small groups of finite morley rank with involutions by analogy with thompson s classification of nonsolvable finite n groups we classify groups of finite morley rank with solvable local subgroups of even and mixed type we also consider several aspects reminiscent of small groups of finite morley rank of odd type

an identification theorem for groups of finite morley rank and even type the paper contains a construction of a definable bn pair in a simple group of finite morley rank and even type with a sufficiently good system of 2 local parabolic subgroups this provides the final identification theorem for simple groups of finite morley rank and even type

mathematics of granular materials this is a short and somewhat informal review on the most mathematical parts of the kinetic theory of granular media intended for physicists and for mathematicians outside the field

solvable groups of finite morley rank abstract we prove the following theorem the commutator subgroup of a solvable connected group of finite morley rank is nilpotent

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

rate of convergence to equilibrium for the spatially homogeneous boltzmann equation with hard potentials for the spatially homogeneous boltzmann equation with hard potentials and grad s cutoff e g hard spheres we give quantitative estimates of exponential convergence to equilibrium and we show that the rate of exponential decay is governed by the spectral gap for the linearized equation on which we provide a lower bound our approach is based on establishing spectral gap like estimates valid near the equilibrium and then connecting the latter to the quantitative nonlinear theory this leads us to an explicit study of the linearized boltzmann collision operator in functional spaces larger than the usual linearization setting

lagrangian numerical approximations to one dimensional convolution diffusion equations this work focuses on the numerical analysis of one dimensional nonlinear diffusion equations involving a convolution product first homogeneous friction equations are considered algorithms follow recent ideas on mass transportation methods and lead to simple schemes which can be proved to be stable to decrease entropy and to converge toward the unique solution of the continuous problem in particular for the first time homogeneous cooling states are displayed numerically further we present results on the more delicate fourth order thin film equation for which a nonnegativity preserving scheme is derived the dead core phenomenon is presented for the hele shaw cell

groups of finite morley rank with a generically sharply multiply transitive action abstract we prove that if g is a group of finite morley rank that acts definably and generically sharply n transitively on a connected abelian group v of morley rank n with no involutions then there is an algebraically closed field f of characteristic 2 such that v has the structure of a vector space of dimension n over f and g acts on v as the group gl n f in its natural action on f n

cercignani s conjecture is sometimes true and always almost true we establish several new functional inequalities comparing boltzmann s entropy production functional with the relative h functional first we prove a longstanding conjecture by cercignani under the nonphysical assumption that the boltzmann collision kernel is superquadratic at infinity the proof rests on the method introduced in 39 combined with a novel use of the blachman stam inequality if the superquadraticity assumption is not satisfied then it is known that cercignani s conjecture is not true however we establish a slightly weakened version of it for all physically relevant collision kernels thus extending previous results from 39 finally we consider the entropy entropy production version of kac s spectral gap problem and obtain estimates about the dependence of the constants with respect to the dimension the first two results are sharp in some sense and the third one is likely to be too they contain all previously known entropy estimates as particular cases this gives a first coherent picture of the study of entropy production according to a program started by carlen and carvalho 12 ten years ago these entropy inequalities are one step in our study of the trend to equilibrium for the boltzmann equation in both its spatially homogeneous and spatially inhomogeneous versions

probabilistic view of explosion in an inelastic kac model let mu cdot t t ge 0 be the family of probability measures corresponding to the solution of the inelastic kac model introduced in pulvirenti and toscani j stat phys 114 1453 1480 2004 it has been proved by gabetta and regazzini j stat phys 147 1007 1019 2012 that the solution converges weakly to equilibrium if and only if a suitable symmetrized form of the initial data belongs to the standard domain of attraction of a specific stable law in the present paper it is shown that for initial data which are heavier tailed than the aforementioned ones the limiting distribution is improper in the sense that it has probability 1 2 adherent to infty and probability 1 2 adherent to infty it is explained in which sense this phenomenon is amenable to a sort of explosion and the main result consists in an explicit expression of the rate of such an explosion the presentation of these statements is preceded by a discussion about the necessity of the assumption under which their validity is proved this gives the chance to make an adjustment to a portion of a proof contained in the above mentioned paper by gabetta and regazzini

groupes de rang de morley fini de type pair avec un sous groupe faiblement inclus resume we show that a minimal counterexample to the cherlin zil ber conjecture of even type and with a weakly embedded subgroup is isomorphic to psl 2 k where k is an algebraically closed field of characteristic 2

development of higher order fdtd schemes with controllable dispersion error a new methodology that facilitates the control of the inherent dispersion error in the case of higher order finite difference time domain fdtd schemes is presented in this paper the basic idea is to define suitable algebraic expressions that reflect numerical inaccuracies reliably then finite difference operators are determined via the minimization of the error estimators at selected frequencies in order to apply this procedure an error expansion in terms of cylindrical harmonic functions is performed which also enables accuracy enhancement for all propagation angles the design process produces a set of two dimensional 2 d fdtd algorithms with optimized frequency response contrary to conventional methodologies the proposed techniques adjust their reliability range according to the requirements of the examined problem and can be therefore more efficient in computationally demanding simulations

contractive metrics for a boltzmann equation for granular gases diffusive equilibria we quantify the long time behavior of a system of partially inelastic particles in a stochastic thermostat by means of the contractivity of a suitable metric in the set of probability measures existence uniqueness boundedness of moments and regularity of a steady state are derived from this basic property the solutions of the kinetic model are proved to converge exponentially as t to this diffusive equilibrium in this distance metrizing the weak convergence of measures then we prove a uniform bound in time on sobolev norms of the solution provided the initial datum has a finite norm in the corresponding sobolev space these results are then combined using interpolation inequalities to obtain exponential convergence to the diffusive equilibrium in the strong l1 norm as well as various sobolev norms

on the generalization of the boltzmann h theorem for a spatially homogeneous maxwell gas sufficient conditions are given for a convex and isotropic functional that are monotonically decreasing in time along the solution of the spatially homogeneous boltzmann equation for maxwellian molecules the conditions are valid for plane velocities and for axially symmetric solutions in any dimension explicit examples are given

an fdtd scheme on a face centered cubic fcc grid for the solution of the wave equation a method is proposed to improve the numerical dispersion characteristics for simulations of the scalar wave equation in 3d using the fdtd method the improvements are realized by choosing a face centered cubic fcc grid instead of the typical cartesian yee grid which exhibits non physical distortions of the wavefront due to the fd stencil fcc grids are the logical extension of hexagonal grids in 2d and have been shown previously to provide optimal sampling of space based on close packing of spheres highest density the difference equations are developed for the wave equation on this alternative grid and the dispersion relationship and stability for grids of equal and non equal aspect ratios are derived a comparison is made between fcc and cartesian formulations based upon having an equal volume density of gridpoints in each method i e the computational storage requirements of each method would be the same for the same simulated space the comparison shows that the fcc grid exhibits a much more isotropic dispersion relation than the cartesian grid of equivalent density furthermore for an equivalent density the fcc method has a more relaxed stability criterion by a factor of approximately 1 35 resulting in a further reduction in computational resources

homogeneous cooling states are not always good approximations to granular flows a widespread belief in the study of granular flow is the existence of homogeneous cooling states i e self similar solutions which would attract all solutions faster than the equilibrium solution does in most cases the existence of these self similar solutions is an open problem here we consider a one dimensional model which has been used for some years and for which simple self similar solutions do exist however we prove that the approximation is quite poor our proof makes use of the powerful and simple tools of mass transportation and exploits the structure of the evolution equation seen as a nonlinear transport equation

fdtd on face centered cubic fcc grids for maxwell s equations the standard fdtd method for maxwell s equations is reformulated on face centered cubic fcc grids as an alternative to the standard cartesian yee grid dispersion and stability for the formulation are also presented and a comparison is made with the standard fdtd formulation using the metric of equal volume density of gridpoints although the new formulation has four times as many lattice points per unit voxel in comparison with the standard formulation the new formulation shows much better grid isotropy furthermore the new formulation has a stability criterion that is approximately 1 4 times more relaxed than that of the standard formulation

the classical involution theorem for groups of finite morley rank abstract this paper gives a partial answer to the cherlin zil ber conjecture which states that every infinite simple group of finite morley rank is isomorphic to an algebraic group over an algebraically closed field the classification of the generic case of tame groups of odd type follows from the main result of this work which is an analogue of aschbacher s classical involution theorem for finite simple groups

stability convergence to self similarity and elastic limit for the boltzmann equation for inelastic hard spheres we consider the spatially homogeneous boltzmann equation for inelastic hard spheres in the framework of so called constant normal restitution coefficients alpha in 0 1 in the physical regime of a small inelasticity that is alpha in alpha 1 for some constructive alpha in 0 1 we prove uniqueness of the self similar profile for given values of the restitution coefficient alpha in alpha 1 the mass and the momentum therefore we deduce the uniqueness of the self similar solution up to a time translation

signalizers and balance in groups of finite morley rank abstract there is a longstanding conjecture due to gregory cherlin and boris zilber that all simple groups of finite morley rank are simple algebraic groups the most successful approach to this conjecture has been borovik s program analyzing a minimal counterexample or simple k group we show that a simple k group of finite morley rank and odd type is either algebraic of else has prufer rank at most two this result signifies a switch from the general methods used to handle large groups to the specilized methods which must be used to identify psl 2 psl 3 psp 4 and g 2

on solvable groups of finite morley rank we investigate solvable groups of finite morley rank we find conditions on g for g to split in g in particular if g is abelian and z g 1 we prove that g g t for some t and the ring z t anng is intepretable in g we exploit the methods used in proving these results to find more information about solvable groups

tame minimal simple groups of finite morley rank abstract we consider tame minimal simple groups of finite morley rank and of odd type we show that the prufer 2 rank of such a group is bounded by 2 we also find all potential nonalgebraic configurations there are essentially four of them and we delineate them with some precision

structure des groupes lineaires definissables dans un corps de rang de morley fini we generalize several properties of linear algebraic groups to the case of the linear groups which are defined in a field of finite morley rank we show three structure theorems and the conjugacy of the borels of the carter subgroups of the maximal tori in conclusion we generalize the results to fields of morley rank n without subgroups of finite rank in contrary to the case of finite morley rank there are known examples of such fields with definable non algebraic subgroups of gln k

the spectral order of accuracy a new unified tool in the design methodology of excitation adaptive wave equation fdtd schemes we define a new concept termed the spectral order of accuracy sooa which is the spectral domain analogue of the familiar order of accuracy ooa the sooa is pivotal in a refined version of a recently introduced methodology for formulating excitation adaptive wave equation fdtd we fdtd schemes described below this concept is the basis for a unified classification for both existing and new schemes both one and two dimensional cases are presented for boundless source free homogeneous isotropic and lossless media the 1 d and 2 d cases are developed in detail for the 3 2m 1 temporal spatial and 3 3 2 d stencils respectively stability analysis is built into the methodology in terms of either analytical conditions or stability maps defined herein the methodology is seen as a generalization of many existing schemes that also provides a unified tool for a systematical design of we fdtd schemes subject to specific requirements in terms of the spectral content of the excitation the computational efficiency for all schemes remains the same for a given stencil since the core of the fdtd code is unchanged between schemes the difference being only in the values of scheme coefficients

cosets genericity and the weyl group abstract in a connected group of finite morley rank in which generically elements belong to connected nilpotent subgroups proper normalizing cosets of definable subgroups are not generous we explain why this is true and what consequences this has for the abstract theory of weyl groups in groups of finite morley rank

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

probabilistic study of the speed of approach to equilibrium for an inelastic kac model this paper deals with a one dimensional model for granular materials which boils down to an inelastic version of the kac kinetic equation with inelasticity parameter p 0 in particular the paper provides bounds for certain distances such as specific weighted distances and the kolmogorov distance between the solution of that equation and the limit it is assumed that the even part of the initial datum which determines the asymptotic properties of the solution belongs to the domain of normal attraction of a symmetric stable distribution with characteristic exponent 2 1 p with such initial data it turns out that the limit exists and is just the aforementioned stable distribution a necessary condition for the relaxation to equilibrium is also proved some bounds are obtained without introducing any extra condition sharper bounds of an exponential type are exhibited in the presence of additional assumptions concerning either the behaviour close to the origin of the initial characteristic function or the behaviour at infinity of the initial probability distribution function

a new approach to quantitative propagation of chaos for drift diffusion and jump processes this paper is devoted the study of the mean field limit for many particle systems undergoing jump drift or diffusion processes as well as combinations of them the main results are quantitative estimates on the decay of fluctuations around the deterministic limit and of correlations between particles as the number of particles goes to infinity to this end we introduce a general functional framework which reduces this question to the one of proving a purely functional estimate on some abstract generator operators consistency estimate together with fine stability estimates on the flow of the limiting nonlinear equation stability estimates then we apply this method to a boltzmann collision jump process for maxwell molecules to a mckean vlasov drift diffusion process and to an inelastic boltzmann collision jump process with stochastic thermal bath to our knowledge our approach yields the first such quantitative results for a combination of jump and diffusion processes

a low dispersion 3 d second order in time fourth order in space fdtd scheme m3d sub 24 a second order in time fourth order in space modified finite difference time domain fdtd scheme for three dimensional electromagnetic problems m3d sub 24 is presented the algorithm enables the numerical phase error to be minimized so that it leads to high accuracy with low resolution grids the advantage of this method is demonstrated by considering the long distance propagation of the wave radiated from a time harmonic elementary dipole using a low resolution grid and comparing the results with other fdtd schemes

self similar blowup solutions to an aggregation equation in r n we present numerical simulations of radially symmetric finite time blowup for the aggregation equation ut u k u where the kernel k x x the dynamics of the blowup exhibits self similar behavior in which zero mass concentrates at the core at the blowup time computations are performed in r n for n between 2 and 10 using a method based on characteristics in all cases studied the self similarity exhibits second kind anomalous scaling

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

groups of mixed type the cherlin zil ber conjecture states that an infinite simple group of finite morley rank is an algebraic group over an algebraically closed field the proofs of the following three conjectures would provide a proof of the cherlin zil ber conjecture conjecture 1 there exist no nonsolvable connected groups of finite morley rank all of whose proper definable connected subgroups are nilpotent such groups are called bad groups 2 conjecture 2 there exists no structure k q a of finite morley rank where k is an algebraically closed field and a is an infinite proper definable subgroup of the multiplicative group of k such structures are called bad fields

a comprehensive new methodology for formulating fdtd schemes with controlled order of accuracy and dispersion numerical dispersion errors in the wave equation finite difference time domain we fdtd method have been treated by higher order schemes coefficient modification schemes dispersion relation preserving and non standard schemes in this work a unified methodology is formulated for the systematic generation of we fdtd schemes tailored to the spectrum of the excitation the methodology enables the scheme designer to gradually trade order of accuracy ooa for lower dispersion errors in a controlled manner at the cost of sacrificing low frequency behavior that is not deemed critical for this type of excitation the methodology is shown to encompass both existing and new schemes stability analysis is carried out concurrently with the generation of each scheme using a stencil size of 3 and 5 temporal and spatial samples respectively long term errors of a scheme designed for a specific pulse are compared with the standard 4 4 scheme that has the same computational complexity via simulation of a modulated pulse that propagates over a million time steps

contractions in the 2 wasserstein length space and thermalization of granular media an algebraic decay rate is derived which bounds the time required for velocities to equilibrate in a spatially homogeneous flow through model representing the continuum limit of a gas of particles interacting through slightly inelastic collisions this rate is obtained by reformulating the dynamical problem as the gradient flow of a convex energy on an infinite dimensional manifold an abstract theory is developed for gradient flows in length spaces which shows how degenerate convexity or even non convexity if uniformly controlled will quantify contractivity limit expansivity of the flow

on groups of finite morley rank with weakly embedded subgroups the present paper is a contribution to the classification of tame stable groups of finite morley rank a group of finite morley rank is called tame if it involves no bad groups and no bad fields in the sense of 1 see also 1 it is conjectured that an infinite simple tame group of finite morley rank is isomorphic to an algebraic group over an algebraically closed field it is known that a minimal counterexample is either of even type meaning that the sylow 2 subgroups are of bounded exponent or of odd type meaning that the sylow 2 subgroups are finite extensions of divisible abelian groups we deal here with tame simple groups of even type having a weakly embedded subgroup a notion related to the notion of strongly embedded subgroup in the finite case but having no exact analog in that case in 10 the classication of simple tame groups with a strongly embedded subgroup was derived in an inductive framework as follows

groupes de type mixte resume we show that a minimal counterexample to the cherlin zil ber conjecture tame or not cannot be of mixed type

fourier analysis of numerical algorithms for the maxwell equations the fourier method is used to analyze the dispersive dissipative and isotropy errors of various spatial and time discretizations applied to the maxwell equations on multi dimensional grids both cartesian grids and non cartesian grids based on hexagons and tetradecahedra are studied and compared the numerical errors are quantitatively determined in terms of phase speed wavenumber propagation direction gridspacings and cfl number the study shows that centered schemes are more efficient and accurate than upwind schemes and the non cartesian grids yield superior isotropy than the cartesian ones for the centered schemes the staggered grids produce less errors than the unstaggered ones a new unstaggered algorithm which has all the best properties is introduced using an optimization technique to determine the nodal weights the new algorithm provides the highest accuracy among all the schemes discussed the study also demonstrates that a proper choice of time discretization can reduce the overall numerical errors due to the spatial discretization

long time asymptotics of kinetic models of granular flows we analyze the long time asymptotics of certain one dimensional kinetic models of granular flows which have been recently introduced in 22 in connection with the quasi elastic limit of a model boltzmann equation with dissipative collisions and variable coefficient of restitution these nonlinear equations classified as nonlinear friction equations split naturally into two classes depending on whether or not the temperature of their similarity solutions homogeneous cooling states reduce to zero in finite time for both classes we show uniqueness of the solution by proving decay to zero in the wasserstein metric of any two solutions with the same mass and mean velocity furthermore if the temperature of the similarity solution decays to zero in finite time we prove by computing explicitly upper bounds for the lifetime of the solution in terms of the length of the support that the temperature of any other solution with initially bounded support must also decay to zero in finite time

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

minimal connected simple groups of finite morley rank with strongly embedded subgroups let g be a simple k group of finite morley rank of odd type which is not algebraic then g has prufer 2 rank at most two this improves on an earlier result of the second and third authors for the tame case in the critical case g is minimal connected simple of odd type with a proper definable strongly embedded subgroup the bulk of the analysis relies on the first author s theory of 0 unipotence and the related 0 sylow subgroup theory as well as the so called bender method adapted to this context

a signalizer functor theorem for groups of finite morley rank there is a longstanding conjecture due to gregory cherlin and boris zilber that all simple groups of finite morley rank are simple algebraic groups towards this end the development of the theory of groups of finite morley rank has achieved a good theory of sylow 2 subgroups it is now common practice to divide the cherlin zilber conjecture into different cases depending on the nature of the connected component of the sylow 2 subgroup known as the sylow 2 subgroup we shall be working with groups whose sylow 2 subgroup is divisible or odd type groups to date the main theorem in the area of odd type groups is borovik s trichotomy theorem the trichotomy here is a case division of the minimal counterexamples within odd type more technically borovik s result represents a major success at transferring signalizer functors and their applications from finite group theory to the finite morley rank setting the major difference between the two settings is the absence of a solvable signalizer functor theorem this forced borovik to work only with nilpotent signalizer functors and the trichotomy theorem ends up depending on the assumption of tameness to assure that the necessary signalizer functors are nilpotent the present paper shows that one may obtain a connected nilpotent signalizer functor from any sufficiently non trivial solvable signalizer functor this result plugs seamlessly into borovik s work to eliminate the assumption of tameness from his trichotomy theorem in the meantime a new approach to the trichotomy theorem has been developed by borovik 7 based on the generic identification theorem of berkman and borovik 5 borovik uses his original signalizer functor arguments and incorporates the result of the present paper the paper is organized as follows the first section will develop a limited characteristic zero notion of unipotence to complement the usual p unipotence theory the section on centralizers and generation which follows will establish some background needed in the rest of the paper in section 4 we prove our main result on signalizer functors and in section 5 we discuss some applications with borovik s kind permission we include a proof of the nilpotent signalizer functor theorem as an appendix the results of section 3 are based in part on a section of an unpublished version of 3

decay rates in probability metrics towards homogeneous cooling states for the inelastic maxwell model we quantify the long time behavior of solutions to the nonlinear boltzmann equation for spatially uniform freely cooling inelastic maxwell molecules by means of the contraction property of a suitable metric in the set of probability measures existence uniqueness and precise estimates of overpopulated high energy tails of the self similar profile proved in ref 9 are revisited and derived from this new liapunov functional for general initial conditions the solutions of the boltzmann equation are then proved to converge with computable rate as t to the self similar solution in this distance which metrizes the weak convergence of measures moreover we can relate this fourier distance to the euclidean wasserstein distance or tanaka functional proving also its exponential convergence towards the homogeneous cooling states the findings are relevant in the understanding of the conjecture formulated by ernst and brito in refs 15 16 and complement and improve recent studies on the same problem of bobylev and cercignani 9 and bobylev cercignani and one of the authors 11

the lobatto cell robust explicit higher order fdtd that handles inhomogeneous media to improve numerical dispersion and second order convergence behavior of the traditional yee finite difference time domain fdtd algorithm various higher order finite difference implementations have been proposed many have difficulty handling interfaces between distinct dielectric media or metallic sheets because the high order taylor expansions on which these codes are based are invalidated by field or derivative discontinuities one sided extensions extrapolations are a common approach to remedy interface discontinuities here another approach based upon whitney form mass lumped finite elements is pursued to develop the ldquolobatto cell rdquo a high order replacement for the yee cell with appropriate continuity behavior at contrasting media interfaces the lobatto cell retains many robust features of the yee method discrete conservation laws a guarantee of conditional stability and straightforward explicit updating but provides higher order interpolation and dispersion error convergence a notable drawback of the method is larger modeling granularity staircasing which is managed here by combining the lobatto cell method with preexisting robust subgridding techniques

propagation of smoothness and the rate of exponential convergence to equilibrium for a spatially homogeneous maxwellian gas we prove an inequality for the gain term in the boltzmann equation for maxwellian molecules that implies a uniform bound on sobolev norms of the solution provided the initial data has a finite norm in the corresponding sobolev space we then prove a sharp bound on the rate of exponential convergence to equilibrium in a weak norm these results are then combined using interpolation inequalities to obtain the optimal rate of exponential convergence in the strong l 1 norm as well as various sobolev norms these results are the first showing that the spectral gap in the linearized collision operator actually does govern the rate of approach to equilibrium for the full non linear boltzmann equation even for initial data that is far from equilibrium

non solvable groups of morley rank 3 abstract we prove the following theorem let g be a connected centerless non solvable group of morley rank 3 whose proper definable subgroups have rank 1 then g has no involutions and for all x g cg x is connected our method permits us to deduce that son r is not stable for n 3

tanaka theorem for inelastic maxwell models we show that the euclidean wasserstein distance is contractive for inelastic homogeneous boltzmann kinetic equations in the maxwellian approximation and its associated kac like caricature this property is as a generalization of the tanaka theorem to inelastic interactions even in the elastic classical boltzmann equation we give a simpler proof of the tanaka theorem than the ones in 29 31 consequences are drawn on the asymptotic behavior of solutions in terms only of the euclidean wasserstein distance

regularity theory for the spatially homogeneous boltzmann equation with cut off we develop the regularity theory of the spatially homogeneous boltzmann equation with cut off and hard potentials for instance hard spheres by i revisiting the lp theory to obtain constructive bounds ii establishing propagation of smoothness and singularities iii obtaining estimates on the decay of the singularities of the initial datum our proofs are based on a detailed study of the regularity of the gain operator an application to the long time behavior is presented

the construction of low dispersive fdtd on hexagon based on the sampling theorem with arbitrary geometry the spatial discretization can be carried out on hexagonal lattices grids instead of rectangular lattices in the two dimensional case with the concept of the directional derivative in aid a new finite difference time domain fdtd method based on hexagonal lattices h fdtd is introduced the coefficients of the h fdtd method are chosen with aim to obtain the low numerical dispersion final numerical experiment shows that the numerical anisotropy of the h fdtd method is lower than that of the conventional yee fdtd method based on rectangular lattices r fdtd in addition some numerical examples are provided to verify its good performance on the low numerical anisotropy

metrics for probability distributions and the trend to equilibrium for solutions of the boltzmann equation this paper deals with the trend to equilibrium of solutions to the spacehomogeneous boltzmann equation for maxwellian molecules with angular cutoff as well as with infinite range forces the solutions are considered as densities of probability distributions the tanaka functional is a metric for the space of probability distributions which has previously been used in connection with the boltzmann equation our main result is that if the initial distribution possesses moments of order 2 e then the convergence to equilibrium in his metric is exponential in time in the proof we study the relation between several metrics for spaces of probability distributions and relate this to the boltzmann equation by proving that the fourier transformed solutions are at least as regular as the fourier transform of the initial data this is also used to prove that even if the initial data only possess a second moment then v r f v t v 2 dv 0 asr and this convergence is uniform in time

on the self similar asymptotics for generalized nonlinear kinetic maxwell models maxwell models for nonlinear kinetic equations have many applications in physics dynamics of granular gases economics etc in the present manuscript we consider such models from a very general point of view including those with arbitrary polynomial non linearities and in any dimension space it is shown that the whole class of generalized maxwell models satisfies properties one of which can be interpreted as an operator generalization of usual lipschitz conditions this property allows to describe in detail a behavior of solutions to the corresponding initial value problem in particular we prove in the most general case an existence of self similar solutions and study the convergence in the sense of probability measures of dynamically scaled solutions to the cauchy problem to those self similar solutions as time goes to infinity a new application of multi linear models to economics and social dynamics is discussed

exponential convergence to equilibrium for the homogeneous boltzmann equation for hard potentials without cut off this paper deals with the long time behavior of solutions to the spatially homogeneous boltzmann equation the interactions considered are the so called non cut off with a moderate angular singularity and non mollified hard potentials we prove an exponential in time convergence towards the equilibrium improving results of villani commun math phys 234 3 455 490 2003 where a polynomial decay to equilibrium is proven the basis of the proof is the study of the linearized equation for which we prove a new spectral gap estimate in a l 1 space with a polynomial weight by taking advantage of the theory of enlargement of the functional space for the semigroup decay developed by gualdani et al http hal archives ouvertes fr ccsd 00495786 2013 we then get our final result by combining this new spectral gap estimate with bilinear estimates on the collisional operator that we establish

groups in supersimple and pseudofinite theories we consider groups g interpretable in a supersimple nite rank theory t such that t eq eliminates 9 1 it is shown that g has a denable soluble radical if g has rank 2 then if g is pseudonite it is soluble by nite and partial results are obtained under weaker hypotheses such as functional unimodularity of the theory a classication is obtained when t is pseudonite and g has a denable and denably

asymptotic properties of the inelastic kac model we discuss the asymptotic behavior of certain models of dissipative systems obtained from a suitable modification of kac caricature of a maxwellian gas it is shown that global equilibria different from concentration are possible if the energy is not finite these equilibria are distributed like stable laws and attract initial densities which belong to the normal domain of attraction if the initial density is assumed of finite energy with higher moments bounded it is shown that the solution converges for large time to a profile with power law tails these tails are heavily dependent of the collision rule

finite difference time domain dispersion reduction schemes the finite difference time domain fdtd although recognized as a flexible robust and simple to implement method for solving complex electromagnetic problems is subject to numerical dispersion errors in addition to the traditional ways for reducing dispersion i e increasing sampling rate and using higher order degrees of accuracy a number of schemes have been proposed recently in this work an unified methodology for deriving new difference schemes is presented it is based on certain modifications of the characteristic equation that accompanies any given discretized version of the wave equation the method is duly compared with existing schemes and verified numerically