linearization and krein like functionals of hypergeometric orthogonal polynomials the krein like r functionals of the hypergeometric orthogonal polynomials pn x with the kernel of the form xs x pm1 x pmr x being x the weight function on the interval r are determined by means of the srivastava linearization method the particular 2 functionals which are particularly relevant in quantum physics are explicitly given in terms of the degrees and the characteristic parameters of the polynomials they include the well known power moments and the novel krein like moments moreover various related types of exponential and logarithmic functionals are also investigated

procedure to reduce the effects of modal truncation in eigensolutionreanalysis the influence of structural perturbations on the eigensolutions of a mechanical system is calculated through the use of a discrete model composed of the first normal modes of the base system it consists of modifying the structural perturbations before the application of the classical sensitivity relationships

clebsch gordan type linearisation relations for the products of laguerre polynomials and hydrogen like functions two series of clebsch gordan type linearisation relations are derived for the most general product of the laguerre polynomials ln1alpha 1 u1x ln2alpha 2 u2x which differ in orders n weights alpha and scaling multipliers u the general form and particular cases of coefficients in the expansion of the polynomial xkln1alpha 1 u1x lnnalpha n unx in terms of the laguerre polynomials are established the applications to hydrogen like functions and morse oscillators are indicated connection with an earlier carlitz expansion the technical links with the hyperspherical harmonics formalism and different approaches to the important koornwinder s positivity theorems are discussed briefly

direct spreading measures of laguerre polynomials the direct spreading measures of the laguerre polynomials l n a x which quantify the distribution of its rakhmanov probability density r n a x 1d n 2x ae x l n a x 2 along the positive real line in various complementary and qualitatively different ways are investigated these measures include the familiar root mean square or standard deviation and the information theoretic lengths of fisher renyi and shannon types the fisher length is explicitly given the renyi length of order q such that email protected n is also found in terms of n a by means of two error free computing approaches one makes use of the lauricella function f a 2 q 1 1q 1q 1 which is based on the srivastava niukkanen linearization relation of laguerre polynomials and another one utilizes the multivariate bell polynomials of combinatorics the shannon length cannot be exactly calculated because of its logarithmic functional form but its asymptotics is provided and sharp bounds are obtained by the use of an information theoretic optimization procedure finally all these spreading measures are mutually compared and computationally analyzed in particular it is found that the apparent quasilinear relation between the shannon length and the standard deviation becomes rigorously linear only asymptotically i e for email protected 1

renyi entropies lq norms and linearization of powers of hypergeometric orthogonal polynomials by means of multivariate special functions the quantification of the spreading of the orthogonal polynomials p n x can be investigated by means of the renyi entropies r q r q being a positive integer number of the associated rakhmanov probability densities r x email protected x p n 2 x where w x is the corresponding weight function the renyi entropies are closely related to the l q norms of the polynomials in this manuscript the l q norms and the associated renyi entropies of the real hypergeometric orthogonal polynomials i e hermite laguerre and jacobi polynomials and the generalized hermite polynomials are expressed in an explicit way in terms of some generalized multivariate special functions of lauricella and srivastava daoust types which are evaluated at some specific values of 2q variables these functions depend on 4q 1 and 6q 2 parameters respectively which are determined by the order q the degree n of the polynomial and the parameters of the orthogonality weight function w x the key idea is based on some extended linearization formulas for these polynomials these results open the way to determine the renyi information entropies of the quantum systems whose wavefunctions are controlled by hypergeometric orthogonal polynomials

upper semicontinuity of attractors for random lattice systems perturbed by small white noises abstract in this paper we first present some conditions for the upper semicontinuity of perturbed random attractors to a limiting random attractor then we apply this result to establish the upper semicontinuity of random attractors for the first order stochastic lattice differential equations with random coupled coefficients and multiplicative additive white noises in the weighted space of infinite sequences as the coefficient of the white noise term tends to zero

existence and upper semicontinuity of bi spatial pullback attractors for smoothing cocycles abstract in this work we establish several criteria for the existence as well as the upper semi continuity of bi spatial attractors under a closedness condition which dramatically weakens the usual requirement on the continuity of the cocycle it is also shown that though the continuity plays a less important role in the study of attractors it is impossible to establish an existence criteria for common attractors for systems without any continuity like properties however for such bad systems one can expect a mini attractor which is shown adequate well to depict the asymptotic behavior of non continuous systems finally we study the l 2 h 0 1 pullback attractor for a stochastic complex ginzburg landau equation a spectrum decomposition method is employed to overcome the lack of sobolev compactness embeddings in h 0 1

design derivatives of eigenvalues and eigenfunctions for self adjoint distributed parameter systems in this paper analytic expressions are obtained for the design derivatives of eigenvalues and eigenfunctions of self adjoint linear distributed parameter systems explicit treatment of boundary conditions is avoided by casting the eigenvalue equation into integral form results are expressed in terms of the linear operators defining the eigenvalue problem and are therefore quite general sufficiency conditions appropriate to structural op timization of eigenvalues are obtained erivatives of eigenvalues and eigenvectors with respect to the design parameters are useful if not essen tial for design sensitivity and structural optimization studies in one of the earlier works to address this question fox and kapoor1 restrict their discussion to self adjoint discrete systems plaut and huseyin2 extend this approach to non self adjoint discrete systems by treating the associated adjoint eigenvalue problem for distributed parameter systems design derivatives of eigenvalues were first encountered in optimization studies prager and taylor3 generalized several earlier studies4 5 using rayleigh s principle this approach however is not suitable for determining the design derivative of eigenfunctions or for calculating higher order derivatives of eigenvalues finally rayleigh s principle is not appropriate for the determina tion of conditions necessary or sufficient to ensure the ex istence of the design derivatives of eigenvalues and eigenfunctions the foregoing considerations were recently addressed by haug and rousselet6 using a functional analysis approach in this paper a relatively simple method is used to determine explicit results for the design derivatives of eigenvalues and eigenfunctions self adjoint operator equations are cast into their integral form using green s function since the design derivative of green s functional is explicitly known 7 it is a straightforward matter to differentiate the eigenvalue equa tion with respect to the design parameters to obtain an equa tion for the variations of the eigenvalues and eigenfunctions the solution to this equation is explicitly obtained by recognizing that the eigenfunctions form a complete set

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

attractors for random dynamical systems a criterion for existence of global random attractors for rds is established existence of invariant markov measures supported by the random attractor is proved for spde this yields invariant measures for the associated markov semigroup the results are applied to reation diffusion equations with additive white noise and to navier stokes equations with multiplicative and with additive white noise

quantum information entropies and orthogonal polynomials this is a survey of the present knowledge on the analytical determination of the shannon information entropies for simple quantum systems single particle systems in central potentials emphasis is made on d dimensional harmonic oscillator and coulombian potentials in both position and momentum spaces first of all these quantities are explicitly shown to be controlled by the entropic integrals of some classical orthogonal polynomials hermite laguerre and gegenbauer then the connection of these integrals with more common mathematical objects such as the logarithmic potential energy and l p norms of orthogonal polynomials is brie y described third its asymptotic behaviour is discussed for both general and varying weights the explicit computation of these integrals is carried out for the chebyshev and gegenbauer polynomials which have a bounded orthogonality interval as well as for hermite polynomials to illustrate the di culties encountered when the interval is unbounded these results have allowed us to nd the position and momentum entropies of the ground and excited states of the physical systems mentioned above c 2001 elsevier science b v all rights reserved

variable step size control in the numerical solution of stochastic differential equations we introduce a variable step size method for the numerical approximation of pathwise solutions to stochastic differential equations sdes the method which is dependent on a representation of bro

entropic integrals of orthogonal hypergeometric polynomials with general supports the boltzmann shannon information entropy of probability measures which involve the continuous hypergeometric type polynomials fpn x g orthogonal with respect to a general weight function x is determined by two integral quantities one with kernel p 2 x x lnp 2 x called as entropy of the polynomial pn x and another one with kernel p 2 x x ln x here an explicit expression for the latter quantity and for a broader family of related integrals is obtained in terms only of the second order dierential equation satised by the involved polynomials for illustration the general formula is applied to evaluate the integrals corresponding to the three classical families of continuous orthogonal polynomials on the real axis of hypergeometric type hermite laguerre and jacobi c 2000 elsevier science b v all rights reserved

spectral gaps in wasserstein distances and the 2d stochastic navier stokes equations we develop a general method to prove the existence of spectral gaps for markov semigroups on banach spaces unlike most previous work the type of norm we consider for this analysis is neither a weighted supremum norm nor an lp type norm but involves the derivative of the observable as well and hence can be seen as a type of 1 wasserstein distance this turns out to be a suitable approach for infinite dimensional spaces where the usual harris or doeblin conditions which are geared toward total variation convergence often fail to hold in the first part of this paper we consider semigroups that have uniform behavior which one can view its the analog of doeblin s condition we then proceed to study situations where the behavior is not so uniform but the system has a suitable lyapunov structure leading to a type of harris condition we finally show that the latter condition is satisfied by the two dimensional stochastic navier stokes equations even in situations where the forcing is extremely de generate using the convergence result we show that the stochastic navier stokes equations invariant measures depend continuously on the viscosity and the structure of the forcing

operational rules and a generalized hermite polynomials in this paper we use operational rules associated with three operators corresponding to a generalized hermite polynomials introduced by szego to derive as far as we know new proofs of some known properties as well as new expansions formulae related to these polynomials

algebraic evaluation of matrix elements in the laguerre function basis the laguerre functions constitute one of the fundamental basis sets for calculations in atomic and molecular electron structure theory with applications in hadronic and nuclear theory as well while similar in form to the coulomb bound state eigenfunctions from the schrodinger eigenproblem or the coulomb sturmian functions from a related sturm liouville problem the laguerre functions unlike these former functions constitute a complete discrete orthonormal set for square integrable functions in three dimensions we construct the su 1 1 so 3 dynamical algebra for the laguerre functions and apply the ideas of factorization or supersymmetric quantum mechanics to derive shift operators for these functions we use the resulting algebraic framework to derive analytic expressions for matrix elements of several basic radial operators involving powers of the radial coordinate and radial derivative in the laguerre function basis we illustrate how matrix elements for more general spherical tensor operators in three dimensional space such as the gradient may then be constructed from these radial matrix elements

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

upper semicontinuity of attractors for lattice systems under singular perturbations abstract consider the following first order lattice system u m 2 u m u m 1 u m 1 m u m f m u m g m m z which is perturbed by the small two order term u m u m 2 u m u m 1 u m 1 m u m f m u m g m m z under certain conditions on f m m and g m the original systems and the small perturbed systems have global attractors a in l 2 and a in l 2 l 2 respectively and a can be naturally embedded into a compact set a 0 in l 2 l 2 we prove the upper semicontinuity of a 0 with respect to the attractors a at zero by showing that for any neighborhood o a 0 of a 0 a enters o a 0 if is small enough

linearization coefficients for orthogonal polynomials using stochastic processes given a basis for a polynomial ring the coefficients in the expansion of a product of some of its elements in terms of this basis are called linearization coefficients these coefficients have combinatorial significance for many classical families of orthogonal polynomials starting with a stochastic process and using the stochastic measures machinery introduced by rota and wallstrom we calculate and give an interpretation of linearization coefficients for a number of polynomial families the processes involved may have independent freely independent or q independent increments the use of noncommutative stochastic processes extends the range of applications significantly allowing us to treat hermite charlier chebyshev free charlier and rogers and continuous big q hermite polynomials we also show that the q poisson process is a markov process

expansions in series of orthogonal hypergeometric polynomials abstract let us consider an arbitrary hypergeometric polynomial qj x and a set of orthogonal hypergeometric polynomials p n x in the domain of orthogonality here the expansion coefficients of xm and xmqj x m n0 in series of the set p n x are found in terms of the polynomials x and x characterizing the second order differential equations satisfied by the involved hypergeometric polynomials the resulting general expressions which are given in an explicit and compact form are used to produce known for checking and unknown expansions for various concrete classical orthogonal polynomials

buckling of rectangular symmetric angle ply laminated plates determined by eigensensitivity analysis closed form approximate solutions for uniaxial and biaxial compressive buckling of rectangular symmetric angle ply laminates are developed from eigensensitivity analysis boundary conditions which consist of the various combinations of clamped and simply supported edges are incorporated through appropriate edge condition parameters calculated results presented for elastic moduli ratios up to 40 compare quite favorably with the corresponding buckling loads obtained from the ritz method

recurrence equations and their classical orthogonal polynomial solutions the classical orthogonal polynomials are given as the polynomial solutions p n x of the differential email protected x y x email protected x y x email protected ny x 0 where s x is a polynomial of at most second degree and t x is a polynomial of first degree in this paper a general method to express the coefficients a n b n and c n of the recurrence equationp n 1 x a nx b n p n x c np n 1 x in terms of the given polynomials s x and t x is used to present an algorithm to determine the classical orthogonal polynomial solutions of any given holonomic three term recurrence equation i e a homogeneous linear three term recurrence equation with polynomial coefficients in a similar way classical discrete orthogonal polynomial solutions of holonomic three term recurrence equations can be determined by considering their corresponding difference email protected x email protected y x email protected x dy x email protected ny x 0 where dy x y x 1 y x and y x y x y x 1 denote the forward and backward difference operators respectively and a similar approach applies to classical q orthogonal polynomials being solutions of the q difference email protected x d qd 1 qy x email protected x d qy x email protected q ny x 0 whered qf x f qx f x q 1 x q 1 denotes the q difference operator

representations of orthogonal polynomials abstract zeilberger s algorithm provides a method to compute recurrence and differential equations from given hypergeometric series representations and an adaption of almquist and zeilberger computers recurrence and differential equations for hyperexponential integrals further versions of this algorithm allow the computation of recurrence and differential equations from rodrigues type formulas and from generating functions in particular these algorithms can be used to compute the differential difference and recurrence equations for the classical continuous and discrete orthogonal polynomials from their hypergeometric representations and from their rodrigues representations and generating functions in recent work we used an explicit formula for the recurrence equation of families of classical continuous and discrete orthogonal polynomials in terms of the coefficients of their differential difference equations to give an algorithm to identify the polynomial system from a given recurrence equation in this article we extend these results by presenting a collection of algorithms with which any of the conversions between the differential difference equation the hypergeometric representation and the recurrence equation is possible the main technique is again to use explicit formulas for structural identities of the given polynomial systems

existence and continuity of bi spatial random attractors and application to stochastic semilinear laplacian equations abstract a concept of a bi spatial random attractor for a random dynamical system is introduced a unified result about existence and upper semi continuity for a family of bi spatial random attractors is obtained if a family of random systems is convergent uniformly absorbing in an initial space and uniformly omega compact in both initial and terminate spaces the upper semi continuity result improves all existing results even for single spatial attractors as an application of the abstract result it is shown that every semilinear laplacian equation on the entire space perturbed by a multiplicative and stochastic noise possesses an l 2 l q random attractor with q 2 moreover it is proved that the family of obtained attractors is upper semi continuous at any density of noises and the family of attractors for the corresponding compact systems is both upper and lower semi continuous at infinity under the topology of both spaces

strong convergence of euler type methods for nonlinear stochastic differential equations traditional finite time convergence theory for numerical methods applied to stochastic differential equations sdes requires a global lipschitz assumption on the drift and diffusion coefficients in practice many important sde models satisfy only a local lipschitz property and since brownian paths can make arbitrarily large excursions the global lipschitz based theory is not directly relevant in this work we prove strong convergence results under less restrictive conditions first we give a convergence result for euler maruyama requiring only that the sde is locally lipschitz and that the pth moments of the exact and numerical solution are bounded for some p 2 as an application of this general theory we show that an implicit variant of euler maruyama converges if the diffusion coefficient is globally lipschitz but the drift coefficient satisfies only a one sided lipschitz condition this is achieved by showing that the implicit method has bounded moments and may be viewed as an euler maruyama approximation to a perturbed sde of the same form second we show that the optimal rate of convergence can be recovered if the drift coefficient is also assumed to behave like a polynomial

a stochastic pitchfork bifurcation in a reaction diffusion equation we study in some detail the structure of the random attractor for the chafee infante reaction diffusion equation perturbed by a multiplicative white noise d u u u u 3 d t u o d w t x d r m first we prove for m 5 a lower bound on the dimension of the random attractor which is of the same order in as the upper bound we derived in an earlier paper and is the same as that obtained in the deterministic case then we show for m 1 that as passes through 1 the first eigenvalue of the negative laplacian from below the system undergoes a stochastic bifurcation of pitchfork type we believe that this is the first example of such a stochastic bifurcation in an infinite dimensional setting central to our approach is the existence of a random unstable manifold

linearization and connection coefficients for hypergeometric type polynomials we consider the problem of finding closed analytical formulas for both the linearization and connection coefficients for hypergeometric type polynomials directly in terms of the corresponding differential equations we illustrate the method by producing explicit formulas for hermite polynomials

general linearization formulae for products of continuous hypergeometric type polynomials the linearization of products of wavefunctions of exactly solvable potentials often reduces to the generalized linearization problem for hypergeometric polynomials hps of a continuous variable which consists of the expansion of the product of two arbitrary hps in series of an orthogonal hp set here this problem is algebraically solved directly in terms of the coefficients of the second order differential equations satisfied by the involved polynomials general expressions for the expansion coefficients are given in integral form and they are applied to derive the connection formulae relating the three classical families of hypergeometric polynomials orthogonal on the real axis hermite laguerre and jacobi as well as several generalized linearization formulae involving these families the connection and linearization coefficients are generally expressed as finite sums of terminating hypergeometric functions which often reduce to a single function of the same type when possible these functions are evaluated in closed form in some cases sign properties of the coefficients such as positivity or non negativity conditions are derived as a by product from their resulting explicit representations

stability of random attractors under perturbation and approximation abstract the comparison of the long time behaviour of dynamical systems and their numerical approximations is not straightforward since in general such methods only converge on bounded time intervals however one can still compare their asymptotic behaviour using the global attractor and this is now standard in the deterministic autonomous case for random dynamical systems there is an additional problem since the convergence of numerical methods for such systems is usually given only on average in this paper the deterministic approach is extended to cover stochastic differential equations giving necessary and sufficient conditions for the random attractor arising from a random dynamical system to be upper semi continuous with respect to a given family of perturbations or approximations

spreading lengths of hermite polynomials the renyi shannon and fisher spreading lengths of the classical or hypergeometric orthogonal polynomials which are quantifiers of their distribution all over the orthogonality interval are defined and investigated these information theoretic measures of the associated rakhmanov probability density which are direct measures of the polynomial spreading in the sense of having the same units as the variable share interesting properties invariance under translations and reflections linear scaling and vanishing in the limit that the variable tends towards a given definite value the expressions of the renyi and fisher lengths for the hermite polynomials are computed in terms of the polynomial degree the combinatorial multivariable bell polynomials which are shown to characterize the finite power of an arbitrary polynomial play a relevant role for the computation of these information theoretic lengths indeed these polynomials allow us to design an error free computing approach for the entropic moments weighted l q norms of hermite polynomials and subsequently for the renyi and tsallis entropies as well as for the renyi spreading lengths sharp bounds for the shannon length of these polynomials are also given by means of an information theoretic based optimization procedure moreover the existence of a linear correlation between the shannon length as well as the second order renyi length and the standard deviation is computationally proved finally the application to the most popular quantum mechanical prototype system the harmonic oscillator is discussed and some relevant asymptotical open issues related to the entropic moments mentioned previously are posed

entropic uncertainty measures for large dimensional hydrogenic systems the entropic moments of the probability density of a quantum system in position and momentum spaces describe not only some fundamental and or experimentally accessible quantities of the system but also the entropic uncertainty measures of renyi type which allow one to find the most relevant mathematical formalizations of the position momentum heisenberg s uncertainty principle the entropic uncertainty relations it is known that the solution of difficult three dimensional problems can be very well approximated by a series development in 1 d in similar systems with a non standard dimensionality d moreover several physical quantities of numerous atomic and molecular systems have been numerically shown to have values in the large d limit comparable to the corresponding ones provided by the three dimensional numerical self consistent field methods the d dimensional hydrogenic atom is the main prototype of the physics of multidimensional many electron systems in this work we rigorously determine the lea

solving connection and linearization problems within the askey scheme and its q analogue via inversion formulas for the polynomial families pn x n belonging to the askey scheme or to its q analogue the hypergeometric represen tation provides a natural expansion of the form pn x n 0 dm n m x where the expanding basism x is in general a product of pochhammer symbols or q shifted factorials in this paper we solve the corresponding inversion problem i e we compute the coe6cients im n in the expansionn x n m 0 im n pm x which are then used as a tool for solving any connection and linearization problem within the askey scheme and its q analogue extensions of this approach for polynomials outside these two schemes are also given c