linearly stable orbits in 3 dimensional billiards we construct linearly stable periodic orbits in a class of billiard systems in 3 dimensional domains with boundaries containing semispheres arbitrarily far apart it shows that the results about planar billiard systems in domains with convex boundaries which have nonvanishing lyapunov exponents cannot be easily extended to 3 dimensions

the transversal homoclinic solutions and chaos for stochastic ordinary differential equations abstract we consider the persistence of a transversal homoclinic solution and chaotic motion for ordinary differential equations with a homoclinic solution to a hyperbolic equilibrium under an unbounded random forcing driven by a brownian force by lyapunov schmidt reduction the persistence of transversal homoclinic solution is reduced to find the zeros of some bifurcation functions defined between two finite spaces it is shown that for almost all sample paths of the brownian motion the perturbed system exhibits chaos

difference equations for the co recursive rth associated orthogonal polynomials of the dq laguerre hahn class we use some relations between the rth associated orthogonal polynomials of the dq laguerre hahn class to derive the fourth order q difference equation satisfied by the co recursive rth associated orthogonal polynomials of the dq laguerre hahn class when r 1 and for q semi classical situations this q difference equation factorizes as product of two second order q difference equations finally we study some classical situations and give some examples relative to the co recursive associated discrete q hermite ii orthogonal polynomials

recurrence relations for the connection coefficients of orthogonal polynomials of a discrete variable on the lattice x s q 2 abstract we give explicitly recurrence relations satisfied by the connection coefficients between two families of the classical orthogonal polynomials of a discrete variable on a non uniform lattice x s q2s i e the q analogues of charlier meixner krawtchouk and hahn polynomials in terms of the coefficients and of the pearson equation satisfied by the weight function and the coefficients of the three term recurrence relation and of two structure relations obeyed by these polynomials

recurrence relations for the connection coefficients of orthogonal polynomials of a discrete variable we give explicitly recurrence relations satisfied by the connection coefficients between two families of the classical orthogonal polynomials of a discrete variable i e associated with the names of charlier meixner krawtchouk and hahn also a recurrence relation is given for the coefficients in the formula expressing the nth associated polynomial in terms of the original polynomials

the existence of transverse homoclinic solutions for higher order equations abstract differential equations are considered which contain a small parameter when the parameter is zero the equation is autonomous with a hyperbolic equilibrium and a homoclinic solution no restriction is placed on the dimension of the phase space or the dimension of intersection of the stable and unstable manifolds sufficient conditions on the perturbation terms for producing a transverse homoclinic solution and hence chaos are obtained the technique follows the work of palmer using exponential dichotomies and the method of lyapunov schmidt

how high dimensional stadia look like we give the affirmative answer to the long standing question whether or not the mechanism of defocusing can produce a chaotic behavior in high dimensional hamiltonian systems to do this we prove that billiards in a class of regions in r n n 2 with focusing and flat boundary components have nonvanishing lyapunov exponents

modified clebsch gordan type expansions for products of discrete hypergeometric polynomials abstract starting from the second order difference hypergeometric equation satisfied by the set of discrete orthogonal polynomials p n we find the analytical expressions of the expansion coefficients of any polynomial rm x and of the product rm x qj x in series of the set p n these coefficients are given in terms of the polynomial coefficients of the second order difference equations satisfied by the involved discrete hypergeometric polynomials here qj x denotes an arbitrary discrete hypergeometric polynomial of degree j the particular cases in which r m corresponds to the non orthogonal families x m the rising factorials or pochhammer polynomials x m and the falling factorial or stirling polynomials x m are considered in detail the connection problem between discrete hypergeometric polynomials which here corresponds to the product case with m 0 is also studied and its complete solution for all the classical discrete orthogonal hypergeometric cdoh polynomials is given also the inversion problems of cdoh polynomials associated to the three aforementioned nonorthogonal families are solved

construction of recurrences for the coefficients of expansions in q classical orthogonal polynomials an algorithmic approach is given to construct recurrence relations for the coefficients of the series expansion of a solution of a linear q difference equation in q classical orthogonal polynomials in the hahn s sense

billiards with pesin region of measure one we give a description of a large class of plane billiards with pesin region of measure one open conditions including properly those founded by wojtkowski w1 forc4 focusing boundaries are obtained lyapunov s forms introduced by lewowicz are used

a transversal fundamental theorem for semi dispersing billiards for billiards with a hyperbolic behavior fundamental theorems ensure an abundance of geometrically nicely situated and sufficiently large stable and unstable invariant manifolds a transversal fundamental theorem has recently been suggested by the present authors to proveglobal ergodicity and then as an easy consequence the k property of semidispersing billiards in particular the global ergodicity of systems ofn 3 elastic hard balls conjectured by the celebratedboltzmann sinai ergodic hypothesis in fact the suggested transversal fundamental theorem has been successfully applied by the authors in the casesn 3 and 4 the theorem generalizes the fundamental theorem of chernov and sinai that was really the fundamental tool to obtainlocal ergodicity of semi dispersing billiards our theorem however is stronger even in their case too since its conditions are simpler and weaker moreover a complete set of conditions is formulated under which the fundamental theorem and its consequences like the zig zag theorem are valid for general semi dispersing billiards beyond the utmost interesting case of systems of elastic hard balls as an application we also give conditions for the ergodicity and consequently the k property of dispersing billiards transversality means the following instead of the stable and unstable foliations occurring in the chernov sinai formulation of the stable version of the fundamental theorem we use the stable foliation and an arbitrary nice one transversal to the stable one

semi focusing billiards hyperbolicity in this paper we answer affirmatively the question concerning the existence of hyperbolic billiards in convex domains of 3 we also prove that a related class of semi focusing billiards has mixed dynamics i e their phase space is an union of two invariant sets of positive measure such that the dynamics is integrable on one set and is hyperbolic on the other these billiards are the first rigorous examples of billiards in domains of 3 with divided phase space

chaotic behavior in differential equations driven by a brownian motion abstract in this paper we investigate the chaotic behavior of ordinary differential equations with a homoclinic orbit to a saddle fixed point under an unbounded random forcing driven by a brownian motion we prove that for almost all sample paths of the brownian motion in the classical wiener space the forced equation admits a topological horseshoe of infinitely many branches this result is then applied to the randomly forced duffing equation and the pendulum equation

conditional proof of the boltzmann sinai ergodic hypothesis we consider the system of n 2 elastically colliding hard balls of masses m1 mn and radius r on the flat unit torus mathbb t nu 2 we prove the so called boltzmann sinai ergodic hypothesis i e the full hyperbolicity and ergodicity of such systems for every selection m1 mn r of the external parameters provided that almost every singular orbit is geometrically hyperbolic sufficient i e the so called chernov sinai ansatz is true the present proof does not use the formerly developed rather involved algebraic techniques instead it employs exclusively dynamical methods and tools from geometric analysis

nowhere dispersing 3d billiards with non vanishing lyapunov exponents we describe a class of 3 dimensional regions with focusing components that generate a billiard system with non vanishing lyapunov exponents to do this we answer affirmatively the long standing question whether or not the chaotic motion caused by defocusing can be produced in more than two dimensions

homoclinic tangencies and non normal jacobians effects of noise in nonhyperbolic chaotic systems abstract a general study of the effects of noise interacting with the deterministic dynamics of nonhyperbolic chaotic 2 d maps and 3 flows is presented noise in these systems can have dramatic effects on the invariant measure of the system in regions of homoclinic tangencies perturbations are transported away from the neighborhood of the attractor leading to deformations which can be one to two orders of magnitude larger than the noise level a qualitative understanding is attained by a study of the universal properties of the unstable and stable foliation of the phase space in the vicinity of homoclinic tangencies through the investigation of nontrivial structures of the tangent space at homoclinic tangencies related to the non normal jacobians we obtain a quantitative description of these noise induced deformations of the attractor local expansion rates which are much larger than the maximal lyapunov exponent can be used as a measure of the system s structural instability under perturbations we exemplify our general results on the henon ikeda and duffing system a new and effective algorithm to calculate the homoclinic tangencies in the entire phase space based on the results of this paper is presented

some connection and linearization problems for polynomials in and beyond the askey scheme the connection problem is considered in a hypergeometric function framework for i the two most general families of polynomials belonging to the askey scheme wilson and racah and ii some generalized laguerre and jacobi polynomials falling outside that scheme sister celine cohen and prabhakar jain which are relevant to the study of quantum mechanical systems and include as particular cases the generalizations of the classical families with sobolev type orthogonality in addition using the same method three new linearization like formulae for the gegenbauer polynomials are also derived a linearization formula that generalizes the m n case of dougall s formula the analogue of the m n case of nielsen s inverse linearization formula for hermite polynomials and a connection formula for the squares closed analytical formulae for the corresponding connection and linearization coe cients are given in terms of hypergeometric functions of unit argument which at times can be further simpli ed and expressed as single hypergeometric terms c 2001 published by elsevier science b v

on the connection and linearization problem for discrete hypergeometric q polynomials abstract in the present paper starting from the second order difference hypergeometric equation on the non uniform lattice x s satisfied by the set of discrete hypergeometric orthogonal q polynomials p n we find analytical expressions of the expansion coefficients of any q polynomial r m x s on x s and of the product r m x s q j x s in series of the set p n these coefficients are given in terms of the polynomial coefficients of the second order difference equations satisfied by the involved discrete hypergeometric q polynomials

q analogue of the krawtchouk and meixner orthogonal polynomials abstract the comparative analysis of krawtchouk polynomials on a uniform grid with wigner d functions for the su 2 group is done as a result the partnership between corresponding properties of the polynomials and d functions is established giving the group theoretical interpretation of the properties of the krawtchouk polynomials in order to extend such an analysis to the quantum groups suq 2 and suq 1 1 q analogues of krawtchouk and meixner polynomials of a discrete variable are studied in detail as solutions of the finite difference equation of hypergeometric type on the nonuniform grid x s q2s however on this grid there are two kinds of krawtchouk and meixner polynomials characterized by different weight functions in the orthogonality relation as a first step to such analysis the simpler polynomials of the first kind are considered the total set of characteristics of these polynomials orthogonality condition normalization factor recurrent relation the explicit analytic expression the rodrigues formula the difference derivative formula various particular cases and values is calculated

homoclinic solutions for autonomous ordinary differential equations with nonautonomous perturbations nonautomonous ordinary differential equations depending on two parameters 1 and 2 are considered in rn it is assumed that when both parameters are zero the differential equation is autonomous with a hyperbolic equilibrium and a homoclinic solution no restriction is placed on the dimension of the phase space rn or on the dimension of intersection of the stable and unstable manifolds by means of the method of lyapunov schmidt a bifurcation function h is constructed between two finite dimensional spaces where the zeros of h correspond to homoclinic solutions at nonzero parameter values the independent variables of h consist of scalars 1 2 and a vector where is a phase angle and corresponds to directions other than along the original homoclinic solution tangent to both the stable and unstable manifolds when is fixed the equation h 0 yields in general several bifurcation curves through the origin in the 1 2 plane along which there exists a homoclinic solution when is varied these become a number of wedge shaped regions the theory is applied to two examples one in r6 where the invariant manifolds meet in dimension three and a second in r4 where these manifolds agree

recurrence relations for connection coefficients between q orthogonal polynomials of discrete variables in the non uniform lattice x s q2s we obtain the structure relations for q orthogonal polynomials in the exponential lattice and from these we construct the recurrence relation for the connection coefficients between two families of polynomials belonging to the classical class of discrete q orthogonal polynomials an explicit example is also given

using integrability to produce chaos billiards with positive entropy a new open class of convex 2 dimensional planar billiards with positive lyapunov exponent almost everywhere is constructed we introduce the notion of a focusing arc and show that such arcs can be used to build billiard systems with positive lyapunov exponents we prove that under smallc6 perturbations focusing arcs remain focusing and thereby show that perturbations of the bunimovich stadium billiard have positive lyapunov exponents

a theorem on ergodicity of two dimensional hyperbolic billiards ergodicity of two dimensional billiards which satisfy some general conditions is proved this theorem is applied to one concrete class of billiards that contains in particular billiards in the stadium

potentials on the two torus for which the hamiltonian flow is ergodic we consider the motion of a test particle in a compound central potential field on a two dimensional torus we discuss three different classes of potentials attracting repelling and mixed that lead to hamiltonian systems which have positive lyapunov exponent almost everywhere and are ergodic included among the mixed potentials are smooth potentials without singularities

many dimensional nowhere dispersing billiards with chaotic behavior abstract we consider some examples of domains in the d dimensional euclidean space d 3 that generate billiards with chaotic behavior time correlations in these billiards tend to zero in the limit t the rate of decay of the correlations is a power law for billiards in discrete time and could be a power law or quasiexponential for billiards with continuous time

efficient algorithms for construction of recurrence relations for the expansion and connection coefficients in series of al salam carlitz i polynomials two formulae expressing explicitly the derivatives and moments of al salam carlitz i polynomials of any degree and for any order in terms of al salam carlitz i themselves are proved two other formulae for the expansion coefficients of general order derivatives d p f x and for the moments xd p f x of an arbitrary function f x in terms of its original expansion coefficients are also obtained application of these formulae for solving q difference equations with varying coefficients by reducing them to recurrence relations in the expansion coefficients of the solution is explained an algebraic symbolic approach using mathematica in order to build and solve recursively for the connection coefficients between al salam carlitz i polynomials and any system of basic hypergeometric orthogonal polynomials belonging to the q hahn class is described

linearly independent homoclinic bifurcations parameterized by a small function abstract in this paper we discuss a small nonautonomous perturbation of an autonomous system on r n which has a homoclinic solution regarding the small perturbation as a parameter in an appropriate space of functions we discuss various situations of co existence of homoclinic orbits those conditions of various co existence actually define bifurcation manifolds in the space of functions for linearly independent homoclinic bifurcations