 ) , then in the frame @xmath3 , the spiral path of the electron tilts with respect to the @xmath4-axis with the angle as shown in fig.[dfig1](b ) . at one instant of time , for example , @xmath5 instant , the spiral path pierces many points at the plane @xmath5 , for example , the points labeled @xmath6 , @xmath7 and @xmath8 in fig.[dfig1](b ) , these points indicate that the electron can appear at many points at the time @xmath9 , in agreement with the concept of probability in quantum mechanics . this situation gives us a hint to approach quantum wave nature from relativistic mechanics . because the electron pierces the plane @xmath5 with 4-vector velocity @xmath10 , at every pierced point we can label a local 4-vector velocity . the pierced points may be numerous if the path winds up itself into a cell about the nucleus ( due to a nonlinear effect in a sense ) , then the 4-vector velocities at the pierced points form a 4-vector velocity field . it is noted that the observation plane selected for the piercing can be taken at an arbitrary orientation , so the 4-vector velocity field may be expressed in general as @xmath11 , i.e. the velocity @xmath10 is a function of 4-vector position . at every point in the reference frame @xmath3 the electron satisfies relativistic newton s second law of motion : @xmath12 the notations consist with the convention@xcite . since the cartesian coordinate system is a frame of reference whose axes are orthogonal to one another , there is no distinction between covariant and contravariant components , only subscripts need be used . here and below , summation over twice repeated indices is implied in all case , greek indices will take on the values 1,2,3,4 , and regarding the rest mass @xmath13 as a constant . as mentioned above , the 4-vector velocity @xmath10 can be regarded as a multi - variable function , then @xmath14 @xmath15 substituting them back into eq.([1 ] ) , and re - arranging these terms , we obtain @xmath16 using the notation @xmath17 eq.([4 ] ) is given by @xmath18 because @xmath19 contains the variables @xmath20 , @xmath21 , @xmath22 and @xmath23 which are independent from @xmath24 , then a main solution satisfying eq.([6 ] ) is given by @xmath25 ( in this paper we do not discuss the special solutions that @xmath26 , if they exist ) . according to green s formula or stokes s theorem , the above equation allows us to introduce a potential function @xmath27 in mathematics , further set @xmath28 , we obtain a very important equation @xmath29 where @xmath30 representing wave nature may be a complex mathematical function , its physical meanings is determined from experiments after the introduction of the planck s constant @xmath31 . the magnitude formula of 4-vector velocity of particle is given in its square form by @xmath32 which is valid at every point in the 4-vector velocity field . multiplying the two sides of the above equation by @xmath33 and using eq.([8 ] ) , we obtain @xmath34 \nonumber \\ & = & ( -i\hbar \partial _ \mu -qa_\mu ) ( -i\hbar \partial _ \mu -qa_\mu ) \psi \nonumber \\ & & -[-i\hbar \psi \partial _ \mu ( mu_\mu ) ] \label{10}\end{aligned}\ ] ] according to the continuity condition for the electron motion @xmath35 we have @xmath36 it is known as the klein - gordon equation . on the condition of non - relativity , schrodinger equation can be derived from the klein - gordon equation @xcite(p.469 ) . however , we must admit that we are careless when we use the continuity condition eq.([11 ] ) , because , from eq.([8 ] ) we obtain @xmath37 where we have used lorentz gauge condition . thus from eq.([9 ] ) to eq.([10 ] ) we obtain @xmath38 this is of a complete wave equation for describing the motion of the electron accurately . the klein - gordon equation is a linear wave equation so that the principle of superposition is valid , however with the addition of the last term of eq.([14 ] ) , eq.([14 ] ) turns to display chaos . in the following we shall show dirac equation from eq.([8 ] ) and eq.([9 ] ) . from eq.([8 ] ) , the wave function can be given in integral form by @xmath39 where @xmath40 is an integral constant , @xmath41 and @xmath42 are the initial and final points of the integral with an arbitrary integral path . since maxwell s equations are gauge invariant , eq.([8 ] ) should preserve invariant form under a gauge transformation specified by @xmath43 where @xmath44 is an arbitrary function . then eq.([15 ] ) under the gauge transformation is given by @xmath45 the situation in which a wave function can be changed in a certain way without leading to any observable effects is precisely what is entailed by a symmetry or invariant principle in quantum mechanics . here we emphasize that the invariance of velocity field is held for the gauge transformation . suppose there is a family of wave functions @xmath46 which correspond to the same velocity field denoted by @xmath47 , they are distinguishable from their different phase angles @xmath40 as in eq.([15 ] ) . then eq.([9 ] ) can be given by @xmath48 suppose there are four matrices @xmath49 which satisfy @xmath50 then eq.([18 ] ) can be rewritten as @xmath51[a_{\mu jk}p_\mu \psi ^{(k)}-i\delta _ { jk}mc\psi ^{(k ) } ] \nonumber \\ & & \label{20}\end{aligned}\ ] ] where @xmath52 is kronecker delta function , @xmath53 . for the above equation there is a special solution given by @xmath54\psi ^{(k)}=0 \label{21}\ ] ] there are many solutions for @xmath49 which satisfy eq.([19 ] ) , we select a set of @xmath49 as @xmath55 where @xmath56 and @xmath57 are the matrices defined in dirac algebra@xcite(p.557 ) . substituting them into eq.([21 ] ) , we obtain @xmath58\psi = 0 \label{23}\ ] ] where @xmath30 is an one - column matrix about @xmath59 . then eq.([23 ] ) is just the dirac equation . the dirac equation is a linear wave equation , the principle of superposition is valid for it . let index @xmath60 denote velocity field , then @xmath61 , whose four component functions correspond to the same velocity field @xmath60 , may be regarded as the eigenfunction of the velocity field @xmath60 ( it may be different from the eigenfunction of energy ) . because the velocity field is an observable in a physical system , in quantum mechanics we know , @xmath62 constitute a complete basis in which arbitrary function @xmath63 can be expanded in terms of them @xmath64 obviously , @xmath63 satisfies eq.([23 ] ) . then eq.([23 ] ) is just the dirac equation suitable for composite wave function . alternatively , another method to show the dirac equation is more traditional : at first , we show the dirac equation of free particle by employing plane waves , we easily obtain eq.([23 ] ) on the condition of @xmath65 ; next , adding electromagnetic field , the plane waves are still valid in any finite small volume with the momentum of eq.([8 ] ) when we regard the field to be uniform in the volume , so the dirac equation eq.([23 ] ) is valid in the volume even if @xmath66 , the plane waves constitute a complete basis in the volume ; third , the finite small volume can be chosen to locate at anywhere , then anywhere have the same complete basis , therefore the dirac equation eq.([23 ] ) is valid at anywhere . of course , on the condition of non - relativity , schrodinger equation can be derived from the dirac equation @xcite(p.479 ) . by further calculation , the dirac equation can arrive at klein - gordon equation with an additional term which represents the effect of spin , this term is just the last term of eq.([14 ] ) approximately . but , do not forget that the dirac equation is a special solution of eq.([20 ] ) , therefore we believe there are some quantum effects beyond the dirac equation . with this consequence , it is easy to understand why some problems of quantum electrodynamics can not been completely explained by the dirac equation . eq.([20 ] ) originates from the magnitude formula of 4-vector velocity of particle , the formula is suitable for any particle , so it satisfies the particle invariance . the dirac equation is regarded as an approximation to eq.([20 ] ) , the approximation brings out many troubles with the spin concept . from the dirac equation we can predict that a composite particle and an its constituent both have their own spins , but this prediction is not true for mesons because pion has zero spin while its constituent quark has 1/2 spin , in other words , due to the approximation the dirac equation does not involve some states such as zero spin state . that is why we want to classify particles into fermions and mesons by spin and use different equations . if we can find a precise solution of eq.([20 ] ) instead of the dirac equation , then the classification is not necessary . it is noted that eq.([20 ] ) is nonlinear while the dirac equation is linear , this reminders us that we can never find any precise solutions in a linear equation which satisfy eq.([20 ] ) . therefore , for this problem , a good solution depends on how much precision we can reach for our requirement . in one hand , it is rather remarkable that klein - gordon equation and dirac equation can be derived from relativistic newton s second law of motion approximately , in another hand , all particles , such as fermions , bosons and classical particles , satisfy the relativistic newton s second law ( it will be further clear later ) , thus it is a natural choice that only the relativistic newton s second law is independent and necessary . only one formalism is necessary for any particle , this is just the particle invariance , we arrive at the aim . as mentioned above , the spin is one feature hidden in the relativistic newton s second law , but more features will turn out from the relativistic newton s second law in the following sections . in this section we discuss how to determine the planck s constant that emerges in the preceding section . in 1900 , m. planck assumed that the energy of a harmornic oscilator can take on only discrete values which are integral multiples of @xmath67 , where @xmath68 is the vibration frequency and @xmath69 is a fundamental constant , now either @xmath69 or @xmath70 is called as planck s constant . the planck s constant next made its appearance in 1905 , when einstein used it to explain the photoelectric effect , he assumed that the energy in an electromagnetic wave of frequency @xmath71 is in the form of discrete quanta ( photons ) each of which has an energy @xmath72 in accordance with planck s assumption . from then , it has been recognized that the planck s constant plays a key role in quantum mechanics . according to the previous section , no mater how to move or when to move in minkowski s space , the motion of a particle is governed by a potential function @xmath27 as @xmath73 for applying eq.([p1 ] ) to specific applications , without loss of generality , we set @xmath74 , then eq.([p1 ] ) is rewritten as @xmath75 the coefficient @xmath76 is subject to the interpretation of @xmath30 . there are three mathematical properties of @xmath30 worth recording here . first , if there is a path @xmath77 joining initial point @xmath41 to final point @xmath78 , then @xmath79 second , the integral of eq.([p3 ] ) is independent from the choice of path . third , the superposition principle is valid for @xmath80 , i.e. , if there are @xmath81 paths from @xmath41 to @xmath42 , then @xmath82 @xmath83 @xmath84 where @xmath85 is the average momentum . to gain further insight into physical meanings of this equations , we shall discuss two applications . as shown in fig.[pfig1 ] , suppose that the electron gun emits a burst of electrons at @xmath41 at time @xmath86 , the electrons arrive at the point @xmath42 on the screen at time @xmath87 . there are two paths for the electron to go to the destination , according to our above statement , @xmath30 is given by @xmath88 where we use @xmath89 and @xmath90 to denote the paths @xmath91 and @xmath92 respectively . multiplying eq.([p7 ] ) by its complex conjugate gives @xmath93 \nonumber \\ & = & 2 + 2\cos [ \frac p\kappa ( l_1-l_2 ) ] \label{p8}\end{aligned}\ ] ] where @xmath94 is the momentum of the electron . we find a typical interference pattern with constructive interference when @xmath95 is an integral multiple of @xmath96 , and destructive interference when it is a half integral multiple . this kind of experiments has been done since a long time age , no mater what kind of particle , the comparison of the experiments to eq.([p8 ] ) leads to two consequences : ( 1 ) the complex function @xmath30 is found to be probability amplitude , i.e. , @xmath97 expresses the probability of finding a particle at location @xmath42 in the minkowski s space . ( 2 ) @xmath76 is the planck s constant . the integral of time component in the above calculation has been autimatically canceled because the experimental pattern is stable . let us consider the modification of the two slit experiment , as shown in fig.[pfig2 ] . between the two slits there is located a tiny solenoid s , designed so that a magnetic field perpendicular to the plane of the figure can be produced in its interior . no magnetic field is allowed outside the solenoid , and the walls of the solenoid are such that no electron can penetrate to the interior . like eq.([p7 ] ) , the amplitude @xmath30 is given by @xmath98 and the probability is given by @xmath99 \nonumber \\ & = & 2 + 2\cos [ \frac p\kappa ( l_1-l_2)+\frac 1\kappa \oint_{(l_1+\overline{l_2}% ) } qa_\mu dx_\mu ] \nonumber \\ & = & 2 + 2\cos [ \frac p\kappa ( l_1-l_2)+\frac{q\phi } \kappa ] \label{p10}\end{aligned}\ ] ] where @xmath100 denotes the inverse path to the path @xmath90 , @xmath101 is the magnetic flux that passes through the surface between the paths @xmath89 and @xmath100 , and it is just the flux inside the solenoid . now , constructive ( or destructive ) interference occurs when @xmath102 where @xmath103 is an integer . when @xmath76 takes the value of the planck s constant , we know that this effect is just the aharonov - bohm effect which was shown experimentally in 1960 . in this section we shall correct a mistake about coulomb s force and gravitational force in physical education , which cluttered the path from classical mechanics to quantum mechanics . we also shall discuss maxwell s equations in detail . in the world , almost every young person was educated to know that the coulomb s force and gravitational force act along the line joining a couple of particles , but this knowledge is incorrect in the theory of relativity . in relativity theory , the 4-vector velocity @xmath10 of a particle has components @xmath104 , the magnitude of the 4-vector velocity @xmath10 is given by @xmath105 the above equation is valid so that any force can never change @xmath10 in the magnitude but can change @xmath10 in the direction . we therefore conclude that the coulomb s force and gravitational force on a particle always act in the direction orthogonal to the 4-vector velocity of the particle in the 4-dimensional space - time , rather than along the line joining a couple of particles . alternatively , any 4-vector force @xmath106 satisfy the following perpendicular or orthogonal relation @xmath107 this simple inference clearly tells us that the forces are not centripetal or centrifugal forces about their sources , even if in 3-dimensional space [ see eq.([e7 ] ) ] , this character provides a internal reason for accounting for the quantum behavior of particle or chaos . thus the derivations in terms of 4-vector velocity field in the preceding section become reasonable . in the present paper , eq.([e02 ] ) has been elevated to an essential requirement for definition of force , which brings out many new aspects for coulomb s force and gravitational force . we assume that coulomb s law remains valid only for two particles both at rest in usual 3-dimensional space . suppose there are two charged particle @xmath108 and @xmath109 locating at positions @xmath42 and @xmath110 in a cartesian coordinate system @xmath1 and moving at 4-vector velocities @xmath10 and @xmath111 respectively , as shown in fig.[afig1 ] , where we use @xmath112 to denote @xmath113 . the coulomb s force @xmath106 acting on particle @xmath108 is perpendicular ( orthogonal ) to the velocity direction of @xmath108 , as illustrated in fig.[afig1 ] , like a centripetal force , the force @xmath106 should make an attempt to rotate itself about its path center , the center may locate at the front or back of the particle @xmath109 , so the force @xmath106 should lie in the plane of @xmath111 and @xmath112 , then @xmath114 where @xmath115 and @xmath116 are unknown coefficients , the possibility of this expansion will be further clear in the next subsection in where the expansion is not an assumption [ see eq.([g3 ] ) ] . using the relation @xmath117 , we get @xmath118 we rewrite eq.([e1 ] ) as @xmath119 \label{e3}\ ] ] it follows from the direction of eq.([e3 ] ) that the unit vector of the coulomb s force direction is given by @xmath120 \label{e4}\ ] ] because @xmath121 \nonumber \\ & = & \frac 1{c^2r}[(u\cdot r)u^{\prime } -(u\cdot u^{\prime } ) r ] \nonumber \\ & = & -[(\widehat{u}\cdot \widehat{r})\widehat{u}^{\prime } -(\widehat{u}\cdot \widehat{u}^{\prime } ) \widehat{r } ] \nonumber \\ & = & -\widehat{u}^{\prime } \cosh \alpha + \widehat{r}\sinh \alpha \label{e5a}\end{aligned}\ ] ] @xmath122 where @xmath123 refers to the angle between @xmath10 and @xmath124 , @xmath125 , @xmath126 , @xmath127 , @xmath128 , @xmath129 . suppose that the magnitude of the force @xmath106 has classical form @xmath130 combination of eq.([e6 ] ) with ( [ e4 ] ) , we obtain a modified coulomb s force @xmath131 \nonumber \\ & = & \frac{kqq^{\prime } } { c^2r^3}[(u\cdot r)u^{\prime } -(u\cdot u^{\prime } ) r ] \label{e7}\end{aligned}\ ] ] this force is in the form of lorentz force for the two particles , relating with the ampere s law and biot - savart - laplace law . is perpendicular to the 4-vector velocity @xmath10 of @xmath108 , and lies in the plane of @xmath111 and @xmath112 with the retardation with respect to @xmath109 . ] it follows from eq.([e7 ] ) that the force can be rewritten in terms of 4-vector components as @xmath132 where we have used the relations @xmath133 from eq.([e8c ] ) , because of @xmath134 , we have @xmath135 it is known as the lorentz gauge condition . to note that @xmath124 has three degrees of freedom on the condition @xmath136 , so we have @xmath137 @xmath138 from eq.([e8b ] ) , we have @xmath139 where we define @xmath140 . from eq.([e8b ] ) , by exchanging the indices and taking the summation of them , we have @xmath141 the eq.([e14 ] ) and ( [ e15 ] ) are known as the maxwell s equations . for continuous media , they are valid as well . from the maxwell s equations , we know there is a retardation time for action to propagate between the two particles , the retardation effect is measured by @xmath142 as illustrated in fig.[afig1 ] . then @xmath143 obviously , eq.([e18 ] ) is known as the lienard - wiechert potential for a moving particle . the above formalism clearly shows that maxwell s equations can be derived from the classical coulomb s force and the perpendicular ( orthogonal ) relation of force and velocity . in other words , the perpendicular relation is hidden in maxwell s equation . specially , eq.([e3 ] ) directly accounts for the geometrical meanings of curl of vector potential , the curl contains the perpendicular relation . since the perpendicular relation of force and velocity is one of the consequences from relativistic newton s second law , it is also one of the features from the particle invariance . the above formalism has a significance on guiding how to develope the theory of gravity . in analogy with the modified coulomb s force of eq.([e7 ] ) , we directly suggest a modified universal gravitational force as @xmath144 \nonumber \\ & = & -\frac{gmm^{\prime } } { c^2r^3}[(u\cdot r)u^{\prime } -(u\cdot u^{\prime } ) r ] \label{g1}\end{aligned}\ ] ] for a couple of particles with masses @xmath13 and @xmath145 respectively . comparing with some incorrect statements about coulomb s force and gravitational force in most textbooks , and for emphasizing its feature , the perpendicular ( orthogonal ) relation of force and velocity was called the direction adaptation nature of force in the author s previous paper@xcite . we emphasize that the perpendicular relation of force and velocity must be valid if gravitational force can be defined as a force . it follows from eq.([g1 ] ) that we can predict that there are gravitational radiation and magnet - like components for the gravitational force . particularly , the magnet - like components will act as a key role in the geophysics and atmosphere physics . if we have not any knowledge but know there exists the classical universal gravitation @xmath146 between two particles @xmath13 and @xmath145 , what form will take the 4-vector gravitational force @xmath106 ? suppose that @xmath147 is at rest at the origin , using @xmath148 , @xmath149 and @xmath150 , we have @xmath151 @xmath152 \nonumber \\ & = & \frac 1{icu_4}[(u^{\prime } \cdot u)\mathbf{f}-(\mathbf{u}\cdot \mathbf{f}% ) u^{\prime } ] \nonumber \\ & = & \frac{|\mathbf{f}|}{icu_4|\mathbf{r}|}[(u^{\prime } \cdot u)\mathbf{r}-(% \mathbf{u}\cdot \mathbf{r})u^{\prime } ] \nonumber \\ & = & \frac{|\mathbf{f}|}{icu_4|\mathbf{r}|}[(u^{\prime } \cdot u)r-(u\cdot r)u^{\prime } ] \label{g3}\end{aligned}\ ] ] where @xmath125 , @xmath153 . if we rotate our frame of reference to make @xmath145 not to be at rest , eq.([g3 ] ) will still be valid because of covariance . then we find the 4-vector gravitational force goes back to the form of eq.([g1 ] ) , like lorentz force , having the magnet - like components . it is noted that the perpendicular relation of force and velocity is valid for any force : strong , electromagnetic , weak and gravitational interactions , therefore there are many new aspects remaining for physics to explore . under the invariance of particle , the most simple model of particle is that all particles are composed of identical constituents , the constituent is regarded as the most elementary and most small particle in the world . since quarks have never been observed , our speculation leads us to propose a better model to organize known data . for this challenging purpose , in the present paper , we introduce a fictitious elementary particle , given a name dollon for our convenience , to assemble other particles such as fermions , mesons or classical particles , the dollon is regarded as the most elementary and most small particle in the world . our work focuses on conceptual development . consider a dollon moving in minkowski s space @xmath154 with 4-vector velocity @xmath148 , the motion of the dollon satisfies the magnitude formula of 4-vector velocity of particle @xmath155 differentiating the above equation with respect to the proper time interval @xmath156 of the dollon gives @xmath157 where the result has been written in the two parts by defining a 3-dimensional vector @xmath146 . defining a 4-vector @xmath158 then from eq.([r1 ] ) we have readily @xmath159 it means that @xmath10 and @xmath106 are orthogonal with each other . consider two particles bob and alice located at @xmath42 and @xmath110 in the 4-dimensional space respectively , they are composed of many dollons , the number of dollons in alics is @xmath160 , and in bob is @xmath13 , when bob and alice move with 4-vector velocities @xmath10 and @xmath111 respectively , following eq.([r2 ] ) , they can be assigned two sets of motion equations as @xmath161 @xmath162 now we have a question : what is the interaction between bob and alice ? obviously , the form of eq.([r5 ] ) seems to be relativistic newton second law for bob , @xmath146 seems to be a 3-vector force , @xmath163 seems to be the rate at which the force does work on bob . for seeking for further answers , we need to recall the newton s first law of motion , the law is valid in theory of relativity and reads _ first law : an object at rest will remain at rest and an object in motion will continue to move in a straight line at constant speed forever unless some net external force acts to change this motion . _ if the object is a composite system composed of many dollons , then we can understand the first law with three consequences . consequence 1 : let @xmath1 denote the number of dollons in a composite system , the average velocity of the system is defined as @xmath164 where @xmath165 is the 4-vector velocity of the ith dollon . the average velocity represents the motion of the center of the system . the fist law only means that the center of the system remains at rest or in motion , i.e. , rotation about its center is permited . consequence 2 : the total number of dollons in the system must be unchanged , i.e. , the conservation of dollon number must be held , otherwise any creation or annihilation of dollon will lead to a sudden shift of the center of the system . consequence 3 : when two bodies are seperated from an infinite distance , the interaction between them must vanish . otherwise , no body can be at rest , bacause a rest body will always be affected by the motion of a far distance body , whereas the far distance bodies are innumerable as a background . now we go back to consider the whole system composed of bob and alics , without loss of generality , suppose that the center is at rest at the origin of the frame of reference , then the center has a 4-vector velocity @xmath166 , the at rest refers to being at rest in usual 3-dimensional space . from eq.([r4 ] ) , the quantity @xmath106 must be orthogonal with the 4-vector velocity @xmath10 of bob , likewise for alice , we have @xmath167 they set up a rule for the interaction between bob and alice in the composite system . we specially choose to study the interaction which happens at such instant that the position vector @xmath112 of bob with respect with alice ( i.e. , @xmath168 ) is orthogonal to @xmath10 and @xmath111 simultaneously . @xmath169 the existence of such instant of time will become clear in the subsection [ x ] . from eq.([r8 ] ) and eq.([r9 ] ) , we get parallel relations @xmath170 for bob , using notation @xmath171 , @xmath172 , vector - multiplying eq.([r5 ] ) by @xmath173 , because @xmath146 parallels @xmath173 , we have @xmath174 it means @xmath175 where @xmath176 is an integral constant . likewise for alice . from eq.([r10 ] ) we can expand @xmath177 in a taylor series in @xmath178 , this gives @xmath179 from eq.([r5 ] ) we obtain @xmath180 where @xmath181 is an integral constant . now consider eq.([r13 ] ) , it means that bob moves around alice ( no matter by attractive or repulsive interaction ) , when @xmath182 , bob may access alice as close as possible at perihelion point , at the perihelion point we find @xmath183|_{perihelion } \nonumber\\ \label{r16}\end{aligned}\ ] ] since @xmath184 are the coefficients that are independent from distance @xmath185 , integral constant @xmath69 and integral constant @xmath181 , they take the same values for various cases which have various man - controlled parameters @xmath186 and @xmath181 . now we consider two extreme cases . _ first case : bob is at rest forever . _ according to the newton s first law of motion , the interaction between them must completely vanishes . since bob speed @xmath187 should not depend on the distance , according to eq.([r15 ] ) , a reasonable solution may be @xmath188 , @xmath189 , @xmath190 and @xmath191 . to note that the values of @xmath192 and @xmath193 do not depend on this extreme case . _ second case : with _ @xmath182 , _ bob passes the perihelion point about alice with a distance _ @xmath194__. _ _ according to eq.([r16 ] ) , a reasonable solution may be that the all coefficients @xmath184 are zero but except @xmath195 . therefore we obtain @xmath196 where the subscript of @xmath195 has been dropped . likewise for alice , we have @xmath197 where @xmath6 and @xmath198 are coefficients . differentiating eq.([r7 ] ) with respect to time interval @xmath199 gives the center acceleration as @xmath200 where we have used @xmath201 , @xmath202 . substituting eq.([r17 ] ) and eq.([r19 ] ) into eq.([r21 ] ) , we get @xmath203 this equation leads to @xmath204 where @xmath205 is a constant . then eq.([r5 ] ) and eq.([r6 ] ) may be rewritten as @xmath206 @xmath207 if @xmath205 takes a negative constant , then , the above equations show that bob is attracted by alice with newton s universal gravitation force . _ but we do not want to make this conclusion at once , because there are still a few problems among them . _ in this subsection , we study coulomb s force by based on our the most simple model : all particles are composed of identical constituents dollons . from the above subsection , now we can manifestly interpret the quantity @xmath106 as the 4-vector force exerting on a dollon of bob . it is a natural idea to think of that dollon has two kinds of charges : positive and negative . if bob and alice are separated by a far distance , and @xmath106 is the force acting on a positive dollon in bob , then @xmath208 is the force acting on a negative dollon in bob . regardless of the internal forces in bob , it follows from eq.([r2 ] ) that the motion of the ith dollon is governed by @xmath209 where @xmath210 , @xmath165 and @xmath211 denote the proper time interval , 4-vector velocity and 4-vector force acting on the ith dollon , respectively . taking sum over all dollons in bob , we get @xmath212=\frac{d(mu_c)}{dt } \label{c2 } \\ \sum\limits_{i=1}^m\frac{icf^{(i)}}{u_4^{(i ) } } & \simeq & q\frac{icf_c}{u_{c4 } } \label{c3}\end{aligned}\ ] ] where @xmath213 is the 4-vector velocity of the center of bob , @xmath214 denotes its 4th component , @xmath108 denotes the net charges of bob , @xmath215 denotes the 4-vector force acting on the dollon located at the center of bob ( this dollon may be virtual one because it features the average action ) . combining eq.([c2 ] ) and eq.([c3 ] ) with eq.([c1 ] ) , we obtain @xmath216 where we neglect the approximation in eq.([c3 ] ) . like that in the above subsection , the first law must be valid for the composite system of bob and alice , in other words , when they are separated from a infinite distance they are isolated , whereas they go to nearest points they should not touch each other , these requirements lead to @xmath217 where @xmath218 @xmath7 and @xmath6 are coefficients . without loss of generality , we have @xmath219 substituting eq.([c5 ] ) and eq.([c6 ] ) into eq.([c7 ] ) , we get @xmath220 where @xmath109 denotes the net charges of alice . this equation leads to @xmath221 where @xmath222 is a constant . then the motions of bob and alice are governed by @xmath223 @xmath224 the 4th component equations corresponding to the above equations express the energy change rates of bob and alice , they are not independent components . the eq.([c10 ] ) and eq.([c11 ] ) are known as the coulomb s forces . if bob and alice are two atoms with neutral net charges , the coulomb s force between them will vanish off . but , precisely , this is not true , the inspection of eq.([c3 ] ) tells us that the net interaction between them still remains when the atoms are considered as composite systems . in this paper , planet , stone , molecule , atom and nucleus are all regarded as composite systems composed of dollons , the constituents of the composite systems move about their centers . if bob and alice are two planets with neutral net charges , then it is reasonable _ to assume that the net force acting on bob is proportional to the number of dollons in bob _ , eq.([c3 ] ) reads @xmath225 where @xmath226 is a very very small proportional coefficient . then the motion of bob is given by @xmath227 in analogy with the above subsections , we may obtain the motion equations of bob and alice , they are governed by @xmath228 @xmath229 where @xmath230 is a constant proportional to @xmath226 . the @xmath13 and @xmath160 has been identified or defined as the masses by employing dollon mass as a unit when we count the dollon numbers in bob or alice . the eq.([c14 ] ) and eq.([c15 ] ) are known as the newton s universal gravitational forces . why is the net force of bob attractive ? this may be explained as that electrons with light masses move always around massive nuclei , the attraction is a little bigger than the repulsion between two atoms separated by a far distance . in this formulation , the gravitational force possesses statistic meanings . we use the most simple model all particles are composed of identical dollons to study nuclear force , to fulfill the conceptual development boosted by the newton s first law of motion . now consider that bob and alice are two nucleons composed of dollons . if bob and alice go closely in a distance comparable with the sizes of them , then it is clear that eq.([c3 ] ) turns to be inadequate , their polarization can provide a strong interaction , while the effect of net charges between their centers becomes to be trivial . the strong interaction is regarded as the nuclear force in this paper . therefore , the strong nuclear force is charge - independent , it only comes into play when the nucleons are very close together , and it drops rapidly to coulomb s force for far distance , we know from experiments that the sensitive distance is about @xmath231 . as mentioned above , the ith dollon in bob is governed by @xmath232 then the motion of bob is given by @xmath233 ^ 2 } \label{c18}\end{aligned}\ ] ] where @xmath234 where @xmath213 can be understood as the velocity of momentum center ( see eq.([c19 ] ) , but @xmath213 is not the _ relativistic _ velocity of the geometrical center of bob , the _ relativistic _ velocity of the geometrical center of bob is defined by using its geometrical center proper time , i.e. , @xmath235 , thus we have to establish their relation by introducing a correcting factor @xmath236 so that @xmath237 , i.e. , @xmath238 in the following we drop the subscript center when without confusion , then above equations can be rewritten as @xmath239 ^ 2 } \label{c22}\end{aligned}\ ] ] to note that the right side of eq.([c22 ] ) is the rate at which the forces do works on bob , then the quantity @xmath240 in the left side should be energy , thus we can define the energy as @xmath241 where @xmath242 , @xmath187 is the classical speed of the geometrical center of bob , @xmath243 is the relativistic mass , while @xmath13 is the rest mass . eq.([c23 ] ) is known as the energy mass relationship . but eq.([c23 ] ) has a little difference from einstein s mass - energy relationship . our energy formula contains a factor @xmath236 that represents the internal motion of dollons in bob , obviously , @xmath244 , this can be seen clearly from eq.([c20 ] ) , in other words , even if the center is at rest , the internal constituents can still have relativistic energies . in dealing with nuclear reaction , in many textbooks , mass defect is understood as the decrease in total relativistic mass , even if all nuclei seem to be at rest before or after the nuclear reaction the total relativistic masses should not have apparent change . we have been puzzled by these statements for a long time . now the reasons are clear , no relativistic masses change but @xmath236 changes in these cases , in other words , the internal energy of particle has changed . @xmath236 is a physical quantity sensitive to the internal structure of a particle , is a criteria for particle being elementary or not . consider that a hadron possesses net charge @xmath108 , we can naturally image that the charge distributes in several parts inside the hadron , assuming three parts , the three parts have net charges denoted by @xmath245 , @xmath246 , and @xmath247 respectively , then @xmath248 comparing with the gell - mann - nishijima relation @xmath249 we can understand the conservations of isospin @xmath250 , baryon number @xmath116 and strangeness number @xmath1 with four remarks : ( 1 ) the three parts inside the hadron are insulated from one another , no charge transports from one to another . ( 2 ) during collision of hadrons , only identical parts impact or touch each other , with exchanging net charges . ( 3 ) the mass of the hadron seems to depend primarily on the masses of the parts inside the hadron , weakly on the net charges of the parts . ( 4 ) if we assign the quantum states of quarks @xmath10 , @xmath60 and @xmath251 to the three parts , the quark model seems to be improved in a manner that we can avoid the fractional charges of the quarks . in the preceding subsections , we have mentioned that the interaction between bob and alice we studied happens at such instant that their relative position in the minkowski s space is denoted by a 4-vector @xmath252 , @xmath112 satisfies the orthogonal relation simultaneously @xmath253 the purpose of choosing this instant is to meet the convenience that @xmath112 parallels to @xmath106 and @xmath254 simultaneously , because @xmath255 see eq.([r8])-([r11 ] ) . eq.([c30 ] ) can be rewritten in the form of inner product of two vectors as @xmath256 this leads to two solutions given by @xmath257 eq.([c33 ] ) again leads to two solutions given by @xmath258 the first solution expresses that the force acting on bob is retarded by time @xmath259 , the second one expresses that the action is preceded . our choice is the first one which gives an effect that follows the cause . we know , this retarded time is just the time needed for the propagation of interaction from alice to bob , the propagation speed is @xmath8 , no mater what kind of interaction . eq.([c35 ] ) represents the orthogonal relationship . therefore , the interaction happens at such instant that either in retarded state or in orthogonal state , or mixture . in preceding sections , we have realized that relativistic newton s second law and forces can be derived from newton s first law and the magnitude formula of 4-vector velocity of particle . the formula is given by @xmath260 in a minkowski s space . it is noted that all particles satisfy the above equation , it then is regarded as the origin of the particle invariance . we wonder at what is the essence of the minkowski s space . in this section we shall discuss the minkowski s space , for this purpose we need to establish a standard method for describing the motion of particle in space - time . our construction follows four steps . suppose alice is a pretty girl being famous for her fast running records , we state some her records here in a story ( in imagination ) . \(1 ) jan . , 1 , 2001 , 10:00 am , sportsground in buaa , beijing . in a time interval @xmath261 alice ran a straight line distance @xmath262 at a constant speed @xmath263 . this data can be given in physical terms by @xmath264 it can be rewritten either as @xmath265 or as @xmath266 where @xmath42 and @xmath267 denote the coordinate system fixed at the sportsground . by defining a imaginary quantity @xmath268 , the data is given by @xmath269 we appreciate the simplicity and beauty of its form . it is also our favorite manner to mark the running process in a graph with three mutually perpendicular axes @xmath270 and @xmath271 . the distance from the starting point to the final point in this coordinate system equals to zero because of eq.([m4 ] ) . this graph we called as 20010101 graph . \(2 ) jan . , 2 , 2001 , 10:00 am , sportsground in buaa , beijing . in a time interval @xmath261 alice ran a straight line distance @xmath272 at a constant speed @xmath273 . this data is given in physical terms by @xmath274 we directly mark this day running process in the yesterday s 20010101 graph , we are lazy to draw a new graph . \(3 ) jan . , 3 , 2001 , 10:00 am , sportsground at buaa , beijing . in a time interval @xmath261 alice ran a straight line distance @xmath275 at a constant speed @xmath276 . we also directly mark the running process in the 20010101 graph . bob was also a good runner , in a time interval @xmath261 bob ran a straight line distance @xmath277 at a constant speed @xmath278 . we also directly mark the running process in the 20010101 graph . in fact , their running records all are marked in the 20010101 graph . because of laziness , we only use the 20010101 graph to record the running data , it has actually become a temporary standard frame , all motions can be marked or calculated in the graph , it is much connivent for describing any movement . to note that @xmath271 , the @xmath279 axis in the graph has involved the speed @xmath280 created by alice on jan 1 , 2001 . thus we find that the geometrical distance @xmath281 from the starting point to the final point in the graph on the second day for alice is not equal to zero . @xmath282 it is clear after comparing with eq.([m5 ] ) . so do for bob , the distance @xmath283 for bob in the 20010101 graph is given by @xmath284 @xmath285 dividing the two sides of the above equation by @xmath286 , we get @xmath287 defining modified velocity @xmath288 where @xmath289 , we have dropped the subscript @xmath7 that indicates bob , then eq.([m9 ] ) is given by @xmath290 the modified velocity of bob in the 20010101graph is based on the alice s best speed record @xmath291 . in fact , any one , any body or any particle , their modified velocity in the 20010101 graph satisfies eq.([m13 ] ) . because of convenience , it has become a habit for us to use the 20010101 graph to mark the all motions of any body . it is sure that not all scientists in the world like alice , then we gradually recognize that we need a permanent runner for establishing a standard graph . now we had to face a new task : to look for a new hero . it was said that the light , an element of the nature , is the fastest runner , whenever and whereever its speed is @xmath292 . we do not hesitate to use the light speed to replace alice s speed , and setup a new frame called standard graph , the standard graph contains four mutually perpendicular axes @xmath42 , @xmath267 , @xmath293 and @xmath294 ( we can draw several partial frames to assemble the whole frame ) . from then , any motions can be described in the standard graph with the space - time @xmath295 or @xmath296 . in analogy with eq.([m10])-([m13 ] ) , defining modified velocity @xmath297 where @xmath298 , we obtain @xmath299 the standard graph is just the minkowski s space , the 4-vector velocity @xmath300 is known as the relativistic velocity . we immediately recognize that physics holds its validity only in the standard graph ( involving with the light speed ) , rather than in the 20010101 graph ( involving with the alice s speed ) , this situation can be explained by the fact that all physical quantities and their measurements are defined on facilities whose principles are based on the light directly or indirectly , for example , the meter and second are defined on the light speed directly . if we do not hope that one graph has advantage over than another , then the transformation between the standard graph and 20010101 graph will be given by @xmath301 where the subscript 1 denotes in the 20010101 graph . it means we need to redefine all physical quantities such as rod and clock in the 20010101 graph , do not use the light . a complete inner product space is called a hilbert space . our experience in the preceding sections tells us that it is an easy thing to put dynamics into the hilbert space if we have an invariant quantity . the formalism of the interaction can be derived from some basic laws , it is strongly based on concrete instances . in the section [ basicf ] , the newton s first law of motion means that the 4-vector average velocity of an isolated system remains at rest or in motion . this explanation is based on the definition of average velocity given by @xmath302 where @xmath1 denotes the number of dollons in the system as in eq.([r7 ] ) . the newton s first law of motion becomes a sort of strong constraint , inevitably leads to action reaction law or momentum conversation law being valid inside the system , for example , for a rest system composed of two dollons alice and bob we have @xmath303 the above equation means that the action and reaction are equal in magnitude and reverse in directions on the line joining the two particle . but we immediately wonder at that alice and bob have to adjust their proper times @xmath304 and @xmath305 from time to time to meet the requirement of eq.([d2 ] ) . that is why we say the first law is a strong constraint for the system . obviously , the geometrical center of a system is defined by @xmath306 the relativistic 4-vector velocity of the geometrical center of the system is given by @xmath307 as mentioned in the section [ nuclf ] , @xmath308 . immediately , we find the newton s first law of motion can be newly explained by based on the relativistic 4-vector velocity of the geometrical center of the system , i.e. , the newton s first law of motion means that the 4-vector velocity of the geometrical center of an isolated system remains at rest or in motion . this new explanation implies that the action reaction law for the relativistic 4-vector forces inside the system are not held [ comparing to eq.([d2 ] ) ] , but the following expansions for alice and bob become possible . @xmath309 where @xmath115 , @xmath116 , @xmath310 and @xmath311 are unknown coefficients . all conclusions we obtained in the preceding sections can be retained or modified by retracing the route of the paper , in accordance with the section [ direction ] . the new explanation seems to be much reasonable , but it is worth further studying the action reaction law and momentum conversation law which are confronting with serious troubles , they need special treatment like that for ampere s force in electromagnetism . another topic we would like to discuss briefly is su(n ) group . each infinitesimal transformation of the su(n ) group can be written in the form @xmath312 as usual , repeated indices must be summed over . where the real parameter @xmath313 are treated as small quantities , @xmath314 and @xmath315 are matrices which satisfy the definition of the group @xmath316 we recall from eq.([20 ] ) that dirac equation was derived from the following equation @xmath317[a_{\mu jk}p_\mu \psi ^{(k)}-i\delta _ { jk}mc\psi ^{(k)}]=0 \label{d9}\ ] ] it is much impressive that eq.([d8 ] ) and eq.([d9 ] ) have a similar form , especially when we let a matrix @xmath318 to absorb the right side of eq.([d8 ] ) , i.e. @xmath319 from this comparison we may understand why the su(n ) group could embed in quantum mechanics in a obscure way . this situation arouses our interest to measure a new group whose matrices satisfy @xmath320 we believe this new group has even more direct relations with quantum mechanics . it is important to recognize that physics must be invariant for composite particles and their constituent particles , only one physical formalism exists for any particle , this requirement is called particle invariance . under the particle invariance , it is rather remarkable to find that klein - gordon equation and dirac equation can be derived from the relativistic newton s second law of motion on different conditions respectively , thus only one formalism is necessary for particle , the relativistic newton s second law is regarded as one which suitable for any kinds of particles . we point out that the coulomb s force and gravitational force on a particle always act in the direction orthogonal to the 4-vector velocity of the particle in 4-dimensional space - time , rather than along the line joining a couple of particles . this inference is obviously supported from the fact that the magnitude of the 4-vector velocity is kept constant . maxwell s equations can be derived from classical coulomb s force and the magnitude formula of 4-vector velocity of particle . our speculation on the quarks model leads to introduce a new elementary particle called dollon to assemble particles such as baryons , mesons and other composite particles . instead of quark model , the dollon model is better in organizing known data , specially in modelling interactions . it is found that relativistic newton s second law and various interactions can be derived from the newton s first law of motion and the magnitude formula of 4-vector velocity of particle . the structure of minkowski s space is discussed in detail , it indicates that the magnitude formula of 4-vector velocity of particle is only a geometrical distance formula ( pythagoras s theorem ) , so that it is completely free from any particle property . any dynamics or dynamical characteristics originated from the magnitude formula of 4-vector velocity of particle will completely preserve the particle invariance , i.e. , the dynamics do not distinguish particle species . thus the magnitude formula of 4-vector velocity of particle is regarded as the origin of the particle invariance .

since the particles such as molecules , atoms and nuclei are composite particles , it is important to recognize that physics must be invariant for the composite particles and their constituent particles , this requirement is called particle invariance in this paper . but difficulties arise immediately because for fermion we use dirac equation , for meson we use klein - gordon equation and for classical particle we use newtonian mechanics , while the connections between these equations are quite indirect . thus if the particle invariance is held in physics , i.e. , only one physical formalism exists for any particle , we can expect to find out the differences between these equations by employing the particle invariance . as the results , several new relationships between them are found , the most important result is that the obstacles that cluttered the path from classical mechanics to quantum mechanics are found , it becomes possible to derive the quantum wave equations from relativistic mechanics after the obstacles are removed . an improved model is proposed to gain a better understanding on elementary particle interactions . this approach offers enormous advantages , not only for giving the first physically reasonable interpretation of quantum mechanics , but also for improving quark model . + + pacs numbers : 11.30.ly , 12.90.+b , 03.65.ta . + +

a large number of physical systems , in particular in plasma physics , are described in terms of a variable evolving in a deterministic velocity field under the effect of a random perturbation . this is described by a stochastic differential equation of the form @xmath0 where @xmath1 is the velocity field and the perturbation is the white noise @xmath2 this general form has been invoked in several applications . more recently , in a series of works devoted to the explanation of the intermittent behavior of the statistical characteristics of the turbulence in magnetically confined plasma it has been developed a formalism based on barrier crossing . previous works that have discussed subcritical excitation of plasma instabilities are refs . @xcite , @xcite , @xcite , @xcite , @xcite , @xcite , @xcite , @xcite , @xcite . we offer a comparative presentation of two functional integral approaches to the determination of the statistical properties of the system s variable @xmath3 for the case where the space dependence of @xmath4 is characterized by the presence of three equilibrium points , @xmath5 . we will take @xmath6 in studying the stochastic processes the functional methods can be very useful and obtain systematic results otherwise less accessible to alternative methods . the method has been developed initially in quantum theory and it is now a basic instrument in condensed matter , field theory , statistical physics , etc . in general it is based on the formulation of the problem in terms of an action functional . there are two distinct advantages from this formulation : ( 1 ) the system s behavior appears to be determined by all classes of trajectories that extremize the action and their contributions are summed after appropriate weights are applied ; ( 2 ) the method naturally includes the contributions from states close to the extrema , so that fluctuations can be accounted for . there are technical limitations to the applicability of this method . in the statistical problems ( including barrier - type problems ) it is simpler to treat cases with white noise , while colored noise can be treated perturbatively . in the latter case , the procedure is however useful since the diagrammatic series can be formulated systematically . the colored noise can be treated by extending the space of variables : the stochastic variable with finite correlation is generated by integration of a new , white noise variable . the one dimensional version can be developed up to final explicit result . since however the barrier type problem is frequently formulated in two - dimensions , one has to look for extrema of the action and ennumerate all possible trajectories . it is however known that , in these cases , the behavior of the system is dominated by a particular path , `` the optimum escape path '' , and a reasonnable approximation is to reduce the problem to a one - dimensional one along this system s trajectory . we will briefly mention the steps of constructiong the msr action functional , in the jensen path integral reformulation . we begin by choosing a particular realisation of the noise @xmath7 . all the functions and derivatives can be discretised on a lattice of points in the time interval @xmath8 $ ] ( actually one can take the limits to be @xmath9 ) . the solution of the equation ( [ eqcanoi ] ) is a `` configuration '' of the field @xmath10 which can be seen as a point in a space of functions . we extend the space of configurations @xmath3 to this space of functions , including all possible forms of @xmath10 , not necessarly solutions . in this space the solution itself will be individualised by a functional dirac @xmath11 function . @xmath12 any functional of the system s real configuration ( _ i.e. _ solutions of the equations ) can be formally expressed by taking as argument an arbitrary functional variable , multiplying by this @xmath11 functional and integrating over the space of all functions . we will skip the discretization and the fourier representation of the @xmath13 functions , followed by reverting to the continuous functions . the result is the following functional @xmath14 \emph{d}\left [ k\left ( t\right ) \right ] \exp \left [ i\int_{-t}^{t}dt\left ( -k\overset{\cdot } { x}+kv\left ( x\right ) + \sqrt{2d}k\xi \right ) \right]\ ] ] the label @xmath15 means that the functional is still defined by a choice of a particular realization of the noise . the generating functional is obtained by averaging over @xmath15 . @xmath16 \emph{d}\left [ k\left ( t\right ) \right ] \exp \left [ i\int dt\left ( -k\overset{\cdot } { x}+kv\left ( x\right ) + idk^{2}\right ) % \right ] \label{zfaraj}\ ] ] we add a formal interaction with two currents @xmath17 \emph{d}\left [ k\left ( t\right ) \right ] \exp \left [ i\int dt\left ( -k\overset{\cdot } { x}% + kv\left ( x\right ) + idk^{2}+j_{1}x+j_{2}k\right ) \right ] \label{zj}\ ] ] in view of future use to the determination of correlations . this functional integral must be determined explicitely . the standard way to proceed to the calculation of @xmath18 is to find the saddle point in the function space and then expand the action around this point to include the fluctuating trajectories . this requires first to solve the euler - lagrange equations @xmath19 the simplest case should be examined first . we assume there is no deterministic velocity ( @xmath20 ) in order to see how the purely diffusive behavior is obtained in this framework @xmath21 the equations can be trivially integrated @xcite , @xcite @xmath22 where @xmath23 \\ \delta _ { 12}\left ( t , t^{\prime } \right ) & = & \theta \left ( t - t^{\prime } \right ) \\ \delta _ { 21}\left ( t , t^{\prime } \right ) & = & \theta \left ( t^{\prime } -t\right ) \\ \delta _ { 22}\left ( t , t^{\prime } \right ) & = & 0\end{aligned}\ ] ] with the symmetry @xmath24 the lowest approximation to the functional integral @xmath18 is obtained form this saddle point solution , by calculating the action along this system s trajectory . we insert this solutions in the expression of the generating functional , for @xmath25 @xmath26 \emph{d}\left [ k\left ( t\right ) \right ] \exp \left [ i\int_{-t}^{t}dt\left ( -k\overset{\cdot } { x}+idk^{2}+j_{1}x+j_{2}k\right ) % \right ] \right| _ { x^{\left ( 0\right ) } , k^{\left ( 0\right ) } } \\ & = & \exp \left [ \frac{1}{2}i\int_{-t}^{t}dt\int_{-t}^{t}dt^{\prime } j_{i}\left ( t\right ) \delta _ { ij}\left ( t , t^{\prime } \right ) j_{j}\left ( t\right ) \right]\end{aligned}\ ] ] the dispersion of the stochastic variable @xmath10 can be obtained by a double functional derivative followed by taking @xmath27 . we obtain @xmath28 \right| _ { j_{1,2}=0 } \\ & = & d\min \left ( t , t^{\prime } \right)\end{aligned}\ ] ] which is the diffusion . the same mechanism will be used in the following , with the difference that the equations can not be solved in explicit form due to the nonlinearity . in general the nonlinearity can be treated by perturbation expansion , if the amplitude can be considered small . this is an analoguous procedure as that used in the field theory and leads to a series of terms represented by feynman diagrams . we can separate in the lagrangian the part that can be explicitely integrated and make a perturbative treatment for the non - quadratic term . this is possible when we assume a particular ( polynomial ) form of the deterministic velocity , @xmath1 . obviously , this term is @xmath29 in eq.([zj ] ) . the functional integral can be written , taking account of this separation @xmath30 \emph{z}_{j}^{\left ( q\right ) } \label{zjnl}\ ] ] where the remaining part in the lagrangian is _ quadratic _ @xmath31 \emph{d}\left [ k\left ( t\right ) \right ] \exp \left [ i\int dt\left ( -k% \overset{\cdot } { x}+akx+idk^{2}+j_{1}x+j_{2}k\right ) \right ] \label{zq}\ ] ] the euler - lagrange equations are @xmath32 the solutions can be expressed as follows @xmath33 with @xmath34 \theta \left ( t^{\prime } -t\right ) + \left [ \exp \left ( -2at^{\prime } \right ) -1\right ] \theta \left ( t - t^{\prime } \right ) \right\ } \notag \\ & & \times \exp \left ( at^{\prime } \right ) \notag \\ \delta _ { 21}\left ( t , t^{\prime } \right ) & = & \exp \left ( -at\right ) \theta \left ( t^{\prime } -t\right ) \exp \left ( at^{\prime } \right ) \notag \\ \delta _ { 12}\left ( t , t^{\prime } \right ) & = & \exp \left ( at\right ) \theta \left ( t - t^{\prime } \right ) \exp \left ( -at^{\prime } \right ) \notag \\ \delta _ { 22}\left ( t , t^{\prime } \right ) & = & 0 \notag\end{aligned}\ ] ] the form of the generating functional derived from the quadratic part is @xmath35 the occurence of the first term in the exponent is the price to pay for not making the expansion around @xmath36 . however , such an expansion would have produced two non - quadratic terms in the lagrangian density : @xmath37 and @xmath38 . this would render the perturbative expansion extremly complicated since we would have to introduce two vertices : one , of order four , is that shown in eq.([zjnl ] ) and another , of order three , related to the first of the nonlinearities mentioned above . even in the present case , the calculation appears very tedious . we have to expand the vertex part of @xmath18 as an exponential , in series of powers of the vertex operator . in the same time we have to expand the exponential in eq.([zjq ] ) as a formal series . then we have to apply term by term the first series on the second series . the individual terms can be represented by diagrams . in this particular case we have a finite contribution even at the zero - loop order ( the `` tree '' graph ) . it is however much more difficult to extract the statistics since we will need at least the diagrams leaving two free ends with currents @xmath39 . in the case we examine here , the perturbative treatment is not particularly useful since the form of the potential ( from which the velocity field is obtained ) supports topologically distinct classes of saddle point solutions and this can not be represented by a series expansion . to make comparison with other approaches , we take @xmath40 and integrate over the functional variable @xmath41 . @xmath42 \emph{d}\left [ k\left ( t\right ) \right ] \exp \left\ { \int_{0}^{t}dt\left [ -dk^{2}+i\left ( -k% \overset{\cdot } { x}+kv\right ) \right ] \right\ } \\ & = & \int \emph{d}\left [ x\left ( t\right ) \right ] \emph{d}\left [ k\left ( t\right ) \right ] \exp \left\ { -\int_{0}^{t}dt\left [ \sqrt{d}k+\frac{i% \overset{\cdot } { x}-iv}{2\sqrt{d}}\right ] ^{2}+\int_{0}^{t}dt\frac{\left ( i% \overset{\cdot } { x}-iv\right ) ^{2}}{4d}\right\}\end{aligned}\ ] ] @xmath43 \exp \left\ { -% \frac{1}{4d}\int_{0}^{t}dt\left ( \overset{\cdot } { x}-v\right ) ^{2}\right\}\ ] ] in other notations @xmath43 \exp \left [ -% \frac{s}{d}\right]\ ] ] where @xmath44 this is eq.(25 ) of the reference lehmann , riemann and hnggi , pre62(2000)6282 . in this reference it is called the onsager - machlup action functional and the analysis is based on this formula . however , we can go further and we will find inconsistencies . we now take account of the fact that the velocity is derived form a potential @xmath45 & = & -\frac{du\left [ x\left ( t\right ) \right ] } { dx } \\ & \equiv & -u^{\prime } \left [ x\left ( t\right ) \right]\end{aligned}\ ] ] @xmath46 \\ & = & \frac{1}{2}\left [ u\left ( t\right ) -u\left ( 0\right ) \right ] + \int_{0}^{t}dt\left [ \frac{1}{4}\left ( \overset{\cdot } { x}^{2}+u^{\prime 2}\right ) \right]\end{aligned}\ ] ] this leads to the form of the generating functional @xmath47 } { \exp \left [ -\frac{u\left ( 0\right ) } { 2d}\right ] } k\left ( x , t;x_{i},t_{i}\right)\ ] ] with @xmath48 \exp \left ( -\frac{s}{d}\right)\ ] ] @xmath49 \label{sfara}\ ] ] these are almost identical to the formulas ( 2 - 5 ) of the reference @xcite ( except that @xmath50 ) . also , it is quite close of the eqs.(7a-7c ) of the ref.@xcite . * however there is an important difference*. there is a term missing in eq.([sfara ] ) which however is present in the two above references . the full form of the action @xmath51 , instead of eq.([sfara ] ) is @xmath52 @xmath53 this term comes from the _ jacobian _ that is hidden in the functional @xmath13 integration . in our approach the most natural way of proceeding with a stochastic differential equation is to use the msr type reasonning in the jensen reformulation . the equation is discretized in space and time and selectd with @xmath11 functions in an ensemble of functions ( actually in sets of arbitrary numbers at every point of discretization ) . the result is a functional integral . there is however a particular aspect that needs careful analysis , as mentioned in the previous subsection . it is the problem of the jacobian associated with the @xmath11 functions . this problem is discussed in ref.@xcite . the equation they analyse is presented in most general form as @xmath54 + \theta _ { j}\ ] ] where the number of stochastic equations is @xmath55 , @xmath56 is functional of the fields , @xmath57 is the streaming term which obeys a current - conserving type relation @xmath58 \exp \left\ { -h% \left [ \phi \right ] \right\ } = 0\ ] ] the noise is @xmath59 . the following generating functional can be written @xmath60 \exp \int dt% \left [ l_{j}\phi _ { j}\left ( t\right ) \right ] \prod_{j , t}\delta \left ( \frac{% \partial \phi _ { j}\left ( t\right ) } { \partial t}+k_{j}\left [ \phi \left ( t\right ) \right ] -\theta _ { j}\right ) j\left [ \phi \right]\ ] ] the functions @xmath61 are currents , @xmath62 \equiv -\left ( \gamma _ { 0}\right ) _ { jk}\frac{\delta h}{\delta \phi _ { k}\left ( t\right ) } + v_{j}\left [ \phi \left ( t\right ) \right]\ ] ] and @xmath63 $ ] is the jacobian associated to the dirac @xmath11 functions in each point of discretization . the jacobian can be written @xmath64 } { \delta \phi _ { k}}\right ) \delta \left ( t - t^{\prime } \right ) \right]\ ] ] up to a multiplicative constant @xmath65 \right)\ ] ] or @xmath66 \right)\ ] ] since the operator @xmath67 is retarded , only the lowest order term survives after taking the trace @xmath68 } { \delta \phi _ { j}\left ( t\right ) } \right]\ ] ] the factor @xmath69 comes from value of the @xmath70 function at zero . in the treatment which preserves the dual function @xmath71 associated to @xmath72 in the functional , there is a part of the action @xmath73\ ] ] then a @xmath71 and a @xmath72 of the same coupling term from @xmath74 $ ] close onto a loop.since @xmath75 is retarded , all these contributions vanish except the one with a single propagator line . this cancels exactly , in all orders , the part coming from the jacobian . then it is used to ignore all such loops and together with the jacobian . we can now see that in our notation this is precisely the term needed in the expression of the action . @xmath76 & \rightarrow & u^{\prime } \left [ x\left ( t\right ) \right ] \\ \frac{\delta k_{j}\left [ \phi \left ( t\right ) \right ] } { \delta \phi _ { j}\left ( t\right ) } & \rightarrow & -u^{\prime \prime } \left [ x\left ( t\right ) \right]\end{aligned}\ ] ] and the action ( [ sfara ] ) is completed with the new term @xmath77 now the generating functional is @xmath47 } { \exp \left [ -\frac{u\left ( 0\right ) } { 2d}\right ] } k\left ( x , t;x_{i},t_{i}\right)\ ] ] with @xmath48 \exp \left ( -\frac{s}{d}\right)\ ] ] and @xmath78 @xmath79 now the two expressions are identical with those in the references cited . this will be the starting point of our analysis . in conclusion we have compared the two starting points in a functional approach : the one that uses _ dual functions _ @xmath10 and @xmath80 , closer in spirit to msr ; and the approach based on onsager - machlup functional , traditionally employed for the determination of the probabilities @xcite , @xcite . either we keep @xmath81 and ignore the jacobian ( the first approach ) or integrate over @xmath80 and include the jacobian . the approaches are equivalent and , as we will show below , lead to the same results . a final observation concerning the choice of one or another method : in the msr method , the trajectories include the diffusion from the direct solution of the euler - lagrange equations . in the onsager - machlup method the paths extremizing the action are deterministic and the diffusion is introduced by integrating on a neighborhood in the space of function , around the deterministic motion . the equations for the saddle point trajectory are in complex so we extend also the variable in complex space @xmath82 the obtain a system of four nonlinear ordinary differential equations which can be integrated numerically . a typical form of the solution @xmath10 is similar to the _ kink instanton _ ( _ i.e. _ the @xmath83 function ) . the function @xmath84 spends very much time in the region close to the equilibrium point ; then it performs a fast transition to the neighbour equilibrium point , where it remains for the rest of the time interval . ( unstable ) equilibrium point in the effective potential @xmath85,width=377 ] ( unstable ) equilibrium point with a position close to the right ( stable ) equilibrium point to in the effective potential @xmath85,width=377 ] the action functional eqs.([news ] ) and ( [ w ] ) leads to the following differential equation ( which replaces eqs.([eqel ] ) ) @xmath86 multiplying by @xmath87 and integrating we have @xmath88 ^{1/2 } \label{eqxpoint}\ ] ] we are interested in the functions @xmath10 that has the following physical property : they stay for very long time stuck to the equilibrium points and perform a fast jump between them at a certain moment of time . then we can take @xmath89 . the solution can be obtained form the integration @xmath90 the upper limit @xmath91 will be specified later . for the next calculation it will be taken as the smallest of the roots of the polynomial under the square root . the details of the calculations in terms of elliptic functions can be found in _ byrd and friedman _ @xcite . the roots of the forth degree polynomial will be noted @xmath92 where @xmath93 such as to have @xmath94 ; then we will use @xmath95 . the notations are @xmath96 the following substitutions are required @xmath97 a new variable is introduced identifying the lower limit of the integral , @xmath98 @xmath99 the integral can be written @xmath100 this integral can be expressed in terms of _ elliptic _ functions . we take @xmath101@xmath102\end{aligned}\ ] ] here the notations are @xmath103 @xmath104\end{aligned}\ ] ] @xmath105\end{aligned}\ ] ] and @xmath106 the symbol @xmath107 represents the legendre s incomplete elliptic integral of the third kind and @xmath108 is the amplitude of @xmath109 . the symbols @xmath110 , @xmath111 , @xmath112 represent the jacobi elliptic functions . there are several well known examples of instantons . they appear in physical systems whose lowest energy state is degenerate and the minima of the action functional ( or the energy , for stationary solutions ) are separated by energy barriers . instantons connect these minima by performing transitions which are only possible in imaginary time ( the theory is expressed in euclidean space , with uniform positive metric ) . it is only by including these instantons that the action functional is correctly calculated and real physical quantities can be determined . from this calculation we can obtain the explicit trajectories that extremize the action functional and in the same time reproduce the jump of the system between the two distant equilibrium positions . these trajectories will be necessary in the calculation of the functional integral . however , since we have eliminated the external currents and integrated over the dual functional variable @xmath113 we can not derive the statistical properties of @xmath10 from a generating functional . in the approach based on the onsager - machlup action the instanton is not used in its explicit form ( elliptic functions ) in the calculation of the action . the reason is that the result can be proved to depend essentially on local properties of the potential @xmath1 . this will be shown later . in the approach with dual functions , one can reduce the instanton to its simplest form , an instantaneous transition between two states , a jump appearing at an arbitrary moment of time . using this form as a first approximation we will calculate the solutions of the euler - lagrange equations and then the action . using the onsager - machlup action we have @xmath114 k\left ( x , t;x_{i},t_{i}\right)\ ] ] the new function @xmath115 has the expression @xmath116 \exp \left ( -\frac{1}{d}\int_{t_{i}}^{t}d\tau \left [ \frac{% \overset{\cdot } { x}^{2}}{4}+w\left ( x\right ) \right ] \right ) \label{kcar}\ ] ] the integrand at the exponent can be considered as the lagrangean density for a particle of mass @xmath69 moving in a potential given by @xmath117 in a semiclassical treatment ( similar to the quantum problem , where @xmath118 is the _ small _ diffusion coefficient @xmath119 of the present problem ) , the most important contribution comes from the neighborhood of the classical trajectories , @xmath120 that extremalizes the action @xmath51 . the `` classical '' equation of motion is @xmath121 to take into account the trajectories in a functional neighborhood around @xmath122 we expand the action to second order introducing the new variables @xmath123 this gives @xmath124 \\ & & \times \int_{y\left ( t_{i}\right ) = 0}^{y\left ( t\right ) = 0}\emph{d}\left [ y\left ( \tau \right ) \right ] \exp \left\ { -\frac{1}{d}\int_{t_{i}}^{t}d\tau % \left [ \frac{1}{4}\overset{\cdot } { y}^{2}+\frac{1}{2}y^{2}w^{\prime \prime } \left ( x_{c}\left ( \tau \right ) \right ) \right ] \right\}\end{aligned}\ ] ] the deviation of the action from that obtained at the extremum @xmath125 , can be rewritten @xmath126 y\left ( \tau \right)\ ] ] the functional integration can be done since it is gaussian and the result is @xmath127 ^{1/2}}\ ] ] in order to calculate the determinant , one needs to solve the eigenvalue problem for this operator @xmath128 y_{n}\left ( \tau \right ) = \lambda _ { n}y_{n}\left ( \tau \right)\ ] ] withe the eigenfunctions verifying the conditions @xmath129 @xmath130 the formal result for @xmath115 is ( also van vleck ) @xmath131\ ] ] where @xmath55 is a constant that will be calculated by normalizing @xmath132 . another way to calculate @xmath55 is to fit this result to the known _ harmonic oscillator _ problem . it has been shown ( coleman ) that the factor arising from the determinant can be written in the form @xmath133 ^{1/2}}\ ] ] where the function @xmath134 is the solution of @xmath135 with the boundary conditions @xmath136 in the case where there are degenerate minima in @xmath137 the particle can travel from one minimum to another . these solutions are called instantons . consider for example the potential with two degenerate maxima of @xmath138 at @xmath139 and with a minimum at @xmath140 . we want to calculate the probability @xmath141 . the classical solution connecting the point @xmath142 to the point @xmath143 is a _ the energy of this solution is exponentially small @xmath144\end{aligned}\ ] ] this solution spends quasi - infinite time in both harmonic regions around @xmath145 where it has very small velocity ; and travels very fast , in a short time @xmath146 between these points ( this is the time - width of the instanton ) . the special effect of the translational symmetry in time is seen in the presence of the parameter representing the center of the instanton . it can be any moment of time between @xmath147 and @xmath148 . this case must be treated separately and we note that this corresponds to the lowest eigenvalue in the spectrum , since the range of variation of the coefficient in the expansion of any solution in terms of eigenfunction is the inverse of the eigenvalue @xmath149 the widest interval , for the variation of the center of the instanton , must be associated with the smallest eigenvalue and this and its eigenfunction must be known explicitely . instead of a precise knowledge of the lowest eigenvalue and its corresponding eigenfunction we will use an approximation , exploiting the fact the function @xmath150 is very close of what we need . we start by noting that @xmath150 is a solution of the eigenvalue problem for the operator of second order functional expansion around the instanton . this eigenfunction corresponds to the eigenvalue @xmath151@xmath152 and has boundary conditions @xmath153 very close to @xmath151 , which is be the exact boundary condition we require from the eigenfunctions of the operator . so the difference between @xmath154 and the true eigenfunction are very small . since @xmath150 corresponds to eigenvalue @xmath151 we conclude that , by continuity , the true eigenfunction will have an eigenvalue @xmath155 very small , exponentially small . then the range of important values of the coefficient @xmath156 is very large and the gaussian expansion is invalid since the departure of such a solution from the classical one ( the instanton ) can not be considered small . the degeneracy in the moment of time where the center of the instanton is placed ( _ i.e. _ the moment of transition ) can be solved treating this parameter as a colective coordinate . the result is @xmath157 ^{1/2}\int_{-t/2}^{t/2}d\theta \left\ { \frac{s\left [ x_{i}\left ( \tau -\theta \right ) \right ] } { 4\pi d}\right\ } ^{1/2 } \\ & & \times \exp \left\ { -\frac{1}{ds\left [ x_{i}\left ( \tau -\theta \right ) % \right ] } \right\}\end{aligned}\ ] ] the parameter @xmath158 in the expression of the instanton solution shows is the current time variable along the solution that is used to calculate the action . the integartion is performed on the intermediate transition moment @xmath159 . we also note that @xmath160 = \underset{t\rightarrow \infty } { \lim } s\left [ x_{i}\left ( \tau \right ) \right ] = s_{0}\ ] ] since , except for the very small intervals ( approx . the width of the instanton ) at the begining and the end of the interval @xmath161 , the value of the action @xmath51 is not sensitive to the position of the transition moment . @xmath162 ^{1/2}\left ( \frac{s_{0}}{4\pi d}% \right ) ^{1/2}t\exp \left ( -\frac{s_{0}}{d}\right)\ ] ] the instanton degeneracy introduces a _ linear _ time dependence of the probability . in general , for a function @xmath163 that has two minima separated by a barrier ( a maximum ) the potential @xmath137 calculated form the action will have three minima and these are not degenerate . the inverse of this potential , @xmath164 , which is appears in the equation of motion , will have three maxima in general nondegenerate and the differences in the values of @xmath164 at these maxima is connected with the presence of the term containing @xmath119 . since we assume that @xmath119 is small , the non - degeneracy is also small . the previous discussion in which the notion of instanton was introduced and @xmath115 was calculated , take into consideration the degenerate maxima and the instanton transition at equal initial and final @xmath165 . let us consider the general shape for @xmath164 with three maxima , at @xmath166 , @xmath151 and @xmath143 . the heigths of these maximas are @xmath167 this is because the extrema of @xmath164 @xmath168 } { dx}=0\ ] ] coincides according to the equation of motion to the pointes where @xmath169 and the constant can not be taken other value but zero @xmath170 then , since we have approximately that @xmath171 ( for small @xmath119 ) then we have that at these extrema of @xmath172 we have @xmath173 and only the second term in the expression of @xmath137 remains . this justifies eq.([dupp ] ) . it will also be assumed that @xmath174 . we want to calculate , on a kramers time scale , @xmath175 the probability @xmath176 we have to find the classical solution of the equation of motion connecting @xmath177 with @xmath178 . this is the trivial solution , particle sitting at @xmath179@xmath180 we have to calculate explicitely the form of the propagator in this case @xmath131\ ] ] we use the formulas given before @xmath133 ^{1/2 } } \label{colem}\ ] ] where the function @xmath134 is the solution of @xmath181 with the boundary conditions @xmath182 we use simply @xmath183 and @xmath184 for the respective functions calculated at the fixed point @xmath179 . we have @xmath185 \\ & \approx & \frac{1}{2}\left ( u^{\prime \prime } \right ) ^{2}\;+\;\text{terms of order } d\end{aligned}\ ] ] the equation for the eigenvalues becomes @xmath186 @xmath187 and it results form the boundary conditions @xmath188 \\ a_{2 } & = & -\frac{1}{2\left| u^{\prime \prime } \right| } \exp \left [ \left| u^{\prime \prime } \right| t_{i}\right]\end{aligned}\ ] ] then @xmath189 -\exp \left [ -\left| u^{\prime \prime } \right| \left ( t - t_{i}\right ) \right ] \right\}\ ] ] * note*. this is the @xmath190 which is obtained in the calculation of the ground level splitting by quantum tunneling for a particle in two - well potential . now we can calculate @xmath191 ^{1/2}}=\frac{1}{\left ( 4\pi d\right ) ^{1/2}}\left [ 2\left| u^{\prime \prime } \right| \right ] ^{1/2}% \frac{\exp \left [ -\frac{1}{2}\left| u^{\prime \prime } \right| \left ( t - t_{i}\right ) \right ] } { \left ( 1-\exp \left [ -2\left| u^{\prime \prime } \right| \left ( t - t_{i}\right ) \right ] \right ) ^{1/2}}\ ] ] it remains to calculate the action for this trivial trajectory @xmath192@xmath193 \\ & = & \int_{t_{i}}^{t}dtw\left ( x_{c}\right ) \\ & = & \int_{t_{i}}^{t}dt\left [ \frac{1}{4}u^{\prime 2}\left ( x_{c}\right ) -% \frac{d}{2}u^{\prime \prime } \left ( x_{c}\right ) \right]\end{aligned}\ ] ] since the position of the extremum of @xmath165 is very close ( to order @xmath119 ) of the position where @xmath1 is zero , and since @xmath194 , we can take with good approximation the first term in the integrand zero . then @xmath195 we have to put together the two factors of the propagator and take into account that at @xmath179 we have @xmath196@xmath197 ^{1/2}}\exp \left [ -\frac{s_{c}}{d}\right ] \\ & = & \left ( \frac{u^{\prime \prime } } { 2\pi d}\right ) ^{1/2}\frac{\exp \left [ -% \frac{1}{2}u^{\prime \prime } \left ( t - t_{i}\right ) \right ] } { \left ( 1-\exp % \left [ -2u^{\prime \prime } \left ( t - t_{i}\right ) \right ] \right ) ^{1/2}}\exp \left ( \frac{1}{2}u^{\prime \prime } \left ( t - t_{i}\right ) \right)\end{aligned}\ ] ] @xmath198 \right ) ^{1/2}}\ ] ] the contribution to the action is @xmath199 ^{1/2 } } \\ & \simeq & \left ( \frac{u_{b}^{\prime \prime } } { 2\pi d}\right ) ^{1/2}\;\;\text{% ( at large } t\text{)}\end{aligned}\ ] ] it should be noticed that no other classical solution exists since there are no turning points permitting the solution to come back to @xmath179 . we apply the procedure described for the purely diffusive case to the case @xmath200 and for this we need the solution of the euler - lagrange equations . an approximation is possible if the diffusion coefficient is small . in this case the diffusion will take place around the equilibrium positions @xmath201 and @xmath202 . taking the equilibrium @xmath203 in the equation for @xmath80 we have @xmath204 where @xmath205 \theta \left ( t^{\prime } -t\right ) \exp \left [ -\left ( -a+3bx_{0}^{2}\right ) t^{\prime } \right ] \label{delta21}\ ] ] the symbol @xmath70 stands for the heaviside function . in the equation for @xmath3 we expand around the equilibrium position @xmath206 and solve the equation @xmath207 taking account of eq.([k1 ] ) @xmath208 where @xmath209 \label{delta11 } \\ & & \hspace*{-1cm}\times \left\ { \frac{1}{2\left ( -a+3bx_{0}^{2}\right ) } % \left ( \exp \left [ 2\left ( -a+3bx_{0}^{2}\right ) t\right ] -1\right ) \theta \left ( t^{\prime } -t\right ) \right . \notag \\ & & \hspace*{-1cm}\left . + \frac{1}{2\left ( -a+3bx_{0}^{2}\right ) } \left ( \exp % \left [ 2\left ( -a+3bx_{0}^{2}\right ) t^{\prime } \right ] -1\right ) \theta \left ( t - t^{\prime } \right ) \right\ } \notag \\ & & \times \exp \left [ \left ( a-3bx_{0}^{2}\right ) t^{\prime } \right ] \notag\end{aligned}\ ] ] and @xmath210 \theta \left ( t - t^{\prime } \right ) \exp \left [ \left ( -a+3bx_{0}^{2}\right ) t^{\prime } \right ] \label{delta12}\ ] ] we note that at the limit where no potential would be present , @xmath211 , the propagators @xmath212 become @xmath213\ ] ] @xmath214 _ i.e. _ the propagators of a purely diffusive process ( see @xcite ) . using the solutions eqs.([k1 ] ) and ( [ x1 ] ) we can calculate the action along this path . @xmath215\ ] ] we will insert the expansion eq.([expx ] ) , perform an integration by parts over the first term and take into account the equations , _ i.e. _ the first line of eq.([eqel ] ) and eq.([eqeps ] ) @xmath216 \varepsilon + j_{2}k\right\}\end{aligned}\ ] ] @xmath217 now we express the two solutions , for @xmath113 and @xmath218 in terms of the propagators @xmath212@xmath219 where summation over @xmath220 is assumed , and @xmath221 . we now dispose of the generating functional of the system _ when this is in a region around _ @xmath222 , the fixed equilibrium points . to see what is the effect of the diffusion in this case we calculate for the variable @xmath10 the average and the dispersion . @xmath223 which was to be expected . and @xmath224 @xmath225 \\ & & \hspace*{-1cm}\times \left [ x_{0}+\frac{1}{2}\int_{0}^{t}dt^{\prime \prime } \delta _ { 11}\left ( t^{\prime } , t^{\prime \prime } \right ) j_{1}\left ( t^{\prime \prime } \right ) + \frac{1}{2}\int_{0}^{t}dt^{\prime \prime } \delta _ { 12}\left ( t^{\prime } , t^{\prime \prime } \right ) j_{2}\left ( t^{\prime \prime } \right ) + \frac{1}{2}\int_{0}^{t}dtj_{2}\left ( t\right ) \delta _ { 21}\left ( t , t^{\prime } \right ) \right ] \\ & & \times \exp \left ( i\emph{s}_{j}\right)\end{aligned}\ ] ] this gives the result @xmath226 d \label{xtxtp } \\ & & \times \left\ { \frac{1}{2\left ( -a+3bx_{0}^{2}\right ) } \left ( \exp \left [ 2\left ( -a+3bx_{0}^{2}\right ) t\right ] -1\right ) \theta \left ( t^{\prime } -t\right ) \right . \notag \\ & & \left . + \frac{1}{2\left ( -a+3bx_{0}^{2}\right ) } \left ( \exp \left [ 2\left ( -a+3bx_{0}^{2}\right ) t^{\prime } \right ] -1\right ) \theta \left ( t - t^{\prime } \right ) \right\ } \notag \\ & & \times \exp \left [ \left ( a-3bx_{0}^{2}\right ) t^{\prime } \right ] \notag\end{aligned}\ ] ] in the absence of the potential @xmath227 and @xmath228 , this is simply @xmath229 _ i.e. _ the diffusion around the position @xmath36 . in the present case , we note that @xmath230 with @xmath231 . fixing the parameters , we take @xmath232 and obtain @xmath233 \exp \left ( -u_{0}^{\prime \prime } t^{\prime } \right)\ ] ] for @xmath234 ( the dispersion ) we obtain ( with @xmath235 ) @xmath236 \label{x2canoi}\ ] ] ) ( continuous line ) and numerical integration of the stochastic equation.,width=377 ] it is straightforward to calculate the higher order statistics for this process , since the functional derivatives can easily be done . we have to remember that this derivation was based on the approximation consisting in taking the equilibrium position in the euler - lagrange equation for @xmath81 . however , since @xmath237 and @xmath238 are small quantities and we can suppose that the instantons , even if they are not exact solutions of the equations of motion , can give a contribution to the action . the instantons connects the points of maximum not of @xmath164 ( because they are not equal ) but of a different potential , a _ corrected _ @xmath164 to order one in @xmath119 that has degenerate maxima . this potential will be called @xmath239 and it will be considered in the calculation of the contribution of the instantons and antiinstantons . @xmath240 where @xmath241 is the point corresponding to the minimum situated between the two maximas . it is introduced the family of trajectories @xmath242 which leaves @xmath166 at time @xmath243 and reach @xmath140 at time @xmath244 . they have all the same energy @xmath245 and the classical action @xmath246 . we now consider the travel from @xmath166 to @xmath140 made by an instanton @xmath247 with the center located at time @xmath248 ; next the return made by an antiinstanton which is actually an instanton @xmath249 starting from @xmath140 and going to @xmath166 with the center located at time @xmath248 . with these two instantons we create a single classical solution @xmath250 the contribution to the action of this _ assambled _ solution is @xmath251 ^{2}}{2}\end{aligned}\ ] ] the contributions in this formula comes from the potential energy and the kinetic energy along the trajectory . we should remember that the action is the integral on time of the density of lagrangian , where there is the kinetic energy term and minus the potential energy . @xmath252 \right\}\ ] ] if the particle would have remained in @xmath166 imobile for all time @xmath253 then the contribution from the potential would have been @xmath254 the instanton spends @xmath255 time in the point @xmath151 before returning to @xmath179 . then it accumulates the action equal with the difference in potential between @xmath179 and @xmath151 multiplied with this time interval . @xmath256 \left ( t_{1}-t_{0}\right ) = \left ( w_{0}-w_{b}\right ) \left ( t_{1}-t_{0}\right)\ ] ] ( since @xmath257 ) . define the kinetic energy and the energy @xmath258 \\ \overset{\cdot } { x}\left ( t\right ) & = & \sqrt{2\left ( e+w\left ( x\right ) \right ) } \end{aligned}\ ] ] then @xmath259 in this formula we have to replace the expression of the trajectory @xmath260 and integrate . in our case the energy has the value of the initial position . here the velocity is zero and the potential is @xmath261@xmath262 and the potential is actually the current value of the zeroth -order potential @xmath263 -\left [ -w^{\left ( 0\right ) } \left ( x\right ) \right ] \\ & = & w^{\left ( 0\right ) } \left ( x\right ) -w\left ( b\right)\end{aligned}\ ] ] if we want to calculate the integral of the _ kinetic energy _ along the trajectory , we have to consider separately the intervals where the kinetic energy is strongly determined by the velocity , _ i.e. _ the region where the instanton transition occurs , from the rest of the trajectory where , the velocity being practically zero , the potential is a better description and can be easily approximated . the approximation will relay on the fact that the particle is practically imobile in @xmath179 or in @xmath151 , after the transition has been made . so we will use both expressions for the kinetic energy @xmath264 ^{1/2}\ ] ] the terms must be considered twice and the interval of integration can be extended for the region of transition , since in any case it is very small @xmath265 ^{1/2}-2\int_{\frac{t_{1}+t_{0}}{2}}^{\infty } d\tau \frac{1}{2}\overset{% \cdot } { x}^{2}\left ( \tau -t_{0}\right)\ ] ] the term @xmath246 is @xmath266 ^{1/2}\ ] ] using an approximation for the form of the instanton , it results @xmath267\end{aligned}\ ] ] where @xmath268 and @xmath269 ^{1/2}}-\frac{1}{\left [ w_{h}^{\left ( 0\right ) } \left ( x\right ) -w_{b}\right ] ^{1/2}}\right\}\ ] ] the local harmonic approximation to @xmath270 is @xmath271@xmath272 * note * since the trajectory @xmath273 is not an _ exact _ solution of the equation of motion in the potential @xmath274 , the expansion of the action @xmath51 will not be limitted to the zeroth and the second order terms . it will also contain a firts order term , @xmath275 . it can be shown that this contribution is negligible in the order @xmath276 . the calculation of the contribution to the functional integral from the second order expansion around @xmath277 is done as usual by finding the eigenvalues of the determinant of the corresponding operator . as before , the product of the eigenvalues should not include the first eigenvalue since this is connected with the translational symmetry of the instanton solution . this time there will be two eigenvalues , one for @xmath248 ( the transition performed by the @xmath228 instanton ) and the second for @xmath278 ( the transition prtformed by the antiinstanton , or the transition @xmath279 ) . another way of expressing this invariance to the two time translations is to say that the pair of instantons has not a determined central moment and , in addition , there is an _ internal _ degree of freedom of the _ breathing _ solution , which actually is this pair instanton - antiinstanton . to take into account these modes , whose eigenvalues are zero , we need to integrate in the functional integral , over the two times , @xmath248 and @xmath278 . the measure of integration for the two translational symmetries @xmath248 and @xmath280 is @xmath281 this quantity is the jacobian of the change of variables in the functional integration over the fluctuations around the instanton solution . the fluctuation that corresponds to the lowest ( almost zero ) eigenvalue is replaced in the measure of integration with the differential of the time variable representing the moment of transition . then it results this jacobian . it can be shown that in the approximation given by exponentially small terms , the contribution to the path integral of the small fluctuations around the classical instanton - antiinstanton @xmath277 solution is the _ product _ of the fluctuation terms around the instanton and antiinstanton separately . @xmath282 ^{1/2}\exp \left [ -\frac{s_{ia}\left ( t;t_{0},t_{1}\right ) } { d}% \right]\end{aligned}\ ] ] where @xmath283 represent the lowest eigenvalue of the operator arising from the second order expansion of the action , defined on the time intervals @xmath284 the notation @xmath285 and respectively @xmath286 represents @xmath287 @xmath288 this corresponds to the formula of coleman which replaces the infinite product of eigenvalues with @xmath289 where @xmath134 is calculated at the end of the interval of time , where @xmath134 verifies the boundary conditions ( [ psibo ] ) . the result is @xmath290 ^{1/2}\frac{\left ( x_{m}-b\right ) \left| x_{m}\right| w_{0}^{\prime \prime } w_{b}^{\prime \prime } } { 2\pi ^{2}d^{2}}\exp \left [ -\frac{\varepsilon _ { 0}^{\left ( b\right ) } t}{d}\right ] \notag \\ & & \times \exp \left [ -\frac{2s_{b0}}{d}+\delta _ { bm}\left ( 2w_{b}^{\prime \prime } \right ) ^{1/2}+\delta _ { bm}\left ( 2w_{0}^{\prime \prime } \right ) ^{1/2}\right ] \notag \\ & & \hspace*{-1cm}\times \int_{-t/2}^{t/2}dt_{0}\int_{t_{0}}^{t/2}dt_{1}\exp \left\ { -\frac{\varepsilon _ { 0}^{\left ( 0\right ) } -\varepsilon _ { 0}^{\left ( b\right ) } } { d}\left ( t_{1}-t_{0}\right ) + \frac{c}{d}\exp \left [ -\left ( 2w_{0}^{\prime \prime } \right ) ^{1/2}\left ( t_{1}-t_{0}\right ) \right ] \right\ } \notag\end{aligned}\ ] ] where @xmath291 is the lowest eigenvalue of the schrodinger equation associated with the fokker - planck diffusion equation in the local harmonic approximation of the potential @xmath165 in the well @xmath292 . @xmath293 \label{constc}\ ] ] and @xmath294 ^{1/2}}-\frac{1}{\left [ w_{h}^{\left ( 0\right ) } \left ( x\right ) -w_{b}\right ] ^{1/2}}\right\}\ ] ] there is a very important problem with this formula : the coefficient @xmath295 is positive and the contribution of this part in the @xmath278 integration comes from time intervals @xmath296 since this is a very small time interval , it results that the contributions are due to states where the instanton and the antiinstanton are very close one of the other , which is unphysical . it will be necessary to calculate in a particular way this part , introducing acontour in the complex @xmath297 plane , with an excursion on the imaginary axis . * the calculation of the propagator * @xmath298 . here @xmath299 means that only one pair of instanton and anti - instanton is considered . we particularize the formula above using the expression for @xmath300 given in terms of @xmath163 . now we have @xmath301 the expression of @xmath246 is @xmath302 @xmath303 where @xmath304\ ] ] we will use the functional approach in the setting that has been developed by us to the calculation of the probability of transition from one minimum to the same minimum with an intermediate stay at the unstable maximum point ( symmetric potential ) . the initial equation is @xmath305 the action functional with external current added is @xmath306 @xmath307 the equations of motion are @xmath308 we have to solve these equations , and replace the solutions @xmath84 and @xmath113 in the action functional . then the functional derivatives to the external current @xmath309 will give us the correlations for the stochastic variable @xmath10 . we can also calculate the probability that the particle , starting from one point at a certain time will be found at another point at other time . this will be done below . the first step is to obtain an analytical solution to the euler - lagrange equation . the method to solve these equation is essentially a successive approximation , as we have done above , for the diffusive motion around a stable position in the potential ( harmonic region ) . we know that the classical trajectory must be of the type of a transition between the initial point , taken here as the left minimum of the potential and the final point , the unstable maximum of the potential . from there , the particle will return to the left minimum by an inverse transition . we have to calculate simultaneously @xmath10 and @xmath113 , but we have sufficient information to find an approximation for @xmath10 , by neglecting the effect of diffusion ( the last term in the differential equation for @xmath10 ) . @xmath310 this gives @xmath311 \right\ } ^{1/2}}\ ] ] this solution is a transition between either of the minima @xmath312 and @xmath151 . it shows the same characteristics as found numerically or by integrating the elliptic form of the equation , in the case of onasger - machlup action . the particle spends long time in the initial and final points and makes a fast transition between them at an arbitrary time @xmath313 . the width of transition is small compared to the rest of quasi - imobile stays in the two points , especially if @xmath143 is large . then we will make an approximation , taking the solution as @xmath314 where @xmath201 and @xmath151 are the initial and final positions , the indice @xmath315 means that this is the calssical solution ( extremum of the action ) and @xmath299 means the first part of the full trajectory , which will also include the inverse transition , from @xmath151 to @xmath201 . the following structure of the total trajectory is examined : the total time interval is between @xmath316 and @xmath317 ( later the parameter @xmath317 will be identified with @xmath147 for comparison with the results from the literature ) . the current time variable is @xmath158 and in the present notations , @xmath318 is any moment of time in the interval @xmath319 . at time @xmath248 the particle makes a jump to the position @xmath140 and remains there until @xmath320 . at @xmath278 it performs a jump to the position @xmath201 , where it remains for the rest of time , until @xmath317 . with this approximative solution of the euler - lagrange equations ( since we have neglected the term with @xmath119 in the equation of @xmath321 ) we return to the equation for @xmath322 . we first calculate @xmath323 we need the integration of this quantity in the inverse direction starting from the end of the motion toward the initial time @xmath324 \theta \left ( t - t_{0}\right ) + \left [ -at+3at_{0}-2at\right ] \theta \left ( t_{0}-t\right)\ ] ] we now introduce the second part of the motion : at time @xmath325 the particle makes the inverse transition @xmath326 with the similar quantities . after explaining the steps of the calculation , we change to work with the full process , assembling the two transitions and the static parts into a single trajectory @xmath327 @xmath328 and @xmath329 we have introduce the notation @xmath330 then the solution of the equation for the dual variable is @xmath331 \left\ { k_{t}+\int_{t}^{\tau } dt^{\prime } \left [ -j_{1}\left ( t^{\prime } \right ) \right ] \exp \left [ w\left ( t^{\prime } \right ) \right ] \right\}\ ] ] according to the procedure explained before we will need to express the solutions as bilinear combinations of currents , so we identify @xmath332 \exp \left [ w\left ( t^{\prime } \right ) % \right ] \label{del21new}\ ] ] and the solution can be rewritten @xmath333 + \int_{-t}^{t}dt^{\prime } \delta _ { 21}\left ( t , t^{\prime } \right ) j_{1}\left ( t^{\prime } \right ) \label{kdel21}\ ] ] using these first approximations for the extremizing path @xmath334 we return to the euler lagrange equations and expend the variable @xmath335 as @xmath336 whose equation is @xmath337 and the solution @xmath338 \left\ { b_{0}+\int_{0}^{t}dt^{\prime } 2idk_{\eta } \left ( t^{\prime } \right ) \exp % \left [ -\int_{0}^{t^{\prime } } dt^{\prime \prime } \left ( \frac{dv}{dx}\right ) _ { x_{\eta } \left ( t^{\prime \prime } \right ) } \right ] \right\}\ ] ] here @xmath339 is a constant to be determined by the condition that @xmath340 vanishes at the final point . we note that here all integrations are performed forward in time . we also notice that this expression will contain the current @xmath39 and we will introduce the propagator @xmath341 . performing the detailed calculations we obtain @xmath342 + \exp % \left [ w\left ( t\right ) \right ] \int_{-t}^{t}dt^{\prime } \left ( 2idk_{t}\right ) \theta \left ( t - t^{\prime } \right ) \exp \left [ -w\left ( t^{\prime } \right ) \right ] \label{delxp } \\ & & + \int_{-t}^{t}dt^{\prime } \delta _ { 11}\left ( t , t^{\prime } \right ) j_{1}\left ( t^{\prime } \right ) \notag\end{aligned}\ ] ] the first two terms depend on constants of integrations and the propagator is @xmath343 \int_{-t}^{t}dt^{\prime \prime } 2id\theta \left ( t - t^{\prime \prime } \right ) \delta _ { 21}\left ( t^{\prime \prime } , t^{\prime } \right ) \exp \left [ -w\left ( t^{\prime \prime } \right ) \right ] \label{del11}\ ] ] using the solutions ( [ xexpa ] ) and ( [ kdel21 ] ) we can calculate the action along this trajectory . @xmath344 \\ & = & i\int_{-t}^{t}dt\left ( \frac{1}{2}x\overset{\cdot } { k}+\frac{1}{2}% kv+j_{1}x\right ) \\ & = & i\int_{-t}^{t}dt\left [ \frac{1}{2}x\left ( -j_{1}+k\frac{dv}{dx}\right ) + % \frac{1}{2}kv+j_{1}x\right ] \\ & = & i\int_{-t}^{t}dt\left\ { \frac{1}{2}j_{1}x+\frac{1}{2}k\left [ v-\left ( \frac{dv}{dx}\right ) x\right ] \right\}\end{aligned}\ ] ] [ it becomes obvious that when the potential is linear ( which means that the lagrangean is quadratic the two functional variables @xmath10 and @xmath113 ) the potential does not contribute to the action along the extremal path ] . using the solutions we have @xmath345 \\ & & + i\int_{-t}^{t}dt\frac{1}{2}\left [ v\left ( x_{\eta } \right ) -\left ( \frac{% dv}{dx}\right ) _ { x_{\eta } } x_{\eta } \right ] i\int_{-t}^{t}dt^{\prime } \delta _ { 21}\left ( t , t^{\prime } \right ) j_{1}\left ( t^{\prime } \right)\end{aligned}\ ] ] it will become clear later that the parts containing constants are not significative for the final answer , the probability . we will focus on the terms containing explicitely the current @xmath39 since the statistical properties are determined by functional derivatives to this parameter . the action is @xmath346 \int_{-t}^{t}dt^{\prime } \delta _ { 21}\left ( t , t^{\prime } \right ) j_{1}\left ( t^{\prime } \right ) \right\}\end{aligned}\ ] ] the generating functional of the correlations ( at any order ) is @xmath347 = \exp \left\ { \emph{s}\left [ j_{1}\right ] \right\ } \label{zands}\ ] ] and we can calculate any quantity by simply parforming functional derivatives and finally taking @xmath348 . instead of that and in order to validate our procedure , we will calculate the probability for the process : a particle in the initial position @xmath349 at time @xmath316 can be found at the final position @xmath350 at the time @xmath351 . ( later we will particularize to @xmath352 and @xmath353 ) . the calculation of this probability @xmath132 can be done in the functional approach in the following way . we have defined the statistical ensemble of possible particle trajectories starting at @xmath354 at @xmath355 and reaching an arbitrary point @xmath335 at time @xmath318 . in the martin - siggia - rose - jensen approach it is derived the generating functional as a functional integration over this statistical ensemble of a weight measure expressed as the exponential of the classical action . now we restrict the statistical ensemble by imposing that the particle is found at time @xmath351 in the point @xmath350 . by integration this will give us precisely the probability required . the condition can be introduced in the functional integration by a dirac @xmath13 function which modifies the functional measure @xmath356 \rightarrow \emph{d}\left [ x\left ( \tau \right ) \right ] \delta \left [ x\left ( t_{c}\right ) -x_{c}\right]\ ] ] and we use the fourier transform of the @xmath11 function @xmath357 \delta \left [ x\left ( t_{c}\right ) -x_{c}\right ] \exp \left ( \emph{s}\right ) \\ & = & \frac{1}{2\pi } \int d\lambda \exp \left ( -i\lambda x_{c}\right ) \int \emph{d}\left [ x\left ( \tau \right ) \right ] \exp \left [ s+i\lambda x\left ( t_{c}\right ) \right]\end{aligned}\ ] ] it is convenient to write @xmath358 + i\int_{-t}^{t}d\tau \lambda x\left ( \tau \right ) \delta \left ( \tau -t_{c}\right ) \\ & = & i\int_{-t}^{t}d\tau \left [ \emph{l}+\lambda x\left ( \tau \right ) \delta \left ( \tau -t_{c}\right ) \right]\end{aligned}\ ] ] we can see that the new term plays the same rle as the external current @xmath359 and this sugests to return to the eq.([zands ] ) and to perform the modification @xmath360 obtaining @xmath361 \right| _ { j_{1}=0}\ ] ] using the previous results we have @xmath362 \right| _ { j_{1}=0 } & = & \exp \left\ { i\frac{1}{2}\lambda x_{\eta } \left ( t_{c}\right ) \right . \label{zlamb1 } \\ & & + i\frac{1}{2}\lambda ^{2}\delta _ { 11}\left ( t_{c},t_{c}\right ) \notag \\ & & \left . + i\frac{1}{2}\left [ v-\left ( \frac{dv}{dx}\right ) x\right ] _ { t = t_{c}}\lambda \delta _ { 21}\left ( t_{c},t_{c}\right ) \right\ } \right| _ { j_{1}=0 } \notag\end{aligned}\ ] ] this expression will have to be calculated at @xmath363 and @xmath364 . @xmath365 so the last term is zero . _ { j_{1}=0}=\left . \exp \left [ -i\frac{1}{2}\lambda \sqrt{\frac{a}{b}}+i\frac{% 1}{2}\lambda ^{2}\delta _ { 11}\left ( t , t\right ) \right ] \right| _ { j_{1}=0 } \label{zdel}\ ] ] we need the propagator @xmath367 in the expression for @xmath341 , @xmath368 \exp \left [ w\left ( t\right ) % \right]\ ] ] @xmath369 then the intration giving the propagator @xmath341 is @xmath370 \exp \left [ -w\left ( t^{\prime } \right ) \right ] \\ & = & 2id\int_{-t}^{t}dt^{\prime } \exp \left [ -2w\left ( t^{\prime } \right ) % \right ] \\ & = & 2id\left\ { \exp \left [ -4at+6at_{1}-6at_{0}\right ] \int_{-t}^{t_{0}}dt% \exp \left ( 4at\right ) \right . \\ & & + \exp \left [ -4at+6at_{1}\right ] \int_{t_{0}}^{t_{1}}dt\exp \left ( -2at\right ) + \\ & & \left . + \exp \left [ -4at\right ] \int_{t_{1}}^{t}dt\exp \left ( 4at\right ) \right\}\end{aligned}\ ] ] we introduce the notation @xmath371 \left\ { \exp \left [ 2a\left ( t_{1}-t_{0}\right ) \right ] -1\right\ } \\ & & -\exp \left [ -8at\right ] \exp \left [ 6a\left ( t_{1}-t_{0}\right ) \right]\end{aligned}\ ] ] and we obtain the expression @xmath372 returning to ( [ zdel ] ) @xmath373 = \exp \left [ -i\frac{% \lambda } { 2}\sqrt{\frac{a}{b}}-\frac{\lambda ^{2}d}{4a}y\right]\ ] ] and the probability is @xmath374 \emph{z}^{\left ( \lambda \right ) } \left [ j_{1}=0\right ] \\ & = & \frac{1}{2\pi } \int_{-\infty } ^{\infty } d\lambda \exp \left [ \frac{1}{2}% i\lambda \sqrt{\frac{a}{b}}-\frac{\lambda ^{2}d}{4a}y\left ( t_{0},t_{1}\right ) \right]\end{aligned}\ ] ] we have made manifest the dependence of the probability on the arbitrary times of jump @xmath375 . integrating on @xmath376@xmath377 this expression contains an arbitrary parameter , which is the duration of stay in the intermediate position , _ i.e. _ at the unstable extremum of the potantial , @xmath140 . integrating over this duration , @xmath378 , adimensionalized with the unit @xmath379 , for all possible values between @xmath151 and all the time interval , @xmath380 , will give the probability . in doing this time integration we will make a simplification to consider that @xmath317 is a large quantity , such that the exponential terms in @xmath381 will vanish . then @xmath381 reduces to @xmath382 and the integration is trivial @xmath383 p\left ( t_{0},t_{1}\right ) \\ & = & 2ta\sqrt{\frac{a}{\pi d}}\exp \left ( -\frac{a^{2}}{4bd}\right)\end{aligned}\ ] ] this is our answer . to compare with the result of caroli et al . , we first adopt their notation @xmath384 and we write the potential as @xmath385 with @xmath386 we note that @xmath387 which allows to write @xmath388\ ] ] which corresponds to the result of caroli et al . for a single instanton - anti - instanton pair , where @xmath389 the only problem is the need to consider the reverse of sign due to the virtual reflection at the position @xmath140 . in the tratment of caroli , it is included by an integration in complex time plane , after the second jump , @xmath280 . the problem arises at the examination of the eq.([k(1)raw ] ) where it is noticed that the constant @xmath295 multiplying the exponential term _ in the exponential _ in the integrations over @xmath248 and @xmath278 is positive ( see eq.([constc ] ) ) . then the dependence of the integrand on @xmath297 makes that the most important contribution to the integration over the variable @xmath278 to come from @xmath390 here the integrand is of the order @xmath391 this would imply , in physical terms , that the most important configurations contributing to the path integral would be pairs of instanton anti - instanton very closely separated , _ i.e. _ transitions with very short time of stay in the point @xmath151 . this is not correct since we expect that in large time regime the trajectories assembled from instantons and anti - instantons with arbitrary large separations ( duration of residence in @xmath151 before returning to @xmath179 ) should contribute to @xmath115 . this problem can be solved by extending the integration over the variable representing the time separation between the transitions @xmath392 in complex . * note*. the quantity @xmath393 is the frequency of the harmonic oscillations in the parabolic approximation of the potential . the contour _ c _ is defined as @xmath394 the form of the trajectory @xmath277 connecting the point @xmath179 to itself , @xmath179 , assembled from one instanton @xmath395 and one anti - instanton @xmath396 can now be described as follows . the trajectory starts in the point @xmath179 and follows the usual form of the instanton solution , which means that for times @xmath397 it practically conicides with the fixed position @xmath179 . at @xmath248 it makes the transition to the point @xmath151 , the duration of the transition ( width of the instanton ) being @xmath146 . then it remains almost fixed at the point @xmath398 . the problem is that there is _ no _ turning point at @xmath140 since there the potential @xmath164 is smaller than at @xmath166 . then we do not have the classical instanton which takes an infinite time to reach exactly @xmath140 , etc . also there is no exact instanton solution which starts from @xmath140 to go back to @xmath166 . in the assambled solution @xmath399 , there is a discontinuity at @xmath140 in the derivative , the velocity simply is forced ( assumed ) to change from almost zero positive ( directed toward @xmath140 ) to almost zero negative ( directed toward @xmath166 ) . what we do is to introduce a turning point , which limits effectively the transition solution , but for this we have to go to complex time , to compensate for the negative value of the potential . making time imaginary changes the sign of the velocity squared ( kinetic energy ) and correspondingly can be seen as a change of sign of the potential which now becomes positive and presents a wall from where the solution can reflect . this is actually a local euclidean - ization of the theory , around the point @xmath140 . at time @xmath400 the trajectory starts moving along the imaginary @xmath401 axis . it reaches an imaginary turning point at @xmath402 and returns to the point @xmath403 , @xmath404 . this trajectory can be considered the limiting form of the trajectory of a fictitious particle that traverses a part of potential ( forbidden classically ) when this part goes to zero . then the expression is @xmath405 where the minus sign is due to the reflection at the turning point . * calculation of the transitions implying the points * @xmath179 and @xmath143 this is the term @xmath406 and corresponds to the transitions @xmath407 again we have to introduce a new potential replacing @xmath165 . the real potential @xmath137 as it results from the langevin equation , depende on @xmath119 which is supposed to be small . then the `` expansion '' in this small parameter @xmath119 has as zerth order the new potential @xmath408 that will be used in the computations below . this is @xmath409 @xmath410 where the new point @xmath411 is in the region of the right - hand minimum of @xmath412 . this point will correspond to the center of the instanton connecting @xmath151 to @xmath143 . we introduce the new family of instanton solution @xmath413 in the potential @xmath414 that leave @xmath140 at @xmath243 and reach @xmath415 at time @xmath244 . with this instanton and the previously defines ones , we assamble a function that provide approximately the transitions conncetin @xmath179 to @xmath179 via @xmath143 . this , obviously , is not a classical solution of the equation of motion , although it is very close to a solution . @xmath416 these trajectories are rather artificially , being assambled form pieces of true solutions . they have singularities : in the middel , at @xmath417 the function is continuous but the derivative is discontinuous ; in the points where one instanton arrives and another must continue ( _ i.e. _ @xmath400 and @xmath418 , both the function and the derivatives are discontinuous . this is corrected by extending the trajectories in complex time plane . one defines the variables @xmath419 and the three integration contours @xmath420 together these contours must assure the continuity of the function @xmath277 in the region @xmath151 , from which the condition arises @xmath421 = -% \widetilde{x}_{i}\left [ \func{re}\left ( \frac{t_{0}-t_{1}}{2}\right ) \right]\ ] ] which gives for large separations @xmath297@xmath422 the exprsssion of @xmath115 at this moment is @xmath423 \\ & & \times \int_{-t/2}^{t/2}dt_{0}\int_{\emph{c}_{1}}dz_{1}\int_{\emph{c}% _ { 2}}dz_{2}\int_{\emph{c}_{3}}dz_{3}\exp \left [ f\left ( z_{1}\right ) + g\left ( z_{2}\right ) + f\left ( z_{3}\right ) \right]\end{aligned}\ ] ] where @xmath424 @xmath425 and @xmath426 @xmath427 it is necessary to carry out the complex time integrals . * the * @xmath428 * * integral**. here we have @xmath429 along the real axis of @xmath401 , we can substitute @xmath430 and obtain @xmath431 this simply results in the constant @xmath432 * the * @xmath433 * * integral * * ; the @xmath433 integral involves only the exponential of @xmath434 and the latter function does not contain a term proportional with @xmath435 ( comapre with ( [ fz ] ) ) , but only the exponential term . the function @xmath434 is very close to @xmath382 as soon as @xmath436 where @xmath437 is the suzuki time for the region @xmath143@xmath438 then , with very good approximation @xmath439 = \int_{0}^{% \frac{t}{2}-t_{1}}dz_{2}\exp \left [ g\left ( z_{2}\right ) \right ] \simeq \frac{t}{2}\ ] ] * the * @xmath440 * integral*. this is similar to the integral over @xmath428 and will imply again @xmath441 as in ( [ rezc3 ] ) . finally we note that only the constant @xmath442 relative to the intermediate point @xmath151 appears , and the one for the farthest point of the trajectory , @xmath143 , is absent . there will be then a time integration over the moment of the first transition , @xmath248 , of an integrand that contains the time @xmath318 arising from the integration on the contour @xmath443 , in the complex plane of the variable @xmath444 . this variable represents the duration between the last transition to @xmath143 and the first transition from @xmath143 , _ i.e. _ is the duration of stay in @xmath143 . this time integration will produce a term with @xmath445 . replacing in the expressions resulting from the integrations on @xmath440 and @xmath446 the constant @xmath442 from ( [ co ] ) the factors @xmath241 and @xmath411 disappear . the result is @xmath447 the objective is to calculate @xmath448 on the basis of the above results . we have to sum over any number of independent pseudomolecules of the four species @xmath449 where the factor which is associated to each pseudomolecule is @xmath450 a particular path containing @xmath451 @xmath452 and @xmath453 @xmath454 has the form @xmath455 it was taken into account that a path @xmath456 contains the same number @xmath457 of @xmath458 and of @xmath459 . the results from the previous calculations are listed below . * the probability that a particle initially at * @xmath460 * will be found at * @xmath178 @xmath461 \right\}\ ] ] * the probability that a particle initially at * @xmath460 * will be found at * @xmath462@xmath463 \right\}\ ] ] * the probability that a particle initially at * @xmath464 * will be found at * @xmath178@xmath465 \right\}\ ] ] * the probability that a particle initially at * @xmath464 * will be found at * @xmath462@xmath466 \right\}\ ] ] * the long time limit of the probability density * @xmath467 the formula has been derived for the case where the two positions @xmath335 and @xmath468 belong to the harmonic regions @xmath469 and @xmath470 . this is because asymptotically these regions will be populated . consider the case where @xmath471 the first trajectory contributing to the path integral is the direct connection between @xmath179 and @xmath143 . the contribution to @xmath115 is @xmath472 since the points of start and/or arrival @xmath473 are different of @xmath474 , there is finite ( _ i.e. _ non exponentially small ) slope of the solution , and there is nomore degeneracy with respect to the translation of the center of the instantons . then there will be not a proportionality with time in the one - instanton term . the terms connecting @xmath36 with @xmath335 with onely one intermediate step in either @xmath179 or @xmath143 have comparable contributions to the action . now the term with two intermediate steps @xmath475 in the very long time regime , @xmath476 can be factorized @xmath477 this equation is independent of @xmath478 and @xmath479 as long as @xmath480 then @xmath481\ ] ] there is also the approximate equality @xmath482 it results that all the contributions are included if in the expression of @xmath483 we replace the middle factor , which has become @xmath484 by the full @xmath485 . it is obtained @xmath486 p\left ( a,\frac{t}{2};b,-\frac{t}{2}% \right ) \\ & = & \exp \left [ -\frac{u\left ( x\right ) -u_{a}}{2d}\right ] \left ( \frac{% u_{a}^{\prime \prime } } { 2\pi d}\right ) ^{1/2}\frac{\alpha _ { b}}{\alpha _ { a}+\alpha _ { b}}\left\ { 1-\exp \left [ -t\left ( \alpha _ { a}+\alpha _ { b}\right ) \right ] \right\}\end{aligned}\ ] ] * the case where both * @xmath335 * and * @xmath36 * belong to the harmonic region * @xmath487 the same calculation shows that @xmath488 \left ( \frac{u_{b}^{\prime \prime } } { 2\pi d}% \right ) ^{1/2}\frac{1}{\alpha _ { a}+\alpha _ { b}}\left\ { \alpha _ { a}+\alpha _ { b}\exp \left [ -t\left ( \alpha _ { a}+\alpha _ { b}\right ) \right ] \right\}\ ] ] in the following we reproduce the graphs of the time - dependent probability distribution of a system governed by the basic langevin equation . various initialisations are considered , showing rapid redistribution of the density of presence of the system . the speed of redistribuition is , naturally , connected to the asymmetry of the potential . each figure consists of a set of graphs : * the potential functions @xmath489 , @xmath490 and @xmath491 * four probabilities of passage from and between the two equilibrium points @xmath143 and @xmath179 ; * the average value of the position of the system , as function of time , for the two most characteristic initializations the two parameters @xmath143 and @xmath119 take different values , for illustration . ( we apologize for the quality of the figures . better but larger ps version can be downloaded from http://florin.spineanu.free.fr/sciarchive/topicalreview.ps ) and @xmath492,width=340 ] and @xmath493,width=415 ] and @xmath494,width=415 ] and @xmath493,width=415 ] and @xmath494,width=415 ] and @xmath493,width=415 ] and @xmath494,width=415 ] and @xmath495,width=415 ] and @xmath494,width=415 ] the stochastic motion is described in terms of the _ conditional probability _ @xmath496 that the particle initially @xmath497 at @xmath498 to be at the point @xmath335 at time @xmath318 . we will use the notation that suppress the @xmath151 as the initial time , @xmath499 . the conditional probability obeys the following fokker - planck equation @xmath500 + d\frac{\partial ^{2}p}{\partial x^{2}}\ ] ] where the velocity function is here derived from the potential @xmath501@xmath502 \equiv -u^{\prime } \left ( x\right)\ ] ] since there is no drive . the initial condition for the probability function is @xmath503 the solution of the fokker - planck equation is given in terms of the following _ path - integral _ @xmath506 } { \exp \left [ -u\left ( x_{i}\right ) /2d\right ] } k\left ( x , t;x_{i}\right)\ ] ] where @xmath507 \exp \left [ -\frac{s}{d}\right ] \label{kfunc}\ ] ] where the functional integration is done over all trajectories that start at @xmath508 and end at @xmath509 . the _ action functional _ is given by @xmath510\ ] ] with the notation @xmath511 in order to calculate explicitely the functional integral we look first for the paths @xmath10 that extremize the action . they are provided by the euler - lagrange equations , which reads @xmath512 with the boundary condition @xmath513 the first thing to do after finding the extremizing paths is to calculate the _ action _ functional along them , @xmath514 . after obtaining these extremum paths we have to consider the contribution to the functional integral of the paths situated in the neighborhood and this is done by expanding the action to second order . the argument of the expansion is the difference between a path from this neigborhood and the extremum path . the functional integration over these differences can be done since it is gaussian and the result is expressed in terms of the _ determinant _ of the operator resulting from the second order expansion of @xmath51 . then the expression ( [ kfunc ] ) becomes @xmath515 where @xmath516 the constant @xmath55 is for normalization . in order to find the determinant , one has to solve the eigenvalue problem @xmath517 \phi _ { n}\left ( \tau \right ) = 0 \label{eig}\ ] ] with boundary conditions for the _ differences _ between the trajectories in the neighborhood and the extremum trajectory @xmath518 there is a fundamental problem concerning the use of the equation ( [ kexp ] ) . it depends on the possibility that all the eigenvalues from the eq.([eig ] ) are determined . however , a path connecting a point in the neigborhood of the maximum of @xmath163 , @xmath521 with a point in the neighborhood of one of the minima @xmath522 is a kinklike solution . this solution ( which will be replaced in the expression of the operator in eq.([eig ] ) contains a parameter , the `` center '' of the kink , @xmath248 . this is an arbitrary parameter since the moment of traversation is arbitrary . the equation for the eigenvalues has therefore a symmetry at time translations of @xmath248 and has as a consequence the appearence of a _ zero _ eigenvalue . then the expression of the determinant would be infinite and the rate of transfer would vanish . actually , the time translation invariance is treated by considering the arbitrary moment @xmath248 as a new variable and performing a change of variable , from the set of functions @xmath523 to the set @xmath524 . the trajectory that extremizes the action is a kinklike solution @xmath525 ( is not exactly a kink since the shape of the potential is not that which produces the @xmath83 solution ) connecting the points : @xmath526 this treatment then consists of considering @xmath248 as a collective coordinate ( * see rajaraman and coleman and gervais & sakita * ) . the new form of eq.([kexp ] ) is @xmath527 where @xmath528 is the action of the path @xmath529 . the fact that the zero eigenvalue has been eliminated is indicated by the @xmath530 . two limits of time are important . the first is @xmath532 for the times @xmath533 for a particle initially in the region of the _ unstable _ state , @xmath534 , the region around the _ stable _ state , @xmath535 , or : @xmath536 is insignificant for the calculation of the probability @xmath537 . then the expression ( [ kexp ] ) can be used . much more important is the subsequent time regime @xmath538 where the particles leave the _ unstable _ point and the equilibrium state with density around the _ stable _ positions @xmath535 is approached . then the general formula ( [ kexpbar ] ) should be used . the energy @xmath520 of the path @xmath120 connecting the points @xmath539 and @xmath509 can be approximated from the expression ( * see caroli * ) @xmath540 ^{1/2}}-\theta _ { 0}\left ( x,0\right)\ ] ] where @xmath541 the notations are @xmath542 @xmath543 the last expressions are the harmonic approximation of @xmath544 and the potential around the _ unstable _ point @xmath140 . the general expression of the function @xmath115 in the case where we include the time regimes beyond the limits given above . it has the form of a convolution @xmath555 here the approximations for the ratio @xmath556 ^{1/2}$ ] are obtained by the same method as before . the relation between the energy and the time is used for the two main regions : around the _ stable _ and the _ unstable _ ( initial ) positions @xmath557 ^{1/2}}-\theta _ { 0}\left ( x_{m},x_{i}\right)\ ] ] @xmath558 ^{1/2}}-\theta _ { 1}\left ( x , x_{m}\right)\ ] ]

we review the analytical methods of solving the stochastic equations for barrier - type dynamical behavior in plasma systems . the path - integral approach is examined as a particularly efficient method of determination of the statistical properties .

quantum coherence is one of the fundamental features which distinguish quantum world from classical realm . it is the origin for extensive quantum phenomena such as interference , laser , superconductivity @xcite and superfluidity @xcite . it is an important subject related to quantum mechanics , from quantum optics @xcite , solid state physics @xcite , thermodynamics @xcite , to quantum biology @xcite . coherence , together with quantum correlations like quantum entanglement @xcite , quantum discord @xcite , are crucial ingredients in quantum computation and information tasks @xcite . coherence shines its quantum merits in quantum metrology @xcite , quantum key distribution @xcite , entanglement creation @xcite , etc . unlike quantum entanglement and other quantum correlations , coherence , regarded as a physical resource @xcite , has been just investigated very recently in establishing the framework of quantifying coherence in the language of quantum information theory @xcite . due to its fundamental role in quantum physics and quantum information theory , it is still necessary to understand how coherence works in information processing and investigate the relations between coherence and quantum correlations . in this paper , we consider the creation of coherence of a quantum system a , which initially has zero coherence , with the help of quantum channels on another quantum system b. such creation of coherence depends on the correlations between a and b , as well as the quantum operations on system b. we establish an explicit relation among the creation of the coherence on system a , the quantum entanglement ( concurrence ) between a and b , and the quantum operations . remote creation of coherence ( rcc ) can be illustrated by a simple example . consider a two - qubit system ab , which is initially in the maximally entangled state @xmath0 . if b undergoes a projective measurement under basis @xmath1 , and tells measurement outcome , for example @xmath2 , to a , the system a s final state would be a superposition of @xmath3 and @xmath4 with some probability , if the basis @xmath2 is neither @xmath3 nor @xmath4 . the same analysis also holds for measurement outcome @xmath5 . namely , system a can gain an averaged coherence over all the outcomes . for general states , we first investigate what kind of states can be used to create remote coherence and present a necessary and sufficient condition . then for pure states , we give a necessary and sufficient condition that the operations must satisfy for nonzero rcc . and finally , we give upper bounds of rcc and investigate the relations between coherence and entanglement in our scenario . since a quantum state s coherence depends on the reference basis , throughout our paper , we fix the system a s reference basis to be the computational basis . a well defined and mostly used coherence measure is the @xmath6 norm coherence @xmath7 @xcite . the @xmath6 norm coherence of a quantum state is defined as the sum of all off - diagonal elements of the state s density matrix under the reference basis , i.e. , @xmath8 where @xmath9 is the absolute value of @xmath10 . in the following , we use the @xmath6 norm coherence when we discuss the upper bound of rcc and its relation to entanglement . let @xmath11 be a bipartite quantum state and @xmath12 a quantum operation acting on the subsystem b. let @xmath13/\operatorname{tr}[(\mathbb{i}\otimes \$ ) { \rho^{ab}}]$ ] be the reduced state of the system a after the operation . concerning the quantum operations used for remote creation of coherence with a general quantum state , we have the following theorem . [ th1 ] given a bipartite quantum state @xmath11 , for any quantum operation @xmath12 acting on the subsystem b , the coherence of the final subsystem a @xmath14 if and only if @xmath11 is an incoherent - quantum state @xmath15 . let @xmath16 be the quantum state of system ab under the computational basis , then the marginal state @xmath17 . after the quantum operation @xmath18 acting on b , @xmath19 becomes , @xmath20 where @xmath21 $ ] is the probability of getting the state @xmath22 . tracing over the system b , we get the final state of system a , @xmath23 where @xmath24 and @xmath25 . `` @xmath26 '' . from equations ( [ cl1 ] ) and ( [ mixrhoap ] ) , for any operation @xmath18 , @xmath27 means that for any @xmath28 and @xmath29 , @xmath30 , i.e. , @xmath31 for the arbitrary hermitian operator @xmath32 ( the operation @xmath18 is aribitray ) , @xmath33 and the diagonal entries of @xmath28 are arbitrary real numbers which are independent of the off diagonal entries . thus from ( [ mix1 ] ) , we get @xmath34 , @xmath35 . on the other hand , for all @xmath29 , @xmath36 set @xmath37 , where @xmath38 is the unit imaginary number . substituting it into the above equation , we get @xmath39=0 . \end{aligned}\ ] ] since @xmath40 @xmath41 is arbitrary , @xmath42 and @xmath43 are all independent . then equation ( [ mix2 ] ) implies that for all @xmath44 , @xmath45 , i.e. , @xmath46 , @xmath47 . therefore @xmath48 , @xmath49 . hence the initial state @xmath11 becomes @xmath50 which actually is an incoherent - quantum state @xcite . `` @xmath51 '' . if @xmath11 is an incoherent - quantum state , @xmath15 . it is easy to check that , for any operation @xmath18 acting on system b , the final state of the system a , @xmath13/\operatorname{tr}[(\mathbb{i}\otimes \$ ) { \rho^{ab}}]$ ] , has zero coherence . from the above proof one can see that @xmath52 also means that the initial coherence of system a is also zero , @xmath53 . the above theorem implies that any non - incoherent quantum state can be used for rcc under certain operations . note that the incoherent - quantum state actually is the classical - quantum correlated state with the fixed reference basis @xcite . thus the states , that can be used to create coherence , are not limited within entangled states . according to theorem [ th1 ] a non - incoherent - quantum separable state , with the system a having zero coherence , can also be used for rcc . interestingly , the condition in theorem [ th1 ] for creating nonzero rcc is the same as the distillable coherence of collaboration in @xcite . however , it should be noticed that our scenario is different from the asymptotic scenario in @xcite . in @xcite , they study the maximal distilled coherence of collaboration under local quantum - incoherent operations and bilateral classical communications for the asymptotic case . while our work investigates the system a s average coherence after system b going through a _ certain _ quantum channel ( see the rest of our paper ) , and the rcc in our scenario only requires one way classical communication and does not involve the maximal process @xcite . the next natural question is , for a non - incoherent quantum state , what is the exact form of the operation acting on system b for the creation of coherence ? obviously for operation @xmath54 acting on system b such that @xmath55 , where @xmath56 is an arbitrary real number bigger than 1 , the coherence can not be created , which can be seen by substituting @xmath57 into equation ( [ mixrhoap ] ) . one can check that the important quantum operations including depolarizing operations , phase flip operations , bit flip operations and bit - phase flip operations all belong to this form . moreover , all the trace preserving quantum channels will not create the coherence remotely , and measurements on the system b are necessary in order to create nonzero coherence . we give the necessary and sufficient condition for rcc in the following . let @xmath58 be a pure bipartite entangled quantum state with zero coherence of the system a. we have [ th2 ] after a quantum operation @xmath12 acts on the system b , the system a , with initial coherence being zero , gains coherence if and only if there is a computational basis @xmath59 with @xmath60\neq 0 $ ] , where @xmath61 is the initial state of ab , @xmath62 and @xmath63=ab - ba$ ] is the lie bracket . with the local computational basis , the state of system ab can be expressed as @xmath64 , with @xmath65 the normalization condition . one has @xmath66 . respect to @xmath53 , the rows of the coefficient matrix @xmath67 are mutually orthogonal . then the singular value decomposition of @xmath68 has a simple form , @xmath69 , where @xmath70 is a diagonal matrix with nonzero singular values @xmath71 ( zero singular values are trivial for our proof ) , @xmath72 is a unitary matrix . thus we get the schmidt decomposition , @xmath73 , @xmath74 . we also have @xmath75 . after the local operation acting on the system b , the final state of the system a has the form , @xmath76/p ' \\ = & \operatorname{tr}_b [ ( \mathbb{i}\otimes \$)\sum _ { ij } \sqrt{\omega_i\omega_j } |i\ra \la j|\otimes |\beta_i\ra \la \beta_j|]/p ' \\ = & \sum_{ij } \sqrt{\omega_i\omega_j } |i\ra \la j| \operatorname{tr}[\sum_n f_n(|\beta_i\ra \la \beta_j|)f_n^\dagger]/p'\\ = & \sum_{ij } \sqrt{\omega_i\omega_j } n_{ji } |i\ra \la j|/p ' , \end{aligned}\ ] ] where @xmath77 , @xmath78 and @xmath79 $ ] , the probability of getting the state @xmath80 . since @xmath81 @xmath82 , all the off diagonal entries of @xmath80 vanish if and only if @xmath83 , which means that @xmath84s are just the eigenvectors of @xmath28 . equivalently , @xmath85=0 $ ] , @xmath82 . thus @xmath86 if and only if there is a computational basis @xmath59 such that @xmath85\neq 0 $ ] , where @xmath87 . for a pure bipartite entangled state @xmath88 with a s initial coherence being zero , we have presented an operational way to determine whether @xmath12 on system b can create system a s coherence . in fact , from the proof we have also given the explicit form of the quantum operation that can not create coherence of system a. such operation @xmath28 has the form : @xmath89 , where @xmath90s are arbitrary real numbers in @xmath91 $ ] , @xmath92 . theorem [ th2 ] shows the necessary and sufficient condition for quantum operations that create nonzero coherence . now we study how much rcc can be created and the relation between rcc and the entanglement between a and b. [ le1 ] under a quantum operation @xmath93 , the remote created coherence for a pure bipartite state @xmath88 is bounded by @xmath94 where @xmath95 is the @xmath6 norm coherence , @xmath96 is the entanglement measure , concurrence , @xmath97 is the probability of getting the state @xmath80 , @xmath98 and @xmath99 is the matrix elements under @xmath88 s schmidt decomposition s basis of system b. from equations ( [ cl1 ] ) and ( [ purerhoap ] ) we have , @xmath100 by cauchy inequality @xmath101 , we have @xmath102 on the other hand , the concurrence of the state ( [ psiab ] ) is given by @xmath103)}=\sqrt{2 \sum_{i\neq j } \omega_i \omega_j}$ ] , which together with ( [ crhoap ] ) and ( [ neq1 ] ) complete the proof . under the operation @xmath18 acting on the system b , one interesting thing is that the upper bound of rcc ( [ bound1 ] ) is proportional to the entanglement while inversely proportional to the probability of getting the final state . here , one should note that @xmath99s depend on the local schmidt decomposition basis , hence the bound ( [ bound1 ] ) not only depends on the entanglement between a and b , but also the local basis of the system b , as the coherence is a basis dependent quantity . next , we investigate the relations among the average rcc under an operation on the system b for maximally entangled states , that for an arbitrarily given pure quantum state and the entanglement of this given state . consider a pure bipartite entangled state @xmath88 and a trace preserving quantum channel @xmath104 , @xmath105 . under the channel @xmath18 on the system b , the state @xmath88 becomes @xmath106 with probability @xmath107 $ ] . bob communicates each outcome to alice such that , under the channel @xmath18 , alice can gain an average coherence over all the outcomes , @xmath108/p_n')$ ] . [ th3 ] for a pure bipartite entangled state @xmath88 with zero coherence of system a , under a trace preserving quantum channel @xmath104 on system b , the average created coherence of system a satisfies the following relation : @xmath109 where @xmath110 is the maximal entangled state in the schmidt decomposition basis of @xmath88 , @xmath111 and @xmath112 are the average coherence of systems a under the channel @xmath18 for states @xmath88 and @xmath110 respectively , @xmath95 is the @xmath6 norm coherence , @xmath113 is the dimension of system a and @xmath96 is the concurrence . by setting @xmath114 , @xmath115 , in ( [ psiab ] ) we get the maximally entangled state with respect to @xmath88 s schmidt basis , latexmath:[\[\label{beta } @xmath113 is the dimension of system a. under the channel @xmath18 on system b , the final states of system a corresponding to @xmath88 and @xmath110 are given by @xmath117/p_n'\ ] ] and @xmath118/p_n'',\ ] ] where @xmath107 $ ] and @xmath119 $ ] are the probabilities of getting the states @xmath120 and @xmath121 respectively . employing equation ( [ purerhoap ] ) , we get the coherence @xmath122 and @xmath123 where @xmath62 and @xmath124 . utilizing the following relation @xmath125 and lemma [ le1 ] , we get the relation @xmath126 since @xmath18 is trace preserving , the average coherence satisfies @xmath127 which gives rise to the relation ( [ bound2 ] ) . theorem 3 shows that for a bipartite state @xmath88 with initial zero coherence of system a , under a tracing preserving channel of system b , the average increasing of the coherence of system a is bounded by the entanglement between systems a and b , and the average coherence of system a for the maximally entangled state @xmath110 under the same channel for the system b. in fact , ( [ bound2 ] ) is also valid for the system going through some non trace - preserving channels with certain probabilities . let @xmath128 be a set of quantum operations such that @xmath129 , i.e. , each @xmath130 is not a trace - preserving channel , but all @xmath130 together is . if the system b of a given state @xmath61 and the maximally entangled state @xmath110 go through the operation @xmath130 with probability @xmath131 $ ] and @xmath132 $ ] , respectively , then one can prove that the average coherence @xmath133/p_k')$ ] and @xmath134/p_k'')$ ] satisfy the relation ( [ bound2 ] ) . here , if each @xmath130 is given by one kraus operator , @xmath135 , then one recovers the result for a trace preserving channel . it should be noticed here that for the remote creation of the averaged coherence , the one - way classical communication is required . otherwise , one would end up with super - luminal signaling . _ remark _ averaging the @xmath130s for equation ( [ bound1 ] ) we can also obtain a tighter bound for the average rcc . @xmath136 where @xmath137 and @xmath138 , with @xmath139 as given in ( [ beta ] ) . this inequality provides a tighter bound than ( [ bound2 ] ) and could be used to estimate the average rcc for a pure entangled state and a set of operations . [ fig1 ] pure states with the form of equation ( [ psiab ] ) in @xmath140 system , of which the average rccs are under the phase damping channel @xmath141 acting on the system b. the average rccs , @xmath111 ( blue dots ) , get larger as the phase damping rate @xmath142 increases ( the different darkness of blue dots stands for different values of @xmath142 ) , while the ratio between the average rcc of a state and the average rcc of its corresponding maximal entangled state , @xmath143 ( red dots ) , is always equal to its entanglement for all @xmath142.,title="fig:",width=264 ] nevertheless , the theorem [ th3 ] gives a more explicit relation among the entanglement , rccs of the given state and the corresponding maximally entangled state . in particular , for two - qubit case , we have the following theorem : [ th4 ] for a pure @xmath144 entangled state @xmath88 with zero coherence of system a , under a trace preserving quantum channel @xmath104 on system b , the average rcc of @xmath88 equals to the product of the entanglement of @xmath88 and the average rcc of @xmath88 s corresponding maximally entangled state @xmath110 , i.e. , @xmath145 the proof for the above can be derived by noting that , in @xmath144 case , the summation in ( [ neq1 ] ) and ( [ neq2 ] ) only contains one entry which leads to that the equality holds in theorem [ th3 ] . the above factorization law manifests that , in a @xmath140 system , if the operations satisfy the nonzero rcc condition in theorem [ th2 ] , then the average rcc of the state is proportional to its entanglement . since the entanglement is smaller than 1 , its average rcc is always smaller than its corresponding maximal entangled state s average rcc . fig . 1 shows the relations between the average rcc and entanglement for @xmath144 pure states under phase damping channel . for higher dimensional systems , while the upper bound of the average rcc for a state is proportional to its entanglement , depending on the operations one chooses , its average rcc can exceeds its corresponding maximally entangled state s average rcc . here it should be noted that the coherence depends on the reference basis , while the entanglement is local unitary invariant . hence we fix the same basis for the maximally entangled state @xmath110 to the one of @xmath88 s schmidt basis . besides quantum entanglement , similar relations like ( [ bound2 ] ) or ( [ bound3 ] ) may also exist for other quantum correlations like quantum discord , as theorem [ th1 ] implies that all the non - incoherent - quantum states are useful for rcc , other quantum correlations could be responsible for rcc either . in conclusion , we have studied the coherence creation for a system a with zero initial coherence , with the help of quantum operations on another system b that is correlated to a and one - way classical communication . we have found that all the non - incoherent quantum states can be used for rcc and all the incoherent - quantum states can not . for pure states , the necessary and sufficient condition of rcc for the quantum operations on system b has been presented . the upper bound of average remote created coherence has been derived , which shows the relation among the entanglement and rcc of the given quantum state , and the rcc of the corresponding maximally entangled state . moreover , for two - qubit systems , a simple factorization law for the average remote created coherence has been given . * acknowledgments * we thank b. chen , y. k. wang and s. h. wang for useful discussions . this work is supported by the nsfc under numbers 11401032 , 11275131 , 11175094 , 11675113 and 91221205 , and the national basic research program of china ( 2015cb921002 ) . f. london and h. london , proc . a * 149 * , 71 ( 1935 ) . a. b. migdal , nucl . phys . * 13 * , 655 ( 1959 ) . l. mandel and e. wolf , _ optical coherence and quantum optics _ ( cambridge university press , cambridge , england , 1995 ) . j. aberg , phys . . lett . * 113 * , 150402 ( 2014 ) . g. s. engel , t. r. calhoun , e. l. read , t. k. ahn , t. manal , y. c. cheng , r. e. blakenship , and g. r. fleming , natrue ( london ) * 446 * , 782 ( 2007 ) . r. horodecki , p. horodecki , m. horodecki , and k. horodecki , rev . phys . * 81 * , 865 ( 2009 ) . k. modi , a. brodutch , h. cable , t. paterek , and v. vedral , rev . phys . * 84 * , 1655 ( 2012 ) . m. a. nielsen and l. chuang , _ quantum computation and quantum information _ ( cambridge university press , cambridge , england , 2000 ) . v. giovannetti , s. lloyd , and l. maccone , science * 306 * , 1330 ( 2004 ) . r. demkowicz - dobrzanski and l. maccone , phys . lett . * 113 * , 250801 ( 2014 ) . t. sashki , y. yamamoto , and m. koashi , natrue * 509 * , 475 ( 2014 ) . j. k. asboth , j. calsamiglia , and h. ritsch , phys . rev . lett . * 94 * , 173602 ( 2005 ) . a. streltsov , u. singh , h. s. dhar , m. n. bera , and g. adesso , phys . lett . * 115 * , 020403 ( 2015 ) . t. baumgratz , m. cramer , and m. b. plenio , phys . lett . * 113 * , 140401 ( 2014 ) . t. r. bromley , m. cianciaruso , and g. adesso , phys . lett . * 114 * , 210401 ( 2015 ) . e. chitambar , a. streltsov , s. rana , m. n. bera , g. adesso , and m. lewenstein , phys . lett . * 116 * , 070402 ( 2016 ) . y. yao , x. xiao , l. ge , and c. p. sun , phys . a * 92 * , 022112 ( 2015 ) . m. n. bera , t. qureshi , m. a. siddiqui , and a. k. pati , phys . a * 92 * , 012118 ( 2015 ) . d. girolami , phys . * 113 * , 170401 ( 2014 ) . a. mani and v. karimipour , phys . a * 92 * , 032331 ( 2015 ) . x. yuan , h. zhou , z. cao , and x. f. ma , phys . a * 92 * , 022124 ( 2015 ) .

we study remote creation of coherence ( rcc ) for a quantum system a with the help of quantum operations on another system b and one way classical communication . we show that all the non - incoherent quantum states are useful for rcc and all the incoherent - quantum states are not . the necessary and sufficient conditions of rcc for the quantum operations on system b are presented for pure states . the upper bound of average rcc is derived , giving a relation among the entanglement ( concurrence ) , the rcc of the given quantum state and the rcc of the corresponding maximally entangled state . moreover , for two - qubit systems we find a simple factorization law for the average remote created coherence .

the cabibbo - kobayashi - maskawa ( ckm ) matrix element @xmath8 , the coupling of the @xmath9 quark to the @xmath10 quark , is a fundamental parameter of the standard model . it is one of the smallest and least known elements of the ckm matrix . with the increasingly precise measurements of decay - time - dependent @xmath11 asymmetries in @xmath12-meson decays , in particular the angle @xmath13 @xcite , improved measurements of the magnitude of @xmath8 will allow for stringent experimental tests of the standard model mechanism for @xmath11 violation @xcite . this is best illustrated in terms of the unitarity triangle , the graphical representation of the unitarity condition for the ckm matrix , for which the length of the side that is opposite to the angle @xmath13 is proportional to @xmath14 . the extraction of @xmath14 is a challenge , both theoretically and experimentally . experimentally , the principal challenge is to separate the signal @xmath15 decays from the 50 times larger @xmath16 background . this can be achieved by selecting regions of phase space in which this background is highly suppressed . in the rest frame of the @xmath12 meson , the kinematic endpoint of the electron spectrum is @xmath17 for the dominant @xmath18 decays and @xmath19 for @xmath20 decays . thus the spectrum above 2.3 @xmath21is dominated by electrons from @xmath20 transitions . this allows for a relatively precise measurement , largely free from background , in a 300 @xmath22interval that covers approximately 10% of the total electron spectrum for charmless semileptonic @xmath12 decays . in the @xmath23 rest frame , the finite momenta of the @xmath12 mesons cause additional spread of the electron momenta of @xmath24@xmath22 , extending the endpoints to higher momenta . the weak decay rate for @xmath25 can be calculated at the parton level . it is proportional to @xmath26 and @xmath27 , where @xmath28 refers to the @xmath9-quark mass . to relate the semileptonic decay rate of the @xmath12 meson to @xmath14 , the parton - level calculations have to be corrected for perturbative and non - perturbative qcd effects . these corrections can be calculated using various techniques : heavy quark expansions ( hqe ) @xcite and qcd factorization @xcite . both approaches separate perturbative from non - perturbative expressions and sort terms in powers of @xmath29 . hqe is appropriate for the calculations of total inclusive @xmath12 decay rates and for partial @xmath12 decay rates integrated over sufficiently large regions of phase space where the mass and momentum of the final state hadron are large compared to @xmath30 . qcd factorization is better suited for calculations of partial rates and spectra near the kinematic boundaries where the hadronic mass is small . in this region the spectra are affected by the distribution of the @xmath9-quark momentum inside the @xmath12 meson @xcite , which can be described by a structure or shape function ( sf ) , in addition to weak annihilation and other non - perturbative effects . extrapolation from the limited momentum range near the endpoint to the full spectrum is a difficult task , because the sf can not be calculated . to leading order , the sf should be universal for all @xmath31 transitions ( here @xmath32 represents a light quark ) @xcite . several functional forms for the sf , which generally depend on two parameters related to the mass and kinetic energy of the @xmath9-quark , @xmath33 or @xmath28 , and @xmath34 or @xmath35 , have been proposed . the values and precise definitions of these parameters depend on the specific ansatz for the sf , the mass renormalization scheme , and the renormalization scale chosen . in this paper , we present a measurement of the inclusive electron momentum spectrum in charmless semileptonic @xmath12 decays , averaged over charged and neutral @xmath12 mesons , near the kinematic endpoint . we report measurements of the partial branching fractions in five overlapping momentum intervals . the upper limit is fixed at 2.6 @xmath21 , while the lower limit varies from 2.0 @xmath21to 2.4 @xmath21 . by extending the interval for the signal extraction down to 2.0 @xmath21 , we capture about 25% of the total signal electron spectrum , but also much larger @xmath36 backgrounds . inclusive measurements of @xmath14 have been performed by several experiments operating at the @xmath37 resonance , namely argus @xcite , cleo @xcite , @xcite , and belle @xcite , and experiments operating at the @xmath38 resonance , namely l3 @xcite , aleph @xcite , delphi @xcite , and opal @xcite . this analysis is based on a method similar to the one used in previous measurements of the lepton spectrum near the kinematic endpoint @xcite . the results presented here supersede those of the preliminary analysis reported by the collaboration @xcite . the extraction of @xmath14 relies on two different theoretical calculations of the differential decay rates for @xmath39 and @xmath5 : the original work by defazio and neubert ( dn ) @xcite , and kagan and neubert @xcite , and the more comprehensive recent calculations by bosch , lange , neubert , and paz ( blnp ) @xcite . the dn calculations allow for the extrapolation of the observed partial @xmath0 decay rate above a certain electron momentum to the total inclusive @xmath0 decay rate using the measured sf parameters and a subsequent translation of the total decay rate to @xmath14 . the theoretical uncertainties on the rate predictions are estimated to be of order 1020% . the blnp authors have presented a systematic treatment of the sf effects , incorporated all known corrections to the differential decay rates , and provided an interpolation between the hqe and the sf regions . they have also performed a detailed analysis of the theoretical uncertainties . the calculations directly relate the partial decay rate to @xmath14 . while the calculations by blnp are to supersede the earlier work by dn , we use both approaches to allow for a direct comparison of the two calculations , and also a comparison with previous measurements based on the dn calculations . we adopt the sf parameters extracted by the collaboration : for the dn method we rely on the photon spectrum in @xmath5 decays @xcite ; for the more recent blnp method , we also use sf parameters derived from the photon spectrum , its moments , the hadron - mass and lepton - energy moments in inclusive @xmath6 decays @xcite , and the combination of all moments measured by the collaboration @xcite . the data used in this analysis were recorded with the detector at the 2 energy - asymmetric @xmath40 collider . the data sample of 88 million events , corresponding to an integrated luminosity of 80.4 @xmath41 , was collected at the resonance . an additional sample of 9.5 @xmath41 was recorded at a center - of - mass ( c.m . ) energy 40 @xmath42below the resonance , _ i.e. _ just below the threshold for production . this off - resonance data sample is used to subtract the non- contributions from the data collected on the resonance . the relative normalization of the two data samples has been derived from luminosity measurements , which are based on the number of detected @xmath43 pairs and the qed cross section for @xmath44 production , adjusted for the small difference in center - of - mass energy . the detector has been described in detail elsewhere @xcite . the most important components for this study are the charged - particle tracking system , consisting of a five - layer silicon detector and a 40-layer drift chamber , and the electromagnetic calorimeter assembled from 6580 csi(tl ) crystals . these detector components operate in a @xmath45-@xmath46 solenoidal magnetic field . electron candidates are selected on the basis of the ratio of the energy detected in the calorimeter to the track momentum , the calorimeter shower shape , the energy loss in the drift chamber , and the angle of the photons reconstructed in a ring - imaging cherenkov detector . the electron identification efficiency and the probabilities to misidentify a pion , kaon , or proton as an electron have been measured @xcite as a function of the laboratory momentum and angles with clean samples of tracks selected from data . within the acceptance of the calorimeter , defined by the polar angle in the laboratory frame , @xmath47 , the average electron identification efficiency is @xmath48 . the average hadron misidentification rate is about 0.1% . we use monte carlo ( mc ) techniques to simulate the production and decay of @xmath12 mesons , and the detector response @xcite , to estimate signal and background efficiencies , and to extract the observed signal and background distributions . the simulated sample of generic events exceeds the data sample by about a factor of three . information from studies of selected control data samples on efficiencies and resolutions is used to improve the accuracy of the simulation . comparisons of data with the mc simulations have revealed small differences in the tracking efficiencies , which have been corrected for . no significant impact of non - gaussian resolution tails has been found for high momentum tracks in the endpoint region . the mc simulations include radiative effects such as bremsstrahlung in the detector material and qed initial and final state radiation @xcite . adjustments for small variations of the beam energy over time have also been included . in the mc simulations the branching fractions for hadronic @xmath12 and @xmath49 decays are based on values reported in the review of particle physics @xcite . the simulation of charmless semileptonic decays , @xmath50 , is based on a heavy quark expansion to @xmath51 @xcite . this calculation produces a continuous spectrum of hadronic states . the hadronization of @xmath52 with masses above @xmath53 is performed by jetset @xcite . the motion of the @xmath9 quark inside the @xmath12 meson is implemented with the sf parameterization given in @xcite . three - body decays to low - mass hadrons , @xmath54 , are simulated separately using the isgw2 model @xcite and mixed with decays to non - resonant and higher mass resonant states @xmath55 , so that the cumulative distributions of the hadron mass , the momentum transfer squared , and the electron momentum reproduce the hqe calculation as closely as possible . the generated electron spectrum is reweighted to accommodate variations due to specific choices of the sf parameters . the mc - generated electron - momentum distributions for @xmath20 decays are shown in fig . [ f : bu_pe ] , for individual decay modes and for their sum . here and throughout the paper , the electron momentum and all other kinematic variables are measured in the rest frame , unless stated otherwise . above 2@xmath21 , the principal signal contributions are from decays involving the light mesons @xmath56 , and @xmath57 , and also some higher mass resonant and non - resonant states @xmath55 . decays : @xmath58 , @xmath59 , @xmath60 , @xmath61 , @xmath62 , the sum of @xmath12-meson decay modes to non - resonant and higher - mass resonance states ( @xmath63 ) , and the sum of all decay modes ( all ) . the spectra are normalized to a total rate of 1.0 . , height=245 ] for the simulation of the dominant @xmath64 decays , we have chosen a variety of models . for @xmath65 and @xmath66 decays we use parameterizations @xcite of the form factors , based on heavy quark effective theory ( hqet ) . decays to pseudoscalar mesons are described by a single form factor @xmath67 , where the variable @xmath68 is the scalar product of the @xmath12 and @xmath49 meson four - vector velocities and is equal to the relativistic boost of the @xmath49 meson in the @xmath12 meson rest frame . the linear slope @xmath69 has been measured by the cleo @xcite and belle @xcite collaborations . we use the average value , @xmath70 . the differential decay rate for @xmath71 can be described by three amplitudes , which depend on three parameters : @xmath72 , @xmath73 , and @xmath74 . we adopt values recently measured by @xcite : @xmath75 , @xmath76 , and @xmath77 . here the parameter @xmath72 is the slope assuming a linear dependence of the form factor on the variable @xmath68 . the quoted errors reflect the statistical and systematic uncertainties . we use the isgw2 @xcite model for various decays to higher - mass @xmath78 resonances . we have adopted a prescription by goity and roberts @xcite for the non - resonant @xmath79 decays . the shapes of the mc - generated electron spectra for individual @xmath64 decays are shown in fig . [ f : sp1 ] . above 2 @xmath21the principal background contributions are from decays involving the lower - mass charm mesons , @xmath80 and @xmath49 . higher - mass and non - resonant charm states are expected to contribute at lower electron momenta . the relative contributions of the individual @xmath64 decay modes are adjusted to match the data by a fit to the observed spectrum ( see below ) . decay modes : @xmath81 , @xmath82 , @xmath83 , @xmath84 , and @xmath20 , and the sum of all decay modes ( all ) . the signal @xmath50 spectrum is shown for comparison . the spectra are normalized to a total rate of 1.0 . , height=245 ] we select events with a semileptonic @xmath12 decay by requiring an electron with momentum @xmath85 . to reject electrons from the decay @xmath86 , we combine the electron candidate with any second electron of opposite charge and reject the combination , if the invariant mass of the pair falls in the interval @xmath87 . to suppress background from non- events , primarily low - multiplicity qed ( including @xmath88 pairs ) and @xmath89 processes ( here @xmath32 represents any of the @xmath90 or @xmath91 quarks ) , we veto events with fewer than four charged tracks . we also require that the ratio of the second to the zeroth fox - wolfram moment @xcite , @xmath92 , not exceed @xmath93 . @xmath92 is calculated including all detected charged particles and photons . for events with an electron in the momentum interval of 2.0 to 2.6 @xmath21 , these two criteria reduce the non- background by a factor of about 6 , while the loss of signal events is less than 20% . in semileptonic @xmath12 decays , the neutrino carries sizable energy . in events in which the only undetected particle is this neutrino , the neutrino four - momentum can be inferred from the missing momentum , @xmath94 , the difference between the four - momentum of the two colliding - beam particles , and the sum of the four - momenta of all detected particles , charged and neutral . to improve the reconstruction of the missing momentum , we impose a number of requirements on the charged and neutral particles . charged tracks are required to have a minimum transverse momentum of 0.2 @xmath21and a maximum momentum of 10 @xmath21 in the laboratory frame . charged tracks are also restricted in polar angle to @xmath95 and they are required to originate close to the beam - beam interaction point . the individual photon energy in the laboratory frame is required to exceed 30 @xmath42 . the selection of semileptonic @xmath12 decays is enhanced by requiring @xmath96 , and that @xmath97 points into the detector fiducial volume , @xmath98 , thereby effectively reducing the impact of particle losses close to the beams . furthermore , since in semileptonic @xmath12 decays with a high - momentum electron , the neutrino and the electron are emitted preferentially in opposite directions , we require that the angle @xmath99 between these two particles fulfill the condition @xmath100 . these requirements for the missing momentum reduce the continuum background from qed processes and @xmath89 production by an additional factor of 3 , while the signal loss is less than 20% . the stated selection criteria result in an efficiency ( including effects of bremsstrahlung ) of @xmath101% for selecting @xmath50 decays ; its dependence on the electron momentum is shown in fig . [ fig : p0 ] . decays as a function of the electron momentum . the error bars represent the statistical errors . , height=264 ] the spectrum of the highest momentum electron in events selected by the criteria described above is shown in fig . [ fig : p1]a , separately for data recorded on and below the resonance . the data collected on the @xmath23 resonance include contributions from events and non- background . the latter is measured using off - resonance data , collected below production threshold , and using on - resonance data above 2.8 @xmath21 , _ i.e. , _ above the endpoint for electrons from @xmath12 decays . the background to the @xmath102 spectrum is estimated from mc simulation , with the normalization of the individual contributions determined by a fit to the total observed spectrum . rest frame : ( a ) on - resonance data ( open circles blue ) , scaled off - resonance data ( solid circles green ) ; the solid line shows the result of the fit to the non- events using both on- and off - resonance data ; ( b ) on - resonance data after subtraction of the fitted non- background ( triangles blue ) compared to simulated background that is adjusted by the combined fit to the on- and off - resonance data ( histogram ) ; ( c ) on - resonance data after subtraction of all backgrounds ( linear vertical scale , data points red ) , compared to the simulated @xmath50 signal spectrum ( histogram ) ; the error bars indicate errors from the fit , which include the uncertainties in the fitted scale factors for non- and @xmath103 backgrounds . the shaded area indicates the momentum interval for which the on - resonance data are combined into a single bin for the purpose of reducing the sensitivity of the fit to the shape of the signal spectrum in this region . , height=321 ] to determine the non- background we perform a @xmath104 fit to the off - resonance data in the momentum interval of 1.1 to 3.5 @xmath21and to on - resonance data in the momentum interval of 2.8 to 3.5 @xmath21 . since the c.m . energy for the off - resonance data is 0.4% lower than for the on - resonance data , we scale the electron momenta for the off - resonance data by the ratio of the c.m . energies . the relative normalization for the two data sets is @xmath105 where @xmath106 and @xmath107 refer to the c.m . energy squared and integrated luminosity of the two data sets . the statistical uncertainty of @xmath108 is determined by the number of detected @xmath43 pairs used for the measurement of the integrated luminosity ; the systematic error of the ratio is estimated to be @xmath109 . the @xmath104 for the fit to the non- events is defined as follows , @xmath110 here @xmath111 and @xmath112 refer to the number of selected events in the off- and on - resonance samples , for the @xmath113-th or @xmath114-th momentum bin ( @xmath115 ) , and @xmath116 is the set of free parameters of the fit . for the function approximating the momentum spectrum , we have chosen an exponential expression of the form @xmath117 the fit describes the data well : @xmath118 for 58 degrees of freedom . above 2.8 @xmath21 , we observe @xmath119 events in the on - resonance data , compared to the fitted number of @xmath120 events . the electron spectrum from @xmath12-meson decays is composed of several contributions , dominated by the various semileptonic decays . hadronic @xmath12 decays contribute mostly via hadron misidentification and secondary electrons from decays of @xmath49 , @xmath121 , and @xmath122 mesons . we estimate the total background by fitting the observed inclusive electron spectrum to the sum of the signal and individual background contributions . for the individual signal and background contributions , we use the mc simulated spectra , and treat their relative normalization factors as free parameters in the fit . the non- background is parameterized by the exponential functions @xmath123 , as described above . we expand the @xmath104 definition as follows , @xmath124 where the first sum is for the off - resonance data and the second sum for the on - resonance data . the electron spectrum is approximated as @xmath125 , where the free parameters @xmath126 are the correction factors to the mc default branching fractions for the six individual contributions @xmath127 representing the signal @xmath128 decays , the background @xmath129 , @xmath130 , @xmath131 , @xmath132 decays , and the sum of other background events with electrons from secondary decays or misidentified hadrons . @xmath133 is the statistical error of the number of simulated events in the @xmath114-th bin . the momentum spectra @xmath127 are histograms taken from mc simulations . lrrrrr @xmath134 ( @xmath21 ) & 2.0 2.6 & 2.1 2.6 & 2.2 2.6 & 2.3 2.6 & 2.4 2.6 + total sample & 609.81 @xmath135 0.78 & 295.76 @xmath135 0.54 & 133.59 @xmath135 0.37 & 65.48 @xmath135 0.26 & 35.38 @xmath135 0.19 + non- background & 142.38 @xmath135 0.63 & 105.20 @xmath135 0.48 & 74.86 @xmath135 0.36 & 50.13 @xmath135 0.25 & 29.96 @xmath135 0.16 + @xmath103 background & 416.22 @xmath135 2.52 & 157.17 @xmath135 1.29 & 38.82 @xmath135 0.47 & 4.00 @xmath135 0.10 & 0.09 @xmath135 0.01 + @xmath121 and @xmath122 & 6.17 @xmath135 0.14 & 4.00 @xmath135 0.10 & 2.33 @xmath135 0.06 & 1.17 @xmath135 0.04 & 0.47 @xmath135 0.02 + other @xmath136 background & 1.61 @xmath135 0.05 & 0.62 @xmath135 0.02 & 0.24 @xmath135 0.01 & 0.08 @xmath135 0.01 & 0.03 @xmath135 0.00 + @xmath137 mis - identification & 1.34 @xmath135 0.04 & 0.98 @xmath135 0.03 & 0.64 @xmath135 0.02 & 0.34 @xmath135 0.02 & 0.10 @xmath135 0.01 + @xmath138 mis - identification & 0.47 @xmath135 0.02 & 0.26 @xmath135 0.01 & 0.13 @xmath135 0.01 & 0.05 @xmath135 0.01 & 0.01 @xmath135 0.00 + other mis - identification & 0.27 @xmath135 0.01 & 0.15 @xmath135 0.01 & 0.08 @xmath135 0.01 & 0.04 @xmath135 0.01 & 0.02 @xmath135 0.00 + @xmath139 background & 1.62 @xmath135 0.10 & 0.66 @xmath135 0.05 & 0.20 @xmath135 0.02 & 0.03 @xmath135 0.01 & 0.01 @xmath135 0.00 + @xmath140 signal & 39.72 @xmath135 2.70 & 26.72 @xmath135 1.49 & 16.31 @xmath135 0.71 & 9.64 @xmath135 0.38 & 4.70 @xmath135 0.25 + @xmath139 efficiency ( % ) & 42.1 @xmath135 0.3 & 41.2 @xmath135 0.4 & 40.2 @xmath135 0.5 & 39.5 @xmath135 0.7 & 37.9 @xmath135 1.0 + the fit is performed simultaneously to the on- and off - resonance electron momentum spectra in the range from 1.1 to 3.5 @xmath21 , in bins of 50@xmath22 . the lower part of the spectrum determines the relative normalization of the various background contributions , allowing for an extrapolation into the endpoint region above 2.0 @xmath21 . to reduce a potential systematic bias from the assumed shape of the signal spectrum , we combine the on - resonance data for the interval from 2.1 to 2.8 @xmath21into a single bin . the lower limit of this bin is chosen so as to retain the sensitivity to the steeply falling background distributions , while containing a large fraction of the signal events in a region where the background is low . the fit results are insensitive to changes in this lower limit in the range of 2.0 to 2.2 @xmath21 . the number of signal events in a given momentum interval is taken as the excess of events above the fitted background . the observed spectra , the fitted non- and backgrounds and the signal are shown and compared to mc simulations in fig . [ fig : p1 ] . the fit has a @xmath104 of @xmath141 for 73 degrees of freedom . above 2.3 @xmath21 , the non- background is dominant , while at low momenta the semileptonic background dominates . contributions from hadron misidentification are small , varying from 6% to 4% as the electron momentum increases . the theoretical prediction for the signal @xmath50 spectrum based on the blnp calculations uses sf parameters extracted from the combined fit @xcite to the moments measured by the collaboration . the fitting procedure was chosen in recognition of the fact that currently the branching fractions for the individual @xmath6 decays are not well enough measured to perform an adequate background subtraction . the mc simulation takes into account the form factor and angular distributions for the @xmath142 and @xmath143 decays . for decays to higher - mass mesons , this information is not available . as a result , we do not consider this fit as a viable method of measuring these individual branching fractions . nevertheless , the fitted branching fractions agree reasonably well with the measured branching fractions @xcite . for the decays to higher - mass states , the ability of the fit to distinguish between decays to @xmath144 and @xmath145 is limited . the sum of the two contributions , however , agrees with current measurements @xcite . table [ table : r1 ] shows a summary of the data , principal backgrounds and the resulting signal . the errors are statistical , but for the non- and @xmath103 background they include the uncertainties of the fitted parameters . the data are shown for five overlapping signal regions , ranging in width from 600 to 200 @xmath22 . we choose 2.6 @xmath21 as the common upper limit of the signal regions because at higher momenta the signal contributions are very small compared to the non- background . as the lower limit is extended to 2.0 @xmath21 , the error on the background subtraction increases . a summary of the systematic errors is given in table [ table : t2 ] for five intervals in the electron momentum . the principal systematic errors originate from the event selection and the background subtraction . the uncertainty in the event simulation and its impact on the momentum dependence of the efficiencies for signal and background are the experimental limitations of the current analysis . the second largest source of uncertainties is the estimate of the background derived from the fit to the observed electron spectrum , primarily due to the uncertainties in the simulated momentum spectra of the various contributions . in addition , there are relatively small corrections to the momentum spectra due to variations in the beam energies , and radiative effects . . summary of the relative systematic errors ( % ) on the partial branching fraction measurements for @xmath50 decays , as a function of @xmath146 , the lower limit of the signal momentum range . the common upper limit is 2.6 @xmath21 . the sensitivity of the signal extraction to the uncertainties in the sf parameters is listed as an additional systematic error , separately for the four sets of sf parameters . [ cols="<,^,^,^,^,^",options="header " , ] the errors listed for @xmath147 and @xmath14 are specified as follows . the first error reflects the error on the measurement of @xmath148 , which includes statistical and experimental systematic uncertainties , except for the uncertainty in the sf parameters . the second error is due to experimental uncertainty of sf parameters affecting both @xmath149 and @xmath148 . the third error is the theoretical uncertainty of @xmath149 . the fourth error on @xmath14 accounts for the theoretical uncertainty in the translation from @xmath147 to @xmath14 , as specified in eq . 5 . this error also depends on the @xmath9-quark mass and thus is correlated with the theoretical uncertainty on the sf . the results for the total branching fraction @xmath150 and @xmath14 obtained from the different momentum intervals are consistent within the experimental and theoretical uncertainties . for intervals extending below 2.3@xmath21 , the total errors on @xmath147 and @xmath14 do not depend very strongly on the chosen momentum interval . while the errors on @xmath148 are smallest above the kinematic endpoint for @xmath36 decays , the dominant uncertainty arises from the determination of the fraction @xmath151 and increases substantially with higher momentum cut - offs . the stated theoretical errors on @xmath151 , acknowledged as being underestimated @xcite , do not include uncertainties from weak annihilation and other power - suppressed corrections . assuming that one can combine the experimental and theoretical errors in quadrature , the best measurement of the total branching fraction is obtained for the momentum interval @xmath2@xmath152 @xmath21 . though the measurement of the photon spectrum @xcite results in the best estimate for the sf parameters , we have also considered sets of sf parameters obtained from photon spectra measured by the cleo @xcite and belle @xcite collaborations . these parameters are listed in table [ table : sf_all ] . in table [ table : comparison ] the results obtained for these different sf parameters based on the semileptonic data and on the dn calculations are listed for the momentum interval @xmath153 . the differences between the sf parameters obtained by the cleo and belle collaborations and the results are comparable to the experimental errors on these parameters . these differences affect the signal spectrum , and thereby the fitted background yield . the effect is small for high momentum region and increases for the signal intervals extending to lower momenta . the impact of the sf parameters on the partial branching fractions is included in the total error ( see table [ table : t2 ] ) . lrll & @xmath154 ( @xmath155 ) & @xmath156 ( @xmath157 ) + ' '' '' & ( spectrum ) @xmath158 @xcite & @xmath159 & @xmath160 + ' '' '' cleo & ( spectrum ) @xmath158 @xcite & @xmath161 & @xmath162 + ' '' '' belle & ( spectrum ) @xmath158 @xcite & @xmath163 & @xmath164 + & @xmath154 ( @xmath155 ) & @xmath165 ( @xmath157 ) + ' '' '' & ( spectrum ) @xmath158 @xcite & @xmath166 & @xmath167 + ' '' '' & ( moments ) @xmath158 @xcite & @xmath168 & @xmath169 + ' '' '' & ( moments ) @xmath170 @xcite & @xmath171 & @xmath172 + ' '' '' & ( comb1 . moments ) @xcite & @xmath173 & @xmath174 + llccc experiment & sf input & @xmath175 & @xmath176 & @xmath177 + ' '' '' & @xmath158 ( spectrum ) & @xmath178 & @xmath179 & @xmath180 + ' '' '' cleo & @xmath158 ( spectrum ) & @xmath181 & @xmath182 & @xmath183 + ' '' '' belle & @xmath158 ( spectrum ) & @xmath184 & @xmath185 & @xmath186 + ' '' '' & @xmath158 ( spectrum ) & @xmath187 & @xmath188 & @xmath189 + ' '' '' & @xmath158 ( moments ) & @xmath190 & @xmath191 & @xmath192 + ' '' '' & @xmath170 ( moments ) & @xmath193 & @xmath194 & @xmath195 + ' '' '' & combined fit to moments & @xmath196 & @xmath197 & @xmath198 + for the four sets of sf parameters ( see table iv ) based on the calculations of blnp , extracted from the photon energy spectrum ( short dash - red ) and from the photon energy moments ( dot - dash - green ) in @xmath5 , from the lepton energy and hadron mass moments in @xmath36 decays ( long dash - black ) , as well as from the combined fit to moments ( solid - blue ) measured by the collaboration . also shown are two straight lines indicting values of the sf parameters , for which the partial branching fraction ( dotted - magenta ) and @xmath14 ( solid - light blue ) are constant . , height=283 ] the second method for extracting @xmath14 is based on recent blnp calculations @xcite . in this framework the partial branching fraction @xmath148 is related directly to @xmath14 : @xmath199 where @xmath200 is the prediction for the partial rate for @xmath39 decays ( in units of @xmath201 ) . in these calculations the leading order sf is constrained by the hqe parameters , obtained either from the @xmath5 or @xmath36 decays , or both . the values of the sf parameters extracted from the analyses of inclusive @xmath202 @xcite , @xmath203 @xcite decays , and the combined fit @xcite to all moments measured by the collaboration are listed in table [ table : sf_all ] . note that the definitions of shape functions and the sf parameters are different for the dn and blnp calculations . the different sf parameters and their measurement errors are also shown in fig . [ fig : p4 ] . the sf parameters based on @xmath5 data only are extracted from either a fit to the photon spectrum or to the first and second moments of this spectrum in the `` shape function '' scheme . the hqe parameters extracted from fits to measured moments in the kinetic mass scheme have been translated into the `` shape function '' scheme at the appropriate scale . specifically , the hqe parameters extracted from the moments in @xmath36 decays have been translated based on two - loop calculations @xcite . the hqe parameters resulting from the combined fit to moments of the photon , lepton , and hadron mass spectra in the kinetic scheme are used to predict the first and second moments of the photon spectrum down to photon energies of 1.6@xmath204 , based on calculations by benson , bigi , and uraltsev @xcite . the lower limit on the photon energy is chosen such that the estimated cut - induced perturbative and non - perturbative corrections to the hqe are negligible . from these predicted moments , the sf parameters are extracted using the next - to - leading order calculations in a framework that is consistent with the one used for the determination of @xmath14 @xcite . the smallest errors on the sf parameters are obtained from the fit to the photon spectrum and the combined fit to all moments . the fit to the photon spectrum is most sensitive to the high end of the photon energy spectrum , and relies on the theoretical prediction for the shape of the spectrum down to low photon energies . since this shape is not directly calculable , several forms of the sf are used to assess the uncertainty of this approach . the use of two sets of the first and second moments of the photon spectrum , above 1.90 and above 2.09 @xmath204 , is less powerful , due to much larger statistical and systematic errors , but insensitive to the theoretical knowledge of the detailed shape of the spectrum . the sf parameters obtained from moments of the photon spectrum above 1.90 @xmath21agree with those obtained from the global fit to the moments , but also have larger errors . nevertheless , the inclusion of the photon energy moments significantly improves the sensitivity of the global fit to more than 30 measured moments . the results for the partial branching fractions @xmath148 and @xmath14 based on the blnp calculations are listed in tables [ table : br2f_bbrxsg_direct ] , [ table : br2f_bbrxsgm_direct ] , [ table : br2f_bbrxcenu_direct ] , and [ table : br2f_cf_direct ] for the four sets of sf parameters . ccrc @xmath205 ( @xmath21 ) & @xmath175 & @xmath206 & @xmath177 + ' '' '' @xmath207 & @xmath208 & @xmath209 & @xmath210 + ' '' '' @xmath211 & @xmath212 & @xmath213 & @xmath214 + ' '' '' @xmath215 & @xmath216 & @xmath217 & @xmath218 + ' '' '' @xmath219 & @xmath220 & @xmath221 & @xmath222 + ' '' '' @xmath223 & @xmath224 & @xmath225 & @xmath226 + ccrc @xmath205 ( @xmath21 ) & @xmath175 & @xmath206 & @xmath177 + ' '' '' @xmath207 & @xmath227 & @xmath228 & @xmath229 + ' '' '' @xmath211 & @xmath230 & @xmath231 & @xmath232 + ' '' '' @xmath215 & @xmath233 & @xmath234 & @xmath235 + ' '' '' @xmath219 & @xmath236 & @xmath237 & @xmath238 + ' '' '' @xmath223 & @xmath224 & @xmath239 & @xmath240 + ccrc @xmath205 ( @xmath21 ) & @xmath175 & @xmath206 & @xmath241 + ' '' '' @xmath207 & @xmath242 & @xmath243 & @xmath244 + ' '' '' @xmath211 & @xmath245 & @xmath246 & @xmath247 + ' '' '' @xmath215 & @xmath248 & @xmath249 & @xmath250 + ' '' '' @xmath219 & @xmath236 & @xmath251 & @xmath252 + ' '' '' @xmath223 & @xmath224 & @xmath253 & @xmath254 + ccrc @xmath205 ( @xmath21 ) & @xmath175 & @xmath255 & @xmath177 + ' '' '' @xmath207 & @xmath256 & @xmath257 & @xmath258 + ' '' '' @xmath211 & @xmath259 & @xmath260 & @xmath261 + ' '' '' @xmath215 & @xmath262 & @xmath263 & @xmath264 + ' '' '' @xmath219 & @xmath236 & @xmath265 & @xmath266 + ' '' '' @xmath223 & @xmath224 & @xmath267 & @xmath268 + ccc @xmath205 ( @xmath21 ) & @xmath149 & @xmath176 + ' '' '' @xmath207 & @xmath269 & @xmath270 + ' '' '' @xmath211 & @xmath271 & @xmath272 + ' '' '' @xmath215 & @xmath273 & @xmath274 + ' '' '' @xmath219 & @xmath275 & @xmath276 + ' '' '' @xmath223 & @xmath277 & @xmath278 + the errors cited in these tables are defined and determined in analogy to those in table [ table : br2f_bbrxsgdfn ] . the first error on the predicted rate @xmath279 accounts for the uncertainty due to the errors in measured parameters of the leading sf , the second error refers to the theoretical uncertainties in the subleading sfs , and variations of scale matching , as well as weak annihilation effects . for @xmath14 , the first error is the experimental error on the partial branching fraction , which includes the statistical and the experimental systematic uncertainty , the second error includes systematic uncertainties on the partial branching fraction and @xmath279 due to the uncertainty of the sf parameters , and the third error is the theoretical uncertainty on @xmath279 , estimated using the prescription suggested by blnp . in table [ table : comparison ] the results obtained for these different sf parameters based on the semileptonic data and on the blnp ( and dn ) calculations are listed for the momentum interval @xmath153 . the observed differences are consistent with the total error stated ; they are largest for the sf parameters extracted from the fit to the photon spectrum as compared to the moments of the photon spectrum . for all four sets of sfs we observe a tendency for the total branching fraction , and therefore also @xmath14 , to be slightly larger at the higher momentum intervals , but the uncertainties in the predicted rates @xmath279 are very large for the highest momentum interval . based on the blnp calculations @xcite of the inclusive lepton spectra , we have also determined the total @xmath280 branching fraction . the results are presented in table [ table : br2f_cf ] . the results for @xmath14 extracted for the blnp calculations are close to those obtained for the dn calculations ( see table [ table : comparison ] ) . in fact , the results based on the fit to the photon spectrum measured by the collaboration are identical for all electron momentum ranges , even though the partial branching fractions differ by one standard deviation of the experimental error ( see tables iii and vi ) . changing the ansatz for the sf from the exponential to a hyperbolic function @xcite has no impact on the results . in summary , we have measured the inclusive electron spectrum in charmless semileptonic @xmath12 decays and derived partial branching fractions in five overlapping electron momentum intervals close to the kinematic endpoint . we have extracted the partial and total branching fractions and the magnitude of the ckm element @xmath14 based on two sets of calculations : the earlier ones by defazio and neubert @xcite and kagan and neubert @xcite , and the more comprehensive calculations by lange , neubert and paz @xcite , as summarized in table v. within the stated errors , the measurements in the different momentum intervals are consistent for both sets of calculations . we adopt the results based on the more recent calculations ( blnp ) @xcite , since they represent a more complete theoretical analysis of the full electron spectrum and relate the sf to the hqe parameters extracted from inclusive @xmath281 and @xmath282 decays . we choose the sf parameters obtained from the combined fit to moments of inclusive distributions measured by the collaboration rather than the single most precise measurement of the sf parameters obtained from the recent measurement @xcite of the semi - inclusive photon spectrum in @xmath281 decays . assuming it is valid to combine the experimental and the estimated theoretical errors in quadrature , and taking into account the fraction of the signal contained in this interval , we conclude that the best measurement can be extracted from the largest momentum interval , @xmath2 to @xmath3 . for this momentum interval the partial branching fraction is @xmath283 here the first error is statistical and the second is the total systematic error , as listed in table [ table : t2 ] . in addition to the systematic uncertainty due to the signal extraction , the normalization , and various small corrections , this error also includes the observed dependence of the extracted signal on the choice of the sf parameters . based on the blnp method , we obtain a total branching fraction of @xmath284 and @xmath285 here the first error represents the total experimental uncertainty , the second refers to the uncertainty in the sf parameters from the combined fit to moments , and the third combines the stated theoretical uncertainties in the extraction of @xmath14 , including uncertainties from the subleading sfs , weak annihilation effects , and various scale - matching uncertainties . no additional uncertainty due to the theoretical assumption of quark - hadron duality has been assigned . the improvement in precision compared to earlier analyses of the lepton spectrum near the kinematic endpoint can be attributed to improvements in experimental techniques , to higher statistics , and in particular , to improved background estimates , as well as significant advances in the theoretical understanding of the sfs and extraction of the sf parameters from inclusive spectra and moments . while earlier measurements were restricted to lepton energies close to the kinematic endpoint for @xmath6 decays at 2.3 @xmath21and covered only 10% of the @xmath286 spectrum , these and other more recent measurements have been extended to lower momenta , including about 25% of the spectrum , and thus have resulted in a significant reduction in the theoretical uncertainties on @xmath14 . the determination of @xmath14 is currently limited primarily by our knowledge of sf parameters . an approximate linear dependence of @xmath14 on these parameters is @xmath287 for @xmath288 and @xmath289 . thus the uncertainty on the @xmath9-quark mass dominates . it should be noted that this dependence on @xmath33 is a factor of two smaller for measurements based on the dn calculations . these results are in excellent agreement with earlier measurements of the inclusive lepton spectrum at the resonance , but their overall precision surpasses them @xcite . the earlier results were based on the dn calculations . we observe that for the same experimental input , i.e. the same measured lepton and photon spectra , the extracted values of @xmath14 based on dn calculations agree very well with those based on bnlp calculations for the various momentum ranges under study , even though the corresponding partial branching fractions may differ by one standard deviation . the results presented here are also comparable in precision to , and fully compatible with , inclusive measurements recently published by the @xcite and belle @xcite collaborations , based on two - dimensional distributions of lepton energy , the momentum transfer squared and the hadronic mass , with sf parameters extracted from @xmath5 and @xmath6 decays . we would like to thank the cleo and belle collaborations for providing detailed information on the extraction of the shape function parameters from the photon spectrum in @xmath290 transitions . we are also indebted to m. neubert and his co - authors b. lange , g. paz , and s. bosch for providing us with detailed information on their calculations . we are grateful for the extraordinary contributions of our 2 colleagues in achieving the excellent luminosity and machine conditions that have made this work possible . the success of this project also relies critically on the expertise and dedication of the computing organizations that support . the collaborating institutions wish to thank slac for its support and the kind hospitality extended to them . this work is supported by the us department of energy and national science foundation , the natural sciences and engineering research council ( canada ) , institute of high energy physics ( china ) , the commissariat lenergie atomique and institut national de physique nuclaire et de physique des particules ( france ) , the bundesministerium fr bildung und forschung and deutsche forschungsgemeinschaft ( germany ) , the istituto nazionale di fisica nucleare ( italy ) , the foundation for fundamental research on matter ( the netherlands ) , the research council of norway , the ministry of science and technology of the russian federation , and the particle physics and astronomy research council ( united kingdom ) . individuals have received support from conacyt ( mexico ) , the a. p. sloan foundation , the research corporation , and the alexander von humboldt foundation . m. shifman and m. voloshin , sov . j. nucl . phys . * 41 * , 120 ( 1985 ) ; j. chay , h. georgi , and b. grinstein , phys . b * 247 * , 399 ( 1990 ) ; i. i. bigi and n. uraltsev , phys . b * 280 * , 271 ( 1992 ) ; a. v. manohar and m. b. wise , phys . d * 49 * , 1310 ( 1994 ) ; b. blok , l. koyrakh , m. shifman and a. i. vainshtein , phys . rev . d * 49 * , 3356 ( 1994 ) . collaboration , b. aubert _ et al . _ , _ measurement of the inclusive electron spectrum in charmless semileptonic b decays , _ contributions to ichep02 , amsterdam ( 2002 ) , hep - ex/0207081 ; collaboration , b. aubert _ et al . _ , _ determination of the partial branching fraction for @xmath291 and of @xmath14 from the inclusive electron spectrum near the kinematic endpoint , _ contribution to ichep04 , beijing ( 2004 ) , hep - ex/0408075 . collaboration , b. aubert _ et al . _ , _ measurement of the partial branching fraction for inclusive charmless semileptonic @xmath12 decays and the extraction of @xmath14 _ , hep - ex/0507017 , contribution to the int . symposium of lepton - photon interactions , uppsala ( 2005 ) .

we present a measurement of the inclusive electron spectrum in @xmath0 decays near the kinematic limit for @xmath1 transitions , using a sample of 88 million pairs recorded by the detector at the resonance . partial branching fraction measurements are performed in five overlapping intervals of the electron momentum ; for the interval of @xmath2@xmath3 we obtain @xmath4 . combining this result with shape function parameters extracted from measurements of moments of the inclusive photon spectrum in @xmath5 decays and moments of the hadron mass and lepton energy spectra in @xmath6 decays we determine @xmath7 . here the first error represents the combined statistical and systematic experimental uncertainties of the partial branching fraction measurement , the second error refers to the uncertainty of the determination of the shape function parameters , and the third error is due to theoretical uncertainties in the qcd calculations . -pub-05/45 , + slac - pub-11499 +

because of its closeness , the sun is the best - observed and the only spatially - resolved x - ray star . it is therefore natural to consider it as a template and a guide to analyze and interpret what we observe of the other stars . in this perspective two previous works ( orlando et al . 2000 , peres et al . 2000a , hereafter paper i and paper ii , respectively ) illustrate a method to put solar x - ray data collected by the _ soft x - ray telescope _ on board yohkoh into the same format and framework as stellar x - ray data . the method allows us to simulate accurately the observation of the sun at stellar distances with a stellar instrument and to apply to the relevant data the same analysis as if they were real stellar data : we can compare homogeneously stellar to solar data and use the latter as a template for stellar observations . paper i focussed on the details of the treatment of yohkoh data and paper ii outlined the method in its generality and showed some representative applications to observations in different phases of the solar cycle plus one flare case . the present work applies this analysis to several solar flares in the perspective to interpret stellar flares and some features of very active stellar coronae . coronal flares are transient , x - ray bright and localized events : they last from few minutes to several hours , they easily overcome the luminosity of the whole solar corona and they occur in relatively small regions , often in single coronal magnetic structures ( loops ) . as such their occurrence and evolution is mostly independent of the structure and evolution of the rest of the corona . since the phenomenology , duration and intensity can be very different from one flare to another it is difficult to take a single flare as representative . in this work we analyze the emission measure distribution vs. temperature and its evolution during some selected solar flares , representative of the wide range of possible events ; as we did in paper ii we then use these em(t ) s to synthesize relevant stellar - like spectra , which are then analyzed with standard analysis tools of stellar coronal physics . together with the general problem of the structure and heating of the solar corona , flares represent an unsolved puzzle of solar physics . their early impulsive phase is so fast that the trigger mechanism remains elusive , in spite of the high quality of the data collected with many instruments , and in particular with the yohkoh / sxt , optimized to observe flares . apart from experimental limitations , intrinsic physical reasons make the diagnostics of the flare heating very difficult : the flare starts when a low density plasma is heated at more than @xmath5 k ; due to the low emission measure , initially the plasma is not easily observable while thermal conduction is so efficient that it smoothes out in a few seconds any trace of local thermal perturbation . the rise of the brightness in the soft x - ray band , where flares are best - observed , does not follow the evolution of the heating ( e.g. reale & peres 1995 ) . while analysing the rise phase is useful to investigate the mechanisms which originate the flares , the decay phase has been shown to be useful in other respects . it has long been known that the duration of the decay is linked to the size of the flaring region , because the thermal conduction cooling time depends on the length of the flaring loops . this has direct implications on stellar flares : from the observed light curve it has been possible to estimate the size of the unresolved stellar flaring loops . on the other hand it has also been shown that the estimates based on simple scalings from the conduction or radiation cooling times only can be largely incorrect : indeed , significant heating can be present ( jakimiec et al . 1992 ) and sustain the solar and stellar flare decays ( reale et al . 1997 , reale & micela 1998 , schmitt and favata 1999 , favata et al . 2000 , maggio et al . 2000 ) longer than expected , therefore leading to very coarse loop lengths overestimates . a useful tool to identify the presence of such a prolonged heating is the flare density - temperature diagram : the longer the heating decay ( compared to the free cooling loop decay time , serio et al . 1991 ) , the smaller is the slope of the decay in this diagram ( sylwester et al . 1993 ) . knowing the decaying light curve and the path in the density - temperature diagram leads us to obtain reliable estimates of the flaring loop length and of the heating in the decay phase ( reale et al . 1997 ) . the analysis of stellar flares have been often based on loop models ( e.g. reale et al . 1988 , van den oord et al . 1988 , van den oord & mewe 1989 , reale & micela 1998 ) . the present study , instead , uses solar flares really observed and spatially resolved by the yohkoh / sxt as templates : we will apply here the method illustrated in papers i and ii to obtain the evolving em(t ) and stellar - like flare spectra as observed by non - solar instruments , in particular the _ solid - state imaging spectrometer _ ( sis ) on board the _ advanced satellite for cosmology and astrophysics _ ( asca ) and the _ position sensitive proportional counter _ ( pspc ) on board the _ rntgen satellit _ ( rosat ) , which have been among the most used x - ray observatories for the observation of stellar coronae . it is interesting to apply the standard stellar spectra analysis to the resulting flare spectra and see analogies and differences from really observed stellar flares . the advantages of this approach are that : i ) it is relatively model - independent ( only spectral models have to be included ) ; ii ) it analyzes the whole flaring region including contributions from structures adjacent but outside the dominant ( if any ) loop , or , in the case of arcade flares , from many loops . spectra of several intense ( and therefore yielding good count statistics ) stellar flares collected by rosat ( e.g. preibisch et al . 1993 , ottman et al . 1996 ) and especially by asca ( e.g. gotthelf et al . 1994 , gdel et al . 1999 , osten et al . 2000 , hamaguchi et al . 2000 , tsuboi et al . 2000 ) have been analyzed in the recent past . in order to yield a count statistics appropriate for a sound analysis , each spectrum is typically integrated on time intervals spanning significant fractions of the flare , therefore averaging on rapidly evolving plasma and emission conditions . in spite of this , spectrum models of isothermal plasma ( 1-t models ) are generally able to describe each spectrum ( subtracted by the quiescent `` background '' spectrum ) with temperatures and emission measures following expected solar - like trends . the peak temperatures of analyzed stellar flares are quite higher , and their emission measures orders of magnitude higher ( @xmath6 and @xmath7 @xmath8 , respectively ) , than typical solar ones . deviations from the single temperature description have been also detected ( ottmann & schmitt 1996 , favata et al . 2000 ) . it is interesting in this context to investigate if there is correspondence with analogous spectra synthesized from solar flares which may then be used as templates to interpret stellar flare spectra . a debated question about stellar flare spectra is the fact that many of them are better fit by allowing the metal abundance to vary in the fitting model . the spectra are acceptably fit with metal abundance increasing in the rising phase and then decreasing gradually in the decay ( ottman et al . 1996 , favata et al . the physical meaning of this result is still far from being settled . wherever possible , it has also been shown that the abundance variations seem to change element by element ( osten et al . 2000 , gdel et al . 1999 ) . although the problem of abundance variations in stellar coronae is intriguing , far from being settled and addressed by several investigations with xmm - newton ( _ x - ray multi - mirror mission _ ) and chandra satellites ( e.g. evidence for an inverse first ionization potential effect has been found on the star hr1099 , brinkman et al . 2001 ) , the approach presented here does not allow to investigate it properly . what we can do is to explore the effect of allowing abundances to vary in fitting spectra originating from parent multi - thermal emission measure distributions . in this study beyond the extrapolation of the solar flares to the stellar environment as isolated , self - standing and evolving events , we will also focus on their possible contribution to make up the emission of very active coronae . in this respect , evidence has been collected from multi - line xuv observations that the emission measure of some active stellar coronae has two peaks , one at a few @xmath9 k and the other at @xmath10 k and it has been proposed that the higher temperature peak is due to a continuous flaring activity ( e.g. gdel 1997 , drake et al . 2000 ) . paper ii addressed , although limitedly , the evidence from extensive rosat observations that solar - like stars cover an extended region in the plane flux / hardness ratio ( schmitt 1997 ) , while the sun , as a whole , spans the low hr - low flux part of the same region . in this context it is worth investigating the region occupied by flares . this work is structured as follows : in section [ sec : data ] , we describe the solar data and our analysis for the derivation of the flare em(t ) s and of the stellar - like spectra ; section [ sec : results ] shows the results obtained for our sample of yohkoh / sxt flares , the related stellar - like spectra , collected with asca / sis and rosat / pspc , and their analysis with standard stellar methods ; in section [ sec : discuss ] we discuss our results and draw our conclusions . in order to study the em(t ) distribution of solar flares , we have selected a sample of flares well - observed for most of their duration by yohkoh / sxt and covering a wide range of flare intensities and physical conditions . in particular we have selected eight flares ranging from relatively weak ( class c5.8 ) to very intense flares ( x9.0 ) . the flares and their sxt observations are listed in table [ tab : fl_list ] , along with the date and time of the flare beginning as measured with the _ geosynchronous operational environmental satellite _ ( goes ) , the goes class , the duration of the flare as in the goes log file , and the start , maximum emission measure and end times of the sxt observations . the observations monitor large fractions of the rise , peak and decay of the thermal phase of the flares . the analyzed data include the sxt data taken in flare mode , and , in particular , in the two filter passbands specific for flare mode observations , i.e. be 119 @xmath11 m and al 11.4 @xmath11 m . as discussed in section [ sec : emt ] , additional data taken in two other softer filters , al.1 and almg , are included to analyze flare 4 . the sxt data have been processed according to the standard yohkoh analysis system . the datasets consist of sequences of frames @xmath12 pixels of 2.5 arcsec side , taken alternatively in the two filter bands , with sampling cadence usually ranging between @xmath13 sec ( typically in the rise phase and around the flare peak ) and @xmath14 sec in late decay phase of long - lasting flares . the analysis then includes the derivation of temperature and em maps , as in papers i and ii , during the flare evolution . since the plasma conditions may change significantly in the time taken to switch between the two filters , , the data in the al 11.4 @xmath11 m filter band have been interpolated to the exact times of the be 119 @xmath11 m data ( as done routinely in the standard yohkoh data analysis ) , in order to improve the accuracy of the temperature and emission measure estimates . the distributions of em vs t ( em(t ) ) are then derived from the t and em maps with the procedure illustrated in paper i and ii . in any of the two filter bands we have screened out saturated pixels , typically in the brightest regions , and pixels which collected photons below a threshold of 10 photons . these pixels would introduce large errors and are therefore critical in the analysis of em vs t for localized events like flares . we have carefully selected for our analysis only those frames with no ( or just very few ) saturated pixels . the temperature bins for the em(t ) are those defined in papers i and ii , i.e. 29 bins uniform in @xmath15 between log t = 5.5 and log t = 8 . from each of the several em@xmath16 distributions of a flare obtained during its evolution we synthesize the relevant x - ray spectrum with the mekal spectral code ( kaastra 1992 ; mewe et al . 1995 ) , and filter it through the instrumental response of non - solar telescopes of interest . the process to generate the stellar - like spectra from an em@xmath16 distribution derived from the solar data is described in paper ii . metal abundances are assumed as in anders & grevesse ( 1989 ) . in this paper , we consider the stellar - like focal plane spectra that _ asca_/sis and _ rosat_/pspc would collect , observing the solar flares selected here at stellar distances . it would be very interesting to perform finely time - resolved spectroscopy , but in real stellar observations this is prevented by the limited counting statistics of the source . analogously to stellar observations , therefore , the flare spectra have to be integrated on time intervals of hundreds of seconds to increase the overall statistics of the fitting process . this means that the flares are binned into few time intervals . since flares evolve on time scales smaller or of the same order , the data collected in a time bin do not represent steady plasma conditions but integrate on significant variations of the plasma temperature and density in the same dataset . therefore when time - resolved flare data are analyzed with steady - state models , one should keep anyhow in mind that this is an _ a priori _ limited representation and description of the data . in order to approach the typical conditions of non - solar flare observations we make the following exercise : 1 . we sample the em(t ) s of a flare at constant time intervals of 60 sec 2 . from each em(t ) we synthesize the corresponding spectrum filtered through the spectral response and effective area of two instruments , namely asca / sis and rosat / pspc 3 . we bin the flare into three ( or more ) long time segments , ( at least ) one including the rise phase , one the flare maximum and one the decay phase . the duration of the time segments varies within each flare and from flare to flare , depending on the flare duration and on the data structure ( the presence of gaps , for instance ) , and span from a minimum of 180 s to a maximum of @xmath17 s. 4 . each spectrum is normalized so as to yield a total number of counts in each time interval typical of good stellar flare observations ( e.g. between 1000 and 10000 counts for asca ) ; to this end we have to assume the distance at which the solar flare would yield the appropriate statistics . we randomize the photon counts according to poisson statistics 5 . we sum all folded and randomized spectra within each time bin and therefore obtain a single spectrum for each time bin . we analyze each spectrum with the standard thermal models used for stellar data analysis . we apply the standard stellar analysis to the binned spectra and use the tools commonly used by the stellar community . in particular , we fit the stellar - like spectra with multiple - isothermal components models . in the rosat / pspc case , we exclude channels with less than three counts from the analysis to grant an appropriate evaluation of @xmath18 , and the first two channels because they are typically affected by systematic errors . in the asca / sis case , the energy channels are grouped so to have at least 20 photon counts per channel ; furthermore the channels with bad quality or empty are discarded . we use the x - ray spectral fitting package xspec v10.0 and the manipulation task ftools v4.0 . most fittings are performed with single temperature components , assuming negligible column density @xmath19 , and standard metal abundances kept fixed . some checks have been done by allowing metal abundance to vary all by the same amount . all flares listed in table [ tab : fl_list ] last between 9 and 90 min , except the huge flare 8 , of class x9 , whose duration is more than 6 hours ( despite the goes log reports less than 1 hour ) . the sxt observation of flare 8 is divided into 3 segments separated by two gaps lasting more than one and two hours , respectively . the first segment begins just around the flare maximum , and therefore most of the flare rise phase is not covered by sxt . we have selected anyhow this flare because it is a rather extreme case of intense flare and the information from the rise phase is not crucial for our findings , as described below . in all the other cases the related sxt observation covers quite well both the rise and the decay phase and the flare maximum is well within the observation . only for flare 7 ( class x1.5 ) the decay phase is monitored for a relatively short time . as part of our analysis we investigate the role of the morphology of the flaring region in determining the em(t ) , and the effect of the heating release , and in particular of its intensity , duration and decay time , on the evolution of the em(t ) . in this perspective we tag each flare with the light curve in both filters of sxt flare - mode , its path in the density - temperature ( hereafter _ n - t _ ) diagram ( the square root of the emission measure is used as proxy for the density ) , and the main morphological features observable in the sxt images . [ fig : lcnt ] and fig . [ fig : lcnt2 ] show the light curves in the al 11.4 @xmath11 m and be 119 @xmath11 m filter bands obtained by summing all the counts in each 64x64 pixels sxt frame , and the corresponding n - t diagrams , obtained from the ratio of the light curves data points . notice that the resulting temperature is a weighted average temperature of the whole region in each frame . flare 2 is particularly well - covered since the total luminosity at the end has decayed to the values at the beginning ; this corresponds to a closed cycle in the n - t diagram . the n - t diagram also shows that this is the only flare in which the temperature changes significantly during the rise phase ( from @xmath20 to @xmath21 ) . this may indicate that the heating which triggers the flare is released more gradually in the rise phase of this flare than in the others ( e.g. sylwester et al . 1993 ) . during the other six flares for which the rise phase has been observed , the temperature is in fact much more constant and stays above @xmath5 k during the rise phase . as for the density , yohkoh / sxt has detected an increase of more than half a decade of em@xmath22 for five flares ( flare 2 , 4 , 5 , 6 and 7 ) ; during the decay of flare 8 the decrease of em@xmath22 is particularly significant , i.e. almost one order of magnitude . table [ tab : fl_par ] shows relevant physical and morphological characteristics of the selected flares obtained with the flare mode filters , i.e. the slope of the flare decay path in the @xmath23 diagram , the decay time of the light curve in the al 11.4 @xmath11 m filter band ( the latter two quantities are used to estimate the flaring loop length according to reale et al . 1997 ) , conservative loop half - length ranges obtained from measuring the loop projections on the images and from applying the method of reale et al . ( 1997 ) , the morphology of the flaring region ( fig . [ fig : frames ] shows one grey - scale image sampled during each flare ) , the maximum temperature and emission measure obtained with the filter ratio method from the ratio of the data points of the light curves in the two flare - mode filter bands . the slope @xmath24 is an indicator of sustained heating during the decay , whenever significantly smaller than @xmath25 ( sylwester et al . 1993 ) . from table [ tab : fl_par ] we see that : * heating is negligible during the decay of three flares ( 1 , 3 and 6 ) , significant during the decay of the other five and in particular of the most intense ones ( 7 and 8) . the slope @xmath24 of the whole decay of flare 8 is below the minimum possible value ( @xmath26 ) predicted by single loop hydrodynamic modeling ( reale et al . 1997 ) , which means that this model is not applicable in this case , and that , therefore , significant magnetic rearrangements and complicated and continued heating release probably occur . * the light curve e - folding decay time is below 30 min for all flares except flare 4 , an arcade flare , and flare 8 . * the loop half - length is in the range 10 mm and 100 mm , typical of active region loops is to be taken as an indicative scale size . the morphology of the flaring regions ranges from simple single loops to multiple loops , an arcade and even more complicated structures . * the maximum values of the average temperatures , as measured with yohkoh filter - band ratio , are in the range @xmath27 , weakly increasing with the goes class . * the maximum total emission measure increases with the flare goes class and spans two orders of magnitudes from @xmath28 to @xmath29 @xmath8 . the above considerations make us confident that this sample of flares is enough representative of flare conditions typically met on the sun and can be used to derive general properties to be compared to those observed in stellar flares . [ fig : emt ] shows the evolution of the em(t ) obtained with the two hardest sxt filters , during the eight selected flares . for each flare we show the emission measure distributions sampled at a constant rate of one every 2 min since the beginning of the data selected as in section [ sec : data ] . the longer the flare , the more are the em(t ) s shown : those shown for the short flares 1 and 7 are much fewer than those shown for flare 8 , by far the longest one . for reference , [ fig : emt ] shows also the em(t ) s obtained in paper ii for the sun near the maximum and the sun near the minimum of the cycle . [ fig : emt ] shows that at any flare phase the em(t ) is typically quite narrow , practically independent of the flaring region morphology , and covers a temperature decade around @xmath10 k. it is much narrower than any em(t ) of the whole sun . exceptions are the em(t ) s obtained for flare 8 , the most intense one , which shows significant amounts of hotter plasma , at temperatures up to @xmath30 k. the em(t ) reaches for this flare maximum values @xmath31 @xmath8 . hot plasma at temperatures above 30 mk is present in flares more intense than m1.1 ( flare 3 ) . in fig . [ fig : emt ] the em(t ) of all flares clearly follows a common evolution path : it starts low but already at a relatively high temperature , centered at @xmath10 k ; it grows toward higher em values , maintaining a more or less constant shape and sometimes shifting slightly rightwards to higher temperatures ( e.g. flares 2 , 4 and 6 ) ; then it decays by gradually cooling ( leftwards ) and decreasing ( downwards ) . we can clearly identify envelopes of the evolving em(t ) s and notice that the slope of the envelopes in the decay phase ( determined by the relative rate of the emission measure decrease and the temperature decrease ) is linked to the slope in the n - t diagram ( see fig . [ fig : lcnt ] and fig . [ fig : lcnt2 ] ) . several em(t ) distributions late during the decay are partially hidden by the higher preceding ones ( e.g. flares 3 , 4 and 5 ) . this clearly shows that the em(t ) mostly evolves inside a common envelope " in the rise and the decay phase , with important implications on the interpretation of some stellar coronal em(t ) s ( see section [ sec : discuss ] ) . the em(t ) s shown in fig . [ fig : emt ] are derived from observations in the hardest sxt filters bands . since these filters are most sensitive to plasma around and above @xmath5 k , one may wonder whether contributions from plasma at lower temperatures , not detected by the two filters , may be important to the flare em(t ) , or not . the observation of flare 4 includes several frames taken in the al.1 and almg filter bands and allows us to investigate this item . [ fig : emt_4 ] shows the em(t ) s of flare 4 at the same times as those shown in fig.[fig : emt ] , but including contributions to the em(t ) s derived from images in the softer filters bands mentioned above . cooler contributions to the em(t ) s are clearly present from comparison with fig.[fig : emt ] . these contributions make each em(t ) flatter at temperatures below the em(t ) maxima ( @xmath10 k ) ; the slope in this rising region approaches the @xmath32 trend expected from loop structures ( see peres et al . 2001 ) . although the presence of such contributions modify the shape of the em(t ) s in the low temperature part , they are anyhow significantly smaller than the dominant components around @xmath5 k , and we have checked that the analysis of stellar spectra synthesized from em(t ) s with and without such contributions does not change significantly , as further discussed in section [ sec : spectra ] . since very few flares yielded good data in softer filters bands ( saturated pixels are much more frequent ) , and contributions from such bands are anyhow affected by uncertainties and contaminations from the coexistence of cool and hot plasma within the same pixel , we prefer to perform the analysis considering expositions only in the two hardest filters and keeping anyhow in mind the limitations that this choice implies . all em(t ) s obtained for flares are well separated and distinct from the em(t ) of the sun at minimum . they are also mostly `` higher '' than the em(t ) of the sun at minimum . the em(t ) s of the first three flares are instead all lower than the em(t ) of the sun at maximum , while the em(t ) s of the remaining five flares are comparable or higher . if we did the exercise , similar to that in paper ii , to build a single em(t ) of the flaring sun at maximum by combining that of the sun at maximum and any one of the latter flares , we would invariably obtain an em(t ) with two distinct peaks , the hotter one being associated with the flare . flare 8 involves emission measures quite higher than the emission measure of the full non - flaring sun at maximum ( see paper ii ) . [ fig : maxemt ] shows the maxima of the em(t ) s of all of the eight flares . this figure indicates that an increasing intensity in the goes class mostly corresponds to an increase in the em(t ) height and much less to an increase in temperature and/or in the em(t ) width . an exception is flare 8 which is significantly hotter and wider . from each em(t ) obtained during the flares we can synthesize the corresponding x - ray parent spectra for an optically thin plasma in thermal equilibrium , as described in paper ii . [ fig : spectra ] shows examples of x - ray spectra in the 0.2 - 20 kev band , synthesized from em(t ) at the beginning , peak and end of flare 2 and at the peak and data end of flare 8 . the maximum luminosities in the whole band result to be @xmath33 erg / s and @xmath34 erg / s , respectively . from fig . [ fig : spectra ] we see concentrations of emission lines around 1 kev , typical of plasma mostly at temperature around 10@xmath35 k , and mostly due to the fe - l complex . the spectrum at the peak of flare 8 shows prominent lines of the fe group at energies around 6 kev , which are sensibly reduced at the end of the data ( lower panel , dashed line ) . notice that the continuum of the spectrum at the peak of flare 8 is considerably flatter than the others shown and indicates that there are contributions of plasma at significantly hotter temperatures than in all other synthesized spectra shown . [ fig : spectra ] shows asca / sis spectra of flare 4 . these spectra have been analyzed by fitting them with spectral models consisting of one isothermal plasma component ( _ 1-t fit _ ) . table [ tab : fl_asca ] shows the results of the spectral analysis applied to four of the eight selected flares ( 2 , 4 , 6 , and 8) . for each flare , the table includes the distance at which we put the flaring sun , and , for each time bin , the phase of the flare it covers , the time range , the number of counts , the best - fit temperature and emission measure , the number of degrees of freedom ( the number of energy channels yielding a significant number of counts minus the parameters of the fitting ) and the reduced @xmath18 of the fitting . table [ tab : fl_asca ] tells us , first of all , that the single thermal component model already provides an acceptable description of the various phases of the flares , in agreement to the rather peaked em(t ) distributions that we obtain from the two hard sxt filters , even averaging over each time bin . in spite of the fast flare evolution , temperature and emission measure variations _ within _ each time bin do not affect the 1-t fitting in the rise phase , mainly for two reasons : i ) the emission measure at the end of the bin in the rise phase is much higher than , and dominates over , that at the beginning ; ii ) generally the temperature does not vary much during the rise phase . conditions are quite stable during the flare maximum and a single temperature therefore describes reasonably well this phase . on the contrary , in spite of the relatively slow evolution , during the decay plasma temperature may vary significantly from the beginning to the end of the same time bin , while the emission measure remains relatively constant ( or slowly decreasing ) . the single temperature component may therefore fail to describe time bins of the decay phase , especially in flares with a steep slope in the n - t diagram . indeed in bins e ) and f ) in the decay phase of flare 6 , the fitting is not as good ( @xmath36 ) as in the other ones or in the other flare decays : the slope of the decay of this flare in the n - t diagram is very high ( @xmath37 ) , higher than in all the other flares . 1-t fitting of spectra taken with asca during stellar flare decays , and subtracted of the spectrum of the background non - flaring corona , have sometimes failed ( e.g. favata et al . 2000 ) , and our results suggest a possible explanation . the best - fit temperature values approximately correspond to the maxima of the em(t ) distribution averaged in the respective time bin and the corresponding emission measure values are approximately proportional to the total emission measure in that bin . the maximum emission measure values are slightly smaller than the values listed in table [ tab : fl_par ] , both because some contributions are excluded by the instrument limited spectral band and because in the respective time bins of table [ tab : fl_asca ] the emission measure is not constantly at maximum . notice that due to scaling with respect to the maximum , the last bin of flare 8 yields only 400 photon counts , since the emission measure is two orders of magnitude smaller than at maximum . the last section of table [ tab : fl_asca ] reports again results for flare 4 but including the components obtained from the softer filters ( see fig.[fig : emt_4 ] ) . comparing to the results without the softer filters , we notice the slightly higher count statistics , the slightly lower best - fit temperature , the slightly larger emission measure , the generally higher @xmath18 . the higher @xmath18 indicates that the single thermal model is not as good as previously , but not to such a point to discriminate the presence of the cooler components , also considering the higher count statistics . finally , we have checked the effect of fitting the simulated asca data with 1-t models in which the global metal abundance is left free to vary , from one spectrum to the other . the resulting best - fit abundance values are generally different from the expected solar value and vary within a factor two , either in excess or in defect , from the central unity value . in the course of a flare we obtain either abundances all smaller than one , or all larger than one , or fluctuating around one . although in a few cases the abundance appears to decrease during the decay , we were not able to identify clear systematic trends of abundance variations during the flare evolution . the best - fit @xmath18 values are improved by 0 - 10% , and in no case by more than 20% , while the temperature values are practically unchanged ( while the emission measure values vary in inverse proportion to the abundance values ) . the detection of such abundance variations are clearly an artifact of the model fitting process , since the synthesized spectra are consistently built by assuming fixed solar metal abundances at any step and can be explained in the following terms : the em(t ) flare distributions are neither sharp enough to be described at best by a simple @xmath38 function ( an isothermal model ) , nor broad enough to require a multi - thermal ( e.g. 2-t ) model . moderate abundance variations are sufficient to account , and adjust the fit , for the presence of minor emission measure components around the dominating maximum component . this exercise tells us that although the limited broadness of the flare em(t ) may favor an improvement of spectral fitting by letting abundance vary , the resulting abundance variations seem to be random ; therefore we can not exclude that systematic abundance variations are to be explained by effects other than the one illustrated here . rosat / pspc has a lower spectral resolution than asca / sis , and is , therefore , less able to discriminate multi - thermal components . in fact , we find that a single thermal component model describes even better the pspc spectra obtained from the flare em(t ) s averaged over the same time bins as those used for asca / sis . we have analysed spectra yielding a maximum between 3000 and 5000 total counts : all fittings were acceptable ( reduced @xmath39 with less than 30 degrees of freedom ) . the temperature and emission measure values are very similar ( with a mean deviation within 10% and 15% , respectively ) to those obtained with asca / sis spectral fitting . on the average , the rosat temperature is slightly systematically lower ( @xmath40% ) than the corresponding asca one , and the emission measure slightly higher ( @xmath41% ) . analogously to paper ii , as an additional piece of analysis of the synthesized rosat / pspc data , we compute the surface x - ray flux @xmath42 , defined as the x - ray luminosity in the rosat / pspc spectral band divided by the pixel area over which the yohkoh / sxt photon count is larger than 10 cts / s , and the hardness ratio defined as in schmitt ( 1997 ) : @xmath43 where @xmath44 are the total counts in the ( soft ) pspc sub - band 0.13 - 0.4 kev and @xmath45 are the total counts in the ( hard ) sub - band 0.55 - 1.95 kev . fig . [ fig : rosat_flux ] shows @xmath42 vs hr for each time bin in which a spectrum has been collected , and for all flares . all data points appear to lie in a relatively thin vertical strip around a hardness ratio value of 0.3 and with @xmath42 ranging from @xmath9 to @xmath46 erg @xmath3 s@xmath4 . the smallest flares evolve mostly in the range @xmath5 to @xmath47 erg @xmath3 s@xmath4 , while the most intense ones seem to move in a wider flux range . the flare evolution has no well defined trend in this plane , mostly because of the very limited variation of the hardness ratio . the almost constant hr is mostly due to two factors : i ) the flare average temperature , of which the hardness ratio is a proxy , does not change much from @xmath48 k , during the flare evolution ; ii ) the hardness ratio , as defined above , is weakly sensitive to temperature variations ( and even multi - valued ) around @xmath49 k. for comparison , fig . [ fig : rosat_flux ] shows the plane region occupied by late - type stars observed by rosat / pspc ( schmitt 1997 ) : solar flares ( isolated from the remaining corona ) , practically in any phase of their evolution , are harder than typical stellar coronal emission and in a region of high surface flux . papers i and ii illustrated a method to use the sun as a template of x - ray stars , converting yohkoh / sxt solar data into corresponding focal plane data collected by non - solar telescopes , such as rosat and asca , from a sun located at stellar distances . this work presents the application of this method to solar flares . in order to sample the various flare conditions , we have selected a set of flares spanning from weak to extremely intense and occurring either in simple loop structures , or in complex regions or arcades . the method is , in general , the same as that used to analyse the full - disk non - flaring observations . however some important differences are in order : * yohkoh sxt flare observations are performed with different characteristics , the so - called flare - mode , which involves mostly the use of two harder filters , double ( full ) spatial resolution and a much higher sampling cadence . * flares evolve on short time scales and the evolution of the emission measure distribution is a mostly important item of the present study . * flares are highly localized in areas of the order of 1/1000 of the solar hemisphere ; although their luminosity is often comparable to that of the whole corona , their evolution is mostly independent of what happens in the rest of the corona , and we study them as self - standing phenomena also from the stellar point of view . the two flare - mode filters are mostly sensitive to plasma at temperatures above @xmath5 k , appropriate for flares ; however their sensitivity to plasma below @xmath5 k is low , and temperature and emission measure diagnostics are not at best . since the plasma is typically stratified inside coronal structures , even during flares , we may then miss contributions from relatively cool plasma components which may be significant , or even dominant , or coexisting with hotter plasma , in some pixels during the flare evolution . the presence of such a cooler plasma has indeed been detected in a flare observed both with flaring mode filters and with standard mode filters ( flare 4 ) : the emission measure distribution during the evolution is not modified for @xmath50 k but additional ( lower ) contributions appear for @xmath51 k making the em(t ) less steep on the cool side of the maximum . this should be kept in mind when discussing the shape of the em(t ) distributions . the slope of the cool side approaches 3/2 , the typical value expected for standard loop structures in equilibrium ( see also peres et al . 2001 ) , coherently to flare evolution being dominated by plasma confined in closed magnetic structures and suggesting that , after the initial ( not bright ) impulsive phases , the dynamics is significantly less important and the flaring plasma is very close to equilibrium conditions . we notice that the shape and evolution of the em(t ) during flares little depend on the detailed geometry of the flare region , as well as on the flare intensity ( except , of course , for the relative em(t ) height ) : in fig.3 one can hardly distinguish flare 4 , an arcade flare , and flare 8 , very intense and complex , from the other flares with simpler geometry . flare 4 and flare 8 last longer than the others , and their temperature decreases very slowly , but these features are not evident in fig.3 . the temperature of the em(t ) maximum is also weakly dependent on the flare intensity : this may depend in part on the flare - mode filters used , which are more sensitive to plasma around that temperature , but also on the strong efficiency of thermal conduction ( @xmath52 ) at higher and higher temperature , which implies a very large energy input even for a small temperature increase , to balance conductive losses to chromosphere . significant amount of plasma at very high temperature ( @xmath53 k ) , comparable to the peak temperatures of intense stellar flares , is detected , in particular during flare 8 . such hot plasma , however , has an overall low emission measure with respect to the dominant relatively cooler plasma at @xmath48 k. our results show that minor components of emission measure hotter than the dominant component described by the single temperature fitting are practically not detectable in low resolution spectra collected by rosat and asca during stellar flares . high resolution spectra collected by chandra and newton may show them ( fig.6 ) . rosat and asca spectra synthesized from our em(t ) s , even integrated over long time segments and at relatively high count statistics , are well - fitted by single isothermal components , at or around the temperature of the em(t ) peak . even when we include additional cooler contributions obtained from the softer sxt filters , 1-t fit is successful , albeit at a slightly lower best - fitting temperature . deviations from isothermal behavior in rosat and asca synthesized spectra are sometimes obtained in the decay phase because of variations of plasma temperature within the same time segment . these results can be used to interpret observations of stellar flares : successful isothermal fittings of time - resolved spectra ( such as those mentioned in section [ sec : intro ] ) detect the dominant component of a multi - component but single - peaked emission measure distribution , and the distributions of the solar flares presented here may be taken for reference . high @xmath18 values during the decay may not be indicative of a multi - temperature distribution of the flaring region , rather of the evolving temperature of the dominant emission measure component . we also notice that the single component fittings , both of rosat and asca spectra , generally well evaluate the total emission measure involved , missing only few percent of the parent emission measure . therefore the emission measure values obtained from the fittings are rather reliable . our approach does not allow us to address exhaustively the problem of metal abundance variations obtained to best - fit the spectra of several stellar flares ; however we made the exercise to fit the stellar - like solar spectra letting the global metal abundance free to vary . we have found that non - solar abundances help to improve somewhat the fitting quality , probably because they better account for the limited broadness of the flare em(t ) s , but they do not seem to vary systematically ( e.g. increase first and then decrease in the decay ) during each flare . therefore , we can not exclude that the evidence of systematic abundance variations requires explanations different from the one suggested by our results . the analysis of the rosat / pspc flare simulated spectra tells us that the solar flares group in the x - ray flux vs hardness ratio diagram , and in particular in a narrow strip spanning two orders of magnitude at relatively high flux values and hardness ratio value between 0.2 and 0.5 . this strip is rightwards ( harder ) , in practice completely outside , of the region occupied by the g stars sampled by schmitt ( 1997 ) , which however pertain to whole coronae , outside flares . this is consistent with both flux and hardness ratio of non - flaring solar - like stellar coronae being well below conditions of typical solar flaring regions , which also agrees with the range of temperatures seen in non - flaring stars vs. stellar or solar flares . we now discuss some major implications of our results on the analysis of stellar coronae . there is evidence that the emission measure distribution of very active stellar coronae , obtained from spectrally resolved xuv observations , is double - peaked ; the first peak is at a few @xmath9 k , and the second peak above @xmath5 k ( griffiths and jordan 1998 ) . this aspect is much debated and still open , but it has been suggested that this hot component may be due to continuous flaring activity ( gdel 1997 , drake et al . 2000 ) : the stellar surface is covered by active regions , flares are very frequent and their light curves overlap , cancelling out any variability due to the single events . in this framework , the present work shows that a double - peaked em(t ) distribution is indeed obtained if one combines the em(t ) of the whole corona to the envelope of the em(t ) s during the flares ( see fig . the two peaks are clearly evident also when summing the em(t ) s of flares from 4 ( m2.8 ) to 8 ( x9 ) to the em(t ) of the sun at maximum of its cycle of activity . this seems to suggest that uninterrupted sequences of overlapping proper flares , whichever their evolution , would not be able to fill the gap between the two em(t ) peaks , which , therefore , would be a permanent feature of the em(t ) of an active star . indeed this second distinct maximum associated with flares is not surprising also on the basis of theoretical modeling . from hydrodynamic modeling of flaring plasma confined in magnetic loops , it has been generally found that the impulsive heating originating the flare first causes a very fast local temperature enhancement above @xmath5 k which propagates along the whole loop in few seconds due to the highly efficient ( even in saturated regimes ) thermal conduction . the loop plasma density ( and therefore emission measure ) increases more slowly and gradually , because determined by evaporation of plasma up from the chromosphere , on typical dynamic time scales ( minutes ) . later , when heating decreases , temperature and density both decay , their relative decay rate dictated by the rate of the heating decrease . _ models never predict that temperature does decrease and density does not _ , which would result into an emission measure distribution maintaining the bell - shape , with little or no change of shape and shifting from high to lower temperature , thus filling the gap between the two em(t ) peaks . this scenario would be at variance with both hydrodynamic models and observations as represented in the density- temperature diagrams shown in fig.1 : for all flares except flare 2 the temperature is high and the emission measure low ( upper left extremes of the paths ) already at the beginning of the observation . in the decay the path is never vertical ( i.e. maintaining a constant em and decreasing t ) , but has a maximum slope ( theoretically found to be less than 2 , jakimiec et al . all this means that the em(t ) during flares , whatever its exact shape , will : 1 . increase starting already from @xmath10 k 2 . cool but also invariably decrease , in the flare decay . the cooling may be much slower than the decrease , if the heating is sustained during the decay , as it seems to occur in many flares ( sylwester et al . 1993 , reale et al . 1997 ) . these considerations and results , in our opinion , show that a major continuous flaring activity on a stellar corona would produce an em(t ) with two distinct peaks , the higher temperature one at @xmath54 k ; the peak will be sharper if flares are heated during their decay . this work provides a key to interpret stellar flare x - ray data in terms of solar ones . in particular : it provides templates of stellar flare spectra and it tells us that the single thermal components which typically fit stellar flares low - resolution spectra are in agreement with single peaked , relatively sharp , emission measure distributions vs temperature . our simulations indicate that the best - fit temperature of flare spectra collected with rosat / pspc and asca / sis corresponds to the maximum of the flare em(t ) , and that the emission measure values obtained from the fitting well reproduce the total parent emission measures . this work also allows us to put solar flares in relationship with the surface emission of stellar coronae and explains why a continuous , moderate , flaring activity may produce a second peak in the emission measure distribution vs temperature . we expect more detailed information , such as the detection of minor emission measure components at very high temperatures , from high resolution spectra of stellar flares collected by chandra and xmm / newton . anders , e. , grevesse , n. , 1989 , geochimica et cosmochimica acta , 53 , 197 . brinkman , a.c . , et al . , 2001 , a&a , 365 , l324 . drake , j.j . , peres , g. , orlando , s. , laming , j.m . , maggio , a. , 2000 , apj in press favata , f. , reale , f. , micela , g. , sciortino , s. , maggio , a. , matsumoto , h. , 2000 , a&a , 353 , 987 . gotthelf , e.v . , jalota , l. , mukai , k. , white , n.e . , 1994 , apjl , 436 , l91 griffiths , n.w . , jordan , c. , 1998 , apj 497 , 883 gdel , m. , 1997 , apj , 480 , l121 . gdel , m. , linsky , j. l. , brown , a. , nagase , f. , 1999 , apj , 511 , 405 hamaguchi , k. , terada , h. , bamba , a. , koyama , k. , 2000 , apj , 532 , 1111 jakimiec , j. , sylwester , b. , sylwester , j. , et al . 1992 , a&a , 253 , 269 . kaastra , j. s. , an x - ray spectral code for optically thin plasmas , internal report , updated version 2.0 ( sron - leiden ) . maggio , a. , pallavicini , r. , reale , f. , tagliaferri , g. , 2000 , a&a , 356 , 627 . mewe , r. , kaastra , j. s. , liedahl , d.a . , 1995 , legacy , 6 , 16 . orlando , s. , peres , g. , reale , f. , 2000 , apj , 528 , 524 ( paper i ) osten , r. a. , brown , a. , ayres , t. r. , linsky , j. l. , drake , s. a. , gagn , m. , stern , r. a. , 2000 , apj , 544 , 953 ottmann , r. , schmitt , j.h.m.m . , 1996 , a&a , 307 , 813 peres , g. , orlando , s. , reale , f. , rosner , r. , hudson , h. , 2000a , apj , 528 , 537 ( paper ii ) peres , g. , orlando , s. , reale , f. , rosner , r. , 2001 , apj , submitted preibisch , t. , zinnecker , h. , schmitt , j.h.m.m . , 1993 , a&a , 279 , l33 reale , f. , betta , r. , peres , g. , serio , s. , mctiernan , j. , 1997 , a&a 325 , 782 reale , f. , micela , g. , 1998 , a&a , 334 , 1028 reale , f. , peres , g. , 1995 , a&a , 299 , 225 schmitt , j. h. m. m. , 1997 , a&a , 318 , 215 . schmitt , j. h. m. m. , favata , f. , 1999 , nat . , 401 , 44 serio , s. , reale , f. , jakimiec , j. , sylwester , b. , sylwester , j. , 1991 , a&a , 241 , 197 . sylwester , b. , sylwester , j. , serio , s. , et al . , 1993 , a&a , 267 , 586 . tsuboi , y. , imanishi , k. , koyama , k. , grosso , n. , montmerle , t. , 2000 , apj , 532 , 1089 van den oord , g.h.j . , mewe , r. , brinkman , a.c . , 1988 , a&a , 205 , 181 . van den oord , g.h.j . , mewe , r. , 1989 , a&a , 213 , 245 .

in previous works we have developed a method to convert solar x - ray data , collected with the yohkoh / sxt , into templates of stellar coronal observations . here we apply the method to several solar flares , for comparison with stellar x - ray flares . eight flares , from weak ( goes class c5.8 ) to very intense ones ( x9 ) are selected as representative of the flaring sun . the emission measure distribution vs. temperature , em(t ) , of the flaring regions is derived from yohkoh / sxt observations in the rise , peak and decay of the flares . the em(t ) is rather peaked and centered around @xmath0 k for most of the time . typically , it grows during the rise phase of the flare , and then it decreases and shifts toward lower temperatures during the decay , more slowly if there is sustained heating . the most intense flare we studied shows emission measure even at very high temperature ( @xmath1 k ) . time - resolved x - ray spectra both unfiltered and filtered through the instrumental responses of the non - solar instruments asca / sis and rosat / pspc are then derived . synthesized asca / sis and rosat / pspc spectra are generally well fitted with single thermal components at temperatures close to that of the em(t ) maximum , albeit two thermal components are needed to fit some flare decays . rosat / pspc spectra show that solar flares are in a two - orders of magnitude flux range ( @xmath2 erg @xmath3 s@xmath4 ) and a narrow pspc hardness ratio range , however higher than that of typical non - flaring solar - like stars .

within the hierarchical model of the universe , galaxy mergers are thought to be the main formation channel for the build up of massive elliptical galaxies . these , and fainter , galaxies are known to host a supermassive black hole ( smbh ) at their centre ( magorrian et al . 1998 ; richstone et al . 1998 ; ferrarese & ford 2005 , and references therein ) . it has therefore been hypothesized that the depleted cores in core - srsic galaxies are due to the action of coalescing black hole binaries which are produced in major , dissipationless ( gas free or `` dry '' ) mergers of galaxies ( e.g. , begelman et al . 1980 ; ebisuzaki et al . 1991 ; milosavljevi & merritt 2001 ; merritt 2006 ) . indeed , observations have found binary smbhs at kpc separation ( e.g. , ngc 6240 , komossa et al . 2003 ; arp 299 , ballo et al . 2004 ; 0402 + 379 , rodriguez et al . 2006 ; mrk 463 , bianchi et al . 2008 ) , and slowly more are being found at closer separations ( e.g. , burke - spolaor 2011 ; ju et al . 2013 ; liu et al . in such a scenario , three - body interactions involving stars and the smbh binary would decay the smbh binary orbit via the slingshot ejection of stars from the centres of the `` dry '' galaxy merger remnant , naturally creating the observed central stellar light ( mass ) deficits in giant elliptical galaxies and bulges ( e.g. , king & minkowski 1966 ; king 1978 ; binney & mamon 1982 ; lauer 1985 ; kormendy 1985 ; crane et al . 1993 ; ferrarese et al . 1994 , 2006 ; lauer et al . 1995 ; byun et al . 1996 ; faber et al . 1997 ; carollo et al . . numerical simulations targeting the evolution of massive black hole binaries have predicted that the central stellar mass deficit , @xmath10 , of a core - srsic galaxy , i.e. , the ejected stellar mass , scales with the mass of the smbh binary and the number of ( equivalent ) major `` dry '' mergers that the galaxy has experienced ( e.g. , milosavljevi & merritt 2001 ; merritt 2006 ) . other theoretical studies have proposed enhanced mass deficits as large as 5 @xmath15 as a result of additional stellar ejections from repetitive core passages of `` recoiled '' smbhs ( e.g. , boylan - kolchin et al . 2004 ; gualandris & merritt 2008 ) , or due to the actions of multiple smbhs from merging galaxies ( kulkarni & loeb 2012 ) . tight scaling relations involving the structural parameters of both core - srsic and srsic galaxies have been shown to exist ( e.g. , graham 2013 and references therein ) . while these correlations can yield clues to the processes of galaxy formation and evolution , the reliability of this approach depends on the robustness of the modelling employed for deriving the structural parameters . although the parameters of the nuker model ( grillmair et al . 1994 ; kormendy et al . 1994 ; lauer et al . 1995 ) are known to vary with the fitted radial extent of the light profile , due to the fact that the straight outer power - law profile of the nuker model fails to capture the curved ( srsic ) surface brightness profiles of galaxies beyond their core region ( graham et al . 2003 ; dullo & graham 2012 ) , a work around has been suggested . applying the nuker model to the light profiles of 120 `` core '' galaxies , lauer et al . ( 2007a ) noted that the nuker model break radii ( @xmath16 ) are only roughly correlated with the galaxy properties . however , they identified a better correlated parameter @xmath17 ( the `` cusp radius '' ) as a measure of the core size ( carollo et al . 1997a ; rest et al . dullo & graham ( 2012 ) subsequently showed that this cusp radius closely matches the break radius of the core - srsic model which we employ here ) are typically two times bigger than the core - srsic model break radii which are defined relative to the inward extrapolation of the outer srsic function . ] . fitting this model additionally enables us to determine a galaxy s global structural parameters such as its luminosity and half light radius , and to measure the central stellar deficit relative to the outer srsic profile . as noted by graham et al . ( 2003 ) , the issue is not only measuring the core sizes of core - srsic galaxies , but also the misclassification of coreless `` srsic '' galaxies as galaxies having partially depleted cores . using the core - srsic model , dullo & graham ( 2012 ) found that 18@xmath18 of their sample of 39 galaxies which were previously alleged to have depleted cores according to the nuker model were actually srsic galaxies with no cores . although lauer ( 2012 ) subsequently reported that the core identification disagreement between the nuker model and the core - srsic model was only at the level of @xmath19 , dullo & graham ( 2013 , their appendix a.2 ) revisited and confirmed the 18% disagreement . in some cases , additional nuclear components show up as an excess relative to the outer srsic profile , yet these components shallow inner profile resulted in the nuker model labelling them as `` core '' galaxies . in this paper we analyse the 26 suspected core - srsic elliptical galaxies , presenting new light profiles which cover a large radial range @xmath20 . our sample selection , data reduction , and light profile extraction technique are discussed in section [ sec2 ] . in section [ sec3 ] we outline our fitting analysis , provide our results , and additionally compare them to those from published works , paying attention to the issue of double , triple and higher srsic model fits . in section [ sec4 ] , we present updated structural parameters and scaling relations , including central and global properties , of core - srsic early - type galaxies . in section [ sec5.1 ] we discuss the connection between the galaxy core size and the black hole mass . in section [ sec5.2 ] , we discuss the methodology that is applied to derive the stellar mass deficits in the core - srsic early - type galaxies , and in section [ sec5.3 ] we then compare our mass deficits with past measurements . finally , section [ sec5.4 ] discusses alternative scenarios which have been presented in the literature for generating cores in luminous galaxies . our main conclusions are summarised in section [ sec6 ] . we include three appendices at the end of this paper . the first presents the core - srsic model fits for all 26 galaxies together with a review on two galaxies with complicated structures . notes on five suspected lenticular galaxies with a bulge plus disc stellar light distribution are given in the second appendix , while the third appendix provides a comparison between this work and modelling by others of common light profiles . in so doing we highlight a number of issues in the literature today that are important but currently poorly recognised . @llcccc@ galaxy&type & @xmath21 & d & @xmath22 & hst + & & ( mag)&(mpc)&(km s@xmath23)&filters + ( 1)&(2)&(3)&(4)&(5)&(6 ) + + ngc 0507@xmath24 & s0 & @xmath25 & @xmath26 & 306&f555w + ngc 0584@xmath24 & e@xmath27 @xmath28 & @xmath29 & @xmath30&206&f555w + ngc 0741@xmath24 & e @xmath28 & @xmath31 & @xmath32&291&f555w + ngc 1016@xmath24 & e @xmath28 & @xmath33 & @xmath34&302&f555w + ngc 1399@xmath35 & e @xmath28 & @xmath36 & @xmath37&342&f475w / f814w + ngc 1700@xmath24 & e @xmath28 & @xmath38 & @xmath39&239&f555w + ngc 2300@xmath24 & s0 @xmath28 & @xmath40 & @xmath41&261&f555w + ngc 3379@xmath24 & e @xmath28 & @xmath42 & @xmath43 & 209&f555w + ngc 3608@xmath24 & e @xmath28 & @xmath44 & @xmath45&192&f555w + ngc 3640@xmath24 & e @xmath28 & @xmath46 & @xmath47 & 182&f555w + ngc 3706@xmath24 & e @xmath28 & @xmath48 & @xmath49 & 270&f555w + ngc 3842@xmath24 & e @xmath28 & @xmath50 & @xmath51 & 314&f555w + ngc 4073@xmath24 & cd@xmath27 @xmath28 & @xmath52 & @xmath53&275&f555w + ngc 4278@xmath35 & e @xmath28 & @xmath54 & @xmath55&237&f475w / f850lp + ngc 4291@xmath24 & e @xmath28 & @xmath56 & @xmath57 & 285&f555w + ngc 4365@xmath35 & e @xmath28 & @xmath58 & @xmath59 & 256&f475w / f850lp + ngc 4382@xmath35 & s0 @xmath28 & @xmath60 & @xmath61 & 179&f475w / f850lp + ngc 4406@xmath35 & e @xmath28 & @xmath62 & @xmath63 & 235&f475w / f850lp + ngc 4472@xmath35 & e@xmath27 @xmath28 & @xmath64 & @xmath65 & 294&f475w / f850lp + ngc 4552@xmath35 & e@xmath27 @xmath28 & @xmath66 & @xmath67&253&f475w / f850lp + ngc 4589@xmath24 & e @xmath28 & @xmath68 & @xmath69 & 224&f555w + ngc 4649@xmath35 & e @xmath28 & @xmath70 & @xmath71&335&f475w / f850lp + ngc 5061@xmath24 & e @xmath28 & @xmath72 & @xmath73 & 186&f555w + ngc 5419@xmath24 & e@xmath27 @xmath28 & @xmath74 & @xmath75&351&f555w + ngc 5557@xmath24 & e @xmath28 & @xmath76 & @xmath77&253&f555w + ngc 5813@xmath24 & s0 @xmath28 & @xmath78 & @xmath79 & 237&f555w + ngc 5982@xmath24 & e @xmath28 & @xmath48 & @xmath80 & 239&f555w + ngc 6849@xmath24 & sb0 @xmath28 & @xmath81 & @xmath82 & 209&f555w + ngc 6876@xmath24 & e @xmath28 & @xmath83 & @xmath84 & 229&f555w + ngc 7619@xmath24 & e @xmath28 & @xmath85 & @xmath86 & 323&f555w + ngc 7785@xmath24 & e @xmath28 & @xmath87 & @xmath88&255&f555w + notes.col . ( 1 ) galaxy name . instrument : @xmath89 _ hst _ wfpc2 ; @xmath90 _ hst _ acs . ( 2 ) morphological classification from the nasa / ipac extragalactic database ( ned ) except for ngc 3706 and ngc 5813 . we adopt an elliptical morphology for ngc 3706 based on the results in dullo & graham ( 2013 ) , while for ngc 5813 we adopt an s0 morphology based on the fitting analysis as well as the photometric profile ( section [ sec3 ] , appendices [ apppa ] and [ apppb ] ) . the superscript @xmath91 shows elliptical galaxies which are classified as disc galaxies in the literature ( appendix [ apppb ] ) . ( 3 ) absolute _ v_-band ( galaxy or bulge ) magnitude obtained from lauer et al . ( 2007b ) . these magnitudes are corrected for galactic extinction and @xmath92 surface brightness dimming , and adjusted using the distances from col . the five bulge magnitude are additionally corrected for inclination and internal dust attenuation ( driver et al . 2008 , their table 1 and eqs . 1 and 2 ) . sources : ( @xmath93 ) tonry et al . ( 2001 ) after reducing their distance moduli by 0.06 mag ( blakeslee et al . 2002 ) ; ( @xmath94 ) from ned ( 3k cmb ) . ( 5 ) central velocity dispersion from hyperleda ( paturel et al . ( 6 ) filters . as noted above , in this paper we have targeted 26 suspected core - srsic elliptical galaxies from dullo & graham ( 2012 ) using more radially extended light profiles . we additionally use the data for the five core - srsic galaxies ( ngc 507 , ngc 2300 , ngc 3706 , ngc 4382 and ngc 6849 ) from dullo & graham ( 2013 ) excluding ngc 3607 because of its dusty nuclear spiral which affected the recovery of the structural parameters as detailed there . updated magnitudes , distances and velocity dispersions pertaining to this combined sample of 31 core - srsic early - type galaxies are presented in table [ tabbb1 ] . the _ hst _ images for the 26 core - srsic elliptical galaxies , observed with the acs and/or wfpc2 cameras were taken from the public hubble legacy archive ( hla ) . although both the acs ( plate scale of @xmath95 ) and the wfpc2 ( plate scale of @xmath96 ) cameras have comparable high spatial resolution , the acs wide field channel ( wfc ) has a larger , rhomboidal field - of - view ( fov ) of @xmath97 compared to that of the wfpc2 ccd array ( 160@xmath98@xmath99160@xmath98 l - shaped fov ) . we therefore prefer the acs / f475w ( @xmath100 sdss @xmath101 ) and acs / f850lp ( @xmath100 sdss @xmath102 ) images when available . for ngc 1399 , these were not available , and we instead used the acs / f475w ( @xmath100 sdss @xmath101 ) and acs / f814w ( similar to the johnson - cousins @xmath103-band ) images . for the other galaxies where the acs images were not available we use the wfpc2 images taken in the f555w filter ( similar to the johnson - cousins v - band ) . table [ tabbb1 ] provides the observation summaries including the programs , instruments and filters used for imaging our sample galaxies . ) . these colour calibrations are from acs @xmath101- and @xmath102-band magnitudes into the wfpc2 @xmath104-band magnitude except for ngc 1399 . for ngc 1399 , we transform the acs @xmath101- and @xmath103-band data into wfpc2 @xmath105-band data . our @xmath101- , @xmath102- and @xmath103-band data points each derived here from same major - axis with equal position angle and ellipticity using the iraf / ellipse fit along with the lauer et al . ( 2005 ) @xmath104- and @xmath105-band surface brightness profiles are used for creating the data points in each panel . each panel shows the least - squares fit and the pertaining equation ( eq ) for the galaxy . shaded regions show the 1@xmath106 uncertainty on the best fits for only one galaxy in which the error associated with the slope of the least - square fit is large . ] as in dullo & graham ( 2013 ) , we build new , composite light profiles for all the 26 core - srsic elliptical galaxies by combining the lauer et al . ( 2005 ) very inner ( @xmath107 ) deconvolved f555w ( @xmath100 @xmath104-band ) light profiles with our new ( calibrated @xmath104-band ) major - axis light profiles ( see below ) . we have chosen to use the inner deconvolved light profile from lauer et al . ( 2005 ) so that differences in the core parameters between our works are not attributed to the use of different treatments of the psf , but rather the application of the nuker versus the core - srsic model . the new outer profiles are extracted using the iraf / ellipse task ( jedrzejewski 1987 ) and cover @xmath108@xmath98 in radius . our data reduction steps , along with the surface brightness profile extraction procedures , are discussed in detail in sections @xmath109 and @xmath110 of dullo & graham ( 2013 ) . in order to match the acs @xmath101 and @xmath102-band data with the very inner lauer et al . ( 2005 ) deconvolved @xmath104-band data , we made color transformations for the six galaxies without @xmath104-band data ( ngc 4278 , ngc 4365 , ngc 4406 , ngc 4472 , ngc 4552 and ngc 4649 ) using eqs.[eqqiii2]@xmath111[eqqiii7 ] which were derived here by applying a least - squares fit to each galaxy s @xmath112 and @xmath113 data ( see also dullo & graham 2013 and references therein for a similar practice ) . for ngc 1399 , lauer et al . ( 2005 ) published this galaxy s f606w ( roughly @xmath105-band ) light profile , thus we calibrate our acs @xmath101- and @xmath103-band data to the @xmath105-band light profile using eq . [ eqqiii1 ] . [ figiii1 ] illustrates these linear fits to the above seven sample galaxies for which the acs wide field channel ( wfc ) images are available , as listed in table [ tabbb1 ] . @xmath114 @xmath115 @xmath116 @xmath117 @xmath118 @xmath119 @xmath120 accurate sky level subtraction is of critical importance when determining the surface brightness profile of a galaxy at large radii . the galaxy flux is often just a few percent of the sky background values in the outermost parts . the automatic hla reduction pipeline subtracts the sky values from the images . thus , poor sky subtraction is a concern for galaxies which extend beyond the _ hst _ wfpc2 and acs fovs . however , reliable sky background determination can be done for galaxies with @xmath121 mag arcsec@xmath122 diameters @xmath123 . not only do the bulk of these galaxies lie within the wfpc2 and acs fovs but their `` counts '' at the edges of the wfpc2/acs ccds are about 10% fainter than the typical _ hst _ @xmath104-band sky value @xmath124 mag arcsec@xmath122 ( lauer et al . eight of the 31 sample galaxies ( ngc 1399 , ngc 3379 , ngc 4365 , ngc 4382 , ngc 4406 , ngc 4472 , ngc 4552 and ngc 4649 ) have major - axis diameter ( at @xmath121 mag arcsec@xmath122 ) @xmath125 ( graham et al . 1998 ; smith et al . 2000 ; trager et al . fortunately , all these galaxies except two ( ngc 1399 and ngc 3379 ) have published composite ( acs plus ground - based ) data ( ferrarese et al . 2006 ; kormendy et al . the ground - based data enabled these authors to better constrain the sky level . for ngc 3379 , schombert & smith ( 2012 ) published an extended ( @xmath126 ) ground - based profile and for ngc 1399 , li et al . ( 2011 ) published a very extended ground - based light profile . our _ hst_-derived light profiles for all eight extended galaxies are in a good agreement with those past published profiles , suggesting only a small or negligible sky subtraction error by the pipeline . for the remaining galaxies , we additionally checked the pipeline sky subtraction by measuring the sky values at the edges of the wfpc2 and acs chips , i.e. , away from the galaxy and free of contaminating sources . as expected , the average of the median of the sky values from several 10 @xmath99 10 pixel boxes is very close to zero for most galaxies . for a few galaxies , we find that the average values are slightly below zero even if these galaxies are well within the fov of the wfpc2 or acs . therefore , we adjust the background level to zero . in general , the three - parameter srsic ( 1968 ) @xmath127 model , a generalization of the de vaucouleurs ( 1948 ) @xmath128 model , is known to provide accurate representations to the stellar light distributions of both elliptical galaxies and the bulges of disc galaxies over a large luminosity and radial range ( e.g. , caon et al . 1993 ; donofrio et al . 1994 ; young & currie 1994 ; andredakis et al . 1995 ; graham et al . however , because the @xmath128 model was very popular , even referred to by many as the @xmath128 law , saglia et al . ( 1997 ) attempted to explain the observed @xmath127 light profiles as the sum of @xmath128 models and exponential discs . while this approach had some merit , in that pressure - supported elliptical galaxies are becoming increasingly rare , and in fact many intermediate luminosity early - type galaxies possess rotating discs ( e.g. , graham et al . 1998 ; emsellem et al . 2011 ; scott et al . 2014 ) , we now know that lenticular galaxies are very well described by an @xmath127 bulge plus an exponential disc which can often additionally contain a bar and/or a lens ( e.g. laurikainen et al . 2013 , and references therein ) . the srsic model s radial intensity distribution can be written as @xmath129 , \label{eqiii8}\ ] ] where @xmath130 is the intensity at the half - light radius @xmath8 . the variable @xmath131 , for @xmath132 ( e.g.,caon et al . 1993 ) , is coupled to the srsic index @xmath94 , and ensures that the half - light radius encloses half of the total luminosity . the luminosity of the srsic model within any radius @xmath105 is given by @xmath133 where @xmath134 is the incomplete gamma function and @xmath135 . the review by graham & driver ( 2005 ) describes the srsic model in greater detail . the srsic model fits the surface brightness profiles of the low- and intermediate - luminosity ( @xmath136 mag ) spheroids all the way to the very inner region . although , additional nuclear components are often present in these galaxies and require their own model ( e.g. , graham & guzmn 2003 ; ct et al . 2006 ; den brok et al . 2014 ) . on the other hand , the inner light profiles of luminous ( @xmath137 mag ) spheroids deviate downward from the inward extrapolation of their outer srsic model fits ( graham et al . 2003 ; trujillo et al . 2004 ; ferrarese et al . 2006 ) . in order to describe such galaxies light distributions , graham et al . ( 2003 ) introduced the core - srsic model which is a combination of an inner power - law with an outer srsic model . this six - parameter model is defined as @xmath138^{\gamma /\alpha } \exp \left[-b\left(\frac{r^{\alpha}+r^{\alpha}_{b}}{r_{e}^{\alpha } } \right)^{1/(\alpha n)}\right ] , \label{eqq10}\ ] ] with @xmath139 . \label{eqq11}\ ] ] @xmath140 is intensity measured at the core break radius @xmath141 , @xmath142 is the slope of the inner power - law profile , and @xmath143 controls the sharpness of the transition between the inner power - law and the outer srsic profile . @xmath8 represents the half - light radius of the outer srsic model , and the quantity @xmath144 has the same meaning as in the srsic model ( eq . [ eqiii8 ] ) . the total core - srsic model luminosity ( trujillo et al . 2004 ; their eq . a19 ) is @xmath145 fig . [ figa1 ] shows our core - srsic model fit to the underlying host galaxy , major - axis , light distributions for all 26 galaxies . the fit residuals together with their root - mean - square ( rms ) values are given for each galaxy in appendix [ apppa ] . we find that the light profile for one of these suspected elliptical galaxies ( ngc 5813 ; see appendices [ apppa ] and [ apppb ] ) is better described with a core - srsic bulge plus an exponential disc model , suggesting an s0 morphology as discussed later . we note that this galaxy s large - scale disc had negligible contribution to the dullo & graham ( 2012 ) @xmath100@xmath147 light profile fit . further , in agreement with dullo & graham ( 2012 ) , we also detect additional nuclear light components ( i.e. , agn or nuclear star clusters ) on the top of the underlying core - srsic light distributions in six galaxies ( ngc 741 , ngc 4278 , ngc 4365 , ngc 4472 , ngc 4552 , and ngc 5419 ) . we account for these nuclear light excesses using a gaussian function . dullo & graham ( 2013 ) presented the core - srsic(+exponential ) model fits , along with the fit parameters , to five galaxies ( ngc 507 , ngc 2300 , ngc 3706 , ngc 4382 and ngc 6849 ) . table [ tabbb2 ] presents the best fit structural parameters for the full ( 26 + 5=)31 galaxy sample obtained by applying our adopted models to the @xmath104-band , major - axis , light profiles probing large radial ranges ( @xmath148 ) . in general , from appendix [ apppa ] it is apparent that the main body of luminous ellipticals can be very well described with the core - srsic model . our fits yield a median root - mean - square ( rms ) residual of @xmath149 mag arcsec@xmath122 . out of the full galaxy sample , only two elliptical galaxies ( ngc 4073 and ngc 6876 ) reveal complicated structures , as such their light profiles , discussed in appendix [ apppa ] , are somewhat poorly matched by the core - srsic model . , table 2 , @xmath104-band ) with previously published break radii measurements ( @xmath150 ) from ( i ) trujillo et al . ( 2004 , @xmath105-band , stars ) , ( ii ) lauer et al . ( 2005 , @xmath104-band , open ellipses ) , ferrarese et al . ( 2006 , g - band , blue ellipses ) , richings et al . ( 2011 , @xmath103- and @xmath151-bands , triangles ) and dullo & graham ( 2012 , @xmath104-band , red circles for ellipticals and blue disc symbols for s0s ) . we also compare the kormendy et al . ( 2009 ) inner most srsic model fitting radii ( @xmath152 ) and our break radii for six common galaxies ( squares).the geometric - mean break radii from ferrarese et al . ( 2006 ) were converted into semi - major axis break radii using their published galaxy ellipticities at @xmath1 . the lauer et al . ( 2005 ) break radii came from their nuker model fits , while all remaining works mentioned above applied the core - srsic model for measuring the galaxies break radii . the break radii from each model is the radius where each model has it s maximum curvature . i.e. where the second derivative of the model s intensity profile is a maximum . the solid line indicates a one - to - one relation while the dashed line is the @xmath153 = 2 @xmath154 relation . a representative error bar is shown at the bottom of the panel . ] for three sample elliptical galaxies ngc 1700 , ngc 3640 and ngc 7785 , while their light profiles are well fit by the core - srsic model with reasonable rms residuals @xmath155 mag arcsec@xmath122 , we find that their cores are unusually small for their brightnesses . the core - srsic model yields break radii of @xmath156 and @xmath157 for ngc 1700 , ngc 3640 , and ngc 7785 , respectively . not surprisingly , in section [ sec4 ] , it can be seen that these three questionable cores are outliers in several galaxy scaling relations involving @xmath1 , @xmath106 , @xmath4 , and @xmath8 and as such they have not been included in the regression analysis . for the interested reader , these galaxies are discussed in more detail in appendix [ apppqes ] . @lllcccccccccccc@ galaxy&type&@xmath158&@xmath159 & @xmath160 & @xmath1 & @xmath161&@xmath143&@xmath94&@xmath8&@xmath8&@xmath162&@xmath163&@xmath164 + & & & & ( arcsec)&(pc)&&&&(arcsec)&(kpc)&(mag)&&(arcsec ) + ( 1)&(2)&(3)&(4)&(5)&(6)&(7)&(8)&(9)&(10)&(11)&(12)&(13)&(14 ) + + ngc 0507@xmath165 & s0&16.16 & 16.38 & 0.33 & 102 & 0.07 & 5 & 2.2 & 5.3 & 1.65 & & 21.03 & 27.69 + ngc 0584 & e@xmath27&13.81 & 14.61 & 0.21 & 21 & 0.47 & 5 & 6.6 & 112.5 & 11.25 & + ngc 0741 & e&16.83 & 17.52 & 0.76 & 267 & 0.19 & 5 & 7.4 & 53.0 & 18.60 & 22.1 + ngc 1016 & e&16.35 & 17.00 & 0.48 & 204 & 0.15 & 2 & 5.2 & 41.7 & 17.79 + ngc 1399@xmath166&e&15.45 & 16.36 & 2.30 & 202 & 0.11 & 2 & 5.6 & 36.6 & 3.22 + ngc 1700 ? & e&13.34 & 13.38 & 0.04 & 11 & 0.19 & 5 & 6.1 & 32.0 & 8.23 + ngc 2300@xmath165&s0 & 16.23 & 16.61 & 0.53 & 70 & 0.08 & 2 & 2.2 & 7.7 & 1.02 & & 20.39 & 21.08 + ngc 3379 & e&14.76 & 15.76 & 1.21 & 102 & 0.19 & 2 & 5.9 & 50.2 & 4.21 + ngc 3608 & e&14.56 & 15.14 & 0.23 & 24 & 0.29 & 5 & 6.4 & 68.7 & 7.28 + ngc 3640 ? & e&14.80 & 14.72 & 0.03 & 4 & -0.01 & 5 & 3.5 & 28.0 & 2.99 + ngc 3706@xmath165&e & 14.15 & 14.16 & 0.11 & 24 & -0.02 & 10 & 6.4 & 42.1 & 9.18 & + ngc 3842 & e&16.73 & 17.42 & 0.72 & 315 & 0.19 & 5 & 6.9 & 102.4 & 45.17 + ngc 4073 & cd@xmath27&16.51 & 16.46 & 0.22 & 90 & -0.06 & 10 & 6.1 & 141.6 & 58.61 & & + ngc 4278 & e&15.07 & 15.84 & 0.83 & 52 & 0.22 & 5 & 3.8 & 20.2 & 1.25 & 19.4 & + ngc 4291 & e&14.87 & 15.14 & 0.30 & 36 & 0.10 & 5 & 4.4 & 13.6 & 1.64 + ngc 4365 & e&16.14 & 16.50 & 1.21 & 127 & 0.00 & 2 & 4.8 & 47.3 & 4.97 & 20.2 + ngc 4382@xmath165&s0&14.82 & 15.01 & 0.27 & 24 & 0.07 & 5 & 2.7 & 11.1 & 0.99 & & 19.50 & 35.07 + ngc 4406 & e&15.86 & 15.97 & 0.76 & 61 & 0.00 & 5 & 5.5 & 145.2 & 11.62 + ngc 4472 & e@xmath27&16.18 & 16.34 & 1.21 & 108 & -0.02 & 2 & 3.0 & 48.8 & 4.34 & 22.0 + ngc 4552 & e@xmath27&14.91 & 15.03 & 0.38 & 17 & 0.03 & 10 & 4.4 & 29.7 & 1.31 & 20.6 + ngc 4589&e&14.80 & 15.33 & 0.20 & 27 & 0.30 & 5 & 5.6 & 70.8 & 9.56 + ngc 4649&e&15.75 & 16.70 & 2.51 & 241 & 0.21 & 2 & 3.6 & 62.8 & 6.02 + ngc 5061&e&13.70 & 14.09 & 0.22 & 34 & 0.16 & 5 & 8.4 & 68.44 & 10.81 + ngc 5419&e@xmath27&17.35 & 17.53 & 1.43 & 416 & -0.06 & 2 & 5.6 & 55.0 & 16.01 & 19.9 + ngc 5557&e&15.05 & 15.46 & 0.23 & 51 & 0.19 & 5 & 4.6 & 30.2 & 6.80 + ngc 5813 & s0&16.04 & 16.11 & 0.35 & 51 & -0.10 & 2 & 2.8 & 7.1 & 1.02 & & 20.30 & 31.28 + ngc 5982&e&15.22 & 15.48 & 0.25 & 51 & 0.09 & 5 & 4.3 & 26.8 & 5.45 + ngc 6849@xmath165 & sb0&16.33 & 16.67 & 0.18 & 69 & 0.20 & 5 & 3.2 & 7.8 & 2.98 & & 20.72 & 16.93 + ngc 6876&e&17.00 & 16.98 & 0.45 & 119 & 0.00 & 10 & 5.9 & 250.0&65.8 + ngc 7619&e&15.41 & 15.93 & 0.49 & 109 & 0.16 & 5 & 7.2 & 72.2 & 16.23 + ngc 7785?&e&14.98 & 14.76 & 0.03 & 5 & 0.00 & 10 & 4.9 & 55.1 & 12.63 + notes.structural parameters from fits to the @xmath104-band major - axis surface brightness profiles ( appendix [ apppa ] ) . the superscript + indicates that we use an @xmath105-band surface brightness profile instead of a @xmath104-band surface brightness profile for ngc 1399 . the superscript * shows ( 4s0s and 1e ) galaxies for which the fit parameters are taken from dullo & graham ( 2013 ) . a `` ? '' is used to indicate three galaxies with questionable core sizes . ( 1 ) galaxy name ( 2 ) adopted morphological classification . the superscript @xmath91 shows elliptical galaxies which are classified as disc galaxies in the literature ( appendix [ apppb ] ) . ( 3)-(11 ) best - fit parameters from the core - srsic model , eq . [ eqq10 ] . col . ( 12 ) central point source apparent magnitude . ( 13 ) disc central surface brightness . col . ( 14 ) disc scale length . the surface brightnesses @xmath167 , @xmath168 and @xmath169 are in units of mag arcsec@xmath122 . ) with previous srsic values ( @xmath170 ) from ( i ) trujillo et al . ( 2004 , stars ) , ( ii ) ferrarese at al . ( 2006 , blue ellipses ) , richings et al . ( 2011 , triangles ) and dullo & graham ( 2012 , red circles for ellipticals and blue disc symbols for s0s ) . 68% of the data resides within -30% and + 25% of perfect agreement , and the outliers are explained and accounted for in section 3.3 . ] here we illustrate two diagrams comparing our values of @xmath1 and @xmath94 with those from similar studies in the literature . the agreement is generally good . we have gone to some effort to identify and explain all notable disagreements with past studies . we have three core - srsic galaxies ( ngc 4291 , ngc 5557 , and ngc 5982 ) in common with trujillo et al . ( 2004 ) who used a radial extent of @xmath100@xmath171 . ngc 1700 is also an overlapping galaxy but its light profile used by trujillo et al . ( 2004 ) extends from @xmath172 @xmath173 to @xmath174 , thus they did not detect the questionably small core ( @xmath175 ) that we potentially measure here . with the exception of ngc 1700 , classified as a srsic galaxy by trujillo et al . ( 2004 ) , there is an excellent agreement between our break radii and those from trujillo et al . ( 2004 ) , see fig . [ figiii4 ] . their srsic indices are also consistent for all four galaxies in common with our study , i.e. including ngc 1700 ( fig . [ figiii5 ] ) . there are six core - srsic galaxies ( ngc 4365 , ngc 4382 , 4406 , ngc 4472 , ngc 4552 and 4649 ) in common with ferrarese et al . ( 2006 ) who used a radial extent of @xmath100@xmath171 . for these galaxies , the ferrarese et al . ( 2006 ) geometric - mean , @xmath101-band , break radii were taken and converted to major - axis values . we prefer their g - band than the z - band data as it more closely matches our @xmath104-band data . the agreement between their break radii and our measurements are good except for three galaxies ( ngc 4382 , ngc 4552 , and ngc 4472 ) . the most discrepant ( by more than 100% ) is the s0 galaxy ngc 4382 but this is because it was modelled with a core - srsic+exponential model by dullo & graham ( 2013 , their fig . ferrarese et al . ( 2006 ) treated this galaxy as a single component system , and thus fit the bulge+disc light with just a core - srsic model , resulting in a systematically higher @xmath1 and @xmath94 value ( figs . [ figiii4 ] and [ figiii5 ] , see also dullo & graham 2012 , their fig . the core - srsic stellar light distributions of the remaining two elliptical galaxies . ] ngc 4552 and ngc 4472 have a broad and an intermediate ( inner core)-to-(outer srsic ) transition region , respectively , which are well described by the core - srsic model @xmath176 and @xmath177 values ( fig . [ figa1 ] ) . this in contrast to ferrarese et al s . sharp transition ( @xmath178 ) core - srsic model fits which poorly match these two galaxies transition regions as can be seen by the systematic bump in their fit residuals . due to this , we find a 70% discrepancy between the ferrarese et al . ( 2006 ) break radii and ours for ngc 4552 and ngc 4472 , much bigger than the @xmath179 uncertainty range of 10% quoted in dullo & graham ( 2012 ) . however , dullo & graham ( 2012 , their fig . 7 ) already discussed the source of this discrepancy for ngc 4552 . omitting the s0 galaxy ngc 4382 , the agreement between the srsic indices of ferrrarese et al . ( 2006 ) and ours is generally good . we note that @xmath180 of the data in fig . [ figiii5 ] have a srsic index ratio within -30% and 25% of perfect agreement . this is in fair agreement with typical uncertainties of 25% reported for the srsic index ( e.g. , caon et al . 1993 ) and is slightly better than the allen et al . ( 2006 ) @xmath179 uncertainty range of @xmath181 36% . it should be remembered that using different filters as well as modelling minor , major and geometric - mean axis profiles can yield different @xmath94 values for a galaxy ( e.g. , caon et al . 1993 ; ferrari et al . 2004 ; kelvin et al . 2012 ) . with past measurements @xmath182 from ( i ) dullo & graham ( 2012 , filled circles and the disc symbols are for their core - srsic ellipticals and s0s , respectively ) and ( ii ) lauer et al . ( 2007a , ellipses ) . these surface brightnesses are where the respective intensity profile models have the maximum value of their second derivative , i.e. where the curvature of the models is greatest . ] we have six core - srsic galaxies ( ngc 3379 , ngc 3608 , ngc 4278 , ngc 4472 , ngc 4552 and ngc 5813 ) in common with richings et al . ( 2011 ) whose data extended to @xmath183 . their break radii for these common galaxies agree with ours except for ngc 5813 . ngc 5813 is a similar case to that of the s0 galaxy ngc 4382 noted above ; it has a core - srsic bulge+exponential disc light distribution which was modelled using only the core - srsic model by richings et al . ( 2011 ) , thus they measured larger @xmath1 and @xmath94 values . our srsic indices agree within 25% with richings et al . ( 2011 ) for half of the six core - srsic galaxies in common , but for the remaining half ( ngc 3379 , ngc 4552 plus the s0 ngc 5813 ) there is more than a 40% discrepancy . for ngc 3379 and ngc 4552 , the origin of this discrepancy appears to be the @xmath100@xmath147 _ nic2 f160w light profiles mag arcsec@xmath122 ) major - axis diameter of @xmath100@xmath184 , extend beyond the nicmos nic2 cdd . ] used by richings et al . . their profiles for these two galaxies may be too limited in radius for the core - srsic model to capture the actual galaxy light distributions . finally , we note that the elliptical galaxy ngc 5982 is also in common with richings et al . ( 2011 ) who classified it as a srsic galaxy based on their @xmath185 srsic model fit to the @xmath100@xmath147 nicmos nic2 f160w profile . it seems that richings et al . ( 2011 ) might have missed the core with their @xmath185 srsic fit to this large elliptical galaxy with @xmath186 km s@xmath23 and @xmath187 mag . kormendy et al . ( 2009 ) adopted graham et al s . ( 2003 ) logic of defining a core as a deficit in light relative to the inward extrapolation of a spheroid s outer srsic profile , but they fit the major - axis light profiles of their core - srsic galaxies using only the srsic model . they advocate fitting the srsic model over the radius range where it fits well by eye and distinguishing the core region in a subjective manner . this exercise assumes no transition region between the inner core and the outer srsic profile , but actual galaxy profiles can have a broad transition region . as with the nuker model , the `` break radius '' is not the outermost boundary of this transition region controlled by the parameter @xmath143but the mid - point of the transition . the outer edge of the transition region is very hard to judge by eye , and highly subjective . we have six core - srsic galaxies ( ngc 4365 , ngc 4382 , ngc 4406 , ngc 4472 , ngc 4552 and ngc 4649 ) in common with kormendy et al . ( 2009 ) . it is worth comparing the inner most srsic model fitting radius ( @xmath188 ) from kormendy et al . ( 2009 ) with our break radius ( @xmath1 ) for these six overlapping galaxies : ngc 4365 , ngc 4382 , ngc 4406 , ngc 4472 , ngc 4552 and ngc 4649 . as shown in fig . [ figiii4 ] , for each core - srsic galaxy in common with kormendy et al . ( 2009 ) , their @xmath188 are much further out than ours and part of the explanation likely arises from their method of not measuring the actual `` break radii '' . fitting the core - srsic model provides this radius , the extent of the transition region , and the central flux deficit from within the outer - edge of the transition region . figs . [ figiii4 ] and [ figiii5a ] additionally compare our break radii and break surface brightnesses , respectively , with those from lauer et al . ( 2005 ) and dullo & graham ( 2012 ) for all 31 galaxies ( including the three with questionable core in appendix [ apppqes ] ) . it is important to note that lauer et al . ( 2005 ) fit the nuker model to their @xmath147 galaxy light profiles , while dullo & graham ( 2012 ) re - modelled these using the core - srsic model . our new break radii , determined from spatially extended ( @xmath20 ) light profiles , are in excellent agreement with those from dullo & graham ( 2012 ) , i.e. , within the uncertainty range , except for ngc 1700 and ngc 2300 . ngc 1700 is an elliptical galaxy with a questionably small core mentioned earlier ( see section [ sec4 ] and section [ sec8.1.1 ] for further details ) , while for the s0 galaxy ngc 2300 , as noted in dullo & graham ( 2013 ) , the contribution of the disc light to the @xmath189 light profile modeled by dullo & graham ( 2012 ) resulted in a bigger break radius and srsic index . given the remarkable agreement between the core - srsic break radii of dullo & graham ( 2012 ) and those from their model - independent estimates ( their fig . 11 ) , it implies that our break radii from this work ( table [ tabbb2 ] ) are also in a very good agreement with the model - independent radii where the slope of the logarithmic profile equals -1/2 ( carollo et al . in addition , as can be seen in fig . [ figiii5a ] , our new break surface brightnesses fully agree with those from dullo & graham ( 2012 ) . on the other hand , in line with previous core - srsic works , we find the nuker model break radii are larger ( fig . [ figiii4 ] ) . the nuker break radii ( e.g. , lauer et al . 2005 , 2007a , b ; krajnovi et al . 2013 ) are on average two times bigger than our core - srsic break radii . in estimating larger break radii , the nuker model consequently estimates the associated surface brightness up to 2 mag arcsec@xmath122 fainter ( fig . [ figiii5a ] ) . hopkins et al . ( 2009a , b ) fit the surface brightness profiles of both srsic and core - srsic elliptical galaxies using a double srsic model . they claimed that these galaxies outer component is an old spheroid ( with @xmath190 ) formed by the violent relaxation of pre - existing stars from a merger event while their inner component was `` excess light '' formed from a dissipative starburst produced by the same `` wet '' ( gas - rich ) merger event ( e.g. , hernquist et al . 1993 ; mihos & hernquist 1994 ) . while this reasonable scenario sounds plausible , we point out two concerns . first , the lower luminosity early - type ( srsic ) galaxies ( @xmath191 mag ) tend to have fast - rotating , outer exponential discs ( emsellem et al . 2011 ; krajnovi et al . 2013 ) , rather than old , outer spheroid - like components . second , the higher luminosity ( core - srsic ) elliptical galaxies have a central deficit of light rather than an excess , and are thought to be formed from dry merger events ( e.g. , faber et al . the violent relaxation simulations yield @xmath190 ( e.g. , van albada 1982 ; mcglynn 1984 ) and therefore can not account for luminous elliptical galaxies with @xmath192 built from dry mergers . in addition , they do not explain how low - luminosity elliptical galaxies with @xmath193 are made , nor why these galaxies follow the same @xmath194 relation as the massive ellipticals with @xmath195 . the hopkins et al . ( 2009a , b ) galaxy sample included lenticular galaxies ( ngc 507 , ngc 1400 , ngc 2778 , ngc 4382 , ngc 4459 , ngc 4476 , ngc 5813 and ngc 4515 ) which were thought to be ellipticals and modelled using an inner srsic model plus an outer srsic model with @xmath196 . our fits in fig . [ figa1 ] and those in dullo & graham ( 2013 ) show that ngc 507 , ngc 4382 and ngc 5813 are core - srsic lenticular galaxies that are well described by a core - srsic bulge plus an exponential disc model with very small rms residuals of 0.027 mag arcsec@xmath122 , 0.016 mag arcsec@xmath122 and 0.015 mag arcsec@xmath122 , respectively . laurikainen et al . ( 2010 , 2011 ) have also detected a weak nuclear bar in the unsharp mask image of ngc 507 . further , all of our core - srsic elliptical galaxies ( table [ tabbb1 ] ) , except for ngc 1016 , ngc 3706 and ngc 4073 , are in common with hopkins et al . ( 2009b ) . as discussed in section [ sec3.2 ] , these galaxies are well described by the core - srsic model , which can be seen from the residual profiles ( fig . [ figa1 ] ) which also have small rms residuals @xmath100 0.045 mag arcsec@xmath122 . this can be compared to the larger rms residuals given by hopkins et al . ( 2009b , their figs . 10 - 14 ) from their double srsic model fits for six core - srsic galaxies in common with our sample ( ngc 4365 , ngc 4382 , ngc 4406 , ngc 4472 , ngc 4552 and ngc 4649 ) . that is , with better fits , we have shown that these galaxies have a central deficit of light , in accord with the dry merging scenario involving supermassive black holes . in stark contrast , hopkins et al . ( 2009b ) argued that these galaxies support a wet merger scenario . similarly , dhar & williams ( 2012 ) argued that both srsic and core - srsic galaxies can be represented well by two or three `` dw '' functions which are the 2d projections of the 3d einasto density model ( dhar & williams 2010 ) . as in hopkins et al . ( 2009a , b ) , all their sample galaxies , including the six common core - srsic galaxies ( ngc 4365 , ngc 4382 , ngc 4406 , ngc 4472 , ngc 4552 and ngc 4649 ) , are presented as having an inner `` extra light '' component that has a half - light radius @xmath8 @xmath197 kpc . this is somewhat similar to the ground - based work by huang et al . ( 2013 ) who argued that elliptical galaxies comprise three distinct components an inner ( @xmath198 kpc ) component , a middle ( @xmath199 kpc ) component and an outer @xmath200 kpc envelope which are all represented by srsic models with @xmath201 , at odds with the traditional picture of violent relaxation producing @xmath202 profiles . in contrast , the fits from our study show that the inner @xmath100 kpc of `` ellipticals '' ( excluding the depleted core ) are not disconnected from , but are rather the simple extensions of , their outer regions . of course disturbed , unrelaxed galaxies , especially those with peculiar morphology , wo nt be well described by a single core - srsic model . these particular galaxies may well appear to have multiple ( srsic ) spheroidal components . in addition to a handful of s0 galaxies ( ic 2006 , ngc 4697 ) , a handful of barred s0 galaxies ( ic 4329 , ngc 6673 ) , and a handful of unrelaxed peculiar galaxies ( ngc 2305 , ngc 4976 ) , huang et al . ( 2013 ) included 15 cd galaxies ( ic 1633 , ic 2597 , ic 4765 , ic 4797 , ngc 596 , ngc 1172 , ngc 1339 , ngc 1427 , ngc 3087 , ngc 4696 , ngc 4786 , ngc 6909 , ngc 6958 , ngc 7192 , ngc 7796 ) in their galaxy sample . the tell - tale signature of a fit which has failed to fully capture the curvature in the radial stellar distribution is evidenced by the pattern in the residual profile . this can be seen in , for example , fig . 36 from huang et al . ( 2013 ) , where , from 150@xmath98 to beyond 400@xmath98 there is a systematic hump in their residual profile for eso 185-g054 . the artificial ring in their residual image also reveals that the fit is not optimal . this is because the 9 parameters of their three fitted srsic models have collectively managed to approximate the light profile out to a radius @xmath203 . in this instance , the use of a fourth , extended srsic model would have enabled a better fit to the outer half of the light profile , as in the case of es0 221-g026 which huang et al . ( 2013 ) fit with 4 srsic models . however , this does not mean that the galaxy eso 185-g054 actually has 4 components , simply that if one uses enough parameters then one can better approximate the light profile . rather than applying a multitude of srsic components , we advocate trying to establish which components are real and then applying the appropriate function , as done by , for example , lsker et al . ( 2014 ) . to continue this important point , but avoid creating too much of a distraction in the main text , in appendix [ apppc ] we provide comparisons between our core - srsic modelling of our galaxy sample and some other recent works which obtained dramatically different results for galaxies with depleted cores . we explore several galaxy structural parameter relations for 28 core - srsic early - type galaxies with carefully acquired core - srsic parameters . the good agreement between the structural parameters from this work and those form our initial study using @xmath100@xmath147 profiles ( dullo & graham 2012 ) suggests that the different correlations that will be shown here agree with those of dullo & graham ( 2012 ) . fig . [ figiii6 ] shows the relation between the core - srsic break radius @xmath1 ( table [ tabbb2 ] ) and central galaxy properties including ( a ) the core - srsic model s central @xmath104-band surface brightness @xmath204 ( table [ tabbb2 ] ) , ( b ) the break surface brightness @xmath205 ( table [ tabbb2 ] ) and ( c ) the velocity dispersion @xmath106 ( table [ tabbb1 ] ) . the solid and dashed lines shown in each panel of figs . [ figiii6 ] to [ figiii9 ] are two distinct linear regression fits obtained with and without the three elliptical galaxies ( ngc 1700 , ngc 3640 and ngc 7785 ) with questionably small cores ( @xmath206 ) . we note that all the relations given in table [ tabbb30 ] are for the galaxy data without these three questionable galaxies . using the ordinary least squares ( ols ) bisector regression from feigelson & babu ( 1992 ) , a fit to the @xmath1 and @xmath204 data gives @xmath207 , while applying the bisector fit to @xmath1 and @xmath205 yields @xmath208 , and the bisector fit to @xmath1 and @xmath106 yields @xmath209 ( table [ tabbb30 ] ) . @llccccccccc@ relation&ols bisector fit&@xmath210 ( vertical scatter ) + + @xmath211 & @xmath212&0.24 dex + @xmath213&@xmath214&0.18 dex + @xmath215&@xmath216&0.29 dex + @xmath217&@xmath218&0.30 dex + @xmath219&@xmath220&0.43 dex + @xmath221&@xmath222&0.45 dex + @xmath223&@xmath224&0.28 + @xmath225 & @xmath226&0.80 + @xmath227&@xmath228&0.80 + @xmath229&@xmath230&1.17 + @xmath231 ( @xmath232 derived @xmath4 for 23 galaxies & @xmath233&0.27 dex + plus 8 direct @xmath4 masses ) & & + @xmath231 ( @xmath234 derived @xmath4 for 23 galaxies & @xmath235&0.27 dex + plus 8 direct @xmath4 masses ) & & + similar to fig . [ figiii6 ] , fig . [ figiii7 ] reveals that the core - srsic break radii @xmath1 correlate with global galaxy properties such as ( a ) the @xmath104-band absolute magnitude @xmath21 ( table [ tabbb1 ] ) and ( b ) the effective radius @xmath8 ( table [ tabbb2 ] ) . the bisector fit gives the near - linear relation between @xmath1 and @xmath21 as @xmath236 , while the fitted relation for the @xmath1 and @xmath8 data is @xmath237 ( table [ tabbb30 ] ) . of all the relations ( figs . [ figiii6 ] and [ figiii7 ] , table [ tabbb30 ] ) , the weakest correlation with a pearson correlation coefficient of @xmath238 is between @xmath1 and @xmath8 , while the for the remaining relations @xmath239 . we note that , intriguingly , the bulges seem to reveal a systematic trend in the @xmath211 , @xmath213 ( figs [ figiii6]a and [ figiii6]b ) and @xmath219 ( fig . [ figiii7]b ) diagrams , although it is more obvious in the @xmath219 plane . for a given break radius , bulges appear to be compact , i.e. , @xmath240 kpc ( see also dullo & graham 2013 and graham 2013 ) and possess somewhat fainter central and break surface brightnesses . thus , in fig [ figiii7](b ) we additionally include the ols bisector fit to the relation between @xmath1 and @xmath8 ( dotted line ) for only the elliptical galaxies , which is given by @xmath241 ( table [ tabbb30 ] ) . combining this relation with the @xmath242 relation ( table [ tabbb30 ] , fig . [ figiii7]a ) gives @xmath243 for elliptical galaxies with @xmath244 mag . this can be compared with the bright end of the curved @xmath9 relation given in section 5.3.1 of graham & worley ( 2008 ) . the linear regression fit to the luminous ( @xmath245 mag ) galaxies in graham & worley ( 2008 , their fig . 11a ) has a slope of 0.9 ( see also liu et al . 2008 , bernardi et al . 2007 ) , and it is @xmath246 steeper at brighter luminosities . while the elliptical galaxies appear to follow the steeper near - linear @xmath1 @xmath247 @xmath248 relation than the @xmath1 @xmath247 @xmath249 relation for the combined ( elliptical+bulge ) sample , the vertical rms scatters for both these relations are large ( table [ tabbb30 ] ) . , but shown here are the correlations between the core - srsic break radius @xmath1 and ( a ) absolute @xmath104-band magnitude of a galaxy or a bulge for a disc galaxy ( table [ tabbb1 ] ) , and ( b ) effective ( half - light ) radius @xmath8 ( table [ tabbb2 ] ) . in fig . [ figiii7](b ) we also include least - squares fit to the @xmath1 and @xmath8 data for just the elliptical galaxies ( dotted line ) . ] the tight correlations seen in the @xmath213 and @xmath215 diagrams ( figs . [ figiii6]b and [ figiii6]c ) were also shown by faber et al . ( 1997 , their fig . 8) , lauer et al . ( 2007a , their figs . 4 and 6 ) and dullo & graham ( 2012 , their fig . similar trends to the @xmath217 distribution ( fig.[figiii7]a ) can be seen in the works by faber et al . ( 1997 , their fig . 4 ) , ravindranath et al . ( 2001 , their fig . 5a , b ) , laine et al . ( 2003 , their fig . 9 ) , trujillo et al . ( 2004 , their fig . 9 ) , de ruiter et al . ( 2005 , their fig . 8) , lauer et al . ( 2007a , their figs . 5 ) and dullo & graham ( 2012 , their fig . the slope of the @xmath250 relation @xmath251 ( table [ tabbb30 ] ) that we find here can be compared to the similar slopes 1.15 , 0.72 , @xmath252 , @xmath253 and @xmath254 published by faber et al . ( 1997 ) , laine et al . ( 2003 ) , de ruiter et al . ( 2005 ) , lauer et al . ( 2007a ) and dullo & graham ( 2012 ) , respectively . it is worth noting that the @xmath250 relations in faber et al . ( 1997 ) , laine et al . ( 2003 ) and de ruiter et al . ( 2005 ) were derived using the nuker break radii , while lauer et al . ( 2007a ) used the `` cusp radius''the radius at which the negative logarithmic slope of the nuker model equals 0.5 . the slopes ( not the intercepts ) of the @xmath250 relations obtained using the core - srsic model and the nuker model break radii can coincidentally be consistent because of the way the nuker model systematically overestimates the break radius in comparison with the core - srsic model . also provided here , achieved using well constrained core - srsic fit parameters , are the @xmath213 and @xmath215 relations which are consistent with dullo & graham ( 2012 , their eqs . 7 , and 5 ) within the errors . due to coupling of @xmath1 and @xmath205 along the light profile ( dullo & graham 2012 , their figs . 17c and 18 ) , our @xmath213 relation agrees with that of lauer et al . ( 2007a , their eq . 17 ) . as shown in fig . [ figiii6](b ) , the core size of a galaxy ( @xmath1 ) and its surface brightness ( @xmath205 ) are closely related . [ figiii8 ] reveals that @xmath205 is thus also tightly correlated with ( a ) the central surface brightness @xmath204 , ( b ) the velocity dispersion @xmath106 , ( c ) the spheroid absolute magnitude @xmath21 , and ( d ) the effective ( half - light ) radius @xmath8 . the ols bisector fits are given in table [ tabbb30 ] . the @xmath225 and @xmath255 relations ( table [ tabbb30 ] ) agree with those in dullo & graham ( 2012 , their eqs . 10 and 9 , respectively ) . lastly , given the disagreement between the `` core '' parameters ( @xmath256 ) of the core - srsic model and the nuker model ( section [ sec3.1 ] ) , it is expected that our galaxy scaling relations ( table [ tabbb30 ] ) may differ from similar relations obtained using the nuker model . however , as mentioned above , some of the slopes ( not the intercepts ) of these scaling relations derived from these two models can agree . moreover , the close agreement between the core - srsic break radius and the lauer et al . ( 2007a ) `` cusp radius '' ( dullo & graham 2012 ) suggests that the scaling relations based on these two core measurements would be consistent . if galaxy core formation proceeds by the orbital decay of black hole binaries , from merging galaxies , as suggested by simulations ( e.g. , ebisuzaki et al . 1991 ; merritt 2006 ) and advocated by faber et al . ( 1997 ) , then a close relation between the core size ( @xmath1 ) and the black hole mass ( @xmath4 ) of a galaxy might be expected . given the well known @xmath257 ( ferrarese & merritt 2000 ; gebhardt et al . 2000 ) and @xmath258 ( marconi & hunt 2003 ; graham & scott 2013 ) relations , the strong @xmath215 and @xmath250 correlations in section [ sec4.1 ] hint at a tight @xmath231 relation . this trend is observed in fig . [ figiii9 ] , and quantified in table [ tabbb30 ] for our sample of 31 galaxies . eight of these galaxies have direct smbh mass measurements , while the remaining smbh masses were predicted using either the graham & scott ( 2013 ) `` non - barred @xmath257 '' relation relation , which has smaller uncertainties , because it is consistent with the core - srsic @xmath232 relation . ] @xmath259 or their @xmath260-band core - srsic @xmath258 relation which is converted here to @xmath104-band using @xmath261 ( fukugita et al . 1995 ; faber et al . 1997 ) , to give @xmath262 following graham et al . ( 2011 , see the discussion in their section 2.1.1 ) , we assumed a 10 % uncertainty on our velocity dispersions in order to estimate the errors on the smbh masses which were predicted using the @xmath257 relation . the predicted masses are given in table [ tabbb3 ] . note that since the resulting @xmath231 distributions , shown in figs . [ figiii9]a and b , are primarily driven by galaxies with predicted smbh masses , the observed trend may simply be due to the existence of the @xmath215 ( fig . [ figiii6]c ) , @xmath250 ( fig . [ figiii7]a ) relations and the @xmath257 , @xmath258 relations , although we find below that this is not the case . ( table [ tabbb2 ] ) and black hole mass . the smbh masses are acquired from direct smbh mass measurements for 8 circled galaxies , while for the remaining 23 galaxies the smbh masses are predicted using either ( a ) the graham & scott ( 2013 ) `` non - barred @xmath263 '' relation ( table [ tabbb3 ] ) or ( b ) their @xmath260-band core - srsic @xmath258 relation which is converted here to the @xmath104-band using @xmath261 ( fukugita et al . symbolic representations are as in fig . [ figiii6 ] . the solid lines are the least - squares fits to our core - srsic data , the shaded regions cover the corresponding 1@xmath106 uncertainties on these regression fits . pearson correlation coefficients , @xmath264 , ( and representative error bars ) are shown at the bottom ( top ) of each panel . ] [ figiii11 ] plots the @xmath231 relation for the eight galaxies with directly measured smbh masses . ngc 1399 has two distinct smbh mass measurements in the literature ( houghton et al.2006 , @xmath265 ; gebhardt et al . 2007 , @xmath266 ) . while this galaxy has a normal core ( 202 pc ) for its absolute magnitude ( @xmath267 mag ) , it appears that the @xmath268 dynamical smbh mass measurement of gebhardt et al . ( 2007 ) _ may _ be too small for its 202 pc core size . this mass estimate makes the galaxy an obvious outlier in the @xmath231 diagram ( fig . [ figiii11 ] ) . using the houghton et al . ( 2006 ) smbh mass for ngc 1399 , the ols bisector fit to the @xmath1 and @xmath4 data for our eight galaxies with direct smbh measurements yields @xmath269 with an rms scatter of @xmath270 dex in the log @xmath4 direction . when using the smbh mass for ngc 1399 from gebhardt et al . ( 2007 ) rather than from houghton et al . ( 2006 ) , @xmath271 with an intercept of 1.96 @xmath181 0.14 ( fig . [ figiii11 ] , dashed line ) . while this relation is in excellent agreement with eq . [ eqq34 ] , the scatter in this distribution is larger ( 0.35 dex in the log @xmath4 direction ) . we note that , as shown in fig . [ figiii11 ] , the near - linear @xmath231 relation established by these eight galaxies ( eq . [ eqq34 ] , solid line ) is consistent with the relations constructed by including the remaining sample galaxies with predicted smbh mass measurements ( figs . [ figiii9]a and b , table [ tabbb30 ] ) . however , the @xmath258 relation ( eq . [ eqq34b ] , graham & scott 2013 ) appears to somewhat overpredict the smbh masses relative to the @xmath257 relation ( eq . [ eqq34a ] , graham & scott 2013 ) for our core - srsic galaxy sample . this can be seen from the smaller intercept of the ( @xmath258)-based @xmath231 relation , 1.75 @xmath181 0.06 ( fig . [ figiii9]b , table [ tabbb30 ] ) , compared to the intercept of the ( @xmath257)-based @xmath231 relation , @xmath272 ( fig . [ figiii9]a , table [ tabbb30 ] ) . this unexpected situation has arisen because of a difference in the @xmath273 relation between the core - srsic galaxy sample in graham & scott ( 2013 ) and that used here . we find a 2.44@xmath106 difference between the intercepts of the @xmath273 relations from these two studies , which largely explains the 3.25@xmath106 difference between the intercepts of the ( @xmath257)-based and ( @xmath274)-based @xmath231 relations ( table [ tabbb30 ] ) . to further appreciate the ( galaxy core)-(smbh mass ) connection we derive additional @xmath231 relations by combining the non - barred @xmath257 relation from graham & scott ( 2013 ) with the @xmath215 relation ( table [ tabbb30 ] ) to obtain the new @xmath231 relation @xmath275 which is in good agreement with eq . [ eqq34 ] . similarly , combining the @xmath276 relation ( table [ tabbb30 ] ) with the core - srsic @xmath258 relation from graham & scott ( 2013 , their table 3 @xmath260-band ) , which is converted here to the @xmath104-band using @xmath277=1.0 , gives @xmath278 although we only have eight galaxies with direct black hole mass measurements , eq . [ eqq34 ] is consistent ( i.e. , overlapping @xmath179 uncertainties ) with the two inferred relations ( eqs . [ eqq35 ] and [ eqq36 ] ) . ( table [ tabbb2 ] ) and black hole mass ( table [ tabbb3 ] ) for eight galaxies with dynamically determined ( i.e. , direct ) black hole mass @xmath4 measurements . for ngc 1399 we include two direct smbh mass measurements : ( i ) ( 4.7@xmath1810.6)@xmath99@xmath279 ( gebhardt et al . 2007 ) and ( ii ) ( 1.2@xmath181@xmath280)@xmath99@xmath281@xmath282 ( houghton et al . 2006 ) . the solid line is the least - squares fit assuming the houghton et al . ( 2006 ) smbh mass for ngc 1399 , while the dashed line uses the gebhardt et al . ( 2007 ) mass , see the text for further detail . the inner shading marks the @xmath283@xmath106 uncertainty on eq . [ eqq34 ] , while the outer shading extends this by 0.26 dex ( the rms scatter ) in the log @xmath4 direction . the dotted and dashed - dotted lines are the fits for the full galaxy sample shown in figs . [ figiii9](a ) and ( b ) , respectively ( table [ tabbb30 ] ) . the pearson correlation coefficient , r , obtained when using the houghton et al . ( 2006 ) smbh mass for ngc 1399 is shown at the bottom of the panel . ] @llcccccc@ galaxy&@xmath284&@xmath285&log ( @xmath286)&log ( @xmath287)&log ( @xmath288)&@xmath289 + & & & & & & + ( 1)&(2)&(3)&(4)&(5)&(6)&(7 ) + + ngc 0507 & 1.40[a ] & 5.5&8.34&9.08&9.24@xmath290[p]&0.69 + ngc 0584 & 1.27[b]&4.5&7.78&8.43 & 8.29@xmath290[p]&1.38 + ngc 0741 & 1.32[a]&5.0&8.96&9.66&9.12@xmath290[p]&3.43 + ngc 1016 & 1.32[a]&5.0&9.10&9.80&9.21@xmath291[p]&3.85 + ngc 1399 & 1.21[a]&4.0&9.57&9.97&8.67 @xmath292 , 9.07 @xmath293[d]&20.0 , 7.90 + ngc 1700 & 1.29[b]&4.8&7.77&8.45&8.65@xmath291[p]&0.63 + ngc 2300 & 1.33[b]&5.0&7.81&8.51&8.86@xmath291[p]&0.45 + ngc 3379 & 1.28[b]&4.6&8.63&9.29&8.60@xmath294[d]&4.90 + ngc 3608 & 1.29[b]&4.8&7.82&8.50&8.30@xmath295[d]&1.57 + ngc 3640&1.22[a]&4.0&6.35&6.96&7.99@xmath291[p]&0.09 + ngc 3706 & 1.34[a]&5.2&8.30&9.01&8.94@xmath290[p]&1.18 + ngc 3842&1.38[a]&5.6&9.20&9.95&9.98@xmath296[d]&0.93 + ngc 4073 & 1.17[a]&3.2&8.50&9.00&8.98@xmath290[p]&1.05 + ngc 4278 & 1.26[b]&4.5&8.31&8.96&8.62@xmath290[p]&2.17 + ngc 4291 & 1.27[a]&4.5&8.15&8.81&8.52@xmath297[d]&1.94 + ngc 4365 & 1.33[b]&5.0&8.68&9.38&8.81@xmath290[p]&3.70 + ngc 4382 & 1.10[b]&2.6&7.68&8.09&7.95@xmath290[p]&1.37 + ngc 4406 & 1.25[b]&4.5&8.28&8.93&8.61@xmath290[p]&2.12 + ngc 4472&1.33[b]&5.0&8.59&9.29&9.15@xmath290[p]&1.40 + ngc 4552 & 1.29[b]&4.8&8.02&8.70&8.67@xmath298[d]&1.08 + ngc 4589 & 1.33[b]&5.0&7.50&8.22&8.49@xmath290[p]&0.54 + ngc 4649&1.34[a]&5.2&9.05&9.76&9.67@xmath299[d]&1.22 + ngc 5061 & 1.24[a]&4.2&8.64&9.27&8.05@xmath291[p]&16.7 + ngc 5419 & 1.35[a]&4.7&9.70&10.37&9.57@xmath300[p]&6.26 + ngc 5557 & 1.18[a]&3.2&8.23&8.74&8.78@xmath290[p]&0.90 + ngc 5813&1.31[b]&5.0&8.24&8.93&8.83@xmath298[d]&1.27 + ngc 5982 & 1.26[b]&4.5&8.24&8.89&8.65@xmath291[p]&1.75 + ngc 6849&1.03[a]&2.4&7.97&8.35&8.04@xmath291[p]&2.03 + ngc 6876&1.26[a]&4.5&8.52&9.17&8.55@xmath290[p]&4.25 + ngc 7619&1.35[b]&4.7&8.99&9.66&9.37@xmath290[p]&1.95 + ngc 7785&1.33[a]&5.0&6.67&7.37&8.80@xmath290[p]&0.04 + notes.col . ( 1 ) galaxy name . ( 2 ) galaxy colour : we use the lauer et al . ( 2005 ) central @xmath284 colours [ b ] when available ; otherwise the @xmath284 colours [ a ] were taken from the hyperleda database . ( 3 ) @xmath104-band stellar mass - to - light ( @xmath301 ) ratios determined using the galaxy colours ( col . 2 ) and the colour - age - metallicity-@xmath302 relation given by graham & spitler ( 2009 , their fig . ( 4 ) central luminosity deficit in terms of @xmath104-band solar luminosity . ( 5 ) central stellar mass deficit determined using col . ( 3 ) and col . ( 6 ) smbh mass . sources : [ p ] supermassive black hole mass predicted using the graham & scott ( 2013 , their table 3 and fig . 2 ) `` non - barred @xmath232 '' relation ( and the `` barred @xmath232 '' relation for ngc 6849 ) ; [ d ] galaxies with dynamically determined smbh mass measurements taken from graham & scott ( 2013 ) . for ngc 1399 , we use two direct ( dynamically determined ) smbh mass measurements taken from gebhardt et al . ( 2007 , @xmath3034.7@xmath304@xmath99@xmath305 @xmath282 ) and houghton et al . ( 2006 , @xmath3031.2@xmath306@xmath99@xmath281@xmath282 ) and adjusted to our distance . we use eq . 4 from graham et al . ( 2011 ) , updated according to the relation in graham & scott ( 2013 ) , as well as the @xmath106 values in table [ tabbb1 ] and assume a 10% uncertainty on @xmath106 to estimate the error on the predicted smbh mass ( see graham et al . 2011 , their section 2.1.1 ) . ( 7 ) ratio between mass deficit and black hole mass . as mentioned before , the central stellar mass deficits of core - srsic galaxies are naturally generated through the gravitational sling - shot ejection of core stars by the inspiraling black hole binaries that that are formed in a merger remnant ( begelman et al . 1980 ; ebisuzaki et al . a key point to note is that high - accuracy simulations ( e.g. , milosavljevi & merritt 2001 ; merritt 2006 ) predicted that multiple dissipationless mergers will have cumulative effects on core formation . merritt ( 2006 ) found that the total stellar mass deficit , @xmath10 , after @xmath307 successive dry major mergers is @xmath308 , with @xmath4 the final smbh mass . past studies have quantified this stellar mass deficit from the difference in luminosity , @xmath309 , between the inward extrapolation of the outer srsic profile ( of the core - srsic model ) and a sharp - transition ( graham et al . core - srsic model ( graham 2004 ; ferrarese et al . 2006 ; hyde et al . 2008 ) . here , we apply the same prescription for @xmath309 as in these past works but we use a smoother transition ( instead of a sharp ) core - srsic model by applying a finite @xmath143 in eq . [ eqq10 ] ( cf . also dullo & graham 2012 , 2013 ) . therefore , the difference in luminosity between the outer srsic model ( eq . [ eqq9 ] ) and the core - srsic model ( eq . [ eqq37 ] ) is the central stellar luminosity deficit @xmath310 . for each galaxy this luminosity deficit is converted into a mass deficit using the @xmath104-band stellar mass - to - light ( @xmath301 ) ratio given in table [ tabbb3 ] . in order to determine these @xmath311 ratios , the central ( if available , otherwise the galaxy ) @xmath284 colours ( table [ tabbb3 ] ) together with the colour - age - metallicity-(@xmath301 ) diagram ( graham & spitler 2009 , their fig . a1 ) were used , assuming a 12 gyr old stellar population . we note that the graham & spitler ( 2009 ) colour - age - metallicity-(@xmath301 ) diagram is constructed using the bruzual & charlot ( 2003 ) stellar population models and the chabrier ( 2003 ) stellar initial mass function ( imf ) . recent works suggest that the imf may vary with velocity dispersion for early - type galaxies but there is a significant scatter in this relation ( e.g. , cappellari et al . 2012 , 2013 ; conroy & van dokkum 2012 ; spiniello et al . 2012 ; wegner et al . 2012 ; ferreras et al . 2013 ; zaritsky et al . galaxies with low - velocity dispersions require a kroupa ( 2001 , or a chabrier 2003 ) imf , while high - velocity dispersion galaxies ( @xmath312 km s@xmath23 ) may prefer a `` bottom - heavy '' imf having a steeper slope than that of salpeter ( 1955 ) , although cappellari et al . ( 2013 , their fig . 15 ) found a shallow ( @xmath313@xmath106 relation for their slow rotators with @xmath314 km s@xmath23 ( see also rusli et al . 2013 ; clauwens , schaye & franx 2014 ) . once these mass - to - light issues are settled , it may be worth trying to refine the stellar mass deficits reported here . versus black hole mass ( @xmath4 ) for the 31 core - srsic galaxies listed in table [ tabbb1 ] . the data for four s0s ( ngc 507 , ngc 2300 , ngc 4382 and ngc 6849 ) and one elliptical ( ngc 3706 ) are taken from dullo & graham ( 2013 ) . the graham & scott ( 2013 , their table 3 ) `` non - barred @xmath315-@xmath106 '' relation was used for estimating the smbh masses of 23 galaxies , while for the remaining 8 galaxies ( enclosed in boxes ) we used their direct smbh mass measurements as given in graham & scott ( 2013 ) . for ngc 1399 , we also plot the larger black hole mass from houghton et al . ( 2006 ) adjusted for our distance of 19.4 mpc . three questionable galaxies with unusually small cores ( ngc 1700 , ngc 3640 and ngc 7785 ) are circled . a representative error bar is shown at the bottom of the panel . ] [ figiii12 ] plots the mass deficits that we derive against the dynamically determined or predicted smbh masses for our 31 core - srsic early - type galaxies ( table [ tabbb3 ] ) , the largest sample of core - srsic galaxies with extended light profiles that has been modelled to date . we find the mass deficits for these galaxies are typically @xmath316 @xmath4 , in agreement with past core - srsic model estimates ( graham 2004 ; ferrarese et al.2006 ; hyde et al . 2008 ; dullo & graham 2012 , 2013 ) . this translates to core - srsic galaxy formation through one to several successive `` dry '' major merger events , consistent with theoretical expectations ( e.g. , haehnelt & kauffmann 2002 ) . in addition , recent observations on close major merger pairs have revealed that massive galaxies have undergone @xmath317 major mergers since @xmath318 ( e.g. , bell et al.2004 , 2006 ; bluck et al . 2012 ; man et al . 2012 ; xu et al . the most massive galaxies , with stellar mass @xmath319 , may have experienced up to six major mergers since @xmath320 ( conselice 2007 ) . four elliptical galaxies ngc 1399 , ngc 3640 , ngc 5061 and ngc 7785 are outliers from the main @xmath321 distribution ( fig . [ figiii12 ] ) . ngc 3640 and ngc 7785 are two of the three galaxies with unusually small depleted cores ( see section 4 and appendix [ apppqes ] ) . they both have small mass deficits ( @xmath322 and @xmath323 ) for their predicted smbh masses ( @xmath324 and @xmath325 ) . the remaining galaxy with a questionable core ( ngc 1700 ) is an outlier in most central galaxy scaling relations ( figs . [ figiii6 ] , [ figiii7 ] and [ figiii8 ] ) but it has a normal mass deficit for its smbh mass . this owes to the fact that ngc 1700 , unlike ngc 3640 and ngc 7785 , has a relatively steep outer srsic profile ( @xmath326 ) . this larger srsic index value helps to compensate for the small core size , taking its estimated mass deficit into the normal range in fig . [ figiii12 ] . in the case of the potentially outlying galaxy ngc 1399 , the discussion given in section [ sec5.1 ] explains the behavior seen here . the smaller dynamical smbh mass determination by gebhardt et al . ( 2007 ) yields an inflated @xmath289 ratio of @xmath327 , while assuming the larger dynamical smbh mass measurement of houghton et al . ( 2006 ) implies a somewhat reasonable value of @xmath328 . the situation with the fourth offset elliptical galaxy ngc 5061 is somewhat unclear given it is not a deviant galaxy in the other galaxy scaling relations ( section [ sec4 ] ) . the core - srsic model fits its light profile very well with a fairly small rms residual of @xmath329 mag arcsec@xmath122 , but from these fit parameters we determine a large mass ratio @xmath330 . its core size @xmath331 pc is a good match to its @xmath104-band absolute magnitude @xmath332 mag ( table [ tabbb1 ] ) and velocity dispersion @xmath333 km s@xmath23 ( table [ tabbb1 ] ) , but its srsic index @xmath334 ( table [ tabbb2 ] ) may be too high for the aforementioned galaxy properties . indeed , ngc 5061 has the largest srsic index from our sample , attributed to its noticeably straight surface brightness profile ( fig . [ figa1 ] ) . this may suggest that the envelope of this galaxy was built via several dry minor and major merging events ( hilz et al . 2013 ) . on the other hand , it has the third smallest velocity dispersion @xmath335 km s@xmath23 ( hyperleda s mean value ) from our sample which seems to underpredict its smbh mass @xmath336 ( table [ tabbb3 ] ) . using the largest reported velocity dispersion value @xmath337 km s@xmath23 ( davies et al . 1987 ) , instead of the mean measurement , increases its predicted smbh mass roughly by a factor of 2 , i.e. , @xmath338 . as such the associated @xmath289 ratio reduces roughly by a factor of two , to give @xmath339 . this latter ratio is marginally consistent with the @xmath289 distribution shown in fig . [ figiii11 ] . in summary , it appears that both the high srsic index and the relatively low velocity dispersion of ngc 5061 _ may _ collectively act to inflate the @xmath289 ratio to 17 . versus cumulative number of `` dry '' major merger events ( @xmath307 ) for the 31 core - srsic galaxies listed in table [ tabbb1 ] . symbolic representations are as in fig . [ figiii12 ] . pearson correlation coefficient , @xmath264 , ( and representative error bar ) are shown at the top ( bottom ) of the panel . ] as shown in fig . [ figiii12b ] , excluding the three galaxies with questionably small / real cores , and using the hougton et al . ( 2006 ) smbh mass for ngc 1399 , the ols bisector regression between @xmath340 and @xmath289 ( @xmath341 @xmath342 , merritt 2006 ) gives @xmath343 in fig . [ figiii12c ] , we explore the behavior of the @xmath344 ratio with @xmath4 . we find that the distribution in this diagram appears largely consistent with the simulations by merritt ( 2006 , his table 2 ) . as in merritt ( 2006 ) , for the first merger ( @xmath345 ) , an object with a supermassive black hole mass of ( 1/3)(@xmath346 ) was added to a system having a black hole mass of ( 2/3)(@xmath346 ) . the same black hole mass of ( 1/3)(@xmath346 ) was then added for each successive merger . these accumulated black hole masses are plotted against @xmath342 to construct each of the three curves shown in fig . [ figiii12c ] . form this figure it is apparent that core - srsic galaxies with the same smbh mass ( or merger history ) can have different merger histories ( smbh masses ) . this is consistent with the notion that the stellar mass deficits of core - srsic galaxies reflect the amount of merging as well as the masses of their smbhs . the mean elliptical galaxy @xmath289 ratio from graham ( 2004 ) is 2.1 @xmath181 1.1 , while ferrarese et al . ( 2006 ) reported a mean @xmath347 ratio of 2.4 @xmath181 0.8 after excluding the s0 galaxy ngc 4382 from their sample . hyde et al . ( 2008 ) found a comparable mean @xmath289 ratio of 2.3 @xmath181 0.67 for their sample . in dullo & graham ( 2012 ) , we modelled @xmath100@xmath147 light profiles and cautioned that the outer srsic parameters might be less constrained than desirable , although the srsic indices were shown to be in a fair agreement with those determined from published fits to larger radial extents . nonetheless , we reported tentative @xmath344 ratios that were some 0.5@xmath348 . in dullo & graham ( 2013 ) we fit the extended light profiles of four core - srsic lenticular galaxies ( ngc 507 ; ngc 2300 ; 4382 and ngc 6849 ) using a core - srsic model for the bulge plus an exponential model for the disc . one suspected s0 galaxy ngc 3706 was found to have a stellar distribution that is best described by the core - srsic model and was thus reclassified as an elliptical galaxy . using these core - srsic fit parameters we reported a robust @xmath349 for these five core - srsic galaxies ( dullo & graham 2013 , their fig . 4 ) , which are also shown here in fig . [ figiii12 ] . to supermassive black hole mass ( @xmath4 ) as a function of @xmath350 for the 31 core - srsic galaxies listed in table [ tabbb1 ] . the three curves are based on the simulations by merritt ( 2006 , his table 2 ) . that is , we started with a certain ( total ) supermassive black hole mass @xmath4 for @xmath345 and a third of this black hole mass was added for each successive merger shown by a star . as such , the cumulative black hole mass @xmath4 increases linearly with the number of merger @xmath307 , but the binary mass ratio decreases with @xmath307 . these black hole masses were then plotted against @xmath342(@xmath341 @xmath289 , merritt 2006 ) to construct each curve , see the text for further detail . ] prior to graham ( 2004 ) who reported @xmath289 ratio of @xmath351 , previous estimates based on nuker model parameters had been an order of magnitudes larger ( e.g. , milosavljevic & merritt 2001 ; milosavljevi et al . 2002 ; ravindranath 2002 ) . graham ( 2004 ) argued that the universe was some ten times less violent , in terms of major galaxy mergers , than previously believed . subsequent works using nuker model parameters ( lauer et al . 2007a ; gltekin et al . 2011 ) or subjectively identifying the core from visual inspections ( kormendy et al . 2009 ) have reported mass deficits up to an order of magnitude larger than typically found here . as discussed in section [ sec3 ] and appendix [ apppc ] , this discrepancy is partly due to the contrasting core sizes measured by these distinct methodologies . using a model - independent analysis of the light profiles , hopkins & hernquist ( 2010 ) confirmed the result of graham ( 2004 ) and reported @xmath352 . the larger mass deficits ( @xmath353 ) of kormendy & bender ( 2009 ) are also partly because they used the dynamical mass - to - light @xmath354 ratios rather than the stellar @xmath301 ratios to derive the mass deficits . there is not much dark matter at the centre of massive spheroids ( e.g. , dekel et al . 2005 ) , and assuming the dynamical @xmath354 for the galaxy (= [ @xmath355 + @xmath356 $ ] / @xmath2 ) overpredicts the central mass deficits . recently , kormendy & ho ( 2013 ) scaled up the smbh masses for the galaxies in kormendy & bender ( 2009 ) by about a factor of two and reported a new lower , mean ratio @xmath357 . alternative mechanisms for the production of enhanced depleted cores in luminous galaxies has been suggested in the literature ( boylan - kolchin et al . 2004 ; gualandris & merritt 2008 ; kulkarni & loeb 2012 ) . gualandris & merritt ( 2008 ) invoked the recurrent core passages of gravitationally kicked and `` recoiled '' ( and then oscillating about the centre ) smbh in a merger remnant to explain the formation of large stellar mass deficits that are up to 5 @xmath15 . kulkarni & loeb ( 2012 ) also suggested that mass deficits as large as @xmath358 could be formed as a result of the action of multiple smbhs from merging galaxies . if these processes have always occurred , then the result we found here , i.e. , @xmath359 @xmath4 for our sample galaxies implies that these galaxies are formed via just one major merger or minor merger events only , at odds with both observations of close galaxy pairs of equal mass , and theoretical expectations ( e.g. , khochfar & burkert 2003 ; bell et al . 2004 ; naab et al . 2006 ; bluck et al . however , spheroids with @xmath360 ( i.e. , @xmath361 probably need some black hole oscillations as 8 or more major mergers seem excessive . we find two elliptical galaxies ( ngc 1399 , ngc 5061 ) with @xmath362 , suggesting that these oversized mass deficits might be partly due to the action of their gravitationally kicked smbhs . it should be noted that several authors have considered alternative ways in which cores can be produced in luminous galaxies . dissipationless collapses in existing dark matter haloes were invoked by nipoti et al . ( 2006 ) as a possible mechanism for forming depleted cores . another suggested alternative was the adiabatic expansion of the core region driven by the rapid mass loss from the effects of supernova and agn feedback ( navarro et al . 1996 ; read & gilmore 2005 ; peirani , kay & silk 2008 ; martizzi et al . 2012 , 2013 ) and krajnovi et al . ( 2013 ) pointed out that this scenario is compatible with the properties of `` core slow rotators '' . in addition , goerdt et al . ( 2010 ) proposed that the energy transferred from sinking massive objects would produce cores that are as large as 3 kpc in size . these suggested mechanisms , however , are not without problems . for example , it is unclear how the cores created by the simulations in nipoti et al . ( 2006 ) are guarded against infalling satellites ( which would replenish the core ) in the absence of a central smbh . also , the oversized ( @xmath363 kpc ) cores produced by the latter two mechanisms ( e.g. , goerdt et al . 2010 ; martizzi et al.2012 , 2013 ) are generally inconsistent with the typical @xmath3640.5 kpc cores observed in galaxies ( e.g. , trujillo et al . 2004 ; ferrarese et al . 2006 ; richings et al . 2011 ; dullo & graham 2012 ) . section 6.1 of dullo & graham ( 2013 ) provides further details , including the merits and weaknesses of these core formation models in the context of the observations . finally , we note that the standard lambda cold dark matter ( @xmath365cdm ) model predicts that elliptical galaxies are built via major mergers ( e.g. , kauffmann et al . 1993 ; khochfar & burkert 2005 ) . in this hierarchical picture , the bulges of lenticular galaxies form early via major mergers while their discs grow later through gas accretion events ( e.g. , steinmetz & navarro 2002 ) . alternatively , the evolutionary transformation of spiral galaxies into s0 galaxies via mechanisms such as ram pressure stripping has been suggested ( e.g. , gunn & gott 1972 ) . comparing various galaxy scaling relations , luarikainen et al . ( 2010 ) , for example , showed that the bulges of s0 galaxies are closely correlated with the bulges of bright spiral galaxies , having @xmath366(bulge ) @xmath36720 mag , than with elliptical galaxies . on the other hand , in dullo & graham ( 2013 ) , we argued that core - srsic s0s with @xmath21(bulge ) @xmath368 mag might be assembled inside - out in two stages : an earlier `` dry '' major merger process involving smbhs forms their bulge component , while the surrounding disc is subsequently formed via cold gas accretion . the bulges of our small s0 galaxy sample tend to have @xmath369 ( dullo & graham 2013 , see also balcells et al . 2003 and laurikainen et al . 2005 ) , however , core - srsic elliptical galaxies have @xmath370 . as we mentioned in section [ sec4.1 ] , for the same core size , these massive bulges tend to be compact ( @xmath240 kpc , fig . [ figiii7]b ) and have somewhat fainter break , and central , surface brightnesses than the elliptical galaxies ( figs . [ figiii6]a and [ figiii6]b ) . furthermore , the @xmath289 ratio for s0s is lower compared to the spread seen in elliptical galaxies ( fig . [ figiii12 ] ) . this implies that the bulges of core - srsic s0s have experienced fewer major mergers than core - srsic elliptical galaxies ( fig . [ figiii12b ] ) . we extracted the major - axis surface brightness profiles from 26 core - srsic early - type galaxies observed with the _ hst _ wfpc2 and acs cameras . we additionally included five core - srsic early - type galaxies ( ngc 507 , ngc 2300 , ngc 3706 , ngc 4382 and ngc 6849 ) from dullo & graham ( 2013 ) . this compilation represents the largest number of core - srsic galaxies modelled to large radii @xmath371 , giving the fitting functions enough radial expanse to robustly measure the galaxy stellar distribution ( see fig . [ figiii5 ] for a comparison of the srsic indices obtained from fits using @xmath372 profiles ) . we fit the extended surface brightness profiles of the 26 core - srsic elliptical galaxies using the core - srsic model , while light profiles of the remaining five core - srsic s0 galaxies were modelled with the core - srsic model for the bulge plus an exponential model for the disc . we accounted for additional nuclear cluster light using the gaussian function . our principal results are summarised as follows : \1 . the global stellar distributions of core - srsic elliptical galaxies are robustly represented with the core - srsic model , while core - srsic lenticular galaxies are accurately described using the core - srsic bulge + exponential disc model . these fits yield a median rms scatter of 0.045 mag arcsec@xmath122 for our sample of 31 core - srsic galaxies , and argue against excessive multi - component srsic models ( section 3.4 ) . we provide updated core - srsic model parameters @xmath373 for 31 core - srsic early - type galaxies with spheroidal components having @xmath374 mag and @xmath375 km s@xmath23 . in general , there is a good agreement with the parameters obtained from our earlier analysis of the publicly available , but radially limited ( @xmath376 ) surface brightness profiles given by lauer et al . ( 2005 ) . the bulges of our core - srsic s0s are compact ( @xmath377 2 kpc ) and have @xmath378 ( dullo & graham 2013 ) , as compared to the core - srsic elliptical galaxies which typically have @xmath379 kpc and @xmath195 . \4 . the core - srsic model break radii are in agreement with both ( i ) the previously published core - srsic break radii and ( ii ) the model - independent break radii which mark the locations where the negative logarithmic slopes of the light profiles equal 0.5 ( carollo et al . 1997a ; dullo & graham 2012 ) . updated structural parameter relations involving both the central and global galaxy properties are provided in section [ sec4 ] . we have found tight correlations involving the central galaxy properties @xmath1 , @xmath204 , @xmath205 , @xmath106 and @xmath4 ( see table [ tabbb30 ] ) . we have also found near - linear relations between the break radius @xmath1 , and the spheroid luminosity @xmath2 and the smbh mass @xmath4 given by @xmath380 and @xmath5 . we additionally found a near - linear relation between @xmath1 and @xmath8 such that @xmath381 but with a large scatter . we have derived central stellar mass deficits in 31 early - type galaxies that are typically 0.5 to 4 times the host galaxy s black hole mass . given published theoretical results , these mass deficits suggest a few dissipationless major mergers for core - srsic galaxies . \7 . as noted in dullo & graham ( 2013 ) , mass deficits in core - srsic s0s suggest a two stage assembly : an earlier `` dry '' major merger event involving smbhs creates the bulges with depleted cores , and the disc subsequently builds up via cold gas accretion events . the relation between the stellar mass deficit @xmath10 and the cumulative number @xmath307 of major `` dry '' mergers that the galaxy has undergone is such that @xmath10 is roughly @xmath247 @xmath382 . the close relation between the galaxy cores and the smbhs supports the popular core depletion hypothesis where cores are thought to be created by sinking binary smbhs that eject stars away from the centres of their host galaxies . the small cores seen in some galaxies , if real , may arise from loss cone regeneration by newly produced stars and/or recent stellar accretion events . alternatively , small galaxy cores can be interpreted as a sign of minor mergers . we have identified two galaxies ( ngc 1399 and ngc 5061 ) which have a high @xmath289 ratio , suggesting that their central smbh may have experienced a kick due to a gravitational - radiation recoil event leading to multiple core passages . this research was supported under the australian research council s funding scheme ( dp110103509 and ft110100263 ) . this research has made use of the nasa / ipac extragalactic database ( ned ) which is operated by the jet propulsion laboratory , california institute of technology , under contract with the national aeronautics and space administration . we acknowledge the usage of the hyperleda database ( http://leda.univ-lyon1.fr . ) . btd is grateful for the supra scholarship offered by swinburne university of technology , and travel support from the astronomical society of australia . figure [ figa1 ] shows core - srsic model fits to the major - axis surface brightness profiles of the 26 ( suspected elliptical ) core - srsic galaxies listed in table [ tabbb1 ] . notes on two of these 26 galaxies ( ngc 4073 and ngc 6876 ) with complicated structures are given below , while their photometric profiles are shown in figs . [ figb1 ] and [ figb2 ] . ngc 4073 is a cd galaxy in the poor mkw 4 cluster ( de vacouleurs et al . it has a double classification in laurikainen et al . ( 2011 , their table 3 ) , i.e. , sab0@xmath383 for the inner regions and e@xmath384 for the outer parts . also , this galaxy has a small bump in its light profile over @xmath385 due to a nuclear ring of stars ( lauer et al . our models are not designed to fit stellar rings , thus we simply exclude data points that are contaminated by the nuclear ring light . further , the core - srsic model fit to ngc 4073 shows an excess of light on top of the core - srsic light distribution over @xmath386 , creating the residual pattern seen in fig . [ figa1 ] . cd galaxies are known to grow via cannibalism of their neighboring , less massive cluster galaxies ( e.g. , ostriker & hausman 1977 ) . if these accreted objects ( or at least their dense cores ) survive , they would be visible as extra light like in ngc 4073 . however , we did not find clear evidence for such a feature in this galaxy s 2d residual image , possibly suggesting a collection of large scale disturbances in the galaxy . [ figb1 ] shows the surface brightness and photometric profiles for ngc 4073 determined using the iraf task ellipse . these profiles are connected with the galaxy s residual structure observed in the model fit ( fig . [ figa1 ] ) . the position angle profile shows an abrupt 90@xmath387 twist at around @xmath388 . the galaxy is also highly flattened ( @xmath389 ) outside @xmath390 , while the ellipticity shows a steady drop from 0.45 ( at @xmath391 ) to 0.1 ( at @xmath392 ) accompanied by negative b@xmath393 values , then becomes nearly circular ( @xmath394 ) inside the core ( @xmath395 ) . fisher , illingworth & franx ( 1995 ) reported ngc 4073 as having a counterrotating stellar core which they attribute to possible cannibalism events . laurikainen et al . ( 2011 ) interpreted the excess light and the associated ellipticity trend over @xmath396 as being due to a weak inner bar . without accounting for its light excess at @xmath397 , and omitting its ring light , the core - srsic model fits the light distribution of ngc 4073 with rms residual of @xmath398 mag arcsec@xmath122 . ngc 6876 is a dominant elliptical galaxy in the pavo group . as shown in fig . [ figb2 ] , this galaxy is only @xmath399 from its smaller companion , the elliptical galaxy ngc 6877 . using multi - wave observations , machacek et al . ( 2005 , 2009 ) showed evidence for interaction between ngc 6876 and the highly disturbed spiral galaxy ngc 6872 . dullo & graham ( 2012 , their fig . 4 ) showed the double optical nucleus in ngc 6876 possibly associated with the dense core of a lesser galaxy or the ends of an inclined ring ( lauer et al . 2002 ) . the residual structure at @xmath400 ( fig . [ figa1 ] ) is associated with a 13@xmath387 twist in the position angle , and the change in the ellipticity and the isophote shape of this galaxy beyond its core regions . * ngc 1700*[sec8.1.1 ] + ngc 1700 , a luminous ( @xmath401 mag ) elliptical galaxy with an estimated ( luminosity weighted ) age of 3@xmath1811 gyr ( brown et al . 2000 ) and a velocity dispersion @xmath402 km s@xmath23 , shows post merger morphological signatures such as shells , and boxy isophote at large radii @xmath403 ( franx et al . 1989 ; statler et al . 1996 ; whitmore et al . 1997 ; brown et al . 2000 ; stratler & mcnamara 2002 ) . it was also reported to be offset from the fundamental plane ( e.g. , statler et al . 1996 ; reda et al . given this galaxy s recent ( wet ) merger history , its small core size may be because of loss cone regeneration via newly produced and/or accreted stars . that is , a pre - existing large core in this galaxy may be partially replenished by these new stars . we do however caution that because this core is defined by just the inner two data points of the deconvolved surface brightness profile , i.e. , within @xmath404 , its validity can be questioned , but it remains unlikely that they are any bigger than reported here . + * ngc 3640 * + ngc 3640 is a fast rotating ( krajnovi et al . 2013 ) early - type galaxy with @xmath405 mag and @xmath406 km s@xmath23 . prugniel et al . ( 1988 , see also michard & prugniel 2004 ) noted that this galaxy is a merger in progress , possibly with a gas - poor disc system . not only did they find low surface brightness structures such as shells and ripples but they also showed that the @xmath407 radial colour profile of this galaxy gets bluer towards the inner region . hibbard & sansom ( 2003 ) however fail to detect neutral hydrogen associated with ngc 3640 . also , tal et al . ( 2009 ) recently reported this galaxy to be highly disturbed system revealing morphological peculiarities . therefore , as in the case of ngc 1700 , this galaxy s undersized core and mass deficit can be interpreted as loss cone replenishment via new star formation and/or stellar accretion not associated with a secondary black hole ( see also krajnovi et al.2013 ) . it is also possible that the observed core , defined by just one inner data point , may not be real . + * ngc 7785 * + our core - srsic model fit to this elliptical galaxy light profile yields a core size of just 5 pc , which is too small for its absolute magnitude @xmath408 mag and velocity dispersion @xmath409 255 km s@xmath23 . there is no evidence for an ongoing or a recent merging event associated with this galaxy in the literature . thus , its small core size as well as undersized mass deficit suggest that one or a few minor ( instead of major ) dissipationless mergers might have taken place in the absence of loss cone refilling . alternatively , the apparent core in this galaxy , also defined by one data point , may be spurious . this section provides a review of five galaxies ( i.e. , excluding ngc 4073 already discussed above ) from our 26 suspected elliptical galaxies which were shown to have a bulge+disc stellar distribution in the literature . this galaxy is classified as an elliptical galaxy in the third reference catalogue , rc3 ( de vacouleurs et al . 1991 ) and as an s0 by sandage & tammann ( 1981 ) . laurikainen et al . ( 2010 , 2011 ) fit a 3-component ( bulge+bar+disc ) model to this galaxy s @xmath100@xmath410 @xmath411-band light profile and noted that it has a large disc - like outer envelope and a weak inner bar . however , these ( bar and disc ) components were not detected in the @xmath100@xmath412 @xmath104-band light profile that we modelled . we do however wish to bring them to the attention of readers . the virgo cluster galaxy ngc 4472 is classified as an elliptical galaxy in the rc3 but as an s0 by sandage & tammann ( 1981 ) . laurikainen et al . ( 2010 , 2011 ) identified a large - scale disc in this galaxy which dominates the light at large radii ( @xmath413 . in contrast , ngc 4472 was considered to be an elliptical galaxy by kormendy et al . ( 2009 ) who fit a single srsic model to its light profile over @xmath414 . we did not detect a disc component in our @xmath171 profile ( see also ferrarese et al . interestingly , however , this galaxy has the lowest srsic value ( @xmath415 ) from our 26 suspected elliptical galaxies ( table 2 ) , possibly suggesting an s0 morphology . it is classified as a slow rotator ( over its inner region ) by the atlas3d team ( cappellari et al . 2011 ) . like ngc 4472 , ngc 4552 is a member of the virgo cluster . it is classified as an elliptical galaxy in the rc3 , but was recognized as an s0 by sandage & tammann ( 1981 ) . while kormendy et al . ( 2009 ) adopted an elliptical morphology for this galaxy , their srsic model fit to its light profile over @xmath416 shows a clear residual structure , in agreement with the s0 morphology adopted by laurikainen et al . ( 2010 , 2011 ) . however , it too is classified as a slow rotator by cappellari et al . ( 2011 ) perhaps because they only sampled the pressure - supported , bulge - dominated inner portion of the galaxy . the disc light ( e.g. , laurikainen et al . 2011 ) , dominant at large radii , does not contribute to our @xmath100@xmath171 light profile ( see also ferrarese et al . 2006 ) . as hinted at in section [ sec3.2 ] , although ngc 5813 is classified as an elliptical galaxy ( e1 - 2 ) in rc3 , we find that its ( @xmath100@xmath412 ) light profile is best fit using the core - srsic bulge+exponential model with a small rms residual of @xmath417 mag arcsec@xmath122 . this suggests the galaxy may be an s0 disc galaxy , consistent with its steadily increasing ellipticity at @xmath418 as well as the kinks in the position angle and @xmath419 profiles at @xmath420 ( fig . [ figiii3 ] ) . the residual image of this galaxy shown in fig . [ figiii3 ] is regular , there is no evidence for a distinct morphological feature . trujillo et al . ( 2004 ) also concluded that ngc 5813 may be an s0 galaxy which is better described using a bulge+disc model . this however appears to disagree with the rotation curve given by efstathiou , ellis & carter ( 1982 ) . using long slit data , these authors showed that the core ( @xmath421 ) of ngc 5813 rotates rapidly , while beyond @xmath172@xmath147 the galaxy shows little rotation : evidence for a kinematically decoupled core . this galaxy is classified as an elliptical galaxy in the rc3 , yet it was identified as an s0 by sandage & tammann ( 1981 ) and also recently by laurikainen et al . we fit a core - srsic bulge plus a gaussian point source model to @xmath100@xmath171 ( @xmath104-band ) light profile , while laurikainen et al . ( 2011 ) fit a three - component ( bulge+lens+outer - disc ) model to a more radially extended @xmath422-band brightness profile . our @xmath100@xmath171 data does not probe this galaxy s large - scale disc . also , the lens component is not seen in our @xmath104-band light profile . recently , krajnovi et al . ( 2013 ) fit the nuker model ( kormendy et al . 1994 ) to the nuclear light profiles of 135 atlas@xmath423 galaxies . while these authors are aware that the nuker model is not a robust parametrization ( i.e. , the parameters are unstable ) , this model was used for the purpose of distinguishing core and coreless galaxies . as in the previous nuker model works ( e.g. , lauer et al . 2005 ) they opt to use @xmath424 ( the slope of the nuker model at @xmath173 ) as a diagnostic tool for separating their galaxies as core ( @xmath425 ) , intermediate ( 0.3 @xmath426 0.5 ) and power - law ( @xmath427 ) type . however , it has been shown ( graham et al . 2003 ; dullo & graham 2012 , their section 5.1 ) that the distance - dependent @xmath424 is not a physically robust or meaningful quantity to use . moreover , galaxies with small srsic indices will have @xmath428 but no central deficit relative to the inward extrapolation of their outer srsic profile . in dullo & graham ( 2012 , see also graham et al . 2003 ) , we highlighted that the nuker model is not ideal for describing central light profiles , and we revealed that seven ( @xmath429 ) of the sample galaxies with low @xmath94 and thus shallow inner profile slopes were misidentified as galaxies having depleted cores by the nuker model . four of these seven galaxies ( ngc 4458 , ngc 4473 , ngc 4478 and ngc 5576 ) are in common with krajnovi et al . krajnovi et al . ( 2013 ) classify three of them ( ngc 4458 , ngc 4478 and ngc 5576 ) as an intermediate type ( i.e. , coreless galaxies based on their criteria ) in agreement with dullo & graham ( 2012 ) and at odds with lauer et al . ( 2005 ) . they , however , reminded the reader that the nuker model actually detected cores in all these galaxies but this detection may change depending on the range over which the fits were done . it is perhaps worth clarifying that the nuker model break radii reported by lauer et al . ( 2005 ) for ngc 4458 , ngc 4478 and ngc 5576 are @xmath430 smaller than those from krajnovi et al.(2013 ) . these differing break radii have resulted in a significant difference in the @xmath424 measurements from these two works . for these three galaxies in question , krajnovi et al . ( 2013 ) found @xmath424 values that are @xmath431 times larger than those from lauer et al . for the remaining common galaxy ngc 4473 , krajnovi et al . ( 2013 ) found @xmath432 and subsequently classified it as a core galaxy in agreement with lauer et al . ( 2005 ) and in contrast to ferrarese et al . ( 2006 ) , kormendy et al . ( 2009 ) and dullo & graham ( 2012 ; 2013 , their section 9.2.2 ) . ngc 4473 does not have a central light deficit but quite the opposite : an embedded stellar disk ( e.g. , krajnovi et al . 2006 ; capellari et al . 2007 ) . 99999 allen , p.d . , driver , s.p . , graham , a.w . , cameron , e. , liske , j. , de propris , r. , 2006 , mnras , 371 , 2 andredakis y. c. , peletier , r. f. , & balcells , m. 1995 , mnras , 275 , 874 balcells , m. , graham , a. w. , domnguez - palmero , l. , & peletier , r. f. 2003 , apjl , 582 , l79 ballo l. , braito v. , della ceca r. , maraschi l. , tavecchio f. , dadina m. , 2004 , apj , 600 , 634 begelman , m. c. , blandford , r. d. , & rees , m. j.1980 , nature , 287 , 307 bell , e.f . , wolf c. , meisenheimer k. , et al . 2004 , apj , 608 , 752 bell e. f. et al . , 2006 , apj , 640 , 241 bianchi , s. , chiaberge , m. , piconcelli , e. , guainazzi , m. , & matt , g. 2008 , mnras , 386 , 105 binney , j. , & mamon , g. a. 1982 , mnras , 200 , 361 blakeslee , j. p. , lucey , j. r. , tonry , j. l. , hudson , m. j. , narayanan , v. k. , harris , b. j. , 2002 , mnras , 330 , 443 bluck , a. f. l. , conselice , c. j. , buitrago , f. , et al . 2012 , apj , 747 , 34 bouch , n. , murphy , m. t. , kacprzak , g. g. , proux , c. , con- tini , t. , martin , c. , & dessauges - zavadsky , m. , 2013 , science , 341 , 50 boylan - kolchin m. , ma c. , quataert e. , 2004 , apj , 613 , l37 brown r. j. n. , forbes d. a. , kissler - patig m. , brodie j. p. , 2000 , mnras , 317 , 406 bruzual g. , charlot s. , 2003 , mnras , 344 , 1000 buitrago f. , trujillo i. , conselice c. j. , haeussler b. , 2013 , mnras , 428 , 1460 burke - spolaor s. , 2011 , mnras , 410 , 2113 byun , y .- i . , et al . 1996 , aj , 111 , 1889 caon n. , capaccioli m. , donofrio m. 1993 , mnras , 265 , 1013 cappellari , m. , mcdermid , r. m. , alatalo , k. , et al . 2012 , natur , 484 , 485 cappellari , m. , mcdermid , r. m. , alatalo , k. , et al . 2013 , mnras , 432 , 1862 carollo , c. m. , franx , m. , illingworth , g. d. , & forbes , d. a. 1997a , apj , 481 , 710 carollo , c. m. , stiavelli , m. , de zeeuw , p. t. , & mack , j. 1997b , aj , 114 , 2366 carter , d. 1978 , mnras , 182 , 797 chabrier g. , 2003 , apj , 586 , l133 cimatti , a. , nipoti , c. , & cassata , p. 2012 , mnras , 422 , l62 clauwens , b. , schaye , j. , franx . , m. , 2014 , mnras , arxiv:1406.0854 conroy , c. , & van dokkum , p. g. 2012 , apj , 760 , 71 conselice c. j. , 2006 , mnras , 373 , 1389 conselice c. j. , 2007 , in combes f. , palous j. , eds , proc . galaxy evolution across the hubble time . cambridge univ . press , cambridge , p. 381 ct , p. , et al . 2007 , apj , 671 , 1456 crane , p. , stiavelli , m. , king , i. r. , et al . 1993 , aj , 106 , 1371 daddi , e. , renzini , a. , pirzkal , n. , et al . 2005 , apj , 626 , 680 davies r. l. , burstein d. , dressler a. , faber s. m. , lynden bell d. , terlevich r. j. , wegner g. , 1987 , apjs , 64 , 581 de ruiter , h. r. , parma , p. , capetti , a. , fanti , r. , morganti , r. , santantonio , l. , 2005 , a&a , 439 , 487 de vaucouleurs g. , 1948 , ann . dastrophys . , 11 , 247 de vaucouleurs g. , de vaucouleurs a. , corwin h. g. , jr . , et al . 1991 , third reference catalogue of bright galaxies ( berlin : springer ) dhar , b. k. , & williams , l. l. r. 2012 , mnras , 427 , 204 djorgovski , s. , & davis , m. 1987 , apj , 313 , 59 dressler a. , lynden - bell d. , burstein d. , davies r. l. , faber s. m. , terlevich r. , wegner g. , 1987 , apj , 313 , 42 driver s. p. , popescu c. c. , tuffs r. j. , graham a. w. , liske j. , baldry i. , 2008 , apj , 678 , l101 dullo b. t. , graham a. w. , 2012 , apj , 755 , 163 dullo b. t. , graham a. w. , 2013 , apj , 768 , 36 ebisuzaki t. , makino j. , & okumura s. k. 1991 , nature , 354 , 212 efstathiou g. , ellis r. s. , carter d. , 1982 , mnras , 201 , 975 emsellem e. et al . , 2011 , mnras , 414 , 888 faber , s. m. , tremaine , s. , ajhar , e. a. , et al . 1997 , aj , 114 , 1771 feigelson , e. d. , & babu , g. j. 1992 , apj , 397 , 55 ferrarese , l. , merritt , d. , 2000 , apj , 539 , l9 ferrarese , l. , ct , p. , jordn , a. , et al . 2006 , apjs , 164 , 334 ferrarese l. , ford h. , 2005 , space sci . rev . , 116 , 523 ferrarese l. , van den bosch f. c. , ford h. c. , jaffe w. , oconnell r. w. , 1994 , aj , 108 , 1598 ferrari , f. , dottori , h. , caon , n. , nobrega , a. , pavani , d.b . 2004 , mnras , 347 , 824 ferreras , i. , la barbera , f. , de la rosa , i. g. , et al . 2013 , mnras , 429 , l15 fisher d. , illingworth g. , franx m. , 1995 , apj , 438 , 539 franx m. , illingworth g. d. , & heckman t. 1989 , aj , 98 , 538 fukugita m. , shimasaku k. , & ichikawa t. 1995 , pasp , 107 , 945 gebhardt k. , bender , r. , bower , g. , et al . 2000 , apj , 539 , l13 gebhardt k. , lauer t. r. , pinkney j. , et al . 2007 , apj , 671 , 1321 glass l. et al . , 2011 , apj , 726 , 31 goerdt t. , moore b. , read j. i. , stadel j. , 2010 , apj , 725 , 1707 graham a. w. , 2004 , apj , 613 , l33 graham , a. w. 2012 , apj , 746 , 113 graham a. w. 2013 , in planets , stars and stellar systems , 6 , ed . t. d. oswalt & w. c. keel ( berlin : springer ) , 91 graham a. w. , colless m. m. , busarello g. , zaggia s. , longo g. , 1998 , a&as , 133 , 325 graham a. w. , & driver s.p . 2005 , publ . astron . soc . australia , 22 , 118 graham a. w. , erwin p. , trujillo i. , & asensio ramos a. 2003 , aj , 125 , 2951 graham a. w. , & guzmn r. 2003 , aj , 125 , 2936 graham a. w. , lauer t.r . , colless m.m . , & postman m. 1996 , apj , 465 , 534 graham a. w. , & scott n. 2013 , apj , 764 , 151 graham a. w. , spitler l. r. , 2009 , mnras , 397 , 2148 grillmair , c. j. , faber , s. m. , lauer , t. r. , et al . 1994 , aj , 108 , 102 gualandris a. , merritt d. , 2008 , apj , 678 , 780 gunn j. e. , gott j. r. , 1972 , apj , 176 , 1 gltekin k. , richstone d. o. , gebhardt k. , et al . 2011 , apj , 741 , 38 haehnelt m.g . , kauffmann h. 2002 , mnras , 336 , l61 hernquist l. , spergel d. n. , & heyl j. s. 1993 , apj , 416 , 415 hibbard j. e. , sansom a. e. , 2003 , aj , 125 , 667 hilz m. , naab t. , ostriker j. p. , 2013 , mnras , 429 , 2924 holley - bockelmann k. , richstone d. o. , 1999 , apj , 517 , 92 houghton r. , magorrian j. , sarzi m. , thatte n. , davies r. , & krajnovic d. 2006 , mnras , 367 , 2 hopkins p. f. , cox t. j. , dutta s. n. , hernquist l. , kormendy j. , lauer t. r. , 2009a , apjs , 181 , 135 hopkins p. f. , hernquist l. , cox t. j. , dutta s. n. , rothberg b. , 2008 , apj , 679 , 156 hopkins p. f. , lauer t. r. , cox t. j. , hernquist l. , kormendy j. , 2009b , apjs , 181 , 486 hopkins p. f. , & hernquist l. 2010 , mnras , 407 , 447 huang , s. , ho , l. c. , peng , c. y. , li , z .- y . , & barth , a. j. 2013 , apj , 766 , 47 hyde j. b. , bernardi m. , fritz a. , sheth r. k. , nichol r. c. , 2008 , mnras , 391 , 1559 jedrzejewski r. i. , 1987 , mnras , 226 , 747 ju et al . , 2013 , arxiv:1306.4987 kauffmann g. , white s. d. m. , guiderdoni b. , 1993 , mnras , 264 , 201 kelvin l. s. et al . , 2012 , mnras , 421 , 1007 king , i. r. 1978 , apj , 222 , 1 king , i. r. , & minkowski , r. 1966 , apj , 143 , 1002 komossa , s. , burwitz , v. , hasinger , g. , et al . 2003 , apjl , 582 , l15 kormendy , j. 1985 , apj , 292 , l9 kormendy j. , & bender r. 2009 , apj , 691 , l142 kormendy , j. , dressler , a. , byun , y. i. , et al . 1994 , in eso / ohp workshop on dwarf galaxies , ed . g. meylan & p. prugniel ( garching : eso ) , 147 kormendy j. , fisher d.b . , cornell m.e . , bender r. 2009 , apjs , 182 , 216 kormendy , j. , & ho , l. c. 2013 , ara&a , 51 , arxiv:1304.7762 kulkarni , g. , & loeb , a. 2012 , mnras , 422 , 1306 krajnovi d. , cappellari m. , de zeeuw p. t. , copin y. , 2006 , mnras , 366 , 787 krajnovi , d. , karick , a. m. , davies , r. l. , et al . 2013 , mnras , 433 , 2812 kroupa p. , 2001 , mnras , 322 , 231 laine , s. , van der marel , r. p. , lauer , t. r. , postman , m. , odea , c. p. , & owen , f. n. 2003 , aj , 125 , 478 lasker r. , ferrarese l. , van de ven g. , 2014 , apj , 780 , 69 lauer , t. r. 1985 , apj , 292 , 104 lauer t. r. , 2012 , apj , 759 , 64 lauer t. r. , ajhar , e. a. , byun , y .- i . , et al . 1995 , aj , 110 , 2622 lauer t. r. , et al . 2005 , aj , 129 , 2138 lauer t. r. , et al . , 2007a , apj , 662 , 808 lauer t. r. , et al . , 2007b , apj , 664 , 226 lauer , t. r. , gebhardt , k. , richstone , d. , et al . 2002 , aj , 124 , 1975 laurikainen e. et al . 2013 , mnras , 430 , 3489 laurikainen e. , salo h. , buta r. , 2005 , mnras , 362 , 1319 laurikainen e. , salo h. , buta r. , knapen j. h. , & comern s. 2010 , mnras , 405 , 1089 laurikainen e. , salo h. , buta r. , knapen j. h. , 2011 , mnras , 418 , 1452 li z .- y . , ho l. c. , barth a. j. , & peng c. y. 2011 , apjs , 197 , 22 liu , x. , shen , y. , bian , f. , loeb , a. , & tremaine , s. apj , 789 , 140 liu f. s. , xia x. y. , mao s. , wu h. , deng z. g. , 2008 , mnras , 385 , 23 magorrian j. , et al . 1998 , aj , 115 , 2285 mcglynn t. , 1984 , apj , 281 , 13 machacek , m. , ashby , m. l. n. , jones , c. , & bastian , n. , 2009 , apj , 691 , 1921 machacek , m. , nulsen , p. e. j. , stirbat , l. , jones , c.,&forman , w. r. 2005 , apj , 630 , 280 man , a. w. s. , toft , s. , zirm , a. w. , wuyts , s. , & van der wel , a. 2012 , apj , 744 , 85 marconi , a. , & hunt , l. k. 2003 , apj , 589 , l21 martizzi , d. , teyssier , r. , & moore , b. 2012 , mnras , 420 , 2859 martizzi , d. , teyssier , r. , & moore , b. 2012 , mnras , 432 , 1947 merritt d. , 2006 , apj , 648 , 976 michard , r. , & prugniel , p. 2004 , a&a , 423 , 833 mihos j. c. , & hernquist l. 1994 , apj , 437 , l47 milosavljevi m. , & merritt d. 2001 , apj , 563 , 34 milosavljevi m. , merritt d. , rest a. , & van den bosch f. c. 2002 , mnras , 331 , l51 naab t. , khochfar s. , burkert a. , 2006 , apj , 636 , l81 navarro j. f. , eke v. r. , frenk c. s. , 1996 , mnras , 283 , l72 ostriker j. p. , hausman m. a. , 1977 , apj , 217 , l125 paturel g. , petit c. , prugniel p. , theureau g. , rousseau j. , brouty m. , dubois p. , cambrsy l. , 2003 , a&a , 412 , 45 peirani s. , kay s. , silk j. , 2008 , a&a , 479 , 123 postman m. , et al . , 2012 , apj , arxiv:1205.3839 postman m. , & geller m. j. 1984 , apj , 281 , 95 prugniel , p. , nieto , j. l. , bender , r. , et al . 1988 , a&a , 204 , 61 quilis v. , moore b. , & bower r. 2000 , science , 288 , 1617 ravindranath s. , ho l. c. , & filippenko a. v. 2002 , apj , 566 , 801 ravindranath , s. , ho , l. c. , peng , c. y. , filippenko , a. v. , & sargent , w. l. w. 2001 , aj , 122 , 653 read j. i. , gilmore g. , 2005 , mnras , 356 , 107 reda f. , forbes d. , hau g. , 2005 , mnras , 360 , 693 davies j. , & schafer j. 2001 , aj , 121 , 2431 rest , a. , van den bosch , f. c. , jaffe , w. , et al . 2001 , aj , 121 , 2431 richings a. j. , uttley p. , k@xmath433rding e. , 2011 , mnras , 415 , 2158 richstone d. , et al . 1998 , nature , 395 , 14 rodriguez , c. , taylor , g. b. , zavala , r. t. , et al . 2006 , apj , 646 , 49 rusli , s. p. , thomas , j. , saglia , r. p. , et al . 2013 , aj , 146 , 45 salpeter e. e. , 1955 , apj , 121 , 161 saglia r. p. , bertschinger e. , baggley g. , burstein d. , colless m. , davies r. l. , mcmahan r. k. , jr , wegner g. , 1997 , apjs , 109 , 79 sandage a. , tammann g. a. , 1981 , revised shapley - ames catalogue of bright galaxies . carnegie institution , washington schombert j. , smith a. , 2012 , pasa , 29 , 174 scott n. , et al . , 2014 , mnras , 441 , 274 seigar , m. s. , graham , a. w. , & jerjen , h. 2007 , mnras , 378 , 1575 srsic j. l. 1968 , atlas de galaxias australes ( cordoba : observatorio astronomico ) smith r. j. , lucey j. r. , hudson m. j. , schlegel d. j. , davies r. l. , 2000 , mnras , 313 , 469 spiniello c. , trager s. c. , koopmans l. v. e. , chen y. p. , 2012 , apj , 753 , l32 statler t. s. , mcnamara b. r. apj 2002;581:1032 statler , t. s. , smecker - hane , t. , & cecil , g. 1996 , aj , 111 , 1512 steinmetz , m. , & navarro , j. f. 2002 , newa , 7 , 155 tal , t. , van dokkum , p. g. , nelan , j. , & bezanson , r. 2009 , aj , 138 , 1417 tonry j. l. , et al . , 2001 , apj , 546 , 681 trager s. c. , faber s. m. , worthey g. , gonzlez j. j. , 2000 , aj , 119 , 1645 3trujillo i. , erwin p. , asensio ramos a. , & graham a. w. 2004 , aj , 127 , 1917 van albada t. , 1982 , mnras , 201 , 939 volonteri m. , haardt f. , madau p. , 2003 , apj , 582 , 559 wegner , g. a. , corsini , e. m. , thomas , j. , et al . 2012 , aj , 144 , 78 whitmore , b. c. , miller , b. w. , schweizer , f. , & fall , s. m. 1997 , aj , 114 , 1797 xu , c. k. , zhao , y. , scoville , n. , et al . 2012 , apj , 747 , 85 young c. k. , & currie m. j. 1994 , mnras , 268 , l11 zaritsky d. , gil de paz a. , bouquin a. y. k. , 2014 , apj , 780 , l1

new surface brightness profiles from 26 early - type galaxies with suspected partially depleted cores have been extracted from the full radial extent of _ hubble space telescope _ images . we have carefully quantified the radial stellar distributions of the elliptical galaxies using the core - srsic model whereas for the lenticular galaxies a core - srsic bulge plus an exponential disc model gives the best representation . we additionally caution about the use of excessive multiple srsic functions for decomposing galaxies and compare with past fits in the literature . the structural parameters obtained from our fitted models are in general , in good agreement with our initial study using radially limited ( @xmath0 ) profiles , and are used here to update several `` central '' as well as `` global '' galaxy scaling relations . we find near - linear relations between the break radius @xmath1 and the spheroid luminosity @xmath2 such that @xmath3 , and with the supermassive black hole mass @xmath4 such that @xmath5 . this is internally consistent with the notion that major , dry mergers add the stellar and black hole mass in equal proportion , i.e. , @xmath6 . in addition , we observe a linear relation @xmath7 for the core - srsic elliptical galaxies where @xmath8 is the galaxies effective half light radii which is collectively consistent with the approximately - linear , bright - end of the curved @xmath9 relation . finally , we measure accurate stellar mass deficits @xmath10 that are in general 0.5@xmath11 @xmath4 , and we identify two galaxies ( ngc 1399 , ngc 5061 ) that , due to their high @xmath12 ratio , may have experienced oscillatory core - passage by a ( gravitational radiation)-kicked black hole . the galaxy scaling relations and stellar mass deficits favor core - srsic galaxy formation through a few `` dry '' major merger events involving supermassive black holes such that @xmath13 , for @xmath14 . [ firstpage ] galaxies : elliptical and lenticular , cd galaxies : fundamental parameters galaxies : nuclei galaxies : photometry galaxies : structure

due to the high frequency of the fastest internal motions in molecular systems , the discrete time step for molecular dynamics simulations must be very small ( of the order of femtoseconds ) , while the actual span of biochemical proceses typically require the choice of relatively long total times for simulations ( e.g. , from microseconds to milliseconds for protein folding processes ) . in addition to this , since biologically interesting molecules ( like proteins @xcite and dna @xcite ) consist of thousands of atoms , their trajectories in configuration space are esentially chaotic , and therefore reliable quantities can be obtained from the simulation only after statistical analysis @xcite . in order to cope with these two requirements , which force the computation of a large number of dynamical steps if predictions want to be made , great efforts are being done both in hardware @xcite and in software @xcite solutions . in fact , only in very recent times , simulations for interesting systems of hundreds of thousand of atoms in the millisecond scale are starting to become affordable , being still , as we mentioned , the main limitation of these computational techniques the large difference between the elemental time step used to integrate the equations of motion and the total time span needed to obtain useful information . in this context , strategies to increase the time step are very valuable . a widely used method to this end is to constrain some of the internal degrees of freedom @xcite of a molecule ( typically bond lengths , sometimes bond angles and rarely dihedral angles . for a verlet - like integrator @xcite , stability requires the time step to be at least about five times smaller than the period of the fastest vibration in the studied system @xcite . here is where constraints come into play . by constraining the hardest degrees of freedom , the fastest vibrational motions are frozen , and thus larger time steps still produce stable simulations . if constraints are imposed on bond lengths involving hydrogens , the time step can typically be increased by a factor of 2 to 3 ( from 1 fs to 2 or 3 fs ) @xcite . constraining additional internal degrees of freedom , such as heavy atoms bond lengths and bond angles , allows even larger timesteps @xcite , but one has to be careful , since , as more and softer degrees of freedom are constrained , the more likely it is that the physical properties of the simulated system could be severely distorted @xcite . the essential ingredient in the calculation of the forces produced by the imposition of constraints are the so - called lagrange multipliers @xcite , and their efficient numerical evaluation is therefore of the utmost importance . in this work , we show that the fact that many interesting biological molecules are esentially linear polymers allows to calculate the lagrange multipliers in order @xmath0 operations ( for a molecule where @xmath0 constraints are imposed ) in an exact ( up to machine precision ) , non - iterative way . moreover , we provide a method to do so which is based in a clever ordering of the constraints indices , and in a recently introduced algorithm for solving linear banded systems @xcite . it is worth mentioning that , in the specialized literature , this possibility has not been considered as far as we are aware ; with some works commenting that solving this kind of linear problems ( or related ones ) is costly ( but not giving further details ) @xcite , and some other works explicitly stating that such a computation must take @xmath2 @xcite or @xmath3 @xcite operations . also , in the field of robot kinematics , many @xmath1 algorithms have been devised to deal with different aspects of constrained physical systems ( robots in this case ) @xcite , but none of them tackles the calculation of the lagrange multipliers themselves . this work is structured as follows . in sec . [ sec_aclm ] , we introduce the basic formalism for the calculation of constraint forces and lagrange multipliers . in sec . [ soc ] , we explain how to index the constraints in order for the resulting linear system of equations to be banded with the minimal bandwidth ( which is essential to solve it efficiently ) . we do this starting by very simple toy systems and building on complexity as we move forward towards the final discussion about dna and proteins ; this way of proceeding is intended to help the reader build the corresponding indexing for molecules not covered in this work . in sec . [ sec : numerical ] , we apply the introduced technique to a polyalanine peptide using the amber molecular dynamics package and we compare the relative efficiency between the calculation of the lagrange multipliers in the traditional way ( @xmath2 ) and in the new way presented here ( @xmath1 ) . finally , in sec . [ sec : conclusions ] , we summarize the main conclusions of this work and outline some possible future applications . if holonomic , rheonomous constraints are imposed on a classical system of @xmath4 atoms , and the dalembert s principle is assumed to hold , its motion is the solution of the following system of differential equations @xcite : [ sistembasico ] @xmath5 where ( [ newton ] ) is the modified newton s second law and ( [ constr ] ) are the equations of the constraints themselves ; @xmath6 are the lagrange multipliers associated with the constraints ; @xmath7 represents the external force acting on atom @xmath8 , @xmath9 is its euclidean position , and @xmath10 colectively denote the set of all such coordinates . we assume @xmath7 to be conservative , i.e. , to come from the gradient of a scalar potential function @xmath11 ; and @xmath12 should be regarded as the _ force of constraint _ acting on atom @xmath8 . also , in the above expression and in this whole document we will use the following notation for the different indices : * @xmath13 ( except if otherwise stated ) for atoms . * @xmath14 ( except if otherwise stated ) for the atoms coordinates when no explicit reference to the atom index needs to be made . * @xmath15 for constrains and the rows and columns of the associated matrices . * @xmath16 as generic indices for products and sums . the existence of @xmath17 constraints turns a system of @xmath18 differential equations with @xmath19 unknowns into a system of @xmath20 algebraic - differential equations with @xmath20 unknowns . the constraints equations in ( [ constr ] ) are the new equations , and the lagrange multipliers are the new unknowns whose value must be found in order to solve the system . if the functions @xmath21 are analytical , the system of equations in ( [ sistembasico ] ) is equivalent to the following one : @xmath22 in this new form , it exists a more direct path to solve for the lagrange multipliers : if we explicitly calculate the second derivative in eq . ( [ constr2 ] ) and then substitute eq . ( [ newton2 ] ) where the accelerations appear , we arrive to @xmath23 where we have implicitly defined [ pq ] @xmath24 and it becomes clear that , at each @xmath25 , the lagrange multipliers @xmath26 are actually a _ known _ function of the positions and the velocities . we shall use the shorthand @xmath27 and , @xmath28 , @xmath29 , and @xmath30 to denote the whole @xmath0-tuples , as usual . now , in order to obtain the lagrange multipliers @xmath31 , we just need to solve @xmath32 this is a linear system of @xmath17 equations and @xmath17 unknowns . in the following , we will prove that the solution to it , when constraints are imposed on typical biological polymers , can be found in @xmath33 operations without the use of any iterative or truncation procedure , i.e. , in an exact way up to machine precision . to show this , first , we will prove that the value of the vectors @xmath29 and @xmath30 can be obtained in @xmath33 operations . then , we will show that the same is true for all the non - zero entries of matrix @xmath34 , and finally we will briefly discuss the results in @xcite , where we introduced an algorithm to solve the system in ( [ lm ] ) also in @xmath33 operations . it is worth remarking at this point that , in this work , we will only consider constraints that hold the distance between pairs of atoms constant , i.e. , @xmath35 where @xmath36 is a constant number , and the fact that we can establish a correspondence between constrained pairs ( @xmath37 ) and the constraints indices has been explicitly indicated by the notation @xmath38 . this can represent a constraint on : * a bond length between atoms @xmath8 and @xmath39 , * a bond angle between atoms @xmath8 , @xmath39 and @xmath40 , if both @xmath8 and @xmath39 are connected to @xmath40 through constrained bond lengths , * a principal dihedral angle involving @xmath8 , @xmath39 , @xmath40 and @xmath41 ( see @xcite for a rigorous definition of the different types of internal coordinates ) , if the bond lengths ( @xmath37 ) , ( @xmath42 ) and ( @xmath43 ) are constrained , as well as the bond angles ( @xmath44 ) and ( @xmath45 ) , * or a phase dihedral angle involving @xmath8 , @xmath39 , @xmath40 and @xmath41 if the bond lengths ( @xmath37 ) , ( @xmath42 ) and ( @xmath46 ) are constrained , as well as the bond angles ( @xmath44 ) and ( @xmath47 ) . this way to constrain degrees of freedom is called _ triangularization_. if no triangularization is desired ( as , for example , if we want to constrain dihedral angles but not bond angles ) , different explicit expressions than those in the following paragraphs must be written down , but the basic concepts introduced here are equally valid and the main conclusions still hold . now , from eq . ( [ sigma_generica ] ) , we obtain @xmath48 inserting this into ( [ pq1 ] ) , we get a simple expression for @xmath49 @xmath50 the calculation of @xmath51 is more involved , but it also results into a simple expression : first , we remember that the indices run as @xmath14 , and @xmath52 , and we produce the following trivial relationship : @xmath53 where @xmath54 , @xmath55 and @xmath56 are the unitary vectors along the @xmath10 , @xmath57 and @xmath58 axes , respectively . therefore , much related to eq . ( [ grad_sigma ] ) , we can compute the first derivative of @xmath59 : @xmath60 \ , \end{aligned}\ ] ] and also the second derivative : @xmath61 \nonumber \\ & & \mbox { } \cdot [ ( \delta_{3\alpha-2,\nu}\hat{i } + \delta_{3\alpha-1,\nu}\hat{j } + \delta_{3\alpha,\nu}\hat{k } ) -(\delta_{3\beta-2,\nu}\hat{i } + \delta_{3\beta-1,\nu}\hat{j } + \delta_{3\beta,\nu}\hat{k } ) ] \nonumber \\ & = & 2 ( \delta_{3\alpha-2,\mu}\delta_{3\alpha-2,\nu } + \delta_{3\beta-2,\mu}\delta_{3\beta-2,\nu } -\delta_{3\alpha-2,\mu}\delta_{3\beta-2,\nu } -\delta_{3\beta-2,\mu}\delta_{3\alpha-2,\nu } \nonumber \\ & & \mbox { } + \delta_{3\alpha-1,\mu}\delta_{3\alpha-1,\nu } + \delta_{3\beta-1,\mu}\delta_{3\beta-1,\nu } -\delta_{3\alpha-1,\mu}\delta_{3\beta-1,\nu } -\delta_{3\beta-1,\mu}\delta_{3\alpha-1,\nu } \nonumber \\ & & \mbox { } + \delta_{3\alpha,\mu}\delta_{3\alpha,\nu } + \delta_{3\beta,\mu}\delta_{3\beta,\nu } -\delta_{3\alpha,\mu}\delta_{3\beta,\nu } -\delta_{3\beta,\mu}\delta_{3\alpha,\nu } ) \ .\end{aligned}\ ] ] taking this into the original expression for @xmath51 in eq . ( [ pq2 ] ) and playing with the sums and the deltas , we arrive to @xmath62 now , eqs . ( [ defo ] ) , ( [ neop ] ) and ( [ neoq ] ) can be gathered together to become @xmath63 where we can see that the calculation of @xmath64 takes always the same number of operations , independently of the number of atoms in our system , @xmath4 , and the number of constraints imposed on it , @xmath17 . therefore , calculating the whole vector @xmath28 in eq . ( [ lm ] ) scales like @xmath17 . in order to obtain an explicit expression for the entries of the matrix @xmath34 , we now introduce eq . ( [ grad_sigma ] ) into its definition in eq . ( [ defr ] ) : @xmath65 where we have used that @xmath66 looking at this expression , we can see that a constant number of operations ( independent of @xmath4 and @xmath17 ) is required to obtain the value of every entry in @xmath34 . the terms proportional to the kroenecker deltas imply that , as we will see later , in a typical biological polymer , the matrix @xmath34 will be sparse ( actually banded if the constraints are appropriately ordered as we describe in the following sections ) , being the number of non - zero entries actually proportional to @xmath0 . more precisely , the entry @xmath67 will only be non - zero if the constraints @xmath68 and @xmath69 share an atom . now , since both the vector @xmath28 and the matrix @xmath34 in eq . ( [ lm ] ) can be computed in @xmath1 operations , it only remains to be proved that the solution of the linear system of equations is also an @xmath1 process , but this is a well - known fact when the matrix defining the system is banded . in @xcite , we introduced a new algorithm to solve this kind of banded systems which is faster and more accurate than existing alternatives . essentially , we shown that the linear system of equations @xmath70 where @xmath71 is a @xmath72 matrix , @xmath10 is the @xmath73 vector of the unknowns , @xmath74 is a given @xmath73 vector and @xmath71 is _ banded _ , i.e. , it satisfies that for known @xmath75 @xmath76 can be directly solved up to machine precision in @xmath77 operations . this can be done using the following set of recursive equations for the auxiliary quantities @xmath78 ( see @xcite for details ) : [ coefsband ] @xmath79 if the matrix @xmath71 is symmetric ( @xmath80 ) , as it is the case with @xmath34 [ see ( [ defr ] ) ] , we can additionally save about one half of the computation time just by using @xmath81 instead of ( [ xifinal_c ] ) . ( [ xifinal_csim ] ) can be obtained from ( [ coefsband ] ) by induction , and we recommend these expressions for the @xmath82 coefficients because other valid ones ( like considering @xmath83 , @xmath84 , which involves square roots ) are computationally more expensive . in the next sections , we show how to index the constraints in such a way that nearby indices correspond to constraints where involved atoms are close to each other and likely participate of the same constraints . in such a case , not only will the matrix @xmath34 in eq . ( [ lm ] ) be banded , allowing to use the method described above , but it will also have a minimal bandwidth @xmath85 , which is also an important point , since the computational cost for solving the linear system scales as @xmath86 ( when the bandwidth is constant ) . in this section we describe how to index the constraints applied to the bond lengths and bond angles of a series of model systems and biological molecules with the already mentioned aim of minimizing the computational cost associated to the obtention of the lagrange multipliers . the presentation begins by deliberately simple systems and proceeds to increasingly more complicated molecules with the intention that the reader is not only able to use the final results presented here , but also to devise appropriate indexings for different molecules not covered in this work . the main idea we have to take into account , as expressed in section [ sec_aclm ] , is to use nearby numbers to index constraints containing the same atoms . if we do so , we will obtain _ banded _ @xmath34 matrices . further computational savings can be obtained if we are able to reduce the number of @xmath82 coefficients in eqs . ( [ coefsband ] ) to be calculated . in more detail , solving a linear system like ( [ lm ] ) where the @xmath34 is @xmath87 and banded with semi - band width ( i.e. , the number of non - zero entries neighbouring the diagonal in one row or column ) @xmath85 requires @xmath88 operations if @xmath85 is a constant . therefore , the lower the value of @xmath85 , the smaller the number of required numerical effort . when the semi - band width @xmath85 is not constant along the whole matrix , things are more complicated and the cost is always between @xmath89 and @xmath90 , depending on how the different rows are arranged . in general , we want to minimize the number of zero fillings in the process of gaussian elimination ( see @xcite for further details ) , which is achieved by not having zeros below non - zero entries . this is easier to understand with an example : consider the following matrices , where @xmath91 and @xmath92 represent different non - zero values for every entry ( i.e. , not all @xmath92 , nor all @xmath91 must take the same value , and different symbols have been chosen only to highlight the main diagonal ) : @xmath93 during the gaussian elimination process that is behind ( [ coefsband ] ) , in @xmath71 , five coefficients @xmath82 above the diagonal are to be calculated , three in the first row and two in the second one , because the entries below non - zero entries become non - zero too as the elimination process advances ( this is what we have called ` zero filling ' ) . on the other hand , in @xmath94 , which contains the same number of non - zero entries as @xmath71 , only three coefficients @xmath82 have to be calculated : two in the first row and one in the second row . whether @xmath34 looks like @xmath71 or like @xmath94 depends on our choice of the constraints ordering . one has also to take into account that no increase in the computational cost occurs if a series of non - zero columns is separated from the diagonal by columns containing all zeros . i.e. , the linear systems associated to the following two matrices require the same numerical effort to be solved : @xmath95 as promised , we start by a simple model of a biomolecule : an open linear chain without any branch . in this case , the atoms should be trivially numbered as in fig . [ fig : lc ] ( any other arrangement would have to be justified indeed ! ) . if we only constrain bond lengths , the fact that only consecutive atoms participate of the same constraints allows us to simplify the notation with respect to eq . ( [ sigma_generica ] ) and establish the following ordering for the constraints indices : @xmath96 with @xmath97 this choice results in a tridiagonal matrix @xmath34 , whose only non - zero entries are those lying in the diagonal and its first neighbours . this is the only case for which an exact calculation of the lagrange multipliers exists in the literature as far as we are aware @xcite . the next step in complexity is to constrain the bond angles of the same linear chain that we discussed above . the atoms are ordered in the same way , as in fig . [ fig : lc ] , and the trick to generate a banded matrix @xmath34 with minimal bandwidth is to alternatively index bond length constraints with odd numbers , @xmath98 and bond angle constraints with even ones , @xmath99 where the regular pattern involving the atom indicies that participate of the same constraints has allowed again to use a lighter notation . the constraints equations in this case are @xmath100 respectively , and , if this indexing is used , @xmath34 is a banded matrix where @xmath85 is 3 and 4 in consecutive rows and columns . therefore , the mean @xmath101 is 3.5 , and the number of @xmath82 coefficients that have to be computed per row in the gaussian elimination process is the same because the matrix contains no zeros that are filled . a further feature of this system ( and other systems where both bond lengths and bond angles are constrained ) can be taken into account in order to reduce the computational cost of calculating lagrange multipliers in a molecular dynamics simulation : a segment of the linear chain with constrained bond lengths and bond angles is represented in fig . [ fig : angs ] , where the dashed lines correspond to the virtual bonds between atoms that , when kept constant , implement the constraints on bond angles ( assuming that the bond lengths , depicted as solid lines , are also constrained ) . due to the fact that all these distances are constant , many of the entries of @xmath34 will remain unchanged during the molecular dynamics simulation . as an example , we can calculate @xmath102 where we have used the law of cosines . the right - hand side does not depend on any time - varying objects ( such as @xmath9 ) , being made of only constant quantities . therefore , the value of @xmath103 ( and many other entries ) needs not to be recalculated in every time step , which allows to save computation time in a molecular dynamics simulation . in order to incrementally complicate the calculations , we now turn to a linear molecule with only one atom connected to the backbone , such the one displayed in figure [ fig : branched1 ] . the corresponding equations of constraint and the ordering in the indices that minimizes the bandwidth of the linear system are [ singlybranched ] @xmath104 where the trick this time has been to alternatively consider atoms in the backbone and atoms in the branches as we proceed along the chain . the matrix @xmath34 of this molecule presents a semi - band width which is alternatively 2 and 1 in consecutive rows / columns , with average @xmath105 and the same number of superdiagonal @xmath82 coefficients to be computed per row . the next molecular topology we will consider is that of an alkane ( a familiy of molecules with a long tradition in the field of constraints @xcite ) , i.e. , a linear backbone with two 1-atom branches attached to each site ( see fig . [ fig : nalkane ] ) . the ordering of the constraints that minimizes the bandwidth of the linear system for this case is @xmath106 where the trick has been in this case to alternatively constrain the bond lengths in the backbone and those connecting the branching atoms to one side or the other . the resulting @xmath34 matrix require the calculation of 2 @xmath82 coefficients per row when solving the linear system . if we want to additionally constrain bond angles in a molecule with the topology in fig . [ fig : branched1 ] , the following ordering is convenient : @xmath107 this ordering produces 16 non - zero entries above the diagonal per each group of 4 rows in the matrix @xmath34 when making the calculations to solve the associated linear system . this is , we will have to calculate a mean of @xmath108 super - diagonal coefficients @xmath82 per row . when we studied the linear molecule with constrained bond lengths and bond angles , this mean was equal to @xmath109 , so including minimal branches in the linear chain makes the calculations just slightly longer . if we now want to add bond angle constraints to the bond length ones described in sec . [ sec : doubly_branched ] for alkanes , the following ordering produces a matrix @xmath34 with a low half - band width : @xmath110 in this case , the average number of @xmath82 coefficients to be calculated per row is approximately 5.7 . if we have cycles in our molecules , the indexing of the constraints is only slightly modified with respect to the open cases in the previous sections . for example , if we have a single - branch cyclic topology , such as the one displayed in fig . [ fig : rings]a , the ordering of the constraints is the following : @xmath111 these equations are the same as those in [ secbl ] , plus a final constraint corresponding to the bond which closes the ring . these constraints produce a matrix @xmath34 where only the diagonal entries , its first neighbours , and the entries in the corners ( @xmath112 and @xmath113 ) are non - zero . in this case , the associated linear system in eq . ( [ lm ] ) can also be solved in @xmath1 operations , as we discuss in @xcite . in general , this is also valid whenever @xmath34 is a sparse matrix with only a few non - zero entries outside of its band , and therefore we can apply the technique introduced in this work to molecular topologies containing more than one cycle . the ordering of the constraints and the resulting linear systems for different cyclic species , such as the one depicted in fig . [ fig : rings]b , can be easily constructed by the reader using the same basic ideas . as we discussed in sec . [ sec : introduction ] , proteins are one of the most important families of molecules from the biological point of view : proteins are the nanomachines that perform most of the complex tasks that need to be done in living organisms , and therefore it is not surprising that they are involved , in one way or another , in most of the diseases that affect animals and human beings . given the efficiency and precision with which proteins carry out their missions , they are also being explored from the technological point of view . the applications of proteins even outside the biological realm are many if we could harness their power @xcite , and molecular dynamics simulations of great complexity and scale are being done in many laboratories around the world as a tool to understand them @xcite . proteins present two topological features that simplify the calculation of the lagrange multipliers associated to constraints imposed on their degrees of freedom : represents the first numbered atom in each residue ( the amino nitrogen ) and @xmath114 is the number of atoms in the side chain . * b ) * indexing of the bond length constraints ; @xmath68 denotes the index of the first constraint imposed on the residue ( the n - h bond length ) and @xmath115 is the variable number of constraints imposed on the side chain.,width=302 ] * they are linear polymers , consisting of a backbone with short ( 17 atoms at most ) groups attached to it @xcite . this produces a banded matrix @xmath34 , thus allowing the solution of the associated linear problem in @xmath1 operations . even in the case that disulfide bridges , or any other covalent linkage that disrupts the linear topology of the molecule , exist , the solution of the problem can still be found efficiently if we recall the ideas discussed in sec . [ ring1 ] . * the monomers that typically make up these biological polymers , i.e. , the residues associated to the proteinogenic aminoacids , are only 20 different molecular structures . therefore , it is convenient to write down explicitly one block of the @xmath34 matrix for each known monomer , and to build the @xmath34 matrix of any protein simply joining together the precalculated blocks associated to the corresponding residues the protein consists of . the structure of a segment of the backbone of a protein chain is depicted in fig . [ fig : prot ] . the green spheres represent the side chains , which are the part of the amino acid residue that can differ from one monomer to the next , and which usually consist of several atoms : from 1 atom in the case of glycine to 17 in arginine or tryptophan . in fig . [ fig : prot]a , we present the numbering of the atoms , which will support the ordering of the constraints , and , in fig . [ fig : prot]b , the indexing of the constraints is presented for the case in which only bond lengths are constrained ( the bond lengths plus bond angles case is left as an exercise for the reader ) . using the same ideas and notation as in the previous sections and denoting by @xmath116 the block of the matrix @xmath34 that corresponds to a given amino acid residue @xmath117 , with @xmath118 , we have that , for the monomer dettached of the rest of the chain , @xmath119 where the explicit non - zero entries are related to the constraints imposed on the backbone and @xmath120 denotes a block associated to those imposed on the bonds that belong to the different sidechains . the dimension of this matrix is @xmath121 and the maximum possible semi - band width is 12 for the bulkiest residues . a protein s global matrix @xmath34 has to be built by joining together blocks like the one above , and adding the non - zero elements related to the imposition of constraints on bond lengths that connect one residue with the next . these extra elements are denoted by @xmath122 and a general scheme of the final matrix is shown in fig . [ fig : proteinmatrix ] . for a protein molecule with @xmath123 residues . in black , we represent the potentially non - zero entries , and each large block in the diagonal is given by ( [ defr]).,width=264 ] the white regions in this scheme correspond to zero entries , and we can easily check that the matrix is banded . in fact , if each one of the diagonal blocks is constructed conveniently , they will contain many zeros themselves and the bandwidth can be reduced further . the size of the @xmath124 blocks will usually be much smaller than that of their neighbour diagonal blocks . for example , in the discussed case in which we constrain all bond lengths , @xmath124 are @xmath125 ( or @xmath126 ) blocks , and the diagonal blocks size is between @xmath127 ( glycine ) and @xmath128 ( tryptophan ) . nucleic acids are another family of very important biological molecules that can be tackled with the techniques described in this work . dna and rna , the two subfamilies of nucleic acids , consist of linear chains made up of a finite set of monomers ( called ` bases ' ) . this means that they share with proteins the two features mentioned in the previous section and therefore the lagrange multipliers associated to the imposition of constraints on their degrees of freedom can be efficiently computed using the same ideas . it is worth mentioning that dna typically appears in the form of two complementary chains whose bases form hydrogen - bonds . since these bonds are much weaker than a covalent bond , imposing bond length constraints on them such as the ones in eq . ( [ sigma_generica ] ) would be too unrealistic for many practical purposes , denotes the index of the first constraint imposed on the nucleotide and @xmath115 is the variable number of constraints imposed on the bonds in the base.,width=340 ] in fig . [ fig : dna_constrs ] , and following the same ideas as in the previous section , we propose a way to index the bond length constraints of a dna strand which produces a banded matrix @xmath34 of low bandwidth . green spheres represent the ( many - atom ) bases ( a , c , t or g ) , and the general path to be followed for consecutive constraint indices is depicted in the upper left corner : first the sugar ring , then the base and finally the rest of the nucleotide , before proceeding to the next one in the chain . this ordering translates into the following form for the block of @xmath34 corresponding to one single nucleotide dettached from the rest of the chain : @xmath129 where @xmath120 is the block associated to the constraints imposed on the bonds that are contained in the base , @xmath130 , @xmath131 , @xmath132 , and @xmath133 are very sparse rectangular blocks with only a few non - zero entries in them , and the form of the diagonal blocks associated to the sugar ring and backbone constraints is the following : [ eq : r11r33 ] @xmath134 analagously to the case of proteins , as many blocks as those in eq . ( [ rpartdna ] ) as nucleotides contains a given dna strand have to be joined to produce the global matrix @xmath34 of the whole molecule , together with the @xmath122 blocks associated to the constraints on the bonds that connect the different monomers . in fig . [ fig : matrixadn ] , a scheme of this global matrix is depicted and we can appreciate that it indeed banded . the construction of the matrix @xmath34 for a rna molecule should follow the same steps and the result will be very similar . for a dna molecule with @xmath135 nucleotides . in black , we represent the potentially non - zero entries , and each large block in the diagonal is given by ( [ rpartdna]).,width=264 ] in this section , we apply the efficient technique introduced in this work to a series of polyalanine molecules in order to calculate the lagrange multipliers when bond length constraints are imposed . we also compare our method , both in terms of accuracy and numerical efficiency , to the traditional inversion of the matrix @xmath34 without taking into account its banded structure . we used the code avogadro @xcite to build polyalanine chains of @xmath1362 , 5 , 12 , 20 , 30 , 40 , 50 , 60 , 80 , 90 and 100 residues , and we chose their initial conformation to be approximately an alpha helix , i.e. , with the values of the ramachandran angles in the backbone @xmath137 and @xmath138 @xcite . next , for each of these chains , we used the molecular dynamics package amber @xcite to produce the atoms positions ( @xmath10 ) , velocities ( @xmath139 ) and external forces ( @xmath140 ) needed to calculate the lagrange multipliers ( see sec . [ sec_aclm ] ) after a short equilibration molecular dynamics simulations . we chose to constrain all bond lengths , but our method is equally valid for any other choice , as the more common constraining only of bonds that involve hydrogens . in order to produce reasonable final conformations , we repeated the following process for each of the chains : * solvation with explicit water molecules . * minimization of the solvent positions holding the polypeptide chain fixed ( 3,000 steps ) . * minimization of all atoms positions ( 3,000 steps ) . * thermalization : changing the temperature from 0 k to 300 k during 10,000 molecular dynamics steps . * stabilization : 20,000 molecular dynamics steps at a constant temperature of 300 k. * measurement of @xmath10 , @xmath139 and @xmath140 . neutralization is not necessary , because our polyalanine chains are themselves neutral . in all calculations we used the force field described in @xcite , chose a cutoff for coulomb interactions of 10 and a time step equal to 0.002 ps , and impose constraints on all bond lengths as mentioned . in the thermostated steps , we used langevin dynamics with a collision frequency of 1 ps@xmath141 . .,width=302 ] using the information obtained and the indexing of the constraints described in this work , we constructed the matrix @xmath34 and the vector @xmath28 and proceeded to find the lagrange multipliers using eq . ( [ lm ] ) . since ( [ lm ] ) is a linear problem , one straightforward way to solve is to use traditional gauss - jordan elimination or lu factorization @xcite . but these methods have a drawback : they scale with the cube of the size of the system . i.e. , if we imposed @xmath0 constraints on our system ( and therefore we needed to obtain @xmath0 lagrange multipliers ) , the number of floating point operations that these methods would require is proportional to @xmath142 . however , as we showed in the previous sections , the fact that many biological molecules , and proteins in particular , are essentially linear , allows to index the constraints in such a way that the matrix @xmath34 in eq . ( [ lm ] ) is banded and use different techniques for solving the problem which require only @xmath1 floating point operations @xcite . [ table : pruebas ] .comparison of numerical complexity and accuracy between a traditional gauss - jordan solver and the banded algorithm described in this work , for the calculation of the lagrange multipliers on a series of polyalanine chains as a function of their number of residues @xmath143 . [ cols="^,^,^,^,^ " , ] in fig . [ fig : pruebas1 ] and table [ table : pruebas ] , we compare both the accuracy and the execution time of the two different methods : gauss - jordan elimination @xcite , and the banded recursive solution advocated here and made possible by the appropriate indexing of the constraints . the calculations have been run on a mac os x laptop with a 2.26 ghz intel core 2 duo processor , and the errors were measured using the normalized deviation of @xmath144 from @xmath145 . i.e. , if we denote by @xmath146 the solution provided by the numerical method , @xmath147 from the obtained results , we can see that both methods produce an error which is very small ( close to machine precision ) , being the accuracy of the banded algorithm advocated in this work slightly higher . regarding the computational cost , as expected , the gauss - jordan method presents an effort that approximately scales with the cube of the number of constraints @xmath0 ( which is approximately proportional to @xmath143 ) , while the banded technique allowed by the particular structure of the matrix @xmath34 follows a rather accurate lineal scaling . although it is typical that , when two such different behaviours meet , there exists a range of system sizes for which the method that scales more rapidly is faster and then , at a given system size , a crossover takes place and the slower scaling method becomes more efficient from there on , in this case , and according to the results obtained , the banded technique is less time - consuming for all the explored molecules , and the crossover should exist at a very small system size ( if it exists at all ) . this is very relevant for any potential uses of the methods introduced in this work . we have shown that , if we are dealing with typical biological polymers , whose covalent connectivity is that of essentially linear objects , the lagrange multipliers that need to be computed when @xmath0 constraints are imposed on their internal degrees of freedom ( such as bond lengths , bond angles , etc . ) can be obtained in @xmath1 steps as long as the constraints are indexed in a convenient way and banded algorithms are used to solve the associated linear system of equations . this path has been traditionally regarded as too costly in the literature @xcite , and , therefore , our showing that it can be implemented efficiently could have profound implications in the design of future molecular dynamics algorithms . since the field of imposition of constraints in moleculary dynamics simulations is dominated by methods that cleverly achieve that the system exactly stays on the constrained subspace as the simulation proceeds by not calculating the exact lagrange multipliers , but a modification of them instead @xcite , we are aware that the application of the new techniques introduced here is not a direct one . however , we are confident that the low cost of the new method and its close relationship with the problem of constrained dynamics could prompt many advances , some of which are already being pursued in our group . among the most promising lines , we can mention a possible improvement of the shake method @xcite by the use of the exact lagrange multipliers as a guess for the iterative procedure that constitutes its most common implementation . also , we are studying the possibility of solving the linear problems that appear either in a different implementation of shake ( mentioned in the original work too @xcite ) or in the lincs method @xcite , and which are defined by matrices which are different from but related to the matrix @xmath34 introduced in this work , being also banded if an appropriate indexing of the constraints is used . finally , we are exploring an extension of the ideas introduced here to the calculation not only of the lagrange multipliers but also of their time derivatives , to be used in higher order integrators than verlet . we would like to thank giovanni ciccotti for illuminating discussions and wise advices , and claudio cavasotto and isaas lans for the help with the setting up and use of amber . the numerical calculations have been performed at the bifi supercomputing facilities ; we thank all the staff there for the help and the technical assistance . this work has been supported by the grants fis2009 - 13364-c02 - 01 ( micinn , spain ) , grupo de excelencia `` biocomputacin y fsica de sistemas complejos '' , e24/3 ( aragn region government , spain ) , araid and ibercaja grant for young researchers ( spain ) . r . is supported by a jae predoc scholarship ( csic , spain ) . , r. ron o. dror , j. salmon , j. grossman , k. mackenzie , j. bank , c. young , b. batson , k. bowers , e. edmond chow , m. eastwood , d. ierardi , j. john l. klepeis , j. jeffrey s. kuskin , r. larson , k. kresten lindorff - larsen , p. maragakis , m. m.a . , s. piana , s. yibing , and b. towles , , in proceedings of the acm / ieee conference on supercomputing ( sc09 ) , acm press , new york , ( 2009 ) . , t. darden , t. e. cheatham , c. simmerling , w. junmei , d. r. e. , r. luo , k. m. merz , m. a. pearlman , m. crowley , r. walker , z. wei , w. bing , s. hayik , a. roitberg , g. seabra , w. kim , f. paesani , w. xiongwu , v. brozell , s. tsui , h. gohlke , y. lijiang , t. chunhu , j. mongan , v. hornak , p. guanglei , c. beroza , d. h. mathews , c. schafmeister , w. s. ross , and p. a. kollman , , university of california : san francisco ( 2006 ) .

in order to accelerate molecular dynamics simulations , it is very common to impose holonomic constraints on their hardest degrees of freedom . in this way , the time step used to integrate the equations of motion can be increased , thus allowing , in principle , to reach longer total simulation times . the imposition of such constraints results in an aditional set of @xmath0 equations ( the equations of constraint ) and unknowns ( their associated lagrange multipliers ) , that must be solved in one way or another at each time step of the dynamics . in this work it is shown that , due to the essentially linear structure of typical biological polymers , such as nucleic acids or proteins , the algebraic equations that need to be solved involve a matrix which is banded if the constraints are indexed in a clever way . this allows to obtain the lagrange multipliers through a non - iterative procedure , which can be considered exact up to machine precision , and which takes @xmath1 operations , instead of the usual @xmath2 for generic molecular systems . we develop the formalism , and describe the appropriate indexing for a number of model molecules and also for alkanes , proteins and dna . finally , we provide a numerical example of the technique in a series of polyalanine peptides of different lengths using the amber molecular dynamics package . + * keywords : * constraints , lagrange multipliers , banded systems , molecular dynamics , proteins , dna +

in @xcite , randall and sundrum ( rs ) proposed a new approach to extra dimensions for space - time to solve the hierarchy problem . in their model , it is assumed that there exists only one extra space - like dimension which is taken homeomorphic with an orbifold @xmath1 . this orbifold has two fixed points at @xmath2 and @xmath3 . at each fixed point , they put a 4-dimensional brane world . one of them which is located at @xmath3 is called the visible world and is assumed we are living on it , and the other one is called hidden world . in rs method same as the arkani - hamed , dimopoulos and dvali ( add ) approach @xcite , it is assumed that except the graviton ( and also axions ) all the standard model ( sm ) fields are confined in these two distinct worlds . the physical laws are the same on these two worlds but the masses and the coupling constants may differ . in this model , the classical action is assumed to be : s&= & s_gravity + s_vis + s_hid , + s_gravity&= & d^4x d \{+ 2 m r } , + s_vis&= & d^4x \{l_vis -v_vis } , + s_hid&= & d^4x \{l_hid -v_hid } , where @xmath4 and @xmath5 are the 5-dimensional planck mass and the cosmological constant respectively , and @xmath6 ( @xmath7 ) is the sm or any effective lagrangian corresponding to matter and force fields except the gravity . the @xmath8 and @xmath9 are vacuum expectations on the branes . the classical solution of einstein equation for the mentioned action is the following metric : ds^2 = e^-2 ( ) _ dx^dx^+ r_c^2 d^2 , with ( ) = r_c and radius of @xmath10 in the orbifold @xmath11 and @xmath12 . by integrating out the fifth dimension , the coupling constant of the effective 4-dimensional action yields the 4-dimensional planck scale @xcite : m_p = m^3 (1-e^-2r_c ) . it is found that a field on visible brane with the fundamental mass parameter @xmath13 will appear to have a physical mass @xmath14 . by taking @xmath15 , the observed scale hierarchy reproduces naturally by exponential factor and no additional large hierarchies arise @xcite . at this stage , it is natural to search for any observable effects of this extra fifth dimension in rs model . many efforts have been done to probe the effects of this extra dimension in ordinary particle interactions @xcite . there would be two kinds of gravitons in this formalism ; the first type is massless ordinary graviton , which is also confined to the 4-dimensional physical space - time , and the others are massive gravitons . in @xcite it is shown that the effects of the massless gravitons in particles interactions are in order of @xmath16 where @xmath17 is the 4-dim planck mass . however , the contributions of the massive ones are considerable and comparable with the weak scale of the standard model . the masses of gravitons come from kaluza - klein compactification of the fifth dimension . due to non - factorization of the geometry , the masses of gravitons are @xmath18 , where @xmath19 s are the roots of @xmath20 , the bessel function of order one . in this paper , we calculate the total cross - sections of scalar - scalar to photon and graviton fields @xmath21 . this process in standard model , without producing gravitons is forbidden by energy - momentum conservation . at the beginning , we consider the scalar electrodynamics as the effective theory of matter and forces in visible and hidden spaces . the action of this theory is : s_vis = s_sed & = & \ { g^(_-i e a_)^(_+i e a _ ) + & -&m_^2 ^-1 4g^g^f_f _ } d^4x , where @xmath23 and @xmath24 is the inverse of the classical metric ( 1.2 ) . the factor @xmath5 for massless graviton is equal to @xmath17 and for the massive gravitons is @xmath25 . inserting @xmath26 in eq . ( 2.1 ) and absorbing the conformal factor @xmath27 in scalar fields and their mass , one can reduce the interaction part of lagrangian up to the first order in @xmath28 to the following terms , [ lagrangian ] l_i&= & i e a^(_^-^_)+ e^2 a^a_^ + & + & _ ^_+ i e a_(_^-^ _ ) + & - & ( ^a__a_-2 ^a__a_+ _ a^_a _ ) + & + & e^2 a_a_^. as it is pointed out in ref . @xcite , we can use @xmath29 as @xmath30 , for this theory . since we are searching for amplitude of @xmath31 , the relevant terms of hamiltonian to this interaction are , [ hamilton ] h_1&=&-i e a^(_^- ^ _ ) , + h_2&=&- _ ^ _ , + h_3&= & -i e a_(_^- ^ _ ) , + h_4&= & ( ^a__a_-2 ^a__a_+ _ a^_a _ ) , which contribute to the following diagrams , ( 0,0)(0,0 ) ( -150,-35 ) ( -100,50)(0,0 ) ( 0,0)(-100,-50 ) ( 0,0)(100,-50)510 ( 0,0)(100,50)5 ( 0,0)3 ( 0,-67)[]the feynman diagram of @xmath32 ( -60,25)[]@xmath33 ( -60,-25)[]@xmath34 ( 58,25)[]@xmath35 ( 58,-25)[]@xmath36 ( 150,-20 ) ( -100,50)(0,0 ) ( 0,0)(100,50)5 ( 0,0)3 ( 0,0)(0,-65 ) ( 0,-65)(-100,-115 ) ( 0,-65)(100,-115)510 ( 0,-65)3 ( 0,-83)[]the feynman diagram of @xmath37 ( -60,25)[]@xmath33 ( -60,-60)[]@xmath34 ( 58,25)[]@xmath35 ( 58,-60)[]@xmath36 3.5 cm ( 0,0)(0,0 ) ( -150,0 ) ( -100,50)(0,-65 ) ( 0,-65)(0,0 ) ( 0,0)(-100,-115 ) ( 0,0)3 ( 0,-65)3 ( 0,0)(100,50)5 ( 0,-65)(100,-115)510 ( 0,-83)[]the feynman diagram of @xmath38 ( -60,25)[]@xmath33 ( -60,-60)[]@xmath34 ( 58,25)[]@xmath35 ( 58,-60)[]@xmath36 ( 115,-20 ) ( -100,50)(0,0 ) ( 0,0)(-110,-50 ) ( 0,0)3 ( 105,0)3 ( 105,0)(205,50)5 ( 0,0)(100,0)510 ( 105,0)(205,-50)510 ( 25,-60)[]the feynman diagram of @xmath39 ( -60,25)[]@xmath33 ( -64,-25)[]@xmath34 ( 112,25)[]@xmath35 ( 112,-25)[]@xmath36 + denoting in - state by @xmath40 where @xmath41 and @xmath42 are the momenta of scalar and anti - scalar particles respectively and out - state by @xmath43 , where @xmath44 and @xmath45 are the momenta due to the massive graviton and photon . now , we are going to derive the @xmath46-matrix elements for the above diagrams in the tree level , & = & < k , q t e^-i_i dt p_1 , p_2 > + & = & i ( 2)^4 ^(4 ) ( q+k - p_1-p_2)m_tot , where @xmath47 is invariant amplitude which is the sum of the following amplitudes ( see figs . ) , [ ampl ] m_1 & = & - _ ( q ) e^ ( k ) ( p_1-p_2 ) _ , + m_2 & = & - ^(q ) e^ ( k ) ( p_1-k)_p_1 ( p_2-p_1+k ) _ , + m_3 & = & - ^(q ) e^ ( k ) ( p_2-k)_p_2 ( p_2-p_1-k ) _ , + m_4&= & ( ( p_1-p_2)__(q(p_1+p_2 ) ) . + & -&.(p_1-p_2)_q_((p_1+p_2))-_(p_1+p_2)_(p_1-p_2)q . + & + & .q_(p_1+p_2)_(p_1-p_2 ) ) . in the above equation @xmath48 and @xmath49 are the polarization of photon and graviton respectively . to calculate the unpolarized cross - section , we should make summation over these polarizations . we have , _ pol.e_(k)e_(k)&=&f_(k ) , + _ pol._(q)_(q)&=&-g _ , in which @xmath50 for a massive graviton is @xcite , & & f_(k)=12\ { g_g_+g_g _ -g_g _ } + & + & 12\ { g_k_k_+ g_k_k_+ g_k_k_+ g_k_k _ } + & + & 2 3(1 2g_-k_k _ ) ( 1 2g_-k_k _ ) , and for a massless graviton , f_(k)=12\ { g_g_+g_g _ -23g_g_}. using the above relations , it is straightforward to calculate the cross - section in the center of mass frame of the incident particles for the massive gravitons . the differential cross - section of the mentioned process in the center of mass frame is @xcite , [ cs ] ( d)_cm= 1 |k| |m_tot|^2 , where @xmath51 , and @xmath52 is the center of mass energy of incident particles . the contribution of the massless graviton can be neglected , as it is pointed out in @xcite . for a massive graviton of mass @xmath53 , we have calculated in the appendix the terms in @xmath54 . to obtain the total cross - section , as a function of @xmath55 , one should integrate the ( [ cs ] ) over the scattering angles and sum over all massive gravitons which their masses are less than @xmath52 . for the calculations the computer program mathematica version 3.0 was used . the final result is the following graph , where the solid curve shows the behavior of the total cross - section versus the energy , @xmath55 . here , we take @xmath56 , @xmath57 and @xmath58 . according to the mass formula , @xmath59 , we obtain the first four gravitons masses , @xmath60 , @xmath61 , @xmath62 and @xmath63 . for @xmath64 , only the first graviton mode contributes to the total cross - section . for @xmath65 , the dashed curve shows the contribution of this first mode to the @xmath66 . the individual behavior of the other graviton modes are similar to this dashed curve which shows a monotonic increasing behavior . this increasing behavior is expected due to the non - renormalizability of the quantum gravity . the peaks on the solid curve show the resonance behavior according to creation of the graviton modes . ( 0,0)(0,0 ) ( 0,0 ) ( -300,0)(300,0)(10,5,-4,0,1.5 ) ( -300,-700)(300,-700)(10,5,4,0,1.5 ) ( -300,0)(-300,-700)(12,5,4,0,1.5 ) ( 300,0)(300,-700)(12,5,-4,0,1.5 ) ( -105,-252)[].8 ( -84,-252)[]1 ( -63,-252)[]1.2 ( -42,-252)[]1.4 ( -21,-252)[]1.6 ( 0,-252)[]1.8 ( 21,-252)[]2 ( 42,-252)[]2.2 ( 63,-252)[]2.4 ( 84,-252)[]2.6 ( 105,-252)[]2.8 ( 0,-270)[]@xmath55 in tev ( -130,-126)[][l]@xmath67 in pb ( -112,-245)[r]0 ( -112,-225)[r]10 ( -112,-205)[r]20 ( -112,-185)[r]30 ( -112,-164.4)[r]40 ( -112,-143.8)[r]50 ( -112,-123.2)[r]60 ( -112,-102.6)[r]70 ( -112,-82)[r]80 ( -112,-61.4)[r]90 ( -112,-40.8)[r]100 ( -112,-20.2)[r]110 ( -112,0)[r]120 ( 0,-295)[]total cross section of @xmath68 versus energy of one incident particles , with ( 0,-310)[]@xmath69 . the peaks correspond to the productions of massive gravitons . we would like to thank a. shafiekhani for his contribution to the primary version of this article . in this appendix , all the ten terms of the scattering diagrams @xmath70 up to the order of @xmath71 for @xmath72 in terms of the mass of graviton , @xmath53 , and enrgy , @xmath55 , have been calculated . _ pol.|m_1|^2&= & , + _ pol.(_1 _ 2^&+&_1 ^_2)= ( ( 4e^2 - m^2 ) ( 32.c^4 - 16.e^6m^2 .. + & - & .. 16.c^3m^2 + 0.64cm^4 + ( 0.0512 + 0.04m^2 ) m^4 + c^2m^2 ( -2.56 + 2.m^2 ) + e^2 ( -64.c^3 + 96.c^2m^2 + ... + & + & ... c ( 2.56 - 16.m^2 ) m^2 + ( -2.88 - 1.m^2 ) m^4 ) + e^4 ( 32.c^2 - 64.cm^2 + m^2 ( 0.64 + 40.m^2 ) ) ) ) , + _ pol.(_1 _ 3^&+&_1 ^_3)= ( 0.000527714 ( 4e^2 - m^2 ) ( -2.c^4 + 1.e^6m^2 .. + & - & c^3m^2 + c^2 ( 0.16 - 0.125m^2 ) m^2 + 0.04cm^4 + ( -0.0032 - 0.0025m^2 ) m^4 + e^4 ( -2.c^2 - 4.cm^2 . + & + & ... ( -0.04 - 2.5m^2 ) m^2 ) + e^2 ( -4.c^3 - 6.c^2m^2 + c ( 0.16 - 1.m^2 ) m^2 + ( 0.18 + 0.0625m^2 ) m^4 ) ) ) , + _ pol.(_1 _ 4^&+&_1 ^_4)= - - + & + & + & + & + & + & , + _ pol.|m_2|^2&= & ( 4096.e^12 + e^10 ( -163.84 - 20480.c + 2048.m^2 ) . + & + & e^8 ( 40960.c^2 + c ( 655.36 - 7168.m^2 ) + ( -450.56 - 256.m^2 ) m^2 ) + & + & e^4 ( 20480.c^4 + c^3 ( 655.36 - 2048.m^2 ) + c^2 ( -1228.8 - 3072.m^2 ) m^2 . + & + & . 640.c ( -0.276264 + m^2 ) m^2 ( 0.148264 + m^2 ) - 16 . ( -3.07951 + m^2 ) m^4 ( 0.199512 + m^2 ) ) + & + & e^2 ( -4096.c^5 + c^4 ( -163.84 - 2048.m^2 ) - 512.c^2 ( -0.492029 + m^2 ) m^2 ( 0.0520294 + m^2 ) . + & + & 8 . ( -0.64 + m^2 ) m^4 ( 0.0626408 + m^2 ) ( 0.817359 + m^2 ) - 16.cm^4 ( 0.0812907 + m^2 ) + & & . ( 5.03871 + m^2 ) + c^3m^2 ( 327.68 + 2560.m^2 ) ) + e^6 ( -40960.c^3 - 256 . ( -0.189783 + m^2 ) m^2 . ( 0.269783 + m^2 ) + cm^2 ( 1310.72 + 1536.m^2 ) + c^2 ( -983.04 + 8192.m^2 ) ) + m^2 ( 1024.c^5 . + & + & c^4 ( 40.96 - 768.m^2 ) - 12.c ( -0.64 + m^2 ) m^4 ( 0.426667 + m^2 ) + 32.c^2 ( -0.0849806 + m^2 ) m^2 + & & .. ( 1.20498 + m^2 ) + c^3m^2 ( -122.88 + 128.m^2 ) + 1.m^4 ( 0.16 + m^2 ) ( 0.4096 - 1.28m^2 + m^4 ) ) ) , + _ pol.(_2 _ 3^&+&_2 ^_3)= + & & ( 256.e^8 + 256.c^4 - 1280.e^6m^2 + c^2 ( -20.48 - 32.m^2 ) m^2 + 1.m^4 ( 0.171487 + m^2 ) . + & & ( 2.38851 + m^2 ) + e^2m^2 ( 1280.c^2 + ( -40.96 - 80.m^2 ) m^2 ) + & + & . e^4 ( -512.c^2 + m^2 ( 40.96 + 864.m^2 ) ) ) , + _ pol.(_2 _ 4^&+&_2 ^_4)= ( -1024.e^12 - 7.1054310 ^ -15c^4m^2 . + & + & 3.5527110 ^ -15c^3m^6 + c^2 ( -1.92 - 4.4408910 ^ -16m^2 ) m^6 + ( 0.0384 + 0.12m^2 ) m^8 + + & + & e^10 ( 40.96 + 2048.c + 2304.m^2 ) + e^8 ( 1.1368710 ^ -13c^2 + c ( -81.92 - 4096.m^2 ) . ( -112.64 - 640.m^2 ) m^2 ) + e^4 ( 1024.c^4 + 1024.c^3m^2 + c^2 ( -30.72 - 512.m^2 ) m^2 . c ( -35.84 - 128.m^2 ) m^4 + 44.m^4 ( 0.0220983 + m^2 ) ( 0.210629 + m^2 ) ) + & + & e^2m^2 ( -256.c^4 - 128.c^3m^2 - 3.m^4 ( 0.106667 + m^2 ) ( 0.8 + m^2 ) + cm^4 ( 1.28 + 8.m^2 ) . + & + & . c^2m^2 ( 12.8 + 64.m^2 ) ) + e^6 ( -2048.c^3 - 96 . ( -0.247773 + m^2 ) m^2 ( 0.0344401 + m^2 ) . + & + & .. c^2 ( 40.96 + 1024.m^2 ) + cm^2 ( 143.36 + 1280.m^2 ) ) ) , + _ pol.|m_3|^2&= & ( 4096.e^12 + e^10 ( -163.84 + 20480.c .. + & + & .2048 . m^2 ) + e^2 ( 4096.c^5 + c^4 ( -163.84 -2048.m^2 ) + c^3 ( -327.68 - 2560.m^2 ) m^2 . + & - & 512.c^2 ( -0.492029 + m^2 ) m^2 ( 0.0520294 + m^2 ) + 8 . ( -0.64 + m^2 ) m^4 ( 0.0626408 + m^2 ) + & & . ( 0.817359 + m^2 ) + 16.cm^4 ( 0.0812907 + m^2 ) ( 5.03871 + m^2 ) ) + e^4 ( 20480.c^4 . + & + & c^2 ( -1228.8 - 3072.m^2 ) m^2 - 640.c ( -0.276264 + m^2 ) m^2 ( 0.148264 + m^2 ) - 16 . + & & . ( -3.07951 + m^2 ) m^4 ( 0.199512 + m^2 ) + c^3 ( -655.36 + 2048.m^2 ) ) + e^8 ( 40960.c^2 . + & - & .. ( 450.56 + 256.m^2 ) m^2 + c ( -655.36 + 7168.m^2 ) ) + e^6 ( 40960.c^3 - c(1310.72 + 1536.m^2 ) m^2 .. + & - & .. 256 . ( -0.189783 + m^2 ) m^2 ( 0.269783 + m^2 ) + c^2 ( -983.04 + 8192.m^2 ) ) + m^2 ( -1024.c^5 .. + & + & c^4 ( 40.96 - 768.m^2 ) + c^3 ( 122.88 - 128.m^2 ) m^2 + 12.c ( -0.64 + m^2 ) m^4 ( 0.426667 + m^2 ) + & + & .. 32.c^2 ( -0.0849806 + m^2 ) m^2 ( 1.20498 + m^2 ) + 1.m^4 ( 0.16 + m^2 ) ( 0.4096 - 1.28m^2 + m^4 ) ) ) , + _ pol.(_3 _ 4^&+&_3 ^_4)= ( + e^4 ( 6 + - ) . + & + & . e^2 ( + - ) + + + + ) + & - & ( c ( 4e^2 - m^2 ) ( -8e^6 ( 2c + 3m^2 ) + m^2 ( -4.c^3 ... + & - & . 2.c^2m^2 + c ( 0.24 - 2.25m^2 ) m^2 + 0.06m^4 ) + e^2 ( -16.c^3 + 8.c^2m^2 + c ( 0.96 - 11.m^2 ) m^2 . + & + & ... ( -0.48 - 1.5m^2 ) m^4 ) + e^4 ( -32.c^2 - 12.cm^2 + m^2 ( 0.96 + 12.m^2 ) ) ) ) + & - & ( ( 4e^2 - m^2 ) ( 4e^2 + m^2 ) ( -8e^6 ( 2c + 3m^2 ) .. + & + & m^2 ( -4.c^3 - 2.c^2m^2 + c ( 0.24 - 2.25m^2 ) m^2 + 0.06m^4 ) + e^2 ( -16.c^3 + 8.c^2m^2 . + & + & ... c ( 0.96 - 11.m^2 ) m^2 + ( -0.48 - 1.5m^2 ) m^4 ) + e^4 ( -32.c^2 - 12.cm^2 + m^2 ( 0.96 + 12.m^2 ) ) ) ) + + & + & ( ( -4e^2 + m^2 ) ^2 ( 32.c^4 - 16.e^6m^2 + 16.c^3m^2 .. + & - & 0.64cm^4 + ( 0.0768 - 0.6m^2 ) m^4 + c^2m^2 ( -3.2 + 18.m^2 ) + e^2 ( 64.c^3 + 96.c^2m^2 . + & + & ... m^4 ( -3.52 + 15.m^2 ) + cm^2 ( -2.56 + 16.m^2 ) ) + e^4 ( 32.c^2 + 64.cm^2 + m^2 ( 0.64 + 40.m^2 ) ) ) ) , + _ pol.|m_4|^2&= & + + & + & + + & + & + & - & + & + & + & + & + & - & + & - & , where @xmath73 . 99 l. randall and r. sundrum , phys . 83 ( 1999 ) 3370 , _ hep - ph/9905221_. l. randall and r. sundrum , phys . rev . 83 ( 1999 ) 4690 , _ hep - th/9906064_. i. antoniadis , n. arkani - hamed , s. dimopoulos and g. dvali , phys . lett . * b436 * ( 1998 ) 257 , _ hep - ph/9804398_. n. arkani - hamed , s. dimopoulos and g. dvali , phys . lett . * b429 * ( 1998 ) 263 , _ hep - ph/9803315 _ ; phys . rev . * d59 * ( 1999 ) 086004 , _ hep - ph/9807344_. e.a . mirabelli , m. perelstein , m.e . peskin , phys . ( 1999 ) 2236 , _ hep - ph/9811337_. h. davoudiasl , j.l . hewett and t.g . rizzo , phys . 84 ( 2000 ) 2080 , _ hep - ph/9909255_. h. davoudiasl , j.l . hewett and t.g . rizzo , phys . b473 * ( 2000 ) 43 , _ hep - ph/9911262_. c. itzykson and j - b . zuber , _ quantum field theory _ , mcgraw - hill , new york , 1980 . m.j.g . veltman , `` quantum theory of gravitation '' in _ methods in field theory _ , les houches , ( 1975 ) 265 . m. e. peskin , d. v. schroeder , _ an introduction to quantum field theory _ , addison - wesley , 1995 .

considering the randall - sundrum background , we calculate the total cross - section for @xmath0 in the framework of the scalar electrodynamics . + l 3 mm 230 mm ipm / p2000/021 + hep - ph/0005166

if cooled below a critical temperature ( @xmath0k in @xmath1he and @xmath2 in at @xmath3hehe we mean the b - phase of @xmath3he ] at saturated vapour pressure ) , liquid helium undergoes bose - einstein condensation @xcite , becoming a quantum fluid and demonstrating superfluidity ( pure inviscid flow ) . besides the lack of viscosity , another major difference between superfluid helium and ordinary ( classical ) fluids such as water or air is that , in helium , vorticity is constrained to vortex line singularities of fixed circulation @xmath4 , where @xmath5 is planck s constant , and @xmath6 is the mass of the relevant boson ( in the most common isotope @xmath1he , @xmath7 , the mass of an atom ; in the rare isotope @xmath3he , @xmath8 , the mass of a cooper pair ) . these vortex lines are essentially one - dimensional space curves , like the vortex lines of fluid dynamics textbooks ; for example , in @xmath1he the vortex core radius @xmath9 m is comparable to the inter atomic distance . this quantisation of the circulation thus results in the appearance of another characteristic length scale : the mean separation between vortex lines , @xmath10 . in typical experiments ( both in @xmath1he and @xmath3he ) @xmath10 is orders of magnitude smaller than the outer scale of turbulence @xmath11 ( the scale of the largest eddies ) but is also orders of magnitudes larger than @xmath12 . there is a growing consensus @xcite that on length scales much larger than @xmath10 the properties of superfluid turbulence are similar to those of classical turbulence if excited similarly , for example by a moving grid . the idea is that motions at scales @xmath13 should involve at least a partial polarization @xcite of vortex lines and their organisation into vortex bundles which , at such large scales , should mimic continuous hydrodynamic eddies . therefore one expects a classical richardson - kolmogorov energy cascade , with larger `` eddies '' breaking into smaller ones . the spectral signature of this classical cascade is indeed observed experimentally in superfluid helium . in the absence of viscosity , in superfluid turbulence the kinetic energy should cascade downscale without loss , until it reaches the small scales where the quantum discreteness of vorticity is important . it is also believed that at this point the richardson - kolmogorov eddy - dominated cascade should be replaced by a second cascade which arises from the nonlinear interaction of kelvin waves ( helical perturbation of the vortex lines ) on individual vortex lines . this kelvin wave cascade should take the energy further downscale where it is radiated away by thermal quasi particles ( phonons and rotons in @xmath1he ) . although this scenario seems quite reasonable , crucial details are yet to be established . our understanding of superfluid turbulence at scales of the order of @xmath10 is still at infancy stage , and what happens at scales below @xmath10 is a question of intensive debates . the quasi - classical " region of scales , @xmath14 , is better understood , but still less than classical hydrodynamic turbulence . the main reason is that at nonzero temperatures ( but still below the critical temperature @xmath15 ) , superfluid helium is a two - fluid system . according to the theory of landau and tisza @xcite , it consists of two inter penetrating components : the inviscid superfluid , of density @xmath16 and velocity @xmath17 ( associated to the quantum ground state ) , and the viscous normal fluid , of density @xmath18 and velocity @xmath19 ( associated to thermal excitations ) . the normal fluid carries the entropy @xmath20 and the viscosity @xmath21 of the entire liquid . in the presence of superfluid vortices these two components interact via a mutual friction force@xcite . the total helium density @xmath22 is practically temperature independent , while the superfluid fraction @xmath23 is zero at @xmath24 , but rapidly increases if @xmath25 is lowered ( it becomes 50% at @xmath26 , 83% at @xmath27 and 95% at @xmath28 @xcite ) . the normal fluid is essentially negligible below @xmath29 . one would therefore expect classical behaviour only in the high temperature limit @xmath30 , where the normal fluid must energetically dominate the dynamics . experiments show that this is not the case , thus raising the interesting problem of double - fluid " turbulence which we study here . the aim of this article is to present the current state of the art in this intriguing problem , clarify common features of turbulence in classical and quantum fluids , and highlight their differences . to achieve our aim we shall overview and combine experimental , theoretical and numerical results in the simplest possible ( and , probably , the most fundamental ) case of homogeneous , isotropic turbulence , away from boundaries and maintained in a statistical steady state by continuous mechanical forcing . the natural tools to study homogeneous isotropic turbulence are spectral , thus we shall consider the velocity spectrum ( also known as the energy spectrum ) and attempt to give a physical explanation for the observed phenomena . we recall that ordinary incompressible viscous flows are described by the navier - stokes equation @xmath31 = - \frac{1}{\rho}{{{\bf{\nabla } } } p}+\nu { { { \bf{\nabla}}}^2 } { { \bf{u}}},\ ] ] and the solenoidal condition @xmath32 for the velocity field @xmath33 , where @xmath34 is the pressure , @xmath35 the density , and @xmath36 the kinematic viscosity . the dimensionless parameter that determines the properties of hydrodynamic turbulence is the reynolds number re@xmath37 . the reynolds number estimates the ratio of nonlinear and viscous terms in eq . at the outer length scale @xmath11 ( typically the size of a streamlined body ) , where @xmath38 is the root mean square turbulent velocity fluctuation . in fully developed turbulence ( re@xmath39 ) the @xmath11-scale eddies are unstable and give birth to smaller scale eddies , which , being unstable , generate further smaller eddies , and so on . this process is the richardson - kolmogorov energy cascade toward eddies of scale @xmath40 , defined as the length scale at which the nonlinear and viscous forces in eq . approximately balance each other . @xmath40-scale eddies are stable and their energy is dissipated into heat by viscous forces . the hallmark feature of fully developed turbulence is the coexistence of eddies of all scales from @xmath11 to @xmath41 with universal statistics ; the range of length scales @xmath42 where both energy pumping and dissipation can be ignored is called the inertial range . in the study of homogeneous turbulence it is customary to consider the energy density per unit mass @xmath43 ( of dimensions @xmath44 ) . in the isotropic case the energy distribution between eddies of scale @xmath45 can be characterized by the one dimensional energy spectrum @xmath46 of dimensions @xmath44 ) with wavenumber defined as @xmath47 ( or as @xmath48 ) , normalized such that + @xmath49 where @xmath50 is volume . in the inviscid limit , @xmath43 is a conserved quantity ( @xmath51 ) , thus @xmath46 satisfies the continuity equation @xmath52 where @xmath53 is the energy flux in spectral space ( of dimensions @xmath54 ) . in the stationary case , energy spectrum and energy flux are @xmath55independent , thus eq . immediately dictates that the energy flux @xmath56 is @xmath57-independent . assuming that this constant @xmath56 is the only relevant characteristics of turbulence in the inertial interval and using dimensional reasoning , in 1941 kolmogorov and ( later ) obukhov suggested that the energy spectrum is @xmath58 where the ( dimensionless ) kolmogorov constant is approximately @xmath59 . this is the celebrated kolmogorov - obukhov @xmath60 law ( ko41 ) , verified in experiments and numerical simulations of eq . ; it states in particular that in incompressible , steady , homogeneous , isotropic turbulence , the distribution of kinetic energy over the wavenumbers is @xmath61 . in the inviscid limit the energy flux goes to smaller and smaller scales , reaching finally the interatomic scale and accumulating there . to describe this effect , leith @xcite suggested to replace the algebraic relation between @xmath62 and @xmath63 by the differential form : @xmath64 this approximation dimensionally coincides with eq . , but the derivative @xmath65/dk$ ] guarantees that @xmath66 if @xmath67 . the numerical factor @xmath68 , suggested in @xcite , fits the experimentally observed value @xmath69 in eq . . a generic energy spectrum with a constant energy flux was found in @xcite as a solution to the equation @xmath70 constant : @xmath71^{2/3}.\ ] ] notice that at low @xmath57 , eq . coincides with ko41 , while for @xmath72 it describes a thermalized part of the spectrum , @xmath73 , with equipartition of energy ( shown by the solid black line at the right of in fig . [ f:3]a , and , underneath in the same figure , by the solid red line , although the latter occurs in slightly different contexts).a , the energy flux @xmath74 is not preserved along the cascade , but continuously decreases due to dissipation and ultimately vanishes at the maximum @xmath57 . ] we shall have also to keep in mind that although eq . is an important result of classical turbulence theory , it presents only the very beginning of the story . in particular , its well known @xcite that in the inertial range , the turbulent velocity field is not self similar , but shows intermittency effects which modify the ko41 scenario . in this paper we apply these ideas to superfluid helium , explain how to overcome technical difficulties to measure the energy spectrum near absolute zero , and draw the attention to three conceptual differences between classical hydrodynamic turbulence and turbulence in superfluid @xmath1he . the first difference is that the quantity which ( historically ) is most easily and most frequently detected in turbulent liquid helium is not the superfluid velocity but rather the vortex line density @xmath75 , defined as the superfluid vortex length per unit volume ; in most experiments ( and numerical simulations ) this volume is the entire cell ( or computational box ) which contains the helium . this scalar quantity @xmath75 has no analogy in classical fluid mechanics and should not be confused with the vorticity , whose spectrum , in the classical ko41 scenario , scales as @xmath76 correspondingly to the @xmath77 scaling of the velocity . notice that in a superfluid the vorticity is zero everywhere except on quantized vortex lines . in order to use as much as possible the toolkit of ideas and methods of classical hydrodynamics , we shall define in the next sections an `` effective '' superfluid vorticity field @xmath78 ; this definition ( which indeed @xcite yields the classical @xmath76 vorticity scaling corresponding to the @xmath77 velocity scaling ) is possible on scales that exceed the mean intervortex scale @xmath10 , provided that the vortex lines contained in a fluid parcel are sufficiently polarized . this procedure opens the way for a possible identification of `` local '' values of @xmath79 with the magnitude @xmath80 of the vector field @xmath78 . the second difference is that liquid helium is a two fluid system , and we expect both superfluid and normal fluid to be turbulent . this makes the problem of superfluid turbulence much richer than classical turbulence , but the analysis becomes more involved . for example , the existence of the intermediate scale @xmath10 makes it impossible to apply arguments of scale invariance to the entire inertial interval and calls for its separation into three ranges . the first is a `` hydrodynamic '' region of scales @xmath81 ( corresponding to @xmath82 in k - space where @xmath83 and @xmath84 ) , which is similar ( but not equal ) to the classical inertial range ; the second is a `` kelvin wave region '' @xmath85 where energy is transferred further to smaller scales in @xmath1he@xcite . ] by interacting kelvin waves ( helix - like deformations of the vortex lines ) . in the third , intermediate region @xmath86 , the energy flux is caused probably by vortex reconnections . finally , the third difference is that mutual friction between normal and superfluid components leads to ( dissipative ) energy exchange between them in either direction . studies of classical turbulence are solidly based on the navier - stokes eq . . unfortunately , there are no well established equations of motion for @xmath1he in the presence of superfluid vortices . we have only models at different levels of description ( for an overview see sec . [ s : theory ] ) . all these issues make the problem of superfluid turbulence very interesting from a fundamental view point , simultaneously creating serious problems in experimental , numerical and analytical studies . [ cols="^,^ " , ] in the absence of superfluid vortices , landau s two - fluid equations@xcite for the superfluid and normal fluid velocities @xmath87 and @xmath88 account for all phenomena observed in he - ii at low velocities , for example second sound and thermal counterflow . in the incompressible limit ( @xmath89 , @xmath90 ) landau s equations are : [ landau ] @xmath91 & = & - { { \bf{\nabla } } } p_s/ \rho_s,\\ \label{landau2 } \big [ ( \partial \,{{\bf{u}}}{_{\text { n}}}/\partial t)+ ( { { \bf{u}}}{_{\text { n}}}\cdot { { \bf{\nabla}}}){{\bf{u}}}{_{\text { n } } } \big]&= & - { { \bf{\nabla } } } p_n /\rho_n + \nu_n \nabla^2 { { { \bf{u}}}{_{\text { n } } } } \,,\end{aligned}\ ] ] where @xmath92 is the kinematic viscosity , and the efficient pressures @xmath93 and @xmath94 are defined by @xmath95 and @xmath96 . on physical ground , laudau argued that the superfluid is irrotational . the main difficulty in developing a theory of superfluid turbulence is the lack of an established set of equations of motion for he - ii in the presence of superfluid vortices . we have only models at different levels of description . at the first , most microscopic level of description , we must account for phenomena at the length scale of the superfluid vortex core , @xmath97 . monte carlo models of the vortex core @xcite , although realistic , are not suitable for the study of the dynamics and turbulent motion . a practical model of a pure superfluid is the gross - pitaevskii equation ( gpe ) for a weakly - interacting bose gas @xcite : @xmath98 where @xmath99 is the complex condensate s wave function , @xmath100 the strength of the interaction between bosons , @xmath101 the chemical potential and @xmath6 the boson mass . the condensate s density @xmath102 and velocity @xmath103 are related to @xmath104 via the madelung transformation @xmath105 , which confirms landau s intuition that the superfluid is irrotational . it can be shown that , at length scales @xmath106 , the gpe reduces to the classical continuity equation and the ( compressible ) euler equation . it must be stressed that , although the gpe accounts for quantum vortices , finite vortex core size ( of the order of @xmath12 ) , vortex nucleation , vortex reconnections , sound emission by accelerating vortices and kelvin waves , it is only a qualitative model of the superfluid component . he - ii is a liquid , not a weakly - interacting gas , and the condensate is only a fraction of the superfluid density @xmath107 . no adjustment of @xmath100 and @xmath101 can fit both the sound speed and the vortex core radius , and the dispersion relation of the uniform solution of eq . lacks the roton s minimum which is characteristic of he - ii @xcite . this is why , strictly speaking , we can not identify @xmath108 with @xmath107 and @xmath109 with @xmath87 . nevertheless , when solved numerically , the gpe is a useful model of superfluid turbulence at low @xmath25 where the normal fluid fraction vanishes , and yields results which can be compared to experiments , as we shall see . far away from the vortex core at length scales larger than @xmath12 and in the zero mach number limit , the gpe describes incompressible euler dynamics . this is the level of description of a second practical model , the vortex filament model ( vfm ) of schwarz@xcite . at this level ( length scales @xmath110 ) we ignore the nature of the vortex core but distinguish individual vortex lines , which we describe as oriented space curves @xmath111 of infinitesimal thickness and circulation @xmath112 , parametrised by arc length @xmath113 . their time evolution is determined by schwarz s equation [ vfm ] @xmath114 where the self - induced velocity @xmath115 is given by the biot - savart law @xcite , and the line integral extends over the vortex configuration . at nonzero temperatures the term @xmath116 accounts for the friction between the vortex lines and the normal fluid@xcite : @xmath117\ , , \quad { { \bf{u } } } { _ { \text { ns}}}= { { { \bf{u } } } { _ { \text { n}}}}-{{{\bf{u } } } { _ { \text { si}}}}\,,\ ] ] where @xmath118 is the unit tangent at @xmath119 , and @xmath120 , @xmath121 are known @xcite temperature - dependent friction coefficients . in the very low temperature limit ( @xmath122 ) , @xmath120 and @xmath121 become negligible@xcite , and we recover the classical result that each point of the vortex line is swept by the velocity field produced by the entire vortex configuration . in numerical simulations based on the vfm , vortex lines are discretized in a lagrangian fashion , biot - savart integrals are desingularised using the vortex core radius @xmath12 , and reconnections are additional artificial _ ad - hoc _ procedures that change the way pairs of discretization points are connected . reconnection criteria are described and discussed in ref . @xcite ; ref . @xcite compares gpe and vfm reconnections with each other and with experiments . simulations at large values of vortex line density are performed using a tree algorithm@xcite which speeds up the evaluations of biot - savart integrals from @xmath123 to @xmath124 where @xmath125 is the number of discretization points . the major drawback of the vfm is that the normal fluid @xmath126 is imposed ( either laminar or turbulent ) , therefore the back - reaction of the vortex lines on @xmath126 is not taken into account . the reason is the computational difficulty : a self - consistent simulation would require the simultaneous integration in time of eq . for the superfluid , and of a navier - stokes equation for the normal fluid ( implemented with suitable friction forcing at vortex lines singularities ) . such self - consistent simulations were carried out only for a single vortex ring @xcite and for the initial growth of a vortex cloud @xcite . this limitation is likely to be particularly important at low and intermediate temperatures ( at high temperatures the normal fluid contains most of the kinetic energy , so it is less likely to be affected by the vortices ) . at a third level of description we do not distinguish individual vortex lines any longer , but rather consider fluid parcels which contain a continuum of vortices . at these length scales @xmath13 we seek to generalise landau s equations to the presence of vortices . in laminar flows the vortex lines ( although curved ) remain locally parallel to each other , so it is possible to define the components of a macroscopic vorticity field @xmath78 by taking a small volume larger than @xmath10 and considering the superfluid circulation in the planes parallel to the cartesian directions ( alternatively , the sum of the oriented vortex lengths in each cartesian direction ) . we obtain the so - called hall - vinen ( or hvbk ) coarse - grained " equations @xcite : [ nse ] @xmath127 = - \frac{1}{\rho_s}{{{\bf{\nabla } } } p_s}-{{\bf{f } } } { _ { \text { ns } } } , \\ \label{nseb } & & \hskip - 0.7 cm \big[\frac{\partial \,{{\bf{u}}}{_{\text { n}}}}{\partial t}+ ( { { \bf{u}}}{_{\text { n}}}\cdot { { \bf{\nabla}}}){{\bf{u}}}{_{\text { n } } } \big]= -\frac{1}{\rho_n } { { { \bf{\nabla } } } p_n } + \nu_n \nabla^2 { { { \bf{u}}}{_{\text { n}}}}+\frac{\rho_s}{\rho } { { \bf{f } } } { _ { \text { ns } } } , \\ \label{nsec } & & \hskip - 0.7 cm { { \bf{f } } } { _ { \text { ns}}}=\alpha { \hat { { \mbox{\boldmath $ \omega$}}}_s } \times ( { { \mbox{\boldmath $ \omega$}}}_s \times { { \bf{u } } } { _ { \text { ns } } } ) + \alpha ' { \hat { { \mbox{\boldmath $ \omega$}}}_s } \times { { \bf{u } } } { _ { \text { ns}}},\end{aligned}\ ] ] where @xmath128 , @xmath129 and @xmath130 is the mutual friction force . these equations have been used with success to predict the taylor - couette flow of he - ii , its stability @xcite and the weakly nonlinear regime @xcite . in these flows , the vortex lines are fully polarised and aligned in the same direction , and their density and orientation may change locally and vary as a function of position ( on length scales larger than @xmath10 ) . the difficulty with applying the hvbk equations to turbulence is that in turbulent flows the vortex lines tend to be randomly oriented with respect to each other , so the components of @xmath131 cancel out to zero ( partially or totally ) , resulting in local vortex length ( hence energy dissipation ) without any effective superfluid vorticity . in this case , the hvbk equations may become a poor approximation and underestimate the mutual friction coupling . nevertheless , they are a useful model of large scale superfluid motion with characteristic scale @xmath14 , particularly because ( unlike the vfm ) they are dynamically self - consistent ( normal fluid and superfluid affect each other ) . we must also keep in mind that eq . do not have physical meaning at length scales smaller than @xmath10 . in the next sections we shall describe numerical simulations of equations as well as shell models and theoretical models based on these equations . in some models the mutual friction force is simplified to @xmath132 where @xmath133 . numerical simulations in the framework of all three approaches , and are shown in figs . they clearly show ko41 scaling , in agreement with the experimental results shown in figs . details of this simulations will be described below . since the pioneering work of schwarz @xcite , numerical experiments have played an important role , allowing the exploration of the consequences of limited sets of physical assumptions in a controlled way , and providing some flow visualization . numerical simulations of the gpe in a three - dimensional periodic box have been performed for decaying turbulence @xcite following an imposed arbitrary initial condition , and for forced turbulence @xcite . besides vortex lines , the gpe describes compressible motions and sound propagation ; therefore , in order to analyse turbulent vortex lines , it is necessary to extract from the total energy of the system ( which is conserved during the evolution ) its incompressible kinetic energy part whose spectrum is relevant to our discussion . to reach a steady state , large - scale external forcing and small - scale damping was added to the gpe @xcite . the resulting turbulent energy spectrum , shown in fig . [ f:3]c , agrees with the ko41 scaling ( shown by cyan dot - dashed line ) , and demonstrates bottleneck energy accumulation near the intervortex scale at zero temperature predicted earlier in @xcite and discussed in sect . the ko41 scaling observed in gpe simulations was found to be consistent with the vfm at zero temperature @xcite and has also been observed when modelling a trapped atomic bose einstein condensate @xcite . the gpe can be extended to take into account finite temperature effects . different models have been proposed @xcite . most vfm calculations have been performed in a cubic box of size @xmath11 with periodic boundary conditions . in all relevant experiments we expect that the normal fluid is turbulent because its reynolds number @xmath134 is large ( where @xmath135 the root mean square normal fluid velocity ) . recent studies thus assumed the form @xcite + @xmath136 where @xmath137 , @xmath138 and @xmath139 are wave vectors and angular frequencies . the random parameters @xmath140 , @xmath141 and @xmath138 are chosen so that the normal fluid s energy spectrum obeys ko41 scaling @xmath142 in the inertial range @xmath143 . this synthetic turbulent flow @xcite is solenoidal , time - dependent , and compares well with lagrangian statistics obtained in experiments and direct numerical simulations of the navier - stokes equation . other vfm models included normal - fluid turbulence generated by the navier stokes equation @xcite and a vortex - tube model @xcite , but , due to limited computational resources , only a snapshot of the normal fluid , frozen in time , was used to drive the superfluid vortices . in all numerical experiments , after a transient from some initial condition , a statistical steady state of superfluid turbulence is achieved , in the form of a vortex tangle ( see fig . [ f:1]-b ) in which the vortex line density @xmath144@xmath145 fluctuates about an average value independent of the initial condition . recent analytical @xcite and numerical studies@xcite of the geometry of the vortex tangle reveal that the vortices are not randomly distributed , but there is a tendency to locally form bundles of co - rotating vortices , which keep forming , vanish and reform somewhere else . these bundles associate with the kolmogorov spectrum : if turbulence is driven by a uniform normal fluid ( as in the original work of schwarz@xcite recently verified in ref . @xcite ) , there are nor kolmogorov scaling nor bundles . laurie et al.@xcite decomposed the vortex tangle in a polarised part ( of density @xmath146 ) and a random part ( of density @xmath147 ) , as argued by roche & barenghi @xcite , and discovered that @xmath146 is responsible for the @xmath77 scaling of the energy spectrum , and @xmath147 for the @xmath148 scaling of the vortex line density fluctuations , as suggested in ref.@xcite . from a computational viewpoint , the hvbk equations are similar to the navier - stokes equation . not surprisingly , standard methods of classical turbulence have been adapted to the hvbk equations , e.g. large eddy simulations @xcite , direct numerical simulations @xcite and eddy damped quasinormal markovian simulations @xcite . the hvbk equations are ideal to study the coupled dynamics of superfluid and normal fluid in the limit of intense turbulence at finite temperature . indeed , by ignoring the details of individual vortices and their fast dynamics , hvbk simulations do not suffer as much as vfm and gpe simulations from the wide separation of space and time scales which characterize superfluid turbulence . moreover , well optimized numerical solvers have been developed for navier - stokes turbulence and they can be easily adapted to the hvbk model . thus , simulations over a wide temperature range ( @xmath149 corresponding to @xmath150 ) evidence a strong locking of superfluid and normal fluid ( @xmath151 ) at large scales , over one decade of inertial range ( @xcite ) . in particular , it was found that even if one single single fluid is forced at large scale ( the dominant one ) , both fluids still get locked very efficiently . [ f:3]a illustrates velocity spectra generated by direct numerical simulation of the hvbk equations , while the red and blue solid lines of fig . [ f:3]b show spectra obtained using a shell model of the same equations ( see paragraph at the end of the section ) . in both case , a clear @xmath77 spectrum is found for both fluid components , at all temperature and large scale . information about the quantization of vortex lines is lost in the coarse graining procedure which leads to eqs . . as discussed in sect . 8b , a quantum constrain can be re - introduced in this model by truncating superfluid phase space for @xmath152 , causing an upward trend of the low temperature velocity spectrum of fig . [ f:3]a which can be interpreted as partial thermalization of superfluid excitations . this procedure also leads to the prediction @xmath153 @xcite which is consistent with experiments and allows to identify the spectrum of @xmath75@xmath154 with the spectrum of the scalar field @xmath155 . it is found that this spectrum is temperature dependent in the inertial range with a flat part at high temperature ( reminiscent of the corresponding spectrum of the magnitude of the vorticity in classical turbulence ) which contrasts the @xmath77 decreases at low temperature ( consistent with experiments @xcite ) . essential simplification of the hvbk eqs . can be achieved with the shell - model approximation@xcite . the complex shell velocities @xmath156 and @xmath157 mimic the statistical behaviour of the fourier components of the turbulent superfluid and normal fluid velocities at wavenumber @xmath57 . the resulting ordinary differential eqs . for @xmath158 capture important aspects of the hvbk eqs . . because of the geometrical spacing of the shells ( @xmath159 ) , this approach allows more decades of @xmath57-space than eqns . ( see fig . [ fig : interim ] with eight decades in @xmath57-space @xcite ) . this extended inertial range allows detailed comparison of intermittency effects in superfluid turbulence and classical turbulence ( see sec . in this section we discuss theoretical models of large scale ( eddy dominated ) motions at wavenumbers @xmath160 which are important at all temperatures from 0 to @xmath161 . these motions can be tackled using the hydrodynamic hvbk eqs . , thus generalising what we know about classical turbulence . we shall start by considering the simpler case of @xmath3he ( sect . [ 6a ] ) , in which there is only one turbulent fluid , then move to more difficult case of @xmath1he ( sect . [ 6b ] ) in which there are two coupled turbulent fluids , and finally discuss intermittency ( sect . [ 6c ] ) . the viscosity of @xmath3he is so large that , in all @xmath3he turbulence experiments , we expect the normal fluid to be at rest ( @xmath162 ) or in solid body rotation in rotating cryostat ( in which case our argument requires a slight modification ) . liquid @xmath3he thus provides us with a simpler turbulence problem ( superfluid turbulence in the presence of linear friction against a stationary normal fluid ) than @xmath1he ( superfluid turbulence in the presence of normal fluid turbulence ) . at scales @xmath13 , we expect eq . to be valid , provided we add a suitable model for the friction . following ref . @xcite , we approximate @xmath130 as + @xmath163with @xmath164 . here @xmath165 , @xmath166 is the characteristic turbulent " superfluid vorticity , estimated via the spectrum @xmath63 . @xmath167 is the energy pumping scale . using the differential approximation for the energy spectrum , the continuity eq . in the stationary case becomes @xmath168 + \gamma e{_{\text { s } } } ( k)=0.\ ] ] analytical solutions of eq . are in good agreement @xcite with the results of numerical simulation of the shell model to the hydrodynamic eq . , providing us with quasi - qualitative description of turbulent energy spectra in @xmath3he over a wide region of temperatures . in @xmath1he , we have to account for motion of the normal component , which has very low viscosity and is turbulent in the relevant experiments . eqs . , and ( now with @xmath169 ) result in a system of energy balance equations for superfluid and normal fluid energy spectra @xmath170 and @xmath171 that generalises eq . @xcite : here @xmath180 and @xmath181 are characteristic interaction frequencies ( or turnover frequencies ) of eddies in the normal and superfluid components . they are related to the well known effective turbulent viscosity @xmath182 by @xmath183 . for large mutual friction or / and small @xmath57 , when @xmath184 and can be neglected , eq . has a physically motivated solution @xmath185 corresponding to full locking @xmath186 . in this case the sum of eq . ( multiplied by @xmath16 ) and eq . ( multiplied by @xmath187 ) yields the navier - stokes equation with effective viscosity @xmath188 . thus , in this region of @xmath57 , one expects classical behaviour of hydrodynamic turbulence with ko41 scaling ( up to intermittency corrections discussed in sec . for small mutual friction or / end large @xmath57 , when @xmath189 , eqs . gives @xcite @xmath190 i.e. full decorrelation of superfluid and normal fluid velocities . in this case normal component will have ko41scaling , @xmath191 up to the kolmogorov micro - scale @xmath192 that can be found from the condition @xmath193 , giving the well known estimate @xmath194 simultaneously , the superfluid spectrum also obeys the same ko41 scaling , @xmath195 . moreover , since at small @xmath57 the two fluid components are locked , we expect that @xmath196 . assuming that the @xmath60 scaling is valid up to wavenumber @xmath197 , we estimate that @xmath198 which is similar to the estimate for @xmath192 with the replacement of @xmath199 with @xmath112 . in he - ii , the numerical values of @xmath200 and @xmath112 are similar , thus we conclude that the viscous cutoff @xmath192 for the normal component and the quantum cutoff @xmath201 for the superfluid component are close to each other . at a given temperature , the decorrelation crossover wave vector @xmath202 between the two regimes described above can be found from the condition @xmath203 using the estimate @xmath204 with @xmath205 . we obtain @xmath206 . the quantity @xmath207 varies between 1.2 and 0.5 @xcite in the temperature range @xmath208 where the motion of the normal fluid is important . we conclude that @xmath209 , which means that normal fluid and superfluid eddies are practically locked over the entire inertial interval . nevertheless , dissipation due to mutual friction can not be completely ignored , leading to intermittency enhancement described next . the first numerical study of intermittent exponents @xcite did not find any intermittent effect peculiar to superfluid turbulence neither at low temperature ( @xmath210 , @xmath211 ) nor at and high temperature ( @xmath212 , @xmath213 ) , in agreement with experiments performed at the same temperatures ( see sect . [ s : exp ] ) . recent shell model simulations @xcite with eight decades of @xmath57-space allowed detailed comparison of classical and superfluid turbulent statistics in the intermediate temperature range corresponding to @xmath214 . the results were the following . for @xmath25 slightly below @xmath161 , when @xmath215 , the statistics of turbulent superfluid @xmath1he appeared similar to that of classical fluids , because the superfluid component can be neglected , see green lines in fig . [ fig : interim ] with @xmath216 . the same result applies to @xmath217 ( @xmath218 ) , as expected due to the inconsequential role played by the normal component , see blue lines with @xmath219 . in agreement with the previous study the intermittent scaling exponents appeared the same in classical and low - temperature superfluid turbulence ( indeed the nonlinear structure of the equation for the superfluid component is the same as of euler equation , and dissipative mechanisms are irrelevant . ) a difference between classical and superfluid intermittent behaviour in a wide ( up to three decades ) interval of scales was found in the range @xmath220 ( @xmath221 ) , as shown by red lines in fig . [ fig : interim ] with @xmath222 . the exponents of higher order correlation functions also deviate further from the ko41 values . what is predicted is thus an enhancement of intermittency in superfluid turbulence compared to the classical turbulence . superfluid ( solid lines ) and normal fluid ( dash lines ) compensated energy spectra @xmath223 ; the compensation factor is the classical energy spectrum with intermittency correction . inset : @xmath224 for @xmath225 . shell model simulation of the hvbk model at @xmath226 k ( green ) , 0.9 ( red ) and 0.9 ( blue ) , corresponding to @xmath227 and 0.9 respectively @xcite . the vertical dash lines indicate @xmath228.,scaledwidth=50.0% ] now we come to the more complicated and more intensively discussed aspect of the superfluid energy spectrum : what happens for @xmath229 , where the quantisation of vortex lines becomes important . this range acquires great importance at low temperatures , typically below 1 k in @xmath1he , and is relevant to turbulence decay experiments . here we shall describe only the basic ideas , avoiding the most debated details . for @xmath230 we neglect the interaction between separate vortex lines ( besides the small regions around vortex reconnection events , which will be discussed later ) . under this reasonable assumption , at large @xmath57 superfluid turbulence can be considered as a system of kelvin waves ( helix - like deformation of vortex lines ) with different wavevectors interacting with each other on the same vortex . the prediction that this interaction results in turbulent energy transfer toward large @xmath57 @xcite was confirmed by numerical simulations in which energy was pumped into kelvin waves at intervortex scales by vortex reconnections @xcite or simply by exciting the vortex lines @xcite . the first analytical theory of kelvin wave turbulence ( propagating along a straight vortex line and in the limit of small amplitude compared to wavelength ) was proposed by kozik and svistunov @xcite ( ks ) , who showed that the leading interaction is a six - wave scattering process ( three incoming waves and three outgoing waves ) . under the additional assumption of locality of the interaction ( that only compatible wave - vectors contribute to most of the energy transfer ) ks found that ( using the same normalisation of other hydrodynamic spectra such as eqs . ) the energy spectrum of kelvin waves is + @xmath231 + here @xmath232 or @xmath233 in typical @xmath1he and @xmath3he experiments , and @xmath234 is the energy flux in three dimensional @xmath235-space . later lvov - nazarenko ( ln ) @xcite criticised the ks assumption of locality and concluded that the leading contribution to the energy transfer comes from a six waves scattering in which two wave vectors ( from the same side ) have wavenumbers of the order of @xmath236 . ln concluded that the spectrum is @xmath237 this ks vs ln controversy triggered an intensive debate ( see e.g. refs @xcite ) , which is outside the scope of this article . we only mention that the three dimensional energy spectrum @xmath238 can be related to the one dimensional amplitude spectrum @xmath239 by @xmath240 where @xmath241 is the angular frequency of a kelvin wave of wavenumber @xmath57 , @xmath242 the energy of one quantum , and @xmath243 the number of quanta ; therefore , in terms of the kelvin waves amplitude spectrum ( which is often reported in the literature and can be numerically computed ) , the two predictions are respectively @xmath244 ( ks ) and @xmath245 ( ln ) . the two predicted exponents ( -3.40 and -3.67 ) are very close to each other ; indeed vfm simulations @xcite could not distinguish them ( probably because the numerics were not in the sufficiently weak regime of the theory in terms of ratio of amplitude to wavelength ) . nevertheless , more recent gpe simulations by krstulovic @xcite based on long time integration of eq . and averaged over initial conditions ( slightly deviating from a straight line ) support the ln spectrum . at finite temperature , it was shown in ref . @xcite that the kelvin wave spectrum is suppressed by mutual friction for @xmath246 , reaching core scale ( @xmath247 ) at @xmath248k and fully disappears at @xmath249k , when @xmath250 . the region of the spectrum near the intervortex scale @xmath251 is difficult because both eddy - type motions and kelvin waves are important , and the discreteness of the superfluid vorticity prevents direct application of the tools of classical hydrodynamic . nevertheless , some progress can be made : sect . [ 8a ] presents a differential model for the @xmath252 limit @xcite , and sect . [ 8b ] describes a complementary truncated hvbk model @xcite designed for the @xmath253 temperature range . the description of superfluid turbulence for @xmath254 is more complicated than @xmath255 because there are no well justified theoretical approaches ( like in the problem of kelvin wave turbulence at @xmath256 ) or even commonly accepted uncontrolled closure approximations . nevertheless , there some qualitative predictions can be tested numerically and experimentally , at least in the zero temperature limit . comparison @xcite of the hydrodynamic spectrum with the kelvin wave spectrum at @xmath257 suggests that the one dimensional nonlinear transfer mechanisms among weakly nonlinear kelvin waves on individual vortex lines is less efficient than the three dimensional , strongly nonlinear eddy - eddy energy transfer . the consequence is an energy cascade stagnation at the crossover between the collective eddy - dominated scales and the single vortex wave - dominated scales . @xcite argued that the superfluid energy spectrum @xmath63 at @xmath254 should be a mixture of three dimensional hydrodynamics modes and one dimensional kelvin waves motions ; the corresponding spectra should be @xmath258e(k)\ .\ ] ] here @xmath259^{-1}\ ] ] is theblending " function which was found @xcite by calculating the energies of correlated and uncorrelated motions produced by a system of @xmath10-spaced wavy vortex lines . the total energy flux , @xmath62 arising from hydrodynamic and kelvin - wave contributions , was modelled @xcite by dimensional reasoning in the differential approximation , similar to eq . : for @xmath260 the energy flux is purely hydrodynamic and @xmath63 is given by eq . , while for @xmath261 it is purely supported by kelvin waves and @xmath63 is given by eq . . this approach leads to the ordinary differential equation @xmath262 constant , which was solved numerically . the predicted energy spectra @xmath63 for different values of @xmath263 are shown in fig . [ f:3]c , exhibit a bottleneck energy accumulation @xmath73 in agreement with eq . . recently a model @xcite has been proposed that accounts for the fact that ( according to numerical evidence @xcite and analytical estimates @xcite ) small scales excitations ( @xmath264 ) , such as kelvin waves and isolated rings , are fully damped for @xmath265k . thus , at these temperatures , the energy flux @xmath266 should be very small at scales @xmath267 . the idea @xcite was to use the hvbk eqs . but truncating the superfluid beyond a cutoff wavenumber @xmath268 , where @xmath269 is a fitting parameter of order one . obviously , a limitation of this model is the abruptness of the truncation ( a more refined model could incorporate a smoother closure which accounts for the dissipation associated with vortex reconnections and the difference between @xmath270 and @xmath271 ) . direct numerical simulations of this truncated hvbk model for temperatures @xmath272 with @xmath273 confirmed the ko41 scaling of the two locked fluids in the range @xmath274 ( see fig . [ f:3]a ) . at smaller scales , an intermediate ( meso ) regime @xmath275 was found that expands as the temperature is lowered @xcite . apparently , superfluid energy , cascading from larger length scales , accumulates beyond @xmath276 . at the lowest temperatures , this energy appears to thermalize , approaching equipartition with @xmath277 , as shown by the red curve of fig . the process saturates when the friction coupling with the normal fluid becomes strong enough to balance the incoming energy flux @xmath278 . in physical space , this mesoscale thermalization should manifest itself as a randomisation of the vortex tangle . the effect is found to be strongly temperature dependent@xcite : @xmath279 . the truncated hydrodynamic model reproduces the decreasing spectrum of the vortex line density fluctuations at small @xmath57 and reduces to the classical spectrum in the @xmath280 limit . this accumulation of thermalized superfluid excitations at small scales and finite temperature was predicted by an earlier model developed to interpret vortex line density spectra @xcite . we conclude that , at large hydrodynamic scales @xmath82 , the evidence for the ko41 @xmath77 scaling of the superfluid energy spectrum which arises from experiments , numerical simulations and theory ( across all models used ) is strong and consistent , and appears to be independent of temperature ( including the limit of zero temperature in the absence of the normal fluid @xcite ) . this direct spectral evidence is also fully consistent with an indirect body of evidence arising from measurements of the kinetic energy dissipation ( @xcite ) and vortex line density decay @xcite in turbulent helium flows . the main open issue is the existence of vortex bundles @xcite predicted by the vfm , for which there is no direct experimental observation yet . intermittency effects , predicted by shell models @xcite , also await for experimental evidence . what happens at mesoscales just above @xmath281 is less understood . the differential model ( at @xmath257 , sect . [ 8a ] ) and the truncated hvbk model ( at finite @xmath25 , sec . [ 8b ] ) , predict an upturning of the spectrum ( temperature - dependent for the latter model ) in this region of @xmath57-space . if confirmed by the experiments and the vfm model , this would signify the striking appearance of quantum effects at scales larger than @xmath10 . further insight could arise from better understanding of fluctuations of the vortex line density . it is worth noticing that similar macroscopic manifestation of the singular nature of the superfluid vorticity was also predicted for the pressure spectrum @xcite . at length scales of the order of @xmath10 and less than @xmath10 the situation is even less clear . this regime is very important at the lowest temperatures , where the kelvin waves are not damped , and energy is transferred from the eddy dominated , three dimensional kolmogorov - richardson cascade into a kelvin wave cascade on individual vortex lines , until the wavenumber is large enough that energy is radiated as sound . the main open issues which call for better understanding concern the cross over and more elaborated description of the bottleneck energy accumulation around @xmath10 in the wide temperature range from 0 to about @xmath161 and the role of vortex reconnections in the strong regime ( large kelvin wave amplitudes compared to wavelength ) of the cascade . at the moment , there is much debate on these problems but no direct experimental evidence for these effects . it is however encouraging that the most recent gpe simulations @xcite in the weak regime ( small amplitude compared to wavelength ) seem to agree with theoretical predictions . fisher ( 2008 ) , _ turbulence experiments in superfluid 3he at very low temperatures _ , in _ vortices and turbulence at very low temperatures _ , edited by c.f . barenghi and y.a . sergeev , cism courses and lectures , vol . 501 , springer verlag ( 2008 ) , 157257 . d. schmoranzer , m. rotter , j. sebek , and l. skrbek ( 2009 ) , _ experimental setup for probing a von karman type flow of normal and superfluid helium _ , experimental fluid mechanics 2009 , proceedings of the international conference , 304 j. salort , b. chabaud , e. lvque , and p .- e . roche ( 2011 ) . investigation of intermittency in superfluid turbulence . in _ j. phys . _ , volume 318 of _ proceedings of the 13th euromech european turbulence conference , sept 12 - 15 , 2011 , warsaw _ , page 042014 . iop publishing . roche ( 2013 ) , _ energy spectra and characteristic scales of quantum turbulence investigated by numerical simulations of the two - fluid model _ , to appear in the proc . of the 14th euromech european turbulence conference , sept 1 - 4 , 2013 , lyon . p. walstrom , j. weisend , j. maddocks , and s. van sciver ( 1988 ) _ turbulent flow pressure drop in various he ii transfer system components . cryogenics ( 28):101 b. rousset , g. claudet , a. gauthier , p. seyfert , a. martinez , p. lebrun , m. marquet , and r. van weelderen ( 1994 ) _ pressure drop and transient heat transport in forced flow single phase helium ii at high reynolds numbers . cryogenics ( 34):317 m. abid , m. e. brachet , j. maurer , c. nore , and p. tabeling ( 1998 ) _ experimental and numerical investigations of low - temperature superfluid turbulence eur . j. mech b - fluid ( 17):665 s. fuzier , b. baudouy , and s. w. van sciver ( 2001 ) _ steady - state pressure drop and heat transfer in he ii forced flow at high reynolds number cryogenics ( 41):453 l. skrbek , j. j. niemela , and k. r. sreenivasan ( 2001 ) _ energy spectrum of grid - generated heii turbulence phys . e , ( 64):067301 j. niemela , k. sreenivasan , and r. donnelly ( 2005 ) _ grid generated turbulence in helium ii j. low temp . ( 138):537 _ _ _ _ _ _

turbulence in superfluid helium is unusual and presents a challenge to fluid dynamicists because it consists of two coupled , inter penetrating turbulent fluids : the first is inviscid with quantised vorticity , the second is viscous with continuous vorticity . despite this double nature , the observed spectra of the superfluid turbulent velocity at sufficiently large length scales are similar to those o ordinary turbulence . we present experimental , numerical and theoretical results which explain these similarities , and illustrate the limits of our present understanding of superfluid turbulence at smaller scales .

the problem of classification of germs of foliations in the complex plane is stated by thom @xcite . he conjectured that the analytic type of a foliation defined in a neighborhood of a singular point is completely determined by its associated separatrix and its corresponding holonomy . r. moussu in @xcite gave a counterexample for this statement and showed that we have to consider the holonomy representation of each irreducible component of the exceptional divisor in a desingularization instead of the economizes . for this new version , thom s problem is proved for cuspidal type singular points @xcite , @xcite and more generally for quasi - homogeneous foliations @xcite . however , the new statement of thom s conjecture was refuted by j.f . mattei by computing the dimension of the space of isoholonomic deformations @xcite , @xcite : there must be other invariants for the non quasi - homogeneous foliations . this conclusion is confirmed by the number of free coefficients in the normal forms in @xcite , @xcite and in the hamiltonian part of the normal forms of vector field in @xcite . by adding a new invariant called _ set of slidings _ this paper solves the problem of strict classification for the non - dicritical foliations whose camacho - sad indices are not rational . here , strict classification means up to diffeomorphism tangent to identity . a germ of singular foliation @xmath0 in @xmath1 is called _ reduced _ if there exists a coordinate system in which it is defined by a @xmath2-form whose linear part is @xmath3 @xmath4 is called the _ camacho - sad index _ of @xmath0 . when @xmath5 , the origin is called a _ saddle - node _ singularity , otherwise it is called _ nondegenerate_. a theorem of a. seidenberg @xcite says that any singular foliation @xmath0 with isolated singularity admits a canonical _ desingularization_. more precisely , there is a holomorphic map @xmath6 obtained as a composition of a finite number of blowing - ups at points such that any point @xmath7 of the _ exceptional divisor _ @xmath8 is either a regular point or a reduced singularity of the strict transform @xmath9 . an intersection of two irreducible components of @xmath10 is called a _ corner_. an irreducible component of @xmath10 is a _ dead branch _ if in this component there is a unique singularity of @xmath11 that is a corner . a _ separatrix _ of @xmath0 is an analytical irreducible invariant curve through the origin of @xmath0 . it is well known that any germ of singular foliation @xmath0 in @xmath1 possesses at least one separatrix @xcite . when the number of separatrices is finite @xmath0 is _ non - dicritical_. otherwise it is called _ dicritical_. denote by @xmath12 the set of all singularities of the strict transform @xmath11 . let @xmath13 be a non - dicritical irreducible component of the exceptional divisor @xmath10 , then @xmath14 is a leaf of @xmath11 . let @xmath7 be a regular point in @xmath15 and @xmath16 a small analytic section through @xmath7 transverse to @xmath11 . for any loop @xmath17 in @xmath15 based on @xmath7 there is a germ of a holomorphic return map @xmath18 which only depends on the homotopy class of @xmath17 in the fundamental group @xmath19 . the map @xmath20 is called the _ vanishing holonomy representation _ of @xmath0 on @xmath13 . let @xmath21 be a foliation that also admits @xmath22 as its desingularization map . assume that @xmath23 where @xmath24 is the set of singularities of the strict transform @xmath25 . denote by @xmath26 in @xmath27 the vanishing holonomy representation of @xmath0 . we say that the vanishing holonomy representation of @xmath0 and @xmath21 on @xmath13 are conjugated if there exists @xmath28 such that @xmath29 . the vanishing holonomy representation of @xmath0 and @xmath21 are called conjugated if they are conjugated on every non - dicritical irreducible component of @xmath10 . [ no1 ] we denote by @xmath30 the set of all non - dicritical foliations @xmath0 defined on @xmath1 such that the camacho - sad index of @xmath11 at each singularity is not rational . if @xmath0 is in @xmath30 then after desingularization all the singularities of @xmath11 are not saddle - node . moreover , the chern class of an irreducible component of divisor , which is an integer , is equal to the sum of camacho - sad indices of the singularities in this component @xcite . therefore , every element in @xmath31 after desingularization admits no dead branch in its exceptional divisor . let @xmath22 as in be a composition of a finite number of blowing - ups at points . a germ of singular holomorphic foliation @xmath32 is said @xmath22-_absolutely dicritical _ if the strict transform @xmath33 is a regular foliation and the exceptional divisor @xmath34 is completely transverse to @xmath35 . when @xmath22 is the standard blowing - up at the origin , we called @xmath32 a _ radial foliation_. at each corner @xmath36 of @xmath10 , the diffeomorphism from @xmath37 to @xmath38 that follows the leaves of @xmath35 is called the _ dulac map _ of @xmath35 at @xmath39 . the existence of such foliations for any given @xmath22 is proved in @xcite . in fact , in @xcite the authors showed that if in each smooth component of @xmath10 we take any two smooth curves transverse to @xmath10 then there is always an absolutely dicritical foliation admitting them as their integral curves . we will denote by @xmath40 if at any point @xmath41 the separatrices of @xmath11 through @xmath39 are transverse to @xmath35 . [ lem1 ] let @xmath0 be a non - dicritical foliation such that @xmath22 is its desingularization map . then there exists a @xmath22-absolutely dicritical foliation @xmath32 satisfying @xmath40 denote by @xmath42 the strict transforms of the separatrices of @xmath0 . on each component @xmath13 of @xmath10 that does not contain any singularity of @xmath11 except the corners we take a smooth curve @xmath43 transverse to @xmath13 . then we have the set of curves @xmath44 such that each component of @xmath10 is transverse to at least one curve @xmath45 . denote by @xmath46 . by @xcite , for each @xmath47 there exists a @xmath22-absolutely dicritical foliation @xmath48 defined by a @xmath2-form @xmath49 verifying that @xmath45 is transverse to @xmath50 . choose a local chart @xmath51 at @xmath52 such that @xmath53 and @xmath54 where `` h.o.t . '' stands for higher order term . write down @xmath49 in the local chart @xmath55 @xmath56 because @xmath50 is transverse to @xmath10 , we have @xmath57 . we define @xmath58 . there always exists a vector @xmath59 such that for @xmath60 , @xmath61 denote by @xmath62 . then , in the local chart @xmath55 , we have @xmath63 because @xmath64 and @xmath49 , @xmath65 , have the same multiplicity on each component of @xmath10 , they have the same vanishing order . since each component of @xmath10 contains at least one point @xmath52 and the strict transform @xmath35 of the foliation @xmath32 defined by @xmath64 is transverse to @xmath10 in each neighborhood of each @xmath52 , @xmath35 is generically transverse to @xmath10 . by @xcite , @xmath32 is absolutely dicritical and satisfies @xmath40 . consider first a nondegenerate reduced foliation @xmath0 in @xmath1 . by @xcite , there exists a coordinate system in which @xmath0 is defined by @xmath66 where @xmath67 . let @xmath32 be a germ regular foliation whose invariant curve through the origin ( we call it the separatrix of @xmath32 ) is transverse to the two separatrices of @xmath0 , which are denoted by @xmath68 and @xmath69 . then we have the following lemma , whose proof is straightforward . [ lem2 ] the tangent curve of @xmath0 and @xmath32 , denoted @xmath70 , is a smooth curve transverse to the two separatrices of @xmath0 . moreover , if the separatrix of @xmath32 is tangent to @xmath71 then @xmath70 is tangent to @xmath72 . and @xmath32 . ] after a standard blowing - up @xmath73 at the origin , the strict transform @xmath74 of @xmath70 is transverse to @xmath11 and cut @xmath75 at @xmath39 . we denote by @xmath76 and @xmath77 the vanishing holonomy representation of @xmath0 . we choose a generator @xmath17 for @xmath78 . then @xmath73 induces @xmath79 we call @xmath80 the _ holonomy on the tangent curve @xmath70_. denote by @xmath81 and @xmath82 the projection by the leaves of @xmath32 from @xmath70 to @xmath68 and @xmath69 respectively . the sliding of a reduced foliation @xmath0 and a regular foliation @xmath32 on @xmath68 ( resp . , @xmath69 ) is the diffeomorphism ( figure [ figure1 ] ) @xmath83 let @xmath84 be the dulac map of @xmath32 ( section [ sec1.2 ] ) . since @xmath85 , it is obvious that @xmath86 now let @xmath0 be a non - dicritical foliation such that after desingularization by the map @xmath22 all singularities of @xmath87 are nondegenerate . by lemma [ lem1 ] there exists a @xmath22-absolutely dicritical foliation @xmath88 such that @xmath89 . [ no5 ] we denote by @xmath90 the set of all @xmath22-absolutely dicritical foliations @xmath32 satisfying the two following properties : * @xmath35 and @xmath91 have the same dulac maps at any corner of @xmath10 . * at each singularity @xmath39 of @xmath11 , the invariant curves of @xmath35 and @xmath91 through @xmath39 are tangent ( figure [ p38 ] ) . of @xmath92 let @xmath32 in @xmath90 and @xmath13 be an irreducible component of @xmath10 . suppose that @xmath93 are the singularities of @xmath11 on @xmath13 . then we denote by @xmath94 where @xmath95 is the sliding of @xmath11 and @xmath35 in a neighborhood of @xmath52 . the sliding of @xmath0 and @xmath32 is @xmath96 where @xmath97 is the set of all irreducible components of @xmath10 . the set of slidings of @xmath0 relative to direction @xmath88 is the set @xmath98 we will prove in corollary [ cor3 ] that @xmath99 is an invariant of @xmath0 : if @xmath0 and @xmath21 are conjugated by @xmath100 then for each @xmath32 in @xmath90 we have @xmath101 . under some conditions for @xmath0 and @xmath21 ( theorem [ thr1 ] ) , we will have @xmath102 . therefore @xmath103 . moreover , @xmath104 is also in @xmath90 . consequently , @xmath105 . [ re7 ] for each singularity @xmath39 of @xmath11 that is a corner , i.e. , @xmath36 , there are two slidings @xmath106 and @xmath107 . however , by , @xmath107 is completely determined by @xmath106 and the dulac map of @xmath35 at @xmath39 . + this invariant is named `` sliding '' because it gives an obstruction for the construction of local conjugacy of two foliations that fixes the points in the exceptional divisor ( corollary [ cor3 ] ) . + the definition of @xmath99 does not depend on choosing a element @xmath88 in @xmath90 . more precisely , if @xmath108 then @xmath109 and @xmath110 + although @xmath111 is a set of local diffeomorphisms , it is not a local invariant . @xmath111 also contains the information of the relation of those local diffeomorphisms because all these local diffeomorphisms are defined by the holonomy projections following the global fibration @xmath32 : in some sense , any fibration @xmath112 plays the role of a global common transversal coordinate on which the slidings invariants are computed all together and at the same time . let us clarify here the role of the sliding invariant in the problem of classification of germs of foliations . suppose that two non - dicritical foliations @xmath0 and @xmath21 satisfy that their separatrices and their vanishing holonomies are conjugated . moreover , after desingularization , all the camacho - sad indices of @xmath0 and @xmath21 are coincide . then after blowing - ups , @xmath0 and @xmath21 are locally conjugated in a neighborhood of their singularities . although we have the conjugation of their vanishing holonomies , in general , we can not glue the local conjugation together . the obstruction is that these local conjugations induce the local diffeomorphisms on the exceptional divisor which we call the slidings . in general , there is no reason for those slidings being parts of a global diffeomorphism of the divisor . let @xmath113 . we say that their _ strict separatrices are tangent _ , denoted @xmath114 , if they have the same desingularization map and the same set of singularities . moreover , at each singularity which is not a corner of the divisor the separatrices of @xmath11 and @xmath25 are tangent . if @xmath114 and @xmath88 is an absolutely dicritical foliation satisfying @xmath89 then @xmath115 and @xmath116 . we denote by @xmath117 the set of camacho - sad indices of @xmath11 at all singularities . we also denote by @xmath118 if at each singularity , @xmath11 and @xmath25 have the same camacho - sad index . [ thr1 ] let @xmath0 and @xmath21 be two foliations in the class @xmath30 ( see notation [ no1 ] ) such that @xmath114 . suppose that @xmath88 is an absolutely dicritical foliation satisfying @xmath89 . let @xmath90 be as in notation [ no5 ] and @xmath99 , @xmath119 the corresponding sets of slidings . then the three following statements are equivalent : 1 . @xmath0 and @xmath21 are strictly analytically conjugated . their vanishing holonomy representations are strictly analytically conjugated , @xmath118 and @xmath105 . their vanishing holonomy representations are strictly analytically conjugated , @xmath118 and @xmath120 . here , a strict conjugacy means a conjugacy tangent to identity . we will prove that the slidings of foliations are finitely determined : [ thr2 ] let @xmath0 be a non - dicritical foliation without saddle - node singularities after desingularization . there exists a natural @xmath121 such that if there is a non - dicritical foliation @xmath21 satisfying the following conditions : 1 . @xmath0 and @xmath21 have the same set of singularities after desingularization and at a neighborhood of each singularity , @xmath11 and @xmath25 are locally strictly analytically conjugated , 2 . there exist @xmath122 in @xmath90 such that @xmath123 , then there exists @xmath124 such that @xmath124 is strictly conjugated with @xmath32 and @xmath125 . here @xmath123 means @xmath126 for all @xmath127 in @xmath111 , @xmath128 in @xmath129 , where @xmath130 stands for the regular part of degree @xmath121 in the taylor expansion of @xmath127 . + these two theorems also give two corollaries on finite determination of the class of isoholonomic non - dicritical foliations and absolutely dicritical foliations that have the same dulac maps ( see corollary [ cor17 ] and [ cor19 ] ) . + this paper is organized as follows : in section 2 , local conjugacy of the pair @xmath131 will be proved . we prove theorem [ thr1 ] in section 3 . section 4 is devoted to prove theorem [ thr2 ] and two corollaries of finite determination of class of isoholonomic non - dicritical foliations and absolutely dicritical foliations that have the same dulac maps . let @xmath0 , @xmath21 be two germs of nondegenerate reduced foliations in @xmath1 . denote by @xmath68 , @xmath69 and @xmath132 , @xmath133 the separatrices of @xmath0 and @xmath21 respectively . let @xmath32 and @xmath134 be two germs of regular foliations such that their separatrices @xmath135 and @xmath136 are transverse to the two separatrices of @xmath0 and @xmath21 respectively . suppose that @xmath100 is a diffeomorphism conjugating @xmath131 and @xmath137 , then the restriction of @xmath100 on the tangent curves commutes with the holonomies on @xmath70 and @xmath138 of @xmath0 and @xmath21 . the converse is also true : [ pro8 ] suppose that @xmath0 and @xmath21 have the same camacho - sad index . if @xmath139 is a diffeomorphism commuting with the holonomies of @xmath0 and @xmath21 then @xmath140 extends to a diffeomorphism @xmath100 of @xmath1 sending @xmath131 to @xmath137 . moreover , if we require that @xmath100 sends @xmath68 ( resp . @xmath69 ) to @xmath132 ( resp . @xmath133 ) then this extension is unique . by lemma [ lem1 ] , the curves @xmath68 , @xmath69 , @xmath135 , @xmath70 ( resp . , @xmath132 , @xmath133 , @xmath136 , @xmath138 ) are four transverse smooth curves . it is well known that there exist two radial foliations @xmath141 and @xmath142 such that @xmath68 , @xmath69 , @xmath135 , @xmath70 and @xmath132 , @xmath133 , @xmath136 , @xmath138 are the invariant curves of @xmath141 and @xmath142 respectively . after a blowing - up at the origin , denote by @xmath143 , @xmath144 , @xmath145 , @xmath146 ( resp . , @xmath147 , @xmath148 , @xmath149 , @xmath150 ) the intersections of strict transforms of @xmath68 , @xmath69 , @xmath135 , @xmath70 ( resp . , @xmath132 , @xmath133 , @xmath136 , @xmath138 ) with @xmath151 . take @xmath152 in @xmath153 that sends @xmath143 , @xmath144 , @xmath145 to @xmath147 , @xmath148 , @xmath149 respectively . by lemma [ lem1 ] , the direction of @xmath70 ( resp . , @xmath138 ) is completely determined by the camacho - sad index and the direction of @xmath135 ( resp . , @xmath136 ) . therefore , @xmath154 . using the path lifting method after a blowing - up @xcite , @xmath140 extends to a diffeomorphism @xmath155 of @xmath1 sending @xmath156 to @xmath157 . denote by @xmath158 . because @xmath159 sends @xmath136 and @xmath138 to @xmath135 and @xmath70 respectively , @xmath135 is also the separatrix of @xmath88 and @xmath160 . we denote by @xmath161 the tangent curve @xmath162 . the proof is reduced to show that there exists a diffeomorphism fixing points in @xmath161 sending @xmath131 to @xmath163 . choose a system of coordinates @xmath164 such that @xmath88 is defined by @xmath165 and @xmath0 is defined by a @xmath2-form @xmath166 then @xmath161 is defined by @xmath167 we claim that there exist a natural @xmath168 and a holomorphic function @xmath169 such that @xmath32 is defined by @xmath170 indeed , assume that @xmath32 is defined by @xmath171 where @xmath172 is invertible . rewrite the equation of @xmath161 as @xmath173 where @xmath174 . because @xmath175 , the maps @xmath176 and @xmath177 are diffeomorphic . hence there exists a diffeomorphism @xmath178 such that @xmath179 this is equivalent to @xmath180 therefore , there exist a natural @xmath181 and a function @xmath169 satisfying @xmath182 such that @xmath183 because @xmath184 is a diffeomorphism , @xmath32 is also defined by @xmath185 . let us prove @xmath168 . we have @xmath186 because @xmath161 is defined by @xmath187 , we have @xmath188 the fact @xmath189 forces to @xmath190 and @xmath191 . this implies @xmath192 . consequently , @xmath168 . now let @xmath193 tangent to @xmath0 . now we will show that there exists @xmath194 such that the diffeomorphism @xmath195x$ ] satisfies @xmath196x(x , y)=\sum_{i\geq 0}\frac{\tau^{i(n-1)}\alpha^i}{i!}\mathrm{ad}^i_{x}(x+y)=f(x , y),\ ] ] where @xmath197 is the adjoint representation . since @xmath198 becomes @xmath199 hence , the existence of @xmath200 comes from the implicit function theorem . + now we will prove the uniqueness of @xmath100 . in fact , we only need to show that if there exists a diffeomorphism @xmath201 that sends @xmath163 to itself , preserves the two separatrices of @xmath0 and fixes the points of @xmath161 then @xmath202 . since @xmath203 , @xmath201 sends every leaf of @xmath0 into itself . by @xcite , there exists @xmath204 such that @xmath205x.\ ] ] because @xmath88 is defined by the function @xmath206 and @xmath201 fixes points in @xmath161 , we get @xmath207x = x+y.\ ] ] decompose @xmath208 into the homogeneous terms @xmath209 since @xmath210 and @xmath211 for all @xmath47 , where @xmath212 , we have @xmath213x&=\sum_{i=0}^{\infty}\frac{\beta_0^i}{i!}x+\sum_{i=0}^{\infty}\frac{\beta_0^i}{i!}((-\lambda)^iy)+h.o.t.\\ & = \exp(\beta_0)x+\exp(-\lambda\beta_0)y+h.o.t .. \end{aligned}\ ] ] so leads to @xmath214 hence , @xmath215x&=\sum_{i=0}^{\infty}\frac{\beta_0^i}{i!}x=\exp(\beta_0 ) x = x , \label{5}\\ y\circ\exp[\beta_0]x&=\sum_{i=0}^{\infty}\frac{\beta_0^i}{i!}\left((-\lambda)^i y+c_i\right)=\exp(-\lambda \beta_0 ) y+c = y+c,\label{6}\end{aligned}\ ] ] where @xmath216 . we claim that @xmath217x=\exp[\beta_0]x\circ\exp[\bar\beta]x.\ ] ] indeed , for any @xmath218 we have @xmath219x\circ\exp[\bar\beta]x & = \left(\sum_{i=0}^{\infty}\frac{\beta_0^i}{i!}\mathrm{ad}^i_x(h)\right)\circ\exp[\bar\beta]x = \sum_{j=0}^{\infty}\frac{\bar\beta^j}{j!}\mathrm{ad}^j_x\left(\sum_{i=0}^{\infty}\frac{\beta_0^i}{i!}\mathrm{ad}^i_x(h)\right)\\ & = \sum_{k=0}^{\infty}\sum_{i+j = k}\frac{\bar\beta^j \beta_0^i}{j!i!}\mathrm{ad}^k_x(h)=\sum_{k=0}^{\infty } \frac{(\bar\beta+\beta_0)^k}{k!}\mathrm{ad}^k_x(h)=h\circ\exp[\beta]x.\end{aligned}\ ] ] we write @xmath220 where @xmath221 . by , , we get @xmath213x&=x\circ\exp[\beta_0]x\circ \exp[\bar\beta]x+ y\circ\exp[\beta_0]x\circ\exp[\bar\beta]x\nonumber\\ & = x\circ\exp[\bar\beta]x+ ( y+c)\circ\exp[\bar\beta]x\nonumber\\ & = \sum_{i=0}^{\infty}\frac{\bar\beta^i}{i!}x+\sum_{i=0}^{\infty}\frac{\bar\beta^i}{i!}\mathrm{ad}^i_x(y+c)\nonumber\\ & = \exp(\bar\beta)x+\exp(-\lambda\bar{\beta})y+ \sum_{i=0}^{\infty}\frac{\bar\beta^i}{i!}d_i\nonumber\\ & = x\prod_{i=1}^{\infty}\exp(\beta_i)+ y\prod_{i=1}^{\infty}\exp(-\lambda\beta_i)+ \sum_{i=0}^{\infty}\frac{\bar\beta^i}{i!}d_i.\label{8}\end{aligned}\ ] ] we will prove @xmath222 by induction . from we have @xmath223x = x(1+\beta_1)+ y(1-\lambda\beta_1)+h.o.t.\ ] ] so forces @xmath224 . suppose that @xmath225 , we have @xmath223x = x(1+\beta_k)+ y(1-\lambda\beta_k)+h.o.t .. \ ] ] then again leads to @xmath226 and consequently @xmath227 . this implies that @xmath228x=(x , y+c).\ ] ] finally , again gives @xmath229 . so @xmath202 . [ cor11 ] suppose that @xmath0 and @xmath21 are two nondegenerate reduced foliations that are analytically conjugated . let @xmath32 and @xmath134 be two regular foliations that are transverse to the two separatrices of @xmath0 and @xmath21 respectively . then there exists a diffeomorphism that sends @xmath131 to @xmath137 . let @xmath201 be the conjugacy of @xmath0 and @xmath21 . denote by @xmath230 . then the restriction @xmath231 commutes with the holonomies of @xmath0 on @xmath70 and @xmath21 on @xmath232 . moreover by the holonomy transport , the holonomies of @xmath21 on @xmath232 and on @xmath138 are conjugated . hence , the holonomies of @xmath0 on @xmath70 and @xmath21 on @xmath138 are conjugated . by proposition [ pro8 ] there exists a diffeomorphism that sends @xmath131 to @xmath137 by projecting on @xmath68 and @xmath69 the holonomies defined on @xmath70 and @xmath138 respectively , we can obtain [ cor3 ] if @xmath100 is a diffeomorphism conjugating @xmath131 and @xmath137 , then @xmath233 reciprocally , if @xmath234 and @xmath235 is a diffeomorphism satisfying @xmath236 then @xmath140 uniquely extends to a diffeomorphism @xmath100 of @xmath1 sending @xmath131 to @xmath137 . because @xmath100 conjugates @xmath131 and @xmath137 , the restriction @xmath237 commutes with the holonomies @xmath80 and @xmath26 of @xmath0 and @xmath21 . denote by @xmath81 ( resp . , @xmath238 ) the projection by the leaves of @xmath32 ( resp . , @xmath134 ) from @xmath70 ( resp . , @xmath138 ) to @xmath68 ( resp . , @xmath132 ) . since @xmath100 sends @xmath131 to @xmath137 , we have @xmath239 therefore @xmath240 reciprocally , suppose @xmath235 is a diffeomorphism commuting with the slidings of @xmath0 and @xmath21 . denote by @xmath241 then @xmath242 by proposition [ pro8 ] , @xmath243 uniquely extends to a diffeomorphism @xmath100 that sends @xmath131 to @xmath137 . in particular , if in corollary [ cor3 ] we have @xmath244 and @xmath245 then there exists a diffeomorphism sending @xmath131 to @xmath137 and fixing points in @xmath68 . this whole section is devoted to prove theorem [ thr1 ] . the direction ( ( ii)@xmath246(iii ) ) is obvious . + ( ( i)@xmath246(ii ) ) since the camacho - sad index is an analytic invariant , it is obvious that @xmath118 . let @xmath100 be the strict conjugacy and @xmath247 be its lifting by @xmath22 . suppose that a non - corner point @xmath7 of @xmath10 is a fixed point of @xmath248 . then the linear map @xmath249 has two eigenvalues . one corresponds to the direction of the divisor . we denote by @xmath250 the other eigenvalue and define @xmath251 for each corner @xmath7 . [ lem9 ] @xmath102 so @xmath252 is a function defined on @xmath10 and moreover @xmath253 . denote by @xmath73 the standard blowing - up at the origin of @xmath1 @xmath254 on @xmath255 , we use the two standard chart @xmath256 and @xmath257 together with the transition functions @xmath258 , @xmath259 . suppose that @xmath260 then in the coordinate system @xmath256 we have @xmath261 where @xmath262 . therefore @xmath263 fixes points in @xmath255 and @xmath264 . let @xmath39 be a non - reduced singularity of @xmath265 on @xmath255 . we will show that @xmath266 and apply the inductive hypothesis for @xmath155 in a neighborhood of @xmath39 . indeed , let @xmath267 be the blowing - up at @xmath39 and @xmath268 . denote by @xmath269 and @xmath270 all invariant curves of @xmath265 and @xmath271 through @xmath39 . because every element in @xmath31 after desingularization admits no dead component in its exceptional divisor , @xmath272 is not a dead component . therefore there is at least one irreducible component @xmath273 of @xmath269 that are not tangent to @xmath255 . because @xmath274 and @xmath114 , @xmath275 has an eigenvector different from the direction of @xmath255 , which is corresponding to the direction of @xmath273 . so @xmath276 has two eigenvectors . since both of their eigenvalues are @xmath2 , we have @xmath266 . now let @xmath277 and denote by @xmath278 . since @xmath102 , the strict transforms @xmath35 and @xmath279 have the same dulac maps . moreover , because @xmath114 , at each singularity @xmath39 of @xmath11 , @xmath280 has two eigenvectors . as @xmath253 and @xmath102 we have @xmath281 . therefore the invariant curves of @xmath35 and @xmath279 through @xmath39 are tangent . this gives @xmath282 . because @xmath283 fixes points in @xmath10 , by corollary [ cor3 ] the identity map commutes with the slides of @xmath0 and @xmath21 . this leads to @xmath284 . consequently , @xmath105 . moreover , the vanishing holonomy representation of @xmath0 and @xmath21 are conjugated by @xmath283 . since @xmath253 this conjugacy is strict . + ( ( iii)@xmath246(i ) ) suppose that @xmath32 , @xmath285 satisfy @xmath284 . by corollary [ cor3 ] , at each singularity @xmath52 , @xmath286 , of @xmath11 there exists a neighborhood @xmath287 of @xmath52 and a local conjugacy @xmath288 such that @xmath289 . let @xmath290 be a neighborhood of @xmath291 such that @xmath290 does not contain any singularity of @xmath11 . note that @xmath290 is not connected and the restriction of @xmath11 and @xmath25 on @xmath290 are regular . the strict conjugacy of the vanishing holonomy representations can be extended by path lifting method to the conjugacy @xmath292 satisfying that the second eigenvalue function @xmath293 is identically @xmath2 . we will show that on each intersection @xmath294 , @xmath295 and @xmath296 coincide . denote by @xmath297 we claim that @xmath298 on @xmath299 . let @xmath300 in @xmath301 . denote by @xmath302 and @xmath303 the invariant curves of @xmath35 through @xmath39 and @xmath304 respectively . as the two maps @xmath305 and @xmath306 are conjugated by the holonomy transport , we have @xmath307 . consequently , @xmath308 is constant on @xmath309 . since @xmath310 , it follows that @xmath311 is constant on @xmath301 . therefore , @xmath311 is constant on @xmath312 . moreover , at the singularity @xmath52 , @xmath313 has three eigenvectors corresponding to the directions of the divisor and the directions of invariant curves of @xmath11 and @xmath35 through @xmath52 . since @xmath313 has also one eigenvalue @xmath2 corresponding to the directions of the divisor , we have @xmath314 . this gives @xmath315 and consequently @xmath298 . now at each point @xmath316 , the map @xmath305 commutes with the holonomy of @xmath11 around @xmath52 . since the camacho - sad index @xmath317 of @xmath11 at @xmath52 is not rational , lemma [ lem10 ] below says that @xmath318 and so @xmath319 . hence we can glue all @xmath295 together and the strict conjugacy we need is the projection of this diffeomorphism on @xmath1 by @xmath22 . [ lem10 ] let @xmath320 such that @xmath321 where @xmath322 . if @xmath323 satisfying @xmath324 and @xmath325 then @xmath326 . since @xmath322 , there is a formal diffeomorphism @xmath140 such that @xmath327 denote by @xmath328 , then @xmath329 and @xmath330 . the proof is reduced to show that @xmath331 . suppose that @xmath332 . then @xmath333 and @xmath334 since @xmath322 , it forces @xmath335 for all @xmath336 . hence @xmath331 . let @xmath337 be a germ of curve at @xmath39 in a surface @xmath338 . we denote by @xmath339 the set of all germs of singular curves having the same desingularization map and having the same singularities as @xmath337 after desingularization . here , the singularities of @xmath337 after desingularization are the singularities of the curve defined by the union of strict transform of @xmath337 and the exceptional divisor . if @xmath337 is smooth , we denote by @xmath340 the set of all holomorphic functions on @xmath337 whose vanishing orders at @xmath39 are at least @xmath341 . [ pro11 ] let @xmath337 be a germ of curve in @xmath1 and @xmath342 be its irreducible components . suppose that @xmath343 is a finite composition of blowing - ups such that all the transformed curves @xmath344 are smooth . then there exists a natural @xmath121 such that if @xmath345 , @xmath346 , then there exists @xmath347 such that @xmath348 . moreover , the same @xmath121 can be chosen for all elements in @xmath339 . we first consider the statement when @xmath337 is irreducible . if @xmath337 is smooth then @xmath349 is diffeomorphic to @xmath337 . so we can suppose that @xmath337 is singular . denote by @xmath350 . choose a coordinate system @xmath351 in a neighborhood of @xmath39 such that @xmath352 and @xmath353 . then @xmath354 is defined by @xmath355 where @xmath356 , @xmath357 and @xmath358 . so we have @xmath359 therefore , there exist a natural @xmath360 and a holomorphic function @xmath169 such that @xmath361 where @xmath362 . we claim that @xmath169 is a unit . indeed , suppose @xmath363 and denote by @xmath364 the curve @xmath365 . let @xmath135 be a curve defined in @xmath1 such that @xmath366 . let @xmath367 be a reduced equation of @xmath135 . by , @xmath368 and @xmath369 . it contradicts @xmath357 . now denote by @xmath370 which is a unit , we have @xmath371 for each @xmath372 there exists @xmath373 such that @xmath374 . thus @xmath375 so implies that if a holomorphic function @xmath376 satisfies @xmath377 then there exists a holomorphic function @xmath378 such that @xmath379 . consequently @xmath380 in the general case , suppose that @xmath381 is defined by @xmath382 . if @xmath383 , @xmath346 , with @xmath121 big enough , there exist @xmath384 , @xmath346 , such that @xmath385 . we will find a holomorphic function @xmath386 such that @xmath387 for all @xmath346 . this is reduced to show that there exists a natural @xmath388 such that the following morphism @xmath389 is surjective @xmath390 indeed , by hilbert s nullstellensatz , there exists a natural @xmath391 such that @xmath392 for all @xmath393 . we will show that for all @xmath346 , @xmath394 the elements @xmath395 , where @xmath396 is in the @xmath397 position , are in @xmath398 and then @xmath388 can be chosen as @xmath399 . we decompose @xmath400 where @xmath401 . by , there exist @xmath402 such that @xmath403 . this implies that @xmath404 now we will show that the same @xmath121 can be chosen for all elements of @xmath339 . in the case @xmath337 is irreducible , let @xmath405 in @xmath339 and @xmath406 be the equation of @xmath407 . we also have @xmath408 where @xmath409 which is a unit . consequently , holds . in the general case , it is sufficient to show that the same @xmath391 in can be chosen for all elements of @xmath339 . let @xmath410 be the smallest natural satisfying @xmath411 we claim that @xmath412 indeed , there exists @xmath413 . let @xmath414 , @xmath415 , be a sequence of monomials such that @xmath416 , @xmath417 and either @xmath418 or @xmath419 . since @xmath420 we have @xmath421 for all @xmath422 . we will show that @xmath423 is independent in the vector space @xmath424 over @xmath425 . suppose that @xmath426 suppose there exists @xmath427 . let @xmath428 be the smallest natural such that @xmath429 . then @xmath430 this implies that @xmath431 in @xmath432 and it is a contradiction . now , it is well known that the intersection number @xmath433 is a topological invariant . it means that if two curves @xmath434 and @xmath435 are topologically conjugated then @xmath436 . consequently , @xmath391 can be chosen as @xmath437 that does nt depend on the elements of @xmath339 . now , we will prove the finite determinacy property of the slidings of foliations . suppose that @xmath438 , @xmath439 where @xmath440 and @xmath441 are irreducible components of @xmath70 and @xmath138 . then the singularities of @xmath11 and @xmath25 are @xmath442 . denote by @xmath443 the holonomy of @xmath11 on @xmath440 . now let @xmath52 be a singularity @xmath11 . we first suppose that @xmath52 is not a corner . denote by @xmath13 the irreducible component of @xmath10 through @xmath52 . because @xmath11 and @xmath25 are strictly conjugated in a neighborhood of @xmath52 , by corollaries [ cor11 ] and [ cor3 ] , there exists a diffeomorphism @xmath444 in @xmath445 tangent to identity such that @xmath446 let @xmath447 ( resp . , @xmath448 ) be the projection from @xmath440 ( resp . , @xmath441 ) to @xmath13 that follows the leaves of @xmath35 ( resp . , @xmath279 ) . denote by @xmath449 then @xmath450 . since @xmath451 , @xmath452 is tangent to identity map at order at least @xmath121 . in the case @xmath52 is a corner , let @xmath13 be one of two irreducible components of @xmath10 through @xmath52 and define @xmath452 as above . we also have that @xmath452 is tangent to identity map at order at least @xmath121 . [ lem17 ] suppose that there exists a diffeomorphism @xmath100 such that the lifting @xmath453 satisfies * @xmath454 , * @xmath455 , * @xmath456 . then @xmath457 satisfies @xmath125 . let @xmath52 be a singularity @xmath11 . in the case @xmath52 is not a corner , we denote @xmath13 , @xmath447 , @xmath448 as above . let @xmath458 be the projection following the leaves of @xmath459 from @xmath440 to @xmath13 , then @xmath460 we have @xmath461 if @xmath52 is a corner , @xmath462 , we also have @xmath463 since @xmath454 the dulac maps of @xmath459 and @xmath279 in a neighborhood of @xmath52 are the same . so remark [ re7 ] leads to @xmath464 now we will prove the existence of @xmath100 in lemma [ lem17 ] for @xmath121 big enough . suppose that @xmath0 and @xmath32 are respectively defined by @xmath465 then the tangent curve @xmath466 is defined by @xmath467 let @xmath468 be a vector field tangent to @xmath161 and @xmath469 be its lifting by @xmath22 . by the implicit function theorem , if @xmath121 is big enough , there exists @xmath470 defined on @xmath440 such that @xmath471\left(\restr{\tilde{x}_{q}}{t_i}\right)=\phi_i.\ ] ] using proposition [ pro11 ] , by choosing @xmath121 big enough , there exists @xmath472 such that @xmath473\restr{\tilde{x}_{q}}{t_i}=\phi_i.\ ] ] for each @xmath474 , denote by @xmath475 it is easy to see that @xmath476 if and only if @xmath477 . for each holomorphic function @xmath478 , we denote by @xmath479x_q.\ ] ] lemma [ lemma18 ] below says that there exists a holomorphic function @xmath172 such that @xmath480 satisfies lemma [ lem17 ] for @xmath121 big enough . moreover , by proposition [ pro11 ] , we can chose @xmath121 that only depends on @xmath0 . [ lemma18 ] if @xmath121 is big enough , for all @xmath478 in @xmath481 there exists a holomorphic function @xmath172 such that @xmath482 . we have @xmath483 similarly , @xmath484 this implies that @xmath485 denote by @xmath486 . then @xmath487 is the tangent curve of @xmath32 and the foliation defined by the level sets of @xmath304 . since at each singularity @xmath52 of @xmath11 , the irreducible component @xmath440 of @xmath161 is transverse to @xmath35 , the irreducible components of the strict transform of @xmath487 at @xmath52 are also transverse to @xmath440 . this implies that @xmath488 and the two curves @xmath489 and @xmath490 are topologically conjugated . by hilbert s nullstellensatz and the proof of proposition [ pro11 ] there exists a natural @xmath388 such that @xmath491 and @xmath492 . this implies that @xmath493 . so if @xmath494 , by we can choose @xmath495 such that @xmath482 . if we replace the condition @xmath11 and @xmath25 are locally strictly analytically conjugated " in theorem [ thr2 ] by the condition `` @xmath0 and @xmath21 are in @xmath31 '' then the conclusion in theorem [ thr2 ] becomes : `` for all natural @xmath496 there exists @xmath497 such that @xmath498 '' . indeed , in that case , because the camacho - sad indices are not rational , @xmath11 and @xmath25 are locally formally conjugated . so we can choose @xmath243 in such that @xmath499 [ cor17 ] let @xmath500 defined by a @xmath2-form @xmath501 then there exists a natural @xmath121 such that if @xmath502 is defined by a @xmath2-form @xmath503 satisfying that @xmath504 and the vanishing holonomy representations of @xmath0 and @xmath21 are strictly analytically conjugated , then @xmath0 and @xmath21 are strictly analytically conjugated . let @xmath505 then @xmath506 where @xmath507 is an increasing function on @xmath121 and @xmath508 when @xmath509 . by theorem [ thr2 ] if @xmath121 is big enough there exists @xmath510 such that @xmath511 . by theorem [ thr1 ] , @xmath0 and @xmath21 are strictly analytically conjugated . this corollary is consistent with the result of j.f . mattei in @xcite which says that the dimension of moduli space of the equisingular unfolding of a foliation is finite . note that the vanishing holonomy representations of two foliations that are jointed by a unfolding are conjugated but the converse is not true . [ cor19 ] let @xmath32 be a @xmath22-absolutely dicritical foliation defined by @xmath2-form @xmath501 . there exists a natural @xmath121 such that if @xmath134 is a @xmath22-absolutely dicritical foliation defined by @xmath503 satisfying @xmath504 and the dulac maps of @xmath35 and @xmath279 are the same then @xmath32 and @xmath134 are strictly analytically conjugated . suppose that @xmath512 where @xmath513 is an irreducible component of @xmath10 . we take a pair of irreducible functions @xmath470 and @xmath514 for each @xmath515 , such that the curve @xmath516 and @xmath517 satisfy the following properties : * the strict transforms @xmath518 and @xmath519 cut @xmath513 at two different points @xmath52 , @xmath520 , respectively , such that none of them is a corner . * @xmath518 , @xmath519 are smooth and transverse to the invariant curve of @xmath35 through @xmath52 , @xmath520 respectively . because @xmath521 $ ] is an infinite field extension , there exists @xmath522 such that @xmath523 now , let us consider the non - dicritical foliation @xmath0 defined by the @xmath2-form @xmath524 then @xmath0 admits @xmath22 as its desingularization map and the singularities of the strict transform @xmath11 are the corners of @xmath10 and @xmath525 , @xmath346 . we claim that at each singularity , the camacho - sad index of @xmath11 is not rational . indeed , denote by @xmath526 the multiplicity of @xmath527 and @xmath528 on @xmath529 . at the corner @xmath530 , we take coordinates @xmath164 such that @xmath531 . in this coordinate system , we can write @xmath532 as @xmath533 where @xmath534 is a unit and @xmath535 is a holomorphic form . so the camacho - sad index of @xmath11 at @xmath536 is @xmath537 similarly , the camacho - sad indices of @xmath11 at @xmath52 and @xmath520 , respectively , are @xmath538 now if @xmath504 then @xmath539 where @xmath540 is an increasing function on @xmath121 and @xmath508 when @xmath509 . moreover if @xmath121 is big enough the invariant curves of @xmath35 and @xmath279 through the singularities of @xmath11 are tangent . by using theorem [ thr2 ] for @xmath541 , there exists a foliation @xmath124 strictly conjugated with @xmath32 such that the two couples @xmath542 and @xmath543 are strictly conjugated . consequently , @xmath32 and @xmath134 are strictly conjugated . this paper is based on the main part of my doctoral thesis at the institute of mathematics of toulouse , france . i would like to thank my advisors , yohann genzmer and emmanuel paul , for having guided me so well over all these years , jean - franois mattei for useful discussions and suggestions about the problem . l. ortiz - bobadilla ; e. rosales - gonzlez ; s. m. voronin - _ thom s problem for the orbital analytic classification of degenerate singular points of holomorphic vector fields on the plane . _ dokl . math . * 82 * no . 2 , ( 2010 ) , 759761 . moussu , r. - _ holonomie venescente des quations diffrentielles dgneres transverses _ , singularities and dynamical systems ( irklion , 1983 ) , 161173 , north - holland math . stud . , 103 , north - holland , amsterdam , 1985 .

by introducing a new invariant called the set of slidings , we give a complete strict classification of the class of germs of non - dicritical holomorphic foliations in the plan whose camacho - sad indices are not rational . moreover , we will show that , in this class , the new invariant is finitely determined . consequently , the finite determination of the class of isoholonomic non - dicritical foliations and absolutely dicritical foliations that have the same dulac maps are proved .

let @xmath0 be a complete finite - area orientable hyperbolic surface with one cusp , and @xmath1 the space of complete geodesic rays in @xmath0 emanating from the puncture . then , there is a natural action of the ( full ) mapping class group @xmath2 of @xmath0 on @xmath3 ( see section [ sb ] ) . the dynamics of the action of an element of @xmath1 plays a key role in the nielsen - thurston theory for surface homeomorphisms . it also plays a crucial role in the variation of mcshane s identity for punctured surface bundles with pseudo - anosov monodromy , established by @xcite and @xcite . it is natural to ask what does the action of the whole group @xmath2 ( or its subgroups ) look like . however , the authors could not find a reference which treats this natural question , though there are various references which study the action of ( subgroups of ) the mapping class groups on the projective measured lamination spaces , which are homeomorphic to higher dimensional spheres ( see for example , @xcite ) . in particular , such an action is minimal ( cf . @xcite ) and moreover ergodic @xcite . the purpose of this paper is to prove that the action of @xmath2 on @xmath1 is almost everywhere wandering ( see theorem [ b1 ] for the precise meaning ) . this forms a sharp contrast to the above result of @xcite . we would like to thank katsuhiko matsuzaki for his helpful comments on the first version of the paper . let @xmath4 be a complete finite - area orientable hyperbolic surface with precisely one cusp , where @xmath5 . let @xmath6 be the space of complete geodesic rays in @xmath7 emanating from the puncture . then @xmath1 is identified with a horocycle , @xmath8 , in the cusp . in fact , a point of @xmath9 determines a geodesic ray in @xmath7 emanating from the puncture , or more precisely , a bi - infinite geodesic path with its positive end going out the cusp and meeting @xmath9 in the given point . any mapping class @xmath10 of @xmath7 maps each geodesic ray to another path which can be straightened out to another geodesic ray , and hence determines another point of @xmath9 . this gives an action of the infinite cyclic group generated by @xmath11 on @xmath12 . a rigorous construction of this action is described as follows . choose a representative , @xmath13 , of @xmath11 , so that its lift @xmath14 to the universal cover @xmath15 is a quasi - isometry . then @xmath14 extends to a self - homeomorphism of the closed disc @xmath16 . for a geodesic ray @xmath17 , let @xmath18 be the closure in @xmath16 of a lift of @xmath19 to @xmath15 . then @xmath20 is an arc properly embedded in @xmath16 , and its endpoints determine a geodesic in @xmath15 , which project to another geodesic ray @xmath21 . thus , we obtain an action of @xmath11 on @xmath6 , by setting @xmath22 . the dynamics of this action plays a key role in @xcite . however , one needs to verify that this action does not depend on the choice of a representative @xmath13 of @xmath11 . in the following , we settle this issue , by using the canonical boundary of a relatively hyperbolic group described in @xcite . though we are really interested here only in the case where the group is the fundamental group of a once - punctured closed orientable surface , and the the peripheral structure is interpreted in the usual way ( as the conjugacy class of the fundamental group of a neighborhood of the puncture ) , we give a discussion in a general setting . let @xmath23 be a non - elementary relatively hyperbolic group with a given peripheral structure @xmath24 , which is a conjugacy invariant collection of infinite subgroups of @xmath23 . by ( * ? * definition 1 ) , @xmath23 admits a properly discontinuous isometric action on a path - metric space , @xmath25 , with the following properties . 1 . @xmath25 is proper ( i.e. , complete and locally compact ) and gromov hyperbolic , 2 . every point of the boundary of @xmath25 is either a conical limit point or a bounded parabolic point , 3 . the peripheral subgroups , i.e. , the elements of @xmath24 , are precisely the maximal parabolic subgroups of @xmath23 , and 4 . every peripheral subgroup is finitely generated . it is proved in ( * ? ? ? * theorem 9.4 ) that the gromov boundary @xmath26 is uniquely determined by @xmath27 , ( even though the quasi - isometry class of the space @xmath25 satisfying the above conditions is not uniquely determined ) . thus the boundary @xmath28 is defined to be @xmath26 . by identifying @xmath23 with an orbit in @xmath25 , we obtain a natural topology on the disjoint union @xmath29 which is compact hausdorff , with @xmath23 discrete and @xmath30 closed . the action of @xmath23 on itself by left multiplication extends to an action on @xmath29 by homeomorphism . this gives us a geometrically finite convergence action of @xmath23 on @xmath30 . let @xmath31 be the subgroup of the automorphism group , @xmath32 , of @xmath23 which respects the peripheral structure @xmath24 . this contains the inner automorphism group , @xmath33 . now , by the naturality of @xmath34 ( ( * ? ? ? * theorem 9.4 ) ) , the action of @xmath31 on @xmath23 also extends to an action on @xmath29 , which is @xmath23-equivariant , i.e. , @xmath35 for every @xmath36 , @xmath37 and @xmath38 . ( in order to avoid confusion , we use @xmath39 to denote group actions , only in this place . ) under the natural epimorphism @xmath40 , this gives rise to the same action on @xmath41 as that induced by left multiplication . the centre of @xmath23 is always finite , and for simplicity , we assume it to be trivial . in this case , we can identify @xmath23 with @xmath33 . suppose that @xmath42 is a parabolic point . its stabiliser , @xmath43 , in @xmath23 is a peripheral subgroup . now @xmath44 acts properly discontinuously cocompactly on @xmath45 , so the quotient @xmath46 is compact hausdorff ( cf . * section 6 ) ) . let @xmath47 be the stabiliser of @xmath48 in @xmath31 . then @xmath44 is a normal subgroup of @xmath49 , and we get an action of @xmath50 on @xmath51 . if there is only one conjugacy class of peripheral subgroups , then the orbit @xmath52 is @xmath31-invariant , and it follows that the group @xmath49 maps isomorphically onto @xmath53 , so in this case we can naturally identify the group @xmath54 with @xmath55 . suppose now that @xmath7 is a once - punctured closed orientable surface , with negative euler characteristic @xmath56 . we write @xmath57 , where @xmath58 , the universal cover , and @xmath59 . let @xmath24 be the peripheral structure of @xmath23 arising from the cusp of @xmath7 , namely @xmath24 consists of the conjugacy class of the fundamental group of a neighbourhood of the end of @xmath7 . then @xmath60 is a relatively hyperbolic group , because if we fix a complete hyperbolic structure on @xmath7 then @xmath61 is identified with @xmath15 and the isometric action of @xmath23 on @xmath62 satisfies the conditions ( 1)(4 ) in the above , namely ( * ? ? ? * definition 1 ) . now @xmath61 admits a natural compactification to a closed disc , @xmath63 , where @xmath64 is the dynamically defined circle at infinity . we can identify @xmath64 with @xmath30 . in fact , if @xmath65 is any point of @xmath61 , then identifying @xmath23 with the orbit @xmath66 , we get an identification of @xmath29 with @xmath67 . as above we get an action of @xmath31 on @xmath64 . if @xmath68 is parabolic , then its stabiliser @xmath44 in @xmath23 is isomorphic to the infinite cyclic group @xmath69 , and we get an action of @xmath55 on the circle @xmath70 . since @xmath55 is identified with the ( full ) mapping class group , @xmath71 , of @xmath7 , we obtain a well defined action of @xmath71 on the circle @xmath51 . we now return to the setting in the beginning of this section , where @xmath4 is endowed with a complete hyperbolic structure . then we can identify the ( dynamically defined ) circle @xmath72 with the horocycle , @xmath9 , in the cusp , which in turn is identified with the space of geodesic rays , @xmath6 . this gives an action of @xmath71 on @xmath6 . since the action of @xmath23 on @xmath15 satisfies the conditions ( 1)-(4 ) in the above ( i.e. , ( * ? ? ? * definition 1 ) ) , we see that , for each mapping class @xmath11 of @xmath7 , its action on @xmath6 , defined via the straightening process presented at the beginning of this section , is identical with the action which is dynamically constructed in the above , independently from the hyperbolic structure . thus the problem raised at the beginning of this section is settled . in order to state the main result , we prepare some terminology . let @xmath73 be a group acting by homeomorphism on a topological space @xmath25 . an open subset , @xmath74 , is said to be _ wandering _ if @xmath75 for all @xmath76 . ( note that this definition is stronger than the usual definition of wandering , where it is only assumed that the number of @xmath77 such that @xmath78 is finite . ) the _ wandering domain _ , @xmath79 is the union of all wandering open sets . its complement , @xmath80 , is the _ non - wandering set_. this is a closed @xmath73-invariant subset of @xmath25 . note that if @xmath81 is a @xmath82-invariant open set , then @xmath83 . if @xmath84 is a normal subgroup , we get an induced action of @xmath85 on @xmath86 . ( in practice , the action of @xmath87 on @xmath25 will be properly discontinuous . ) one checks easily that @xmath88 with equality if @xmath89 . note that any hyperbolic structure on @xmath7 induces a euclidian metric on @xmath51 ( via the horocycle @xmath9 ) . if one changes the hyperbolic metric , the induced euclidian metrics on @xmath51 are related by a quasisymmetry . however , they are completely singular with respect to each other ( see @xcite ) . ( that is , there is a set which has zero measure in one structure , but full measure in the other . ) in general , this gives little control over how the hausdorff dimension of a subset can change . we say that a subset , @xmath90 is _ small _ if it has hausdorff dimension stricty less than 1 with respect to any hyperbolic structure on @xmath7 . now we can state our main theorem . [ b1 ] let @xmath7 be a once - punctured closed orientable surface , with @xmath91 , and consider the action of @xmath71 on the circle @xmath51 , defined in the above . then the non - wandering set in @xmath51 with respect to the action of @xmath71 is small . in particular , the non - wandering set has measure 0 with respect to any hyperbolic structure , and so has empty interior . given that two different hyperbolic structures give rise to quasisymmetically related metrics on @xmath51 , it is natural to ask if there is a more natural way to express this . for example , is there a property of ( closed ) subsets of @xmath51 , invariant under quasisymmetry and satisfied by the non - wandering set , which implies hausdorff dimension less than 1 ( or measure 0 ) ? let @xmath4 be a complete finite - area orientable hyperbolic surface with precisely one cusp , where @xmath5 . thus the universal cover @xmath58 is identified with the hyperbolic plane @xmath15 . write @xmath64 for the ideal boundary of @xmath61 , which we consider equipped with a preferred orientation . thus @xmath23 acts on @xmath64 as a geometrically finite convergence group . let @xmath92 be the set of parabolic points of @xmath23 . given @xmath93 , let @xmath94 be the generator of @xmath95 which acts on @xmath96 as a translation in the positive direction . given distinct @xmath97 , let @xmath98 \subseteq d \cup c $ ] denote the oriented geodesic from @xmath65 to @xmath99 . if @xmath37 is hyperbolic , write @xmath100 , @xmath101 respectively , for its attracting and repelling fixed points ; @xmath102 $ ] for its axis ; and @xmath103 for the oriented closed geodesic in @xmath0 corresponding to @xmath104 , i.e. , the image of @xmath105 in @xmath0 . if @xmath97 are distinct , then @xmath98 \cap d $ ] projects to an oriented bi - infinite geodesic path , @xmath106 , in @xmath7 . if @xmath107 , then this is a proper geodesic path , with a finite number , @xmath108 , of self - intersections . let @xmath109 , i.e. , @xmath110 consists of pairs @xmath111 of parabolic points such that @xmath112 is a proper geodesic arc . ( by an _ arc _ , we mean an embedded path . ) given @xmath93 , write @xmath113 . pick an element @xmath114 . then the proper arc @xmath112 intersects a sufficiently small horocycle , @xmath9 , in precisely two points . let @xmath115 be the horocircle centred at @xmath48 which is a connected component of the inverse image of @xmath9 , and let @xmath116 be the inverse image of the two points in @xmath117 , located in this order , such that @xmath118\cap \tilde\tau $ ] and @xmath119 . then there is a unique element @xmath120 such that @xmath121 and @xmath122\cap \tilde \tau = s_1 $ ] . namely , @xmath122 $ ] is the closure of the lift of @xmath112 with endpoint @xmath48 which is closest to @xmath123 $ ] , among the lifts of @xmath112 with endpoint @xmath48 , with respect to the preferred orientation of @xmath117 . ( see figure [ figure1 ] . ) in the quotient surface @xmath7 , the oriented closed geodesic @xmath124 is homotopic to the simple oriented loop obtained by shortcutting the oriented arc @xmath112 by the horocyclic arc which is the image of the subarc of @xmath125 bounded by @xmath126 and @xmath127 . thus @xmath124 is a simple closed geodesic disjoint from the proper geodesic arc @xmath128 . in particular , @xmath129\cap \alpha(g(p , q))=\emptyset$ ] . in fact , the map @xmath130 : \delta \longrightarrow \gamma $ ] is characterised by the following properties : for all @xmath131 , we have @xmath132 , @xmath133 , and @xmath134 \cap \alpha(g(p , q ) ) = \emptyset $ ] . and @xmath135 of the hyperbolic transformations @xmath136 and @xmath137 respectively . the blue arcs with thin arrows represent the oriented geodesic @xmath123 $ ] and its images by the infinite cyclic groups @xmath138 and @xmath139 . the three intersection points of the blue arcs and the horocircle @xmath117 centred at @xmath48 are @xmath140 , @xmath141 and @xmath142 , from left to right . [ figure1 ] ] write @xmath143 and @xmath144 . then the points @xmath48 , @xmath145 , @xmath146 , @xmath147 , @xmath148 , @xmath149 occur in this order around @xmath64 . let @xmath150 , @xmath151 and @xmath152 be open intervals in @xmath64 . thus @xmath153 , @xmath154 for all @xmath155 , and @xmath156 for all @xmath157 . in the quotient surface @xmath7 , the oriented simple closed geodesics @xmath124 and @xmath158 cut off a punctured annulus containing the geodesic arc @xmath128 , in which the simple geodesic rays @xmath159 and @xmath160 emanating from the puncture spiral to @xmath124 and @xmath158 , respectively . thus , each of @xmath161 projects homeomorphically onto a _ gap _ in the horocircle @xmath8 , in the sense of @xcite . in fact , each of @xmath161 is a maximal connected subset of @xmath162 consisting of points @xmath163 such that the geodesic ray @xmath164 is non - simple . moreover , if @xmath164 is non - simple , then @xmath163 is contained in @xmath161 for some @xmath165 ( see @xcite ) . . then we obtain the following as a consequence of ( * ? ? ? * corollary 5 ) and @xcite ( see also ( * ? ? ? * section 5 ) ) : [ c1 ] the elements of @xmath167 are pairwise disjoint . the complement , @xmath168 , is a cantor set of hausdorff dimension @xmath169 . here , of course , the hausdorff dimension is taken with respect to the euclidean metric on the horocycle , @xmath9 . up to a scale factor , this is the same as the euclidean metric in the upper - half - space model with @xmath48 at @xmath170 . ( note that we could equally well use the circular metric on the boundary , @xmath64 , induced by the poincar model , since all the transition functions are mbius , and in particular , smooth . ) write @xmath171 . this is a closed set , whose complementary components are precisely the intervals @xmath172 for @xmath173 . thus the set @xmath174 is characterised by the following property : a point @xmath175 belongs to @xmath174 if and only if @xmath176 and the geodesic ray @xmath164 in @xmath0 is simple . for @xmath177 , we define maps @xmath178 , @xmath179 and @xmath180 from @xmath181 to @xmath182 , @xmath183 and @xmath184 , respectively , by the following rule . if @xmath185 , then @xmath186 for some unique @xmath187 and @xmath173 . define @xmath188 , @xmath189 , and @xmath190 or @xmath191 according to whether @xmath192 or @xmath193 . note that the definition is symmetric under simultaneously reversing the orientation on @xmath64 and swapping @xmath194 with @xmath195 . it should be noted that if @xmath185 , then , in the quotient surface @xmath7 , the geodesic ray @xmath196 is obtained from the non - simple geodesic ray @xmath197 by cutting a loop , homotopic to @xmath198 , and straightening the resulting piecewise geodesic ( see figure [ figure2 ] ) . ( in the quotient , we are allowing ourselves to cut out any peripheral loops that occur at the beginning . ) in particular , if @xmath199 , then both @xmath197 and @xmath200 are proper geodesic paths in @xmath7 , and their self - intersection numbers satisfy the inequality @xmath201 . and so @xmath190 . [ figure2 ] ] by repeatedly applying these maps , we associate for a given @xmath202 , a sequence @xmath203 in @xmath23 , @xmath204 in @xmath205 , and @xmath206 in @xmath182 as follows . * step 0 . * pick a parabolic point @xmath177 , and define @xmath207 . thus , @xmath208 is independent of @xmath202 . * step 1 . * if @xmath209 , we stop with the 1-element sequence @xmath208 , and define @xmath203 and @xmath206 to be the empty sequence . if @xmath210 , set @xmath211 , @xmath212 , @xmath213 , and continue to the next step . ( the sequences @xmath203 and @xmath206 begin with index @xmath214 . ) * step 2 . * if @xmath215 , we stop with the 1-element sequences @xmath216 and @xmath217 and @xmath218-element sequence @xmath219 . if @xmath220 , set @xmath221 , @xmath222 and @xmath223 . we continue this process , forever or until we stop . we call the resulting sequences @xmath203 , @xmath204 and @xmath206 the _ derived sequences _ for @xmath163 . more specifically , we call @xmath203 and @xmath204 the _ derived @xmath23-sequence _ and the _ derived @xmath224-sequence _ for @xmath163 , respectively . [ c2 ] let @xmath225 , and let @xmath203 , @xmath204 and @xmath206 be the derived sequences for @xmath163 . then the following hold . \(1 ) the sequences @xmath204 and @xmath206 are determined by the sequence @xmath203 by the following rule : @xmath226 where @xmath227 , and @xmath228 or @xmath195 according to whether @xmath229 or @xmath230 . \(2 ) a point @xmath231 has the derived @xmath23-sequence beginning with @xmath232 for some @xmath233 , if and only if @xmath234 . \(3 ) set @xmath235 . if @xmath236 , then the derived @xmath23-sequence @xmath203 is infinite . \(4 ) if @xmath237 , then the derived @xmath23-sequence @xmath203 is finite . ( 1 ) , ( 2 ) and ( 3 ) follow directly from the definition of the derived sequences . to prove ( 4 ) , let @xmath65 be a point in @xmath224 . if @xmath238 , then @xmath203 is the empty sequence . so we may assume @xmath199 . then by repeatedly using the observation made prior to the construction of the derived sequences , we see that the self - intersection number @xmath239 of the proper geodesic path @xmath240 is strictly decreasing . hence @xmath241 for some @xmath242 . this means that @xmath243 and so the derived sequences terminate at @xmath242 . the following is an immediate consequence of lemma [ c2](2 ) . [ c3 ] suppose that @xmath202 has derived @xmath23-sequence beginning with @xmath244 for some @xmath245 . then there is an open set , @xmath246 , containing @xmath65 , such that if @xmath247 , then @xmath244 is also an initial segment of the derived @xmath23-sequence for @xmath99 . recall from section [ sb ] that @xmath248 denotes the subgroup of @xmath32 preserving @xmath224 setwise and fixing @xmath93 . [ c4 ] let @xmath249 be an element of @xmath250 with @xmath251 . then the following holds for every point @xmath202 . if @xmath203 , @xmath204 and @xmath206 are the derived sequences for @xmath65 , then the derived sequences for @xmath252 are @xmath253 , @xmath254 and @xmath255 . this can be proved through induction , by using the fact that the following hold for each @xmath256 . 1 . @xmath257 . 2 . for any @xmath165 , we have : 1 . if @xmath249 is orientation - preserving , then @xmath258 , @xmath259 , @xmath260 , and @xmath261 . 2 . if @xmath249 is orientation - reversing , then @xmath262 , @xmath263 , @xmath264 , and @xmath265 . let @xmath65 be a point in @xmath64 and @xmath266 the ( finite or infinite ) derived @xmath205-sequence for @xmath163 . write @xmath267 for the projection of @xmath268 \cap d $ ] to @xmath7 . this is a proper geodesic arc in @xmath7 . we call the sequence @xmath269 the _ derived sequence of arcs _ for @xmath163 . we say that @xmath65 is _ filling _ if the arcs @xmath270 eventually fill @xmath7 , namely , there is some @xmath157 such that @xmath271 is a union of open discs . let @xmath272 be the subset of @xmath273 consisting of points which are filling . in this section , we prove the following proposition . [ d1 ] the set @xmath272 is open in @xmath64 , and its complement has hausdorff dimension strictly less than @xmath274 . in particular , @xmath272 has full measure . we begin with some preparation . let @xmath275 be a simple closed geodesic in @xmath7 , and let @xmath276 be the path - metric completion of the component of @xmath277 containing the cusp . then we can identify @xmath276 as @xmath278 , where @xmath279 is a subgroup of @xmath23 containing @xmath280 , and @xmath281 is the convex hull of the limit set @xmath282 . in other words , @xmath276 is the `` convex core '' of the hyperbolic surface @xmath283 . note that @xmath284 and @xmath285 . let @xmath286 be the closure of a component of @xmath287 . this is a bi - infinite geodesic in @xmath63 . let @xmath288 be the component of @xmath289 not containing @xmath48 . thus , @xmath290 is an open interval in @xmath64 , which is a component of the discontinuity domain of @xmath73 . note in particular , that @xmath291 . [ d2 ] suppose @xmath292 , and let @xmath293 , @xmath294 and @xmath295 . then , if @xmath296 , we have @xmath297 . in particular , @xmath298 for every @xmath299 . to simplify notation we can assume ( via the orientation reversing symmetry of the construction ) that @xmath192 . note that @xmath300 , so @xmath301 \subseteq h(g ) $ ] . also @xmath302 and @xmath303 . it follows that @xmath301 $ ] , @xmath304 and @xmath286 are pairwise disjoint . thus , @xmath290 lies in a component of @xmath305 . since @xmath192 , the four points , @xmath306 are located in @xmath64 in this cyclic order , and so @xmath307 is a component of @xmath308 . since @xmath290 and @xmath309 share the point @xmath65 , we obtain the first assertion that @xmath297 with @xmath192 . the second assertion follows from the first assertion and the definition of @xmath310 . [ d3 ] suppose that @xmath311 and that the derived @xmath23-sequence @xmath203 for @xmath65 is infinite . then there is some @xmath312 such that @xmath313 . suppose , for contradiction , that @xmath314 for all @xmath312 . it follows that @xmath315 for all @xmath312 , and so @xmath316 for all @xmath312 . by lemma [ d2 ] , we have @xmath317 for all @xmath299 . ( here @xmath204 is the derived @xmath224-sequence for @xmath65 and @xmath318 . ) now , applying lemma [ d2 ] with @xmath319 in place of @xmath48 , we get that @xmath320 . continuing inductively we get that @xmath321 for all @xmath312 . in other words , the derived @xmath23-sequence for @xmath99 is identical to that for @xmath65 , and so , in particular , it must be infinite . we now get a contradiction by applying lemma [ c2](4 ) to any point @xmath322 . if we take @xmath323 to be a standard horoball neighbourhood of the cusp , then @xmath324 for all simple closed geodesic in @xmath0 , and so we can identify @xmath323 with a neighbourhood of the cusp in any @xmath276 . [ d4 ] there is some @xmath325 such that for each simple closed geodesic , @xmath275 , the hausdorff dimension of @xmath326 is at most @xmath327 . this is an immediate consequence of ( * ? ? * theorem 3.11 ) ( see also ( * ? ? ? * theorem 1 ) ) which refines the result of @xcite , on observing that the groups @xmath328 are uniformly `` geometrically tight '' , as defined in that paper . here , this amounts to saying that there is some fixed @xmath329 ( independent of @xmath275 ) such that the convex core , @xmath330 , is the union of @xmath323 and the @xmath331-neighbourhood of the geodesic boundary of the convex core . from the earlier discussion , we see that @xmath331 is bounded above by the diameter of @xmath332 , and so in particular , independent of @xmath275 . let @xmath333 be the union of the limit sets @xmath334 as @xmath335 ranges over all subgroups of @xmath184 obtained from a simple closed geodesic @xmath336 in @xmath0 . applying lemma [ d4 ] , we see that @xmath337 is a @xmath23-invariant subset of @xmath273 of hausdorff dimension strictly less than 1 . this is because it is a countable union of the limit sets @xmath334 whose hausdorff dimensions are uniformly bounded by a constant @xmath338 . recall the set @xmath235 defined in lemma [ c2](3 ) . then @xmath339 is also @xmath23-invariant and has hausdorff dimension zero by theorem [ c1 ] . [ d5 ] if @xmath340 , then @xmath163 is filling . namely , @xmath341 . suppose , for contradiction , that some @xmath340 is not filling . then there must be some simple closed geodesic , @xmath275 , in @xmath7 , which is disjoint from every @xmath342 , where @xmath269 is the derived sequence of arcs for the point @xmath65 . consider the hyperbolic surface @xmath343 and its fundamental group @xmath344 , as described at the beginning of this section . by hypothesis , @xmath345 , and so @xmath65 lies in some component , @xmath290 , of the discontinuity domain of @xmath73 . by lemma [ d3 ] , there must be some @xmath346 with @xmath347 . choose the minimal such @xmath312 . thus , @xmath348 but @xmath349 , where @xmath350 . we have @xmath351 and @xmath352 . ( the latter assertion can be seen as follows . if @xmath353 then @xmath354 is a parabolic fixed point of @xmath73 . since @xmath276 has a single cusp , there is an element @xmath355 such that @xmath356 . since @xmath357 , we have @xmath358 . this implies @xmath359 , a contradiction . ) therefore @xmath268 $ ] meets @xmath360 , giving the contradiction that @xmath342 crosses @xmath275 in @xmath7 . by lemma [ d5 ] , we have @xmath361 . since @xmath362 and @xmath337 both have hausdorff dimension strictly less than @xmath274 , the same is true of @xmath363 . thus , we have only to show that @xmath272 is open . pick an element @xmath364 . then there is some @xmath157 such that @xmath271 is a union of open discs , where @xmath270 is a derived sequence of arcs for @xmath163 . by corollary [ c3 ] , there is an open neighbourhood @xmath365 of @xmath65 in @xmath64 such that every @xmath366 shares the same initial derived @xmath184-sequence @xmath367 with @xmath65 . thus , every @xmath366 shares the same beginning derived sequence of arcs @xmath368 with @xmath65 . hence every @xmath366 is filling , i.e. , @xmath369 . recall that @xmath71 is identified with @xmath370 , where @xmath47 and @xmath371 , respectively , are the stabilisers of @xmath48 in @xmath31 and @xmath23 . as described in section [ sb ] , @xmath49 acts on @xmath96 , and @xmath372 acts on the circle @xmath373 . the wandering domain @xmath374 is equal to @xmath375 , because @xmath376 . ( see the general remark on the wandering domain given in section [ sb ] . ) we want to show that any @xmath378 has a wandering neighbourhood . by assumption , some initial segment , @xmath379 , of the derived sequence of arcs for @xmath65 fills @xmath7 . by corollary [ c3 ] , there is an open neighbourhood , @xmath365 , of @xmath65 , such that for every @xmath366 , the initial segment of length @xmath242 of the derived sequence of arcs is identical with @xmath379 . suppose that @xmath380 for some non - trivial element @xmath249 of @xmath381 . pick a point @xmath382 and set @xmath383 . by assumption , the derived sequences of arcs for both @xmath65 and @xmath99 begin with @xmath384 . on the other hand , lemma [ c4 ] implies that the derived sequence of arcs for @xmath385 is equal to the image of that for @xmath65 by @xmath249 . hence we see that @xmath386 for all @xmath387 . it follows by lemma [ e2 ] below , that @xmath249 is the trivial element of @xmath71 , a contradiction . [ e2 ] let @xmath379 be a set of proper oriented arcs in @xmath7 which together fill @xmath7 . suppose that @xmath11 is a mapping class on @xmath7 fixing the proper homotopy class of each @xmath342 . then @xmath11 is trivial . by proposition [ d1 ] , @xmath272 is an open set of @xmath96 whose complement has hausdorff dimension strictly less than @xmath388 . since @xmath389 contains @xmath272 by lemma [ e1 ] , its complement in @xmath96 also has hausdorff dimension strictly less than @xmath388 . since @xmath390 , this implies that the non - wandering set , @xmath391 , has hausdorff dimension strictly less than @xmath388 . fix any complete finite - area hyperbolic structure on @xmath7 , and use it to identify @xmath392 with @xmath393 . construct a graph , @xmath54 , as follows . the vertex set , @xmath394 , is the set of bi - infinite geodesics which are lifts of the arcs @xmath342 for all @xmath312 . two arcs @xmath395 are deemed adjacent in @xmath54 if either ( 1 ) they cross ( that is , meet in @xmath393 ) , or ( 2 ) they have a common ideal point in @xmath396 , and there is no other arc in @xmath394 which separates @xmath397 and @xmath398 . one readily checks that @xmath54 is locally finite . moreover , the statement that the arcs @xmath342 fill @xmath7 is equivalent to the statement that @xmath54 is connected . note that @xmath399 acts on @xmath54 with finite quotient . note also that @xmath54 can be defined formally in terms of ordered pairs of points in @xmath400 ( that is corresponding to the endpoints of the geodesics , and where crossing is interpreted as linking of pairs ) . the action of @xmath23 on @xmath54 is then induced by the dynamically defined action of @xmath23 on @xmath401 . now suppose that @xmath402 . lifting some representative of @xmath11 and extending to the ideal circle gives us a homomorphism of @xmath401 , equivariant via the corresponding automorphism of @xmath23 . suppose that @xmath11 preserves each arc @xmath342 , as in the hypotheses . then @xmath11 induces an automorphism , @xmath403 . given some @xmath404 , by choosing a suitable lift of @xmath11 , we can assume that @xmath405 . we claim that this implies that @xmath13 is the identity on @xmath54 . to see this , first let @xmath406 be the set of vertices adjacent to @xmath397 . this is permuted by @xmath13 . consider the order on @xmath407 defined as follows . let @xmath408 and @xmath409 , respectively , be the closed intervals of @xmath401 bounded by @xmath410 which lies to the right and left of @xmath397 . orient each of @xmath408 and @xmath409 so that the initial / terminal points of @xmath397 , respectively , are those of the oriented @xmath408 and @xmath409 . each @xmath411 determines a unique pair @xmath412 such that @xmath413 and @xmath414 are the endpoints of @xmath19 . now we define the order @xmath415 on @xmath407 , by declaring that @xmath416 if either ( i ) @xmath417 or ( ii ) @xmath418 and @xmath419 . this order must be respected by @xmath13 , because @xmath13 preserves the orders on @xmath408 and @xmath409 . since @xmath407 is finite , we see that @xmath420 is the identity . the claim now follows by induction , given that @xmath54 is connected . it now follows that the lift of @xmath11 is the identity on the set of all endpoints of elements of @xmath394 . since this set is dense in @xmath401 , it follows that it is the identity on @xmath401 , and we deduce that @xmath11 is the trivial mapping class as required . h. akiyoshi , h. miyachi , m. sakuma , _ variations of mcshane s identity for punctured surface groups _ : proceedings of the workshop `` spaces of kleinian groups and hyperbolic 3-manifolds '' , london math . soc . , lecture note series * 329 * ( 2006 ) 151185 . a. fathi , f. laudenbach , v. ponaru , et al , _ thurston s work on surfaces _ : translated from the 1979 french original by d. m. kim and d. margalit , mathematical notes , * 48 * , princeton university press , princeton , 2012 .

let @xmath0 be a complete finite - area orientable hyperbolic surface with one cusp , and let @xmath1 be the space of complete geodesic rays in @xmath0 emanating from the puncture . then there is a natural action of the mapping class group of @xmath0 on @xmath1 . we show that this action is almost everywhere wandering .

traffic problems have been attracting not only engineers but also physicists @xcite . especially it has been widely accepted that the phase transition from free to congested traffic flow can be understood using methods from statistical physics @xcite . in order to study the transition in detail , we need a realistic model of traffic flow which should be minimal to clarify the underlying mechanisms . in recent years cellular automata ( ca ) @xcite have been used extensively to study traffic flow in this context . due to their simplicity , ca models have also been applied by engineers , e.g. for the simulation of complex traffic systems with junctions and traffic signals @xcite . many traffic ca models have been proposed so far @xcite , and among these ca , the deterministic rule-184 ca model ( r184 ) , which is one of the elementary ca classified by wolfram @xcite , is the prototype of all traffic ca models . r184 is known to represent the minimum movement of vehicles in one lane and shows a simple phase transition from free to congested state of traffic flow . in a previous paper @xcite , using the ultra - discrete method @xcite , the burgers ca ( bca ) has been derived from the burgers equation @xmath0 which was interpreted as a macroscopic traffic model @xcite . the bca is written using the minimum function @xmath1 by @xmath2 where @xmath3 denotes the number of vehicles at the site @xmath4 and time @xmath5 . if we put the restriction @xmath6 , it can be easily shown that the bca is equivalent to r184 . thus we have clarified the connection between the burgers equation and r184 , which offers better understanding of the relation between macroscopic and microscopic models . the bca given above is considered as the _ euler _ representation of traffic flow . as in hydrodynamics there is an another representation , called _ lagrange _ representation @xcite , which is specifically used for car - following models . the lagrange version of the bca is given by @xcite @xmath7 where @xmath8 and @xmath9 is the position of @xmath10-th car at time @xmath5 . note that in ( [ lagbca ] ) @xmath11 corresponds a `` perspective '' or anticipation parameter @xcite which represents the number of cars that a driver sees in front , and @xmath12 is the maximum velocity of cars . ( [ lagbca ] ) is derived from the bca mathematically by using an euler - lagrange ( el ) transformation @xcite which is a discrete version of the well - known el transformation in hydrodynamics . in this paper , we will develop the bca ( [ lagbca ] ) to a more realistic model by introducing slow - to - start ( s2s ) effects @xcite and a driver s perspective @xmath11 . moreover , a stochastic generalization is also considered by combining it with the nagel - schreckenberg ( ns ) model @xcite . first , let us extend ( [ lagbca ] ) to the case @xmath13 and combine it with the s2s model . the s2s model @xcite is written in lagrange form as @xmath14 note that the inertia effect of cars is taken into account in this model . comparing ( [ eqs2s ] ) and ( [ lagbca ] ) , we see that , in the s2s model , the velocity of a car depends not only on the present headway @xmath15 , but also on the past headway @xmath16 . this rule has only a nontrivial effect if @xmath17 and @xmath18 , i.e. if the leading car has started to move in the previous time step . in this case the following car is not allowed to move immediately ( s2s ) . before combining ( [ lagbca ] ) and ( [ eqs2s ] ) , it is worth pointing out that we can choose the perspective parameter as @xmath19 in the model according to observed data . we define the size of a cell as 7.5 m and @xmath20 in our model according to the ns model . since @xmath12 corresponds to about 100 km / hour in reality , then one time step in the ca model becomes 1.3 s. moreover , the gradient of the free line and jamming line in the fundamental diagram , which is the dependence of the traffic flow @xmath21 on density @xmath22 , is known to be about 100 km / hour and @xmath23 km / hour @xcite according to many observed data ( see fig . [ fig1 ] ) @xcite . these values correspond to the typical free velocity and the jam velocity , respectively . ) and jamming line ( @xmath24 ) is known to be about 100 km / hour and @xmath23 km / hour . we also see that there is a wide scattering area near the phase transition region from free to jamming state . , scaledwidth=80.0% ] thus , considering the fact that the positive and negative gradient of each line is given by @xmath20 and @xmath25 , respectively , in the ca model @xcite , we should choose @xmath26 in the ca model . since only integer numbers for @xmath11 are allowed in this model , we will simply choose @xmath19 for studying the effect of the perspective of drivers . it is noted that other possibilities , such as velocity - dependent @xmath11 or stochastic choice of @xmath11 , are also possible . now by combining ( [ lagbca ] ) and ( [ eqs2s ] ) we propose a new lagrange model with @xmath19 which is defined by the rules listed below : let @xmath27 be the velocity of the @xmath10-th car at a time @xmath5 . the update procedure from @xmath5 to @xmath28 is divided into five stages : 1 . _ accerelation _ @xmath29 2 . _ slow - to - accelerate effect _ @xmath30 3 . _ deceleration due to other vehicles _ @xmath31 4 . _ avoidance of collision _ @xmath32 5 . _ vehicle movement _ @xmath33 the velocity @xmath34 is used as @xmath27 in the next time step . ( [ avoid ] ) is the condition that the @xmath10-th car does not overtake its preceding @xmath35-th car , including anticipation . the accerelation ( [ acdet ] ) is the same as in the ns model , which is needed for a mild accerelating behaviour of cars . in the step 2 , we call ( [ s2a ] ) as `` slow - to - accelerate '' instead of s2s . this is because this rule affects not only the behaviour of standing cars but also that of moving cars , which is considered to be a generalization of s2s rule . it is not difficult to write down the new model in a single equation for general @xmath11 . the result is @xmath36 where the last term represents the collision - free condition explained in fig . [ fig2 ] , and @xmath37 the condition that there is no collision between the @xmath10-th and @xmath38-th cars ( @xmath39 ) is given by @xmath40 for @xmath41 ( if @xmath42 then we simply put @xmath43 ) , which is identical to the last term in ( [ lagfm ] ) . in contrast to the ns model , the velocity of the preceeding car is taken into account in the calculation of the safe velocity in the step 4 , i.e. our model also includes anticipation effects . -th and @xmath38-th car . , next , we investigate the fundamental diagram of this new hybrid model . in fig . [ fig3 ] , we observe a complex phase transition from a free to congested state near the critical density 0.2 @xmath44 0.4 . there are many metastable branches in the diagram , similar to our previous models in euler form @xcite or in other models with anticipation @xcite . we also point out that there is a wide scattering area near the critical density in the observed data ( fig . [ fig1 ] ) which may be related to these metastable branches . as we will discuss later , these branches may account for some aspects of the scattering area observed empirically . first , we discuss properties of the state in the metastable branches . in all cases it consists of _ pairs of vehicles _ that move coherently with vanishing headway ( see fig . [ fig4 ] ) . cars are represented by black squares , and the direction of the road is horizontal right and time axis is vertical down . the corresponding velocity distributions are also given in fig . we see that there are stopping cars which velocity are zero only in the case of the lowest branch given in the state in fig . [ fig4 ] ( e ) . at density @xmath45 with different strength of perturbation : ( a ) very weak , ( b ) less weak , ( c ) medium , ( d ) stronger and finally ( e ) strongest perturbation . the details of these perturbations are all explained in detail in the text . the stationary state of these five cases correspond to a state in each metastable branch appearing in fig . [ fig3 ] , although the branch corresponding to ( a ) does not appear in the numerical simulations with random initial conditions . , title="fig:",scaledwidth=40.0% ] at density @xmath45 with different strength of perturbation : ( a ) very weak , ( b ) less weak , ( c ) medium , ( d ) stronger and finally ( e ) strongest perturbation . the details of these perturbations are all explained in detail in the text . the stationary state of these five cases correspond to a state in each metastable branch appearing in fig . [ fig3 ] , although the branch corresponding to ( a ) does not appear in the numerical simulations with random initial conditions . , title="fig:",scaledwidth=40.0% ] + ( a)(b ) + at density @xmath45 with different strength of perturbation : ( a ) very weak , ( b ) less weak , ( c ) medium , ( d ) stronger and finally ( e ) strongest perturbation . the details of these perturbations are all explained in detail in the text . the stationary state of these five cases correspond to a state in each metastable branch appearing in fig . [ fig3 ] , although the branch corresponding to ( a ) does not appear in the numerical simulations with random initial conditions . , title="fig:",scaledwidth=40.0% ] at density @xmath45 with different strength of perturbation : ( a ) very weak , ( b ) less weak , ( c ) medium , ( d ) stronger and finally ( e ) strongest perturbation . the details of these perturbations are all explained in detail in the text . the stationary state of these five cases correspond to a state in each metastable branch appearing in fig . [ fig3 ] , although the branch corresponding to ( a ) does not appear in the numerical simulations with random initial conditions . , title="fig:",scaledwidth=40.0% ] + ( c)(d ) + at density @xmath45 with different strength of perturbation : ( a ) very weak , ( b ) less weak , ( c ) medium , ( d ) stronger and finally ( e ) strongest perturbation . the details of these perturbations are all explained in detail in the text . the stationary state of these five cases correspond to a state in each metastable branch appearing in fig . [ fig3 ] , although the branch corresponding to ( a ) does not appear in the numerical simulations with random initial conditions . , title="fig:",scaledwidth=40.0% ] + ( e ) , corresponding to the states given in fig . [ fig4 ] . , title="fig:",scaledwidth=40.0% ] , corresponding to the states given in fig . , title="fig:",scaledwidth=40.0% ] + ( a)(b ) + , corresponding to the states given in fig . [ fig4 ] . , title="fig:",scaledwidth=40.0% ] , corresponding to the states given in fig . , title="fig:",scaledwidth=40.0% ] + ( c)(d ) + , corresponding to the states given in fig . [ fig4 ] . , title="fig:",scaledwidth=40.0% ] + ( e ) next let us calculate the flow - density relation for each branch . in the metastable branches we find phase separation into a free - flow and a jamming region . in the former , pairs move with velocity @xmath46 and a headway of @xmath47 empty cells between consecutive pairs . in the jammed region , the velocity of the pairs is @xmath48 and the headway @xmath49 . @xmath50 and @xmath51 are the numbers of cars in the jamming cluster and the free uniform flow , respectively . we assume @xmath51 and @xmath50 to be even so that there are @xmath52 and @xmath53 pairs , respectively . then the total number of cars @xmath54 is given by @xmath55 and the total length of the system becomes @xmath56 . since the average velocity is @xmath57 and density and flow of the system are given by @xmath58 and @xmath59 , we obtain the flow - density relation as @xmath60 from the stationary states in fig . [ fig4 ] we have @xmath61 therefore the resulting equations for each branch are @xmath62 where @xmath63 , which correspond to the branches @xmath64 and @xmath65 in fig . [ fig6 ] , respectively . end points of the branches are found to be given by @xmath66 and @xmath67 , where @xmath68 is the point that the metastable branches intersect the free flow branch , and @xmath69 is the maximal possible density in the metastable branches . note that all @xmath69 lie on the line @xmath70 , which is indicated as the broken line in fig . [ fig6 ] . and @xmath71 ) and the jamming line ( @xmath65 ) in the new model . the highest flow state is represented by @xmath72 , which is quite unstable and easy to go down to the lower flow state @xmath73 according to the magnitude of the perturbation . , scaledwidth=60.0% ] next let us now study the stability of each metastable branch . we mainly consider the density @xmath45 and , in particular , we will focus on the uniform flow represented by @xmath74 , which shows the highest flow given in the point @xmath72 in fig . [ fig6 ] . spatio - temporal patterns due to various kinds of perturbations are already seen in fig . perturbation in this case means that some cars are shifted backwards at the initial configuration . the initial conditions for fig . [ fig4](a)-(e ) are given as follows : 1 . very weak perturbation ( one car is shifted one site backwards)@xmath75 2 . weak perturbation ( one car is shifted two sites backwards)@xmath76 3 . moderate perturbation ( one car is shifted three sites backwards)@xmath77 4 . strong perturbation ( one car is shifted five sites backwards)@xmath78 5 . strongest perturbation ( three cars are shifted backwards ) @xmath79 the stationary state of ( a ) , ( b ) , ( c ) , ( d ) and ( e ) are given by the points @xmath80 , @xmath81 , @xmath82 , @xmath83 and @xmath84 in fig . [ fig6 ] , respectively . that is , if the system in @xmath72 is perturbated , then the flow easily goes down to a lower branch in the course of time depending on the magnitude of the perturbation . since the density does not change due to the perturbation , we obtain @xmath85 , @xmath86 , @xmath87 , @xmath88 and @xmath89 by substituting @xmath45 into eq . ( [ metaq ] ) . we see a jamming cluster propagating backwards in the cases of ( d ) and ( e ) in fig . in other cases the jamming cluster propagates forward ( ( a ) and ( b ) ) or does not move ( c ) . these facts are related to the gradient of the metastable branches which are given by @xmath90 according to eq . ( [ metaq ] ) . finally we will combine the above model with the ns model in order to take into account the randomness of drivers . the ns model is written in lagrange form as @xmath91 where @xmath92 with probability @xmath93 and @xmath94 with probability @xmath95 . the last term in the mininum in ( [ eqns ] ) represents the acceleration of cars . the randomness in this model is considered as a kind of random braking effect , which is known to be responsible for spontaneous jam formation often observed in real traffic @xcite . we also consider random _ accerelation _ in this model which is not taken into account in the ns model . thus a stochastic generalization of the hybrid model in the case of @xmath19 is similarly given by the following set of rules : 1 . _ random accerelation _ @xmath96 where @xmath97 with the probability @xmath98 and @xmath99 with @xmath100 . slow - to - accelerate effect _ @xmath101 3 . _ deceleration due to other vehicles _ @xmath102 4 . _ random braking _ @xmath103 where @xmath104 with the probability @xmath105 and @xmath106 with @xmath107 . _ avoidance of collision _ @xmath108 with @xmath109 , which is an iterative equation that has to be applied until @xmath110 converges to @xmath111 . _ vehicle movement _ @xmath112 again the velocity @xmath113 is used as @xmath27 in the next time step . step 5 must be applied to each car iteratively until its velocity does not change any more , which ensures that this model is free from collisions . this is the difference between the deterministic and stochastic case . in the deterministic model it is sufficient to apply the avoidance of collision stage only once in each update , while in the stochastic case generically it has to be applied a few times in order to avoid collisions between successive cars . and @xmath19 . upper two figures are the case of @xmath114 and @xmath115 , middle ones are @xmath116 and @xmath117 , and the bottom ones are @xmath116 and @xmath115 . , title="fig:",scaledwidth=40.0% ] and @xmath19 . upper two figures are the case of @xmath114 and @xmath115 , middle ones are @xmath116 and @xmath117 , and the bottom ones are @xmath116 and @xmath115 . , title="fig:",scaledwidth=40.0% ] + and @xmath19 . upper two figures are the case of @xmath114 and @xmath115 , middle ones are @xmath116 and @xmath117 , and the bottom ones are @xmath116 and @xmath115 . , title="fig:",scaledwidth=40.0% ] and @xmath19 . upper two figures are the case of @xmath114 and @xmath115 , middle ones are @xmath116 and @xmath117 , and the bottom ones are @xmath116 and @xmath115 . , title="fig:",scaledwidth=40.0% ] + and @xmath19 . upper two figures are the case of @xmath114 and @xmath115 , middle ones are @xmath116 and @xmath117 , and the bottom ones are @xmath116 and @xmath115 . , title="fig:",scaledwidth=40.0% ] and @xmath19 . upper two figures are the case of @xmath114 and @xmath115 , middle ones are @xmath116 and @xmath117 , and the bottom ones are @xmath116 and @xmath115 . , title="fig:",scaledwidth=40.0% ] the fundamental diagrams of this stochastic model for some values of @xmath98 and @xmath105 are given in fig . the randomization effect can be considered as a sort of perturbation to the deterministic model . hence some unstable branches seen in the deterministic case disappear in the stochastic case , especially if we consider the random braking effect as seen in fig . random accerelation itself does not significantly destroy the metastable branches . moreover , from the spatio - temporal pattern it is found that spontaneous jam formation is observed only if we allow random braking . random accerelation alone is not sufficient to produce spontaneous jamming . we also note that a wider scattering area appears if we introduce both random accerelation and braking . in this paper we have proposed a new hybrid model of traffic flow of lagrange type which is a combination of the bca and the s2s model . its stochastic extension is also proposed by further incorporating stochastic elements of the ns model and random accerelation . the model shows several metastable branches around the critical density in its fundamental diagram . the upper branches are unstable and will decrease its flow under perturbations . it is shown that the magnitude of a perturbation determines the final value of flow in the stationary state . moreover , introduction of stochasticity in the model makes the metastable branches dilute and hence produces a wide scattering area in the fundamental diagram . we would like to point out that this metastable region around the phase transition density is similar to so - called synchronized flow proposed by @xcite . our investigation shows that one possible origin of such a region is the occurance of many intermediate congested states near the critical density . if some of them are unstable due to perturbation or randomness , then a dense scattering area near the critical density is formed around the metastable branches . this is in some sense in between the two cases of a fundamental diagram based approach ( with unique flow - density relation ) and the so - called 3-phase model of @xcite which exhibits a full two - dimensional region of allowed states even in the deterministic limit . this work is supported in part by a grant - in - aid from the japan ministry of education , science and culture .

a new stochastic cellular automaton ( ca ) model of traffic flow , which includes slow - to - start effects and a driver s perspective , is proposed by extending the burgers ca and the nagel - schreckenberg ca model . the flow - density relation of this model shows multiple metastable branches near the transition density from free to congested traffic , which form a wide scattering area in the fundamental diagram . the stability of these branches and their velocity distributions are explicitly studied by numerical simulations .

puga@xmath0 appears in two crystal structures with similar electronic structures but different magnetic structures . et al . _ noted in 1964 the appearance of the two crystal structures and identified the low - temperature ( low-@xmath3 ) form as hexagonal and isostructural with ni@xmath0sn;@xcite the following year larson _ et al . _ identified the high - temperature ( high-@xmath3 ) form as a 12-layer rhombohedral close - packed structure with space group r@xmath4m.@xcite four decades later , magnetic measurements revealed the low-@xmath3 form to be an antiferromagnet ( @xmath5 k ) and the high-@xmath3 form to be a ferromagnet ( @xmath6 k ) , and specific heat measurements suggest for both phases an electronic structure with heavy fermion character.@xcite these characteristics place puga@xmath0 in compelling relation to other heavy fermion systems of significant interest . the electronic specific heat coefficient @xmath7 , a measure of the electronic density of states at the fermi energy , has values ( 220 and 100 mj / mol k@xmath8 for low-@xmath3 and high-@xmath3 , respectively ) similar to the heavy fermion superconductors pucoga@xmath9 and purhga@xmath9.@xcite neither of these are magnetic experimentally , though electronic structure calculations favor antiferromagnetic order.@xcite with rather delocalized @xmath1 electrons , puga@xmath0 lies between pucoga@xmath9 and @xmath10-pu,@xcite which has more localized @xmath1 electrons but also shows no localized magnetic moments.@xcite the combination of magnetic structure and heavy fermion behavior in puga@xmath0 suggest a challenging system for electronic structure calculations . the electronic structure of pu , many pu compounds , and some other actinide systems requires special attention to be paid to the strong @xmath1 electron correlation . calculations with `` standard '' density functional theory ( dft ) methods , which involve limited approximate treatments of the electronic correlation , favor an antiferromagnetic structure,@xcite and some aspects are even better modeled with disordered local moments or approximations thereof.@xcite experimentally , pure pu shows no signs of magnetic moments.@xcite the breaking of spin symmetry in dft calculations delivers a static approximation of the spatial separation experienced by dynamically correlated electrons . as a result , calculations allowing a localized magnetic moment can be used to explore nonmagnetic aspects of pu and pu compounds without introducing material - dependent parameters . the existence of magnetic structures in puga@xmath0 entangles the magnetic moments and the electronic correlation , which , along with their entanglement with the observed crystal structures , motivates this study . the crystal structures of puga@xmath0 can be viewed as close - packed puga@xmath0 planes with different stacking sequences.@xcite the low-@xmath3 structure follows an ab sequence , as does hexagonal close packed ( hcp ) ; stacking in the high-@xmath3 structure progresses as ababcacabcbc ( with some in - plane distortions away from the perfect close - packed planar structures ) . this layered , close - packed nature already appears in crystal structures of pure pu : the face - centered cubic structure of @xmath10-pu exhibits abc stacking , the crocker pseudostructure for @xmath11-pu follows from the @xmath11 structure s repeating two planes of a distorted hexagonal structure,@xcite and the orthorhombic structure of @xmath7-pu exhibits close - packed pu planes stacked such that pu atoms in one plane sit above bonds in the plane underneath ( giving rise to an abcd stacking pattern ) . the close - packed puga@xmath0 planes correspond to the close - packed pu planes with ordered substitutional placing of ga . these stacking sequences of close - packed planes ( excluding that of @xmath7-pu ) can also be written as sequences of shifts between planes : ab , bc , ca being shifts to the right ( r ) and ac , cb , ba being shifts to the left ( l ) . abc stacking always shifts in the same direction ( rrrr ) , ab stacking alternates between the two directions ( rlrl ) , and ababcacabcbc stacking , rewritten as ( abca)(cabc)(bcab ) , reverses direction once every four planes ( rrrl ) . the missing unique pattern with four shifts , rrll , is abcb stacking , which corresponds to double hcp ( dhcp ) , exemplified by @xmath11-la . while the r and l shifts are equivalent , a stark difference exists between a plane that links two shifts with the same direction and one that sits at a reversal in the direction . the local environment of the atomic sites in the ideal close - packed lattices has twelve nearest neighbors in both cases . sites in a plane between two identical shifts have the inversion symmetry , while those between two opposite shifts do not . the lack of inversion symmetry disrupts an otherwise straight line of bonding oriented 60@xmath12 to the planes . a natural order of the four crystal structures arises : abc stacking has no disruptions , ab stacking has disruptions in every plane , and the remaining two stacking sequences lie in between . the work presented here applies dft to reveal the interplay between crystal structures based on these four structural patterns , a series of magnetic structures , and the resulting electronic structures . starting from `` standard '' dft in the generalized gradient approximation ( gga ) , calculations furthermore explore the effects of adding either spin - orbit coupling or a hubbard @xmath2 ( in the gga+u method ) . the calculations presented here set aside thermal effects , in particular those due to phonons . preliminary calculations of the phonons and their contribution to the free energy suggest they can not make the low-@xmath3 phase more favorable in the gga to dft without specifically addressing @xmath13 electron correlation . the dft calculations employ the vasp package.@xcite they make use of the generalized gradient approximation ( gga ) of perdew , burke , and ernzerhof.@xcite the pu(@xmath14 ) and ga(@xmath15 ) electrons are treated in the valence using a plane - wave basis and with projector - augmented wave potentials.@xcite the calculations employ methfessel - paxton smearing ( with width 0.1 ev ) , a k - point mesh of density 40 @xmath16 , and an energy cutoff of 400 ev . the self - consistent cycles are converged to within 10@xmath17 ev . calculations aimed at improving the treatment of the the on - site coulomb repulsion between 5@xmath13 electrons use an effective hubbard parameter @xmath2 in the rotationally invariant form of dudarev _ et al._.@xcite in this form the hubbard parameter @xmath2 and the exchange parameter @xmath18 appear only in the difference @xmath19 , throughout this report the difference is referred to as @xmath2 . calculations that include the effects of spin - orbit coupling do so in the noncollinear mode of vasp,@xcite the implementation follows the approach of kleinman and macdonald , picket , and koelling.@xcite the calculations optimize crystal structures that start as ideal close - packed planes with one pu and three ga atoms , stacked according to one of the four patterns described above . relaxation of the structures retains the overall layered structure , but displacements within the planes make initially equivalent planes lose their exact equality . the size of the unit cell , in particular the number of planes ( between two and twelve ) , follows from the particular pattern and the magnetic structure used to seed the calculations . the latter either has all spins in the same direction for the ferromagnetic ( fm ) structure , or spins that switch direction every one , two , three , four , or six planes . these arrangements define spin wave structures with wave vectors @xmath20 of magnitude @xmath21 , @xmath22 , @xmath23 , @xmath24 , @xmath25 , and , in the fm case , @xmath26 , scaled by @xmath27 , where @xmath28 represents the interplanar spacing . additional magnetic structure within the close - packed plane affects the results , but these are not reported here , other than to note that their energy lies above that of the antiferromagnetic ( afm , @xmath29 ) state . the results from three approaches appear in the following three subsections . sections [ gga ] and [ ggaplussoc ] report the results of dft calculations in the gga without and with spin - orbit coupling , respectively , for the four crystal structures in a sequence of magnetic states . section [ ggaplusu ] focuses on results of the gga+u method applied to the low-@xmath3 and cubic structures in the afm state . table [ tab : opt ] summarizes the energies , volumes and @xmath30 ratios calculated in the three approaches for the four crystal structures in the afm state . figure [ fig : abcafvary ] shows the interplay between the four crystal structures and the magnetic structures using the gga to dft . all four stacking sequence patterns show a preference for the magnetic structure with the shortest spin wave length , the afm state . the ordering of energies of the crystal structures in the afm state correlates with the order arising from the number of changes in r and l shifts mentioned in the introduction . with this magnetic structure , the experimentally observed low-@xmath3 phase lies highest , 117 mev / puga@xmath0 above the favored structure with abc stacking . this cubic structure is observed for puin@xmath0 and is often considered a building block for the layered superconductors pucoga@xmath9 , cept@xmath31in@xmath32 , ce@xmath31rhin@xmath33 , etc . the preference for this cubic structure appears only for the magnetic structure with the shortest spin wave length ; for longer spin wave lengths it lies higher than the other states ( albeit by small amounts ) . among the fm states the high-@xmath3 phase lies lowest , the slight 13 mev / puga@xmath0 difference to the low-@xmath3 phase suggests the importance of thermal effects . ( color online ) calculated dependence of energy on stacking and magnetic structure for puga@xmath0 using gga ( @xmath34 ) . stacking denotes initial crystal structure ; upon relaxation the planes with the same letter are no longer necessarily equivalent . stacking direction corresponds to body diagonal of the conventional aucu@xmath0 crystal structure unit cell ; the lowest energy appears for g - type antiferromagnetism ( afm(g ) ) . dashed lines serve to guide the eye . , width=321 ] the optimized structures agree reasonably well with experimental volumes , while the optimized @xmath30 ratios consistently lie above the experimental values . the afm volume calculated for the low-@xmath3 structure is only 1% smaller , but the @xmath30 ratio is close to 6% larger than the experimental value ( see table [ tab : opt ] ) . the afm volume calculated for the high-@xmath3 structure is 0.25% smaller than the experimental value , and the @xmath30 ratio is close to 4% larger than the experimental value . the fm volume calculated for the high-@xmath3 structure is 1% larger , and the @xmath30 ratio is 3% larger than the experimental value . the distances from pu to nearest ga atoms ( located in adjacent planes ) differ by negligible amounts between the calculated and experimental high-@xmath3 structures . the larger calculated @xmath30 ratio does affect the angle spanned by a pu atom and two ga atoms in adjacent planes , decreasing it by as much as 13% . figure [ fig : ggaedos1](a ) compares calculated total electronic densities of states ( dos ) and suggests why the cubic structure appears more favorable in the afm state . the low-@xmath3 , high-@xmath3 , and cubic structure differ significantly in the highest occupied states . the low-@xmath3 and cubic structure both exhibit a single peak , but the cubic structure has it almost 0.3 ev further below the fermi level @xmath35 . the high-@xmath3 structure exhibits a double peak centered between the other two structures . while the band energy is only one part of the total energy , this ability of the structures to push states down and away from @xmath35 corresponds to their order in total energy . figure [ fig : ggaedos1](b ) plots the analogous comparison for the three structures in the fm state . compared to the afm state , the peaks appear much more similar for the three structures than in the afm state . accordingly , the total energies for the fm state differ by smaller amounts compared to the afm state . the peaks sit closer to @xmath35 in the fm state , concurring with the energies of the fm state lying higher than those of the afm state . ( color online ) calculated electronic densities of states ( dos ) near the fermi energy @xmath35 using the gga to dft ( @xmath34 ) for low-@xmath3 , high-@xmath3 , and aucu@xmath0 crystal structures with ( a ) afm and ( b ) fm structure . only the dos for one spin orientation appears for afm . , width=321 ] ( color online ) calculated electronic dos near @xmath35 projected on a pu site with @xmath13 character using the gga to dft ( @xmath34 ) for low-@xmath3 and aucu@xmath0 crystal structures with afm magnetic structure . the plotted dos represent the majority spin on the pu site . , width=321 ] figure [ fig : fdos1 ] shows the @xmath13 symmetry character ( projected out on a pu site ) of the electronic dos calculated for the low-@xmath3 and aucu@xmath0 crystal structures with afm magnetic structure . the @xmath13-projected peaks correspond to the peaks in fig . [ fig : ggaedos1](a ) . the projected dos are identical for all sites in each case , as expected given the sites identical environments : each site has the same structural environment and nearest neighbors with opposite spin . the structural environment differs between the two cases , the aucu@xmath0 crystal structure s inversion symmetry allows the @xmath13-projected peaks to be pushed down lower . the less symmetric local environment in the low-@xmath3 structure makes it less atomic - like , requiring the @xmath13 electrons to hybridize more . ( color online ) calculated electronic dos near @xmath35 projected on pu sites with @xmath13 character using gga ( @xmath34 ) for low-@xmath3 and aucu@xmath0 crystal structures with magnetic structure that has spin wave length spanning twelve planes . the plotted dos represent the majority spin on each site . in terms of geometry , all sites are equivalent for each crystal structure . they differ depending on where they sit within the magnetic structure : adjacent to the spin flip ( `` edge '' ) , one layer farther in ( `` intermediate '' ) , or most distant to the spin flip ( `` center '' ) . , width=321 ] figure [ fig : fdos2 ] compares the @xmath13 symmetry character projected out on pu sites from the electronic dos calculated for the low-@xmath3 and aucu@xmath0 crystal structures in the magnetic state with spin wave vector magnitude @xmath36 . this choice of spin wave vector stems from the differences it reveals among the pu sites , unlike the ferromagnetic structure where all sites ( within each crystal structure ) remain equivalent . for the low-@xmath3 structure the projected electronic dos differs only slightly between the three types of sites , a slight shift down from @xmath35 occurs closer to the edge of the magnetic subdivision . the cubic structure shows dramatic differences between the three types of sites : all show a projected electronic dos hugging @xmath35 from below , and only the site at the edge of the magnetic subdivision appears able to spread a significant amount down several tenths of an ev . in the fm state , the @xmath13-projected dos on any of the sites closely resembles the @xmath13-projected dos shown here for center atoms . ( color online ) calculated dependence of energy on stacking and magnetic structure for puga@xmath0 using gga ( @xmath34 ) and including spin - orbit coupling . dashed lines serve to guide the eye . notation follows fig . [ fig : abcafvary ] . , width=321 ] figure [ fig : abcafvarysoc ] shows the interplay between the four crystal structures and magnetic structures using the gga to dft and including spin - orbit coupling . the inclusion of spin - orbit coupling reduces the energy differences overall , hence figure [ fig : abcafvarysoc ] appears much like a scaled version of fig . [ fig : abcafvary ] . the aucu@xmath0 crystal structure in the afm state remains the most favored , in the fm state it remains the least favored . results from calculations that include spin - orbit coupling repeat the correlation between which structure is energetically favored and its ability to push electronic states down and away from @xmath35 . with spin - orbit coupling , the electronic dos of the low-@xmath3 and aucu@xmath0 crystal structure differ from one another less than in fig . [ fig : ggaedos1 ] , but the more favored aucu@xmath0 crystal structure still succeeds better at pushing electronic states to lower energies . table [ tab : opt ] shows how treating the on - site coulomb repulsion between @xmath1 electrons with a hubbard u changes the ranking of crystal structures . setting @xmath37 ev reverses the sequence in energy from the gga result ( with or without spin - orbit coupling ) : the low-@xmath3 crystal structure becomes most favored while the aucu@xmath0 crystal structure becomes the least favored . the high-@xmath3 and `` @xmath11-la '' structures remain in between and switch their order as well . comparison of the energies for the different crystal structures only has meaning for each value of @xmath2 individually , which is somewhat unsatisfactory since the different crystal structures would be better described with different values ( differences in the electronic specific heat coefficient @xmath7 and in the pu - ga distances in the low-@xmath3 and high-@xmath3 crystal structure suggest different degrees of @xmath1 delocalization,@xcite implying incompatible values of @xmath2 ) . c | c c c c | c structure & 4@xmath2 ( ev ) & soc & 0 & 1 & 2 & 3 & ( @xmath34)6relative energies ( mev / puga@xmath0 ) low-@xmath3 & 0 & 0 & 0 & 0 & 0@xmath11-la " & -49 & -9 & 6 & 109 & -26 high-@xmath3 & -72 & -53 & -26 & 120 & -47 aucu@xmath0 & -117 & -67 & 2 & 168 & -62 6volumes ( @xmath38/puga@xmath0 ) low-@xmath3 & 77.37 & 78.09 & 80.00 & 81.28 & 77.99 @xmath11-la " & 77.59 & 78.58 & 79.78 & 80.84 & 77.52 high-@xmath3 & 77.50 & 78.22 & 79.68 & 80.90 & 77.50 aucu@xmath0 & 77.72 & 77.72 & 79.88 & 80.77 & 77.67 6@xmath30 ratio low-@xmath3 & 0.38 & 0.38 & 0.37 & 0.36 & 0.37 @xmath11-la " & 0.40 & 0.39 & 0.39 & 0.39 & 0.39 high-@xmath3 & 0.39 & 0.39 & 0.39 & 0.38 & 0.39 aucu@xmath0 & 0.41 & 0.41 & 0.41 & 0.41 & 0.41 figure [ fig : edosabaf1 ] plots the electronic dos for the low-@xmath3 crystal structure in the afm state calculated with the gga+u method . as @xmath2 increases , the dominant peaks , dft s rendering of the upper and lower hubbard bands,@xcite increasingly separate . this separation pushes the occupied states down from @xmath35 more than it pushes the unoccupied states up . the symmetry between up and down spin remains intact , and , based on site - projected dos ( not shown here ) , the equivalence among sites with the same spin remains . figure [ fig : edosabcaf1 ] plots the electronic dos for the aucu@xmath0 crystal structure in the afm state calculated with the gga+u method . again the increasing @xmath2 drives the dominant peaks apart , but for this crystal structure the separation occurs mainly by pushing up the unoccupied states . the occupied states change little as @xmath2 increases from 0 ev to 1 ev . increasing @xmath2 from 1 ev to 2 ev pushes the occupied states down . setting @xmath37 ev breaks the symmetries of up and down spins as well as the equivalence among sites with the same spin . ( color online ) calculated electronic dos with varying hubbard @xmath2 for the low-@xmath3 crystal structure with afm magnetic structure at the experimental volume . the dos for the two spin orientations appear as positive and negative , respectively . , width=321 ] ( color online ) calculated electronic dos with varying hubbard @xmath2 for the cubic crystal structure with afm magnetic structure at the experimental volume . the dos for the two spin orientations appear as positive and negative , respectively . for u=@xmath39 ev and above the symmetry between pu sites is broken and variations of up to 0.6% appear in the site - projected charge and of up to 5% appear in the site - projected magnetic moments . , width=321 ] comparison of figs . [ fig : edosabaf1 ] and [ fig : edosabcaf1 ] for each value of @xmath2 correlates well with the energy differences in table [ tab : opt ] . for @xmath40 ev the aucu@xmath0 crystal structure retains the peak around 0.5 ev below @xmath35 while the low-@xmath3 crystal structure has its main peak shifted lower than for @xmath34 . for @xmath41 ev both crystal structures have shifted ( and broadened ) the peak to around 1 ev below @xmath35 . for @xmath37 ev the overall dos changes somewhat for the aucu@xmath0 crystal structure while for the low-@xmath3 crystal structure a dramatic shift downward occurs . systematically studying the crystal and magnetic structures of puga@xmath0 reveals how they affect the electronic structure and how the three types of structure are entangled . the key to understanding the entanglement lies in the position of the @xmath1 electron states relative to the fermi level @xmath35 in the electronic dos . the position relative to @xmath35 is determined by both the symmetry of the crystal structure and the imposed magnetic structure . how far the @xmath1 peak sits below @xmath35 dovetails with how favorable the system in question is in terms of calculated total energy . magnetic structure affects the energy more strongly than crystal structure . with or without spin - orbit coupling , the calculations favor the afm state over the fm state for all crystal structures . spin density waves with wave lengths between those of the fm state ( infinity ) and of the afm state ( twice the spacing between pu planes ) give total energies between the two limiting values . pu sites neighboring a junction between up and down spins have their @xmath1 electron states farther below @xmath35 than other pu sites . each such junction gives the sites sandwiching the junction less hybridization of @xmath1 states with neighbors on the other side of the junction . in the limiting case of afm , every site has the least hybridization because pu sites in neighboring planes have opposite spin . calculations using standard gga result in the wrong crystal structure ( aucu@xmath0 ) having the lowest energy in the favored afm state . the @xmath1 electron states in the cubic structure sit farther below @xmath35 than they do in the experimentally observed ni@xmath0sn crystal structure , because the inversion symmetry at sites in the cubic structure requires less hybridization between the pu @xmath1 states and other states . adding a hubbard u to treat the strong @xmath1 electron correlation results in the correct crystal structure having the lowest energy . the @xmath2 raises and lowers the potential acting on the unoccupied and occupied @xmath1 states , respectively , but the effect of @xmath2 on the positions of the @xmath1 states relative to @xmath35 depends on how they are hybridized.@xcite increasing the value of @xmath2 proves more effective at lowering the energy of the @xmath1 electron peak for the ni@xmath0sn crystal structure , making it most favored for @xmath37 ev . allowing localized magnetic moments to simulate correlation effects fails for puga@xmath0 . the strong preference for the cubic crystal structure over the hexagonal crystal structure suggests the failure stems not from the actual presence of a magnetic structure ( observed in experiment ) , but from the symmetry at pu sites in the hexagonal crystal structure being much lower than in the cubic crystal structure . the use of allowing localized magnetic moments to simulate correlation does so by permitting the @xmath1 electrons on the same pu site to occupy more orbitals that differ spatially . the inversion symmetry present in the cubic crystal structure makes the localized magnetic moments most effective at simulating correlation effects . in the hexagonal crystal structure the lower symmetry prevents an adequate decoupling of @xmath13 states from hybridization and their energy can not be lowered sufficiently to make the crystal structure most favorable . these results suggest a explanation for the effectiveness of using magnetism to approximate correlation effects in @xmath10-pu . the crystal structure of @xmath10-pu is face - centered cubic , and all sites exhibit the inversion symmetry shown here to be important in the closely - related aucu@xmath0 structure . given the similarities , the preference for an afm state in @xmath10-pu does not surprise . nor does the additional effectiveness of modeling correlation effects with disordered local moments astonish , since such a magnetic `` structure '' reduces also the in - plane hybridization between @xmath13 electrons on neighboring sites . analogous to the relation between @xmath10-pu and puga@xmath0 in the aucu@xmath0 crystal structure , @xmath11-pu relates to puga@xmath0 in the ni@xmath0sn crystal structure . the crystal structures of both @xmath11-pu and the low-@xmath3 phase of puga@xmath0 are the most stable and both have an ab stacking pattern . the @xmath11-pu crystal structure stacks distorted close - packed pu planes ; replacing three of four pu atoms with ga removes the distortion to restore the symmetry in the close packed planes of puga@xmath0 , which could relate to the stabilization of @xmath10-pu to low temperatures by adding a small amount of ga.@xcite the electronic specific heat coefficient @xmath7 differs dramatically between the low-@xmath3 phase of puga@xmath0 , where @xmath42 mj / mol k@xmath8 , and @xmath11-pu , where @xmath43 mj / mol k@xmath8 was measured.@xcite correspondingly , @xmath11-pu can be well described by standard dft methods,@xcite while the work presented here shows that the low-@xmath3 phase of puga@xmath0 requires special attention be paid to the strong @xmath1 electron correlation , and allowing spin polarization does not suffice to describe the effects of the strong correlation . this research was supported by the los alamos national laboratory , under the auspices of the national nuclear security agency , by the u.s . department of energy under grant no . ldrd - dr 20120024 ( `` pu-242 : a national resource for the fundamental understanding of the 5f electrons of pu '' ) . many thanks go to in particular eric chisolm , anders niklasson , and john wills as well as eric bauer , john joyce , and paul tobash , for helpful and encouraging discussions . the author expresses a deep gratitude to neil henson for assistance with the andulu computational facility . last , but not least , fond thanks go to lucia lin and anna lan for spurring alternative approaches to understanding . 26ifxundefined [ 1 ] ifx#1 ifnum [ 1 ] # 1firstoftwo secondoftwo ifx [ 1 ] # 1firstoftwo secondoftwo `` `` # 1''''@noop [ 0]secondoftwosanitize@url [ 0 ] + 12$12 & 12#1212_12%12@startlink[1]@endlink[0]@bib@innerbibempty @noop * * , ( ) @noop * * , ( ) @noop * * , ( ) link:\doibase 10.1103/physrevb.75.184501 [ * * , ( ) ] link:\doibase 10.1103/physrevb.70.104504 [ * * , ( ) ] @noop * * , ( ) @noop * * , ( ) @noop * * , ( ) @noop * * , ( ) @noop * * , ( ) @noop * * , ( ) @noop * * , ( ) @noop * * , ( ) @noop * * , ( ) @noop * * , ( ) @noop * * , ( ) @noop * * , ( ) @noop * * , ( ) @noop * * , ( ) link:\doibase 10.1103/physrevb.21.2630 [ * * , ( ) ] http://stacks.iop.org/0022-3719/13/i=14/a=009 [ * * , ( ) ] @noop * * , ( ) @noop * * , ( ) @noop _ _ ( , , ) @noop * * , ( ) @noop * * , ( )

systematically studying the crystal , magnetic , and electronic structures of puga@xmath0 with density functional theory ( dft ) reveals the entanglement of the three types of structure . magnetic structure affects the energy more strongly than crystal structure . for dft to correctly order the crystal structures in agreement with experiment requires special treatment of the electronic correlation in the @xmath1 states , exemplified here by the gga+u approach . the upper and lower hubbard bands change with increasing @xmath2 in very dissimilar ways for the two most different crystal structures . the results suggest the effectiveness of using magnetic structure to simulate correlation effects in the actinides depends on both the magnetic and the crystal structure .

a benzenoid system is determined with all the hexagons lying inside cycle @xmath0 of the hexagonal lattice . they represent molecules called benzenoid hydrocarbons . these graphs are also known as the hexagonal systems and form one of the most extensively studied family of chemical graphs . for fundamental properties of benzenoid systems see @xcite , while some recent results can be found in @xcite . if we embed benzenoid systems on a surface of a cylinder and join some edges we obtain structures called open - ended single - walled carbon nanotubes also called tubulenes . carbon nanotubes are carbon compounds with a cylindrical structure and they were first observed in 1991 @xcite . if we close a carbon nanotube with two caps composed of pentagons and hexagons , we obtain a fullerene . more exactly , a fullerene is a molecule of carbon in the form of a hollow sphere , ellipsoid , tube , or many other shapes . the first fullerene molecule was discovered @xmath1 years ago . in graph theory , a fullerene is a @xmath2-regular plane graph consisting only of pentagonal and hexagonal faces . the overview of some results on fullerene graphs can be found in @xcite . papers @xcite present a sample of recent investigations . the concept of the resonance graph appears quite naturally in the study of perfect matchings of molecular graphs of hydrocabons that represent kekul ' e structures of corresponding hydrocarbon molecules . therefore , it is not surprising that it has been independently introduced in the chemical @xcite as well as in the mathematical literature @xcite ( under the name @xmath3-transformation graph ) and then later rediscovered in @xcite . the equivalence of the zhang - zhang polynomial of the molecular graph and the cube polynomial of its resonance graph was established for benzenoid systems @xcite , tubulenes @xcite , and fullerenes @xcite . the zhang - zhang polynomial counts clar covers with given number of hexagons , i.e.conjugated 6-cycles . for some recent research on the zhang - zhang polynomial see @xcite the resonance energy is a theoretical quantity which is used for predicting the aromatic stability of conjugated systems . in the conjugated - circuit model , the resonance energy is determined with conjugated cycles of different lengths ( see @xcite ) , not only with 6-cycles . among them , only 6-cycles and 10-cycles have uniquely determined structure . therefore , we introduce the concept of the generalized zhang - zhang polynomial , which considers both of them . in this paper we prove the equivalence of the generalized zhang - zhang polynomial of a molecular graph and the generalized cube polynomial of the corresponding resonance graph . a _ benzenoid system _ consists of a cycle @xmath0 of the infinite hexagonal lattice together with all hexagons inside @xmath0 . a _ benzenoid graph _ is the underlying graph of a benzenoid system . next we formally define open - ended carbon nanotubes , also called _ tubulenes _ @xcite . choose any lattice point in the hexagonal lattice as the origin @xmath4 . let @xmath5 and @xmath6 be the two basic lattice vectors . choose a vector @xmath7 such that @xmath8 and @xmath9 are two integers and @xmath10 , @xmath11 . draw two straight lines @xmath12 and @xmath13 passing through @xmath4 and @xmath14 perpendicular to @xmath15 , respectively . by rolling up the hexagonal strip between @xmath12 and @xmath13 and gluing @xmath12 and @xmath13 such that @xmath14 and @xmath4 superimpose , we can obtain a hexagonal tessellation @xmath16 of the cylinder . @xmath12 and @xmath13 indicate the direction of the axis of the cylinder . using the terminology of graph theory , a _ tubulene _ @xmath17 is defined to be the finite graph induced by all the hexagons of @xmath18 that lie between @xmath19 and @xmath20 , where @xmath19 and @xmath20 are two vertex - disjoint cycles of @xmath18 encircling the axis of the cylinder . the vector @xmath21 is called the _ chiral vector _ of @xmath17 and the cycles @xmath19 and @xmath20 are the two open - ends of @xmath17 . -type tubulene . ] for any tubulene @xmath17 , if its chiral vector is @xmath22 , @xmath17 will be called an @xmath23-type tubulene , see figure [ fig - nano ] . a _ fullerene _ @xmath24 is a @xmath2-connected 3-regular plane graph such that every face is bounded by either a pentagon or a hexagon . by euler s formula , it follows that the number of pentagonal faces of a fullerene is exactly @xmath25 . a _ 1-factor _ of a graph @xmath24 is a spanning subgraph of @xmath24 such that every vertex has degree one . the edge set of a 1-factor is called a _ perfect matching _ of @xmath24 , which is a set of independent edges covering all vertices of @xmath24 . in chemical literature , perfect matchings are known as kekul structures ( see @xcite for more details ) . petersen s theorem states that every bridgeless @xmath2-regular graph always has a perfect matching @xcite . therefore , a fullerene always has at least one perfect matching . a hexagon of @xmath24 with exactly 3 edges in a perfect matching @xmath26 of @xmath24 is called a _ sextet_. let @xmath24 be a benzenoid system , a tubulene or a fullerene with a perfect matching . the _ resonance graph _ @xmath27 is the graph whose vertices are the perfect matchings of @xmath24 , and two perfect matchings are adjacent whenever their symmetric difference forms a hexagon of @xmath24 . the _ hypercube _ @xmath28 of dimension @xmath8 is defined in the following way : all vertices of @xmath28 are presented as @xmath8-tuples @xmath29 where @xmath30 for each @xmath31 and two vertices of @xmath28 are adjacent if the corresponding @xmath8-tuples differ in precisely one coordinate . a _ convex subgraph _ @xmath32 of a graph @xmath24 is a subgraph of @xmath24 such that every shortest path between two vertices of @xmath32 is contained in @xmath32 . let @xmath24 be a benzenoid system , a tubulene or a fullerene . clar cover _ is a spanning subgraph of @xmath24 such that every component of it is either @xmath33 or @xmath34 . zhang - zhang polynomial _ of @xmath24 is defined in the following way : @xmath35 where @xmath36 is the number of clar covers of @xmath24 with @xmath37 hexagons . a _ generalized clar cover _ is a spanning subgraph of @xmath24 such that every component of it is either @xmath33 , @xmath38 or @xmath34 . see figure [ clar_cover ] for an example . . ] the _ generalized zhang - zhang polynomial _ of @xmath24 is defined in the following way : @xmath39 where @xmath40 is the number of generalized clar covers of @xmath24 with @xmath37 cycles @xmath41 and @xmath42 cycles @xmath38 . note that for a graph @xmath24 number @xmath43 equals the number of vertices of @xmath27 and @xmath44 equals the number of edges of @xmath27 . furthermore , number @xmath45 represents the number of clar covers with @xmath37 hexagons . let @xmath32 be a graph . the _ cube polynomial _ of @xmath32 is defined as follows : @xmath46 where @xmath47 denotes the number of induced subgraphs of @xmath32 that are isomorphic to the @xmath37-dimensional hypercube . let @xmath24 be a graph and @xmath48 an integer . then by @xmath49 we denote the cartesian product of @xmath50 copies of @xmath24 , i.e. @xmath51 . also , @xmath52 . furthermore , for any @xmath53 we define @xmath54 , where @xmath55 and @xmath56 are paths on 2 and 3 vertices , respectively . obviously , @xmath57 is the @xmath37-dimensional hypercube . moreover , if @xmath58 , vertices of the graph @xmath59 can be presented as @xmath60-tuples @xmath61 , where @xmath62 if @xmath63 and @xmath64 if @xmath65 . in such representation two vertices @xmath61 and @xmath66 are adjacent if and only if there is @xmath67 such that @xmath68 and @xmath69 for any @xmath70 . let @xmath32 be a graph . the _ generalized cube polynomial _ of @xmath32 is defined as follows : @xmath71 where @xmath72 denotes the number of induced convex subgraphs of @xmath32 that are isomorphic to the graph @xmath59 . in this section we prove that the generalized zhang - zhang polynomial of every benzenoid system , tubulene or fullerene equals the generalized cube polynomial of its resonance graph . let @xmath24 be a benzenoid system , a tubulene or a fullerene with a perfect matching . then the generalized zhang - zhang polynomial of @xmath24 equals the generalized cube polynomial of its resonance graph @xmath27 , i.e. @xmath73 [ main ] let @xmath37 and @xmath42 be nonnegative integers . for a graph @xmath24 we denote by @xmath74 the set of all generalized clar covers of @xmath24 with exactly @xmath37 cycles @xmath33 and @xmath42 cycles @xmath38 . on the other hand , consider a graph @xmath32 ; the set of induced convex subgraphs of @xmath32 that are isomorphic to a graph @xmath59 is denoted by @xmath75 . let us define a mapping @xmath76 from the set of generalized clar covers of @xmath24 with @xmath37 cycles @xmath33 and @xmath42 cycles @xmath38 to the set of induced convex subgraphs of the resonance graph @xmath27 isomorphic to the graph @xmath59 @xmath77 in the following way : for a generalized clar cover @xmath78 consider all perfect matchings @xmath79 , @xmath80 , @xmath81 , @xmath82 of @xmath24 such that : * if cycle @xmath33 in @xmath0 , then @xmath83 for all @xmath84 , * if cycle @xmath38 of @xmath0 is composed of two hexagons , @xmath85 and @xmath86 , then @xmath87 for all @xmath84 , * each isolated edge of @xmath0 is in @xmath88 for all @xmath84 . finally , assign @xmath89 as an induced subgraph of @xmath27 with vertices @xmath79 , @xmath80,@xmath81 , @xmath82 . note first that in case when @xmath90 and @xmath91 generalized clar covers are the perfect matchings of @xmath24 and if @xmath0 is such a generalized clar cover then @xmath89 is a vertex of the resonance graph and the mapping is obviously bijective . so from now on at least one of @xmath37 and @xmath42 will be positive . we first show that @xmath76 is a well - defined function . for each generalized clar cover @xmath78 it follows that @xmath92 . [ lema1 ] first we show that @xmath89 is isomorphic to the graph @xmath59 . let @xmath19 , @xmath20,@xmath81,@xmath93 be the hexagons of @xmath0 and let @xmath94 be cycles @xmath38 that are in @xmath0 . obviously , every hexagon of @xmath0 has two possible perfect matchings . let us call these possibility 0 " and possibility 1 " . moreover , for every cycle @xmath38 in @xmath0 we obtain tree possible perfect matchings of graph @xmath95 , which will be denoted as possibility 0 " , possibility 1 " , and possibility 2 " . also , if cycle @xmath38 is composed of hexagons @xmath85 and @xmath86 , possibility 1 " denotes the perfect matching containing the common edge of @xmath85 and @xmath86 . for any vertex @xmath26 of @xmath89 let @xmath96 , where @xmath97 if on @xmath98 possibility @xmath50 is selected it is obvious that @xmath99 is a bijection . let @xmath100 for @xmath101 . if @xmath26 and @xmath102 are adjacent in @xmath89 , then @xmath103 for a hexagon @xmath104 of some @xmath105 , where @xmath106 . therefore , @xmath107 for each @xmath70 and @xmath108 , which implies that @xmath109 and @xmath110 are adjacent in @xmath59 . conversely , if @xmath111 and @xmath112 are adjacent in @xmath59 , it follows that @xmath26 and @xmath102 are adjacent in @xmath89 . hence @xmath113 is an isomorphism between @xmath89 and @xmath59 . to complete the proof we have to show that @xmath89 is a convex subgraph of @xmath27 therefore , let @xmath26 and @xmath102 be two vertices of @xmath89 . obviously , perfect matchings @xmath26 and @xmath102 can differ only in the edges of hexagons that belong to cycles of @xmath0 . therefore , any shortest path between @xmath26 and @xmath102 in @xmath27 contains perfect matchings that are vertices of @xmath89 . it follows that @xmath89 is convex in @xmath27 . the following lemma shows that @xmath76 is injective . the mapping @xmath114 is injective for any integers @xmath115 . [ lema2 ] let @xmath0 and @xmath116 be distinct generalized clar covers in @xmath74 . if @xmath0 and @xmath116 contain the same set of cycles , then the isolated edges of @xmath0 and @xmath116 are distinct . therefore , @xmath89 and @xmath117 are disjoint induced subgraphs of @xmath27 and thus @xmath118 . therefore , suppose that @xmath0 and @xmath116 contain different sets of cycles . without loss of generality we can assume that there is hexagon @xmath104 such that @xmath104 has at least five edges in @xmath0 and @xmath104 has at most three edges in @xmath116 . hence at least one edge @xmath119 of @xmath104 does not belong to @xmath116 . from the definition of the function @xmath76 , @xmath119 is thus unsaturated by those perfect matchings that correspond to the vertices in @xmath117 . however , there obviously exists perfect matching @xmath120 such that @xmath121 . as a result , @xmath122 and @xmath123 . the next lemma was proved in @xcite for benzenoid systems . the same proof can be applied in the case of tubulenes or fullerenes . @xcite [ stiri ] let @xmath24 be a benzenoid systems , a tubulene , or a fullerene with a perfect matching . if the resonance graph @xmath27 contains a 4-cycle @xmath124 , then @xmath125 and @xmath126 are disjoint hexagons . also , we have @xmath127 and @xmath128 . the following lemma shows that @xmath76 is surjective . the mapping @xmath114 is surjective for any integers @xmath115 . [ lema3 ] let @xmath115 be integers and @xmath129 . then the vertices of @xmath130 can be identified with strings @xmath61 , where @xmath62 if @xmath63 or @xmath64 if @xmath65 , so that two vertices of @xmath130 are adjacent in @xmath130 if and only if their strings @xmath113 and @xmath131 differ in precisely one position @xmath50 , such that @xmath132 . let @xmath133 , @xmath134 , @xmath135 , , @xmath136 be the vertices of @xmath130 . it is obvious that @xmath137 is an edge of @xmath27 for every @xmath138 . by definition of @xmath27 , the symmetric difference of perfect matchings @xmath26 and @xmath139 is the edge set of a hexagon of @xmath24 . we denote this hexagon by @xmath140 and we obtain the set of hexagons @xmath141 of graph @xmath24 . if two of these hexagons were the same , for example if @xmath142 for @xmath143 and @xmath144 , then @xmath145 - a contradiction . hence , we have the set of @xmath146 distinct hexagons . in the next claim we show that these hexagons are pairwise disjoint . the hexagons @xmath140 , @xmath106 , are pairwise disjoint . let @xmath143 and @xmath144 . let @xmath147 be a vertex of @xmath130 having exactly two @xmath148 s ( and these are in the @xmath50th and @xmath149th position ) and @xmath150 at every other position . obviously , @xmath151 is a 4-cycle and therefore , by lemma [ stiri ] , @xmath140 and @xmath152 are disjoint hexagons . next , we consider the vertices @xmath153 , @xmath154 , such that @xmath155 has @xmath156 in the @xmath50th position and @xmath150 in every other position . obviously , @xmath157 is the edge of @xmath27 for any @xmath154 . let @xmath158 be the hexagon of @xmath24 corresponding to the edge @xmath157 . if @xmath154 , the hexagon @xmath158 has exactly one common edge with @xmath140 . it is easy to see that @xmath159 ( otherwise @xmath160 ) . therefore , suppose that @xmath140 and @xmath158 are disjoint . since they are both sextets in the perfect matching @xmath139 , there is a vertex @xmath161 of @xmath27 , @xmath162 , which is adjacent to @xmath26 and @xmath155 . if @xmath163 , the string of @xmath161 must differ from @xmath26 for @xmath148 in exactly one position and must differ from @xmath155 for @xmath148 in exactly one position , which means @xmath164 - a contradiction therefore , @xmath161 is not in @xmath130 . since @xmath165 is a shortest path between @xmath26 and @xmath155 , @xmath130 is not convex subgraph of @xmath27 , which is a contradiction . hence , @xmath140 and @xmath158 have exactly one common edge . let @xmath154 . then the hexagon @xmath158 is disjoint with every @xmath152 , @xmath166 . let @xmath161 be a vertex in @xmath130 with @xmath156 in the @xmath50th position , @xmath148 in the @xmath149th position and @xmath150 in every other position . furthermore , let @xmath167 be a vertex in @xmath130 with @xmath148 in the @xmath50th position , @xmath148 in the @xmath149th position and @xmath150 in every other position . obviously , @xmath168 ( otherwise @xmath169 , which is a contradiction ) . since @xmath170 is a @xmath171-cycle such that @xmath158 corresponds to the edge @xmath157 and @xmath152 corresponds to the edge @xmath172 , it follows from lemma [ stiri ] that hexagons @xmath158 and @xmath152 are disjoint . let @xmath154 . then the hexagon @xmath158 is disjoint with every @xmath173 , @xmath174 . define the following vertices in @xmath130 : * @xmath175 has @xmath148 in the @xmath50th position , @xmath148 in the @xmath149th position and @xmath150 in every other position , * @xmath176 has @xmath148 in the @xmath50th position , @xmath156 in the @xmath149th position and @xmath150 in every other position , * @xmath177 has @xmath156 in the @xmath50th position , @xmath148 in the @xmath149th position and @xmath150 in every other position , * @xmath178 has @xmath156 in the @xmath50th position , @xmath156 in the @xmath149th position and @xmath150 in every other position . using lemma [ stiri ] we can easily see that hexagon @xmath173 corresponds to the edge @xmath179 and hexagon @xmath158 corresponds to the edge @xmath180 . since @xmath181 is a @xmath171-cycle in the resonance graph , lemma [ stiri ] again implies that @xmath158 and @xmath173 are disjoint and the proof is complete . let @xmath182 , @xmath65 be a @xmath183-cycle formed by @xmath140 and @xmath158 . moreover , let @xmath0 be a spanning subgraph of @xmath24 such that @xmath184 . therefore , @xmath0 is a generalized clar cover with @xmath37 hexagons and @xmath42 10-cycles . it is obvious that every edge in @xmath130 corresponds to some hexagon @xmath140 , @xmath185 or @xmath158 , @xmath65 . therefore , @xmath186 . since both @xmath130 and @xmath89 are induced subgraphs of the resonance graph , it follows @xmath187 . we have proved that @xmath76 is bijective function and hence , @xmath188 . therefore , the proof is complete . in this final section we give an example of a benzenoid system @xmath24 and calculate the generalized zhang - zhang polynomial of @xmath24 , i.e. the generalized cube polynomial of the resonance graph of @xmath24 . see figures [ ben_sistem ] and [ resonancni ] . . ] . ] the polynomials are @xmath189 @xmath190 for example , the coefficient in front of @xmath191 is 3 , since there are 3 generalized clar covers in @xmath24 with two @xmath33 and one @xmath38 . on the other hand , this coefficient counts the number of induced convex subgraphs of @xmath27 isomorphic to the graph @xmath192 . supported in part by the ministry of science of slovenia under grant @xmath193 . chou , j .- s . kang , h. a. witek , _ closed - form formulas for the zhang - zhang polynomials of benzenoid structures : prolate rectangles and their generalizations_. discrete appl . math . * 198 * ( 2016 ) , 101108 . d. plavi ' c , s. nikoli ' c , n. trinajsti ' c , _ the conjugated - circuit model : application to nonalternant hydrocarbons and a comparison with some other theoretical models of aromaticity_. j. mol . struct . ( theochem ) * 277 * ( 1992 ) , 213237 .

in this paper we study the resonance graphs of benzenoid systems , tubulenes , and fullerenes . the resonance graph reflects the interactions between the kekul ' e structures of a molecule . the equivalence of the zhang - zhang polynomial ( which counts clar covers ) of the molecular graph and the cube polynomial ( which counts hypercubes ) of its resonance graph is known for all three families of molecular graphs . instead of considering only interactions between 6-cycles ( clar covers ) , we also consider 10-cycles , which contribute to the resonance energy of a molecule as well . therefore , we generalize the concepts of the zhang - zhang polynomial and the cube polynomial and prove the equality of these two polynomials .

let @xmath1 be an oval , i.e. , a planar , closed , regular , simple , oriented counterclockwise , @xmath2 curve , @xmath3 , parameterized by @xmath4 and such that its radius of curvature @xmath5 . the billiard problem on @xmath1 consists in the free motion of a point particle on the planar region enclosed by @xmath1 , being reflected elastically at the impacts on the boundary . the trajectories will be polygonals on this planar region . since the motion is free inside the region , it is completely determined by the points of impacts at @xmath1 , and the direction of motion immediately after each reflection , defined by the angle @xmath6 between it and the oriented tangent to the boundary at the reflection point . therefor , to each oval @xmath1 is associated a billiard map @xmath7 from the cylinder @xmath8 into itself which to each initial condition @xmath9 associates the next impact and direction : @xmath10 . the map @xmath7 has some very well known properties ( see for instance @xcite , @xcite , @xcite , @xcite , @xcite ) : it is a @xmath11-diffeomorphism , preserving the measure @xmath12 , where @xmath13 is the arclength parameter for @xmath1 . it is reversible with respect to the reversing symmetry @xmath14 , which satisfies @xmath15 and @xmath16 . it is a monotone twist map . so , the billiard map defines a discrete reversible conservative bidimensional dynamical system in the cylinder @xmath8 . this map has no fixed points but birkhoff @xcite proved that it has periodic orbits of any period greater or equal to 2 . in this paper we show that , under certain generic conditions , billiards on ovals have only a finite number of periodic orbits , for each period @xmath0 , all non - degenerate and at least one of them is hyperbolic . moreover , the invariant curves of two hyperbolic points are transversal . the generic existence , in the analytic case , of at least one nondegenerate periodic orbit for each period , was proved by kozlov and treschev @xcite . our results generalize theirs in the sense that we work with curves at least @xmath17 , prove the finiteness and study the heteroclinic connections . once we have a hyperbolic periodic orbit , which unstable manifold project to @xmath18 , we can construct the instability region associated to it from the closure of the invariant manifold . the dynamical object we obtain is the same as the one constructed by le calvez in @xcite and thus shares the same dynamical properties . billiard maps are a special kind of diffeomorphisms and @xmath19 perturbations of the map may produce diffeomorphisms which are not billiards . in order to assure that the perturbed diffeomorphism remains a billiard , we have to perturb the boundary curve instead of the map itself . as planar rigid motions and homotheties do not change the geometrical features of the curve , they do not change the dynamical characteristics of the associated billiard map and so it is useful to work on the set of curves modulo this equivalence relation . let @xmath20 be the set of equivalence classes of @xmath17 ovals . given a representative @xmath1 of an equivalence class @xmath21\in{\cal c}$ ] , the normal bundle @xmath22 is @xmath19 . for @xmath23 , a tubular neighbourhood of @xmath1 is given by @xmath24 . @xmath25\in{\cal c}$ ] is @xmath26-close to @xmath21\in{\cal c}$ ] if there exist representatives @xmath1 in @xmath21 $ ] and @xmath27 in @xmath25 $ ] such that the image of @xmath27 is in @xmath28 , and its canonical projection onto the image of @xmath1 is a diffeomorphism . as a consequence , @xmath27 can be written as @xmath29 , with @xmath30 at least @xmath17 and periodic . @xmath25\in{\cal c}$ ] is @xmath26-@xmath17-close to @xmath21\in{\cal c}$ ] if @xmath25 $ ] is @xmath26-close to @xmath21 $ ] and @xmath31 . [ prop : baire ] @xmath20 is a baire space . @xmath32 with the @xmath17-topology is a baire space . by sard s theorem , the subset @xmath33 of immersions is open and dense in @xmath32 , and so also a baire space . for @xmath34 let @xmath35 be its radius of curvature on each @xmath36 and let @xmath37 be the total curvature divided by @xmath38 . let @xmath39 and @xmath40 such that @xmath41 and @xmath42 . note that @xmath43 is exactly the set of closed , regular , simple , oriented , @xmath17-curves with positive radius of curvature . it is clear that @xmath44 is closed and @xmath43 is open and dense in @xmath44 , so @xmath43 , the set of ovals , is baire with the @xmath17-topology . cutting by the equivalence relation and taking the induced topology , we have that @xmath20 is a baire space . as we have showed in @xcite , by perturbing the classes of curves we get nearby diffeomorphisms in the @xmath19-topology : [ prop : difeo ] if @xmath25\in{\cal c}$ ] is @xmath26-@xmath17-close to @xmath21\in{\cal c}$ ] then there exist representatives @xmath1 in @xmath21 $ ] and @xmath27 in @xmath25 $ ] such that the associated billiard maps @xmath45 and @xmath46 are @xmath19-close . to lighten the notation , in what follows we will denote both the curve @xmath1 and the equivalence class @xmath21 $ ] by @xmath1 , unless where confusion may happen let @xmath47 . since the radius of curvature @xmath35 is strictly positive , @xmath1 can be reparameterized by @xmath48 , the positive oriented angle between the tangent vector @xmath49 and a fixed direction ( say , the x - axis ) . let @xmath50 be the associated billiard map . a point @xmath51 is @xmath52-periodic if @xmath52 is the smallest positive integer such that @xmath53 . such a point defines an @xmath52-periodic orbit @xmath54 an @xmath52-periodic orbit is nondegenerate if each of its points @xmath55 is a nondegenerate fixed point of @xmath56 . otherwise it is degenerate . let @xmath57 the set of ovals @xmath58 such that for each @xmath59 divisor of @xmath0 , the associated billiard map @xmath7 has only nondegenerate @xmath52-periodic orbits . if @xmath60 then for each @xmath59 divisor of @xmath0 , @xmath7 has only a finite number of @xmath52-periodic orbits . let @xmath61 be an @xmath52-periodic orbit of @xmath45 . then @xmath62 , @xmath63 , ... , @xmath64 are the vertices of a polygon inscribed in @xmath1 . let @xmath65 be its internal angles and @xmath66 if @xmath67 or @xmath68 if @xmath69 . as @xmath70 , @xmath71 and then @xmath72 . so there is at least one @xmath73 such that @xmath74 . this means that if @xmath60 then each @xmath52-periodic orbit of @xmath7 , with @xmath59 divisor of @xmath0 , has at least one point in the compact cylinder @xmath75 $ ] . since all the points of @xmath76 are nondegenerate , there is only a finite number of them on the compact cylinder and so , only a finite number of @xmath52-periodic orbits . for diffeomorphisms @xmath19 on @xmath77 , with the @xmath19-topology , having only a finite number of nondegenerate fixed points on a compact subset is an open property . taking the restriction to billiard maps , we get @xmath57 is open on @xmath78 . [ prop : open ] by birkhoff s theorem , any @xmath79 has periodic orbits of any period . so @xmath1 will be outside @xmath57 if it has ( finitely or infinitely many ) degenerate @xmath52-periodic orbits for @xmath59 , divisor of @xmath0 . the following lemma provides the basic tool for proving the density of @xmath57 . [ lem : perturb ] suppose that @xmath80 is a degenerate @xmath52-periodic orbit for the billiard map @xmath7 associated to a curve @xmath79 . then there is a curve @xmath81 , @xmath17-close to @xmath1 , such that @xmath82 is a nondegenerate @xmath52-periodic orbit for the associated billiard map @xmath83 . let @xmath84 and @xmath85 . the jacobian matrix of @xmath7 at @xmath86 is given by : @xmath87 and , for any point @xmath88 of @xmath89 , det@xmath90 . it follows that @xmath89 is degenerate if @xmath91 , for any point of the orbit . we have that @xmath92 where @xmath93 let us isolate the terms that depend on , for instance , @xmath94 . only @xmath95 and @xmath96 have entries with @xmath94 and @xmath97 then @xmath98\\ & = & \frac{1}{x_1}\mbox{tr}\left(a_{n-1}a_{n-2} ... a_2b_1\right)+ \mbox{tr}\left(a_{n-1}a_{n-2} ... a_2c_{1,0}\right)\\ & = & \frac{b_1}{x_1}+c_1\end{aligned}\ ] ] where neither @xmath99 nor @xmath100 depend on @xmath94 . if @xmath101 , let @xmath33 be an interval such that @xmath102 and @xmath103 for @xmath104 . let @xmath105 , where @xmath30 is a @xmath17 periodic function satisfying @xmath106 if @xmath107 , @xmath108 , @xmath109 and @xmath110 small enough to guarantee that @xmath111 , the radius of curvature of @xmath112 , is strictly positive . the perturbed curve @xmath112 and the original one @xmath1 coincide , except on a neighbourhood of @xmath113 and , at @xmath114 , they have a contact of order one . so the polygonal trajectory that corresponds , on the billiard table , to the periodic orbit is unchanged and @xmath82 is also a @xmath52-periodic orbit for @xmath83 . moreover @xmath115 , the radius of curvature @xmath116 if @xmath104 and @xmath117 . then @xmath118 and we can choose @xmath30 as small as we want such that @xmath119 and @xmath82 is a nondegenerate @xmath52-periodic orbit for @xmath83 . if @xmath120 then @xmath121 with @xmath122 where @xmath123 has the same form of @xmath124 , replacing @xmath94 by @xmath125 and @xmath126 by @xmath127 . then @xmath128 where neither @xmath129 nor @xmath130 depend on @xmath94 and @xmath131 . if @xmath132 then we can make the normal perturbation on a neighbourhood of @xmath133 as above . if @xmath134 we continue the process until finding a @xmath135 and then making the normal perturbation at @xmath136 or to end up with all @xmath137 s @xmath138 , for @xmath139 in which case , as @xmath140 , we will have @xmath141 as @xmath142 is the perimeter of the polygonal trajectory and then different from 0 , we can perform the normal perturbation on a neighbourhood of @xmath143 as above concluding the proof of the lemma . @xmath57 is dense on @xmath78 . [ prop : dense ] given @xmath144 , let @xmath145 be the set of all fixed points of @xmath146 and @xmath147 be the projection on the first coordinate . as @xmath144 , there is a @xmath148 such that @xmath149 is a degenerate @xmath52-periodic orbit of @xmath7 for @xmath59 , divisor of @xmath0 . by lemma [ lem : perturb ] we can find a curve @xmath150 , close to @xmath1 , such that @xmath149 is a non - degenerate @xmath52-periodic orbit of @xmath151 and so there are intervals @xmath152 such that @xmath153 is the unique point of @xmath154 in @xmath155 . as @xmath18 is compact , we can construct after a finite number of steps , a curve @xmath156 , as close as we want to @xmath1 , such that @xmath157 and intervals @xmath158 , with @xmath159 , and such that each @xmath160 is the unique point of @xmath161 in @xmath162 and so each fiber @xmath163 contains at least one nondegenerate fixed point and maybe other degenerate or nondegenerate fixed points . if @xmath164 has only non - degenerate fixed points , the problem is solved . otherwise , let @xmath165 and @xmath166 be two different fixed points of @xmath164 , with @xmath165 non - degenerate and @xmath167 $ ] degenerate . if @xmath168 then they are both non - degenerate , so suppose that they do not belong to the same orbit . let @xmath52 and @xmath169 be the periods of the orbits , respectively . then @xmath170 and @xmath171 . as in the proof of lemma [ lem : perturb ] , there is an @xmath172 such that @xmath173 with @xmath174 and then a normal perturbation in the interval containing @xmath175 will produce a curve @xmath27 , @xmath17-close to @xmath112 , such that both of @xmath165 and @xmath166 are nondegenerate fixed points of @xmath176 . to finish the proof of the proposition , we remark that @xmath177 $ ] is a closed interval . so , after a finite number of perturbations we can construct a curve in @xmath57 , arbitrarily close to the given @xmath178 . taking the intersection of the open and dense sets @xmath57s and remembering that @xmath20 is a baire space it follows that generically , for billiards on ovals , there is only a finite number of periodic orbits , for each period @xmath0 , and they are all non - degenerate . let @xmath1 be a@xmath2 oval , @xmath7 the associated billiard map and @xmath179 a lift of @xmath7 . if @xmath180 is an @xmath52-periodic orbit of @xmath7 then there is an @xmath169 , @xmath181 such that @xmath182 . we will say that @xmath183 is an @xmath184-periodic orbit . given @xmath52 and @xmath169 , @xmath181 , let @xmath185 @xmath180 is a non - degenerate @xmath184-periodic orbit if and only if @xmath186 , @xmath187 , @xmath188 is a non - degenerate singularity of @xmath189 . by the mackay - meiss criterion @xcite , non - degenerate minima of @xmath189 correspond to hyperbolic orbits and non - degenerate maxima to elliptic or reverse hyperbolic ones . if @xmath60 then all the @xmath52-periodic orbits , @xmath59 divisor of @xmath0 , are nondegenerate singularities for the appropriate @xmath189 and then are hyperbolic or elliptic . actually , for each @xmath52 and each @xmath169 such that gcd@xmath190 , the first step in the proof of birkhoff s theorem is the existence of a global minimum of @xmath189 . so there is at least one hyperbolic @xmath52-periodic for @xmath7 . the stability of the other periodic orbits is strongly related to the geometry of the oval @xmath1 , and they can even be all hyperbolic , like the 2-periodic orbits in the examples given in @xcite or @xcite . let @xmath191 be one @xmath52-periodic hyperbolic orbit . associated to each point @xmath55 there are two @xmath11 invariant curves @xmath192 and @xmath193 called , respectively , unstable and stable curves of @xmath55 . a point @xmath194 is hetero(homo)clinic if @xmath195 for @xmath196 ( @xmath197 ) . for the hyperbolic periodic orbits corresponding to global minima of @xmath189 , bangert s results @xcite assures the existence of hetero and homoclinic points . other hyperbolic periodic orbits may have only homoclinic points , as the 2-periodic orbit plotted in the figure bellow . we do nt know if , for a generic oval @xmath1 there exists a hyperbolic periodic orbit without hetero or homoclinic points . in particular , @xmath198 is an open property for @xmath19-diffeomorphisms , and so will be open for billiards on ovals . an heteroclinic or homoclinic point @xmath194 is called transversal ( tangent ) if the invariant stable and unstable curves meet transversally ( tangentially ) at @xmath194 . transversal intersection of stable and unstable curves is also an open property for @xmath19-diffeomorphisms , and so it will be open for billiards on ovals . in the next lemma we will prove that any billiard with a tangent hetero(homo)clinic point can be approached by billiards with a transversal one . we use the techniques introduced in @xcite and used in @xcite . [ lem : levallois ] let @xmath60 and @xmath199 be an @xmath52-periodic hyperbolic orbit such that a stable and an unstable curve , say @xmath200 and @xmath201 intersect tangentially at @xmath194 . then @xmath1 can be approximated by curves in @xmath57 such that @xmath194 is a transverse heteroclinic ( or homoclinic ) point of the associated billiard map . since @xmath202 there are sequences @xmath203 and @xmath204 as @xmath205 . then , there exists an interval @xmath206 such that @xmath207 , @xmath208 and @xmath209 . as @xmath7 is a @xmath11-diffeomorphism , each @xmath210 is also a heteroclinic tangent point for every @xmath211 . moreover , as @xmath7 has the twist property , it is not possible to the tangency at every @xmath212 to be vertical . so we can suppose that the stable and unstable curves are local graphs over the @xmath213-axis at a neighbourhood of , for instance , @xmath194 . so there is an interval @xmath214 containing @xmath215 such that @xmath200 and @xmath201 are given locally by the graphs of @xmath216 and @xmath217 , with @xmath218 and @xmath219 . those graphs define two pencils of rays that focuses forward and backward at the distances ( see , for instance @xcite ) @xmath220 where @xmath221 is the radius of curvature of @xmath1 at @xmath215 . let @xmath222 and @xmath112 be a normal perturbation @xmath223 where @xmath106 if @xmath107 , @xmath224 , @xmath225 and @xmath226 is sufficiently small in order that @xmath227 and @xmath7 and @xmath83 are @xmath19-close . the two curves @xmath1 and @xmath112 have a contact of order 1 at @xmath215 and the radius of curvature of @xmath112 at this contact point is @xmath228 . as @xmath229 and @xmath1 differs only on @xmath230 , @xmath231 is also a @xmath52-periodic hyperbolic orbit for @xmath83 . moreover , every trajectory not hitting @xmath232 is the same both for @xmath1 and for @xmath112 . let @xmath233 and @xmath234 be the stable and unstable curves of @xmath235 and @xmath55 under @xmath83 . we can choose @xmath226 sufficiently small such that both are also local graphs over @xmath33 given by @xmath236 and @xmath237 , where @xmath238 is the angular parameter of @xmath229 with @xmath239 . the pencil of rays defined by @xmath240 will focuses backward at the distance @xmath241 and the pencil @xmath242 will focuses forward at the distance @xmath243 with @xmath244 as the contact of @xmath1 and @xmath112 on @xmath215 is of order 1 , preserving the point and the tangent , the trajectory of @xmath194 is also the same for both billiards . then @xmath194 is a heteroclinic point for @xmath83 and @xmath245 . as the curve is unchanged outside @xmath33 , the beam of trajectories given by @xmath240 remains the same until it hits @xmath246 , implying that @xmath247 and then @xmath248 applying the same construction for the stable curves , with @xmath249 and @xmath250 gives @xmath251 and @xmath252 implies that the invariant curves for @xmath83 will intersect transversally . the same reasoning also works for the invariant curves of points on different hyperbolic orbits . as we have only a countable number of hyperbolic orbits , each one with a finite number of points , we can conclude that : generically , for billiards on ovals , the invariant curves of two hyperbolic points are transversal . we remark that we do not prove that every homo / heteroclinic orbit is transversal . we do know that generically two invariant stable and unstable curves either do not intersect or have at least one transversal homoclinic orbit , but there can also be tangent orbits . a closed , simple , continuous curve @xmath253 which is not homotopic to a point is called a rotational curve . it is invariant if @xmath254 . the phase - space of the circular billiard , for instance , is foliated by rotational invariant curves . on the other hand , there are billiards on ovals with no rotational invariant curves at all , as showed in @xcite . however , for sufficiently differentiable ovals ( @xcite , @xcite ) , the twist property implies that @xmath7 has rotational invariant curves in any small neighbourhood of the boundaries @xmath255 and @xmath256 of the cylinder @xmath77 . for generic oval billiards the rotation number of any rotational invariant curve is irrational . a rotational invariant curve @xmath257 is a lipschitz graph over @xmath258 @xcite . so @xmath259 , where @xmath6 is continuous and there exists @xmath260 such that @xmath261 , @xmath262 . let @xmath263 and @xmath264 . then @xmath265 is a homeomorphism of the circle and so its degree is @xmath266 . suppose that the rotation number of @xmath257 is rational . if deg@xmath267 then @xmath265 has periodic orbits , all with the same period . if deg@xmath268 then @xmath269 has periodic orbits , all with the same period . as @xmath7 is generic , there is only a finite number of periodic orbits , for each period , and they are nondegenerate . hence there exists an @xmath0 such that @xmath270 is a homeomorphism with a finite number of fixed points , all nondegenerate . let @xmath271 be two consecutive fixed points . then for every @xmath272 , @xmath273 , @xmath274 as @xmath275 and @xmath276 as @xmath277 . let @xmath23 be such that @xmath278 and let @xmath279 and @xmath280 . clearly @xmath281 . let us suppose that @xmath282 is an elliptic periodic point of @xmath7 . then @xmath283 is a rotation of angle @xmath284 . there exists @xmath285 such that @xmath286 . as @xmath7 is at least @xmath19 then there is @xmath287 such that @xmath282 is the unique fixed point of @xmath288 in @xmath289 and @xmath290 . but this is impossible since @xmath291 . then all periodic points in @xmath257 must be hyperbolic . but this is also impossible in the generic case since @xmath257 will be the union of the periodic points and saddle connections . so , the rotation number must be irrational . from the proof of the above proposition we have the following results : [ lem : irrac ] generically , rotational invariant curves can not cross the middle segment @xmath292 . let , as before , @xmath293 be the reversing symmetry . clearly @xmath257 is a rotational invariant curve if and only if @xmath294 is also a rotational invariant curve . suppose that a point @xmath295 . then @xmath296 and , since they are invariant , @xmath297 and @xmath298 is a 2-periodic orbit in @xmath257 , which is impossible . then , for a generic oval billiard , if @xmath299 and @xmath300 ( resp . @xmath301 ) then @xmath302 ( resp . @xmath303 ) for all @xmath48 or , in other words , the orbits on @xmath257 respect the order of the circle @xmath258 ( respec . reverse the order ) . [ lem : rot ] let @xmath257 be a rotational invariant curve and @xmath304 be a hyperbolic periodic orbit , with unstable manifold @xmath305 and stable manifold @xmath306 . then @xmath307 and @xmath308 . suppose that @xmath309 . then @xmath310 and then @xmath311 , since @xmath257 is a continuous invariant curve . but @xmath257 has irrational rotation number and then do not contain any periodic orbit . the argument for @xmath305 is analogous . birkhoff @xcite called the region between two invariant rotational curves , with no other invariant rotational curves inside , an instability region . in this subsection we will characterize the instability regions for a generic oval billiard . a cylinder @xmath312 $ ] is a non - empty closed connected set such that @xmath313 and whose boundaries are two continuous rotational curves @xmath314 and @xmath315 . note that we are not asking neither the cylinder nor the boundary curves to be invariant under @xmath45 . [ teo : inst ] let @xmath7 be generic and @xmath304 be a hyperbolic periodic orbit such that its unstable manifold @xmath305 satisfies @xmath316 . then the smallest cylinder containing @xmath317 is an instability region . before proving this theorem , let us remark that for a generic oval billiard and for each period @xmath52 , there is only a finite number of @xmath52-periodic hyperbolic orbits and at least one of them is a global minimizer of the action @xmath189 , for a suitable @xmath169 . this implies that it has heteroclinic connections @xcite and then @xmath316 . hence the projection hypothesis applies to at least one orbit of each period . let @xmath318 be a hyperbolic periodic point , @xmath304 its orbit and @xmath305 the unstable manifold of @xmath304 , with @xmath316 . first , assume that there is an invariant rotational curve @xmath259 such that @xmath304 lies in the cylinder @xmath319 bounded by @xmath257 and @xmath294 . since , by lemma [ lem : irrac ] , @xmath257 has irrational rotation number we have that @xmath320 and @xmath321 . moreover , @xmath319 is invariant and contains @xmath317 . let then , @xmath322 be the smallest cylinder containing @xmath317 . since @xmath323 is also a cylinder and @xmath324 , @xmath325 , and we have that @xmath326 . so , @xmath327 is also a cylinder containing @xmath317 . and as @xmath328 is the smallest cylinder , we must have @xmath329 . however , @xmath45 is area - preserving , so we conclude that @xmath330 , i.e , @xmath328 is an invariant cylinder . as @xmath45 is a diffeomorphism , @xmath331 and @xmath332 . from birkhoff s theorem it follows that @xmath333 is the union of two rotational invariant curves @xmath100 and @xmath130 . in fact , these two curves @xmath100 and @xmath130 are contained in @xmath317 . to prove this fact , let us suppose that there is a point @xmath334 . then there is an open ball @xmath43 , centered at @xmath335 such that @xmath336 . we can then construct a new cylinder @xmath337 with @xmath338 where @xmath339 outside @xmath340 and @xmath341 otherwise . we have then a new cylinder satisfying @xmath342 and so @xmath328 is not the smallest one , which is a contradiction . now , as @xmath305 is the union of continuous curves , its closure contains the boundaries @xmath100 and @xmath130 and @xmath307 for every @xmath257 rotational and invariant , it follows that there is no invariant curve in @xmath343 . we have proved that @xmath328 is an invariant cylinder , its boundaries are rotational invariant curves and there are no other rotational invariant curves inside it . so , it is the instability region containing @xmath317 . suppose now that there is no invariant rotational curves curves bounding @xmath305 from bellow , i.e , @xmath344 for all rotational invariant curves @xmath259 . we can extend @xmath7 continuously to the circle @xmath255 , observing that @xmath345 . now , any cylinder with boundaries @xmath346 and a rotational invariant curve contains @xmath317 . the smallest one , @xmath328 , will have boundaries @xmath346 and @xmath130 , because if there were a new boundary @xmath100 then it will be a rotational invariant curve with @xmath347 . reasoning as above we conclude that @xmath328 is the instability region that contains @xmath317 . the case @xmath348 for all rotational invariant curves follows from the reversibility property of @xmath7 . finally , if @xmath45 has no rotational invariant curves , the instability region containing @xmath317 is the entire cylinder @xmath349 $ ] . obviously , we can make a similar construction and obtain the same result for the stable manifold @xmath306 . [ prop : ilha ] let @xmath350 be the complement of @xmath317 in the instability region @xmath328 . then @xmath351 , where all the @xmath352 s are disjoint open sets , homeomorphic to discs . dynamically , each @xmath352 returns over itself . @xmath350 is open and is the union of its disjoint open connected components . let @xmath353 be one of them . @xmath353 can not divide @xmath328 into two disjoint sets because @xmath354 contains @xmath305 , which projects over @xmath258 and converges to the two rotational curves of @xmath333 . so @xmath353 is homotopically trivial . as @xmath355 $ ] has finite area and each connected component has positive area . so there can not exist a non countable number of them . clearly , @xmath356 . as @xmath45 is area preserving then , given @xmath352 , there exists a smallest @xmath357 such that @xmath358 . but , as each @xmath352 is a connected component @xmath350 , we must have @xmath359 . in other words , each component @xmath352 of @xmath360 is periodic . the above proposition suggests the definition given @xmath211 , let @xmath357 be the smallest integer such that @xmath359 . we call @xmath352 an island of stability and @xmath357 its period . the invariant set @xmath361 will be called an archipel . remark that , as each island @xmath352 is invariant by @xmath362 and homeomorphic to a ball then the boundary @xmath363 is a closed connected set , with empty interior , contained in @xmath317 and invariant by @xmath362 . an easy consequence of proposition [ prop : ilha ] is every periodic point in an island of stability has a period multiple of the period of the island . more strongly , we have that each island of stability contains a periodic point with the same period as the island . let @xmath352 be an island of stability with period @xmath357 . since the restriction @xmath364 is area preserving and conjugated to a diffeomorphism of the plane which is orientation preserving , it follows from brouwer s translation theorem that @xmath364 has a fixed point . so there exists a point @xmath365 such that @xmath366 and so a periodic orbit @xmath367 with the smallest possible period . remark that , as billiards have no fixed points then @xmath368 and @xmath369 . all points in an island of stability have the same rational rotation type . [ prop : rotnum ] let , as above , @xmath352 be an island of stability with period @xmath357 . following @xcite we observe that if @xmath370 is the lift of @xmath352 to the universal cover and @xmath371 is a lift of @xmath7 , then there exists an integer @xmath169 such that @xmath372 . since @xmath370 is bounded , this implies that that for all @xmath373 if @xmath374 denotes a lift of @xmath375 , then there exists @xmath376 , where @xmath377 is the projetion on the first factor . for the sake of completeness , we conclude by describing some well known dynamical consequences of the area preserving and the twist properties . let @xmath7 be generic , in the sense of the previous sections , and @xmath304 be a hyperbolic periodic orbit such that its unstable manifold @xmath305 satisfies @xmath316 . let @xmath328 be the instability region containing @xmath317 . following the terminology of franks in @xcite we call @xmath378 the instability set . clearly , the instability set @xmath354 does not contain any moser stable periodic orbit , since they have open islands around them . for hyperbolic periodic orbits we have let @xmath379 be a hyperbolic periodic orbit in @xmath354 and let @xmath380 and @xmath381 be its invariant stable and unstable manifolds . then either @xmath382 and @xmath383 or @xmath384 , ie , the boundary of an island . moreover , for @xmath385 or @xmath386 , if @xmath387 and @xmath388 then @xmath389 ; or if @xmath384 then two branches of the invariant curves of @xmath390 and/or @xmath391 are contained in @xmath363 . suppose that there exists @xmath392 such that @xmath393 , @xmath394 . then by lemma 3.1 of @xcite , the branch of @xmath395 which contains @xmath375 is entirely contained in @xmath378 . so if each branch of @xmath395 intersects @xmath378 then @xmath396 and obviously , @xmath397 . let us now suppose that a branch @xmath257 of @xmath395 is not in @xmath378 . then @xmath257 is contained in an island of stability @xmath352 or is outside the region of instability @xmath328 . as the @xmath398-limit of @xmath257 is @xmath399 , either @xmath384 or @xmath400 or @xmath130 , the boundary curves of @xmath328 . but this last case is impossible in the generic case . so @xmath384 . but in this case , it is enough to remark that @xmath363 is an invariant closed connected set and that the only sets with these properties containing @xmath399 are the invariant manifolds to conclude that two branches of the invariant curves of @xmath390 and/or @xmath391 are contained in @xmath363 . if @xmath388 then @xmath401 and , by theorem [ teo : inst ] , @xmath389 . concerning non - periodic orbits and as a consequence of proposition [ prop : rotnum ] we have that any aubry- mather set of irrational type contained in @xmath328 is actually contained in the instability set @xmath378 . and as it was proven by le calvez @xcite the instability set @xmath378 contains the closure of the orbits which @xmath1 and/or @xmath402-limits are @xmath100 and @xmath130 . so far we have obtained a rough description of the dynamics in the instability region , very similar to what hapens for generic area preserving twist maps . it is made of the closure of an hyperbolic orbit and the union of periodic islands . in order to proceed with this description it is necessary to address two basic related themes . the first one has to do with the islands : the existence of a finite number of them , the existence of a lower bound for the period of an island ( e.g. there are examples with no period two islands ) and the existence of connecting orbits between the boundaries of the islands . a very interesting question is if it is possible to have an instability region without any islands . a starting point is the instability region that contains the period two orbits . in this region , the reversibility symmetry of the phase space , together with geometric properties of the boundary of the billiard may allow us to obtain new results about the above questions . * acknowledgments * : the authors thank conselho nacional de desenvolvimento cientfico ( cnpq ) and fundao de amparo a pesquisa de minas gerais , brazilian agencies , for financial support . spc gratefully acknowledges the hospitality of laboratoire emile picard , universit paul sabatier ( toulouse iii ) , where part of this work was done , under the financial support of coordenao de aperfeioamento de pessoal de nvel superios ( capes ) . v.bangert : mather sets for twist maps and geodesics on tori , dynamics reported vol 1 , 1 - 57 , 1988 . g.d.birkhoff : _ dynamical systems_. providence , ri : a. m. s. colloquium publications , 1966 , ( original ed . 1927 , ) g.d.birkhoff : sur quelques courbes fermes remarquables , bull . smf , * 60 * , 1 - 26 , 1932 . ( also in _ collected math . papers of g ; d ; birkhoff _ , vol ii , 444 - 461 . ) m.j.dias carneiro , s.oliffson kamphorst , s.pinto de carvalho : elliptic islands in strictly convex billiards , erg.th.dyn.sys . , * 23/3 * , 799 - 812 , 2003 . v.j.donnay : creating transverse homoclinic connections in planar billiards , jr . math . , * 128/2 * , 2747 - 2753 , 2005 . r.douady : applications du thorme des tores invariants . thse de 3me cycle , univ . paris vii , 1982 . j.franks : rotation numbers and instability sets , bull . ams new series ) , * 40/3 * , 263 - 279 , 2003 . b.halpern : strange billiard tables , trans . * 232 * , 297 - 305 , 1977 . a.katok , b.hasselblat : _ introduction to the modern theory of dynamical systems_. univ . cambridge press , 1997 . v.v.kozlov , d.v.treshchev : _ billiards - a genetic introduction to the dynamics of systems with impacts _ , transl.math.monog . , ams , vol 89 , 1980 . v.v.kozlov : two - link billiard trajectories : extremal properties and stability , j.appl . maths mechs , * 64/6 * , 903 - 907 , 2000 . v.f.lazutkin : the existence of caustics for a billiard problem in a convex domain , math.ussr izvestija , * 7/1 * , 185 - 214 , 1973 . p.le calvez : proprits dynamiques des rgions dinstabilit , ann.sci . , tomme 20 , 443 - 464 , 1987 . p.levallois , m.b.tabanov : sparation des sparatrices du billard elliptique pour une prturbation algbrique et symmtrique de lllipse , c.r.ac.sci.paris -sr i - math , * 316/3 * , 589 - 592 , 1993 . r. mackay , j. meiss : linear stability of periodic orbits in lagrangian systems , phys.lett . a , * 98 * , 92 - 94 , 1983 . r.markarian , s.oliffson kamphorst e s.pinto de carvalho : chaotic properties of the elliptical stadium , comm . phys . * 174 * , 661 - 679 , 1996 . c.g.ragazzo , m.j.dias carneiro , s.addas zanata : introduo dinmica de aplicaes do tipo twist . pub.mat . , impa , 2005 . s.tabachnikov : _ billiards _ , s m f panorama et syntheses 1 , 1995 .

in this paper we show that , under certain generic conditions , billiards on ovals have only a finite number of periodic orbits , for each period @xmath0 , all non - degenerate and at least one of them is hyperbolic . moreover , the invariant curves of two hyperbolic points are transversal . we explore these properties to give some dynamical consequences specially about the dynamics in the instability regions .

a useful formulation of gauge theories , both from the conceptual and methodological point of view , is the one in terms of gauge invariant excitations or string - like objects . the so - called _ p - representation _ @xcite , consisting of a hilbert space of path labeled states , has been used on the lattice to perform analytical hamiltonian calculations . a cluster approximation allowed to provide qualitatively good results for the @xmath0 qed @xcite and the @xmath1 qed @xcite with staggered fermions . a description in terms of paths or strings , besides the general advantage of only involving gauge invariant excitations , is appealing because all the gauge invariant operators have a simple geometrical meaning when realized in the path space . however , the computational method implemented , up to now , on a formally _ infinite _ lattice , has the serious drawback of the explosive proliferation of clusters with the order of the approximation . in order to tackle this difficulty we propose in this paper to explore the previous method implemented now on a _ finite lattice_. as a first test , we choose the simplest lattice gauge theory with dynamical fermions , the schwinger model or ( 1 + 1 ) qed . this massless model can be exactly solved in the continuum and it is rich enough to share relevant features with 4-dimensional qcd as confinement or chiral symmetry breaking with an axial anomaly @xcite . for this reason it has been extensively used as a laboratory to analyze the previous phenomena . the lattice schwinger model also become a popular benchmark to test different techniques to handle dynamical fermions @xcite-@xcite . this article is organized into four sections . in section 2 we show the formulation of the model in the p - representation . the electric and interaction components of the hamiltonian operator are realized in this basis of electromeson " states . in section 3 , first , we describe the finite lattice hamiltonian approach . second , we show the calculation of the ground - state energy , the mass gap and chiral condensate . these results are discussed in the concluding section . the p - representation offers a gauge invariant description of physical states in terms of kets @xmath2 , where @xmath3 labels a set of connected paths @xmath4 with ends @xmath5 and @xmath6 in a lattice of spacing @xmath7 . in the continuum , the connection between the p - representation and the ordinary representation ( configuration " representation ) , in terms of the fermion fields @xmath8 and the gauge fields @xmath9 $ ] , can be performed considering the natural gauge invariant object constructed from them : @xmath10 where @xmath11 $ ] ( @xmath12 denote the links ) . the immediate problem we face is that @xmath13 is not purely an object belonging to the `` configuration '' basis because it includes the canonical conjugate momentum of @xmath8 , @xmath14 . the lattice offers a solution to this problem consisting in the decomposition of the fermionic degrees of freedom . let us consider the hilbert space of kets @xmath15 , where @xmath16 corresponds to the @xmath17 part of the dirac spinor and @xmath18 to the @xmath19 part . those kets are well defined in terms of `` configuration '' variables ( the canonical conjugate momenta of @xmath20 and @xmath21 are @xmath22 and @xmath23 respectively . ) then , the internal product of one of such kets with one of the path dependent representation ( characterized by a lattice path @xmath4 with ends @xmath5 and @xmath6 ) is given by @xmath24 @xmath25 where @xmath26 and @xmath27 denote a component of the spinor @xmath16 and @xmath18 respectively . thus , it seems that the choice of staggered fermions is the natural one in order to build the lattice p - representation . therefore , the lattice paths @xmath4 start in sites @xmath5 of a given parity and end in sites @xmath6 with opposite parity . the one spinor component at each site can be described in terms of the susskind s @xmath28 single grassmann fields @xcite . the path creation operator @xmath29 in the space of kets @xmath30 of a path with ends @xmath5 and @xmath6 is defined as @xmath31 its adjoint operator @xmath32 acts in two possible ways @xcite : annhilating the path @xmath33 or joining two existing paths in @xmath34 one ending at @xmath5 and the other starting at @xmath6 . the schwinger hamiltonian is given by @xmath35 @xmath36 @xmath37 @xmath38 where @xmath5 labels sites , @xmath39 the spatial links pointing along the spatial unit vector @xmath40 , @xmath41 is the electric field operator , the kets @xmath42 are eigenvectors of this operator @xmath43 where the eigenvalue @xmath44 is the number of times that the link @xmath45 appears in the set of paths @xmath3 . the @xmath46 are displacement " operators corresponding to the quantity defined in ( [ eq : phi ] ) for the case of a one - link path i.e. @xmath47 . the realization of both hamiltonian terms in this representation is as follows @xcite : by ( [ eq : e ] ) the action of the electric hamiltonian is given by @xmath48 the interaction term @xmath49 is realized in the loop space as @xmath50 where the factor @xmath51 is 0 or @xmath52 dictated by the algebra of the operators . the different actions of operators @xmath46 over path - states @xmath53 and their corresponding @xmath51 are schematically summarized in fig.1 . -6 mm -5 mm our method of calculation works assuming a lattice of some fixed even number of sites @xmath54 and periodic boundary conditions ( pbc ) . starting with the zero - path state @xmath55 ( infinite coupling vacuum ) , then a collection of new states @xmath56 are generated by applying successively the non - diagonal @xmath57 interaction hamiltonian operator whose action is to add or to eliminate links to to the path @xmath58 as it was described in the previous section up to order @xmath59 . the traslational symmetry can be exploited in order to reduce the dimension of the space tacking only one representative @xmath60 of each class of translationally equivalent paths @xmath61 . the hamiltonian matrix , with all the transitions between the different states @xmath62 , is then built for the scalar and vector sectors and their eigenvalues @xmath63 are numerically evaluated . in order to perform the generation and recognition of diagrams ( the elementary lattice paths ) as well as the computation of transitions between them , we resorted to the prolog language which is very suitable to carry out the symbolic manipulations . the calculations of the ground - state energy , mass gap and chiral order parameter were performed on lattices ranging from size @xmath64 to @xmath65 and at least up to order @xmath66 in each case . our aim is to extrapolate these results to the continuum limit : @xmath67 , @xmath68 ( @xmath69 . ) it is clear from the plots ( figs . 2 to 5 ) that the lattice results show convergence to the expected continuum values . this convergence is , however , non - uniform and for @xmath70 large enough the plots show deviation from the continuum values although the region of assimptotic regime becomes larger when the size is increased . it is patent that for a fixed lattice size @xmath54 the best results for the vacuum energy and the chiral condensate are achieved for order @xmath71 . this appears to be the order at which the finite size effects are minimized . this is not the case with the mass gap which always gets closer to the continumm value when the order increases . _ ground state energy _ in the continuum limit the ground - state energy density is known exactly @xcite : @xmath72 when the order increases @xmath73 tends to a fixed value . for a fixed size @xmath54 the closer value to ( [ eq : e - exact ] ) is given by order @xmath71 . the value for size @xmath65 and order @xmath74 at @xmath75 is @xmath76 , so the discrepancy from the exact value is less than 0.05 % . the approximations converge with considerable rapidity . 2 shows @xmath77 for orders @xmath78 for @xmath70 ranging from 0 to 100 on a lattice of size @xmath79 . -13 mm -10 mm in order to obtain a result in a consistent way we compute the energy for two large values of @xmath70 , for three correlative large orders and for three correlative large sizes . then , for fixed size and order we first extrapolate to @xmath80 assuming the behaviour @xmath81 . second , for fixed size we extrapolate to infinite order assuming exponential dependence . finally we extrapolate to infinite size assuming exponential behaviour . the results are given in table 1 . the error using lattice sizes up to @xmath65 is 0.17% . @l@rrrr & & & & + @xmath82 & @xmath83 & @xmath84 & @xmath85 & + @xmath86 & @xmath87 & @xmath88 & @xmath89 & + @xmath90 & @xmath91 & @xmath92 & @xmath93 & + @xmath94 & @xmath95 & @xmath96 & @xmath97 & @xmath98 + _ mass gap _ the mass gap for the massless continuum schwinger model can be computed exactly @xcite : @xmath99 the lattice mass gap is computed as : @xmath100 comparing our results with those of crewther and hamer @xcite obtained by a similar method , although they use a different representation ( jordan - wigner transformation ) , we find complete agreement for given values of @xmath54 and @xmath59 . when we reach larger @xmath54 we observe that the value of the mass gap improves substantially . for instance , in fig . 3 we show a plot of the mass gap for @xmath101 for several orders . as it can be seen in the region @xmath102 , the mass gap values decrease with the order and the size approaching the continuum result . given the non - uniformity of the convergence it is more difficult to extrapolate to the limit although values @xmath103 are obtained at the modest size of @xmath104 . _ chiral order parameter _ an interesting quantity to compute is the vacuum expectation of the chiral condensate per - lattice - site @xmath105 , defined as @xmath106 , \label{eq : chir - cond}\ ] ] where @xmath107 is the number of lattice sites . the corresponding operator is realized in the p - representation and thus we get for the chiral condensate : @xmath108 where @xmath109 is the number of connected paths in @xmath3 . to compute @xmath105 the @xmath110 hamiltonian is modifyed as @xmath111 , \label{eq : w'}\ ] ] where @xmath112 is an arbitrary parameter . thus , @xmath105 is obtained in the standard way as @xmath113 the massless continuum schwinger model undergoes a breaking of chiral symmetry with @xmath114 where @xmath115 is the euler constant . this non - zero value of the chiral condensate is one of the main efects of the axial anomaly . in fig . 4 we report the value of the chiral condensate per - lattice - site for lattice sizes ranging from @xmath64 to @xmath101 . fig . 5 shows this chiral order parameter for different lattice sizes up to order @xmath90 for each size . notice that the results in the weak coupling region converge to the corresponding continuum value ( [ eq : chiral - cont ] ) as long as @xmath54 increases while for a fixed @xmath54 the value improves with the order @xmath59 till the value @xmath90 is reached . our general proposal is to to show that the p - representation is a valuable and alternative computational tool for gauge theories with dynamical fermions . in particular , in this work , we wanted to test the hamiltonian approach on finite lattices . with tis aim , we chose the simplest model : ( 1 + 1 ) qed . this also enables us to compare with the corresponding numerical simulations @xcite using the lagrangian counterpart of the p - representation or the socalled _ worldsheet formulation _ @xcite . this comparison shows that , for this case of one spatial dimension , the hamiltonian method is less time consuming . the results are very good and confirm the belief of hamer et al @xcite in obtaining with considerable accuracy the observables working on lattices of moderate size . consequently , this procedure is appealing because one can run all the needed computations in small machines obtaining quite fair results .

the schwinger model is studied in a finite lattice by means of the p - representation . the vacuum energy , mass gap and chiral condensate are evaluated showing good agreement with the expected values in the continuum limit .

recently , barger _ @xcite addressed astophysical constraints on extra dimensions by computing energy loss rates from the sun , red giant stars and ( type ii ) supernovae due to possible excitation of graviton modes , @xmath1 , in the case that the extra dimensions are compactified @xcite . the processes @xmath2 ( in the static nucleon approximation ) , and @xmath3 where considered . they worked in the zero density approximation , varying only the temperature . their calculation neglected plasma effects and they anticipated that this neglect should not be important because of the high power dependence on @xmath4 , the inverse of the compactification dimension . the purpose of this paper is to address the extent to which the process @xmath5 ( and the crossed process @xmath6 ) is affected by a non - zero charged particle density and the presence of both longitudinal and transverse plasmons . our aims are , in brief , to find numerical values for the energy loss rate per @xmath0 for densities from @xmath7 to @xmath8gm/@xmath0 and temperatures from @xmath7 to @xmath9 ev , to confirm the barger _ . _ expectation of density insensitivity , to show the extent to which the systems they considered are close to the border in density at which their expectation fails , to determine the relative contributions of longitudinal and transverse plasmon processes , to address the size of the contribution of the crossed process , and , importantly , to note the ambiguities in the form of a covariant interaction between plasmons and kaluza - klein modes . it is clear that the expectation of barger et al . @xcite must fail at sufficiently high density , for fixed temperature , since the energy loss rate goes as an integral over the bose - einstein distribution , @xmath10 , in which the frequencies are given by a dispersion relation with an effective photon mass that grows with density . it is essentially a numerical question as to the point at which suppression sets in and the rate ( in density ) at which it proceeds . intuition is hampered by the fact that the natural parameter is the electron chemical potential which is not simply related to the particle density . we give the numerical results over the temperature and density ranges cited above . an important problem in carrying out this work is the question of the appropriate lagrangian . the free space coupling between the electromagnetic field and gravitons can be found in textbooks , for example @xcite , and has been generalized to the case of higher dimensional kaluza - klein excitations @xcite . however , we have not found a parallel literature for the case in which the free space photon is replaced by a plasmon satisfying a non - trivial dispersion relation . this difficulty is addressed in section 2 . we adopt a diagrammatic approach and also make an approximation that we test numerically . also given in section 2 is the formalism used for the numerical calculations of section 3 . we conclude in section 4 with a brief summary . in a medium with nonzero temperature and density , radiation satisfies the dispersion relation @xmath11 @xmath12 is the transverse or longitudinal component of the polarization tensor , @xmath13 where @xmath14 ( @xmath15 ) are the polarization vectors and the polarization tensor @xmath16 is the photon self energy in the medium . the contribution to this self energy from fermions in the medium is calculated in the medium rest frame by adding a term @xmath17 to the usual ( vacuum ) progagator . here , @xmath18 is the temperature , @xmath19 and @xmath20 is the electron chemical potential , which is related to the electron number density @xmath21 by @xmath22= n_e\ ] ] to lowest order in the fine structure constant @xmath23 , the polarization tensor is given by @xmath24\nonumber \\[4pt ] & & \qquad\times\frac{(p\dot k)^2g^{\mu\nu}+k^2p^{\mu}p^{\nu}-p\dot k(k^{\mu}p^{\nu}+ k^{\nu}p^{\mu})}{(p\dot k)^2}\,.\end{aligned}\ ] ] it turns out that @xmath25 and @xmath26 can be approximated to within 1% for all temperatures and densities @xcite by @xmath27+v_*^2|\vec{k}|^2-|\vec{k}|^2\ , , \\ [ 4pt ] \pi_t & = & \omega_p^2\left[1+\frac{1}{2}g(v_*^2|\vec{k}|^2/\omega^2)\right]\,,\end{aligned}\ ] ] where @xmath28 is an average value of @xmath29 for the electron ( the only fermion which contributes for stellar temperatures and densities ) . explicitly , @xmath30 with @xmath31 given by @xmath32 where @xmath33 is the sum of the electron and positron distributions ( the square bracket in eq.([pimunu ] ) above ) . the plasma frequency @xmath34 is given by @xmath35 and the function @xmath36 is @xmath37\,.\ ] ] it will be important below to note that , for transverse photons , @xmath38 is @xmath39 for @xmath40 and _ increases _ as @xmath41 increases , while , for longitudinal photons , @xmath42 at @xmath40 and _ decreases _ as @xmath41 increases . integration over @xmath41 for longitudinal photons must be cut off at the point where @xmath43 becomes negative , @xmath44\,.\ ] ] we include in our numerical evaluations the renormalization constants @xmath45 although this is inconsistent with calculating only to the lowest order in @xmath23 . it is a check on our results that they do not change significantly when the @xmath46 are set to unity . the @xmath46 are given by @xcite @xmath47 + ( \omega^2+|\vec{k}|^2)(\omega^2-v_*^2|\vec{k}|^2 ) } \\ [ 4pt ] z_l & = & \frac{2\omega^2(\omega^2-v_*^2|\vec{k}|^2)}{3\omega_p^2-\omega^2+v_*^2|\vec{k}|^2 } \frac{\omega^2}{\omega^2-|\vec{k}|^2}\,.\end{aligned}\ ] ] the rate of graviton emission can be calculated using the lagrangian for the coupling of kaluza - klein field @xmath48 , corresponding to the mass excitation @xmath49 , to the photon energy - momentum tensor @xmath50 . neglecting gauge terms , this coupling is @xcite @xmath51 & = & \frac{\kappa}{2}\left({\cal g}_{\vec{n}}^{\mu\nu}f_{\mu}^{\la}f_{\nu\la}-\frac{1}{4 } { \cal g}_{\vec{n},\mu}^{\mu}f^{\la\ro}f_{\la\ro}\right)\,,\end{aligned}\ ] ] where @xmath52 is the electromagnetic field tensor . we consider only the coupling of the spin-2 component of the kaluza - klein field ; the spin-0 component does not couple to photons . the matrix element for @xmath53 obtained from eq.([lint ] ) is @xcite @xmath54 where @xmath55 with @xmath56 d_{\mu\nu,\la\ro}(k_1,k_2 ) & = & \eta_{\mu\nu}k_{1\,\ro}k_{2\,\la } - \left[\eta_{\mu\ro}k_{1\,\nu}k_{2\,\la}+\eta_{\mu\la}k_{1\,\la}k_{2\,\nu } + ( \mu\leftrightarrow\nu)\right]\,.\end{aligned}\ ] ] the sum over polarizations of the kaluza - klein state is @xcite @xmath57 with @xmath58 e_{\mu\nu } & = & \eta_{\mu\nu}-\frac{k_{\mu}k_{\nu}}{m_{\vec{n}}^2}\,.\label{projop}\end{aligned}\ ] ] the coupling eq.([coupling ] ) is gauge invariant even if @xmath59 and @xmath60 are not zero , e.g. @xmath61 . however , it is not conserved , @xmath62 , if @xmath59 and/or @xmath60 differs from zero . we can not write a conserved coupling by using the energy - momentum tensor for a massive vector field because @xmath59 is not necessarily equal to @xmath60 . this means that if we square @xmath63 of eq.([matrixel ] ) , and use eq.([spinsum ] ) , we get extra terms of the form @xmath64 or @xmath65 from the second term in eq.([projop ] ) . to have a conserved amplitude with @xmath66 , we must include all the diagrams of fig.([diags ] ) . the feynman rules for the , height=288 ] fermion - fermion-@xmath67 coupling and the fermion - fermion - photon-@xmath67 are given in refs.@xcite , and the loops are calculated by using eq.([prop ] ) for one of the legs . we have shown the the sum of these diagrams is gauge invariant and conserved for arbitrary @xmath59 and @xmath60 at finite temperature and density . however , this was done without actually evaluating the diagrams . in particular , diagram ( d ) is very tedious and we have not computed it . instead , we have used only diagram ( a ) ( eq.([matrixel ] ) above ) but have evaluated every energy loss twice - once including the @xmath64 and @xmath65 terms and once omitting them . in every case , the results were almost identical . while this proves nothing , it does seem to indicate that performing the complete one - loop calculation would not give a substantially different answer . the reaction rate must be summed over the kaluza - klein states , which is done by integrating over @xmath68 where @xmath69 is the number of extra dimensions . @xmath4 is the string scale which is related to the compactification scale @xmath70 and newton s constant @xmath71 . specifically , we use @xmath72 our definition of @xmath73 differs from that of @xcite by a factor of @xmath74 , i.e. their @xmath4 is larger by a factor @xmath75 . as a consequence , values of the energy loss per unit volume obtained from our tables must be multiplied by 2 when comparing with barger _ et al._@xcite . for 2 particles @xmath76 1 particle reactions there remains a delta function from phase space which identifies @xmath77 with the center of mass squared energy @xmath78 . thus , the integral over @xmath77 , eq.([dmsq ] ) , replaces @xmath77 by @xmath78 and our results depend on @xmath69 through the factor @xmath79 . the rate of energy loss per unit volume is given by the standard expression @xmath80 where @xmath81 denotes the cross section times the relative velocity @xcite . the initial photons have @xmath82 , @xmath83 and can be transverse , @xmath84 , longitudinal , @xmath85 or mixed , e.g. @xmath86 . the factor @xmath87 gives the number of spin states : @xmath88 for @xmath89 , @xmath90 or @xmath91 . for longitudinal photons , the @xmath41 integrals are cut off at @xmath92 given by eq.([kmax ] ) . the corresponding expression for the energy loss in the decay @xmath93 is @xmath94 & & \qquad\times\frac{(2\pi)^4\left((k_t - k_l)^2\right)^{(n-2)/2}}{\kappa^2m_s^{n+2 } } \,|{\cal m}|^2 \,,\end{aligned}\ ] ] where @xmath63 is given by eq.([matrixel ] ) with @xmath95 . the first step in the calculations is to obtain @xmath96 from eq.([density ] ) . in doing this we assume that the electron number density , @xmath21 is related to the mass density @xmath97 by @xmath98 where @xmath99 is the proton mass . this is useful for comparison purposes and is a reasonable order of magnitude approximation but needs correction ( by less than an order of magnitude ) for a supernova or a neutron star . the results of the calculation of @xmath20 are given in tables [ mu ] and [ mutilde ] for the matrix of @xmath97 and @xmath18 values : @xmath100 gm/@xmath0 and @xmath101 . two tables are given ( @xmath102 and @xmath103 ) in order to make clear both the deviation of @xmath20 from @xmath104 ( taken as 0.51 mev ) at low temperature and its deviation from zero at high . note the rapid variation of @xmath20 ( for the lowest densities ) at the temperature @xmath18 around @xmath105 where pair production first begins to be copious , and the slower but similar variation at higher densities @xmath97 . the variation is slower for higher densities because the electron - positron density difference needs to have a large value . these variations are illustrated in fig.[mu_1 ] , where , for display purposes , the lowest value of @xmath20 shown , @xmath106 , is an upper limit on the exact numbers in table [ mu ] . is plotted for the range of density @xmath97 and temperature @xmath18 given in the text . here , for display purposes , the lowest value of @xmath20 shown , @xmath106 , is an upper limit on the exact numbers in table [ mu].[mu_1],height=288 ] in calculating @xmath20 we used an iteration procedure and required that the output density value equal the input to better than a percent . we now pass on to the results of calculating @xmath107 for the case of @xmath108 extra dimensions . it was possible to evaluate the integral in eq.([dedtdv ] ) over the cosine of the angle between the two plasmons analytically so that , for all the processes under consideration , only two integrals remain in finding the energy loss rate - the integrals over the two plasmon momenta . it should be noted that there are only four processes to consider : ( 1 ) @xmath109 , ( 2 ) @xmath110 , ( 3 ) @xmath111 , and ( 4 ) @xmath112 . this is because , as the plasmon momentum @xmath41 increases , the effective mass of a transverse plasmon increases while the effective mass of a longitudinal plasmon decreases @xcite . thus the _ missing _ processes , @xmath113 and @xmath114 are forbidden by energy - momentum conservation . the assertion is clear for the first process since @xmath115 . for the second , we note that , in the rest frame of the decaying longitudinal plasmon , @xmath116 and conservation of energy and momentum implies that the graviton mass , @xmath117 , satisfies @xmath118 where @xmath119 is the energy of the final plasmon . using the dispersion relation for longitudinal plasmons , it can be shown that the right side of eq.([lneqlg ] ) is less than zero for @xmath120 . the results of the calculations are given in tables [ tt]-[tot2 ] . these have the energy loss rates for the four processes , and for the sum , for a matrix of density and temperature values - @xmath121 to @xmath8 gm/@xmath0 for density @xmath97 and @xmath7 to @xmath122 ev for temperature @xmath18 - in both cases in factors of @xmath123 increments . in these tables , @xmath18 increases from left to right while @xmath97 increases from top to bottom . the entries are logs to the base 10 of the energy loss rate in ergs per @xmath0-s . note that barger _ @xcite give results per unit mass , but results per unit volume are better for our purposes since they show more clearly the way in which the zero - density approximation breaks down as the density increases . we give the results for @xmath124 tev . two additional tables , [ dom ] and [ frac ] , give respectively the number of the process that dominates for the reaction ( zero if the rate is zero , i.e. below @xmath125 ) and the fraction of the total represented by the dominant contribution . in fig.[tttog](@xmath126 ) , we see at a glance the effect cited in the for the process @xmath127 is plotted for the range of density @xmath97 and temperature @xmath18 given in the text . [ tttog],height=288 ] introduction : for fixed @xmath18 the energy loss rate is independent of the density until @xmath97 increases to a point where the effective photon mass and plasmon density are sufficiently high that the rate drops exponentially . the numerical values are given in table[tt ] . for the process @xmath128 is plotted for the range of density @xmath97 and temperature @xmath18 given in the text . [ lltog],height=288 ] the analogous plot for @xmath112 is shown in fig.[lltog ] . here , the the rate grows slightly for a fixed @xmath18 before dropping exponentially in the mass of the longitudinal plasmon with increasing @xmath41 . again , numerical values are found in table[ll ] . the sun , a red giant , and a type ii supernova are , in the ( @xmath129 ) plane , given by barger _ _ @xcite to be at ( 156 gm/@xmath0,1.3 kev ) , ( 10@xmath130 gm/@xmath0 , 8.6 kev ) , and ( 10@xmath131 gm/@xmath0 , 30 mev ) respectively . we see from table [ tt ] that the sun is in a low density region where the zero - density approximation holds while the supernova is on the edge of a constant density region . we also see from table 6 that the other processes contribute little in these two cases . the red giant ( rg ) case is more interesting . in table [ tt ] , we see from the @xmath132 and @xmath133 columns that the rg is in the gentle fall off region for the former , but the steep fall off region for the latter . examining the region between 1 and 10 kev more closely gives , with t varying in 1.0 kev increments , for the log of the tt energy loss rate : -16.7 , -5.1 , -0.91 , 1.36 , 2.85 , 3.94 , 4.78 , 5.48 , 6.07 , 6.57 . the total energy loss rate varies as : -15.96 , -4.14 , 0.37 , 2.65 , 3.77 , 4.59 , 5.25 , 5.82 , 6.31 , 6.77 . in short , at the rg , the tt energy loss rate , 1.4 erg / g@xmath134s , is down by a factor 4.5 from its zero - density approximation @xcite , but the other processes bring the total rate up to 2.6 erg / g@xmath134s , or within about a factor 2 of the zero density result . finally we turn to the dependence of the emission rates on the number n of ( large ) extra dimensions . we give , in tables [ tot3 ] and [ tot4 ] , the results for the total rates for n=3 and n=4 . one sees , in the low - density limit , the @xmath135 behavior pointed out by barger _ @xcite for the @xmath136 rate which dominates for @xmath137 in the same places as in the @xmath108 case . the approximations of zero density and purely transverse ( @xmath126 ) photon annihilation into gravitons of @xcite must fail , for fixed temperature , at high densities . we have computed a first estimate of finite density corrections to two - plasmon production , for both transverse and longitudinal plasmons of kaluza - klein excitations , as well as the decay process @xmath138 as a function of plasma density and temperature over a wide range of interest in both parameters . our conclusion is that the zero - density , pure transverse approximation is satisfactory for the sun , marginal for supernovae , and fails by about a factor of 5 for red giants . it is interesting to note that , while very little of the @xmath139 plane is occupied , astrophysical systems appear to be preferentially located relatively close to the boundaries ( in @xmath97 ) at which the transverse photon approximation begins to fail . our calculation is approximate in that we omit most of the diagrams of fig.([diags ] ) . however , we believe our results indicate that the full calculation of all the diagrams is unlikely to modify our conclusions . one of us ( v. t. ) wishes to thank r. mohapatra for helpful conversations . this work was supported in part by the national science foundation under grant phy-9802439 and by the department of energy under contract nos . de - fg03 - 93er40757 and de - fg03 - 95er40908 . 99 v. barger , t. han , c. kao and r .- j . zhang , phys . b461 * , 34 ( 1999 ) ; hep - ph/9905474 ( 1999 ) . n. arkani - hamed , s. dimopoulos and g. dvali , phys . b * 429 * , 263 ( 1998 ) ; i. antoniadis , n. arkani - hamed , s. dimopoulos and g. dvali _ ibid . _ , * 436 * , 506 ( 1998 ) . s. weinberg , _ gravitation and cosmology _ , john wiley and sons , new york , 1972 . g. f. guidice , r. ratazzi and j. d. wells , nucl . b * 544 * , 3 ( 1999 ) . t. han , j. d. lykken and r .- j . zhang , phys . d * 59 * , 105006 ( 1999 ) . e. bratten and d. segel , phys . d * 48 * , 1478 ( 1993 ) . georg g. raffelt , _ stars as laboratories for fundamental physics _ , the university of chicago press , 1996 . .the entries are @xmath140 for the process ( 1 ) ( @xmath127 ) . the rows are labeled by the density @xmath97 in gm/@xmath0 , and the columns by the temperature @xmath18 in ev . a blank entry indicates that @xmath141erg/@xmath0-s . the number of compact dimensions is @xmath108.[tt ] [ cols="^,^,^,^,^,^,^,^,^,^,^",options="header " , ]

recently , barger _ et al . _ computed energy losses into kaluza klein modes from astrophysical plasmas in the approximation of zero density for the plasmas . we extend their work by considering the effects of finite density for two plasmon processes . our results show that , for fixed temperature , the energy loss rate per @xmath0 is constant up to some critical density and then falls exponentially . this is true for transverse and longitudinal plasmons in both the direct and crossed channels over a wide range of temperature and density . a difficulty in deriving the appropriate covariant interaction energy at finite density and temperature is addressed . we find that , for the cases considered by barger _ et al . _ , the zero density approximation and the neglect of other plasmon processes is justified to better than an order of magnitude .

the renormalization group equations ( rges ) for the rigid couplings and soft parameters in susy gauge theories play a crucial role in applications . actually , all predictions of the mssm are based on solutions to these equations in leading and next - to - leading orders @xcite . typically , one has three gauge couplings , one or three yukawa couplings ( for the case of low or high @xmath0 , respectively ) and a set of soft couplings . in leading order , solutions to the rge for the gauge couplings are simple ; however , already for yukawa couplings , they are known in an analytical form only for the low @xmath0 case , where only the top coupling is left . moreover , even in this case solutions for the soft terms look rather cumbersome and difficult to explore @xcite . in a recent paper @xcite , it has been shown that solutions to the rge for the soft couplings follow from those for the rigid ones in a straightforward way . one takes the solution for the rigid coupling ( gauge or yukawa ) , substitute instead of the initial conditions their modified expressions @xmath1 where @xmath2 , @xmath3 and @xmath4 are the grassmannian parameters , and expand over these parameters . this gives the solution to the rges for the soft couplings . hereafter the following notation is used : @xmath5 where @xmath6 and @xmath7 are the gauge and yukawa couplings , respectively , and @xmath8 are the soft masses associated with each scalar field . this procedure , however , assumes that one knows solutions to the rge for the rigid couplings in the analytic form . for instance , in the case of the mssm in the low @xmath0 regime this allows one to get solutions for the soft couplings and masses simpler than those known in the literature ( see @xcite ) . at the same time , in many cases such solutions are unknown . actual examples are the mssm with high @xmath0 and nmssm . one is bound to solve the rges numerically when the number of coupled equations increases dramatically with the soft terms being included . below we propose simple analytical formulae which give an approximate solution to the rge for yukawa couplings in an arbitrary susy theory with the accuracy of a few per cent . performing the grassmannian expansion in these approximate solutions one can get those for the soft couplings in a straightforward way . as an illustration we consider the mssm in the high @xmath0 regime . one can immediately see that approximate solutions obtained in this way possess infrared quasi - fixed points @xcite which can be found analytically . they appear in the limit when the initial values of the yukawa couplings are much larger than those for the gauge ones . then , one can analytically trace how the initial conditions for the soft terms disappear from their solutions in the above mentioned limit . the paper is organized as follows . in sect . 2 , we consider the mssm in the low @xmath0 regime , where all solutions are known analytically and describe briefly the grassmannian expansion . in sect . 3 , we present our approximate solutions for the yukawa couplings and obtain those for the soft terms . we also present numerical illustration and compare approximate solutions with the numerical ones . the fixed point behaviour is discussed . section 4 contains our conclusions . the explicit formulae for the soft couplings and masses are given in appendices . consider the mssm in the low @xmath0 regime . one has three gauge and one yukawa coupling . the one - loop rg equations are @xmath9 with the initial conditions : @xmath10 , and @xmath11 . their solutions are given by @xcite @xmath12 where @xmath13 to get solutions for the soft terms , it is enough to perform the substitution @xmath14 and @xmath15 for the initial conditions in ( [ sol ] ) and expand over @xmath16 and @xmath17 . expanding the gauge coupling in ( [ sol ] ) up to @xmath18 one has ( hereafter we assume @xmath19 ) @xmath20 performing the same expansion for the yukawa coupling and using the relations @xmath21 one finds the well - known expression @xcite @xmath22 to get the solution for the @xmath23 term , one has to make expansion over @xmath18 and @xmath17 . this can be done with the help of the following relations : @xmath24 as a result one has @xcite @xmath25\!.\ , \label{si}\end{aligned}\ ] ] which is much simpler than what one finds in the literature @xcite , though coinciding with it after some cumbersome algebra . one can also write down solutions for the individual masses using the grassmannian expansion of those for the corresponding superfield propagators . for the first two generations one has @xmath26 where @xmath27 the third generation masses get a contribution from the top yukawa coupling @xmath28 where @xmath29 is related to @xmath30 ( [ si ] ) by @xmath31 \\ & = & \frac{\sigma_0-a_0 ^ 2}{1 + 6y_0f}+\frac{(a_0-m_{1/2}6y_0(te - f))^2}{(1 + 6y_0f)^2 } -m_{1/2}^2\frac{6y_0}{1 + 6y_0f}t^2\frac{de}{dt}- \sigma_0.\end{aligned}\ ] ] with analytic solutions ( [ sol],[a],[si ] ) one can analyze asymptotic and , in particular , find infrared quasi - fixed points @xcite which correspond to @xmath32 @xmath33\!. \label{sf}\end{aligned}\ ] ] one can clearly see that the dependence on @xmath34 and @xmath35 disappears from ( [ yf])-([sf ] ) . some residual dependence on @xmath36 is left for the soft masses and partially cancels with that of @xmath29 . below we demonstrate how the same procedure works in the case of approximate solutions . as a realistic example we take the mssm in the high @xmath0 regime . the one - loop rge for the yukawa couplings in this case look like @xmath37 since the exact solution is absent and might be too cumbersome , we look for an approximate one in a simple form similar to that of ( [ sol ] ) . in choosing approximate solutions we follow the idea of @xcite where an approximate solution for @xmath38 and @xmath39 ignoring @xmath40 has been proposed . our suggestion is to consider separate brackets for each propagator entering into the yukawa vertex . then , one has the following expressions for the yukawa couplings : @xmath41^{1/a } [ 1 + 2by_{t0}f_t]^{1/b } [ 1 + 3cy_{t0}f_t]^{1/c } } , \quad \frac 1a \!+\ ! \frac 1b \!+\ ! \frac 1c\!=\!1 \\ y_b&=&\frac{\displaystyle y_{b0}e_b}{\displaystyle [ 1+a(y_{t0}f_t+y_{b0}f_b)]^{1/a } [ 1 + 2by_{b0}f_b]^{1/b } [ 1+c(3y_{b0}f_b+y_{\tau 0}f_\tau ) ] ^{1/c } } , \\ y_\tau & = & \frac{\displaystyle y_{\tau 0}e_\tau } { \displaystyle [ 1+a^{\prime } y_{\tau 0}f_\tau ] ^{1/a^{\prime } } [ 1 + 2b^{\prime } y_{\tau 0}f_\tau]^{1/b^{\prime } } [ 1+c(3y_{b0}f_b+y_{\tau 0}f_\tau ) ] ^{1/c}},\quad \frac1{a^{\prime } } \!+\!\frac 1{b^{\prime } } \!+\!\frac 1c\!=\!1\end{aligned}\ ] ] where the brackets correspond to the @xmath42 , @xmath43 and @xmath44 propagators , respectively . here @xmath45 and @xmath46 are given by ( [ e ] ) and ( [ f ] ) and @xmath47 and @xmath48 have the same form but with @xmath49 and @xmath50 , respectively . the brackets are organized so that they reproduce the contributions of particular diagrams to the corresponding anomalous dimensions . the coefficients @xmath51 and @xmath52 are arbitrary and their precise values are not so important . when yukawa couplings @xmath53 are small enough , one can make expansion in each bracket , and the dependence of these coefficients disappears . however , for large couplings , which are of interest for us because of the fixed points , we have some residual dependence . the requirement that the sum of exponents equals 1 follows from a comparison with rges . solutions are close to the exact ones when the brackets are roughly equal to each other . apparently , since @xmath54 and @xmath55 one can not completely satisfy this requirement . our choice of the coefficients @xmath51 and @xmath52 is dictated mainly by simplicity . in the following we choose them as @xmath56 this gives approximate solutions like @xmath57^{2/7}\!\ ! \left[1 + 7y_{t0}f_t\right]^{5/7 } } , \label{y1 } \\ ! \frac{\displaystyle y_{b0}e_b } { \displaystyle \left[1+\frac{7}{2}(y_{t0}f_t+y_{b0}f_b)\right]^{2/7}\!\ ! \left[1 + 7y_{b0}f_b\right]^{2/7}\left[1+\frac{7}{3}(3y_{b0}f_b+y_{\tau 0}f_\tau ) \right]^{3/7 } } \label{y2 } , \\ y_\tau \!&\approx & \!\!\frac{\displaystyle y_{\tau 0}e_\tau } { \displaystyle \left[1\!+\!\frac{21}{4}y_{\tau 0}f_\tau \right]^{4/7}\!\ ! \left[1\!+\!\frac{7}{3}(3y_{b0}f_b\!+\!y_{\tau 0}f_\tau ) \right]^{3/7 } } . \label{y3}\,\end{aligned}\ ] ] solutions for @xmath58 and @xmath59 can be obtained by grassmannian expansion with the initial conditions @xmath60 these initial conditions correspond to the so - called universality hypothesis which we follow in our numerical illustration for simplicity . however , one can choose arbitrary initial conditions for the soft terms when needed . it leads to an obvious modification of the formulae . one can get also the corresponding solutions for the individual soft masses . this can be achieved either by grassmannian expansion of the corresponding brackets in ( [ y1])-([y3 ] ) , or by expressing the masses through the @xmath23s in an exact way . the second way gives a slightly better agreement with numerical solutions ( see below ) . we present the explicit expressions for the soft terms and masses in appendix a. we start by investigating the precision of approximate solutions for the yukawa couplings . to estimate the accuracy , we introduce a relative error which is defined as @xmath61 and corresponds to the @xmath62 scale ( @xmath63 ) at the end of the integration range . the accuracy for the solutions of soft terms is defined in the same way . let us take at the beginning all three yukawa couplings to be equal at the gut scale and to have their common value @xmath64 in the range @xmath65 . the upper limit is taken in order not to leave the perturbativity regime . we find that for @xmath66 the approximation errors are less than @xmath67 for all @xmath68 s . while for @xmath38 it remains smaller than @xmath69 over the whole range of initial values at the gut scale , for @xmath39 the error increases up to @xmath70 and for @xmath71 up to @xmath72 ( for large values of @xmath64 ) . it is worth mentioning that for small @xmath64 ( around @xmath73 and below ) the accuracy is very good ( fractions of per cent or better ) . consider now @xmath74 and let the top yukawa coupling vary within the limits @xmath75 in order to examine the applicability of our formulae . in this case the accuracy it is spoilt a little bit with increasing initial values . namely , the error for @xmath38 increases up to @xmath76 , and for @xmath39 and @xmath40 up to @xmath77 . however , if one keeps @xmath64 in the range @xmath78 the accuracy for @xmath38 remains better than @xmath67 , and for @xmath39 and @xmath71 better than @xmath76 . the particular case considered above seems to have the worst accuracy . this is not surprising since our approximate formulae are supposed to work best of all when all three yukawa couplings are nearly equal . if we keep @xmath79 and the relative ratios less than @xmath80 , we get an average error of less than @xmath81 for @xmath38 , about @xmath81 for @xmath39 and @xmath76 for @xmath71 . this statement is illustrated in fig.[errandm ] . for each yukawa coupling we have plotted the error as a function of @xmath82 in the range @xmath83 . the ratios are kept within the region @xmath84 and @xmath85 . further on , we narrow the range of initial values up to @xmath86 because the errors ( defined as in ( [ defer ] ) ) come to an asymptotic value for @xmath87 and almost vanish for @xmath88 . the comparison of numerical and approximated solutions is shown in fig.[evays ] for three different sets of @xmath64 s . the approximate solutions follow the numerical ones quite well , preserving their shape , and they have a high accuracy , especially in the case of equal yukawa couplings at the gut scale . however , as can be seen from the top of fig . [ evays ] , one can take arbitrary initial conditions for the yukawa couplings , in particular those which are needed to fit the @xmath89 masses , and to use our approximate solutions for these purposes . for the soft couplings , @xmath90 , we take the initial values at the gut scale to be @xmath91 and leave @xmath64s in the narrow range as above . then , we get an accuracy of @xmath92 for @xmath93 and @xmath94 . for @xmath95 the approximation is worse when @xmath96 is taken to be negative or smaller than @xmath97 ( see fig . [ evays ] ) , but things go better for large initial values of @xmath96 and we get an accuracy of about @xmath98 . again it should be mentioned that this is an accuracy at the end point where @xmath99 itself is close to 0 and the accuracy defined as ( [ defer ] ) merely gives an odd hint of the precision . along the curves the accuracy is much better . in fig.[evays ] we have plotted the behaviour of @xmath93 , @xmath94 and @xmath99 for three different initial values of @xmath96 , namely @xmath100 and for one set of @xmath64s . as for the @xmath23 s , keeping the range of parameter space for @xmath64 and @xmath96 as above , we get an accuracy of typically @xmath101 for @xmath102 ( even better for fairly equal @xmath64s ) . for @xmath103 the precision is around @xmath70 . with @xmath104 we get into the same troubles as for @xmath95 . the approximation becomes good ( about @xmath76 ) only for a large enough ratio of @xmath105 . the approximation errors for @xmath90 s and @xmath23 s are linked with those for @xmath68 . if one considers only the sets of small initial values for @xmath64 ( less than @xmath73 ) , then @xmath23 s are approximated with a precision better than @xmath106 , regardless of the @xmath96 values . the precision for @xmath23 increases with @xmath96 , but this dependence is not so striking as the one on @xmath64 . the approximate formulae for the soft masses may be derived from @xmath23 using ( [ a1])-([a7 ] ) . in this case the approximate solutions give an accuracy of about @xmath107 for @xmath108 , @xmath109 and @xmath92 for @xmath110 . for the higgs masses we get a good approximation ( of about @xmath81 on average ) for @xmath111 , and a satisfactory one for @xmath112 ( typically @xmath76 ) . this accuracy is almost insensitive to the @xmath96 variation ( we took it to be in the range @xmath113 ) and on the ratio @xmath114 ( taken to be @xmath115 ) . the slepton masses ( see fig . [ errandm ] ) are not approximated properly in an analogous way . this is mainly due to the less accurate approximation of @xmath116 . as a concluding remark on numerical analysis , it should be mentioned that one has a rather good approximation for small ( less than @xmath73 ) initial values of the yukawa couplings . for larger values of @xmath68 s one has a good approximation especially in the case of unification of the yukawa couplings . one can easily see that solutions ( [ y1])-([y3 ] ) exhibit the quasi - fixed point behaviour when the initial values @xmath117 . in this case , one can drop 1 in the denominator and the resulting expressions become independent of the initial conditions @xmath118^{2/7}\!\ ! \left[7 f_t\right]^{5/7 } } \label{yfp1},\\ y_b^{fp}&\approx&\frac{\displaystyle e_b}{\displaystyle \left[\frac{7}{2}(f_t+f_b)\right]^{2/7}\!\!\left[7f_b\right]^{2/7}\!\ ! \left[\frac{7}{3}(3f_b+ f_\tau ) \right]^{3/7 } } \label{yfp2},\\ y_\tau^{fp } & \approx & \frac{\displaystyle e_\tau } { \displaystyle \left[\frac{21}{4}f_\tau \right]^{4/7}\!\ ! \left[\frac{7}{3}(3f_b+f_\tau ) \right]^{3/7}}. \label{yfp3}\end{aligned}\ ] ] these expressions being expanded over the grassmannian variables give the quasi - fixed points for the soft terms and masses . the explicit expressions are presented in appendix b. we see that the irqfp behaviour is sharply expressed for @xmath38 and @xmath39 ( see fig . [ fpays ] ) , and our approximate solution describes the fixed point line well . the same takes place for the corresponding @xmath90s and @xmath23s . for @xmath119 and @xmath120 the accuracy is worse , however , the solution is still reliable . the soft mass terms there exhibit the same irqfp behaviour , though some residual dependence on the initial conditions is left in full analogy with the exact solutions in the low @xmath0 case . the approximate solutions allow one to calculate the irqfp analytically . one can see that the fixed points for the soft terms naturally follow from the grassmannian expansion of our approximate solutions ( [ yfp1])-([yfp3 ] ) and they inherit their stability properties , as has been shown in @xcite . in particular , the behaviour of @xmath23s essentially repeats that of the yukawa couplings in agreement with @xcite . the existence of the irqfps allows one to make predictions for the soft masses without exact knowledge of the initial conditions . this property has been widely used ( see , for example , @xcite ) and though the irqfps give a slightly larger top mass when imposing @xmath121 unification , it is still possible to fit the quark masses within the error - bars and to make predictions for the higgs and sparticle spectrum @xcite . this explains general interest in the irqfps . we hope to convince the reader that the approximate solutions presented above reproduce the behaviour of the yukawa couplings with good precision in the whole integration region and for a large range of initial values . relative accuracy is typically a few per cent and is worse only at the end of the integration region mainly due to the smallness of the quantities themselves . moreover , we have shown how the approximate solutions for the soft terms and masses follow from those for the rigid couplings . this demonstrates how the grassmannian expansion , advocated in @xcite , works in the case of approximate solutions as well . for illustration we have considered universal initial conditions for the soft terms . in recent time there appeared some interest in non - universal boundary conditions . non - universality can also be included in our formulae at the expense of changing ( [ tayl ] ) and ( [ init ] ) using the same substitution rules , see ( [ g ] ) and ( [ y ] ) . since the form of our approximate solutions has been `` guessed '' ad hoc starting from some reasonable arguments , there is no direct way to improve them . however , one can imagine more constructive derivation of those solutions which would allow one to make corrections . needless to say that it is enough to construct a solution for the rigid terms . solutions for the soft terms will follow automatically . * acknowledgement * we would like to thank a.v.gladyshev for valuable discussions . financial support from rfbr grants # 99 - 02 - 16650 and # 96 - 15 - 96030 is kindly acknowledged . we here present approximate expressions for the soft couplings and masses corresponding to ( [ y1])-([y3 ] ) : @xmath122 to find the individual soft masses one can formally perform integration of the rg equations and express the masses through @xmath23s solving a system of linear algebraic equations . this gives @xmath123 the masses of squarks and sleptons of the first two generations are given by ( [ me])-([mq ] ) . we present here the irqfps for the soft couplings and masses . they are obtained via grassmannian expansion of ( [ yfp1])-([yfp3 ] ) . @xmath124 ^ 2 } { ( f_t+f_b ) ^2 } + \frac 57\ \frac{(te_t - f_t)^2 } { f_t^2 } \right)\\ \sigma_b^{fp}\!&\approx & \ ! m_{1/2}^2\left(-\ ! \frac 27\!\ \frac{t^2\frac{de_t}{dt}\!+\!t^2\frac{de_b}{dt } } { ( f_t\!+\!f_b ) } \!-\!\frac 27\ ! \ \frac{t^2\frac{de_b}{dt}}{f_b}\ ! -\ ! \frac{3t^2\frac{de_b}{dt } \!+\!t^2\frac{de_\tau } { dt } } { ( 3f_b\!+\!f_\tau ) } \right . \\ & & \left . + \frac 27 \ \frac{[(te_t - f_t)+(te_b - f_b)]^2 } { ( f_t+f_b ) ^2 } + \frac 27\ \frac{(te_b - f_b)^2 } { f_b^2 } \right . \\ & & \left . + \frac 37 \ \frac{[3(te_b - f_b)+(te_\tau -f_\tau)]^2 } { ( 3f_b+ f_\tau ) ^2 } + \frac{d}{dt}\left ( \frac{t^2}{e_b}\frac{de_b}{dt}\right)\right ) \\ \sigma_\tau^{fp } \!&\approx&\ ! m_{1/2}^2\!\left ( -\frac 47\!\ \frac{t^2\frac{de_\tau } { dt } } { f_\tau } -\frac 37\!\ \frac{3t^2\frac{de_b}{dt } + t^2\frac{de_\tau}{dt } } { ( 3f_b+\!f_\tau ) } + \ ! \frac{d}{dt}\!\left ( \frac{t^2}{e_\tau } \frac{de_\tau}{dt } \right ) \right . \\ & & \left . + \frac 47\ ! \frac{(te_\tau -f_\tau)^2}{f_\tau^2 } + \frac 37\ \frac{[3(te_b - f_b)+(te_\tau -f_\tau)]^2}{(3f_b+f_\tau)^2 } \right)\\\end{aligned}\ ] ] 99 v. barger , m. s. berger and p. ohmann , phys.rev . * d47 * , 1093 ( 1993 ) , hep - ph/9209232 + m. carena , m. olechowski , s. pokorski and c.e.m . wagner , nucl.phys . * b419 * , 213 ( 1994 ) , cern - th.7060/93 , hep - ph/9311222 + w. de boer , r. ehret and d.i . kazakov , z.phys . * c67 * , 647 ( 1995 ) , hep - ph/9405342 ; z.phys . * c71 * , 415 ( 1996 ) , hep - ph/9603350 + damien m. pierce , jonathan a. bagger , konstantin t. matchev , ren - jie zhang , nucl.phys . * b491 * , 3 ( 1997 ) , hep - ph/9606211 l.e . ibez , c. lpez and c. muoz , nucl.phys . * b256 * , 218 ( 1985 ) . + m. carena , p. chankowski , m. olechowski , s. pokorski and c.e.m . wagner , nucl.phys . * b491 * , 103 ( 1997 ) , hep - ph/9612261 d.i . kazakov , phys . lett . * b449 * , 201 ( 1999 ) , l.a . avdeev , d.i . kazakov and i.n . kondrashuk , nucl . phys . * b510 * , 289 ( 1998 ) , hep - ph/9709397 i. jack and d.r.t . jones , phys.lett . * b415 * , 383 ( 1997 ) , hep - ph/9709364 g.f . giudice and r. rattazzi , nucl.phys . * b511 * , 25 ( 1998 ) , hep - ph/9706540 c.t . hill , phys.rev . , * d24 * , 691 ( 1981 ) , + c.t . hill , c.n . leung and s. rao , nucl . phys . , * b262 * , 517 ( 1985 ) . m. carena and c.e.m . wagner , proc . of _ 2_nd ift workshop on yukawa couplings and the origins of mass , 1994 , gainesville ; cern - th.7320/94 and /9407208 i. jack and d.r.t . jones , phys . lett . * b443 * , 177 ( 1998 ) , hep - ph/9809250 i. jack and d.r.t . jones , phys . lett . , * b426 * , 73 ( 1998 ) , hep - ph/9712542 + t. kobayashi , j.kubo and g. zoupanos , phys.lett . * b427 * , 291 ( 1998 ) , hep - ph/9802267 m. carena , m. olechowski , s. pokorski and c.e.m . wagner , nucl.phys . * b426 * , 269 ( 1994 ) . m. jurcisin and d. kazakov , mod.phys.lett . * a14 * , 671 ( 1999 ) . hep - ph/9902290

we present simple analytical formulae which describe solutions of the rg equations for yukawa couplings in susy gauge theories with the accuracy of a few per cent . performing the grassmannian expansion in these solutions , one finds those for all the soft couplings and masses . the solutions clearly exhibit the fixed point behaviour which can be calculated analytically . a comparison with numerical solutions is made .

the p - wave spin - triplet charmonium states were originally observed @xcite in radiative decays of the @xmath1 soon after the discovery of the @xmath16 and @xmath1 resonances . a number of decay modes of these states have been observed and branching fractions reported @xcite . most of the existing results are from the mark i experiment , which had a data sample of 0.33 million @xmath1 decays @xcite . because the photon capabilities of the mark i detector were limited , the detection of the photon from the @xmath17 process was not required , and one constraint kinematic fits were used to reconstruct the final states . recently there has been a renewed interest in the p - wave charmonium states . since in lowest - order perturbative qcd the @xmath12 and @xmath18 decay via the annihilation of their constituent @xmath19 quarks into two gluons , followed by the hadronization of the gluons into light mesons and baryons , these decays are expected to be similar to those of a bound @xmath20 state ; a detailed knowledge of the hadronic decays of the @xmath12 and @xmath18 may provide an understanding of the decay patterns of glueball states that will help in their identification . the mass differences between the three @xmath21 states provide information on the spin - orbit and tensor interactions in non - relativistic potential models and lattice qcd calculations . the masses of the @xmath22 and @xmath18 have been precisely determined ( to a level of @xmath23 mev ) by fermilab experiment e760 @xcite using the line shape measured in the @xmath24 formation reaction . in contrast , the @xmath12 mass is much more poorly known ; the pdg average for @xmath25 has an uncertainty of @xmath26 mev @xcite . in this paper , we report the analyses of all - charged - track final states from @xmath27 decays , including @xmath28 , @xmath29 , @xmath30 , @xmath31 , @xmath32 and @xmath33 . the results for @xmath27 decays into @xmath34 , @xmath35 and @xmath36 have been reported elsewhere @xcite . we use the combined invariant mass distribution from all of the channels under study to determine the @xmath12 mass with improved precision . a byproduct of this analysis is a determination of the mass of the @xmath37 . this is of interest because the @xmath38 mass difference measures the strength of the hyperfine splitting term in heavy quark interactions . however , in spite of a number of measurements , the current experimental value of @xmath39 remains ambiguous : the pdg @xcite average is based on a fit to seven measurements with poor internal consistency @xcite and the confidence level of the fit is only 0.001 . a recent measurement from e760 @xcite disagrees with the value reported by the dm2 group @xcite by almost four standard deviations . additional measurements may help clarify the situation . the data used for the analysis reported here were taken with the bes detector at the bepc storage ring at a center - of - mass energy corresponding to @xmath40 . the data sample corresponds to a total of @xmath41 @xmath1 decays , as determined from the observed number of inclusive @xmath42 decays @xcite . bes is a conventional solenoidal magnet detector that is described in detail in ref . a four - layer central drift chamber ( cdc ) surrounding the beampipe provides trigger information . a forty - layer cylindrical main drift chamber ( mdc ) , located radially outside the cdc , provides trajectory and energy loss ( @xmath43 ) information for charged tracks over @xmath44 of the total solid angle . the momentum resolution is @xmath45 ( @xmath46 in @xmath47 ) , and the @xmath43 resolution for hadron tracks is @xmath48 . an array of 48 scintillation counters surrounding the mdc measures the time - of - flight ( tof ) of charged tracks with a resolution of @xmath49 ps for hadrons . radially outside of the tof system is a 12 radiation length thick , lead - gas barrel shower counter ( bsc ) operating in the limited streamer mode . this device covers @xmath50 of the total solid angle and measures the energies of electrons and photons with an energy resolution of @xmath51 ( @xmath52 in gev ) . outside the bsc is a solenoid , which provides a 0.4 tesla magnetic field over the tracking volume . an iron flux return is instrumented with three double layers of counters that identify muons of momentum greater than 0.5 gev / c . we use monte carlo simulated events to determine the detection efficiency ( @xmath53 ) and the mass resolution ( @xmath54 ) for each channel analyzed . the monte carlo program ( mc ) generates events of the type @xmath55 under the assumption that these processes are pure @xmath56 transitions @xcite : the photon polar angle distributions are @xmath57 , @xmath58 and @xmath59 . multihadronic @xmath60 decays are simulated using phase space distributions . for each channel , either 10000 or 5000 events are generated , depending on the numbers of events for the corresponding mode that are observed in the data sample . a neutral cluster is considered to be a photon candidate when the angle in the @xmath61 plane between the nearest charged track and the cluster is greater than @xmath62 , the first hit is in the beginning 6 radiation lengths , and the difference between the angle of the cluster development direction in the bsc and the photon emission direction is less than @xmath63 . when these selection criteria are applied to kinematically selected samples of @xmath64 and @xmath65 events , fewer than 20% of the events have @xmath66 candidates , which indicates that the fake - photon rejection ability is adequate ( see fig . [ ngm ] ) . the number of photon candidates in an event is limited to four or less . the photon candidate with the largest energy deposit in the bsc is treated as the photon radiated from @xmath1 and used in a four - constraint kinematic fit to the hypothesis @xmath67 . each charged track is required to be well fit to a three - dimensional helix and be in the polar angle region @xmath68 . for each track , the tof and @xmath43 measurements are used to calculate @xmath69 values and the corresponding confidence levels to the hypotheses that the particle is a pion , kaon and proton ( @xmath70 ) . the reliability of the confidence level assignments is verified using a sample of @xmath71 , @xmath72 and @xmath73 events , where the particle identification confidence levels ( @xmath74 ) of the tracks in different momentum ranges are found to be distributed uniformly between zero and one as expected @xcite . typically the @xmath74 value of each track for a given decay hypothesis is required to be greater than 1% in our analysis . for all decay channels , the candidate events are required to satisfy the following selection criteria : 1 . the number of charged tracks is required to be four or six with net charge zero . the maximum number of neutral clusters in an event is eight , and the number of photon candidates remaining after the application of the photon selection is required to be four or less . the sum of the momenta of the lowest momentum @xmath75 and @xmath76 tracks is required to be greater than 550 mev ; this removes contamination from @xmath71 events . the @xmath69 probability for a four - constraint kinematic fit to the decay hypothesis is greater than 0.01 . the particle identification assignment of each charged track is @xmath77 . a combined probability of the four - constraint kinematic fit and particle identification information is used to separate @xmath78 and the different particle assignments for the @xmath79 final states . this combined probability , @xmath80 , is defined as @xmath81 where @xmath82 is the sum of the @xmath69 values from the four - constraint kinematic fit and those from each of the four particle identification assignments , and @xmath83 is the corresponding total number of degrees of the freedom used in the @xmath69 determinations . the particle assignment with the largest @xmath80 is selected , and further cuts on the kinematic fit probability and particle identification probability are imposed . figure [ mks - mks ] shows a scatterplot of @xmath84 vs @xmath84 invariant masses for events with a @xmath28 mass between 3.2 and 3.6 gev . the cluster of events in the lower left - hand corner indicates the presence of a @xmath85 signal . a fit of a gaussian function to the @xmath86 mass distribution gives a peak mass at @xmath87 mev and a width @xmath88 mev that is consistent with the mc expectation for the mass resolution . we select @xmath89 candidates by requiring the mass of both @xmath84 combinations in the event to be within @xmath90 of the nominal @xmath91 mass . the invariant mass distributions for the @xmath28 , @xmath29 and @xmath85 events that survive all the selection requirements are shown in figs . [ mallpppp ] , [ mallppkk ] and [ mallksks ] . there are peaks corresponding to the @xmath27 states in each of the plots . ( the high mass peaks in figs . [ mallpppp ] and [ mallppkk ] correspond to the @xmath1 decays to all charged tracks final states that are kinematically fit with a fake low - energy photon . ) we fit the @xmath28 , @xmath29 or @xmath85 invariant mass distribution between 3.20 and 3.65 gev with three breit - wigner resonances convoluted with gaussian mass resolution functions and a linear background shape using an unbinned maximum likelihood method . in the fit , the mass resolutions are fixed to their mc - determined values and the widths of the @xmath22 and @xmath18 are fixed to the pdg average values of of 0.88 and 2.00 mev @xcite , respectively . the results of the fit are listed in table [ tab - ppkk - res ] and shown in figs . [ mallpppp ] , [ mallppkk ] and [ mallksks ] . table [ tab - ppkk - res ] also lists the mc - determined efficiencies and mass resolutions . if one of the four tracks is identified as a proton or antiproton , the event is assumed to be @xmath92 . we assign probabilities to the remaining particle assignment using the same technique that was used for @xmath29 decays ; the combination with the highest probability is selected . the @xmath30 invariant mass distribution for the selected events is shown in fig . [ mallpppr ] . here clear signals for all three @xmath27 states are apparent . we fit the mass spectrum using the same method described in the previous section ; the results are listed in table [ tab - pppr - res ] and shown as the smooth curve in fig . [ mallpppr ] . for the case where all the tracks are kaons , the contamination from @xmath94 is not an important background , and the requirement on total momentum of the lowest momentum @xmath75 and @xmath76 tracks , which is aimed at removing these events , is not used . the @xmath31 invariant mass distribution is shown in fig . [ mallkkkk ] . figure [ mphi - mphi ] shows a scatterplot of @xmath35 vs @xmath35 invariant masses for the events with @xmath31 mass between 3.2 and 3.6 gev . the concentration of events in the lower left - hand corner of the plot indicates the presence of @xmath95 final states . a fit to the @xmath35 mass distribution with a gaussian function gives a peak mass of @xmath96 mev and a width @xmath97 mev , consistent with mc expectations . events where the mass of two @xmath35 combinations are in the range @xmath98 gev are identified as @xmath99 candidates . the @xmath95 mass distribution for these events is shown in fig . [ mallphiphi ] , where there are clear signals for the @xmath12 and @xmath18 . the @xmath31 mass and @xmath95 mass plots are fitted with three breit - wigner resonances and two breit - wigner resonances , respectively , as described previously . the results of the fit are listed in table [ tab - kkkk - res ] and are shown as smooth curves in figs . [ mallkkkk ] and [ mallphiphi ] . because of the large fraction of @xmath100 intermediate events observed in the @xmath31 mode and the significant difference between the detection efficiency for phase - space events and those coming from @xmath100 decays , the detection efficiency for the @xmath12 and @xmath101 channels is a weighted average of the phase space and @xmath100 efficiency . the detection efficiencies and mass resolutions are listed in table [ tab - kkkk - res ] . the @xmath103 decay channels have serious potential backgrounds from @xmath104 ( including @xmath105 ) and @xmath106 final states . to eliminate these backgrounds , we exploit the feature that there is one and only one @xmath91 with a secondary vertex in real @xmath32 events . in each event , we determine @xmath107 , the number of two charged track combinations with net charge zero and effective mass within @xmath108 mev of @xmath109 , when the tracks are assigned a pion mass . the combination with mass closest to @xmath110 is considered to be a @xmath111 candidate . the @xmath111 vertex is defined as the point of closest approach of these two tracks ; the primary vertex is defined as the point of closest approach of the other two charged tracks in the event . two parameters are used to identify the @xmath111 : the distance between primary vertex and secondary vertex in the @xmath61 plane , @xmath112 , and the cosine of the angle between the @xmath111 momentum vector and its vertex direction @xmath113 , which is expected to be very near unity for a real @xmath111 event . candidate @xmath114 events are selected by requiring the mass of the @xmath111 candidate determined from the track four - vectors returned by the 4c - fit to be within @xmath115 of the nominal @xmath116 mass , @xmath117 , @xmath118 mm , and @xmath119 . in the invariant mass distribution of the selected events , shown in fig . [ mallkskp ] , only a @xmath120 signal is prominent . the mc simulation indicates that the numbers of events in the the @xmath121 and @xmath122 mass region are consistent with residual backgrounds from @xmath104 , @xmath105 and @xmath106 final states . we set upper limits on the branching fractions of @xmath121 and @xmath122 . the @xmath32 invariant mass distribution between 3.20 and 3.65 gev are fitted with the procedure described above . the mass resolutions are fixed at their mc - determined values , the width of the @xmath12 is fixed at the recent bes value of 14.3 mev @xcite and those of the @xmath22 and @xmath18 at their pdg values @xcite . the mass of the three @xmath21 states are also fixed at their pdg @xcite values . the fit results are listed in table [ tab - kskp - res ] and are shown as a smooth curve in fig . [ mallkskp ] . after the selections based on the kinematic fit and particle i d , the main background to the @xmath124 decays comes from the decay chain @xmath125 . the requirement on the total momentum of the lowest momentum @xmath75 and @xmath76 tracks removes one third of the mc - simulated events while rejecting almost all the @xmath94 background . the @xmath33 invariant mass distribution for the selected events is shown in fig . [ malltpp ] , where prominent signals for all three @xmath27 states can be seen . the smooth curve in the figure is the result of the fitting procedure described above . the results of the fit and the mc - determined efficiencies and resolutions are listed in table [ tab - tpp - res ] . we determine branching fractions from the relation @xmath126 where the values for @xmath127 are taken from the pdg tables @xcite . for the @xmath85 [ @xmath95 ] channel , a factor of @xmath129 [ @xmath130 is included in the denominator . systematic errors common to all modes include the uncertainties in the total number of @xmath131 events ( 8.2% ) and the @xmath132 branching fractions ( 8.6% , 9.2% and 10.3% for @xmath121 , @xmath120 and @xmath122 , respectively ) . other sources of systematic errors were considered . the variation of our results for different choices of the selection criteria range from 10% for high statistics channels to 25% for those with low statistics . the systematic errors due to the statistical precision of the mc event samples range from 2% to 5% depending on the detection efficiencies of the channels . changes in the detection efficiency when the phase space event generator is replaced by one using possible intermediate resonant states indicate that the systematic error on the efficiency due to the unknown dynamics of the decay processes is 15% . the variation of the numbers of observed events due to shifts of the mass resolutions and the total widths of the @xmath60 states is 7% ; that coming from changes in the shape used for the background function is less than 5% . the total systematic error is taken as the quadrature sum of the individual errors and ranges from 25% to 35% , depending on the channel . the branching fraction results are listed in table [ chic - result ] , where all bes results for @xmath27 branching fractions are given , including those for the two - charged track modes reported in ref . @xcite . in each case , the first error listed is statistical and the second is systematic . for comparison , we also provide the previous world averages for those channels when they exist @xcite . our branching fractions for @xmath2 , @xmath3 , @xmath4 , @xmath5 , @xmath6 and @xmath7 ( j=0,1,2 ) are the first reported measurements for these decays . the results for @xmath121 and @xmath4 are in agreement with the isospin prediction of the @xmath27 decays compared with the corresponding @xmath35 branching ratios . for the other decay modes , signals with large statistics are observed and the corresponding branching fractions are determined with precisions that are significantly better than those of existing measurements . note that our results are consistently lower than the previous measurements , sometimes by as much as a factor of two or more . we can find no obvious explanation for these discrepancies . we determine @xmath133 by fitting the combined invariant mass distribution of all of the channels discussed above to three resolution - broadened breit - wigner functions with the resolution fixed at the value of 13.8 mev , which is determined from fits to the @xmath22 and @xmath18 , and the total widths of the @xmath22 and @xmath18 fixed at the pdg values @xcite . the masses of all three @xmath27 states and the total width of the @xmath12 are left as free parameters . the results of the fit for @xmath134 ( @xmath135 mev ) and @xmath136 ( @xmath137 mev ) agree with the pdg values within errors . the fit value for @xmath133 is @xmath138 mev , where the error is statistical . the fit gives a total width for the @xmath12 that is in good agreement with the recently reported bes result @xcite . figure [ etac_mass ] shows the combined invariant mass distribution for the @xmath139 , @xmath140 , @xmath141 , and @xmath32 channels in the region of the @xmath13 , where an @xmath13 signal is evident . superimposed on the plot is a fit to the spectrum using a resolution - smeared breit - wigner line shape with a mass that is allowed to vary , a total width fixed at the pdg value of @xmath142 mev @xcite , and a fourth - order polynomial background function . the fit gives a total of @xmath143 events in the peak and has a @xmath144 , which corresponds to a confidence level of 27.9% . the mass value from the fit is @xmath145 mev , where the error is statistical . ( a fit with only the background function and no @xmath13 has a confidence level of 0.8% . ) the systematic error on the mass determination includes a possible uncertainty in the overall mass scale ( @xmath146 mev ) , which is determined from the rms average of the differences between the fitted values for @xmath134 and @xmath136 and their pdg values . the systematic errors associated with uncertainties is the particle s total widths and the experimental resolutions ( @xmath147 mev for @xmath39 and less than @xmath148 mev for @xmath133 ) are added in quadrature . the resulting masses and errors are : @xmath149 and @xmath150 the precision of our @xmath133 measurement represents a substantial improvement on the existing pdg value of @xmath151 mev @xcite . our result for @xmath39 agrees with the dm2 group s value of @xmath152 mev @xcite and is 2.4 standard deviations below the e760 group s result of @xmath153 mev @xcite . events of the type @xmath17 in a @xmath154 @xmath1 event sample are used to determine branching fractions for @xmath27 decays to four and six charged particle final states . our results for @xmath32 , @xmath85 , @xmath95 , and @xmath31 are the first measurements for these decays . the branching fractions for @xmath155 , @xmath29 , @xmath30 , and @xmath33 final states are measured with better precision and found to be consistently lower than previous measurements . @xmath133 and @xmath39 were determined using the same data sample . we thank the staffs of the bepc accelerator and the computing center at the institute of high energy physics , beijing , for their outstanding scientific efforts . this project was partly supported by china postdoctoral science foundation . the work of the bes collaboration was supported in part by the national natural science foundation of china under contract no . 19290400 and the chinese academy of sciences under contract no . h-10 and e-01 ( ihep ) , and by the department of energy under contract nos . de - fg03 - 92er40701 ( caltech ) , de - fg03 - 93er40788 ( colorado state university ) , de - ac03 - 76sf00515 ( slac ) , de - fg03 - 91er40679 ( uc irvine ) , de - fg03 - 94er40833 ( u hawaii ) , de - fg03 - 95er40925 ( ut dallas ) . r. m. baltrusaitis _ et al . _ ( mark iii collab . ) , phys . * d33 * , 629 ( 1986 ) ; j. e. gaiser _ et al . _ ( crystal ball collab . ) , phys . rev . * d34 * , 711 ( 1986 ) ; c. bagelin _ et al . _ , phys . lett . * b231 * , 557 ( 1989 ) ; and z. bai _ et al . _ ( mark iii collab . ) , phys . lett . * 65 * , 1309 ( 1990 ) . j. z. bai _ et al . _ ( bes collab . ) , phys . rev . * d58 * , 092006 ( 1998 ) . in the determination of the number of @xmath1 events , the branching ratio @xmath156 ( r. m. barnett _ et al . _ , ( particle data group ) , phys . rev . * d54 * part i ( 1996 ) ) was used . using the @xmath157 events sample and the particle identification procedures described in the text , we determine the branching fraction @xmath158 which is in good agreement with the world average @xcite .

hadronic decays of the p - wave spin - triplet charmonium states @xmath0 are studied using a sample of @xmath1 decays collected by the bes detector operating at the bepc storage ring . branching fractions for the decays @xmath2 , @xmath3 , @xmath4 , @xmath5 , @xmath6 and @xmath7 are measured for the first time , and those for @xmath8 , @xmath9 , @xmath10 and @xmath11 are measured with improved precision . in addition , we determine the masses of the @xmath12 and @xmath13 to be @xmath14 mev and @xmath15 mev . 0.1 cm plus 1pt minus 1pt

one of the interesting characteristics of the percolation model is the existence of universal quantities , which are independent of the microscopic qualities of the system . critical exponents and amplitude ratios are examples of one class of universal quantities that are only dependent on the dimensionality of the system . shape - dependent universal quantites , which depend on the shape of the boundary as well as the dimensionality of the system , have been identified in the ising model by privman and fisher @xcite , mller @xcite and kamieniarz and blte @xcite . in percolation , recent research has been focused on shape - dependent universal quantities such as the crossing probabilities @xcite and the excess number of clusters @xcite . the excess number of clusters @xmath0 on a two dimensional system is defined by @xcite @xmath8 as @xmath9 , where @xmath10 is the total number of the clusters in a system of area @xmath11 , and @xmath1 is the number of clusters per unit area in an infinite system . in ( [ definition ] ) , it is assumed that the system is at criticality , and that there are no boundaries to the system . the average density of clusters @xmath1 is a non - universal quantity , meaning that it depends on the microscopic qualities of the system and will have different values for different systems . in two - dimensional percolation , exact values of @xmath1 have been determined theoretically for bond percolation on two lattices @xcite . also , simulations have provided precise values of @xmath1 for 2d @xcite and 3d @xcite lattice percolation . while @xmath1 is a system - dependent quantity , @xmath0 is a universal quantity that is independent of the type of percolation but does depend on shape of the system boundary and the dimensionality of the system . previous work has demonstrated the shape dependence of @xmath0 for toroidal systems using @xmath12 sytems with periodic boundary conditions , for which @xmath13 has been found exactly @xcite . for an @xmath2 system , the theoretical prediction is @xmath14 @xcite , which agrees with the numerical value @xmath15 @xcite found for site and bond percolation on square and triangular lattices . for 3d lattice percolation , theoretical predictions for @xmath16 ( the excess number per unit length ) do not exist ; however , the universality of @xmath16 has been confirmed using numerical values for different @xmath17 cubic systems @xcite . in the present paper , we are interested in finding @xmath0 for percolation on the surface of a sphere , another two dimensional surface with no boundaries , for which however no theoretical prediction exists . in order to do this , we use the 2d continuum `` swiss cheese '' model of percolation , for which the critical density has recently been found to high accuracy @xcite . in addition to the spherical systems , we are interested in using this continuum model to detemine @xmath0 for @xmath2 toroidal systems and comparing it to the theoretical prediction . in order to find @xmath0 , we must also find @xmath1 , the number of clusters per unit area for the continuum percolation model . we note that continuum percolation on the surface of a sphere ( and hypersphere ) has also been studied in a recent publication @xcite , in the context of diffusion on fractal clusters . in the following section , we describe the simulations that were used to model the square and spherical system . then we present and summarize our results for the excess number of clusters and the average density of clusters in these systems . the basic `` cluster counting '' algorithm that was used for determining the number of clusters in the @xmath2 toroidal and spherical continuum systems is identical to the algorithm we used to study similar problems in lattice percolation . however , the implementation of this algorithm was quite different for the two continuum systems . the @xmath2 system with periodic boundary conditions was first divided into squares of unit area . discs , whose radius @xmath18 is equal to 0.5 , were distributed into each of the unit squares in the plane using a poisson function , where the probability @xmath19 that there are @xmath20 particles in a given volume @xmath21 is given by @xmath22 here @xmath23 is the critical density of discs ( @xmath24 for 2d continuum percolation of discs @xcite ) and @xmath25 is the volume of each of the unit squares . [ note : the critical density of discs is often referred to as @xmath1 in the continuum percolation literature , but in order to differentiate from the average density @xmath1 , we use @xmath23 here . ] the algorithms in ref.@xcite were used to generate numbers with this distribution . for each unit square , a random number @xmath20 was generated and if @xmath26 , then @xmath20 discs were placed within that square . the @xmath27- and @xmath28- coordinates of each disc were stored in two one - dimensional arrays , which were indexed by the order that the discs were distributed ( i. e. , the first sphere placed is numbered 0 , the second is numbered 1 , @xmath29 ) . the index of the first and last disc distributed in the square were also stored in two one - dimensional pointer arrays . after discs were placed in each unit square , a search was made for clusters , starting with the first disc placed in the first unit square . the search checked only the neighboring eight unit squares , as opposed to the entire system , for discs . if the distance between any two discs was less than or equal to 1 , then the two were considered to be in the same cluster . the coordinates of each disc in the cluster were stored in two one - dimensional list ( `` growth '' ) arrays , which were indexed by the order that the discs were determined to be part of the cluster . after the first disc was checked for overlapping neighbors , then subsequent discs on the list arrays were checked , in the order that they were placed on the list , for overlapping neighbors . this process was continued until a cluster stopped growing ; at which time , the same search for each of the unchecked discs in the current square was performed . after each of the discs in a square were checked , the search moved to the next square that had unchecked discs remaining . this cluster search was continued until all discs within the system were determined to be part of a cluster . when simulating the spherical systems , we encountered two areas of difficulty that were not present when using the @xmath2 system . first , we were nt able to divide the surface into smaller sub - sections because of the difficulty in producing equal area sub - sections . instead , the discs were distributed on the surface of the entire sphere , using ( [ poisson ] ) to determine the number , and the @xmath30-coordinate and the cosine of the @xmath31-coordinate for each disc were stored in two one - dimensional arrays . the entire list was searched for each neighbor check , resulting in a much slower simulation for larger spherical systems compared to the toroidal system . we began our cluster search algorithm with the first disc that was placed on the sphere . the `` great circle '' distance between two discs @xmath32 , whose coordinates are ( @xmath33 ) and ( @xmath34 ) , is given by @xmath35 which is the arc length between two points on the surface of a sphere of radius @xmath36 . two discs were considered to be overlapping if @xmath37 . the cluster search algorithm was applied to the spherical system until every disc on the sphere was checked . the second area of difficulty we encountered was in the selection of a criterion for the critical threshold that is consistant with the criterion used for a @xmath2 system . different choices ( density of discs , surface area coverage , etc . ) lead to different numbers of discs . as a result of this ambiguity , we considered two different methods to determine the average number of discs that were required . in general , the total number of the discs @xmath38 is defined as @xmath39 where @xmath40 is the area covered by each disc , @xmath41 is the coverage of the discs , and @xmath11 is the total area of the system . for a finite system , @xmath42 , where @xmath23 is the critical density of discs . first , we used `` type 1 '' discs , where we chose the number of discs per unit area on the sphere to have the same value as the @xmath2 critical system . therefore , the number of discs is @xmath43 , where @xmath36 is the radius of the sphere and @xmath24 is the critical density of discs for a flat system . for the second type of discs ( `` type 2 '' discs ) , we determined the number of discs required to keep the critical coverage the same as it was for a flat system . the area covered by each disc is calculated by @xmath44 where @xmath36 is the radius of the sphere , @xmath31 is the angle between the horizontal axis of the sphere and a point on the surface of the sphere , and @xmath45 is the angle between the edge of the disc and the horizontal axis as shown in fig . [ geo ] . in this case @xmath46 , therefore , the area covered by each type 2 disc is @xmath47 the number @xmath38 of these discs that would be required to achieve the same coverage as the @xmath2 system ( @xmath48 ) was determined using ( [ coverage ] ) , with the result that @xmath49 where @xmath36 is the radius of the sphere , where @xmath50 . using these two simulations , we were able to study planar @xmath2 systems , where @xmath51 8 , 16 , 32 , 64 , and 128 , and spherical systems with radius @xmath52 5 , 6 , 10 , and 15 . the simulations counted the number of clusters that existed within the system . numerous realizations ( @xmath53 for spherical systems and @xmath54 for planar systems ) were averaged over in order to calculate the density of clusters within each system . the random numbers used in these simulations were generated by the four - tap shift - register rule @xmath55 , where @xmath56 is the exclusive - or operation @xcite . the number of clusters @xmath10 present in a system which has area @xmath11 is expected to follow ( [ definition ] ) . using our simulations , we were able to calculate the total number of clusters present and then determine the overall density of clusters @xmath57 for each system size , which by rearranging ( [ definition ] ) is @xmath58 , or @xmath59 \\ \label{excesssphere } & n = n_c + b/(4\pi r^2 ) + \ldots \quad[{\rm spherical\ system } ] . \end{aligned}\ ] ] figure [ xs ] is a plot of @xmath20 vs. @xmath60 for each of the three systems . by fitting these plots with equations ( [ excessll ] ) and ( [ excesssphere ] ) , we were able to determine the values of @xmath0 and @xmath1 for these systems from the slope and the intercept , respectively , which are summarized in table [ results ] . several observations can be made from these results . first , the average density of clusters for the spherical systems and for the @xmath2 system is , as expected , the same ( within numerical error ) . also , the excess number of clusters @xmath0 for the planar @xmath3 continuum system is consistent with the theoretical value ( @xmath61 ) @xcite and the simulation value ( @xmath15 ) @xcite found for the @xmath3 lattice percolation system , which further confirms the universality of this quantity . finally , the values of the excess number of clusters @xmath0 for the spherical systems are different than that found for the planar system , which is expected because the shape and topology of the boundary is different . however , there is a difference in the value of @xmath0 for the two spherical systems , which is a result of the different definitions for the critical number of discs placed on the surface of the sphere . when the total number of discs @xmath38 that are placed on the sphere in each case are compared , the difference in @xmath0 becomes more understandable : @xmath62 \nonumber\\ & m = \rho_c ( 4\pi r^{2 } ) + \rho_c ( \pi/12 ) + o(1/r^{2 } \ldots ) \quad [ { \rm type\ 2\ discs } ] . \label{comparenumber}\end{aligned}\ ] ] wherethe second expression follows from a taylor series expansion of ( [ number3 ] ) . the first term in the expression for type 2 discs is exactly the number of type 1 discs present on the surface of the sphere . however , the second term shows that of the order of one more disc is present on the surface of the sphere than type 1 disc at the same ( critical ) density . these relatively small differences in the number of discs present on the sphere can cause a significant difference in the density of clusters . in general , about the critical point , one expects the density of clusters to behave as @xcite @xmath63 where @xmath64 in two dimensions , @xmath65 , and @xmath66 . then the number of clusters @xmath10 is @xmath67 therefore , if @xmath38 changes by an amount of order 1 , then @xmath10 will also change by an amount of order 1 , which is the same order as the @xmath0 term in ( [ definition ] ) . thus , the slight differences in the two definitions of the total number of discs @xmath38 even though they are asymptotically identical for large @xmath36 lead to non - zero differences in the value of @xmath0 . the universality of the excess number of clusters @xmath0 has been demonstrated using 2-d and 3-d lattice percolation , but it had never been tested with a continuum model . our result @xmath68 for the @xmath3 planar system is the same as found for the @xmath2 lattice percolation system @xcite , thus it further confirms the universality of this quantity . the excess number of clusters had never been calculated for spherical systems . by using the swiss cheese continuum model , we were able to study percolation on the surface of sphere . we found that the excess number of clusters ( @xmath69 , @xmath70 ) is slightly dependent upon the definition of the critical density of discs for the spherical system . we believe that the type 2 definition is the most reasonable as it is based upon the idea that the surface coverage is the same as the planar system , which is consistent with the fact that the surface coverage is invariant when the @xmath2 periodic system is deformed to a torus . therefore , we propose the value @xmath71 for the spherical surface . while theoretical results exist for @xmath0 for the @xmath72 periodic system , there is no prediction for the sphere . v. privman and m. e. fisher , phys . b * 30 * , 322 ( 1984 ) . b. mller , int . j. mod . c * 9 * , 1 ( 1998 ) . g. kamieniarz and h. w. j. blte , j. phys . a * 26 * , 201 ( 1993 ) . r. p. langlands , c. pichet , p. pouliot , and y. saint - aubin , j. stat . 67 * , 553 ( 1992 ) . r. p. langlands , p. pouliot , and y. saint - aubin , bull . ams * 30 * , 1 ( 1994 ) . m. aizenman , _ the i m a volumes in mathematics and its applications _ , springer , berlin , 1997 . j. l. cardy , j. phys . a * 25 * , l201 ( 1992 ) . r. m. ziff , phys . * 69 * , 2670 ( 1992 ) . hovi and a. aharony , phys . e * 53 * , 235 ( 1996 ) . hu , and c .- y . lin , phys . * 75 * , 193 ( 1995 ) . hu , c .- y . lin and j .- a . chen , phys . lett . * 77 * , 8 ( 1996 ) . g. m. t. watts , j. phys . a * 29 * , l363 , ( 1996 ) . h. t. pinson , j. stat . phys . * 75 * , 1167 ( 1994 ) . r. m. ziff , c. d. lorenz , and p. kleban , physica a * 266 * , 17 ( 1999 ) . c. d. lorenz and r. m. ziff , j. phys . a * 31 * , 8147 ( 1998 ) . r. m. ziff , s. r. finch , and v. s. adamchik , phys . * 79 * , 3447 ( 1997 ) . p. kleban and r. m. ziff , phys . b * 57 * , r8075 ( 1998 ) . h. n. v. temperley and e. h. lieb , proc . r. soc . a * 322 * , 251 ( 1971 ) . r. j. baxter , h. n. v. temperley , and s. e. ashley , proc . r. soc . a * 358 * , 535 ( 1978 ) . j. quintanilla , s. torquato , and r. m. ziff , j. phys . a * 33 * , l399 ( 2000 ) . p. jund , r. jullien , and i. campbell , cond - mat/0011494 . w. h. press , s. a. teukolsky , w. t. vetterling , and b. p. flannery , _ numerical recipes in c _ , revised 2nd ed . ( cambridge university press , 1992 ) . r. m. ziff , computers in physics * 12 * , 385 ( 1998 ) . d. stauffer and a. aharony , _ an introduction to percolation theory _ , revised 2nd . ed . ( taylor and francis , london , 1994 ) . .values of the critical average density of clusters @xmath1 and the excess number of clusters @xmath0 for two - dimensional `` swiss cheese '' model . numbers in parenthesis represent the error in the last digit . [ cols="<,<,<",options="header " , ]

monte carlo simulations were performed in order to determine the excess number of clusters @xmath0 and the average density of clusters @xmath1 for the two - dimensional `` swiss cheese '' continuum percolation model on a planar @xmath2 system and on the surface of a sphere . the excess number of clusters for the @xmath3 system was confirmed to be a universal quantity with a value @xmath4 as previously predicted and verified only for lattice percolation . the excess number of clusters on the surface of a sphere was found to have the value @xmath5 for discs with the same coverage as the flat critical system . finally , the average critical density of clusters was calculated for continuum systems @xmath6 . #

pseudoknots and pseudolinks are knots and links about which we have incomplete information . this incompleteness is expressed in diagrams by the appearance of _ precrossings _ that have no over or under designation . in @xcite , pseudoknots are defined as equivalence classes of knot or link diagrams ( called _ pseudodiagrams _ in @xcite ) where some crossing information may be missing . we picture the precrossings that lack definitive over / under information , as undecorated self - intersections . these unknown crossings can be involved in reidemeister - like moves in predictable ways . ( see figure [ rmoves ] . ) in terms of their behavior , pseudoknots act like 4-valent rigid vertex spatial graphs ( as in @xcite ) or singular knots ( as in @xcite ) . the one important difference between pseduoknots and these other objects is entirely characterized by the pr1 move . the pr1 move allows us to eliminate a single unknown crossing at a curl . the reason this move is reasonable for pseudoknots is that , regardless of whether we replace the precrossing with a positive or negative classical crossing , it can be removed with an r1 move . on the other hand , vertices or singularities are not removable in rigid vertex spatial graphs or singular knots . while several new pseduoknot invariants have been introduced @xcite , we introduce a schema for a large collection of new invariants . these invariants of pseudoknots , defined by replacing precrossings with tangles , can be adapted to serve as invariants for 4-valent rigid vertex spatial graphs and singular knots as well . in @xcite , vassiliev showed how to extend an invariant @xmath0 of links to the class of singular links . he did so by defining the _ derivative _ of an invariant as follows : @xmath1@xmath2@xmath3@xmath4 the derivative of an invariant is an invariant of singular links with a single double - point . in general the @xmath5th derivative , defined recursively as follows , is an invariant of a singular link with @xmath5 double - points . note that when @xmath6 , @xmath7 is the first derivative @xmath8 , and @xmath9 . @xmath10@xmath11@xmath12@xmath4 this idea of extending invariants by taking linear combinations of invariant values for diagrams that are related by tangle replacements can be generalized ( see @xcite ) . for instance , we could define @xmath13 recursively as follows . @xmath14@xmath15@xmath16@xmath17@xmath4 if we are careful about which invariants @xmath0 we use and how we choose our coefficients @xmath18 , @xmath19 , and @xmath20 , this equation can be used to define an invariant of singular links , rigid vertex spatial graphs , or pseudoknots and links . note that any choice of coefficients that yields an invariant of singular links is also an invariant of pseudoknots if it satisfies the following additional relation . in general , suppose @xmath0 is a polynomial or integer , real , or complex - valued invariant of knots and links that behaves nicely under and connected sum . in particular , suppose @xmath0 obeys the following property . @xmath21 here , @xmath22 and @xmath23 denote arbitrary pseudoknot or link diagrams , and @xmath24 indicates a connected sum . we observe that the jones and alexander polynomials satisfy this property , among other commonly used invariants . once a suitable link invariant has been chosen , we choose a set of oriented , rational ( 2,2)-tangles , @xmath25 . recall that , since each @xmath26 is a rational tangle , @xmath27 satisfies the symmetries shown in figure [ tangle_sym_free ] , by the flip theorem for rational tangles @xcite . some examples of oriented rational tangles are shown in figure [ example_t ] . + * vertical flip * + + * horizontal flip * + ( a)(b)(c ) + + ( d)(e)(f ) now that we have our desired link invariant and set of tangles , we define @xmath28 recursively as follows for an oriented pseudodiagram @xmath29 containing precrossing @xmath20 . @xmath30 in this definition , @xmath31 denotes the pseudodiagram where tangle @xmath32 is inserted at precrossing @xmath20 ( respecting the orientations of the tangles and the pseudodiagram ) , and the symbol @xmath33 is a variable . if a given pseudodiagram @xmath34 contains no precrossings , we define @xmath35 . the restrictions on our allowable invariants and tangles guarantee that @xmath36 is an invariant of singular links and 4-valent rigid vertex spatial graphs . to guarantee invariance under _ all _ pseudo - reidemeister moves , we need to further impose a relation among our coefficients . @xmath37 we take @xmath28 modulo a linear combination of the values of the link invariant of the denominator closures of each of our tangles to ensure that @xmath38 is invariant under pr1 , and hence , is an invariant of oriented pseudoknots and links . note that we are implicitly assuming that the value of our chosen invariant is nonzero on the denominator closure of at least one of our tangles @xmath39 . to gain an understanding of why each of the restrictions imposed above is necessary , let us prove that @xmath38 is an invariant of oriented pseudolinks . we begin by acknowledging that , since @xmath0 is a link invariant , invariance of @xmath13 for classical reidemeister moves is immediate . pr1 invariance is shown in figure [ pr1_pf ] . line 1 follows from the definition of @xmath13 , line 2 follows from our required connected sum property of the link invariant @xmath0 ( which ensures that @xmath13 behaves similarly under connected sum ) , line 3 is an algebraic distribution , and line 4 is a consequence of the relation @xmath40 . the proof of invariance under the pr2 move illustrates why our tangles @xmath26 are chosen to have the symmetries of rational tangles . a flype together with a rational tangle symmetry is required to show invariance of @xmath13 under each oriented pr2 move , pictured in figure [ pr2_pf ] . finally , pr3 invariance is straightforward . figure [ pr3_pf ] illustrates how @xmath13 is invariant under pr3 since a strand that lies entirely above or entirely below a tangle can be moved freely past the tangle . this completes our proof that @xmath13 is an invariant of pseudoknots . note that we also proved that @xmath36 is a singular link and 4-valent rigid vertex spatial graph invariant . + * this tangle insertion method can be generalized beyond rational tangle insertion by asking that the tangles satisfy the symmetries shown in figure [ tangle_sym_free ] . we shall take up this aspect of the construction in another paper . + one obvious question to ask now that we have created a schema for creating pseudoknot invariants is : how does this schema relate to known pseudoknot invariants ? one of the simplest , yet most powerful invariants of pseudoknots is the _ weighted resolution set _ , or _ were - set _ , introduced in @xcite . the were - set of a pseudodiagram is the set @xmath41 of pairs @xmath42 of knot types @xmath43 that can be realized by some choice of crossing information for the diagram s precrossings . the number @xmath44 is the probability that knot @xmath43 will be produced if crossing information is randomly chosen , where positive and negative crossings are both equally likely . it was proven that the were - set is indeed a pseudoknot invariant . how does this invariant relate to our schema ? first , we notice that choosing crossing information for a precrossing is equivalent to inserting a basic + 1 or -1 rational tangle ( i.e. tangles ( a ) and ( b ) in figure [ example_t ] ) , so let our tangle set @xmath45 consist of these two tangles . next , let @xmath0 be a knot invariant such that @xmath46 if @xmath47 is the unknot ( such as the jones polynomial or the alexander polynomial ) . then choosing the coefficients @xmath48 will satisfy the relation @xmath40 , since this amounts to @xmath49!{\hcross } ! { \hcap[1 ] } [ l]!{\hcap[-1 ] } } ) + \frac{1}{2}\mathcal{i}(\xygraph { ! { 0;/r1.0pc/ : } [ u(.5)]!{\htwist } ! { \hcap[1 ] } [ l]!{\hcap[-1 ] } } ) = \frac{1}{2}+\frac{1}{2}=1,\ ] ] so an invariant @xmath13 is determined by our ingredients , following the recipe above . the invariant we have just created with our schema is equivalent to the following composition of the were - set with @xmath0 . @xmath50 note that , if we make a different coefficient choice , we no longer recover the were - set . for instance , making the choice @xmath51 and @xmath52 would correspond to a distinct variant of the were - set where positive crossings are chosen with probability @xmath53 and negative crossings are chosen with probability @xmath54 . let us return to our original motivation for tangle insertion invariants : the derivative of an invariant . in our new framework , the tangles @xmath55 that are used to define the derivative are tangles ( a ) and ( b ) in figure [ example_t ] , respectively . their denominator closures are both the unknot , @xmath47 . the coefficient @xmath56 is 1 and @xmath57 . notice , then , that our imposed relation @xmath40 states that @xmath58 . but clearly @xmath59 . so , oddly enough , our original motivating example is not an instance of our scheme after all . this is one of the reasons why the generalization is so valuable . it allows for the creation of many new invariants of pseudoknots just as our old singular knot invariants fail to be useful . instead , let us consider the other example we mentioned above , namely : @xmath14@xmath15@xmath16@xmath17@xmath4 here , we take tangles ( a ) , ( b ) , and ( c ) from figure [ example_t ] to form the tangle set @xmath45 , and we do nt yet specify values for our coefficients . @xmath60!{\hcross } ! { \hcap[1 ] } [ l]!{\hcap[-1 ] } } ) + b\overline{\mathcal{i}}(\xygraph { ! { 0;/r1.0pc/ : } [ u(.5)]!{\htwist } ! { \hcap[1 ] } [ l]!{\hcap[-1 ] } } ) + c\overline{\mathcal{i}}(\xygraph { ! { 0;/r1.0pc/ : } [ u(0.5)r]!{\hcap[1 ] } ! { \hcap[-1 ] } [ ll]!{\hcap[1 ] } ! { \hcap[-1 ] } } ) = 1\ ] ] suppose we choose our link invariant @xmath0 to be the jones polynomial , @xmath61 . we will compute @xmath61 using the bracket polynomial , @xmath62 , via the relation @xmath63 where @xmath64 is the writhe of the link @xmath23 as in @xcite , and @xmath65 is a variable . then any choice for @xmath18 , @xmath19 , and @xmath20 satisfying @xmath66 will do , where @xmath67 is the polynomial such that @xmath68 . in particular , we could choose @xmath69 and @xmath70 . note that this choice is equivalent to simply starting with the singleton tangle set consisting of the 0-tangle , ( c ) . let us apply this invariant to a pseudo - trefoil and its mirror image , shown in figure [ tref ] . when we insert tangle ( c ) into both pseudodiagrams , we get the links shown in figure [ tref_insert ] . the link on the left is l2a1@xmath71 and the link on the right is l2a1@xmath72 . both links have the same bracket polynomial value , @xmath73 , since they are the same as unoriented links . ( see @xcite , p. 151 for this computation . ) but the writhe of l2a1@xmath71 is 2 while the writhe of l2a1@xmath72 is -2 . hence , the jones polynomial of l2a1@xmath71 is @xmath74 while the jones polynomial of l2a1@xmath72 is @xmath75 thus , the two pseudoknots shown in figure [ tref ] are distinct , since their @xmath13 values are @xmath76 and @xmath77 , respectively . another interesting pair of examples , pictured in figure [ perko ] , is derived from the famous perko pair , pictured in figure [ perko_orig ] . let us use the alexander polynomial and take our tangle set @xmath45 to be the set consisting of tangles ( a ) and ( b ) . if we insert tangle ( b ) at both precrossings in both diagrams , we recover the perko pair itself , which were shown by perko to be equivalent . on the other hand , the knot @xmath78 is obtained by inserting tangle ( a ) at one precrossing and tangle ( b ) at the other precrossing ( regardless of the order of insertion ) in both diagrams . finally , if tangle ( a ) is inserted at both precrossings in both diagrams as in figure [ perko_pos ] , the two resulting knots are distinct . one knot is determined by the alexander polynomial to be the knot @xmath79 , while the other is knot @xmath80 . hence , the invariant @xmath13 derived from the alexander polynomial and tangle set @xmath45 ( where , say , @xmath48 ) can distinguish these two pseudoknots . + * remark . * notice that , because our pseudoknot invariant was defined using the alexander polynomial and tangles ( a ) and ( b ) with the coefficient choice @xmath48 , the were - set together with the alexander polynomial can alternatively be used to distinguish the perko pseudoknots . + .,height=124 ] in our next example , we consider the pseudodiagram related to the borromean rings pictured in figure [ borr_insert ] ( on the left ) . here , we use the tangle insertion invariant @xmath13 that is defined using the jones polynomial ( computed via the bracket , with variable @xmath65 ) and take our tangle set to be the singleton set containing the tangle ( c ) . as shown in figure [ borr_insert ] , inserting the tangle ( c ) produces the whitehead link . since this link has jones polynomial @xmath81 @xmath13 can be used to prove that the original borromean pseudoknot is nontrivial . note that , if we had used a linking number invariant rather than the jones polynomial to define our pseudoknot invariant , we would not have been able to detect the nontriviality of this example since the linking number of the whitehead link is 0 . .,height=115 ] these are just a few examples to illustrate how invariants of pseudoknots can be derived from classical link invariants and tangle sets . we intend to consider many more examples in future work . given this framework for developing pseudoknot invariants , there are many questions that have yet to be be explored . we provide the reader with an initial list of open questions . 1 . are there examples of pseudoknot pairs that can only be distinguished using a tangle set @xmath45 that contains more complex tangles than the basic tangles ( a ) , ( b ) , and ( c ) in figure [ example_t ] ? for instance , can tangles ( d ) , ( e ) , or ( f ) be used to construct more powerful pseudoknot invariants ? 2 . given two arbitrary distinct pseudoknots @xmath82 and @xmath83 , does there exist a classical link invariant @xmath0 and tangle set @xmath45 such that the corresponding pseudoknot invariant @xmath13 distinguishes @xmath82 and @xmath83 ? 3 . can our invariant schema be generalized to include invariants of other sorts , e.g. the matrix of linking numbers ? ( this can be done for the case of a single precrossing . ) 4 . is there a relationship between the gauss - diagrammatic pseudoknot invariants defined in @xcite and the invariant schema presented here ? 5 . can we determine if a given reduced pseudodiagram has the fewest number of precrossings among all equivalent diagrams ? specifically , can our tangle insertion invariants be used for this purpose ? l. h. kauffman and s. lambropoulou . classifying and applying rational knots and rational tangles . _ physical knots : knotting , linking , and folding geometric objects in _ @xmath84 ( las vegas , nv , 2001 ) . 223259 , contemp . soc . , providence , ri _ ( 2002 ) .

_ the notion of a pseudoknot is defined as an equivalence class of knot diagrams that may be missing some crossing information . we provide here a topological invariant schema for pseudoknots and their relatives , 4-valent rigid vertex spatial graphs and singular knots , that is obtained by replacing unknown crossings or vertices by rational tangles . _

mass loss via stellar winds is thought to play a dominant role in the evolution of massive o - type stars , because of the loss of _ mass _ , as winds `` peel off '' the star s outermost layers ( conti 1976 ) , as well as through the associated loss of _ angular momentum _ ( e.g. langer 1998 , meynet & maeder 2002 ) . however , during the last decade , large uncertainty has been pointed out regarding our quantitative knowledge of the mass - loss rates of massive stars , as stellar winds have been revealed to be clumped , resulting in empirical rates that have been overestimated . although it had been known for decades that o - type winds are clumped ( lupie & nordsieck 1987 , eversberg et al . 1998 ) , the severity did not appear to be fully recognized until bouret et al . ( 2005 ) and fullerton et al . ( 2006 ) claimed mass - loss reductions of factors @xmath23 - 7 and @xmath220 - 130 respectively in comparison to unclumped h@xmath3 and radio mass - loss rates ( e.g. lamers & leitherer 1993 ) . the h@xmath3 diagnostics depends on the density squared , and are thus sensitive to clumping , whilst ultraviolet p cygni lines such as pv are _ in_sensitive to clumping as these depend linearly on the density . the above - mentioned bouret et al . and fullerton et al . analyses were based on models where the wind is divided into a portion of the wind containing all the material with a volume filling factor @xmath4 ( the reciprocal of the clumping factor ) , whilst the remainder of the wind is assumed to be void . this pure _ micro_-clumping approach is probably an oversimplification of the real situation , but it provides interesting insights into the potential mass - loss rate reductions . in reality , clumped winds are likely porous , with a range of clump sizes , masses , and optical depths . _ macro_-clumping and porosity have been investigated with respect to both the spectral analyses ( e.g. oskinova et al . 2007 , sunqvist et al . 2010 , surlan et al . 2012 ) as well as the radiative driving ( muijres et al . the upshot from these studies is that o star mass - loss rates may only be reduced by a moderate factor of @xmath23 ( repolust et al . 2004 , puls et al . 2008 ) , which would bring their clumping properties in agreement with those of wolf - rayet ( wr ) winds , for which similar moderate clumping factors have been derived ( hamann & koesterke 1998 ) . the latter are based on the analysis of emission line wings due to electron scattering , which have the advantage that they do not depend on detailed ionization fractions and abundances of trace elements . these moderate clumping factors would imply that massive star evolution modelling is not affected , as current state - of - the - art rotating stellar models ( e.g. georgy et al . 2011 ; brott et al . 2011 ) already employ moderately reduced rates via the theoretical relations of vink et al . ( 2000 ) . in light of the severe mass - loss reductions claimed e.g. by fullerton et al . ( 2006 ) , smith & owocki ( 2006 ) argued that the integrated mass loss from stationary stellar winds for very massive stars ( vms ) above @xmath050 may be vastly insufficient to explain their role as the progenitors of wr stars and stripped - envelope ibc supernovae . instead , smith & owocki argued that the bulk of vms mass loss is likely of an eruptive rather than a stationary nature . in particular , they highlighted the alternative option of eruptive mass loss during the luminous blue variable ( lbv ) phase . in view of the new porosity results , the arguments of smith & owocki ( 2006 ) however seem to have lost weight . furthermore , quantitative estimates on the integrated amount of eruptive mass loss are hard to obtain as both the eruption frequency , and the amounts of mass lost per eruption span a wide range ( of a factor 100 ) with lbv nebular mass estimates varying from @xmath20.1 in pcygni to @xmath210 in @xmath5car , as discussed by smith & owocki ( 2006 ) . moreover , the energies required to produce such giant mass eruptions are very high ( @xmath6erg ) , and their energy source is unknown . soker ( 2004 ) discussed that the energy and angular momentum required for @xmath5car great eruption can not be explained within a single - star scenario . whilst stationary winds in o and wr stars are ubiquitous , it is not at all clear if lbv - type objects like @xmath5car have encountered a special evolution ( such as a merger ) or if all massive stars go through eruptive mass - loss phases . on the other hand , for the most massive main - sequence wnh stars ( crowther et al . 2010 , bestenlehner et al . 2011 ) there is both theoretical and empirical evidence for strong eddington parameter @xmath7-dependent ( see definition eq.[eq_gamma ] ) mass loss ( grfener et al . 2011 ) . for vms the role of stationary mass loss has thus increased rather than decreased in recent years . in summary , the relevant roles of eruptive versus stationary mass loss seem rather uncertain and unsettled at the current time . there are ongoing debates as to whether wind clumping reduces the mass - loss rates by moderate factors of @xmath02 - 3 , such that stellar evolution would not be affected , or by more severe factors of order @xmath210 . in the latter case , line - driving would become negligible and alternatives such as eruptive mass loss would need to be considered . in order to address the relative role of wind versus eruptive mass loss , it would be beneficial to be able to calibrate either one of them . at the moment both stellar wind and eruptive mass loss could be inaccurate by factors of 10 , and possibly even more . in this letter , we attempt to alleviate this problem by presenting a methodology that involves a model - independent mass - loss indicator , the transition mass - loss rate @xmath1 located right at the transition from optically thin to optically thick stellar winds . martins et al . ( 2008 ) found two mass - loss relations for vms arches cluster stars , one for the of stars and one for the late - type wnh stars respectively . the fact that wolf - rayet stars with wnh spectral classification have optical depth larger than one has already been discussed in literature ( e.g. grfener & hamann 2008 ) , and one might thus expect to witness a transition from optically thin o - type winds to optically thick wolf - rayet winds . vink et al . ( 2011 ) discovered a sudden change in the slope of the mass - loss versus @xmath7 relation at the transition from o - type ( optically thin ) to wr - type ( optically thick ) winds . interestingly , this transition was found to occur for a wind efficiency parameter @xmath5 @xmath8 @xmath9 @xmath10/(@xmath11 ) of order unity . this key result from monte carlo modelling that the transition from o to wr - type mass loss coincides with @xmath12 , can also be found analytically ( sect.[sec_anal ] ) . and the result can be utilized to `` calibrate '' wind mass loss in an almost model independent manner ( sect.[sec_arches ] ) . netzer & elitzur ( 1993 ) and lamers & cassinelli ( 1999 ; hereafter lc99 ) give general momentum considerations for dust - driven winds ( see lc99 pages 152 - 153 ) that can also be applied to line - driven winds . the integral form of the momentum equation contains four terms ( eq . 7.5 of lc99 ) . because hydrostatic equilibrium is a good approximation for the subsonic part of the wind , and the gas pressure gradient is small beyond the sonic point , lc99 argue that the second and third terms are negligible compared to the first and fourth , resulting in : @xmath13 employing the mass - continuity equation @xmath14 , one obtains @xmath15 where @xmath16 denotes the sonic radius and @xmath17 the eddington factor with respect to the total flux - mean opacity @xmath18 : @xmath19 using the wind optical depth @xmath20 , one obtains @xmath21 where it is assumed that @xmath7 is significantly larger than one , and the factor @xmath22 is thus close to unity ( lc99 s second assumption ) , resulting in @xmath23 one can now derive a key condition for the wind efficiency number @xmath5 @xmath24 the key point of our letter is that we can employ the unique condition @xmath25 right at the transition from optically thin o - star winds to optically - thick wr winds , and obtain a _ model - independent _ @xmath26 . in other words , if we were to have an empirical data - set available that contains luminosity determinations for o and wr stars , we can obtain the transition mass - loss rate @xmath1 simply by considering the transition luminosity @xmath27 and the terminal velocity @xmath28 representing the transition point from o to wr stars : @xmath29 we note that this transition point can be obtained by purely spectroscopic means , _ independent _ of any assumptions regarding wind clumping . [ tab_models ] in the above analysis we made two assumptions that we wish to check with numerical tests involving sophisticated hydrodynamic wind models ( from grfener & hamann 2008 ) and simpler @xmath30-type velocity laws , commonly used in o / wr wind modelling . the results are compiled in tab.1 . we first confirm lc99 s first assumption through the comparison of @xmath5 , determined from @xmath26 and @xmath31 , to the approximate @xmath32 values as computed from the right - hand - side of integral in eq.[eq_lc99 ] , where @xmath33 . evidently , the values of @xmath5 and @xmath32 agree at the few percent level , and the first lc99 approximation is verified . second , we investigate the assumption that the term @xmath34 in eq.[eq_gaga ] is close to unity by numerical integration of @xmath35 . we obtain a correction factor @xmath36 , which we define by @xmath37 with this definition eq.[eq_eta ] becomes @xmath38 to compute the integral numerically , we need to obtain the density @xmath39 , and the flux - mean opacity @xmath40 in the stellar wind ( @xmath17 follows from eq.([eq_gamma ] ) . the hydrodynamic wind models of grfener & hamann ( 2008 ) have these quantities directly available . we have performed a direct computation for the first model ( hyd ) in tab.1 for the galactic wnh star wr22 ( grfener & hamann 2008 ) . for this model we obtain @xmath41 . this value is lower than , but of the order of , unity . the terminal wind speed in this model is significantly lower than the observed value for wr22 ( 980km / s vs. 1785km / s ) . consequently , our derived @xmath36 is likely on the low side . we expect @xmath42 to be connected to the ratio @xmath43 , and @xmath44 . here we follow a model - independent approach , adopting @xmath30-type velocity laws . the mean opacity @xmath45 then follows from the resulting radiative acceleration @xmath46 @xmath47 @xmath46 follows from the prescribed density @xmath39 and velocity structures @xmath48 via the equation of motion @xmath49 where we assume a grey temperature structure to compute the gas pressure @xmath50 . we note that these results are completely independent of any assumptions regarding wind porosity , or the chemical composition of the wind material . the _ only _ assumption that goes into these considerations is that the winds are radiatively driven . the resulting mean opacity @xmath45 consequently captures all physical effects that could potentially affect the radiative driving . the obtained values for the correction factor @xmath36 are summarized in tab.1 . the first three models in tab.1 represent a consistency test with the hydrodynamic model for wr22 . using a beta law with @xmath51 , and the same @xmath31 , we obtain almost exactly the same @xmath36 as for the hydrodynamic model , justifying our @xmath30-law approach , which we employ in the following . now employing the _ observed _ and therefore likely close to correct value of @xmath31 ( and a correspondingly increased @xmath26 , we obtain @xmath52 . to get a handle on the overall behaviour of this factor @xmath36 , we computed a series of wind models for a range of stellar parameters @xmath53 and @xmath54 , with wind efficiencies around the transition region ( @xmath55 ) . remarkably , the resulting values of @xmath36 depend _ only _ on the adopted values of @xmath56 and @xmath30 . for @xmath57 , we obtain @xmath58 0.7 , 0.6 respectively for @xmath59 , 1.0 , 1.5 , where the last value is probably most appropriate ( vink et al . overall , we derive values of @xmath60 in the range 0.4 - 0.8 , with a mean value of 0.6 . we note that the error on this number @xmath60 is within the uncertainty of the luminosity determinations described in the next section . for transition objects with @xmath61 we thus expect that @xmath62 , i.e. the transition between o and wr spectral types should occur at mass - loss rates of @xmath63 the fact that the correction factor is within a factor of two of our idealized approach ( eq . [ eq_transm ] ) is highly encouraging . we stress that this number is independent of any potential model deficiencies , as we have used the observed values of @xmath31 in this analysis . [ tab_arches ] llllllc ' '' '' star & subtype & @xmath64 & @xmath65 & @xmath31 & @xmath66 + & & [ @xmath67 & [ @xmath68 & [ @xmath69 & [ @xmath68 + ' '' '' f9 & wn8 - 9 & 6.35 & @xmath704.78 & 1800 & @xmath704.60 + f1 & wn8 - 9 & 6.30 & @xmath704.70 & 1400 & @xmath704.54 + f14 & wn8 - 9 & 6.00 & @xmath705.00 & 1400 & @xmath704.84 + b1 & wn8 - 9 & 5.95 & @xmath705.00 & 1600 & @xmath704.95 + f16 & wn8 - 9 & 5.90 & @xmath705.11 & 1400 & @xmath704.94 + ' '' '' f15 & o4 - 6if+ & 6.15 & @xmath705.10 & 2400 & @xmath704.92 & + f10 & o4 - 6if+ & 5.95 & @xmath705.30 & 1600 & @xmath704.95 & @xmath1 + ' '' '' f18 & o4 - 6i & 6.05 & @xmath705.35 & 2150 & @xmath704.98 + f21 & o4 - 6i & 5.95 & @xmath705.49 & 2200 & @xmath705.09 + f28 & o4 - 6i & 5.95 & @xmath705.70 & 2750 & @xmath705.18 + f20 & o4 - 6i & 5.90 & @xmath705.42 & 2850 & @xmath705.25 + f26 & o4 - 6i & 5.85 & @xmath705.73 & 2600 & @xmath705.26 + f32 & o4 - 6i & 5.85 & @xmath705.90 & 2400 & @xmath705.22 + f33 & o4 - 6i & 5.85 & @xmath705.73 & 2600 & @xmath705.26 + f22 & o4 - 6i & 5.80 & @xmath705.70 & 1900 & @xmath705.17 + f23 & o4 - 6i & 5.80 & @xmath705.65 & 1900 & @xmath705.17 + f29 & o4 - 6i & 5.75 & @xmath705.60 & 2900 & @xmath705.41 + f34 & o4 - 6i & 5.75 & @xmath705.77 & 1750 & @xmath705.19 + f40 & o4 - 6i & 5.75 & @xmath705.75 & 2450 & @xmath705.33 + f35 & o4 - 6i & 5.70 & @xmath705.76 & 2150 & @xmath705.33 + designations , subtypes , luminosities ( @xmath71 ) , mass - loss rates ( @xmath26 ) , and terminal wind velocities ( @xmath31 ) according to martins et al . the 6th column indicates the mass - loss rate where @xmath5=1 . for the arches cluster , we obtain @xmath72 . martins et al . ( 2008 ) analyzed 28 vms in the arches cluster , with equal numbers of o - type supergiants and nitrogen - rich wolf - rayet ( wnh ) stars ( sometimes called `` o stars on steroids '' ) . for the o - type supergiants , we expect the winds to be optically thin , whilst the wnh stars should have optically thick winds . here we postulate that the o4 - 6if@xmath73 represent the transition point where the optical depth crosses unity . in tab.2 , we compiled a subset of 20 stars , skipping those objects with a he - enriched surface composition . the objects are sorted with respect to their spectral subtypes , and within each subtype bin with respect to their luminosity . together with the basic stellar and wind parameters derived by martins et al . ( 2008 ) , we list the mass - loss rate for which @xmath74 if the stars would have a wind efficiency of exactly 1 . the values listed in tab.2 show that there is a transition between o and wr spectral types . the _ spectroscopic _ transition for spectral subtypes o4 - 6if+ occurs at @xmath75 and @xmath76 . this is the resulting transition mass - loss rate for the arches cluster stars . its determined value does not depend on model uncertainties involving issues such as wind clumping . the only remaining uncertainties are due to uncertainties in the terminal velocity and the stellar luminosity @xmath71 . the latter results from errors in the distances and reddening parameters , as well as the determination of effective temperatures from non - lte model atmospheres . if the derived value for @xmath1 is compared with empirically determined mass - loss rates , the uncertainties in distances and reddening parameters nearly cancel , as for empirical mass - loss rates based on recombination line analyses @xmath77 , while @xmath78 . to estimate uncertainties in the effective temperature scale for o stars we can use historical values from the last four decades ( e.g. panagia et al . 1973 , martins et al . 2005 ) as an indicator for potential systematic errors in the inclusion / neglect of certain micro - physics ( line blanketing , wind effects , etc . ) , the best error estimate is @xmath210% in effective temperature , leading to potential errors in the luminosity of at most @xmath240% . this is several factors smaller than the order - of - magnitude uncertainties in mass - loss rates due to clumping and porosity . in other words our simple equation ( eq.[eq_transm ] ) is of tremendous value in calibrating stellar wind mass loss , and assessing its role in the mass loss during the evolution of massive stars ( sect.[sec_disc ] ) . how does our determination of the transition mass - loss rate compare to other methods ? let us first compare our transition mass - loss value to the mass - loss rates of martins et al . martins et al . use the non - lte cmfgen code by hillier & miller ( 1998 ) , employing a micro - clumping approach with a volume filling factor @xmath79 for their k - band analysis . their values are in good agreement with our transition mass - loss rates for the objects at the boundary between o and wr ( see their table 2 ) . this is unlikely to be a coincidence . according to our findings in sect.[sec_mod ] , we expect mass - loss rates of the order of @xmath80 for the transition objects , i.e. @xmath81 . we also compare the transition mass - loss rate to the oft - used theoretical mass - loss relation of vink et al . ( 2000 ) , for which we find @xmath82 for an assumed stellar mass @xmath83 . this number is within 0.2 dex from the transition mass - loss rate @xmath84 . the comparison is hardly compromised as a result of the vink et al . dependence on stellar mass , as for masses in the range 40 - 80@xmath85 , the vink et al . mass - loss rate varies by at most 0.04 dex . in summary , we have three independent mass - loss rate determinations that agree within a factor of two . this means that our concept of the transition mass - loss rate has indeed been able to test the accuracy of current mass - loss estimates by stellar winds . now that we have calibrated stellar wind mass loss in the high mass and luminosity regime , we assess the role of stellar wind mass loss for massive star evolution . for a 60@xmath85 star , the main sequence lifetime is @xmath03 myrs ( e.g. weidner & vink 2010 and references therein ) . with a stationary mass - loss rate of @xmath86 @xmath87 as derived for the transition mass - loss rate in sect.[sec_arches ] , this means such an object will lose @xmath030 , i.e. half its initial mass , already on the main sequence during core hydrogen burning . we have not yet addressed the new concept of @xmath7 dependent mass loss , nor any additional stellar wind mass loss during the subsequent core - helium burning wr phase . in other words , solar - metallicity 60@xmath85 stars are expected to lose the bulk of their initial masses through stellar winds , leaving very little ( if any ) space for additional ( e.g. eruptive ) mass loss . it is plausible that the strong - winded vms remain on the blue side of the hr diagram , without ever entering an eruptive luminous blue variable ( lbv ) or red supergiant ( rsg ) phase . current wisdom thus suggests that solar metallicity vms likely `` evaporate '' primarily as the result of stationary wind mass loss , without the _ necessity _ of additional eruptive mass loss . however , finding out if eruptive mass loss might play an additional role remains an interesting exercise , especially for the lower initial mass and sub - solar metallicity ranges , as their story might be expected to be different . moreover , we know @xmath5car analogs and supernova impostors exist in external galaxies ( e.g. van dyck et al . 2005 , pastorello et al . 2010 , kochanek et al . 2012 ) . contrary to the most massive stars , stars below @xmath240 likely evolve into the rsg , yellow super / hypergaint , and/or lbv regimes of the stellar hr diagram ( see e.g. vink 2009 ) . we note that the vink et al . ( 2000 ) main - sequence mass - loss rates currently in use in stellar models ( e.g. brott et al . 2011 ) for lower mass `` normal '' 20 - 60@xmath85 o stars are also already reduced by a factor 2 - 3 in comparison to previous unclumped empirical rates . there is currently no particular reason to assume they are still overestimated . the results presented here certainly boost confidence in the mass - loss rates currently in use , although they remain uncalibrated for the lower mass regime . one should also realize when working down the mass range , starting from our 60@xmath85 calibrator star , that the mass - loss rates drop significantly below @xmath86 @xmath87 . its effects on stellar evolution remain significant due to the longer evolutionary timescales for lower mass objects and the fact that it is the multiplication of the mass - loss rate times the duration that is relevant . this is especially relevant for angular momentum evolution , possibly down to stellar masses as low as 10 - 15 ( vink et al . 2010 ) .

a debate has arisen regarding the importance of stationary versus eruptive mass loss for massive star evolution . the reason is that stellar winds have been found to be clumped , which results in the reduction of unclumped empirical mass - loss rates . most stellar evolution models employ theoretical mass - loss rates which are _ already _ reduced by a moderate factor of @xmath02 - 3 compared to non - corrected empirical rates . a key question is whether these reduced rates are of the correct order of magnitude , or if they should be reduced even further , which would mean that the alternative of eruptive mass loss becomes necessary . here we introduce the transition mass - loss rate @xmath1 between o and wolf - rayet ( wr ) stars . its novelty is that it is model independent . all that is required is postulating the _ spectroscopic _ transition point in a given data - set , and determining the stellar luminosity , which is far less model dependent than the mass - loss rate . the transition mass - loss rate is subsequently used to calibrate stellar wind strength by its application to the of / wnh stars in the arches cluster . good agreement is found with two alternative modelling / theoretical results , suggesting that the rates provided by current theoretical models are of the right order of magnitude in the @xmath250 mass range . our results do not confirm the specific _ need _ for eruptive mass loss as luminous blue variables , and current stellar evolution modelling for galactic massive stars seems sound . mass loss through alternative mechanisms might still become necessary at lower masses , and/or metallicities , and the _ quantification _ of alternative mass loss is desirable .

in what follows @xmath9 is a variable vector in @xmath1 ( @xmath10 ) . as usual , @xmath11:={\ensuremath{\mathbb{r}}}[x_1,\ldots , x_d]$ ] denotes the ring of polynomials in variables @xmath12 and coefficients in @xmath13 a subset @xmath0 of @xmath1 which can be represented by @xmath14 for @xmath15 $ ] ( @xmath3 ) is said to be an _ elementary closed semi - algebraic set _ in @xmath16 clearly , the number @xmath17 from is not uniquely determined by @xmath7 let us denote by @xmath18 the minimal @xmath17 such that is fulfilled for appropriate @xmath15.$ ] analogously , a subset @xmath19 of @xmath1 which can be represented by @xmath20 for some @xmath21 $ ] ( @xmath3 ) is said to be an _ elementary open semi - algebraic set _ in @xmath16 the quantity @xmath22 associated to @xmath19 is introduced analogously to @xmath23 the system of polynomials @xmath2 from ( resp . ) is said to be a _ polynomial representation _ of @xmath0 ( resp . @xmath19 ) . from the well - known theorem of brcker and scheiderer ( see ( * ? ? ? * chapter 5 ) , and and the references therein ) it follows that , for @xmath0 and @xmath19 as above , the following inequalities are fulfilled : @xmath24 both of these inequalities are sharp . it should be emphasized that all known proofs of and are highly non - constructive . the main aim of this paper is to provide constructive upper bounds for @xmath18 and @xmath22 for certain classes of @xmath0 and @xmath19 ; see also @xcite , , , @xcite , , and for previous results on this topic . we also mention that constructive results on polynomial representations of special semi - algebraic sets are related to polynomial optimization ; see @xcite , @xcite , @xcite , @xcite , and @xcite . let @xmath15 $ ] and let @xmath25 be non - empty . the assumptions of our main theorems are formulated in terms of the following functionals , which depend on @xmath2 . the functional @xmath26 determines the set of constraints defining @xmath0 which are `` active '' in @xmath27 furthermore , we define @xmath28 where @xmath29 stands for the cardinality . the geometric meaning of @xmath30 and @xmath31 can be illustrated by the following special situation . let @xmath0 be a @xmath32-dimensional polytope with @xmath17 facets ( see for information on polytopes ) . then @xmath0 can be given by with all @xmath33 having degree one ( the so - called _ h - representation _ ) . in this case @xmath30 is the maximal number of facets of @xmath0 having a common vertex and @xmath31 is the set consisting of those vertices of @xmath0 which are contained in the maximal number of facets of @xmath7 if the polytope @xmath0 is _ simple _ ( that is , each vertex of @xmath0 lies in precisely @xmath32 facets ) , then @xmath34 and @xmath31 is the set of all vertices of @xmath7 now we are ready to formulate our main results . [ main : n+1 ] let @xmath21 $ ] , @xmath35 , and @xmath36 assume that @xmath0 is non - empty and bounded , and @xmath37 then the following inequalities are fulfilled : @xmath38 furthermore , there exists an algorithm that gets @xmath2 and returns @xmath8 polynomials @xmath39 $ ] satisfying @xmath40 and @xmath41 @xmath42 in the case when @xmath31 is finite theorem [ main : n+1 ] can be improved . [ main : n ] let @xmath21 $ ] , @xmath35 , and @xmath36 assume that @xmath0 is non - empty and bounded , @xmath43 is finite , and @xmath44 . then the following inequalities are fulfilled : @xmath45 furthermore , there exists an algorithm that gets @xmath2 and @xmath46 and returns @xmath5 polynomials @xmath47 satisfying @xmath48 and @xmath49 @xmath42 below we discuss existing results and problems related to theorems [ main : n+1 ] and [ main : n ] . let @xmath0 be a convex polygon in @xmath50 with @xmath17 edges , which is given by with all @xmath33 having degree one . bernig showed that setting @xmath51 one can construct a strictly concave polynomial @xmath52 vanishing on all vertices of @xmath0 which satisfies @xmath53 ; see fig . [ simp : polyt:2d : fig ] . as it will be seen from the proof of theorem [ main : n ] , for the case @xmath54 and @xmath0 as in theorem [ main : n ] we also set @xmath55 and choose @xmath56 in such a way that it vanishes on each point of @xmath57 and the set @xmath58 approximates @xmath0 sufficiently well ; see fig . [ fig : semi - alg:2d ] . however , since @xmath0 from theorem [ main : n ] is in general not convex , the construction of @xmath56 requires a different idea . the statement of theorem [ main : n ] concerned with @xmath19 and restricted to the cases @xmath59 and @xmath60 , @xmath61 ( with slightly different assumptions on @xmath19 ) was obtained by bernig . c0.6 mm ( 200,50 ) ( 60,2 ) ( 0,2 ) ( 150,2 ) ( 125,25 ) ( 20,25)@xmath58 ( 80,25)@xmath62 ( 170,25)@xmath0 + c0.7 mm ( 200,50 ) ( 60,2 ) ( 0,2 ) ( 150,2 ) ( 125,25 ) ( 20,25)@xmath58 ( 80,25)@xmath62 ( 170,25)@xmath0 + the study of @xmath18 for the case when @xmath0 is a polyhedron of an arbitrary dimension was initiated by grtschel and henk . in it was noticed that @xmath63 for every @xmath32-dimensional polytope @xmath7 on the other hand , bosse , grtschel , and henk gave an upper bound for @xmath18 which is linear in @xmath32 for the case of an arbitrary @xmath32-dimensional polyhedron @xmath7 in particular , they showed that @xmath64 if @xmath0 is @xmath32-dimensional polytope . in the following conjecture was announced . [ bgh : conj ] for every @xmath32-dimensional polytope @xmath0 in @xmath1 the equality @xmath65 holds . @xmath42 this conjecture has recently been confirmed for all simple @xmath32-dimensional polytopes ; see . [ avehenkthm ] let @xmath0 be a @xmath32-dimensional simple polytope then @xmath66 furthermore , there exists an algorithm that gets polynomials @xmath2 ( @xmath3 ) of degree one satisfying @xmath67 and returns @xmath32 polynomials @xmath68 satisfying @xmath69 @xmath42 elementary closed semi - algebraic sets @xmath25 with @xmath34 can be viewed as natural extensions of simple polytopes in the framework of real algebraic geometry . thus , we can see that theorem [ avehenkthm ] is a consequence of theorem [ main : n ] . [ cube : p : rep ] illustrates theorem [ avehenkthm ] for the case when @xmath0 is a three - dimensional cube . this figure can also serve as an illustration of theorem [ main : n ] with the only difference that in theorem [ main : n ] the set @xmath70 does not have to be convex anymore . c0.5 mm ( 132,160 ) ( 35,55 ) ( -15,30 ) ( 90,30 ) ( 35,120 ) ( -15,90 ) ( 90,90 ) ( 35,0 ) ( 5,10 ) ( 45,55)@xmath0 ( -5,30)@xmath71 ( 100,30)@xmath62 ( 45,120)@xmath72 ( -5,90)@xmath73 ( 100,90)@xmath74 ( 45,0)@xmath75 + while proving our main theorems we derive the following approximation results which can be of independent interest . the _ hausdorff distance _ @xmath76 is a metric defined on the space of non - empty compact subsets of @xmath1 by the equality @xmath77 see . [ approx : thm1 ] let @xmath15,$ ] @xmath25 , and @xmath78 assume that @xmath0 is non - empty and bounded . then there exists an algorithm that gets @xmath2 and @xmath79 and returns a polynomial @xmath80 $ ] such that @xmath81 , @xmath82 , and the hausdorff distance from @xmath0 to @xmath83 is at most @xmath84 @xmath42 [ approx : thm2 ] let @xmath15,$ ] @xmath25 , and @xmath78 assume that @xmath0 is non - empty and bounded , @xmath85 is finite , and @xmath86 then there exists an algorithm that gets @xmath2 , @xmath57 , and @xmath79 and returns a polynomial @xmath80 $ ] such that @xmath81 , @xmath82 , the hausdorff distance from @xmath0 to @xmath83 is at most @xmath87 , and @xmath88 for every @xmath89 @xmath42 we note that some further results on approximation by sublevel sets of polynomials can be found in @xcite , @xcite , and . the paper has the following structure . section [ prelim : sect ] contains preliminaries from real algebraic geometry . in section [ approx : sect ] we obtain approximation results ( including theorems [ approx : thm1 ] and [ approx : thm2 ] ) . finally , in section [ main : proofs : sect ] the proofs of theorems [ main : n+1 ] and [ main : n ] are presented . in the beginning of the proofs of theorems [ main : n+1 ] and [ main : n ] one can find the formulas defining the polynomials @xmath90 ( see and ) as well as sketches of the main arguments . the origin and the euclidean norm in @xmath1 are denoted by @xmath91 and @xmath92 respectively . we endow @xmath1 with its euclidean topology . by @xmath93 we denote the closed euclidean ball in @xmath1 with center at @xmath94 and radius @xmath95 the interior ( of a set ) is abbreviated by @xmath96 we also define @xmath97 where @xmath98 is the set of all natural numbers . a set @xmath99 given by @xmath100 where @xmath101 and @xmath102 $ ] , is called _ semi - algebraic_. an expression @xmath103 is called a _ first - order formula over the language of ordered fields with coefficients in @xmath104 _ if @xmath103 is a formula built with a finite number of conjunctions , disjunctions , negations , and universal or existential quantifier on variables , starting from formulas of the form @xmath105 or @xmath106 with @xmath107 $ ] ; see . the _ free variables _ of @xmath103 are those variables , which are not quantified . a formula with no free variables is called a _ sentence_. each sentence is is either true or false . the following proposition is well - known ; see also and . [ semialg : over : formula ] let @xmath103 be a first - order formula over the language of ordered fields with coefficients in @xmath104 and free variables @xmath108 then the set @xmath109 consisting of all @xmath110 for which @xmath103 is true , is semi - algebraic . @xmath42 a real valued function @xmath111 defined on a semi - algebraic set @xmath112 is said to be a _ semi - algebraic function _ if its graph is a semi - algebraic set in @xmath113 the following theorem presents _ ojasiewicz s inequality _ ; see @xcite and . [ loj ] let @xmath112 be non - empty , bounded , and closed semi - algebraic set in @xmath16 let @xmath114 and @xmath115 be continuous , semi - algebraic functions defined on @xmath112 and such that @xmath116 then there exist @xmath117 and @xmath118 such that @xmath119 for every @xmath120 @xmath42 considering algorithmic questions we use the following standard settings ; see ( * ? ? ? * chapter 8.1 ) . it is assumed that a polynomial in @xmath11 $ ] is given by its coefficients and that a finite list of real coefficients occupies finite memory space . furthermore , arithmetic and comparison operations over reals are assumed to be atomic , i.e. , computable in one step . the following well - known result is relevant for the constructive part of our theorems ; see . [ dec : problem ] let @xmath103 be a sentence over the language of ordered fields with coefficients in @xmath104 . then there exists an algorithm that gets @xmath103 and decides whether @xmath103 is true or false . the following proposition ( see ) presents a characterization of the convergence with respect to the hausdorff distance . [ hausd : convergence ] a sequence @xmath121 of compact convex sets in @xmath1 converges to a compact set @xmath112 in the hausdorff distance if and only if the following conditions are fulfilled : 1 . every point of @xmath112 is a limit of a sequence @xmath122 satisfying @xmath123 for every @xmath124 2 . if @xmath125 is a strictly increasing sequence of natural numbers and @xmath126 is a convergent sequence satisfying @xmath127 ( @xmath128 ) , then @xmath129 converges to a point of @xmath112 , as @xmath130 3 . the set @xmath131 is bounded . @xmath42 let @xmath15.$ ] the following theorem states that for the case when @xmath25 is non - empty and bounded , appropriately relaxing the inequalities @xmath132 , which define @xmath0 , we get a bounded semi - algebraic set that approximates @xmath0 arbitrarily well . let us define @xmath133 with @xmath134 and @xmath135 [ semi : approx ] let @xmath15 $ ] , @xmath25 , and @xmath78 assume that @xmath0 is non - empty and bounded . then there exists an algorithm that gets @xmath2 and returns values @xmath134 and @xmath136 such that the following conditions are fulfilled : 1 . [ boundedness : part ] @xmath137 is bounded for @xmath138 2 . [ convergence : part ] @xmath139,$ ] converges to @xmath0 in the hausdorff distance , as @xmath140 @xmath42 first we show the existence of @xmath141 and @xmath142 from the assertion , and after this we show that these two quantities are constructible . let us derive the existence of @xmath141 and @xmath142 satisfying condition [ boundedness : part ] . since @xmath0 is bounded , after replacing @xmath0 by an appropriate homothetical copy , we may assume that @xmath143 by proposition [ semialg : over : formula ] , the function @xmath144 is semi - algebraic . we also have @xmath145 for all @xmath146 with @xmath147 furthermore , the set @xmath137 can be expressed with the help of @xmath111 by @xmath148 for @xmath149 the function @xmath150 is positive and non - increasing . using proposition [ hausd : convergence ] it can be shown that @xmath151 is continuous . moreover , in view of proposition [ semialg : over : formula ] , we see that @xmath151 is semi - algebraic . in the case @xmath152 condition [ boundedness : part ] is fulfilled for @xmath153 and @xmath154 in the opposite case we have @xmath155 , as @xmath156 then @xmath157 is a continuous semi - algebraic function on @xmath158 $ ] with @xmath159 if and only if @xmath160 thus , applying theorem [ loj ] to the functions @xmath161 and @xmath162 defined on @xmath158 $ ] , we see that there exist @xmath134 and @xmath163 such that @xmath164 for every @xmath165.$ ] consequently @xmath166 for every @xmath167 the latter implies that @xmath168 and condition [ boundedness : part ] is fulfilled for @xmath141 as above and @xmath169 now we show that condition [ boundedness : part ] implies condition [ convergence : part ] . assume that condition [ boundedness : part ] is fulfilled . then the set @xmath137 is bounded for all @xmath170.$ ] hence @xmath171 is well defined for all @xmath170.$ ] consider an arbitrary sequence @xmath172 with @xmath173 $ ] and @xmath174 as @xmath175 using proposition [ hausd : convergence ] we can see that @xmath176 as @xmath130 consequently , condition [ convergence : part ] is fulfilled . finally we show that @xmath142 and @xmath141 are constructible . for determination of @xmath141 one can use the following `` brute force '' procedure . procedure : : : determination of @xmath177 input : : : @xmath178.$ ] output : : : a number @xmath134 such that for some @xmath179 the set @xmath180 is bounded . 1 . set @xmath181 2 . for @xmath182 introduce the first - order formula @xmath183 with free variables @xmath184 3 . test the existence of @xmath179 for which @xmath180 is bounded . more precisely , determine whether the sentence @xmath185 is true or false ( cf . theorem [ dec : problem ] ) . 4 . if @xmath186 is true , return @xmath141 and stop . otherwise set @xmath187 and go to step 2 . in view of the conclusions made in the proof , the above procedure terminates after a finite number of iterations . for determination of @xmath142 we can use a similar procedure . we start with @xmath188 and assign @xmath189 at each new iteration , terminating the cycle as long as @xmath180 is bounded . [ bounded : extension : remark ] we wish to show theorem [ semi : approx ] can not be improved by setting @xmath190 since @xmath191 may be unbounded for all @xmath135 let us consider the following example . let @xmath192 @xmath193 @xmath194 and @xmath195 then the set @xmath196 is bounded . in fact , if @xmath197 then the term @xmath198 , appearing in the definition of @xmath199 , is positive . but the remaining terms @xmath200 and @xmath201 can not vanish simultaneously . hence , @xmath202 for every @xmath203 with @xmath197 which shows that @xmath204 furthermore , since @xmath205 we see that @xmath0 has non - empty interior ( which shows that our example is non - degenerate enough ) . let us show that @xmath206 is unbounded for every @xmath135 for @xmath207 with @xmath208 one has @xmath209 and @xmath210 as @xmath211 ; see also fig . [ bd : ubd : fig ] . this implies unboundedness of @xmath212 throughout the rest of the paper we shall use the following polynomials associated to @xmath15.$ ] for @xmath134 , @xmath213 , and @xmath214 we define @xmath215 if @xmath85 is finite , we define @xmath216 where @xmath217 [ sublevel : approx : part ] let @xmath15 $ ] , @xmath25 , and @xmath78 assume that @xmath0 is non - empty and bounded . then for every @xmath79 , @xmath134 , @xmath218 and @xmath214 satisfying @xmath219 the polynomial @xmath220 fulfills the relations @xmath221 furthermore , there exists an algorithm that gets @xmath2 , @xmath79 , and @xmath134 and constructs @xmath222 $ ] satisfying and . inclusions @xmath223 and @xmath224 follow from . it remains to show the inclusion @xmath225 assume that @xmath226 then @xmath227 consequently @xmath228 or equivalently , @xmath229 . hence @xmath230 now let us discuss the constructibility of @xmath231 it suffices to show the constructibility of @xmath232 satisfying . for determination of @xmath232 we iterate starting with @xmath233 , set @xmath234 at each new step , and use , reformulated as a first - order formula , as a condition for terminating the cycle . c0.9 mm ( 80,70 ) ( 0,2 ) ( 70,10)@xmath235 ( 10,70)@xmath236 ( 36,38)@xmath0 + one can see that theorem [ approx : thm1 ] from the introduction is a direct consequence of theorem [ semi : approx ] and lemma [ sublevel : approx : part ] . [ 07.12.10,11:53 ] let @xmath237 , @xmath25 , and @xmath238 . assume that @xmath0 is non - empty and bounded , @xmath43 is finite , and @xmath37 then there exists an algorithm that gets @xmath239 @xmath240 @xmath134 , and @xmath79 and returns @xmath80 $ ] fulfilling the relations @xmath241 furthermore , @xmath242 can be defined by @xmath243 where @xmath244 , @xmath213 , and @xmath217 @xmath42 analogously to the proof of theorem [ semi : approx ] , we first show the existence of @xmath242 from the assertion and then we derive the constructive part of the theorem . we fix @xmath232 and @xmath245 satisfying and and set @xmath246 let us derive the inclusions @xmath81 and @xmath82 . first we show that @xmath247 let @xmath248 since @xmath249 for every @xmath250 the set @xmath251 is properly contained in @xmath252 . consequently , for every @xmath250 we get @xmath253 thus , is fulfilled . therefore we can fix @xmath254 with @xmath255 in view of and the finiteness of @xmath256 we can fix @xmath257 such that @xmath258 and @xmath259 for all @xmath260 with @xmath261 let us consider an arbitrary @xmath262 we show that , for an appropriate choice of @xmath263 and @xmath264 we have @xmath265 and the latter inequality is strict for @xmath266 _ case a : _ @xmath267 let us fix @xmath268 such that @xmath269 since @xmath250 , we have @xmath270 furthermore , due to the choice of @xmath271 equality is attained if and only if @xmath272 let @xmath273 be an arbitrary scalar satisfying @xmath274 applying theorem [ loj ] to the functions @xmath275 and @xmath276 restricted to @xmath277 , we have @xmath278 for appropriate parameters @xmath279 and @xmath280 independent of @xmath27 in view of the choice of @xmath281 we deduce @xmath282 where @xmath283 and @xmath284 we have @xmath285 in view of , for all sufficiently large @xmath263 the inequality @xmath286 is fulfilled . assuming that holds , and taking into account , we have @xmath287 now assume that @xmath203 lies in @xmath288 then , if @xmath289 satisfies , we get @xmath290 _ case b : _ @xmath291 then @xmath292 for every @xmath293 from the definition of elementary symmetric functions and the assumptions it easily follows that @xmath294 let us choose @xmath295 with @xmath296 thus , we get the bounds @xmath297 and @xmath298 in view of , for all sufficiently large @xmath263 the inequality @xmath299 is fulfilled . assuming that is fulfilled , we obtain @xmath290 now we show the inclusion @xmath300 . consider an arbitrary @xmath301 then @xmath302 which is equivalent to @xmath303 the latter implies that @xmath304 and therefore @xmath305 we have @xmath306 the above estimate for @xmath307 together with the estimate @xmath308 and implies that @xmath309 if @xmath289 fulfills the inequality @xmath310 since @xmath311 , is fulfilled if @xmath263 is large enough . thus , we obtain that the inequality @xmath312 holds for all sufficiently large @xmath313 now we show the constructive part of the assertion . we present a sketch of a possible procedure that determines @xmath314 it suffices to evaluate the parameters @xmath315 and @xmath281 involved in the definition of @xmath242 . constructibility of @xmath232 and @xmath245 follows from lemma [ sublevel : approx : part ] . let us apply theorem [ dec : problem ] in the same way as in the previous proofs . determine the following parameters in the given sequence . we can determine @xmath316 satisfying for an appropriate @xmath317 and all @xmath318 using the same idea as in the procedure for determination of @xmath141 in the proof of theorem [ semi : approx ] . a parameter @xmath281 satisfying is constructible in view of theorem [ dec : problem ] ( by means of iteration procedure which we also used in the previous proofs ) . an appropriate @xmath289 can be easily found from inequalities , , and . thus , for evaluation of @xmath289 we should first find the parameters @xmath319 and @xmath320 appearing in , , and . the parameters @xmath254 , @xmath321 , and @xmath295 are determined by means of , , and . one can see that theorem [ approx : thm2 ] from the introduction is a straightforward consequence of theorem [ semi : approx ] and theorem [ 07.12.10,11:53 ] . the parameters @xmath322 involved in the statements of this section were computed with the help of the theorem [ dec : problem ] . in contrast to this , in general it is not possible to compute @xmath46 exactly , since evaluation of @xmath46 would involve solving a polynomial system of equations . this explains why in the statement of theorem [ 07.12.10,11:53 ] the set @xmath46 is taken as a part of the input . the parameters @xmath232 and @xmath281 from lemma [ sublevel : approx : part ] and theorem [ 07.12.10,11:53 ] , respectively , are upper bounds for certain polynomial programs . in fact , by the parameter @xmath213 is a common upper bound for the optimal solutions of @xmath17 non - linear programs @xmath323 with constraints @xmath324 , @xmath325 from the proof of theorem [ 07.12.10,11:53 ] we see that @xmath281 can be any number satisfying @xmath326 hence @xmath327 is an upper bound for the optimal solution of the polynomial program @xmath328 with @xmath329 unknowns ( which are coordinates of @xmath330 and @xmath331 ) and the @xmath332 constraints @xmath333 and @xmath334 , @xmath335 the same observations apply also to the parameters @xmath254 and @xmath295 from the proof of theorem [ 07.12.10,11:53 ] , which are used for determination of @xmath289 . in this respect we notice that upper bounds of polynomial programs can be determined using convex relaxation methods ; see @xcite , @xcite , and @xcite . given @xmath3 , @xmath336 and @xmath337 the _ @xmath245-th elementary symmetric function _ in variables @xmath338 is defined by @xmath339 we also put @xmath340 [ berniglemma ] let @xmath337 with @xmath341 then the following statements hold : a. [ nonstrict : part ] @xmath342 if and only if @xmath343 . b. [ strict : part ] @xmath344 if and only if @xmath345 . the necessities of both of the parts are trivial . let us prove the sufficiencies . we introduce the polynomial @xmath346 whose roots are the the values @xmath347 by _ vieta s formulas _ , we have @xmath348 thus , if @xmath349 for every @xmath350 , then all coefficients of @xmath351 are non - negative , while the coefficient at @xmath352 is equal to one . it follows that @xmath351 can not have strictly positive roots . hence @xmath353 for all @xmath354 which shows the sufficiency of part [ nonstrict : part ] . now assume that the strict inequality @xmath355 holds for every @xmath356 then @xmath357 i.e. , zero is not a root of @xmath358 and , using the sufficiency of part [ nonstrict : part ] , we arrive a the strict inequalities @xmath359 this shows the sufficiency in part [ strict : part ] . proposition [ berniglemma ] was noticed by bernig , who derived it from _ descartes rule of signs_. our elementary proof ( slightly ) extends the arguments given in . [ determ : n ] let @xmath15 $ ] and @xmath360 assume that @xmath0 is non - empty and bounded . then there exists an algorithm which gets @xmath2 and returns @xmath361 since @xmath0 is bounded , we have @xmath362 we suggest the following procedure for evaluation of @xmath361 procedure : : : evaluation of @xmath30 input : : : @xmath15.$ ] output : : : @xmath30 1 . for @xmath363 introduce the formula @xmath364 with free variables @xmath365 2 . set @xmath366 3 . [ iter : begin ] introduce the formula @xmath367 with free variables @xmath365 4 . verify whether the sentence @xmath368 is true or not . if @xmath186 is true and @xmath249 set @xmath369 and go to step [ iter : begin ] . 6 . if @xmath186 is true and @xmath370 return @xmath5 and stop . if @xmath186 is false , set @xmath371 , return @xmath5 , and stop it is not hard to see that the above procedure terminates in a finite number of steps and returns @xmath361 as in the previous proofs , we first show the existence of @xmath372 from the assertion and then discuss the algorithmic part . we define @xmath373 by the formula @xmath374 where @xmath214 , @xmath134 , and @xmath213 will be fixed later . ( we recall that @xmath375 is defined by . ) let us first present a brief sketch of our arguments . it turns out that the polynomials @xmath376 which are defined with the help of elementary symmetric functions , represent @xmath0 locally , that is , @xmath0 and @xmath377 coincide in a neighborhood of @xmath7 in order to pass to the global representation , the additional polynomial @xmath378 is chosen in such a way that the sublevel set @xmath379 approximates @xmath0 sufficiently well . given @xmath79 let us consider the set @xmath137 defined by . by theorem [ semi : approx ] there exist @xmath134 and @xmath179 such that @xmath180 is bounded . since @xmath380 it follows that @xmath381 for all @xmath250 and @xmath382 thus , the above strict inequalities hold also for @xmath203 in a small neighborhood of @xmath7 consequently , by theorem [ semi : approx ] , we can fix an @xmath170 $ ] such that @xmath381 for all @xmath383 and @xmath382 we define the sets @xmath384 let us consider an arbitrary @xmath262 obviously , @xmath385 for @xmath386 where all inequalities are strict if @xmath266 assume that @xmath232 and @xmath245 satisfy and . then , by lemma [ sublevel : approx : part ] , @xmath387 where the inequality is strict if @xmath266 hence @xmath388 and @xmath389 let us show the reverse inclusions . let @xmath390 then , by the definition of @xmath391 we have @xmath392 for @xmath393 and @xmath394 but , by the choice of @xmath87 and @xmath375 , we also have @xmath392 for @xmath382 thus , @xmath392 for @xmath395 and , in view of proposition [ berniglemma]([strict : part ] ) , we have @xmath396 for @xmath335 this shows the inclusion @xmath397 the inclusion @xmath398 can shown analogously ( by means of proposition [ berniglemma]([nonstrict : part ] ) ) . finally we discuss the constructive part of the statement . by lemma [ determ : n ] , @xmath5 is computable . consequently , the polynomials @xmath399 are also computable , since they are arithmetic expressions in @xmath400 the computability of @xmath378 follows from directly from theorem [ semi : approx ] . the polynomials @xmath401 will be defined by @xmath402 where @xmath403 will be fixed below . we give a rough description of the arguments . we start with the same remark as in the proof of theorem [ main : n+1 ] . namely , polynomials @xmath404 with @xmath405 represent @xmath0 locally . we shall disturb the polynomial @xmath275 by subtracting an appropriate non - negative polynomial @xmath406 which is small on @xmath0 , has high order zeros at the points of @xmath240 and is large for all points @xmath203 sufficiently far away from @xmath7 see also fig . [ fig : semi - alg:2d ] for an illustration of theorem [ main : n ] in the case @xmath407 we first show the existence of @xmath399 from the assertion . given @xmath408 let us consider the set @xmath137 defined by . by theorem [ semi : approx ] there exist @xmath134 and @xmath179 such that @xmath180 is bounded . since @xmath380 it follows that @xmath381 for all @xmath250 and @xmath382 thus , the above strict inequalities hold also for @xmath203 in a small neighborhood of @xmath7 consequently , by theorem [ semi : approx ] , we can fix @xmath409 $ ] such that @xmath381 for all @xmath410 and @xmath382 let us borrow the notations from the statements of theorems [ semi : approx ] and [ 07.12.10,11:53 ] . we set @xmath411 with @xmath80 $ ] as in theorem [ 07.12.10,11:53 ] . define the semi - algebraic sets @xmath412 let us consider an arbitrary @xmath262 obviously , @xmath385 for @xmath413 where all inequalities are strict if @xmath266 furthermore , by theorem [ 07.12.10,11:53 ] we also have @xmath414 and this inequality is strict if @xmath266 thus , we get the inclusions @xmath388 and @xmath389 it remains to verify the inclusions @xmath398 and @xmath397 let us consider an arbitrary @xmath415 that is , for some @xmath416 one has @xmath417 if @xmath418 then , by the choice of @xmath419 @xmath392 for all @xmath382 but , on the other hand , by proposition [ berniglemma]([strict : part ] ) , @xmath420 for some @xmath325 hence we necessarily have @xmath421 , and we get that @xmath422 consequently @xmath423 now assume @xmath424 then , by proposition [ berniglemma]([nonstrict : part ] ) , @xmath425 for some @xmath325 but , in the same way as we showed above , we deduce that @xmath426 hence @xmath427 which means that @xmath428 if @xmath429 then , by theorem [ 07.12.10,11:53 ] , one has @xmath430 and by this @xmath431 as for the algorithmic part of the assertion , we notice that @xmath432 can be easily computed from @xmath433 the computability of @xmath56 follows from theorem [ 07.12.10,11:53 ] . we mention that the `` combinatorial component '' of our proofs ( dealing with elementary symmetric functions ) resembles in part the proof of theorem [ avehenkthm ] . however , the crucial parts of the proofs of theorems [ main : n+1 ] and theorem [ main : n ] concerning the approximation of @xmath0 are based on different ideas . the polynomials @xmath68 from theorem [ avehenkthm ] can be computed in a rather straightforward way ; see . in contrast to this , the constructive parts of the proofs of theorems [ main : n+1 ] and [ main : n ] use decidability of the first order logic over reals and , by this , lead to algorithms of extremely high complexity . even though theorem [ loj ] and proposition [ berniglemma ] were also used in , our proofs can not be viewed as extensions of the proofs from . i am indebted to prof . martin henk for his support during the preparation of the manuscript . the examples in remark [ bounded : extension : remark ] arose from a discussion with prof . claus scheiderer . c. andradas , l. brcker , and j. m. ruiz , _ constructible sets in real geometry _ , ergebnisse der mathematik und ihrer grenzgebiete ( 3 ) [ results in mathematics and related areas ( 3 ) ] , vol . 33 , springer - verlag , berlin , 1996 . mr 98e:14056 j. bochnak , m. coste , and m .- f . roy , _ real algebraic geometry _ , ergebnisse der mathematik und ihrer grenzgebiete ( 3 ) [ results in mathematics and related areas ( 3 ) ] , vol . 36 , springer - verlag , berlin , 1998 , translated from the 1987 french original , revised by the authors .

let @xmath0 be an elementary closed semi - algebraic set in @xmath1 , i.e. , there exist real polynomials @xmath2 ( @xmath3 ) such that @xmath4 ; in this case @xmath2 are said to represent @xmath0 . denote by @xmath5 the maximal number of the polynomials from @xmath6 that vanish in a point of @xmath7 if @xmath0 is non - empty and bounded , we show that it is possible to construct @xmath8 polynomials representing @xmath7 furthermore , the number @xmath8 can be reduced to @xmath5 in the case when the set of points of @xmath0 in which @xmath5 polynomials from @xmath6 vanish is finite . analogous statements are also obtained for elementary open semi - algebraic sets . primary : 14p10 , secondary : 14q99 , 03c10 , 90c26 approximation , elementary symmetric function , ojasiewicz s inequality , polynomial optimization , semi - algebraic set , theorem of brcker and scheiderer

the quantum mechanical description of neutrino oscillations @xcite has been the subject of much discussion and debate in the recent literature . the ` standard ' oscillation formula @xcite , yielding an oscillation phase or @xmath1 . ] , at distance @xmath2 from the neutrino source , between neutrinos , of mass @xmath3 and @xmath4 and momentum @xmath5 , of : are used throughout . ] @xmath6 is derived on the assumption of equal momentum and equal production times of the two neutrino mass eigenstates . other authors have proposed , instead , equal energies @xcite or velocities @xcite at production , confirming , in both cases , the result of the standard formula . the latter reference claims , however , that the standard expression for @xmath7 should be multiplied by a factor of two in the case of the equal energy or equal momentum hypotheses when different production times are allowed for the two mass eigenstates . however , the equal momentum , energy or velocity assumptions are all incompatible with energy - momentum conservation in the neutrino production process @xcite . in two recent calculations @xcite a covariant formalism was used in which exact energy - momentum conservation was imposed . these calculations used the invariant feynman propagator @xcite to describe the space - time evolution of the neutrino mass eigenstates . in ref . @xcite a formula for the neutrino oscillation phase differing by a factor of 9.9 from eqn(1.1 ) was found for the case of pion decay at rest , and it was predicted that correlated spatial oscillations in the detection probability of neutrinos and the recoiling decay muons could be observed . however , the author of ref . @xcite as well as others @xcite claimed that muon oscillations would either be completely suppressed , or essentially impossible to observe . the present paper calculates the probabilities of oscillation of neutrinos and muons produced by pions decaying both at rest and in flight , as well as the probabilities of neutrino oscillation following muon decay or @xmath8decay of a nucleus at rest . the calculations , which are fully covariant , are based on feynman s reformulation of quantum mechanics @xcite in terms of interfering amplitudes associated with classical space - time particle trajectories . the essential interpretational formula of this approach , though motivated by the seminal paper of dirac on the lagrangian formulation of quantum mechanics @xcite , and much developed later in the work of feynman and other authors @xcite , was actually already given by heisenberg in 1930 @xcite . the application of the path amplitude formalism to neutrino or muon oscillations is particularly staightforward , since , in the covariant formulation of quantum mechanics , energy and momentum are exactly conserved at all vertices and due to the macroscopic propagation distances of the neutrinos and muons all these particles follow essentially classical trajectories ( _ i.e. _ corresponding to the minima of the classical action ) which are rectilinear paths with constant velocities . the essential formula of feynman s version of quantum mechanics , to be employed in the calculations presented below , is @xcite : @xmath9 where @xmath10 is the probability to observe a final state @xmath11 , given an initial state @xmath12 , and @xmath13 are ( unobserved ) intermediate quantum states . in the applications to be described in this paper , which , for simplicity , are limited to the case of the first two generations of leptons , eqn(1.2 ) specialises to : @xmath14 for the case of neutrino oscillations and @xmath15 for the case of muon oscillations . @xmath16 is the probability to observe the charged current neutrino interaction : @xmath17 following the decay : @xmath18 , while @xmath19 is the probability to observe the decay @xmath20 , after the same decay process . in eqns.(1.3),(1.4 ) @xmath21 are neutrino mass eigenstates while @xmath22 are the corresponding recoil muon states from pion decay . @xmath23 , @xmath24 and @xmath25 denote invariant decay amplitudes , @xmath26 is the invariant amplitude of the charged current neutrino interaction , @xmath27 is the invariant space - time propagator of particle @xmath28 between the space - time points @xmath29 and @xmath30 and @xmath31 is an invariant amplitude describing the production of the @xmath32 by the source @xmath33 and its space - time propagation to the space - time point @xmath34 . an important feature of the amplitudes appearing in eqns(1.3 ) and ( 1.4 ) is that they are completely defined in terms of the physical neutrino mass eigenstate wavefunctions @xmath35 . this point will be further discussed in section 5 below . the difference of the approach used in the present paper to previous calculations presented in the literature can be seen immediately on inspection of eqns(1.3 ) and ( 1.4 ) . the initial state contributes only a multiplicative constant to the transition probabilites . the initial state can then just as well be defined as ` pion at @xmath34 ' , rather than @xmath36 . this is done in the calculations presented in section 2 below . ] is a pion at space time - point @xmath34 , the final state an @xmath37 or @xmath38 produced at space - time point @xmath39 . these are unique points , for any given event and do not depend in any way on the masses of the unobserved neutrino eigenstates propagating from @xmath40 to @xmath39 in eqn(1.3 ) . on the other hand the ( unobserved ) space - time points @xmath40 at which the neutrinos and muons are produced _ do _ depend on @xmath41 . indeed , because of the different velocities of the propagating neutrino eigenstates , only in this case can both neutrinos and muons ( representing _ alternative _ classical histories of the decaying pion ) both arrive simultanously at the unique point @xmath39 where the neutrino interaction occurs ( eqn(1.3 ) ) or the muon decays ( eqn(1.4 ) ) . the crucial point in the above discussion is that the decaying pion , _ via _ the different path amplitudes in eqns(1.3 ) and ( 1.4 ) , _ interferes with itself_. to modify very slightly dirac s famous statement : ` each pion then interferes only with itself . interference between two different pions never occurs ' . because of the different possible decay times of the pion in the two interfering path amplitudes , the pion propagators @xmath42 in eqns(1.3 ) and ( 1.4 ) above give important contributions to the interference phase . to the author s best knowledge , this effect has not been taken into account in any previously published calculation of neutrino oscillations . the results found for the oscillation phase are , for pion decays at rest : @xmath43 and for pion decays in flight : @xmath44 where @xmath45 the superscripts indicate the particles whose propagators contribute to the interference phase . also @xmath46 , @xmath47 and @xmath48 are the energies of the parent @xmath49 and the decay @xmath50 and @xmath51 and @xmath52 , @xmath53 the angles between the pion and the neutrino , muon flight directions . in eqns.(1.5 ) to ( 1.7 ) terms of order @xmath54 , @xmath55 , and higher , are neglected , and in eqns.(1.6 ) and ( 1.7 ) ultrarelativistic kinematics with @xmath56 is assumed . formulae for the oscillation phase of neutrino oscillations following muon decays or nuclear @xmath0-decays at rest , calculated in a similar manner to eqn(1.5 ) , are given in section 3 below . a brief comment is now made on the generality and the covariant nature of the calculations presented in this paper . although the fundamental formula ( 1.2 ) is valid in both relativistic and non - relativistic quantum mechanics , it was developed in detail by feynman @xcite only for the non - relativistic case . for the conditions of the calculations performed in the present paper ( propagation of particles in free space ) the invariant space - time propagator can either be derived ( for fermions ) from the dirac equation , as originally done by feynman @xcite or , more generally , from the covariant feynman path integral for an arbitary massive particle , as recently done in ref . @xcite . in the latter case , the invariant propagator for any stable particle with pole mass @xmath57 , between space - time points @xmath58 and @xmath59 in free space is given by the path integral @xcite : @xmath60 \exp \left \ { -\frac{im}{2 } \int_{x_i}^{x_f } \left ( \frac{dx}{d\tau } \cdot \frac{dx}{d\tau}+1 \right)d \tau\right \}\ ] ] where @xmath61 is the proper time of the particle . by splitting the integral over @xmath62 on the right side of eqn(1.8 ) into the product of a series of infinitesimal amplitudes corresponding to small segments , @xmath63 , gaussian integration may be performed over the intermediate space - time points . finally , integrating over the proper time @xmath61 , the analytical form of the propagator is found to be @xcite : @xmath64 where : @xmath65 and @xmath66 is a first order hankel function of the second kind , in agreement with ref . @xcite . in the asymptotic region where @xmath67 , or for the propagation of on - shell particles @xcite , the hankel function reduces to an exponential and yields the configuration space propagator @xmath68 of eqn(2.11 ) below . it is also shown in ref . @xcite that energy and momentum is exactly conserved in the interactions and decays of all such ` asymptotically propagating ' particles . the use of quasi - classical particle trajectories and the requirement of exact energy - momentum conservation are crucial ingredients of the calculations presented below . the structure of the paper is as follows . in the following section the case of neutrino or muon oscillations following pion decay at rest is treated . full account is taken of the momentum wave - packets of the progagating neutrinos and muons resulting from the breit - wigner amplitudes describing the distributions of the physical masses of the decaying pion and daughter muon . the corresponding oscillation damping corrections and phase shifts are found to be very small , indicating that the quasi - classical ( constant velocity ) approximation used to describe the neutrino and muon trajectories is a very good one . the incoherent effects , of random thermal motion of the source pion , and of finite source and detector sizes , on the oscillation probabilities and the oscillation phases , are also calculated . these corrections are found to be small in typical experiments , but much larger than those generated by the coherent momentum wave packets . in section 3 , formulae are derived to describe neutrino oscillations following muon decay at rest or the @xmath0-decay of radioactive nuclei . these are written down by direct analogy with those derived in the previous section for pion decay at rest . in section 4 , the case of neutrino and muon oscillations following pion decay in flight is treated . in this case the two - dimensional spatial geometry of the particle trajectories must be related to the decay kinematics of the production process . due to the non - applicability of the ultrarelativistic approximation to the kinematics of the muon in the pion rest frame , the calculation , although straightforward , is rather tedious and lengthy for the case of muon oscillations , so the details are relegated to an appendix . finally , in section 5 , the positive aspects and shortcomings of previous treatments in the literature of the quantum mechanics of neutrino and muon oscillations are discussed in comparison with the method and results of the present paper . to understand clearly the different physical hypotheses and approximations underlying the calculation of the particle oscillation effects it is convenient to analyse a precise experiment . this ideal experiment is , however , very similar to lnsd @xcite and karmen @xcite except that neutrinos are produced from pion , rather than muon , decay at rest . the different space - time events that must be considered in order to construct the probability amplitudes for the case of neutrino oscillations following pion decay at rest are shown in fig.1 . a @xmath32 passes through the counter c@xmath69 , where the time @xmath70 is recorded , and comes to rest in a thin stopping target t ( fig.1a ) ) . for simplicity , the case of only two neutrino mass eigenstates @xmath71 and @xmath72 of masses @xmath3 and @xmath4 ( @xmath3 @xmath73 @xmath4 ) is considered . the pion at rest constitutes the initial state of the quantum mechanical probability amplitudes . the final state is an @xmath74 system produced , at time @xmath75 , via the process @xmath76 at a distance @xmath2 from the decaying @xmath32 ( fig.1d ) ) . two different physical processes may produce the observed @xmath74 final state , as shown in fig.1b ) and 1c ) , where the pion decays either at time @xmath77 into @xmath71 or at time @xmath78 into @xmath72 . the probability amplitudes for these processes are , up to an arbitary normalisation constant , and neglecting solid angle factors in the propagators : @xmath79 note that following the conventional ` @xmath80 ' ( final , initial ) ordering of the indices of matrix elements in quantum mechanics , the path amplitude is written from right to left in order of increasing time . this ensures also correct matching of ` bra ' and ` ket ' symbols in the amplitudes . in eqn(2.1 ) , @xmath81 , @xmath82 are ` reduced ' invariant amplitudes of the @xmath50 charged current scattering and pion decay processes , respectively , @xmath83 , @xmath84 and @xmath85 , @xmath86 are relativistic breit - wigner amplitudes , @xmath87 and @xmath88 are elements of the unitary maki - nagagawa - sakata ( mns ) matrix @xcite , @xmath89 , describing the charged current coupling of a charged lepton , @xmath90 , ( @xmath91 ) to the neutrino mass eigenstate @xmath12 . the reduced invariant amplitudes are defined by factoring out the mns matrix element from the amplitude for the process . for example : @xmath92 . since the purely kinematical effects of the non - vanishing neutrino masses are expected to be very small , the reduced matrix elements may be assumed to be lepton flavour independent : @xmath93 where @xmath94 denotes a massless neutrino . in eqn(2.1 ) , @xmath95 is the lorentz - invariant configuration space propagator @xcite of the pion or neutrino . the pole masses and total decay widths of the pion and muon are denoted by @xmath96 , @xmath97 and @xmath98 , @xmath99 respectively . for simplicity , phase space factors accounting for different observed final states are omitted in eqn(2.1 ) and subsequent formulae . because the amplitudes and propagators in eqn(2.1 ) are calculated using relativistic quantum field theory , and the neutrinos propagate over macroscopic distances , it is a good approximation , as already discussed in the previous section , to assume exact energy - momentum conservation in the pion decay process , and that the neutrinos are on their mass shells , _ i.e. _ @xmath100 , where @xmath101 is the neutrino energy - momentum four - vector . in these circumstances the neutrino propagators correspond to classical , rectilinear , particle trajectories . the pion and muon are unstable particles whose physical masses @xmath102 and @xmath103 differ from the pole masses @xmath104 and @xmath98 appearing in the breit - wigner amplitudes and covariant space - time propagators in eqn(2.1 ) . the neutrino momentum @xmath105 will depend on these physical masses according to the relation : @xmath106[w_{\pi}^2-(m_i - w_{\mu(i)})^2 ] \right]^{\frac{1}{2 } } } { 2 w_{\pi}}\ ] ] note that , because the initial state pion is the same in the two path amplitudes in eqn(2.1 ) @xmath102 does not depend on the neutrino mass index @xmath12 . however , since the pion decays resulting in the production of @xmath71 and @xmath72 are independent physical processes , the physical masses of the unobserved muons , @xmath107 , recoiling against the two neutrino mass eigenstates are not , in general , the same . in the following kinematical calculations sufficient accuracy is achieved by retaining only quadratic terms in the neutrino masses , and terms linear in the small quantities : @xmath108 , @xmath109 . this allows simplification of the relativistic breit - wigner amplitudes : @xmath110 developing eqn(2.2 ) up to first order in @xmath111 , @xmath112 and @xmath113 yields the relation : @xmath114 \end{aligned}\ ] ] where @xmath115 the term @xmath116 which is also included in eqn(2.4 ) gives a negligible o(@xmath117 ) contribution to the neutrino oscillation formula . for muon oscillations , however , it gives a term of o(@xmath111 ) in the interference term , as discussed below . similarly , the exact formula for the neutrino energy : @xmath118 in combination with eqn(2.4 ) gives , for the neutrino velocity : @xmath119~+~o(m_i^4,\delta_{\pi}^2,\delta_i^2)\ ] ] this formula will be used below to calculate the neutrino times - of - flight : @xmath120 . from the unitarity of the mns matrix , the elements @xmath89 may be expressed in terms of a single real angular parameter @xmath121 : @xmath122 the parts of the amplitudes requiring the most careful discussion are the invariant space - time propagators @xmath95 , as it is mainly their treatment that leads to the different result for the neutrino oscillation phase found in the present paper , as compared to those having previously appeared in the literature . in the limit of large time - like separations , the propagator may be written as @xcite : @xmath123\ ] ] @xmath95 is the amplitude for a particle , originally at a space - time point ( @xmath124 , @xmath125 ) , to be found at ( @xmath126 , @xmath127 ) and @xmath128 , @xmath129 . in the following , according to the geometry of the experiment shown in fig.1 , only one spatial coordinate will be considered ( @xmath130 ) and only the exponential factor in eqn(2.10 ) , containing the essential phase information for particle oscillations will be retained in the amplitudes . solid angle correction factors , taken into account by the factor in large brackets in eqn(2.10 ) , are here neglected , but are easily included in the final oscillation formulae . writing then @xmath131 = \exp[-im\delta \tau ] \equiv \exp[-i\delta \phi]\ ] ] it can be seen that the increment in phase of the propagator , @xmath132 , when the particle undergoes the space - time displacement ( @xmath133 , @xmath134 ) is a lorentz invariant quantity equal to the product of the particle mass and the increment , @xmath63 , of proper time . using the relativistic time dilatation formula : @xmath135 and also the relation , corresponding to a classical , rectilinear , particle trajectory : @xmath136 gives , for the phase increments corresponding to the paths of the neutrinos and the pion in fig.1 : @xmath137 \\ \delta \phi_i^{\pi } & = & m_{\pi } ( t_i - t_0 ) = m_{\pi } ( t_d - t_0)-\frac{m_{\pi}l}{v_i } \nonumber \\ & = & m_{\pi } ( t_d - t_0)-m_{\pi}l\left\{1+\frac{m_i^2}{2 p_0 ^ 2}\left[1- \frac{2\delta_{\pi}}{m_{\pi}}\frac{(m_{\pi}^2+m_{\mu}^2)}{(m_{\pi}^2-m_{\mu}^2 ) } + \frac{4\delta_i m_{\mu}}{m_{\pi}^2-m_{\mu}^2}\right]\right\ } \end{aligned}\ ] ] where terms of o(@xmath117 ) and higher are neglected . to perform the integral over @xmath112 in eqn(2.16 ) it is convenient to approximate the modulus squared of the breit - wigner amplitude by a gaussian , _ via _ the substitution : @xmath145 where the width of the gaussian is chosen so that it has approximately the same full width at half maximum , @xmath146 , as the breit - wigner function . after the substitution ( 2.21 ) , the integral over @xmath112 in eqn(2.16 ) is easily evaluated by a change of variable to ` complete the square ' in the argument of the exponential , with the result : @xmath147\ ] ] eqn(2.18 ) gives , for @xmath148 , the numerical value : @xmath149 for typical physically interesting values ( see below ) of @xmath150 ev and @xmath151 m , @xmath148 takes the value 0.48 mev@xmath152 , so that @xmath153 then , to very good accuracy , @xmath154 , independently of the neutrino mass . it follows that for neutrino oscillations , the muon mass dependence of the amplitudes may be neglected for any physically interesting values of @xmath155 and @xmath2 . from eqns ( 2.16 ) and ( 2.22 ) the probability to observe the reactions @xmath76 at distance @xmath2 from the pion decay point and at time @xmath75 is : @xmath156\delta_{\pi}\right]\right\ } \nonumber\end{aligned}\ ] ] the time dependent exponential factors in the curly brackets of eqn(2.24 ) are easily understood . if @xmath157 then @xmath158 . this implies that the neutrino of mass @xmath3 results from an earlier decay than the neutrino of mass @xmath4 , in order to be detected at the same time . because of the exponential decrease with time of the pion decay amplitude , the contribution to the probability of the squared amplitude for the neutrino of mass @xmath3 is larger . the interference term resulting from the product of the decay amplitudes of the two neutrinos of different mass , has an exponential factor that is the harmonic mean of those of the squared amplitudes for each neutrino mass eigenstate , and so is also suppressed relative to the squared amplitude for the neutrino of mass @xmath3 . the integral over the physical pion mass is readily performed by replacing the breit - wigner function by a gaussian as in eqn(2.21 ) . this leads to an overall multiplicative constant @xmath159 and a factor : @xmath160 multiplying the interference term . for @xmath161 and @xmath151 m the numerical value of this factor is @xmath162 . this tiny correction is neglected in the following equations . integrating over @xmath75 gives the average probability to observe the @xmath74 final state at distance @xmath2 : @xmath163 \cos\frac{2 m_{\pi } m_{\mu}^2 \delta m^2 } { ( m_{\pi}^2- m_{\mu}^2)^2}l\right\ } \end{aligned}\ ] ] where all kinematical quantities are expressed in terms of @xmath164 , @xmath96 and @xmath98 . note that the minimum value of @xmath75 is @xmath165 , @xmath166 and @xmath165 for the squared amplitude terms of neutrinos of mass @xmath3 , @xmath4 and the interference term , respectively . on integrating over @xmath75 , the squared amplitude terms give equal contributions , the larger amplitude for mass @xmath3 being exactly compensated by a smaller range of integration . the exponential damping factor in the interference term in eqn(2.26 ) is derived using the relations : @xmath167 and @xmath168 to obtain @xmath169 the damping factor arises because the difference in the times - of - flight of the two neutrino paths is limited by the pion lifetime . it will be seen below , however , that for distances @xmath2 of practical interest for the observation of neutrino oscillations , the damping effect is tiny . the part of the oscillation phase in eqn(2.24 ) originating from the neutrino propagators ( the term associated with the ` 1 ' within the large curved brackets ) differs by a factor two from the corresponding expression in the standard formula . the contribution to the oscillation phase of the propagator of the decaying pion ( the term associated with @xmath170 in the large curved brackets of eqn(2.24 ) ) has not been taken into account in any published calculation known to the author of the present paper . the oscillation phase in eqn(2.26 ) is @xmath171 times larger than that given by the standard formula ( 1.1 ) . for @xmath172 m , as in the lnsd experiment , the first oscillation maximum occurs for @xmath173 . denoting by @xmath174 the phase of the cosine interference term in eqn(2.26 ) , the pion lifetime damping factor can be written as : @xmath175 so the damping effect is vanishingly small when @xmath176 . the oscillation formula ( 2.26 ) is calculated on the assumption that the decaying pion is at rest at the precisely defined position @xmath58 . in fact , the positive pion does not bind with the atoms of the target , but will rather undergo random thermal motion . this has three effects : an uncertainy in the value of @xmath58 , a doppler shift of the neutrino energy and a time dilatation correction correction factor of @xmath177 in the equation ( 2.15 ) for the pion phase increment . assuming that the target is at room temperature ( t= @xmath178 k ) , the mean kinetic energy of @xmath179 correponds to a mean pion momentum of @xmath180 mev and a mean velocity of @xmath181 km / sec . the pion will move , in one mean lifetime ( @xmath182 sec ) , a distance of 146@xmath183 . this is negligible as compared to @xmath2 ( typically @xmath184 30 m ) and so eqn(2.26 ) requires no modification to account for this effect . the correction factor due to the doppler effect and time dilatation is readily calculated on the assumption of a maxwell - boltzmann distribution of the pion momentum : @xmath185 here @xmath186 mev . details of the calculation are given in appendix a. the interference term in eqn(2.26 ) is modified by a damping factor : @xmath187l\right)^2 \right\}\ ] ] while the argument of the cosine term acquires an additional phase factor : @xmath188l\ ] ] for @xmath189 , @xmath190 and @xmath191 . if the target in which the pion stops is of thickness @xmath192 , then the effect of different stopping points of the @xmath49 ( assumed uniformly distributed ) is to multiply the interference term in ( 2.26 ) by the factor : @xmath193 if the position of the neutrino interaction point within the target has an uncertainy of @xmath194 a similar correction factor is found , with the replacement @xmath195 in eqn(2.34 ) . the calculation of this correction factor is also described in appendix a. if the target t in which the pion comes to rest ( fig.1a ) ) is chosen to be sufficiently thin , the pion decay process may be detected by observing the recoil muon in the counter c@xmath69 at times @xmath77 or @xmath78 ( fig.1b ) or 1c ) ) . a sufficiently accurate measurement of the times @xmath77 , @xmath78 and @xmath75 would , in principle , enable separation of the different processes in figs.1b ) and 1c ) by the observation of separated peaks in the time - of - flight distribution at @xmath196 and @xmath197 . in this case the interference term in eqn(2.26 ) vanishes as the two alternative space - time paths leading to the neutrino interaction are distinguishable . however , for @xmath172 m and @xmath198(ev)@xmath199 the difference in the times of flight is only @xmath200 sec , more than ten orders of magnitude smaller than can be measured with existing techniques . as discussed in section 5 below , the momentum smearing due to the doppler effect at room temperature is some eleven orders of magnitude larger than the shift due to a neutrino mass difference with @xmath198(ev)@xmath199 . thus , even with infinitely good time resolution , separation of such neutrino mass eigenstates by time - of - flight is not possible . the ideal experiment , described above to study neutrino oscillations , is easily adapted to the case of oscillations in the decay probability of muons produced by charged pion decay at rest . as previously pointed out in ref . @xcite , such oscillations will occur if neutrinos with different masses exist . as before , the pion stops in the target t at time @xmath70 ( see fig.2a ) ) . at time @xmath77 the pion decays into @xmath71 and the corresponding recoil muon ( @xmath201 ) , whose passage is recorded in the counter c@xmath202 ( fig.2b ) ) . similarly , a decay into @xmath72 and @xmath203 may occur at time @xmath78 ( fig.2c ) ) . with a suitable choice of the times @xmath77 and @xmath78 , such that muons following the alternative paths both arrive at the same time @xmath75 at the point @xmath59 , interference occurs between the path amplitudes when muon decay occurs at the space - time point ( @xmath59 , @xmath75 ) in the detector d ( fig.2d ) ) . the probability for two _ classical _ trajectories to arrive at _ exactly _ the same space - time point of course vanishes . the correct way to consider the quantum mechanical calculation is rather to ask _ given that _ the muon decay occurs at the point ( @xmath204 , @xmath75 ) , does the muon recoil against @xmath71 or @xmath72 ? if these two possiblities are not distinguished by the measurement of the decay process , the corresponding probability _ amplitudes _ ( not probabilities ) must be added in the calculation of the probability of the observed decay process . the path amplitudes corresponding to muons recoiling against neutrinos of mass @xmath3 and @xmath4 are : @xmath205 the various factors in these equations are defined , _ mutatis mutandis _ , as in eqn(2.1 ) . with the same approximations , concerning the neutrino masses and the physical pion and muon masses , as those made above , the velocity of the muon recoiling against the neutrino mass eigenstate @xmath206 is : @xmath207\end{aligned}\ ] ] where @xmath208 comparing with eqn(2.7 ) , it can be seen that for the muon case , unlike that where neutrino interactions are observed , there are pion and muon mass dependent correction terms that are independent of the neutrino masses , implying a velocity smearing effect due to the physical pion and muon masses that is @xmath209 larger than for the case of neutrino oscillations . the phase increments corresponding to the paths of the muons and the pion in fig . 2 are , using ( 2.4 ) . ] and ( 2.12)-(2.15 ) and ( 2.36 ) : @xmath210 \end{aligned}\ ] ] @xmath211 @xmath212 @xmath213\ ] ] where @xmath214 using eqns(2.11),(2.38 ) and ( 2.39 ) to re - write the space - time propagators in eqn(2.35 ) , as well as eqn(2.3 ) for the breit - wigner amplitudes gives : @xmath215 d \delta_i~~i=1,2 \end{aligned}\ ] ] where @xmath216 @xmath217 @xmath218\ ] ] and @xmath219 where , as in eqns(2.18 ) and ( 2.19 ) , imaginary parts of order @xmath220 are neglected . making the substitution ( 2.21 ) and performing the integral over @xmath112 according to eqn(2.22 ) , the following formula is found for the probability for muon decay at distance @xmath2 and time @xmath75 . @xmath221 ^ 2 |<e^+\nu_k\overline{\nu}_{l}|t|\mu^+>|^2 e^{-\frac{\gamma_{\mu}v_0^{\mu}}{\gamma_0^{\mu}}l } \nonumber \\ & & |<\nu_0 \mu^+|t_r|\pi^+>|^2 e^{-\gamma_{\pi}(t_d - t_0 ) } \frac{\gamma_{\pi}^2}{4(\delta_{\pi}^2+\frac{\gamma_{\pi}^2}{4 } ) } \nonumber \\ & & \left\{\sin^2\theta e^{\gamma_{\pi } t_{\mu(1)}^f } + \cos^2 \theta e^{\gamma_{\pi } t_{\mu(2)}^f } \right . \nonumber \\ & & -2\sin \theta \cos \theta e^{\frac{\gamma_{\pi}}{2}(t_{\mu(1)}^f + t_{\mu(2)}^f ) } \nonumber \\ & & \left . { \it re } \exp i \left [ \frac{m_{\mu}^2 \delta m^2}{2 p_0 ^ 3}\left(1-\frac { e_0^{\mu}}{m_{\pi}}\right)l + [ \alpha_{\pi}^{\mu}(1)-\alpha_{\pi}^{\mu}(2)]\delta_{\pi } \right ] \right\}\end{aligned}\ ] ] where the effect of the non - zero neutrino masses are neglected in the reduced pion decay amplitude so that @xmath222 and this amplitude is a common factor in both path amplitudes . the muon path difference yields the term associated with @xmath223 in the interference phase in eqn(2.46 ) while the pion path is associated with ` 1 ' in the large round brackets . the numerical value of the damping factor : @xmath224 [ 1-\frac{\alpha^{\mu } \gamma_{\mu}}{3}]^2\ ] ] resulting from the integral over the physical muon mass is , for @xmath151 m , 0.774 , so , unlike for the case of neutrino oscillations , the correction is by no means negligible . this is because , in the muon oscillation case , the leading term of @xmath225 is not proportional to the neutrino mass squared . the non - leading terms proportional to @xmath111 have been neglected in eqn(2.43 ) . this correction however effects only the overall normalisation of the oscillation formula , not the functional dependence on @xmath2 arising from the interference term . integrating over @xmath113 using eqns(2.21 ) and ( 2.22 ) , as well as over @xmath75 , gives the probability of muon decay , into the final state @xmath226 , at distance , @xmath2 , from the production point , where all kinematical quantities are expressed in terms of @xmath164 , @xmath96 and @xmath98 : @xmath227[1- \frac{4}{3 } \frac{\gamma_{\mu } m_{\mu } m_{\pi } } { m_{\pi}^2- m_{\mu}^2}l]^2 \nonumber \\ & & |<e^+\nu_k\overline{\nu}_{l}|t|\mu^+>|^2 \exp \left[-\frac{2 \gamma_{\mu } m_{\pi } m_{\mu } ( m_{\pi}^2- m_{\mu}^2)}{(m_{\pi}^2 + m_{\mu}^2)^3 } l \right]|<\nu_0 \mu^+|t_r|\pi^+>|^2 \nonumber \\ & & \left\{1-\sin 2 \theta\exp[-\frac{2 \gamma_{\pi } m_{\pi}^2 m_{\mu}^2 \delta m^2 } { ( m_{\pi}^2- m_{\mu}^2)^3}l ] \cos \frac{2 m_{\pi } m_{\mu}^2 \delta m^2 } { ( m_{\pi}^2- m_{\mu}^2)^2 } l \right\}\end{aligned}\ ] ] in this expression the correction due to the damping factor of the interference term : @xmath228\ ] ] arising from the integral over the physical pion mass has been neglected . for @xmath161 and @xmath151 m the numerical value of this factor is @xmath229 . denoting by @xmath230 the argument of the cosine in eqn(2.48 ) , the exponential damping factor due to the pion lifetime may be written as : @xmath231 for @xmath232 , @xmath233 $ ] so , as in the neutrino oscillation case , the pion lifetime damping of the interference term is very small . introducing the reduced muon decay amplitude : @xmath234 the total muon decay probability is given by the incoherent sum over the four possible final states containing massive neutrinos : @xmath235 where the unitarity of the mns matrix has been used . @xmath236 is given by the replacement of @xmath237 by @xmath238 in eqn(2.48 ) . eqn(2.52 ) shows that the muon decay width is independent of the values of the mns matrix elements . corrections due to time dilatation and the doppler effect are calculated in a similar way to the neutrino oscillation case with the results ( see appendix a ) : @xmath239l\right)^2 \right\}\ ] ] and @xmath240l\ ] ] as for neutrino oscillations , the corresponding corrections are very small for oscillation phases of order unity . the phase of the cosine in the interference term is the same in neutrino and muon oscillations , as can be seen by comparing eqns(2.26 ) and ( 2.48 ) . it follows that the target or detector size correction ( eqn(2.34 ) ) is the same in both cases . neutrino and muon oscillations from pion decay at rest then have an identical oscillation phase for given values of @xmath164 and @xmath2 . in view of the much larger event rate that is possible , it is clearly very advantageous in this case to observe muons rather than neutrinos , since the rate of neutrino oscillation events is severely limited by the very small neutrino interaction cross section . in fact , it it not necessary to observe muon decay , as in the example discussed above . the oscillation formula applies equally well if the muons are observed at the distance @xmath2 using any high efficiency detector such as , for example , a scintillation counter . according to eqn(1.2 ) , interference between the path amplitudes must occur if the muon detection device does not discriminate muons recoiling against @xmath71 from those recoiling against @xmath72 . the formula describing ` @xmath241 neutrino oscillations ' appearence ' . as discussed in more detail in section 5 below , ` @xmath242 ' and ` @xmath243 ' do not exist , as physical states , if neutrinos are massive and the mns matrix is non - diagonal . it is still , however , current practice in the literature to use the symbols ` @xmath244 ' , ` @xmath245 ' and ` @xmath246 ' to refer to massive neutrinos . this is still a useful and meaningful procedure if it is employed only to identify , in a concise manner , the type of charged current interaction by which the neutrinos are produced or detected , i.e. ` @xmath247 ' means neutrinos ( actually several , with different generation numbers ) produced together with the charged lepton @xmath248 or detected by observation of the charged lepton @xmath249 . it should not be forgotten however that only the wavefunctions of the mass eigenstates @xmath206 occur in the amplitudes of standard model processes . ] following the decay at rest of a @xmath250 , @xmath251 is easily derived from the similar formula for @xmath32 decay at rest , ( 2.25 ) . because the neutrino momentum spectrum is continous , smearing effects due to the finite muon lifetime may be neglected from the outset . the phase increment associated with the neutrino path is then given by eqn(2.14 ) with the replacements @xmath252 and @xmath253 , where @xmath254 is the antineutrino momentum . the phase increment of the decaying muon is given by the same replacements in eqn(2.15 ) with , in addition , @xmath255 and @xmath256 . the formula , analogous to eqn(2.26 ) , for the time - averaged probability to detect the process @xmath257 at a distance l from the muon decay point is then : @xmath258 \cos \left[\frac{\delta m^2 } { p_{\overline{\nu}}}\left ( \frac{m_{\mu}}{2 p_{\overline{\nu}}}-1\right)l \right ] \right\}\end{aligned}\ ] ] the standard neutrino oscillation formula , hitherto used in the analysis of all experiments , has instead the expression @xmath259 for the argument of the cosine term in eqn(3.1 ) . denoting my @xmath260 the value of @xmath164 obtained using the standard formula , and @xmath261 that obtained using the feynman path ( fp ) formula ( 3.1 ) then : @xmath262 for a typical value of @xmath263 of 45 mev , eqn(3.2 ) implies that @xmath264 . thus the @xmath243 oscillation signal from @xmath250 decays at rest reported by the lnsd collaboration @xcite corresponding to @xmath265 ( ev)@xmath199 for @xmath266 implies @xmath267 ( ev)@xmath199 for a similar mixing angle . for the case of @xmath0-decay : @xmath268 @xmath96 in the first line of eqn(2.15 ) is replaced by @xmath269 , the total energy release in the @xmath0-decay process : @xmath270 where @xmath271 and @xmath272 are the masses of the parent and daughter nuclei . that the phase advance of an unstable state , over a time , @xmath134 , is given by @xmath273 where @xmath274 is the excitation energy of the state , is readily shown by the application of time - dependent perturbation theory to the schrdinger equation @xcite . a more intuitive derivation of this result has also been given in ref . @xcite . in the present case , @xmath275 . omitting the lifetime damping correction , which is about eight orders of magnitude smaller than for pion decay , given a typical @xmath0-decay lifetime of a few seconds , the time - averaged probabilty to detect @xmath276 via the process @xmath277 , at distance @xmath2 from the decay point is given by the formula , derived in a similar way to eqns(2.26 ) and ( 3.1 ) : @xmath278 \right\}\end{aligned}\ ] ] where @xmath279 the lifetime of the unstable nucleus @xmath280 . until now , all experiments have used the standard expression @xmath259 for the neutrino oscillation phase . the values of @xmath164 found should be scaled by the factor @xmath281 , suitably averaged over @xmath263 , to obtain the @xmath164 given by the feynman path formula ( 3.4 ) . in this section , the decays in flight of a @xmath32 beam with energy @xmath282 into @xmath283 are considered . as the analysis of the effects of the physical pion and muon masses have been shown above to give negligible corrections to the @xmath2 dependence of the oscillation formulae , for the case of decays at rest , such effects will be neglected in this discussion of in - flight decays . the overall structure of the path amplitudes for neutrinos and muons is the same as for decays at rest ( see eqns(2.1 ) and ( 2.35 ) ) . however , for in - flight decays , in order to calculate the interfering paths originating at different and terminating at common space - time points , the two - dimensional spatial geometry of the problem must be properly taken into account . in fig.3 a pion decays at a into the 1 mass eigenstate , the neutrino being emitted at an angle @xmath284 in the lab system relative to the pion flight direction . if @xmath285 a later pion decay into the 2 mass eigenstate at the angle @xmath286 may give a path such that both eigenstates arrive at the point b at the same time . a neutrino interaction @xmath287 occuring at this space - time point will than be sensitive to interference between amplitudes corresponding to the paths ab and acb . the geometry of the triangle abc and the condition that the 1 and 2 neutrino mass eigenstates arrive at b at the same time gives the following condition on their velocities : @xmath288 expanding to first order in the small quantity @xmath289 , rearranging , and neglecting terms of @xmath290 , gives : @xmath291\ ] ] where the relation : @xmath292 has been used . rearranging eqn(4.2 ) : @xmath293\ ] ] the difference in phase of the neutrino paths ab and cb is ( see eqn(2.14 ) ) : @xmath294 since the angle @xmath295 is @xmath296 , the difference between ab and cb is of the same order , and so : @xmath297 where @xmath298 , the measured neutrino energy . from the geometry of the triangle abc : @xmath299 so , to first order in @xmath295 , and using eqn(4.4 ) : @xmath300 and @xmath301 eqns(4.8 ) and ( 4.9 ) give , for the phase increment of the pion path : @xmath302 in eqn(4.10 ) , the lorentz invariant character of the propagator phase is used . setting @xmath303 and @xmath304 , gives for @xmath305 a prediction consistent with that obtained from eqn(2.15 ) . eqns(4.6 ) and ( 4.10 ) give , for the total phase difference of the paths ab , acb : @xmath306\ ] ] using the expressions , valid in the ultra - relativistic ( ur ) limit where @xmath307 : @xmath308 and @xmath309 where @xmath310 is the angle between the directions of the pion and neutrino momentum vectors in the pion rest frame , eqn(4.11 ) may be rewritten as : @xmath311 for @xmath312 the oscillation phase is the same as for pion decay at rest ( see eqn(2.26 ) ) since in the latter case , @xmath313 . using eqn(4.14 ) , the probability to observe a neutrino interaction , at point b , produced by the decay product of a pion decay occuring within a region of length , @xmath314 , ( @xmath315 ) centered at the point a , in a beam of energy @xmath46 , is given by a formula analagous to eqn(2.26 ) : @xmath316 as in the case of pion decay at rest , eqn(2.26 ) , the oscillation phase differs by the factor @xmath171 from that given by the standard formula . the derivation of the formula describing muon oscillations following pion decays in flight is very similar to that just given for neutrino oscillations . the condition on the velocities so that the muons recoiling against the different neutrino mass eigenstates arrive at the point b ( see fig.3 ) at the same time , is given by a formula analagous to ( 4.2 ) : @xmath317 } { v_{\pi}\sin\theta_1}\delta\ ] ] the formula relating the neutrino masses to the muon velocities is , however , more difficult to derive than the corresponding relation for neutrinos , ( 4.3 ) , as the decay muons are not ultra - relativistic in the pion rest frame . the details of this calculation are given in appendix b. the result is : @xmath318 using eqn(4.16 ) and the relation , valid to first order in @xmath295 , : @xmath319 where @xmath134 is the flight time of the pion from a to c in fig.3 ( and also the difference in the times - of - flight of the muons recoiling against the two neutrino eigenstates ) , the angle @xmath295 may be eliminated to yield : @xmath320}\ ] ] using now the kinematical relation ( see appendix b ) : @xmath321 and the expression for the phase difference of the paths ab and acb : @xmath322 together with eqn(4.19 ) , it is found , taking the ur limit , where @xmath323 , that @xmath324\ ] ] the probability of detecting a muon decay at b is then : @xmath325 |<\nu_{0 } \mu^+|t_r|\pi^+>|^2 \nonumber \\ & & \times \left\{1-\sin 2 \theta \cos \frac { 2 m_{\mu}^2 \delta m^2}{e_{\mu}^2 ( m_{\pi}^2-m_{\mu}^2)^2 ) } \left[\frac{(m_{\mu}^2 e_{\pi}- m_{\pi}^2 e_{\mu})l}{\cos\theta_1}\right ] \right\ } \end{aligned}\ ] ] where @xmath314 is defined in the same way as in eqn(4.15 ) . the quantum mechanics of neutrino oscillations has been surveyed in recent review articles @xcite , where further extensive lists of references may be found . in this section , the essential differences between the calculations presented in the present paper and all previous treatments in the literature of the quantum mechanics of neutrino oscillations , as cited in the above review articles , will be summarised . a critical review of the existing literature will then be given . hitherto , it has been assumed that the neutrino source produces a ` lepton flavour eigenstate ' that is a superposition of mass eigenstates , at some fixed time . in this paper it is , instead , assumed following shrock @xcite that the neutrino mass eigenstates are produced incoherently in different physical processes . this follows from the structure of the leptonic charged current in the electroweak standard model : @xmath326 only the wavefunctions of the physical neutrino mass eigenstates @xmath206 appear in this current , and hence in the initial or final states of any physical process . a consequence is that the neutrino mass eigenstates can be produced at different times in the path amplitudes corresponding to different mass eigenstates . it has recently been shown that experimental measurements of the decay width ratio ; @xmath327 and of the mns matrix elements are inconsistent with the production of a coherent ` lepton flavour eigenstate ' in pion decay @xcite and that the the ` equal time ' or ` equal velocity ' hypothesis resulting from this assumption underestimates , by a factor of two , the contribution of neutrino propagation to the oscillation phase @xcite . as demonstrated above , allowing for the possibility of different production times of the neutrinos results in an important , decay process dependent , contribution to the oscillation phase from the propagator of the source particle . the non - diagonal elements of @xmath89 in eqn(5.1 ) describe violation of lepton flavour ( or generation number ) by the weak charged current . for massless neutrinos , the mns matrix becomes diagonal ; lepton flavour is conserved within each generation , and the familiar ` lepton flavour eigenstates ' are give by the replacements : @xmath328 , @xmath329 , @xmath330 . only in this case are the lepton flavour eigenstates physical , being all mass eigenstates of vanishing mass . the standard derivation of the neutrino oscillation phase will now be considered , following the treatment of ref . @xcite , but using the notation of the present paper . the calculation is performed assuming an initial ` lepton lavour eigenstate of the neutrino ' that is a superposition of the mass eigenstates @xmath331 and @xmath332 : @xmath333 where @xmath334 are mass eigenstates of fixed momentum @xmath335 . this flavour eigenstate is assumed to evolve with laboratory time , @xmath336 , according to fixed energy solutions of the non - relativistic schrdinger equation into the mixed flavour state @xmath337 : @xmath338 where @xmath339 , @xmath340 are the laboratory energies of the neutrino mass eigenstates . the amplitude for transition into the ` electron flavour eigenstate ' : @xmath341 at time @xmath336 is then , using eqns(5.3 ) , ( 5.4 ) : @xmath342 because it is assumed that the neutrinos have the same momentum but different energies : @xmath343 and using ( 5.5 ) and ( 5.6 ) , the probability of the flavour state @xmath244 at time @xmath336 is found to be : @xmath344 \right)\ ] ] finally , since the velocity difference of the neutrino mass eigenstates is o(@xmath164 ) , then , to the same order in the oscillation phase , the replacement @xmath345 can be made in eqn(5.7 ) to yield the standard oscillation phase of eqn(1.1 ) . * the time evolution of the neutrino mass eigenstates in eqn(5.3 ) according to the schrdinger equation yields a non - lorentz - invariant phase @xmath346 , to be compared with the lorentz - invariant phase @xmath347 given in eqn(2.14 ) above . although the two expressions agree in the non - relativistic limit @xmath348 it is clearly inappropriate to use this limit for the description of neutrino oscillation experiments . it may be noted that the lorentz - invariant phase is robust relative to different kinematical approximations . the same result is obtained to order @xmath349 for the phase of spatial oscillations independent of whether the neutrinos are assumed to have equal momenta or energies . this is not true in the non - relativistic limit . assuming equal momenta gives the standard result of eqn(1.1 ) , whereas the equal energy hypothesis results in a vanishing oscillation phase . a contrast may be noted here with the standard treatment of neutral kaon oscillations , which follows closely the derivation in eqns(5.2 ) to ( 5.7 ) above , except that the particle phases are assumed to evolve with time according , to the lorentz invariant expression , @xmath350 $ ] , where @xmath57 is the particle mass and @xmath61 is its proper time , in agreement with eqn(2.11 ) . * as pointed out in ref . @xcite , the different neutrino mass eigenstates do not have equal momenta as assumed in eqns(5.2 ) and ( 5.6 ) . the approximation of assuming equal momenta might be justified if the fractional change in the momentum of the neutrino due to a non - vanishing mass were much less than that of the energy . however , in the case of pion decay as is readily shown from eqns(2.4 ) and ( 2.6 ) above , the ratio of the fractional shift in momentum to that in energy is actually @xmath351 ; so , in fact , the opposite is the case . * the derivation of eqn(5.7 ) is carried out in the abstract hilbert space of the neutrino mass eigenstates or ` lepton flavour eigenstates ' without any reference to the production or detection processes necessary for the complete description of an experiment in which ` neutrino oscillations ' may be observed . in this calculation the ` mass ' and ` flavour ' bases are treated as physically equivalent . however in standard model amplitudes only states of the mass basis appear . also it has been pointed out that ` flavour momentum eigenstates ' can not be defined in a theoretically consistent manner @xcite . their existence is , in any case , excluded by experiment for the case of pion decay @xcite . * what are the physical meanings of @xmath336 , @xmath335 in eqn(5.3 ) ? in this equation it is assumed that the neutrino mass eigenstates are both produced , and both detected , at the same times . thus both have the same time - of - flight @xmath336 . the momentum @xmath335 can not be the same for both eigenstates , as assumed in eqn(5.6 ) , if both energy and momentum are conserved in the decay process . for any given value of the laboratory time @xmath336 the different neutrino mass eigenstates must be at different space - time positions because they have different velocities , if it is assumed that both mass eigenstates are produced at the same time . it then follows that the different mass eigenstates can not be probed , at the same space - time point , by a neutrino interaction , whereas the latter must occur at a definite space - time point in every detection event . in fact , there is an inconsistent treatment of the velocity of the neutrinos . equal production times imply equal space - time velocities , whereas it is assumed thast ` kinematical velocities ' defined as @xmath352 are different for the different mass eigenstates . * the historical development of the calculation of the neutrino oscillation phase is of some interest . the first published prediction @xcite actually obtained a phase a factor two larger than eqn(1.1 ) _ i.e. _ in agreement with the contribution from neutrino propagation found in the present paper . this prediction was later used , for example , in ref.@xcite . the derivation sketched above , leading to the standard result of eqn(1.1 ) was later given in ref.@xcite . a subsequent paper @xcite by the authors of ref.@xcite , published shortly afterwards , cited both ref.@xcite and ref.@xcite , but used now the prediction of the latter paper . no comment was made on the factor of two difference in the two calculations . in a later review article , @xcite , by the authors of ref.@xcite a calculation similiar to that of ref.@xcite was presented in detail . subsequently , all neutrino oscillation experiments have been analysed on the assumption of the standard oscillation phase of eqn(1.1 ) . it may be thought that the kinematical and geometrical inconsistencies mentioned in points ( ii ) and ( iv ) above result from a too classical approach to the problem . after all , what does it mean , in quantum mechanics , to talk about the ` position ' and ` velocity ' of a particle , in view of the heisenberg uncertainty relations @xcite ? this point will become clear later in the present discussion , but first , following the original suggestion of ref . @xcite , and , as done in almost all subsequent work on the quantum mechanics of neutrino oscillations , the ` wave packet ' description of the neutrino mass eigenstates will be considered . in this approach , both the ` source ' and also possibly the ` detector ' in the neutrino oscillation experiment are described by coherent spatial wave packets . here the ` source ' wavepacket treatment in the covariant approach of ref . @xcite will be briefly sketched . after discussing the results obtained , and comparing them with those of the present paper , the general consistency of the wave packet approach with the fundamental quantum mechanical formula ( 1.2 ) will be examined . a further discussion of wave packets as applied to neutrino oscillations can be found in ref . @xcite . the basic idea of the wave packet approach of ref . @xcite is to replace the neutrino propagator ( 2.11 ) in the path amplitude by a four - dimensional convolution of the propagator with a ` source wave packet ' which , presumably , describes the space - time position of the decaying pion . for mathematical convenience , this wave packet is taken to have a gaussian form with spatial and temporal widths @xmath353 and @xmath354 respectively . thus , the neutrino propagator @xmath95 is replaced by @xmath355 where : @xmath356 where @xmath58 and @xmath59 are 4-vectors that specify the neutrino production and detection positions , respectively , and : @xmath357\ ] ] the integral in ( 5.8 ) was performed by the stationary phase method , yielding the result ( up to multiplicative and particle flux factors ) , and here assuming , for simplicity , one dimensional spatial geometry : @xmath358\ ] ] where @xmath359 and @xmath360 . for the case of ` @xmath361 oscillations ' , following @xmath32 decay at rest , the probability to observe flavour @xmath244 at time @xmath336 and distance @xmath204 is given by : @xmath362 performing the integral over @xmath134 and making the ultra - relativistic approximation @xmath363 yields finally , with @xmath364 : it can be seen that the oscillation phase in eqn(5.12 ) is the same as standard one of eqn(1.1 ) . this is a consequence of the ` equal production time ' hypothesis implicit in eqn(5.10 ) , where @xmath134 does not depend on the mass eigenstate label @xmath366 . the exponential damping factors in eqn(5.12 ) are the same as those originally found in ref . @xcite for spatial wave packets ( @xmath367 ) . considering now only spatial wave packets and using the property @xmath368 derived from the fourier transform of a gaussian , the two terms in the exponential damping factor may be written as : @xmath369\ ] ] and @xmath370\ ] ] the spatial damping factor @xmath371 is usually interpreted in terms of a ` coherence length ' @xcite . if @xmath372 then @xmath373 and the neutrino oscillation term is strongly suppressed . eqn(5.15 ) expresses the condition that oscillations are only observed provided that the wave packets overlap . since @xmath374 the separation of the wave packets is @xmath375 , so that eqn(5.15 ) is equivalent to @xmath376 ( no wave packet overlap ) . the damping factor @xmath377 is typically interpreted @xcite in terms of the ` heisenberg uncertainty principle ' . this factor is small , unless the difference in mass of the eigenstates is much less than @xmath378 , so it is argued that only for wide momentum wave packets can neutrino oscillations be observed , whereas in the contrary case , when the mass eigenstates are distinguishable , the interference effect vanishes . in the case of pion decay the difference in momentum of the two interfering mass eigenstates comes only from the @xmath112 term in eqn(2.4 ) , as @xmath113 , being a property of the common initial state , is the same for both eigenstates . the neutrino momentum smearing in pion decay is then estimated from eqn(2.4 ) as : @xmath379 for @xmath380 the value of @xmath377 is found to be @xmath381 giving complete suppression of neutrino oscillations . this prediction is in clear contradiction with the tiny damping corrections found in the path amplitude analysis in of section 2 above . the preceding discussion of the derivation of the standard formula for the oscillation phase ( 1.1 ) in terms of ` flavour eigenstates ' revealed contradictions and inconsistencies if the neutrinos are assumed to follow classical space - time trajectories . the ` source wave packet ' treatment gives the standard result for the oscillation phase and predicts that the interference term will be more or less damped depending on the widths in space - time and momentum space of the wave packets . so do wave packets actually play a role in the correct quantum mechanical description of neutrino oscillations , as suggested in ref.@xcite ? are the packets actually constrained by the heisenberg uncertainty relations ? how do the properties of the wave packet effect the possibility to observe neutrino oscillations ? the answers to all these questions are contained in the results of the calculations presented in section 2 above . they are now reviewed , with special emphasis on the basic assumptions made and the physical interpretation of the equations . referring again to fig.1 , in a ) a single pion comes to rest in the stopping target t. the time of its passage is recorded by the counter c@xmath69 , which thus defines the initial state as a @xmath32 at rest at time @xmath70 . this pion , being an unstable particle , has a mass @xmath102 that is , in general , different from its most likely value which is the pole mass @xmath96 . what are shown in fig.1b ) and fig.1c ) are two different classical histories of this _ very same pion_. in b ) it decays into the neutrino mass eigenstate @xmath71 at time @xmath77 , and in c ) it decays into the neutrino mass eigenstate @xmath72 at time @xmath78 . because these are independent classical histories , the physical masses , @xmath382 and @xmath383 , of the muons recoiling against the mass eigenstates @xmath71 and @xmath72 , respectively , are , in general , not equal . taking into account , now , exact energy - momentum conservation in the decay processes ( appropriate because of the covariant formulation used throughout ) the eigenstates @xmath71 and @xmath72 will have momenta which depend on @xmath102 and @xmath382 and @xmath102 and @xmath383 respectively . these momenta are calculated in eqn(2.4 ) . the distributions of @xmath102 and @xmath103 are determined by breit - wigner amplitudes ( that are the fourier transforms of the corresponding exponential decay laws ) in terms of the decay widths @xmath97 and @xmath99 respectively . in accordance with eqn(1.2 ) , only the breit - wigner amplitudes corresponding to the physical muon masses are ( coherently ) integrated over at the amplitude level . the integral over @xmath102 ( a property of the initial state ) is performed ( incoherently ) at the level of the transition probability . because of the long lifetimes of the @xmath49 and @xmath51 , the corresponding momentum wave packets are very narrow , so that all corrections resulting from integration over the resulting momentum spectra are found to give vanishingly small corrections . indeed , as is evident from the discussion of eqn(5.10 ) above , the width of the momentum wave packet is much smaller than the difference in the momenta of the eigenstates expected for experimentally interesting values of the neutrino mass difference ( say , @xmath384 ) . contrary to the prediction of eqn(5.10 ) , this does not at all prevent the observation of neutrino oscillations . this is because the oscillations result from interference between amplitudes corresponding to different propagation times , not different momenta , of the neutrino mass eigenstates . table 1 shows contributions to the fractional smearing of the neutrino momentum from different sources : * neutrino mass difference @xmath384 ( eqn(2.4 ) ) * coherent effect due to the physical muon mass ( @xmath385 in eqn(2.4 ) ) * incoherent effect due to the physical pion mass ( @xmath386 in eqn(2.4 ) ) * incoherent doppler effect assuming @xmath387 the pion mass effect is of the same order of magnitude as the neutrino mass shift . the muon mass effect is two orders of magnitude smaller , while the doppler effect at room temperature gives a shift eleven orders of magnitude larger than a ( 1ev)@xmath199 neutrino mass difference squared . according to the usual interpretation of the heisenberg uncertainty principle , the neutrinos from pion decay , which , as has been shown above , correspond to very narrow momentum wave packets , would be expected to have a very large spatial uncertainty . indeed , interpreting the width of the coherent momentum wave packet generated by the spread in @xmath388 according to the the momentum - space uncertainty relation @xmath389 gives @xmath390 km . does this accurately represent the knowledge of the position of a decay neutrino obtainable in the experiment shown in fig.1 ? without any experimental difficulty , the decay time of the pion can be measured with a precision of @xmath391sec , by detecting the decay muon . thus , at any later time , the distance of the neutrino from the decay point is known with a precision of @xmath392 cm . this is a factor 4@xmath393 more precise than the ` uncertainty ' given by the heisenberg relation . it is clear that the experimental knowledge obtainable on the position of the neutrino is essentially classical , in agreement with the theoretical description in terms of classical particle trajectories in space - time . in this case the momentum - position uncertainty relation evidently does not reflect the possible experimental knowledge of the position and momentum of the neutrino . this is because it does not take into account the prior knowledge that the mass of the neutrino is much less than that of the pion or muon , so that its velocity is , with negligible uncertainty , c. in spite of this , a heisenberg uncertainty relation is indeed respected in the pion decay process . the breit - wigner amplitude that determines the coherent spread of neutrino momentum is just the fourier transform of the exponential decay law of the muon . the width parameter of the breit - wigner amplitude and the muon mean lifetime do indeed respect the _ energy - time _ uncertainty relation @xmath394 . it is then clear , from this careful analysis of neutrino oscillations following pion decay at rest , that , in contradiction to what has been almost universally assumed until now , _ the neutrinos are not described by a coherent spatial wave packet_. there is a coherent momentum wave packet , but it is only a kinematical consequence of a breit wigner amplitude . if for mathematical convenience , the momentum wave packet is represented by a gaussian , a conjugate ( and spurious ) gaussian spatial wave packet will be generated by fourier transformation . indeed , in the majority of wave packet treatments that have appeared in the literature , gaussian momentum and spatial wave packets related by a fourier transform with widths satisfying the ` uncertainty relation ' @xmath395 have been introduced . in fact , it is evident , by inspection , that the space - time wavepacket of eqn(5.10 ) does not correctly reflect the known space - time structure of the sequence of events corresponding to pion decay . since @xmath364 , the fixed source - detector distance , the integral of @xmath134 over the range from @xmath396 to @xmath397 assumed in order to derive eqn(5.12 ) implies that the neutrino velocities @xmath398 vary also in the range : @xmath399 . this unphysical range of integration results in an average neutrino momentum that is less than the kinematical velocity , @xmath352 , of either mass eigenstate , due to contributions to the integral from negative values of @xmath134 @xcite . also the production time dependence is known to be exponential , not gaussian . there are indeed ` heisenberg uncertainties ' in the production times of the neutrinos , due to the finite source lifetime , but once the neutrinos are produced their motion in space - time is well approximated , in each alternative history , by that of a free classical particle . the above discussion shows clearly that the _ ad hoc _ gaussian wave packet introduced in eqns(5.8 ) and ( 5.9 ) does not correspond to the actual sequence of the space - time events that constitute realistic neutrino oscillation experiments . the limitation on the detection distance @xmath2 , for observation of neutrino oscillations , given by the damping factor @xmath400 of eqn(2.30 ) is easily understood in terms of the classical particle trajectories shown in fig.1 . for a given velocity difference , the time difference @xmath401 becomes very large when both neutrinos , in the alternative classical histories , are required to arrive simultaneously at a far distant detector . because of the finite pion lifetime , however , the amplitude for pion decay at time @xmath78 is smaller than that at @xmath77 by the factor @xmath402 $ ] . integrating over all decay times results in the exponential damping factor @xmath400 of eqn(2.30 ) . it is clear that , contrary to the damping factor @xmath371 of eqn(5.14 ) , the physical origin of the @xmath2 dependent damping factor is quite unrelated to ` wavepacket overlap ' . for a given value of @xmath401 the coherent neutrino momentum spread originating in the breit wigner amplitude for @xmath388 produces a corresponding velocity smearing that reduces the number of possible classical trajectories arriving at the detection event . this effect is taken into account in the integral shown in eqn(2.22 ) . the effect is shown , in section 2 above , to be much smaller than the ( already tiny ) pion lifetime damping described above , and it is neglected in eqn(2.26 ) . as described by eqn(2.25 ) , the ( incoherent ) integration over the breit wigner function containing @xmath102 gives an additional damping correction to the interference term that is also very tiny compared to that due to the pion lifetime . a final remark is now made on the physical interpretation of the damping factors ( 5.13 ) and ( 5.14 ) that have often been derived and discussed in the literature . @xmath371 is replaced , in the path amplitude calculations , by @xmath400 and @xmath377 by the factor resulting from the coherent integration over the physical muon mass @xmath388 . the reason for the huge suppression factor predicted by eqn(5.13 ) , and the tiny one found in the path amplitude calculation , is that , in deriving ( 5.13 ) and ( 5.14 ) , it is assumed that the neutrino eigenstates are both produced and detected at equal times . this will only be possible if both the hypothetical ` wave packet overlap ' is appreciable ( eqn(5.14 ) ) and also the momentum smearing is sufficiently large that the time - of - flight differences due to the different neutrino masses are washed out ( eqn(5.13 ) . in the path amplitude calculation interference and hence oscillations are made possible by different _ decay times _ of the source pion , and the damping factors analagous to @xmath371 and @xmath377 turn out to give vanishingly small corrections to the oscillation term . it may be remarked that the physical interpretation of ` neutrino oscillations ' provided by the path amplitude description is different from the conventional one in terms of ` flavour eigenstates ' . in the latter the amplitudes of different flavours in the neutrino are supposed to vary harmonically as a function of time . this may be done , for example , by changing the basis states , in eqn(5.3 ) above , from the mass to the flavour basis by using the inverses of eqns(5.2 ) and ( 5.4 ) . in the amplitudes for the different physical processes in the path amplitude treatment there is , instead , no variation of the ` lepton flavour ' in the propagating neutrinos . if the mass eigenstates are represented as superpositions of ` lepton flavour eigenstates ' there is evidently no temporal variation of the lepton flavour composition within each interfering amplitude . indeed , the neutrinos in general occupy different spatial positions at any given time , making it impossible to project out a ` flavour eigenstate ' at any time by using the inverses of eqns(5.2 ) and ( 5.4 ) . only _ in the detection process itself _ where the different neutrino histories occupy the same space time point are the ` lepton flavour eigenstates ' projected out , and the interference effect occurs that is described as ` neutrino oscillations ' . in the case of the observation of decay products of the recoil muons no such projection on to a ` lepton flavour eigenstate ' takes place , but exactly similar interference effects are predicted to occur . as previously emphasised @xcite , the ` flavour oscillations ' of neutrinos , neutral kaons and b - mesons are just special examples of the universal phenomenon of quantum mechanical superposition described by eqn(1.2 ) , that also describes all the interference effects of physical optics . by far the most widespread difference from the path amplitude treatment of the present paper is the non - respect of the basic quantum mechanical formula , ( 1.2 ) , by the introduction of wave packets to describe ` source ' and/or ` detector ' particles[7,8,10,11,30 - 46 ] . since in any practical neutrino oscillation experiment a single initial or final quantum state , as specified by eqn(1.2 ) , is not defined , but rather sets of initial and final states @xmath403 and @xmath404 determined by experimental conditions , eqn(1.2 ) may be generalised to : @xmath405 to be contrasted with the formula used in the references cited above : @xmath406 here , @xmath407 and @xmath408 are ` source ' and ` detector ' wave packets respectively . in eqn(5.17 ) the initial states @xmath409 and final states @xmath410 of eqn(5.16 ) , that correspond to different spatial positions , and also , possibly , different kinematic properties , of the source particle or detection event , are convoluted with _ ad hoc _ spatial and/or temporal ` wave packets ' , that are , in the opinion of the present writer , for the reasons given above , devoid of any physical significance . a possible reason for the widespread use of eqn(5.17 ) instead of ( 5.16 ) may be understood following a remark of the author of ref . @xcite concerning a ` paradox ' of the complete quantum field theory calculation that takes into account , by a single invariant amplitude , production , propagation and detection of the neutrinos . it was noticed that , if the amplitude for the complete chain of processes is considered to correspond to one big feynman diagram , then integration over the space - time coordinates of the initial and final states particles will reduce the exponential factors , containing the essential information on the interference phase , to energy - momentum conserving delta functions , and so no oscillations will be possible . a related remark was made by the authors of ref . @xcite who stated that , as they were assuming exact energy - momentum conservation , the integration over the space - time coordinates could be omitted . they still , however , ( quite inconsistently , in view of the previous remark ) retained the exponential factors containing the interference phase information . these considerations indicate a general confusion between momentum space feynman diagram calculations , where it is indeed legitimate to integrate , at the amplitude level , over the unobserved space - time positions of the initial and final state particles , and the case of neutrino oscillations , where the amplitude is defined with initial and final states corresponding to space - time positions . in the latter case , it is the unobserved momenta of the propagating particles that should be integrated over , as is done in eqns(2.1 ) and ( 2.35 ) above , and not the space - time positions of the ` source ' or ` detector ' particles , as in eqn(5.17 ) . it is clear that exact energy - momentum conservation plays a crucial role in the path amplitude calculation . this is valid only in a fully covariant theory . still , several authors , in spite of the ultra - relativistic nature of neutrinos , used a non - relativistic theory to describe the production , propagation and detection of neutrinos @xcite . as is well known , in such ` old fashioned perturbation theory ' @xcite energy is not conserved at the level of propagators and so no precise analysis of the kinematics and the space - time configurations of the production and detection events , essential in the covariant path amplitude analysis , is possible . even when , in some cases , the complete production , propagation and detection process of the neutrinos were described @xcite , equal neutrino energies @xcite , equal neutrino momenta @xcite or either @xcite were assumed , in contradiction with energy - momentum conservation and a consistent space - time description of the production and detection events . as follows directly from eqn(5.6 ) ( or the similar formula , for the neutrino momentum , obtained by assuming equal neutrino energies ) , the standard formula ( 1.1 ) for the oscillation phase was obtained in all the above cited references . as shown in reference @xcite this is a consequence of the universal equal production time ( or equal velocity ) hypothesis . in fact , the assumptions of equal momenta , equal energies or exact energy - momentum conservation give only negligible , o(@xmath411 ) , changes in the oscillation phase @xcite . an interesting discussion of the interplay between different kinematical assumptions ( not respecting energy - momentum conservation ) and the space - time description of the production and detection events was provided in ref . this treatment was based on the lorentz - invariant propagator phase of eqn(2.11 ) . by assuming either equal momentum or equal energy for the propagating neutrinos , but allowing _ different times of propagation _ for the two mass eigenstates , values of @xmath412 agreeing with eqns(2.14 ) and ( 2.24 ) above were found , _ i.e. _ differing by a factor two from the standard formula . alteratively , assuming equal velocities , ( and hence equal propagation times ) the standard result ( 1.1 ) was obtained . in this latter case , however , the masses , momenta and energies of the neutrinos must be related , up to corrections of o(@xmath111 ) , according to : @xmath413 since the ratio of the neutrino masses may take , in general , any value , so must then the ratio of their momenta . for the case of neutrino production from pion decay at rest with @xmath414 the relation ( 5.18 ) is clearly incompatible with eqn(2.4 ) which gives : @xmath415 on the other hand , eqn(5.19 ) is clearly compatible ( up to corrections of o(@xmath164 ) ) with the ` equal momentum ' hypothesis . there is similar compatiblity with the ` equal energy ' hypothesis . even so , the authors of ref . @xcite recommended the use of the equal velocity hypothesis . with the hindsight provided by the path amplitude analysis , in which the two neutrino mass eigenstates do indeed have different propagation times , it can be seen that the kinematically consistent ` equal momentum ' and ` equal energy ' choices are good approximations and the neutrino oscillation phase , resulting from the propagation of the neutrinos alone , is indeed a factor two larger than the prediction of the standard formula . a more detailled discussion of the effects of the ` equal momentum ' ` equal energy ' and ` equal velocity ' hypotheses may be found in reference @xcite the author of ref . @xcite included the propagator of the decaying pion in the complete production - propagation - detection amplitude ; compare eqn(8 ) of ref . @xcite with eqn(2.1 ) above . however , no detailed space - time analysis of production and detection events was performed . square spatial wave packets for the ` source ' and ` detector ' were convoluted at amplitude level as in eqn(5.17 ) . as the neutrinos were assumed to have a common production time , no contribution to the oscillation phase from the source particle was possible , and so the standard result for the oscillation phase was obtained . although , following ref . @xcite , most recent studies of the quantum mechanics of neutrino oscillations have considered the complete production - propagation - detection process , some authors still use , in spite of the criticisms of ref . @xcite , the ` flavour eigenstate ' description @xcite . in the last two of these references a ` quantum field theory ' approach ia adopted , leading to an ` exact ' oscillation formula @xcite that does not make use of the usual ultra - relativistic approximation . this work included neither exact kinematics , nor an analysis of the space - time structure of production and detection events . in ref . @xcite the equal energy hypothesis was used and in ref . @xcite the equal momentum hypothesis . in all three cases the standard result was found for the oscillation phase in the ultra - relativistic limit , as a consequence of the assumption of equal production times . correlated production and detection of neutrinos and muons produced in pion decay were considered in ref . the introduction to this paper contains a valuable discussion of the universality of the ` particle oscillation ' phenomenon . it is pointed out that this is a consequence of the general principle of amplitude superposition in quantum mechanics , and so is not a special property of the @xmath416 , @xmath417 and neutrino systems which are usually discussed in this context . this paper used a covariant formalism that employed the ` energy representation ' of the space - time propagator . in the introduction , the important difference between eqns(5.16 ) and ( 5.17 ) was also touched upon : even so , in the amplitude for the correlated detection of the muon and neutrino ( eqn(2.10 ) of ref . @xcite ) not only are the space - time positions of the production points of the neutrino and muon integrated over , but they are assumed to be at _ different _ space - time points . the propagator of the decaying pion is not included in the amplitude , and although exact energy - momentum conservation is imposed , no space - time analysis of the production and decay points is performed . correlated spatial oscillations of neutrinos and muons are predicted , though with interference phases different from the results of both the present paper and the standard formula . pion and muon lifetime effects were mentioned in ref . @xcite , but neither the role of the pion lifetime in enabling different propagation times for the neutrinos nor the momentum smearing , induced by the fourier - transform - related breit wigner amplitudes , were discussed . the claim of ref . @xcite ) that correlated neutrino - muon oscillations should be observable in pion decay was questioned in ref . the authors of the latter paper attempted to draw conclusions on the possibility , or otherwise , of particle oscillations by using ` plane waves ' , _ i.e. _ energy - momentum eigenfunctions . as is well known , such wavefunctions are not square integrable , and so can yield no spatial information . the probability to find a particle described by such a wave function in any finite spatial volume is zero . due to the omission of the ( infinite ) normalisation constants of the wavefunctions many of the equations in ref . @xcite are , as previously pointed out @xcite , dimensionally incorrect . momentum wavepackets for the decaying pion convoluted at amplitude level as in eqn(5.17 ) were also discussed in ref . @xcite . although exact energy - momentum conservation constraints were used , it was assumed , as in ref . @xcite , that the muons and the different neutrino mass eigenstates are both produced and detected at common points ( eqn(35 ) of ref . the latter assumption implies equal velocities , yielding the standard neutrino oscillation phase as well as the inconsistent kinematical relation ( 5.18 ) . the authors of ref . @xcite concluded that : * correlated @xmath418 oscillations of the type discussed in ref . @xcite could be observed , though with different oscillation phases . * oscillations would not be observed if only the muon is detected * neutrino oscillations can be observed even if the muon is not detected . conclusion ( b ) is a correct consequence of the ( incorrect ) assumption that the muons recoiling against the different neutrino mass eigenstates have the same velocity . as both muons have the same mass they will have equal proper time increments . so according to eqn(2.12 ) the phase increments will also be equal and the interference term will vanish . the conclusion ( c ) is in agreement with the prediction of eqn(3.22 ) of ref . the path amplitude calculation of the present paper shows that conclusion ( b ) is no longer valid when the different possible times of propagation of the recoiling muons are taken into account . observation of neutrino oscillations following pion decay , using a covariant formalism ( schwinger s parametric integral representation of the space - time propagator ) was considered in ref . exact energy - momentum conservation was imposed , and integration over the pion spatial position at amplitude level , as in eqn(5.17 ) , was done . the propagator of the pion source was included in the amplitudes , but as the different mass eigenstates were produced and detected at the same space - time points , equal propagation velocities were implicitly assumed , so that just as in refs@xcite , where the same assumption was made , the standard neutrino oscillation phase was obtained . in the conclusion of this paper the almost classical nature of the space - time trajectories followed by the neutrinos was stressed , although this was not taken to its logical conclusion in the previous discussion , _ e.g. _ the kinematical inconsistency of the equal velocity hypothesis that requires the evidently impossible condition ( 5.18 ) to be satisfied . in a recent paper @xcite , the standard neutrino oscillation formula with oscillation phase given by eqn(1.1 ) was compared with a neutrino decoherence model . in order to take into account incertainties in the position of the source and the neutrino energy , an average was made over the quantity @xmath419 , assuming that it is distributed according to a gaussian with mean value @xmath249 and width @xmath420 . the average was performed in an incoherent manner . thus the calculation is closely analagous to those for the effects of target or detector length or of thermal motion of the neutrino source , presented in the appendix a of the present paper . perhaps uniquely then , in the published literature , in ref . @xcite the effects of source position and motion are taken into account correctly , according to eqn(5.16 ) instead of eqn(5.17 ) . however , the source of the neutrino energy uncertainty is not specified . in as far as it is generated from source motion the calculation is , in principle , correct . there is however also the ( typically much smaller , see table 1 above , for the case of pion decay at rest ) coherent contribution originating from the variation in the physical masses of the unstable particles produced in association with the neutrino , as discussed in detail above . it was concluded in ref . @xcite that the gaussian averaging procedure used gave equivalent results to the decoherence model for a suitable choice of parameters . it is clearly of great interest to apply the calculational method developed in the present paper to the case of neutral kaon and b - meson oscillations . indeed the use of the invariant path amplitude formalism has previously been recommended @xcite for experiments involving correlated pairs of neutral kaons . here , just a few remarks will be made on the main differences to be expected from the case of neutrino or muon oscillations . a further discussion can be found in ref . @xcite and a more detailed treatment will be presented elsewhere@xcite . in the case of neutrino and muon oscillations , the interference effect is possible as the different neutrino eigenstates can be produced at different times . this is because the decay lifetimes of all interesting sources ( pions , muons , @xmath0-decaying nuclei ) are much longer than the time difference beween the paths corresponding to the interfering amplitudes . to see if a similar situation holds in the case of @xmath421 oscillations , three specific examples will be considered with widely differing momenta of the neutral kaons : these correspond to neutral kaon centre - of - mass momenta of 108 mev , 750 mev and 5 gev respectively . in each case the time difference ( @xmath426 ) of production of @xmath427 and @xmath428 mesons , in order that they arrive at the same time at a point distant @xmath429 ( where @xmath430 is the usual relativistic parameter ) from the source in the centre - of - mass frame is calculated . exact relativistic kinematics is assumed and only leading terms in the mass difference @xmath431 are retained . taking the value of @xmath432 and the various particle masses from ref . @xcite the following results are found for @xmath426 in the three cases : ( i ) 2.93@xmath433sec , ( ii ) 8.3@xmath434sec and ( iii ) 6.4@xmath435sec . for comparison , for neutrino oscillations following pion decay at rest , with @xmath436 and @xmath437 m , eqn(2.29 ) gives @xmath438sec . the result ( i ) may be compared with the mean life of the @xmath439 meson of 1.5@xmath440sec @xcite . thus the @xmath439 lifetime is a factor of about 27 larger than @xmath426 indicating that @xmath421 interference should be possible by a similar mechanism to neutrino oscillations following pion decays , _ i.e. _ without invoking velocity smearing of the neutral kaon mass eigenstates . in cases ( i ) and ( ii ) the interference effects observed will depend on the ` characteristic time ' of the non resonant ( and hence incoherent ) strong interaction process , a quantity that has , hitherto , not been susceptible to experimental investigation . if this time is much less than , or comparable to , @xmath426 , essentially equal velocities ( and therefore appreciable velocity smearing ) of the eigenstates will be necessary for interference to occur . since @xmath432 and @xmath441 are comparable in size , velocity smearing effects are expected to be , in any case , much larger than for neutrino oscillations following pion decay . these effects may be roughly estimated by using the gaussian approximation ( 2.22 ) of the present paper . the main contribution to the velocity smearing is due to the variation of the physical mass of the @xmath427 rather than those of the @xmath428 or @xmath442 . for the @xmath443 oscillation case , analagous to ( i ) above , @xmath444 , where @xmath445 mev , the value of @xmath446 is found to be @xmath447sec , to be compared with @xmath448sec @xcite , which is a factor 3 smaller . thus , velocity smearing effects are expected to play an important role in @xmath443 oscillations . this is possible , since the neutral b - meson decay width ( @xmath449 mev ) , and mass difference ( @xmath450 mev ) , have similar sizes . in closing , it is interesting to mention two types of atomic physics experiments where interference effects similar to the conjectured ( and perhaps observed @xcite ) neutrino oscillations have aleady been clearly seen . the first is quantum beat spectroscopy @xcite . this type of experiment , which has previously been discussed in connection with neutrino oscillations @xcite , corresponds closely to the gedanken experiment used by heisenberg @xcite to exemplify the the fundamental law of quantum mechanics , eqn(1.2 ) . the atoms of an atomic beam are excited by passage through a thin foil or a laser beam . the quantum phase of an atom with excitation energy @xmath274 evolves with time according to : @xmath451 ( see the discussion after eqn(3.3 ) above ) . if decay photons from two nearby states with excitation energies @xmath452 and @xmath453 are detected after a time interval @xmath134 ( for example by placing a photon detector beside the beam at a variable distance @xmath454 from the excitation foil ) a cosine interference term with phase : @xmath455 where @xmath456 is the average velocity of the atoms in the beam , is observed @xcite . an atom in the beam , before excitation , corresponds to the neutrino source pion . the excitation process corresponds to the decay of the pion . the propagation of the two different excited states , _ alternative _ histories of the initial atom , correspond to the _ alternative _ propagation of the two neutrino mass eigenstates . finally the dexcitation of the atoms and the detection of a _ single _ photon corresponds to the neutrino detection process . the particular importance of this experiment for the path amplitude calculations presented in the present paper , is that it demonstrates , experimentally , the important contribution to the interference phase of the space - time propagators of excited atoms , in direct analogy to the similar contributions of unstable pions , muons and nuclei discussed above . an even closer analogy to neutrino osillations following pion decay is provided by the recently observed process of photodetachement of an electron by laser excitation : the ` photodetachment microscope ' @xcite . a laser photon ejects the electron from , for example , an @xmath457 ion in a beam . the photodetached electron is emitted in an s - wave ( isotropically ) and with a fixed initial energy . it then moves in a constant , vertical , electric field that is perpendicular to the direction of the ion beam and almost parallel to the laser beam . an upward moving electron that is decelerated by the field eventually undergoes ` reflection ' before being accelerated towards a planar position - sensitive electron detector situated below the beam and perpendicular to the electric field direction ( see fig.1 of ref . @xcite ) . in these circumstances , it can be shown @xcite that , just two classical electron trajectories link the production point to any point in the kinematically allowed region of the detection plane . typical parameters for @xmath457 are @xcite : initial electron kinetic energy , 102 @xmath51ev : detector distance , 51.4 cm ; average time - of - flight , 117 ns ; difference in emission times for the electrons to arrive in spatial - temporal coincidence at the detector plane , 160 ps . an interference pattern is generated by the phase difference between the amplitudes corresponding to the two allowed trajectories . the phase difference , derived by performing the feynman path integral of the classical action along the classical trajectories @xcite , gives a very good description of the observed interference pattern . the extremely close analogy between this experiment and the neutrino oscillation experiments described in sections 2 and 3 above is evident . notice that the neutrinos , like the electrons in the photodetachement experiment , must be emitted at different times , in the alternative paths , for interference to be possible . this is the crucial point that was not understood in all previous treatments of the quantum mechanics of neutrino oscillations . actually , ref . @xcite contains , in section iv , a path amplitude calculation for electrons in free space that is geometrically identical to the discussion of pion decays in flight presented in section 4 above ( compare fig.3 of the present paper with fig.3 of ref . @xcite ) the conclusion of ref . @xcite is that , in this case , no interference effects are possible for electrons that are mononergetic in the source rest frame . as is shown in section 4 above , if these electrons are replaced either by neutrinos of different masses from pion decay , or muons recoiling against such neutrinos , observable interference effects are indeed to be expected . random thermal motion of the decaying pion in the target has two distinct physical effects on the phase of neutrino oscillations , @xmath458 ( where the first and second terms in eqn(a1 ) give the contributions of the neutrino and pion paths respectively ) : * the observed neutrino momentum , @xmath459 , is no longer equal to @xmath460 , due to the boost from the pion rest frame to the laboratory system . ( doppler effect or lorentz boost ) * the time increment of the pion path @xmath461 ( see eqn(2.16 ) ) no longer corresponds to the pion proper time . ( relativistic time dilatation ) taking into account ( 1 ) and ( 2 ) gives , for the neutrino oscillation phase : @xmath462 where : @xmath463 here , @xmath464 is the angle between the neutrino momentum vector and the pion flight direction in the pion rest frame . developing @xmath465 and @xmath466 in terms of the small quantity @xmath467 , eqn(a2 ) may be written as : @xmath468 \cos\theta_{\nu}^{\ast } + \left(\frac{p_{\pi}}{m_{\pi}}\right)^2 \frac{\delta m^2 l}{2 p_0}\left[1-\frac{3 m_{\pi}}{2 p_0}\right ] ~~~~~~~(a3)\ ] ] performing now the average of the interference term over the isotropic distribution in @xmath469 : @xmath470 d \cos \theta_{\nu}^{\ast } \nonumber \\ & = & \frac{1}{2 } { \it re } \exp \left\{i \phi_{12}^{\nu,\pi}(0)+ \left(\frac{p_{\pi}}{m_{\pi}}\right)^2\left(i\frac{\delta m^2 l}{2 p_0 } \left[1-\frac{3 m_{\pi}}{2 p_0}\right ] \right . . \nonumber \\ & & \left . ~~~~~~~~~~-\frac{1}{6}\left(\frac{\delta m^2 l}{p_0 } \left[1-\frac{m_{\pi}}{p_0}\right ] \right)^2\right ) \right\}~~~~~~~~~~~~~~~~~~(a4 ) \nonumber\end{aligned}\ ] ] in deriving eqn(a4 ) the following approximate formula is used : @xmath471 = \frac{\sin \alpha}{\alpha } \simeq 1-\frac{\alpha^2}{6}~~~~~~~~~~~~~~~~~~~~~~~~~(a5)\ ] ] where @xmath472 \ll 1\ ] ] the average over the maxwell - boltzmann distribution ( 2.31 ) is readily performed by ` completing the square ' in the exponential , with the result : @xmath473\right)^2 \right . \nonumber \\ & & + \left . i\left[\phi_{12}^{\nu,\pi}(0)+ \frac{3}{4}\left(\frac{\overline{p}_{\pi}}{m_{\pi}}\right)^2\left(\frac{\delta m^2 l}{p_0 } \left[\frac{3 m_{\pi}}{2 p_0}-1\right]\right)\right]\right\ } \nonumber \\ & \equiv & f^{\nu}(dop)\cos [ \phi_{12}^{\nu,\pi}(0)+\phi^{\nu}(dop ) ] \nonumber~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~(a6)\end{aligned}\ ] ] leading to eqns(2.32 ) and ( 2.33 ) for the doppler damping factor @xmath474 and phase shift @xmath475 , respectively . the correction for the effect of thermal motion in the case of muon oscillations @xmath475 , respectively . is performed in a similar way . the oscillation phase : @xmath476 is modified by the lorentz boost of the muon momentum and energy , and the relativistic time dilatation of the phase increment of the pion path , to : @xmath477 where @xmath478 and @xmath479 is given by eqn(2.37 ) . developing , as above , in terms of @xmath467 , gives : @xmath480 \cos\theta_{\mu}^{\ast } + \left(\frac{p_{\pi}}{m_{\pi}}\right)^2 \frac{m_{\mu}^2 \delta m^2 l}{p_0 ^ 3 } \left[\frac{e_0^{\mu}}{2 m_{\pi}}-1\right ] ( a9)\ ] ] performing the averages over @xmath481 and @xmath482 then leads to eqns(2.53 ) and ( 2.54 ) for the damping factor @xmath483 and phase shift @xmath484 , respectively . the effect of the finite longitudinal dimensions of the target or detector is calculated by an appropriate weighting of the interference term according to the value of the distance @xmath485 between the decay and detection points ( see fig.1 ) . writing the interference phase as @xmath486 , and assuming a uniform distribution of decay points within the target of thickness @xmath192 : @xmath487 substituting the value of @xmath0 appropriate to neutrino oscillations yields eqn(2.34 ) . since the value of @xmath0 is the same for neutrino and muon oscillations , the same formula is also valid in the latter case . the same correction factor , with the replacement @xmath195 describes the effect of a finite detection region of length @xmath488 : @xmath489 the first step in the derivation of eqn(4.17 ) relating @xmath490 to @xmath164 is to calculate the angle @xmath491 , in the centre - of - mass ( cm ) system of the decaying pion , corresponding to @xmath295 in the laboratory ( lab ) system ( see fig.3 ) . it is assumed , throughout , that the pion and muon are ultra - relativistic in the latter system , so that : @xmath492 . the lorentz transformation relating the cm and lab systems gives the relation : @xmath493 the starred quantities refer to the pion cm system . making the substitutions : @xmath494 , @xmath495 , eqns(b1 ) may be solved to obtain , up to first order in @xmath295 , @xmath491 and @xmath164 : @xmath496 where , ( c.f . eqn(2.37 ) : @xmath497 using eqn(2.36 ) @xmath498 may be expressed in terms of the neutrino mass difference : @xmath499 eliminating now @xmath498 between ( b2 ) and ( b4 ) gives a relation between @xmath295 , @xmath491 and @xmath164 : @xmath500 in the lab system , and in the ur limit , the difference of the velocities of the muons recoiling against the two neutrino mass eigenstates is : @xmath501~~~~~~~~~~~~~~~~~(b6)\ ] ] where @xmath48 is the muon energy in the lab system for vanishing neutrino masses . making the lorentz transformation of the muon energy from the pion cm to the lab frames , and using eqns(2.4 ) and ( 2.36 ) to retain only terms linear in @xmath164 and @xmath491 , enables eqn(b6 ) to be re - written as : @xmath502 ~~~~~~~~~~~~~~~~~~~~~~~(b7)\ ] ] where @xmath46 is the energy of the pion beam . by combining the geometrical constraint equation for the muon velocities , ( 4.16 ) with ( b5 ) and ( b7 ) the angles @xmath295 and @xmath491 may be eliminated to yield the equation for lab frame velocity difference : @xmath503 where @xmath504 @xmath505 to simplify ( b8 ) , the quantity @xmath506 is now expressed in terms of kinematic quantities in the pion cm system . within the ur approximation used , @xmath507 so that @xmath508 writing eqn(b1 ) to first order in @xmath284 , and neglecting terms of @xmath509 : @xmath510 using eqn(b12 ) , and expressing @xmath48 in terms of pion cm quantities , eqn(b11 ) may be written as : @xmath511 expressing the rhs of ( b13 ) in terms of @xmath46 and @xmath48 , using the relation : @xmath512 gives eqn(4.20 ) of the text . on substituting ( b13 ) into the rhs of ( b10 ) , it can be seen that the factor in the large curly brackets is the same in ( b9 ) and ( b10 ) , and so cancels in the ratio @xmath513 in eqn(b8 ) . it follows that : @xmath514 finally , using ( b3 ) and ( b14 ) to express the factor in large brackets in eqn(b15 ) in terms of @xmath48 and @xmath46 , eqn(4.17 ) of the text is obtained . 99 b.pontecorvo , jetp * 33 * 599 ( 1957 ) , [ sov . phys . jetp * 6 * 429 ( 1958 ) ] ; jetp * 34 * 247 ( 1958 ) , [ sov . jetp * 7 * 172 ( 1958 ) ] . s.m.bilenky and b.pontecorvo , phys . rep . * 41 * 225 ( 1978 ) . s.m.bilenky and s.t.petcov , rev . phys . * 59 * 671 ( 1987 ) . y.grossman and h.j.lipkin , phys . rev . * d55 * 2760 ( 1997 ) . s. de leo , g.ducati and p.rotelli,mod . phys . lett . * a 15 * 2057 ( 2000 ) . r.g.winter , lettere al nuovo cimento * 30 * 101 ( 1981 ) . s.mohanty , _ covariant treatment of flavour oscillations _ , hep - ph/9702424 . y.srivastava , a.widom and e.sassaroli , eur . j * c2 * 769 ( 1998 ) . r.p.feynman , phys . rev . * 76 * 749 ( 1949 ) . et al . _ , phys . rev . * d59 * 054018 ( 1999 ) . a.d.dolgov _ et al . _ , b502 * 3 ( 1997 ) . r.p.feynman , rev . phys . * 20 * 367 ( 1948 ) . p.a.m.dirac , physikalische zeitschrift der sowjetunion band 3 , heft 1 ( 1933 ) . reprinted in _ selected papers on quantum electrodynamics _ j.schwinger , dover , new york , ( 1958 ) p312 ; see also ref . @xcite , chapter v , section 32 . r.p.feynman and a.r.hibbs , _ quantum mechanics and path integrals _ , mcgraw hill , new york , ( 1965 ) ; c.grosche and f.steiner _ handbook of path integrals _ , springer tracts in modern physics , springer - verlag , berlin ( 1998 ) . w.heisenberg , _ the physical principals of the quantum theory _ , english translation by c.eckart and f.c.hoyt , university of chicago press , chicago , 1930 . chapter iv , section 2 . p.a.m.dirac , _ the principles of quantum mechanics _ , fourth edition , o.u.p . , london ( 1958 ) p9 . c. athanassopoulos _ et al . _ , phys . rev . * c58 * 2489 ( 1998 ) ; a.aguilar _ et al . _ , hep - ex/0104049 . b.zeitnitz _ et al . _ , progress in particle and nuclear physics , * 40 * 169(1998 ) ; j.kleinfeller nucl . b ( proc . suppl . ) * 87 * 281 ( 2000 ) . z.maki , m.nakagawa and s.sakata , prog . theor * 28 * 870 ( 1962 ) . l.i.schiff , _ quantum mechanics _ , 2nd edition , mcgraw hill , new york , ( 1955 ) , chapter viii . j.m.lvy-leblond and f.balibar , _ quantics , rudiments of quantum physics , _ north - holland , amsterdam , ( 1990 ) , chapter v. m.zralek , acta . polon . , * b29 * 3925 ( 1998 ) . c.giunti and c.w.kim , found . phys . lett.*14 * 213 ( 2001 ) . b.kayser , neutrino physics as explored by flavour change in phys . rev . * d66 * ( 2002 ) 010001 ( review of particle properties ) . r.e.shrock , phys . lett . * b96 * 159 ( 1980 ) . r.e.shrock , phys . rev . * d24 * 1232 ( 1981 ) ; * d24 * 1275 ( 1981 ) . j.h.field , _ lepton flavour eigenstates do not exist if neutrinos are massive : ` neutrino oscillations ' reconsidered _ hep - ph/0301231 . j.h.field , _ a covariant path amplitude description of flavour oscillations : the gribov - pontecorvo phase for neutrino vacuum propagation is right _ , hep - ph/0211199 . c.giunti , c.w.kim and u.w.lee , phys . rev . * d45 * 2414 ( 1992 ) . v.gribov and b.pontecorvo , phys . lett . * b28 * 493 ( 1969 ) . s.m.bilenky and b.pontecorvo , phys b61 * 248 ( 1976 ) . h.fritsch and p.minkowski , phys b62 * 72 ( 1976 ) . s.m.bilenky and b.pontecorvo , lettere al nuovo cimento * 17 * 569 ( 1976 ) . s.m.bilenky and b.pontecorvo , physics reports * 41 * 225 ( 1978 ) . see ref . @xcite , chapter ii . b.kayser , phys . rev . * d24 * 110 ( 1981 ) . c.giunti , c.w.kim and u.w.lee , phys . rev . * d44 * 3635 ( 1991 ) . s.nussinov , phys . lett . * b63 * 201 ( 1976 ) . c.giunti , c.w.kim and u.w.lee , phys . lett . * b274 * 87 ( 1992 ) . k.kiers , s.nussinov and n.weiss , phys . rev . * d53 * 537 ( 1996 ) . m.beuthe , phys . 375 105 ( 2003 ) . c.giunti , _ coherence and wave packets in neutrino oscillations _ , hep - ph/0302026 . et al . _ , phys . rev . * d48 * 4310 ( 1993 ) . j.rich , phys . rev . * d48 * 4318 ( 1993 ) . w.grimus and p.stockinger , phys * d54 * 3414 ( 1996 ) . j.e.campagne , phys . lett . * b400 * 135 ( 1997 ) . k.kiers and n.weiss , phys . rev . * d57 * 3091 ( 1998 ) . yu.v.shtanov , phys . rev . * d57 * 4418 ( 1998 ) . c.giunti and c.w.kim , phys . rev . * d58 * 017301 ( 1998 ) . c.giunti , c.w.kim and u.w.lee , phys . lett . * b421 * 237 ( 1998 ) . w.grimus , s.mohanty and p.stockinger , phys . rev . * d59 * 013011 ( 1999 ) . w.grimus , s.mohanty and p.stockinger , phys . rev . * d61 * 033001 ( 1999 ) . c.y.cardell , phys . rev . * d61 * 073006 ( 1999 ) . f.halzen and a.d.martin , _ quarks and leptons : an introductory course in particle physics _ , john wiley and sons , new york 1984 , section 4.8 . m.blasone and g.vitello , ann . ( new york ) * 244 * 283 ( 1995 ) ; e * 249 * 363 ( 1995 ) . m.blasone , p.a.henning and g.vitello , phys . lett . * b451 * 140 ( 1999 ) . y.n.srivastava and a.widom , _ of course muons can oscillate _ , hep - ph/9707268 . t.ohlsson , phys . lett . * b502 * 159 ( 2001 ) . b.kayser and l.stodolsky , phys . lett . * b359 * 343 ( 1995 ) . j.h.field , _ a covariant feynman path amplitude calculation of neutral kaon and b - meson oscillations _ , paper in preparation . , d.e.groom _ et al . j * c15 * , 1 ( 2000 ) . s.deleo and p.rotelli , jetp lett . * 76 * 56 ( 2002 ) . t.kajita and y.totsuka , rev . phys . * 73 * 85 ( 2001 ) . j.n.bahcall , m.h.pinsonneault and s.basu , astrophys . * j 555 * 990 ( 2001 ) . h.g.berry and j.l.subtil , phys . * 27 * 1103 ( 1971 ) . c.blondel , c.delsart and f.dulieu , phys . lett . * 77 * 3755 ( 1996 ) . _ , am j. phys . * 66 * 38 ( 1998 ) . c.blondel , s.berge and c.delsart , am j. phys . * 69 * 810 ( 2001 ) .

feynman s path amplitude formulation of quantum mechanics is used to analyse the production of charged leptons from charged current weak interaction processes . for neutrino induced reactions the interference effects predicted are usually called ` neutrino oscillations ' . similar effects in the detection of muons from pion decay are here termed ` muon oscillations ' . processes considered include pion decay ( at rest and in flight ) , and muon decay and nuclear @xmath0-decay at rest . in all cases studied , a neutrino oscillation phase different from the conventionally used one is found . a concise critical review is made of previous treatments of the quantum mechanics of neutrino and muon oscillations . 24.5 cm -5pt -5pt -50pt addtoresetequationsection * j.h.field * dpartement de physique nuclaire et corpusculaire universit de genve . 24 , quai ernest - ansermet ch-1211 genve 4 . pacs 03.65.bz , 14.60.pq , 14.60.lm , 13.20.cz quantum mechanics , neutrino oscillations .

a long outstanding challenge in modern astrophysics is the intriguing observational behavior of the galactic rotation curves , and the mass discrepancy in clusters of galaxies . both these observations suggest the existence of a ( non or weakly interacting ) form of dark matter at galactic and extra - galactic scales . indeed , according to newton s theory of gravitation , at the boundary of the luminous matter , the rotation curves of test particles gravitating around galaxies or galaxy clusters should show a keplerian decrease of the tangential rotational speed @xmath1 with the distance @xmath2 , so that @xmath3 , where @xmath4 is the dynamical mass within the radius @xmath2 . however , the observational evidence indicates rather flat rotation curves @xcite . the tangential rotational velocities @xmath1 increase near the galactic center , as expected , but then intriguingly remain approximately constant at a value of @xmath5 km / s . therefore , observations provide a general mass profile of the form @xmath6 @xcite . consequently , the mass around a galaxy increases linearly with the distance from the center , even at large distances , where very little or no luminous matter can be observed . as mentioned above , the observed behavior of the galactic rotation curves is explained by assuming the existence of some dark ( invisible ) matter , distributed in a spherical halo around the galaxies , and interacting only gravitationally with ordinary matter . the dark matter is usually described as a pressureless and cold medium . a recently proposed model considered the possibility that dark matter is a mixture of two non - interacting perfect fluids , with different four - velocities and thermodynamic parameters . the two - fluid model can be described as an effective single anisotropic fluid , with distinct radial and tangential pressures @xcite . in fact , many possible candidates for non - luminous dark matter have been proposed in the literature ranging from @xmath7 solar mass black holes , running through low mass stars to @xmath8 ev axions , although the most popular being the weakly interacting massive particles ( wimps ) ( for a review of the particle physics aspects of dark matter see @xcite ) . indeed , the interaction cross section of wimps with normal baryonic matter , although practically negligible , is expected to be non - zero , and therefore there is a possibility of detecting them directly . nevertheless , despite several decades of intense experimental and observational effort , there is presently still no direct evidence of dark matter particles @xcite . however , it is important to emphasize that the masses in galaxies and clusters of galaxies are deduced from the observed distances and velocities of the system under consideration . these relationships are based on newton s laws of dynamics , thus the newtonian dynamical masses of galactic systems are not consistent with the observed masses . indeed , newton s laws have proven extremely reliable in describing local phenomena , that there is an overwhelming tendency to apply them in the intermediate galactic scales . note that the mass discrepancy is interpreted as evidence for the existence of a _ missing mass _ , i.e. , dark matter in galactic systems . therefore , one can not _ a priori _ exclude the possibility that einstein s ( and newtonian ) gravity breaks down at the galactic or extra - galactic scales . indeed , a very promising way to explain the recent observational data @xcite on the recent acceleration of the universe and on dark matter is to assume that at large scales einstein s general relativity , as well as its newtonian limit , breaks down , and a more general action describes the gravitational field . several theoretical models , based on a modification of newton s law or of general relativity , including modified newtonian orbital dynamics ( mond ) @xcite , scalar fields or long range coherent fields coupled to gravity @xcite , brane world models @xcite , bose - einstein condensates @xcite , modified gravity with geometry - matter coupling @xcite , non - symmetric theories of gravity @xcite , and eddington - inspired born - infeld gravity @xcite have been used to model galactic `` dark matter '' . from a theoretical point of view a very attractive possibility is to modify the form of the einstein - hilbert lagrangian , so that such a modification could naturally explain dark matter and dark energy , without resorting to any exotic forms of matter . the simplest extension of the einstein - hilbert action consists in modifying the geometric part of the action , through the substitution of the ricci scalar with a generic function @xmath0 . this change in the action introduces higher order terms in the gravitational field action . the so - called @xmath0 gravitational theories were first proposed in @xcite , and later used to find a non - singular isotropic de sitter type cosmological solution @xcite . detailed reviews of @xmath0 theories can be found , for instance , in @xcite . the most serious difficulty of @xmath0 theories is that , in general , these theories can not pass the standard solar system tests @xcite . however , there exists some classes of theories that can solve this problem @xcite . using phase space analysis of the specific involved gravitational model , it was shown that @xmath0 theories , in general , can explain the evolution of the universe , from a matter dominated early epoch up to the present , late - time self accelerating phase @xcite . @xmath0 type gravity theories can be generalized by including the function @xmath0 in the bulk action of the brane - world theories @xcite . it has been shown in @xcite that this type of generalized brane world theories can describe a universe beginning with a matter - dominated era , and ending in an accelerated expanding phase . the classical tests of this theory were considered in @xcite . on the other hand , @xmath0 theories have also been studied in the palatini approach , where the metric and the connection are regarded as independent fields @xcite . in fact , these approaches are certainly equivalent in the context of general relativity , i.e. , in the case of the linear einstein - hilbert action . on the other hand , for a general @xmath0 term in the action , they seem to provide completely different theories , with very distinct field equations . the palatini variational approach , for instance , leads to second order differential field equations , while the resulting field equations in the metric approach are fourth order coupled differential equations . these differences also extend to the observational aspects . all the palatini @xmath0 models aimed at explaining the cosmic speedup studied so far lead to microscopic matter instabilities , and to unacceptable features in the evolution patterns of cosmological perturbations @xcite . hence , in order to cure some of the pathologies of the @xmath0 gravity models in both their metric and palatini formulation , a novel approach was recently proposed @xcite , that consists of adding to the einstein - hilbert lagrangian an @xmath0 term constructed within the framework of the palatini formalism . using the respective dynamically equivalent scalar - tensor representation , even if the scalar field is very light , the theory can pass the solar system observational constraints . therefore the long - range scalar field is able to modify the cosmological and galactic dynamics , but leaves the solar system unaffected . the absence of instabilities in perturbations was also verified , and explicit models , which are consistent with local tests and lead to the late - time cosmic acceleration , were also found . furthermore , the cosmological applications of the hybrid metric - palatini gravitational theory were investigated in @xcite , where specific criteria to obtain the cosmic acceleration were analyzed , and the field equations were formulated as a dynamical system . indeed , several classes of dynamical cosmological solutions , depending on the functional form of the effective scalar field potential , describing both accelerating and decelerating universes , were explicitly obtained . the cosmological perturbation equations were also derived and applied to uncover the nature of the propagating scalar degree of freedom and the signatures these models predict in the large - scale structure . the general conditions for wormhole solutions according to the null energy condition violation at the throat in the hybrid metric - palatini gravitational theory were also presented in @xcite . a new approach to modified gravity which generalizes the hybrid metric - palatini gravity was introduced in @xcite . the gravitational action is taken to depend on a general function of both the metric and palatini curvature scalars . the dynamical equivalence with a non - minimally coupled bi - scalar field gravitational theory was proved . the evolution of the cosmological solutions in this model was studied by using dynamical systems techniques . in @xcite a method was developed to analyse the field content of `` hybrid '' gravity theories whose actions involve both the independent palatini connection and the metric levi - civita connection , and , in particular , to determine whether the propagating degrees of freedom are ghosts or tachyons . new types of second , fourth and sixth order derivative gravity theories were investigated , and from this analysis it follows that the metric - palatini theory is singled out as a viable class of `` hybrid '' extensions of general relativity . in addition to this , the stability of the einstein static universe was analysed in @xcite . in the latter , by considering linear homogeneous perturbations , the stability regions of the einstein static universe were parameterized by the first and second derivatives of the scalar potential , and it was explicitly shown that a large class of stable solutions exists in the respective parameter space . for a brief review of the hybrid metric - palatini theory , we refer the reader to @xcite . thus , the hybrid metric - palatini theory opens up new possibilities to approach , in the same theoretical framework , the problems of both dark energy and dark matter . in @xcite , the generalized virial theorem in the scalar - tensor representation of the hybrid metric - palatini gravity was analysed . more specifically , taking into account the relativistic collisionless boltzmann equation , it was shown that the supplementary geometric terms in the gravitational field equations provide an effective contribution to the gravitational potential energy . indeed , it was shown that the total virial mass is proportional to the effective mass associated with the new terms generated by the effective scalar field , and the baryonic mass . this shows that the geometric origin in the generalized virial theorem may account for the well - known virial theorem mass discrepancy in clusters of galaxies . in addition to this , the astrophysical applications of the model were considered and it was shown that the model predicts that the mass associated to the scalar field and its effects extend beyond the virial radius of the clusters of galaxies . in the context of the galaxy cluster velocity dispersion profiles predicted by the hybrid metric - palatini model , the generalized virial theorem can be an efficient tool in observationally testing the viability of this class of generalized gravity models . thus , hybrid metric - palatini gravity provides an effective alternative to the dark matter paradigm of present day cosmology and astrophysics . in this latter context , it is the purpose of the present paper to investigate the possibility that the observed properties of the galactic rotation curves could be explained in the framework of hybrid metric - palatini gravity , without postulating the existence of the hypothetical dark matter . as a first step in this study , we obtain the expression of the tangential velocity of test particles in stable circular orbits around galaxies . since we assume that the test particles move on the geodesic lines of the space - time , their tangential velocity is determined only by the radial distance to the galactic center , and the metric , through the derivative of the @xmath9 metric component with respect to the radial coordinate . the metric in the outer regions of the galaxy is largely shaped by the energy contained in the effective scalar field of the hybrid metric - palatini gravitational theory . therefore the behavior of the neutral hydrogen gas clouds outside the galaxies , and their flat rotation curves , can be explained by the presence of the scalar field generated in the model . by using the weak field limit of the gravitational field equations in the hybrid metric - palatini model we obtain the explicit form of the tangential velocity , and show that the existence of a constant velocity region is possible for some specific values of the model parameters . since the observations on the galactic rotation curves are obtained from the doppler frequency shifts , we generalize the expression of the frequency shifts by including the effect of the scalar field . we also consider the velocity dispersion of the stars in the galaxy , and obtain the stellar velocity dispersion as a function of the scalar field . thus , at least in principle , all the basic parameters of the model can be obtained directly from astronomical observations . the knowledge of the tangential velocity allows the complete determination of the functional form of the scalar field from the observational data . the basic parameters of the model can be immediately obtained in the flat rotation curves region , which is determined by the derivative of the scalar field potential . hence the functional form of the scalar field can be obtained exactly , within the weak field limit of the model , for the entire galactic space - time , and tested at the galactic scale . therefore , all the physical parameters of the hybrid metric - palatini gravitational theory can be either obtained directly , or severely constrained by astronomical observations . the present paper is organized as follows . in section [ sec : b ] , the field equations of the hybrid metric - palatini gravity model , as well as the general fluid representation of the stress - energy tensor are presented . in section [ sect3 ] , the tangential velocity of test particles in stable circular orbits are derived and in section [ sect4 ] , the tangential velocities of test particles in the galactic halos in the hybrid model are discussed . the comparison of the theoretical predictions for the rotational velocity and the observational data for four low surface brightness galaxies is considered in section [ sect5 ] . the velocity dispersion of the stars in the galaxy , representing important observational tests of the model , as well as the red and blue shifts of the electromagnetic radiation emitted by the gas clouds are also investigated . we discuss and conclude our results in section [ sect6 ] . in this paper , we use the landau - lifshitz @xcite sign conventions and definitions of the geometric quantities . in this section , we briefly present the basic formalism , for self - completeness and self - consistency , and the field equations of the hybrid metric - palatini gravitational theory within the equivalent scalar - tensor representation ( we refer the reader to @xcite for more details ) , and furthermore obtain the perfect fluid form of the stress - energy tensor of the scalar field . the action for hybrid metric - palatini gravity is obtained by adding an @xmath11 term , constructed within the framework of the palatini formalism , to the metric einstein - hilbert lagrangian @xcite , and is given by @xmath12 + s_m \ , \ ] ] where @xmath13 is the gravitational coupling constant , and the scalar curvature @xmath14 , depending on both the metric and an independent dynamical connection @xmath15 , is defined as @xmath16 @xmath17 is the ricci tensor obtained from the connection @xmath15 . the hybrid metric - palatini theory may be expressed in a purely scalar - tensor representation , by the following action @xmath18 + s_m \,,\ ] ] which differs fundamentally from the @xmath19 brans - dicke theory in the coupling of the scalar to the scalar curvature , where @xmath20 is the dimensionless brans - dicke coupling constant . the variation of this action with respect to the metric tensor gives the field equations [ einstein_phi ] g_=^2(t _ + t_^ ( ) ) , where @xmath21 is the ordinary matter stress - energy tensor , and t_^()&= & , is the stress - energy tensor of the scalar field . the variation of the action with respect to the scalar field gives [ variation_phi ] r - + _ ^- = 0 . moreover , one can show that the identity @xmath22 also holds , and that the scalar field @xmath23 is governed by the second - order evolution equation @xmath24=\frac{\phi\kappa^2}{3}t\,,\ ] ] with @xmath25 , which is an effective klein - gordon equation . the stress - energy tensor of a fluid can be generally represented as t_=(+p)u_u_-pg_+q_u_+q_u_+s _ , where @xmath26 is the four - velocity of the fluid , @xmath27 and @xmath28 are the energy density and isotropic pressure , respectively , @xmath29 is the heat flux , and @xmath30 is the tensor of the anisotropic dissipative stresses . the heat flux four - vector and the anisotropic stress tensor satisfy the conditions @xmath31 , @xmath32 , and @xmath33 , respectively . the four - velocity is normalized so that @xmath34 , and @xmath35 . by introducing the projection tensor @xmath36 , with the properties @xmath37 , @xmath38 , and @xmath39 , the thermodynamic parameters of the fluid can be obtained from the stress - energy tensor as [ ex ] & & = u^u^t _ , p =- h^t _ , + & & q_=u^h^_t _ , s_=h^_h^_t_+ph_. in order to obtain the perfect fluid representation of the stress - energy tensor of the scalar field in hybrid metric - palatini gravity , we introduce first the four - velocity of the scalar field as u^ _ ( ) = , which satisfies the relation @xmath40 . therefore , with the use of eqs . ( [ ex ] ) we obtain the effective energy density @xmath41 and pressure @xmath42 in the scalar field description of hybrid metric - palatini gravity as _ & = & \{^_(_^)-_^ + & & -+t - v ( ) } , p_&=&\{--_^ + & & + -t+v ( ) } , q^()=u^ _ ( ) _ u^ _ ( ) , s^()&= & + & & , where @xmath43 is the expansion of the fluid . it is interesting to note that the fluid - equivalent stress - energy tensor in hybrid metric - palatini gravity is not of a perfect fluid form , but contains `` heat transfer '' terms , as well as an anisotropic dissipative component . therefore , the stress - energy of the scalar field can be written in a form equivalent to a general fluid as t^()&=&(_+p_)u^_()u^_()-p_g^+ + & & q^()u^_()+q^()u^ _ ( ) + s^ ( ) , which will be useful in determining the galactic geometry in the context of the tangential velocity curves analysis outlined below . the most direct method for studying the gravitational field inside a spiral galaxy is provided by the galactic rotation curves . they are obtained by measuring the frequency shifts @xmath44 of the 21-cm radiation emission from the neutral hydrogen gas clouds . the 21-cm radiation also originates from stars . the 21-cm background from the epoch of reionization is a promising cosmological probe : line - of - sight velocity fluctuations distort redshift , so brightness fluctuations in fourier space depend upon angle , which linear theory shows can separate cosmological from astrophysical information ( for a recent review see @xcite ) . instead of using @xmath44 the resulting redshift is presented by astronomers in terms of a velocity field @xmath1 @xcite . in the following , we will assume that the gas clouds behave like test particles , moving in the static and spherically symmetric geometry around the galaxy . without a significant loss of generality , we assume that the gas clouds move in the galactic plane @xmath45 , so that their four - velocity is given by @xmath46 , where the overdot stands for derivation with respect to the affine parameter @xmath47 . the static spherically symmetric metric outside the galactic baryonic mass distribution is given by the following line element @xmath48 where the metric coefficients @xmath49 and @xmath50 are functions of the radial coordinate @xmath2 only . the motion of a test particle in the gravitational field with the metric given by eq . ( [ line ] ) , is described by the lagrangian @xcite @xmath51 , \ ] ] where @xmath52 , which simplifies to @xmath53 along the galactic plane @xmath54 . from the lagrange equations it follows that we have two constants of motion , namely , the energy @xmath55 per unit mass , and the angular momentum @xmath56 per unit mass , given by @xmath57 and @xmath58 , respectively . the normalization condition for the four - velocity @xmath59 gives @xmath60 , from which , with the use of the constants of motion , we obtain the energy of the particle as @xmath61 from eq . ( [ energy ] ) it follows that the radial motion of the test particles is analogous to that of particles in newtonian mechanics , having a velocity @xmath62 , a position dependent effective mass @xmath63 , and an energy @xmath64 , respectively . in addition to this , the test particles move in an effective potential provided by the following relationship @xmath65 the conditions for circular orbits , namely , @xmath66 and @xmath67 lead to @xmath68 and @xmath69 respectively . note that the spatial three - dimensional velocity is given by @xcite @xmath70 .\ ] ] for a stable circular orbit @xmath71 , and the tangential velocity of the test particle can be expressed as @xmath72 in terms of the conserved quantities , and along the galactic plane @xmath45 , the angular velocity is given by @xmath73 and taking into account eqs . ( [ cons1 ] ) and ( [ cons2 ] ) , we finally obtain the following relationship @xcite @xmath74 therefore , once the tangential velocity of test particles is known , the metric function @xmath75 outside the galaxy can be obtained as ( r)=2 . [ metricnu ] the tangential velocity @xmath76 of gas clouds moving like test particles around the center of a galaxy is not directly measurable , but can be inferred from the redshift @xmath77 observed at spatial infinity , for which @xmath78 \left ( 1\pm v_{tg}/c\right ) /\sqrt{1-v_{tg}^{2}/c^2}$ ] @xcite . due to the non - relativistic velocities of the gas clouds , with @xmath79 , we observe that @xmath80 , as the first part of a geometric series . the observations show that at distances large enough from the galactic center the tangential velocities assume a constant value , i.e. , @xmath81 constant @xcite . in the latter regions of the constant tangential velocities , ( [ metricnu ] ) can be readily integrated to provide the following metric tensor component [ nu ] e^=()^2v_tg^2/c^21 + 2 ( ) , where @xmath82 is an arbitrary constant of integration . if we match the metric given by eq . ( [ nu ] ) with the schwarzschild metric on the surface of the galactic baryonic matter distribution , having a radius @xmath83 , @xmath84 , we obtain the following relationship r_=. an important physical requirement for the circular orbits of the test particle around galaxies is that they must be stable . let @xmath85 be the radius of a circular orbit and consider a perturbation of it of the form @xmath86 , where @xmath87 @xcite . taking expansions of @xmath88 and @xmath89 about @xmath90 , it follows from eq . ( [ energy ] ) that @xmath91 the condition for stability of the simple circular orbits requires @xmath92 @xcite . hence , with the use of the condition @xmath93 , we obtain the condition of the stability of the orbits as @xmath94|_{r = r_0}$ ] . by taking into account eq . ( [ vtg ] ) , it immediately follows that for massive test particles whose velocities are determined by the @xmath9 component of the metric tensor only the stability condition of the circular orbits is always satisfied . the rotation curves only determine one , namely @xmath49 , of the two unknown metric functions , @xmath49 and @xmath50 , which are required to describe the gravitational field of the galaxy . hence , in order to determine @xmath95 we proceed to solve the gravitational field equations for the hybrid metric - palatini gravitational theory outside the baryonic matter distribution . this allows us to take all the components of the ordinary matter stress - energy tensor as being zero . taking into account the stress - energy tensor for the equivalent scalar field representation of hybrid metric - palatini gravity the gravitational field equations describing the geometry of the galactic halo take the form [ f1b ] -e^- ( - ) + & = & ( _ + p_)u _ ( ) tu^t_()-p _ + & & + q^()tu_()t+q^()_tu_()^t+s^()t_t , [ f2b ] e^- ( + ) - & = & -(_+p_)u _ ( ) ru^r_()+p _ + & & -q^()ru_()r - q^()_ru_()^r - s^()r_r , & & e^- ( ^++- ) = - s^ ( ) _ + & & -(_+p_)u _ ( ) u^_()+p_- q^()u_()-q^()_u_()^ , where there is no summation upon the pair of indices @xmath96 . the weak field limit of the gravitational theories at the solar system level is usually obtained by using isotropic coordinates . however , it is useful to apply schwarzschild coordinates in studying exact solutions and in the context of galactic dynamics . in the following , we will adopt in our analysis the schwarzschild coordinate system . we assume that the gravitational field inside the halo is weak , so that @xmath97 , which allows us to linearise the gravitational field equations retaining only terms linear in @xmath98 . moreover , we assume that the scalar field @xmath99 is also weak , so that @xmath100 . by representing the scalar field as @xmath101 , where @xmath102 is a small perturbation around the background value @xmath103 of the field , in the first order of perturbation , the scalar field potential @xmath104 and its derivative with respect to @xmath99 can be represented as v()=v(_0+)v(_0)+v(_0)+ .... , and v()v(_0)+ v(_0 ) , respectively . in the linear approximation we have @xmath105 . therefore the effective klein - gordon type equation of the scalar field , eq . ( [ eq : evol - phi ] ) takes the form [ yuk ] ( ^2 -)=0 , where a constant on the right - hand side of this equation has been absorbed into a redefinition of @xmath106 , and the following parameter has been defined for notational simplicity = . from a physical point of view @xmath107 represents ( in natural units ) the inverse of the mass @xmath108 of the particle associated with the scalar field , @xmath109 . the hybrid metric - palatini gravity theory can pass the solar system observational constraints even if the scalar field is very light , that is , @xmath108 is very small @xcite . within this linear approximation the stress - energy tensor of the scalar field is given by t_^()= , where @xmath110 and @xmath111 are defined by = - v(_0 ) , = -v(_0 ) . therefore the linearized gravitational field equations take the form [ f1 ] ( r)=+&=&^(eff ) , + [ f2 ] -+=++&=&-p_r^(eff ) , + [ f3 ] -(+)=+&=&-p_^(eff ) . using spherical symmetry , eq . ( [ yuk ] ) takes the form r-=0 , which yields the following general solution ( r)=_0 , where @xmath112 is an integration constant . comparing this expression with the results obtained in @xcite for the weak - field limit ( taking into account the transformation from isotropic to schwarzschild coordinates ) , we find that [ psi0 ] _ 0=-_0<0 , where @xmath113 and @xmath83 are the mass and the radius of the galactic baryonic distribution , respectively . ( [ f1 ] ) can be immediately integrated to provide ( r)&=&+^r(+)^2d + & = & + r^2-(1 + ) , where @xmath114 is an integration constant . comparing again with the results obtained in @xcite for the weak - field limit , we find that @xmath115 . the tangential velocity of the test particles in stable circular orbits moving in the galactic halo can be derived immediately from eq . ( [ f2 ] ) , and is given by = = -r^2-r^2-r^2 , which in terms of the solutions found above becomes & = & r^2 + - + & & , where @xmath116 . the term proportional to @xmath117 corresponds to the cosmological background , namely the de sitter geometry , and we assume that it has a negligible contribution on the tangential velocity of the test particles at the galactic level . on the surface of the baryonic matter distribution the tangential velocity must satisfy the boundary condition , which , with the use of eq . ( [ psi0 ] ) , gives the following constraint on the parameters of the model , ( 1+)(2+r_^2)+(1+r_^2)0 . in order to satisfy the above condition would require that @xmath118 , or , equivalently , v(_0)>0 , and 2<v(_0)r_^2<3 , respectively . in the regions near the galactic baryonic matter distribution , where @xmath119 , we have @xmath120 , to a very good approximation . hence in this region the tangential velocity can be approximated as & & -_0 + & & -(1+r_^2)r , r_br r_. if the parameters of the model satisfy the condition 2gm_b - c^2_0(r_^2 + 2)0 , the term proportional to @xmath121 becomes negligible , while for small values of @xmath112 , and @xmath122 , the term proportional to @xmath2 can also be neglected . therefore for the tangential velocity of test particles rotating in the galactic halo we obtain -_0- , r_br r_. since according to our assumptions , @xmath123 , the coefficient @xmath124 can be approximated as @xmath125 , which provides for the rotation curve , in the constant velocity region , the following expression , r_br r_. since @xmath126 , the scalar field potential must satisfy the condition @xmath127 . in the first order of approximation , with @xmath128 , for the tangential velocity we obtain the expression + r+r^2 . alternatively , in general we can write the tangential velocity as follows , [ vfin ] & = & r^2 + \{1+e^}. as compared to our previous results , in this representation we have @xmath129 instead of @xmath130 . since we are working in a regime in which @xmath131 , the choice of the constants @xmath83 or @xmath113 does not seem very relevant , since it just amounts to a rescaling of @xmath106 . from now on we will also assume that @xmath132 . from the above equation we want to find the constraints on the model parameters that arise from the expected behavior at different scales . for that purpose , it is convenient to write the equation , equivalently , as follows : & = & + + & & ( 2+r_^2 ) e^-+ ( 1+r_^2 ) ( ) e^- . at intermediate scales , the asymptotic tangential velocity tends to a constant . if we expand the exponential as @xmath133 , then we obtain the following three constraints on the free parameters of the model , & & a ) 1+(2+r_^2 ) 0 , + & & b ) ( 2+r_^2 ) ( 1-)c= constant , + & & c ) ( ) |c| . with increasing @xmath2 , and by assuming that the condition @xmath134 still holds , the rotation curves will decay , at very large distances from the galactic center , to the zero value . in the present section , we will present some observational possibilities of directly checking the validity of the hybrid metric - palatini gravitational model . more specifically , we will first compare the theoretical predictions of the model with a sample of rotation curves of low surface brightness galaxies , respectively . then we consider the possibility of observationally determining the functional form of the scalar field @xmath135 by using the velocity dispersion of stars in galaxies , and the red and blue shifts of gas clouds moving in the galactic halo . in order to test our results we compare the predictions of our model with the observational data on the galactic rotation curves , obtained for a sample of low surface luminosity galaxies in @xcite . generally , in a realistic situation , a galaxy consists of a distribution of baryonic ( normal ) matter , consisting of stars of mass @xmath136 , ionized gas of mass @xmath137 , neutral hydrogen of mass @xmath138 etc . , and the `` dark matter '' of mass @xmath139 , which , in the present model , is generated by the extra contributions to the total energy - momentum , due to the contribution of the effective scalar field . hence the total mass of the galactic baryonic matter is @xmath140 . as for the distribution of the baryonic mass , we assume that it is concentrated into an inner core of radius @xmath141 , and that its mass profile @xmath142 can be described by the simple relation m_b ( r ) = m_b()^3 , rr_b , r_b r_c , where @xmath143 for high surface brightness galaxies ( hsb ) and @xmath144 for low surface brightness ( lsb ) and dwarf galaxies , respectively @xcite . for @xmath145 we have @xmath146 . by representing the coefficient @xmath124 as @xmath147 , where @xmath148 , from eq . ( [ vfin ] ) we obtain the tangential velocity of the massive particles in stable galactic circular orbits as v_tg^2=v_kepl^2+v_kepl^2e^-r / r_(1++),rr_b , where @xmath149 . hence , by also taking into account the baryonic matter contribution , we obtain the total tangential velocity of a massive test particle as [ comp ] v_tgtot^2(km / s)&=&4.3310 ^ 4 ( ) ^6 + & & + 2.7710 ^ 4 _ 0 _ 0 + & & ( 1++ ) , rr_b , where we have used the value @xmath150 . in order to compare the prediction of eq . ( [ comp ] ) with the observed rotation curves of the lsb galaxies in the following we assume @xmath127 , and we fix the numerical values of the universal parameters @xmath151 as @xmath152 . then any variability in the behavior of the rotation curves is due to the variation of the baryonic mass of the galaxy @xmath113 , and of its baryonic mass distribution in the core , described by @xmath141 . in fig . [ fig1 ] , we have compared the predictions of eq . ( [ comp ] ) for the behavior of the rotation curves in the `` dark matter '' region for four lsb galaxies , ddo189 , ugc1281 , ugc711 , and ugc10310 , respectively @xcite . the comparison of the predictions of the theoretical model with the observational results show that the contribution of the scalar field energy density to the tangential velocity of the test particles can explain the existence of a constant rotational velocity region around the baryonic matter , without requiring the presence of the dark matter . of course , in order to gain a better understanding of the behavior of the galactic rotation curves in the hybrid metric - palatini gravity model , the qualitative approach considered in the present section must be reconsidered by taking into account more realistic galactic baryonic matter distributions , and a much larger sample of galaxies having different morphologies . in hybrid metric - palatini gravity one can formally associate an approximate `` dark matter '' mass profile @xmath153 to the tangential velocity profile , which taking into account eq . ( [ f1 ] ) , is given by @xmath154 so that the metric tensor component @xmath95 can be written as ( r)=. the effective `` dark matter '' density profile @xmath155 is obtained as _ dm(r)==(+ ) . in order to observationally constrain @xmath139 and @xmath155 , we assume that each galaxy consists of a single , pressure - supported stellar population that is in dynamic equilibrium and traces an underlying gravitational potential , which is created due to the presence of the scalar field @xmath135 . by assuming spherical symmetry , the equivalent mass profile induced by the scalar field ( the mass profile of the `` effective dark matter '' halo ) can be obtained from the moments of the stellar distribution function via the jeans equation @xcite @xmath156 + \frac{2\rho _ { s}\left ( r\right ) \beta _ { an}(r)}{r}=-\frac{g\rho _ { s}m_{dm}(r)}{r^{2}},\]]where @xmath157 , @xmath158 , and @xmath159 describe the three - dimensional density , the radial velocity dispersion , and the orbital anisotropy of the stellar component , where @xmath160 is the tangential velocity dispersion . with the assumption of constant anisotropy , @xmath161 , the jeans equation can be solved to give @xmath162 as @xcite @xmath163 with the use of eq . ( [ darkmass ] ) we obtain for the stellar velocity dispersion the equation [ integral ] _ sv_r^2 & & r^-2_an_r^s^2(2-_an)_s ( s ) + & & \{_r_b^s^2 d}ds . the `` effective dark matter '' mass profile can be related through the projection along the line of sight to two observable quantities , the projected stellar density @xmath164 , and to the stellar velocity dispersion @xmath165 , respectively , according to the relation @xcite @xmath166 given a projected stellar density model @xmath164 , one recovers the three - dimensional stellar density from @xcite @xmath167 therefore , once the stellar density profile @xmath164 , the stellar velocity dispersion @xmath168 , and the quantities @xmath124 , @xmath169 , @xmath83 and @xmath113 , determining the geometry of the space - time outside the baryonic matter distribution , are known , with the use of the integral equation eq . ( [ integral ] ) one can constrain the explicit functional form of the scalar field @xmath170 , the two free parameters of the model , @xmath112 and @xmath171 , as well as the equivalent mass and density profiles induced by the presence of the scalar field . the simplest analytic projected density profile is the plummer profile @xcite , given by @xmath172 , where @xmath173 is the total luminosity , and @xmath174 is the projected half - light radius ( the radius of the cylinder that encloses half of the total luminosity ) . the rotation curves of spiral galaxies are inferred from the astrophysical observations of the red and blue shifts of the radiation emitted by gas clouds moving in circular orbits on both sides of the central region in the galactic plane . the light signal travels on null geodesics in the galactic geometry with tangent @xmath175 . we may , without a significant loss of generality , restrict @xmath175 to lie in the equatorial plane @xmath45 , and evaluate the frequency shift for a light signal emitted from the observer @xmath176 in circular orbit in the galactic halo , and detected by the observer @xmath177 situated at infinity . the frequency shift associated to the emission and detection of the light signal from the gas cloud is defined as @xmath178 where @xmath179 , and the index @xmath180 refers to emission ( @xmath181 ) or detection ( @xmath182 ) at the corresponding space - time point @xcite . we can associate with light propagation two frequency shifts , corresponding to maximum and minimum values , in the same and opposite direction of motion of the emitter , respectively . from an astrophysical point of view such shifts are frequency shifts of a receding or approaching gas cloud , respectively . in terms of the tetrads @xmath183 , @xmath184 , @xmath185 , @xmath186 , the frequency shifts can be represented as @xcite @xmath187 /2}\left ( 1\mp v\right ) \gamma , \ ] ] where @xmath188^{1/2}$ ] , with @xmath189 the components of the particle s four velocity along the tetrad ( i.e. , the velocity measured by an eulerian observer whose world line is tangent to the static killing field ) . in eq . ( [ 60 ] ) , @xmath190 is the usual lorentz factor , and @xmath191 represents the value of @xmath192 $ ] for @xmath193 . in the case of circular orbits in the @xmath45 plane , we obtain @xmath194 /2}\frac{% 1\mp \sqrt{r \nu ^{\prime } /2 } } { \sqrt{1-r \nu ^{\prime } /2 } } .\ ] ] it is convenient to define two other quantities , @xmath195 , giving the differences in the doppler shifts for the receding and approaching gas clouds , and @xmath196 , representing the mean value of the doppler shifts , respectively @xcite . these redshift factors are given by @xmath197 /2}\frac{\sqrt{r\nu ^{\prime } /2 } } { \sqrt{1-r \nu ^{\prime } /2 } } , \ ] ] and @xmath198 /2}}{\sqrt{1-r \nu ^{\prime } /2 } } , \ ] ] respectively , and they can be easily connected to the astrophysical observations @xcite . @xmath199 and @xmath200 satisfy the relation @xmath201 $ ] , and thus in principle , the metric tensor component @xmath202 $ ] can be directly determined from observations . from eq . ( [ f2 ] ) we obtain r=-(++)r^2 , and ( r)=_r_b^rdr , respectively . by substituting these expressions of the metric tensor and of its derivative in eqs . ( [ zd ] ) and ( [ za ] ) , in principle we obtain a direct observational test of the galactic geometry , of the functional form of the scalar field , and , implicitly , of the hybrid metric - palatini gravitational model . the behavior of the galactic rotation curves , especially their constancy , and the mass deficit in clusters of galaxies , continues to pose a major challenge to present day physics . it is essential to have a better understanding of some of the intriguing phenomena associated with them , such as their universality , the very good correlation between the amount of dark matter and the luminous matter in the galaxy , as well as the nature of the dark matter particle , if it really does exist . to explain these intriguing observations , the commonly adopted models are based on exotic , beyond the standard model , particle physics in the framework of newtonian gravity , or of some extensions of general relativity . in the present paper , we have considered the observational implications of the model proposed in @xcite , and proposed an alternative view to the dark matter problem , namely , the possibility that the galactic rotation curves and the mass discrepancy in galaxies can naturally be explained in gravitational models in which an @xmath11 term , constructed within the framework of the palatini formalism , is added to the metric einstein - hilbert lagrangian . the extra - terms in the gravitational field equations , which can be described as a function of an equivalent scalar field , modify , through the metric tensor components , the equations of motion of test particles , and induce a supplementary gravitational interaction , which can account for the observed behavior of the galactic rotation curves . due to the presence of the scalar field , the rotation curves show a constant velocity region , which decay to zero at large distances from the galactic center , a behavior which is perfectly consistent with the observational data @xcite , and is usually attributed to the existence of dark matter . by using the weak field limit of the gravitational field equations , the rotation curves can be completely reconstructed as functions of the scalar field , without any supplementary assumption . if the galactic rotation velocity profiles are known from observations , the galactic metric can be derived theoretically , and the scalar field function can be reconstructed exactly over the entire mass distribution of the galactic halo . the formalism developed in the present paper could also be extended to the case of the galaxy clusters . the latter are cosmological structures consisting of hundreds or thousands of galaxies . we emphasize that the analysis of the geometric properties of the galaxy clusters can also be done in weak field approximation considered in the present paper . the comparison of the observed velocity dispersion profiles of the galaxy clusters and the velocity dispersion profiles predicted by the hybrid metric - palatini gravity model can provide a powerful method for the observational test of the theory , and for observationally discriminating between the different modified gravity theoretical models . the nature and dynamics of the cosmological evolution can be investigated by using a variety of cosmological observations . one of the important methods for the study of the cosmic history relies on extracting the baryon acoustic oscillations ( bao ) in the high-@xmath44 galaxy power spectrum . the baryon acoustic oscillations imprint a characteristic scale on the galaxy distribution that acts as a standard ruler . the origin of the bao in the matter power spectrum can be understood as the velocity fluctuation of the baryonic fluid at the decoupling time . the characteristic scale of the baryon oscillation is determined by the sound speed and horizon at decoupling , which is a function of the total matter and baryon densities @xcite . since this scale can be measured in both the transverse and radial directions , the bao yields both the angular diameter distance , and the hubble parameter at that redshift . therefore , the precise measurement of the bao scale from the galaxy power spectrum can impose important constraints on the cosmic expansion history . different expansion histories in modified gravity models shifts the peak positions of oscillations relative to the @xmath203cdm model @xcite . therefore the predicted shifts in the bao can potentially be used to distinguish between the @xmath204cdm models and modified gravity models . thus , by using the bao analysis it can be shown that the original dvali - gabadadze - porrati model @xcite is disfavored by observations , unless the matter density parameter exceeds 0.3 @xcite . the recently released planck satellite data @xcite , as combined with the bao measurements @xcite provide strong constraints on the modified @xmath0 gravity model @xcite . in the @xmath0 modified gravity models the lensing amplitude return to be compatible with @xmath205 at 68% confidence limit ( c.l . ) if one consider the planck or planck combined with the hubble space telescope measurement data , and even at 95% c.l . if we consider planck data combined with bao data moreover , in the framework of the considered @xmath0 models the standard value of the lensing amplitude @xmath205 is in agreement with the planck measurements , oppositely to what happens in the @xmath203cdm scenario . the study of the bao and of the cmb data can provide very powerful and high precision constraints in discriminating between the hybrid metric - palatini gravity model , and alternative gravity models , as well as the standard dark matter model . weak gravitational lensing , whereby galaxy images are altered due to the gravitational field influence of the mass along the line of sight , is a powerful probe of the dark matter in cosmology , with promising results obtained in recent years @xcite . in standard general relativity , the weak lensing distortion field provides a direct tracer of the underlying matter distribution . however , modifications to gravity theory can conceivably alter the way that mass curves spacetime , and thus the way that null geodesics behave in a given matter distribution . the possibilities of constraining modified gravity theories with weak lensing were considered recently in @xcite . due to its sensitivity to the growth rate of the structure , weak lensing can be very useful to constrain modified gravity theories , and to distinguish between various modified gravity and standard dark matter models , when combined with cmb observations . future weak lensing surveys as euclid can constrain modified gravity models , as those predicted by scalar - tensor and @xmath0 theories . since the hybrid metric - palatini gravity model can be formulated in terms of an equivalent scalar - tensor theory , the analysis of the future weak lensing observational data , such as those provided by the euclid mission , may provide a powerful method to observationally constrain the free parameters of this theoretical model . even that the effective scalar field of the hybrid metric - palatini gravity model provides a gravitational `` mass '' equivalent to the dark matter , due to the specific functional form and numerical values of the model parameters , its imprint on the weak lensing properties on a cosmological scale is different from that of the standard dark matter . in the present model all the relevant physical quantities , including the `` dark mass '' associated to the equivalent scalar - tensor description , and which plays the role of dark matter , its corresponding density profile , as well as the scalar field and its potential , are expressed in terms of observable parameters the tangential velocity , the baryonic ( luminous ) mass , the doppler frequency shifts of test particles moving around the galaxy , and the velocity dispersions of the stars . therefore , this opens the possibility of directly testing the modified gravitational models with palatini type @xmath11 terms added to the gravitational action by using direct astronomical and astrophysical observations at the galactic or extra - galactic scale . in this paper we have provided some basic theoretical ideas , which , together with the virial theorem considered in @xcite , are the necessary tools for the in depth comparison of the predictions of the hybrid metric - palatini gravity model with the observational results . we would like to thank to the anonymous referee for comments and suggestions that helped us to significantly improve our manuscript . sc is supported by infn ( iniziative specifiche qgsky and teongrav ) . tsk is supported by the research council of norway . fsnl acknowledges financial support of the fundao para a cincia e tecnologia through the grants cern / fp/123615/2011 and cern / fp/123618/2011 . gjo is supported by the spanish grant fis2011 - 29813-c02 - 02 , the consolider programme cpan ( csd2007 - 00042 ) , and the jae - doc program of the spanish research council ( csic ) . by convention , the weak - field limit of theories of gravity in the solar system is discussed using isotropic coordinates . however , exact solutions and galactic dynamics are usually considered in terms of the schwarzschild coordinates . since we have already obtained the form of the weak - field limit in @xcite , in order to translate those results to the current problem we just need to transform our metric from isotropic to schwarzschild coordinates . this will give coherence to these series of papers . what we need to do is just to compare the schwarzschild - like line element used in the study of the galactic dynamics with the isotropic results , and find the change of coordinates . + in schwarzschild coordinates the linearized line element is ds^2&=&e^dt^2-e^dr^2-r^2d^2 + & & ( 1+)dt^2-(1+)dr^2-r^2d^2 . the comparison of the coordinates gives the relations @xmath209 the second of these equations allows to express @xmath210 and @xmath211 in terms of the schwarzschild coordinate @xmath2 . it also allows us to find an expression for @xmath212 as follows , @xmath213 ^ 2.\ ] ] inserting this result in eq . ( [ 81 ] ) , and expanding to leading order , we find @xmath214.\ ] ] therefore , @xmath215 and @xmath216 . taking into account the explicit form of @xmath217 and @xmath204 @xcite , @xmath218 with @xmath219 we find that @xmath220.\end{aligned}\ ] ] here we can consider some simplifications . as a first approximation we can assume @xmath221 to be very small , which is equivalent to consider scales much smaller than the cosmological horizon . we can also assume @xmath106 small , but we keep @xmath222 non - negligible . in this case , we find the same relation between coordinates as in gr , namely , r^2 ^ 2(1 + ) , which leads to r(1+)=gm+ . with this result , we find @xmath223\right|_{\rho = r - gm}.\nonumber\\\end{aligned}\ ] ] note that even though at galactic scales we have @xmath224 and @xmath225 , the exponential corrections @xmath226 experience a magnification of the _ apparent _ value of @xmath106 by an exponential factor : @xmath227 , i.e. , @xmath228 . v. c. rubin , w. k. ford , and n. thonnard , astrophys . j. * 238 * , 471 ( 1980 ) ; m. persic , p. salucci , and f. stel , mon . not . r. astron . soc . * 281 * , 27 ( 1996 ) ; a. boriello and p. salucci , mon . not . * 323 * , 285 ( 2001 ) . a. g. riess et al . j. * 116 * , 1009 ( 1998 ) ; s. perlmutter et al . , astrophys . j. * 517 * , 565 ( 1999 ) ; p. de bernardis et al . , nature * 404 * , 955 ( 2000 ) ; s. hanany et al . , astrophys . j. * 545 * , l5 ( 2000 ) . e. w. mielke and f. e. schunck , , 023503 ( 2002 ) ; j. p. mbelek , astron . astrophys . * 424 * , 761 ( 2004 ) ; x. hernndez , t. matos , r. a. sussman , and y. verbin , , 043537 ( 2004 ) ; b. fuchs and e. w. mielke , mon . not . 350 * , 707 ( 2004 ) ; d. bertacca , n. bartolo and s. matarrese , jcap * 0805 * , 005 ( 2008 ) ; j. l. cervantes - cota , m. a. rodriguez - meza , and d. nunez , , 064011 ( 2009 ) . m. k. mak and t. harko , , 024010 ( 2004 ) ; t. harko and k. s. cheng , , 044013 ( 2007 ) ; t. harko and k. s. cheng , astrophys . j. * 636 * , 8 ( 2005 ) ; l. @xmath229 . gergely , t. harko , m. dwornik , g. kupi , and z. keresztes , mon . not . . soc . * 415 * , 3275b ( 2011 ) ; k. c. wong , t. harko , k. s. cheng , and l. @xmath229 . gergely , phys . d * 86 * , 044038 ( 2012 ) . c. g. boehmer and t. harko , jcap * 0706 * , 025 ( 2007 ) ; j .- w . lee , phys . b * 681 * , 118 ( 2009 ) ; j .- w . lee and s. lim , jcap * 1001 * , 007 ( 2010 ) ; t. harko , jcap * 1105 * , 022 ( 2011 ) ; t. harko , mon . not . royal astron.soc . * 413 * , 3095 ( 2011 ) ; t. harko and e. j. m. madarassy , jcap * 01 * , 020 ( 2012 ) . o. bertolami , c. g. boehmer , t. harko and f. s. n. lobo , * 75 * , 104016 ( 2007 ) ; t. harko , phys . b * 669 * , 376 ( 2008 ) ; t. harko and f. s. n. lobo , eur . j. c * 70 * , 373 ( 2010 ) ; t. harko , f. s. n. lobo , and o. minazzoli , phys . d * 87 * , 047501 ( 2013 ) . h. a. buchdahl , mon . not . soc . * 150 * , 1 ( 1970 ) . a. a. starobinsky , phys . b * 91 * , 99 ( 1980 ) . a. de felice and s. tsujikawa , living rev * 13 * , 3 ( 2010 ) ; t. p. sotiriou and v. faraoni , rev . phys . * 82 * , 451 ( 2010 ) ; s. nojiri and s. d. odintsov , phys . rept . * 505 * , 59 ( 2011 ) ; f. s. n. lobo , arxiv:0807.1640 [ gr - qc ] ; t. clifton , p. g. ferreira , a. padilla and c. skordis , phys . rep . * 513 * , 1 ( 2012 ) . t. chiba , phys * b 575 * , 1 ( 2003 ) ; g. j. olmo , phys . lett . * 95 * , 261102 ( 2005 ) ; g. j. olmo , phys . rev . * d 75 * , 023511 ( 2007 ) . l. amendola , r. gannouji , d. polarski and s. tsujikawa , phys . d 75 * , 083504 ( 2007 ) ; n. goheer , j. larena and p. dunsby , phys . d * 80 * , 061301 ( 2009 ) ; ; a. abebe , m. abdelwahab , a. cruz - dombriz and p. dunsby , class . * 29 * , 135011 ( 2012 ) ; a. abebe , a. cruz - dombriz and p. dunsby , arxiv:1304.3462 ( 2013 ) . x. meng and p. wang , gen . * 36 * , 1947 ( 2004 ) ; a.e . dominguez d.e . barraco , phys . d * 70 * , 043505 ( 2004 ) ; v. faraoni , phys . rev . d * 74 * , 023529 ( 2006 ) ; t. koivisto , h. kurki - suonio , class . quant . grav . * 23 * , 2355 ( 2006 ) ; t. koivisto , phys . d * 73 * , 083517 ( 2006 ) ; t. p. sotiriou , gen . . gravit . * 38 * , 1407 ( 2006 ) ; g. j. olmo , phys . 98 * , 061101 ( 2007 ) ; g. j. olmo , phys . d * 77 * , 084021 ( 2008 ) ; g. j. olmo , int . j. mod d * 20 * , 413 ( 2011 ) . t. harko , t. s. koivisto , f. s. n. lobo and g. j. olmo , phys . d * 85 * , 084016 ( 2012 ) . s. capozziello , t. harko , t. s. koivisto , f. s. n. lobo and g. j. olmo , jcap * 04 * , 011 ( 2013 ) . s. capozziello , t. harko , t. s. koivisto , f. s. n. lobo and g. j. olmo , phys . d * 86 * , 127504 ( 2012 ) . n. tamanini and c. g. boehmer , phys . d * 87 * 084031 ( 2013 ) . j. w. moffat and i. y. sokolov , phys . b * 378 * , 59 ( 1996 ) ; j. r. brownstein and j. w. moffat , astrophys . j. * 636 * , 721 ( 2006 ) ; j. r. brownstein and j. w. moffat , mon . not . * 367 * , 527 ( 2006 ) . f. beutler , c. blake , m. colless , d. h. jones , l. staveleysmith , l. campbell , q. parker , w. saunders et al . , mon . not . soc . * 416 * , 3017 ( 2011 ) ; w. j. percival et al . [ sdss collaboration ] , mon . not . . soc . * 401 * , 2148 ( 2010 ) ; n. padmanabhan , x. xu , d. j. eisenstein , r. scalzo , a. j. cuesta , k. t. mehta and e. kazin , arxiv:1202.0090 ( 2012 ) ; c. blake , e. kazin , f. beutler , t. davis , d. parkinson , s. brough , m. colless and c. contreras et al . , mon . not . astron . soc . 418 , 1707 ( 2011 ) ; l. anderson , e. aubourg , s. bailey , d. bizyaev , m. blanton , a. s. bolton , j. brinkmann , j. r. brownstein et al . , mon . not . astron . soc . * 427 * 3435 ( 2013 ) . m. martinelli , e. calabrese , f. de bernardis , a. melchiorri , l. pagano , and r. scaramella , phys . d * 83 * , 023012 ( 2011 ) ; s. a. thomas , s. a. appleby , and j. weller , jcap * 03 * , 036 ( 2011 ) ; i. tereno , e. semboloni , and t. schrabback , astron . astrophys . * 530 * , a68 ( 2011 ) ; r. ali vanderveld , r. r. caldwell , and j. rhodes , phys d * 84 * , 123510 ( 2011 ) ; s. asaba , c. hikage , k. koyama , g .- b . zhao , a. hojjati , and l. pogosian , jcap * 08 * , 029 ( 2013 ) .

generally , the dynamics of test particles around galaxies , as well as the corresponding mass deficit , is explained by postulating the existence of a hypothetical dark matter . in fact , the behavior of the rotation curves shows the existence of a constant velocity region , near the baryonic matter distribution , followed by a quick decay at large distances . in this work , we consider the possibility that the behavior of the rotational velocities of test particles gravitating around galaxies can be explained within the framework of the recently proposed hybrid metric - palatini gravitational theory . the latter is constructed by modifying the metric einstein - hilbert action with an @xmath0 term in the palatini formalism . it was shown that the theory unifies local constraints and the late - time cosmic acceleration , even if the scalar field is very light . in the intermediate galactic scale , we show explicitly that in the hybrid metric - palatini model the tangential velocity can be explicitly obtained as a function of the scalar field of the equivalent scalar - tensor description . the model predictions are compared model with a small sample of rotation curves of low surface brightness galaxies , respectively , and a good agreement between the theoretical rotation curves and the observational data is found . the possibility of constraining the form of the scalar field and the parameters of the model by using the stellar velocity dispersions is also analyzed . furthermore , the doppler velocity shifts are also obtained in terms of the scalar field . all the physical and geometrical quantities and the numerical parameters in the hybrid metric - palatini model can be expressed in terms of observable / measurable parameters , such as the tangential velocity , the baryonic mass of the galaxy , the doppler frequency shifts , and the stellar dispersion velocity , respectively . therefore , the obtained results open the possibility of testing the hybrid metric - palatini gravitational models at the galactic or extra - galactic scale by using direct astronomical and astrophysical observations . + + * keywords * : modified gravity : galactic rotation curves : dark matter :

it is generally accepted that supernova remnants ( snrs ) are the dominant source of galactic cosmic rays , at least for energies up to 3@xmath310@xmath12 ev . these objects provide a significant fraction of the mechanical energy that heats , compresses and chemically enriches the interstellar medium ( ism ) . therefore , snrs can be used to investigate global properties of the galaxy as well as the local environment where they evolve . thanks to significant advances in the angular resolution capabilities of modern x - ray observatories such as xmm-_newton _ and _ chandra _ , important progress has been made concerning the detection of new and well - known snrs ( sasaki et al . 2004 ; bocchino et al . 2005 ; combi et al . 2010a ; combi et al . at present , 30% of the radio snrs display x - ray emission and more than a dozen of snrs were originally discovered through their x - ray emission ( e.g. , schwentker 1994 ; aschenbach 1998 ; bamba et al . 2003 ; yamaguchi et al . 2004 ) . studying the connexion between radio , infrared and x - ray emission of snrs enables us to explore how stars end their lives and better understand the evolutionary process in this kind of fascinating sources . the southern galactic snr g296.8 - 0.3 ( 1156 - 62 ) lies in the direction to the scutum - crux arm of our galaxy . it was first detected at radio frequencies by large & vaughan ( 1972 ) . initially , it was considered as a shell - type remnant despite its unusual and ill - defined shape ( shaver & goss 1970 ; goss & shaver 1970 ) . subsequent higher - resolution radio observations at 843 mhz ( whiteoak & green 1996 ) showed a complicated multi - ringed structure , with its diffuse interior emission being brightest in its northwest side . hi observations carried out by gaensler et al . ( 1997 ) allowed to obtain lower and upper limits on its systemic velocity in the range + 15 to + 30 km s@xmath13 . this corresponds to a distance of 9.6@xmath140.6 kpc . a flux density of 7.0@xmath14 0.3 jy was measured at 1.3 ghz for the total radio structure . throughout this paper , a mean distance of 9 kpc is assumed . it corresponds to an angular size of @xmath2 31 pc . at x - ray energies , an exploratory study was carried out with the rosat satellite by hwang & markert ( 1994 ) . these authors noted marginal x - ray emission ( 4 @xmath15 detection ) near the peak of radio emission to the northwest part of the remnant . however , the poor statistics and limited x - ray energy range of the rosat telescope did not allow to observe a clear well - defined morphology of the x - ray emission . as part of a program addressed to study the x - ray emission of supernova remnants , in this paper we report xmm@xmath0_newton _ observations of g296.8 - 0.3 , and infrared data obtained with the spitzer space telescope in order to study the physical characteristics of the object and the surrounding ism where it evolves . the structure of the paper is as follows : in sect . 2 , we describe the xmm-_newton _ observations , data reduction and present our x - ray analysis . ir results are reported in sect . 3 . in sect . 4 , we discuss the implications of our results . finally in sect . 5 , we summarize our main conclusions . g296.8 - 0.3 was observed by the xmm - newton x - ray satellite in two separate pointings . these were performed on 2008 february 16 ( obsid 0503780301 ) and 2008 august 16 ( obsid 0550170101 ) , with the epic mos ( turner et al . 2001 ) and epic pn ( strder et al . 2001 ) cameras . both observations have similar pointing coordinates ( @xmath17=11@xmath1858@xmath1930.0 , @xmath20=@xmath062@xmath2135@xmath2200.0 ; j2000 ) , and were placed at the ccd center . the xmm-_newton _ data were calibrated and analyzed with the xmm science analysis system ( sas ) version 10.0.0 . to exclude high background activity , which can affect the observations , we extracted light curves of photons above 10 kev from the entire field - of - view of the cameras , and excluded time intervals with count rate higher than 3 @xmath15 above average to produce a gti file . unfortunately , the observation obsid 0503780301 was affected by a high and variable soft proton background level ( lumb et al . 2002 ) , whereas the other one obsid 0550170101 is unaffected by background fluctuations . in order to avoid contamination for high background patterns , hereafter our analysis concerns only the second observation . after the time filtering , 45.2 ks of useful data for mos1 , 45.1 ks for mos2 , and 46.5 ks for the pn cameras are available for further data analysis , which is @xmath280@xmath23 of the total exposure . to create images , spectra , and light curves , we selected events with flag @xmath24 0 and pattern @xmath25 12 for mos1/2 and pattern @xmath25 4 for pn . figure 1 ( left panel ) shows a color composite xmm-@xmath16 image g296.8 - 0.3 in three energy bands : 0.5 - 1.0 ( red ) , 1.0 - 2.0 ( green ) , and 2.0 - 5.0 kev ( blue ) , for the combined mos1/2 cameras . images were corrected for the spatial dependent exposure , and the instrumental background was also substracted . at x - ray energies @xmath6 2.6 kev the snr is not detected . the contour map overplotted on figure 1 is the most 843 mhz radio continuum map ( whiteoak & green 1996 ) , which can help us to identify the origin of the x - rays . the x - ray structure of the snr is complex and covers @xmath2 30% of the total radio extent . it shows three different components : interior diffuse emission coincident with the unusual rectangular strip ( gaensler et al . 1998 ) running through its center seen at radio frequencies ( indicated with the dashed line in fig.1 , left panel ) , a bright soft shell - like feature with an angular size of @xmath2 8 , coincident with the northwest radio shell , and at least 9 point - like sources . all of them are numbered in fig.1 , right panel . these point sources were detected by using the source detection meta - task ` edetect - chain ' . sources 1,2,3,4,5,6,7 and 9 display medium and hard x - ray emission and could be background agns . the remaining one , source 8 , displays soft and medium x - ray emission . as can be seen , source 8 is located close to the geometrical center of the radio structure ( @xmath17=11@xmath1858@xmath1936.2 , @xmath20=@xmath062@xmath2135@xmath2220.0 ; j2000 ) . the source is catalogued as 2xmmi j15836.1 - 623516 in the xmm-_newton _ serendipitous source catalogue 2xmmi - dr3 ( watson et al . 2009 ) . the interior of the radio remnant is filled with x - ray emitting material , and the spatial coincidence of x - ray and radio emission suggests that the physical conditions of the terminal shock region are very similar to those found at the outer shocks of ordinary snrs . from the x - ray image we see enhanced x - ray emission on the northwest part of the snr , which is probably caused by density enhancements in the medium in which the shock propagates and forms a more or less continuous structure . we notice that the x - ray emission is entirely contained within the boundaries of the radio shell . this bright filamentary x - ray structure is centered at ( @xmath17=11@xmath1857@xmath1944.5 , @xmath20=@xmath062@xmath2133@xmath2200.0 ; j2000 ) and has an angular size of 8 arcmin on the plane of the sky . the x - ray emission of the snr fades out to the southeast , lacking a clearly defined edge . in order to study the physical properties of the plasma in the remnant , a x - ray spectrum of g298.6 - 0.3 was extracted from the epic cameras using an elliptical region ( shown in fig . 1 , right panel ) that comprises @xmath2 50@xmath23 of the central ccd of the mos cameras and spread part of 4 ccds of the pn camera . for such purpose we use the sas task ` evselect ' with suitable parameters for the mos 1/2 and pn cameras . the background spectrum of the snr was also taken from an elliptical region located within the central ccd of the mos cameras . these regions were defined exclusively on the central chip to avoid having to account for chip - to - chip variations . the spectral analysis was performed with the xspec package ( arnaud , 1996 ) . the x - ray spectrum of the snr is shown in fig . 2a ( left panel ) . the global spectrum has two components . the diffuse x - ray emission of the snr ( dominant in the energy range of 0.5@xmath03.0 kev ) , plus the contribution of point - like sources ( mainly contributing in energy range of 2.5@xmath05.0 kev ) . in order to obtain the x - ray spectrum of the snr , we have excluded all the point - like sources taking circular regions with a radius of 15@xmath26 . the extracted epic mos1/2 and pn spectra were grouped with a minimum of 30 counts per spectral bin , and the @xmath27 statistics was used . ancillary response files ( arfs ) and redistribution matrix files ( rmfs ) were calculated . the x - ray emission of the snr peaks in the 0.53.0 kev energy range and it is clearly dominated by thermal emission . thus , we used a pshock model affected by an absorption interstellar model ( phabs ; balucinska - church and mccammon 1992 ) to fit it . the x - ray parameters for the best - fit of the diffuse emission are given in table 1 . [ cols="<,<,^ " , ] normalization is defined as 10@xmath28/4@xmath29d@xmath30@xmath31 , where @xmath32 is distance in [ cm ] , n@xmath33 is the hydrogen density [ cm @xmath34 , @xmath35 is the electron density [ @xmath9 ] , and @xmath37 is the volume [ @xmath38 . the flux in the three energy ranges 0.5 - 1.0 , 1.0 - 2.0 and 2.0 - 3.0 kev , is absorption - corrected . values in parentheses are the single parameter 90% confidence interval . the abundance parameter is given relative to the solar values of anders & grevesse ( 1989 ) . since the point - like source is surrounded by diffuse x - ray emission of the snr , we extracted its spectrum from a circular region with a radius of 6 arcsec ( ee@xmath2 50% ) and grouped it with a minimun of 18 counts per spectral bin . the background spectrum was estimated from an annular region with radii of 8 and 25 arcsec . the spectrum of the source is shown in figure 2 ( right panel ) . in order to study the x - ray properties of this compact source we fitted its spectrum with several spectral models . however , due to the low - photons statistic , the most representative one is a simple power - law ( pl ) model that yields a neutral hydrogen absorption column @xmath1=0.55 @xmath140.1 @xmath5 , an index @xmath39=4.3 @xmath140.7 and a normalization of 6.1(@xmath142.5)@xmath310@xmath40 @xmath5 . the absorption corrected x - ray flux is f@xmath41=9.9 ( @xmath140.1 ) @xmath310@xmath28 erg s@xmath13 @xmath5 in the 0.3@xmath03.0 kev band . the fit is acceptable in terms of the minimum @xmath27 ( @xmath42= 1.1 for 24 d.o.f ) . as can be seen , this model provides a good value of @xmath27 and the value of @xmath1 is similar to the value obtained for the snr . this fact , supports the possibility that the point source detected at the geometrical centre of the snr has a real physical connexion with g296.8 - 0.3 . in addition , we found no significant pulsed signal with a period greater than twice the read - out time of the epic - pn camera in the ff mode ( 73.3 ms ) , which corresponds to a nyquist limit of 0.146 s. using radio observations performed with the most radiotelescope at 0.843 ghz , and infrared spitzer - mips ( rieke et al . 2004 ) observations of the snr , we have investigated the positional correlation between all the detected emissions . the mips basic calibrated data ( bcd ) were downloaded from the spitzer archive . these images were re - processed with the regular mips pipeline ( version s18.7.0 ) , and then mosaicked using mopex ( version 18.3.1 ) and the standard mips 24@xmath11 m mosaic pipeline . in fig . 3 , we show the mips image at 24 @xmath11 m with the radio contours superimposed . the 24 @xmath11 m emission is strongly correlated with the radio shells , with several faint filaments coincident on the northwest and southeast parts of the snr . this emission is generally interpreted as thermal emission from dust grains that have been swept up and shock - heated by the supernova blast wave ( tappe et al . the extraction regions used for computing the infrared fluxes on the shell - like boundaries are indicated in green . the mid - infrared fluxes at 24 @xmath11 m of the northwest and southeast regions are 30.2@xmath140.2 jy and 14.3@xmath140.1 jy , respectively . the x - ray emission detected on the northwest radio shell follows the infrared emission very well , which demonstrates the connection between the x - ray emitting plasma and the heated dust grains . this result seems to indicate that the enhanced x - ray emission is caused by the expansion through a dense ism with a density gradient toward the northwest side of the remnant . under these conditions , it is possible to roughly make quantitative estimates of the swept - up ism dust mass in the northwest and southeast rims using the formulae introduced by whittet ( 2003 ) , @xmath43,\ ] ] in this equation we have adopted a density @xmath44 for silicate / graphite grains , a distance @xmath45 for the snr , a dust temperature @xmath46 @xmath2 100 k at 24@xmath11 m , and an average ratio of grain radius @xmath47 over emissivity @xmath48 of @xmath49 ( see , tappe et al . 2006 ) . using the infrared flux densities @xmath50 computed at 24@xmath11 m and the planck function @xmath51(@xmath52 ) at 24 @xmath11 m , we derived a dust mass of @xmath2 0.010 @xmath53 and @xmath2 0.006 @xmath53 for the northwest and southeast rims , respectively . 1 is an approximation assuming spherical dust grains of uniform size , composition , and in thermal equilibrium . in order to inspect if there exists some infrared counterpart to 2xmmi j115836.1 - 623516 , we show in fig . 4 a _ spitzer_/irac deep image of the central region of g296.9 - 0.3 in the 3.6 @xmath11 m band . the extraction circle with a radius of 6 arcsec , used in the x - ray spectral analysis , is overplotted . as can be seen only one infrared source lies near the edge of the encircled region . the position of the x - ray source is given in the 2xmm catalogue ( watson et al . , 2009 ) with a relative precision of one arcsec . however , it is well - known from cross - correlations with other astrometric catalogues that the absolute accuracy in the position of _ xmm - newton _ sources ranges between 5 and 10 arcsec ( della ceca et al . 2004 , lpez - santiago et al . 2007 , combi et al . 2011 ) . therefore , we can not assure if the infrared source observed near the edge of the encircle region is physically associated or not with the x - ray object . m image of g296.8 - 0.3 with the radio contours at 843 mhz ( in white ) overlaid . the extraction regions used for computing the infrared fluxes are indicated in green color.,width=317 ] m band . we have overplotted the 6 arcsec extraction region used for the spectral x - ray analysis.,width=302 ] two possible evolutionary scenarios to explain the unusual morphology observed at radio frequencies in snr g296.8@xmath00.3 , were initially studied by gaensler et al . these authors suggested that the biannular appearance is induced by axial symmetry in the progenitor wind , or the snr morphology resulted from the inhomogeneous ism into which it is expanding . they conclude that the second possibility is more probable and that g296.8@xmath00.3 seems consistent with being the remnant of a single explosion , where the unusual linear feature running north - south through the remnant may represent a low density tunnel which has been re - energized by an encounter with the sn shock . it is clear that g296.8 - 0.3 is found in a complex area in the scutum - crux arm of our galaxy , where the ism is particularly inhomogeneous and density variations in the pre - shock medium are present . the radio , infrared and x - ray observations of g296.8 - 0.3 here analyzed , can provide crucial information about the origin and evolution of the snr , as well as regarding the age , energetics , ambient conditions and the presence of heated dust . with all this information in mind , we could outline a possible framework that allows to explain the characteristics of the emissions observed from the snr . the xmm observations reveal that there is diffuse x - ray emission in the interior of the snr well - correlated with the unusual rectangular strip running through its center seen at radio frequencies , a bright soft shell - like feature coincident with the internal northwest radio shell , and several hard point - like sources ( possibly background agns ) . moreover , the 24 @xmath11 m observations show two limb - brightened shell - like structures on the northwest and southeast parts of the snr , and faint filaments strongly correlated with the radio shells . the emission measure ( em ) computed for the global region of the snr can allow us to estimate the corresponding density of the x - ray emitting gas . from the x - ray image , we can roughly assume that the x - ray emission fills an ellipsoid with radii of @xmath2 3 @xmath3 4 arcmin and estimate the volume @xmath37 of the x - ray emitting plasma . at a distance of 9 kpc , the snr defines an x - ray emitting volume @xmath54= 6.1@xmath310@xmath55 @xmath56 . based on the em determined by the spectral fitting ( see table 1 ) , we can estimate the electron density of the plasma , @xmath57 , by @xmath57=@xmath58 , which results in @xmath57@xmath2 0.18 @xmath9 . in this case , the number density of the nucleons was simply assumed to be the same as that of electrons . the age @xmath59 is determined from the ionization timescale , @xmath60 , by @xmath59=@xmath60/@xmath57 . therefore , the elapsed time after the plasma was heated is @xmath59 @xmath2 1.0@xmath310@xmath61 yr . this result shows that g296.8@xmath00.3 is a middle - aged snr . assuming that the snr is in the adiabatic ( sedov - taylor ) phase , gaensler et al . ( 1998 ) obtained several physical parameters for the object . taking into account a kinetic energy of the initial explosion of @xmath62= 10@xmath63 erg , and that the snr expands in an ism with a density of 0.2 @xmath9 , these authors computed an age of @xmath59=(10@xmath142)@xmath310@xmath64 yr for g296.8 - 0.3 . this value agrees very well with the age obtained by us above using the x - ray information . concerning the point - like x - ray source located close to the geometrical center of the radio structure , we see no significant variability that disfavors an accreting binary origin , a soft thermal spectrum that eliminates a background active nucleus , lack of radio counterpart , and absence of a surrounding pulsar wind nebula . at first sight , we can see that the source displays some characteristics of the so - called cco ( see pavlov et al . 2004 , for a review ) , a new population of isolated neutron stars ( nss ) with clear differences from isolated rotation - powered pulsars and accretion - powered x - ray pulsars in close binary systems . the nature of these objects is still unclear . it is thought that the x - ray emission from ccos is generally due to the thermal cooling of the ns ( e.g zavlin , trumper & pavlov 1999 ) , with typical temperatures of a few 10@xmath65 k , as inferred from their thermal - like spectra . they have x - ray luminosities ( @xmath66 ) in the range of 10@xmath67 - 10@xmath68 erg s@xmath13 and display x - ray spectra characterized by a blackbody model with temperatures ( @xmath7 ) in the range of 0.2 - 0.5 kev or a power - law model with very steep index @xmath39 ( see pavlov et al . halpern & gotthelf ( 2010 ) have recently suggested that these objects could be weakly magnetized nss ( @xmath69 g ) , i.e. , a kind of `` anti - magnetars '' . in order to check the characteristics of 2xmmi j115836.1 - 623516 , we computed its @xmath66 and spin - down luminosity @xmath70 , to compare with other well - known ccos ( pavlov et al . 2003 ) . adopting a mean distance of 9 kpc and a total unabsorbed x - ray flux of @xmath71=9.9@xmath310@xmath28 ergs @xmath5 s@xmath13 , we obtain an unabsorbed luminosity @xmath66= 1.0@xmath310@xmath67 ergs s@xmath13 . a rough estimate of the spin - down luminosity can be derived using the empirical formula by seward & wang ( 1988 ) , log @xmath66 ( ergs s@xmath13)= 1.39 log @xmath70 - 16.6 , which gives @xmath70= 3.9@xmath310@xmath72 ergs s@xmath13 . @xmath66 lies within the range suggested by pavlov et al . ( 2004 ) and by halpern & gotthelf ( 2010 ) for cco objects . the second quantity , @xmath70 , falls below the empirical threshold for generating bright wind nebulae of @xmath73 @xmath74 4@xmath310@xmath75 ergs s@xmath13 . these results suggest that the system g296.8 - 0.3/2xmmi j115836.1 - 623516 is a thermal snr with , possibly , a nondetected ns . plausible reasons for the nondetection of a ns are the low - photon statistic , a short rotation period or unfavorable geometrical conditions . we have analyzed radio , infrared and x - ray data of the snr g296.8 - 0.3 to investigate the origin of the radiative process involved in the generation of the emission observed . the diffuse x - ray emission is clearly correlated with the unusual rectangular strip running through its center seen at radio frequencies , where we found a region with significant lower density , and a bright x - ray shell - like feature at the northwest part of the snr coincident with the internal boundary of the radio shell . the 24 @xmath11 m observations show two limb - brightened shell - like structures on the northwest and southeast parts of the snr , and faint filaments strongly correlated with the radio shells . the spectral study confirms that the x - ray diffuse emission is thermal and the column density of the snr is high ( @xmath1@xmath20.64@xmath310@xmath4 @xmath5 ) supporting distant location ( @xmath769 kpc ) for the snr . in addition , a compact x - ray source was also detected close to the geometrical center of the snr . the object presents some characteristics of the ccos , and the neutral hydrogen absorption column @xmath1 is consistent with that of the snr . although these results support a physical connexion with the snr , high - resolution x - ray observations carry out with the chandra satellite are necessary to better understand the nature of the x - ray source . the authors acknowledge support by dgi of the spanish ministerio de educacin y ciencia under grants aya2010 - 21782-c03 - 03 , feder funds , plan andaluz de investigacin desarrollo e innovacin ( paidi ) of junta de andaluca as research group fqm-322 and the excellence fund fqm-5418 . j.a.c . and j.f.a.c . are researchers of conicet . j.f.a.c was supported by grant pict 2007 - 02177 ( secyt ) . j.a.c was supported by grant pict 07 - 00848 bid 1728/oc - ar ( anpcyt ) and pip 2010 - 0078 ( conicet ) . j.l.s . acknowledges support by the spanish ministerio de innovacin y tecnologa under grant aya2008 - 06423-c03 - 03 .

we report xmm-_newton _ observations of the galactic supernova remnant g296.8@xmath00.3 , together with complementary radio and infrared data . the spatial and spectral properties of the x - ray emission , detected towards g296.8@xmath00.3 , was investigated in order to explore the possible evolutionary scenarios and the physical connexion with its unusual morphology detected at radio frequencies . g296.8@xmath00.3 displays diffuse x - ray emission correlated with the peculiar radio morphology detected in the interior of the remnant and with the shell - like radio structure observed to the northwest side of the object . the x - ray emission peaks in the soft / medium energy range ( 0.5 - 3.0 kev ) . the x - ray spectral analysis confirms that the column density is high ( @xmath1@xmath20.64@xmath310@xmath4 @xmath5 ) which supports a distant location ( d@xmath69 kpc ) for the snr . its x - ray spectrum can be well represented by a thermal ( pshock ) model , with @xmath7 @xmath2 0.86 kev , an ionization timescale of 6.1@xmath310@xmath8 @xmath9 s , and low abundance ( @xmath2 0.12 @xmath10 ) . the 24 @xmath11 m observations show shell - like emission correlated with part of the northwest and southeast boundaries of the snr . in addition a point - like x - ray source is also detected close to the geometrical center of the radio snr . the object presents some characteristics of the so - called compact central objects ( cco ) . its x - ray spectrum is consistent with those found at other ccos and the value of @xmath1 is consistent with that of g296.8@xmath00.3 , which suggests a physical connexion with the snr .

two of the most intriguing questions in particle physics are the the ewsb mechanism and the origin of fermion masses . although the sm remains a successful theory when compared with all the available data , it lacks predictability in the higgs sector , which determines the masses of gauge bosons , as well as of fermions through _ ad hoc _ yukawa couplings . this suggests the possibility that new physics beyond the sm might be associated with either of these questions . in general , the energy scales and dynamics behind the ewsb sector and the fermion masses may be unrelated . in order to avoid fine - tuning , the scale associated with ewsb can not be much higher than a few tev , whereas the scales where light fermion masses are generated could be much higher . if the mechanism responsible for the breaking of the electroweak symmetry involves some new strong dynamics , deviations from the sm might be observable in low energy signals even at energies much smaller than the scale of new physics @xmath0 . reaching this new frontier by direct observation of new physical states or even of tree - level effects in the couplings of sm particles , may require not only very large energies but also some previous knowledge of what ( and what not ) to expect . thus , low energy measurements might be of paramount importance in planning experiments and search strategies at high energy machines . among these low energy signals are electroweak measurements such as those at lep and the tevatron . on the other hand , processes involving flavor changing neutral currents ( fcnc ) can play a complementary role , since the fact that these processes are largely suppressed or forbidden in the sm may compensate the suppression by factors of @xmath1 ( with @xmath2 the low energy scale , e.g. @xmath3 , @xmath4 , etc . ) . here we address the potential of rare @xmath5 and @xmath6 decays as a complement to other low energy measurements in constraining models where strong dynamics is associated to either the ewsb sector and/or the origin of fermion masses . in the absence of a completely satisfactory theory of dynamical symmetry breaking and fermion masses , it is convenient to carry out a model - independent analysis that makes maximum use of the known properties of the electroweak interactions . this is the case with the ewsb sector , where an effective lagrangian approach allows us to parameterize the effects of the new strong dynamics in very much the same way chiral perturbation theory parameterizes low energy qcd . on the other hand , the effects from fermion mass generation can also be addressed by a general operator analysis . however , in addition , most theories predict the existence of relatively light states ( scalars , pseudo - goldstone bosons , etc . ) which generally couple to mass in one way or another . to exemplify the effects of such states ( which can not be integrated out ) we work with a particular set of models known as topcolor - assisted technicolor ( tatc ) . this provides a current example of how strong dynamics model building deals with the large top - quark mass and illustrates the distinct low energy phenomenology emerging from non - standard ewsb scenarios . in the absence of a light higgs boson the symmetry breaking sector is represented by a non - renormalizable effective lagrangian corresponding to the non - linear realization of the @xmath7 model . the essential feature is the spontaneous breaking of the global symmetry @xmath8 . to leading order the interactions involving the goldstone bosons associated with this mechanism and the gauge fields are described by the effective lagrangian @xcite @xmath9 + \frac{v^2}{4}{{\rm tr}}\left[d_\mu u^\dagger d^\mu u\right ] , \label{lo}\ ] ] where @xmath10 and @xmath11 $ ] are the the @xmath12 and @xmath13 field strengths respectively , the electroweak scale is @xmath14 gev and the goldstone bosons enter through the matrices @xmath15 . the covariant derivative acting on @xmath16 is given by @xmath17 . to this order there are no free parameters once the gauge boson masses are fixed . the dependence on the dynamics underlying the strong symmetry breaking sector appears at next to leading order . to this order , a complete set of operators includes one operator of dimension two and nineteen operators of dimension four @xcite . the effective lagrangian to next to leading order in the basis of ref . @xcite is given by @xmath18 where @xmath19 is a dimension two custodial - symmetry violating term absent in the heavy higgs limit of the sm . if we restrict ourselves to cp invariant structures , there remain fifteen operators of dimension four . the coefficients of some of these operators are constrained by low energy observables . for instance precision electroweak observables constrain the coefficient of @xmath19 , which gives a contribution to the electroweak parameter @xmath20 . the @xmath21 limit requires @xmath22 the combinations @xmath23 and @xmath24 contribute to the electroweak parameters @xmath25 and @xmath26 . for instance , the constraint on @xmath25 translates into @xmath27 in addition , the coefficients @xmath28 , @xmath29 , @xmath30 and @xmath31 modify the triple gauge - boson couplings ( tgc ) and will be probed at lepii and the tevatron at the few percent level @xcite . the remaining operators contribute to oblique corrections only to one loop and , in some cases , only starting at two loops . to the last group belong @xmath32 and @xmath33 given that their contributions to the gauge boson two - point functions only affect the longitudinal piece of the propagators . of particular interest is the operator @xmath32 defined by @xcite @xmath34 , \label{defo11}\ ] ] with @xmath35 and the covariant derivative acting on @xmath36 defined by @xmath37 $ ] . the equations of motion for the @xmath38 field strength imply @xcite @xmath39 where the @xmath13 current is @xmath40 , @xmath41 denote the left - handed fermion doublets . the dominant effect appears in the quark sector due to the presence of terms proportional to @xmath42 . after the quark fields are rotated to the mass eigenstate basis , the operator @xmath32 can be written as @xcite @xmath43 where @xmath44 , the @xmath45 are cabibbo - kobayashi - maskawa ( ckm ) matrix elements and the dots stand for terms suppressed by small fermion masses . from the above discussion we see that the leading effects of the ewsb sector in fcnc processes are coming from the insertion of anomalous tgc vertices and four - fermion operators like ( [ ffo11 ] ) . in the rest of this section , we review the status and future impact of these constraints on the symmetry breaking sector . the effects of the four - fermion operators in ( [ ffo11 ] ) in rare b and k decays were considered in ref . the loop insertion will result in contributions to several fcnc processes , that are controlled by both the coefficient @xmath46 of the effective lagrangian ( [ lnlo ] ) as well as by the high energy scale @xmath0 . to one loop , only one parameter is needed , namely @xmath47 this parameter also governs the contributions of ( [ ffo11 ] ) to other neutral processes , both flavor changing and flavor conserving . for instance , the @xmath48 term in ( [ ffo11 ] ) gives a contribution to @xmath49 , whereas the terms like @xmath50 appear in @xmath51 mixing @xcite . thus the measurements of @xmath52 and the rate of @xmath5 mixing ( together with all other ckm information ) can be used to derive a bound on @xmath53 . although the bound carries some uncertainty mainly associated with ckm quantities like @xmath54 and @xmath55 , we will take it to be , approximately @xcite @xmath56 next , we use this as the allowed range for @xmath53 in order to explore the possible impact of this physics in rare @xmath5 and @xmath6 decays . the one - loop insertion of the terms @xmath57 induces new contributions to various fcnc vertices in @xmath5 decays ( the first two terms in ( [ rareo11 ] ) ) , as well as in @xmath6 decays ( third term in ( [ rareo11 ] ) ) . first , let us consider @xmath58 processes leading , for instance , to the inclusive @xmath59 , since this rate has been recently measured @xcite . the one - loop insertion of the operator @xmath32 does not give a contribution to these processes given that it does not mix with the operator @xmath60 responsible for the on - shell photon amplitude . mixing only occurs at two loops , when qcd corrections are taken into account . as a result the effect , in all @xmath58 transitions is expected to be only a few percent of the sm branching ratios @xcite . on the other hand , the off - shell amplitudes for photons , @xmath61 s and gluons are non - zero at one loop . they generate contributions to processes such as @xmath62 , @xmath63 , @xmath64 ; as well as to similar rare kaon decays like @xmath65 , etc . in order to asses the potential effects we define @xmath66 which is plotted in fig . 1 as a function of the parameter @xmath53 defined in ( [ ydef ] ) , for the allowed range of @xmath53 ( [ ybound ] ) . analogously , we can define the ratio @xmath67 , which tracks the effects in @xmath68 decays ; whereas the contribution to gluon penguin processes such as @xmath69 is represented by the ratio @xmath70 . as it is clear from fig . [ fig1 ] , the effects of the operator @xmath32 are very similar in all three types of @xmath5 decays . we see that , even with the @xmath52 and @xmath71 mixing constraints , large deviations from the sm predictions for these modes are possible . the current experimental bounds on these processes are still not binding on @xmath53 . however , sensitivity to sm branching ratios will be reached in the next round of experiments at the various @xmath5 factories at cornell , kek , slac and fermilab . the distinct feature of this effect is that no significant deviation is expected in @xmath72 , even when large deviations are observed in all the other modes . the effects are very similar in rare @xmath6 decays such as @xmath73 and @xmath74 , etc . in fig . 2 we plot @xmath75 , a quantity analogous to @xmath76 in ( [ defrat ] ) . again , large effects of up to factors of @xmath77 deviations , are allowed . the recently reported @xcite observation of one event in @xmath73 roughly translates into @xmath78 , which is still not constraining . although in this model - independent approach we can not , as a matter of principle , calculate the size of the coefficients @xmath79 , we can use general arguments to estimate their approximate value . using naive dimensional analysis @xcite we have @xmath80 with the scale of new physics obeying @xmath81 . for instance , taking @xmath82 , one would obtain @xmath83 . on the other hand , if @xmath84 , one has @xmath85 . in any case , these are meant to be order of magnitude estimates . therefore , the experimental relevance of the effect strongly depends on details of the dynamics we are not able to compute in a model - independent fashion . finally , we should note that rare @xmath5 and @xmath6 decays are the most sensitive signals for this effect . this is due to the fact that four - lepton operators are suppressed by the lepton masses , and that @xmath32 does not mix quarks and leptons . imposing @xmath86 and @xmath87 conservation , the most general form of the @xmath88 ( @xmath89 ) couplings can be written as @xcite @xmath90 with the conventional choices being @xmath91 and @xmath92 @xcite . in principle , there are six free parameters . making contact with the electroweak lagrangian ( [ lnlo ] ) , these parameters can be expressed in terms of the next - to - leading order coefficients @xcite @xmath93 and @xmath31 . conservation of the electromagnetic charge implies @xmath94 . furthermore , to this order in the energy expansion ( [ lnlo ] ) @xmath95 . then we are left with @xmath96 , @xmath97 and @xmath98 . finally , when considering rare @xmath5 and @xmath6 decays , we can neglect the contribution of @xmath97 since it will be suppressed by powers of the small external momenta over @xmath99 . thus , in this simplistic approach , there are only two parameters relevant at very low energies . the sm predicts @xmath100 . the effects of anomalous tgc have been previously studied in the literature @xcite . however , this hierarchical approach to the couplings has not been the one used in the various analyses and a more comprehensive study is needed . the experiments at lep ii and the next tevatron run are going to be sensitive to deviations from the sm prediction at the @xmath101 level @xcite . effects of this size might be also observed in rare @xmath5 and @xmath6 decays . for instance , @xmath102 can produce enhancements in the branching ratios of @xmath103 decay modes of up to @xmath104 @xcite . in the near future , @xmath5 factory experiments will have sensitivity to these processes at the sm level , turning these low energy measurements into an excellent complement of direct probes of the tgc . up to now , we have only considered the effects of the dynamics associated with the ewsb . these are encoded in the effective lagrangian ( [ lnlo ] ) , which only involves the goldstone boson and gauge fields . additionally , it is possible that the new strong dynamics may also affect some or all fermions . we first comment on the effective lagrangian approach for non - sm couplings of fermions to gauge bosons , and then examine the effects of a prototypical class of theories ( topcolor ) where the dynamical generation of fermion masses imply the existence of relatively light new states . the effects of new dynamics on the couplings of fermions with the sm gauge bosons can be , in principle , also studied in an effective lagrangian approach . for instance , if in analogy with the situation in qcd , fermion masses are dynamically generated in association with ewsb , residual interactions of fermions with goldstone bosons could be important @xcite if the @xmath105 . thus residual , non - universal interactions of the third generation quarks with gauge bosons could carry interesting information about both the origin of the top quark mass and ewsb . in a very general parameterization , the anomalous couplings of third generation quarks can be written as @xmath106 where the parameters @xmath107 , @xmath108 contain the residual , non - universal effects associated with the new dynamics , perhaps responsible for the large top quark mass . then , if we assume that the new couplings are cp conserving , there are six new parameters . they are constrained at low energies by a variety of experimental information , mostly from electroweak precision measurements and the rate of @xmath72 . several simplifications are usually made in order to reduce the number of free parameters . for instance , in most of the literature , it is assumed that @xmath109 @xcite . a stringent bound on the right - handed charged coupling is obtained from @xmath72 @xcite : @xmath110 . the bounds obtained on a particular coupling from electroweak observables such as @xmath25 , @xmath20 , @xmath26 and @xmath52 generally strongly depend on assumptions about the other couplings . for example , if @xmath111 , then the combination @xmath112 is strongly constrained since it contributes to @xmath20 . on the other hand , if @xmath113 , then @xmath114 @xcite since it is the only ( linear ) contribution to @xmath20 . thus , although in general most parameters are confined to a few percent , some of them are allowed to be as large as @xmath115 under certain conditions . this `` model - dependent '' situation requires more experimental information . a global analysis of the effects of the couplings of eqn . ( [ lfer ] ) in rare @xmath5 and @xmath6 processes such as @xmath103 , @xmath65 , etc . may help disentangle the various possible effects and perhaps will give constraints that may be of importance in interpreting data from higher energy experiments @xcite . the description of the residual effects of strong dynamics at low energies on fermion couplings by using ( [ lfer ] ) corresponds to cases where the states associated with the new physics are heavy compared to the weak scale . thus , integrating out the heavy states , leaves us with effective couplings which might be generated at tree level or through loops in the full theory . however , most theories in which electroweak symmetry and/or fermion masses have a dynamical origin also contain states with masses comparable to the weak scale . such is the case , for instance , in technicolor models where the breaking of large chiral symmetries imply the presence of pseudo - goldstone bosons with masses of at most a few hundred gev . it is also the case in topcolor - assisted technicolor ( tatc ) models @xcite , where a top - condensation mechanism generated by the topcolor interactions is responsible for the large dynamical top quark mass , whereas technicolor breaks the electroweak symmetry giving ( most of ) the @xmath116 and @xmath61 masses . the tatc scenario is designed to relief the problems of extended technicolor ( etc ) in generating a heavy top @xcite . although the new gauge bosons associated with the tatc gauge group are heavier than @xmath117tev , the presence of several scalar and pseudo - scalar states with masses in the few - hundred gev range , forces us to take these into account directly in our calculations . from the point of view of their impact in low energy observables , the most important of these states are the top - pions @xmath118 , the triplet of goldstone bosons associated with the breaking of the top chiral symmetry . since top condensation does not fully break the electroweak symmetry ( @xmath119 ) , after mixing with the techni - pions , there will be a triplet of physical top - pions in the spectrum , with a coupling to third generation quarks given by @xmath120 they acquire masses of a few hundred gev due to explicit etc quark mass terms . additionally , in most models there are scalar and pseudo - scalar bound states due to the strong ( although sub - critical ) effective coupling of right - handed @xmath121-quarks . the closer the effective couplings are from criticality , the lighter these bound states tend to be . the spectrum and properties of these states , unlike those of top - pions , are not determined by model - independent features of the symmetry breaking pattern but depend on details of the model . finally , in all tatc models there will be pseudo - goldstone bosons from the breaking of techni - fermion chiral symmetries . however , their couplings to third generation quarks are reduced with respect to ( [ tpcop ] ) by @xmath122 , where @xmath123 is a small etc mass of the order of @xmath117gev . the presence of the relatively light top - pions , as well as the additional bound states , imposes severe constraints on topcolor models due to their potential loop effects in low energy observables , most notably @xmath52 and rare @xmath5 and @xmath6 decays . [ [ top - pion - effects - in - r_b ] ] top - pion effects in @xmath52 + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + : the one - loop contributions of top - pions to the @xmath124 process were studied in ref . there it was shown that they shift @xmath52 negatively by an amount controlled by @xmath125 and @xmath126 . for instance , for @xmath127gev the correction is about @xmath128 for @xmath129gev , and top - pions with masses in the expected @xmath130gev range give unacceptably large deviations . this value of the top - pion decay constant is obtained by using the pagels - stokar formula , which gives @xmath126 a logarithmic dependence on the topcolor energy scale , chosen here to be a few tev . potentially cancelling contributions by other states , such as the scalar and pseudo - scalar bound states , topcolor vector and axial - vector mesons , etc . , are either of the wrong sign or not large enough . possible ways out of this constraint are : larger top - pion masses or larger values of @xmath126 . the larger @xmath126 is , the smaller the coupling , and the top - pions are more goldstone - boson - like . for @xmath131gev , for instance , the shift of @xmath52 is well within the experimentally allowed region even for @xmath132gev . however , in order to obtain such an enhancement in the decay constant we must either assume large corrections to the pagels - stokar expression or introduce new and exotic fermion states . [ [ rare - b - and - k - decays ] ] rare @xmath5 and @xmath6 decays + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + : the top - pions and other scalar states , give one - loop contributions to fcnc processes . these depend not only on @xmath126 and @xmath125 but typically also on one or more elements of the quark rotation matrices necessary to diagonalize the quark yukawa couplings . the contributions of top - pions , as well as `` b - pions '' ( scalar and pseudo - scalar bound states in models where @xmath133 couples to the topcolor interaction ) to @xmath72 depend on @xmath134 , the @xmath135 element in the left or right down rotation matrix . furthermore , the two contributions tend to cancel . thus , the freedom in this model - dependent aspects of the prediction makes it possible to have quite low masses and still satisfy the bound from the experimental measurement of @xmath59 @xcite . the situation changes drastically in @xmath103 processes , where the cancellations are much less efficient . although experiments have not yet reached sensitivity to sm branching ratios @xcite , it will be soon achieved at both hadron and lepton @xmath5 factories . as an example , we plot in fig . 3 the br@xmath136 as a function of the top - pion mass with no other contributions , for @xmath137gev . the @xmath72 constraint is in this case @xmath138 . however , one can see that , even for heavier top - pions the effect can still be a @xmath139 enhancement over the sm prediction of @xmath140 . on the other hand , in the presence of a @xmath141gev charged b - pion the curve changes little , but the @xmath72 bound is now @xmath142 . finally , to compare the potential of these fcnc transitions with the @xmath52 constraints , let us say that if we take @xmath131gev ( which avoids conflict with @xmath52 measurements ) , then the effect of a @xmath141gev top - pion in @xmath103 is still an enhancement of more than @xmath143 with respect to sm expectations . thus , the observation of these modes will further constrain topcolor models beyond the @xmath52 bounds . we expect similar effects due to top - pions and/or b - pions to be present in kaon processes such as @xmath73 . we have seen that a complete , model - independent analysis of the effects of strong dynamics in rare @xmath5 and @xmath6 decays could shed light on the nature of the ewsb mechanism and the origin of fermion masses . the signals are also likely to be important in models where relatively light scalars couple strongly to mass , like in the case of tatc . in most cases , the next round of experiments will have sensitivity to sm branching ratios . this will be the case , for instance , for the tevatron experiments , as well the kek and slac @xmath5 factories in the @xmath144 modes . it will also be the situation in the next generation of kaon experiments for @xmath73 and @xmath74 . the amount and variety of experimental information from these processes is such that suggests a parallel to the role of electroweak measurements at the @xmath61 pole as not only a constraint on new physics sources but also as guidance in the searches to be carried out at high energy machines such as the tevatron in run ii , the lhc and eventually the nlc and/or the muon collider . it is possible to imagine a scenario where deviations from the sm in @xmath5 and/or @xmath6 decays point to a particular source , e.g. corrections to goldstone boson propagators given by @xmath32 , anomalous tgc or anomalous couplings of third generation quarks to gauge bosons as in ( [ lfer ] ) . the nature of the deviation might dictate the road to follow at high energies . as an example , if the source of an effect is in one the top quark couplings @xmath145 , there would be a strong case for a lepton collider running at @xmath146 threshold . other scenarios may not be so clear , and may require a comprehensive and careful analysis of all the data to come ( including issues like hadronic uncertainties in @xmath5 decays ) . this , however , constitutes a very well defined research program . a. longhitano , _ phys . * d*22 , 1166 ( 1980 ) , _ nucl . * b*188 , 118 ( 1981 ) . t. appelquist and g. wu , _ phys . rev . _ * d*48 , 3235 ( 1993 ) . k. hagiwara , k. hikasa , r. d. peccei and d. zeppenfeld , _ nucl . phys . _ * b*282 , 253 ( 1987 ) ; k. hagiwara , s. ishiara , r. szalapski and d. zeppenfeld , _ phys . lett . _ * b*283 , 353 ( 1992 ) and _ phys . * d*48 , 2182 ( 1993 ) . f. feruglio , _ int . phys . _ * a*8 , 4937 ( 1993 ) . j. bernabu , d. comelli , a. pich and a. santamaria , _ phys . lett . _ * 78 * , 2902 ( 1997 ) . g. burdman , _ phys . lett . _ * b*409 , 443 ( 1997 ) . r. balest _ et al . _ , the cleo collaboration , _ phys . _ 74 , 2885 ( 1995 ) . s. adler _ _ , the bnl 787 collaboration , _ phys . rev . _ 79 , 2204 ( 1997 ) . a. manohar and h. georgi , _ nucl . phys . _ * b*234 , 189 ( 1984 ) ; h. georgi , `` _ weak interactions and modern particle theory _ '' , benjamin / cummings , menlo park , california , 1984 . t. barklow _ et al . _ , slac - pub-7366 , in the proceedings of new directions for high - energy physics ( snowmass 96 ) , snowmass , co , 25 jun - 12 jul 1996 . s. p. chia , _ phys . lett . _ * b*240 , 465 ( 1990 ) ; k. a. peterson , _ phys . lett . _ * b*282 , 207 ( 1992 ) ; g. baillie , z. phys . * c*61 , 667 ( 1994 ) . r. d. peccei and x. zhang , _ nucl . * b*337 , 269 ( 1990 ) ; r. d. peccei , s. peris and x. zhang , _ nucl . phys . _ * b*349 , 305 ( 1991 ) . for a treatment of cp violating effects in @xmath5 decays from these couplings see a. abd el - hady and g. valencia , _ phys . lett . _ * b*414 , 173 ( 1997 ) . e. malkawi and c. p. yuan , _ phys . _ * d*50 , 4462 ( 1994 ) . j. hewett and t. rizzo , _ phys . rev . _ * d*49 , 319 ( 1994 ) ; k. fujikawa and a. yamada , _ phys . rev . _ * d*49 , 5890 ( 1994 ) ; b. dobrescu and j. terning , _ phys . _ * b*416 , 129 ( 1998 ) . g. burdman , in preparation . c. t. hill , _ phys . lett . _ * b*345 , 483 ( 1995 ) . e. eichten and k. lane , _ phys . lett . _ * b*352 , 382 ( 1995 ) . e. eichten and k. lane , _ phys . lett . _ * b*222 , 274 ( 1989 ) ; _ ibid _ , * 388 * , 803 ( 1996 ) . g. burdman and d. kominis , _ phys . * b*403 , 101 ( 1997 ) . g. buchalla , g. burdman , c. t. hill and d. kominis , _ phys . rev . _ * d*53 , 5185 ( 1996 ) . c. albajar _ , the ua1 collaboration , _ phys . _ * b*262 , 163 ( 1991 ) ; s. glenn _ _ , the cleo collaboration , clns-97/1514 , hep - ex/9710003 ; b. abbot _ _ , the d0 collaboration , hep - ex/9801027 .

we discuss the constraints from rare b and k decays on the electroweak symmetry breaking ( ewsb ) sector , as well as on theories of fermion masses . we focus on models involving new strong dynamics and show that transitions involving flavor changing neutral currents ( fcnc ) play an important role in disentangling the physics in these scenarios . in a model - independent approach to the ewsb sector , the information from rare decays is complementary to precision electroweak observables in bounding the contributions to the effective lagrangian . we compare the pattern of deviations from the standard model ( sm ) that results from these sources , with the deviations associated with the mechanism for generating fermion masses . madph-98 - 1039 +

despite years of intensive research , the pairing mechanism responsible for d - wave superconductivity ( dsc ) in the high-@xmath0 cuprates remains a puzzle . it is generally believed that the conventional phonon mechanism is inconsistent with d - wave pairing symmetry and not strong enough to explain transition temperatures higher than 100 k. most investigations in this direction have been focused on the pure electronic mechanism , but no consensus has been reached so far . recently , accurate experiments displayed pronounced phonon and electron - lattice effects in these materials , which are manifested by a large softening and broadening of certain phonon modes in the whole doping region . in particular , the in - plane copper - oxygen bond - stretching phonon , apical oxygen phonon ( aop ) , and oxygen @xmath1 buckling phonon are shown to be strongly coupled to charge carriers @xcite . moreover , photoemission - spectroscopy - resolved kink structures @xcite are probably caused by coupling of quasiparticles to phonon modes . these findings suggest that phonons are important for the physical properties of high-@xmath0 cuprates . various theoretical attempts have been made to understand the role of phonons in high-@xmath0 superconductivity ( htsc ) @xcite , but the answer remains unclear . in a functional renormalization group study @xcite , honerkamp _ et al . _ found that the @xmath1 buckling phonon enhances the d - wave pairing instability in the hubbard model . more recently , an exact diagonalization ( ed ) study of the @xmath2 model coupled to phonons shows that coupling to the buckling mode stabilizes d - wave pairing while coupling to the breathing mode favors a p - wave pairing @xcite . on the other hand , based on dynamical cluster monte carlo calculations of the hubbard model coupled to holstein , buckling and breathing phonons , macridin _ et al . _ found that while these phonons can indeed enhance pairing , a strong phonon - induced reduction of quasiparticle weight leads to a suppression of dsc @xcite . in this paper , we study the effect of aop on dsc in the more realistic three - band hubbard model . our work is motivated by recent experiments showing that the distance between apical oxygen and the @xmath3 plane @xcite , the disorder around apical oxygen @xcite , and the apical hole state @xcite have significant effects on @xmath0 . basically , there are two routes for apical oxygen to affect htsc : one is to tune the electronic structure of @xmath3 plane , leading to a change of @xmath0 @xcite ; the other one is to directly couple apical oxygen vibrations to charge carriers on the conducting @xmath3 plane @xcite . here , we focus on the second route and study a strongly anharmonic vibration of apical oxygen in a double - well potential , which is evidenced in the x - ray absorption spectroscopy of several typical high-@xmath0 compounds @xcite . for a strongly anharmonic motion in the double - well potential , the first excitation energy @xmath4 is much smaller than the ones excited to higher energy levels @xmath5 . in this case one can take into account only the lowest quantum states @xmath6 and @xmath7 with energies @xmath8 and @xmath9 and model the low - energy motion of apical oxygen by a local two - level system represented by a pseudospin @xcite degree of freedom . our main results , obtained by ed and constrained - path monte carlo ( cpmc ) methods , are presented in figs . [ sum2x2](b ) , [ binde](a ) and [ pdvdu246 ] . fig . [ binde](a ) clearly shows that the coupling to the aop induces a strong enhancement of hole binding energy , and this enhancement effect grows as the coulomb repulsion @xmath10 on the copper site is increased . an analysis of the contribution of different energies to the hole binding energy reveals a novel potential - energy - driven pairing mechanism that involves reduction of both electronic potential energy and phonon related energy . as a combination of increasing pairing interaction and quasiparticle weight ( see fig . [ sum2x2](b ) ) , the d - wave pairing correlations are found to be strongly enhanced by the electron - phonon ( el - ph ) coupling ( see fig . [ pdvdu246 ] ) . our paper is organized as follows : in section [ model ] , we define the hamiltonian and the physical quantities calculated and discuss the choice of model parameters . in section [ results ] , we present our numerical results and discuss the physical mechanism responsible for the aop - induced enhancement of dsc . finally , in section [ conclusions ] , we discuss in detail our main conclusions . to model the electronic structure of @xmath3 plane and the coupling of holes to the anharmonic aop , we adopt the following hamiltonian proposed in ref . @xcite , @xmath11 where @xmath12 , @xmath13 , and @xmath14 stand for the kinetic motion of holes , the potential energy for holes , and phonon related energy , respectively . they are expressed in the form : @xmath15 @xmath16 and @xmath17 here , the operator @xmath18 creates a hole at a cu @xmath19 orbital and @xmath20 creates a hole in an o@xmath21 or @xmath22 orbital . @xmath23 and @xmath24 are the pseudospin operators for @xmath25 . @xmath10 denotes the coulomb energy at the cu sites . @xmath26 and @xmath27 are the cu - o and o - o hybridizations , respectively , with the cu and o orbital phase factors included in the sign . the charge - transfer energy is @xmath28 , i.e. the oxygen orbital energy . @xmath29 and @xmath30 in @xmath14 denote the strength of el - ph coupling . @xmath31 stands for the tunneling frequency of the two - level system . @xmath32 is the vector connecting cu and its nearest - neighbor ( nn ) o. according to quantum cluster calculations @xcite , the parameters lie in the range : @xmath33 , @xmath34 , @xmath35 , and @xmath36 . in units of @xmath37 , we choose a parameter set @xmath38 and @xmath39 , while @xmath10 is varied from weak to strong coupling , including the physical value @xmath40 . @xmath41 and @xmath42 are assumed for the results presented below , except explicitly noted otherwise . our calculations are performed on clusters of @xmath43 , @xmath44 , @xmath45 and @xmath46 unit cells with periodic boundary conditions using the ed and cpmc methods @xcite . in the cpmc method , we follow refs . and to use the worldline representation for the pseudospins and projected the ground state @xmath47 of el - ph interacting system from a trial wave function @xmath48 represent the hole and phonon parts , respectively . the cpmc algorithm has been checked against ed on the @xmath49 cluster , and the difference for the electronic kinetic energy , as well as for the charge and magnetic moment at the copper sites , is less than @xmath50 up to @xmath51 . the hole binding energy is defined as : @xmath52 with @xmath53 the ground - state energy for n doped holes . the @xmath54 pairing correlation is defined by , @xmath55 where @xmath56\\ + & [ & p^x_{\vec{r}\uparrow}p^x_{\vec{r}+\vec{\delta}\downarrow } -p^x_{\vec{r}\downarrow}p^x_{\vec{r}+\vec{\delta}\uparrow } ] \\ + & [ & p^y_{\vec{r}\uparrow}p^y_{\vec{r}+\vec{\delta}\downarrow } -p^y_{\vec{r}\downarrow}p^y_{\vec{r}+\vec{\delta}\uparrow } ] \}\end{aligned}\ ] ] with @xmath57 . @xmath58 for @xmath59 and @xmath60 for @xmath61 . we calculate also the vertex contribution to the correlations defined as follows : @xmath62 where @xmath63 is the bubble contribution obtained with the dressed ( interacting ) propagator @xcite . first , we show ed results for the @xmath43 cluster with one hole doped beyond half filling . the electronic kinetic energy @xmath64 , the peak value @xmath65 of the single - particle spectral function @xmath66 at the fermi energy ( @xmath67 ) , the charge @xmath68 and the magnetic moment @xmath69 at the copper sites are displayed in figs . [ sum2x2](a)-[sum2x2](d ) . here , @xmath70 with @xmath71 and @xmath72 denoting the ground - state wave function and its energy for one - hole doping . the index @xmath73 corresponds either to the @xmath54 or to the @xmath74 orbitals , and @xmath75 . one can clearly see that @xmath76 is lowered with increasing the el - ph coupling @xmath77 at all coulomb energies . meanwhile , an increase of @xmath65 with increasing @xmath77 indicates that the quasiparticle weight is increased by the el - ph coupling . in contrast , previous studies of harmonic phonons in the holstein - hubbard model found that the kinetic energy of electrons is increased with increasing el - ph coupling , accompanying a reduction of quasiparticle weight @xcite . to explore the physical reasons for lowering @xmath76 , we switch off the coupling of apical phonon either to copper or to in - plane oxygen , i.e. , we set @xmath78 or @xmath79 in eq.([hph ] ) . it is found that @xmath76 is lowered for the former case , but increased for the latter case . these results demonstrate that the special coupling of apical oxygen phonon to in - plane oxygen is responsible for lowering the electronic kinetic energy . .@xmath77 dependence of @xmath76 ( per unit cell ) , @xmath80 , @xmath81 , and nn @xmath82 spin correlation @xmath83 on the @xmath84 cluster at @xmath51 . the number of holes @xmath85 , corresponding to a hole doping density @xmath86 . statistical errors are in the last digit and shown in the parentheses . [ cols="<,<,<,<,<",options="header " , ] [ tab6x6 ] from fig . [ sum2x2](c ) , we notice that the charge is transferred from copper to oxygen sites , which , in combination with a phonon - mediated retarded attraction between holes with opposite spins , results in a reduction of magnetic moment at @xmath87 sites , as shown in fig . [ sum2x2](d ) . similar effects of aop on @xmath76 , @xmath80 and @xmath81 are also observed for larger clusters obtained by cpmc simulations , and representative results on the @xmath45 cluster are shown in table [ tab6x6 ] . the last column in tab . [ tab6x6 ] shows that the value of nn @xmath82 spin correlation @xmath88 becomes less negative with increasing @xmath77 , implying a suppression of antiferromagnetic ( afm ) spin correlation . the hole binding energy @xmath89 is shown in fig . [ binde](a ) as a function of the coulomb energy @xmath10 at different @xmath77 . at all @xmath10 , the binding energy is decreased by switching on the el - ph coupling , signaling an enhancement of hole pairing interaction . it is remarkable that this enhancement effect becomes stronger with increasing @xmath10 , which is particularly evident in the region @xmath90 . in order to identify the physical origin for this enhancement , the contributions to @xmath89 from the hole kinetic energy @xmath76 , the hole potential energy @xmath91 , and the phonon related energy @xmath92 are depicted in figs . [ binde ] ( b)-(d ) , respectively . here , @xmath93 , @xmath94 and @xmath95 have similar definitions to @xmath89 , with @xmath96 in eq.([binding ] ) replaced with @xmath76 , @xmath97 and @xmath98 , respectively . these quantities represent the gain in the corresponding energy when the second hole is doped in the vicinity of the first one . although the kinetic energy of holes is reduced upon increasing the el - ph coupling ( see fig . [ sum2x2](a ) ) , an increase of @xmath93 with increasing @xmath77 displayed in fig . [ binde](b ) indicates that the kinetic energy gain for two doped holes is reduced by the el - ph coupling . as seen in fig . [ binde](c ) and fig . [ binde](d ) , a decrease of @xmath94 and @xmath95 with increasing @xmath77 reveals that it is the reduction of electronic potential energy and phonon related energy between two doped holes that enhances the hole binding energy @xmath89 . in addition , the decrease of @xmath94 and @xmath95 becomes more pronounced as @xmath10 is increased , leading to a stronger enhancement of the hole binding energy in the strong correlation regime ( see fig . [ binde](a ) ) . [ binde2 ] displays the hole binding energy and different contributions to @xmath89 on the @xmath49 cluster with antiperiodic and mixed ( periodic in the x direction and antiperiodic in the y direction ) boundary conditions . a comparison of the results in figs . [ binde ] and [ binde2 ] shows that although the boundary condition has strong effects on the amplitude of hole binding energy , the aop - induced enhancement of hole binding energy is qualitatively similar for different boundary conditions . this demonstrates that our findings reflect the intrinsic effects of aop in the studied model . based on the ed results on the small cluster , we can conclude that the coupling of aop to holes can enhance superconductivity on the @xmath3 plane . the question arising is whether the d - wave pairing symmetry is enhanced ? this issue can be addressed by examining the behavior of the d - wave pairing correlation @xmath99 in eq.([pairing ] ) . in figs . [ pdvdu246](a)-(c ) we show @xmath100 as a function of @xmath101 for the @xmath45 cluster at @xmath102 , @xmath103 and @xmath104 , respectively . the corresponding vertex contribution is also displayed in figs . [ pdvdu246](d)-(f ) . in the weak - correlation case ( @xmath102 ) , we observe that @xmath100 is modified slightly by the el - ph coupling . however , when @xmath105 is increased to @xmath103 and @xmath104 , the pairing correlation is enhanced at all long - range distances for @xmath106 . at @xmath107 , the average enhancement of the long - range part of @xmath108 is estimated to be about @xmath109 and @xmath110 for @xmath111 and @xmath51 , respectively . this increasing enhancement of @xmath112 with @xmath10 is consistent with our findings for the binding energy and quasiparticle weight . a dramatic increment of the vertex contribution with increasing @xmath77 , as shown in fig . [ pdvdu246](e ) and fig . [ pdvdu246](f ) , further provides strong evidence that the d - wave pairing interaction is actually increased by the el - ph coupling . we also study the effect of the tunneling frequency @xmath31 on the pairing correlations . in the inset of fig . [ pdvdu246 ] ( c ) , the average of long - range d - wave pairing correlation , @xmath113 where @xmath114 is the number of hole pairs with @xmath115 , is plotted as a function of @xmath31 for @xmath51 and @xmath116 . we notice that its frequency dependence is rather weak , suggesting that the isotope effect of apical oxygen on the superconductivity is very small . this is in good agreement with the site - selected oxygen isotope effect in @xmath117 @xcite . to illustrate the effect of hole doping on the phonon - induced enhancement of dsc , in fig . [ pdave ] we show @xmath118 as a function of hole doping density @xmath119 at @xmath51 for the @xmath120 , @xmath44 , and @xmath46 clusters . the combination of the results on the three clusters shows that in a wide hole doping region @xmath121 , the d - wave pairing correlation is enhanced by the el - ph coupling , and as the hole doping density is increased from underdoping to optimal doping and then to overdoping , the enhancement of @xmath118 is weakened monotonically . the reduced enhancement of @xmath118 with larger hole doping , together with the stronger enhancement of @xmath89 and @xmath108 with increasing @xmath10 , demonstrates that strong electronic correlations and/or afm fluctuations play a crucial role in the phonon - induced enhancement of dsc . in summary , our numerical simulations show that the hole binding energy is strongly enhanced by an aop - induced reduction of electronic potential energy and phonon related energy . as a combination of two concurring effects , i.e. the enhancement of hole pairing interaction and the increase of quasiparticle weight , the long - range part of d - wave pairing correlations is dramatically enhanced with increasing the el - ph coupling strength . our results also show that strong electronic correlations and/or afm fluctuations are crucial for this phonon - induced enhancement effect . the consistent behavior of our results on different clusters suggests that the phonon - induced enhancement of dsc could survive in the thermodynamic limit . we thank w. hanke , f. f. assaad . , and a. s. mishchenko for enlightening discussions . this work was supported by nsfc under grant nos . 10674043 and 10974047 . hql acknowledges support from hkrgc 402109 . ea was supported by the austrian science fund ( fwf ) under grant p18551-n16 . 99 l. pintschovius , phys . ( b ) * 242 * , 30 ( 2005 ) . l. pintschovius , d. reznik , and k. yamada , phys . b * 74 * , 174514 ( 2006 ) . a. lanzara , p. v. bogdanov , x. j. zhou , s. a. keller , d. l. feng , e. d. lu , t. yoshida , h. eisaki , a. fujimori , k. kishio , j .- shimoyama , t. noda , s. uchida , z. hussain , and z .- x . shen , nature * 412 * , 510 ( 2001 ) . t. cuk , f. baumberger , d. h. lu , n. ingle , x. j. zhou , h. eisaki , n. kaneko , z. hussain , t. p. devereaux , n. nagaosa , z .- x . shen , phys . lett . * 93 * , 117003 ( 2004 ) . z. b. huang , w. hanke , e. arrigoni , and d. j. scalapino , phys . b * 68 * , 220507(r ) ( 2003 ) . c. honerkamp , h. c. fu , and d. h. lee , phys . rev . b * 75 * , 014503 ( 2007 ) . l. vidmar , j. bonca , s. maekawa , and t. tohyama , phys . lett . * 103 * , 186401 ( 2009 ) . a. macridin , b. moritz , m. jarrell , and t. maier , phys . lett . * 97 * , 056402 ( 2006 ) ; cond - mat/0611067 . j. a. slezak , j. h. lee , m. wang , k. mcelroy , k. fujita , b. m. andersen , p. j. hirschfeld , h. eisaki , s. uchida , and j. c. davis , proc . , * 105 * , 3203 ( 2008 ) . h. hobou , s. ishida , k. fujita , m. ishikado , k. m. kojima , h. eisaki , and s. uchida , phys . b * 79 * , 064507 ( 2009 ) . w. b. gao , q. q. liu , l. x. yang , y. yu , f. y. li , c. q. jin , and s. uchida , phys . b * 80 * , 094523 ( 2009 ) . m. merz , n. nucker , p. schweiss , s. schuppler , c. t. chen , v. chakarian , j. freeland , y. u. idzerda , m. klaser , g. muller - vogt , and th . wolf , lett . * 80 * , 5192 ( 1998 ) . m. mori , g. khaliullin , t. tohyama , and s. maekawa , phys . * 101 * , 247003 ( 2008 ) ; e. pavarini , i. dasgupta , t. saha - dasgupta , o. jepsen , and o. k. andersen , phys . lett . * 87 * , 047003 ( 2001 ) . m. frick , i. morgenstern , and w. von der linden , z. phys . b - condensed matter * 82 * , 339 ( 1991 ) . m. frick , w. von der linden , i. morgenstern , and h. de raedt , z. phys . b - condensed matter * 81 * , 327 ( 1990 ) . d. haskel , e. a. stern , d. g. hinks , a. w. mitchell , and j. d. jorgensen , phys . b * 56 * , r521 ( 1997 ) . s. d. conradson , i. d. raistrick , and a. r. bishop , science * 248 * , 1394 ( 1990 ) . j. mustre de leon , s. d. conradson , i. batistic , and a. r. bishop , phys . * 65 * , 1675 ( 1990 ) . n. m. plakida , physica scripta * t29 * , 77 ( 1989 ) . n. m. plakida , v. l. aksenov , and s. l. drechsler , europhys . * 4 * , 1309 ( 1987 ) . r. l. martin , phys . b * 53 * , 15501 ( 1996 ) . z. b. huang , h. q. lin , and j. e. gubernatis , phys . b * 63 * , 115112 ( 2001 ) . s. w. zhang , j. carlson and j. e. gubernatis , phys . * 74 * , 3652 ( 1995 ) ; phys . b * 55 * 7464 ( 1997 ) . s. r. white , d. j. scalapino , r. l. sugar , n. e. bickers , r. t. scalettar , phys . rev . b * 39 * , 839 ( 1989 ) . g. sangiovanni , o. gunnarsson , e. koch , c. castellani , and m. capone , phys . * 97 * , 046404 ( 2006 ) . d. zech , h. keller , k. conder , e. kaldis , e. liarokapis , n. poulakis , and k. a. muller , nature * 371 * , 681 ( 1994 ) .

we study the hole binding energy and pairing correlations in the three - band hubbard model coupled to an apical oxygen phonon , by exact diagonalization and constrained - path monte carlo simulations . in the physically relevant charge - transfer regime , we find that the hole binding energy is strongly enhanced by the electron - phonon interaction , which is due to a novel potential - energy - driven pairing mechanism involving reduction of both electronic potential energy and phonon related energy . the enhancement of hole binding energy , in combination with a phonon - induced increase of quasiparticle weight , leads to a dramatic enhancement of the long - range part of d - wave pairing correlations . our results indicate that the apical oxygen phonon plays a significant role in the superconductivity of high-@xmath0 cuprates .

in this note we deal with the uniqueness of distributional solutions to the continuity equation with a sobolev vector field and with the property of being a lagrangian solution , i.e. transported by a flow of the associated ordinary differential equation . let us first recall the by now classical diperna - lions theory @xcite . we fix @xmath0 and @xmath1 and we consider a vector field @xmath2;w_{\mathrm{loc}}^{1,p}({\mathbb r}^{n};{\mathbb r}^{n})\right ) \ , , \qquad \operatorname{div}{\boldsymbol b}\in l^{1 } \left ( [ 0,t];l^\infty({\mathbb r}^n ) \right ) \ , , \\ & \frac{|{\boldsymbol b}(t,{\boldsymbol x})|}{1+|{\boldsymbol x}| } \in l^{1}\left ( [ 0,t ] ; l^1({\mathbb r}^{n } ) \right ) + l^{1}\left ( [ 0,t ] ; l^\infty({\mathbb r}^{n } ) \right)\ , . \end{aligned}\ ] ] given an initial datum @xmath3 , we consider distributional solutions to the cauchy problem for the continuity equation @xmath4 defined as usual by a formal `` integration by parts '' after testing the equation with lipschitz test functions . given a vector field @xmath5 as in , the diperna - lions theory @xcite guarantees uniqueness of distributional solutions @xmath6 ; l^q({\mathbb r}^n ) \right)\ ] ] to the problem , where @xmath7 is the conjugate exponent of @xmath8 , that is , @xmath9 . if @xmath10 the existence of solutions in this class can be proved by an easy approximation procedure . moreover , such unique solution is transported by the unique regular lagrangian flow associated to @xmath5 ( see definition [ d : rlf ] ) . we remark that the theory of @xcite has been extended to vector fields with bounded variation by ambrosio @xcite . the need for considering solutions in the class follows from the strategy of proof in @xcite , which consists in showing the renormalization property for distributional solutions . to this aim , the authors prove the convergence to zero of a suitable commutator , that can be rewritten as an integral expression involving essentially the product of @xmath11 and @xmath12 . however , distributional solutions to the cauchy problem can be defined as long as the product @xmath13 \times { \mathbb r}^n)$ ] . therefore , the theory in @xcite leaves open the question whether uniqueness holds for solutions with less integrability than . ideally , the `` extreme '' case would be that of @xmath14 and @xmath15 , both locally in space . our main result in this direction is the following : [ t : main ] let @xmath5 be a vector field as in , with @xmath16 . assume in addition that @xmath17 is continuous for @xmath18-a.e . @xmath19 $ ] , with modulus of continuity on compact sets which is uniform in time . then , given an initial datum @xmath20 , the cauchy problem for has a unique solution @xmath21\times{\mathbb r}^{n}\right ) \,.\ ] ] such unique solution is lagrangian and renormalized . the continuity assumption on the vector field in theorem [ t : main ] is satisfied for example when @xmath22;w_{\mathrm{loc}}^{1,p}({\mathbb r}^{n})\right)$ ] with @xmath23 . theorem [ t : main ] can be easily extended to the case where a source term or a linear term of zero order are present in the continuity equation , under suitable integrability conditions on the coefficients . in particular , we can also deal with the transport equation @xmath24 instead of the continuity equation . let us describe in few words the strategy of the proof of theorem [ t : main ] . given a distributional solution @xmath25\times{\mathbb r}^{n}\right)$ ] of the cauchy problem we aim at proving that it is transported by the regular lagrangian flow @xmath26 associated to @xmath5 . to this aim , we change variable using the flow in the distributional formulation of . however , due to the lack of lipschitz regularity of the flow with respect to the space variable , we do not obtain yet the lagrangian formulation in distributional sense : after the change of variable we do not obtain the full class of test functions . nevertheless some regularity of the flow `` on large sets '' is in fact available ( see theorem [ t : regularity ] ) . this guarantees that the test function we obtain is lipschitz on a `` large flow tube '' , although with a possibly large lipschitz constant . we need to extend this function to a globally lipschitz test function . the key remark is that , in order to estimate the error resulting from this extension , only the lipschitz constant along the characteristics is relevant , not the global lipschitz constant . we then implement a `` directional extension lemma '' ( lemma [ l : extension ] ) , stating that we can construct an extension which is both globally lipschitz and directionally lipschitz along the flow , and the directional lipschitz constant can be estimated quantitatively . this allows to conclude the proof . after presenting in [ s : prelim ] some background material , in [ s : proof ] we give a complete proof of theorem [ t : main ] , under the additional assumption [ a : srfe ] on the existence of a directional lipschitz extension . in [ s : lemmalip ] we sketch a proof of the validity of assumption [ a : srfe ] under the continuity assumptions on the vector field in theorem [ t : main ] . a complete proof is deferred to the follow up paper @xcite . in the non smooth context the suitable notion of flow of a vector field is that of regular lagrangian flow , introduced in the following form in @xcite : [ d : rlf ] we say that a map @xmath27^{2}\times{\mathbb r}^{n}\to{\mathbb r}^{n}$ ] is a regular lagrangian flow associated to the vector field @xmath5 if 1 . [ item:1rlf ] for @xmath28 ^ 2 $ ] we have @xmath29 . 2 . for @xmath30-a.e . @xmath31 the map @xmath32 satisfies the ordinary differential equation @xmath33 we notice that @xmath34 for later use we set @xmath35 and observe that by definition [ d : rlf](i ) we have @xmath36 the theory in @xcite guarantees that , given a vector field @xmath5 as in , there exists a unique regular lagrangian flow associated to it . moreover , in @xcite the following regularity of the regular lagrangian flow has been proved : [ t : regularity ] let @xmath5 be a vector field as in and let @xmath26 be the associated regular lagrangian flow . assume that @xmath16 . then , for all @xmath37 and @xmath38 there exists a compact set @xmath39 such that 1 . @xmath40 is lipschitz continuous on @xmath41 , uniformly w.r.t . @xmath42 $ ] . 2 . @xmath43 . the restriction to the case @xmath44 in theorem [ t : regularity ] and therefore in theorem [ t : main ] is due to the use of some harmonic analysis estimates in its proof . we finally introduce the following concept of directional lipschitz continuity : [ d : directional ] let @xmath45 be defined on a borel set @xmath46\times{\mathbb r}^n$ ] and let @xmath47\times a\to { \mathbb r}^{n}$ ] be a borel map , where @xmath48 is a borel set . we say that the function @xmath45 is @xmath49-directionally lipschitz continuous if for all @xmath50 $ ] and for all @xmath51 such that @xmath52 there holds @xmath53 we focus in this paper only on directional lipschitz continuity in the specific case @xmath54 , where @xmath26 is a regular lagrangian flow . in this section we give a complete proof of theorem [ t : main ] , under the additional assumption [ a : srfe ] on the existence of a directional lipschitz extension that we introduce in step 2 here below . a proof of assumption [ a : srfe ] is sketched in [ s : lemmalip ] below and a full proof deferred to @xcite . [ [ step-0 . ] ] * step 0 . * + + + + + + + + + by the linearity of the continuity equation , it is enough to prove that @xmath55 implies @xmath56 . we do this by showing that every distributional solution @xmath12 of satisfies a lagrangian formulation . in this context this amounts to the fact that the function @xmath57 solves in distributional sense the equation @xmath58=0 $ ] , where @xmath59 is defined in , with initial datum @xmath60 , that is @xmath61 where @xmath62 and @xmath63 , the spaces of lipschitz functions with compact support , and of essentially bounded functions with compact support , respectively . notice that the validity of implies that @xmath64 , and thus with we obtain @xmath56 . since @xmath65 is dense in @xmath66 with respect to the weak star topology of @xmath67 , we reduced the proof of theorem [ t : main ] to the proof of the following claim : [ c : claim ] the lagrangian formulation holds for every @xmath68 . we fix @xmath69 we prove in the next steps that claim [ c : claim ] holds . [ [ step-1 . ] ] * step 1 . * + + + + + + + + + fix @xmath38 and consider a compact set of the form @xmath70\times b_{r}({\boldsymbol 0})$ ] which contains the support of the function @xmath71 fixed in . we use theorem [ t : regularity ] to find a compact subset @xmath39 on which the regular lagrangian flow @xmath40 is uniformly lipschitz continuous . [ l : sfsdva ] on the compact flow tube @xmath72}$ ] starting from @xmath73 the function @xmath74 is lipschitz continuous and @xmath75-directionally lipschitz continuous , with @xmath76 as in . we start by proving the @xmath75-directional lipschitz continuity . let @xmath77 and thus by we get @xmath78 we now prove the lipschitz continuity of @xmath79 on @xmath72}$ ] . given @xmath80 one has @xmath81 when comparing two points @xmath82 and @xmath83 , for some @xmath84 , we simply define @xmath85 and we estimate @xmath86 where in the last inequality we applied @xmath87 * step 2 . * + + + + + + + + + we can proceed with the proof under the following assumption . [ a : srfe ] given @xmath38 let @xmath79 be as in . we assume that there exists @xmath88 \times { \mathbb r}^{n}\to{\mathbb r}$ ] which is an extension of @xmath79 and in addition is 1 . lipschitz continuous , and 2 . @xmath89-directionally lipschitz continuous , where @xmath90 does not depend on @xmath91 . in fact , we are able to prove that assumption [ a : srfe ] holds when the vector field @xmath5 satisfies the continuity condition assumed in theorem [ t : main ] . in section [ s : lemmalip ] we give a sketch of the proof of this fact , and we defer a complete proof to a next paper . [ [ step-3 . ] ] * step 3 . * + + + + + + + + + we now derive some consequences of assumption [ a : srfe ] in the @xmath92-variables . we define @xmath93 $ and $ { \boldsymbol x}\in { \mathbb r}^n$}\ ] ] and we observe that 1 . [ i : szvds ] @xmath94 is @xmath95-lipschitz continuous for all @xmath96 . this follows from assumption [ a : srfe](ii ) and from the definition of directional lipschitz continuity ( definition [ d : directional ] ) . 2 . @xmath97 for every @xmath98 and every @xmath19 $ ] . in particular , we can test @xmath58(t,{\boldsymbol y})$ ] agains @xmath99 : by the definitions in and we obtain @xmath100\,dtd{\boldsymbol y}\,.\end{aligned}\ ] ] we now apply the change of variable @xmath101 , obtaining @xmath102 \,dtd{\boldsymbol x}=0\,,\ ] ] because @xmath12 is a distributional solution of with zero initial datum . we stress that the first equality in follows by the definition of push - forward measure because the results in @xcite establish that the regular lagrangian flow @xmath26 satisfies the absolute continuity estimate in definition [ d : rlf ] . this is a very important brick in this disintegration strategy , and in other settings it requires to be proved ad hoc , see for instance @xcite . [ [ step-4 . ] ] * step 4 . * + + + + + + + + + we conclude the proof of claim [ c : claim ] , thus establishing theorem [ t : main ] under assumption [ a : srfe ] . the main observation is that equation gives the validity of claim [ c : claim ] with the test function @xmath71 replaced by the approximation @xmath103 defined in . therefore , we simply estimate the integral containing @xmath71 with the integral containing @xmath104 plus an error , and we only need to show that the error converges to zero as @xmath105 . indeed , we compute as follows : @xmath106\,dtd{\boldsymbol y}\\ & = \cancel{\int_{0}^{t}\int_{k_{\varepsilon } } \frac{u}{r}\,\partial_{t}\left[\psi-\psi_{\varepsilon}\right]\,dtd{\boldsymbol y } } + \int_{0}^{t}\int _ { b_{r}({\boldsymbol 0})\setminus k_{\varepsilon } } \ ! \frac{u}{r}\,\partial_{t}\left[\psi-\psi_{\varepsilon}\right]\,dtd{\boldsymbol y}\,,\end{aligned}\ ] ] where @xmath107 is as in step 1 , and by construction @xmath108 on @xmath70 \times k_\varepsilon$ ] . since @xmath109 is @xmath76-lipschitz continuous by definition and each @xmath94 is @xmath95-lipschitz continuous by step 3([i : szvds ] ) , we finally get @xmath110 using and the fact that the function @xmath111 in belongs to @xmath112 \times { \mathbb r}^n)$ ] . this concludes the proof of theorem [ t : main ] under assumption [ a : srfe ] . we finally briefly sketch the strategy of proof of the following lemma . a full proof in a more general context is deferred to @xcite . [ l : extension ] let @xmath5 be a vector field as in , with @xmath16 . assume in addition that @xmath17 is continuous for @xmath18-a.e . @xmath19 $ ] , with modulus of continuity on compact sets which is uniform in time . then assumption [ a : srfe ] holds . in the above lemma one can as well require that @xmath113 . we start by noticing that a function @xmath79 is @xmath75-directionally lipschitz continuous according to definition [ d : directional ] if and only if @xmath79 is @xmath76-lipschitz continuous for the following degenerate distance @xmath114 : @xmath115 moreover , we denote by @xmath116 the usual euclidean distance in @xmath70 \times { \mathbb r}^{n}$ ] . consider a lipschitz continuous function @xmath79 defined on the compact flow tube @xmath117}$ ] of assumption [ a : srfe ] . we remind that we assume that @xmath79 is @xmath75-directionally lipschitz continuous , and that we need to extend @xmath79 to @xmath70 \times { \mathbb r}^n$ ] in such a way that the extension is 1 . @xmath75-directionally lipschitz continuous , i.e. @xmath95-lipschitz continuous for @xmath114 , with @xmath95 depending on @xmath76 , and 2 . lipschitz continuous for the euclidean distance , i.e. lipschitz continuous for @xmath116 . in other words , we need to prove a lipschitz extension theorem with respect to two non equivalent distances at the same time : to the best of our knowledge , this is a new and non trivial task . notice that for our purposes we need that the lipschitz constant for @xmath118 only depends on @xmath76 , while we do not need a quantitative control on the lipschitz constant for @xmath119 . we now give a rough idea of the proof of lemma [ l : extension ] . for @xmath120 we introduce a family of distances @xmath121 , each of them equivalent to the euclidean distance @xmath119 . the distance @xmath121 penalizes with a factor @xmath122 displacements which are not along the flow . moreover , the distances @xmath123 converge to the degenerate distance @xmath114 , i.e. @xmath124 as @xmath125 . in particular , a function which is @xmath95-lipschitz continuous for @xmath121 is also @xmath95-lipschitz continuous for @xmath114 . the key point in the proof of lemma [ l : extension ] is the fact that , when @xmath125 , the lipschitz constant of @xmath79 for @xmath121 converges to the lipschitz constant @xmath76 of @xmath79 for @xmath114 : @xmath126 using this property , we choose @xmath127 small enough so that @xmath128 is close to @xmath76 . we extend @xmath79 by using mcshane extension theorem for the distance @xmath129 . in this way , we get an extension which is 1 . @xmath130-directionally lipschitz continuous , and @xmath131 is close to @xmath76 , and 2 . lipschitz continuous for the euclidean distance @xmath116 , since @xmath129 is equivalent to @xmath119 . in the above procedure , we are currently able to prove only assuming that the vector field @xmath5 is continuous for @xmath18-a.e . @xmath19 $ ] , with modulus of continuity on compact sets which is uniform in time . this work was started during a visit of lc at the university of basel and carried on during a visit of gc at the university of padova as a visiting scientist . the authors gratefully acknowledge the support and the hospitality of both institutions . lc is a member of the gruppo nazionale per lanalisi matematica , la probabilit e le loro applicazioni ( gnampa ) of the istituto nazionale di alta matematica ( indam ) . gc is partially supported by the erc starting grant 676675 flirt .

we deal with the uniqueness of distributional solutions to the continuity equation with a sobolev vector field and with the property of being a lagrangian solution , i.e. transported by a flow of the associated ordinary differential equation . we work in a framework of lack of local integrability of the solution , in which the classical diperna - lions theory of uniqueness and lagrangianity of distributional solutions does not apply due to the insufficient integrability of the commutator . we introduce a general principle to prove that a solution is lagrangian : we rely on a disintegration along the unique flow and on a new directional lipschitz extension lemma , used to construct a large class of test functions in the lagrangian distributional formulation of the continuity equation . 0.5 * rsum * 0.5*unicit et proprit lagrangienne des solutions manquant dintgrabilit de lquation de continuit . * on tudie lunicit des solutions distributionnelles de lquation de continuit avec des champs de vecteurs sobolev et la proprit dtre une solution lagrangienne , cest - - dire une solution transporte par le flot de lquation diffrentielle ordinaire associe au champ de vecteurs . on travaille dans un cadre o les solutions considres manquent dintgrabilit locale et o on ne peut pas appliquer la thorie classique de diperna - lions dunicit des solutions distributionnelles et de la proprit dtre lagrangienne parce que on na pas assez dintgrabilit pour le commutateur . on introduit un principe gnral pour dmontrer la proprit dtre une solution lagrangienne : notre technique se base sur une desintgration le long le flot unique et sur un lemme dextension lipschitzienne directionnelle qui nous permet de construire une vaste famille de fonction test pour la formulation distributionnelle lagrangienne de lquation de continuit . , received * * * * * ; accepted after revision + + + + + + presented by

in this paper , we continue our general study of the _ uniform thickness property _ ( utp ) in the context of iterated torus knots that are embedded in @xmath0 with the standard tight contact structure . as stated in a previous paper , _ studying uniform thickness i _ @xcite , our goal in this study is to determine the extent to which iterated torus knot types fail to satisfy the utp , and the extent to which this failure leads to cablings that are legendrian or transversally non - simple . motivation for this study is due to the work of etnyre and honda @xcite , who showed that the failure of the utp is a necessary condition for transversal non - simplicity in the class of iterated torus knots . they also established that the @xmath1 torus knot fails the utp and supports a transversally non - simple cabling . in @xcite we extended this study of the utp by establishing new necessary conditions for both the failure of the utp and transversal non - simplicity in the class of iterated torus knots ; in so doing we obtained new families of legendrian simple iterated torus knots . the specific goal of this note is to fully answer the first motivating question of our study by providing a complete utp classification of iterated torus knots , that is , determining which iterated torus knot types satisfy the utp , and which fail the utp . we will also address the second motivating question of our study by proving that failure of the utp for an iterated torus knot type is a sufficient condition for the existence of transversally non - simple cablings of that knot . specifically , we have the following two theorems and corollary : [ main theorem ] let @xmath2 be an iterated torus knot , where the @xmath3 s are measured in the standard preferred framing , and @xmath4 for all @xmath5 . then @xmath6 fails the utp if and only if @xmath7 for all @xmath5 , where @xmath8 . in the second theorem , @xmath9 is the euler characteristic of a minimal genus seifert surface for a knot @xmath10 : [ second theorem ] if @xmath6 is an iterated torus knot that fails the utp , then it supports infinitely many transversally non - simple cablings @xmath11 of the form @xmath12 , where @xmath13 ranges over an infinite subset of positive integers . to state our corollary to theorem [ main theorem ] , recall that if @xmath10 is a fibered knot , then there is an associated open book decomposition of @xmath0 that supports a contact structure , denoted @xmath14 ( see @xcite ) . iterated torus knots are fibered knots , and hedden has shown that the subclass of iterated torus knots where each iteration is a positive cabling , i.e. @xmath7 for all @xmath5 , is precisely the subclass of iterated torus knots where @xmath15 is isotopic to @xmath16 @xcite . we thus obtain the following corollary : an iterated torus knot @xmath6 fails the utp if and only if @xmath17 . we make a few remarks about these theorems . first , it will be shown that these transversally non - simple cablings will all have two legendrian isotopy classes at the same rotation number and maximal thurston - bennequin number @xmath18 , and thus they will exhibit legendrian non - simplicity at @xmath18 . second , in the class of iterated torus knots there are certainly more transversally non - simple cablings than those in theorem [ second theorem ] , as evidenced by etnyre and honda s example of the transversally non - simple @xmath1-cabling of a @xmath1-torus knot . however , we present just the class of transversally non - simple cablings in theorem [ second theorem ] , and leave a more complete legendrian and transversal classification of iterated torus knots as an open question . we now present a conjectural generalization of the above two theorems and corollary . to this end , recall that hedden has shown that for general fibered knots @xmath10 in @xmath0 , @xmath19 precisely when @xmath10 is a fibered strongly quasipositive knot @xcite ; he also shows that for these knots , the maximal self - linking number is @xmath20 @xcite . furthermore , from the work of etnyre and van horn - morris @xcite , we know that for fibered knots @xmath10 in @xmath0 that support the standard contact structure there is a unique transversal isotopy class at @xmath21 . in the present paper , all of these ideas are brought to bear on the class of iterated torus knots , and this motivates the following conjecture concerning general fibered knots : let @xmath10 be a fibered knot in @xmath0 ; then @xmath10 fails the utp if and only if @xmath19 , and hence if and only if @xmath10 is fibered strongly quasipositive . moreover , if a topologically non - trivial fibered knot @xmath10 fails the utp , then it supports cablings that are transversally non - simple . we also ask the following question of non - fibered knots in @xmath0 : if @xmath10 is a non - fibered strongly quasipositive knot , does @xmath10 fail the utp and support transversally non - simple cablings ? we will be using tools developed by giroux , kanda , and honda , and used by etnyre and honda in their work , namely convex tori and annuli , the classification of tight contact structures on solid tori and thickened tori , and the legendrian classificaton of torus knots . most of the results we use can be found in @xcite , @xcite , @xcite , or @xcite , and if we use a lemma , proposition , or theorem from one of these works , it will be specifically referenced . the plan of the note is as follows . in 2 we recall definitions , notation , and identities used in @xcite and @xcite . in 3 we outline a strategy of proof of theorem [ main theorem ] that yields the statement of two key lemmas . in 4 and 5 we prove the first lemma . in 6 we prove the second lemma and complete the proof of theorem [ main theorem ] . in 7 we prove theorem [ second theorem ] . _ iterated torus knots _ , as topological knot types , can be defined recursively . let 1-iterated torus knots be simply torus knots @xmath22 with @xmath23 and @xmath24 co - prime nonzero integers , and @xmath25 . here @xmath23 is the algebraic intersection with a longitude , and @xmath24 is the algebraic intersection with a meridian in the preferred framing for a torus representing the unknot . then for each @xmath22 torus knot , take a solid torus regular neighborhood @xmath26 ; the boundary of this is a torus , and given a framing we can describe simple closed curves on that torus as co - prime pairs @xmath27 , with @xmath28 . in this way we obtain all 2-iterated torus knots , which we represent as ordered pairs , @xmath29 . recursively , suppose the @xmath30-iterated torus knots are defined ; we can then take regular neighborhoods of all of these , choose a framing , and form the @xmath31-iterated torus knots as ordered @xmath31-tuples @xmath32 , again with @xmath33 and @xmath34 co - prime , and @xmath35 . for ease of notation , if we are looking at a general @xmath31-iterated torus knot type , we will refer to it as @xmath6 ; a legendrian representative will usually be written as @xmath36 . note that we will use the letter @xmath31 both for the rotation number ( see below ) and as an index for our iterated torus knots ; context will distinguish between the two uses . we will study iterated torus knots using two framings . the first is the standard framing for a torus , where the meridian bounds a disc inside the solid torus , and we use the preferred longitude which bounds a surface in the complement of the solid torus . we will refer to this framing as @xmath37 . the second framing is a non - standard framing using a different longitude that comes from the cabling torus . more precisely , to identify this non - standard longitude on @xmath38 , we first look at @xmath6 as it is embedded in @xmath39 . we take a small neighborhood @xmath40 such that @xmath38 intersects @xmath39 in two parallel simple closed curves . these curves are longitudes on @xmath38 in this second framing , which we will refer to as @xmath41 . note that this @xmath41 framing is well - defined for any cabled knot type . moreover , for purpose of calculations there is an easy way to change between the two framings , which will be reviewed below . given a simple closed curve @xmath42 on a torus , measured in some framing as having @xmath43 meridians and @xmath44 longitudes , we will say this curve has slope of @xmath45 ; i.e. , longitudes over meridians . therefore we will refer to the longitude in the @xmath41 framing as @xmath46 , and the longitude in the @xmath37 framing as @xmath47 . the meridian in both framings will have slope @xmath48 . we will also use a convention that meridians in the standard @xmath37 framing , that is , algebraic intersection with @xmath47 , will be denoted by upper - case @xmath49 s . on the other hand , meridians in the non - standard @xmath50 framing , that is , algebraic intersection with @xmath51 , will be denoted by lower - case @xmath52 s . given a curve @xmath53 on a torus @xmath54 , there is then a relationship between the framings @xmath50 and @xmath37 on @xmath54 . in terms of a change of basis , we get from @xmath50 to @xmath37 by multiplying on the left by the matrix @xmath55 . given an iterated torus knot type @xmath56 where the @xmath57 s are measured in the @xmath50 framing , we define two quantities . the two quantities are : @xmath58 note here we use a convention that @xmath59 . also , if we restrict to the first @xmath5 iterations , that is , to @xmath60 , we have an associated @xmath61 and @xmath62 . for example , @xmath63 . finally , from @xcite we obtain four useful identities which we will apply extensively throughout this note : @xmath64 recall that for legendrian knots embedded in @xmath0 with the standard tight contact structure , there are two classical invariants of legendrian isotopy classes , namely the thurston - bennequin number , @xmath65 , and the rotation number , @xmath31 . for a given topological knot type , if the ordered pair @xmath66 completely determines the legendrian isotopy classes , then that knot type is said to be _ legendrian simple_. for transversal knots there is one classical invariant , the self - linking number @xmath67 ; for a given topological knot type , if the value of @xmath67 completely determines the transversal isotopy classes , then that knot type is said to be _ transversally simple_. for a given topological knot type , if we plot legendrian isotopy classes at points @xmath66 , we obtain a plot of points that takes the form of a _ legendrian mountain range _ for that knot type . we will be examining legendrian knots which are embedded in convex tori . recall that the characteristic foliation induced by the contact structure on a convex torus can be assumed to have a standard form , where there are @xmath68 parallel _ legendrian divides _ and a one - parameter family of _ legendrian rulings_. parallel push - offs of the legendrian divides gives a family of @xmath68 _ dividing curves _ , referred to as @xmath69 . for a particular convex torus , the slope of components of @xmath69 is fixed and is called the _ boundary slope _ of any solid torus which it bounds ; however , the legendrian rulings can take on any slope other than that of the dividing curves by giroux s flexibility theorem @xcite . a _ standard neighborhood _ of a legendrian knot @xmath70 will have two dividing curves and a boundary slope of @xmath71 . we can now state the definition of the _ uniform thickness property _ as given by etnyre and honda @xcite . for a knot type @xmath10 , define the _ contact width _ of @xmath10 to be @xmath72 in this equation the @xmath73 are solid tori having representatives of @xmath10 as their cores ; slopes are measured using the preferred framing where the longitude has slope @xmath47 ; the supremum is taken over all solid tori @xmath73 representing @xmath10 where @xmath74 is convex . a knot type @xmath10 then satisfies the utp if the following hold : * @xmath75 , where @xmath18 is the maximal thurston - bennequin number for @xmath10 . * every solid torus @xmath73 representing @xmath10 can be thickened to a standard neighborhood of a maximal @xmath65 legendrian knot . for a topological knot type @xmath10 , if @xmath73 is a solid torus having a representative of @xmath10 as its core and convex boundary , then @xmath73 _ fails to thicken _ if for all @xmath76 , we have @xmath77 . if we define @xmath78 to be the twisting of the contact planes along @xmath70 with respect to the @xmath50 framing on @xmath79 , equation 2.1 in @xcite gives us : @xmath80 observe that @xmath81 is also the twisting of the contact planes with respect to the framing given by @xmath74 , and so is equal to @xmath82 times the geometric intersection number of @xmath70 with @xmath83 . @xmath84 will denote the maximal twisting number with respect to this framing . we also had two definitions introduced in @xcite that will be useful in this note . _ let @xmath73 be a solid torus with convex boundary in standard form , and with @xmath85 in some framing . if @xmath86 is the geometric intersection of the dividing set @xmath69 with a longitude ruling in that framing , then we will call @xmath87 the _ intersection boundary slope__. note that when we have an intersection boundary slope @xmath87 , then @xmath88 is the number of dividing curves . _ for @xmath89 and positive integer @xmath13 , define @xmath90 to be any solid torus representing @xmath6 with intersection boundary slope of @xmath91 , as measured in the @xmath50 framing . also define the integer @xmath92 . _ note that @xmath90 has @xmath93 dividing curves . note also that the above definition is only for @xmath94 . however , we will also define @xmath95 to be a standard neighborhood of a @xmath96 representative , and thus have this as the @xmath97 case . finally , recall that if @xmath98 is a convex annulus with legendrian boundary components , then dividing curves are arcs with endpoints on either one or both of the boundary components . dividing curves that are boundary parallel are called _ bypasses _ ; an annulus with no bypasses is said to be _ standard convex_. recall that a contact structure @xmath99 on a 3-manifold @xmath100 is said to be _ overtwisted _ if there exists an embedded disc @xmath101 which is tangent to @xmath99 everywhere along @xmath102 , and a contact structure is _ tight _ if it is not overtwisted . moreover , one can further analyze tight contact 3-manifolds @xmath103 by looking at what happens to @xmath99 when pulled back to the universal cover @xmath104 via the covering map @xmath105 . in particular , if the pullback of @xmath99 remains tight , then @xmath103 is said to be _ universally tight_. the classification of universally tight contact structures on solid tori is known from the work of honda . specifically , from proposition 5.1 in @xcite , we know there are exactly two universally tight contact structures on @xmath106 with boundary torus having two dividing curves and slope @xmath107 in some framing . these are such that a convex meridional disc has boundary - parallel dividing curves that separate half - discs all of the same sign , and thus the two contact structures differ by @xmath108 . ( if @xmath109 , there is only one tight contact structure , and it is universally tight . ) also from the work of honda , we know that if @xmath99 is a contact structure which is everywhere transverse to the fibers of a circle bundle @xmath100 over a closed oriented surface @xmath110 , then @xmath99 is universally tight . this is the content of lemma 3.9 in @xcite , and such a transverse contact structure is said to be _ horizontal_. given a legendrian knot @xmath70 , recall that there are well - defined _ positive and negative transverse push - offs _ , denoted by @xmath111 and @xmath112 , respectively . moreover , the self - linking numbers of these transverse push - offs are given by the formula @xmath113 in this section we present a strategy of proof for theorem [ main theorem ] . we begin with a theorem that in previous works has in effect been proved , but not stated . in this theorem @xmath10 is a knot type and @xmath114 is the @xmath53-cabling of @xmath10 . [ utp preserved ] if @xmath10 satisfies the utp , then @xmath114 also satisfies the utp . the case where the cabling fraction @xmath115 is the content of theorem 1.3 in @xcite . for the case where @xmath116 , the proof follows from examining the proofs of theorem 3.2 @xcite and theorem 1.1 in @xcite and observing that legendrian simplicity of @xmath10 is not needed to preserve the utp . with this theorem in mind , we will prove theorem [ main theorem ] by way of two lemmas , one of which uses induction . for this purpose we make the following inductive hypothesis , which from here on we will refer to as _ the inductive hypothesis_. we will need to justify its veracity for the base case of positive torus knots . + * inductive hypothesis : * let @xmath117 be an iterated torus knot , as measured in the standard @xmath37 framing . assume that the following hold : * @xmath7 for all @xmath5 , where @xmath8 . ( thus @xmath118 for all @xmath5 as well . ) * @xmath119 . ( thus @xmath120 . ) * any solid torus @xmath121 representing @xmath6 thickens to some @xmath90 ( including @xmath95 which is a standard neighborhood of a @xmath18 representative ) . * if @xmath121 fails to thicken then it is an @xmath90 , and it has at least @xmath93 dividing curves . * the candidate non - thickenable @xmath90 exist and actually fail to thicken for @xmath122 , where @xmath123 is some positive integer that varies according to the knot type @xmath6 . moreover , these @xmath90 that fail to thicken have contact structures that are universally tight , with convex meridian discs containing bypasses all of the same sign . also , a legendrian ruling preferred longitude on these @xmath124 has rotation number zero for @xmath125 . + our first key lemma used in proving theorem [ main theorem ] is the following , which along with the base case of positive torus knots , will show that if @xmath126 is such that @xmath7 for all @xmath5 , then @xmath6 fails the utp . [ indhyppreserved ] suppose @xmath6 satisfies the inductive hypothesis , and @xmath11 is a cabling where @xmath127 ; then @xmath11 satisfies the inductive hypothesis . the main idea in the argument used to prove this lemma will be that since @xmath6 satisfies the inductive hypothesis , there is an infinite collection of non - thickenable solid tori whose boundary slopes form an increasing sequence converging to @xmath128 in the @xmath50 framing ( which is @xmath47 in the @xmath37 framing ) . as a consequence , it will be shown that cabling slopes with @xmath127 in the @xmath37 framing will have a similar sequence of non - thickenable solid tori . our second key lemma is the following , which along with theorem [ utp preserved ] and the fact that negative torus knots satisfy the utp , will show that if at least one of the @xmath129 , then @xmath6 satisfies the utp . [ negativeutp ] suppose @xmath6 satisfies the inductive hypothesis , and @xmath11 is a cabling where @xmath130 ; then @xmath11 satisfies the utp . our outline for the next three sections is as follows . in the next section , 4 , we establish the truth of the inductive hypothesis for the base case of positive torus knots . in 5 we prove lemma [ indhyppreserved ] , and in 6 we prove lemma [ negativeutp ] . in this section we show that positive torus knots @xmath22 satisfy the inductive hypothesis described in 3 . from lemma 4.3 in @xcite , we know that items 1 - 4 of the inductive hypothesis are satisfied ; it remains to establish item 5 , that each solid torus candidate @xmath131 actually exists with a universally tight contact structure and the appropriate complement in @xmath0 , and that these @xmath131 indeed fail to thicken ( for all @xmath132 in this case of positive torus knots ) . to establish item 5 , we employ arguments similar to those used in @xcite for solid tori representing the @xmath1 torus knot , specifically lemmas 5.2 and 5.3 in @xcite . from lemma 4.3 in @xcite , we know that if @xmath131 fails to thicken , its complement @xmath133 must be contactomorphic to the manifold obtained by taking a neighborhood of a hopf link @xmath134 and a standard convex annulus @xmath98 joining the two neighborhoods of the hopf link , where @xmath98 has boundary components that are legendrian ruling representatives of @xmath135 . moreover , we know that the two components of the hopf link must have @xmath65 values equal to @xmath136 and @xmath137 , respectively , for @xmath132 . we first show that the candidate @xmath131 have universally tight contact structures . if @xmath131 fails to thicken , then its contact structure is universally tight ; moreover , for @xmath138 , a convex meridian disc contains bypasses that all bound half - discs of the same sign . also , a legendrian ruling preferred longitude on @xmath139 has rotation number zero for @xmath125 . the lemma is immediately true for @xmath97 , so we may assume that @xmath138 . to fix notation , let @xmath140 be the legendrian unknot with @xmath141 and let @xmath142 be the unknot with @xmath143 . then @xmath144 thickens outward to @xmath145 ; we denote @xmath146 and @xmath147 . since @xmath148 and @xmath149 are convex , we can take @xmath150$]-invariant neighborhoods of each ; our convention will be that the two @xmath151 will bound a thickened torus that contains the two @xmath152 . now @xmath153 is a convex torus with dividing curves that divide the torus into two annuli , @xmath154 and @xmath155 . we locate a ( topological ) meridian curve @xmath43 on @xmath153 that intersects each dividing curve efficiently @xmath156 times , and so that @xmath157 consists of @xmath24 arcs which intersect @xmath154 and @xmath155 at least @xmath13 times each . we then can realize @xmath43 as a legendrian ruling using theorem 3.7 in @xcite . we then examine a horizontal convex annulus @xmath158 in the space @xmath159 , bounded by meridian rulings on @xmath151 . this horizontal convex annulus @xmath158 has two dividing curves that connect its two boundary components ; the other @xmath160 bypasses have endpoints on @xmath153 . by lemma 4.14 in @xcite , we may assume that all of these bypasses are boundary compressible , meaning there are no nested bypasses . the two dividing curves connecting the two boundary components of @xmath158 thus divide @xmath158 into two discs , one containing all bypasses of positive sign , the other disc containing all negative bypasses . we will show that in fact all bypasses on @xmath158 must be of the same sign . to this end , let @xmath98 be the standard convex annulus with @xmath22 legendrian rulings as its boundary components on @xmath152 . we first examine @xmath161 , which is @xmath24 arcs with endpoints on the two @xmath162 . at first glance , it is possible that there may be points of intersection between these @xmath24 arcs and the boundary - parallel dividing curves on @xmath158 . however , up to a choice of contact vector field for the convex annulus @xmath158 , we may assume that all boundary - parallel dividing curves for @xmath158 are in a collar neighborhood of @xmath153 and avoid @xmath98 . this contact vector field may also be chosen so that the two non - separating dividing curves on @xmath158 intersect @xmath98 transversely . now @xmath163 is one of the annuli that forms @xmath139 , and the intersection of this annulus with @xmath158 will be @xmath24 arcs , which we denote as @xmath164 for @xmath165 . by the above considerations we thus have that , as a collection , the @xmath164 have support that intersects all of the @xmath160 bypasses on @xmath158 . see figure [ fig : universallytightsetupbb ] . . the thick gray arcs represent the intersection of @xmath98 with @xmath158 . some of the @xmath160 bypasses are shown ; in the figure , @xmath166.,scaledwidth=55.0% ] we next perform edge - rounding for the curves of intersection of @xmath167 and the two annuli coming from @xmath168 ; after edge - rounding we obtain @xmath139 . thus @xmath139 intersects @xmath158 in @xmath24 ( topological ) meridian curves for @xmath131 ; this set of curves , call it @xmath169 , is _ nonisolating _ in the sense of section 3.3.1 in @xcite , meaning that each curve is transverse to the dividing set of @xmath158 and every component of @xmath170 has boundary that intersects @xmath69 . moreover , @xmath169 is also a nonisolating set of curves on @xmath139 . then by theorem 3.7 in @xcite , we can realize the @xmath24 topological meridian curves as legendrian meridian curves for @xmath139 . now these legendrian meridian curves may not have efficient geometric intersection with @xmath171 . however , by the construction of @xmath139 , any holonomy of dividing curves on the two annuli coming from the two sides of @xmath172 cancels each other out . thus we can destabilize these legendrian meridian curves on the surface @xmath139 so that they do have geometric intersection @xmath173 with @xmath171 , and we can do so _ away from the @xmath164_. these destabilizations can thus be accomplished by attachment of bypasses off of the @xmath24 convex meridian discs , but these ( attached ) bypasses will avoid the original @xmath160 bypasses along the @xmath164 . the resulting @xmath24 convex meridian discs therefore inherit the bypasses of @xmath158 . by construction , it is possible that one of the @xmath24 convex meridian discs may inherit @xmath174 bypasses from @xmath158 ; if this is the case , however , these bypasses must all be of the same sign , and we have the desired conclusion . so we may assume that each of the @xmath24 meridian discs intersects @xmath13 bypasses of @xmath158 . so suppose , for contradiction , that the @xmath160 bypasses on @xmath158 have mixed sign , meaning some are negative and some are positive . since each of the @xmath24 meridian discs is a convex meridian disc for @xmath131 , then by the classification of tight contact structures on solid tori we know that if one of the discs has a negative bypass , then all of them must ; the same is true for positive bypasses . but since the negative bypasses on @xmath158 are grouped in succession , and since we may assume @xmath175 , this forces one of the discs to inherit only negative bypasses , contradicting the fact that it is supposed to also have positive bypasses . thus all of the bypasses on @xmath158 must be of the same sign , as must be all of the bypasses on a convex meridian disc for @xmath131 . as a result the contact structure for @xmath131 is universally tight . we can now calculate the rotation number for the @xmath22 ruling on @xmath144 . since @xmath159 is universally tight , one can show that if @xmath176 is a convex seifert surface for the longitude on @xmath177 , we must have @xmath178 . by lemma 2.2 in @xcite , we have that the @xmath22 ruling on @xmath177 has rotation number equal to @xmath179 this yields @xmath180 . we now let @xmath101 be a meridian disc for @xmath131 and @xmath110 be a seifert surface for the preferred longitude on @xmath139 . we know @xmath181 , and we know that the @xmath22 torus knot , which is @xmath51 on @xmath139 , is actually a @xmath182 knot on @xmath139 in the preferred framing . so using a similar equation from above , we obtain that @xmath183 . thus @xmath184 . we note that there are two universally tight contact structures , diffeomorphic by @xmath108 , which satisfy the conditions set by the above lemma . we now show that these appropriate @xmath131 and associated @xmath185 actually exist in @xmath0 . the standard tight contact structure on @xmath0 splits into a universally tight contact structure on @xmath131 and @xmath185 . the idea is to build @xmath0 . to begin , choose one of the above two universally tight candidates for @xmath131 . we then claim we can join @xmath131 to itself by a standard convex annulus @xmath186 with boundary @xmath51 rulings so that @xmath187 is a ( universally tight ) thickened torus with boundary @xmath188 having associated boundary slopes of @xmath189 and @xmath190 and two dividing curves . one way to see this is that we can think of @xmath139 as being composed of four annuli , one from @xmath191 , one from @xmath192 , and two from @xmath193 $ ] . since we are constructing the thickened torus , with a suitable choice of holonomy of @xmath186 , we can assure that the dividing curves on @xmath194 have only one longitude , and two components . since we know the twisting of @xmath51 on @xmath131 is equal to @xmath195 , a calculation shows that the dividing curves on @xmath194 must have slope @xmath190 . but then the slopes of the dividing curves on @xmath194 and @xmath124 are determined , making the slope of dividing curves on @xmath149 equal to @xmath189 based on equation 8 in lemma 4.3 in @xcite . now as in the proof of lemma 5.2 in @xcite , the contact structure on @xmath196 can be isotoped to be transverse to the fibers of @xmath196 , which are parallel copies of @xmath197 , while preserving the dividing set on @xmath198 . such a horizontal contact structure is universally tight . we then use the classification of tight contact structures on @xmath0 , solid tori , and thickened tori to conclude that any tight contact structure on @xmath199 $ ] with boundary conditions being tori with two dividing curves and slopes @xmath189 and @xmath190 glues together with standard neighborhoods of unknots with those boundary slopes to give the tight contact structure on @xmath0 . we now show that these @xmath131 with complements @xmath185 fail to thicken . the @xmath131 with complement @xmath185 fail to thicken . by inequality 14 in @xcite , it suffices to show that @xmath131 does not thicken to any @xmath200 for @xmath201 . so to this end , observe that the @xmath22 positive torus knot is a fibered knot over @xmath202 with fiber a seifert surface @xmath110 of genus @xmath203 ( see @xcite ) . moreover , the monodromy is periodic with period @xmath204 . thus , @xmath185 has a @xmath204-fold cover @xmath205 . if one thinks of @xmath185 as @xmath206 $ ] modulo the relation @xmath207 for monodromy @xmath208 , then one can view @xmath209 as @xmath204 copies of @xmath206 $ ] cyclically identified via the same monodromy . now note that downstairs in @xmath185 , @xmath51 intersects any given seifert surface @xmath204 times efficiently . it is therefore evident that we can view @xmath185 as a seifert fibered space with base space @xmath110 and two singular fibers ( the components of the hopf link ) . the regular fibers are topological copies of @xmath51 , which itself is a legendrian ruling on @xmath139 with twisting @xmath210 . in fact , the regular fibers can be assumed to be legendrian isotopic to the @xmath139-fibers except for small neighborhoods around the singular fibers . we claim the pullback of the tight contact structure to @xmath209 admits an isotopy where the @xmath202 fibers are all legendrian and have twisting number @xmath210 with respect to the product framing . this isotopy can be accomplished because in @xmath209 , the lifts of the singular fibers have tight neighborhoods with convex boundary tori which have dividing curves with one longitude and where @xmath51 has twisting @xmath210 . thus these neighborhoods of the lifts of the singular fibers are in fact standard neighborhoods of a legendrian fiber with twisting @xmath210 ; the contact structure can then be isotoped so that every fiber inside these neighborhoods is legendrian with twisting @xmath210 . so , if @xmath131 can be thickened to @xmath200 , then there exists a legendrian curve topologically isotopic to the regular fiber of the seifert fibered space @xmath185 with twisting number greater than @xmath210 , measured with respect to the seifert fibration . pulling back to the @xmath204-fold cover @xmath209 , we have a legendrian knot which is topologically isotopic to a fiber but has twisting greater than @xmath210 . we will obtain a contradiction , thus proving that @xmath131 can not be thickened to @xmath200 . to obtain our contradiction , we let @xmath211 be the projection map onto the base space . thus the hypothesis that @xmath131 can be thickened to @xmath200 yields a knot @xmath212 which is isotopic to @xmath213 for some @xmath214 , but where @xmath215 . thus there is a continuous isotopy @xmath216 where @xmath217 and @xmath218 . now look at @xmath219 . then this is a continous map , and since @xmath220 , we actually obtain @xmath221 . this means that @xmath222 is contained inside a tight @xmath106 that is fibered by legendrian fibers with twisting @xmath210 , and is thus a solid torus neighborhood of a legendrian knot with twisting @xmath210 . by the classification of tight contact structures on solid tori , such a @xmath222 can not exist . this is our contradiction . now that we know that the base case holds for positive torus knots , we begin to prove lemma [ indhyppreserved ] for the bulk of this section we will thus have that @xmath127 , @xmath6 satisfies the inductive hypothesis , and we work to show that @xmath11 satisfies the inductive hypothesis . we will need to break the proof of lemma 3.2 into two cases , case i being where @xmath223 , and case ii being where @xmath224 . however , before we do that , we prove two general lemmas concerning iterated cablings that begin with positive torus knots . if @xmath6 is an iterated torus knot with @xmath225 , then @xmath226 . we use induction . @xmath227 is evident from equation 1 above . then inductively , @xmath228 . we now use the above lemma to prove the following . let @xmath6 be an iterated torus knot with @xmath225 . if @xmath229 and both @xmath230 , then @xmath231 . we have that @xmath231 if and only if @xmath232 . but this is true if and only if @xmath233 , which is true . we now directly address the two different cases in two different subsections . + + we work through proving items 2 - 5 in the inductive hypothesis via a series of lemmas . the following lemma begins to address item 2 . if @xmath223 , then @xmath234 . the proof is similar to that of lemma 3.3 in @xcite . we first claim that @xmath235 . if not , there exists a legendrian @xmath236 with @xmath237 and a solid torus @xmath121 with @xmath236 as a legendrian divide . but then we would have a boundary slope of @xmath223 in the @xmath37 framing , which can not occur . so since @xmath235 , any legendrian @xmath236 must be a ruling on a convex @xmath238 with slope @xmath239 in the @xmath50 framing . but then if @xmath240 , we have that @xmath241 . this shows that @xmath242 is achieved by a legendrian ruling on a convex torus having slope @xmath243 in the standard @xmath37 framing . finally , note that @xmath244 . with the following lemma we prove that items 3 and 4 of the inductive hypothesis hold for @xmath11 . [ nonthickening inductive step ] if @xmath223 , let @xmath245 be a solid torus representing @xmath11 , for @xmath89 . then @xmath245 can be thickened to an @xmath246 for some nonnegative integer @xmath247 . moreover , if @xmath245 fails to thicken , then it has the same boundary slope as some @xmath246 , as well as at least @xmath248 dividing curves . in this case , for the @xmath50 framing , we have either @xmath249 or @xmath250 ( the latter being relevant only if @xmath251 ) . the proof in this case is nearly identical to the proof of lemma 4.4 in @xcite ; we will include some of the details , however , as certain particular calculations differ . moreover , we will use modifications of this argument in case ii and thus will be able to refer to the details here . let @xmath245 be a solid torus representing @xmath11 . let @xmath36 be a legendrian representative of @xmath6 in @xmath252 and such that we can join @xmath253 to @xmath254 by a convex annulus @xmath255 whose boundaries are @xmath256 and @xmath51 rulings on @xmath253 and @xmath254 , respectively . then topologically isotop @xmath257 in the complement of @xmath245 so that it maximizes @xmath65 over all such isotopies ; this will induce an ambient topological isotopy of @xmath255 , where we still can assume @xmath255 is convex . a picture is shown in ( a ) in figure [ fig : non - thickening1b ] . in the @xmath50 framing we will have @xmath258 where @xmath259 , since @xmath260 . now if @xmath261 , then there will be no bypasses on the @xmath253-edge of @xmath255 , since the @xmath256 ruling would be at maximal twisting . on the other hand , if @xmath262 , then there will still be no bypasses on the @xmath253-edge of @xmath255 , since such a bypass would induce a destabilization of @xmath257 , thus increasing its @xmath65 by one see lemma 4.4 in @xcite . to satisfy the conditions of this lemma , we are using the fact that either @xmath249 or @xmath250 . furthermore , we can thicken @xmath245 through any bypasses on the @xmath254-edge , and thus assume @xmath255 is standard convex . now let @xmath263 . inductively we can thicken @xmath121 to an @xmath90 with intersection boundary slope @xmath91 where @xmath13 is minimized over all such thickenings ( if we have @xmath97 , then we will have @xmath245 thickening to a standard neighborhood of a knot at @xmath18 see the proof of theorem 1.1 in @xcite ; so we can assume @xmath138 ) . then consider a convex annulus @xmath264 from @xmath253 to @xmath265 , such that @xmath266 is in the complement of @xmath267 and @xmath268 consists of @xmath256 rulings . a picture is shown in ( b ) in figure [ fig : non - thickening1b ] . by an argument identical to that used in lemma 4.4 in @xcite , @xmath266 is standard convex ; in brief , if @xmath266 was not standard convex , either a bypass would occur on its @xmath253-edge , or @xmath13 would not be minimized , neither of which is true . [ nonthickening1 ] is the larger solid torus in gray ; @xmath269 is the smaller solid torus in gray.,title="fig : " ] now four annuli compose the boundary of a solid torus @xmath270 containing @xmath245 : the two sides of a thickened @xmath266 ; @xmath271 ; and @xmath272 . we can compute the intersection boundary slope of this solid torus . to this end , recall that @xmath258 where @xmath273 ( @xmath274 would be the @xmath84 case which we have take care of above ) . to determine @xmath275 we note that the geometric intersection of @xmath256 with @xmath69 on @xmath265 and @xmath253 must be equal , yielding the equality @xmath276 these equal quantities are greater than zero , since @xmath277 we note here that this will yield @xmath278 for the calculations below . in the meantime , however , the above equation gives @xmath279 we define the integer @xmath280 . we now choose @xmath281 to be a curve on these two tori such that @xmath282 , and we change coordinates to a framing @xmath283 via the map @xmath284 . under this map we obtain @xmath285 @xmath286 we then obtain in the @xmath50 framing , after edge - rounding , that the intersection boundary slope of @xmath270 is @xmath287 this shows that any @xmath245 representing @xmath11 can be thickened to one of the @xmath246 , and if @xmath245 fails to thicken , then it has the same boundary slope as some @xmath246 . we now note that if @xmath245 fails to thicken , and if it has the minimum number of dividing curves over all such @xmath245 which fail to thicken and have the same boundary slope as @xmath246 , then @xmath245 is actually an @xmath246 , by an argument identical to that used in lemma 4.4 in @xcite . in brief , if @xmath245 fails to thicken and is at minimum number of dividing curves , then taking @xmath288 gives an @xmath90 ; one then concludes that @xmath245 is an @xmath246 . we now finish the proof of item 2 of the inductive hypothesis . if @xmath223 , then @xmath289 . we show that @xmath290 for any candidate @xmath246 . as a consequence , since any @xmath245 thickens to some @xmath246 ( including @xmath291 ) , we have , to prevent overtwisting , that @xmath289 . now note that our intended inequality is automatically true if @xmath292 ; thus we may assume that @xmath235 . we have that @xmath290 holds if and only if @xmath293 inductively we know that @xmath294 where @xmath295 . this implies that @xmath296 we can now prove inequality 11 ; we begin with @xmath297 . we have : @xmath298 we conclude this subsection by proving item 5 of the inductive hypothesis . if @xmath223 , the candidate @xmath246 exist and actually fail to thicken for @xmath299 , where @xmath300 is some positive integer . moreover , these @xmath246 have contact structures that are universally tight and have convex meridian discs whose bypasses bound half - discs all of the same sign . also , the preferred longitude on @xmath301 has rotation number zero for @xmath302 . we first prove that the contact structure on a candidate @xmath246 which fails to thicken is universally tight . to see this note that from lemma 5.2 above , and the inductive hypothesis , such a candidate @xmath246 is embedded inside a @xmath90 with a universally tight contact structure . now there is a @xmath303-fold cover of @xmath90 that maps @xmath303 lifts @xmath304 to @xmath246 , the lifts themselves each being an @xmath106 . this cover in turn has a universal cover @xmath305 that contains @xmath303 copies of a universal cover @xmath305 of @xmath246 . since , by the inductive hypothesis , the universal cover of @xmath90 has a tight contact structure , a tight contact structure is thus induced on the universal cover of @xmath246 . to see that a meridian disc for @xmath246 contains bypasses all of the same sign , note that this is immediate if @xmath301 has two dividing curves . for the case of @xmath68 dividing curves where @xmath306 , we argue in a similar fashion to lemma 4.1 . specifically , since a meridian disc for @xmath90 inductively has bypasses all of the same sign , a horizontal annulus @xmath158 with boundary on @xmath124 and @xmath307 will have @xmath308 bypasses all of the same sign . thus , as in lemma 4.1 , a meridian disc for @xmath246 will inherit @xmath309 bypasses all of the same sign . to show that the preferred longitude on @xmath301 has rotation number zero , we use an argument similar to that used in lemma 4.1 . we call the meridian disc for @xmath90 , @xmath310 , and the seifert surface for the preferred longitude on @xmath124 , @xmath311 . if we then look at the @xmath312 cable on @xmath124 , we can calculate its rotation number as @xmath313 but then since this same knot is a @xmath314 cable on @xmath301 , we have that @xmath315 @xmath316 , where @xmath110 is a seifert surface for the preferred longitude on @xmath301 . this implies that @xmath184 . now we know inductively that there exists a @xmath123 such that if @xmath317 , then the @xmath90 exist and fail to thicken . so suppose @xmath318 for some @xmath122 . we will show that @xmath246 exists and fails to thicken for @xmath319 . then @xmath300 will be the least such @xmath318 . we take one of the two universally tight candidate @xmath246 , and as in lemma 4.2 above we construct a universally tight @xmath320 and glue in an appropriate solid torus neighborhood of a legendrian knot @xmath36 to obtain a universally tight @xmath90 , which then glues into @xmath0 inductively . this shows that @xmath246 exists . to show that @xmath246 fails to thicken , by lemma 5.2 it suffices to show that @xmath246 does not thicken to any @xmath321 , where @xmath322 . inductively , we can assume @xmath323 fails to thicken ; in particular , the @xmath324 that contains @xmath246 fails to thicken . thus , if @xmath246 admits a non - trivial thickening , it must do so inside of @xmath324 . define @xmath325 ; then @xmath100 is a seifert fibered space with one singular fiber , @xmath257 , and with regular fibers that are topologically isotopic to the legendrian copies of @xmath11 on the boundary of @xmath100 . @xmath100 has a @xmath303-fold cover , @xmath104 , that is a @xmath303-punctured disc times @xmath202 , where the tight contact structure admits an isotopy so that all the @xmath202 fibers are legendrian with twisting @xmath326 with respect to the product framing . we can then glue in @xmath303 standard neighborhoods of fibers with twisting @xmath326 to obtain an @xmath106 which itself is a standard neighborhood of a knot with twisting @xmath326 . but then , if @xmath246 thickens to a @xmath321 , where @xmath322 , that means that in this cover there will be a knot isotopic to one of the fibers , but with twisting greater than @xmath326 , contradicting the classification of tight contact structures on solid tori . + as in case i , we work through proving items 2 - 5 in the inductive hypothesis via a series of lemmas . we begin by proving item 2 in the inductive hypothesis . if @xmath224 , then @xmath327 . the proof is almost identical to that of step 1 in theorem 1.5 in @xcite ; we will include the details , though , since certain key aspects differ . we first examine representatives of @xmath11 at @xmath18 . since there exists a convex torus representing @xmath6 with legendrian divides that are @xmath256 cablings ( inside of the solid torus representing @xmath6 with @xmath328 ) we know that @xmath329 . to show that @xmath330 , we show that @xmath331 by showing that the contact width @xmath332 , since this will yield @xmath333 . so suppose , for contradiction , that some @xmath245 has convex boundary with @xmath334 , as measured in the @xmath50 framing , and two dividing curves . after shrinking @xmath245 if necessary , we may assume that @xmath335 is a large positive integer . then let @xmath98 be a convex annulus from @xmath254 to itself having boundary curves with slope @xmath51 . taking a neighborhood of @xmath336 yields a thickened torus @xmath320 with boundary tori @xmath148 and @xmath149 , arranged so that @xmath148 is inside the solid torus @xmath121 representing @xmath6 bounded by @xmath149 . now there are no boundary parallel dividing curves on @xmath98 , for otherwise , we could pass through the bypass and increase @xmath335 to @xmath51 , yielding excessive twisting inside @xmath245 . hence @xmath98 is in standard form , and consists of two parallel nonseparating arcs . we now choose a new framing @xmath283 for @xmath121 where @xmath337 ; then choose @xmath338 so that @xmath339 and such that @xmath340 and @xmath341 . as mentioned in [ eh1 ] , this is possible since @xmath342 is obtained from @xmath343 by @xmath344 right - handed dehn twists . then note that in the @xmath41 framing , we have that @xmath345 , and @xmath346 and @xmath347 are connected by an arc in the farey tessellation of the hyperbolic disc ( see section 3.4.3 in [ h ] ) . thus , since @xmath348 is connected by an arc to @xmath349 in the farey tessellation , we must have that @xmath350 . thus we can thicken @xmath121 to one of the solid tori with @xmath351 which fails to thicken . then , just as in claim 4.2 in [ eh1 ] , we have ( i ) inside @xmath320 there exists a convex torus parallel to @xmath352 with slope @xmath346 ; ( ii ) @xmath320 can thus be decomposed into two layered _ basic slices _ ; ( iii ) the tight contact structure on @xmath320 must have _ mixing of sign _ in the poincar@xmath353 duals of the relative half - euler classes for the layered basic slices ; and ( iv ) this mixing of sign can not happen inside the universally tight solid torus which fails to thicken . this last statement is due to the proof of proposition 5.1 in @xcite , where it is shown that mixing of sign will imply an overtwisted disc in the universal cover of the solid torus . thus we have contradicted @xmath354 . so @xmath355 . with the following lemma we prove that items 3 and 4 of the inductive hypothesis hold for @xmath11 . [ nonthickening inductive step 2 ] if @xmath224 , let @xmath245 be a solid torus representing @xmath11 , for @xmath89 . then @xmath245 can be thickened to an @xmath246 for some nonnegative integer @xmath247 . moreover , if @xmath245 fails to thicken , then it has the same boundary slope as some @xmath246 , as well as at least @xmath248 dividing curves . this is the case where @xmath356 but @xmath357 ; we have that @xmath331 . we begin as we did in case i. if @xmath245 is a solid torus representing @xmath11 , as before choose @xmath36 in @xmath252 such that @xmath307 is joined to @xmath254 by an annulus @xmath255 , and with @xmath358 maximized over topological isotopies in the space @xmath252 . now suppose @xmath258 where @xmath359 . then inside @xmath360 is an @xmath121 with boundary slope @xmath346 . but then we can extend @xmath255 to an annulus that has no twisting on one edge , and we can thus thicken @xmath245 so it has boundary slope @xmath51 . moreover , since there is twisting inside @xmath360 , we can assure there are two dividing curves on the thickened @xmath245 . so this situation yields no nontrivial solid tori @xmath245 which fail to thicken . alternatively , suppose @xmath361 . furthermore , for the moment suppose @xmath362 . then we can use lemma 4.4 in @xcite to conclude that there are no bypasses on the @xmath307-edge of @xmath255 , and so we can thicken @xmath245 through bypasses so that @xmath255 is standard convex . then the calculation of the boundary slope goes through as above in lemma 5.4 , and we conclude that @xmath245 thickens to some @xmath246 . the @xmath90 that is used for this will have @xmath363 ; note that such @xmath90 exist since @xmath364 as @xmath13 increases . for the remaining case , suppose @xmath361 and @xmath275 is the least positive integer satisfying this inequality . thus @xmath365 . again look at the @xmath307-edge of @xmath255 . we claim that this edge has no bypasses . so , for contradiction , suppose it does . then we can thicken @xmath360 to a solid torus where the ( efficient ) geometric intersection of @xmath256 with dividing curves is less than @xmath366 . suppose the slope of this new solid torus is @xmath367 , where @xmath368 since @xmath275 is minimized in the complement of @xmath245 . we do some calculations . note first that if @xmath369 , then @xmath370 , which means @xmath371 , which implies @xmath372 , which can not happen , again since @xmath275 is minimized in the complement of @xmath245 . thus we must have @xmath373 . but then the geometric intersection of @xmath256 with @xmath374 is @xmath375 = p_{r+1 } + mq_{r+1}$ ] . this is a contradiction . thus there are no bypasses on the @xmath307-edge of @xmath255 , and we can thicken @xmath245 through any bypasses so that @xmath255 is standard convex . the calculations that show @xmath245 thickens to @xmath246 go through as above in lemma 5.4 . this shows that any @xmath245 representing @xmath11 can be thickened to one of the @xmath246 , and if @xmath245 fails to thicken , then it has the same boundary slope as some @xmath246 . we now show that if @xmath245 fails to thicken , and if it has the minimum number of dividing curves over all such @xmath245 which fail to thicken and have the same boundary slope as @xmath246 , then @xmath245 is actually an @xmath246 . to see this , as above we can choose a legendrian @xmath257 that maximizes @xmath65 in the complement of @xmath245 and such that we can join @xmath253 to @xmath254 by a convex annulus @xmath255 whose boundaries are @xmath256 and @xmath51 rulings on @xmath253 and @xmath254 , respectively . now since @xmath245 fails to thicken , we can assume that @xmath277 and that there are no bypasses on the @xmath253-edge , and in this case we have no bypasses on the @xmath254-edge since @xmath245 fails to thicken and is at minimum number of dividing curves . as above , let @xmath376 . we claim this @xmath267 fails to thicken the proof proceeds identically as above in lemma 5.4 , as does the proof that @xmath245 is in fact an @xmath246 . the following proof of item 5 of the inductive hypothesis is similar to that of case i. if @xmath224 , the candidate @xmath246 exist and actually fail to thicken for @xmath299 , where @xmath300 is some positive integer . moreover , these @xmath246 have contact structures that are universally tight and have convex meridian discs whose bypasses bound half - discs all of the same sign . also , the preferred longitude on @xmath301 has rotation number zero for @xmath302 . the proof that the contact structure on a candidate @xmath246 which fails to thicken is universally tight is identical to the argument in case i , as is the proof that their convex meridian discs have bypasses all of the same sign , as well as the proof that the rotation number of the preferred longitude is zero . now we know inductively that there exists a @xmath123 such that if @xmath122 , then the @xmath90 exist and fail to thicken . so suppose @xmath318 for some @xmath122 . also assume that @xmath363 ; we know such a @xmath13 exists since @xmath364 as @xmath13 increases . then @xmath246 exists and fails to thicken as in the argument for case i for @xmath319 , and @xmath300 will be the least such @xmath318 . we provide below the proof of lemma 3.3 , which is really just a matter of referencing a previous proof . this is the case where @xmath377 , we know @xmath6 satisfies the inductive hypothesis , and we wish to show that @xmath11 satisfies the utp . the proof is identical to that of steps 1 and 2 in the proof of theorem 1.5 from @xcite , the key being that since @xmath377 , this cabling slope is shielded from any @xmath90 that fail to thicken . we have completed the utp classification of iterated torus knots ; it now remains to show that in the class of iterated torus knots , failing the utp is a sufficient condition for supporting transversally non - simple cablings . to this end , in this section we prove theorem [ second theorem ] ; we do so by working through a series of lemmas . these lemmas will first give us information about just a piece of the legendrian mountain range for @xmath117 where @xmath7 for all @xmath5 ; we will then use this information to obtain enough information about the legendrian mountain ranges of certain cables @xmath11 to conclude that these cables are transversally non - simple . we will therefore not be completing the legendrian or transversal classification of these iterated torus knots . a formula for @xmath379 is given at the end of the proof of corollary 3 in @xcite . in the notation used in that paper , the formula is @xmath380 , since in our case all the @xmath381 as we are cabling positively at each iteration . however , note that our @xmath382 corresponds to @xmath383 in @xcite for @xmath384 . [ sliceofmtnrange ] suppose @xmath117 is an iterated torus knot where @xmath7 for all @xmath5 . then there exists legendrian representatives @xmath391 with @xmath392 and @xmath393 ; also , @xmath391 destabilizes . the lemma is true for positive torus knots @xcite , so we inductively assume it is true for @xmath389 . then look at legendrian rulings @xmath394 on standard neighborhoods of the inductive @xmath395 . in the @xmath50 framing the boundary slope of these @xmath396 is @xmath397 , and so a calculation shows that @xmath398 ; hence @xmath399 . to calculate the rotation number of @xmath394 , we use the following formula from @xcite , where @xmath101 is a convex meridian disc for @xmath396 and @xmath110 is a seifert surface for the preferred longitude on @xmath400 : this , along with lemma [ eulerchar ] , shows us that @xmath404 is on the right - most slope of the legendrian mountain range of @xmath6 , and @xmath405 is on the left - most edge . to the former we can perform positive stabilizations to reach @xmath406 at @xmath407 and @xmath408 ; to the latter we can perform negative stabilizations to reach @xmath409 at @xmath407 and @xmath410 we know such stabilizations can be performed since @xmath411 . so suppose @xmath6 is an iterated torus knot that fails the utp ( which is precisely when @xmath7 for all @xmath5 ) . then we know that for @xmath122 there exist non - thickenable solid tori @xmath90 having intersection boundary slopes of @xmath91 , where these slopes are measured in the @xmath50 framing . switching to the standard @xmath37 framing , these intersection boundary slopes are @xmath412 . now as @xmath413 , there are infinitely many values of @xmath174 which are prime and greater than @xmath414 . as a consequence , there are infinitely many @xmath90 with two dividing curves . based on this observation , we make the following definition : _ suppose @xmath117 is an iterated torus knot where @xmath7 for all @xmath5 . let @xmath415 be a cabling of @xmath6 with @xmath50 slope @xmath91 , where @xmath416 and there is an @xmath90 with two dividing curves that fails to thicken . _ so given @xmath6 , there are infinitely many such cabling knot types @xmath415 , all of these being cablings of the form @xmath417 as measured in the preferred framing . the following lemma will then prove theorem [ second theorem ] . we now look at the two universally tight non - thickenable @xmath90 that have representatives of @xmath415 as legendrian divides . these legendrian divides have @xmath420 . to calculate rotation numbers , we have two possibilities , depending on which boundary of the two universally tight @xmath90 the legendrian divides reside . using the formula from @xcite , we obtain we will call the two legendrian divides corresponding to @xmath422 , @xmath423 respectively . we can calculate the self - linking number for the negative transverse push - off of @xmath424 to be @xmath425 . this shows that @xmath424 is on the right - most edge of the legendrian mountain range and is at @xmath18 . similarly , @xmath426 is on the left - most edge of the legendrian mountain range and is at @xmath18 . we now look at solid tori @xmath427 with intersection boundary slope @xmath91 , but which _ thicken _ to solid tori with intersection boundary slopes @xmath428 . such tori @xmath429 are embedded in universally tight basic slices bounded by tori with dividing curves of slope @xmath428 and @xmath128 . legendrian divides on such @xmath427 have @xmath430 ; to calculate possible rotation numbers for these legendrian divides , we recall the procedure used in the proof of theorem 1.5 in @xcite . there we used a formula for the rotation numbers from @xcite , where the range of rotation numbers was given by the following ( substituting @xmath431 for @xmath432 ) : now from lemma [ sliceofmtnrange ] we know that there is an @xmath36 with @xmath434 and @xmath435 . plugging this value of the rotation number into the expression above yields @xmath436 . we will call the legendrian divides having these rotation numbers @xmath437 , respectively . important for our purposes is that @xmath437 have the same values of @xmath65 and @xmath31 as @xmath423 . consider first @xmath442 . it is in fact one of the dividing curves on @xmath124 , and is also at maximal self - linking number for @xmath415 . similarly , @xmath443 is one of the dividing curves on @xmath429 , and is also at maximal self - linking number . now from @xcite we know that @xmath415 is a fibered knot that supports the standard contact structure , since it is an iterated torus knot obtained by cabling positively at each iteration . as a consequence , from @xcite , we also know that @xmath415 has a unique transversal isotopy class at @xmath21 . hence we know that @xmath442 and @xmath443 are transversally isotopic . thus there is a transverse isotopy ( inducing an ambient contact isotopy ) that takes these two dividing curves on the two different tori to each other . thus we may assume that @xmath124 and @xmath429 intersect along one component of the dividing curves ; we call this component @xmath444 . now suppose , for contradiction , that @xmath440 is transversally isotopic to @xmath441 . these transverse knots are represented by the other two non - intersecting dividing curves on @xmath124 and @xmath429 , respectively , and there is a transverse isotopy taking one to the other . we claim that this transverse isotopy can be performed relative to @xmath444 . to see this , note that associated to @xmath445 is an open book decomposition of @xmath0 , with pages being seifert surfaces @xmath110 for the knot @xmath444 . moreover , the standard contact structure is supported by this open book decomposition . thus the transverse isotopy taking @xmath440 to @xmath441 will induce an ambient isotopy of open book decompositions supporting the standard contact structure , all with a transversal representative of @xmath444 on the binding . since @xmath124 is incompressible in @xmath445 , it is therefore evident that the isotopy taking @xmath440 to @xmath441 can be accomplished simply as an isotopy of @xmath124 relative to @xmath444 . thus we may assume that after a contact isotopy of @xmath0 , @xmath124 and @xmath429 intersect along their two dividing curves , which we denote as @xmath444 and @xmath446 , and we observe that there is an isotopy ( not necessarily a contact isotopy ) of @xmath90 to @xmath427 relative to @xmath444 and @xmath446 . we claim that as a result @xmath427 can not thicken , thus obtaining our contradiction . we do this by noting that the isotopy of @xmath90 to @xmath427 relative to @xmath444 and @xmath446 may be accomplished by the attachment of successive bypasses . since these bypasses are attached in the complement of the two dividing curves , none of these bypass attachments can change the boundary slope . however , they may increase or decrease the number of dividing curves . starting with @xmath447 , we make the following inductive hypothesis , which we will prove is maintained after bypass attachments : * @xmath448 is a convex torus which contains @xmath444 and @xmath446 , and thus has slope @xmath91 . * @xmath448 is a boundary - parallel torus in a @xmath150$]-invariant @xmath449 $ ] with @xmath450 , where the boundary tori have two dividing curves . * there is a contact diffeomorphism @xmath451 which takes @xmath449 $ ] to a standard @xmath452-invariant neighborhood of @xmath124 and matches up their complements . the argument that follows is similar to lemma 6.8 in @xcite . first note that item 1 is preserved after a bypass attachment , since such a bypass is in the complement of @xmath444 and @xmath446 , and thus can not change the slope of the dividing curves . to see that items 2 and 3 are preserved , suppose that @xmath453 is obtained from @xmath448 by a single bypass . since the slope was not changed , such a ( non - trivial ) bypass must either increase or decrease the number of dividing curves by 2 . suppose first that the bypass is attached from the inside , so that @xmath454 , where @xmath73 is the solid torus bounded by @xmath448 . for convenience , suppose @xmath455 inside the @xmath449 $ ] satisfying items 2 and 3 of the inductive hypothesis . then we form the new @xmath456 $ ] by taking the old @xmath456 $ ] and adjoining the thickened torus between @xmath448 and @xmath453 . now @xmath453 bounds a solid torus @xmath457 , and , by the classification of tight contact structures on solid tori , we can factor a nonrotative outer layer which is the new @xmath458 $ ] . alternatively , if @xmath459 , then we know that @xmath457 thickens to an @xmath90 , and thus there exists a nonrotative outer layer @xmath456 $ ] for @xmath460 , where @xmath148 has two dividing curves . thus the proof is done , for after enough bypass attachments we will obtain @xmath461 , with @xmath427 non - thickenable . but this is a contradiction , since @xmath427 does thicken .

we prove that an iterated torus knot type fails the uniform thickness property ( utp ) if and only if all of its iterations are positive cablings , which is precisely when an iterated torus knot type supports the standard contact structure . we also show that all iterated torus knots that fail the utp support cabling knot types that are transversally non - simple . v c u _ sl(2,c ) ps . psl(2,c ) _

an intense electromagnetic field makes possible the processes which are forbidden in a vacuum such as the neutrino production of an electron positron pair @xmath0 . the list of papers devoted to an analysis of this process and the collection of the results obtained could be found e.g. in ref . @xcite . in most cases , calculations of this kind were made either in the crossed field approximation , or in the limit of a superstrong field much greater than the critical value of @xmath1 g , when the electrons and positrons were born in states corresponding to the ground landau level . however , there are physical situations of the so - called moderately strong magnetic field , @xmath2 is the electron mass , and @xmath3 is the elementary charge . ] , @xmath4 , when electrons and positrons mainly occupy the ground landau level , however , a noticeable fraction may be produced at the next levels . the indicated hierarchy of physical parameters corresponds to the conditions of the kerr black hole accretion disk , regarded by experts as the most likely source of a short cosmological gamma - ray burst . the disc is a source of copious neutrinos and anti - neutrinos , which partially annihilate above the disc and turn into @xmath5 pairs , @xmath6 . this process was proposed and investigated in many details , see e.g. ref . @xcite and the papers cited therein , as a possible mechanism for creating relativistic , @xmath5-dominated jets that could power observed gamma - ray bursts . in ref . @xcite , in addition to @xmath7 annihilation , the contribution of the magnetic field - induced process @xmath0 to the neutrino energy deposition rate around the black hole was also included . however , in calculations of the efficiency of the electron - positron plasma production by neutrino through the process @xmath0 in those physical conditions @xcite ( @xmath8 to 180 @xmath9 , @xmath10 to 25 mev ) , it should be kept in mind that approximations of both the crossed and superstrong field have a limited applicability here . we know a limited number of papers @xcite , where the probability of neutrino - electron processes was investigated , as the sum over the landau levels of electrons ( positrons ) . in the papers @xcite , only the neutrino - electron scattering channel in a dense magnetized plasma was studied , which was the crossed process to the considered here neutrino creation of electron - positron pairs . in the paper @xcite , also devoted to the study of the process @xmath0 , the analytical calculations were presented in a rather cumbersome form , caused by the choice of solutions of the dirac equation . the final results for the process probability were obtained by numerical calculations for some set of landau levels occupied by electrons and positrons . in astrophysical applications , there exists probably more interesting value than the process probability , namely , the mean value of the neutrino energy loss , caused by the influence of an external magnetic field . thus , the aim of this paper is the study of the process @xmath0 in the physical conditions of the moderately strong magnetic field , where the electrons and positrons would be born in the states corresponding to the excited landau levels , and the theoretical description would contain a relatively simple analytical formulas for the mean value of the neutrino energy loss , for a wide range of landau levels . more details of the analysis can be found in our recent paper @xcite . we use the standard calculation technics , see e.g. ref . @xcite . the effective local lagrangian of the process can be written in the form @xmath11 \ , \big [ \bar \nu \gamma^{\alpha } ( 1 - \gamma_5 ) \nu \big ] \ , , \label{eq : l}\ ] ] where the electron field operators are constructed on a base of the solutions of the dirac equation in the presence of an external magnetic field . the constants @xmath12 and @xmath13 for different neutrino types are : @xmath14 where @xmath15 is the weinberg angle . the total probability of the process @xmath16 where the electron and the positron are created in the states corresponding to @xmath17th and @xmath18th landau levels correspondingly , is , in a general case , the sum of the probabilities of the four polarization channels : @xmath19 for each of the channels , the differential probability over the final neutrino momentum per unit time can be written as @xmath20 where @xmath21 is the total interaction time , @xmath22 is the total volume of the interaction region , @xmath23 is the @xmath24-matrix element constructed with the effective lagrangian ( [ eq : l ] ) , and the elements of the phase volume are introduced for the electron and the positron ( the magnetic field is directed along the @xmath25 axis ) : @xmath26 in the integration over the momenta of the electron and the positron , a condition arises : @xmath27 which determines the range of integration over the final neutrino momentum . here , @xmath28 is the change of the four - vector of the neutrino momentum equal to the four - momentum of the @xmath29 pair , @xmath30 and @xmath31 are the four - momenta of the initial and final neutrinos . in turn , the condition ( [ eq : cond ] ) can be satisfied when the energy of the initial neutrino exceeds a certain threshold value . in the reference frame , hereafter called @xmath32 , where the momentum of the initial neutrino directed at an angle @xmath33 to the magnetic field , the threshold energy is given by : @xmath34 in fig . [ fig : open_levels ] , the landau levels of @xmath35 are shown , similarly to fig . 1 of ref . @xcite , to be excited accoring to the condition ( [ eq : conde ] ) , at @xmath36 mev , and at @xmath37 and @xmath38 . to be excited when @xmath39 , at @xmath40 mev , and at @xmath37 ( left ) and @xmath38 ( right ) . ] to be excited when @xmath39 , at @xmath40 mev , and at @xmath37 ( left ) and @xmath38 ( right ) . ] it is convenient to perform further integration over the final neutrino momentum , without loss of generality , not in an arbitrary reference frame @xmath32 , but in the special frame @xmath41 , where the initial neutrino momentum is perpendicular to the magnetic field , @xmath42 . one can then return from @xmath41 to @xmath32 by the lorentz transformation along the field ( recall that the field is invariant with respect to this transformation ) . it is convenient to use the dimensionless cylindrical coordinates in the space of the final neutrino momentum vector @xmath43 : @xmath44 & & r = e'/e_{\mprp } = \sqrt{\rho^2 + z^2 } \ , . \label{eq : variab}\end{aligned}\ ] ] here , @xmath45 is the energy of the initial neutrino in the frame @xmath41 , which is connected with its energy @xmath10 in an arbitrary frame @xmath32 by the relation @xmath46 . we do not present here the set of expressions for the probability of the process @xmath16 , which can be found in the paper @xcite . these probabilities evaluated numerically as the functions of the initial neutrino energy and of the magnetic field strength for all channels considered in ref . @xcite , where the electron and positron are created in the lower landau levels , are in a good agreement with the results of that paper . the probability of the @xmath0 process defines its partial contribution into the neutrino opacity of the medium . the estimation of the neutrino mean free path with respect to this process gives the result which is too large @xcite compared with the typical size of a compact astrophysical object , e.g. the supernova remnant , where a strong magnetic field could exist . however , a mean free path does not exhaust the neutrino physics in a medium . in astrophysical applications , we could consider the values that probably are more essential , namely , the mean values of the neutrino energy and momentum losses , caused by the influence of an external magnetic field . these values can be described by the four - vector of losses @xmath47 , @xmath48 where @xmath49 is the total differential probability of the process @xmath0 . the zeroth component of @xmath47 is connected with the mean energy lost by a neutrino per unit time due to the process considered , @xmath50 . the space components of the four - vector ( [ eq : q0 ] ) are similarly connected with the mean neutrino momentum loss per unit time , @xmath51 . it should be noted that the four - vector of losses @xmath47 can be used for evaluating the integral effect of neutrinos on plasma in the conditions of not very dense plasma , e.g. of a supernova envelope , when an one - interaction approximation of a neutrino with plasma is valid @xcite . in ref . @xcite , the formula ( 10 ) for the energy deposition rate was taken , which was calculated in the crossed field limit @xcite . by the way , the value @xmath52 defined by eq . ( 10 ) of ref . @xcite is not the 4-vector while the value @xmath53 is . however , in the region of the physical parameters used in ref . @xcite ( @xmath8 to 180 @xmath9 , @xmath10 to 25 mev ) , the approximation of a crossed field is poorly applicable , as well as the approximation of a superstrong field when @xmath29 are created in the ground landau level . the contribution of the next landau levels which can be also excited , should be taken into account . we present here the results of our calculation of the mean neutrino energy losses caused by the process @xmath0 in a moderately strong magnetic field , i.e. in the conditions of the kerr black hole accretion disk . we parametrize the energy deposition rate as : @xmath54 where @xmath55 , and the dependence on the initial neutrino energy and the field strength is described by the function @xmath56 . this function calculated in ref . @xcite in the crossed field limit had the form @xmath57 on the other hand , in the strong field limit when both electron and positron are born in the ground landau level , the function @xmath58 was also calculated in ref . @xcite and can be presented in the form @xmath59 where @xmath60 is the modified bessel function . in conditions of moderately strong magnetic field , when the electron and the positron are created in the process @xmath16 in the @xmath17th and @xmath18th landau levels , the result has more complicated form . it is significantly simplified when one of the particles , electron or positron , is born in the ground landau level , and if an approximation @xmath61 is used . we obtain the contribution of the channels @xmath62 and @xmath63 to the function @xmath58 as follows : @xmath64 & & \times \left[(1 - \rho^2)^2 + 4 r^2 - 2 r ( 1+\rho^2 ) \right ] \int\limits_0^{2 \pi } \frac{{\mathrm{d}}\phi}{2 \pi } ( r-\rho \cos \phi ) \nonumber\\[2 mm ] & & \times \ , ( 1 - 2 \rho \cos \phi + \rho^2)^{n-1 } \exp \ ! \left ( - \frac{y^2 ( 1 - 2 \rho \cos \phi + \rho^2)}{2 \eta } \right ) , \label{eq : f_(0n)}\end{aligned}\ ] ] where @xmath65 in figs . [ fig : function180][fig : function50 ] , the function @xmath58 obtained in different approximations is shown at @xmath66 . it can be seen that the crossed field limit gives the overstated result which is in orders of magnitude greater than the sum of the contributions of lower excited landau levels . on the other hand , the results with @xmath29 created at the ground landau level give the main contribution to the energy deposition rate , and almost exhaust it at @xmath67 . this would mean that the conclusion @xcite that the contribution of the process @xmath0 to the efficiency of the electron - positron plasma production by neutrino exceeds the contribution of the annihilation channel @xmath6 , and that the first process dominates the energy deposition rate , does not have a sufficient basis . a new analysis of the efficiency of energy deposition by neutrinos through both processes , @xmath6 and @xmath0 , in a hyper - accretion disc around a black hole should be performed , with taking account of our results for the process @xmath0 presented here . in the paper , a calculation is performed of the mean value of the neutrino energy loss due to the process of electron - positron pair production , @xmath0 , in the magnetic field of an arbitrary strength at which the electrons and positrons can be produced in the states corresponding to the excited landau levels , which could be essential in astrophysical applications . the results obtained should be used for calculations of the efficiency of the electron - positron plasma production by neutrinos in the conditions of the kerr black hole accretion disk , regarded by experts as the most likely source of a short cosmological gamma - ray burst . in those conditions , the crossed field limit used in the previous calculations led to the overstated result which was in orders of magnitude greater than the sum of the lower landau levels . the study may be also useful for further development of computational techniques for the analysis of quantum processes in an external active environment , particularly in conditions of moderately strong magnetic field , when the allowance for the contribution of only the ground landau level is insufficient . we dedicate this paper to the blessed memory of our teacher , colleague , and friend nickolay vladimirovich mikheev , who passed away on june 19 , 2014 . for @xmath68 obtained in the crossed field limit ( dotted line ) , with @xmath29 created at the ground ( 0,0 ) landau level ( dashed line ) , and for the sum of all lower landau levels which are excited in this energy interval according to the condition ( [ eq : conde])(solid line).,scaledwidth=87.0% ] the study was performed with the support by the project no . 92 within the base part of the state assignment for the yaroslavl university scientific research , and was supported in part by the russian foundation for basic research ( project no . ) . kuznetsov and n.v . mikheev , _ electroweak processes in external active media _ ( berlin , heidelberg : springer - verlag , 2013 ) . r. birkl , m.a . aloy , h .- th . janka and e. m " uller , _ astron . astrophys . _ * 463 * , 51 ( 2007 ) . i. zalamea and a.m. beloborodov , _ mon . not . r. astron . soc . _ * 410 * , 2302 ( 2011 ) . bezchastnov and p. haensel , _ phys . rev . d _ * 54 * , 3706 ( 1996 ) . mikheev and e.n . narynskaya , _ mod . lett . a _ * 15 * , 1551 ( 2000 ) ; _ centr . j. phys . _ * 1 * , 145 ( 2003 ) . kuznetsov , d.a . rumyantsev and v.n . savin , _ int . j. mod . phys . a _ * 29 * ( 2014 ) , in press , e - print arxiv:1406.3904 [ hep - ph ] . j.k . daugherty and a.k . harding , _ astrophys . j. _ * 273 * , 761 ( 1983 ) . m. ruffert , h .- th . janka , k. takahashi and g. sch " afer , _ astron . _ * 319 * , 122 ( 1997 ) .

the process of neutrino production of electron positron pairs in a magnetic field of arbitrary strength , where electrons and positrons can be created in the states corresponding to excited landau levels , is analysed . the mean value of the neutrino energy loss due to the process @xmath0 is calculated . the result can be applied for calculating the efficiency of the electron - positron plasma production by neutrinos in the conditions of the kerr black hole accretion disc considered by experts as the most possible source of a short cosmological gamma burst . the presented research can be also useful for further development of the calculation technic for an analysis of quantum processes in external active medium , and in part in the conditions of moderately strong magnetic field , when taking account of the ground landau level appears to be insufficient .

in recent years , research has illuminated a link between active galactic nuclei ( agns ) and the evolution of their host galaxies . studies have found that a galaxy s central black hole mass is correlated with the mass of the central bulge ( e.g. , * ? ? ? * ; * ? ? ? * ; * ? ? ? * ; * ? ? ? * ; * ? ? ? * ; * ? ? ? * ) . in the local universe , high - luminosity agns are preferentially found in early - type galaxies with young mean stellar ages @xcite . as the redshift increases , the cosmic rates of both star formation and agn activity increase @xcite , and , at high redshift , intensely star - forming submillimeter galaxies have been found to have an agn fraction of @xmath620 - 30@xmath7 @xcite . the connection between agns and host evolution is also observed in optical colors . studies have shown that galaxies are organized based on rest - frame color , separating into a ` red sequence ' and a ` blue cloud ' on a color - magnitude diagram ( cmd ; e.g. , * ? ? ? * ; * ? ? ? * ; * ? ? ? * ) . the sparsely populated ` green valley ' between these regions is thought to be a transitional area , where blue cloud galaxies are rapidly migrating onto the red sequence after the cessation of star formation @xcite . a proposed mechanism for this quick transition is agn feedback , where the agn activity , driven by major mergers or tidal interactions , causes a truncation of star formation and leads to the color evolution onto the red sequence @xcite . this theory is supported by a number of studies which find an association between agn activity and the green valley @xcite . however , some studies using mass - selected samples have found a more uniform color distribution of agn hosts @xcite . in addition , @xcite have found that @xmath8of green valley agn hosts are dust - reddened members of the blue cloud , although these results are apparently in conflict with the more recent studies of @xcite . studies examining agns for recent merger activity have also found mixed results ( e.g. , * ? ? ? * ; * ? ? ? * ; * ? ? ? * ; * ? ? ? * ; * ? ? ? @xcite offer an alternative to major merger driven agns , where agns are instead triggered by minor mergers , mainly in red galaxies . star formation in this scenario would only be briefly re - ignited at the cores of many of these systems and the galaxies would return to the red sequence after its cessation . these agns would then represent a red sequence population evolving in luminosity space rather than blue cloud galaxies migrating onto the red sequence . in support of this theory , several studies have found agn activity associated with blue early - type galaxies or red galaxies with blue cores @xcite . additionally , @xcite propose that recycled stellar material in elliptical galaxies could fuel central starbursts and black hole accretion . without precise observations of morphologies , the effects of these two models would appear similar . other plausible models exist as well , some important only in certain regimes . for example , at high redshift ( @xmath9 ) , @xcite propose that violent disk instabilities could lead to powerful nuclear activity , while @xcite present a model where lower luminosity agns could be triggered by mild disk instabilities or gas funneled through bars . note that the properties of individual agns are widely varying , and it could be the case that any number of modes contribute to agn triggering , with different contributions in different mass and/or redshift regimes . to better understand how agn are triggered and the role they play in galaxy evolution , it is useful to study large - scale structures ( lsss ) at high redshift . these environments contain a large number of galaxies in the process of transitioning from actively star - forming to passive , and there is evidence that nuclear activity increases at higher redshifts @xcite . to this end , we have examined the x - ray selected agn population within five lsss at redshifts of @xmath10 - 0.9 . these structures are in varying states of evolution and include complex superclusters , an interacting supergroup , and isolated clusters . each has been studied extensively by the observations of redshift evolution in large - scale environments ( orelse ) survey , which is searching for lsss in the vicinities of 20 known clusters between @xmath11 - 1.3 . the survey has compiled extensive multi - wavelength datasets for each structure , which include multi - band optical , radio , and x - ray imaging , as well as thousands of spectroscopic redshifts @xcite . in this paper , we present analyses of x - ray point sources and the properties of the agn population within the following five lsss : the cl1604 and cl1324 superclusters at @xmath12 and @xmath13 , respectively , the cl0023 + 0423 supergroup at @xmath14 , and two x - ray selected and relatively relaxed , isolated clusters , rxj1757.3 + 6631 at @xmath15 and rxj1821.6 + 6827 at @xmath16 . we study this sample with a combination of chandra x - ray data , optical imaging , and near - ir and optical spectroscopy . for our cosmological model , we assume @xmath17 , @xmath18 , and @xmath19 km s@xmath5 mpc@xmath5 . we discuss the clusters and superclusters in our survey in section [ sec : samp ] . observations , data reduction , and techniques are discussed in section [ sec : red ] . the global properties of our sample are discussed in section [ sec : globchar ] . the statistical measurements of cumulative source counts are covered in section [ sec : csc ] . analysis of the agns is presented in section [ sec : agn ] . in this section we describe the five structures in our sample , which are succintly summarized in table [ strsumtab ] . the redshift boundaries used in the following analyses were determined by visually examining each structure s redshift histogram . delineating where structures end is not straightforward , with some having associated filaments or possible nearby sheets . the redshift boundaries are chosen with the aim to include all galaxies which could be part of each overall lss . lccccccccc cl1604 & 16 04 15 & + 43 16 24 & 0.90 & 0.84 & 0.96 & 10 & 300 - 800 & 531 & 10 + cl0023 & 00 23 51 & + 04 22 55 & 0.84 & 0.82 & 0.87 & 4 & 200 - 500 & 244 & 7 + cl1324 & 13 24 45 & + 30 34 18 & 0.76 & 0.65 & 0.79 & 10 & 200 - 900 & 393 & 6 + rxj1821 & 18 21 32.4 & + 68 27 56 & 0.82 & 0.80 & 0.84 & 1 & @xmath20 & 90 & 3 + rxj1757 & 17 57 19.4 & + 66 31 29 & 0.69 & 0.68 & 0.71 & 1 & @xmath21 & 42 & 1 + [ strsumtab ] the cl1604 supercluster at @xmath22 is one of the largest structures studied at high redshift . it consists of at least 10 clusters and groups and spans @xmath23mpc along the line of sight and @xmath24mpc in the plane of the sky @xcite . the massive member clusters cl1604 + 4304 and cl1604 + 4321 were first discovered in the optical survey of @xcite . the proximity of the clusters suggested that they were components of a larger structure . further wide field imaging has revealed 10 distinct red galaxy overdensities , suggesting the existence of a supercluster @xcite . spectroscopic observations have confirmed four of the overdensities to be clusters with velocity dispersions in excess of 500 km s@xmath5 , while four others were confirmed to be poor clusters or groups with dispersions in the range 300 - 500 km s@xmath5 @xcite . the two most massive clusters in cl1604 have associated diffuse x - ray emission . cl1604 + 4304 and cl1604 + 4314 , hereafter clusters a and b , have measured bolometric x - ray luminosities of @xmath25 and @xmath26erg s@xmath27 and x - ray temperatures of @xmath28 and @xmath29kev , respectively @xcite . while these values place cluster a on the @xmath30-@xmath31 curve for virialized clusters , cluster b is well off from it , suggesting that cluster a is relaxed while cluster b is not . all other clusters have only an upper limit on their bolometric luminosity of @xmath32erg s@xmath27@xcite . while many galaxies in cl1604 have substantial [ ] emission , near - infrared spectroscopy has shown that a significant portion of this emission is due to contributions from low - ionization nuclear emission - line regions ( liners ) and seyferts @xcite . also , @xcite studied 24@xmath33 m selected galaxies in and around three clusters and three groups in cl1604 using the multiband imaging photometer for _ spitzer _ ( mips ; * ? ? ? * ) and found evidence for recent starburst activity and an infalling population . analysis of the morphologies using the advanced camera for surveys ( acs ; * ? ? ? * ) on the hubble space telescope ( hst ) revealed that many of these 24@xmath33 m bright galaxies were disturbed , indicating mergers and interactions were likely responsible for starburst activity . we refer the reader to @xcite , @xcite , and @xcite , for more details on the data processing , supercluster properties , and observations . the cl0023 + 0423 structure at @xmath34 , hereafter cl0023 , was also discovered as an overdensity in the optical survey of @xcite . the structure was later observed by @xcite using the low - resolution imaging spectrograph ( lris ; * ? ? ? * ) on the keck 10 m telescope , where the overdensity was resolved into two structures . further study has shown that the structure consists of four merging galaxy groups separated by approximately 3000 km s@xmath5 in radial velocity and @xmath35 mpc on the plane of the sky @xcite . the constituent groups have measured velocity dispersions within @xmath36mpc of 480@xmath37 , 430@xmath38 , 290@xmath39 , and 210@xmath40 km s@xmath5 @xcite . simulations suggest that the groups will merge to form a cluster of mass @xmath41 within @xmath61 gyr @xcite . @xcite found cl0023 to have a large blue population , with @xmath42of galaxies bluer than their red galaxy color - color cut , down to an @xmath43-band magnitude of 24.5 . spectroscopic analysis found that @xmath44of galaxies had measurable [ ] emission , which , because of the large blue population , is most likely due to ongoing star formation for a discussion of [ ] emission . ] . we refer the reader to @xcite and @xcite for more details on the supergroup properties and observations . the cl1324 supercluster is a lss at @xmath45 . the two most massive clusters in the structure , cl1324 + 3011 at @xmath46 and cl1324 + 3059 at @xmath15 , were first discovered in the optical survey of @xcite . because of the proximity of the clusters on the sky and in redshift space , this structure was chosen for the orelse survey to investigate the possible existence of structure in the field . wide - field imaging has revealed a total of ten clusters and groups , detected through red galaxy overdensities , and , despite extensive spectroscopy , only four have been spectroscopically confirmed as constituent clusters or groups ( see gal et al . 2012 , in preparation ) . lcc cl0023 & 7500 - 7850 & 6200 - 9150 + cl1604 & 7700 & 6385 - 9015 + cl1324 & 7200 & 5900 - 8500 + rxj1821 & 7500 - 7800 & 6200 - 9100 + rxj1757 & 7000 - 7100 & 5700 - 8400 + [ speccovtab ] cl1324 + 3011 was previously studied in @xcite and @xcite , where a velocity dispersion of @xmath47 km s@xmath5 and a temperature of @xmath48kev , using xmm - newton , were measured for the cluster . according to these measurements , the cluster does not fall close to the @xmath30-@xmath31 curve for virialized clusters , which would imply that it is not well relaxed . new chandra results for cl1324 are presented in n. rumbaugh et al . ( 2012 , in preparation ) , including new x - ray temperatures for cl1324 + 3011 and cl1324 + 3059 . in addition , we present here updated velocity dispersions for these two clusters of @xmath49 and @xmath50 km s@xmath5 , respectively . the new measurements place cl1324 + 3011 closer to the @xmath30-t curve for virialized clusters , only offset by @xmath51 . while the older measurements suggested the cluster was not relaxed , the new measurements are more consistent with virialization . similarly to cl1324 + 3011 , cl1324 + 3059 is offset from the curve , but still by less than 1@xmath30 . the photometric and spectroscopic observations of the cl1324 supercluster will be covered in full in r. r. gal et al . ( 2012 , in preparation ) . in this paper , we present redshift histograms of the full structure and velocity dispersions for the four confirmed groups and clusters , as well as the chandra observations . lcccccc cl1604 & 2465 & 1785 & 158(213 ) & 112(128 ) & 43(48 ) & 38(42 ) + cl0023 & 1136 & 892 & 94(133 ) & 58(72 ) & 39(49 ) & 26(32 ) + cl1324 & 1419 & 1155 & 174(217 ) & 126(133 ) & 38(40 ) & 28(30 ) + rxj1821 & 351 & 306 & 102(132 ) & 64(72 ) & 15(18 ) & 10(13 ) + rxj1757 & 549 & 421 & 87(107 ) & 57(62 ) & 18(19 ) & 9(9 ) + [ srcsum ] the x - ray - selected cluster rxj1821.6 + 6827 , hereafter rxj1821 , at @xmath52 , was the highest redshift cluster discovered in the rosat north ecliptic pole ( nep ) survey , where it is also referred to as nep5281 @xcite . using xmm - newton data , the cluster was found to have slightly elongated diffuse x - ray emission with a measured bolometric luminosity of @xmath53 and a temperature of @xmath54kev @xcite . the same study measured a velocity dispersion of @xmath55 km s@xmath5 using 20 cluster members . later analysis by @xcite used 40 galaxies within 1 mpc to measure a velocity dispersion of @xmath56 km s@xmath5 . redshift histograms of rxj1821 are characteristic of a single , isolated structure , although a small kinematically associated group has been detected to the south @xcite . while the temperature and dispersion measurements place the cluster near the @xmath30-@xmath31 relation for virialized clusters , the elongated x - ray emission could be indicative of still ongoing formation of the cluster . @xcite measured a blue fraction of only @xmath57for rxj1821 , down to a magnitude limit of @xmath58 . they found a population dominated by massive , old ( formation epoch of @xmath59 ) galaxies , along with fainter galaxies that were quenched more recently . they also found that @xmath60of galaxies had detectable [ ] emission . near - ir spectroscopy of a subset of these [ ] emitting galaxies suggests that some of the emission is due to liner or agn activity @xcite . we refer the reader to @xcite for more details on the data processing , cluster properties , and observations . the @xmath15 cluster rxj1757.3 + 6631 , hereafter rxj1757 , was discovered as part of the rosat nep survey , where it is also identified as nep200 @xcite . @xcite found the structure to have an x - ray luminosity of 8.6@xmath61erg s@xmath5 in the 0.5 - 2.0 kev band . the structure is dominated by a single , large cluster . in this paper , we present a velocity dispersion , redshift histograms , and analysis of x - ray point sources for rxj1757 , none of which have been previously published . ground - based optical imaging data were obtained with the large format camera ( lfc ; * ? ? ? * ) on the palomar 5 m telescope . observations were taken using the sloan digital sky survey ( sdss ) @xmath62 , @xmath43 , and @xmath63 filters . the 5 @xmath30 point source limiting magnitudes for the five fields ranged from 25.5 - 25.1 , 25.0 - 24.5 , and 23.6 - 23.3 , in the @xmath64 , @xmath65 , and @xmath66 bands , respectively . cl1604 was also imaged using acs . the hst imaging for cl1604 consists of 17 acs pointings designed to image nine of the ten galaxy density peaks in the field . observations were taken using the f606w and f814w bands . these bands roughly correspond to broadband v and i , respectively . our photometric catalog is complemented by an unprecedented amount of spectroscopic data . for this part of the study , we used the deep imaging multi - object spectrograph ( deimos ; * ? ? ? * ) on the keck ii 10 m telescope . in addition , cl1604 and rxj1821 have some lris coverage ( see * ? ? ? * ; * ? ? ? * ; * ? ? ? deimos has a wide field of view ( @xmath67 ) , high efficiency , and is able to position over 120 targets per slit mask , which makes the instrument ideal for establishing an extensive spectroscopic catalog . we targeted objects down to an @xmath43-band magnitude of 24.5 . on deimos , we used the 1200 line mm@xmath5 grating , blazed at 7500 , and 1@xmath68 slits . these specifications create a pixel scale of 0.33 pixel@xmath5 and a fwhm resolution of @xmath69 , or 68 km s@xmath5 . the central wavelength was varied from structure to structure and sometimes between different masks for the same field . central wavelengths for the spectroscopic observations for the five fields and the approximate spectral coverages are displayed in table [ speccovtab ] . when more than one central wavelength was used per field , a range is given . total exposure times for the observations are in the range of 1 - 4 hours per mask . spectroscopic targets were chosen based on color and magnitude . the number of spectroscopic targets in each field is shown in table [ srcsum ] . redshifts were determined or measured for all targets and given a quality flag value , @xmath70 , where @xmath71 indicates that we could not determine a secure redshift , @xmath72 means a redshift was obtained using features that were only marginally detected , @xmath73 means one secure and one marginal feature were used to calculate the redshift , and @xmath74 means at least two secure features were used . those sources determined to be stars were given a flag of @xmath75 . see @xcite for more details on quality flags and the spectral targeting method . for our analysis , redshifts with @xmath76,3,4 were deemed satisfactory , and the number of such sources in each field is shown in table [ srcsum ] . spectroscopic data have been previously presented for the cl1604 supercluster , cl0023 , and rxj1821 as part of the orelse survey . we present new data for each of these structures , as well as for the cl1324 supercluster and rxj1757 . we include ten deimos masks for cl1324 , with exposure times ranging from 6635 s to 10800 s , and four deimos masks for rxj1757 , with exposure times ranging from 7200 s to 14,730 s. we include a total of 18 spectroscopic masks for cl1604 , six more than what were included in @xcite . we include nine total masks for cl0023 , four more than in @xcite . we include three masks for rxj1821 , one more than in @xcite . many of the new targets were optical counterparts to x - ray sources . from our measured redshifts with @xmath77 or 4 from the new masks , we find 114 new galaxies in cl1604 within @xmath78 , 104 new galaxies in cl0023 within @xmath79 here , @xcite use @xmath80 , excluding a sheet of galaxies at @xmath81 - 0.87 . in the smaller range of @xmath80 , we add 96 new galaxies . we use a wider redshift range in order to be consistently liberal in our structure boundaries . ] , and five new galaxies in rxj1821 within @xmath82 . all x - ray imaging of the clusters was conducted with the advanced ccd imaging spectrometer ( acis ) of the chandra x - ray observatory , using the acis - i array ( pi : l. m. lubin ) . this array has a @xmath83 field of view . some of the five structures were imaged with one pointing and some with two , but every pointing had the same approximate total exposure time of 50 ks . cl0023 , rxj1821 , and rxj1757 were each imaged with one pointing of the array . cl1604 and cl1324 , with angular sizes in excess of 20@xmath84 , were observed with two pointings each . for cl1604 , the two pointings are meant to cover as much of the structure as possible , and there is a small overlap ( @xmath85 arcminutes@xmath86 ) . for cl1324 , the two pointings are centered near the two largest and originally discovered clusters , cl1324 + 3011 and cl1324 + 3059 . there is an approximately 13@xmath84 gap between the north and south pointings . in this paper , we present new chandra data for cl1324 , rxj1821 , and rxj1757 . lccc cl1604 & 1.2 & 6.12 & 22.2 + cl0023 & 2.7 & 6.37 & 22.3 + cl1324 & 1.1 & 6.16 & 22.2 + rxj1821 & 5.6 & 8.23 & 22.3 + rxj1757 & 4.0 & 6.60 & 22.2 + [ nhc2ftab ] the reduction of the data was conducted using the _ chandra _ interactive analysis of observations 4.2 software ( ciao ; * ? ? ? each observation was filtered by energy into three bands : 0.5 - 2 kev ( soft ) , 2 - 8 kev ( hard ) , and 0.5 - 8 kev ( full ) . data were checked for flares using _ dmextract _ and the _ chandra _ imaging and plotting system ( chips ) routine _ lc_clean_. exposure maps were created using the routine _ merge_all_. for vignetting correction , exposure maps were normalized to their maximum value , then images were divided by this normalized exposure map . to locate point sources , the routine _ wavdetect _ was run on each observation , without vignetting correction , using wavelet scales of @xmath87 pixels , with @xmath88 ranging from 0 to 8 . a threshold significance of @xmath89 was used , which would imply fewer than one spurious detection per acis chip , which has dimensions of @xmath90 pixels ( @xmath91per pixel ) . however , this assumes a uniform background , which is almost certainly not the case . to measure realistic detection significances , we instead used photometric results explained below . point source detection was carried out on images in each of the soft , hard , and full bands separately . for cl1604 and cl1324 , _ wavdetect _ was run on each of the two pointings separately . output object positions from the three different bands were cross - correlated to create one final composite list for each field . we carried out follow up photometry on the point sources . circular apertures containing 95@xmath7 of the flux were created for each point source using the point - spread function ( psf ) libraries in the _ chandra _ calibration database . for _ chandra _ , the psf depends on both energy and off - axis angle . for the soft and hard bands , respectively , we used the psf libraries for energies of 1.497 and 4.510 kev . photometry was carried out on the vignetting - corrected images in the soft and hard bands . background counts for each source were calculated in annuli with inner and outer radii of @xmath92 and @xmath93 , where @xmath94 is the radius of the circular aperture containing 95@xmath7 of the flux . since only 95@xmath7 of the flux is enclosed , net counts calculated from the apertures were multiplied by 1/0.95 to recover all the counts . full band counts were calculated by summing those from the soft and hard bands . the results of the photometry were used to calculate detection significances for the sources in each of the three bands using @xmath95 where @xmath96 is the net photon counts from the source and @xmath97 is the background counts @xcite . x - ray sources with significances @xmath98 were rejected as spurious . with @xmath99of accepted sources having detection significances @xmath100 , and the remaining @xmath101with detection significances between @xmath102 and @xmath103 , we expect a spurious detection rate of @xmath104 , based on a normal distribution . due to the low number of photons observed for many sources , we opted to normalize a power - law spectral model to the net count rate of individual point sources to determine fluxes . we assumed a photon index of @xmath105 , which is the approximate slope of the x - ray background in the hard band @xcite . count rates were calculated by dividing net counts by the nominal exposure time at the aimpoint of the appropriate observation . the galactic neutral hydrogen column density was calculated at the aim point of each observation using the colden tool from the _ chandra _ proposal toolkit , which uses the dataset of @xcite . hydrogen column densities and derived net count rate to unabsorbed flux conversion factors for each field are summarized in table [ nhc2ftab ] . conversion factors were determined separately for the different pointings of cl1604 and cl1324 , but did not differ to the three significant figures listed in the table . llllll cl1324 + 3011 & a & 13 24 48.7 & + 30 11 48 & @xmath49 + cl1324 + 3059 & i & 13 24 50.5 & + 30 58 19 & @xmath106 + cl1324 + 3013 & b & 13 24 21.5 & + 30 13 10 & @xmath107 + cl1324 + 3025 & c & 13 24 01.8 & + 30 25 05 & @xmath108 + [ 1324clustab ] to search for agns within the individual clusters , we matched x - ray and optical sources . in order to increase our completeness , the input to the matching included all point sources detected by _ wavdetect _ , regardless of significance in any of the three bands . we used the maximum likelihood ratio technique described in @xcite , which was developed by @xcite and also used by @xcite and @xcite . our technique is similar to @xcite , but with a few key differences . the main statistic calculated in each case is the likelihood ratio ( lr ) , which estimates the probability that a given optical source is the genuine match to a given point source relative to the arrangement of the two sources arising by chance . the lr is given by the equation @xmath109 here , @xmath110 is the separation between objects @xmath88 and @xmath111 , @xmath112 is the positional error of object @xmath111 , and @xmath113 is the inverse of the number density of optical sources with magnitude brighter than @xmath114 . the inclusion of the latter quantity is designed to weight against matching to fainter optical objects . however , in our analysis , we found that this particular weighting , used by @xcite , of @xmath115 gave too much favor to bright objects , even when they were much farther from an x - ray object than a faint source . we adjusted the weighting to @xmath116 . we found that this change did not have a large overall effect but changed some borderline cases where a bright object with a large separation from the point had been chosen over a dimmer , much closer object . this includes one case , which prompted the adjustment , where spectroscopy showed that an m - type star was matched instead of a probable agn , even though the m star was three times farther from the x - ray source . for each field , except cl1604 , @xmath117 was measured using @xmath43 magnitudes from our lfc catalogs . for cl1604 , acs data were also available , but these observations did not cover the entire field . all objects were matched to the lfc catalogs . when possible , objects were also matched using the f814w magnitude from the acs catalogs and matches to acs objects took precedence over matches to lfc objects . from the lr , we use monte carlo simulations to derive the probability that a given match is genuine . we ran 10,000 trials for each x - ray object . in each trial , the object s position was randomized , and the lr was calculated based on nearby optical sources . the lr for a given x - ray object to optical counterpart pairing , @xmath118 , was compared against the distribution of lrs from the 10,000 monte carlo trials . we calculated the reliability as : @xmath119 where @xmath120 is the number of matches , to any optical source , across all 10,000 trials for that x - ray object with lr greater than @xmath118 . @xmath121 can be interpreted as the probability that optical source @xmath88 is the true match of x - ray source @xmath111 , in the case of only one optical candidate . when there are multiple candidates , we used the method of @xcite to calculate the probability that optical source @xmath88 is the true match of x - ray source @xmath111 as @xmath122 the probability that no optical source is the true match is : @xmath123 where n is the total number of optical candidates and s is a normalization factor defined so that @xmath124 . for an x - ray source with a single optical counterpart , a match was considered genuine if @xmath125 . for x - ray sources with multiple optical counterparts , a genuine match was chosen if @xmath126 ( which is equivalent to @xmath125 ) and @xmath127 for any one object @xmath88 . if the first condition was true , but the second was not , all objects with @xmath128 were considered as matches . in subsequent sections , only one optical counterpart was considered for each x - ray source . in almost all cases , the highest probability match was used . however , in several cases , spectra of the primary and secondary matches indicated that the secondary match was an agn , and thus more likely to be a genuine match . note that our threshold is a deviation from @xcite . they used @xmath129 instead of 0.15 . we decided to use the more stringent threshold of 0.15 , which has been used by others @xcite , to better limit the number of false matches . the more restrictive cutoff omitted @xmath130 sources per field . we determined this threshold through visual inspection of potential matches . this calibration entailed determining at what approximate level of @xmath131 most matches visually seemed spurious . however , optical candidates above our threshold were also visually scrutinized ( @xmath132 of the total ) , and some were accepted after this inspection where we felt the matching algorithm had failed . note that setting a threshold for genuine matches is not entirely objective , and a precedent has been set for accomplishing this with visual inspection ( e.g. , * ? ? ? * ) . in table [ srcsum ] , we list , for each field , the number of x - ray sources detected at @xmath1333@xmath30 ( @xmath1332@xmath30 ) in one of the three bands , as well as the number of those sources matched to optical counterparts . the five structures in our sample span a range of evolutionary states . they include cl0023 , whose four constituent groups are still in the process of merging to form a single cluster ; the two isolated x - ray selected clusters , rxj1757 and rxj1821 , which appear to be in a more evolved and relaxed state ; and the two superclusters , cl1604 and cl1324 . we would like to compare the agns within our sample based on the evolutionary states of the structures to which they belong , which could shed light on how the agns in these systems are being triggered . in order to make such a comparison , we first present the global properties of the five orelse structures . the cl1604 supercluster , cl0023 , and rxj1821 have all been studied previously as part of the orelse survey ( see section [ sec : samp ] for individual references ) , although we have gathered new data on each , as described in section [ sec : optobs ] . while individual clusters in cl1324 have been studied , the properties of the supercluster as a whole have not . in this section we do a preliminary exploration of this structure , which will be covered more thoroughly in an upcoming paper ( r. r. gal et al . 2012 , in preparation ) . the cluster rxj1757 was studied only as a part of the rosat nep survey @xcite , in little detail . here , we present new redshift histograms of these last two structures derived from our orelse data . figure [ 1324 + 1757.rshists](a ) shows all confirmed redshifts in the spatial vicinity of cl1324 . we can see two peaks in the histogram , at @xmath134 and @xmath135 . these peaks coincide with the two largest clusters in the structure , cl1324 + 3011 and cl1324 + 3059 . from the distribution of red galaxies , we find ten overdensities in the supercluster , some of which can be observed in the redshift histogram . so far , we have confirmed four clusters and groups to be constituents , shown in table [ 1324clustab ] , with coordinates , redshifts , and measured velocity dispersions given . additional multiobject spectroscopy to confirm the nature of the other red galaxy overdensities is planned . the redshift histograms for rxj1757 are displayed in figures [ 1324 + 1757.rshists](c ) and [ 1324 + 1757.rshists](d ) . at @xmath136 and @xmath12 , we see two peaks in the distribution . the former is the overdensity associated with rxj1757 . when we examine the spatial distribution of the higher redshift peak , we find that the galaxies in its vicinity are distributed nearly uniformly across the field of view , implying a sheet of galaxies . looking at the redshift distribution of confirmed galaxies within the bounds of rxj1757 ( figure [ 1324 + 1757.rshists](d ) ) , we can see the distribution is reasonably consistent with a gaussian , confirmed by a kolmogorov - smirnov ( k - s ) test at a @xmath137 level , suggesting there is no significant substructure . however , we caution that we have a smaller sample of confirmed redshifts compared to the other fields . figure [ cmds ] shows cmds for all five fields . all spectroscopically confirmed supercluster / cluster members are shown . squares indicate the confirmed x - ray agns within each structure , which are analyzed in section [ sec : agn ] . the red sequence for each field is delineated by dotted lines . red sequence fits for each field were calculated using a linear fitting and @xmath30-clipping technique . first , a fit to a linear model , of the form @xmath138 where @xmath96 is either @xmath139 or f606w - f814w and @xmath97 is either @xmath43 or f814w , was carried out on member galaxies within a chosen magnitude and color range using a @xmath140 minimization @xcite . the fit was initialized with a color range chosen `` by eye '' to conform to the apparent width of the red sequence of the structure . the magnitude bounds were defined as the range where the photometric errors were small ( @xmath141 ) . after the initial fit , colors were normalized to remove the slope . the color distribution was then fit to a single gaussian using iterative @xmath103 clipping . at the conclusion of the algorithm , the boundaries of the red sequence were defined by a 3@xmath142offset from the center , except for cl1604 and cl1324 . for the two superclusters , the color dispersion was inflated due to the large redshift extent of these structures , and 2@xmath30 offsets were used to achieve reasonable boundaries . for every field except cl1604 , the lfc @xmath143 color magnitude diagrams are shown . for cl1604 , acs data were available and were used in place of lfc data because of their superior precision . note , however , that two of the agns in cl1604 are outside our acs pointings , so that our analysis using acs data only includes 8 agns in the cl1604 supercluster . lcc cl0023 & 0.47@xmath1440.06 & 0.51@xmath1440.13 + cl1604 & 0.57@xmath1440.05 & @xmath145 + cl1324 & 0.42@xmath1440.04 & 0.42@xmath1440.12 + rxj1821 & 0.35@xmath1440.08 & 0.42@xmath1440.15 + rxj1757 & 0.17@xmath1440.08 & 0.09@xmath1440.10 + [ bftab ] while all the structures show a substantial number of galaxies on the red sequence , there are large differences in the blue populations . qualitatively , we observe a lower blue fraction in the cmds of rxj1821 and rxj1757 than in those of cl0023 and cl1604 . this is quantified in table [ bftab ] , where the blue fraction for all confirmed cluster / supercluster members with @xmath43 or f814w magnitudes brighter than 23.5 are displayed in the first column . we can see that cl0023 and cl1604 have the bluest galaxy populations , while rxj1757 has the highest fraction of galaxies on the red sequence of all the fields . since this could be due to the low completeness of spectroscopic coverage in this field , we attempted to make corrections with two approaches : ( 1 ) using our efficiency of spectroscopically confirming structure members to estimate the total number of member galaxies , and ( 2 ) correcting using measurements of the background galaxy density . all efforts to statistically estimate the true blue fraction yielded large errors making the measurements highly uncertain . instead , we created a blue fraction measurement metric that could be compared between fields ( see also * ? ? ? * ) . the different fields have a varying number of spectral masks , with several different spectroscopic priorities . however , the first several masks , excluding those designed to specifically target x - ray matched sources , have similar priorities for choosing targets . therefore , we chose to compare blue fractions only among sources in the first two masks for each field , since these sources should represent similar populations . we chose two masks because this is the number of masks for rxj1757 excluding those where objects matched to x - ray point sources were preferentially targeted . we confirmed that this sampling is representative of the entire galaxy population by recreating composite spectra using only the first two spectral masks . since there are no large differences between the average spectral features , the method should be accurate . the results of this comparison are displayed in the second column of table [ bftab ] . however , we did not calculate a blue fraction for cl1604 in this way , for two reasons . first , there are a total of 24 spectroscopic masks for cl1604 , from several different telescopes . choosing which ones to include is difficult and it may be impossible to select a population congruous with any of the other fields in this manner . second , our spectroscopy for cl1604 is relatively complete , so we are confident in the blue fraction measured down to @xmath146 . examining the results , we can see the same color hierarchy in the structures for both methods of measuring the blue fraction , with rxj1757 having the smallest fraction and cl1604 and cl0023 the largest . these large blue fractions , in particular those for cl1604 and cl0023 , are consistent with the butcher - oemler effect @xcite . we also note that the isolated x - ray - selected clusters , rxj1757 and rxj1821 , are the reddest structures , suggestive of a more advanced , dynamically relaxed evolutionary state . the color hierarchy suggests a similar ranking of galactic star formation in the five structures , which we can explore with our spectroscopic data . lccc cl0023 & @xmath147 & @xmath148 & @xmath149 + cl1604 & @xmath150 & @xmath151 & @xmath152 + cl1324 & @xmath153 & @xmath154 & @xmath155 + rxj1821 & @xmath156 & @xmath157 & @xmath158 + rxj1757 & @xmath159 & @xmath160 & @xmath161 + [ globchartab ] using our spectroscopic data , we examine the typical star formation history of the galaxies in each structure . we formed composite spectra by co - adding the individual spectra of galaxies within each structure , according to the method of @xcite and @xcite . we analyze these spectra in terms of two important features relevant to star formation : the [ ] and h@xmath2 lines . the h@xmath2 absorption is indicative of a population of a and b stars , which disappears @xmath162 gyr after the cessation of star formation within a galaxy , due to the lifetime of a stars @xcite . if star formation is ongoing , a population of o stars , which have weaker hydrogen features , can dominate the continuum and wash out this absorption line . infilling can also occur from balmer emission from regions . the [ ] emission line has been used as an indicator of star formation , especially as a proxy for the h@xmath163 emission line at higher redshifts when h@xmath163 has shifted out of the optical range @xcite . however , recent analysis using near - ir spectroscopy of sources from the cl1604 supercluster and rxj1821 has compared [ ] and h@xmath163 emission , finding that a significant portion of [ ] emission can come from liner or seyfert related processes @xcite . these results are supported by those of @xcite , albeit using a lower redshift sample . in light of this , caution must be exercised when interpreting [ ] measurements . for an additional diagnostic , we measure the d@xmath164(4000 ) strength which is an indicator of mean stellar age @xcite . figure [ oiivshd](a ) shows [ ] vs. h@xmath2 equivalent widths for the composite spectra of members of the five structures . the dotted line represents the average spectral properties for a cluster population composed of various fractions of `` normal '' star - forming and quiescent galaxies @xcite , based on data from the two - degree field ( 2df ) galaxy redshift survey @xcite . asterisks on this line represent a cluster population composed of ( from left to right ) 20% , 40% , 60% , 80% , and 100% star - forming galaxies . for a cluster whose average galaxy lies above this line , the h@xmath2 line is too strong to be produced by normal star formation , requiring some contribution from starbursting or post - starbursting galaxies . the dashed lines enclose 95% of normal star - forming galaxies @xcite . the shaded regions , which are based on the spectral types of @xcite and @xcite , denote the region of this phase space inhabited by ( starting from the upper right and moving counter - clockwise ) starburst ( dark blue ) , post - starburst ( green ) , quiescent ( red ) , and normal star - forming galaxies ( light blue ) . examining the positions of the structures in our sample on this plot , we can see that rxj1757 and rxj1821 have mostly quiescent populations , with each cluster having @xmath165 normal star - forming galaxies . because rxj1757 is offset from the 2df line , there may be some contribution from post - starburst galaxies , but the fractional contribution is low . the stronger [ ] emission in the cl0023 and cl1604 composite spectra suggests that these structures have a higher fraction ( @xmath166 ) of continuously star - forming galaxies . these structures have larger blue fractions than the others , so it is unlikely that the increased ew ( [ ] ) in their average spectra is due to liner processes , which are primarily associated with red sequence galaxies . the ew(h@xmath2 ) measured from the cl0023 and cl1604 composite spectra are significantly in excess of the 2df line , suggesting a substantial contribution from starbursting or post - starburst galaxies . cl1324 is in an intermediate range , with @xmath167 normal star - forming galaxies and an observed ew(h@xmath2 ) for its galaxy population smaller than that of cl0023 and cl1604 . our conclusions based on the [ ] and h@xmath2 lines are supported by the corresponding d@xmath164(4000 ) measurements ( see table [ globchartab ] ) . these results are illustrated in figure [ oiivshd](b ) , where average measurements of ew(h@xmath2 ) and d@xmath164(4000 ) are plotted for the five structures . for comparison , we also indicate ranges of ew(h@xmath2)d@xmath164(4000 ) phase space spanned by four different bruzual & charlot @xcite models for various times after the starburst . the four models include a single burst ( @xmath168 ) and a secondary burst of 20% , 10% , and 5% by mass which occurs 2 gyr after the initial burst . all models are solar metallicity and are corrected for extinction using e(b - v ) @xmath169 and a @xcite extinction law . because bruzual & charlot models only incorporate stellar light , emission infill corrections were made for all ew(h@xmath2 ) measurements using relationships between h@xmath163 and [ ] taken from @xcite and h@xmath163 and h@xmath2 from @xcite . although these corrections are not perfect ( e.g. , see the measurement for region 1 ) , these values should be accurate to within @xmath170 . we can see that rxj1757 and rxj1821 have the largest continuum break strengths , indicating that they possess the oldest average stellar populations of the five structures . the average galaxy in this structure has had 1 - 3 gyr since its last starburst , according to the @xcite model . the other structures have smaller average d@xmath164(4000 ) measurements , consistent with the results of figure [ oiivshd](a ) showing larger fractions of star - forming galaxies and younger galaxy populations . according to the @xcite model , the average galaxy in cl1324 , cl0023 , and cl1604 has had a progressively shorter time since the last starburst . altogether , these spectral results parallel what was found with the blue fractions , in that the reddest structures are also the ones with the most evolved stellar populations and the lowest fraction of star - forming galaxies . despite our extensive spectroscopic sample , there are too few x - ray agns in any individual cluster ( and even supercluster ) to draw statistical conclusions . hence , we divide our structure sample into two categories . the first contains cl0023 and cl1604 which have the highest level of ongoing star formation and starburst activity , as shown in the preceding sections . the second category , consisting of cl1324 , rxj1757 , and rxj1821 , contains structures whose member galaxies are typically quiescent or forming stars at a lower rate . we refer to these two categories as `` unevolved '' and `` evolved '' , respectively , as a means of describing their typical galaxy populations . while we acknowledge that the terms `` more evolved '' and `` less evolved '' would be more appropriate , we adopt the less accurate denominations for brevity . in addition , these terms do not necessarily imply differing levels of cluster dynamical evolution or even a clear temporal sequence from one category to the other . we note that , while some clusters or groups may not fit well with the global characteristics of their parent supercluster ( i.e. cluster a of cl1604 ) , we can not examine all of the agns on a cluster - by - cluster basis . as we will discuss in section [ sec : spatdist ] , many of the agns in the superclusters and the supergroup are not associated with any one cluster or group . so , when analyzing these agns , we take the parent supercluster or supergroup as a whole . the particular segregation of our structure sample is motivated by the clear distinction between the structures shown in figure [ oiivshd ] . the abundant star formation in the unevolved structures suggests the presence of a large gas reservoir in many of their member galaxies . we might expect that this same gas is available for agn fueling conversely , the typical galaxy in the evolved sample has likely consumed most of the available gas in prior star formation episodes , leaving less fuel for the agns . in the following sections , we examine whether the properties of the agn sin the two categories reflects this distinction , and what we can can learn about the relationship of star formation to agn activity in lsss . in this section , we examine the frequency of agn activity within the five structures to determine if there are associated excesses of x - ray point sources . x - ray point source photometry was used to calculate cumulative x - ray source number counts , @xmath171 , using the method of @xcite : @xmath172 where @xmath173 is the number of point sources with fluxes greater than @xmath174 and @xmath175 is the area of the sky in which the _ i_th point source could have been detected at a 3@xmath30 level or higher . variance in @xmath171 was calculated using @xmath176 . to calculate @xmath175 , we used the method of @xcite , which is similar to those used in other literature @xcite . all point sources detected by _ wavdetect _ , in all bands and without significance cuts , were removed and replaced with an estimate of the background using the ciao tool _ dmfilth_. to create a map of the background emission , these images were binned into 32@xmath68 bins . according to equation ( [ eq : detsig ] ) , the flux limit corresponding to a 3@xmath30 detection in this analysis to be consistent with previous work . the inclusion of sources with detection significances between @xmath102 and @xmath103 has a significant effect only on fluxes where sky coverage is below @xmath177 , and the results are generally considered unreliable . ] occuring in one of these binned pixels is given by @xmath178 where @xmath97 is the net counts in a pixel , @xmath179 is the area of a pixel , @xmath94 is the radius of the aperture enclosing 95@xmath7 of an x - ray source s flux , as described in section [ sec : datared ] , @xmath180 is the conversion factor between photon count rate and x - ray flux , also described in section [ sec : datared ] , and @xmath181 is the exposure time of the image . for a given source with flux , @xmath174 , @xmath175 is then equal to the total number of binned pixels for which @xmath174 is greater than @xmath182 , multiplied by the area of a pixel . the cumulative source counts for all the fields in the soft band and the 2 - 10 kev band , hereafter the hard@xmath183 band , are shown in figure [ lognlogsall ] . the latter was extrapolated , field by field , from the 2 - 8 kev band by fitting a power law spectral model with exponent @xmath184to each detected point source . for cl1604 and cl1324 , the source counts for the two pointings were combined . also shown are cumulative source counts measurements from the chandra deep field north and south @xcite . one chandra observation from each deep field was used ( observation i d 582 and 2232 ) . both have exposure times of roughly 130 ks . we re - analyzed these observations using the same reduction pipeline that we used for the orelse fields . we used the combined source counts from these two fields ( hereafter cdf ) to estimate the blank - field counts for comparison with our data . we can see that , in the soft band , cl0023 and rxj1757 appear to be consistent with no overdensity compared to cdf . the other three fields are all overdense to some degree . in the range 3@xmath185 to 10@xmath186 erg s@xmath5 @xmath187 , rxj1821 , cl1324 , and cl1604 have average overdensities of 0.5@xmath30 , 1.0@xmath30 , and 1.5 @xmath30 , respectively , with @xmath30 calculated using the cumulative source count errors from our data and those of cdf . in the hard@xmath183 band , rxj1821 , rxj1757 , and cl0023 all have approximate overdensities of 0.5@xmath30 in the range 7@xmath188 to @xmath189 erg s@xmath5 @xmath187 . rxj1757 appears to be consistent with no overdensity . cl1604 and cl1324 also appear to be overdense , with average overdensities in the flux range @xmath190 to @xmath189 erg s@xmath5 @xmath187 of 1@xmath30 and 1.5 @xmath30 , respectively . it should be noted that the results for cl0023 differ from those presented in the previous work of @xcite . the 2 - 8 kev band cumulative source counts were unintentionally presented as the 2 - 10 kev band counts in that paper . however , we have also found changes due to the different versions of the ciao software used between earlier papers @xcite and this paper , version 3.3 compared to 4.2 . we believe this to be an effect of updated response functions at off - axis angles . the net result for our data is mainly an approximate 5% increase in flux , with some very minor differences in point source detection , the latter being almost entirely below the 3@xmath30 level . in figure [ lognlogscomp ] , co - added cumulative source counts for all five fields are shown . to accomplish this co - addition , all point source lists were first combined into one composite source list . for the calculation of the effective sky area weighting factors , @xmath175 , the combined background of every field was used . with @xmath191 , we can see that the error in @xmath171 will be reduced from its value in any individual field . in the soft band , the composite source counts have an approximate average overdensity of 1@xmath30 in the range @xmath192 to @xmath193 erg s@xmath5 @xmath187 , which falls off to zero brighter than 10@xmath186 erg s@xmath5 @xmath187 . in the hard@xmath183 band , the composite counts have an approximate average overdensity of 1@xmath30 between @xmath194 and @xmath195 erg s@xmath5 @xmath187 , but an approximate underdensity of 0.5@xmath30 between @xmath196 and @xmath197 erg s@xmath5 @xmath187 . while there is substantial variation in the five cumulative source count measurements for the individual fields , we can see from the composite measurement that , on average , the fields of the five structures studied here have a density of x - ray point sources in excess of the control field , though not significantly so . * , hereafter c05 ) have found a positive correlation between cluster redshift and source count overdensity . to compare to this , and our earlier results of @xcite , we first fit the cumulative source counts to a power law of the form @xmath198 , using the maximum likelihood method of @xcite and @xcite . this method fixes the dimensionless variable @xmath180 while fitting for @xmath163 . @xmath180 is calculated by requiring consistency between the model and the data at the flux point @xmath199 . we use @xmath200 erg s@xmath5 @xmath187 and @xmath201 erg s@xmath5 @xmath187 for @xmath202 in the soft and hard@xmath183 bands , respectively , which are the values used by @xcite . c05 measured overdensities using the ratio of @xmath180 in a given field to @xmath180 in a reference set of five blank fields . for these fields , they found @xmath203 and @xmath204 . we calculated @xmath180 for the five fields individually , as well as the composite of all five fields . it should be noted that the composite measurement is not a simple average of the five fields ( see above for details on its creation ) and that significant variation in the individual @xmath180 values come from calculating @xmath180 at a single flux point . our results are shown in figure [ capplot ] for the hard@xmath183 band , where our data are overplotted on that of c05 and the linear fit to their data is shown . just as in @xcite , our data are consistent with their fit . once again ( refer to earlier explanation ) , it is posssible that the issue with the version of the ciao software may have created a systematic offset from c05 s data set . we would expect this offset to increase the value of @xmath180 for the co5 data by about 5@xmath7 . even without this correction , our results are still consistent with c05 within our errors . however , we note that it is difficult to use these overdensities to interpret the actual agn activity in an individual structure , even with our large spectroscopic sample ( see section [ sec : agn ] ) . using optical sources with redshifts with quality flags of @xmath77 or 4 and the results of our optical matching , we were able to identify x - ray sources that are members of the clusters or superclusters in our sample ( see table [ srcsum ] ) . in summary , we found ten confirmed agns in the bounds of cl1604 , seven in cl0023 , six in cl1324 , three in rxj1821 , and one in rxj1757 . note that these numbers include four sources , one in each structure except rxj1757 , that were detected at a @xmath2053@xmath30 ( but at a @xmath1332@xmath30 ) level in at least one of the three x - ray passbands ( see table [ agntab ] ) . we show in the following that these low - significance detections do not bias our results . + + lcccccccccc cl0023 & 1 & 00 24 10.9 & + 04 29 23 & 0.823 & 5.6 & 46.7 & 52.3 & 12.2 & ... & -1.86 + cl0023 & 2 & 00 24 15.5 & + 04 23 09 & 0.829 & 7.8 & & 7.8 & 10.1 & 1.38 & 0.67 + cl0023 & 3 & 00 23 54.9 & + 04 25 24 & 0.830 & 0.5 & 5.1 & 5.6 & 2.6 & 1.00 & 0.33 + cl0023 & 4 & 00 24 09.4 & + 04 22 41 & 0.841 & 24.3 & 35.8 & 60.1 & 39.9 & 0.71 & -6.08 + cl0023 & 5 & 00 23 52.2 & + 04 22 59 & 0.844 & 24.8 & 83.8 & 108.6 & 63.7 & 0.25 & -2.85 + cl0023 & 6 & 00 23 45.6 & + 04 22 59 & 0.850 & 10.8 & 17.0 & 27.8 & 23.2 & 0.48 & -4.70 + cl0023 & 7 & 00 24 07.6 & + 04 27 26 & 0.854 & 3.1 & 3.4 & 6.5 & 3.9 & 2.45@xmath206 & -1.72 + cl1604 & 1 & 16 04 23.9 & + 43 11 26 & 0.867 & 14.5 & 22.3 & 36.7 & 27.8 & 1.23 & -4.43 + cl1604 & 2 & 16 04 25.9 & + 43 12 45 & 0.871 & 5.5 & 8.7 & 14.2 & 10.4 & 0.66 & -1.48 + cl1604 & 3 & 16 04 15.6 & + 43 10 16 & 0.900 & 17.0 & 31.4 & 48.3 & 31.2 & 1.90@xmath206 & -1.84 + cl1604 & 4 & 16 04 37.6 & + 43 08 58 & 0.900 & 0.9 & 4.4 & 5.3 & 2.4 & 2.44 & ... + cl1604 & 5 & 16 04 06.1 & + 43 18 07 & 0.913 & 18.3 & 29.1 & 47.4 & 19.3 & 1.08@xmath206 & -1.02 + cl1604 & 6 & 16 04 36.7 & + 43 21 41 & 0.923 & 6.3 & 18.1 & 24.4 & 9.8 & 0.36 & -3.64 + cl1604 & 7 & 16 04 01.3 & + 43 13 51 & 0.927 & 12.3 & 25.0 & 37.3 & 18.6 & 1.03 & ... + cl1604 & 8 & 16 04 05.1 & + 43 15 19 & 0.934 & 3.6 & 4.7 & 8.4 & 4.3 & 0.29 & -1.07 + cl1604 & 9 & 16 04 10.9 & + 43 21 11 & 0.935 & 0.9 & 10.7 & 11.6 & 4.0 & 1.87@xmath206 & 0.97 + cl1604 & 10 & 16 04 08.2 & + 43 17 36 & 0.937 & 5.0 & 24.9 & 30.0 & 7.1 & 0.84 & -2.47 + cl1324 & 1 & 13 24 51.4 & + 30 12 39 & 0.660 & 1.6 & 0.7 & 2.3 & 4.1 & 0.44@xmath206 & -2.85 + cl1324 & 2 & 13 24 36.4 & + 30 23 16 & 0.662 & 1.0 & 6.8 & 7.8 & 4.2 & ... & -2.66 + cl1324 & 3 & 13 25 04.5 & + 30 22 07 & 0.696 & 1.2 & 0.6 & 1.8 & 2.1 & ... & -2.53 + cl1324 & 4 & 13 24 52.9 & + 30 52 18 & 0.697 & 1.4 & & 1.4 & 3.3 & 2.60 & -1.79 + cl1324 & 5 & 13 24 52.0 & + 30 50 51 & 0.700 & & 10.9 & 10.9 & 6.2 & ... & -1.23 + cl1324 & 6 & 13 24 28.8 & + 30 53 20 & 0.778 & 1.6 & 0.8 & 2.4 & 3.0 & 2.90@xmath206 & -0.24 + rxj1821 & 1 & 18 21 07.7 & + 68 23 38 & 0.813 & 1.9 & 2.5 & 4.4 & 2.7 & 2.17 & -0.51 + rxj1821 & 2 & 18 21 23.9 & + 68 26 33 & 0.822 & 3.4 & 4.1 & 7.5 & 5.5 & 0.67 & -1.20 + rxj1821 & 3 & 18 21 27.0 & + 68 32 34 & 0.824 & 8.6 & 10.6 & 19.3 & 10.3 & 2.13 & -0.66 + rxj1757 & 1 & 17 57 25.2 & + 66 31 50 & 0.693 & 1.9 & 5.1 & 7.1 & 6.1 & 0.24 & -2.11 + [ agntab ] examining the spatial distribution of agns located within each cluster can give insight into what processes triggered their nuclear activity . in figures [ allspats0 ] , [ allspats ] , and [ allspats2 ] , we show the spatial distributions on the sky and redshift distributions of the five structures studied here . the agns are marked in red , and their positions and characteristics are given in table [ agntab ] . of particular note is the lack of agns in dense cluster centers . indeed , we find @xmath207 of agns in cluster cores , defined as being within a projected distance of 0.5 mpc to the nearest cluster or group . an additional @xmath208 lie on the outskirts of clusters ( projected distances between 0.5 and 1.5 mpc ) , and a majority ( @xmath209 ) of agn host galaxies lie more than 1.5 mpc in projected distance from the nearest cluster or group . these results are consistent with previous work studying the spatial distribution of agns , in that the agns tend to be located outside of clusters or in their outskirts . first , the distribution of agns in cl1604 , previously studied by @xcite , is consistent with the other four structures here . our results are also consistent with those of @xcite , who found that agns in the a901/902 supercluster at @xmath210 tend to avoid the densest areas . several other studies have also found that x - ray agns tend to reside in regions of moderate density similar to group environments , up to @xmath211 @xcite . it is thought that regions of intermediate density , such as the outskirts of clusters , are the most conducive to galaxy - galaxy interactions because of the elevated densities , compared to the field , but relatively low velocities @xcite . since we find more of the agns in these areas , this lends support to the theory that mergers or tidal interactions are one of the main instigators of agn activity . since only @xmath212 inhabit dense cluster cores , processes which preferentially occur in these regions , such as ram pressure stripping , are probably not responsible for triggering agns in cluster galaxies . however , the association between agns and these regions could also be related to higher gas availability in galaxies farther from cluster cores . although we find that most of the x - ray agns do not reside in the cluster cores , a number of studies have measured the fraction of cluster galaxies that host agn ( e.g. , * ? ? ? therefore , we attempt to measure the agn fraction for the individual clusters within the five structures in our sample . because we are limited by the spectral completeness of our sample , we make a composite measurement of the most massive , well - sampled clusters : cl1604a , cl1604b , cl1604c , cl1604d , cl1324a , cl1324i , and rxj1821 . we compare to the results of @xcite for low - redshift clusters ( @xmath213 ) , who used galaxies with @xmath214 , within approximately 2000 km s@xmath5 of the mean redshift of cluster members for each system , and within the field of view of their _ chandra _ observations , which ranged from 1.2 to 4.5 mpc in width . to approximate these criteria , we adopt a magnitude cutoff that roughly corresponds to @xmath215 and use galaxies within 1 mpc and @xmath216 of the cluster centers and redshifts . we measure the combined agn fraction for the seven clusters listed above to be 0.012@xmath217 . this is consistent with the results of @xcite , who measured @xmath218 . we note , however , that a number of arbitrary definitions went into our measurement . in addition , our spectroscopy of optical counterparts to x - ray sources is significantly incomplete , in contrast to @xcite . correcting for this incompleteness contributes the largest source of error to our measurement . because of the large uncertainties , we refrain from drawing any conclusions from our measurement . figure [ cmds ] presents cmds , which are described in section [ sec : globchar ] , for all five fields . confirmed agn members of the structures are shown with blue squares . while the agns in cl1324 , rxj1821 , and rxj1757 preferentially reside within the bounds of , or very close to , the red sequence , the agn hosts in cl0023 and cl1604 are more spread out . previous work , including @xcite and a number of wide - field surveys @xcite , has found an association between agn activity in galaxies and the transition onto the red sequence ( possibly for a second time ) in the green valley . while these galaxies could be evolving from the blue cloud onto the red sequence , it is also possible that they could have moved down off the red sequence after a tidal interaction or merger and are evolving back @xcite . we note , however , that some studies using mass - selected samples have found that agn hosts have a color distribution more similar to that of normal galaxies @xcite . in addition , @xcite , using a sample of galaxies with redshifts @xmath219 , have found that many green valley agn hosts are dust - reddened blue cloud members , so that agn host colors acquire the bimodality apparent in the general galaxy population . however , @xcite have also examined the impact of extinction on the colors of agn hosts and did not find a significant impact for galaxies in the redshift range @xmath219 , although bimodality may be introduced at higher redshifts . to address this issue for our study , we are planning to implement spectral energy distribution ( sed ) fitting to evaluate the impact of extinction on the broadband colors of our sample . preliminary results from sed fitting of the cl1604 hosts suggest that extinction levels in our sample are not as drastic as those presented by @xcite . it is also possible that agn host colors are contaminated by the agns themselves . however , this is unlikely because ( 1 ) almost all agn hosts in our sample have rest - frame x - ray luminosities below the quasi - stellar object ( qso ) level of @xmath220 erg s@xmath5 ( see sec [ sec : xlum ] ) and ( 2 ) @xcite found that , in cl1604 , agn hosts with blue cores did not have a rising blue continuum indicative of qso activity . therefore , we proceed to investigate the agn association with the transition zone and to explore differences in the evolutionary states of their host galaxies in each field by examining color offsets of the agn hosts from the red sequence . histograms of offsets from the center of the red sequence are shown in figure [ rsoffsets ] . the first two panels present offsets in terms of color . in the top panel , only the structures where lfc data were used are shown , which is every field except cl1604 . the middle panel shows only cl1604 , for which we used acs colors . in the bottom panel , all five structures are shown , with normalized offsets . in order to compare the acs and lfc data , we scale by the red sequence width , @xmath221 . we define @xmath221 as the distance from the center of the red sequence fit to its boundary ( see section [ trsatbp ] and figure [ cmds ] ) . on this plot , agn hosts on the red sequence will then be located between -1.0 and 1.0 . with red sequence offsets , we can quantitatively examine the green valley . in figure [ rsoffsets](b ) , we plot a histogram of rs offsets , measured from the acs data , for the agns in the cl1604 supercluster . for comparison , we overplot a scaled distribution of rs offsets for all spectroscopically confirmed supercluster members with acs photometry . in the scaled histogram , we can clearly see an area of reduced number density between the red sequence and blue cloud . for cl1604 , the green valley can be approximated as the region @xmath222 , where @xmath223 is the offset of an agn host from the center of the red sequence . only @xmath224of all confirmed supercluster members with acs data fall within this region . however , five out of eight of the cl1604 agn hosts with acs data reside within it . while it is unclear how well this definition of the green valley extends to lfc data , because of larger photometric errors , we can see in figure [ rsoffsets]c that @xmath1 of all host galaxies have @xmath225 and @xmath226 are in the range @xmath227 . while many galaxies in both the evolved and unevolved structures lie in the ` green valley ' region , the percentage of agns on the red sequence is somewhat higher in the evolved structures compared to the unevolved structures , with 30@xmath7 and @xmath177 , respectively . examining figure [ rsoffsets ] , we can see the distribution of agn hosts in evolved structures is clustered closer to the red sequence , while in the unevolved structures , this distribution has a large tail extending into the blue cloud . indeed , none of the agn host galaxies in the evolved structures have @xmath228c @xmath229 , whereas four of the x - ray agn hosts in the unevolved structures have red - sequence offsets below this limit . although these results are suggestive ( and unaffected by our inclusion of the @xmath205 3@xmath30 x - ray sources ) , the two distributions are not statistically different based on the k - s test . morphological analysis by @xcite has shown that @xmath230 of the x - ray agns in cl1604 have had recent mergers or tidal interactions , which could fuel star formation through starburst events . more recent mergers or interactions are one possible explanation for some of the color differences that we see between the agn host galaxies in the evolved and unevolved structures . in particular , we find that nine of the agn host galaxies are members of a kinematic close pair with a relative line - of - sight velocity of @xmath231 km s@xmath5 and projected physical separation ( on the plane of the sky ) of @xmath232 h@xmath233 kpc ( e.g. , * ? ? ? * ) . two are in cl0023 , five in cl1604 , one in cl1324 , and one in rxj1821 ( see table [ agntab ] ) . those agn hosts in pairs include three out of the four galaxies with the largest red sequence offsets ( i.e. , the bluest ) , all of which are members of the unevolved structures . based on their @xmath63 magnitudes or measured stellar masses ( in the case of the cl1604 members ; see @xcite ) , seven out of the nine kinematic pairs have flux or mass ratios of @xmath234 , implying a major merger scenario . the differences in color and , perhaps , merger activity are likely related to the increased level of star formation and starburst activity in the unevolved compared to evolved structures ( see section [ sec : specprop ] ) . to explore the connection between the agn and star formation history , we can use our high - resolution spectroscopy to examine the average spectral properties of their host galaxies . we measure the average spectral properties of the agn host galaxies in the five orelse structures using three composite spectra : one comprised of six agns from cl0023 km s@xmath5 ) which would dominate a composite spectrum and was therefore not included . while the other has a very broad line , the other features are narrow and do not dominate the composite spectrum . as a result , it was included . ] , one comprised of all ten agns from cl1604 , and one comprised of the ten agns from the combined fields of cl1324 , rxj1821 , and rxj1757 . the last spectrum combines all the evolved structures , necessitated by the low number of agns in rxj1821 and rxj1757 compared with the other structures . these three composite spectra are shown in figure [ csplot ] , and measurements of spectral features are listed in table [ co - addtab ] . first , we can see that the agn hosts in all three groupings have substantial [ ] emission . in cl1604 and the evolved structures , most of this emission is probably from the agns , rather than star formation . six of the ten agn host galaxies in our cl1604 sample were analyzed by @xcite using the keck ii near - infrared echelle spectrograph ( nirspec ; * ? ? ? * ) . for five out of the six targets , @xcite found that the [ ] /h@xmath163 flux ratio was too large for a normal star - forming galaxy , which implies that agns are the dominant contributor to [ ] emission @xcite . the cl1324 , rxj1821 , and rxj1757 structures have agns mostly near or on the red sequence . in addition , we found in section [ sec : globchar ] that star formation is low in all three structures ( @xmath235of all galaxies are star - forming ) . because of this low star formation rate , and because @xcite found that @xmath236 of red [ ] emitters are dominated by liner / seyfert emission , it is likely that most of the [ ] emission from the agn host galaxies in the evolved structures comes from the agns as well . deciphering the origin of the [ ] emission in the agn host galaxies in cl0023 is not as straightforward . while some agn host galaxies in cl0023 are on the red sequence , the structure also has the bluest host galaxies in our sample also , as discussed in section [ sec : globchar ] , there is significant star formation in the general population of cl0023 . while this is also true of cl1604 , the nirspec results of @xcite showed most of the [ ] emission in the agn host galaxies in that structure comes from the agns . however , we do not have any near - ir spectroscopic data for the cl0023 agns , so we must use other means to determine the emission source . the average [ ] / [ ] ratio of the cl0023 x - ray agn hosts is 0.429 , typical of type-2 agn emission , emission from metal - poor star formation , or a superposition of the two processes @xcite . note that cl1604 and the evolved structures have values of the [ ] / [ ] ratio of 0.549 and 0.229 , respectively , also typical of type-2 agn emission . combining this result with the blue colors of the cl0023 hosts and the high fraction of star - forming galaxies , it is likely that the observed [ ] emission of the agn hosts in cl0023 is due to a combination of normal star formation and type-2 agn activity . examination of the balmer features reveals further insight into the star formation histories of the agn host galaxies . specifically , based on a single stellar population model , the ew(h@xmath2 ) rises quickly from zero after a starburst in a galaxy , peaks after about 300500 myr , and then declines back to approximately zero at @xmath3 gyr after the burst @xcite . infill can complicate the interpretation of this feature , so care must be taken when measuring the line strength . for the composite spectrum of the cl0023 agn hosts , the equivalent width of h@xmath2 , attempting to correct for infill , is consistent with zero . however , we observe strong emission from other balmer features , ew@xmath237 line , figure [ csplot ] was not drawn out to this range . however , approximately two - thirds of agns in the cl0023 structure do have spectral coverage for h@xmath238 , and the measurement presented here represents the average value for these galaxies . ] and ew@xmath239 , suggesting that emission infill has a significant effect on the measured ew(h@xmath2 ) . this infill could be due to emission from regions , emission from agns or some other liner processes , or from continuum emission produced by o stars . since we observe h@xmath238 and h@xmath240 in emission , it is unlikely that o stars are solely responsible for the observed ew(h@xmath2 ) . the balmer emission lines observed in the cl0023 agn hosts are not broadkm s@xmath5 . ] and are quite strong . in the average type-2 agn , a large fraction of the balmer emission originates from star formation @xcite , which suggests ongoing star formation in the cl0023 hosts . compared to cl0023 , h@xmath240 emission from the cl1604 hosts is low , with ew(h@xmath240 ) = -0.69@xmath241 , which is consistent with a lower level of star formation in the supercluster ) could not be measured for a majority of galaxies in the cl1604 host sample . ] . balmer lines for the evolved structures are in absorption , ew(h@xmath238 ) = 0.48@xmath242 and ew(h@xmath240 ) = 1.41@xmath243 , consistent with an even lower level of star formation . these results combined with the earlier result analyzing the average [ ] / [ ] ratios of the agn hosts strongly indicates that star formation is occurring in the cl0023 galaxies coevally with agn activity , while less star formation activity is occuring in the other structures . lcrccc cl0023 & 6 & @xmath244 & @xmath245 & @xmath246 & @xmath247 + cl1604 & 10 & @xmath248 & @xmath249 & @xmath250 & @xmath251 + evolved sys . & 10 & @xmath252 & @xmath253 & @xmath254 & @xmath255 [ co - addtab ] the h@xmath2 equivalent widths , combined with measurements of the 4000 break , suggest that starbursts have occurred more recently in the average cl1604 agn host compared to those in the evolved structures . larger values of d@xmath164(4000 ) indicate a more passive galaxy , with an older average stellar population , which could mean that more time has passed since the cessation of star formation @xcite . the ew(h@xmath2 ) measured from the cl1604 composite is @xmath256 , roughly half of that from the evolved structures composite . similarly , the cl1604 composite has a lower value for d@xmath164(4000 ) than those in the evolved structures , indicating that the average cl1604 host is more actively star - forming or has a younger stellar population on average . this result is supported by the bluer colors of the cl1604 agn hosts compared to those in the evolved structures . all of these results imply the average cl1604 host has had a starburst more recently than the average agn host in the evolved structures . the agn host composite spectra for cl0023 has a particularly low d@xmath164(4000 ) measurement ( see table [ co - addtab ] ) . since we found that this composite spectra ( which excluded one broadline source ) was consistent with type-2 agns , the agns themselves should not contribute most of the blue continuum . this points to a stellar source , particulary o and b stars . we would then expect significant star formation in the cl0023 hosts within the last 10 - 100 myr , as indicated by the other spectral features as well . related to the d@xmath164(4000 ) measurement , the h+h@xmath257 and the k lines also provide information on star formation . for f , g , and k stars , the ratio of these lines is constant , while the h+h@xmath257/ k ratio increases for a and b stars as the overall strength decreases and the h@xmath257 strength increases @xcite . the h+h@xmath257/ k ratio is @xmath251 for cl1604 and @xmath255 for the evolved structures ( see table [ co - addtab ] ) , consistent with the evolved structures having ( on average ) older stellar populations . we do see a decrease in the overall strengths of both lines in the average spectrum of the cl0023 host galaxies relative to the other structures ; however , we actually measure a dramatic decrease in the ratio for cl0023 ( @xmath247 ) , the opposite of what is expected from a population of a and b stars . the most likely explanation is significant h@xmath257 emission , which would be in concert with the other observed balmer emission . the h@xmath257 emission could be coming from some combination of agns and regions , which would support previous conclusions about the level of activity in the cl0023 hosts . altogether , the composite spectra of the agn hosts in all three bins of figure [ csplot ] show that the average host galaxy has ongoing star formation or has had star formation within the last @xmath162 gyr . however , the hosts in cl1604 and the evolved structures each have , on average , less ongoing star formation than cl0023 , as evidenced by larger values of d@xmath164(4000 ) and the absence of the balmer emission that is observed in cl0023 . these differences suggest a progression in the temporal proximity of the last starburst event , with the hosts in cl0023 having significant ongoing star formation characteristic of a current starburst , to those in cl1604 and the evolved structures each having successively more time since the last significant starburst event . we have confirmed that none of our results based on the composite spectra are changed by removing the four lowest significance ( @xmath2053 @xmath30 ) x - ray sources from our sample , with most spectral measurements remaining the same within the errors . with our analysis of the composite spectra of agn host galaxies in the different structures , we can compare the average properties of these hosts with the average properties of all spectroscopically confirmed galaxies within the same structures . in section [ sec : globchar ] , we found that the galaxy populations in the evolved structures were largely quiescent , with little or no contribution from starburst or post - starburst galaxies . in contrast , the populations in the unevolved structures were comprised of large fractions of star - forming galaxies , with a more significant contribution from starburst or post - starburst galaxies . when comparing these results to the average spectral properties of the agn hosts , we find that , in _ all _ cases , the average agn host galaxy has a _ younger _ stellar population than the average galaxy in the parent structure , irregardless of the evolved or unevolved classification . this result holds even when comparing to member galaxies outside the dense cluster cores ( @xmath258mpc ) , where the vast majority of x - ray agns reside . the most prominent difference comes from the evolved structures where their average agn host galaxy has significantly larger ew(h@xmath2 ) and smaller d@xmath164(4000 ) than the average structure member , indicative of a post - starburst galaxy with a substantial star - formation event within the last @xmath259gyr . such galaxies make a small contribution to the overall population in the evolved structures , which is largely quiescent . while we do observe clear differences between the spectra of the agn hosts , with those in cl0023 having significant ongoing star formation to those in the evolved structures having the most time since the last significant starburst event , these differences are not nearly as pronounced as the differences between the average galaxy in each structure . this suggests that agn activity has a common origin associated with current or recent star - formation . in this section , we explore the differences between the structures and agn host properties based on the x - ray luminosities of the confirmed agns . we calculate rest - frame luminosities for x - ray point sources with known redshifts . k - corrections were carried out using the power law spectral models for sources , with a photon index of @xmath260 , described in section [ sec : red ] . luminosities are measured in the x - ray soft , hard , and full bands . a histogram of full - band rest - frame luminosity , binned by evolved and unevolved structures , is shown in figure [ xlums ] . the luminosity distributions in the soft and hard bands are similar to the one shown . in the left panel of figure [ xlums ] , we can see that the agns in the unevolved structures have higher x - ray luminosities than those in the more evolved structures . k - s tests show that the distributions of the two bins are different at the 99@xmath7 level in each of the three bands . this statistically - significant result is independent of our inclusion of the four low - significance ( @xmath205 3@xmath30 ) x - ray sources . while 10 out of 17 agns in the unevolved structures have full band luminosities above @xmath261 ergs s@xmath5 , there are _ no _ agns above this limit in cl1324 , rxj1757 , or rxj1821 . we follow a bayesian approach , using poisson statistics to calculate the likelihoods , to estimate the probability of finding no high @xmath262 agns in the evolved structures ( @xmath263 ) given the detection rate in the unevolved structures . specifically , we calculate @xmath264 . here , @xmath265 is the number of high @xmath266 agns that are spectroscopically confirmed members in the unevolved structures , and @xmath267 and @xmath268 are the total number of high @xmath266 agns that were targeted for spectroscopy in the unevolved and evolved structures , respectively . based on this calculation , the probability of finding no high @xmath262 agns in the evolved structures is only 0.25% . this result is likely related to the smaller fractions of blue cloud galaxies ( and overall more quiescent populations ) in cl1324 , rxj1757 , or rxj1821 and , thus , the unavailability of large gas reservoirs . at the faint end , there are four sources in the evolved structures below @xmath269 ergs s@xmath5 , all of which are members of cl1324 . however , the x - ray source counts , in all five fields , are significantly incomplete at these luminosity levels with only @xmath270 of optically matched x - ray sources below this limit , most of which are @xmath205 3@xmath30 detections . therefore , we can not say anything definitive about the lack of faint sources in the unevolved structures ; however , if we remove the four low - luminosity sources , the difference between the luminosity distributions in the evolved and unevolved samples is still significant at a 95% level according to the k - s test . the reasonably high significance is clearly due to the lack of high - luminosity sources in the evolved structures . in the right panel of figure [ xlums ] , we plot the full - band rest - frame luminosity versus the red - sequence offset , scaled by rs width ( see section [ sec : hgalcolan ] ) . from this figure , we can see that all agns with host galaxies on the red - sequence have lower x - ray luminosities , all below @xmath271 ergs s@xmath5 ( region 4 ) . this result is not unexpected since agn activity should diminish as the host galaxy moves onto the red sequence . we can also see that 60% of the confirmed agns lie in the green valley ( @xmath272 ; regions 2 and 3 ) . most interestingly , in the green valley there are almost two orders of magnitude variation in x - ray luminosity , with the unevolved structures having all of the highest @xmath273 sources . we try to decipher the origin of these variations in section [ sec : xlums ] . we also do not detect any low @xmath266 blue agns ( region 5 ) or any high @xmath266 red agns ( region 6 ) , not necessarily unexpected given their expected gas contents . however , it is difficult to say for certain if these null results are significant . we do sample optically matched x - ray sources in these regions . specifically , in the five fields , there are a total of 85 sources ( at @xmath133 3@xmath30 ) in region 5 ( @xmath274 and @xmath275 ergs s@xmath5 ) , of which we have targeted 24 ( 28% ) . here , we estimate the x - ray luminosities by assuming that all sources in a particular field are at the mean redshift of the structure , and we choose the lower limit of @xmath276 ergs s@xmath5 so as not to be adversely affected by incompleteness ( see above ) . based on @xcite , the x - ray luminosity function of field agns at similar redshifts in `` optically normal '' galaxies ( comparable to the vast majority of our agn hosts ) shows increasing number densities down to @xmath277 ergs s@xmath5 . as a result , we would naively expect to detect a larger number of fainter x - ray sources in the blue cloud . using the bayesian approach described in section [ sec : xlum ] , the probability of finding no agn in region 5 is 3% , given our success rate of confirming cluster members for the high @xmath266 blue galaxies ( region 1 ) . this formal probability may , in fact , be an _ upper limit _ as we would expect a higher success rate given the larger number densities at fainter x - ray luminosities . the fact that we observe no low @xmath266 blue galaxies in our sample may suggest that , in high - density environments compared to the field , either ( 1 ) the time to reach the highest x - ray luminosities is shorter after agn turn - on or ( 2 ) the host galaxies are transformed more quickly , moving to redder colors by the time their x - ray luminosities drop to lower levels . similarly , there are 25 sources in region 6 ( @xmath278 and @xmath279 ergs s@xmath5 ) , of which we have targeted 7 ( 28% ) . if we assume our success rate for confirming cluster members as measured from all high @xmath266 agns ( regions 1 and 2 ) , the probability of finding no agn in region 6 is 18% . although not a significant result , the conclusions for this region are , of course , more obvious as we do not expect any true red - sequence ( i.e. , non - dust reddened ) galaxies to have enough cold gas to fuel a luminous agn . to explore the origin of the variations observed in figure [ xlums ] , we examine the average spectral properties of the host galaxies within the four distinct regions . specifically , the sample is split by x - ray luminosity at @xmath280 and red - sequence offset , scaled by rs width , to delineate regions containing high @xmath266 blue cloud ( region 1 ) , high @xmath266 ( region 2 ) and low @xmath266 ( region 3 ) green valley , and low @xmath266 red - sequence ( region 4 ) host galaxies . in figure [ specreg ] , we plot the measured ew(h@xmath2 ) versus d@xmath164(4000 ) from the spectral composites in the four regions . for comparison , we also plot post - starburst temporal regimes derived from four @xcite models , described in section [ sec : specprop ] . the small d@xmath281(4000 ) and h@xmath2 in emission indicate that the high @xmath266 blue hosts are coeval with the starburst or ongoing star formation . as we examine galaxies in regions 2 to 4 going from the high to low @xmath266 hosts in the green valley to the low @xmath266 hosts in the red sequence , the time since the burst gets progressively larger . while there is some degeneracy between time since burst and burst strength , it is clear that the low @xmath266 green valley hosts are either ( 1 ) further along since the burst than their high @xmath266 counterparts or ( 2 ) had a weaker initial burst which could explain their lower x - ray luminosities as less gas would likely be funneled to the center . our results are robust to removing the four lowest significance ( @xmath205 3@xmath30 ) x - ray sources , as well as the four x - ray sources below @xmath276 ergs s@xmath5 , where in both cases we are highly incomplete . the most striking results from this spectral analysis are , first , that the average agn host in _ every _ region is either in the process of having a starburst or has had one within last @xmath282 gyr . this global finding clearly demonstrates the close connection between starburst and agn activity as normal star formation does not typically produce the h@xmath2 values seen in these host galaxies . second , we do not detect high x - ray luminosity , young ( as indicated by time since starburst ) galaxies in the evolved structures . this result implies that the entire galaxy population in these structures ( certainly in the isolated , x - ray selected ones ) are more advanced , suggesting that the peak of gas consumption , seen in both star formation and agn activity , occurred at an earlier time . we studied agn activity in five high - redshift clusters and superclusters in the redshift range @xmath283 . before identifying individual agns , we analyzed the structures using the statistical measure of cumulative source counts . we found every structure to have x - ray excesses of 0.5 - 1.5 @xmath30 with respect to the cdfn and cdfs control fields . the method is highly dependent on the field used as a sky estimate , which makes comparing results between studies difficult . @xcite measured cumulative source counts in a number of structures in the range @xmath284 and found a dependence on redshift . while our data are consistent within the errors with the results of c05 , our redshift range is not large enough to evaluate the relation . we note that it is difficult to use these overdensities to interpret the actual agn activity in an individual structure , even with extensive spectroscopy , as we have attempted to do here with our large spectroscopic sample . we recommend caution in using this technique , as its precision can easily be overestimated . we employed a maximum likelihood technique to match x - ray sources to optical counterparts . with an extensive deimos optical spectroscopic campaign with @xmath285 targets , accurate redshifts have been obtained for 126 of these x - ray sources , allowing us to identify a total of 27 agns within all of the structures . these results show that significant spectroscopy is needed to confirm even small numbers of agn members . we find that the spatial distribution of the agn is largely consistent with previous work at lower redshift . across all five structures , we find that agn host galaxies tend to be located away from dense cores ( within 0.5 mpc of a cluster or group center ) , with many instead located on the outskirts of clusters or poorer groups km s@xmath5 . ] . previous studies have found similar results up to @xmath211 @xcite . these intermediate environments are thought to be conducive to galaxy - galaxy interactions , because of the relatively high densities compared to the field and low velocity dispersions compared to cluster cores @xcite . our results would then lend support to these interactions as the trigger of x - ray agn activity in the environments of lsss . however , these may also be regions where gas availability is higher in the member galaxies , which could lead to increased agn activity . with optical counterparts to x - ray sources identified , we were able to analyze the color properties of agn hosts . our analysis showed that agn host galaxies are overrepresented in the green valley . in cl1604 , only @xmath286 of all the supercluster members with acs data were within @xmath287 of the lower boundary of the red sequence , where @xmath221 is the width of the red sequence , defined in section [ sec : globchar ] . however , five out of the eight agn hosts in our acs pointings were in this range . in fact , @xmath1 of host galaxies in all five structures lie within @xmath221 of the lower boundary of the red sequence and @xmath226 are within @xmath287 . our results are supported by other studies which have found an overabundance of agn activity in the green valley @xcite . since the green valley is thought to be a transitional region for galaxies @xcite , these results suggest x - ray agns in lss are a transitional population between blue star - forming and red quiescent galaxies . however , we note that , while our sample is magnitude - limited , several studies using mass - selected samples have found less overrepresentation of agn hosts in the green valley @xcite . additionally , @xcite have found that @xmath8of agn hosts in the green valley are dust - reddened blue cloud galaxies , although these results conflict with the recent studies of @xcite . our results from preliminary sed fittings to agn hosts suggest that the effect is not as drastic in our sample . the five structures studied in this paper occupy a range of evolutionary states . based on the [ ] and h@xmath2 features of the composite spectra for each structure , we grouped our sample into the least evolved structures ( `` unevolved '' ) , cl1604 and cl0023 , and the most evolved structures ( `` evolved '' ) , cl1324 , rxj1821 , and rxj1757 . this distinction is based on the average stellar populations and the presence , or lack thereof , of current star formation . with these two categories , we sought to explore differences in agn activity between structures with different galaxy populations . we did not find any significant differences between the five structures when examining the cumulative source counts or the spatial distributions of the agn hosts . however , the agn host galaxies in the unevolved structures were skewed more towards bluer colors , although this was not at a statistically significant level for this sample size . we did , however , find significant differences between the subsets when examining the x - ray luminosities of the agns and the optical spectra of their hosts . we found that agns in the unevolved structures tend to have higher full band ( 0.58 kev ) x - ray luminosities relative to those in the evolved structures at a @xmath137 level , with all of the most luminous agns ( @xmath279 erg s@xmath5 ) found in the unevolved structures . while _ all _ agn host galaxies either have on - going star formation or have had a starburst within the last @xmath282 gyr , the host galaxies in the unevolved structures are distinctly younger than those in the evolved structures , with shorter times since the last starburst as indicated by smaller average ew@xmath288 and d@xmath164(4000 ) in their composite spectra . the average cl0023 host has current star formation , and the average cl1604 host has had a burst within the last @xmath289 myr . we do not detect any of these young , high x - ray luminosity agns in the evolved structures , implying that the peak of both star formation and agn activity occurred at an earlier time . we note that , regardless of whether they are members of the evolved or unevolved structures , _ all _ agn host galaxies are younger than the average galaxy in their parent population . we also find a large ( two orders of magnitude ) variation in x - ray luminosity for agns within the green valley , while agns in the red sequence have consistently lower luminosities ( @xmath271 erg s@xmath5 ) . as we move from the high to low @xmath266 green - valley hosts to low @xmath266 red - sequence hosts , the time since starburst gets progressively longer . although there is some degeneracy between burst strength and time since burst , the low @xmath266 green valley hosts are either further along since the burst than their high @xmath266 counterparts or have had a weaker initial burst , which may explain their lower x - ray luminosities . the higher agn x - ray luminosities in the unevolved structures are most likely related to their bluer colors and , hence , larger reservoirs of gas that could lead to higher levels of black hole accretion and the higher x - ray luminosities . in addition , both the x - ray and spectral results can be explained if these galaxies had more recently undergone merger - induced , or other , starburst events . specifically , simulations and observations have found that agn activity peaks soon after maximum star formation in a starburst event ( @xmath60.1 - 0.25 gyr ) @xcite . as the agn x - ray luminosity declines after reaching its peak , the star formation rate should be declining as well . this could explain why the most x - ray luminous agns are in the least evolved structures . agn host galaxies in cl0023 were also found to have significant ongoing star formation , which could mean these galaxies have had the most recent merger events , where star formation is still near its peak . the x - ray luminosity differences between the agns in the evolved and unevolved structures could also be viewed as a transition from quasar mode emission towards radio mode emission , as defined in @xcite and @xcite , which is also related to the star formation rate . since agn feedback deters gas from falling into the core and depletes the cold gas in a galaxy , the fuel for both star formation and further agn emission is decreased . ultimately , the dominant fuel for black hole accretion transitions from cold gas funneled to the galactic core from the starburst event to hot halo gas , which leads to a much more quiescent state ( * ? ? ? * ; * ? ? ? * ; for a similar model , see also , e.g. , @xcite , @xcite ) . in our sample , the cl0023 agns are emitting more similarly to the quasar mode , accompanied by substantial star formation . the agn host galaxies in cl1604 and the evolved structures have lower star formation rates than those in cl0023 , while the evolved structures have agns with lower x - ray luminosities , which suggests that they are sequentially further along the track leading toward domination of radio mode agn emission . if the agns in the evolved structures were found to be radio emitters , it would support these conclusions . existing vla b - array observations ( at 1.4 ghz ) of these five structures are currently being analyzed to explore this connection ( c. d. fassnacht et al . 2012 , in preparation ) . altogether , many of our results could support several potential agn triggering scenarios . two possibilities are that : ( 1 ) agns in these structures represent a transitional population where hosts are evolving from the blue cloud onto the red sequence or ( 2 ) agns represent a population evolving mainly in mass space , where red sequence hosts have undergone episodic nuclear star formation induced by minor mergers . additionally , many observable effects of the latter would appear similar to models in which recycled stellar material fuels central starbursts and nuclear activity in elliptical galaxies , which tend to be located on the red sequence . in the first case , agns are triggered by mergers or strong tidal interactions which lead to a starburst . the feedback from the agn quenches star formation , leading to a rapid evolution across the green valley and onto the red sequence @xcite . in the second case , red sequence galaxies undergo minor mergers , which funnel gas into the galactic core , creating a burst of nuclear star formation and fueling the agn @xcite . alternatively , recycled stellar material could create a nuclear starburst and a central instability , leading to black hole accretion @xcite . in either case , the galaxies could evolve into the green valley before agn feedback brought them onto the red sequence again . once on the red sequence , there would not be a large net change in color , in contrast to the dramatic color evoltion of the first possibility . in the case of minor mergers , this process would mainly entail an evolution in mass . in support of agns as a transitional population , many of the host galaxies across all five structures are located close to the red sequence , where the green valley should lie , although it is difficult to determine the location of the green valley in the four structures without precise acs data . in support of all scenarios involving significant starbursts , the average agn host in both the evolved structures and the cl1604 supercluster has substantial h@xmath2 absorption , which is a sign of recently quenched star formation . this is expected for green valley galaxies evolving onto the red sequence . previous morphological analysis of part of our sample by @xcite found that two - thirds of cl1604 host galaxies studied showed signs of recent or pending mergers or tidal interactions , which is expected in the context of both major and minor merger theories . in addition , we find that at least nine of the 27 agn host galaxies are part of a kinematic close pair . in seven of these cases , the companion galaxy has a similar stellar mass or @xmath63 magnitude to the agn host . while being far from conclusive due to sampling and selection effects , these results could point to a major merger scenario . @xcite examined eight of the agns in cl1604 and found that half of the hosts had blue cores in an otherwise red galaxy . other studies have also found blue cores or blue early type galaxies @xcite . these blue cores are predicted by minor merger simulations @xcite , as well as the recycled gas models of @xcite . in addition simulations show that central black hole accretion is most highly correlated with star formation in the nucleus , as opposed to the entire galaxy @xcite . during a major merger , star formation peaks at later times closer to the galactic core @xcite . this could potentially create bluer cores as well . from our data , it seems our results are ambiguous with regards to the various scenarios . however , with evidence supporting more than one possibility , our results could indicate a combination of different triggering mechanisms . these different agn triggering mechanisms could also potentially explain some of the differences that we see between the evolved and unevolved structures . the second agn mode , involving episodic nuclear activity in red sequence galaxies fueled by minor mergers , or recycled gas , is expected to involve accretion rates at a much lower eddington ratio than in major merger driven agn activity @xcite . while agn luminosity depends on both the eddington ratio and the black hole mass @xcite , lower eddington ratios will , on average , correspond to lower luminosities . a difference in eddington ratios could potentially explain the lower x - ray luminosities that we observe in the evolved structures , if our sample of host galaxies consists of a mix of the two agn modes . in the unevolved structures , we would expect to observe more major mergers between blue cloud galaxies due to the larger fraction of blue galaxies , fueling the brightest agns in our sample . since all of our structures have a substantial number of red - sequence galaxies we would expect all structures to have galaxies undergoing the second mode of agn activity , whether induced by minor mergers or , perhaps , recycled gas in ellipticals . we would then expect the x - ray luminosity distributions of the evolved and unevolved structures to look similar except for a tail of higher luminosity objects in the unevolved structures , which is roughly what we observe . while a combination of the two agn modes considered could explain our results , they could also be explained if major mergers were the primary driver , and the quasar mode is less dominant for the agns in the evolved structures , because of the larger red fraction therein . with our current data , we still lack the ability to distinguish between the different triggering mechanisms . breaking the degeneracy will require high - resolution imaging to examine the morphologies of the agn hosts and the colors of their cores in the structures other than cl1604 , as well as reliable stellar masses and hst data , for measuring bulge - to - disk - ratios , with which we could reliably calculate eddington ratios . in summary , we find that most x - ray agn hosts , across all five structures , avoid the dense cluster cores , in agreement with a number of previous studies at a range of redshifts . we interpret this to mean that x - ray agn activity is preferentially triggered in intermediate - density environments , such as the outskirts of clusters . we also find many agn host galaxies in or near the green valley , with 36@xmath7 within one red - sequence width of the lower boundary of the red sequence and 60@xmath7 within two red - sequence widths . with numerous other studies finding a similar connection , this implies that there is an association between this transitional region and agn activity . we divided our sample of five structures into two groups : the more and less evolved structures , which we separated using composite spectra made of all of their spectroscopically - confirmed member galaxies . we define the more evolved structures as those with member galaxies that exhibit , on average , less [ ] emission and less h@xmath290absorption , where the [ ] and h@xmath2 lines are taken as indicators of ongoing and recent star formation , respectively . the more evolved structures also have galaxy populations with a higher red fraction than the less evolved structures . our spectral results indicate that the agn hosts in the less evolved structures have more ongoing star formation , while those in the more evolved structures have stronger average h@xmath2 features . stronger h@xmath2 lines are indicative of star formation within the past @xmath61 gyr , and our results indicate starbursts occurred more recently in the agn host galaxies in the less evolved structures . however , all of the agn hosts , regardless of whether they are members of the more evolved or less evolved structures , are younger than the average galaxy in their parent population . we also found that agns in the less evolved structures had more luminous x - ray emission . this may be expected , since these structures contain the highest fraction of blue galaxies which are likely to have larger reservoirs of cool gas to fuel nuclear activity . if agns were triggered more recently in the less evolved structures , as the spectral data suggests , the difference in luminosity could also be related to a transition from `` quasar mode '' emission in newly triggered agns to `` radio mode '' emission at later times as supplies of inflowing cool gas are shut off . we consider several scenarios for agn triggering that are in agreement with our results . agns are triggered by major mergers or tidal interactions between blue cloud galaxies and/or agns are triggered episodically in red - sequence galaxies , fueled by recycled stellar material or induced by minor mergers . each of these scenarios could explain the association of agn hosts with the green valley . also , the h@xmath290absorption we observe could be indicative of the quenching of star formation that drives galaxies across the green valley . a previous study of the cl1604 agns by @xcite found that a majority of hosts had recent or pending mergers . half of the hosts had blue cores in otherwise red galaxies , which could support the second two scenarios , although other explanations are possible . some of our results , such as the difference in x - ray luminosities , could be explained if the agns in our sample were triggered by a mix of these modes , with a larger fraction of agns in the less evolved structures triggered by major mergers . however , we can not distinguish between the triggering scenarios with our data , so the cause of the agn activity is still ambiguous . future work investigating agn host morphologies , examining for blue cores , and calculating eddington ratios could potentially break the degeneracy . the authors thank phil marshall and robert lupton for useful conversations . this work is supported by the _ chandra _ general observing program under award numbers go6 - 7114x , go7 - 8126x , go8 - 9123a , and go9 - 0139a . in addition , we acknowledge support by the national science foundation under grant no . the spectrographic data presented herein were obtained at the w.m . keck observatory , which is operated as a scientific partnership among the california institute of technology , the university of california , and the national aeronautics and space administration . the observatory was made possible by the generous financial support of the w.m . keck foundation . as always , we thank the indigenous hawaiian community for allowing us to be guests on their sacred mountain . we are most fortunate to be able to conduct observations from this site . baldry , i. k. , glazebrook , k. , brinkmann , j. , et al . 2004 , , 600 , 681 balogh , m. l. , morris , s. l. , yee , h. k. c. , carlberg , r. g. , & ellingson , e. 1999 , , 527 , 54 barger , a. j. , cowie , l. l. , mushotzky , r. f. , et al . 2005 , , 129 , 578 bluck , a. f. l. , conselice , c. j. , almaini , o. , et al . 2011 , , 410 , 1174 bournaud , f. , chapon , d. , teyssier , r. , et al . 2011 , , 730 , 4 bower , r. g. , benson , a. j. , malbon , r. , et al . 2006 , , 370 , 645 boyle , b. j. , & terlevich , r. j. 1998 , , 293 , l49 brandt , w. n. , alexander , d. m. , hornschemeier , a. e. , et al . 2001 , , 122 , 2810 bruzual , g. 2007 , from stars to galaxies : building the pieces to build up the universe , 374 , 303 butcher , h. , & oemler , a. , jr . 1984 , , 285 , 426 calzetti , d. , armus , l. , bohlin , r. c. , et al . 2000 , , 533 , 682 cappelluti , n. , cappi , m. , dadina , m. , et al . 2005 , , 430 , 39 cardamone , c. n. , urry , c. m. , schawinski , k. , et al . 2010 , , 721 , l38 cavaliere , a. , colafrancesco , s. , & menci , n. 1992 , , 392 , 41 ciotti , l. , & ostriker , j. p. 2007 , , 665 , 1038 coldwell , g. v. , martnez , h. j. , & lambas , d. g. 2002 , , 336 , 207 colless , m. , et al . 2001 , , 328 , 1039 crawford , d. f. , jauncey , d. l. , & murdoch , h. s. 1970 , , 162 , 405 croton , d. j. , springel , v. , white , s. d. m. , et al . 2006 , , 365 , 11 davies , r. i. , mller snchez , f. , genzel , r. , et al . 2007 , , 671 , 1388 diamond - stanic , a. m. , & rieke , g. h. 2011 , , in press dickey , j. m. , & lockman , f. j. 1990 , , 28 , 215 dressler , a. , oemler , a. , jr . , poggianti , b. m. , et al . 2004 , , 617 , 867 dressler , a. , & shectman , s. a. 1988 , , 95 , 284 dressler , a. , smail , i. , poggianti , b. m. , et al . 1999 , , 122 , 51 eastman , j. , martini , p. , sivakoff , g. , et al . 2007 , , 664 , l9 faber , s. m. , phillips , a. c. , kibrick , r. i. , et al . 2003 , , 4841 , 1657 faber , s. m. , willmer , c. n. a. , wolf , c. , et al . 2007 , , 665 , 265 fanidakis , n. , baugh , c. m. , benson , a. j. , et al . 2011 , , 410 , 53 ferrarese , l. , & merritt , d. 2000 , , 539 , l9 fisher , d. , fabricant , d. , franx , m. , & van dokkum , p. 1998 , , 498 , 195 ford , h. c. , clampin , m. , hartig , g. f. , et al . 2003 , , 4854 , 81 fruscione , a. , mcdowell , j. c. , allen , g. e. , et al . 2006 , , 6270 , 60 gal , r. r. , lemaux , b. c. , lubin , l. m. , kocevski , d. , & squires , g. k. 2008 , , 684 , 933 gal , r. r. , & lubin , l. m. 2004 , , 607 , l1 gal , r. r. , lubin , l. m. , & squires , g. k. 2005 , , 129 , 1827 gebhardt , k. , bender , r. , bower , g. , et al . 2000 , , 539 , l13 gehrels , n. 1986 , , 303 , 336 georgakakis , a. , nandra , k. , yan , r. , et al . 2008 , , 385 , 2049 georgantopoulos , i. , rovilos , e. , & comastri , a. 2011 , , 526 , a46 gilmour , r. , gray , m. e. , almaini , o. , et al . 2007 , , 380 , 1467 gioia , i. m. , henry , j. p. , mullis , c. r. , et al . 2003 , , 149 , 29 gioia , i. m. , maccacaro , t. , schild , r. e. , et al . 1990 , , 72 , 567 gioia , i. m. , wolter , a. , mullis , c. r. , et al . 2004 , , 428 , 867 gladders , m. d. , lopez - cruz , o. , yee , h. k. c. , & kodama , t. 1998 , , 501 , 571 gunn , j. e. , hoessel , j. g. , & oke , j. b. 1986 , , 306 , 30 hring , n. , & rix , h .- w . 2004 , , 604 , l89 hasinger , g. 2008 , , 490 , 905 heckman , t. m. , kauffmann , g. , brinchmann , j. , et al . 2004 , , 613 , 109 henry , j. p. , mullis , c. r. , voges , w. , et al . 2006 , , 162 , 304 hickox , r. c. , jones , c. , forman , w. r. , et al . 2009 , , 696 , 89 hopkins , p. f. 2011 , , in press hopkins , p. f. , bundy , k. , hernquist , l. , & ellis , r. s. 2007 , , 659 , 976 hopkins , p. f. , & hernquist , l. 2006 , , 166 , 1 hopkins , p. f. , hernquist , l. , martini , p. , et al . 2005 , , 625 , l71 hopkins , p. f. , & quataert , e. 2010 , , 407 , 1529 johnson , o. , best , p. n. , & almaini , o. 2003 , , 343 , 924 kartaltepe , j. s. , sanders , d. b. , scoville , n. z. , et al . 2007 , , 172 , 320 kauffmann , g. , & haehnelt , m. 2000 , , 311 , 576 kauffmann , g. , heckman , t. m. , tremonti , c. , et al . 2003 , , 346 , 1055 kauffmann , g. , white , s. d. m. , heckman , t. m. , et al 2004 , , 353 , 713 kim , m. , kim , d .- w . , wilkes , b. j. , et al . 2007 , , 169 , 401 kocevski , d. d. , faber , s. m. , mozena , m. , et al . 2012 , , 744 , 148 kocevski , d. d. , lemaux , b. c. , lubin , l. m. , et al . 2011 , , 736 , 38 kocevski , d. d. , lubin , l. m. , gal , r. , et al . 2009a , , 690 , 295 kocevski , d. d. , lubin , l. m. , lemaux , b. c. , et al . 2009b , , 700 , 901 kocevski , d. d. , lubin , l. m. , lemaux , b. c. , et al . 2009c , , 703 , l33 kushino , a. , ishisaki , y. , morita , u. , et al . 2002 , , 54 , 327 laird , e. s. , nandra , k. , pope , a. , & scott , d. 2010 , , 401 , 2763 lee , j. h. , lee , m. g. , & hwang , h. s. 2006 , , 650 , 148 lemaux , b. c. , gal , r. r. , lubin , l. m. , et al . 2012 , , 745 , 106 lemaux , b. c. , lubin , l. m. , sawicki , m. , et al . 2009 , , 700 , 20 lemaux , b. c. , lubin , l. m. , shapley , et al . 2010 , , 716 , 970 lin , l. , koo , d. c. , weiner , b. j. , et al . 2007 , , 660 , l51 lubin , l. m. , brunner , r. , metzger , m. r. , postman , m. , & oke , j. b. 2000 , , 531 , l5 lubin , l. m. , gal , r. r. , lemaux , b. c. , kocevski , d. d. , & squires , g. k. 2009 , , 137 , 4867 lubin , l. m. , mulchaey , j. s. , & postman , m. 2004 , , 601 , l9 lubin , l. m. , oke , j. b. , & postman , m. 2002 , , 124 , 1905 lubin , l. m. , postman , m. , & oke , j. b. 1998 , , 116 , 643 mann , r. g. , oliver , s. , carballo , r. , et al . 2002 , , 332 , 549 mann , r. g. , oliver , s. j. , serjeant , s. b. g. , et al . 1997 , , 289 , 482 marconi , a. , & hunt , l. k. 2003 , , 59 , l21 marconi , a. , risaliti , g. , gilli , r. , et al . 2004 , , 351 , 169 martel , a. r. , menanteau , f. , tozzi , p. , ford , h. c. , & infante , l. 2007 , , 168 , 19 martini , p. , mulchaey , j. s. , & kelson , d. d. 2007 , , 664 , 761 mclean , i. s. , becklin , e. e. , bendiksen , o. , et al . 1998 , , 3354 , 566 mcnamara , b. r. , & nulsen , p. e. j. 2007 , , 45 , 117 menanteau , f. , jimenez , r. , & matteucci , f. 2001 , , 562 , l23 merloni , a. , & heinz , s. 2008 , , 388 , 1011 mihos , j. c. , & hernquist , l. 1996 , , 464 , 641 murdoch , h. s. , crawford , d. f. , & jauncey , d. l. 1973 , , 183 , 1 nagao , t. , maiolino , r. , & marconi , a. 2006 , , 459 , 85 nandra , k. , georgakakis , a. , willmer , c. n. a. , et al . 2007 , , 660 , l11 oemler , a. , jr . , dressler , a. , kelson , d. , et al . 2009 , , 693 , 152 oke , j. b. , cohen , j. g. , carr , m. , et al . 1995 , , 107 , 375 oke , j. b. , postman , m. , & lubin , l. m. 1998 , , 116 , 549 pierce , c. m. , lotz , j. m. , laird , e. s. , et al . 2007 , , 660 , l19 poggianti , b. m. , & barbaro , g. 1997 , , 325 , 1025 poggianti , b. m. , smail , i. , dressler , a. , et al . 1999 , , 518 , 576 postman , m. , lubin , l. m. , & oke , j. b. 1998 , , 116 , 560 postman , m. , lubin , l. m. , & oke , j. b. 2001 , , 122 , 1125 rieke , g. h. , young , e. t. , engelbracht , c. w. , et al . 2004 , , 154 , 25 rosario , d. j. , mozena , m. , wuyts , s. , et al . 2011 , arxiv:1110.3816 rosati , p. , tozzi , p. , giacconi , r. , et al . 2002 , , 566 , 667 rose , j. a. 1985 , , 90 , 1927 rutledge , r. e. , brunner , r. j. , prince , t. a. , & lonsdale , c. 2000 , , 131 , 335 snchez , s. f. , becker , t. , garcia - lorenzo , b. , et al . 2005 , , 429 , l21 snchez , s. f. , jahnke , k. , wisotzki , l. , et al . 2004 , , 614 , 586 schawinski , k. , thomas , d. , sarzi , m. , et al . 2007 , , 382 , 1415 schawinski , k. , virani , s. , simmons , b. , et al . 2009 , , 692 , l19 schiavon , r. p. , faber , s. m. , konidaris , n. , et al . 2006 , , 651 , l93 shields , g. a. , gebhardt , k. , salviander , s. , et al . 2003 , , 583 , 124 silverman , j. d. , kova , k. , knobel , c. , et al . 2009a , , 695 , 171 silverman , j. d. , lamareille , f. , maier , c. , et al . 2009b , , 696 , 396 silverman , j. d. , mainieri , v. , lehmer , b. d. , et al . 2008 , , 675 , 1025 simcoe , r. a. , metzger , m. r. , small , t. a. , & araya , g. 2000 , bulletin of the american astronomical society , 32 , 758 somerville , r. s. , hopkins , p. f. , cox , t. j. , robertson , b. e. , & hernquist , l. 2008 , , 391 , 481 springel , v. , di matteo , t. , & hernquist , l. 2005 , , 361 , 776 stott , j. p. , pimbblet , k. a. , edge , a. c. , smith , g. p. , & wardlow , j. l. 2009 , , 394 , 2098 strateva , i. , ivezic , z. , knapp , g. r. , et al . 2001 , , 122 , 1861 sutherland , w. , & saunders , w. 1992 , , 259 , 413 taylor , e. l. , mann , r. g. , efstathiou , a. n. , et al . 2005 , , 361 , 1352 tozzi , p. , rosati , p. , nonino , m. , et al . 2001 , , 562 , 42 tremaine , s. , gebhardt , k. , bender , r. , et al . 2002 , , 574 , 740 trouille , l. , barger , a. j. , & tremonti , c. 2011 , , 742 , 46 weiner , b. j. , phillips , a. c .. faber , s. m. , et al . 2005 , , 620 , 595 wild , v. , heckman , t. , & charlot , s. 2010 , , 405 , 933 xue , y. q. , brandt , w. n. , luo , b. , et al . 2010 , , 720 , 368 yan , r. , newman , j. a. , faber , s. m. , et al . 2006 , , 648 , 281

we use deep chandra imaging and an extensive optical spectroscopy campaign on the keck 10-m telescopes to study the properties of x - ray point sources in two isolated x - ray selected clusters , two superclusters , and one ` supergroup ' at redshifts of @xmath0 . we first study x - ray point sources using the statistical measure of cumulative source counts , finding that the measured overdensities are consistent with previous results , but we recommend caution in overestimating the precision of the technique . optical spectroscopy of objects matched to x - ray point sources confirms a total of 27 agns within the five structures , and we find that their host galaxies tend to be located away from dense cluster cores . more than @xmath1 of the host galaxies are located in the ` green valley ' on a color magnitude diagram , which suggests they are a transitional population . based on analysis of [ ] and h@xmath2 line strengths , the average spectral properties of the agn host galaxies in all structures indicate either on - going star formation or a starburst within @xmath3 gyr , and that the host galaxies are younger than the average galaxy in the parent population . these results indicate a clear connection between starburst and nuclear activity . we use composite spectra of the spectroscopically confirmed members in each structure ( cluster , supergroup , or supercluster ) to separate them based on a measure of the overall evolutionary state of their constituent galaxies . we define structures as having more evolved populations if their average galaxy has lower ew ( [ ] ) and ew(h@xmath2 ) . the agns in the more evolved structures have lower rest - frame 0.58 kev x - ray luminosities ( all below @xmath4 erg s@xmath5 ) and longer times since a starburst than those in the unevolved structures , suggesting that the peak of both star formation and agn activity has occurred at earlier times . with the wide range of evolutionary states and timeframes in the structures , we use our results to analyze the evolution of x - ray agns and evaluate potential triggering mechanisms .

semitauonic @xmath11 meson decays with @xmath12 @xcite transitions are sensitive to new physics ( np ) beyond the standard model ( sm ) involving non - universal coupling to heavy fermions . one prominent candidate for np is the two higgs doublet model ( 2hdm ) @xcite , which has an additional higgs doublet and therefore introduces two neutral and two charged higgs bosons in addition to the sm higgs boson . the charged higgs bosons may contribute to the @xmath13 process , modifying its branching fraction and decay kinematics . exclusive semitauonic decays of the type @xmath14 have been studied by belle @xcite , babar @xcite and lhcb @xcite . the experiments typically measure the ratios of branching fractions , @xmath15 where the denominator is the average for @xmath16 . the ratio cancels uncertainties common to the numerator and the denominator . these include the cabibbo - kobayashi - maskawa matrix element @xmath17 and many of the theoretical uncertainties on hadronic form factors and experimental reconstruction effects . the current averages of the three experiments @xcite are @xmath18 and @xmath19 , which are within @xmath20 and @xmath21 @xcite of the sm predictions of @xmath22 @xcite or @xmath23 @xcite and @xmath24 @xcite , respectively . here , @xmath25 represents the standard deviation . in addition to @xmath26 , the polarization of the @xmath0 lepton and the @xmath27 meson is also sensitive to np @xcite . the polarization of the @xmath0 lepton ( @xmath28 ) is defined by @xmath29 where @xmath30 denotes the decay rate of @xmath14 with a @xmath0 helicity of @xmath31 . the sm predicts @xmath32 for @xmath33 @xcite and @xmath34 for @xmath35 @xcite . the @xmath0 polarization is accessible in two - body hadronic @xmath0 decays with the following formulae @xcite : @xmath36 where @xmath37 , @xmath38 and @xmath39 are , respectively , the decay rate of @xmath40 and the masses of the @xmath0 lepton and the vector meson from the @xmath0 decay . the helicity angle , @xmath41 , is the opening angle between the momentum vectors of the virtual @xmath42 boson and of the @xmath0-daughter meson in the rest frame of the @xmath0 . the parameter @xmath43 describes the sensitivity to @xmath28 for each @xmath0-decay mode ; in particular , @xmath44 for the decay @xmath5 . in this paper , we report a new measurement of @xmath45 in the hadronic @xmath0 decay modes @xmath4 and @xmath46 . this measurement is statistically independent of the previous belle measurements @xcite , with a different background composition . we also report the first measurement of @xmath28 for the decay @xmath1 . we use the full @xmath47 data sample containing @xmath48 pairs recorded with the belle detector @xcite at the asymmetric - beam - energy @xmath49 collider kekb @xcite . the belle detector is a large - solid - angle magnetic spectrometer that consists of a silicon vertex detector ( svd ) , a 50-layer central drift chamber ( cdc ) , an array of aerogel threshold cherenkov counters ( acc ) , a barrel - like arrangement of time - of - flight scintillation counters ( tof ) , and an electromagnetic calorimeter ( ecl ) comprised of csi(tl ) crystals located inside a superconducting solenoid coil that provides a 1.5 t magnetic field . an iron flux - return located outside of the coil is instrumented to detect @xmath50 mesons and to identify muons ( klm ) . the detector is described in detail elsewhere @xcite . two inner detector configurations were used . a 2.0 cm radius beampipe and a 3-layer silicon vertex detector was used for the first sample of @xmath51 pairs , while a 1.5 cm radius beampipe , a 4-layer silicon detector and a small - cell inner drift chamber were used to record the remaining @xmath52 pairs @xcite . the signal selection criteria and the signal and background probability density functions ( pdfs ) used in this measurement rely on the use of monte carlo ( mc ) simulation samples . these samples are generated by the software packages evtgen @xcite and pythia @xcite ; final - state radiation is generated by photos @xcite . detector responses are fully simulated with the belle detector simulator based on geant3 @xcite . the signal decay @xmath1 ( signal mode ) is generated with a decay model based on the heavy quark effective theory ( hqet ) @xcite . we use the current world - average values for the form - factor parameters @xmath53 , @xmath54 and @xmath55 @xcite , which are based on the parameterization in ref . decays of the type @xmath56 ( normalization mode ) , which are used for the denominator of @xmath45 , are also modeled with hqet using the above form - factor values . background from semileptonic decays to orbitally - excited charmed mesons @xmath57 , where @xmath58 denotes @xmath59 , @xmath60 , @xmath61 or @xmath62 , are generated with the isgw model @xcite with their kinematic distributions reweighted to match the dynamics predicted by the llsw model @xcite . additionally , theoretically - predicted radial excitation states @xmath63 are assumed to fill the gap between the inclusively - measured and the sum of the exclusively - measured branching fractions of @xmath64 @xcite . the mc sample of @xmath65 is produced with the isgw model . the branching fractions are assigned according to their theoretical estimates @xcite . the remaining background mc samples are comprised of mostly hadronic @xmath11 meson decays and light quark production processes ( @xmath66 ) . the sample sizes of the signal , @xmath67 , @xmath68 , other @xmath7 , and @xmath69 processes are 40 , 40 , 400 , 10 and 5 times larger , respectively , than the full belle data sample . we first identify events where one of the two @xmath11 mesons ( @xmath70 ) is reconstructed in one of 1149 exclusive hadronic @xmath11 decays using a hierarchical multivariate algorithm @xcite based on the neurobayes package @xcite . more than 100 variables are used in this algorithm , including the difference @xmath71 between the energy of the reconstructed @xmath70 candidate and the kekb beam energy in the @xmath49 center - of - mass system as well as the event shape variables for suppression of @xmath72 background . we further require the beam - energy - constrained mass of the @xmath70 candidate @xmath73 , where @xmath74 denotes the reconstructed @xmath70 three - momentum in the @xmath75 center - of - mass system , to be greater than 5.272 gev@xmath76 and @xmath77 to lie between @xmath78 and @xmath79 . if there are two or more @xmath70 candidates retained after the selection criteria , we select the one with the highest neurobayes output value , which is related to the probability that the @xmath70 candidate is correctly reconstructed . due to limited knowledge of hadronic @xmath11 decays , the branching ratios of the @xmath70 decay modes are not perfectly modeled in the mc . it is therefore essential to calibrate the @xmath70 reconstruction efficiency ( tagging efficiency ) with control data samples . we determine a scale factor for each @xmath70 decay using the method described in ref . @xcite based on events where the signal - side @xmath11 meson candidate ( @xmath80 ) is reconstructed in @xmath81 modes . the ratio of measured to expected rates in each decay mode ranges from 0.2 to 1.4 , depending on the @xmath70 decay mode , and is 0.68 on average . after the efficiency calibration , the tagging efficiencies are about 0.20% for charged @xmath11 mesons and 0.15% for neutral @xmath11 mesons . after @xmath70 selection , we form @xmath80 candidates from the remaining particles not associated with the @xmath70 candidate . charged particles used to form @xmath80 candidates are reconstructed using information from the svd and the cdc . the tracks that are not used in @xmath82 reconstruction are required to have impact parameters to the interaction point ( ip ) of less than 0.5 cm ( 2.0 cm ) in the direction perpendicular ( parallel ) to the @xmath83 beam axis . charged - particle types are identified by a likelihood ratio based on the response of the sub - detector systems . identification of @xmath84 and @xmath85 candidates is done by combining measurements of specific ionization ( @xmath86 ) in the cdc , the time of flight from the ip to the tof counter and the photon yield in the acc . for @xmath0-daughter @xmath85 candidates , an additional proton veto is required in order to reduce background from the baryonic @xmath11 decays @xmath87 . the ecl electromagnetic shower shape , track - to - cluster matching at the inner surface of the ecl , the photon yield in the acc and the ratio of the cluster energy in the ecl to the track momentum measured with the svd and cdc are used to identify @xmath88 candidates . muon candidates are selected based on the comparison of the projected cdc track with interactions in the klm . to form @xmath82 candidates , we combine a pair of oppositely - charged tracks , treated as pions . three requirements are applied : the reconstructed vertex must be detached from the ip , the momentum vector must point back to the ip , and the invariant mass must be within @xmath8930 mev/@xmath90 of the nominal @xmath82 mass @xcite , which corresponds to about 8@xmath25 . ( in this section , @xmath25 denotes the corresponding mass resolution . ) photons are reconstructed using ecl clusters not matched to charged tracks . photon energy thresholds of 50 , 100 and 150 mev are used in the barrel , forward - endcap and backward - endcap regions , respectively , of the ecl to reject low - energy background photons , such as those originating from the @xmath49 beams , and hadronic interactions of particles with material in the detector . neutral pions are reconstructed in the decay @xmath91 . for @xmath92 candidates from @xmath93 or @xmath94 decay , we impose the same photon energy thresholds described above . the @xmath92 candidate s invariant mass must lie between 115 and 150 mev/@xmath90 , corresponding to about @xmath95 around the nominal @xmath92 mass @xcite . in order to reduce the number of fake @xmath92 candidates , we apply the following @xmath92 candidate - selection procedure . the @xmath92 candidates are sorted in descending order according to the energy of the most energetic daughter . if a given photon is the most energetic daughter for two or more candidates , they are sorted by the energy of the lower - energy daughter . we then retain the @xmath92 candidates whose daughter photons are not shared with a higher - ranked candidate . the remaining @xmath92 candidates are used for @xmath93 or @xmath94 reconstruction described later . for the soft @xmath92 from @xmath27 decay , we impose a relaxed photon energy threshold of 22 mev in all ecl regions , the same requirement for the invariant mass of the two photons , and an energy - asymmetry @xmath96 less than 0.6 , where @xmath97 and @xmath98 are the energies of the high- and low - energy photon daughters in the laboratory frame . we do not apply the above candidate - selection procedure for the soft @xmath92 candidates . after reconstructing the final - state particles and light mesons , we reconstruct the @xmath99 candidates using 15 @xmath93 decay modes : @xmath100 , @xmath101 , @xmath102 , @xmath103 , @xmath104 , @xmath105 , @xmath106 , @xmath107 , @xmath108 , @xmath109 , @xmath110 , @xmath111 , @xmath112 , @xmath113 , @xmath114 , and four @xmath27 decay modes : @xmath115 , @xmath116 , @xmath117 and @xmath118 . the @xmath93 invariant mass requirements are optimized for each decay mode . for the @xmath119 modes in the @xmath120 candidates , the invariant masses ( @xmath121 ) are required to be within @xmath122 ( @xmath123 ) of the nominal @xmath119 meson mass @xcite for the high ( low ) signal - to - noise ratio ( snr ) modes . for @xmath124 candidates , the @xmath121 requirements are loosened to @xmath125 and @xmath122 for the high- and low - snr modes , respectively . the requirements for the @xmath126 candidates are @xmath127 for the high - snr modes and @xmath123 for the low - snr modes around the nominal @xmath126 meson mass @xcite . here , the high - snr modes are @xmath100 , @xmath102 , @xmath103 , @xmath105 , @xmath107 , @xmath108 , @xmath109 , @xmath111 ; the low - snr modes are all remaining @xmath93 modes . we reconstruct a @xmath27 candidate by combining a @xmath93 candidate with a @xmath85 , @xmath128 or soft @xmath92 . the @xmath27 candidates are selected based on the mass difference @xmath129 , where @xmath130 denotes the invariant mass of the @xmath27 candidate . the @xmath115 , @xmath131 , @xmath117 and @xmath124 candidates are required to have a @xmath132 within @xmath123 , @xmath122 , @xmath122 and @xmath133 of the nominal @xmath132 . for the @xmath134 candidates , the @xmath94 candidate is formed from the combination of a @xmath85 and a @xmath92 with an invariant mass between 0.66 and 0.96 mev@xmath76 . we then associate a @xmath85 or a @xmath135 candidate ( one charged lepton ) with the @xmath27 candidate to form signal ( normalization ) candidates . for the signal mode , square of the momentum transfer , @xmath136 is required to be greater than 4 gev@xmath137 , where @xmath138 and @xmath139 denote the energy and the three - momentum specified by the subscript . the subscripts `` @xmath49 '' , `` tag '' and `` @xmath27 '' stand for the colliding @xmath83 and @xmath140 , the @xmath70 candidate and the @xmath27 candidate , respectively . due to the kinematic constraint in @xmath35 that @xmath141 is always greater than the square of the @xmath0 mass , almost no signal events exist with @xmath141 below 4 gev@xmath137 . finally , we require that there be no remaining charged tracks nor @xmath92 candidates ( except for soft @xmath92 ) in the event . after the @xmath80 reconstruction procedure is completed , the probability to have multiple candidates ( the number of retained candidates ) per event is about 9% ( 1.09 ) for the charged @xmath11 mesons and 3% ( 1.03 ) for the neutral @xmath11 mesons . most of the multiple - candidate events are due to the existence of two or more @xmath27 candidates in an event . for the @xmath120 candidates in the charged @xmath11 meson sample , about 2% of the events are reconstructed both in the @xmath115 and @xmath131 modes . since the latter mode has a much higher branching fraction , we assign these events to the @xmath131 sample . the contribution of this type of multiple - candidate events is negligibly small in the @xmath142 mode . we then select the most signal - like event as follows . for the @xmath115 events , we select the candidate with the most energetic photon associated with the @xmath119 . for the @xmath131 and @xmath117 events , we select the candidate with the soft @xmath92 having the invariant mass nearest the nominal @xmath92 mass . for the @xmath124 events , we select one candidate at random since the multiple - candidate probability is only @xmath143 . after the @xmath27 candidate selection , roughly 2% of the retained events are reconstructed both in the @xmath4 sample and the @xmath5 sample . according to the mc study , about 80% of such signal events are actually @xmath5 events . we therefore assign these events to the @xmath5 sample . distribution for the @xmath144 mc sample.,width=377 ] in order to measure @xmath28 , the @xmath145 distribution must be reconstructed . this is challenging , as the @xmath0 momentum vector is not fully determined . instead of @xmath145 , we measure the cosine of the angle @xmath146 between the momenta of the @xmath0 lepton and its daughter meson in the rest frame of the virtual @xmath42 boson , @xmath147 as shown in fig . [ fig - kinematics ] ( a ) . here , @xmath138 and @xmath139 denote the energy and the three - momentum of a particle specified by the subscript , where @xmath0 and @xmath148 represent the @xmath0 lepton and its daughter meson , respectively . this angle is equivalent to @xmath41 in this frame . the rest frame of the virtual @xmath42 is obtained from its three - momentum @xmath149 in this frame of reference , the magnitude of the @xmath0 momentum is determined only by @xmath141 since the @xmath0 lepton is emitted in the two - body decay of the static virtual @xmath42 boson . therefore @xmath150 is calculated as @xmath151 we only accept events , for which @xmath152 . here , more than 97% of the reconstructed signal events are retained . due to limited kinematic constraints , one degree of freedom of the @xmath0 momentum direction is not determined . however , the cone around @xmath153 with an angle of @xmath146 , on which @xmath154 lies , is rotationally symmetric and therefore all directions on this cone are equivalent . with this in mind , as shown in fig . [ fig - kinematics ] ( b ) , we take the new right - handed @xmath155 coordinate such that the @xmath156-axis corresponds to the direction of the @xmath153 , and set @xmath157 . the system is boosted to the pseudo @xmath0 rest frame with @xmath158 , where the correct value of @xmath145 is obtained . as shown in fig . [ fig - coshel ] , there are many @xmath159 background events that peak around @xmath160 , which corresponds to @xmath161 , in the @xmath4 sample . this peak arises from low - momentum muons that do not reach the klm and are therefore misreconstructed as pions . to mitigate this background , we only use the region @xmath162 in the fit , in which 94% ( 81% ) of signal events are contained with the sm @xmath28 of @xmath163 ( maximum @xmath28 of @xmath164 ) . in order to separate signal and normalization events from background , we use the variable @xmath165 , the summed energy of ecl clusters not used in the reconstruction of the @xmath80 and @xmath70 candidates . this is a useful variable for the signal extraction since the @xmath165 shape is less affected by changes in kinematics due to np . the variable @xmath166 is additionally used for normalization events , and is defined as @xmath167 where @xmath168 and @xmath169 are the energy and the three - momentum , respectively , of the charged lepton ; the other variables in this formula are defined in eq . [ eq - q2 ] . due to its narrow concentration near @xmath170 , this variable is ideal for measuring normalization events . in the fit , we use the distributions obtained from mc for the pdfs . the @xmath165 shape for the signal component is validated using the normalization sample , which is more than 20 times larger than the signal sample . in this comparison , we find good agreement between the data and the mc distributions . in the @xmath166 comparison for the normalization sample , the @xmath166 resolution in the data sample is slightly worse than in the mc sample . we therefore broaden the @xmath166 peak width of the pdfs to match that of the data sample . the most significant background contribution is from events with incorrectly - reconstructed @xmath27 ( denoted `` fake @xmath27 events '' ) . since the combinatorial fake @xmath27 background processes are difficult to be modeled precisely in the mc , we compare the pdf shapes of these events in @xmath132 sideband regions . the sideband regions 50500 mev@xmath76 , 135190 mev@xmath76 , 135190 mev@xmath76 and 140500 mev@xmath76 are chosen for @xmath115 , @xmath131 , @xmath117 and @xmath124 , respectively , while excluding about @xmath133 around the nominal @xmath132 . these sideband regions contain 550 times more events than the signal region . while we find good agreement of the @xmath165 shapes between the data and the mc for the signal sample , we observe a slight discrepancy in the @xmath166 distributions of the @xmath115 and @xmath131 modes for the normalization sample . the @xmath166 discrepancy is therefore corrected based on this comparison . in both samples , since up to 20% of the yield discrepancies are observed , the fake @xmath27 yields are scaled by the yield ratios of the data to the mc in the @xmath132 sideband regions . semileptonic decays to excited charm modes , @xmath67 and @xmath68 , comprise an important background category as they have a similar decay topology to the signal events . in addition , background events from various types of hadronic @xmath11 decays wherein some particles are not reconstructed are significant in this analysis since there are only hadrons and two neutrinos in the final state of the signal mode . because there are many unmeasured exclusive decay modes of @xmath67 , @xmath68 and hadronic @xmath11 decays , we determine their yields in the fit . with one exception , we sum all the exclusive decays of these background categories into common yield parameters . the exception is the decay to two @xmath93 mesons , such as @xmath171 and @xmath172 , since these are experimentally well measured : we fix their yields based on the world - average branching fractions @xcite . in addition to the yield determination , the pdf shape of these background must be taken into account , as a change in the @xmath11 decay composition may modify the @xmath165 shape and thereby introduce biases in the measurement of @xmath45 and @xmath28 . if a background @xmath11 decay contains a @xmath50 in the final state , it may peak in the @xmath165 signal region . we correct the branching fractions of the @xmath173 and @xmath174 modes in the mc using the measured values @xcite . we do not apply branching fraction corrections for the other decays with @xmath50 since they are relatively minor . however , we change the relative yield from 0% to 200% to estimate systematic uncertainties , as discussed in sec . [ sec - syst ] . .list of the calibration factors for each calibration sample , which are used to correct the amount of each hadronic @xmath11 background in the mc . these calibration factors are obtained from the yield comparison between the data and the mc with the @xmath175 or @xmath176 distributions . the errors on the calibration factors arise from statistics of the calibration samples . [ cols="<,^,^",options="header " , ] [ tab - calib ] other types of hadronic @xmath11 decay background often contain neutral particles such as @xmath92 or @xmath177 or pairs of charged particles . we calibrate the amount of hadronic @xmath11 decays in the mc based on control data samples by reconstructing seven final states with the signal - side particles : @xmath178 , @xmath179 , @xmath180 , @xmath181 , @xmath182 , @xmath183 and @xmath184 . candidate @xmath177 mesons are reconstructed using pairs of photons with an invariant mass ranging from 500 to 600 mev@xmath76 . we then take the yield ratios between the data and the mc for @xmath185 and @xmath186 , which is the same requirement as in the signal sample , with the signal - side energy difference @xmath175 or the beam - energy - constrained mass @xmath176 of the @xmath80 candidate . these ratios are used as yield calibration factors . if there is no observed event in the calibration sample , we assign a 68% confidence level upper limit on the yield . the obtained factors are summarized in table [ tab - calib ] . additionally , we correct the branching fractions of the decays @xmath187 , @xmath188 and @xmath87 based on refs . @xcite . about 80% of the hadronic @xmath11 background is covered by the calibrations discussed above . we discuss the systematic uncertainties on our observables due to the uncertainties of the calibration factors in sec . [ sec - syst ] . we perform an extended binned maximum likelihood fit to the @xmath165 and the @xmath166 distributions for the signal- and the normalization - candidate samples , respectively . in order to extract @xmath28 , we divide the signal sample into two @xmath145 regions : @xmath189 ( forward ) and @xmath190 ( backward ) . according to eq . [ eq - coshel ] , the asymmetry of the number of signal events between the forward and the backward regions is proportional to @xmath28 . in the fit , we divide the @xmath35 component into three groups . signal : : + correctly - reconstructed signal events , which originate from @xmath191 events reconstructed correctly as the @xmath191 sample , are categorized in this component , and are used for the determination of @xmath45 and @xmath28 . @xmath192 cross feed : : + cross - feed events where the decay @xmath5 is reconstructed in the @xmath4 mode due to the misreconstruction of one @xmath92 , or events where the decay @xmath4 is reconstructed in the @xmath5 mode by adding a random @xmath92 , comprise this component . as these events originate from @xmath35 , they can be used in the @xmath45 determination . they also have some sensitivity to @xmath28 ; however , @xmath145 is distorted . the measured @xmath28 from the distorted @xmath145 distribution is mapped to the correct value of @xmath28 using mc information . other @xmath0 cross feed : : + events from other @xmath0 decays also can contribute to the signal sample . they originate mainly from @xmath193 with two missing @xmath92 mesons and @xmath194 with a low - momentum muon . the fraction of these two cross - feed components are , respectively , 11% and 73% in the @xmath4 mode and 69% and 14% in the @xmath5 mode . these modes are less sensitive to @xmath28 since the heavy @xmath195 mass makes the @xmath43 in eq . ( [ eq - alpha ] ) almost equal to 0 , while events with two neutrinos in the @xmath194 mode wash out the @xmath28 information . the relative contribution from the three @xmath35 components are fixed using the mc simulation sample , which contains 40 times more events than the full belle data sample . the parameterization of @xmath45 is @xmath196 where @xmath197 denotes the branching fraction of @xmath4 or @xmath5 , and @xmath198 and @xmath199 are the efficiencies for the signal and the normalization mode , respectively . the observed yields are expressed by @xmath200 and @xmath201 for the signal in the forward ( backward ) region and the normalization , respectively . the polarization is represented by @xmath202 due to detector effects , the extracted value deviates from the true @xmath28 . this detector bias is taken into account with a linear function that relates the true @xmath28 to the extracted @xmath28 . the linear function , which is called the @xmath28 correction function in this paper , is determined using several mc sets of type - ii 2hdm @xcite . in this model , @xmath28 varies between @xmath203 and @xmath204 as a function of the theoretical parameter @xmath205 , where @xmath206 denotes the ratio of the vacuum expectation values of the two higgs doublets coupling to up - type and down - type quarks and @xmath207 is the mass of the charged higgs boson . we then extrapolate the obtained @xmath28 correction function to @xmath208 . for the background , we have four components . @xmath159 : : + the decay @xmath159 contaminates the signal sample due to the misassignment of the lepton as a pion . we fix the @xmath159 yield in the signal sample from the fit to the @xmath166 distribution in the normalization sample . @xmath67 and hadronic @xmath11 decays : : + as discussed in the previous section , we float the sum of the yields of @xmath67 and hadronic @xmath11 decays except for the well - determined two - body @xmath93 final states in the fit . the yield parameters are independent for each sample : @xmath209 . continuum : : + continuum events from @xmath72 process provide a minor contribution . as the size of the contribution is only @xmath210 , we fix the yield using the mc expectation . fake @xmath27 : : + all events containing fake @xmath27 candidates are categorized in this component . the yield is fixed from a comparison of the data and the mc in the @xmath132 sideband regions . @l@|@c@@c@ source & @xmath45 & @xmath28 + hadronic @xmath11 composition & @xmath211 & @xmath212 + mc statistics for each pdf shape & @xmath213 & @xmath214 + fake @xmath27 pdf shape & 3.0% & 0.010 + fake @xmath27 yield & 1.7% & 0.016 + @xmath67 & 2.1% & 0.051 + @xmath68 & 1.1% & 0.003 + @xmath159 & 2.4% & 0.008 + @xmath0 daughter and @xmath3 efficiency & 2.1% & 0.018 + mc statistics for efficiency calculation & 1.0% & 0.018 + evtgen decay model & @xmath215 & @xmath216 + fit bias & 0.3% & 0.008 + @xmath217 and @xmath218 & 0.3% & 0.002 + @xmath28 correction function & 0.1% & 0.018 + + tagging efficiency correction & 1.4% & 0.014 + @xmath27 reconstruction & 1.3% & 0.007 + @xmath93 sub - decay branching fractions & 0.7% & 0.005 + number of @xmath7 & 0.4% & 0.005 + total systematic uncertainty & @xmath219 & @xmath220 + [ tab - syst ] we estimate systematic uncertainties by varying each possible uncertainty source such as the pdf shape and the signal reconstruction efficiency with the assumption of a gaussian error , unless otherwise stated . in several trials , we change each parameter at random , repeat the fit , and then take the mean shifts of @xmath45 and @xmath28 from all such trials as the corresponding systematic uncertainty that is enumerated in table [ tab - syst ] . the most significant systematic uncertainty , arising from the hadronic @xmath11 decay composition , is estimated as follows . uncertainties of each @xmath11 decay fraction in the hadronic @xmath11 decay background are taken from the experimentally - measured branching fractions or estimated from the uncertainties in the calibration factors discussed in sec . [ sec - background ] . for components with no experimentally - measured branching fractions and not covered by the control samples , we vary their contribution continuously from @xmath221 to @xmath222 of the mc expectation and take the maximum shifts of @xmath45 and @xmath28 as the systematic uncertainties . the limited mc sample size used in the construction of the pdfs is also a major systematic uncertainty source . we estimate this by regenerating the pdfs for each component and each sample using a toy mc approach based on the original pdf shapes . the same number of events are generated to account for the statistical fluctuation . the pdf shape of the fake @xmath27 component has been validated by comparing the data and the mc in the @xmath132 sideband region . however , a slight fluctuation from the decay @xmath33 , which is a peaking background in the fake @xmath27 component , may have a significant impact on the signal yield as this component has almost the same shape as the signal mode . to be conservative , we incorporate an additional uncertainty by varying the contribution of the @xmath33 component within the current uncertainties of the experimental averages @xcite : @xmath223 for @xmath224 and @xmath225 for @xmath226 . we take the theoretical uncertainty on the @xmath0 polarization of the @xmath33 mode into account , which is found to be 0.002 for @xmath28 and negligibly small . in addition , we estimate a systematic uncertainty due to the small @xmath166 shape correction for the fake @xmath27 component of the normalization sample : this is 0.15% ( 0.001 ) for @xmath45 ( @xmath28 ) . the fake @xmath27 yield , fixed using the @xmath132 sideband , has an uncertainty that arises from the statistical uncertainties of the yield scale factors . the uncertainty of the decays @xmath67 are twofold : the indeterminate composition of each @xmath58 state and the uncertainty in the form - factor parameters used for the mc sample production . the composition uncertainty is estimated based on uncertainties of the branching fractions : @xmath227 for @xmath228 , @xmath229 for @xmath230 , @xmath231 for @xmath232 and @xmath233 for @xmath234 . if the experimentally - measured branching fractions are not applicable , we vary the branching fractions continuously from @xmath221 to @xmath222 in the mc expectation . we estimate an uncertainty arising from the llsw model parameters by changing the correction factors within the parameter uncertainties and obtain 0.5% and 0.016 for @xmath45 and @xmath28 , respectively . the uncertainty due to limited knowledge of the decays @xmath68 is estimated separately by varying the branching fractions . since there are no experimental measurements of these decays , the branching fractions are varied continuously from 0% to 200% in the mc expectation . as @xmath235 is constrained by the branching fraction of the inclusive semitauonic decay @xmath236 @xcite , which is smaller than the sum of the branching fractions @xmath237 and @xmath238 , we conclude that our assumption is sufficiently conservative . the uncertainties due to the hqet form - factor parameters in the normalization mode @xmath159 are estimated using the uncertainties in the world - average values @xcite . in addition , the uncertainty arising from the small @xmath166 shape correction for the normalization sample is estimated as an uncertainty related to @xmath159 : 0.4% ( 0.008 ) for @xmath45 ( @xmath28 ) . the uncertainties on the reconstruction efficiencies of the @xmath0-daughter particles in the signal sample and the charged leptons in the normalization sample are also considered . here , the uncertainties on the particle identification efficiencies for @xmath85 and @xmath239 and the reconstruction efficiency for @xmath92 are measured with control samples : the @xmath240 sample for @xmath85 , the @xmath241 sample for @xmath92 and the @xmath242 for charged leptons . the sample @xmath243 from @xmath11 decays is also used in order to account for the difference in multiplicity between two - photon events and @xmath11 decay events . estimated uncertainties are as follows : 0.5% and 0.003 for @xmath85 , 0.6% and 0.004 for @xmath92 , 1.6% and 0.004 for charged leptons ( the first and the second values corresponding to uncertainties of @xmath45 and @xmath28 , respectively ) . reconstruction efficiencies of the three @xmath35 components are estimated using mc . the efficiencies have uncertainties arising from the statistics of the signal mc and are varied independently for each component . here , the uncertainty of the fraction of the three @xmath35 components are also taken into account . the @xmath35 decay generator of evtgen , based on the hqet form factors and implemented by belle , neglects the interference between the amplitudes of left- and right - handed @xmath0 leptons . this mis - models the decay topology and so affects the signal reconstruction efficiency . we compare the extracted @xmath45 and @xmath28 between this model and an alternate model based on the isgw form factors , and take differences of @xmath45 and @xmath28 as systematic uncertainties . other small uncertainties arise due to fit bias arising from the @xmath165 bin width selection , which is estimated by comparing the extracted values of @xmath45 and @xmath28 in the nominal fit ( with a bin width of 0.05 gev ) with the values obtained in a wide - bin fit ( bin width of 0.1 gev ) ; the branching fractions of the @xmath0 lepton decays ; and errors on the parameters of the @xmath28 correction function . axis is illustrated . the red - hatched `` @xmath0 cross feed '' combines the @xmath192 cross - feed and the other @xmath0 cross - feed components.,title="fig:",width=566 ] axis is illustrated . the red - hatched `` @xmath0 cross feed '' combines the @xmath192 cross - feed and the other @xmath0 cross - feed components.,title="fig:",width=566 ] axis is illustrated . the red - hatched `` @xmath0 cross feed '' combines the @xmath192 cross - feed and the other @xmath0 cross - feed components.,title="fig:",width=566 ] in addition , common uncertainty sources between the signal sample and the normalization sample are also estimated in this analysis , although they largely cancel at first order in the branching fraction ratio . this is due to the fact that the background yields are partially fixed from the mc expectation . here , uncertainties on the number of @xmath7 ( 1.9% ) events , tagging efficiencies ( 4.7% ) , branching fractions of the @xmath93 decays ( 3.4% ) and @xmath27 reconstruction efficiency ( 4.8% ) are evaluated for their impact on the final measurements . for the @xmath27 reconstruction efficiency , the uncertainty originates from reconstruction efficiencies of @xmath82 , @xmath92 , @xmath84 and @xmath85 , and is therefore correlated with the efficiency uncertainty of the @xmath0-daughter particles containing @xmath85 and @xmath92 . this correlation is taken into account in the total systematic uncertainties shown in table [ tab - syst ] . figure [ fig - fitresult ] shows the fits to the signal and the normalization samples . ( the figures in the forward and backward regions are shown in the appendix [ app - fwdbwd ] . ) the @xmath145 distribution is shown in fig . [ fig - fitcoshel ] . the observed signal and normalization yields are summarized in table [ tab - obtained - yield ] . the @xmath244-values are found to be 15% for the normalization fit and 29% for the signal fit . from the fit , we obtain @xmath245 the signal significance is 9.7@xmath25 ( statistical error only ) or 7.1@xmath25 ( including the systematic uncertainty ) . the significance is taken from @xmath246 , where @xmath247 and @xmath248 are the likelihood with the nominal fit and the null hypothesis , respectively . @l@|@c@@c@ sample & + + & ( signal ) & ( @xmath0 cross feed ) + @xmath249 & @xmath250 & @xmath251 + @xmath252 & @xmath253 & @xmath254 + @xmath255 & @xmath256 & @xmath257 + @xmath258 & @xmath259 & @xmath260 + + @xmath261 & + @xmath262 & + [ tab - obtained - yield ] figure [ fig - significance ] shows a comparison of our result with the theoretical prediction based on the sm @xcite in the @xmath263 plane . the consistency of our result with the sm is @xmath264 . and @xmath265 , respectively ) with the sm prediction ( red triangle ) . the gray region shows the average of the experimental results as of march 2016 @xcite.,width=415 ] we report the measurement of @xmath45 with hadronic @xmath0 decay modes @xmath4 and @xmath5 and the first measurement of @xmath28 in the decay @xmath35 , using @xmath6 @xmath266 data accumulated with the belle detector . our preliminary results are @xmath267 which is consistent with the sm prediction within 0.6@xmath25 . we thank the kekb group for the excellent operation of the accelerator ; the kek cryogenics group for the efficient operation of the solenoid ; and the kek computer group , the national institute of informatics , and the pnnl / emsl computing group for valuable computing and sinet4 network support . we acknowledge support from the ministry of education , culture , sports , science , and technology ( mext ) of japan , the japan society for the promotion of science ( jsps ) , and the tau - lepton physics research center of nagoya university ; the australian research council ; austrian science fund under grant no . p 22742-n16 and p 26794-n20 ; the national natural science foundation of china under contracts no . 10575109 , no . 10775142 , no . 10875115 , no . 11175187 , no . 11475187 and no . 11575017 ; the chinese academy of science center for excellence in particle physics ; the ministry of education , youth and sports of the czech republic under contract no . lg14034 ; the carl zeiss foundation , the deutsche forschungsgemeinschaft , the excellence cluster universe , and the volkswagenstiftung ; the department of science and technology of india ; the istituto nazionale di fisica nucleare of italy ; the wcu program of the ministry of education , national research foundation ( nrf ) of korea grants no . 2011 - 0029457 , no . 2012 - 0008143 , no . 2012r1a1a2008330 , no . 2013r1a1a3007772 , no . 2014r1a2a2a01005286 , no . 2014r1a2a2a01002734 , no . 2015r1a2a2a01003280 , no . 2015h1a2a1033649 ; the basic research lab program under nrf grant no . krf-2011 - 0020333 , center for korean j - parc users , no . nrf-2013k1a3a7a06056592 ; the brain korea 21-plus program and radiation science research institute ; the polish ministry of science and higher education and the national science center ; the ministry of education and science of the russian federation and the russian foundation for basic research ; the slovenian research agency ; ikerbasque , basque foundation for science and the euskal herriko unibertsitatea ( upv / ehu ) under program ufi 11/55 ( spain ) ; the swiss national science foundation ; the ministry of education and the ministry of science and technology of taiwan ; and the u.s . department of energy and the national science foundation . this work is supported by a grant - in - aid from mext for science research in a priority area ( `` new development of flavor physics '' ) , from jsps for creative scientific research ( `` evolution of tau - lepton physics '' ) , a grant - in - aid for scientific research ( s ) `` probing new physics with tau - lepton '' ( no . 26220706 ) and was partly supported by a grant - in - aid for jsps fellows ( no . 25.3096 ) . we thank y. sakaki , r. watanabe and m. tanaka for their invaluable suggestions and helps . , respectively.,title="fig:",width=566 ] , respectively.,title="fig:",width=566 ]

we report the first measurement of the @xmath0 lepton polarization in the decay @xmath1 as well as a new measurement of the ratio of the branching fractions @xmath2 , where @xmath3 denotes an electron or a muon , with the decays @xmath4 and @xmath5 . we use the full data sample of @xmath6 @xmath7 pairs accumulated with the belle detector at the kekb electron - positron collider . our preliminary results , @xmath8 and @xmath9 , are consistent with the theoretical predictions of the standard model within @xmath10 standard deviation .

density functional theory ( dft ) has been the subject of remarkable developments since its original formulation by hohenberg and kohn ( hk ) @xcite . after formal improvements , extensions , and an uncountable number of applications to a wide variety of physical problems , this theoretical approach has become the most efficient , albeit not infallible , method of determining the electronic properties of matter from first principles @xcite . the most important innovation of dft , which is actually at the origin of its breakthrough , is to replace the wave function by the electronic density @xmath18 as the fundamental variable of the many - body problem . in practice , density functional ( df ) calculations are largely based on the kohn - sham ( ks ) scheme that reduces the many - body @xmath19-particle problem to the solution of a set of self - consistent single - particle equations @xcite . although this transformation is formally exact , the implementations always require approximations , since the ks equations involve functional derivatives of the unknown interaction energy @xmath20 $ ] , usually expressed in terms of the exchange and correlation ( xc ) energy @xmath21 $ ] . therefore , understanding the functional dependence of @xmath21 $ ] and improving its approximations are central to the development of df methods . the currently most widespread _ anstze _ for @xmath21 $ ] the local density approximation ( lda ) @xcite with spin polarized @xcite and gradient corrected extensions @xcite were originally derived from exact results for the homogeneous electron gas . it is one of the purposes of this paper to investigate the properties of the interaction - energy functional from an intrinsically inhomogeneous point of view , namely , by considering exactly solvable many - body lattice models . despite the remarkable success of the local spin density approximation , present dft fails systematically in accounting for phenomena where strong electron correlations play a central role , for example , in heavy - fermion materials or high-@xmath22 superconductors . these systems are usually described by simplifying the low - energy electron dynamics using parameterized lattice models such as pariser - parr - pople,@xcite hubbard,@xcite or anderson@xcite models and related hamiltonians @xcite . being in principle an exact theory , the limitations of the df approach have to be ascribed to the approximations used for exchange and correlation and not to the underlying hks formalism . it would be therefore very interesting to extend the range of applicability of dft to strongly correlated systems and to characterize the properties @xmath23 in the limit of strong correlations . studies of the xc functional on simple models should provide useful insights for future extensions to realistic hamiltonians . moreover , taking into account the demonstrated power of the df approach in _ ab initio _ calculations , one may also expect that a dft with an appropriate @xmath23 could become an efficient tool for studying many - body models , a subject of theoretical interest on its own . several properties of dft on lattice models have been already studied in previous works@xcite . gunnarsson and schnhammer were , to our knowledge , the first to propose a df approach on a semiconductor model in order to study the band - gap problem@xcite . in this case the local site occupancies were treated as the basic variables . some years later schindlmayr and godby @xcite provided a different formulation of dft on a lattice by considering as basic variables both diagonal elements @xmath24 and off - diagonal elements @xmath0 of the single - particle density matrix ( see also @xcite ) . then derived a more general framework that unifies the two previous approaches @xcite . using levy s constrained search method @xcite they showed that different basic variables and different @xmath25 functionals can be considered depending on the type of model or perturbation under study . site occupations alone may be used as basic variables , if only the orbital energies are varied ( i.e. , if all hopping integrals @xmath26 are kept constant for @xmath27 ) . however , off - diagonal elements of the single - particle density matrix must be included explicitly if the functional @xmath25 is intended to be applied to more general situations involving different values of @xmath26 , for example , the hubbard model on various lattice structures or for different interaction regimes , i.e. , different @xmath28 . in this paper we investigate the properties of levy s interaction - energy functional @xmath25 as a function of @xmath0 by solving the constrained search minimization problem exactly . in sec . [ sec : teo ] the basic formalism of density - matrix functional theory ( dmft ) on lattice models is recalled and the equations for determining @xmath1 $ ] are derived . [ sec : res ] presents and discusses exact results for the correlation energy @xmath29 of the hubbard model , which is given by the difference between @xmath25 and the hartree - fock energy @xmath30 . these are obtained , either numerically for finite clusters with different lattice structures , or from the bethe - ansatz solution for the one - dimensional chain . finally , sec . [ sec : conc ] summarizes our conclusions and points out some relevant extensions . in sec . [ sec : teolat ] the main results of levy s formulation of dmft are presented in a form that is appropriate for the study of model hamiltonians such as the hubbard model . here , the hopping integrals @xmath26 between sites ( or orbitals ) @xmath6 and @xmath7 play the role given in conventional dft to the external potential @xmath31 . consequently , the single - particle density matrix @xmath0 replaces the density @xmath18 as basic variable @xcite . in sec . [ sec : teoex ] , we derive equations that allow to determine levy s interaction - energy functional @xmath1 $ ] in terms of the ground - state energy of a many - body hamiltonian with effective hopping integrals @xmath32 that depend implicitly on @xmath0 . we consider the many - body hamiltonian @xmath33 where @xmath34 ( @xmath35 ) is the usual creation ( annihilation ) operator for an electron with spin @xmath36 at site ( or orbital ) @xmath6 . @xmath37 can be regarded as the second quantization of schrdinger s equation on a basis @xcite . however , in the present paper , the hopping integrals @xmath26 and the interaction matrix elements @xmath38 are taken as parameters to be varied independently . the matrix @xmath26 defines the lattice ( e.g. , one dimensional chains , square or triangular two - dimensional lattices ) and the range of single - particle interactions ( e.g. , up to first or second neighbors ) . from the _ ab initio _ perspective @xmath26 is given by the external potential and by the choice of the basis @xcite . @xmath38 defines the type of many - body interactions which may be repulsive ( coulomb like ) or attractive ( in order to simulate electronic pairing ) and which are usually approximated as short ranged ( e.g. , intra - atomic ) . ( [ eq : hamgen ] ) is mainly used in this section to derive general results which can then be applied to various specific models by simplifying the interactions . a particularly relevant example , to be considered in some detail in sec . [ sec : res ] , is the single - band hubbard model with nearest neighbor ( nn ) hoppings @xcite , which can be obtained from eq . ( [ eq : hamgen ] ) by setting @xmath39 for @xmath6 and @xmath7 nn s , @xmath40 otherwise , and @xmath41 @xcite . in order to apply dmft to model hamiltonians of the form ( [ eq : hamgen ] ) we follow levy s constrained search procedure @xcite as proposed by schindlmayr and godby @xcite . the ground - state energy is determined by minimizing the functional @xmath42 = e_k[\gamma_{ij } ] + w [ \gamma_{ij}]\ ] ] with respect to the single - particle density matrix @xmath0 . @xmath43 $ ] is physically defined for all density matrices that can be written as @xmath44 for all @xmath6 and @xmath7 , where @xmath45 is an @xmath19-particle state . in other words , @xmath0 must derive from a physical state . it is then said to be pure - state @xmath19-representable @xcite . the first term in eq . ( [ eq : efun ] ) is given by @xmath46 it includes all single - particle contributions and is usually regarded as the kinetic energy associated with the electronic motion in the lattice . notice that eq . ( [ eq : ek ] ) yields the exact kinetic energy for a given @xmath0 . there are no corrections on @xmath47 to be included in other parts of the functional as in the ks approach . the second term in eq . ( [ eq : efun ] ) is the interaction - energy functional given by @xcite @xmath48 = min \left[\frac{1}{2 } \sum _ { nmkl \atop { \sigma \sigma ' } } v_{nmkl } \ ; \langle \psi [ \gamma_{ij } ] | \ ; \hat c_{n \sigma } ^{\dagger } \hat c_{k \sigma ' } ^{\dagger } \hat c_{l \sigma ' } \hat c_{m \sigma } \ ; |\psi[\gamma_{ij}]\rangle \right ] \ ; .\ ] ] the minimization in eq . ( [ eq : w ] ) implies a search over all @xmath19-particles states @xmath49 \rangle$ ] that satisfy @xmath50 | \ ; \sum_\sigma \hat c_{i \sigma } ^{\dagger}\hat c_{j \sigma } \ ; |\psi [ \gamma_{ij } ] \rangle = \gamma_{ij}$ ] for all @xmath6 and @xmath7 . therefore , @xmath1 $ ] represents the minimum value of the interaction energy compatible with a given density matrix @xmath0 . @xmath25 is usually expressed in terms of the hartree - fock energy @xmath51=\frac{1}{2}\sum_{ijkl \atop{\sigma \sigma ' } } v_{ijkl } \left(\gamma_{ij \sigma } \gamma_{kl \sigma ' } - \delta_{\sigma\sigma ' } \gamma_{il \sigma } \gamma_{kj \sigma } \right)\ ] ] and the correlation energy @xmath3 $ ] as @xmath52 = e_{\rm hf}[\gamma_{ij } ] + e_{\rm c}[\gamma_{ij } ] \;.\ ] ] @xmath25 and @xmath29 are universal functionals of @xmath0 in the sense that they are independent of @xmath26 , i.e. , of the system under study . they depend on the considered interactions or model , as defined by @xmath38 , on the number of electrons @xmath9 , and on the structure of the many - body hilbert space , as given by @xmath9 and the number of orbitals or sites @xmath8 . notice that @xmath29 in eq . ( [ eq : xc ] ) does not include any exchange contributions . given @xmath0 ( @xmath53 in nonmagnetic cases ) there is no need to approximate the exchange term , which is taken into account exactly by @xmath30 [ eq . ( [ eq : hf ] ) ] . nevertheless , if useful in practice , it is of course possible to split @xmath25 in the hartree energy @xmath54 and the exchange and correlation energy @xmath23 is a similar way as in the ks approach . the variational principle results from the following two relations @xcite : @xmath55\ ] ] for all pure - state @xmath19-representable @xmath0 @xcite , and @xmath56 \;,\ ] ] where @xmath57 refers to the ground - state energy and @xmath58 to the ground - state single - particle density matrix . as already pointed out in previous works @xcite , @xmath25 and @xmath29 depend in general on both diagonal elements @xmath24 and off - diagonal elements @xmath0 of the density - matrix , since the hopping integrals @xmath26 are non local in the sites the situation is similar to the df approach proposed by gilbert for the study of non - local potentials @xmath59 as those appearing in the theory of pseudo - potentials @xcite . a formulation of dft on a lattice only in terms of @xmath24 would be possible if one would restrict oneself to a family of models with constant @xmath26 for @xmath27 however , in this case the functional @xmath60 $ ] would depend on the actual value of @xmath26 for @xmath27 @xcite . the functional @xmath1 $ ] , valid for all lattice structures and for all types of hybridizations , can be simplified at the expense of universality if the hopping integrals are short ranged . for example , if only nn hoppings are considered , the kinetic energy @xmath47 is independent of the density - matrix elements between sites that are not nn s . therefore , the constrained search in eq . ( [ eq : w ] ) may restricted to the @xmath49 \rangle$ ] that satisfy @xmath50 |\ ; \sum_\sigma \hat c_{i \sigma } ^{\dagger}\hat c_{j \sigma } \;|\psi [ \gamma_{ij } ] \rangle = \gamma_{ij}$ ] only for @xmath61 and for nn @xmath62 . in this way the number of variables in @xmath1 $ ] is reduced significantly rendering the interpretation of the functional dependence far simpler . while this is a great practical advantage , it also implies that @xmath25 and @xmath29 lose their universal character since the dependence on the nn @xmath0 is now different for different lattices . in sec . [ sec : res ] results for one- , two- , and three - dimensional lattices with nn hoppings are compared in order to quantify this effect . for the applications in sec . [ sec : res ] we shall consider the single - band hubbard model with nn hoppings , which in the usual notation is given by@xcite @xmath63 in this case the interaction energy functional reads @xmath64 = min \left [ u \sum_l \langle \psi [ \gamma_{ij } ] |\ ; \hat n_{l\uparrow } \hat n_{l\downarrow } \;|\psi [ \gamma_{ij } ] \rangle \right ] \ ; , \ ] ] where the minimization is performed with respect to all @xmath19-particle @xmath65\rangle$ ] satisfying @xmath50 | \sum_\sigma \hat c_{i \sigma } ^{\dagger}\hat c_{j \sigma } | \psi [ \gamma_{ij } ] \rangle = \gamma_{ij}$ ] for @xmath6 and @xmath7 nn s . if the interactions are repulsive ( @xmath16 ) @xmath1 $ ] represents the minimum average number of double occupations which can be obtained for a given degree of electron delocalization , i.e. , for a given value of @xmath0 . for attractive interactions ( @xmath17 ) double occupations are favored and @xmath1 $ ] corresponds to the maximum of @xmath66 for a given @xmath0 . in order to determine @xmath3 $ ] and @xmath1 $ ] we look for the extremes of @xmath67 \ ; + \;\varepsilon \ ; \big(1 - \langle \psi |\psi \rangle \big ) \ ; \nonumber \\ & + & \sum_{i , j } \lambda_{ij } \ ; \big ( \langle\psi| \sum_\sigma \hat c^\dagger_{i\sigma } \hat c_{j \sigma } | \psi\rangle - \gamma_{ij } \big ) \end{aligned}\ ] ] with respect to @xmath68 . lagrange multipliers @xmath69 and @xmath32 have been introduced to enforce the normalization of @xmath70 and the conditions on the representability of @xmath0 . derivation with respect to @xmath71 , @xmath69 and @xmath32 yields the eigenvalue equations @xmath72 and the auxiliary conditions @xmath73 and @xmath74 . the lagrange multipliers @xmath32 play the role of hopping integrals to be chosen in order that @xmath70 yields the given @xmath0 . the pure - state representability of @xmath0 ensures that there is always a solution @xcite . in practice , however , one usually varies @xmath32 in order to scan the domain of representability of @xmath0 . for given @xmath32 , the eigenstate @xmath75 corresponding to the lowest eigenvalue of eq . ( [ eq : evgen ] ) yields the minimum @xmath1 $ ] for @xmath76 . any other @xmath70 satisfying @xmath74 would have higher @xmath69 and thus higher @xmath25 . the subset of @xmath0 which are representable by a ground - state of eq . ( [ eq : evgen ] ) is the physically relevant one , since it necessarily includes the absolute minimum @xmath77 of @xmath43 $ ] . nevertheless , it should be noted that pure - state representable @xmath0 may be considered that can only be represented by excited states or by linear combinations of eigenstates of eq . ( [ eq : evgen ] ) . in the later case , @xmath78 , and @xmath79 is an eigenstate of the interaction term with lowest eigenvalue . examples shall be discussed in sec . [ sec : res ] . for the hubbard model eq . ( [ eq : evgen ] ) reduces to @xmath80 this eigenvalue problem can be solved numerically for clusters with different lattice structures and periodic boundary conditions . in this case we expand @xmath49 \rangle$ ] in a complete set of basis states @xmath81 which have definite occupation numbers @xmath82 at all orbitals @xmath83 ( @xmath84 with @xmath85 or @xmath86 ) . the values of @xmath82 satisfy the usual conservation of the number of electrons @xmath87 and of the @xmath88 component of the total spin @xmath89 , where @xmath90 . for not too large clusters , the lowest energy @xmath91 \rangle$ ] the ground state of eq . ( [ eq : evhub]) can be determined by sparse - matrix diagonalization procedures , for example , by using lanczos iterative method @xcite . @xmath92 \rangle$ ] is calculated in the subspace of minimal @xmath93 since this ensures that there are no _ a priori _ restrictions on the total spin @xmath94 . in addition , spin - projector operators may be used to study the dependence of @xmath4 on @xmath94 . for a one - dimensional ( 1d ) chain with nn hoppings @xmath95 , translational symmetry implies equal density - matrix elements @xmath0 between nn s . therefore , one may set @xmath96 for all nn @xmath62 , and then eq . ( [ eq : evhub ] ) has the same form as the 1d hubbard model for which lieb and wu s exact solution is available @xcite . in this case the lowest eigenvalue @xmath69 is determined following the work by shiba @xcite . the coupled bethe - ansatz equations are solved as a function of @xmath97 , band - filling @xmath98 , and for positive and negative @xmath99 , by means of a simple iterative procedure . in this section we present and discuss exact results for the correlation energy functional @xmath3 $ ] of the single - band hubbard hamiltonian with nearest neighbor hoppings @xcite . given the lattice structure , @xmath8 and @xmath9 , the model is characterized by the dimensionless parameter @xmath28 which measures the competition between kinetic and interaction energies [ see eq . ( [ eq : hamhub ] ) ] . @xmath16 corresponds to the usual intra - atomic repulsive coulomb interaction , while the attractive case ( @xmath17 ) simulates intra - atomic pairing of electrons . in fig . [ fig : xcanft ] the correlation energy @xmath29 of the one - dimensional ( 1d ) hubbard model is shown for half - band filling ( @xmath100 ) as a function of the density - matrix element or bond order @xmath101 between nn s . @xmath102 for all nn s @xmath6 and @xmath7 . results are given for rings of finite length @xmath8 as well as for the infinite chain . several general qualitative features may be identified . first of all we observe that on bipartite lattices @xcite @xmath103 , since the sign of the nn bond order can be changed without affecting the interaction energy @xmath104 by changing the phase of the local orbitals at one of the sublattices ( @xmath105 for @xmath106 and @xmath107 unchanged for @xmath108 , where @xmath109 and @xmath110 refer to the sublattices ) . let us recall that the domain of definition of @xmath4 is limited by the pure - state representability of @xmath0 . the upper bound @xmath111 and the lower bound @xmath112 for @xmath101 ( @xmath113 on bipartite lattices ) are the extreme values of the bond order between nn s on a given lattice and for given @xmath8 and @xmath9 ( @xmath102 for all nn @xmath62 ) . they represent the maximum degree of electron delocalization . @xmath111 and @xmath112 correspond to the extremes of the kinetic energy @xmath47 [ @xmath114 , where @xmath88 is the coordination number ] and thus to the ground state of the hubbard model for @xmath115 [ @xmath111 for @xmath116 and @xmath112 for @xmath117 , see eq . ( [ eq : hamhub ] ) ] . for @xmath118 the underlying electronic state @xmath79 is usually a single slater determinant and therefore @xmath119 . in other words , the correlation energy vanishes as expected in the fully delocalized limit @xcite . as @xmath120 decreases @xmath29 decreases ( @xmath121 ) since correlations can reduce the coulomb energy more and more efficiently as the electrons localize . @xmath29 is minimum in the strongly correlated limit @xmath122 . for half - band filling this corresponds to a fully localized electronic state ( @xmath123 ) . here , @xmath29 cancels out the hartree - fock energy @xmath30 and the coulomb energy @xmath25 vanishes ( @xmath124 ) @xcite . the ground - state values of @xmath125 and @xmath126 for a given @xmath28 result from the competition between lowering @xmath29 by decreasing @xmath101 and lowering @xmath47 by increasing it ( @xmath116 ) . the divergence of @xmath127 for @xmath128 is a necessary condition in order that @xmath129 for arbitrary small @xmath16 . on the other side , for small @xmath101 , we observe that @xmath130 . this implies that for @xmath131 , @xmath132 and @xmath133 , a well known result in the heisenberg limit of the hubbard model ( @xmath134 ) @xcite . a more quantitative analysis of @xmath4 and in particular the comparison of results for different @xmath8 is complicated by the size dependence of @xmath135 and @xmath30 . it is therefore useful to measure @xmath29 in units of the hartree - fock energy and to bring the domains of representability to a common range by considering @xmath136 as a function of @xmath137 . fig . [ fig : nxcanft ] shows that @xmath15 has approximately the same behavior for all considered @xmath8 . finite size effects are small except for the very small sizes . the largest deviations from the common trend are found for @xmath138 . here we observe a discontinuous drop of @xmath139 for @xmath140 ( @xmath141 ) which is due to the degeneracy of the single - particle spectrum . in fact in this case two of the four electrons occupy a doubly degenerate state in the uncorrelated limit and the minimum interaction energy @xmath104 does not correspond to a single - slater - determinant state even for @xmath142 @xcite . as @xmath8 increases @xmath15 approaches the infinite - length limit with alternations around the @xmath143 curve . the strong similarity between @xmath15 for small @xmath8 and for @xmath143 is a remarkable result . it suggests that good approximations for @xmath4 in extended systems could be derived from finite cluster calculations . [ fig : xcanfl1 ] shows the band - filling dependence of @xmath4 in a 10-site 1d hubbard ring . results are given for @xmath144 , since for @xmath145 , @xmath146 as a result of electron - hole symmetry @xcite . although @xmath4 depends strongly on @xmath9 , several qualitative properties are shared by all band fillings : ( i ) as in the half - filled band case , the domain of representability of @xmath101 is bound by the bond orders in the uncorrelated limits . in fact , @xmath147 , where @xmath111 ( @xmath112 ) corresponds to the ground state of the @xmath115 tight - binding model for @xmath116 ( @xmath117 ) . on bipartite lattices @xmath148 . notice that @xmath135 increases monotonously with @xmath9 as the single - particle band is filled up . this is an important contribution to the band - filling dependence of @xmath29 ( see fig . [ fig : xcanfl1 ] ) . ( ii ) in the delocalized limit , @xmath119 for all the @xmath9 for which @xmath149 derives from a single slater determinant @xcite . moreover , the divergence of @xmath127 for @xmath150 indicates that @xmath129 for arbitrary small @xmath16 , as expected from perturbation theory . ( iii ) starting from @xmath128 , @xmath4 decreases with decreasing @xmath101 reaching its lowest possible value @xmath124 for @xmath151 ( @xmath144 ) . the same behavior is of course observed for @xmath152 . in particular , @xmath153 also for @xmath154 . as shown in fig . [ fig : xcanfl1 ] , @xmath155 decreases rapidly with increasing @xmath9 , since @xmath30 increases quadratically with the electron density @xcite . ( iv ) on bipartite lattices @xmath156 , while on non - bipartite structures one generally has @xmath157 , since the single - particle spectrum is different for positive and negative energies . the decrease of @xmath29 with decreasing @xmath120 shows that the reduction of the coulomb energy due to correlations is done at the expense of kinetic energy or electron delocalization , as already discussed for @xmath100 ( fig . [ fig : xcanft ] ) . ( v ) @xmath158 for all @xmath159 ( @xmath160 for @xmath100 ) . @xmath161 represents the largest nn bond order that can be constructed under the constraint of vanishing coulomb repulsion energy . a lower bound for @xmath161 is given by the bond order @xmath162 in the fully - polarized ferromagnetic state ( @xmath163 ) . this is obtained by occupying the lowest single - particle states with all electrons of the same spin ( @xmath144 ) . therefore , @xmath162 increases with @xmath9 for @xmath164 and then decreases for @xmath165 reaching @xmath166 at half - band filling ( @xmath167 for @xmath168 ) . in this way the non - monotonous dependence of @xmath161 on @xmath9 can be explained ( see fig . [ fig : xcanfl1 ] ) . ( vi ) the correlation energy is constant and equal to @xmath169 for @xmath170 . these values of @xmath101 can never correspond to the ground - state energy of the hubbard model , since in this range increasing @xmath101 always lowers the kinetic energy ( @xmath116 ) without increasing the coulomb repulsion ( @xmath171 ) . for @xmath172 , @xmath101 can not be represented by a ground state of eq . ( [ eq : evhub ] ) . in this range @xmath101 can be derived from a linear combination of states having minimal coulomb repulsion @xcite . in order to compare the functional dependences of the correlation energy for different band fillings , it is useful to scale @xmath29 in units of the hartree - fock energy and to bring the relevant domains @xmath173 of different @xmath9 to a common range . in fig . [ fig : nxcanfl1 ] , @xmath174 is shown as a function of @xmath175 . we observe that the results for @xmath15 are remarkably similar for all band - fillings . the largest deviations from the common trend are found for @xmath176 . as already discussed for @xmath177 , this anomalous behavior is related to the degeneracy of the single - particle spectrum and to the finite size of system . fig . [ fig : nxcanfl1 ] shows that for the hubbard model the largest part of the dependence of @xmath4 on band filling comes from @xmath30 , @xmath178 and @xmath161 . similar conclusions are derived from the results for the infinite 1d chain presented in fig . [ fig:1dinf ] . for a given @xmath179 , @xmath15 depends weakly on @xmath180 if the carrier density is low ( @xmath181 ) , and tends to increase as we approach half - band filling [ see fig . [ fig:1dinf](b ) ] . for high carrier densities it become comparatively more difficult to minimize the coulomb energy for a given degree of delocalization @xmath179 . the effect is most pronounced for @xmath182@xmath183 , i.e. , close to the uncorrelated limit . as we approach the strongly correlated limit ( @xmath184 ) the dependence of @xmath139 on @xmath180 is very weak even for @xmath185 . one concludes that @xmath15 is a useful basis for introducing practical approximations on more complex systems . the correlation energy @xmath29 is a universal functional of the complete single - particle density matrix @xmath0 . @xmath3 $ ] and @xmath1 $ ] may depend on @xmath8 and @xmath9 but are independent of @xmath26 and in particular of the lattice structure . the functional @xmath4 considered in this paper depends by definition on the type of lattice , since the constraints imposed in the minimization only apply to nn bonds . in order to investigate this problem we have determined @xmath4 for 2d and 3d finite clusters having @xmath186 sites and periodic boundary conditions . in fig . [ fig : xcfdima ] we compare these results with those of the 1d @xmath187-site periodic ring . as shown in the inset figure , the qualitative behavior is in all cases very similar . the main quantitative differences come from the domain of representability of @xmath101 , i.e. , from the values of @xmath111 and @xmath112 ( @xmath188 ) . once scaled as a function of @xmath189 , @xmath29 depends rather weakly on the lattice structure . notice that the hartree - fock energy @xmath190 is the same for all structures . however , for the bcc structure we obtain @xmath191 , i.e. , @xmath192 , due to degeneracies in the single - particle spectrum of the considered finite cluster [ see inset fig . [ fig : xcfdima](b ) ] . in order to correct for this finite size effect it is here more appropriate to consider @xmath193 / w(\gamma_{12}^0)$ ] . still , the differences in @xmath139 between bcc and fcc structures appear to be more important than between square and triangular 2d lattices . this is probably related to the degeneracies in the spectrum of the bcc cluster , as already observed for rings with @xmath194 [ figs . [ fig : nxcanft ] and [ fig : nxcanfl1](a ) ] . the largest changes in @xmath139 for different lattice structures are observed for intermediate degree of delocalization ( @xmath195@xmath183 , see fig . [ fig : xcfdima ] ) . note that there is no monotonic trend as a function of the lattice dimension . for example , for @xmath196@xmath183 , @xmath139 first increases somewhat as we go from 1d to 2d lattices , but it then decreases coming close to the 1d curve for the 3d fcc lattice [ @xmath197 for @xmath198 . finally , it is worth noting that in the strongly correlated limit ( @xmath199 ) the results for @xmath15 are nearly the same for all considered lattice structures ( see fig . [ fig : xcfdima ] ) . this should be useful in order to develop simple general approximations to @xmath4 in this limit . the attractive hubbard model describes itinerant electrons with local intra - atomic pairing ( @xmath17 ) . the electronic correlations are very different from those found in the repulsive case discussed so far . in particular levy s interaction energy functional @xmath200 now correspond to the maximum average number of double occupation for a given @xmath0 [ see eq . ( [ eq : whub ] ) ] . therefore , it is very interesting to investigate the properties of the correlation energy functional @xmath201 also for @xmath17 and to contrast them with the results of the previous section . in fig . [ fig : xcsuft ] the correlation energy @xmath4 of the attractive hubbard model is given at half - band filling for various finite rings ( @xmath202 ) and for the infinite 1d chain ( @xmath134 ) . the band - filling dependence of @xmath4 is shown in fig . [ fig : xcsufl ] for a finite @xmath187-site ring ( @xmath203 ) . as in the repulsive case , @xmath204 since the domain of representability of @xmath101 is independent of the form or type of the interaction . moreover , @xmath205 due to the electron - hole symmetry of bipartite lattices @xcite . starting from @xmath111 or @xmath112 ( @xmath206 on bipartite lattices ) , @xmath4 decreases with decreasing @xmath120 reaching the minimum @xmath207 for @xmath208 and for @xmath154 ( @xmath209 in this case ) . for @xmath9 even , @xmath210 , and for @xmath9 odd , @xmath211 , which correspond to the maximum number of electron pairs that can be formed . for @xmath9 even , the minimum @xmath212 $ ] is achieved only for a complete electron localization ( i.e , @xmath160 ) . in contrast , for odd @xmath9 a finite - size effect is observed . in this case , one of the electrons remains unpaired even in the limit of strong electron correlations and the minimum of @xmath29 is @xmath213[1-(n_e+1)/(2n_a)]$ ] . moreover , non - vanishing @xmath161 are obtained as a result of the delocalization of the unpaired electron . @xmath161 represents the maximum bond order that can be obtained when @xmath214 electron pairs are formed ( @xmath215 for @xmath216 , @xmath9 odd ) . notice that in all cases the ground state @xmath125 is found in the interval @xmath217 . it is interesting to observe that @xmath4 can be appropriately scaled in a similar way as for @xmath16 . in fig . [ fig : xcsufl](b ) , @xmath218 is shown as a function of the degree of delocalization @xmath219 . @xmath15 presents a pseudo - universal behavior in the sense that it depends weakly on @xmath8 and @xmath9 . the main deviations from the common trend are found for @xmath220 . as already discussed for @xmath16 , this is a consequence of degeneracies in the single - particle spectrum . in this case , the wave function corresponding to the minimum in levy s functional for @xmath221 [ eq . ( [ eq : whub ] ) ] can not be described by a single slater determinant and @xmath222 . density - matrix functional theory has been applied to lattice hamiltonians taking the hubbard model as a particularly relevant example . in this framework the basic variable is the single - particle density matrix @xmath0 and the key unknown is the correlation energy functional @xmath3 $ ] . the challenge is therefore to determine @xmath3 $ ] or to provide with useful accurate approximations for it . in this paper we presented a systematic study of the functional dependence of @xmath4 on periodic lattices , where @xmath101 is the density - matrix element between nearest neighbors ( @xmath223 for all nn @xmath62 ) . based on finite - cluster exact diagonalizations and on the bethe - ansatz solution of the 1d chain , we derived rigorous results for @xmath4 of the hubbard model as a function of the number of sites @xmath8 , band filling @xmath180 and lattice structure . a basis for applications of density - matrix functional theory to many - body lattice models is thereby provided . the observed pseudo - universal behavior of @xmath224 as a function of @xmath225 encourages transferring @xmath15 from finite - size systems to infinite lattices or even to different lattice geometries . in fact , the exact @xmath4 of the hubbard dimer has been recently used to infer a simple general ansatz for @xmath4 @xcite . with this approximation to @xmath4 the ground - state energies and charge - excitation gaps of 1d and 2d lattices have been determined successfully in the whole range of @xmath28 . further investigations , for example , by considering magnetic impurity models or more complex multiband hamiltonians , are certainly worthwhile . langreth and m.j . mehl , phys . b * 28 * , 1809 ( 1983 ) ; a.d . becke , phys . a * 38 * , 3098 ( 1988 ) ; j.p . perdew , k. burke and m. ernzerhof , phys . lett . * 77 * , 3865 ( 1996 ) and references therein . given a basis set @xmath226 , the hopping integrals are expressed as @xmath227 \phi_j ( \vec r ) \ ; , \ ] ] and the matrix elements of the coulomb interaction as @xmath228 a single - particle density matrix @xmath0 is said to be pure - state @xmath19-representable if an @xmath19-particle state @xmath45 exists such that @xmath229 for all @xmath6 and @xmath7 . an extension of the definition domain of @xmath43 $ ] to ensemble - representable density matrices @xmath230 is straightforward following the work by valone ( see ref . ensemble density matrices are written as @xmath231 . in practice , the @xmath230 are much easier to characterize than pure - state density matrices . c. lanczos 45 * , 255 ( 1950 ) ; b.n . parlett , _ the symmetric eigenvalue problem _ , ( prentice - hall , engelwood cliffs , nj , 1980 ) ; j.k . collum and r.a . willoughby , _ lanczos algorithms for large symmetric eigenvalue computations _ , ( birkhauser , boston , 1985 ) , vol . i. the electron - hole transformation @xmath232 leaves the hubbard hamiltonian and eq . ( [ eq : evhub ] ) formally unchanged , except for an additive constant and a change of sign in the hopping integrals @xmath26 [ or of @xmath32 in eq . ( [ eq : evhub ] ) ] . @xmath233 , since the form of interaction term is unaffected by changing electrons to holes , and since the density matrix for holes satisfies @xmath234 . moreover , on bipartite lattices with only nn hoppings @xmath235 , since the sign of the nn bond order may be changed by changing the sign of @xmath236 on one of the sublattices . a lattice is _ bipartite _ if two distinct subsets of lattice sites @xmath109 and @xmath110 can be defined such that every lattice site belongs either to a or to b , and that there is no pair of nn s belonging to the same subset . all nn bonds ( or hoppings ) connect a site in @xmath109 with a site in @xmath110 . in the presence of degeneracies in the single - particle spectrum of finite systems one may find that the minimum of @xmath1 $ ] does not correspond to a single slater determinant , and that @xmath237 . such a behavior is observed , for example , in rings with @xmath238 . this is a finite - size effect which importance decreases with increasing @xmath8 : @xmath239 , @xmath240 and @xmath241 for rings with @xmath220 , @xmath242 and @xmath187 , respectively . @xmath243 . in the nonmagnetic case the hartree - fock energy of the hubbard model is @xmath244 for @xmath9 even and @xmath245 $ ] for @xmath9 odd . notice that in this model the difference between @xmath30 and the hartree energy @xmath246 is only the self interaction . if @xmath247 , i.e. , @xmath248 , it is more appropriate to consider @xmath249 / w(\gamma_{12}^0)$ ] in order to correct for this finite size effect . in this way @xmath15 is less sensitive to the details of the lattice structure or cluster size . using a linear combination @xmath70 between two @xmath19-particles states @xmath250 and @xmath251 satisfying @xmath252 and latexmath:[$\langle\psi_a| \sum_{\sigma } \hat c_{i \sigma}^{\dagger } \hat c_{j \sigma } @xmath94 or @xmath93 ) , one may represent the nn matrix elements @xmath254 in the interval @xmath255 , where @xmath256 and @xmath257 are the bond orders corresponding to @xmath258 and @xmath259 ( @xmath260 ) . for example , one may take @xmath258 as a fully localized state with maximal @xmath93 ( @xmath261 , @xmath262 ) and @xmath251 as the state representing @xmath161 for @xmath263 .

a density functional theory for many - body lattice models is considered in which the single - particle density matrix @xmath0 is the basic variable . eigenvalue equations are derived for solving levy s constrained search of the interaction energy functional @xmath1 $ ] . @xmath1 $ ] is expressed as the sum of hartree - fock energy @xmath2 $ ] and the correlation energy @xmath3 $ ] . exact results are obtained for @xmath4 of the hubbard model on various periodic lattices , where @xmath5 for all nearest neighbors @xmath6 and @xmath7 . the functional dependence of @xmath4 is analyzed by varying the number of sites @xmath8 , band filling @xmath9 and lattice structure . the infinite one - dimensional chain and one- , two- , or three - dimensional finite clusters with periodic boundary conditions are considered . the properties of @xmath4 are discussed in the limits of weak ( @xmath10 ) and strong ( @xmath11 ) electronic correlations , and in the crossover region ( @xmath12 ) . using an appropriate scaling we observe that @xmath13 has a pseudo - universal behavior as a function of @xmath14 . the fact that @xmath15 depends weakly on @xmath8 , @xmath9 and lattice structure suggests that the correlation energy of extended systems could be obtained quite accurately from finite cluster calculations . finally , the behavior of @xmath4 for repulsive ( @xmath16 ) and attractive ( @xmath17 ) interactions are contrasted . 2

the magnetocaloric effect , _ i.e. _ , a temperature change induced by an adiabatic change of an external magnetic field was discovered in iron by warburg in 1881 @xcite . this effect also has a long history in cooling applications . for example , adiabatic demagnetization of paramagnetic salts was the first method to reach temperatures below 1k @xcite . among the many experimental but also theoretical investigations to follow , we would like to mention in particular the computation of the field variation of the entropy in a quantum spin-1/2 @xmath2 chain @xcite . this investigation is remarkable in so far as it is one of the very first numerical computations for an interacting quantum many - body system . the relation of the entropy of quantum spin chains in a magnetic field to magnetic cooling was also noted @xcite and even one exact computation was performed , albeit for a chain with a strong easy - axis anisotropy @xcite . the magnetocaloric effect in quantum spin systems has recently attracted renewed attention . the reason for this is two - fold . on the one hand , field - induced quantum phase transitions lead to universal responses when the applied field is varied adiabatically @xcite . on the other hand , it was observed that the magnetocaloric effect is enhanced by geometric frustration @xcite , promising improved efficiency in low - temperature cooling applications . in this context , the isotropic spin-1/2 heisenberg chain , the isotropic and anisotropic @xmath3 chain in a transverse field @xcite , and the quantum ising chain in a transverse field @xcite were also revisited . one further example of a non - frustrated quantum spin chain where the magnetocaloric effect was computed is the ferrimagnetic spin-@xmath4-@xmath5 chain @xcite . finally , the adiabatic cooling rate has recently been measured on the spin-1/2 heisenberg chain compound [ cu(@xmath6-c@xmath7o@xmath8)(4-aminopyridine)@xmath7(h@xmath7o)]@xmath9 @xcite and relatedly the magnetocaloric effect of the spin-1/2 isotropic heisenberg and @xmath2 chains was computed exactly @xcite . in this paper we will investigate magnetocaloric properties of quantum spin-@xmath0 heisenberg chains given by the hamiltonians @xmath10 @xmath11 is the antiferromagnetic exchange constant , @xmath12 an external magnetic field , @xmath13 the length of the chain , and @xmath14 are quantum spin-@xmath0 operators at site @xmath15 . we will use periodic boundary conditions , _ i.e. _ , @xmath16 . in section [ sec : methods ] we will first summarize the methods which we are going to use , namely exact diagonalization ( ed ) and quantum monte carlo ( qmc ) simulations using the stochastic series expansion ( sse ) framework . then we will apply these methods to the @xmath1 chain in section [ sec : s1chain ] . the spin @xmath1 heisenberg chain is famous for the presence of the so - called ` haldane ' gap at @xmath17 @xcite . in particular , we will illustrate with the @xmath1 chain that one can exploit the presence of a spin gap for cooling by adiabatic magnetization at low magnetic fields . next , we discuss the scaling with spin quantum number @xmath0 in section [ sec : sscal ] . in particular , we will show that there is a quantum scaling regime at low temperatures close to the saturation field where one has _ linear _ scaling with @xmath0 . finally , in section [ sec : summary ] we will conclude with a summary . thermodynamic quantities can be computed using spectral representations . for example , the entropy can be written as @xcite @xmath18 where @xmath19 is the partition function and @xmath20 are the eigenvalues of the hamiltonian . in order to evaluate spectral representations like ( [ eq : specs ] ) , one needs to diagonalize the hamiltonian . if this is done numerically exactly , this is called exact diagonalization ( ed ) . first , one should perform a symmetry reduction . we have used translations and @xmath21-conservation as well as reflections and spin inversion where appropriate . one can then use a library routine to perform a full diagonalization . such an approach is very much in the spirit of classic work @xcite . the main technical differences of our computations and @xcite are : ( i ) we have exploited @xmath22-symmetry to reconstruct the @xmath23 sector from the @xmath24 sectors and the spin - inversion resolved @xmath25 sector , ( ii ) we have used improved implementations of the diagonalization routines @xcite , ( iii ) and we have substantially more powerful computers at our disposal . the combination of these factors enables us to compute full spectra for bigger systems than in the 1960s . if we are interested only in low - energy properties , we can use iterative diagonalization algorithms like the lanczos method @xcite . the basic lanczos algorithm @xcite for a hermitian matrix @xmath26 proceeds as follows . for a given normalized start vector @xmath27 , one defines a sequence of normalized lanczos vectors @xmath28 via the recurrence relations @xmath29 with the initial conditions @xmath30 and @xmath31 . the real coefficients @xmath32 and @xmath33 define the so - called lanczos matrices @xmath34 the crucial point of the lanczos algorithm is that the eigenvalues of @xmath35 yield a good approximation of the extremal eigenvalues of @xmath26 already for @xmath36 much smaller than the total hilbert space dimension . the lanczos algorithm suffers from one practical problem : the recursion relations ( [ eq : lancrec ] ) are supposed to guarantee mutual orthogonality of the vectors @xmath37 , _ i.e. _ , @xmath38 for @xmath39 . however , this fails to be correct when the computations are carried out numerically , leading to undesired spurious states , called ghost states @xcite . several strategies have been proposed @xcite in order to deal with the ghost problem during the computation of excited eigenvalues . here we suggest another method to compute excited states with controlled accuracy and correct multiplicity . we start by performing a fixed number @xmath36 of lanczos iterations with a given start vector @xmath27 . next we compute the orthogonal transformation @xmath40 , @xmath41 diagonalizing the lanczos matrix ( [ eq : lancmat ] ) . then we fix a number @xmath42 and repeat the lanczos procedure with the same start vector @xmath27 . during this second lanczos pass we construct @xmath43 vectors @xmath44 the second lanczos pass is needed in order to avoid storing the @xmath45 lanczos vectors @xmath28 . the vectors @xmath46 given by ( [ eq : lancapproxev ] ) yield approximations to the eigenvectors of @xmath26 . however , due to the ghost problem , there are spurious vectors which need to be eliminated . we perform this in two steps . first , using the observation that the vectors @xmath46 are supposed to be mutually orthogonal , we can eliminate those @xmath46 which have big projections on the @xmath47 with @xmath48 . second , we reorthogonalize the remaining vectors yielding @xmath49 orthonormal vectors @xmath50 . finally , we project the matrix @xmath26 onto this subspace via @xmath51 a full diagonalization of the @xmath52 matrix @xmath53 and application of the resulting basis transformation to the vectors @xmath50 yields @xmath54 _ orthogonal _ vectors @xmath55 which approximate the eigenvectors of @xmath26 . the eigenvalues @xmath56 associated to the vectors @xmath55 and their accuracy can be estimated according to @xmath57 according to our experience , there are neither any missing nor any spurious eigenvalues among the converged ones . it should be mentioned that there is no guarantee that the lanczos procedure yields the complete spectrum for a given start vector @xmath27 , in particular if one has not performed a complete symmetry decomposition . nevertheless , numerical noise seems to prevent this from happening , at least for sufficiently generic start vectors @xmath27 . for our procedure one needs to choose the number of lanczos iterations @xmath36 and the dimension of the subspace @xmath58 in advance and the whole procedure needs to be repeated if it turns out that less than the desired number of converged eigenvalues have been obtained . in practice , however , one obtains results of a comparable quality for a class of problems where the choice of @xmath36 and @xmath43 has been adjusted for one representative case . in the examples to be reported below , we have been able to obtain about 100 eigenvalues in a given symmetry sector with a relative accuracy of @xmath59 or better using @xmath60 lanczos iterations and @xmath61 vectors @xmath46 . since the system sizes accessible by either full diagonalization or lanczos diagonalization are limited , it is desirable to have other methods at our disposal . for non - frustrated spin models like the spin-@xmath0 heisenberg chain ( [ eq : ham ] ) , one can in principle use quantum monte carlo ( qmc ) simulations . there is , however , one problem : we are particularly interested in the entropy which is usually obtained with large statistical errors from integrating a monte - carlo result for the specific heat . in this subsection we will summarize how one can circumvent this problem . in order to obtain a qmc estimate of the temperature dependence of the entropy , we employed an extended ensemble , broad - histogram method @xcite within the stochastic series expansion ( sse ) framework @xcite . based on this approach , thermodynamic quantities can be obtained over a broad range of temperatures from a single qmc simulation that provides estimates of the expansion coefficients @xmath62 of the system s partition function @xmath63 in a high - temperature series expansion : @xmath64 here , @xmath65 is the inverse temperature . in terms of the expansion coefficients @xmath62 , the internal energy is obtained using @xmath66 and the free energy from @xmath67 finally , the entropy can be calculated using @xmath68 . in the qmc simulation , estimates of the first @xmath69 coefficients @xmath70 are obtained by performing a random walk in the expansion order @xmath36 , such as to sample efficiently all expansion orders from @xmath71 ( where @xmath72 for a spin-@xmath0 chain of @xmath13 sites is known exactly ) to @xmath73 . this is accomplished using an extension of the wang - landau flat - histogram sampling algorithm @xcite to the quantum case , as detailed previously in refs . knowledge of the first @xmath69 coefficients @xmath62 allows for the calculation of thermodynamic quantities from @xmath74 down to a temperature @xmath75 for a system of @xmath13 sites , where @xmath69 scales proportional to @xmath76 @xcite . in particular , for the spin @xmath1 heisenberg chain , we were able to treat systems with up to @xmath77 sites , accessing temperatures down to @xmath78 , which required an already large value of @xmath79 . the data shown in this contribution ( and also those in @xcite ) were obtained based on the original version of the algorithm @xcite , while recently an improved sampling strategy was proposed , based on optimizing the broad histogram quantum monte carlo ensemble @xcite . of the @xmath1 heisenberg chain as a function of temperature @xmath80 for @xmath81 . lines are obtained by ed for @xmath82 and @xmath83 sites , symbols show qmc results for @xmath77 sites . ] we test and compare the aforementioned numerical methods using the example of the @xmath1 heisenberg chain ( [ eq : ham ] ) with a finite magnetization . [ figs1enth3 ] shows the entropy per site @xmath84 as a function of temperature @xmath80 for a fixed magnetic field @xmath81 . for @xmath82 , we have performed a full diagonalization . the corresponding ed curve can therefore be considered as the exact result for @xmath82 . a full determination of the spectrum is clearly out of reach already for @xmath85 . in this case , we therefore had to perform a severe truncation to low energies . thus , the @xmath85 ed curve is only a low - temperature approximation . in this paper , the precise temperature range is fixed as follows : let @xmath86 be the highest energy until which the spectrum is definitely complete . then we restrict to temperatures @xmath87 . this choice ensures that missing states are suppressed by a boltzmann factor @xmath88 . this may seem a small number , but since we are discarding many states , extending truncated data to higher temperatures would yield artifacts which are clearly visible on the figure(s ) . qmc ( denoted by sse in fig . [ figs1enth3 ] ) is able to treat the larger system size @xmath77 . this method , however , is best suited for high temperatures . indeed , at @xmath89 one can see deviations caused by statistical errors in the sse result of fig . [ figs1enth3 ] which prevented us from going to lower temperatures . finite - size effects can be observed in fig . [ figs1enth3 ] in the @xmath82 ed curve for @xmath90 and in the @xmath85 ed result . otherwise it is reassuring that we observe good overall agreement between all three methods . in passing we note that , for the value of the magnetic field used in fig . [ figs1enth3 ] , the low - energy physics of the @xmath1 chain is described by a luttinger liquid ( see ref . @xcite and references therein ) . it is well known that the specific heat @xmath91 of a luttinger liquid is linear in @xmath80 @xcite . due to the relation @xmath92 and because of @xmath93 , the entropy of a luttinger liquid is identical to its specific heat and in particular also linear in @xmath80 . indeed , [ figs1enth3 ] is consistent with a linear behavior @xmath94 at sufficiently large @xmath13 and low @xmath80 . heisenberg chain as obtained from ed and qmc for different chain lengths . the corresponding values of the entropy are @xmath95 , @xmath96 , @xmath97 , @xmath98 , @xmath99 , @xmath100 ( bottom to top ) . ] in this section we will discuss the entropy of the @xmath1 heisenberg chain in more detail . first , we show results for the value of the entropy per site @xmath84 as a function of @xmath12 and @xmath80 in fig . [ figs1isent ] . the ed curves have been obtained by computing the entropy @xmath101 on a mesh in the @xmath12-@xmath80-plane and determining the constant entropy curves , _ i.e. _ , the isentropes from this data . this leads to some discretization artifacts at the cusps in the the @xmath102 curves ( the ed curves at the lowest temperature ) in fig . [ figs1isent ] . conversely , the qmc had problems to resolve the large low - temperature entropy around the saturation field @xmath103 . accordingly , the @xmath77 sse data points with @xmath102 are missing at @xmath104 and @xmath105 . from our qmc simulations we can just conclude that for an @xmath77 chain these two points are in the region @xmath106 and @xmath107 for @xmath108 and @xmath109 , respectively . the ed curves with @xmath102 in fig . [ figs1isent ] have marked finite - size wiggles . also at higher temperatures ( entropies ) , finite - size effects can be observed . there are , however , two regions where finite - size effects are evidently small . one is the low - temperature regime for fields @xmath12 smaller than the haldane gap @xcite @xmath110 also in the gapped high - field regime above the saturation field @xmath111 one observes essentially no finite - size effects at any temperature . this renders ed the method of choice at high fields @xmath112 , in particular at low temperatures , while qmc is preferable otherwise because bigger system sizes @xmath13 are accessible . one can read off from fig . [ figs1isent ] pronounced temperature changes in two regimes . firstly , one can obtain cooling by adiabatic demagnetization from a high magnetic field @xmath113 as one lowers the field to the saturation field . secondly , at low temperatures , one can also cool by adiabatic magnetization from @xmath17 to @xmath114 . this demonstrates that one can use the gap - closing at a generic field - induced quantum phase transition for cooling purposes . in the regime @xmath115 , the spectrum is gapless ( compare fig . [ figs1enth3 ] and the related discussion in section [ sec : test ] ) . accordingly , one observes only small temperature changes induced by adiabatic ( de)magnetization in this field range . heisenberg chain at zero magnetic field @xmath17 as a function of temperature . note that the temperature axis is scaled as @xmath116 and the entropy axis logarithmically . the dashed line is a fit to ( [ eqentgap ] ) with @xmath117 and @xmath118 . ] we now focus on the low - field regime . firstly , we consider the behavior of the entropy at @xmath17 which is shown in fig . [ figs1enth0 ] . one observes that finite - size effects are small for @xmath119 : the @xmath82 curve is almost within the error margins of the @xmath77 qmc curve even at low temperatures . this can be attributed to a finite correlation length @xmath120 which is evidently sufficiently smaller than @xmath121 for all temperatures at @xmath17 . indeed , the correlation length of the @xmath1 heisenberg chain is known to be @xmath122 at @xmath17 and @xmath123 @xcite . because of the presence of a gap @xmath124 at @xmath17 , we expect the entropy to be exponentially activated as a function of temperature . accordingly , the low - temperature asymptotic behavior at @xmath17 should follow the form @xmath125 if we fix the gap to the known value @xmath118 @xcite , the only free parameter in the low - temperature asymptotic behavior ( [ eqentgap ] ) is the prefactor @xmath126 . a fit of the @xmath77 qmc results yields @xmath127 . the dashed line in fig . [ figs1enth0 ] shows that the formula ( [ eqentgap ] ) describes the low - temperature regime quite well with the given parameters . heisenberg chain at @xmath128 , corresponding to the haldane gap . note the doubly logarithmic scale . the dashed line shows the square - root behavior ( [ eqentsqrt ] ) with @xmath129 . ] secondly , we consider a magnetic field exactly equal to the haldane gap @xmath128 . [ figs1enthgap ] shows a doubly - logarithmic plot of the entropy @xmath101 as a function of temperature @xmath80 for this value of the magnetic field . since we are now sitting at a field - induced quantum phase transition , we expect a large number of low - lying states and relatedly an infinite correlation length @xmath130 . on a finite system , these low - lying states will be pushed to higher energies . indeed , we observe pronounced finite - size effects at low temperatures in fig . [ figs1enthgap ] . at a quantum phase transition in one dimension which preserves a @xmath131-symmetry , the entropy @xmath101 is expected @xcite to vary asymptotically as a square root of temperature : @xmath132 as before , the amplitude @xmath133 is the only free parameter . a fit to the @xmath77 qmc data with @xmath134 yields @xmath135 . the dashed line in fig . [ figs1enthgap ] shows that the @xmath77 qmc data is indeed consistent with the asymptotic square root ( [ eqentsqrt ] ) in the range @xmath136 . however , because of finite - size effects , in this case we really need to go to @xmath77 to recover the asymptotic behavior . furthermore , we need to restrict to @xmath137 since one can see from fig . [ figs1enthgap ] that a pure power law is no longer a good description for @xmath138 . comparison of ( [ eqentgap ] ) and ( [ eqentsqrt ] ) shows that one can reach exponentially small temperatures @xmath139 at @xmath114 by adiabatic magnetization of a spin-@xmath4 heisenberg chain from an initial temperature @xmath140 at @xmath17 . for an initial temperature @xmath141 , the final temperature @xmath139 can be quantitatively estimated from the following combination of ( [ eqentgap ] ) and ( [ eqentsqrt ] ) @xmath142 with the parameters @xmath143 , @xmath126 , and @xmath133 given above . we will now turn to the high - field region and consider in particular scaling with the spin quantum number @xmath0 close to the saturation field . heisenberg chains as obtained by ed for different @xmath0 . note that a truncation of the energy spectrum has been used for @xmath144 . the values of the entropy per site are @xmath102 , @xmath145 , @xmath97 , @xmath98 , and @xmath99 ( bottom to top ) . ] it is straightforward to compute the excitation energy @xmath146 of a single flipped spin propagating above a ferromagnetically polarized background ( see , e.g. , @xcite ) : @xmath147 the minimum of this one - magnon dispersion is located at @xmath148 . the saturation field @xmath149 of the spin-@xmath0 heisenberg chain ( [ eq : ham ] ) is given by the one - magnon instability @xcite . accordingly , it can be determined by inserting the condition @xmath150 into ( [ eq : singlemagnon ] ) : @xmath151 the fact that the single - particle energy ( [ eq : singlemagnon ] ) scales linearly with @xmath0 suggests to scale all energies close to the saturation field and at low temperatures linearly with @xmath0 . in order to test this scenario , we have first computed the isentropes for a fixed chain length @xmath85 and @xmath152 . the @xmath153 curves are obtained by a full diagonalization of the hamiltonian @xcite while we have performed additional ed computations for @xmath1 , @xmath154 , and @xmath155 using the truncation procedure described in section [ sec : methods ] . [ figmad ] shows the resulting isentropes for @xmath102 , @xmath145 , @xmath97 , @xmath98 , and @xmath99 close to the saturation field ( [ hsat ] ) . we observe that at low temperatures , scaling both @xmath12 and @xmath80 by @xmath0 leads to a nice collapse of the isentropes around and in particular above the saturation field . ( left column ) and magnetic susceptibility per site @xmath156 ( right column ) as obtained by ed for different @xmath0 . note that a truncation of the energy spectrum has been used for @xmath144 . the values of the magnetic field are chosen somewhat above the saturation field ( top row ) , exactly at the the saturation field ( middle row ) , and somewhat below the saturation field ( bottom row).,title="fig : " ] ( left column ) and magnetic susceptibility per site @xmath156 ( right column ) as obtained by ed for different @xmath0 . note that a truncation of the energy spectrum has been used for @xmath144 . the values of the magnetic field are chosen somewhat above the saturation field ( top row ) , exactly at the the saturation field ( middle row ) , and somewhat below the saturation field ( bottom row).,title="fig : " ] + ( left column ) and magnetic susceptibility per site @xmath156 ( right column ) as obtained by ed for different @xmath0 . note that a truncation of the energy spectrum has been used for @xmath144 . the values of the magnetic field are chosen somewhat above the saturation field ( top row ) , exactly at the the saturation field ( middle row ) , and somewhat below the saturation field ( bottom row).,title="fig : " ] ( left column ) and magnetic susceptibility per site @xmath156 ( right column ) as obtained by ed for different @xmath0 . note that a truncation of the energy spectrum has been used for @xmath144 . the values of the magnetic field are chosen somewhat above the saturation field ( top row ) , exactly at the the saturation field ( middle row ) , and somewhat below the saturation field ( bottom row).,title="fig : " ] + ( left column ) and magnetic susceptibility per site @xmath156 ( right column ) as obtained by ed for different @xmath0 . note that a truncation of the energy spectrum has been used for @xmath144 . the values of the magnetic field are chosen somewhat above the saturation field ( top row ) , exactly at the the saturation field ( middle row ) , and somewhat below the saturation field ( bottom row).,title="fig : " ] ( left column ) and magnetic susceptibility per site @xmath156 ( right column ) as obtained by ed for different @xmath0 . note that a truncation of the energy spectrum has been used for @xmath144 . the values of the magnetic field are chosen somewhat above the saturation field ( top row ) , exactly at the the saturation field ( middle row ) , and somewhat below the saturation field ( bottom row).,title="fig : " ] further details can be read off from the temperature scans at a fixed magnetic field shown in fig . [ fig : scalecchi ] . in this figure , we have chosen to show the specific heat @xmath91 as a representative for energy - related quantities is equivalent to @xmath101 since in the present case @xmath157 for @xmath158 . ] and the magnetic susceptibility @xmath159 as a representative for magnetic quantities . the top row of fig . [ fig : scalecchi ] corresponds to a magnetic field @xmath160 . here we observe a nice collapse with @xmath0 for @xmath161 . the middle row of fig . [ fig : scalecchi ] corresponds to a magnetic field exactly equal to the saturation field . here we observe a good scaling collapse for @xmath162 . finally , the bottom row of fig . [ fig : scalecchi ] corresponds to a magnetic field @xmath163 . in this case , one expects the scaling region to be pushed to even lower temperatures . however , for @xmath164 the finite size @xmath85 leads to artifacts in the low - temperature behavior . therefore it is difficult to make definite statements for this case . to summarize this section , we have provided evidence of _ linear _ scaling with @xmath0 for thermodynamic quantities of spin-@xmath0 heisenberg chains close to the saturation field . note that this is very different from the _ quadratic _ scaling , e.g. of @xmath80 with @xmath165 @xcite or with @xmath166 @xcite , which is needed to approach the classical limit . in this paper we have illustrated the numerical computation of thermodynamic quantities and in particular the entropy for spin-@xmath0 heisenberg chains . from a technical point of view , we have described in section [ sec : ed ] how to perform a reliable computation of a large number of low - lying states using the lanczos method and in section [ sec : qmc ] how to compute the entropy directly by a qmc simulation . in section [ sec : s1chain ] we have then focused on the @xmath1 heisenberg chain and shown in particular that one can cool with an adiabatic magnetization process during which the haldane gap @xmath124 is closed . many previous investigations ( e.g. , @xcite ) have focused on the saturation field . the reason for this is that the saturation field is a field - induced quantum phase transition at a known value of the magnetic field which also gives rise to technical simplifications . however , the scenario of quantum phase transitions @xcite is universal and not restricted to the saturation field . indeed , fig . [ figs1enthgap ] is consistent with the same universal square - root behavior of the entropy @xmath101 at @xmath167 in the spin-1 heisenberg chain as observed previously for @xmath153 chains exactly at the saturation field @xcite . this may also be important from an experimental point of view since a possible spin gap may be accessible by laboratory magnetic fields even if the saturation field is out of reach . in fact , cooling by adiabatic magnetization when closing a spin gap has presumably been indirectly observed in pulse - field magnetization experiments on srcu@xmath7(bo@xmath168)@xmath7 @xcite . in section [ sec : sscal ] we then moved to the saturation field and investigated scaling with the spin quantum number @xmath0 . in contradistinction to the classical scaling regime where temperature should scale quadratically with @xmath0 @xcite , there is a quantum scaling regime around the saturation field where one observes a collapse using the linearly scaled parameters @xmath169 and @xmath170 . also in higher dimensions the one - magnon dispersion typically scales with @xmath0 . by the same arguments as in section [ sec : sscal ] , we therefore expect linear scaling with @xmath0 at any continuous transition to saturation . this expectation could be tested numerically with the methods of the present paper . we are grateful to m.e . zhitomirsky for useful discussions . acknowledges support by the deutsche forschungsgemeinschaft through a heisenberg fellowship ( project ho 2325/4 - 1 ) . s.w . acknowledges hlrs stuttgart and nic jlich for allocation of computing time . some of our numerical simulations were based on the alps libraries @xcite . warburg e. , ann . , 1881 , * 13 * , 141 . giauque w.f . , macdougall d.p . , 1933 , * 43 * , 768 . bonner j.c . , fisher m.e . , soc . , 1962 , * 80 * , 508 . bonner j.c . , nagle j.f . , phys . a , 1972 , * 5 * , 2293 . bonner j.c . , johnson j.d . , physica , 1977 , * 86 - 88b * , 653 . zhu l. , garst m. , rosch a. , si q. , phys . lett . , 2003 , * 91 * , 066404 . garst m. , rosch a. , phys . b , 2005 , * 72 * , 205129 . tachiki m. , yamada t. , j. phys . jpn . , 1970 , * 28 * , 1413 . zhitomirsky m.e . b , 2003 , * 67 * , 104421 . zhitomirsky m.e . , honecker a. , j. stat . : theor . exp . , 2004 , p07012 . zhitomirsky m.e . , tsunetsugu h. , phys . rev . b , 2004 , * 70 * , 100403 . derzhko o. , richter j. , phys . b , 2004 , * 70 * , 104415 . sosin s.s . , prozorova l.a . , smirnov a.i . , golov a.i . , berkutov i.b . , petrenko o.a . , balakrishnan g. , zhitomirsky m.e . , b , 2005 , * 71 * , 094413 . honecker a. , wessel s. , physica b , 2006 , * 378 - 380 * , 1098 . derzhko o. , richter j. , eur . j. b , 2006 , * 52 * , 23 . anov l. , streka j. , jaur m. , j. phys . : condens . matter , 2006 , * 18 * , 4967 . gencer h. , int . j. mod . b , 2006 , * 20 * , 2527 . schmidt b. , shannon n. , thalmeier p. , j. phys . ser . , 2006 , * 51 * , 207 . schmidt b. , thalmeier p. , shannon n. , phys . b , 2007 , * 76 * , 125113 . schnack j. , schmidt r. , richter j. , phys . b , 2007 , * 76 * , 054413 . radu t. , tokiwa y. , coldea r. , gegenwart p. , tylczynski , z. , steglich f. , sci . technol . adv . , 2007 , * 8 * , 406 . pereira m.s.s . , de moura f.a.b.f . , b , 2009 , * 79 * , 054427 . honecker a. , zhitomirsky m.e . , j. phys . ser . , 2009 , * 145 * , 012082 . boyarchenkov a.s . , bostrem i.g . , ovchinnikov a.s . b , 2007 , * 76 * , 224410 . tsui y. , wolf b. , jaiswal - nagar d. , tutsch u. , honecker a. , removi - langer k. , prokofiev a. , assmus w. , donath g. , lang m. , in preparation . klmper a. , honecker a. , ohanyan v. , trippe c. , in preparation . haldane f.d.m . , phys . rev . lett . , 1983 , * 50 * , 1153 . haldane f.d.m . a , 1983 , * 93 * , 464 . white s.r , huse d.a . , phys . b , 1993 , * 48 * , 3844 . golinelli o. , jolicoeur th . , lacaze r. , phys . b , 1994 , * 50 * , 3037 . bonner j.c . , fisher m.e . , 1964 , * 135 * , a640 . bonner j.c . , numerical studies on the linear ising - heisenberg chain . phd thesis , university of london , 1968 . greenbaum a. , dongarra j. , experiments with qr / ql methods for the symmetric tridiagonal eigenproblem . lapack working note 17 , 1989 . honecker a. , schle j. , openmp implementation of the householder reduction for large complex hermitian eigenvalue problems . in : advances in parallel computing * 15 * , lippert t. , mohr b. , peters f. ( eds . ) , ios press , 2008 , lanczos c. , j. res . standards , 1950 , * 45 * , 255 . cullum j.k . , willoughby , r.a . , lanczos algorithms for large symmetric eigenvalue computations . i : theory . siam classics in applied mathematics * 41 * , 2002 . bai z. , demmel j. , dongarra j. , ruhe a. , van der vorst h. , templates for the solution of the algebraic eigenvalue problems . a practical guide . siam , philadelphia , 2000 . troyer m. , wessel s. , alet f. , phys . lett . , 2003 , * 90 * , 120201 . troyer m. , alet f. , wessel s. , braz . j. of phys . , 2004 , * 34 * , 377 . wessel s. , stoop n. , gull e. , trebst s. , troyer m. , j. stat . mech . , 2007 , sandvik a.w . , kurkijrvi j. , phys . b , 1991 * 43 * , 7 . sandvik a.w . b , 1999 , * 59 * , r14157 . syljusen o.f . , sandvik a.w . e , 2002 , * 66 * , 046701 . alet f. , wessel s. , troyer m. , phys . e , 2005 , * 71 * , 036706 . wang f. , landau d.p . lett . , 2001 , * 86 * , 2050 . wang f. , landau d.p . e , 2001 , * 64 * , 056101 . fth g. , phys . rev . b , 2003 , * 68 * , 134445 . giamarchi t. , quantum physics in one dimension . clarendon press , oxford , 2004 . hodgson r.p . , parkinson j.b . , j. phys . c : solid state phys . , 1985 , * 18 * , 6385 . parkinson j.b . , bonner j.c . , phys . b , 1985 , * 32 * , 4703 . kim y.j . , greven m. , wiese u.j . , birgeneau r.j . j. b , 1998 , * 4 * , 291 . levy f. , sheikin i. , berthier c. , horvati m. , takigawa m. , kageyama h. , waki t. , ueda y. , europhys . lett . , 2008 , * 81 * , 67004 . alet f. _ et al . _ , j. phys . suppl . , 2005 , * 74 * , 30 . albuquerque a.f . et al . _ , j. magn . mater . , 2007 , * 310 * , 1187 .

we compute the entropy of antiferromagnetic quantum spin-@xmath0 chains in an external magnetic field using exact diagonalization and quantum monte carlo simulations . the magnetocaloric effect , _ i.e. _ , temperature variations during adiabatic field changes , can be derived from the isentropes . first , we focus on the example of the spin-@xmath1 chain and show that one can cool by closing the haldane gap with a magnetic field . we then move to quantum spin-@xmath0 chains and demonstrate linear scaling with @xmath0 close to the saturation field . in passing , we propose a new method to compute many low - lying excited states using the lanczos recursion . quantum spin chains , magnetocaloric effect , entropy , exact diagonalization , quantum monte carlo 75.10.pq ; 75.30.sg ; 75.50.ee ; 02.70.-c

as expected , the most interesting combinatorial structures are those bearing some kind of symmetry and/or regularity . in fact , in general , high symmetry imply high regularity , but the converse does not necessarily holds . moreover , symmetric structures suggest definitions of new structures obtained , either by relaxing the conditions of symmetry , or having the same regularity properties as the original ones . in turn , the latter can give rise to new definitions by relaxing the mentioned symmetry conditions . in graph theory , a good example of the above are the distance - transitive graphs , with automorphism group having orbits constituted by all vertices at a given distance . attending to their symmetry , related concepts are the vertex - symmetric , symmetric , and orbit polynomial graphs @xcite . besides , concerning regularity , distance - transitive graphs can be generalized to distance - regular graphs @xcite , distance - polynomial graphs , and degree - regular graphs @xcite . in this paper , we introduce the concept of a quotient - polynomial graph , which could be thought of as the regular counterpart of orbit polynomial graphs . in a quotient - polynomial graph , every vertex @xmath0 induces the same regular partition around @xmath0 , with the additional condition that all vertices of each cell are equidistant from @xmath0 . some properties and characterizations of such graphs are studied . for instance , all quotient - polynomial graphs are walk - regular and distance - polynomial . our study allows us to provide a characterization of those distance - polynomial and vertex - transitive graphs which are distance - regular . also , we show that every quotient - polynomial graph generates a ( symmetric ) association scheme . throughout this paper , @xmath1 denotes a ( connected ) graph with vertex set @xmath2 , edge set @xmath3 , and diameter @xmath4 . for every @xmath5 and @xmath6 , let @xmath7 denote the set of vertices at distance @xmath8 from @xmath0 , with @xmath9 , and let @xmath10 be the characteristic ( @xmath0-th unitary ) vector of @xmath11 . eccentricity _ of @xmath0 , denoted by @xmath12 , is the maximum distance between @xmath0 and any other vertex @xmath13 of @xmath1 . let @xmath14 be the @xmath8-th distance matrix , so that @xmath15 is the adjacency matrix of @xmath1 , with spectrum @xmath16 , where @xmath17 , and the superscripts @xmath18 stand for the multiplicities . let @xmath19 , @xmath20 be the minimal idempotents representing the orthogonal projections on the @xmath21-eigenspaces . let @xmath22 $ ] be the _ adjacency algebra _ of @xmath1 , that is , the algebra of all polynomials in @xmath23 with real coefficients . following fiol , garriga and yebra @xcite , the @xmath24-entry of @xmath19 is referred to as the _ crossed @xmath25-@xmath26local multiplicity _ of the eigenvalue @xmath21 , and it is denoted by @xmath27 . in particular , for a regular graph on @xmath28 vertices , @xmath29 and , hence , @xmath30 for every @xmath31 . since @xmath32 , the number of walks of length @xmath33 between two vertices @xmath34 is @xmath35 in particular , the _ @xmath36-@xmath26local multiplicities _ are @xmath37 , @xmath38 , and satisfy @xmath39 and @xmath40 , @xmath38 . a graph @xmath1 with diameter @xmath4 is called _ @xmath41-punctually walk - regular _ , for some @xmath42 , when the number of walks @xmath43 for any pair of vertices @xmath34 at distance @xmath41 only depends on @xmath33 . from the above , this means that the crossed local multiplicities @xmath27 only depend on @xmath21 and we write it as @xmath44 ( see dalf , van dam , fiol , garriga , and gorissen @xcite for more details ) . notice that , in particular , a @xmath45-punctually walk - regular graph is the same as a walk - regular graph , a concept introduced by godsil and mckay @xcite . a partition @xmath46 of the vertex set @xmath2 is called _ regular _ ( or _ equitable _ ) whenever for any @xmath47 , the _ intersection numbers _ @xmath48 , where @xmath49 , do not depend on the vertex @xmath0 but only on the subsets ( _ classes _ or _ cells _ ) @xmath50 and @xmath51 . in this case , such numbers are simply written as @xmath52 , and the @xmath53 matrix @xmath54 is referred to as the _ quotient matrix _ of @xmath23 with respect to @xmath55 . the _ characteristic matrix _ of ( any ) partition @xmath55 is the @xmath56 matrix @xmath57 whose @xmath8-th column is the characteristic vector of @xmath50 , that is , @xmath58 if @xmath49 , and @xmath59 otherwise . in terms of such a matrix , it is known that @xmath55 is regular if and only if there exists an @xmath53 matrix @xmath60 such that @xmath61 moreover , in this case , @xmath62 , the quotient matrix of @xmath23 with respect to @xmath55 . then , using this it easily follows that all the eigenvalues of @xmath63 are also eigenvalues of @xmath23 . for more details , see godsil @xcite . in this section we introduce several types of partitions bearing some regularity properties with respect to a given vertex @xmath0 . we begin by considering those partitions where all vertices of the same class are equidistant from @xmath0 . let @xmath1 have diameter @xmath4 . given a vertex @xmath0 , a _ ( @xmath0-)distance - faithful _ partition around @xmath0 , denoted by @xmath64 , is a partition @xmath65 , with @xmath66 such that @xmath67 and , for @xmath68 , every pair of vertices @xmath69 are at the same distance from @xmath0 : @xmath70 . thus , in particular , @xmath64 is a _ distance partition around _ @xmath0 whenever @xmath69 if and only if @xmath71 . in other words , @xmath72 for every @xmath73 and , hence , @xmath74 . for every pair of vertices @xmath31 we consider the vectors of crossed local multiplicities @xmath75 and numbers of @xmath33-walks for @xmath76 between @xmath0 and @xmath13 @xmath77 the following result is an immediate consequence of ( see , for instance , @xcite ) . given some vertices @xmath78 , we have @xmath79 if and only if @xmath80 . then , we can define the two following equivalent concepts : a partition @xmath81 is _ @xmath36-@xmath26walk - regular _ @xmath82or _ @xmath36-@xmath26-spectrum - regular_@xmath26 around a vertex @xmath5 if , for every @xmath73 the set @xmath83 is constituted by all vertices @xmath84 with the same vector @xmath85 @xmath82or , equivalently , @xmath86 . now we will prove that , if a partition is both regular and @xmath0-distance - faithful , then it is also @xmath0-walk - regular . before that , we have the following straightforward lemma . [ wr->df ] every walk - regular partition @xmath64 around a vertex @xmath5 is also distance - faithful around the same vertex . by contradiction , assume that @xmath87 and @xmath88 . then , we would have @xmath89 but @xmath90 , against the hypothesis of @xmath0-walk - regularity . let @xmath91 be a @xmath0-distance - faithful and regular partition . then @xmath92 defines a @xmath0-walk - regular partition @xmath93 with @xmath94 ( by the union of some sets @xmath50 , if necessary ) . to prove that the number of walks @xmath95 , with @xmath96 , only depend on @xmath8 and @xmath33 , we use induction on @xmath33 . the result is clear for @xmath97 . now suppose that the result holds for some @xmath98 . then , for a given @xmath96 , @xmath99 and , hence , @xmath100 does not depend on @xmath13 . finally , if there are sets @xmath101 with vertices @xmath13 having the same vector @xmath85 , we consider their union @xmath102 to form the claimed @xmath0-walk - regular partition . to prove the converse , we need an extra hypothesis , which in fact leads to a stronger result in terms of the new concept defined below . given a vertex @xmath0 of a graph @xmath103 , the so - called _ @xmath0-local spectrum _ is constituted by those eigenvalues @xmath104 of @xmath103 such that @xmath105 ( that is , with nonzero @xmath0-local multiplicity @xmath106 ) . moreover , these are referred to as the _ @xmath0-local eigenvalues_. let us consider the vector space @xmath107 spanned by the vectors @xmath108 , @xmath76 . then , it is known that @xmath107 has dimension @xmath109 and basis @xmath110 ; see e.g. @xcite . [ defi1 ] let @xmath0 be a vertex with @xmath109 distinct local eigenvalues . let @xmath93 be a @xmath0-walk - regular partition , with @xmath111 being the characteristic vector of @xmath83 @xmath82note that @xmath112@xmath26 . then @xmath64 is said to be _ quotient - polynomial _ whenever @xmath113 for every @xmath114 . in the following result the walk - regular partitions that are quotient - polynomial ( and regular ) are characterized . [ ineq&equ ] let @xmath0 be a vertex with @xmath109 distinct local eigenvalues . let @xmath93 be a @xmath0-walk - regular partition . then , @xmath115 with equality if and only if @xmath64 is a quotient - polynomial partition . moreover , in this case @xmath64 is also regular . for @xmath73 , let @xmath116 the common value of the number of @xmath33-walks from @xmath0 to every @xmath117 , and let us consider the vector space @xmath118 . then , as @xmath119 we have that @xmath120 and , hence , @xmath121 which proves . of course , the same conclusion can be reached by considering the common value @xmath122 of the crossed local multiplicities @xmath27 , for every @xmath117 . then , @xmath123 ( only @xmath109 of the above equations are not trivially null ) . if @xmath124 , we have that @xmath125 , that is , every vector @xmath111 is a linear combination of the vectors @xmath108 for @xmath126 , and @xmath64 is a quotient - polynomial partition . conversely , if @xmath64 is quotient - polynomial , we have @xmath127 . hence , @xmath128 which , together with ( @xmath64 is also walk - regular ) , leads to @xmath124 . in fact , the constants of the above linear combinations , which are the coefficients of the polynomials @xmath129 , can be computed in the following way : the @xmath130 first equations in are , in matrix form , @xmath131 but the coefficient matrix @xmath132 with entries @xmath133 , for @xmath134 , is a change - of - basis matrix and , hence , it is invertible . as a consequence , for every @xmath135 , the coefficients @xmath136 , @xmath136 , , @xmath136 of @xmath137 correspond to the @xmath8-th row of @xmath138 . finally , to prove that @xmath64 is regular , let us choose one vertex @xmath139 in each @xmath83 , @xmath135 and consider the @xmath140 matrices @xmath63 and @xmath141 , with entries @xmath142 and @xmath143 , @xmath144 , respectively . then , the @xmath130 equations of can be written as @xmath145 hence , the entries @xmath146 of the matrix @xmath147 do not depend on the chosen vertices @xmath148 , and the partition is regular with quotient matrix @xmath54 . all the above result can be summarized in the followin theorem . let @xmath0 be a vertex of a graph @xmath1 . let @xmath64 be a partition with @xmath130 classes around @xmath0 having @xmath109 distinct local eigenvalues @xmath149 . then , the following assertions are equivalent : * the partition @xmath64 is quotient - polynomial . * the partition @xmath64 is @xmath0-walk - regular with @xmath124 . * there exist polynomials @xmath137 with @xmath150 , @xmath73 , such that @xmath151 . * @xmath125 with basis @xmath152 . by theorem [ ineq&equ ] , we only need to prove @xmath153 : from @xmath154 we have that @xmath155 and hence @xmath128 . also , as @xmath156 , where @xmath157 , we get @xmath158 , where the left - hand side is a polynomial with degree at most @xmath130 . moreover , @xmath159 , with @xmath160 , is the polynomial of minimum degree satisfying @xmath161 ( see @xcite ) . therefore , @xmath162 and , hence , @xmath124 , @xmath125 , and @xmath163 is a basis . in this section we study the graphs having the same ( i.e. with the same parameters ) quotient - polynomial partition around each of their vertices . with this aim , we now follow a global approach . let @xmath1 be a graph with vertex set @xmath2 , @xmath164 distinct eigenvalues , and adjacency algebra @xmath165 . then a partition @xmath166 of @xmath167 is called _ walk - regular _ whenever each @xmath168 is the set with elements @xmath169 having identical vector @xmath170 ( or @xmath85 ) . so , from lemma [ wr->df ] , all pairs of vertices in a given @xmath168 are at the same distance , and we assume that the pairs in @xmath171 are of the form @xmath172 ( distance zero ) . let @xmath173 , @xmath73 , the @xmath174 matrices , indexed by the vertices of @xmath1 , representing the equivalence classes @xmath168 , that is , @xmath175 let @xmath176 be an equivalence class with elements @xmath169 satisfying @xmath177 . then , @xmath178 if and only if @xmath1 is @xmath41-punctually walk - regular and , in particular , @xmath179 if and only if @xmath1 is walk - regular . from these matrices we can now define our main concept : a graph @xmath1 , with walk - regular partition @xmath166 and adjacency algebra @xmath180 , is _ quotient - polynomial _ if @xmath181 for every @xmath73 . thus , @xmath1 is quotient - polynomial if and only if there exist polynomials @xmath137 , with @xmath182 , such that @xmath183 , @xmath73 ( this inspired our definition ) . in fact , the following result shows that this only happens when @xmath184 . we omit its proof since it goes along the same lines of reasoning as that of theorem [ ineq&equ ] . [ ineq&equ ] let @xmath1 be a graph as above . let @xmath185 be a walk - regular partition . then , @xmath186 and equality occurs if and only if @xmath1 is quotient - polynomial . since @xmath187 , the all-1 matrix , the sum polynomial @xmath188 is the hoffman polynomial satisfying @xmath189 . therefore , a quotient - polynomial graph is connected and regular ( see hoffman @xcite ) . moreover , the same reasoning used in @xcite[th . 2.4 ] to prove that every orbit polynomial graph is vertex transitive , shows that every quotient - polynomial graph is walk - regular , that is , @xmath179 . indeed , if @xmath190 , the equality @xmath191 ( @xmath180 is a commutative algebra ) leads to a contradicion because @xmath1 is connected . thus , for any vertex @xmath0 , the _ induced partition _ @xmath64 of @xmath2 , with characteristic vectors @xmath10 , @xmath192 , , @xmath193 is quotient - polynomial since @xmath194 then , we can summarize all the above results in the ` global analogue ' of theorem [ ineq&equ ] . [ main - theo ] let @xmath1 be a graph with vertex set @xmath2 , and @xmath164 distinct eigenvalues . let @xmath195 be a partition of @xmath167 with @xmath196 . then , the following assertions are equivalent : * @xmath1 is a quotient - polynomial graph . * the partition @xmath197 is walk - regular with @xmath184 . * there exist polynomials @xmath137 with @xmath150 , @xmath73 , such that @xmath198 . * for every vertex @xmath0 , the induced partition @xmath64 of @xmath2 is quotient - polynomial with the same polynomials @xmath137 . * @xmath199 with basis @xmath200.@xmath201 ( notice that the first equality in @xmath202 implies @xmath184 . ) let us consider the following example of quotient - polynomial graph : the circulant graph @xmath203 has vertices @xmath204 and vertex @xmath0 is adjacent to vertices @xmath205 and @xmath206 . then , @xmath1 is a 4-regular vertex - transitive graph with diameter @xmath207 , and spectrum ( with numbers rounded to three decimals ) @xmath208 .,title="fig : " ] -2.5 cm as shown in fig . [ fig1 ] , the walk - regular partition around vertex @xmath45 has clases @xmath209 , @xmath210 , @xmath211 , @xmath212 , and @xmath213 . ( the corresponding intersection diagram is shown in fig . 2 . ) then the matrices @xmath132 and @xmath141 of numbers of walks from @xmath45 to a vertex of @xmath83 are @xmath214 then , from the inverse of @xmath132 we obtain the quotient polynomials : @xmath215 whereas the transpose of the intersection matrix turns out to be @xmath216 as in the case of distance - regular graph , the entries @xmath217 ( shown also in fig . 2 ) can also be calculated by using the formulas @xmath218 where we use the scalar product @xmath219 note that , since @xmath220 for @xmath221 , the quotient polynomials @xmath137 are orthogonal with respect to such a product . ( 1,3 ) circle [ radius=0.6 ] ; at ( 0.4,3.7 ) @xmath222 ; at ( 1.75,3.25 ) @xmath223 ; ( 3.5,3 ) circle [ radius=0.6 ] ; at ( 2.9,3.7 ) @xmath224 ; at ( 2.75,3.25 ) @xmath225 ; at ( 4,3.8 ) @xmath225 ; at ( 4,2.2 ) @xmath226 ; ( 5.5,1 ) circle [ radius=0.6 ] ; at ( 4.75,0.5 ) @xmath227 ; at ( 4.75,1.45 ) @xmath226 ; at ( 5.65,1.85 ) @xmath225 ; at ( 6.25,1.45 ) @xmath225 ; at ( 5.5,0.20 ) @xmath228 ; ( 5.5,5 ) circle [ radius=0.6 ] ; at ( 4.75,5.55 ) @xmath229 ; at ( 4.75,4.5 ) @xmath225 ; at ( 5.65,4.15 ) @xmath225 ; at ( 6.25,4.5 ) @xmath225 ; at ( 5.5,5.85 ) @xmath225 ; ( 7.5,3 ) circle [ radius=0.6 ] ; at ( 6.6,3 ) @xmath230 ; at ( 7,3.8 ) @xmath225 ; at ( 7,2.2 ) @xmath225 ; at ( 8.3,3 ) @xmath226 ; ( 1.6,3)(2.9,3 ) ; ( 3.93,2.57)(5.07,1.43 ) ; ( 3.93,3.43)(5.07,4.57 ) ; ( 5.5,1.6)(5.5,4.4 ) ; ( 5.93,1.43)(7.07,2.57 ) ; ( 5.93,4.57)(7.07,3.43 ) ; at ( 1,3 ) @xmath225 ; at ( 3.5,3 ) @xmath223 ; at ( 5.5,1 ) @xmath223 ; at ( 5.5,5 ) @xmath223 ; at ( 7.5,3 ) @xmath223 ; [ fig2 ] in this section we study the relationships of quotient - polynomial graphs with other known combinatorial structures . delsarte @xcite proved that a graph @xmath1 with @xmath164 distinct eigenvalues is distance - regular if and only if , for every @xmath38 , @xmath231 for some polynomial @xmath137 of degree @xmath8 ; see also weischel @xcite . then , as @xmath232 , the following result is clear . inspired by the above characterization of distance - regular graphs , weichsel @xcite defined a graph @xmath1 with diameter @xmath4 to be _ distance - polynomial _ if @xmath235 . ( that is @xmath231 for every @xmath6 with no condition on the degrees of the polynomials @xmath137 . ) then any distance - regular graph is also distance - polynomial , but the converse does not hold . for instance , any regular graph with diameter two is easily proved to be distance - polynomial ( but not necessarily strongly regular ) . the next result gives a condition for having the equivalence . since every distance - regular graph is distance - polynomial and has diameter @xmath236 , necessity is clear . to prove sufficiency , assume that @xmath1 is distance - polynomial with @xmath236 . then @xmath237 which , as it was proved in @xcite , implies that @xmath1 is distance - regular . let @xmath1 have the walk - regular partition @xmath238 with corresponding matrices @xmath239 for some polynomials @xmath240 , @xmath20 . then , the polynomials @xmath241 with @xmath242 , clearly satisfy @xmath243 for @xmath6 , and the result follows . however , the converse result does not hold , even if we require @xmath1 to have the same equitable walk - regular partition around each of its vertices . a counterexample is the so - called ` chordal ring ' @xmath245 or prism @xmath246 shown in fig . this ( bipartite ) graph has @xmath247 and it has the walk - regular ( and regular ) partition with intersection diagram shown in fig . [ fig4 ] . cccccccccccc 0 & 0 & 0 & 0 & 0 & 0 & 1 & 0 & 0 & 0 & 0 & 0 + 0 & 0 & 0 & 0 & 0 & 0 & 0 & 1 & 0 & 0 & 0 & 0 + 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 1 & 0 & 0 & 0 + 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 1 & 0 & 0 + 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 1 & 0 + 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 1 + 1 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 + 0 & 1 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 + 0 & 0 & 1 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 + 0 & 0 & 0 & 1 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 + 0 & 0 & 0 & 0 & 1 & 0 & 0 & 0 & 0 & 0 & 0 & 0 + 0 & 0 & 0 & 0 & 0 & 1 & 0 & 0 & 0 & 0 & 0 & 0 the definition of orbit polynomial graph @xmath1 , due to beezer @xcite is similar to that of quotient polynomial graph , but now the classes @xmath254 of the partition of @xmath167 are the orbits of the action of the automorphism group @xmath255 on such a set . namely @xmath169 and @xmath256 are in the same equivalence class @xmath168 if and only if there is some @xmath257 such that @xmath258 . then , the @xmath174 matrices , @xmath259 , representing the equivalence classes @xmath168 , are defined as in , and @xmath1 is said to be orbit polynomial whenever @xmath260 for all @xmath73 . recall that a ( symmetric ) association scheme @xmath261 with @xmath262 classes can be defined as a set of @xmath262 graphs @xmath263 , @xmath264 , on the same vertex set @xmath2 , with adjacency matrices @xmath14 satisfying @xmath265 , with @xmath266 , and @xmath267 , for some integers @xmath268 , @xmath269 . then , following godsil [ 24 ] , we say that the graph @xmath270 is the @xmath8-th class of the scheme , and so we indistinctly use the words graph " or class " to mean the same thing . ( for more details , see @xcite . ) a @xmath262-class association scheme @xmath261 is said to be _ generated _ by one of its matrices @xmath14 ( or graph @xmath270 ) if it determines the other relations , that is , the powers @xmath271 and @xmath272 span the bose - mesner algebra of @xmath261 . in particular , if @xmath270 is connected , then it generates the whole scheme if and only if it has @xmath164 distinct eigenvalues . then , since a quotient polynomial graph @xmath1 is connected , theorem [ main - theo]@xmath202 yields : let @xmath1 be a graph with @xmath164 distinct eigenvalues . then , @xmath1 is the connected generating graph of a @xmath262-class association scheme @xmath261 if and only if @xmath1 is a quotient - polynomial graph.@xmath201 if @xmath1 is quotient polynomial , it is connected and the matrices @xmath273 clearly satisfy the conditions for being an association scheme @xmath261 with @xmath262 classes . conversely , if @xmath1 is a connected graph that generates a @xmath262-class association scheme @xmath261 , its adjacency matrix @xmath23 has @xmath164 distinct eigenvalues @xmath274 , and @xmath261 has bases @xmath275 and @xmath276 ( the minimal idempotents ) related by the equalities @xmath277 ( with @xmath278 being the eigenvalues of the scheme , @xmath279 ) . thus , the polynomial @xmath280 $ ] satisfying @xmath281 satisfies @xmath243 for every @xmath38 , and @xmath1 is quotient polynomial by theorem [ main - theo ] . * acknowledgments . * the author acknowledges the useful comments and suggestions of e. garriga and j.l.a . this research is supported by the _ ministerio de ciencia e innovacin _ ( spain ) and the _ european regional development fund _ under project mtm2011 - 28800-c02 - 01 , and the _ catalan research council _ under project 2009sgr1387 .

as a generalization of orbit - polynomial and distance - regular graphs , we introduce the concept of a quotient - polynomial graph . in these graphs every vertex @xmath0 induces the same regular partition around @xmath0 , where all vertices of each cell are equidistant from @xmath0 . some properties and characterizations of such graphs are studied . for instance , all quotient - polynomial graphs are walk - regular and distance - polynomial . also , we show that every quotient - polynomial graph generates a ( symmetric ) association scheme . _ mathematics subject classifications : _ 05e30 , 05c50 . _ keywords : _ graph quotient ; distance - faithful partition ; walk - regular partition ; quotient - polynomial graph ; distance - regular graph ; eigenvalues ; orthogonal polynomials ; intersection numbers .

small submicron particles are well - suited as fillers in non - conducting polymer matrices to obtain a conducting composite with a percolation threshold ( far ) below 1 % . the low percolation threshold is due to the formation of airy aggregates of conducting particles , in which the particles are grown together by diffusion - limited cluster aggregation , creating a network with a fractal dimension around 1.7 . @xcite these airy aggregates can be thought of as conducting spheres forming a 3-dimensional percolating network around the expected aggregate filling fraction 0.16 . as a consequence of the fractal structure within the aggregates , the filler fraction of the particles at the percolation point is much lower . even in case the particles touch , the dc conductivity ( @xmath0 ) of these composites at high filling fractions turns out to be orders of magnitude lower than of the bulk material , as was recently illustrated for a particular crosslinked epoxy composite with filler particles of phthalcon-11 , @xcite co phthalocyanine crystallites of 100 nm size , and explained by purely structural arguments.@xcite when crystalline particles with a diameter of less than 10 nm instead of 100 nm are used , the small size of the particles may impose another important restriction to the maximal possible composite conductivity , which is due to the density of states ( dos ) involved in the dc conductivity through the network of particle contacts . compared to larger crystallites , this dos can be strongly reduced by the charging energy.@xcite we show how those microscopic parameters , which govern the charge - transport process across many decades of length scales , can accurately and consistently be determined by ac ( alternating current ) dielectric spectroscopy from a few hz to infrared frequencies . in particular we can address the parameters for mott variable - range hopping , for heterogeneity - induced enhanced ac response , for phonon- or photon - assisted nearest - neighbor hopping , and for the drude response of individual nanocrystals . due to these quantitative results we can unambiguously determine also the role of the nanocrystal charging energy in limiting the hopping process . we apply the method to antimony - doped tin - oxide ( ato ) crystallites of 7 nm diameter and to 100 nm sized crystallites of phthalcon-11 . it turns out that in densely packed crystallites of ato , due to the strong influence of the charging energy on the dos , @xmath0 at room temperature is four orders of magnitude lower than the dc conductivity extrapolated from the drude plasma frequency ( @xmath1 ) of the crystallites - a result with obvious implications for the design of conducting composites . the dielectric method is well suited for a variety of systems with restricted geometries , as we will illustrate by a short discussion of phase - change materials @xcite and granular oxides . @xcite . for randomly placed conducting spheres in an insulating matrix , the relation between @xmath0 and the fraction @xmath2 of spheres is known from percolation theory . @xciteabove the percolation threshold @xmath3 where @xmath4 is the percolation threshold , @xmath5 and @xmath6 is approximately equal to the conductivity of the spheres . @xcite when the building blocks of the network are fractal aggregates instead of solid spheres , @xmath6 has to be replaced by the aggregate conductivity @xmath7 and depends on the particle conductivity and , via the non - linear relation @xmath8 with @xmath9 , on the real percolation threshold @xmath10 of the particles.@xcite the value of the exponent @xmath11 is related to the random - walk dimension and the fractal dimension @xmath12 , and is maximally @xmath13 . this shows that on purely geometrical grounds for a network with @xmath14 , at the highest filling fraction of the aggregates @xmath0 will be three to four orders of magnitude lower than in the pure filler powder . @xcite as remarked in the introduction , when nanosized particles are used as fillers , charging energies ( and quantum size effects ) impose a further important restriction to the maximal possible composite conductivity.@xcite this effect can be conveniently studied in densely packed powders of filler material by dielectric spectroscopy . in the ohmic regime , if there is a non - negligible density of states around the chemical potential , and the temperature @xmath15 is high enough that also the coulomb interaction can be neglected ( @xmath16 ) , @xmath0 will obey mott s equation for conduction via variable - range hopping ( vrh ) : @xcite @xmath17 \label{esm}\ ] ] with @xmath18 and @xmath19 where @xmath20 denotes the decay length of the electron density , @xmath21 the boltzmann constant , and @xmath22 the density of states relevant in the hopping process . @xcite for randomly packed spheres of radius @xmath23 and spacing @xmath24 the localization length @xmath25 will be enlarged , @xcite and can be approximated by @xcite @xmath26 in the following we drop the tilde . below @xmath27 the @xmath15 dependence of the conductivity will be dominated by a soft coulomb gap , leading to so - called efros - shklovskii ( es ) vrh : @xcite @xmath28 . \label{eses}\ ] ] in the es vrh model in the dilute limit of a large distance between the particles @xmath29 is given by @xmath30 with @xmath31 the electron charge , @xmath32 the vacuum dielectric constant , and @xmath33 the relative dielectric constant of the medium . @xmath27 is given by @xmath34 $ ] and ( for @xmath35 ) the charging energy by @xmath36 . for densely packed small particles , at high temperatures but still in the regime , where coulomb interactions are important ( @xmath37 ) , es vrh behavior will evolve into nearest - neighbor hopping at a temperature @xmath38 . above @xmath38 the conduction is thermally activated with an activation energy @xmath39 of the order of the charging energy , and @xmath40 . the experiments of yu _ et al . _ @xcite on thin films of highly monodispersed semiconducting nanocrystals of cdse of 6 nm diameter , slightly smaller than the ato crystallites discussed here , showed good agreement between the theoretical and experimental value of @xmath29 , @xmath27 and @xmath41 . @xcite at sufficiently low frequencies the conductivity will be frequency independent and equal to its dc value , because the inhomogeneities are averaged out by the motion of the charge carriers . the minimal length scale for homogeneity is referred to as @xmath42 , @xmath43 where @xmath44 is the density of the carriers involved in the hopping process at the border of the homogenous regime , and the onset frequency @xmath45 for the frequency dependence of @xmath46 is divided by @xmath47 , with @xmath48 the dimension of the system . @xcite at high enough frequencies , when during half a period of the oscillation of the applied field electrons can hop solely between nearest - neighbors , the major contribution to the conductivity will be due to tunnelling between localized states at neighboring sites ( the pair limit).@xcite this incoherent process can be either by phonon - assisted or photon - assisted hopping , where in the latter case the energy difference between the sites is supplied by photons instead of phonons . @xcite the phonon - assisted contribution to the conductivity is given by @xmath49 with @xmath50 the decay length of the electronic state outside the conducting particles , @xmath22 the relevant dos at the fermi energy @xmath51 , and @xmath52 the phonon ` attempt ' frequency.@xcite this formula is valid when @xmath53 ; at higher @xmath54 , where the contribution of phonon - assisted hopping to @xmath46 becomes constant , photon - assisted processes usually take over , with a conductivity @xmath55 given by @xmath56 the energy @xmath57 in eq . ( [ esphon ] ) is in eq . ( [ esphot ] ) replaced by @xmath58 and the phonon attempt frequency @xmath52 by @xmath59 , with @xmath60 being the ` overlap ' pre - factor for the energy levels of two neighboring sites . in analogy with @xmath52 , @xmath61 can be interpreted as the attempt frequency for photon - assisted hopping . equation ( [ esphot ] ) is only valid when @xmath62 . as in phonon - assisted hopping , @xmath46 passes over into a plateau at high @xmath54 . at high frequencies ( for ato in the infrared regime ) the short period of the electromagnetic field will restrict the motion of the carriers to the nanocrystallite , and the dielectric response characterized by the complex relative dielectric constant @xmath63 $ ] will be drude - like , with @xmath64 the drude plasma resonance frequency and @xmath65 the damping rate . in practice the constant 1 has to be replaced by @xmath66 due to other contributions in this frequency regime , like the polarization of the ion cores . @xcite the drude plasma frequency is related to the number of carriers per unit of volume @xmath67 and the effective mass @xmath68 as @xmath69 for damping rates comparable to the drude plasma frequency , the real plasma frequency ( where the dielectric constant becomes zero ) will be larger than @xmath1 . @xmath70 is determined by the boundaries of the nanoparticle and additional ( ionized impurity ) scattering : @xmath71 where @xmath72 is the sum of the inverse size of the particle and the inverse phonon scattering length . measurements were performed on sb - doped tin - oxide nanoparticles with [ sb]/([sn]+[sb ] ) equal to 0 , 2 , 5 , 7 , 9 , and 13 at.% . the particles are monocrystalline and spherical with diameters close to 7 nm . @xcite the diameter of the 7% doped crystallite is 7.1 nm . sb is incorporated in the casserite sno@xmath73 lattice by replacing sn@xmath74 . at the doping level of 7% , sb is mainly present as sb@xmath75 , resulting in n - type conductivity of the ato particles according to nutz _ et al._. @xcite the amount of sb@xmath76 present in the particles is negligibly small . @xcite the followed experimental procedures for the dc conductivity and dielectric measurements are described in refs . and . the thickness of the samples was typically a few mm . the dc conductivity measurements were performed in the dark under helium atmosphere . the @xmath15 dependence is given in fig . [ svst ] , and the frequency dependence at temperatures down to 7 k in fig . all data shown are for 7%-doped ato . similar results were obtained at other doping levels , be it with different absolute values . the data were taken in the ohmic regime . for a densely packed powder of 7%-doped ato . for @xmath77 k the data can be fitted by eq . ( [ eses ] ) ( a ) , while for @xmath78 k the @xmath15 dependence is activated ( b).,width=302 ] .transport parameters obtained for ato and indium tin oxide ( ito ) ( second row ) at doping levels of @xmath79 to @xmath80 per @xmath81 . the room - temperature dc conductivities are given in s / cm , the drude frequencies and damping rate in s@xmath82 , and the effective mass in free - electron masses . the first and second row are obtained for films of ato @xcite and ito @xcite resp . , the last two rows contain the data on powders of 6% doped ato particles of ntz _ et al._@xcite and our data ( labelled as pw for present work ) on samples with 7% sb doping . for a bulk material with a drude frequency of @xmath83 hz , a scattering time of @xmath84 s and a carrier mass of 0.3 @xmath85 a dc conductivity is expected of @xmath86 s / cm [ cols="^,^,^,^,^,^,^",options="header " , ] ) .,width=302 ] the infrared ( ir ) transmittance was measured on a pellet of kbr mixed with a small amount of ato . for the ir - reflectance we used a precipitated film of ato with a thickness of about 1 mm . the data are shown in fig . [ trir ] . for the analysis we also used the sub - thz transmittance and phase data ( only shown in fig . [ sfit ] ) . .,width=302 ] in table [ numbers ] we summarize our data on densely packed 7-nm - sized ato crystallites and compare them with measurements on doped tin oxide published in the literature . the values of @xmath1 agree within a factor 2 , while the spread in the scattering rates is larger . in the analysis we first show the procedure to extract the parameter values from the data in the different frequency regimes and to check their consistency . we also make a comparison to the parameter values of phthalcon-11 , for which the data are published elsewhere . @xcite then we concentrate on the density of states ; the latter being important for the dc conductivity . subsequently , the implications for the use of the particles as fillers in nanocomposites are discussed . regarding the @xmath15 dependence of @xmath46 ( fig . [ svst ] ) , the data can be fitted with @xmath87 ( eq . ( [ esm ] ) ) if the fit is restricted to @xmath88 k and with @xmath89 ( eq . ( [ eses ] ) ) for @xmath90 k. the @xmath87 fit gives an activation energy of @xmath86 k , while the exponent @xmath89 at low @xmath15 gives @xmath91 k. the localization length from @xmath92 , see eq . ( [ et0es ] ) , is calculated to be @xmath20 = 3 nm . using eq . ( [ ea ] ) and @xmath93 estimated from the packing density , we find @xmath94 nm , in good agreement with the value calculated from @xmath29 . the onset of the frequency dependence of the conductivity ( see fig . [ svsf ] ) signals that the carrier starts to feel the inhomogeneity of the underlying structure . ( [ esphon ] ) , the typical length scale @xmath42 at the onset can be found . for 7%-doped ato at 300 k , the onset frequency @xmath95 hz and @xmath96 s / cm give a value of @xmath97 cm@xmath82 . the linear frequency dependence of the conductivity at 7 k in the double logarithmic plot of fig . [ svsf ] , is in agreement with phonon - assisted tunneling , see eq . ( [ esphon ] ) . in the range of 10 - 100 cm@xmath82 photon - assisted processes take over.@xcite applying eq . ( [ esphon ] ) to the conductivity data at 293 k and taking the usual value for the phonon frequency in solids @xmath99 s@xmath82,@xcite we find @xmath100 ev@xmath101cm@xmath82 , see fig . [ sfit ] . turning next to the high - frequency data presented in fig . [ trir ] , we performed a simple drude analysis . the fit ( @xmath1 = 11000 cm@xmath82 and @xmath102 = 3300 cm@xmath82 , together with a dielectric constant of 4.0 ) reproduces the main features of the increase of the transmission and the level of the reflectance ( the oscillations in the fit to the transmittance are an artefact because the effective ato film thickness of 0.005 mm is much smaller than the real thickness of the pressed kbr pellet ) . the number of carriers @xmath67 of @xmath80 @xmath103 is directly derived from the drude frequency and is slightly lower than obtained from a simple interpretation of the chemical composition . the bulk dc conductivity calculated from the drude parameters is @xmath86 s / cm . the fit parameters of the present samples are given in table [ numbers ] and agree well with the literature . [ sfit ] shows the reconstructed conductivity of ato as function of frequency due to the processes discussed above . for ato the important values for the dc conductivity can be deduced from the combination of variables that we found from the previous analysis ( i ) @xmath104 cm@xmath82 , ( ii ) @xmath105 j@xmath101m@xmath82 or @xmath106 @xmath107 for photon - assisted hopping and @xmath108 j@xmath101m@xmath82 for the phonon - fit to the data at 7 k , and ( iii ) @xmath109 @xmath103 and @xmath110 s. using ( iii ) the ` extrapolated ' dc conductivity is @xmath86 s / cm , a factor @xmath111 larger than the found value of @xmath112 s / cm . the estimated fermi energy @xmath51 is around 2 ev , and from @xmath113 ( valid for free electrons ) we get @xmath114 ev@xmath82@xmath103 . from ( ii ) with @xmath115 of 3 nm , we find for @xmath116 ev@xmath82@xmath103 , a factor @xmath117 lower than @xmath118 . note that this is an averaged density of states involved in photon assisted hopping . due to the curvature of the density of states around the chemical potential , @xmath119 will be lower at lower energies . for example , for the phonon - fit at 7 k @xmath120 is equal to @xmath121 ev@xmath82@xmath103 . the values for @xmath102 and @xmath42 can be used as a consistency check . the combination of the estimated fermi velocity of @xmath122 cm / s , with the crystallite size of 7 nm and @xmath123,@xcite predicts a surface scattering rate of @xmath124 s@xmath82 , in agreement with the found value of @xmath70 . next , from @xmath125 ev@xmath82@xmath103 we now can estimate @xmath44 at @xmath126 k as @xmath127 @xmath103 . using ( i ) and @xmath128 we find @xmath129 0.3 @xmath130 m . in short , the dielectric data of ato allow a consistent picture of the conduction process . in these densely packed crystallites the localization length is enhanced by a factor 10 and the density of states involved in the dc conductivity is more than a factor @xmath117 smaller than that in the conduction within the crystallites . the relatively large length scale for homogeneity is indicative for the presence of aggregates . indeed , like in ketjen - black , @xcite nanoparticles of ato are known to form chemically bonded aggregates that survive the preparation stage.@xcite due to the nature of the chemical bond , the conductivities between neighboring crystallites in and outside the aggregates are expected to be only slightly different . note that also the value of @xmath131 has to be seen as an average , as inhomogeneities in the doping of ato might be present as well.@xcite for the studied phthalcon-11 crystallites @xmath132 s@xmath82 and @xmath133 s leading to @xmath134 @xmath103 , i.e. about 1 charge per crystallite.@xcite the other values found for phthalcon-11 are : ( i ) @xmath135 cm@xmath82 , ( ii ) @xmath136 j@xmath101m@xmath82 or @xmath137 @xmath107 . in these organic crystals with such a low carrier density , the charge carriers can be seen as an electron gas with an energy scale set by @xmath57 , and @xmath119 can be estimated from @xmath138 to be @xmath139 @xmath140 . this value of @xmath119 is the upper limit for @xmath22 and @xmath141 . from @xmath142 @xmath140 , we find a decay length @xmath20 of 3 nm , as expected from the packing . the phthalcon-11 parameters show that the crystals are semiconducting crystals with a low number of charge carriers . all charges participating in the conductivity within the crystal also contribute to the dc conductivity . as for ato the obtained conduction parameters for phthalcon-11 from the dielectric scans give a consistent picture . for ato , the differences between the density of states involved in the hopping process @xmath143 ev@xmath82@xmath103 and the drude conduction within the crystallites @xmath114 ev@xmath82@xmath103 are clearly significant . the result is as anticipated from the estimated charging energy of the order of 50 mev , and shows its importance for the dc powder conductivity . for phthalcon-11 the very low number of carriers involved in the hopping process is similar to the number of carriers that determines the drude contribution in the crystallites . since the mean size of the particles is 20 times larger than for ato , the charging energies will be of the order of 3 mev , and hence are expected to be negligible at room temperature . in polymer nanocomposites with building blocks formed by diffusion - limited cluster aggregation , the airy structure of the particle network gives a strong reduction in conductivity of the composite compared to the filler ( for the phthalcon11/polymer composite a factor @xmath111 ) . @xcite this effect can be compensated by using better conducting particles . particles of ato or ito seem to be well - suited as the material is known to be very well - conducting . in addition , ato crystallites are relatively easily obtained in sizes around 7 nm , and when properly dispersed can give polymer composites with a low percolation threshold.@xcite however , even if the filler nanoparticles in the composite touch , they will not be in better contact than in a densely packed powder . as shown here for ato , for these small crystallites the dos involved in @xmath0 is dramatically reduced due to the shift of the energy levels away from the fermi level by coulomb charging effects . as a consequence , an additional four orders of magnitude in @xmath0 are lost compared to the bulk value . other systems where size restrictions are expected to be present might be conveniently studied in a similar way . for example several chalcogenide alloys exhibit a pronounced contrast between the optical absorption in the metastable rocksalt after the intense laser pulse and in the initial amorphous phase . @xcite as shown by extended x - ray absorption fine structure spectroscopy ( exafs ) the resistive change after the intense laser recording pulse goes together with a crystallization process , where also small domains are inherently present . our dielectric method might visualize to what extent the domain walls after crystallization limit the conductivity and have consequences for the band structure calculations . if the walls become real barriers quantum size effects in the small domains will invalidate the use of periodic boundary conditions in the calculations . also the glassy behavior in the conductance of deposited indium - tin oxide samples in the insulating regime , @xcite and of quench - condensed insulating granular metals @xcite might be further clarified by the use of our dielectric approach and analysis . scanning the frequency will reveal the evolution of the length scales and dos involved in the relaxation processes . by combining data of sub - thz transmission with infrared transmission and reflection we were able to explain the full frequency response of densely - packed nanosized crystallites using the parameters for mott variable - range hopping , for heterogeneity - induced enhanced ac response , for phonon- or photon - assisted nearest - neighbor hopping , and for the drude response of individual nanocrystals . for 7 nm antimony - doped tin - oxide particles the analysis unambiguously quantified the reduction of the density of states involved in the dc conduction compared to the value extrapolated from the drude response at infrared frequencies . dielectric scans with a similar analysis will also be revealing in other systems where size limitations are expected to play a role . it is a pleasure to acknowledge roel van de belt of nano specials ( geleen , the netherlands ) , who made the ato samples available , and matthias wuttig from the physikalisches institut of the rwth aachen university in germany for fruitful discussions about phase - change materials . this work forms part of the research program of the dutch polymer institute ( dpi ) , project dpi435 .

conducting submicron particles are well - suited as filler particles in non - conducting polymer matrices to obtain a conducting composite with a low percolation threshold . going to nanometer - sized filler particles imposes a restriction to the conductivity of the composite , due to the reduction of the density of states involved in the hopping process between the particles , compared to its value within the crystallites . we show how those microscopic parameters that govern the charge - transport processes across many decades of length scales , can accurately and consistently be determined by a range of dielectric - spectroscopy techniques from a few hz to infrared frequencies . the method , which is suited for a variety of systems with restricted geometries , is applied to densely packed 7-nm - sized tin - oxide crystalline particles with various degree of antimony doping and the quantitative results unambiguously show the role of the nanocrystal charging energy in limiting the hopping process .