a forward in time advection scheme and adaptive multilevel flow solver for nearly incompressible atmospheric flow this paper presents a new forward in time advection method for nearly incompressible flow mu and its application to an adaptive multilevel flow solver for atmospheric flows mu is a modification of leonardet al sutopia scheme mu like utopia is based on third order accurate semi lagrangian multidimensional upwinding for constant velocity flows for varying velocity fields mu is a second order conservative method mu has greater stability and accuracy than utopia and naturally decomposes into a monotone low order method and a higher order accurate correction for use with flux limiting its stability and accuracy make it a computationally efficient alternative to current finite difference advection methods we present a fully second order accurate flow solver for the anelastic equations a prototypical low mach number flow the flow solver is based on mu which is used for both momentum and scalar transport equations this flow solver can also be implemented with any forward in time advection scheme the multilevel flow solver conserves discrete global integrals of advected quantities and includes adaptive mesh refinement its second order accuracy is verified using a nonlinear energy conservation integral for the anelastic equations for a typical geophysical problem in which the flow is most rapidly varying in a small part of the domain the multilevel flow solver achieves global accuracy comparable to a uniform resolution simulation for 10 of the computational cost

generalized hermite polynomials associated with functions of parabolic cylinder associated hermite polynomials he n n z which generalize the usual scaled hermite polynomials he n z corresponding to n 0 are introduced for the purpose to represent the raising and lowering of the indices of functions of the parabolic cylinder d n z in finite integer steps properties of these polynomials such as recursion relations explicit representations and the differential equation are derived the generation of the associated hermite polynomials from the usual hermite polynomials by differential operators representable by means of the confluent hypergeometric function is given an application for the explicit calculation of the functions of the parabolic cylinder for negative integer indices is discussed other applications are visible for the investigation of the zeros of the functions of the parabolic cylinder

convergence analysis and numerical implementation of a second order numerical scheme for the three dimensional phase field crystal equation abstract in this paper we analyze and implement a second order in time numerical scheme for the three dimensional phase field crystal pfc equation the numerical scheme was proposed in hu et al 2009 with the unique solvability and unconditional energy stability established however its convergence analysis remains open we present a detailed convergence analysis in this article in which the maximum norm estimate of the numerical solution over grid points plays an essential role moreover we outline the detailed multigrid method to solve the highly nonlinear numerical scheme over a cubic domain and various three dimensional numerical results are presented including the numerical convergence test complexity test of the multigrid solver and the polycrystal growth simulation

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

interbasis expansions in the zernike system the differential equation with free boundary conditions on the unit disk that was proposed by frits zernike in 1934 to find jacobi polynomial solutions indicated as i serves to define a classical system and a quantum system which have been found to be superintegrable we have determined two new orthogonal polynomial solutions indicated as ii and iii that are separable and involve legendre and gegenbauer polynomials here we report on their three interbasis expansion coefficients between the i ii and i iii bases they are given by f23 1 polynomials that are also special su 2 clebsch gordan coefficients and hahn polynomials between the ii iii bases we find an expansion expressed by f34 1 s and racah polynomials that are related to the wigner 6j coefficients

superintegrable classical zernike system we consider the differential equation that zernike proposed to classify aberrations of wavefronts in a circular pupil as if it were a classical hamiltonian with a non standard potential the trajectories turn out to be closed ellipses we show that this is due to the existence of higher order invariants that close into a cubic higgs algebra the zernike classical system thus belongs to the class of superintegrable systems its hamilton jacobi action separates into three vertical projections of polar coordinates of sphere polar and equidistant coordinates on half hyperboloids and also in elliptic coordinates on the sphere

stable and efficient finite difference nonlinear multigrid schemes for the phase field crystal equation in this paper we present and compare two unconditionally energy stable finite difference schemes for the phase field crystal equation the first is a one step scheme based on a convex splitting of a discrete energy by wise et al s m wise c wang j s lowengrub an energy stable and convergent finite difference scheme for the phase field crystal equation siam j numer anal in press in this scheme which is first order in time and second order in space the discrete energy is non increasing for any time step the second scheme we consider is a new fully second order two step algorithm in the new scheme the discrete energy is bounded by its initial value for any time step in both methods the equations at the implicit time level are nonlinear but represent the gradients of strictly convex functions and are thus uniquely solvable regardless of time step size we solve the nonlinear equations using an efficient nonlinear multigrid method numerical simulations are presented and confirm the stability efficiency and accuracy of the schemes

dyson s crank of a partition holds he was thus led to conjecture the existence of some other partition statistic which he called the crank this unknown statistic should provide a combinatorial interpretation of p lln 6 in the same way that 1 1 and 1 2 treat the primes 5 and 7 in 4 5 one of us was able to find a crank relative to vector partitions as follows for a partition 7r let 7r be the number of parts of ir and cr n be the sum of the parts of ir or the number ir is partitioning with the convention 0 for the empty partition 0 of 0 let

spectral properties of the cauchy transform on l2 c e z 2λ z let hm p z m p z z be the landau orthogonal basis of the hilbert space on l2 c e z 2d z where z is the usual lebesgue measure on the complex plane in this paper we give some spectral properties of the cauchy transform on the orthogonal complement of bargmann space 0 c in l2 c e z 2d z in particular for m fixed we consider the orthogonal projection operator on the hilbert subspace spanned by hm p z p 0 1 2 and we give explicitly the sequence of singular values of its composition with the cauchy transform in l2 c e z 2d z as application of these of the cauchy transform we get some identities for special functions which could be of independent interest

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

operational rules and arbitrary order hermite generating functions operational identities often exploited in the theory of exponential operators are used to derive generating functions of hermite and kampe de feriet polynomials the method we propose allows us to obtain unilateral bilinear and bilateral generating functions by using the same procedure possible extensions of the technique are also discussed

efficient computations for generalized zernike moments and image recovery abstract zernike moments are a set of orthogonal moments which have been successfully applied in the fields of image processing and pattern recognition an innovative calculation method for zernike moments named generalized zernike moments is presented in this study the generalized zernike moment is a variant of zernike moment in this paper we are proposing methods to calculate high order generalized zernike moments two kinds of recurrence for calculating generalized zernike moments were introduced with rigorous proofs through the usage of the symmetries operated by the dihedral group of order eight the proposed method is fast and stable the experimental results show that of the proposed method took 4 206s to compute the top 500 order generalized zernike moments of an image with 512 by 512 pixels furthermore by choosing the extra parameter in the recurrence the method enhanced the accuracy remarkably compared to the regular zernike moments its normalized mean square error is 0 00144067 when was set to 66 and the top 500 order moments were used to reconstruct the image this error is 40 47 smaller than the one obtained by using the regular zernike moments

non radial solutions for some semilinear elliptic equations on the disk abstract starting with approximate solutions of the equation u w u 3 on the disk with zero boundary conditions we prove that there exist true solutions nearby one of the challenges here lies in the fact that we need simultaneous and accurate control of both the inverse dirichlet laplacean and nonlinearities we achieve this with the aid of a computer using a banach algebra of real analytic functions based on zernike polynomials besides proving existence and symmetry properties we also determine the morse index of the solutions

the role of jacobi polynomials in the theory of hermite and laguerre 2d polynomials using an alternative definition of usual hermite polynomials two problems in the theory of general hermite and laguerre 2d polynomials can be separated with advantage for the further treatment the introduction of a general 2d matrix in the linear transformation of powers of the components of a 2d vector and the generation of hermite or laguerre polynomials by applying an integral operator to these powers the jacobi polynomials appear in the finite dimensional irreducible representations of the two dimensional general linear group gl 2 c

efficient spectral galerkin methods for systems of coupled second order equations and their applications we construct in this paper two efficient spectral galerkin algorithms for solving systems of n coupled second order equations the computational complexity of these algorithms is essentially n times the cost of solving one second order equation we present numerical results which illustrate the accuracy and flexibility of these algorithms as well as several interesting and challenging applications including in particular a number of high order nonlinear parabolic type equations

a noncommutative discrete hypergroup associated with q disk polynomials abstract starting from the addition formula for q disk polynomials we prove nonnegativity of the linearization coefficients for these polynomials which allows us to realize a djs hypergroup structure on the space z 2 this hypergroup is discrete and noncommutative

solving the regularized strongly anisotropic cahn hilliard equation by an adaptive nonlinear multigrid method abstract we present efficient second order accurate and adaptive finite difference methods to solve the regularized strongly anisotropic cahn hilliard equation in 2d and 3d when the surface energy anisotropy is sufficiently strong there are missing orientations in the equilibrium level curves of the diffuse interface solutions corresponding to those missing from the sharp interface wulff shape and the anisotropic cahn hilliard equation becomes ill posed to regularize the equation a higher order derivative term is added to the energy this leads to a sixth order nonlinear parabolic equation for the order parameter an implicit time discretization is used to remove the high order time step stability constraints dynamic block structured cartesian mesh refinement is used to highly resolve narrow interfacial layers a multilevel nonlinear multigrid method is used to solve the nonlinear equations at the implicit time level one of the keys to the success of the method is the treatment of the anisotropic term this term is discretized in conservation form in space and is discretized fully implicitly in time numerical simulations are presented that confirm the accuracy efficiency and stability of the scheme we study the dynamics of interfaces under strong anisotropy and compare near equilibrium diffuse interface solutions to the sharp interface wulff shapes in 2d and 3d we also simulate large scale coarsening of a corrugated surface in 3d evolving by anisotropic surface diffusion we show the emergence of long range order during coarsening and an interesting mechanism of ordered coarsening

an energy stable and convergent finite difference scheme for the phase field crystal equation we present an unconditionally energy stable finite difference scheme for the phase field crystal equation the method is based on a convex splitting of a discrete energy and is semi implicit the equation at the implicit time level is nonlinear but represents the gradient of a strictly convex function and is thus uniquely solvable regardless of time step size we present local in time error estimates that ensure the convergence of the scheme while this paper is primarily concerned with the phase field crystal equation most of the theoretical results hold for the related swift hohenberg equation as well

coherent states and hermite polynomials on quaternionic hilbert spaces coherent states similar to the canonical coherent states of the harmonic oscillator labeled by quaternions are established on the right and left quaternionic hilbert spaces on the left quaternionic hilbert space reproducing kernels are established as was claimed by adler and millard j math phys 1997 38 2117 26 it is proved that these coherent states cannot be realized as a group action on a quaternionic hilbert space as an extension of the complex hermite polynomials quaternionic hermite polynomials are defined and these polynomials are associated with an operator as eigenfunctions

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

the fast adaptive composite grid fac method for elliptic equation the fast adaptive composite grid fag method is a systematic process for solving differential boundary value problems fac uses global and local uniform grids both to define the composite grid problem and to interact for its fast solution it can with little added cost substantially improve accuracy of the coarse grid solution and is very suitable for vector and parallel computation this paper develops both the theoretical and practical aspects of fac as it applies to elliptic problems 1 introduction the need for local resolution in physical models occurs frequently in practice special local features of the forcing function operator coefficients boundary and boundary conditions can demand resolution in restricted regions of the domain that is much finer than the required global resolution it is important that the discretization and solution processes account for this locally that is that the local phenomena do not precipitate a dramatic increase in the overall computation unfortunately this objective of efficiently adapting to local features is often in conflict with the solution process equation solvers can degrade or even fail to apply in the presence of varying discretization scales data structures that account for irregular grids can be cumbersome the computer architecture may not be able to effectively account for grid irregularity e g vectorizability may be inhibited etc in fact even the discretization process itself may find difficulty with this objective for finite differences it is problematic to develop accurate difference formulae for irregular grids for finite elements this objective is reflected in the substantial overhead costs needed to automate the discretization the fast adaptive composite grid method fac 11 is a discretization and solution method designed to achieve efficient local resolution by constructing the

zernike functions rigged hilbert spaces and potential applications we revise the symmetries of the zernike polynomials that determine the lie algebra su 1 1 su 1 1 we show how they induce discrete as well as continuous bases that coexist in the framework of rigged hilbert spaces we also discuss some other interesting properties of zernike polynomials and zernike functions one of the areas of interest of zernike functions has been their applications in optics here we suggest that operators on the spaces of zernike functions may play a role in optical image processing we revise the symmetries of the zernike polynomials that determine the lie algebra su 1 1 su 1 1 we show how they induce discrete as well as continuous bases that coexist in the framework of rigged hilbert spaces we also discuss some other interesting properties of zernike polynomials and zernike functions one of the areas of interest of zernike functions has been their applications in optics here we suggest that operators on the spaces of zernike functions may play a role in optical image processing

asymptotic formulae for two variable hermite polynomials new asymptotic formulae for the two dimensional hermite polynomials with large values of indices are found the applications to the photon distribution functions in squeezed one mode mixed quantum states are considered

generalized zernike or disc polynomials an application in quantum optics an application of the generalized zernike or disc polynomials recently introduced in the literature is shown resorting to the lie algebra based investigation of the dynamics of quantum systems driven by two mode interaction hamiltonians further properties of the associated disc functions are deduced also a generalization of the disc polynomials towards the hahn polynomials is suggested

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

image analysis by generalized chebyshev fourier and generalized pseudo jacobi fourier moments in this paper we present two new sets named the generalized chebyshev fourier radial polynomials and the generalized pseudo jacobi fourier radial polynomials which are orthogonal over the unit circle these generalized radial polynomials are then scaled to define two new types of continuous orthogonal moments which are invariant to rotation the classical chebyshev fourier and pseudo jacobi fourier moments are the particular cases of the proposed moments with parameter 0 the relationships among the proposed two generalized radial polynomials and jacobi polynomials shift jacobi polynomials and the hypergeometric functions are derived in detail and some interesting properties are discussed two recursive methods are developed for computing radial polynomials so that it is possible to improve computation speed and to avoid numerical instability simulation results are provided to validate the proposed moment functions and to compare their performance with previous works highlightspaper presents two generalized radial polynomials that are orthogonal over the unit circle paper constructs two new generalized descriptors using the scaled radial polynomials two recursive strategies for the computation of proposed radial polynomials are presented the proposed polynomials are related to jacobi shift jacobi polynomials and hypergeometric functions the distribution of zeroes of the proposed polynomials can be controlled by free parameter

a projection method for locally refined grids a numerical method for the solution of the two dimensional euler equatons for incompressible flow on locally refined grids is presented the method is a second order godunov projection method adapted from bell colella and glaz second order accuracy of the numerical method in time and space is established through numerical experiments the main contributions of this work concern the formulation and implementation of a projection for refined grids a discussion of the adjointness relation between gradient and divergence operators for a refined grid mac projection is presented and a refined grid approximate projection is developed an efficient multigrid method which exactly solves the projection is developed and a method for casting certain approximate projections as mac projections on refined grids is presented

combinatorial and analytic properties of the n dimensional hermite polynomials abstract we investigate some combinatorial and analytic properties of the n dimensional hermite polynomials introduced by hermite in the late 19 th century we derive combinatorial interpretations and recurrence relations for these polynomials we also establish a new linear generating function and a kibble slepian formula for the n dimensional hermite polynomials which generalize the kibble slepian formula for the univariate hermite polynomials and the poisson kernel mehler formula for the n dimensional hermite polynomials

existence of pearled patterns in the planar functionalized cahn hilliard equation the functionalized cahn hilliard fch equation supports planar and circular bilayer interfaces as equilibria which may lose their stability through the pearling bifurcation a periodic high frequency in plane modulation of the bilayer thickness in two spatial dimensions we employ spatial dynamics and a center manifold reduction to reduce the fch equation to an 8th order ode system a normal form analysis and a xed point theorem argument show that the reduced system admits a degenerate 1 1 resonant normal form from which we deduce that the onset of the pearling bifurcation coincides with the creation of a two parameter family of pearled equilibria which are periodic in the in plane direction and exponentially localized in the transverse direction

general form for obtaining unit disc based generalized orthogonal moments the rotation invariance of the classical disc based moments such as zernike moments zms pseudo zms pzms and orthogonal fourier mellin moments ofmms makes them attractive as descriptors for the purpose of recognition tasks however less work has been performed for the generalization of these moment functions in this paper four general forms are developed to obtain a class of disc based generalized radial polynomials that are orthogonal over the unit circle these radial polynomials are scaled to ensure numerical stability and some useful properties are discussed for potential applications they could be used in then these scaled radial polynomials are used as kernel functions to construct a series of unit disc based generalized orthogonal moments dgms the variation of parameters in dgms can form various types of orthogonal moments 1 generalized zms 2 generalized pzms and 3 generalized ofmms the classical zms pzms and ofmms correspond to a special case of these three generalized moments for which the free parameter alpha 0 each member of this family will share some excellent properties for image representation and recognition tasks such as orthogonality and rotation invariance in addition we have also developed two algorithms the so called m recursive and n recursive methods for the computation of these proposed radial polynomials to improve the numerical stability experimental results show that the proposed methods are superior to the classical disc based moments in terms of image representation capability and classification accuracy

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

quantum superintegrable zernike system we consider the differential equation that zernike proposed to classify aberrations of wavefronts in a circular pupil whose value at the boundary can be nonzero on this account the quantum zernike system where that differential equation is seen as a schrodinger equation with a potential is special in that it has a potential and a boundary condition that are not standard in quantum mechanics we project the disk on a half sphere and there we find that in addition to polar coordinates this system separates into two additional coordinate systems non orthogonal on the pupil disk which lead to schrodinger type equations with poschl teller potentials whose eigen solutions involve legendre gegenbauer and jacobi polynomials this provides new expressions for separated polynomial solutions of the original zernike system that are real the operators which provide the separation constants are found to participate in a superintegrable cubic higgs algebra

differentiability of bizonal positive definite kernels on complex spheres abstract we prove that any continuous function with domain z c z 1 that generates a bizonal positive definite kernel on the unit sphere in c q q 3 is continuously differentiable in z c z 1 up to order q 2 with respect to both z and z in particular the partial derivatives of the function with respect to x re z and y im z exist and are continuous in z c z 1 up to the same order

a mass conservative adaptive fas multigrid solver for cell centered finite difference methods on block structured locally cartesian grids abstract we present a mass conservative full approximation storage fas multigrid solver for cell centered finite difference methods on block structured locally cartesian grids the algorithm is essentially a standard adaptive fas afas scheme but with a simple modification that comes in the form of a mass conservative correction to the coarse level force this correction is facilitated by the creation of a zombie variable analogous to a ghost variable but defined on the coarse grid and lying under the fine grid refinement patch we show that a number of different types of fine level ghost cell interpolation strategies could be used in our framework including low order linear interpolation in our approach the smoother prolongation and restriction operations need never be aware of the mass conservation conditions at the coarse fine interface to maintain global mass conservation we need only modify the usual fas algorithm by correcting the coarse level force function at points adjacent to the coarse fine interface we demonstrate through simulations that the solver converges geometrically at a rate that is h independent and we show the generality of the solver applying it to several nonlinear time dependent and multi dimensional problems in several tests we show that second order asymptotic h 0 convergence is observed for the discretizations provided that 1 at least linear interpolation of the ghost variables is employed and 2 the mass conservation corrections are applied to the coarse level force term

recursive computation of generalised zernike polynomials abstract an algorithmic approach for generating generalised zernike polynomials by differential operators and connection matrices is proposed this is done by introducing a new order of generalised zernike polynomials such that it collects all the polynomials of the same total degree in a column vector the connection matrices between these column vectors composed by the generalised zernike polynomials and a family of polynomials generated by a rodrigues formula are given explicitly this yields a rodrigues type formula for the generalised zernike polynomials themselves with properly defined differential operators another consequence of our approach is the fact that the generalised zernike polynomials obey a rather simple partial differential equation we recall also how to define hermite zernike polynomials

quantum interference effects on the superposition of n displaced number states we describe properties of a quantum system prepared in a superposition of n displaced number states with emphasis on the interference effects among the state components explicit expressions for wavefunctions in the coordinate and fock state representations photon number distribution variances and quantum correlation coefficient of the quadrature operators and quasiprobability distributions are calculated in detail nonclassical effects due to such interference and the analogy with diffraction patterns arising in an n slit young type experiment are the main results in the present work in particular we show that interference and correlation effects in phase space are connected with the nondiagonal term of the quasiprobability distributions providing further insight into the phenomenon

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

curvature driven flow of bi layer interfaces abstract we introduce the functionalized cahn hilliard fch energy a negative multiple of the cahn hilliard energy balanced against the square of its own variational derivative as a finite width regularization of the sharp interface canham helfrich energy mass preserving gradient flows associated to the fch energy are higher order phase field models which develop not only single layer or front type interfaces but also bi layer or homoclinic interfaces with associated endcap and multi junction structures the single layer interfaces manifest a fingering instability which grows into endcapped bi layers the meandering growth of the bi layer interfaces and the subsequent merging lead to a multi junction dominated network that bears a striking similarity to the phase separated domains of both perfluorosulfonic membranes and amphiphilic di block co polymer solutions the bi layers generated by the gradient flows of the fch energy have an interfacial width which scales with e 1 however for fixed e there is a class of bi layers parameterized by width and background state our primary result is the asymptotic derivation of the normal velocity of a closed bi layer hypersurface in r d d 2 coupled to the evolution for the surface width curvature and background state we also show the convergence of the fch energy to a scaled canham helfrich type energy for both single and bi layer interfaces with the surface area coefficient of the limiting canham helfrich energy coupling to the bi layer width thus the bi layer networks grow to maximize surface area while minimizing the square of curvature up to the point that the increase in surface area stretches the bi layers too thin

fock expansion of multimode pure gaussian states the fock expansion of multimode pure gaussian states is derived starting from their representation as displaced and squeezed multimode vacuum states the approach is new and appears to be simpler and more general than previous ones starting from the phase space representation given by the characteristic or wigner function fock expansion is performed in terms of easily evaluable two variable hermite kampe de feriet polynomials a relatively simple and compact expression for the joint statistical distribution of the photon numbers in the different modes is obtained in particular this result enables one to give a simple characterization of separable and entangled states as shown for two mode and three mode gaussian states

new orthogonality relations for the hermite polynomials and related hilbert spaces the hermite polynomials give rise to orthonormal bases in bagmann like hilbert spaces xa 0 a 1 consisting of entire functions these hilbert spaces are related to the gelfand shilov space s1212 viz s1212 0 a 1 xa

general hermite and laguerre two dimensional polynomials general hermite and laguerre two dimensional 2d polynomials which form a complex three parameter unification of the special hermite and laguerre 2d polynomials are defined and investigated the general hermite 2d polynomials are related to the two variable hermite polynomials but are not the same the advantage of the newly introduced hermite and laguerre 2d polynomials is that they satisfy orthogonality relations in a direct way whereas for the purpose of orthonormalization of the two variable hermite polynomials two different sets of such polynomials are introduced which are biorthonormal to each other the matrix which plays a role in the new definition of hermite and laguerre 2d polynomials is in a considered sense the square root of the matrix which plays a role in the definition of two variable hermite polynomials two essentially different explicit representations of the hermite and laguerre 2d polynomials are derived where the first involves jacobi polynomials as coefficients in superpositions of special hermite or laguerre 2d polynomials and the second is a superposition of products of two hermite polynomials with decreasing indices and with coefficients related to the special laguerre 2d polynomials generating functions are derived for the hermite and laguerre 2d polynomials

a cell centered adaptive projection method for the incompressible euler equations we present an algorithm to compute adaptive solutions for incompressible flows using block structured local refinement in both space and time this method uses a projection formulation based on a cell centered approximate projection which allows the use of a single set of cell centered solvers because of refinement in time additional steps are taken to accurately discretize the advection and projection operators at grid refinement boundaries using composite operators which span the coarse and refined grids this ensures that the method is approximately freestream preserving and satisfies an appropriate form of the divergence constraint

on the dimensions of oscillator algebras induced by orthogonal polynomials there is a generalized oscillator algebra associated with every class of orthogonal polynomials n x n 0 on the real line satisfying a three term recurrence relation x n x bn n 1 x bn 1 n 1 x 0 x 1 b 1 0 this note presents necessary and sufficient conditions on bn for such algebras to be of finite dimension as examples we discuss the dimensions of oscillator algebras associated with hermite legendre and gegenbauer polynomials some remarks on the dimensions of oscillator algebras associated with multi boson systems are also presented

composite overlapping meshes for the solution of partial differential equations abstract we discuss the generation of curvilinear composite overlapping grids and the numerical solution of partial differential equations on them a composite overlapping grid consists of a set of curvilinear component grids that cover a region and overlap where they meet continuity conditions interpolation are imposed at the overlapping boundaries the principal advantage of composite grids is in the generation of grids for regions with complicated geometries the grid construction program cmpgrd is used to create composite grids with any number of component grids we describe some techniques for the solution of elliptic and time dependent pdes on composite meshes applications to the solution of the compressible navier stokes equations are presented

an invariant class of wave packets for the wigner transform abstract generalised hagedorn wave packets appear as exact solutions of schrodinger equations with quadratic possibly complex potential and are given by a polynomial times a gaussian we show that the wigner transform of generalised hagedorn wave packets is a wave packet of the same type in phase space the proofs build on a parametrisation via lagrangian frames and a detailed analysis of the polynomial prefactors including a novel laguerre connection our findings directly imply the recently found tensor product structure of the wigner transform of hagedorn wave packets

discrete spherical harmonic oscillator transforms on the cartesian grids using transformation coefficients the analog harmonic oscillators are well studied in quantum physics including their energy states wavefunctions orthogonal properties and eigenfunctions of the fourier transform in addition the continuous solutions in different dimension and coordinate systems are known some discrete equivalents of the 1d wavefunctions were also studied however in the 3d spherical coordinate system the discrete equivalents of the 3d wavefunctions are not established in this paper we focus on the spherical harmonic oscillator wavefunctions shows the spherical harmonic oscillator transforms shots and their discrete implementation the shows can be synthesized by linear combinations of the hermite gaussian functions with proper transformation coefficients we find that computing the coefficients can be speeded up using the fast fourier transforms or some recursive relations these coefficients relate the hermite transforms with the shots some applications of the discrete shows and the discrete shots are introduced first the shows are exactly the eigenfunctions of the 3d dft also the shots can be used to derive the spherical harmonic oscillator descriptor shod which is a rotational invariant descriptor we find that the shod is not only compatible with the existing rotational descriptors for the spherically sampled data but also outperforms the existing rotational descriptors for 3d cartesian sampled bandlimited input data besides the shots can be used to decompose 3d signals into spherical components hence 3d signal reconstruction is done using partially chosen spherical component and 3d data compression for mri data is demonstrated using shots for medical applications

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

spherical geometry zernike s separability and interbasis expansion coefficients free motion on a 3 sphere properly projected on the 2 dimensional manifold of a disk yields the zernike system which exhibits the fundamental properties of superintegrability these include separability in a variety of coordinate systems polynomial solutions and a particular subset of clebsch gordan coefficients as interbasis expansion coefficients that are higher orthogonal polynomials from the askey scheme deriving these results from the initial formulation in spherical geometry provides the zernike system with interest beyond its optical applications

complex and real hermite polynomials and related quantizations it is known that the anti wick or standard coherent state quantization of the complex plane produces both canonical commutation rule and quantum spectrum of the harmonic oscillator up to the addition of a constant in this work we show that these two issues are not necessarily coupled there exists a family of separable hilbert spaces including the usual fock bargmann space and in each element in this family there exists an overcomplete set of unit norm states resolving the unity with the exception of the fock bargmann case they all produce non canonical commutation relation whereas the quantum spectrum of the harmonic oscillator remains the same up to the addition of a constant the statistical aspects of these non equivalent coherent state quantizations are investigated we also explore the localization aspects in the real line yielded by similar quantizations based on real hermite polynomials

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

polyharmonic bergman spaces and bargmann type transforms abstract in the main result of the paper we prove the decomposition of polyharmonic bergman spaces over the upper half plane into spaces of polyanalytic functions then we introduce the decomposition of polyharmonic bergman spaces into the orthogonal sum of its true polyharmonic bergman subspaces and we state isometric isomorphisms between the different true polyharmonic bergman spaces this allows us to define the k th harmonic hilbert component of a polyharmonic bergman function and to prove closed formulas for the reproducing kernel functions of the true polyharmonic and the polyharmonic bergman spaces the harmonic complex fourier transform is introduced in order to give an explicit description of the cartesian and the laguerre harmonic components of the images of a bargmann type transform for the true polyharmonic bergman spaces finally it is proved that the polyharmonic bergman space of order j is isometric isomorphic to 2j copies of the corresponding hardy space

operational identities and properties of ordinary and generalized special functions the theory of hermite laguerre and of the associated generating functions is reformulated within the framework of an operational formalism this point of view provides more efficient tools which allow the straightforward derivation of a wealth of new and old identities in this paper a central role is played by negative derivative operators and by their link with the tricomi functions and the generalized laguerre polynomials

modes in n dimensional first order systems properties of the extended hermite polynomials are studied it is proved apparently for the first time that these functions times gaussian functions are the modes of an n dimensional first order system thus generalizing an earlier result due to hardy sheng and siegman for one dimension misalignment nonlosslessness and the exact normalization constant are explicitedly included in the treatment phase conjugators are also analyzed

generalized zernike or disc polynomials we investigate generalized zernike or disc polynomials p m n z z which are orthogonal 2d polynomials in the unit disc 0 zz 1 is a free parameter these polynomials can be expressed by jacobi polynomials of transformed arguments in connection with a simple angle dependence a limiting procedure leads to laguerre 2d polynomials lm n z z furthermore we introduce the corresponding orthonormalized disc functions the disc polynomials and disc functions obey two differential equations a first order and a second order one with a certain degree of freedom and the operators of lowering and raising of the indices are found these operators can be closed to a lie algebra su 1 1 su 1 1 new generating functions are derived from an operational representation which is alternative to the rodrigues type representation the one dimensional analogue of the disc polynomials which are orthogonal polynomials in the interval 0 r 1 with weight factors 1 r2 x are ultraspherical or gegenbauer polynomials in a new standardization the lowering and raising operators to the corresponding orthonormalized functions form a simple su 1 1 lie algebra this is given in the appendix in sketched form

multigrid elliptic equation solver with adaptive mesh refinement in this paper we describe in detail the computational algorithm used by our parallel multigrid elliptic equation solver with adaptive mesh refinement our code uses truncation error estimates to adaptively refine the grid as part of the solution process the presentation includes a discussion of the orders of accuracy that we use for prolongation and restriction operators to ensure second order accurate results and to minimize computational work code tests are presented that confirm the overall second order accuracy and demonstrate the savings in computational resources provided by adaptive mesh refinement

two variable extensions of the laguerre and disc polynomials abstract this work contains a detailed study of a one parameter generalization of the 2 d hermite polynomials and a two parameter extension of zernike s disc polynomials we derive linear and bilinear generating functions and explicit formulas for our generalizations and study integrals of products of some of these 2 d orthogonal polynomials we also establish a combinatorial inequality involving elementary symmetric functions and solve the connection coefficient problem for our polynomials

a cell centred finite volume method for the poisson problem on non graded quadtrees with second order accurate gradients this paper introduces a two dimensional cell centred finite volume discretization of the poisson problem on adaptive cartesian quadtree grids which exhibits second order accuracy in both the solution and its gradients and requires no grading condition between adjacent cells at t junction configurations which occur wherever resolution differs between neighboring cells use of the standard centred difference gradient stencil requires that ghost values be constructed by interpolation to properly recover second order accuracy in the resulting numerical gradients prior work addressing block structured grids and graded trees has shown that quadratic rather than linear interpolation is required the gradients otherwise exhibit only first order convergence which limits potential applications such as fluid flow however previous schemes fail or lose accuracy in the presence of the more complex t junction geometries arising in the case of general non graded quadtrees which place no restrictions on the resolution of neighboring cells we therefore propose novel quadratic interpolant constructions for this case that enable second order convergence by relying on stencils oriented diagonally and applied recursively as needed the method handles complex tree topologies and large resolution jumps between neighboring cells even along the domain boundary and both dirichlet and neumann boundary conditions are supported numerical experiments confirm the overall second order accuracy of the method in the l norm

a presentation for the stabiliser of an element in a free product let g i l gi be the free product of finitely many indecomposable groups none of which are infinite cyclic with natural assumptions on the factors gi the stabiliser in the automorphism group aut g of any element of g is finitely presented

eigenvalue problems for the schrodinger operator with the magnetic field on a compact riemannian manifold abstract we consider the eigenvalue problem of the schrodinger operator with the magnetic field on a compact riemannian manifold first we discuss the least eigenvalue we give a representation of the least eigenvalue by the variational formula and give a relation to the least eigenvalue of the schrodinger operator without the magnetic field second we discuss the asymptotic distribution of eigenvalues by obtaining the asymptotic expansion of the kernel of semigroup here we use the theory of asymptotic expansion for wiener functionals

corrigendum general hermite and laguerre two dimensional polynomials by comparing formula 5 5 with formula a 4 in appendix a a difference in the order of indices v xy in 5 5 with the corresponding v 21 in a 4 was overlooked this has an influence on the orthogonality and completeness relations of the hermite 2d functions which possess the following correct form with an additional transposition notation of u 1 and analogously for the laguerre 2d functions by substitution where is the contragredient matrix to u in order to get full compatibility with appendix a one may substitute in section 5 matrix v by its transposed v i e v v vxy vyx and vzz vz z

holomorphic hermite polynomials in two variables abstract generalizations of the hermite polynomials to many variables and or to the complex domain have been located in mathematical and physical literature for some decades polynomials traditionally called complex hermite ones are mostly understood as polynomials in z and z which in fact makes them polynomials in two real variables with complex coefficients the present paper proposes to investigate for the first time holomorphic hermite polynomials in two variables their algebraic and analytic properties are developed here while the algebraic properties do not differ too much for those considered so far their analytic features are based on a kind of non rotational orthogonality invented by van eijndhoven and meyers inspired by their invention we merely follow the idea of bargmann s seminal paper 1961 giving explicit construction of reproducing kernel hilbert spaces based on those polynomials homotopic behaviour of our new formation culminates in comparing it to the very classical bargmann space of two variables on one edge and the aforementioned hermite polynomials in z and z on the other unlike in the case of bargmann s basis our hermite polynomials are not product ones but factorize to it when bonded together with the first case of limit properties leading both to the bargmann basis and suitable form of the reproducing kernel also in the second limit we recover standard results obeyed by hermite polynomials in z and z

a conservative adaptive projection method for the variable density incompressible navier stokes equations in this paper we present a method for solving the equations governing time dependent variable density incompressible flow in two or three dimensions on an adaptive hierarchy of grids the method is based on a projection formulation in which we first solve advection diffusion equations to predict intermediate velocities and then project these velocities onto a space of approximately divergence free vector fields our treatment of the first step uses a specialized second order upwind method for differencing the nonlinear convection terms that provides a robust treatment of these terms suitable for inviscid and high reynolds number flow density and other scalars are advected in such a way as to maintain conservation if appropriate and free stream preservation our approach to adaptive refinement uses a nested hierarchy of logically rectangular girds with simultaneous refinement of the girds in both space and time the integration algorithm on the grid hierarchy is a recursive procedure in which coarse grids are advanced in time fine grids are advanced multiple steps to reach the same time as the coarse grids and the data at different levels are then synchronized the single grid algorithm is described briefly but the emphasis here is on the time stepping procedure for the adaptive hierarchy numerical examples are presented to demonstrate the algorithms s accuracy and convergence properties and illustrate the behavior of the method an additional example demonstrates the performance of the method on a more realistic problem namely a three dimensional variable density shear layer

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

multivariable orthogonal polynomials associated with tensor products of the oscillator algebra b 1 it is well known that the tensor product of n 1 lowest weight representations of a generalization of the oscillator algebra b 1 can be depicted by a binary coupling tree with each such binary coupling tree we associate a polynomial rl h x that will turn out to be an n variable hermite polynomial we prove that the polynomials rl h x associated with a fixed binary coupling tree are orthogonal over the n dimensional real space for some weight function which is independent of the shape of the considered binary coupling tree the connection coefficients expressing a polynomial associated with a given binary coupling in terms of polynomials associated with another binary coupling tree are the 3nj coefficients of b 1

lie theoretic generating relations of hermite 2d polynomials in this paper we derive some generating relations involving hermite 2d polynomials h2dp hm n u x y of two variables with an arbitrary 2d matrix u as parameter using lie theoretic approach certain known or new generating relations for the polynomials related to h2dp are also obtained as special cases

an adaptive multigrid technique for the incompressible navier stokes equations an automatic adaptive refinement technique has been coupled to the multigrid approach to produce an efficient and stable solution strategy for solving the steady state incompressible navier stokes equations solutions have been obtained for the driven cavity and flow over a backward facing step for reynolds numbers up to 5000 and 800 respectively the refinement criterion is based on the local truncation error the solution error is monitored and automatic refinement can continue until it is reduced to a satisfactory level for driven cavity flow at re 1000 the adaptive refinement approach reduced the computer memory and cpu time to 20 and 40 of the requirements of the pure multigrid method the primitive variable formulation of the navier stokes equations is used so the method can be extended easily to three dimensions application to other nonlinear elliptic problems is equally possible

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

hermite and laguerre 2d polynomials abstract we define hermite 2d polynomials hm n u x y and laguerre 2d polynomials l m n u z z as functions of two variables with an arbitrary 2d matrix u as parameter and discuss their properties and their explicit representation recursion relations and generating functions for these polynomials are derived the advantage of the introduced hermite and laguerre 2d polynomials in comparison to the related usual two variable hermite polynomials is that they satisfy orthogonality relations in a direct way whereas for the purpose of orthonormalization of the last one has to introduce two different kinds of such polynomials which are biorthogonal to each other

block implicit multigrid solution of navier stokes equations in primitive variables abstract a calculation procedure for rapid computation of steady multidimensional viscous flows is presented the method solves the navier stokes equations in primitive variables using a coupled block implicit multigrid procedure the procedure is applicable to finite difference formulations using staggered locations of the flow variables a smoothing technique called symmetrical coupled gauss seidel scgs is proposed and is empirically observed to provide good smoothing rates the viscous flow in a square cavity with a moving top wall is calculated for a range of reynolds numbers calculations with finite difference grids as large as 321 321 nodes have been made to test the accuracy and efficiency of the calculation scheme the cpu times for these calculations are observed to be significantly smaller than other solution algorithms with primitive variable formulation the calculated flow fields in the cavity are in good agreement with earlier studies of the same flow situation