stability analysis of particle methods with corrected derivatives abstract the stability of discretizations by particle methods with corrected derivatives is analyzed it is shown that the standard particle method which is equivalent to the element free galerkin method with an eulerian kernel and nodal quadrature has two sources of instability 1 rank deficiency of the discrete equations and 2 distortion of the material instability the latter leads to the so called tensile instability it is shown that a lagrangian kernel with the addition of stress points eliminates both instabilities examples that verify the stability of the new formulation are given

a complex variable boundary point interpolation method for the nonlinear signorini problem abstract a complex variable boundary point interpolation method cvbpim is presented for numerical analysis of the nonlinear signorini problem to reduce the computational cost and to improve the stability of the point interpolation method a complex variable point interpolation method cvpim is developed to construct meshless shape functions with interpolation properties by using a projection technique to deal with the nonlinear inequality boundary conditions and combining boundary integral equations with the cvpim to establish discrete linear algebraic systems the cvbpim is an easy to implement and boundary only meshless method for nonlinear signorini problems numerical results are presented to show the accuracy and efficiency of the method

holomorphic quantization on the torus and finite quantum mechanics we explicitly construct the quantization of classical linear maps of on toroidal phase space of arbitrary modulus using the holomorphic chiral version of the metaplectic representation we show that finite quantum mechanics fqm on tori of arbitrary integer discretization is a consistent restriction of the holomorphic quantization of to the subgroup being the principal congruent subgroup on a finite dimensional hilbert space the generators of the rotation group for arbitrary values of l are determined as well as their quantum mechanical eigenvalues and eigenstates

error analysis for moving least squares approximation in 2d space abstract in this paper we provide a theoretical analysis of the moving least squares mls approximation which belongs to the family of meshless methods first the non matrix form of the mls shape function in two dimensional space is obtained by using consistency conditions the error estimates for mls approximation in sobolev space are presented when u x y c m 1 and u x y w m 1 q respectively we establish the error estimates for interpolating element free galerkin iefg method when it is used for solving poisson s equation the error bound is related to the radii of the weight functions and the bound of the norm of derivatives of shape functions three numerical examples are selected to confirm our analysis

moyal like form of the star product for generalized su 2 stratonovich weyl symbols the star product is the basic tool used in the phase space formulation of quantum mechanics we find a differential form of the star product for a class of mbox s parametrized su 2 stratonovich weyl symbols which appear in the phase space representation of spin like systems the limit of large spin is considered and the asymptotic form of the differential operator defining the star product is obtained

finite two dimensional oscillator ii the radial model a finite two dimensional radial oscillator of n 1 2 points is proposed with the dynamical lie algebra so 4 su 2 x su 2 y examined in part i of this work but reduced by a subalgebra chain so 4 so 3 so 2 as before there are a finite number of energies and angular momenta the casimir spectrum of the new chain provides the integer radii 0 n and the 2 1 discrete angles on each circle are obtained from the finite fourier transform of angular momenta the wavefunctions of the finite radial oscillator are so 3 clebsch gordan coefficients we define here the hankel hahn transforms with dual hahn polynomials as finite n unitary approximations to hankel integral transforms with bessel functions obtained in the contraction limit n

digital computation of linear canonical transforms we deal with the problem of efficient and accurate digital computation of the samples of the linear canonical transform lct of a function from the samples of the original function two approaches are presented and compared the first is based on decomposition of the lct into chirp multiplication fourier transformation and scaling operations the second is based on decomposition of the lct into a fractional fourier transform followed by scaling and chirp multiplication both algorithms take n log n time where n is the time bandwidth product of the signals the only essential deviation from exactness arises from the approximation of a continuous fourier transform with the discrete fourier transform thus the algorithms compute lcts with a performance similar to that of the fast fourier transform algorithm in computing the fourier transform both in terms of speed and accuracy

error analysis in sobolev spaces for the improved moving least square approximation and the improved element free galerkin method the improved moving least square imls approximation is a method to form shape functions in meshless methods for the application of imls based meshless methods to the numerical solution of boundary value problems it is fundamental to analyze error of the imls approximation in sobolev spaces this paper begins by discussing properties of the imls shape function under appropriate assumption on weight functions error estimates for the imls approximation are then established in sobolev spaces in multiple dimensions the improved element free galerkin iefg method is a typical meshless galerkin method based on coupling the imls approximation and galerkin weak form error analysis of the iefg method is also provided numerical examples are finally presented to prove the theoretical error results

error bounds for gmls derivatives approximations of sobolev functions this paper provides the error estimates for generalized moving least squares gmls derivatives approximations of a sobolev function in l p norms and extends them for local weak forms of dmlpg methods sometimes they are called diffuse or uncertain derivatives but precisely they are direct approximants of exact derivatives which possess the optimal rates of convergence gmls derivatives approximations are different from the standard derivatives of mls approximation while they are much easier to evaluate at considerably lower cost in this paper the same orders of convergence with comparison to the standard derivatives are obtained for them

quantum algebraic structures compatible with the harmonic oscillator newton equation we study some of the algebraic structures that are compatible with the quantization of the harmonic oscillator through its newton equation examples of such structures are given they include undeformed and q deformed oscillators as well as the su 2 and the deformed suq 2 lie algebras which appear in a variety of physical models

equipartition of the eigenfunctions of quantized ergodic maps on the torus we give a simple proof of the equipartition of the eigenfunctions of a class of quantized ergodic area preserving maps on the torus examples are the irrational translations the skew translations the hyperbolic automorphisms and some of their perturbations

wigner functions for curved spaces ii on spheres the form of the wigner distribution function for hamiltonian systems in spaces of constant negative curvature i e hyperboloids proposed in m a alonso g s pogosyan and k b wolf wigner functions for curved spaces i on hyperboloids j math phys 43 5857 2002 is extended here to spaces whose curvature is constant and positive i e spheres an essential part of this construction is the use of the functions of sherman and volobuyev which are an overcomplete set of plane wave like solutions of the laplace beltrami equation for this space rotations that displace the poles transform these functions with a multiplier factor and their momentum direction becomes formally complex the covariance properties of the proposed wigner function are understood in these terms as an example for the one dimensional case we consider the energy eigenstates of the oscillator on the circle in a poschl teller potential the standard theory of quantum oscillators is regained in the contraction limit to the space of zero curvature

analysis of an augmented moving least squares approximation and the associated localized method of fundamental solutions abstract the localized method of fundamental solutions lmfs is an efficient meshless collocation method that combines the concept of localization and the method of fundamental solutions mfs the resultant system of linear algebraic equations in the lmfs is sparse and banded and thus drastically reduces the storage and computational burden of the mfs in the lmfs the moving least square mls approximation based on fundamental solutions is used to construct approximate solution at each node in this paper this fundamental solutions based mls approximation named as an augmented mls amls approximation is generalized to any point in the computational domain computational formulas theoretical properties and error estimates of the amls approximation are derived then taking laplace equation as an example this paper sets up a framework for the theoretical error analysis of the lmfs finally numerical results are presented to verify the efficiency and theoretical results of the amls approximation and the lmfs convergence and comparison researches are conducted to validate the accuracy convergence and efficiency

invariant inner products on spaces of solutions of the klein gordon and helmholtz equations we construct sesquilinear forms which are invariant under the similarity groups of the klein gordon and helmholtz equations these give rise to positive definite inner products on subspaces of solutions of these equations for the klein gordon case the known poincare invariant inner product for positive energy solutions is recovered for the helmholtz case a new euclidean invariant inner product is presented which involves the function and its normal derivative to a line integrated nonlocally over that line

exact evolution equations for su 2 quasidistribution functions we derive an exact differential evolution equation for a class of su 2 quasiprobability distribution functions linear and quadratic cases are considered as well as the quasiclassical limit of the large dimension of representation s 1

the approximation power of moving least squares a general method for near best approximations to functionals on r d using scattered data information is discussed the method is actually the moving least squares method presented by the backus gilbert approach it is shown that the method works very well for interpolation smoothing and derivatives approximations for the interpolation problem this approach gives mclain s method the method is near best in the sense that the local error is bounded in terms of the error of a local best polynomial approximation the interpolation approximation in r d is shown to be a c function and an approximation order result is proven for quasi uniform sets of data points

jackson type inequalities for h p clouds and error estimates abstract the interest in meshfree methods for solving boundary value problems has grown rapidly in recent years a meshless method that has attracted much interest in the community of computational mechanics is the h p clouds method for this kind of applications it is fundamental to analyze the orders of approximation in this paper we prove jackson type inequalities for h p cloud functions these inequalities set up a general framework for the theoretical analysis of high order error estimates of the h p clouds method with the same remarkable features of finite element theory

continuous vs discrete fractional fourier transforms abstract we compare the finite fourier exponential and fourier kravchuk transforms both are discrete finite versions of the fourier integral transform the latter is a canonical transform whose fractionalization is well defined we examine the harmonic oscillator wavefunctions and their finite counterparts mehta s basis functions and the kravchuk functions the fractionalized fourier kravchuk transform was proposed in j opt soc amer a 14 1997 1467 1477 and is here subject of numerical analysis in particular we follow the harmonic motions of coherent states within a finite discrete optical model of a shallow multimodal waveguide

error analysis of method of lines mol via generalized interpolating moving least squares gimls approximation this paper gives an error analysis for the method of lines mol using generalized interpolating moving least squares gimls approximation in this study error bound for the time dependent linear and nonlinear second order differential equations in d dimension will be obtained when the gimls method is used for approximating the spatial variables also the well known courantfriedrichslewy cfl condition will be derived in both cases linear and nonlinear equations finally numerical examples will be reported to confirm the ability of the proposed technique

analysis of the element free galerkin method for signorini problems abstract an efficient element free galerkin efg method is developed to solve signorini boundary value problems the nonlinear inequality signorini boundary conditions are reduced to a sequence of linear boundary conditions by using a projection iterative scheme then the efg method is used to deduce discrete system equations convergence of the proposed method is discussed numerical examples are finally provided to verify the convergence and high computational accuracy of the method

interpolating stabilized moving least squares mls approximation for 2d elliptic interface problems abstract the main aim of the current paper is to propose a new truly meshless numerical technique to solve the one and two dimensional elliptic interface problems the employed meshless approach is based on a new class of mls approximation the numerical procedure is based on the interpolating stabilized mls ismls approximation the new shape functions that have been made by the ismls technique have the kronecker property thus the dirichlet boundary conditions can be applied directly in the current investigation we propose a meshless collocation method for solving elliptic interface problems based on the shape functions of ismls technique several test problems with sufficient complexity have been studied to check the accuracy and efficiency of the new numerical procedure moreover test problems show the acceptable accuracy and efficiency of the proposed scheme

jacobi ϑ functions and discrete fourier transforms properties of the jacobi 3 function and its derivatives under discrete fourier transforms are investigated and several interesting results are obtained the role of modulo n equivalence classes in the theory of functions is stressed an important conjecture is studied

surfaces generated by moving least squares methods an analysis of moving least squares m l s methods for smoothing and interpolating scattered data is presented in particular theorems are proved concerning the smoothness of interpolants and the description of m l s processes as projection methods some properties of compositions of the m l s projector with projectors associated with finiteelement schemes are also considered the analysis is accompanied by examples of univariate and bivariate problems

a generalized element free galerkin method for stokes problem abstract a generalized element free galerkin gefg method is developed in this paper for solving stokes problem in primitive variable form to obtain stable numerical results for both velocity and pressure extended terms are only introduced into the approximate space of velocity in a special way as that in the generalized finite element method theoretical analysis shows that the gefg method implies a stabilized formulation similar to that in the variational multiscale element free galerkin vmefg method numerical results show the efficiency of the present method and reveal that both computational errors and cpu times of the present method are less than those of the vmefg and the finite element methods

analysis of moving least squares approximation revisited in this article the error estimation of the moving least squares approximation is provided for functions in fractional order sobolev spaces the analysis presented in this paper extends the previous estimations and explains some unnoticed mathematical details an application to galerkin method for partial differential equations is also supplied

completion and continuation of nonlinear traffic time series a probabilistic approach when dealing with nonlinear time series of car traffic on highways one of the outstanding problems to be solved is completion and continuation of data in space and time to this end the underlying process is decomposed into stochastic and deterministic components the former is approximated by gaussian white noise while the latter refers apart from always existing trends to the space and time dependent jam propagation process jams are modelled in terms of dynamical bayesian networks with radial basis functions involved the models developed are used to tackle travel time estimation and prediction results are obtained for one of the most crowded traffic areas of europe namely the ring like highway around cologne

non linear version of stabilized conforming nodal integration for galerkin mesh free methods a stabilized conforming sc nodal integration which meets the integration constraint in the galerkin meshfree approximation is generalized for nonlinear problems in sc nodal integration a strain smoothing stabilization is introduced and applied to the deformation gradient using a lagrangian meshfree discretization the assumed strain method expressed in terms of deformation gradient is employed to formulate an efficient meshfree discrete equation for nonlinear problems a significant gain in computational efficiency is achieved as well as enhanced accuracy in comparison to the solution of meshfree method using gauss integration method the performance of the proposed method appears to be quite robust when dealing with irregular discretization

predicting time series using a neural network as a method of distinguishing chaos from noise a neural network approach is presented for making short term predictions on time series the neural network does better at short term predictions of a chaotic signal than does an optimum autoregressive model also the neural network is clearly capable of distinguishing between chaos and additive noise

wigner distribution function for euclidean systems euclidean systems include poly and monochromatic wide angle optics acoustics and also infinite discrete data sets we use a recently defined wigner operator and quasiprobability distribution function to set up and study the phase space evolution of these models subject to differential and difference equations respectively infinite data sets and two dimensional monochromatic helmholtz fields are thus shown by their wigner function on a cylinder of direction and location the wigner function for polychromatic wavefields has c number coordinates of two dimensional wavenumber and position

wigner distributions and quantum mechanics on lie groups the case of the regular representation we consider the problem of setting up the wigner distribution for states of a quantum system whose configuration space is a lie group the basic properties of wigner distributions in the familiar cartesian case are systematically generalized to accommodate new features which arise when the configuration space changes from n dimensional euclidean space rn to a lie group g the notion of canonical momentum is carefully analyzed and the meanings of marginal probability distributions and their recovery from the wigner distribution are clarified for the case of compact g an explicit definition of the wigner distribution is proposed possessing all the required properties geodesic curves in g which help introduce a notion of the midpoint of two group elements play a central role in the construction

a generalized wigner function for quantum systems with the su 2 dynamical symmetry group we introduce a wigner like quasidistribution function to describe quantum systems with the su 2 dynamic symmetry group this function is defined in a three dimensional group manifold and can be used to represent the states defined in several su 2 invariant subspaces the explicit differential moyal like form of the star product is found and analyzed in the semiclassical limit

quantizations on the circle and coherent states we present a possible construction of coherent states on the unit circle as configuration space our approach is based on borel quantizations on s1 including the aharonov bohm type quantum description coherent states are constructed by perelomov s method as group related coherent states generated by weyl operators on the quantum phase space because of the duality of canonical coordinates and momenta i e the angular variable and the integers this formulation can also be interpreted as coherent states over an infinite periodic chain for the construction we use the analogy with our quantization and coherent states over a finite periodic chain where the quantum phase space was the coherent states constructed in this work are shown to satisfy the resolution of unity to compare them with canonical coherent states some of their further properties are also studied demonstrating similarities as well as substantial differences this article is part of a special issue of journal of physics a mathematical and theoretical devoted to coherent states mathematical and physical aspects

continuum quantum systems as limits of discrete quantum systems i state vectors abstract dynamical systems on continuum hilbert spaces may be realized as limits of dynamical systems on discrete possibly finite dimensional hilbert spaces in this first of four papers on the topic the continuum and discrete spaces are interfaced to one another algebraically convergence of vectors is defined in such a way as to preserve inner products and a necessary and sufficient coordinate wise criterion for convergence is proved

analysis of the element free galerkin method with penalty for general second order elliptic problems abstract in this paper an element free galerkin efg method with penalty for solving general second order elliptic problems with mixed boundary conditions is presented a priori approximate estimations of the penalty method are obtained under some proper assumptions which ensure the availability of the penalty method by defining a special norm the existence and uniqueness for the weak solution of the penalized continuous variational formula are proved which guarantee the validity of the efg discretization error estimates of the efg method are provided in l2 and h1 norms for the dirichlet and the mixed boundaries respectively numerical examples are finally performed to illustrate the theoretical results

the element free galerkin method for the nonlinear p laplacian equation abstract the element free galerkin efg method is developed in this paper for solving the nonlinear p laplacian equation the moving least squares approximation is used to generate meshless shape functions the penalty approach is adopted to enforce the dirichlet boundary condition the galerkin weak form is employed to obtain the system of discrete equations and two iterative procedures are developed to deal with the strong nonlinearity then the computational formulas of the efg method for the p laplacian equation are established numerical results are finally given to verify the convergence and high computational precision of the method

nonlinear component analysis as a kernel eigenvalue problem a new method for performing a nonlinear form of principal component analysis is proposed by the use of integral operator kernel functions one can efficiently compute principal components in high dimensional feature spaces related to input space by some nonlinear map for instance the space of all possible five pixel products in 16 16 images we give the derivation of the method and present experimental results on polynomial feature extraction for pattern recognition

discrete repulsive oscillator wavefunctions for the study of infinite discrete systems on phase space the three dimensional lorentz algebra and group so 2 1 and so 2 1 provide a discrete model of the repulsive oscillator its eigenfunctions are found in the principal irreducible representation series where the compact generator that we identify with the position operator has the infinite discrete spectrum of the integers z while the spectrum of energies is a double continuum the right and left moving wavefunctions are given by hypergeometric functions that form a dirac basis for 2 z under contraction the discrete system limits to the well known quantum repulsive oscillator numerical computations of finite approximations raise further questions on the use of dirac bases for infinite discrete systems

finite two dimensional oscillator i the cartesian model a finite two dimensional oscillator is built as the direct product of two finite one dimensional oscillators using the dynamical lie algebra su 2 x su 2 y the position space in this model is a square grid of points while the ordinary continuous two dimensional quantum oscillator has a symmetry algebra u 2 the symmetry algebra of the finite model is only u 1 x u 1 y because it lacks rotations in the position and momentum plane we show how to import an so 2 group of rotations from the continuum model that transforms unitarily the finite wavefunctions on the fixed square grid we thus propose a finite analogue for fractional u 2 fourier transforms

general approach to mathfrak su n quasi distribution functions we propose an operational form for the kernel of a mapping between an operator acting in a hilbert space of a quantum system with an symmetry group and its symbol in the corresponding classical phase space for symmetric irreps of this mapping is bijective we briefly discuss complications that will occur in the general case

time series prediction by feedforward neural networks is it difficult the difficulties that a neural network faces when trying to learn from a quasiperiodic time series are studied analytically using a teacher student scenario where the random input is divided into two macroscopic regions with different variances 1 and 1 2 1 t he generalization error is found to decrease

simulation of the phase field cahn hilliard and tumor growth models via a numerical scheme element free galerkin method abstract the main aim of this research work is to find the numerical solution based on a meshless technique for both the time dependent cahn hilliard and tumor growth partial differential equations the temporal variable is discretized using a second order method based on semi implicit backward differential formula and the stabilized term is added to the considered time discretization also an adaptive time algorithm is used to reduce the number of iterations of the proposed time discretization besides to approximate the spatial variables the element free galerkin method is considered for both mathematical models the result of full discrete schemes in studied models is solved via biconjugate gradient stabilized algorithm some numerical simulations are reported to show the capability of the numerical scheme presented here

improved time series prediction with a new method for selection of model parameters a new method for model selection in prediction of time series is proposed apart from the conventional criterion of minimizing rms error the method also minimizes the error on the distribution of singularities evaluated through the local holder estimates and its probability density spectrum predictions of two simulated and one real time series have been done using kernel principal component regression kpcr and model parameters of kpcr have been selected employing the proposed as well as the conventional method results obtained demonstrate that the proposed method takes into account the sharp changes in a time series and improves the generalization capability of the kpcr model for better prediction of the unseen test data

boundary element free method befm and its application to two dimensional elasticity problems in this study we first discuss the moving least square approximation mls method in some cases the mls may form an ill conditioned system of equations so that the solution cannot be correctly obtained hence in this paper we propose an improved moving least square approximation imls method in the imls method the orthogonal function system with a weight function is used as the basis function the imls has higher computational efficiency and precision than the mls and will not lead to an ill conditioned system of equations combining the boundary integral equation bie method and the imls approximation method a direct meshless bie method the boundary element free method befm for two dimensional elasticity is presented compared to other meshless bie methods befm is a direct numerical method in which the basic unknown quantity is the real solution of the nodal variables and the boundary conditions can be applied easily hence it has higher computational precision for demonstration purpose selected numerical examples are given

continuum quantum systems as limits of discrete quantum systems iv affine canonical transforms affine canonical transforms complex order fourier transforms and their associated coherent states appear in two scenarios finite discrete and continuum we examine the relationship between the two scenarios making systematic use of inductive limits which were developed in the preceding articles in this series

the boundary element free method for 2d interior and exterior helmholtz problems abstract the boundary element free method befm is developed in this paper for numerical solutions of 2d interior and exterior helmholtz problems with mixed boundary conditions of dirichlet and neumann types a unified boundary integral equation is established for both interior and exterior problems by using the improved interpolating moving least squares method to form meshless shape functions mixed boundary conditions in the befm can be satisfied directly and easily detailed computational formulas are derived to compute weakly and strongly singular integrals over linear and higher order integration cells three numerical integration procedures are developed for the computation of strongly singular integrals numerical examples involving acoustic scattering and radiation problems are presented to show the accuracy and efficiency of the meshless method

an adaptive orthogonal improved interpolating moving least square method and a new boundary element free method abstract the improved interpolating moving least square iimls method has been widely used in data fitting and meshfree methods and the obtained shape functions have the property of the delta function compared with those obtained by the moving least square mls method however the moment matrix in iimls may be singular or ill conditioned because of the ill quality of the point sets used in this paper the weighted orthogonal basis functions are applied in iimls to obtain a diagonal moment matrix which can overcome the difficulty caused by directly inversing singular or ill conditioned matrices however the weighted orthogonal basis functions cannot change the nature of the singular or ill conditioned moment matrix since the diagonal elements of the new moment matrix may be zero or close to zero thus an adaptive scheme is further employed to resolve this problem by ignoring the contribution from the zero or very small diagonal elements in the diagonal moment matrix combined with shifted and scaled polynomial basis functions a stabilized adaptive orthogonal iimls sao iimls approximation is obtained based on this approximation a new boundary element free method is proposed for solving elasticity problems numerical results for curve fitting surface fitting and the new boundary element free method have shown that the proposed sao iimls approximation is accurate stable and performs well for ill quality point sets

wavelet transform based multifractal formalism in outlier detection and localisation for financial time series we present a method of detecting and localising outliers in financial time series and other stochastic processes the method checks the internal consistency of the scaling behaviour of the process within the paradigm of the multifractal spectrum deviation from the expected spectrum is interpreted as the potential presence of outliers the detection part of the method is then supplemented by the localisation analysis part using the local scaling properties of the time series localised outliers can then be removed one by one with the possibility of dynamic verification of spectral properties both the multifractal spectrum formalism and the local scaling properties of the time series are implemented on the wavelet transform modulus maxima tree

discrete time discrete frequency time frequency analysis a formulation of a discrete time discrete frequency wigner distribution for analysis of discrete time periodic signals is given using an approach involving group representation theory this approach is motivated by a well known connection between group theory and the continuous wigner distribution the advantage of this approach is that the resulting discrete distribution satisfies mathematical properties analogous to those satisfied by the continuous distribution after outlining the relationship between group representation theory and time frequency analysis we derive the discrete distribution and exhibit many of its mathematical properties these include time and frequency marginals the weyl correspondence and covariance in particular the interpretation of covariance for the discrete distribution is shown to be different than that for the continuous distribution finally we note some unusual features of this discrete distribution which are a consequence of the group theoretic derivation

linear transformations and aberrations in continuous and finite systems in geometric optics there is a natural distinction between the paraxial and aberration regimes which contain respectively the linear and nonlinear canonical transformations of position and momentum in the phase space in the lie theoretical presentation linear inhomogeneous transformations are generated by linear and quadratic functions of the phase space while aberrations of increasing order are generated by homogeneous functions with higher powers of these coordinates in a way parallel but distinct from the schrodinger quantization of continuous classical systems we quantize the geometric optical model into discrete finite dimensional systems based on the lie algebra whose wavefunctions are n point signals phase space is a sphere and transformations are represented by the n n unitary matrices that form the group we factor this group into linear and nonlinear unitary transformations of phase space and classify all its n2 4 aberrations this offers a new parametrization of based on a chosen subgroup

error analysis of the meshless finite point method abstract the finite point method fpm is a notable truly meshless method based on the moving least squares mls approximation and the point collocation technique in this paper the error of the fpm is analyzed theoretically theoretical results show that the present error bound is directly related to the nodal spacing and the order of basis functions used in the mls approximation the present error estimation is independent of the condition number of the coefficient matrix and improves the previously reported estimations numerical examples with more than 160000 nodes are given to confirm the theoretical result

convergence analysis and error estimates of the element free galerkin method for the second kind of elliptic variational inequalities abstract this paper is presented for the convergence analysis of the element free galerkin method for the second kind of elliptic variational inequalities of which the dirichlet boundary conditions are imposed by the penalty method the error estimates illustrate that the convergence order depends not only on the nodal spacing and the number of basis functions in the moving least squares approximation but also the penalty factor a numerical example verifies the theoretical results of the element free galerkin method

on the stability of the moving least squares approximation and the element free galerkin method in this paper the stability of the moving least squares mls approximation and a stabilized mls approximation is analyzed theoretically and verified numerically it is shown that the stability of the mls approximation deteriorates severely as the nodal spacing decreases while the stability of the stabilized mls approximation is not affected by the nodal spacing the stabilized mls approximation is introduced into the element free galerkin efg method to produce a stabilized efg method theoretical error analysis of the stabilized efg method is provided for boundary value problems with mixed boundary conditions of dirichlet and robin type numerical examples confirm the theoretical results and show that the stabilized efg method has higher computational precision and better stability than the original efg method

quantum discrete phase space dynamics and its continuous limit the main aspects of a discrete phase space formalism are presented and the discrete dynamical bracket suitable for the description of time evolution in finite dimensional spaces is discussed a set of operator bases is defined in such a way that the weyl wigner formalism is shown to be obtained as a limiting case in the same form the moyal bracket is shown to be the limiting case of the discrete dynamical bracket the dynamics in quantum discrete phase spaces is shown not to be attained from discretization of the continuous case

variational multiscale interpolating element free galerkin method for the nonlinear darcy forchheimer model abstract in this paper the variational multiscale interpolating element free galerkin vmiefg method is developed to obtain the numerical solution of the nonlinear darcy forchheimer model we use the interpolating moving least squares method instead of the moving least squares approximation to construct meshless shape functions with delta function properties then the flux boundary condition of the darcy forchheimer model can be handled easily hughes variational multiscale hvm method is applied to overcome the numerical oscillation caused by equal order basis for the velocity and pressure moreover the hvm ensures that the resultant formulation in the vmiefg method is consistent and the stabilization parameter or tensor appears naturally consequently the stabilization parameter is free of user defined the fixed point iteration method is used to deal with the nonlinear term some numerical examples are provided to illustrate the stability and performance of the proposed method for solving the nonlinear darcy forchheimer model

tomograms and other transforms a unified view a general framework is presented which unifies the treatment of wavelet like quasidistribution and tomographic transforms explicit formulae relating the three types of transforms are obtained the case of transforms associated with the symplectic and affine groups is treated in some detail special emphasis is given to the properties of the scale time and scale frequency tomograms tomograms are interpreted as a tool to sample the signal space by a family of curves or as the matrix element of a projector

position and momentum bases for the monochromatic maxwell fish eye and the sphere in geometric optics the maxwell fish eye is a medium where light rays follow circles while in scalar wave optics this medium can only trap fields of certain discrete frequencies in the monochromatic case characterized by a positive integer l there are independent fields we identify two bases of functions one known as the sherman volobuyev functions is characterized as of most definite momenta the other is new and composed of most definite positions and normal derivatives for the fish eye scalar wavefields their construction uses the stereographic projection of the sphere and their identification is corroborated in the contraction limit to the homogeneous helmholtz medium

multifractal formalism for fractal signals the structure function approach versus the wavelet transform modulus maxima method several attempts have been made recently to generalize the multifractal formalism originally introduced for singular measures to fractal signals we report on a systematic comparison between the structure function approach pioneered by parisi and frisch in 2 proceedings of the international school on turbulence and predictability in geophysical fluid dynamics and climate dynamics edited by m ghil r benzi and g parisi north holland amsterdam 1985 p 84 to account for the multifractal nature of fully developed turbulent signals and an alternative method we have developed within the framework of the wavelet transform analysis we comment on the intrinsic limitations of the structure function approach this technique has fundamental drawbacks and does not provide a full characterization of the singularities of a signal in many cases we demonstrate that our method based on the wavelet transform modulus maxima representation works in most situations and is likely to be the ground of a unified multifractal description of self affine distributions our theoretical considerations are both illustrated on pedagogical examples and supported by numerical simulations

wigner distribution function for finite systems we construct a wigner distribution function for finite data sets it is based on a finite optical system a linear wave guide where the finite number of discrete sensors is equal to the number of modes which the guide can carry the dynamical group for this model is su 2 and the wave functions are sets of n 2l 1 data points the wigner distribution function assigns classical c numbers to the operators of position momentum and wave guide mode

perturbation and stability analysis of strong form collocation with reproducing kernel approximation solving partial differential equations using strong form collocation with nonlocal approximation functions such as orthogonal polynomials and radial basis functions offers an exponential convergence but with the cost of a dense and ill conditioned linear system in this work the local approximation functions based on reproducing kernel approximation are introduced for strong form collocation method called the reproducing kernel collocation method rkcm we perform the perturbation and stability analysis of rkcm and estimate the condition numbers of the discrete equation our stability analyses validated with numerical tests show that this approach yields a well conditioned and stable linear system similar to that in the finite element method we also introduce an effective condition number where the properties of both matrix and right hand side vector of a linear system are taken into consideration in the measure of conditioning we first derive the effective condition number of the linear systems resulting from rkcm and show that using the effective condition number offers a tighter estimation of stability of a linear system the mathematical analysis also suggests that the effective condition number of rkpm does not grow with model refinement the numerical results are also presented to validate the mathematical analysis copyright 2011 john wiley sons ltd

a new implementation of the element free galerkin method abstract element free galerkin methods efg are methods for solving partial differential equations with moving least squares interpolants efg methods require only nodal data no element connectivity is needed in a previous implementation of the efg method lagrange multipliers were used to enforce the essential boundary condition however the use of lagrange multipliers increases the cost of solving the linear algebraic equations a new implementation is developed based on a modified variational principle in which the lagrange multipliers are replaced at the outset by their physical meaning so that the discrete equations are banded in addition weighted orthogonal basis functions are constructed so the need for solving equations at each quadrature point is eliminated numerical examples show that the present implementation effectively computes stress concentrations and stress intensity factors at cracks with very irregular arrangements of nodes the latter makes it very advantageous for modelling progressive cracking

the boundary node method for potential problems the element free galerkin efg method allows one to use a nodal data structure usually with an underlying cell structure within the domain of a body of arbitrary shape the usual efg combines moving least squares mls interpolants with a variational principle weak form and has been used to solve two dimensional 2 d boundary value problems in mechanics such as in potential theory elasticity and fracture this paper proposes a combination of mls interpolants with boundary integral equations bie in order to retain both the meshless attribute of the former and the dimensionality advantage of the latter this new method called the boundary node method bnm only requires a nodal data structure on the bounding surface of a body whose dimension is one less than that of the domain itself an underlying cell structure is again used for numerical integration in principle the bnm for 3 d problems should be extremely powerful since one would only need to put nodes points on the surface of a solid model for an object numerical results are presented in this paper for the solution of laplace s equation in 2 d dirichlet neumann and mixed problems have been solved some on bodies with piecewise straight and others with curved boundaries results from these numerical examples are extremely encouraging 1997 by john wiley sons ltd

regularized moving least square method and regularized improved interpolating moving least square method with nonsingular moment matrices abstract the moving least square mls method has been popular applied in surface construction and meshless methods however the moment matrix in mls method may be singular for ill quality point sets and the computation of the inverse of the singular moment matrix is difficult to overcome this problem a regularized moving least square method with nonsingular moment matrix is proposed the shape functions obtained from the regularized mls method still do not have the delta function property and may result in difficulty for imposing boundary conditions in regularized mls based meshless method to overcome this problem a regularized improved interpolating moving least square iimls method based on the iimls method is also proposed compared with the regularized mls method the regularized iimls not only has nonsingular moment matrices but also obtains shape functions with delta function property shape functions of the proposed methods are compared in 1d and 2d cases and the methods have been applied in curve fitting surface fitting and meshless method in numerical examples

holder exponent spectra for human gait abstract the stride interval time series in normal human gait is not strictly constant but fluctuates from step to step in a complex manner more precisely it has been shown that the control process for human gait is a fractal random phenomenon that is one with a long term memory herein we study the holder exponent spectra for the slow normal and fast gaits of 10 young healthy men in both free and metronomically triggered conditions and establish that the stride interval time series is more complex than a monofractal phenomenon a slightly multifractal and non stationary time series under the three different gait conditions emerges

continuum quantum systems as limits of discrete quantum systems ii state functions in this second of four papers on the eponymous topic pointwise convergence of a discrete state function to a continuum state function is shown to imply the algebraic criterion for convergence that was introduced in the prequel as examples and as a prerequisite for the sequels the normal approximation theorem and the convergence of the kravchuk functions to the hermite gaussians are expressed in terms of the algebraic notion of convergence

a galerkin boundary node method and its convergence analysis the boundary node method bnm exploits the dimensionality of the boundary integral equation bie and the meshless attribute of the moving least square mls approximations however since mls shape functions lack the property of a delta function it is difficult to exactly satisfy boundary conditions in bnm besides the system matrices of bnm are non symmetric a galerkin boundary node method gbnm is proposed in this paper for solving boundary value problems in this approach an equivalent variational form of a bie is used for representing the governing equation and the trial and test functions of the variational formulation are generated by the mls approximation as a result boundary conditions can be implemented accurately and the system matrices are symmetric total details of numerical implementation and error analysis are given for a general bie taking the dirichlet problem of laplace equation as an example we set up a framework for error estimates of gbnm some numerical examples are also given to demonstrate the efficacity of the method

continuum quantum systems as limits of discrete quantum systems iii operators convergence of a discrete operator to a continuum operator is defined as examples the circular rotor the one dimensional box the harmonic oscillator and the fractional fourier transform are realized as limits of finite dimensional quantum systems limits thus defined preserve algebraic structure the results prepare for a sequel in which some affine canonical transforms will be discretized

stable particle methods based on lagrangian kernels abstract a large deformation particle method based on the krongauz belytschko corrected gradient meshfree method with lagrangian kernels is developed in this form the gradient is corrected by a linear transformation so that linear completeness is satisfied for the test functions shepard functions are used this guarantees that the patch test is met lagrangian kernels are introduced to eliminate spurious distortions of the domain of material stability a mass allocation scheme is developed that captures correct reflection of waves without any explicit application of traction boundary conditions in addition the lagrangian kernel versions of various forms of smooth particle methods sph including the standard forms and the randles libersky modification are presented and studied results are obtained for a variety of problems that compare this method to standard forms of sph the randles libersky correction and large deformation versions of the element free galerkin method

a posteriori error estimates for nonlinear problems l r o t p ω error estimates for finite element discretizations of parabolic equations using the abstract framework of 9 we analyze a residual a posteriori error estimator for space time finite element discretizations of quasilinear parabolic pdes the estimator gives global upper and local lower bounds on the error of the numerical solution the finite element discretizations in particular cover the so called 0 scheme which includes the implicit and explicit euler methods and the crank nicholson scheme

wigner weyl isomorphism for quantum mechanics on lie groups the wigner weyl isomorphism for quantum mechanics on a compact simple lie group g is developed in detail several features are shown to arise which have no counterparts in the familiar cartesian case notable among these is the notion of a semiquantized phase space a structure on which the weyl symbols of operators turn out to be naturally defined and figuratively speaking located midway between the classical phase space t g and the hilbert space of square integrable functions on g general expressions for the star product for weyl symbols are presented and explicitly worked out for the angle angular momentum case

meshless projection iterative analysis of signorini problems using a boundary element free method this paper presents a meshless method based on coupling a boundary element free method befm with a new projection iterative scheme for signorini problems using the new iterative scheme the nonlinear complementary signorini boundary conditions are transformed into neumann boundary conditions then mixed elliptic problems are formed and are solved by the befm in this method only a nodal data structure on the boundary is required and the coefficient matrices of the discrete system equations only have to be computed once in the whole iterative process convergence of the iterative scheme is discussed theoretically numerical examples involving beach percolation and an electropaint process are given to illustrate the convergence rate and high computational efficiency of the method

relaxation and phase space singularities in time series of human magnetoencephalograms as indicator of photosensitive epilepsy to analyze the crucial role of fluctuation and relaxation effects for the function of the human brain we studied some statistical quantifiers that support the information characteristics of neuromagnetic brain responses magnetoencephalogram meg the signals to a flickering stimulus of different color combinations have been obtained from a group of control subjects which is then contrasted with those of a patient suffering photosensitive epilepsy pse we found that the existence of the specific stratification of the phase clouds and the concomitant relaxation singularities of the corresponding nonequilibrium dynamics of the chaotic behavior of the signals in separate areas in a patient provide likely indicators for the zones which are responsible for the appearance of pse

finite q oscillator the finite q oscillator is a model that obeys the dynamics of the harmonic oscillator with the operators of position momentum and hamiltonian being functions of elements of the q algebra suq 2 the spectrum of position in this discrete system in a fixed representation j consists of 2j 1 sensor points and similarly for the momentum observable the spectrum of energies is finite and equally spaced so the system supports coherent states the wavefunctions involve dual q kravchuk polynomials which are solutions to a finite difference schrodinger equation time evolution times a phase defines the fractional fourier q kravchuk transform in the classical limit as q 1 we recover the finite oscillator lie algebra the n 2j limit returns the macfarlane biedenharn q oscillator and both limits contract the generators to the standard quantum mechanical harmonic oscillator