global boundedness of moments of solutions of the boltzmann equation for forces of infinite range this paper proves the global boundedness in time of higher moments to a weak solution of the non linear space homogeneous boltzmann equation for inverse kth power forces with k 5 in the course of the proof a new collisional estimate of v s v s v s v s is obtained where v v are the velocities before and v v are the velocities after a binary collision

a maxwellian lower bound for solutions to the boltzmann equation we prove that the solution of the spatially homogeneous boltzmann equation is bounded pointwise from below by a maxwellian i e a function of the formc1 exp c2v2 this holds for any initial data with bounded mass energy and entropy and for any positive timet t0 the constantsc1 andc2 depend on the mass energy and entropy of the initial data and ont0 0 only

strict entropy production bounds and stability of the rate of convergence to equilibrium for the boltzmann equation we first consider the boltzmann equation with a collision kernel such that all kinematically possible collisions are run at equal rates this is the simplest boltzmann equation having the compressible euler equations as a scaling limit for it we prove a stability result for theh theorem which says that when the entropy production is small the solution of the spatially homogeneous boltzmann equation is necessarily close to equilibrium in the entropie sense and therefore strongl 1 sense we use this to prove that solutions to the spatially homogeneous boltzmann equation converge to equilibrium in the entropie sense with a rate of convergence which is uniform in the initial condition for all initial conditions belonging to certain natural regularity classes every initial condition with finite entropy andp th velocity moment for some p 2 belongs to such a class we then extend these results by a simple monotonicity argument to the case where the collision rate is uniformly bounded below which covers a wide class of slightly modified physical collision kernels these results are the basis of a study of the relation between scaling limits of solutions of the boltzmann equation and hydrodynamics which will be developed in subsequent papers the program is described here

on classes of normalized matrices abstract the integer m n matrices a a ij b b ij are said to be equivalent if a ij u i b ij v j for all i 1 m j 1 n for some u 1 u m v 1 v n z for an integer matrix x the symbol s x denotes the set of all nonnegative integer matrices equivalent to x having a zero element in each row and each column we develop algorithms to solve the problems of the following type for a given class s x decide whether a s x find the largest possible value for each position among all matrices in s x find a matrix in s x with prescribed values of a specified entry or row column find a matrix in s x with term rank 2 a complete description of those s x containing only zero one matrices is presented

l estimates for the space homogeneous boltzmann equation this paper studies the boundedness of solutionsf of the initial value problem for the space homogeneous boltzmann equation for inverse kth power forces whenk 5 and under angular cutoff the main result is that if the initial value isf0 0 with 1 2 0 el1 and 1 sf0e l for somes 2 then 1 s ftel fort 0 and ess tsup 1 s f t 5

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

cooling process for inelastic boltzmann equations for hard spheres part i the cauchy problem we develop the cauchy theory of the spatially homogeneous inelastic boltzmann equation for hard spheres for a general form of collision rate which includes in particular variable restitution coefficients depending on the kinetic energy and the relative velocity as well as the sticky particles model we prove local in time non concentration estimates in orlicz spaces from which we deduce weak stability and existence theorem strong stability together with uniqueness and instantaneous appearance of exponential moments are proved under additional smoothness assumption on the initial datum for a restricted class of collision rates concerning the long time behaviour we give conditions for the cooling process to occur or not in finite time

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

revisiting the 0 1 assignment problem abstract the hungarian method is able to transform an assignment problem into many equivalent matrices in general we characterize these matrices for the 0 1 assignment problem examples and an application to the 0 1 traveling salesman problem are provided

entropy production estimates for boltzmann equations with physically realistic collision kernels we establish strict entropy production bounds for the boltzmann equation with the hard sphere collision kernel using these entropy production bounds we prove results asserting that the rate at which strongl 1 convergence to equilibrium occurs is uniform in wide classes of initial data this extends our previous results in this direction which applied only to a very special collision kernel moreover the present results provide computable lower bounds compactness arguments are entirely avoided the uniformity is an important ingredient in our study of scaling limits of solutions of the non spatially homogeneous boltzmann equation and is the main focus of this paper however the results obtained here provide the only framework known to us in which one can obtain computable estimates on the time it takes a solution of the spatially homogeneous boltzmann equation with initial data far from equilibrium to reach any given small strongl 1 neighborhood of equilibrium

global existence of solutions of the boltzmann equation for bose einstein particles with anisotropic initial data abstract in this paper we prove the global in time existence and uniqueness of solutions of the spatially homogeneous boltzmann equation for bose einstein particles for the hard sphere model for bounded anisotropic initial data the main idea of our proof is as follows we first establish an intermediate equation which is closely related to the original equation and is relatively easily proven to have global in time and unique solutions then we use the multi step iterations of the collision gain operator to obtain a desired uniform l bound for solutions of the intermediate equation so that if an initial datum is sufficiently small relative to the inverse of the planck constant which belongs to the case of very high temperature then the corresponding solution of the intermediate equation becomes the solution of the original equation

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

classes of matrices associated with the optimal assignment problem abstract let e e ij be a matrix with integral elements and let x be an indeterminate defined over the rational field q we investigate matrices of the form x x e ij i 1 m j 1 n m n we may multiply the lines rows or columns of the matrix x by suitable integral powers of x in various ways and thereby transform x into a matrix y x f ij such that the f ij are nonnegative integers and each line of y contains at least one element x 0 1 we call y a normalized form of x and we denote by s x the class of all normalized forms associated with a given matrix x the classes s x have a fascinating combinatorial structure and the present paper is a natural outgrowth and extension of an earlier study we introduce new concepts such as an elementary transformation called an interchange we prove for example that two matrices in the same class are transformable into one another by interchanges our analysis of the class s x also yields new insights into the structure of the optimal assignments of the matrix e by way of the diagonal products of the matrix x

on the boltzmann equation for diffusively excited granular media we study the boltzmann equation for a space homogeneous gas of inelastic hard spheres with a diffusive term representing a random background forcing under the assumption that the initial datum is a nonnegative l 2 n function with bounded mass and kinetic energy second moment we prove the existence of a solution to this model which instantaneously becomes smooth and rapidly decaying under a weak additional assumption of bounded third moment the solution is shown to be unique we also establish the existence but not uniqueness of a stationary solution in addition we show that the high velocity tails of both the stationary and time dependent particle distribution functions are overpopulated with respect to the maxwellian distribution as conjectured by previous authors and we prove pointwise lower estimates for the solutions

some applications of the method of moments for the homogeneous boltzmann and kac equations using the method of moments we prove that any polynomial moment of the solution of the homogeneous boltzmann equation with hard potentials or hard spheres is bounded provided that a moment of order strictly higher than 2 exists initially we also give partial results of convergence towards the maxwellian equilibrium in the case of soft potentials finally exponential as well as maxwellian estimates are introduced for the kac equation

entropy dissipation and moment production for the boltzmann equation leth f m flog f m dv be the relative entropy off and the maxwellian with the same mass momentum and energy and denote the corresponding entropy dissipation term in the boltzmann equation byd f q f f logf dv an example is presented which shows that d f h f m can be arbitrarily small this example is a sequence of isotropic functions and the estimates are very explicitly given by a simple formula ford which holds for such functions the paper also gives a simplified proof of the so called povzner inequality which is a geometric inequality for the magnitudes of the velocities before and after an elastic collision that inequality is then used to prove that f v v s dt c t wheref is the solution of the spatially homogeneous boltzmann equation herec t is an explicitly given function dependings and the mass energy and entropy of the initial data

classes of matrices for the traveling salesman problem abstract some of the foundational structure for the traveling salesman problem is addressed we investigate square matrices a aij where the aij are from the computable real field and the aii may be assigned values as needed we may transform a into a nonnegative matrix and apply the hungarian algorithm to obtain an equivalent matrix which we call a hungarian form of a we denote by h a the class of all hungarian forms associated with a we determine some of the combinatorial structure of the classes h a which yields new insights about traveling salesman problems for each matrix in the class h a we show how to obtain its triangular block form the transformation from the original matrix a to its triangular block form can be accomplished in polynomial time the original matrix a and its triangular block form have the same set of optimal tours up to a renumbering of the cities

a new look at the optimal assignment problem abstract let e e ij be a matrix with integral elements we study matrices of the form x e e ij where x is an indeterminate defined over the rational field q there is a fascinating interplay between the combinatorial structures of the matrices e and x for example in the case of square matrices the number of optimal assignments in the matrix e is precisely the coefficient of the term of highest degree in the polynomial per x we allow the multiplication of rows and columns of x by arbitrary integral powers of x and we study topics such as the diagonal products of x we apply certain familiar results to this exponential setting and we also introduce some new concepts such as the class of matrices s x generated by a matrix x and the notion of an invariant 1 within such a class

global l p properties for the spatially homogeneous boltzmann equation this paper studies the lp behavior for 1 p of solutions of the nonlinear spatially homogeneous boltzmann equation for a class of collision kernels including inverse kth power forces with k 5 and angular cut off the following topics are treated differentiability in lp together with global boundedness in time for lp moments that exist initially translation continuity in lp uniformly in time and strong convergence to equilibrium