on the reconstruction of toeplitz matrix inverses from columns abstract in this paper we discuss the problem whether and how the inverse of a toeplitz matrix can be recovered from some of its columns or parts of columns under the requirement that only 2 n 1 parameters are involved the results generalize and strengthen earlier findings by trench gohberg semencul krupnik ben artzi shalom labahn rodman and others special attention is paid to symmetric skewsymmetric and hermitian toeplitz matrix inverses and the question whether such a matrix can be retrieved from a single column

inversion of toeplitz matrices with only two standard equations abstract it is shown that the invertibility of a toeplitz matrix can be determined through the solvability of two standard equations the inverse matrix is represented by two of its columns which are the solutions of the two standard equations and the entries of the original toeplitz matrix

almost planar homoclinic loops in r3 in this paper we study homoclinic loops of vector fields in 3 dimensional space when the two principal eigenvalues are real of opposite sign which we call almost planar we are interested to have a theory for higher codimension bifurcations almost planar homoclinic loop bifurcations generically occur in two versions non twisted and twisted loops we consider high codimension homoclinic loop bifurcations under generic conditions the generic condition forces the existence of a 2 dimensional topological invariant ring non necessarily unique which is a topological cylinder in the non twisted case and a topological mobius band in the twisted case if the third eigenvalue is negative resp positive the limit set resp limit set of trajectories starting in a neighborhood of the loop is contained in the invariant ring the dynamics is then given by fixed points or periodic points of period 2 of a 1 dimensional map which admits a nice asymptotic expansion allowing to treat homoclinic loop bifurcations of arbitrarily high codimension in the non twisted case

displacement operator based decompositions of matrices using circulants or other group matrices abstract we show how an arbitrary square matrix can be expressed as sums of products of circulant and upper or lower triangular toeplitz matrices and as sums of products of matrices derived from finite groups group matrices and matrices which are close to group matrices the results obtained are interesting from several points of view they lead to different methods for computing linear transforms using ffts or fast convolution algorithms to faster methods for solving toeplitz systems to potential methods for mapping linear transforms to parallel computer architectures with interconnection scheme given by the group graph of a finite group and possibly to matrix theoretic methods for expressing relationship between finite groups

generalized hopf bifurcation and its dual generalized homoclinic bifurcation we show the duality between the generalized hopf bifurcation ghb and the generalized homoclinic bifurcation ghb this duality is twofold 1 the poincare normal forms at a weak focus and at a weak saddle 2 the bifurcation diagrams of the ghb and the ghb since the ghb is well known the ghb may be considered as the heart of the article

the generalized homoclinic bifurcation abstract in this article we study a family x of vector fields having at 0 a homoclinic loop with multiplicity n we give conditions for the displacement function to have the same zeros as all those of p x 0 1 x n 1 xn 1 xn in a positive neighborhood of x 0 where is continuous these conditions determine the versality of the family x in a neighborhood of the loop to obtain the result we use the concept of chebychev systems

skew cyclic displacements and inversions of two innovative patterned matrices in this paper we deal mainly with a class of column upper plus lower cupl toeplitz matrices without toeplitz structure which are close to the toeplitz matrices in the sense that their 1 1 cyclic displacements coincide with cyclic displacement of some toeplitz matrices by constructing the corresponding displacement of the matrices we derive the formulas on representation of the inverses of the cupl toeplitz matrices in the form of sums of products of factor 1 1 circulants and 1 1 circulants furthermore through the relation between the cupl toeplitz matrices and the cupl hankel matrices the inverses of the cupl hankel matrices can be obtained as well

extended sources radiation and laplace type integral representation application to wave propagation above and within layered media abstract in has been shown that the sound field reflected by the plane boundary of a layered ground can always be described by a specularly reflected wave and layer potentials despite its generality this representation is not quite suitable for numerical computation dealing with a very simple case weyl and later on ingard and thomasson proposed a representation of the solution in which the layer potential terms are replaced by the sum of surface wave and a laplace type integral such an integral is very convenient for numerical purposes in this paper it is shown that this kind of representation can be obtained for a very wide class of sound propagation problems above or within layered media a half space bounded by a locally reacting surface a finite layer of porous medium a porous medium with depth varying porosity a thin elastic plate wave propagation in shallow water with an impedance bottom many other applications could be developed

chaos near a resonant inclination flip abstract horseshoes play a central role in dynamical systems and are observed in many chaotic systems however most points in a neighborhood of the horseshoe escape after finitely many iterations in this work we construct a new model by re injecting the points that escape the horseshoe we show that this model can be realized within an attractor of a flow arising from a three dimensional vector field after perturbation of an inclination flip homoclinic orbit with a resonance the dynamics of this model without considering the re injection often contains a cuspidal horseshoe with positive entropy and we show that for a computational example the dynamics with re injection can have more complexity than the cuspidal horseshoe alone

on inversion of toeplitz matrices in 1992 labahn and shalom showed that the inverse of a toeplitz matrix can be reconstructed by two of its columns and by some entries of the original toeplitz matrix we here present a modification of this result and another shorter proof

hilbert s 16th problem for quadratic vector fields and cyclicity of graphics abstract we discuss here a method of proof of the existential part of hilbert s 16th problem for quadratic vector fields initiated in 1991 7 the program consists in proving the finite cyclicity of 121 graphics appearing among quadratic systems we briefly summarize some of the methods introduced for that purpose and describe some new developments obtained with wirszcz and oldek 20 for graphics surrounding a center

zeroes of complete elliptic integrals for 1 2 resonance abstract we give an estimate for the number of zeroes of the elliptic integral i h a x y dx b x y dy along the ovals h x y h h x y y 2 2 x 2 x 4 in terms of the degrees of the symmetric polynomials a and b

sound radiation by baffled plates and related boundary integral equations abstract the displacement of the plate is described by its green s representation involving the infinite fluid loaded plate kernel the boundary conditions lead to a system of two integral equations along the plate boundary from a numerical point of view the method is efficient because this kernel is represented in terms of hankel functions and laplace type integrals which can be computed very fast numerical methods are described and an example is given

weighted toeplitz regularized least squares computation for image restoration the main aim of this paper is to develop a fast algorithm for solving weighted toeplitz regularized least squares problems arising from image restoration based on augmented system formulation we develop new hermitian and skew hermitian splitting hss preconditioners for solving such linear systems the advantage of the proposed preconditioner is that the blurring matrix weighting matrix and regularization matrix can be decoupled such that the resulting preconditioner is not expensive to use we show that for a preconditioned system that is derived from a saddle point structure of size m n times m n the preconditioned matrix has an eigenvalue at 1 with multiplicity n and the other m eigenvalues of the form 1 lambda with lambda 1 we also study how to choose the hss parameter to minimize the magnitude of lambda and therefore the krylov subspace method applied to solving the preconditioned system converges very quickly experimental results for image restoration problems are re

on inversion of toeplitz and close to toeplitz matrices abstract it is proved that three columns of an inverse of a scalar toeplitz matrix when properly chosen are always enough to reconstruct it a parametrization of invertible toeplitz matrices in terms of the first and a combination of two consecutive columns of the inverse matrix is also given

multiple homoclinic orbits in conservative and reversible systems we study dynamics near multiple homoclinic orbits to saddles in conservative and reversible flows we consider the existence of two homoclinic orbits in the bellows configuration where the homoclinic orbits approach the equilibrium along the same direction for positive and negative times in conservative systems one finds one parameter families of suspended horseshoes parameterized by the level of the first integral a somewhat similar picture occurs in reversible systems with two homoclinic orbits that are both symmetric the lack of a first integral implies that complete horseshoes do not exist we provide a description of orbits that necessarily do exist a second possible configuration in reversible systems occurs if a non symmetric homoclinic orbit exists and forms a bellows together with its symmetric image we describe the nonwandering set in an unfolding the nonwandering set is shown to simultaneously contain one parameter families of periodic orbits hyperbolic periodic orbits of different index and heteroclinic cycles between these periodic orbits

inversion of toeplitz structured matrices using only standard equations formulas for the inverse of layered or striped toeplitz matrices in terms of solutions of standard equations are given these results are also generalized in the generic case to mosaic toeplitz matrices and also to toeplitz plus hankel matrices

regular representations of semisimple algebras separable field extensions group characters generalized circulants and generalized cyclic codes let a be a semisimple n dimensional commutative algebra over a field f fix a basis b of a and denote by m a b the transpose of the matrix over f that represents a a regularly with respect to b it is easy to see that the set m a b a a can be simultaneously diagonalized over many fields including all perfect fields we use this fact in order to give an elementary proof that such an algebra over an infinite field is generated by a single element and to describe the subalgebras of a in terms of certain partitions of the set 1 2 3 n several applications of these results are shown 1 we give a new proof for the theorem stating that every finite dimensional separable field extension has a primitive element 2 we show that every finite group g has a character such that every other generalized character of g is a polynomial in with rational coefficients this is true for brauer characters as well 3 we give a necessary condition for two generalized characters or brauer characters and that forces the field of values of to contain that of 4 many collections of patterned matrices over a field f such as circulant matrices and some of their generalizations are known to be algebras generated by a single matrix we observe that each subalgebra of such a collection is also generated by a single matrix also if a and b are two elements of such a collection we give a necessary and sufficient condition in terms of the eigenvalue pattern of a and b for a to be a polynomial in b with coefficients in f 5 we show that if a is a generalized cyclic code then the eigenvalues of m a b are the so called matteson solomon coefficients of the codeword a other applications to coding to groups and to field extensions are discussed as well

explicit group inverse of an innovative patterned matrix in this paper we present an innovative patterned matrix rfpl toeplitz matrix is neither the extension of toeplitz matrix nor its special case we show that the group inverse of this new patterned matrix can be represented as the sum of products of lower and upper triangular toeplitz matrices first the explicit expression of the group inverse of an rfpl toeplitz matrix is obtained second the decomposition of the group inverse is given finally an example demonstrates availability of the two methods for the group inverse

saddle quantities and applications abstract in this paper we make the connection between the theoretical study of the generalized homoclinic loop bifurcation ghb and the practical computational aspects for this purpose we first compare the dulac normal form with the joyal normal form these forms were both used to prove the ghb theorem but the second one is far more practical from the algorithmic point of view we then show that the information carried by these normal forms can be computed in a much simpler way using what we shall call dual lyapunov constants the coefficients of a normal form or the dual lyapunov quantities are particular cases of what we shall call saddle quantities we calculate the saddle quantities for quadratic systems and we show that no more than three limit cycles can appear in a homoclinic loop bifurcation we also study the homoclinic loop bifurcation of order 5 appearing in a 6 parameter family close to a hamiltonian system to our knowledge this is the first time that one can find a complete description of a ghb of such high order finally we calculate the saddle quantities for a symmetric cubic vector field and we deduce a bound for the number of limit cycles that appear in a ghb

block toeplitz operators with rational symbols ii in this paper we derive a formula for the rank of the self commutator of hyponormal block toeplitz operators t with matrix valued rational symbols in via the classical hermite fej r interpolation problem

explicit inverse matrices of tribonacci skew circulant type matrices the determinants and inverses of tribonacci skew circulant type matrices are discussed in the paper firstly tribonacci skew circulant type matrices are defined in addition we show the invertibility of the tribonacci skew circulant matrix and present the determinant and the inverse matrix based on constructing the transformation matrices by utilizing the relation between circulant and left circulant the invertibility of the tribonacci skew left circulant are also discussed finally the determinants and the inverse matrices of these matrices are given respectively

a note on sound radiation by baffled plates and related boundary integral equations in a 1985 paper the second author presented a solution for the radiation problem of a fluid loaded baffled plate driven by a time harmonic force by using the green s kernel for the infinite fluid loaded plate equation written in a form suitable for efficient and accurate computations a coupled system of two integral equations along the boundary of the plate was obtained from the boundary conditions the advantage of this approach is obvious it deals with the coupling phenomenon directly by analytical methods important conclusions on the acoustic properties of fluid loaded baffled plates may be drawn from the analytical solution the aim of this letter is to point out the incorrectness of the expressions for the far field radiation pattern included in reference 1 the correct expressions are presented here for the two dimensional case of a line driven strip which was treated in reference 1 in some detail furthermore the low frequency approximations are also discussed briefly