a new algebraization of the lameequation we develop a new way of writing the lamehamiltonian in lie algebraic form this yields in a natural way an explicit formula for both the lamepolynomials and the classical non meromorphic lame functions in terms of chebyshev polynomials and of a certain family of weakly orthogonal polynomials

shape from sharp and motion blurred image pair motion blur due to camera shake is a common occurrence during image capture the apparent motion of a scene point in the image plane varies according to both camera motion and scene structure our objective is to infer the camera motion and the depth map of static scenes using motion blur as a cue to this end we use an unblurred blurred image pair initially we develop a technique to estimate the transformation spread function tsf which symbolizes the camera shake this technique uses blur kernels estimated at different points across the image based on the estimated tsf we recover the complete depth map of the scene within a regularization framework

practical computation of matrix functions abstract several new representations for an analytic function f a of a complex matrix a and in particular for e at and a t are derived which also are numerically useful in that they avoid the computation of eigenvalues of a

two step darboux transformations and exceptional laguerre polynomials it has been recently discovered that exceptional families of sturm liouville orthogonal polynomials exist that generalize in some sense the classical polynomials of hermite laguerre and jacobi in this paper we show how new families of exceptional orthogonal polynomials can be constructed by means of multiple step algebraic darboux transformations the construction is illustrated with an example of a 2 step darboux transformation of the classical laguerre polynomials which gives rise to a new orthogonal polynomial system indexed by two integer parameters for particular values of these parameters the classical laguerre and the type ii x l laguerre polynomials are recovered

some explicit formulas for the polynomial decomposition of the matrix exponential and applications abstract this paper is devoted to the study of some formulas for polynomial decomposition of the exponential of a square matrix a more precisely we suppose that the minimal polynomial ma x of a is known and has degree m therefore eta is given in terms of p0 a pm 1 a where the pj a are polynomials in a of degree less than m and some explicit analytic functions examples and applications are given in particular the two cases m 5 and m 6 are considered

solutions of linear recurrence equations solutions are given to general homogeneous and non homogeneous recurrence equations defined on the set of integers solutions to homogeneous recurrence equations are given as i an infinite series of terms involving the partial ordinary bell polynomial ii an infinite series of terms involving the complete ordinary bell polynomial iii a weighted finite sum of terms involving powers solutions to non homogeneous recurrence equations are given as i a finite series of terms involving the complete ordinary bell polynomial ii a weighted infinite sum of terms involving powers computational issues of these solutions are also discussed

modeling the performance of image restoration from motion blur when dealing with motion blur there is an inevitable tradeoff between the amount of blur and the amount of noise in the acquired images the effectiveness of any restoration algorithm typically depends on these amounts and it is difficult to find their best balance in order to ease the restoration task to face this problem we provide a methodology for deriving a statistical model of the restoration performance of a given deblurring algorithm in case of arbitrary motion each restoration error model allows us to investigate how the restoration performance of the corresponding algorithm varies as the blur due to motion develops our modeling treats the point spread function trajectories as random processes and following a monte carlo approach expresses the restoration performance as the expectation of the restoration error conditioned on some motion randomness descriptors and on the exposure time this allows us to coherently encompass various imaging scenarios including camera shake and uniform rectilinear motion and for each of these identify the specific exposure time that maximizes the image quality after deblurring

blow up for quasi linear wave equations in three space dimensions we consider equations of the form u phi x t u u u 1a for a function u u x 1 x n t u x t here denotes the d alembertian square frac partial 2 partial t 2 vartriangle 1b and u represents the vector of first derivatives u that of second derivatives of u with respect to the x k and t on linearizing 1a shall go over into the classical wave equation u 0 this means that and its first derivatives with respect to u u u shall vanish for u u u 0

pole assignment of linear uncertain systems in a sector via a lyapunov type approach the problem of designing robust control laws in performance and in stability for uncertain linear systems is considered performances are taken into account using root clustering of the closed loop dynamic matrix in a sector of the complex plane a synthesis procedure based on a sufficient condition for quadratic stabilization and root clustering such as stabilizability is given using an auxiliary convex problem the results are illustrated by a significant example from the literature

quasi exactly solvable extensions of the lame equation quasi exactly solvable equations are spectral equations which possess a finite number of algebraic eigenvectors for a few of these equations such as the lame equation the number of algebraic solutions is larger than predicted by the general theory a class of quasi exactly solvable equations in one variable is considered and the conditions under which a rich algebraization occurs is discussed families of equations which by themselves are not quasi exactly solvable but can be transformed into systems of coupled quasi exactly solvable equations are discussed these results suggest a scheme for the classification of quasi exactly solvable systems in one variable

exact and quasiexact solvability of second order superintegrable quantum systems i euclidean space preliminaries we show that second order superintegrable systems in two dimensional and three dimensional euclidean space generate both exactly solvable es and quasiexactly solvable qes problems in quantum mechanics via separation of variables and demonstrate the increased insight into the structure of such problems provided by superintegrability a principal advantage of our analysis using nondegenerate superintegrable systems is that they are multiseparable most past separation of variables treatments of qes problems via partial differential equations have only incorporated separability not multiseparability also we propose another definition of es and qes the quantum mechanical problem is called es if the solution of schrodinger equation can be expressed in terms of hypergeometric functions fnm and is qes if the schrodinger equation admits polynomial solutions with coefficients necessarily satisfying a three term or higher order of recurrence relations in three dimensions we give an example of a system tha

asymptotic and interlacing properties of zeros of exceptional jacobi and laguerre polynomials abstract in this paper we state and prove some properties of the zeros of exceptional jacobi and laguerre polynomials generically the zeros of exceptional polynomials fall into two classes the regular zeros which lie in the interval of orthogonality and the exceptional zeros which lie outside that interval we show that the regular zeros have two interlacing properties one is the natural interlacing between zeros of consecutive polynomials as a consequence of their sturm liouville character while the other one shows interlacing between the zeros of exceptional and classical polynomials a heine mehler type formula is provided for the exceptional polynomials which allows to derive the asymptotic behaviour of their regular zeros for large degree n and fixed codimension m we also describe the location and the asymptotic behaviour of the m exceptional zeros which converge for large n to fixed values

linear recurrence relations in the algebra of matrices and applications abstract we study here some linear recurrence relations in the algebra of square matrices with the aid of the cayley hamilton theorem we derive some explicit formulas for a n n r and e ta for every r r matrix a in terms of the coefficients of its characteristic polynomial and matrices a j where 0 j r 1

deblurring saturated night image with function form kernel deblurring saturated night images are a challenging problem because such images have low contrast combined with heavy noise and saturated regions unlike the deblurring schemes that discard saturated regions when estimating blur kernels this paper proposes a novel scheme to deduce blur kernels from saturated regions via a novel kernel representation and advanced algorithms our key technical contribution is the proposed function form representation of blur kernels which regularizes existing matrix form kernels using three functional components 1 trajectory 2 intensity and 3 expansion from automatically detected saturated regions their skeleton brightness and width are fitted into the corresponding three functional components of blur kernels such regularization significantly improves the quality of kernels deduced from saturated regions second we propose an energy minimizing algorithm to select and assign the deduced function form kernels to partitioned image regions as the initialization for non uniform deblurring finally we convert the assigned function form kernels into matrix form for more detailed estimation in a multi scale deconvolution experimental results show that our scheme outperforms existing schemes on challenging real examples

multiple algebraisations of an elliptic calogero sutherland model recently gomez ullate et al phys lett b 511 112 2001 have studied an n particle quantum problem with elliptic function potentials they have shown that the hamiltonian operator preserves a finite dimensional space of functions and as such is quasi exactly solvable qes in this article we show that other types of invariant function spaces exist which are in close relation to the algebraic properties of the elliptic functions accordingly series of new algebraic eigenfunctions can be constructed

quasi exact solvability and the direct approach to invariant subspaces we propose a more direct approach to constructing differential operators that preserve polynomial subspaces than the one based on considering elements of the enveloping algebra of this approach is used here to construct new exactly solvable and quasi exactly solvable quantum hamiltonians on the line which are not lie algebraic it is also applied to generate potentials with multiple algebraic sectors we discuss two illustrative examples of these two applications we show that the generalized lame potential possesses four algebraic sectors and describe a quasi exactly solvable deformation of the morse potential which is not lie algebraic

on the matrix powers and exponential by the r generalized fibonacci sequences methods the companion matrix case properties of some r generalized fibonacci sequences in the algebra of square matrices gl r c allow us to derive some explicit formulas for the matrix powers an n 0 and exponential eta for every a gl r c connection with the verde star method is displayed for a companion matrix a we present some explicit combinatorial formulas for an n 0 and eta furthermore the chen louck s theorem is derived

exceptional orthogonal polynomials and the darboux transformation we adapt the notion of the darboux transformation to the context of polynomial sturm liouville problems as an application we characterize the recently described xm laguerre polynomials in terms of an isospectral darboux transformation we also show that the shape invariance of these new polynomial families is a direct consequence of the permutability property of the darboux crum transformation

more explicit formulas for the matrix exponential abstract the matrix exponential plays an important role in the study of dynamical systems and linear systems an elementary method is discussed for computing eta for general a c n n it is shown that this method can be applied to generate explicit formulas for eta in various forms using this method some well known formulas are rederived and some new formulas are also derived as applications explicit formulas for n 3 or n 4 are given in each case a characterization of eta is also made based on the entries of a alone

new quasi exactly solvable hamiltonians in two dimensions quasi exactly solvable schrodinger operators have the remarkable property that a part of their spectrum can be computed by algebraic methods such operators lie in the enveloping algebra of a finite dimensional lie algebra of first order differential operators the hidden symmetry algebra in this paper we develop some general techniques for constructing quasi exactly solvable operators our methods are applied to provide a wide variety of new explicit two dimensional examples on both flat and curved spaces of quasi exactly solvable hamiltonians corresponding to both semisimple and more general classes of lie algebras

type b 3 fold supersymmetry and non polynomial invariant subspaces we obtain the most general type b 3 fold supersymmetry by solving directly the intertwining relation we then show that it is a necessary and sufficient condition for a second order linear differential operator to have three linearly independent local analytic solutions we find that there are eight linearly independent non trivial linear differential operators of this kind as a by product we find new quasi solvable second order operators preserving a monomial or polynomial subspace one in type b two in type c and four in type x2 all of which have been missed in the existing literature in addition we show that type a type b and type c 3 fold supersymmetries are connected continuously via one parameter a few new quasi solvable models are also presented

richardson lucy deblurring for scenes under a projective motion path this paper addresses how to model and correct image blur that arises when a camera undergoes ego motion while observing a distant scene in particular we discuss how the blurred image can be modeled as an integration of the clear scene under a sequence of planar projective transformations i e homographies that describe the camera s path this projective motion path blur model is more effective at modeling the spatially varying motion blur exhibited by ego motion than conventional methods based on space invariant blur kernels to correct the blurred image we describe how to modify the richardson lucy rl algorithm to incorporate this new blur model in addition we show that our projective motion rl algorithm can incorporate state of the art regularization priors to improve the deblurred results the projective motion path blur model along with the modified rl algorithm is detailed together with experimental results demonstrating its overall effectiveness statistical analysis on the algorithm s convergence properties and robustness to noise is also provided

some explicit formulas for the matrix exponential formulas are derived for the exponential of an arbitrary 2 2 matrix in terms of either its eigenvalues or entries these results are then applied to the second order mechanical vibration equation with weak or strong damping some formulas for the exponential of n n matrices are given for matrices that satisfy an arbitrary quadratic polynomial besides the above mentioned 2 2 matrices these results encompass involutory rank 1 and idempotent matrices consideration is then given to n n matrices that satisfy a special cubic polynomial these results are applied to the case of a 3 3 skew symmetric matrix whose exponential represents the constant rotation of a rigid body about a fixed axis

matrix exponentials another approach the exponential of a matrix and the spectral decomposition of a matrix can be computed knowing nothing more than the eigenvalues of the matrix and the cayley hamilton theorem the arrangement of the ideas in this paper is simple enough to be taught to beginning students of odes

invariant modules and the reduction of nonlinear partial differential equations to dynamical systems abstract we completely characterize all nonlinear partial differential equations leaving a given finite dimensional vector space of analytic functions invariant existence of an invariant subspace leads to a reduction of the associated dynamical partial differential equations to a system of ordinary differential equations and provides a nonlinear counterpart to quasi exactly solvable quantum hamiltonians these results rely on a useful extension of the classical wronskian determinant condition for linear independence of functions in addition new approaches to the characterization of the annihilating differential operators for spaces of analytic functions are presented

the combinatorial power of the companion matrix abstract using combinatorial methods we obtain the explicit polynomials for all elements in an arbitrary power of the companion matrix depending on n variables and provide some interesting applications and relationships to waring s formula on symmetric functions the general solution to homogeneous linear recurrence relations the multiplicative inverse of formal power series the generating function of compositions of numbers a unified approach to chebyshev polynomials including two recently discovered classes that satisfy analogous smallest norm and orthogonality properties subject to different weight functions dickson polynomials of various kinds arising from the theory of finite fields combinatorial expansions of toeplitz matrices and the recent notion of cycle dissections involving a bijective study of waring s formula

computations with infinite toeplitz matrices and polynomials we relate polynomial computations with operations involving infinite band toeplitz matrices and show applications to the numerical solution of markov chains of nonlinear matrix equations to spectral factorizations and to the solution of finite toeplitz systems in particular two matrix versions of graeffe s iteration are introduced and their convergence properties are analyzed correlations between graeffe s iteration for matrix polynomials and cyclic reduction for block toeplitz matrices are pointed out the paper contains a systematic treatment of known topics and presentation of new results improvements and extensions

maximum likelihood reconstruction for emission tomography previous models for emission tomography et do not distinguish the physics of et from that of transmission tomography we give a more accurate general mathematical model for et where an unknown emission density x y z generates and is to be reconstructed from the number of counts n d in each of d detector units d within the model we give an algorithm for determining an estimate of which maximizes the probability p n of observing the actual detector count data n over all possible densities let independent poisson variables n b with unknown means b b 1 b represent the number of unobserved emissions in each of b boxes pixels partitioning an object containing an emitter suppose each emission in box b is detected in detector unit d with probability p b d d 1 d with p b d a one step transition matrix assumed known we observe the total number n n d of emissions in each detector unit d and want to estimate the unknown b b 1 b for each the observed data n has probability or likelihood p n the em algorithm of mathematical statistics starts with an initial estimate 0 and gives the following simple iterative procedure for obtaining a new estimate new from an old estimate old to obtain k k 1 2 new b old b dd 1 n d p b d old b p b d b 1 b

two novel classes of solvable many body problems of goldfish type with constraints two novel classes of many body models with nonlinear interactions of goldfish type are introduced they are solvable provided the initial data satisfy a single constraint in one case in the other two constraints i e for such initial data the solution of their initial value problem can be achieved via algebraic operations such as finding the eigenvalues of given matrices or equivalently the zeros of known polynomials entirely isochronous versions of some of these models are also exhibited i e versions of these models whose nonsingular solutions are all completely periodic with the same period

functions and jordan canonical forms of riordan matrices abstract this paper collects results about riordan arrays in the framework of matrix functions actually the following methodology applies to any square matrix m m with exactly one eigenvalue of algebraic multiplicity m n generalized lagrange bases are used to construct hermite polynomials that interpolate a family of functions moreover we show a parallel application of such functions via jordan canonical forms and case studies are given

deblurring images via dark channel prior we present an effective blind image deblurring algorithm based on the dark channel prior the motivation of this work is an interesting observation that the dark channel of blurred images is less sparse while most patches in a clean image contain some dark pixels this is not the case when they are averaged with neighboring ones by motion blur this change in sparsity of the dark channel pixels is an inherent property of the motion blur process which we prove mathematically and validate using image data enforcing sparsity of the dark channel thus helps blind deblurring in various scenarios such as natural face text and low illumination images however imposing sparsity of the dark channel introduces a non convex non linear optimization problem in this work we introduce a linear approximation to address this issue extensive experiments demonstrate that the proposed deblurring algorithm achieves the state of the art results on natural images and performs favorably against methods designed for specific scenarios in addition we show that the proposed method can be applied to image dehazing

almost global existence to nonlinear wave equations in three space dimensions on considere l existence a t grand de solutions de petite amplitude pour des equations d onde d ordre 2 non lineaires a 4 dimensions d espace temps

on explicit formulas of the principal matrix pth root by polynomial decompositions abstract we present some explicit formulas for calculating the principal p th root of a square matrix the main tools are based on various polynomial decompositions of the principal matrix p th root and well known properties of the linear recursive sequences

normalizability of one dimensional quasi exactly solvable schrodinger operators we completely determine necessary and sufficient conditions for the normalizability of the wave functions giving the algebraic part of the spectrum of a quasi exactly solvable schrodinger operator on the line methods from classical invariant theory are employed to provide a complete list of canonical forms for normalizable quasi exactly solvable hamiltonians and explicit normalizability conditions in general coordinate systems

rational extensions of the quantum harmonic oscillator and exceptional hermite polynomials we prove that every rational extension of the quantum harmonic oscillator that is exactly solvable by polynomials is monodromy free and therefore can be obtained by applying a finite number of state deleting darboux transformations on the harmonic oscillator equivalently every exceptional orthogonal polynomial system of hermite type can be obtained by applying a darboux crum transformation to the classical hermite polynomials exceptional hermite polynomial systems only exist for even codimension 2m and they are indexed by the partitions of m we provide explicit expressions for their corresponding orthogonality weights and differential operators and a separate proof of their completeness exceptional hermite polynomials satisfy a 2l 3 recurrence relation where l is the length of the partition explicit expressions for such recurrence relations are given

single image motion deblurring using an adaptive image prior blind deblurring is the restoration of a sharp image from a blurred image when the blur kernel is unknown most image deblurring algorithms impose a uniform sparse gradient prior on the whole image and reconstruct the image with piecewise smooth characteristics although the sparse gradient prior removes ringing and noise artifacts it inevitably removes mid frequency structures leading to poor visual quality the gradient profile of fractal like structures is close to a gaussian distribution and small gradients from such regions are severely penalized by the sparse gradient prior in this paper we introduce an image deblurring algorithm that adapts the image prior to the underlying detailed structures the statistics of a local detailed structure can be different from those of the global structure by identifying the correct image prior for each pixel in the image our approach models the spatially varying motion blur exhibited by camera motion more effectively than conventional methods based on space invariant blur kernels using different priors for the local region and the motion blur kernel we derive a minimization energy function that alternates between blur kernel estimation and deblurring image restoration until convergence experimental results demonstrate that the proposed approach is efficient and effective in reducing motion blur in an arbitrary direction in a single image

on explicit formulas for the principal matrix logarithm we describe a method for evaluating both the fibonacci horner and the polynomial decomposition of the principal matrix logarithm with a view to solve the lifting problem of its explicit computation the binet formula for linear recursive sequences serves as a triggering factor for giving the exact formula we supply some illustrative examples

global existence of radial solutions for general semilinear hyperbolic systems in 3d abstract we study the well posedness of radial solutions for general nonlinear hyperbolic systems in three dimensions we give a proof of the global existence of radial solutions for general semilinear hyperbolic systems in 3d under null condition with small scaling invariant w 2 1 r 3 data we obtain a bilinear estimate that is effective for hyperbolic systems which do not have any time decay it allows us to achieve the boundedness of the weighted bv norm of the radial solutions

improving formulas for the eigenvalues of finite block toeplitz tridiagonal matrices abstract after a short overview improvements based on the kronecker product are proposed for the eigenvalues of n n block toeplitz tridiagonal block tt matrices with k k matrix entries common in applications some extensions of the spectral properties of the toeplitz tridiagonal matrices are pointed out the eigenvalues of diagonalizable symmetric and skew symmetric block tt matrices are studied besides if certain matrix square root is well defined it is proved that every block tt matrix with commuting matrix entries is isospectral to a related symmetric block tt one further insight about the eigenvalues of hierarchical hermitian block tt matrices of use in the solution of pdes is also achieved

formation of singularities in three dimensional compressible fluids presented are several results on the formation of singularities in solutions to the three dimensional euler equations for a polytropic ideal fluid under various assumptions on the initial data in particular it is shown that a localized fluid which is initially compressed and outgoing on average will develop singularities regardless of the size of the initial disturbance

some exact results for mid band and zero band gap states of associated lame potentials applying certain known theorems about one dimensional periodic potentials we show that the energy spectrum of the associated lame potentials a a 1 msn2 x m b b 1 m cn2 x m dn2 x m consists of a finite number of bound bands followed by a continuum band when both a and b take integer values further if a and b are unequal integers we show that there must exist some zero band gap states i e doubly degenerate states with the same number of nodes more generally in case a and b are not integers but either a b or a b is an integer a b we again show that several of the band gaps vanish due to degeneracy of states with the same number of nodes finally when either a or b is an integer and the other takes a half integral value we obtain exact analytic solutions for several mid band states

algebraic solution of the lindblad equation for a collection of multilevel systems coupled to independent environments we consider the lindblad equation for a collection of multilevel systems coupled to independent environments the equation is symmetric under the exchange of the labels associated with each system and thus the open system dynamics takes place in the permutation symmetric subspace of the operator space the dimension of this space grows polynomially with the number of systems we construct a basis of this space and a set of superoperators whose action on this basis is easily specified for a given number of levels m these superoperators are written in terms of a bosonic realization of the generators of the lie algebra in some cases these results enable finding an analytic solution of the master equation using known lie algebraic methods to demonstrate this we obtain an analytic expression for the state operator of a collection of three level atoms coupled to independent radiation baths when analytic solutions are difficult to find the basis and the superoperators can be used to considerably reduce the computational resources required for simulations

nineteen dubious ways to compute the exponential of a matrix twenty five years later in principle the exponential of a matrix could be computed in many ways methods involving approximation theory differential equations the matrix eigenvalues and the matrix characteristic polyn

blowup phenomena of solutions to the euler equations for compressible fluid flow abstract the blowup phenomena of solutions is investigated for the euler equations of compressible fluid flow the approach is to construct special explicit solutions with spherical symmetry to study certain blowup behavior of multi dimensional solutions in particular the special solutions with velocity of the form c t x are constructed to show the expanding and blowup properties the solution with velocity of the form a t x a t for 1 and for any space dimensions is obtained as a corollary another conclusion is that there is only trivial solution with velocity of the form c t x 1 x for 1 and multi space dimensions

lie algebras of differential operators and lie algebraic potentials abstract an explicit characterisation of all second order differential operators on the line which can be written as bilinear combinations of the generators of a finite dimensional lie algebra of first order differential operators is found solving a problem arising in the lie algebraic approach to scattering theory and molecular dynamics one dimensional potentials corresponding to these lie algebras are explicitly classified which include the harmonic oscillator morse one soliton poschl teller mathieu lame confluent hypergeometric and bessel potentials

special cases of a matrix exponential formula abstract some low dimension or low rank cases of a formula for a product of exponentials of matrices are studied using eigenvalue considerations counterexamples to certain natural conjectures involving a product of matrix exponentials are also given and comments relating to certain recently discovered exponential identities are included

determination of a matrix function using the divided difference method of newton and the interpolation technique of hermite computing a function f a of an n by n matrix a is a frequently occurring problem in control theory and other applications in this paper we introduce an effective approach for the determination of matrix function f a we propose a new technique which is based on the extension of newton divided difference and the interpolation technique of hermite and using the eigenvalues of the given matrix a the new algorithm is tested on several problems to show the efficiency of the presented method finally the application of this method in control theory is highlighted

functions of matrices abstract we use the cayley hamilton theorem and the sequence of horner polynomials associated with a polynomial w z to obtain explicit formulas for functions of the form f ta where f is defined by a convergent power series and a is a square matrix we use a well known explicit formula for the resolvent of a and show that f ta is the hadamard product of f t and the function i ta 1 which is easily obtained from the resolvent

solving some generalized vandermonde systems and inverse of their associate matrices via new approaches for the binet formula the aim of this paper is to exhibit new expressions for the binet formula without solving any type of vandermonde systems this allows us to establish a tractable formula for generalized fibonacci sequences as a consequence new solutions of the vandermonde systems and some special generalized vandermonde systems are provided moreover the inverses of their associated matrices are explicated some new results are formulated also others are recovered and generalized finally comparisons with other methods are broached and some numerical examples are supplied

robust image deblurring with an inaccurate blur kernel most existing nonblind image deblurring methods assume that the blur kernel is free of error however it is often unavoidable in practice that the input blur kernel is erroneous to some extent sometimes the error could be severe e g for images degraded by nonuniform motion blurring when an inaccurate blur kernel is used as the input significant distortions will appear in the image recovered by existing methods in this paper we present a novel convex minimization model that explicitly takes account of error in the blur kernel the resulting minimization problem can be efficiently solved by the so called accelerated proximal gradient method in addition a new boundary extension scheme is incorporated in the proposed model to further improve the results the experiments on both synthesized and real images showed the efficiency and robustness of our algorithm to both the image noise and the model error in the blur kernel

a representation of the solution of the nth order linear difference equation with variable coefficients abstract the linear difference equation of the n th order with variable coefficients and a related difference equation are considered it is shown that for each problem the determinants of submatrices of a single solution matrix provides the solution sequence applications to sequences and series are shown results on toeplitz matrices involving the roots of associated polynomial equations are obtained

the darboux transformation and algebraic deformations of shape invariant potentials we investigate the backward darboux transformations addition of the lowest bound state of shape invariant potentials on the line and classify the subclass of algebraic deformations those for which the potential and the bound states are simple elementary functions a countable family m 0 1 2 of deformations exists for each family of shape invariant potentials we prove that the m th deformation is exactly solvable by polynomials meaning that it leaves invariant an infinite flag of polynomial modules p m m p m 1 m where p n m is a codimension m subspace of in particular we prove that the first m 1 algebraic deformation of the shape invariant class is precisely the class of operators preserving the infinite flag of exceptional monomial modules p n 1 by construction these algebraically deformed hamiltonians do not have an sl 2 hidden symmetry algebra structure

quasi exactly solvable problems and sl 2 algebra recently discovered quasi exactly solvable problems of quantum mechanics are shown to be related to the existence of the finite dimensional representations of the groupsl 2 q whereq r c it is proven that the bilinear formh a j j b j stand for the generators allows one to generate a set of quasi exactly solvable problems of different types including those that are already known we get in particular problems in which the spectral riemannian surface containing an infinite number of sheets is split off one or two finite sheet pieces in the general case the transitionh h d2 dx2 v x is realized with the aim of the elliptic functions all known exactly solvable quantum problems with known spectrum and factorized riemannian surface can be obtained in this approach

uniform decay estimates and the lorentz invariance of the classical wave equation on etudie le comportement asymptotique des solutions de u 0 ou t 2 1 2 n 2 pour des conditions initiales u 0 u t g x en t 0 avec g reguliere a support compact dans r n

global solutions of nonlinear hyperbolic equations for small initial data on considere des systemes quasilineaires d equations hyperboliques d ordre 2 qui sont des deformations non lineaires de l equation d onde on etudie l existence globale et le comportement asymptotique des solutions du probleme de cauchy quand la donnee initiale est suffisamment petite