shifted simplicial complexes are laplacian integral we show that the combinatorial laplace operators associated to the boundary maps in a shifted simplicial complex have all integer spectra we give a simple combinatorial interpretation for the spectra in terms of vertex degree sequences generalizing a theorem of merris for graphs we also conjecture a majorization inequality for the spectra of these laplace operators in an arbitrary simplicial complex with equality achieved if and only if the complex is shifted this generalizes a conjecture of grone and merris for graphs

some results involving hermite base polynomials and functions using operational methods this paper is an attempt to stress the usefulness of the operational methods in the theory of special functions using operational methods we derive summation formulae and generating relations involving various forms of hermite base polynomials and functions

cones of closed alternating walks and trails abstract consider a graph whose edges have been colored red and blue assign a nonnegative real weight to every edge so that at every vertex the sum of the weights of the incident red edges equals the sum of the weights of the incident blue edges the set of all such assignments forms a convex polyhedral cone in the edge space called the alternating cone the integral respectively 0 1 vectors in the alternating cone are sums of characteristic vectors of closed alternating walks respectively trails we study the basic properties of the alternating cone determine its dimension and extreme rays and relate its dimension to the majorization order on degree sequences we consider whether the alternating cone has integral vectors in a given box and use residual graph techniques to reduce this problem to the one of searching for an alternating trail connecting two given vertices the latter problem called alternating reachability is solved in a companion paper along with related results

operational methods fractional operators and special polynomials we use methods of operational nature to deal with families of partial differential equations of evolution type to treat problems involving fractional differential operators we also discuss the properties of families of special polynomials or special functions like the riemann z function naturally associated with the proposed formalism

the polytope of degree sequences abstract a nonnegative integer sequence d 1 d 2 d n is called a degree sequence if there exists a simple graph on the vertex set v 1 2 n such that deg i d i for all i the degree sequence of a threshold graph is a threshold sequence let d n convex hull x 1 x 2 x n x 1 x n is a degree sequence it is proved that 1 a degree sequence f is an extreme point of d n if and only if f is a threshold sequence 2 two threshold sequences f and g are adjacent extreme points of d n if and only if f can be obtained from g by either adding 1 to two components of g or subtracting 1 from two components of g 3 d n is determined bythe following system of inequalities i x i i t x i s n 1 t for all sets s t with s t 1 2 n s t moreover this system is totally dual integral furthermore for n 4 s t determines a facet of d n if and only if either s t 1 or else s t s t n 1 n 2 4 f is a threshold sequence if and only if the only degree sequences majorizing f in the sense of hardy littlewood and polya are the rearrangements of f consequently every degree sequence is a convex combination of isomorphic threshold sequences i e threshold sequences that are rearrangements of each other

mode competition in cylindrical flows driven by sidewall oscillations the transition from a two dimensional to three dimensional flow in systems with spatial o 2 symmetry and spatiotemporal z 2 symmetry happens in many fluid systems like wakes or periodically forced flows in most of these systems the dynamics after the first bifurcation is very complex and involves cascades of bifurcations in a very narrow parameter range a numerical study of a flow in an enclosed cylindrical cavity driven by axial oscillations of the sidewall which allows a detailed study of the secondary bifurcations and the corresponding mode interactions is presented the study focuses on a codimension 2 point that acts as the organizing center of the dynamics for moderate values of the forcing frequency the unraveled dynamics is very rich including slow fast dynamics and hysteresis and may help understand the bifurcation cascades in more complex systems

operational methods and differential equations with applications to initial value problems the authors present and develop a general method of operational nature in order to explore and investigate such families of orthogonal polynomials and functions as the classical and generalized hermite and laguerre polynomials as well as their higher order and multi index cases they also consider several applications of these operational techniques to the solutions of a wide class of partial differential equations including for example the heat wave and transport equations some of which involve second order laguerre derivatives

bifurcation from periodic solutions with spatiotemporal symmetry including resonances and mode interactions abstract we study local bifurcation in equivariant dynamical systems from periodic solutions with a mixture of spatial and spatiotemporal symmetries in previous work we focused primarily on codimension one bifurcations in this paper we show that the techniques used in the codimension one analysis can be extended to understand also higher codimension bifurcations including resonant bifurcations and mode interactions in particular we present a general reduction scheme by which we relate bifurcations from periodic solutions to bifurcations from fixed points of twisted equivariant diffeomorphisms which in turn are linked via normal form theory to bifurcations from equilibria of equivariant vector fields we also obtain a general theory for bifurcation from relative periodic solutions and we show how to incorporate time reversal symmetries into our framework

when the telegrapher s equation furnishes a better approximation to the transport equation than the diffusion approximation it has been suggested that a solution to the transport equation which includes anisotropic scattering can be approximated by the solution to a telegrapher s equation a j ishimaru appl opt 28 2210 1989 we show that in one dimension the telegrapher s equation furnishes an exact solution to the transport equation in two dimensions we show that since the solution can become negative the telegrapher s equation will not furnish a usable approximation a comparison between simulated data in three dimensions indicates that the solution to the telegrapher s equation is a good approximation to that of the full transport equation at the times at which the diffusion equation furnishes an equally good approximation s1063 651x 97 04205 0

new complex function space related to both entangled state representation and spin coherent state a new complex function space whose basis is the single variable hermite polynomial h2j 2 is constructed which is related to both entangled state representation and spin coherent state in schwinger bosonic realization new binomial theorem involving two variable hermite polynomial is derived which helps to constitute the new complex function space we also present a new integration transformation of the basis h2j 2 with its reciprocal transformation which is useful to deriving some operator identities

laguerre 2d functions and their application in quantum optics a set of orthonormalized and complete functions of two real variables in complex representation involving laguerre polynomials as a substantial part is introduced and is referred to as the set of laguerre two dimensional 2d functions the properties of the set of laguerre 2d functions are discussed and the fourier and radon transforms of these functions are calculated the laguerre 2d functions form a basis for a realization of the five dimensional lie algebra to the heisenberg weyl group for a two mode system real representation of this set of functions by a sum over products of hermite functions involving jacobi polynomials at the zero argument as coefficients is derived and it leads to new connections between laguerre and hermite polynomials in both directions the set of laguerre 2d functions is the most appropriate set of functions for the fock state representation of quasi probabilities in quantum optics the wigner quasi probability in fock state representation is up to a factor and an argument scaling directly given for each matrix element by a corresponding laguerre 2d function the properties of orthonormality and completeness of the laguerre 2d functions provide the fock state matrix elements of the density operator directly from the quasi probabilities the perina mista representation of the glauber sudarshan quasi probability can be represented with advantage by the laguerre 2d functions the fock state matrix elements of the displacement operator and the scalar product of displaced fock states are closely related to laguerre 2d functions

phase space dynamics and hermite polynomials of two variables and two indices the theory of hermite polynomials of two variables and two indices is discussed herein within the context of phase space formulation of classical and quantum mechanics they play the same role as conventional hermite polynomials in ordinary quantum mechanics finally their extension to m variables and m indices is analyzed

bifurcation theory for three dimensional flow in the wake of a circular cylinder department of mathematics university of houston houston texas 77204 received 20 september 1999 a bifurcation scenario is presented for three dimensional vortex shedding in the wake of a circular cylinderfor reynolds numbers up to 300 amplitude equations are proposed to describe the nonlinear interactionbetween two three dimensional modes of shedding with different spanwise wave numbers and different spa tiotemporal symmetries the amplitude equations explain many features of the transition scenario observedexperimentally pacs number s 47 27 vf 47 15 fe 47 54 1ri introduction

incomplete 2d hermite polynomials properties and applications incomplete forms of two variable two index hermite polynomials are introduced their link with laguerre polynomials is discussed and it is shown that they are a useful tool to study quantum mechanical harmonic oscillator entangled states the possibility of developing the theory of complete 2d hermite polynomials from the point of view of the incomplete forms is analyzed too the orthogonality properties of the associated harmonic oscillator functions are finally discussed

generating relations of tricomi and hermite tricomi functions using lie algebra representation abstract this paper is an attempt to stress the usefulness of the multi variable special functions in this paper we derive generating relations involving multi variable tricomi and hermite tricomi functions by using lie algebraic method further we derive certain new and known generating relations involving other forms of tricomi hermite tricomi and bessel functions as applications

high re solutions for incompressible flow using the navier stokes equations and a multigrid method the vorticity stream function formulation of the two dimensional incompressible navierstokes equations is used to study the effectiveness of the coupled strongly implicit multigrid csi mg method in the determination of high re fine mesh flow solutions the driven flow in a square cavity is used as the model problem solutions are obtained for configurations with reynolds number as high as 10 000 and meshes consisting of as many as 257 x 257 points for re 1000 the 129 x 129 grid solution required 1 5 minutes of cpu time on the amdahl 470 v 6 computer because of the appearance of one or more secondary vortices in the flow field uniform mesh refinement was preferred to the use of one dimensional gridclustering coordinate transformations the past decade has witnessed a great deal of progress in the area of computational fluid dynamics developments in computer technology hardware as well as in advanced numerical algorithms have enabled attempts to be made towards analysis and numerical solution of highly complex flow problems for some of these applications the use of simple iterative techniques to solve the navier stokes equations leads to a rather slow convergence rate for the solutions the solution convergence rate can be seriously affected if the coupling among the various governing differential equations is not properly honored either in the interior of the solution domain or at its boundaries the rate of convergence is also generally strongly dependent on such problem parameters as the reynolds number the mesh size and the total number of computational points this has led several researchers to examine carefully the recently emerging multigrid mg technique as a useful means for enhancing the convergence rate of iterative numerical methods for solving discretized equations at a number of computational grid points so large as to be considered impractical previously

evolution of non spreading airy wavepackets in time dependent linear potentials abstract we report on the use of the algebraic methods to obtain the explicit form of the solution of the schrodinger equation with a linear potential we consider the case of the explicitly time dependent hamiltonian and formulate the general conditions that allow for the solutions to be found that are expressed in terms of airy functions yielding non spreading wave packets the relevant physical meaning of these solutions is analyzed and the examples of their applications are given the role played by the airy transform and its relevance to the problems involving linear potentials is discussed eventually we present a thorough discussion on the analogy between the airy and the gauss weierstrass transform often employed in the solutions of the heat type equations

analytical solutions for heat diffusion beyond fourier law we obtain solutions for differential equations describing a broad range of physical problems by the operational method with recourse to inverse differential operators integral transforms and operational exponent generalized families of orthogonal polynomials and special functions are also employed with recourse to their operational definitions the evolutional type problems for heat transfer in various heat conduction models are studied exact analytical solutions for guyer krumhansl hyperbolic heat equation are obtained and compared with those of fourier and cattaneo equations modelling heat pulse propagation from a laser source is performed in the framework of fourier cattaneo and guyer krumhansl heat transfer models compliance of obtained solutions with the maximum principle is studied

on limit relations between some families of bivariate hypergeometric orthogonal polynomials in this paper we deal with limit relations between bivariate hypergeometric polynomials we analyze the limit relation from trinomial distribution to bivariate gaussian distribution obtaining the limit transition from the second order partial difference equation satisfied by bivariate hypergeometric kravchuk polynomials to the second order partial differential equation verified by bivariate hypergeometric hermite polynomials as a consequence the limit relation between both families of orthogonal polynomials is established a similar analysis between bivariate hahn and bivariate appell orthogonal polynomials is also presented

bifurcation from discrete rotating waves discrete rotating waves are periodic solutions that have discrete spatiotemporal symmetries in addition to their purely spatial symmetries we present a systematic approach to the study of local bifurcation from discrete rotating waves the approach centers around the analysis of diffeomorphisms that are equivariant with respect to distinct group actions in the domain and the range our results are valid for dynamical systems with finite symmetry group and more generally for bifurcations from isolated discrete rotating waves in dynamical systems with compact symmetry group

operational solution for some types of second order differential equations and for relevant physical problems abstract we present an operational method to obtain solutions for differential equations describing a broad range of physical problems including ordinary non integer order and high order partial differential equations inverse differential operators are proposed to solve a variety of differential equations integral transforms and the operational exponent are used to obtain the solutions generalized families of orthogonal polynomials and special functions are also employed with recourse to their operational definitions examples of solutions of physical problems related to propagation of the heat and other quantities are demonstrated by the developed operational technique in particular the evolution type problems the generalizations of the black scholes of the heat conduction of the fokker planck equations are considered as well as equations involving the laguerre derivative operator

orthogonality properties of the hermite and related polynomials the authors present a general method of operational nature with a view to investigating the orthogonality properties of several different families of the hermite and related polynomials in particular the classical hermite polynomials and some of their higher order and multi index generalizations are considered here

a class of generalized complex hermite polynomials a class of generalized complex polynomials of hermite type suggested by a special magnetic schrodinger operator is introduced and some related basic properties are discussed

implicit summation formulae for hermite and related polynomials in this article we derive some implicit summation formulae for hermite and related polynomials by using different analytical means on their respective generating functions

the polytope of degree sequences of hypergraphs abstract let d n r denote the convex hull of degree sequences of simple r uniform hypergraphs on the vertex set 1 2 n the polytope d n 2 is a well studied object its extreme points are the threshold sequences i e degree sequences of threshold graphs and its facets are given by the erdos gallai inequalities in this paper we study the polytopes d n r and obtain some partial information our approach also yields new simple proofs of some basic results on d n 2 our main results concern the extreme points and facets of d n r we characterize adjacency of extreme points of d n r and in the case r 2 determine the distance between two given vertices in the graph of d n 2 we give a characterization of when a linear inequality determines a facet of d n r and use it to bound the sizes of the coefficients appearing in the facet defining inequalities give a new short proof for the facets of d n 2 find an explicit family of erdos gallai type facets of d n r and describe a simple lifting procedure that produces a facet of d n 1 r from one of d n r

transformations of laguerre 2d polynomials with applications to quasiprobabilities laguerre 2d polynomials are defined and their properties are investigated the laguerre 2d functions introduced in 1 2 are related to the laguerre 2d polynomials in such a way that they also include the weight function for the orthonormalization of the laguerre 2d polynomials a one parameter group of transformations applicable to certain classes of polynomials and discrete sets of functions is investigated and applied in particular to hermite polynomials and to laguerre 2d polynomials these transformations allow us to represent the polynomials of the corresponding classes by superpositions of the same kind of polynomials with stretched arguments they contain limiting cases with delta functions and their derivatives and lead to regularized representations of the delta functions and their derivatives as demonstrated for hermite and laguerre 2d polynomials applications of the laguerre 2d polynomials and 2d functions and their transformations to problems of quantum optics as the representation of quasiprobabilities in the fock state basis and by normally and otherwise ordered moments are considered the inversion of these representations is obtained in all cases a restricted design of quasiprobabilities should become possible

generalized polynomials operational identities and their applications abstract it is shown that an appropriate combination of methods relevant to generalized operational calculus and to special functions can be a very useful tool to treat a large body of problems both in physics and mathematics we discuss operational methods associated with multivariable hermite laguerre legendre and other polynomials to derive a wealth of identities useful in quantum mechanics electromagnetism optics etc or to derive new identities between special functions as e g mehler or mixed type generating functions

unoriented laplacian maximizing graphs are degree maximal abstract a connected graph is said to be unoriented laplacian maximizing if the spectral radius of its unoriented laplacian matrix attains the maximum among all connected graphs with the same number of vertices and the same number of edges a graph is said to be threshold maximal if its degree sequence is not majorized by the degree sequence of any other graph and in addition the graph is connected it is proved that an unoriented laplacian maximizing graph is maximal and also that there are precisely two unoriented laplacian maximizing graphs of a given order and with nullity 3 our treatment depends on the following known characterization a graph g is threshold maximal if and only if for every pair of vertices u v of g the sets n u v n v u where n u denotes the neighbor set of u in g are comparable with respect to the inclusion relation and in addition the graph is connected a conjecture about graphs that maximize the unoriented laplacian matrix among all graphs with the same number of vertices and the same number of edges is also posed

graphs with maximal signless laplacian spectral radius abstract by the signless laplacian of a simple graph g we mean the matrix q g d g a g where a g d g denote respectively the adjacency matrix and the diagonal matrix of vertex degrees of g it is known that connected graphs g that maximize the signless laplacian spectral radius q g over all connected graphs with given numbers of vertices and edges are degree maximal for a maximal graph g with n vertices and r distinct vertex degrees r r 1 1 it is proved that q g q h for some maximal graph h with n 1 respectively n vertices and the same number of edges as g if either g has precisely two dominating vertices or there exists an integer i 2 i r 2 respectively if there exist positive integers i l with l 2 i r 2 such that i r 1 i n 1 respectively i r 1 i l r l 1 graphs that maximize q g over the class of graphs with m edges and m k vertices for k 0 1 2 3 are completely determined

the realization graph of a degree sequence with majorization gap 1 is hamiltonian abstract it is known that the degree sequences of threshold graphs are characterized by the property that they are not majorized strictly by any degree sequence consequently every degree sequence d can be transformed into a threshold sequence by repeated operations consisting of subtracting i from a degree and adding 1 to a larger or equal degree the minimum number of these operations required to transform d into a threshold sequence is called the majorization gap of d a realization of a degree sequence d of length n is a graph on the vertices 1 n where the degree of vertex i is d i the realization graph plane1d 4a2 d of a degree sequence d has as vertices the realizations of d and two realizations are neighbors in plane1d 4a2 d if one can be obtained from the other by deleting two existing edges a b c d and adding two new edges a d b c for some distinct vertices a b c d it is known that plane1d 4a2 d is connected we show that if d has a majorization gap of 1 then plane1d 4a2 d is hamiltonian

generalized bessel functions and generalized hermite polynomials abstract in this note we show that generalized hermite polynomials of the gould hopper type are linked to the multivariables generalized bessel functions we show that such a formalism encompasses both first and second kind gould hopper polynomials and that their recurrence properties are straightforwardly derived from those of generalized bessel functions

new relations for two dimensional hermite polynomials the effective formulas reducing the two dimensional hermite polynomials to the classical one dimensional orthogonal polynomials by jacobi gegenbauer legendre laguerre and hermite are given new one parameter generating functions for the diagonal multidimensional hermite polynomials are derived the factorial moments and cumulants of the distribution functions related to the hermite polynomials of two variables with equal indices are expressed in terms of the legendre and chebyshev polynomials asymptotical formulas for the two dimensional polynomials with large values of indices and zero arguments are found the applications to the squeezed one mode states and to the time dependent quantum harmonic oscillator are considered

degree sequences and majorization a classical result concerning majorization is given two nonnegative integer sequences a and b such that a majorizes b a rearrangement of b can be obtained from a by a sequence of unit transformations a recent result says that a degree sequence is a threshold sequence degree sequence of a threshold graph if and only if it is not strictly majorized by any degree sequence motivated by this we define the majorization gap of a degree sequence to be the minimum number of successive reverse unit transformations required to transform it into a threshold sequence we derive a formula for the majorization gap by establishing a lower bound for it and exhibiting reverse unit transformations achieving the bound we also discuss the relationship between the majorization gap and the threshold gap introduced elsewhere and show that they are equal the degree sequences having the maximum majorization gap for a fixed number of edges or vertices are characterized

bivariate raising and lowering differential operators for eigenfunctions of a 2d fourier transform we define a two dimensional 2d fourier transform that self reproduces a one parameter family of bivariate hermite functions these are eigenfunctions of a hamiltonian differential operator of second order whose exponential is that transform we find explicit forms of the bivariate raising and lowering partial differential operators of first degree for the eigenfunctions of this 2d fourier transform

bifurcations in systems with z2 spatio temporal and o 2 spatial symmetry this work analyzes the o 2 symmetry breaking bifurcations in systems with an z2 o 2 symmetry group wherez2 and o 2 are respectively spatio temporal and spatial symmetries that are responsible for the transitions from two dimensional to three dimensional hydrodynamic states this symmetry group describes for example two dimensional time periodic flows past bodies which have reflection symmetry across a wake center plane such as symmetrical airfoils circular and square cylinders normal form analysis of these systems is based on a joint representation of the monodromy matrix for the half period flip map a composition of a half period temporal evolution with a spatial reflection and the spatial o 2 symmetry there are exactly two kinds of codimension one synchronous bifurcations in these systems one preserves the z2 spatio temporal symmetry while the other breaks it when the floquet multipliers occur in complex conjugate pairs non resonant with the periodic basic state there is a single codimension one bifurcation and at the bifurcation point two different kind of solutions appear simultaneously a pair of modulated traveling waves and a circle of modulated standing waves at most one of these two types has stable solutions the symmetries of the system also admit period doubling bifurcations but these are codimension two and the normal form analysis permits specific conclusions regarding these there are also a number of other codimension two bifurcations leading to mixed modes and the strong 1 1 and 1 2 resonances all the codimension one bifurcations are illustrated with reference to a concrete physical example 2003 elsevier b v all rights reserved