reproducing formulas for generalized translation invariant systems on locally compact abelian groups in this paper we connect the well established discrete frame theory of generalized shift invariant systems to a continuous frame theory to do so we let j j 2 j be a countable family of closed co compact subgroups of a sec ond countable locally compact abelian group g and study systems of the form j2jfgj p g 2 j p2pj with generators gj p in l 2 g and with each pj being a countable or an uncountable index set we refer to systems of this form as generalized translation invariant gti systems many of the familiar transforms e g the wavelet shearlet and gabor transform both their discrete and continu ous variants are gti systems under a technical local integrability condition lic we characterize when gti systems constitute tight and dual frames that yield reproducing formulas for l 2 g this generalizes results on generalized shift invariant systems where each pj is assumed to be countable and each j is a uni form lattice in g to the case of uncountably many generators and not necessarily discrete closed co compact subgroups furthermore even in the case of uniform lattices j our characterizations improve known results since the class of gti systems satisfying the lic is strictly larger than the class of gti systems sat isfying the previously used local integrability condition as an application of our characterization results we obtain new characterizations of translation invariant continuous frames and gabor frames for l 2 g in addition we will see that the admissibility conditions for the continuous and discrete wavelet and gabor trans form in l 2 r n are special cases of the same general characterizing equations

the discrete zak transform application to time frequency analysis and synthesis of nonstationary signals algorithms to compute coefficients of the finite double sum expansion of time varying nonstationary signals and to synthesize them from a finite number of expansion coefficients are presented the algorithms are based on the computation of the discrete zak transform dzt fast algorithms to compute dzt are presented modifications to the algorithms which increase robustness are given

the undersampled discrete gabor transform conventional studies on discrete gabor transforms have generally been confined to the cases of critical sampling and oversampling in which the gabor families span the whole signal space in this paper we investigate undersampled discrete gabor transforms for an undersampled gabor triple g a b i e a spl middot b n we show that the associated generalized dual gabor window gdgw function is the same as the one associated with the oversampled g n b n a except for the constant factor ab n computations of undersampled gabor transforms are made possible by applying the methods algorithms developed in oversampled settings the undersampled gdgw is determined then we are able to obtain the best approximation of a signal x by linear combinations of vectors in the gabor family

computation of the fundamental singular subspace of a large matrix abstract we compare the block lanczos and the davidson methods for computing a basis of a singular subspace associated with the smallest singular values of large matrices we introduce a simple modification on the preconditioning step of davidson s method which appears to be efficient on a range of large sparse matrices

periodic solution and stationary distribution of stochastic sir epidemic models with higher order perturbation abstract in this paper we investigate two stochastic sir epidemic models with higher order perturbation for the nonautonomous periodic case of the model by using has minskii s theory of periodic solution we show that the system has at least one nontrivial positive t periodic solution for the system disturbed by both the white noise and telephone noise we establish sufficient conditions for positive recurrence and the existence of ergodic stationary distribution of the positive solution

decroissance exponentielle du noyau de la chaleur sur la diagonale i we give examples based upon large deviation s theory where the heat kernel of a degenerate diffusion has an exponential decay over the diagonal using malliavin calculus we give conditions for a more generalized heat kernel to have an exponential decay over the diagonal we give lower bound in some particular case by using the bismut s condition

an algorithmic introduction to numerical simulation of stochastic differential equations a practical and accessible introduction to numerical methods for stochastic differential equations is given the reader is assumed to be familiar with euler s method for deterministic differential equations and to have at least an intuitive feel for the concept of a random variable however no knowledge of advanced probability theory or stochastic processes is assumed the article is built around 10 matlab programs and the topics covered include stochastic integration the euler maruyama method milstein s method strong and weak convergence linear stability and the stochastic chain rule

gabor representation of generalized functions this paper deals with gabor representation of generalized functions by this we mean that a given function is to be expanded in a series involving gabor functions that are located in the points of a given lattice in the time frequency plane we shall mainly consider lattices in which the area of the elementary cells of the lattice equals 1 in 1946 gabor used these expansions for the simultaneous analysis of signals in time and frequency cf 181 gabor stated that the above mentioned expansions exist for every reasonable signal and that the coefficients in the expansion are uniquely determined by the signal this statement is true when interpreted carefully and the aim of this paper is to find out what kind of generalized functions are sufficiently well behaved to allow a development in a gabor series we shall also consider the question of the uniqueness of the coefficients in the expansions as well as questions concerning convergence it turns out that the uniqueness questions can be handled with the aid of the main results of io in particular 2 12 and 2 13 the questions on the existence of expansions are harder to answer and require special care depending on the particular function we consider lpfunctions 1 p 2 in detail and show existence of gabor representation for tempered distribution in general

the partial total least squares algorithm abstract the partial total least squares ptls subroutine solves the total least squares tls problem ax b by using a partial singular value decomposition psvd hereby improving the computational efficiency of the classical tls algorithm a factor 2 approximately the computational improvement of ptls is based on the observation that the tls solution can be computed from any orthogonal basis of the right singular subspace corresponding to the smallest singular values of the augmented matrix a b the dimension of this subspace may be determined by the rank of the tls approximation of a b or depend on a given upper bound e g a perturbation level for those smallest singular values the routine ptls allows more than one vector in b it also solves determined and underdetermined sets of equations by computing the minimum norm solution

chebyshev acceleration techniques for solving nonsymmetric eigenvalue problems abstract the present paper deals with the problem of computing a few of the eigenvalues with largest or smallest real parts of a large sparse nonsymmetric matrix we present a general acceleration technique based on chebyshev polynomials and discuss its practical application to arnoldi s method and the subspace iteration method the resulting algorithms are compared with the classical ones in a few experiments which exhibit a sharp superiority of the arnoldi chebyshev approach

complete gabor transformation for signal representation properties of the gabor transformation used for image representation are discussed the properties can be expressed in matrix notation and the complete gabor coefficients can be found by multiplying the inverse of the gabor 1946 matrix and the signal vector the gabor matrix can be decomposed into the product of a sparse constant complex matrix and another sparse matrix that depends only on the window function a fast algorithm is suggested to compute the inverse of the window function matrix enabling discrete signals to be transformed into generalized nonorthogonal gabor representations efficiently a comparison is made between this method and the analytical method the relation between the window function matrix and the biorthogonal functions is demonstrated a numerical computation method for the biorthogonal functions is proposed

persistence and extinction in general non autonomous logistic model with delays and stochastic perturbation this is a continuation of the paper liu and wang 2012 14 firstly we consider a general non autonomous logistic model with delays and stochastic perturbation then sufficient conditions for extinction are established as well as nonpersistence in the mean weak persistence and stochastic permanence the threshold between weak persistence and extinction is obtained finally numerical simulations are introduced to support the theoretical analysis results

discrete gabor transform a feasible algorithm for implementing the gabor expansion the coefficients of which are computed by the discrete gabor transform dgt is presented for a given synthesis window and sampling pattern computing the auxiliary biorthogonal function of the dgt is nothing more than solving a linear system the dgt presented applies for both finite as well as infinite sequences by exploiting the nonuniqueness of the auxiliary biorthogonal function at oversampling an orthogonal like dgt is obtained as the discrete fourier transform dft is a discrete realization of the continuous time fourier transform similarly the dgt introduced provides a feasible vehicle to implement the useful gabor expansion

some results on discrete gabor transforms for finite periodic sequences in this correspondence using linear system theory and frame theory we address computation methods for analysis and synthesis sequences of the discrete gabor transform dgt where all sequences are periodic with the same period l and show that the minimum energy solution to the wexler and raz 1990 biorthogonal condition and the solution to the frame operator are the same via linear system theory and frame theory

stochastic stabilization and destabilization it is shown in this paper that any nonlinear systems dot x t f x t t in rd can be stabilized by brownian motion provided f hook x t k x for some k 0 on the other hand this system can also be destabilized by brownian motion if the dimension d 2 similar results are also obtained for any given stochastic differential equation dx t f hook x t t g x t t dw t

an efficient and reliable algorithm for computing the singular subspace of a matrix associated with its smallest singular values abstract in this paper an improved algorithm psvd for computing the singular subspace of a matrix corresponding to its smallest singular values is presented as only a basis of the desired singular subspace is needed the classical singular value decomposition svd algorithm can be modified in three ways first the householder transformations of the bidiagonalization need only to be applied on the base vectors of the desired singular subspace second the bidiagonal must only be partially diagonalized and third the convergence rate of the iterative diagonalization can be improved by an appropriate choice between qr and ql iteration steps an analysis of the operation counts as well as computational results show the relative efficiency of psvd with respect to the classical svd algorithm depending on the gap the desired numerical accuracy and the dimension of the desired subspace psvd can be three times faster than the classical svd algorithm while the same accuracy can be maintained the new algorithm can be successfully used in total least squares applications in the computation of the null space of a matrix and in solving non homogeneous linear equations based on psvd a very efficient and reliable algorithm is also derived for solving nonhomogeneous equations

bargmann transform zak transform and coherent states it is well known that completeness properties of sets of coherent states associated with lattices in the phase plane can be proved by using the bargmann representation or by using the kq representation which was introduced by j zak in this paper both methods are considered in particular in connection with expansions of generalized functions in what are called gabor series the setting consists of two spaces of generalized functions tempered distributions and elements of the class s which appear in a natural way in the context of the bargmann transform also a thorough mathematical investigation of the zak transform is given this paper contains many comments and complements on existing literature in particular connections with the theory of interpolation of entire functions over the gaussian integers are given

a jacobi davidson type svd method we discuss a new method for the iterative computation of a portion of the singular values and vectors of a large sparse matrix similar to the jacobi davidson method for the eigenvalue problem we compute in each step a correction by approximately solving a correction equation we give a few variants of this jacobi davidson svd jdsvd method with their theoretical properties it is shown that the jdsvd can be seen as an accelerated inexact newton scheme we experimentally compare the method with some other iterative svd methods

a stochastic differential equation sis epidemic model in this paper we extend the classical susceptible infected susceptible epidemic model from a deterministic framework to a stochastic one and formulate it as a stochastic differential equation sde for the number of infectious individuals i t we then prove that this sde has a unique global positive solution i t and establish conditions for extinction and persistence of i t we discuss perturbation by stochastic noise in the case of persistence we show the existence of a stationary distribution and derive expressions for its mean and variance the results are illustrated by computer simulations including two examples based on real life diseases

analysis of a general stochastic non autonomous logistic model with delays and levy jumps abstract in this paper a general stochastic non autonomous logistic model with delays and levy jumps is proposed and studied sufficient criteria for extinction non persistence in the mean and weak persistence of the model are established the threshold between weak persistence and extinction is also obtained finally numerical simulations are introduced to illustrate the theoretical analysis results our results illustrate that both persistence and extinction have close relationships with time dependent delay and levy noise

the generalized gabor transform the generalized gabor transform for image representation is discussed for a given function f t t spl isin r the generalized gabor transform finds a set of coefficients a sub mr such that f t spl sigma sub m spl infin sup spl infin spl sigma sub r spl infin sup spl infin spl alpha sub mr g t mt exp i2 spl pi rt t the original gabor transform proposed by d gabor 1946 is the special case of t t the computation of the generalized gabor transform with biorthogonal functions is discussed the optimal biorthogonal functions are discussed a relation between a window function and its optimal biorthogonal function is presented based on the zak 1967 transform when t t is rational the finite discrete generalized gabor transform is also derived methods of computation for the biorthogonal function are discussed the relation between a window function and its optimal biorthogonal function derived for the continuous variable generalized gabor transform can be extended to the finite discrete case efficient algorithms for the optimal biorthogonal function and generalized gabor transform for the finite discrete case are proposed

stability of markov semigroups and applications to parabolic systems abstract a new theorem for asymptotic stability of markov semigroups is proved this result is applied to semigroups generated by parabolic systems describing the evolution of densities of two state diffusion processes

a fast algorithm for the recursive calculation of dominant singular subspaces in many engineering applications it is required to compute the dominant subspace of a matrix a of dimension mxn with m n often the matrix a is produced incrementally so all the columns are not available simultaneously this problem arises e g in image processing where each column of the matrix a represents an image of a given sequence leading to a singular value decomposition based compression s chandrasekaran b s manjunath y f wang j winkeler h zhang an eigenspace update algorithm for image analysis graphical models and image process 59 5 1997 321 332 furthermore the so called proper orthogonal decomposition approximation uses the left dominant subspace of a matrix a where a column consists of a time instance of the solution of an evolution equation e g the flow field from a fluid dynamics simulation since these flow fields tend to be very large only a small number can be stored efficiently during the simulation and therefore an incremental approach is useful p van dooren gramian based model reduction of large scale dynamical systems in numerical analysis 1999 chapman hall crc press london boca raton fl 2000 pp 231 247 in this paper an algorithm for computing an approximation of the left dominant subspace of size k of a r m x n with k m n is proposed requiring at each iteration o mk k 2 floating point operations moreover the proposed algorithm exhibits a lot of parallelism that can be exploited for a suitable implementation on a parallel computer

iterative algorithms for computing the singular subspace of a matrix associated with its smallest singular values abstract many problems in science require the computation of only one singular vector or more generally a singular subspace of a matrix instead of computing the complete singular value decomposition iterative methods can be used to improve the computational speed in particular when a priori information about the solution is available this paper deals with the computation of a basis of a singular subspace of a matrix associated with its smallest singular values three iterative methods inverse chebyshev and inverse chebyshev iteration are compared by analyzing their convergence properties based on the convergence rate and the operation counts per iteration step it is shown in which problems a particular iterative algorithm is most efficient if the gap between the singular values associated with the desired and undesired singular subspace is large inverse iteration is clearly the best choice if not convergence can be accelerated by applying inverse chebyshev iteration provided sufficiently tight bounds delineating the undesired singular value spectrum are known the smaller the gap the larger the gain in speed unless the matrix is structured or sparse this method is shown to be always more efficient than ordinary chebyshev iteration

brief paper dynamics of a multigroup sir epidemic model with stochastic perturbation in this paper we introduce stochasticity into a multigroup sir susceptible infective and recovered model the stochasticity in the model is introduced by parameter perturbation which is a standard technique in stochastic population modeling in the deterministic models the basic reproduction number r 0 is a threshold which completely determines the persistence or extinction of the disease we carry out a detailed analysis on the asymptotic behavior of the stochastic model also regarding of the value of r 0 if r email protected 1 the solution of the model is oscillating around a steady state which is the disease free equilibrium of the corresponding deterministic model whereas if r 0 1 there is a stationary distribution which means that the disease will prevail

long time behavior of a stochastic logistic equation with distributed delay and nonlinear perturbation abstract in this paper we study a stochastic logistic model with distributed delay and nonlinear perturbation we first use the linear chain technique to transfer the one dimensional stochastic model with strong kernel or weak kernel into an equivalent system with degenerate diffusion then we establish sufficient and necessary criteria for extinction of the population with probability one moreover since the diffusion matrix is degenerate the uniform ellipticity condition is not satisfied the markov semigroup theory is used to obtain the existence and uniqueness of a stable stationary distribution we show that the densities of the distributions of the solutions can converge in l 1 to an invariant density the results show that the larger white noise can lead to the extinction of the population while the smaller white noise can guarantee the existence of a stable stationary distribution which leads to the stochastic persistence of the population numerical simulations are provided to demonstrate the theoretical results

preconditioning techniques for nonsymmetric and indefinite linear systems abstract the standard preconditioning techniques for conjugate gradient methods often fail for matrices that are indefinite and or strongly nonsymmetric the most common alterative considered for these cases are either to use expensive direct solvers or to resort to one of many techniques based on the normal equations this paper examines several such alternatives and compares them in particular an incomplete lq factorization is proposed and some of its implementation details are described a number of experiments are reported to compare these methods

continuous and discrete wavelet transforms this paper is an expository survey of results on integral representations and discrete sum expansions of functions in l 2 bf r in terms of coherent states two types of coherent states are considered weyl heisenberg coherent states which arise from translations and modulations of a single function and affine coherent states called wavelets which arise as translations and dilations of a single function in each case it is shown how to represent any function in l 2 bf r as a sum or integral of these states most of the paper is a survey of literature most notably the work of i daubechies a grossmann and j morlet a few results of the authors are included

dynamics of a novel nonlinear stochastic sis epidemic model with double epidemic hypothesis abstract in this paper we propose new mathematical models with nonlinear incidence rate and double epidemic hypothesis then we dedicate to develop a method to obtain the threshold of the stochastic sis epidemic model to this end first we investigate the stability of the equilibria of the deterministic system and obtain the conditions for the extinction and the permanence of two epidemic diseases second we explore and obtain the threshold of a stochastic sis system for the extinction and the permanence in mean of two epidemic diseases the results show that a large stochastic disturbance can cause infectious diseases to go to extinction in other words the persistent infectious disease of a deterministic system can become extinct due to the white noise stochastic disturbance this implies that the stochastic disturbance is conducive to epidemic diseases control to illustrate the performance of the theoretical results we present a series of numerical simulations of these cases with respect to different noise disturbance coefficients

computing multiple roots of inexact polynomials we present a combination of two algorithms that accurately calculate multiple roots of general polynomials algorithm i transforms the singular root finding into a regular nonlinear least squares problem on a pejorative manifold and it calculates multiple roots simultaneously from a given multiplicity structure and initial root approximations to fulfill the input requirement of algorithm i we develop a numerical gcd finder containing a successive singular value updating and an iterative gcd refinement as the main engine of algorithm ii that calculates the multiplicity structure and the initial root approximation while limitations exist in certain situations the combined method calculates multiple roots with high accuracy and consistency in practice without using multiprecision arithmetic even if the coefficients are inexact this is perhaps the first blackbox type root finder with such capabilities to measure the sensitivity of the multiple roots a structure preserving condition number is proposed and error bounds are established according to our computational experiments and error analysis a polynomial being ill conditioned in the conventional sense can be well conditioned with the multiplicity structure being preserved and its multiple roots can be computed with high accuracy

on the sliding window representation in digital signal processing the short time fourier transform of a discrete time signal which is the fourier transform of a windowed version of the signal is interpreted as a sliding window spectrum this sliding window spectrum is a function of two variables a discrete time index which represents the position of the window and a continuous frequency variable it is shown that the signal can be reconstructed from the sampled sliding window spectrum i e from the values at the points of a certain time frequency lattice this sampling lattice is rectangular and the rectangular cells occupy an area of 2 in the time frequency domain it is shown that an elegant way to represent the signal directly in terms of the sample values of the sliding window spectrum is in the form of gabor s signal representation therefore a reciprocal window is introduced and it is shown how the window and the reciprocal window are related gabor s signal representation then expands the signal in terms of properly shifted and modulated versions of the reciprocal window and the expansion coefficients are just the values of the sampled sliding window spectrum

a stability analysis of incomplete lu factorizations abstract the combination of iterative methods with preconditionings based on incomplete lu factorizations constitutes an effective class of methods for solving the sparse linear systems arising from the discretization of elliptic partial differential equations in this paper we show that there are some settings in which the incomplete lu preconditioners are not effective and we demonstrate that their poor performance is due to numerical instability our analysis consists of an analytic and numerical study of a sample two dimensional non self adjoint elliptic problem discretized by several finite difference schemes author

computation of pseudospectra via spectral projectors abstract in this note we discuss new techniques for analyzing the pseudospectra of matrices and propose a numerical method for computing the spectral projector associated with a group of eigenvalues enclosed by a polygonal curve numerical tests are reported

a remark on a stochastic logistic model with levy jumps this note is concerned with a famous stochastic logistic equation with levy noises sufficient and necessary conditions for extinction and permanence are established the results reveal that the levy noise may change the properties of population dynamics significantly the results also reveal an important property of the levy noise it is unfavorable for the permanence of the population some numerical simulations are introduced to validate the analytical results

painless nonorthogonal expansions in a hilbert space h discrete families of vectors hj with the property that f j hj f hj for every f in h are considered this expansion formula is obviously true if the family is an orthonormal basis of h but also can hold in situations where the hj are not mutually orthogonal and are overcomplete the two classes of examples studied here are i appropriate sets of weyl heisenberg coherent states based on certain non gaussian fiducial vectors and ii analogous families of affine coherent states it is believed that such quasiorthogonal expansions will be a useful tool in many areas of theoretical physics and applied mathematics