rotating chemical waves in the gray scott model a set of reaction diffusion equations is considered known as the gray scott model defined on a circle and the stability of rotating wave solutions formed via hopf bifurcations that break the circular o 2 symmetry is investigated using a hybrid numerical analytical technique center manifold normal form reductions are performed to analyze symmetry breaking hopf bifurcations degenerate hopf bifurcations and hopf hopf mode interactions it is found that stable rotating waves exist over broad ranges of parameter values and that the bifurcation behavior of this relatively simple model can be quite complex e g two and three frequency motions exist

exclusion theorems and the perturbation analysis of the generalized eigenvalue problem a bauer fike type theorem is proved for the eigenvalue problem ax lambda bx a generalization is then applied to obtain perturbation bounds for clusters of eigenvalues the conditioning of an eigenvalue finite or infinite is proved to be dependent on a jordan type condition number its eigenvector deficiency and the regularity of the matrix pencil a lambda b an example illustrating the perturbation bounds is given

perturbation of eigenvalues for periodic matrix pairs via the bauer fike theorems abstract in earlier papers the bauer fike technique was applied to the ordinary eigenvalue problem a x x the generalized eigenvalue problem a x b x and the matrix polynomial eigenvalue problem k 0 m k a k x 0 general multiple eigenvalues were dealt with and condition numbers were obtained for individual as well as clusters of eigenvalues in this paper we shall generalize the technique to the eigenvalue problem for regular periodic matrix pairs perturbation to eigenvalues simple or multiple finite or infinite will be considered for perturbations of any size for small perturbations clusters of eigenvalues will be considered

on q extended eigenvectors of the integral and finite fourier transforms mehta has shown that eigenvectors of the n n finite fourier transform can be written in terms of the standard hermite eigenfunctions of the quantum harmonic oscillator 1987 j math phys 28 781 here we construct a one parameter family of q extensions of these eigenvectors based on the continuous q hermite polynomials of rogers in the limit when q 1 these q extensions coincide with mehta s eigenvectors and in the continuum limit as n they give rise to q extensions of eigenfunctions of the fourier integral transform

on eigenfunctions of continuous and discrete anti symmetric multivariate fourier transforms we derive various full sets of eigenfunctions of fourier transforms continuous and discrete sine and cosine fourier transforms in one variable discrete symmetric and antisymmetric multivariate exponentials sine and cosine fourier transforms and continuous symmetric and antisymmetric multivariate sine and cosine fourier transforms associated with hermite polynomials in one variable symmetric and antisymmetric multivariate exponential fourier transforms f satisfy the relation f4 1 whereas symmetric and antisymmetric multivariate sine and cosine fourier transforms f satisfy the relation f2 1

discrete fractional fourier transform based on new nearly tridiagonal commuting matrices based on discrete hermite gaussian like functions a discrete fractional fourier transform dfrft which provides sample approximations of the continuous fractional fourier transform was defined and investigated recently in this paper we propose a new nearly tridiagonal matrix which commutes with the discrete fourier transform dft matrix the eigenvectors of the new nearly tridiagonal matrix are shown to be dft eigenvectors which are more similar to the continuous hermite gaussian functions than those developed before rigorous discussions on the relations between the eigendecomposition of the newly proposed nearly tridiagonal matrix and the dft matrix are described furthermore by appropriately combining two linearly independent matrices that both commute with the dft matrix we develop a method to obtain dft eigenvectors even more similar to the continuous hermite gaussian functions hgfs then new versions of dfrft produce their transform outputs closer to the samples of the continuous fractional fourier transform and their applications are described related computer experiments are performed to illustrate the validity of the works in this paper

the action angle wigner function a discrete finite and algebraic phase space formalism the action angle representation in quantum mechanics is conceptually quite different from its classical counterpart and motivates a canonical discretization of the phase space in this work a discrete and finite dimensional phase space formalism in which the phase space variables are discrete and the time is continuous is developed and the fundamental properties of the discrete weyl wigner moyal quantization are derived the action angle wigner function is shown to exist in the semi discrete limit of this quantization scheme a comparison with other formalisms which are not explicitly based on canonical discretization is made fundamental properties that an action angle phase space distribution respects are derived the dynamical properties of the action angle wigner function are analysed for discrete and finite dimensional model hamiltonians the limit of the discrete and finite dimensional formalism including a discrete analogue of the gaussian wavefunction spread viz the binomial wavepacket is examined and shown by examples that standard continuum quantum mechanical results can be obtained as the dimension of the discrete phase space is extended to infinity

perturbation analysis for the generalized singular value problem this paper discusses perturbation bounds for generalized singular values and for subspaces associated with the generalized singular value decomposition of two matrices having the same number of columns by a limiting procedure from our results we can derive those for the ordinary singular value problem which were proved by l mirsky quart j math oxford 11 1960 pp 50 59 and p a wedin bit 12 1972 pp 99 111

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

existence of traveling wave trains in nerve axon equations we establish the existence of two discrete classes of traveling wave solutions to nerve axon equations under conditions on the linearization of the system about the rest state each solution in one class consists of a finite train of pulses while each solution in the other class consists of an infinite train that arises from the rest state in the leading direction the solutions are distinguished by the spacing of the pulses in the train and indexed by n tuples and sequences of integers respectively in the two classes this result is obtained from study of the resulting parametrized dynamical system which has a a single homoclinic orbit for one parameter value these assumptions were previously used to show the existence of the double pulse solutions and are known to hold in for some nerve models

a note on stewart s theorem for definite matrix pairs abstract let a and b be n n hermitian matrices the matrix pair a b is called definite pair and the corresponding eigenvalue problem ax bx is definite if c a b inf x 1 h a ib x 0 in this note we develop a uniform upper bound for differences of corresponding eigenvalues of two definite pairs and so improve a result which is obtained by g w stewart 2 moreover we prove that this upper bound is a projective metric in the set of n n definite pairs

the discrete harmonic oscillator harper s equation and the discrete fractional fourier transform certain solutions to harper s equation are discrete analogues of and approximations to the hermite gaussian functions they are the energy eigenfunctions of a discrete algebraic analogue of the harmonic oscillator and they lead to a definition of a discrete fractional fourier transform ft the discrete fractional ft is essentially the time evolution operator of the discrete harmonic oscillator

tridiagonal commuting matrices and fractionalizations of dct and dst matrices of types i iv v and viii in this paper we first establish new relationships in matrix forms among discrete fourier transform dft generalized dft gdft and various types of discrete cosine transform dct and discrete sine transform dst matrices two new independent tridiagonal commuting matrices for each of dct and dst matrices of types i iv v and viii are then derived from the existing commuting matrices of dft and gdft with these new commuting matrices the orthonormal sets of hermite like eigenvectors for dct and dst matrices can be determined and the discrete fractional cosine transform dfrct and the discrete fractional sine transform dfrst are defined the relationships among the discrete fractional fourier transform dfrft fractional gdft and various types of dfrct and dfrst are developed to reduce computations for dfrft and fractional gdft

a remark on hilbert s matrix abstract we exhibit a jacobi matrix t which has simple spectrum and integer entries and 0 commutes with hilbert s matrix as an application we replace the computation of the eigenvectors of hilbert s matrix a very ill conditioned problem by the computation of the eigenvectors of t a nicely stable numerical problem

generalized commuting matrices and their eigenvectors for dfts offset dfts and other periodic operations it is well known that some matrices such as dickinson steiglitz s matrix can commute with the discrete fourier transform dft and that one can use them to derive the complete and orthogonal dft eigenvector set recently candan found the general form of the dft commuting matrix in this paper we further extend the previous work and find the general form of the commuting matrix for any periodic quasi periodic and offset quasi periodic operations using the general commuting matrix we can derive the complete and orthogonal eigenvector sets for offset dfts dcts of types 1 4 5 and 8 dsts of types 1 4 5 and 8 discrete hartley transforms of types 1 and 4 the walsh transform and the projection operation the operation that maps a whole vector space into a subspace successfully moreover several novel ways of finding dft eigenfunctions are also proposed furthermore we also extend our theories to the continuous case i e if a continuous transform is periodic quasi periodic or offset quasi periodic such as the ft and some cyclic operations in optics we can find the general form of the commuting operation and then find the complete and orthogonal eigenfunctions set for the continuous transform

pencils of complex and real symmetric and skew matrices abstract this expository paper establishes the canonical forms under congruence for pairs of complex or real symmetric or skew matrices the treatment is in the spirit of the well known book of gantmacher on matrix theory and may be regarded as a supplement to gantmacher s chapters on pencils of matrices

on perturbations of some constrained subspaces perturbation bounds of subspaces such as eigen spaces singular subspaces and canonical subspaces have been extensively studied in the literature in this paper we study perturbations of some constrained subspaces of 1x2 2x1 and 2x2 block matrices in which only one of the sub matrices can be changed such problems rise from the least squares total least squares problem the constrained least squares problem and the constrained total least squares problem

relative perturbation theory iv sin 2θ theorems abstract the double angle theorems of davis and kahan bound the change in an invariant subspace when a hermitian matrix a is subject to an additive perturbation a a a a this paper supplies analogous results when a is subject to a congruential or multiplicative perturbation a a d ad the relative gaps that appear in the bounds involve the spectrum of only one matrix either a or a in contrast to the gaps that appear in the single angle bounds the double angle theorems do not directly bound the difference between the old invariant subspace s and the new one s but instead bound the difference between s and its reflection j s where the mirror is s and j reverses s the orthogonal complement of s the double angle bounds are proportional to the departure from the identity and from orthogonality of the matrix d def d 1 jdj note that d is invariant under the transformation d d for 0 whereas the single angle theorems give bounds proportional to d s departure from the identity and from orthogonality the corresponding results for the singular value problem when a nonsquare matrix b is perturbed to b d 1 bd 2 are also presented

a perturbation bound for definite pencils we attempt to generalize a well known result on spectral variations of a hermitian matrix due to mirsky to the definite generalized eigenvalue problem we also point out that some results on perturbations of definite pencils due to stewart can be slightly improved

homoclinic orbits in a parametrized saddle focus system abstract we consider a parametrized dynamical system with a saddle focus equilibrium point and which for one value of the parameter has a homoclinic orbit conditions on the eigenvalues for the equilibrium point together with transversality conditions imply the existence of an infinite discrete set of parameter values for which the system has a homoclinic orbit such systems arise in the study of nerve axon equations where a homoclinic orbit corresponds to a finite train of nerve impulses traveling at a velocity identified by the associated parameter value

evans functions and bifurcations of nonlinear waves of some nonlinear reaction diffusion equations abstract the main purposes of this paper are to accomplish the existence stability instability and bifurcation of the nonlinear waves of the nonlinear system of reaction diffusion equations u t u x x h u u w w t e u w and to establish the existence stability instability and bifurcation of the nonlinear waves of the nonlinear scalar reaction diffusion equation u t u x x h u u under different conditions on the model constants to establish the bifurcation for the system we will study the existence and instability of a standing pulse solution if 0 2 1 the existence and stability of two standing wave fronts if 2 1 and 2 e 1 the existence and instability of two standing wave fronts if 2 1 and 0 2 e 1 the existence and instability of an upside down standing pulse solution if 0 1 2 1 to establish the bifurcation for the scalar equation we will study the existence and stability of a traveling wave front as well as the existence and instability of a standing pulse solution if 0 2 the existence and stability of two standing wave fronts if 2 the existence and stability of a traveling wave front as well as the existence and instability of an upside down standing pulse solution if 0 2 by the way we will also study the existence and stability of a traveling wave back of the nonlinear scalar reaction diffusion equation u t u x x h u u w 0 where w 0 2 0 is a positive constant if 0 2 to achieve the main goals we will make complete use of the special structures of the model equations and we will construct evans functions and apply them to study the eigenvalues and eigenfunctions of several eigenvalue problems associated with several linear differential operators it turns out that a complex number 0 is an eigenvalue of the linear differential operator if and only if 0 is a zero of the evans function the stability instability and bifurcations of the nonlinear waves follow from the zeros of the evans functions a very important motivation to study the existence stability instability and bifurcations of the nonlinear waves is to study the existence and stability instability of infinitely many fast slow multiple traveling pulse solutions of the nonlinear system of reaction diffusion equations the existence and stability of infinitely many fast multiple traveling pulse solutions are of great interests in mathematical neuroscience

discrete fourier transform of prime order eigenvectors with small support we show how to construct an eigenvector basis of the discrete fourier transform of odd prime order the special feature of the new basis is that the basis vectors have small support

the condition number of equivalence transformations that block diagonalize matrix pencils how ill conditioned must a matrix s be if its columns are constrained to span certain subspaces we answer this question in order to find nearly best conditioned matrices sr and sl that block diagonalize a given matrix pencil t a b i e s l 1 tsr is bloc diagonal we show that the best conditioned sr has a condition number approximately equal to the cosecant of the smallest angle between right subspaces belonging to different diagonal blocks of thus the more nearly the right subspaces overlap the more ill conditioned sr must be the same is true of sl and the left subspaces for the standard eigenproblem t a i sl sr and the cosecant of the angle between subspaces turns out equal to an earlier estimate of the smallest condition number namely the norm of the projection matrix associated with one of the subspaces we apply this result to bound the error in an algorithm to compute analytic functions of matrices for instance exp t

signal scaling by centered discrete dilated hermite functions we present a novel method to generate centered discrete dilated hermite functions by utilizing hermite gaussian differential equation and dft commuting matrix proposed by pei compared with the conventional methods our method provides much higher accuracy and wider working range based on expansion of compact support signals into discrete dilated hermite basis several examples of signal approximation and scaling are demonstrated our results in ecg signal compression can achieve lesser approximation error in comparison with discrete daubechies wavelet transform also our proposed discrete dilated hermite functions have better ability to do signal scaling than the existing method

relative eigenvalues of a definite matrix pair abstract let a and b be hermitian matrices and let c a b min x 2 1 xh a ib x the matrix pair a b is called a definite pair and the corresponding eigenvalue problem ax bx is definite if c a b 0 the relationship between the eigenvalues of a b and those of a definite pair of the form xh1ax1 xh1bx1 is studied in this paper the eigenvalues of xh1ax1 xh1bx1 are said to be the relative eigenvalues of a b with respect to the subspace r x1 from the main result of this paper one can deduce a corresponding conclusion on hermitian matrices

perturbation theorems for the generalized eigenvalue problem abstract given a matrix pair z a b the perturbation of its eigenvalues is studied considering two pairs z w as points of the grassman manifold g n 2 n and its eigenvalues as points in g 1 2 the projective complex plane the distance of the spectra measured in the chordal metric in g 1 2 is bounded by some distance of the matrix pairs in g n 2 n analogs of the bauer fike theorem henrici s theorem and the hoffman weilandt theorem are obtained from which the classical results can be derived

on perturbations of matrix pencils with real spectra perturbation bounds for the generalized eigenvalue problem of a diagonalizable matrix pencil a ab with real spectrum are developed it is shown how the chordal distances between the generalized eigenvalues and the angular distances between the generalized eigenspaces can be bounded in terms of the angular distances between the matrices the applications of these bounds to the spectral variations of definite pencils are conducted in such a way that extra attention is paid to their peculiarities so as to derive more sophisticated perturbation bounds our results for generalized eigenvalues are counterparts of some celebrated theorems for the spectral variations of hermitian matrices such as the weyl lidskii theorem and the hoffman wielandt theorem and those for generalized eigenspaces are counterparts of the celebrated davis kahan sin 0 sin 20 theorems for the eigenspace variations of hermitian matrices the paper consists of two parts part i is for generalized eigenvalue perturbations while part ii deals with generalized eigenspace perturbations

on continuous q hermite polynomials and the classical fourier transform we prove that the classical fourier transform operator intertwines two q difference equations for the continuous q hermite polynomials hn x q of rogers which are associated with the two distinct sets of values for the parameter q 0 q 1 and 1 q

the perturbation bounds for eigenspaces of a definite matrix pair leta andb be hermitian matrices the matrix pair a b is called definite pair and the corresponding eigenvalue problem ax bx is definite if c a b equiv mathop min limits left x right 1 x h a ib x 0 the perturbation bounds for eigenspaces of a definite pair on every unitary invariant matrix norm were obtained by imposing additional restrictions on the location of the generalized eigenvalues thus it gives a positive answer for an open question proposed by stewart 7 the famous davis kahan sin theorems and sin 2 theorem 2 can also be deduced from the present results

trace minimization principles for positive semi definite pencils abstract this paper is concerned with x h ax subject to x h bx j for a hermitian matrix pencil a b where j is diagonal and j 2 i the identity matrix of apt size the same problem was investigated earlier by kovac striko and veselic linear algebra appl 216 1995 139 158 for the case in which b is assumed nonsingular but in this paper b is no longer assumed nonsingular and in fact a b is even allowed to be a singular pencil it is proved among others that the infimum is finite if and only if a b is a positive semi definite pencil in the sense that there is a real number 0 such that a 0 b is positive semi definite the infimum when finite can be expressed in terms of the finite eigenvalues of a b sufficient and necessary conditions for the attainability of the infimum are also obtained

dft commuting matrix with arbitrary or infinite order second derivative approximation recently candan introduced higher order dft commuting matrices whose eigenvectors are better approximations to the continuous hermite gaussian functions hgfs however the highest order 2k of the o h2k ntimesn dft commuting matrices proposed by candan is restricted by 2k 1 les n in this paper we remove this order upper bound restriction by developing two methods to construct arbitrary order dft commuting matrices computer experimental results show that the hermite gaussian like hgl eigenvectors of the new proposed dft commuting matrices outperform those of candan in addition the hgl eigenvectors of the infinite order dft commuting matrix are shown to be the same as those of the n2 dft commuting matrix recently discovered in the literature

on perturbations of matrix pencils with real spectra ii a well known result on spectral variation of a hermitian matrix due to mirsky is the following let a and a be two n x n hermitian matrices and let 1 n and 1 n be their eigenvalues arranged in ascending order then diag 1 1 n n a a for any unitarily invariant norm in this paper we generalize this to the perturbation theory for diagonalizable matrix pencils with real spectra the much studied case of definite pencils is included in this

about two trigonometric matrices abstract let r s be coprime integers s 1 and odd the characteristic polynomials of the matrices sin rmn s 0 and sin rmn s 0 mn s 1 are determined