generalized multiple scale reproducing kernel particle methods abstract an approach to unify reproducing kernel methods under one large umbrella and an extension to include time and spatial shifting are proposed the study is divided into three major topics the groundwork is set by revisiting the fourier analysis of discrete systems the multiresolution concept and its significance in devising the reproducing kernel methods and its discrete counterpart reproducing kernel particle methods are explained an edge detection technique based on multiresolution analysis is developed this wavelet approach together with particle methods gives rise to a straightforward h adaptivity algorithm by using this framework a hermite reproducing kernel method is also proposed and its relation to wavelet methods is presented it is also shown that the new approach generalizes existing kernel methods and it can easily be degenerated into other widely used methods such as partition of unity moving least square interpolants smooth particle hydrodynamics scaling functions and wavelets and multiple scale analysis furthermore the hermite reproducing kernel particle method a particle based discrete version of the hermite reproducing kernel method is developed finally multiple scale methods based on frequency and wave number shifting techniques are presented a stability analysis is also presented for newmark time integration schemes for the low frequency equation numerical examples are presented throughout the paper to illustrate the flexibility and accuracy of this class of multiple scale methods

spherical harmonics and integration in superspace the study of spherical harmonics in superspace introduced in de bie and sommen 2007 j phys a math theor 40 7193 is further elaborated a detailed description of spherical harmonics of degree k is given in terms of bosonic and fermionic pieces which also determines the irreducible pieces under the action of so m x sp 2n in the second part of the paper this decomposition is used to describe all possible integrations over the supersphere it is then shown that only one possibility yields the orthogonality of spherical harmonics of different degrees this is the so called pizzetti integral of which it was shown in de bie and sommen 2007 j phys a math theor 40 7193 that it leads to the berezin integral

on the efetov wegner terms by diagonalizing a hermitian supermatrix the diagonalization of hermitian supermatrices is studied such a change of coordinates is inevitable if one wishes to find certain structures in random matrix theory however it still poses serious problems since until now the calculation of all rothstein contributions known as efetov wegner terms in physics was quite cumbersome we derive the supermatrix bessel function with all efetov wegner terms for an arbitrary rotation invariant probability density function as applications we consider representations of generating functions for hermitian random matrices with and without an external field as integrals over eigenvalues of hermitian supermatrices all results are obtained with all efetov wegner terms which were previously unknown in such an explicit and compact representation

optimal control problem for viscous cahn hilliard equation abstract this paper is concerned with the viscous cahn hilliard equation which arises in the dynamics of viscous first order phase transitions in cooling binary solutions the optimal control under boundary condition is given and the existence of optimal solution to the equation is proved

a supersymmetry approach to billiards with randomly distributed scatterers the density of states for a chaotic billiard with randomly distributed point like scatterers is calculated doubly averaged over the positions of the impurities and shape of the billiard truncating the billiard hamiltonian to an n n matrix an explicit analytic expression is obtained for the case of broken time reversal symmetry depending on the rank n of the matrix number l of scatterers and strength of the scattering potential in the strong coupling limit a discontinuous change is observed in the density of states as soon as l exceeds n

moving least square reproducing kernel method part ii fourier analysis in part i of this work the moving least square reproducing kernel mlsrk method is formulated and implemented based on its generic construction an m consistency structure is discovered and the convergence theorems are established in this part of the work a systematic fourier analysis is employed to evaluate and further establish the method the preliminary fourier analysis reveals that the mlsrk method is stable for sufficiently dense non degenerated particle distribution in the sense that the kernel function family satisfies the riesz bound one of the novelties of the current approach is to treat the mlsrk method as a variant of the standard finite element method and depart from there to make a connection with the multiresolution approximation in the spirits of multiresolution analysis we propose the following mlsrk transformation fm k hu i 1npu k ik h x xi x w1 the highlight of this paper is to embrace the mlsrk formulation with the notion of the controlled fp approximation based on its characterization the strang fix condition for example a systematic procedure is proposed to design new window functions so they can enhance the computational performance of the mlsrk algorithm the main effort here is to obtain a constant correction function in the interior region of a general domain i e c h 1 this can create a leap in the approximation order of the mlsrk algorithm significantly if a highly smooth window function is embedded within the kernel one consequence of this development is the synchronized convergence phenomenon a unique convergence mechanism for the mlsrk method i e by properly tuning the dilation parameter the convergence rate of higher order error norms will approach the same order convergence rate of the l2 error norm they are synchronized

supersymmetry for systems with unitary disorder circular ensembles a generalized hubbard stratonovitch transformation relating an integral over random unitary matrices to an integral over efetov s unitary model manifold is introduced this transformation adapts the supersymmetry method to disordered and chaotic systems that are modelled not by a hamiltonian but by their scattering matrix or time evolution operator in contrast to the standard method no saddle point approximation is made and no massive modes have to be eliminated this first paper on the subject applies the generalized hubbard stratonovitch transformation to dyson s circular unitary ensemble it is shown how a supersymmetric variant of the harish chandra itzykson zuber formula can be used to compute in the large n limit the n level correlation function for any n non trivial applications to random network models quantum chaotic maps and lattice gauge theory are expected

discretization errors associated with reproducing kernel methods one dimensional domains abstract the reproducing kernel particle method rkpm is a discretization technique for partial differential equations that uses the method of weighted residuals classical reproducing kernel theory and modified kernels to produce either mesh free or mesh full methods although rkpm has many appealing attributes the method is new and its numerical performance is just beginning to be quantified in order to address the numerical performance of rkpm von neumann analysis is performed for semi discretizations of three model one dimensional pdes the von neumann analyses results are used to examine the global and asymptotic behavior of the semi discretizations the model pdes considered for this analysis include the parabolic and hyperbolic first and second order wave equations numerical diffusivity for the former and phase speed for the latter are presented over the range of discrete wavenumbers and in an asymptotic sense as the particle spacing tends to zero group speed is also presented for the hyperbolic problems excellent diffusive and dispersive characteristics are observed when a consistent mass matrix formulation is used with the proper choice of refinement parameter in contrast the row sum lumped mass matrix formulation severely degraded performance the asymptotic analysis indicates that very good rates of convergence o x 6 o x 8 are possible when the consistent mass matrix formulation is used with an appropriate choice of refinement parameter and quadrature rule

superbosonization of invariant random matrix ensembles superbosonization is a new variant of the method of commuting and anti commuting variables as used in studying random matrix models of disordered and chaotic quantum systems we here give a concise mathematical exposition of the key formulas of superbosonization conceived by analogy with the bosonization technique for dirac fermions the new method differs from the traditional one in that the superbosonization field is dual to the usual hubbard stratonovich field the present paper addresses invariant random matrix ensembles with symmetry group un on or uspn giving precise definitions and conditions of validity in each case the method is illustrated at the example of wegner s n orbital model superbosonization promises to become a powerful tool for investigating the universality of spectral correlation functions for a broad class of random matrix ensembles of non gaussian and or non invariant type

an improved reproducing kernel particle method for nearly incompressible finite elasticity the previously developed reproducing kernel particle method rkpm employs a high order quadrature rule for desired domain integration accuracy this leads to an over constrained condition in the limit of incompressibility and volumetric locking and pressure oscillation were encountered the employment of a large support size in the reproducing kernel shape function increases the dependency in the discrete constraint equations at quadrature points and thereby relieves locking however this approach consumes high cpu and it cannot effectively resolve pressure oscillation difficulty in this paper a pressure projection method is introduced by locally projecting the pressure onto a lower order space to reduce the number of independent discrete constraint equations this approach relieves the over constrained condition and thus eliminates volumetric locking and pressure oscillation without the expense of employing large support size in rkpm the method is developed in a general framework of nearly incompressible finite elasticity and therefore linear problems are also applicable to further reduce the computational cost a stabilized reduced integration method based on an assumed strain approach on the gradient matrix associated with the deformation gradient is also introduced the resulting stiffness matrix and force vector of rkpm are obtained explicitly without numerical integration

critique of the replica trick it is shown that the replica trick fails to give the correct non perturbative result for the two point function s2 of the gaussian unitary ensemble of n n random matrices the failure arises from an incorrect description of the symmetries of the random matrix system in the limit n to infinity the correct description which involves integration over both non compact and compact degrees of freedom is obtained by using the method of superfields some implications for the localisation transition in disordered electronic systems and the theory of the quantised hall effect are suggested

from gumbel to tracy widom the tracy widom distribution that has been much studied in recent years can be thought of as an extreme value distribution we discuss interpolation between the classical extreme value distribution exp exp x the gumbel distribution and the tracy widom distribution there is a family of determinantal processes whose edge behaviour interpolates between a poisson process with density exp x and the airy kernel point process this process can be obtained as a scaling limit of a grand canonical version of a random matrix model introduced by moshe neuberger and shapiro we also consider the deformed gue ensemble m m 0 sqrt 2s v with m 0 diagonal with independent elements and v from gue here we do not see a transition from tracy widom to gumbel but rather a transition from tracy widom to gaussian

forced vibrations in the medium frequency range solved by a partition of unity method with local information a new approach for the computation of the forced vibrations up to the medium frequency range is formulated for thin plates it is based on the partition of unity method pum first proposed by babuska and used here to solve the elastodynamic problem the paper focuses on the introduction of local information in the basis of the pum coming from previous approximations in order to enhance the accuracy of the solution the method may be iterative and generates a pum approximation leading to smaller models compared with the finite element ones required for a same accuracy level it shows very promising results in terms of frequency range accuracy and computational time copyright 2004 john wiley sons ltd

results of von neumann analyses for reproducing kernel semi discretizations the reproducing kernel particle method rkpm has many attractive properties that make it ideal for treating a broad class of physical problems rkpm may be implemented in a mesh full or a mesh free manner and provides the ability to tune the method via the selection of a window function and its associated dilation parameter in order to achieve the requisite numerical performance rkpm also provides a framework for performing hierarchical computations making it an ideal candidate for simulating multi scale problems although the method has many appealing attributes it is quite new and its numerical performance is still being quantified with respect to more traditional discretization techniques in order to assess the numerical performance of rkpm detailed studies of the method on a series of model partial differential equations has been undertaken the results of von neumann analyses for rkpm semi discretizations of one and two dimensional first and second order wave equations are presented in the form of phase and group errors excellent dispersion characteristics are found for the consistent mass matrix with the proper choice of dilation parameter in contrast row sum lumping the mass matrix is demonstrated to introduce severe lagging phase errors a higher order mass matrix improves the dispersion characteristics relative to the lumped mass matrix but also yields significant lagging phase errors relative to the fully integrated consistent mass matrix published in 2000 by john wiley sons ltd

maximum entropy methods for time harmonic acoustics abstract this paper explores the application of maximum entropy methods max ent to time harmonic acoustic problems max ent basis functions are mesh free approximants that are constructed observing an equivalence between basis functions and discrete probability distributions and applying jaynes s maximum entropy principle they are c continuous and therefore they are particularly suited for the resolution of helmholtz problems where classical finite element methods show a poor accuracy in the high frequency region in addition it was recently shown that max ent approximants can be blended with isogeometric basis functions on the boundary of the domain this preserves the correct representation of the boundary like in isogeometric analysis with the advantage that the discretization of the interior of the domain is straightforward in this paper the max ent mathematical formulation is reviewed and then some numerical applications are studied including a 2d car cavity geometry defined by b spline curves in all cases if the same nodal discretization is used finite elements results are significantly improved

moving least square reproducing kernel methods i methodology and convergence this paper formulates the moving least square interpolation scheme in a framework of the so called moving least square reproducing kernel mlsrk representation in this study the procedure of constructing moving least square interpolation function is facilitated by using the notion of reproducing kernel formulation which as a generalization of the early discrete approach establishes a continuous basis for a partition of unity this new formulation possesses the quality of simplicity and it is easy to implement moreover the reproducing kernel formula proposed is not only able to reproduce any mth order polynomial exactly on an irregular particle distribution but also serves as a projection operator that can approximate any smooth function globally with an optimal accuracy in this contribution a generic m consistency relation has been found which is the essential property of the mlsrk approximation an interpolation error estimate is given to assess the convergence rate of the approximation it is shown that for sufficiently smooth function the interpolant expansion in terms of sampled values will converge to the original function in the sobolev norms as a meshless method the convergence rate is measured by a new control variable dilation parameter of the window function instead of the mesh size h as usually done in the finite element analysis to illustrate the procedure convergence has been shown for the numerical solution of the second order elliptic differential equations in a galerkin procedure invoked with this interpolant in the numerical example a two point boundary problem is solved by using the method and an optimal convergence rate is observed with respect to various norms

an iterative defect correction type meshless method for acoustics accurate numerical simulation of acoustic wave propagation is still an open problem particularly for medium frequencies we have thus formulated a new numerical method better suited to the acoustical problem the element free galerkin method efgm improved by appropriate basis functions computed by a defect correction approach one of the efgm advantages is that the shape functions are customizable indeed we can construct the basis of the approximation with terms that are suited to the problem which has to be solved acoustical problems in cavities with boundary t are governed by the helmholtz equation completed with appropriate boundary conditions as the pressure p x y is a complex variable it can always be expressed as a function of cos x y and sin x y where x y is the phase of the wave in each point x y if the exact distribution x y of the phase is known and if a meshless basis 1 cos x y sin x y is used then the exact solution of the acoustic problem can be obtained obviously in real life cases the distribution of the phase is unknown the aim of our work is to resolve as a first step the acoustic problem by using a polynomial basis to obtain a first approximation of the pressure field p x y as a second step from p x y we compute the distribution of the phase x y and we introduce it in the meshless basis in order to compute a second approximated pressure field p x y from p x y a new distribution of the phase is computed in order to obtain a third approximated pressure field and so on until a convergence criterion concerning the pressure or the phase is obtained so an iterative defect correction type meshless method has been developed to compute the pressure field in this work will show the efficiency of this meshless method in terms of accuracy and in terms of computational time we will also compare the performance of this method with the classical finite element method copyright 2003 john wiley sons ltd

dyson s correlation functions and graded symmetry a new derivation of dyson s k level correlation functions of the gaussian unitary ensemble gue is given the method uses matrices with graded symmetry the number of integrations needed for the ensemble average becomes independent of the level number n for arbitrary level number n the k level correlation function is expressed as an integral involving the eigenvalues of a 2k 2k graded matrix the limit of infinitely many levels n is calculated by a simple saddle point approximation of this integral avoiding the introduction of hermite polynomials and oscillator wave functions

the k point random matrix kernels obtained from one point supermatrix models the k point correlation functions of the gaussian random matrix ensembles are certain determinants of functions which depend on only two arguments they are referred to as kernels since they are the building blocks of all correlations we show that the kernels are obtained for arbitrary level number directly from supermatrix models for one point functions more precisely the generating functions of the one point functions are equivalent to the kernels this is surprising because it implies that already the one point generating function holds essential information about the k point correlations this also establishes a link to the averaged ratios of spectral determinants i e of characteristic polynomials

a meshless galerkin method for stokes problems using boundary integral equations a meshless galerkin scheme for the simulation of two dimensional incompressible viscous fluid flows in primitive variables is described in this paper this method combines a boundary integral formulation for the stokes equation with the moving least squares mls approximations for construction of trial and test functions for galerkin approximations unlike the domain type method this scheme requires only a nodal structure on the bounding surface of a body for approximation of boundary unknowns thus it is especially suitable for the exterior problems compared to other meshless methods such as the boundary node method and the element free galerkin method in which the mls is also introduced boundary conditions do not present any difficulty in using this meshless method the convergence and error estimates of this approach are presented numerical examples are also given to show the efficiency of the method