solving singularly perturbed boundary value problems by spline in tension the difference scheme via spline in tension for the problem y p x y f x p x s 0 y 0 0 y 1 1 is derived the error of the form o h min h is obtained when p x p const the corresponding spline in tension achieves the second order of the global uniform convergence

an efficient mixed asymptotic numerical scheme for singularly perturbed convection diffusion problems abstract in this paper we present and analyze a mixed asymptotic numerical method for the solution of singularly perturbed convection diffusion problems the technique used in this work is the careful factorization of original problem into two explicit problems which are independent of perturbation parameter contaminating the solution the first problem so obtained is the degenerate equation which corresponds to the outer solution while the second problem is obtained by making use of suitable stretching transformation and it corresponds to the inner solution the degenerate equation is solved numerically using q stage runge kutta method the second problem which retains the order of the original problem is solved analytically possible extensions of the present technique to differential equations with delay as well as advance and to the nonlinear problems are also discussed a comparative study of the present method with some state of art existing numerical schemes is carried out by means of several linear and nonlinear test examples that demonstrates the effectiveness and the potential of the present approach

the goursat problem for maxwell s equations using the spinor formalism of electromagnetism the goursat problem for maxell s equations is defined and some examples of solutions are given

on the finite element method for singularly perturbed reaction diffusion problems in two and one dimensions second order elliptic boundary value problems which are allowed to degenerate into zero order equations are considered the behavior of the ordinary galerkin finite element method without special arrangements to treat singularities is studied as the problem ranges from true second order to singularly perturbed

solving singularly perturbed problems by a weak form integral equation with exponential trial functions abstract the second order singularly perturbed problem is transformed to a singularly perturbed parabolic type partial differential equation by using a fictitious time technique then we use green s second identity to derive a boundary integral equation in terms of the adjoint trefftz test functions namely a weak form integral equation method wfiem it accompanying with the exponential trial functions which are designed to satisfy the boundary conditions automatically can provide very accurate numerical solutions of linear and nonlinear singularly perturbed problems for the latter problem the iterative procedure is convergent very fast

a modified chebyshev pseudospectral dd algorithm for the gbh equation in this paper a chebyshev spectral collocation domain decomposition dd semi discretization by using a grid mapping derived by kosloff and tal ezer in space is applied to the numerical solution of the generalized burger s huxley gbh equation to reduce roundoff error in computing derivatives we use the above mentioned grid mapping in this work we compose the chebyshev spectral collocation domain decomposition and kosloff and tal ezer grid mapping elaborately firstly the theory of application of the chebyshev spectral collocation method with grid mapping and dd on the gbh equation is presented this method yields a system of ordinary differential algebraic equations daes secondly we use a fourth order runge kutta formula for the numerical integration of the system of daes application of this modified method to the gbh equation show that this method m dd is faster and more accurate than the standard chebyshev spectral collocation dd s dd method

collocation method using artificial viscosity for solving stiff singularly perturbed turning point problem having twin boundary layers a numerical scheme is proposed to solve singularly perturbed two point boundary value problems with a turning point exhibiting twin boundary layers the scheme comprises a b spline collocation method on a uniform mesh which leads to a tridiagonal linear system asymptotic bounds are established for the derivative of the analytical solution of a turning point problem the analysis is done on a uniform mesh which permits its extension to the case of adaptive meshes which may be used to improve the solution the design of an artificial viscosity parameter is confirmed to be a crucial ingredient for simulating the solution of the problem some relevant numerical examples are also illustrated to verify computationally the theoretical aspects

the main diagonal of a permutation matrix abstract by counting 1 s in the right half of 2 w consecutive rows we locate the main diagonal of any doubly infinite permutation matrix with bandwidth w then the matrix can be correctly centered and factored into block diagonal permutation matrices part ii of the paper discusses the same questions for the much larger class of band dominated matrices the main diagonal is determined by the fredholm index of a singly infinite submatrix thus the main diagonal is determined at infinity in general but from only 2 w rows for banded permutations

some perspectives on the eigenvalue problem this expository paper explores the relationships among a number of algorithms for solving eigenvalue problems including the power method subspace iteration the qr algorithm and the arnoldi and symmetric lanczos algorithms the symmetric lanczos algorithm is shown to be identical to the three term recursion stieltjes procedure for computing orthogonal polynomials with respect to a measure on the real line the connection between measures on the line and symmetric tridiagonal jacobi matrices is investigated if such a matrix is transformed by a step of the qr algorithm there is a corresponding transformation in the measure the tridiagonal matrices are also exploited for the construction of gaussian quadrature formulas for measures on the line the developments on the real line are replicated with suitable modifications on the unit circle via lanczos like procedures for unitary operators the best known procedure of this type is the recursion of szego for computing orthogonal polynomials on the

finite dimensional wiener hopf equations and factorizations of matrices abstract let a be the algebra of all n n matrices over the real or complex numbers let a be the subalgebra of upper triangular matrices and a the subalgebra of strictly lower triangular matrices denote by p the projection of a onto a with kernel a in this paper we investigate the wiener hopf equation p ax y where y a is given and x a is a solution

positive definite toeplitz matrices the arnoldi process for isometric operators and gaussian quadrature on the unit circle abstract we show that the well known levinson algorithm for computing the inverse cholesky factorization of positive definite toeplitz matrices can be viewed as a special case of a more general process the latter process provides a very efficient implementation of the arnoldi process when the underlying operator is isometric this is analogous with the case of hermitian operators where the hessenberg matrix becomes tridiagonal and results in the hermitian lanczos process we investigate the structure of the hessenberg matrices in the isometric case and show that simple modifications of them move all their eigenvalues to the unit circle these eigenvalues are then interpreted as abscissas for analogs of gaussian quadrature now on the unit circle instead of the real line the trapezoidal rule appears as the analog of the gauss legendre formula

some matrix factorization identities for discrete inverse scattering this note analyzes the relationships between several inverse scattering methods and points out that all classical approaches implicitly construct a lower upper triangular factorization of a given positive definite toeplitz matrix it is shown that the various inversion methods implicitly obtain the factorization by solving different nested sets of linear equations and expressing the factors in terms of the solutions obtained the fact that the triangular factorization of a matrix with nonzero leading minors is unique immediately yields various identities e g involving the so called central mass sequence that were derived in the literature with considerably more manipulation in fact our basic factorization results are somewhat more general since they do not require the toeplitz assumption

an exponentially fitted special second order finite difference method for solving singular perturbation problems an exponentially fitted special second order finite difference method is presented for solving singularly perturbed two point boundary value problems with the boundary layer at one end left or right point a fitting factor is introduced in a tri diagonal finite difference scheme and is obtained from the theory of singular perturbations thomas algorithm is used to solve the system and its stability is investigated to demonstrate the applicability of the method we have solved several linear and non linear problems from the results it is observed that the present method approximates the exact solution very well

a uniformly accurate finite element method for a singularly perturbed one dimensional reaction diffusion application d une methode a elements finis avec des elements de base exponentiels a un probleme autoadjoint a perturbation singuliere et a deux points limites generation d un schema aux differences tridiagonal uniformement precis du second ordre

a numerical solution of the generalized burger s huxley equation by pseudospectral method and darvishi s preconditioning abstract in this paper we present a new method for solving of the generalized burger s huxley equation by using the collocation formula for calculating spectral differentiation matrix for chebyshev gauss lobatto point to reduce round off error in spectral collocation method we use darvishi s precondotioning firstly theory of application of spectral collocation method on burger s huxley equation presented this method yields burger s huxley equation to a system of ordinary differential equations odes secondly we use forth order runge kutta formula for the numerical integration of the system of ode the numerical results obtained by this way have been compared with the exact solution to show the efficiency of the method

minimal representations of unitary operators and orthogonal polynomials on the unit circle abstract in this paper we prove that the simplest band representations of unitary operators on a hilbert space are five diagonal orthogonal polynomials on the unit circle play an essential role in the development of this result and also provide a parameterization of such five diagonal representations which shows specially simple and interesting decomposition and factorization properties as an application we get the reduction of the spectral problem of any unitary hessenberg matrix to the spectral problem of a five diagonal one two applications of these results to the study of orthogonal polynomials on the unit circle are presented the first one concerns krein s theorem the second one deals with the movement of mass points of the orthogonality measure under mono parametric perturbations of the schur parameters

an algorithm for determining if the inverse of a strictly diagonally dominant stieltjes matrix is strictly ultrametric it was recently shown that the inverse of a strictly ultrametric matrix is a strictly diagonally dominant stieltjes matrix on the other hand as it is well known that the inverse of a strictly diagonally dominant stieltjes matrix is a real symmetric matrix with nonnegative entries it is natural to ask conversely if every strictly diagonally dominant stieltjes matrix has a strictly ultrametric inverse examples show however that the converse is not true in general i e there are strictly diagonally dominant stieltjes matrices in n n for everyn 3 whose inverses are not strictly ultrametric matrices then the question naturally arises if one can determine which strictly diagonally dominant stieltjes matrices in n n n 3 have inverses which are strictly ultrametric here we develop an algorithm based on graph theory which determines if a given strictly diagonally dominant stieltjes matrixa has a strictly ultrametric inverse where the algorithm is applied toa and requires no computation of inverse moreover if this given strictly diagonally dominant stieltjes matrix has a strictly ultrametric inverse our algorithm uniquely determines this inverse as a special sum of rank one matrices

eigenstructure of order one quasiseparable matrices three term and two term recurrence relations abstract this paper presents explicit formulas and algorithms to compute the eigenvalues and eigenvectors of order one quasiseparable matrices various recursive relations for characteristic polynomials of their principal submatrices are derived the cost of evaluating the characteristic polynomial of an n n matrix and its derivative is only o n this leads immediately to several versions of a fast quasiseparable newton iteration algorithm in the hermitian case we extend the sturm property to the characteristic polynomials of order one quasiseparable matrices which yields to several versions of a fast quasiseparable bisection algorithm conditions guaranteeing that an eigenvalue of a order one quasiseparable matrix is simple are obtained and an explicit formula for the corresponding eigenvector is derived the method is further extended to the case when these conditions are not fulfilled several particular examples with tridiagonal almost unitary hessenberg and toeplitz matrices are considered the algorithms are based on new three term and two term recurrence relations for the characteristic polynomials of principal submatrices of order one quasiseparable matrices r it turns out that the latter new class of polynomials generalizes and includes two classical families i polynomials orthogonal on the real line that play a crucial role in a number of classical algorithms in numerical linear algebra and ii the szego polynomials that play a significant role in signal processing moreover new formulas can be seen as generalizations of the classical three term recurrence relations for the real orthogonal polynomials and of the two term recurrence relations for the szego polynomials

adomian decomposition method for burger s huxley and burger s fisher equations the approximate solutions for the burger s huxley and burger s fisher equations are obtained by using the adomian decomposition method solving frontier problems of physics the decomposition method kluwer boston 1994 the algorithm is illustrated by studying an initial value problem the obtained results are presented and only few terms of the expansion are required to obtain the approximate solution which is found to be accurate and efficient

a modification of the dewilde van der veen method for inversion of finite structured matrices abstract we study a class of block structured matrices r r ij i j 1 n with a property that the solution of the corresponding system rx y of linear algebraic equations may be performed for o n arithmetic operations in this paper for finite invertible matrices we analyze in detail factorization and inversion algorithms these algorithms are related to those suggested by p m dewilde and a j van der veen time varying systems and computations kluwer academic publishers new york 1998 for a class of finite and infinite matrices with a small hankel rank the algorithms presented here are more transparent and are a modification of the algorithms from the above reference the approach and the proofs are essentially different from those in the above mentioned reference the paper contains also analysis of complexity and results of numerical experiments

fast transforms banded matrices with banded inverses it is unusual for both a and a 1 to be banded but this can be a valuable property in applications block diagonal matrices f are the simplest examples wavelet transforms are more subtle we show that every example can be factored into a f1 fn where n is controlled by the bandwidths of a and a 1 but not by their size so this extends to infinite matrices and leads to new matrix groups

a class of simple exponential b splines and their application to numerical solution to singular perturbation problems we shall consider an application of simple exponential splines to the numerical solution of singular perturbation problem the computational effort involved in our collocation method is less than that required for the other methods of exponential type

discrete least squares finite element methods abstract a finite element methodology for large classes of variational boundary value problems is defined which involves discretizing two linear operators 1 the differential operator defining the spatial boundary value problem and 2 a riesz map on the test space the resulting linear system is overdetermined two different approaches for solving the system are suggested although others are discussed 1 solving the associated normal equation with linear solvers for symmetric positive definite systems e g cholesky factorization and 2 solving the overdetermined system with orthogonalization algorithms e g qr factorization the finite element assembly algorithm for each of these approaches is described in detail the normal equation approach is usually faster for direct solvers and requires less storage the second approach reduces the condition number of the system by a power of two and is less sensitive to round off error the rectangular stiffness matrix of second approach is demonstrated to have condition number o h 1 for a variety of formulations of poisson s equation the stiffness matrix from the normal equation approach is demonstrated to be related to the monolithic stiffness matrices of least squares finite element methods and it is proved that the two are identical in some cases an example with poisson s equation indicates that the solutions of these two different linear systems can be nearly indistinguishable if round off error is not an issue and rapidly converge to each other the orthogonalization approach is suggested to be beneficial for problems which induce poorly conditioned linear systems experiments with poisson s equation in single precision arithmetic as well as the linear acoustics problem near resonance in double precision arithmetic verify this conclusion the methodology described here is an outgrowth of the discontinuous petrov galerkin dpg methodology of demkowicz and gopalakrishnan 2015 the strength of dpg is highlighted throughout however the majority of theory presented is more general extensions to constrained minimization principles are also considered throughout but are not analyzed in experiments

a uniformly convergent numerical method on non uniform mesh for singularly perturbed unsteady burger huxley equation abstract in this paper we present a numerical method which deals with the solvability of initial boundary value problem and related general properties of singularly perturbed burger huxley equation singular perturbation problems are the problems in which a very small positive parameter is multiplied to the highest order derivative term when this parameter approaches the limiting value of interest i e 0 the problem exhibits boundary layers and most of the conventional methods fails to capture this effect thus the quest for some new numerical techniques that may handle the difficulties occurring due to the presence of perturbation parameter and nonlinearity in the problem earns relevance in this paper one such parameter robust numerical scheme is constructed the first step in this direction is the discretization of the time variable using the euler implicit method with a constant time step this produces a set of nonlinear stationary singularly perturbed semidiscrete problem class which is linearized further using the quasi linearization process followed by spatial discretization using upwind finite difference operator on a piecewise uniform mesh an extensive amount of analysis is carried out that uses a suitable decomposition of the error into smooth and singular component and a comparison principle combined with appropriate barrier functions the error estimates are obtained which ensures uniform convergence of the method a set of numerical experiments is carried out in support of the predicted theory which validates the theoretical results computationally

a finite element method for a singularly perturbed boundary value problem we examine the problem eu a x u b x u f x for 0 0 b x 2 4e 0 a b andf inc 2 0 1 e in 0 1 u 0 andu 1 given using finite elements and a discretized green s function we show that the el mistikawy and werle difference scheme on an equidistant mesh of widthh is uniformly second order accurate for this problem i e the nodal errors are bounded bych 2 wherec is independent ofh and e with a natural choice of trial functions uniform first order accuracy is obtained in thel 0 1 norm on choosing piecewise linear trial functions hat functions uniform first order accuracy is obtained in thel 1 0 1 norm

solitary wave solutions of the generalised burgers huxley equation exact solitary wave solutions of the generalised burgers huxley equation delta u delta t alpha udelta delta u delta chi delta 2u delta chi 2 beta u 1 udelta udelta gamma are obtained by using the relevant nonlinear transformations the results obtained are the generalisation of former work the method in this paper can also be applied to the burgers fisher equation

fitted fourth order tridiagonal finite difference method for singular perturbation problems in this paper a fitted fourth order tridiagonal finite difference scheme is presented for solving singularly perturbed two point boundary value problems with the boundary layer at one end left or right point we have taken a fourth order tridiagonal finite difference scheme by m m chawla a fourth order tridiagonal finite difference method for general nonlinear two point boundary value problems with mixed boundary conditions j inst maths appl 21 1978 83 93 and introduced a fitting factor the fitting factor is obtained from the theory of singular perturbations thomas algorithm is used to solve the system to demonstrate the applicability of the present method we have solved five linear problems three with left end and two with right end boundary layers solutions of these problems using the present fitted method are compared with chawla s solutions from the results it is observed that the present method is stable and has better approximation to the exact solution

on an inverse eigenvalue problem for unitary hessenberg matrices abstract we show that a unitary upper hessenberg matrix with positive subdiagonal elements is uniquely determined by its eigenvalues and the eigenvalues of a modified principal submatrix this provides an analog of a well known result for jacobi matrices

a note on companion matrices abstract we show that the usual companion matrix of a polynomial of degree n can be factored into a product of n matrices n 1 of them being the identity matrix in which a 2 2 identity submatrix in two consecutive rows and columns is replaced by an appropriate 2 2 matrix the remaining being the identity matrix with the last entry replaced by possibly different entry by a certain similarity transformation we obtain a simple new companion matrix in a pentadiagonal form some generalizations are also possible

green s matrices abstract our goal is to identify and understand matrices a that share essential properties of the unitary hessenberg matrices m that are fundamental for szego s orthogonal polynomials those properties include i recurrence relations connect characteristic polynomials rk x of principal minors of a ii a is determined by generators parameters generalizing reflection coefficients of unitary hessenberg theory iii polynomials rk x correspond not only to a but also to a certain cmv like five diagonal matrix iv the five diagonal matrix factors into a product bc of block diagonal matrices with 2 2 blocks v submatrices above and below the main diagonal of a have rank 1 vi a is a multiplication operator in the appropriate basis of laurent polynomials vii eigenvectors of a can be expressed in terms of those polynomials conditions v connects our analysis to the study of quasi separable matrices but the factorization requirement iv narrows it to the subclass of green s matrices that share properties i vii the key tool is twist transformations that provide 2n matrices all sharing characteristic polynomials of principal minors with a one such twist transformation connects unitary hessenberg to cmv another twist transformation explains findings of fiedler who noticed that companion matrices give examples outside the unitary hessenberg framework we mention briefly the further example of a daubechies wavelet matrix infinite matrices are included

a quasiseparable approach to five diagonal cmv and fiedler matrices abstract recent work in the characterization of structured matrices in terms of characteristic polynomials of principal submatrices is furthered in this paper some classical classes of matrices with quasiseparable structure include tridiagonal related to real orthogonal polynomials and banded matrices unitary hessenberg matrices related to szego polynomials and semiseparable matrices as well as others hence working with the class of quasiseparable matrices provides new results which generalize and unify classical results previous work has focused on characterizing h 1 quasiseparable matrices matrices with order one quasiseparable structure that are also upper hessenberg in this paper the authors introduce the concept of a twist transformation and use such transformations to explain the relationship between h 1 quasiseparable matrices and the subclass of 1 1 quasiseparable matrices without the upper hessenberg restriction which are related to the same systems of polynomials these results generalize the discoveries of cantero fiedler kimura moral and velazquez of five diagonal matrices related to horner and szego polynomials in the context of quasiseparable matrices

parabolic wave equation approximations in heterogenous media the properties of different variants of parabolic approximations of scalar wave equations are analyzed these equations are of general form which includes those used in seismology underwater acoustics and other applications a new version of the parabolic approximation is derived for heterogeneous media it has optimal properties with respect to wave reflection at material interfaces the amplitudes of the reflected and transmitted waves depend continuously on the interface existence uniqueness and energy estimates are proved

an exponentially fitted finite difference method for singular perturbation problems in this paper an exponentially fitted finite difference method is presented for solving singularly perturbed two point boundary value problems with the boundary layer at one end left or right point a fitting factor is introduced in a tridiagonal finite difference scheme and is obtained from the theory of singular perturbations thomas algorithm is used to solve the system the stability of the algorithm is investigated several linear and nonlinear problems are solved to demonstrate the applicability of the method it is observed that the present method approximates the exact solution very well

numerical treatment of singularly perturbed two point boundary value problems we propose a method for numerically solving linear singularly perturbed two point boundary value problems in ordinary differential equations with a boundary layer on the left end of the underlying interval this is a practical method and can be easily implemented on a computer the original problem is divided into inner and outer region differential equation systems the reduced problem is solved to obtain the terminal boundary condition then a new inner region problem is created and solved as a two point boundary value problem tpbvp in turn the outer region problem is also solved as a tpbvp both these tpbvps are efficiently treated by employing a slightly modified classical finite difference scheme coupled with discrete invariant imbedding algorithm to obtain the numerical solutions the stability of some recurrence relations involved in the algorithm is investigated the proposed method is iterative on the terminal point some numerical examples are included and the computational results are compared with exact solutions it is observed that the accuracy predicted can always be achieved with very little computational effort

lattice filter parameterization and modeling of nonstationary processes a general theory of constant parameter modular lattice models for discrete time nonstationary second order processes is presented a complete parametrization of such processes in terms of schur and congruence coefficients is derived by developing a natural connection between the displacement structure of a covariance matrix and schur s test for positive definiteness of matrices schur coeffieients provide a simple solution to problems of covariance extension and rational spectral approximation for nonstationary covariances and they coincide with the well known reflection or parcor coefficients when the covariance is stationary the congruence coefficients provide the time varying gains of a tapped delay line realization of the whitening filter for the process a constant parameter realization of the same filter is derived by combining a lattice filter structure with a tapped delay line both with time invariant gains this configuration also provides a recursive relation for the congruence coefficients namely a generalized levinson szego recursion the tapped delay line part of the realization can be eliminated by introducing the concept of admissibility admissibility also reduces the parametrization of a nonstationary process to schur coefficients alone in analogy to stationary processes which are completely characterized by their parcor coefficients

inverse scattering for discrete transmission line models this paper presents several methods for the identification of the impedance and reflection coefficient profile of a nonuniform discrete transmission line from its response to a given forcing function this problem is the prototype for a wealth of one dimensional inverse scattering problems arising in various fields a unified straightforward approach is presented providing all known inversion procedures and some novel ones the derivations readily follow from causality of signal propagation on the transmission line it is shown that a direct exploitation of the signal propagation model leads to recursive computationally efficient and reliable inversion algorithms in particular the relationships between layer peeling schur type difference equation and layer adjoining levinson type integral equation algorithms are clearly displayed

nondispersive waves interpretation and causality it is proved that the focus wave modes are only a subclass of a larger class of nondispersive waves then these nondispersive waves are interpreted in the framework of goursat s theory of boundary value problems where data are given on the characteristics of the wave equation finally from the remark that causality requires only that no effect precede the cause it is proved that there is no casual dilemma for the nondispersive waves

b spline collocation method for the singular perturbation problem using artificial viscosity in this paper we develop a b spline collocation method using artificial viscosity for solving singularly perturbed equations given by eu x a x u x b x u x f x a x a 0 b x b 0 0 1 u 0 a u 1 b we use the artificial viscosity to capture the exponential features of the exact solution on a uniform mesh and use b spline collocation method which leads to a tridiagonal linear system the convergence analysis is given and the method is shown to have uniform convergence of second order the design of artificial viscosity parameter is confirmed to be a crucial ingredient for simulating the solution of the problem known test problems have been studied to demonstrate the accuracy of the method numerical results show the behaviour of the method with emphasis on treatment of boundary conditions results shown by the method are found to be in good agreement with the exact solution

a note on the derivation of paraxial equation in nonhomogeneous media the paraxial approximation of the two dimensional and three dimensional wave equations in nonhomogeneous media is discussed it is shown that the preferred direction of propagation is along the bicharacteristics of the wave equation the paraxial equation is obtained by assuming that in this direction the variation of the propagating wave is slow enough to make negligible its second derivative the best way to get bicharacteristics is to consider them as the null geodesics of some riemannian space

non classical symmetries and the singular manifold method the burgers and the burgers huxley equations in this paper a generalization of the direct method of clarkson and kruskal 1989 for finding similarity reductions of partial differential equations is found and discussed for the burgers and burgers huxley equations the generalization incorporates the singular manifold method largely based upon the painleve property this singular manifold can be used as a reduced variable furthermore a sort of inverse procedure is hereby developed through which we find the equations that yield the vector field components associated to the symmetries of the pde this procedure also displays the profound relationship among the symmetries and the singular manifold as a reduced variable the symmetries found in this way are shown to be those corresponding to the so called non classical symmetries by bluman and cole 1974 and olver and rosenau 1986

cmv matrices in random matrix theory and integrable systems a survey we present a survey of recent results concerning a remarkable class of unitary matrices the cmv matrices we are particularly interested in the role they play in the theory of random matrices and integrable systems throughout the paper we also emphasize the analogies and connections to jacobi matrices

orthonormal bases of compactly supported wavelets we construct orthonormal bases of compactly supported wavelets with arbitrarily high regularity the order of regularity increases linearly with the support width we start by reviewing the concept of multiresolution analysis as well as several algorithms in vision decomposition and reconstruction the construction then follows from a synthesis of these different approaches

nonexistence of global solutions for a class of two time nonlinear evolution equations nonexistence results of global solutions for a class of ultraparabolic equations and systems involving the fractional laplacian operator are derived the proofs are based on the choice of a suitable test function in the weak formulation of the considered problems the considered equations and systems have their applications in diffusion theory in porous media

five diagonal matrices and zeros of orthogonal polynomials on the unit circle it is shown that monic orthogonal polynomials on the unit circle are the characteristic polynomials of certain five diagonal matrices depending on the schur parameters this result is achieved through the study of orthogonal laurent polynomials on the unit circle more precisely it is a consequence of the five term recurrence relation obtained for these orthogonal laurent polynomials and the one to one correspondence established between them and the orthogonal polynomials on the unit circle as an application some results relating the behaviour of the zeros of orthogonal polynomials and the location of schur parameters are obtained

a brief survey on numerical methods for solving singularly perturbed problems in the present paper a brief survey on computational techniques for the different classes of singularly perturbed problems is given this survey is a continuation of work performed earlier by the first author and contains the literature of the work done by the researchers during the years 2000 2009 however some older important relevant papers are also included in this survey we also mentioned those papers which are not surveyed in the previous survey papers by the first author of this paper see appl math comput 30 1989 223 259 130 2002 457 510 134 2003 371 429 for details thus this survey paper contains a surprisingly large amount of literature on singularly perturbed problems and indeed can serve as an introduction to some of the ideas and methods for the singular perturbation problems

the continuing influence of fiedler s work on companion matrices this is a reconstruction in article like form of a talk given at the minisymposium in honor of miroslav fiedler at the 17th ilas conference held at tu braunschweig germany on thurs 25 aug 2011

the interplay of ranks of submatrices a banded invertible matrix t has a remarkable inverse all upper and lower submatrices of t 1 have low rank depending on the bandwidth in t the exact rank condition is known and it allows fast multiplication by full matrices that arise in the boundary element method we look for the right proof of this property of t 1 ultimately it reduces to a fact that deserves to be better known complementary submatrices of any t and t 1 have the same nullity the last figure in the paper when t is tridiagonal shows that on and above the diagonal of t 1 all rows are proportional

eigenvector computation for almost unitary hessenberg matrices and inversion of szegő vandermonde matrices via discrete transmission lines abstract in this paper we use a discrete transmission line model known to geophysicists as a layered earth model to derive several computationally efficient solutions for the following three problems i as is well known a hessenberg matrix capturing recurrence relations for szeg polynomials differs from unitary only by its last column hence the first problem is how to rapidly evaluate the eigenvectors of this almost unitary hessenberg matrix ii the second problem is to design a fast o n 2 algorithm for inversion of szeg vandermonde matrices generalizing the well known traub algorithm for inversion of the usual vandermonde matrices iii finally the third problem is to extend the well known horner rule to evaluate a polynomial represented in the basis of szeg polynomials as we shall see all three problems are closely related and their solutions can be computed by the same family of fast algorithms although all the results can be derived algebraically here we reveal a connection to system theory to deduce these algorithms via elementary operations on signal flow graphs for digital filter structures including the celebrated markel gray filter widely used in speech processing and certain other filter structures this choice not only clarifies the derivation and suggests a variety of possible computational schemes but it also makes an interesting connection to many other results related to szeg polynomials which have already been interpreted via signal flow graphs for generalized lattice filter structures including the formulas of the gohberg semencul type for inversion of toeplitz like matrices schur type and levinson type algorithms etc for example this connection allows us to show that moment matrices corresponding to what we called horner szeg polynomials though not toeplitz are quasi toeplitz i e they have a certain shift invariance property

the goursat problem for the homogeneous wave equation the goursat problem for the 3 d homogeneous wave equation is presented and some methods to solve it are discussed