on measure solutions of the boltzmann equation part i moment production and stability estimates the spatially homogeneous boltzmann equation with hard potentials is considered for measure valued initial data having finite mass and energy we prove the existence of emph weak measure solutions with and without angular cutoff on the collision kernel the proof in particular makes use of an approximation argument based on the mehler transform moment production estimates in the usual form and in the exponential form are obtained for these solutions finally for the grad angular cutoff we also establish uniqueness and strong stability estimate on these solutions

measurability of optimal transportation and convergence rate for landau type interacting particle systems in this paper we consider nonlinear diffusion processes driven by space time white noises which have an interpretation in terms of partial differential equations for a specific choice of coefficients they correspond to the landau equation arising in kinetic theory the main goal of the paper is to construct an easily simulable diffusive interacting particle system converging towards this nonlinear process and to obtain an explicit pathwise rate this requires to find a significant coupling between finitely many brownian motions and the infinite dimensional white noise process the key idea will be to construct the right brownian motions by pushing forward the white noise processes through the brenier map realizing the optimal transport between the law of the nonlinear process and the empirical measure of independent copies of it a crucial problem will then be to establish the joint measurability of this optimal transport map with respect to the space variable and the parameter time randomness that makes the marginals vary to overcome this point we shall prove a general measurability result for the mass transportation problem and for the supports of the optimal transfer plans in the sense of set valued mappings this will allow us to construct the coupling and to obtain explicit convergence rates

entropy dissipation and long range interactions we study boltzmann s collision operator for long range interactions i e without grad s angular cut off assumption we establish a functional inequality showing that the entropy dissipation controls smoothness of the distribution function in a precise sense our estimate is optimal and gives a unified treatment of both the linear and the nonlinear cases we also give simple and self contained proofs of several useful results that were scattered in previous works as an application we obtain several helpful estimates for the cauchy problem and for the landau approximation in plasma physics

character expansion method for supergroups and extended superversions of the leutwyler smilga and berezin karpelevich integrals we introduce an extension of the character expansion method to the case of supergroups this method allows us to calculate a superversion of the leutwyler smilga integral which to the best of our knowledge has not been calculated before we also use the method to generalize a previously calculated superversion of the berezin karpelevich integral our character expansion method should also allow for the calculation of other supergroup integrals

asymptotic of grazing collisions and particle approximation for the kac equation without cutoff the subject of this article is the kac equation without cutoff we first show that in the asymptotic of grazing collisions the kac equation can be approximated by a fokker planck equation the convergence is uniform in time and we give an explicit rate of convergence next we replace the small collisions by a small diffusion term in order to approximate the solution of the kac equation and study the resulting error we finally build a system of stochastic particles undergoing collisions and diffusion that we can easily simulate which approximates the solution of the kac equation without cutoff we give some estimates on the rate of convergence

differential variational inequalities this paper introduces and studies the class of differential variational inequalities dvis in a finite dimensional euclidean space the dvi provides a powerful modeling paradigm for many applied problems in which dynamics inequalities and discontinuities are present examples of such problems include constrained time dependent physical systems with unilateral constraints differential nash games and hybrid engineering systems with variable structures the dvi unifies several mathematical problem classes that include ordinary differential equations odes with smooth and discontinuous right hand sides differential algebraic equations daes dynamic complementarity systems and evolutionary variational inequalities conditions are presented under which the dvi can be converted either locally or globally to an equivalent ode with a lipschitz continuous right hand function for dvis that cannot be so converted we consider their numerical resolution via an euler time stepping procedure which involves the solution of a sequence of finite dimensional variational inequalities borrowing results from differential inclusions dis with upper semicontinuous closed and convex valued multifunctions we establish the convergence of such a procedure for solving initial value dvis we also present a class of dvis for which the theory of dis is not directly applicable and yet similar convergence can be established finally we extend the method to a boundary value dvi and provide conditions for the convergence of the method the results in this paper pertain exclusively to systems with index not exceeding two and which have absolutely continuous solutions

angular gelfand tzetlin coordinates for the supergroup uosp k 1 2k 2 we construct gelfand tzetlin coordinates for the unitary orthosymplectic supergroup uosp k1 2k2 this extends a previous construction for the unitary supergroup u k1 k2 we focus on the angular gelfand tzetlin coordinates i e our coordinates stay in the space of the supergroup we also present a generalized gelfand pattern for the supergroup uosp k1 2k2 and discuss various implications for representation theory

on the rate of convergence in wasserstein distance of the empirical measure let mu n be the empirical measure associated to a n sample of a given probability distribution mu on mathbb r d we are interested in the rate of convergence of mu n to mu when measured in the wasserstein distance of order p 0 we provide some satisfying non asymptotic l p bounds and concentration inequalities for any values of p 0 and d ge 1 we extend also the non asymptotic l p bounds to stationary rho mixing sequences markov chains and to some interacting particle systems

classical and quantum superintegrability with applications a superintegrable system is roughly speaking a system that allows more integrals of motion than degrees of freedom this review is devoted to finite dimensional classical and quantum superintegrable systems with scalar potentials and integrals of motion that are polynomials in the momenta we present a classification of second order superintegrable systems in two dimensional riemannian and pseudo riemannian spaces it is based on the study of the quadratic algebras of the integrals of motion and on the equivalence of different systems under coupling constant metamorphosis the determining equations for the existence of integrals of motion of arbitrary order in real euclidean space e2 are presented and partially solved for the case of third order integrals a systematic exposition is given of systems in two and higher dimensional space that allow integrals of arbitrary order the algebras of integrals of motions are not necessarily quadratic but close polynomially or rationally the relation between superintegrability and the classification of orthogonal polynomials is analyzed

on the efetov wegner terms by diagonalizing a hermitian supermatrix the diagonalization of hermitian supermatrices is studied such a change of coordinates is inevitable if one wishes to find certain structures in random matrix theory however it still poses serious problems since until now the calculation of all rothstein contributions known as efetov wegner terms in physics was quite cumbersome we derive the supermatrix bessel function with all efetov wegner terms for an arbitrary rotation invariant probability density function as applications we consider representations of generating functions for hermitian random matrices with and without an external field as integrals over eigenvalues of hermitian supermatrices all results are obtained with all efetov wegner terms which were previously unknown in such an explicit and compact representation

gelfand tzetlin coordinates for the unitary supergroup the gelfand tzetlin method provides explicit coordinates on the parameter space of the unitary groupu k which make direct evaluations of group integrals possible it is closely related to the gelfand construction of finite dimensional irreducible representations we generalize the gelfand tzetlin method to the unitary supergroupu k1 k2 the coordinates on the parameter space for supergroup integrals and the invariant haar measure are evaluated as an example the supersymmetric harish chandra itzykson zuber integral is calculated a generalized gelfand pattern containing anticommuting variables is introduced which determines the representation

recursive construction for a class of radial functions ii superspace we extend the recursion formula for matrix bessel functions which we obtained previously to superspace it is sufficient to do this for the unitary orthosymplectic supergroup by direct computations we show that fairly explicit results can be obtained at least up to dimension 8 8 for the supermatrices since we introduce a new technique we discuss various of its aspects in some detail

derivation of determinantal structures for random matrix ensembles in a new way there are several methods to treat ensembles of random matrices in symmetric spaces circular matrices chiral matrices and others orthogonal polynomials and the supersymmetry method are particular powerful techniques here we present a new approach to calculate averages over ratios of characteristic polynomials at first sight paradoxically one can call our approach supersymmetry without supersymmetry because we use structures from supersymmetry without actually mapping onto superspaces we address two kinds of integrals which cover a wide range of applications for random matrix ensembles for probability densities factorizing in the eigenvalues we find determinantal structures in a unifying way as a new application we derive an expression for the k point correlation function of an arbitrary rotation invariant probability density over the hermitian matrices in the presence of an external field

orthogonality relations and supercharacter formulas of u m n representations in this paper we obtain the orthogonality relations for the supergroup u m n which are remarkably different from the ones for the u n case we extend our results for ordinary representations obtained some time ago to the case of complex conjugated and mixed representations our results are expressed in terms of the young tableaux notation for irreducible representations we use the supersymmetric harish chandra itzykson zuber integral and the character expansion technique as mathematical tools for deriving these relations as a byproduct we also obtain closed expressions for the supercharacters and dimensions of some particular irreducible u m n representations a new way of labeling the u m n irreducible representations in terms of m n numbers is proposed finally as a corollary of our results new identities among the dimensions of the irreducible representations of the unitary group u n are presented

moment inequalities for the boltzmann equation and applications to spatially homogeneous problems some inequalities for the boltzmann collision integral are proved these inequalities can be considered as a generalization of the well known povzner inequality the inequalities are used to obtain estimates of moments of the solution to the spatially homogeneous boltzmann equation for a wide class of intermolecular forces we obtain simple necessary and sufficient conditions on the potential for the uniform boundedness of all moments for potentials with compact support the following statement is proved if all moments of the initial distribution function are bounded by the corresponding moments of the maxwelliana exp bv2 then all moments of the solution are bounded by the corresponding moments of the other maxwelliana1 exp b1 t v2 for anyt 0 moreoverb t const for hard spheres an estimate for a collision frequency is also obtained

rate of convergence of the nanbu particle system for hard potentials and maxwell molecules we consider the numerically motivated nanbu stochastic particle system associated to the spatially homogeneous boltzmann equation for true hard potentials we establish a rate of propagation of chaos of the particle system to the unique solution of the boltzmann equation more precisely we estimate the expectation of the squared wasserstein distance with quadratic cost between the empirical measure of the particle system and the solution the rate we obtain is almost optimal as a function of the number of particles but is not uniform in time

a qualitative mathematical analysis of a class of linear variational inequalities via semi complementarity problems applications in electronics the main object of this paper is to present a general mathematical theory applicable to the study of a large class of linear variational inequalities arising in electronics our approach uses recession tools so as to define a new class of problems that we call semi complementarity problems then we show that the study of semi complementarity problems can be used to prove new qualitative results applicable to the study of linear variational inequalities of the second kind

arbitrary rotation invariant random matrix ensembles and supersymmetry orthogonal and unitary symplectic case recently the supersymmetry method was extended from gaussian ensembles to arbitrary unitarily invariant matrix ensembles by generalizing the hubbard stratonovich transformation here we complete this extension by including arbitrary orthogonally and unitary symplectically invariant matrix ensembles the results are equivalent to but the approach is different from the superbosonization formula we express our results in a unifying way we also give explicit expressions for all one point functions and discuss features of the higher order correlations

supersymmetry for systems with unitary disorder circular ensembles a generalized hubbard stratonovitch transformation relating an integral over random unitary matrices to an integral over efetov s unitary model manifold is introduced this transformation adapts the supersymmetry method to disordered and chaotic systems that are modelled not by a hamiltonian but by their scattering matrix or time evolution operator in contrast to the standard method no saddle point approximation is made and no massive modes have to be eliminated this first paper on the subject applies the generalized hubbard stratonovitch transformation to dyson s circular unitary ensemble it is shown how a supersymmetric variant of the harish chandra itzykson zuber formula can be used to compute in the large n limit the n level correlation function for any n non trivial applications to random network models quantum chaotic maps and lattice gauge theory are expected

new correlation functions for random matrices and integrals over supergroups the averages of ratios of characteristic polynomials det x of n n random matrices x are investigated in the large n limit for the gue goe and gse ensembles the density of states and the two point correlation function are derived from these ratios the method relies on an extension of the harish chandra itzykson zuber integrals to the goe ensemble and to supergroups which are explicitly computed through heat kernel differential equations an external matrix source linearly coupled to the random matrices may also be added to the gaussian distribution and allows for a discussion of the universality of the goe results in the large n limit

reformulations of measure differential inclusions and their closed graph property abstract new formulations are given for measure differential inclusions dx dt k t r n where x is a function of bounded variation and k is a set valued map with closed convex values and has closed graph measure differential inclusions were first named by j j moreau for studying rigid body with impacts shocks and coulomb friction and assumed that k t is always a cone the new formulations are used to show that the graph of the solution operator is closed under point wise convergence of the solutions x and weak convergence of the differential measures dx provided that the minimum norm points of k t are bounded and the asymptotic cones k t are always pointed

conservation of energy entropy identity and local stability for the spatially homogeneous boltzmann equation for nonsoft potential collision kernels with angular cutoff we prove that under the initial condition f0 v 1 v 2 logf0 v l1 r3 the classical formal entropy identity holds for all nonnegative solutions of the spatially homogeneous boltzmann equation in the class l 0 l12 r3 c1 0 l1 r3 where l1s r3 f f v 1 v 2 s 2 l1 r3 and in this class the nonincrease of energy always implies the conservation of energy and therefore the solutions obtained all conserve energy moreover for hard potentials and the hard sphere model a local stability result for conservative solutions i e satisfying the conservation of mass momentum and energy is obtained as an application of the local stability a sufficient and necessary condition on the initial data f0 such that the conservative solutions f belong to l1loc 0 l12 r3 is also given

integration of grassmann variables over invariant functions on flat superspaces we study integration over functions on superspaces these functions are invariant under a transformation which maps the whole superspace onto the part of the superspace which only comprises purely commuting variables we get a compact expression for the differential operator with respect to the commuting variables which results from berezin integration over all grassmann variables also we derive cauchy like integral theorems for invariant functions on supervectors and symmetric supermatrices this extends theorems partly derived by other authors as a physical application we calculate the generating function of the one point correlation function in random matrix theory furthermore we give another derivation of supermatrix bessel functions for u k1 k2

estimates of optimal storage conditions in neural network memories based on random matrix theory the authors formulate a method for estimating the critical conditions for storage of sets of data in neural network memory this variational method is based on random matrix theory and calculating the average spectrum of a matrix whose elements are given by overlaps of the stored patterns several generic cases of random overlap matrices are considered they investigate the cases of simply uncorrelated random patterns and spatially and semantically correlated ones they obtain bounds of the critical curve in the control parameters space which determine the stability of the stored data sets

on the well posedness of the spatially homogeneous boltzmann equation with a moderate angular singularity we prove an inequality on the kantorovich rubinstein distance which can be seen as a particular case of a wasserstein metric between two solutions of the spatially homogeneous boltzmann equation without angular cutoff but with a moderate angular singularity our method is in the spirit of 7 we deduce some well posedness and stability results in the physically relevant cases of hard and moderately soft potentials in the case of hard potentials we relax the regularity assumption of 6 but we need stronger assumptions on the tail of the distribution namely some exponential decay we thus obtain the first uniqueness result for measure initial data in the case of moderately soft potentials we prove existence and uniqueness assuming only that the initial datum has finite energy and entropy for very moderately soft potentials plus sometimes an additionnal moment condition we thus improve significantly on all previous results where weighted sobolev spaces were involved

optimization based simulation of nonsmooth rigid multibody dynamics we present a time stepping method to simulate rigid multibody dynamics with inelastic collision contact and friction the method progresses with fixed time step without backtracking for collision and solves at every step a strictly convex quadratic program we prove that a solution sequence of the method converges to the solution of a measure differential inclusion we present numerical results for a few examples and we illustrate the difference between the results from our scheme and previous linear complementarity based time stepping schemes

general random matrix approach to account for the effect of static disorder on the spectral properties of light harvesting systems we develop a random matrix model approach to study static disorder in pigment protein complexes in photosynthetic organisms as a case study we examine the ring of b850 bacteriochlorophylls in the peripheral light harvesting complex of rhodospirillum molischianum formulated in terms of an effective hamiltonian describing the collective electronic excitations of the system we numerically examine and compare various models of disorder and observe that both the density of states and the absorption spectrum of the model show remarkable spectral universality for the case of unitary disorder we develop a method to analytically evaluate the density of states of the ensemble using the supersymmetric formulation of random matrix theory succinct formulas that can be readily applied in future studies are provided in an appendix

convergence of a class of semi implicit time stepping schemes for nonsmooth rigid multibody dynamics in this work we present a framework for the convergence analysis in a measure differential inclusion sense of a class of time stepping schemes for multibody dynamics with contacts joints and friction this class of methods solves one linear complementarity problem per step and contains the semi implicit euler method as well as trapezoidal like methods for which second order convergence was recently proved under certain conditions by using the concept of a reduced friction cone the analysis includes for the first time a convergence result for the case that includes joints an unexpected intermediary result is that we are able to define a discrete velocity function of bounded variation although the natural discrete velocity function produced by our algorithm may have unbounded variation

asymptotic coincidence of the statistics for degenerate and non degenerate correlated real wishart ensembles the correlated wishart model provides the standard benchmark when analyzing time series of any kind unfortunately the real case which is the most relevant one in applications poses serious challenges for analytical calculations often these challenges are due to square root singularities which cannot be handled using common random matrix techniques we present a new way to tackle this issue using supersymmetry we carry out an anlaytical study which we support by numerical simulations for large but finite matrix dimensions we show that statistical properties of the fully correlated real wishart model generically approach those of a correlated real wishart model with doubled matrix dimensions and doubly degenerate empirical eigenvalues this holds for the local and global spectral statistics with monte carlo simulations we show that this is even approximately true for small matrix dimensions we explicitly investigate the k point correlation function as well as the distribution of the largest eigenvalue for which we find a surprisingly compact formula in the doubly degenerate case moreover we show that on the local scale the k point correlation function exhibits the sine and the airy kernel in the bulk and at the soft edges respectively we also address the positions and the fluctuations of the possible outliers in the data

kac s program in kinetic theory this paper is devoted to the study of propagation of chaos and mean field limits for systems of indistinguishable particles undergoing collision processes the prime examples we will consider are the many particle jump processes of kac and mckean kac in proceedings of the third berkeley symposium on mathematical statistics and probability 1954 1955 vol iii pp 171 197 1956 mckean in j comb theory 2 358 382 1967 giving rise to the boltzmann equation we solve the conjecture raised by kac proceedings of the third berkeley symposium on mathematical statistics and probability 1954 1955 vol iii pp 171 197 1956 motivating his program on the rigorous connection between the long time behavior of a collisional many particle system and the one of its mean field limit for bounded as well as unbounded collision rates

probability metrics and uniqueness of the solution to the boltzmann equation for a maxwell gas we consider a metric for probability densities with finite variance on d and compare it with other metrics we use it for several applications both in probability and in kinetic theory the main application in kinetic theory is a uniqueness result for the solution of the spatially homogeneous boltzmann equation for a gas of true maxwell molecules

step size adjustment and extrapolation for time stepping schemes in non smooth dynamics in this paper we use step size adjustment and extrapolation methods to improve moreau s time stepping scheme for the numerical integration of non smooth mechanical systems i e systems with impact and friction the scheme yields a system of inclusions which is transformed into a system of projective equations these equations are solved iteratively switching points are time instants for which the structure of the mechanical system changes for example time instants for which a sticking friction element begins to slide we show how switching points can be localized and how these points can be resolved by choosing a minimal step size in order to improve the integration of non smooth systems in the smooth parts we show how the time stepping method can be used as a base integration scheme for extrapolation methods which allow for an increase in the integration order switching points are processed by a small time step while time intervals during which the structure of the system does not change are computed with a larger step size and improved integration order the overall algorithm which consists of a time stepping module an extrapolation module and a step size adjustment module is discussed in detail and some examples are given

on kac s chaos and related problems this paper is devoted to establish quantitative and qualitative estimates related to the notion of chaos as firstly formulated by m kac cite kac1956 in his study of mean field limit for systems of n undistinguishable particles as n to infty first we quantitatively liken three usual measures of it kac s chaos some involving the all n variables other involving a finite fixed number of variables the cornerstone of the proof is a new representation of the monge kantorovich wasserstein mkw distance for symmetric n particle probability measures in terms of the distance between the law of the associated empirical measures on the one hand and a new estimate on some mkw distance on probability measures spaces endowed with a suitable hilbert norm taking advantage of the associated good algebraic structure next we define the notion of it entropy chaos and it fisher information chaos in a similar way as defined by carlen et al cite ccllv we show that it fisher information chaos is stronger than it entropy chaos which in turn is stronger than it kac s chaos more importantly with the help of the hwi inequality of otto villani we establish a quantitative estimate between these quantities which in particular asserts that it kac s chaos plus it fisher information bound implies it entropy chaos we then extend the above quantitative and qualitative results about chaos in the framework of probability measures with support on the it kac s spheres revisiting cite ccllv and giving a possible answer to cite open problem 11 ccllv additionally to the above mentioned tool we use and prove an optimal rate local clt in l infty norm for distributions with finite 6 th moment and finite l p norm for some p 1 last we investigate how our techniques can be used without assuming chaos in the context of probability measures mixtures introduced by de finetti hewitt and savage in particular we define the level 3 fisher information for mixtures and prove that it is l s c and affine as that was done in cite rr for the level 3 boltzmann s entropy

the integral theorem for supersymmetric invariants a supersymmetric integral theorem that extends results of parisi sourlas efetov wegner and others is rigorously proved in particular arbitrary generators are allowed in the integrand instead of canonical ones and the invariance condition is very much relaxed the connection with cauchy s integral formula is made transparent in passing the unitary lie supergroup is studied by using elementary methods applications in the theory of disordered systems are discussed

character expansions for the orthogonal and symplectic groups formulas for the expansion of arbitrary invariant group functions in terms of the characters for the sp 2n so 2n 1 and so 2n groups are derived using a combinatorial method the method is similar to one used by balantekin to expand group functions over the characters of the u n group all three expansions have been checked for all n by using them to calculate the known expansions of the generating function of the homogeneous symmetric functions an expansion of the exponential of the traces of group elements appearing in the finite volume gauge field partition functions is worked out for the orthogonal and symplectic groups

stochastic particle approximations for generalized boltzmann models and convergence estimates fied boltzmann equation with general motion between collisions and nonlinear bounded jump collision operator and give the nonlinear martingale problem it solves we consider various linear interacting particle systems in order to approximate this nonlinear process we prove propagation of chaos in variation norm on path space with a precise rate of convergence using coupling and interaction graph techniques and a representation of the nonlinear process on a boltzmann tree no regularity nor uniqueness assumption is needed we then consider a nonlinear equation with both vlasov and boltzmann terms and give a weak pathwise propagation of chaos result using a compactness uniqueness method which necessitates some regularity these results imply functional laws of large numbers and extend to multitype models we give algorithms simulating or approximating the particle systems 1 framework and main results the boltzmann equation describes the evolution of the limit density f t x v of molecules of a rarefied gas at time t position x and speed v f is positive and f f t x v dx dv 1 it is given by

impulsive functional differential inclusions and fuzzy population models in this paper we shall establish some existence results for the impulsive functional differential inclusion and the fuzzy impulsive functional differential equation with some conditions and study the properties of the solution set and the attainable set finally the results will be used to fuzzy population models

replica method for wide correlators in gaussian orthogonal unitary and symplectic random matrix ensembles we calculate connected correlators in gaussian orthogonal unitary and symplectic random matrix ensembles by the replica method in the 1 n expansion we obtain averaged one point green s functions up to the next to leading order wide two level correlators up to the first non trivial order and wide three level correlators up to the first non trivial order by carefully treating fluctuations in saddle point evaluation

arbitrary unitarily invariant random matrix ensembles and supersymmetry we generalize the supersymmetry method in random matrix theory to ensembles which are unitarily invariant but otherwise arbitrary our exact approach extends a previous contribution in which we constructed a supersymmetric representation for the class of norm dependent random matrix ensembles here we derive a supersymmetric formulation under very general circumstances a reduced probability density and a projector are identified that map the probability density from ordinary to superspace furthermore it is demonstrated that setting up the theory in fourier superspace has considerable advantages general and exact expressions for the correlation functions are given we also show how the use of hyperbolic symmetry can be circumvented in the present context in which the nonlinear model is not used we construct exact supersymmetric integral representations of the correlation functions for arbitrary positions of the imaginary increments in the green s functions

a supersymmetry approach to billiards with randomly distributed scatterers ii correlations in a previous contribution stockmann h j 2002 j phys a math gen 35 5165 the density of states was calculated for a billiard with randomly distributed delta like scatterers doubly averaged over the positions of the impurities and the billiard shape this result is now extended to the k point correlation function using supersymmetric methods we show that the correlations in the bulk are always identical to those of the gaussian unitary ensemble gue of random matrices in passing from the band centre to the tail states the density of states is depleted considerably and the two point correlation function shows a gradual change from the gue behaviour to that found for completely uncorrelated eigenvalues this can be viewed as similar to a mobility edge

the itzykson zuber integral for u m n the itzykson zuber iz integral is computed for the superunitary group u m n as a consequence the nonzero correlations of superunitary matrices can be found

a method of integration over matrix variables the integral over twon n hermitan matricesz g c dadbexp tr a 2 b 2 2cab g n a 4 b 4 is evaluated in the limit of largen for this purpose use is made of the theory of diffusion equation and that of orthogonal polynomials with a non local weight the above integral arises in the study of the planar approximation to quantum field theory

dyson s correlation functions and graded symmetry a new derivation of dyson s k level correlation functions of the gaussian unitary ensemble gue is given the method uses matrices with graded symmetry the number of integrations needed for the ensemble average becomes independent of the level number n for arbitrary level number n the k level correlation function is expressed as an integral involving the eigenvalues of a 2k 2k graded matrix the limit of infinitely many levels n is calculated by a simple saddle point approximation of this integral avoiding the introduction of hermite polynomials and oscillator wave functions

convergence of a time stepping scheme for rigid body dynamics and resolution of painleve s problem this paper gives convergence theory for a new implicit time stepping scheme for general rigid body dynamics with coulomb friction and purely inelastic collisions and shocks an important consequence of this work is the proof of existence of solutions of rigid body problems which include the famous counterexamples of painleve the mathematical basis for this work is the formulation of the rigid body problem in terms of measure differential inclusions of moreau and monteiro marques the implicit time stepping method is based on complementarity problems and is essentially a particular case of the algorithm described in anitescu potra 2 which in turn is based on the formulation in stewart trinkle 47

on the graded group u 1 1 a noncanonical parametrization for the graded group u 1 1 is introduced similar to the euler angles for the ordinary group su 2 two differential representations for the underlying algebra of u 1 1 are constructed on the full and the restricted i e coset parameter space respectively a space of functions living on the latter is found exhibiting close formal similarities to a hilbert space remarkably the indices of those functions and thus the orthogonality and completeness relations involve anticommuting variables using this hilbert space a representation of u 1 1 is evaluated which shows analogies to the wigner functions for su 2

the k point random matrix kernels obtained from one point supermatrix models the k point correlation functions of the gaussian random matrix ensembles are certain determinants of functions which depend on only two arguments they are referred to as kernels since they are the building blocks of all correlations we show that the kernels are obtained for arbitrary level number directly from supermatrix models for one point functions more precisely the generating functions of the one point functions are equivalent to the kernels this is surprising because it implies that already the one point generating function holds essential information about the k point correlations this also establishes a link to the averaged ratios of spectral determinants i e of characteristic polynomials

norm dependent random matrix ensembles in external field and supersymmetry the class of norm dependent random matrix ensembles is studied in the presence of an external field the probability density of the ensemble depends on the trace of the squared random matrices but is otherwise arbitrary an exact mapping to superspace is performed a transformation formula is derived which gives the probability density in superspace as a single integral over the probability density in ordinary space this is done for orthogonal unitary and symplectic symmetries in the case of unitary symmetry some explicit results for the correlation functions are derived

the calogero moser equation system and the ensemble average in the gaussian ensembles from random matrix theory it is known that for special values of the coupling constant the calogero moser cm equation system is nothing but the radial part of a generalized harmonic oscillator schrodinger equation this allows an immediate construction of the solutions by means of a rodriguez relation the results are easily generalized to arbitrary values of the coupling constant by this the cm equations become nearly trivial as an application an expansion for eitr xy in terms of eigenfunctions of the cm equation system is obtained where x and y are matrices taken from one of the gaussian ensembles and the brackets denote an average over the angular variables

wigner dyson statistics from the replica method we compute the correlation functions of the eigenvalues in the gaussian unitary ensemble using the fermionic replica method we show that non trivial saddle points which break replica symmetry must be included in the calculation in order to correctly reproduce the exact asymptotic behaviour of the correlation functions at a large distance

time stepping for three dimensional rigid body dynamics traditional methods for simulating rigid body dynamics involves determining the current contact arrangement e g each contact is either a rolling or sliding contact the development of this approach is most clearly seen in the work of haug et al mech machine theory 21 1986 401 425 and pfeiffer and glocker multibody dynamics with unilateral contacts wiley 1996 however there has been a controversy about the status of rigid body dynamics as a theory due to simple problems in the area which do not appear to have solutions the most famous if not the earliest is due to paul painleve c r acad sci paris 121 1895 112 115 recently a number of time stepping methods have been developed to overcome these difficulties these time stepping methods use integrals of the forces over time steps rather than the actual forces this allows impulsive forces without the need for a separate formulation or special procedures to cover this case the newest of these methods are developed in terms of complementarity problems the complementarity problems that define the time stepping procedure are solvable unlike previous methods for simulating rigid body dynamics with friction proof of the existence of solutions to the continuous problem can be shown in the sense of measure differential inclusions in terms of these methods in this paper a number of these variants will be discussed and their essential properties proven

stability and uniqueness for the spatially homogeneous boltzmann equation with long range interactions in this paper we prove some a priori stability estimates in weighted sobolev spaces for the spatially homogeneous boltzmann equation without angular cutoff covering all physical collision kernels these estimates are conditional on some regularity estimates on the solutions and therefore reduce the stability and uniqueness issue to one of proving suitable regularity bounds on the solutions we then prove such regularity bounds for a class of interactions including the so called non cutoff and non mollified hard potentials and moderately soft potentials in particular we obtain the first result of global existence and uniqueness for these long range interactions

fourier bessel analysis for ordinary and graded 2 2 hermitian matrices chainlike integrals in matrix spaces play an important part in high energy and solid state physics and in general random matrix theory in the special case of ordinary and graded 2 2 hermitian matrices a method is proposed to integrate out all angular variables the essence of this method is a fourier bessel analysis in these matrix spaces which is formulated in this paper close formal similarities are found between the ordinary and the graded case the main differences arise from the fact that the ordinary case can be reduced to the study of a vector space whereas no analogous feature is present in the graded case

spectral correlations in the crossover between gue and poisson regularity on the identification of scales motivated by questions of present interest in nuclear and condensed matter physics we consider the superposition of a diagonal matrix with independent random entries and a gue the relative strength of the two contributions is determined by a parameter suitably defined on the unfolded scale using results for the spectral two point correlator of this model obtained in the framework of the supersymmetry method we focus attention on two different regimes for 1 the correlations are given by dawson s integral while for 1 we derive a novel analytical formula for the two point function in both cases the energy scales in units of the mean level spacing at which deviations from pure gue behavior become noticeable can be identified we also derive an exact expansion of the local level density for a finite level number