an inverse problem for gyroscopic systems abstract a gyroscopic system is a hermitian matrix valued function of the form l m 2 i g c where m g c r n n with m 0 positive definite g t g 0 c t c and may be indefinite here we study factorizations of the form l i b m i a where a b c n n and use them to construct gyroscopic systems with specified right divisor i a we examine the constraints on the choice of a

lattice sums for off axis electromagnetic scattering by gratings we consider both the spatial domain and spectral domain forms of the green s function appropriate in the electromagnetic diffraction of a plane wave incident at a general angle in the xy plane on a singly periodic structure or grating oriented along the x axis we equate the spatial and spectral forms of the green s function and so establish expressions from which grating lattice sums can be evaluated for oblique incidence we also obtain a set of identities among the lattice sums we use these lattice sums in an expression for the green s function which we show to be computationally fast if knowledge of this function at several points is required for small values of y

on nonsymmetric saddle point matrices that allow conjugate gradient iterations linear systems in saddle point form are usually highly indefinite which often slows down iterative solvers such as krylov subspace methods it has been noted by several authors that negating the second block row of a symmetric indefinite saddle point matrix leads to a nonsymmetric matrix mathcal a whose spectrum is entirely contained in the right half plane in this paper we study conditions so that mathcal a is diagonalizable with a real and positive spectrum these conditions are based on necessary and sufficient conditions for positive definiteness of a certain bilinear form with respect to which mathcal a is symmetric in case the latter conditions are satisfied there exists a well defined conjugate gradient cg method for solving linear systems with mathcal a we give an efficient implementation of this method discuss practical issues such as error bounds and present numerical experiments

on the sign characteristics of hermitian matrix polynomials the sign characteristics of hermitian matrix polynomials are discussed and in particular an appropriate definition of the sign characteristics associated with the eigenvalue infinity the concept of sign characteristic arises in different forms in many scientific fields and is essential for the stability analysis in hamiltonian systems or the perturbation behavior of eigenvalues under structured perturbations we extend classical results by gohberg lancaster and rodman to the case of infinite eigenvalues we derive a systematic approach studying how sign characteristics behave after an analytic change of variables including the important special case of mobius transformations and we prove a signature constraint theorem we also show that the sign characteristic at infinity stays invariant in a neighborhood under perturbations for even degree hermitian matrix polynomials while it may change for odd degree matrix polynomials we argue that the non uniformity can be resolved by introducing an extra zero leading matrix coefficient

large scale complex eigenvalue problems abstract the nonsymmetric eigenvalue problem ax bx is discussed with special emphasis on linear algebra theory on algorithms implemented for solving large scale systems and on interpreting complex spectra obtained in applications from physics and engineering present day supercomputers make the required matrix computations feasible especially in view of the increase in memory

quadratic realizability of palindromic matrix polynomials let cl cl 1 cl 2 be a list consisting of a sublist cl 1 of powers of irreducible monic scalar polynomials over an algebraically closed field ff and a sublist cl 2 of nonnegative integers for an arbitrary such list cl we give easily verifiable necessary and sufficient conditions for cl to be the list of elementary divisors and minimal indices of some t palindromic quadratic matrix polynomial with entries in the field ff for cl satisfying these conditions we show how to explicitly construct a t palindromic quadratic matrix polynomial having cl as its structural data that is we provide a t palindromic quadratic realization of cl our construction of t palindromic realizations is accomplished by taking a direct sum of low bandwidth t palindromic blocks closely resembling the kronecker canonical form of matrix pencils an immediate consequence of our in depth study of the structure of t palindromic quadratic polynomials is that all even grade t palindromic matrix polynomials have a t palindromic strong quadratification finally using a particular m o bius transformation we show how all of our results can be easily extended to quadratic matrix polynomials with t even structure

resurgence flows in porous media porous media with resurgences can be described by a double structure namely a continuous porous medium and capillaries with impermeable walls which relate distant points of the continuous medium the resurgences can be either punctual or extended the equations for flow in such media are derived some general properties of the resulting system which involves nonlocal aspects are deduced a dilute approximation is detailed for punctual resurgences in two dimensional media and is illustrated by a few examples

structure of the scalar field around unidirectional circular cylinders we study transport properties of equal infinite unidirectional circular cylinders that are arbitrarily distributed in a uniform host the problem is reduced to a conjugation problem for the two dimensional laplace equation the flux is written exactly in the form of a power series in bergman s contrast parameter asymptotic formulae for the flux are deduced for densely packed inclusions these formulae involve two small parameters the ratio of the distance between the close discs to their radius and a dimensionless parameter that characterizes the degree of perfect conductor isolator validity of the structural approximation is discussed

transport properties of touching cylinder pairs and of the square array of touching cylinders we consider the solution of the scalar transport problem for a pair of touching cylinders of high conductivity we obtain a closed form expression for the set of multipole coefficients characterizing the potential function in this problem we show that this expression permits the estimation of truncation errors occurring in the matrix solution of the transport problem for the square array of touching cylinders consequently we are able to study the array conductivity for arbitrarily high cylinder conductivities and confirm an asymptotic expression due to o brien we derive a perturbation formula giving the effect of small cylinder separations on array conductivity

the transport properties of layers of elliptical cylinders the rayleigh method has been used in previous work to determine the effective transport properties of a rectangular array of elliptical cylinders for all cases where the ellipses are non intersecting however the calculation of the elliptic lattice sums that naturally arose was numerically inefficient and analytically cumbersome in this present work we use integral transforms to obtain rapidly convergent series for one dimensional elliptical lattice sums and use these lattice sums to determine the transport properties of composites constructed from multiple layers of elliptical cylinders we study the convergence of effective transport properties as a function of the number of layers and show that this convergence is extremely rapid further we use the integral transform representation to exhibit the simplest form of the interior addition formula for harmonic functions in elliptical coordinates

trace minimization and definiteness of symmetric pencils abstract a symmetric matrix pencil a b of order n is called positive definite if there is a such that the matrix a b is positive definite we consider the case with b nonsingular and show that the definiteness is closely related to the existence of min tr x t ax under the condition x t bx j 1 where j 1 is a given diagonal matrix of order n and j 2 1 i we also prove an analog of the cauchy interlacing theorem for some such pencils

low rank perturbations of strongly definitizable transformations and matrix polynomials abstract let a 0 be a transformation on a finite dimensional hilbert space which is self adjoint in an indefinite scalar product generated by g 0 g 0 and invertible the spectrum of a 0 is real when a 0 is g 0 strongly definitizable the problems considered here concern the number of real eigenvalues of a g self adjoint transformation a where a and g are low rank perturbations of a 0 and g 0 a notion called the order of neutrality of a with respect to g is introduced which is relevant to this problem area using linearization as well as direct methods results are obtained concerning self adjoint matrix polynomials which are low rank perturbationsof suitably defined definitizable matrix polynomials applications are made to quadratic matrix polynomials arising in the study of damped systems and gyroscopic systems

the boundary of the numerical range of matrix polynomials abstract some algebraic properties of the sharp points of the numerical range of matrix polynomials are the main subject of this paper we also consider isolated points of the numerical range and the location of the numerical range in a circular annulus

on computing the generalized crawford number of a matrix abstract we use methods of geometric computing combined with hermitean matrix eigenvalue eigenvector evaluations to find the generalized crawford number of a pair of hermitean matrices s 1 and s 2 quickly and to high precision the classical crawford number is defined as the minimal distance from zero in c to the field of values of the associated complex matrix a s 1 is 2 if zero lies outside the field of values of a we describe test and compare a geometry based matlab code for finding for the generalized crawford number that measures the smallest distance of zero from the boundary of the field of values of a even if zero lies inside

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

improving various reversible data hiding schemes via optimal codes for binary covers in reversible data hiding rdh the original cover can be losslessly restored after the embedded information is extracted kalker and willems established a rate distortion model for rdh in which they proved out the rate distortion bound and proposed a recursive code construction in our previous paper we improved the recursive construction to approach the rate distortion bound in this paper we generalize the method in our previous paper using a decompression algorithm as the coding scheme for embedding data and prove that the generalized codes can reach the rate distortion bound as long as the compression algorithm reaches entropy by the proposed binary codes we improve three rdh schemes that use binary feature sequence as covers i e an rs scheme for spatial images one scheme for jpeg images and a pattern substitution scheme for binary images the experimental results show that the novel codes can significantly reduce the embedding distortion furthermore by modifying the histogram shift hs manner we also apply this coding method to one scheme that uses hs showing that the proposed codes can be also exploited to improve integer operation based schemes

detecting a definite hermitian pair and a hyperbolic or elliptic quadratic eigenvalue problem and associated nearness problems an important class of generalized eigenvalue problems ax lambdabx is those in which a and b are hermitian and some real linear combination of them is definite for the quadratic eigenvalue problem qep lambda 2 a lambdab c x 0 with hermitian a b and c and positive definite a particular interest focuses on problems in which x bx 2 4 x ax x cx is one signed for all non zero x for the positive sign these problems are called hyperbolic and for the negative sign elliptic the important class of overdamped problems arising in mechanics is a sub class of the hyperbolic problems for each of these classes of generalized and quadratic eigenvalue problems we show how to check that a putative member has the required properties and we derive the distance to the nearest problem outside the class for definite pairs a b the distance is the crawford number and we derive bisection and level set algorithms both for testing its positivity and for computing it testing hyperbolicity of a qep is shown to reduce to testing a related pair for definiteness the distance to the nearest non hyperbolic or non elliptic n x n qep is shown to be the solution of a global minimization problem with n 1 dearees of freedom numerical results are given to illustrate the theory and algorithms c 2002 published by elsevier science inc

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

deflating quadratic matrix polynomials with structure preserving transformations given a pair of distinct e s l 1 l 2 of an nbyn quadratic matrix polynomial q l with nonsingular leading coefficient and their corresponding ev s we show how to transform q l into a quadratic of the form twobytwoa qd l 0 0 q l having the same e s as q l with qd l an n 1 times n 1 quadratic matrix py and q l a scalar quadratic py with roots l 1 and l 2 this block diagonalization cannot be achieved by a similarity transformation applied directly to q l unless the ev s corresponding to l 1 and l 2 are parallel we identify conditions under which we can construct a family of 2n times 2n elementary similarity transformations that a are rank two modifications of the identity matrix b act on linearizations of q l c preserve the block structure of a large class of block symmetric linearizations of q l thereby defining new quadratic matrix polynomials q 1 l that have the same e s as q l d yield quadratics q 1 l with the property that their ev s associated with l 1 and l 2 are parallel and hence can subsequently be deflated by a similarity applied directly to q 1 l this is the first attempt at building elementary transformations that preserve the block structure of widely used linearizations and which have a specific action

hermitian matrix polynomials with real eigenvalues of definite type part i classification the spectral properties of hermitian matrix polynomials with real eigenvalues have been extensively studied through classes such as the definite or definitizable pencils definite hyperbolic or quasihyperbolic matrix polynomials and overdamped or gyroscopically stabilized quadratics we give a unified treatment of these and related classes that uses the eigenvalue type or sign characteristic as a common thread equivalent conditions are given for each class in a consistent format we show that these classes form a hierarchy all of which are contained in the new class of quasidefinite matrix polynomials as well as collecting and unifying existing results we make several new contributions we propose a new characterization of hyperbolicity in terms of the distribution of the eigenvalue types on the real line by analyzing their effect on eigenvalue type we show that homogeneous rotations allow results for matrix polynomials with nonsingular or definite leading coefficient to be translated into results with no such requirement on the leading coefficient which is important for treating definite and quasidefinite polynomials we also give a sufficient condition for a quasihyperbolic matrix polynomial to be diagonalizable by structure preserving congruence and show that this condition is always satisfied in the quadratic case and for any hyperbolic matrix polynomial thereby identifying an important new class of diagonalizable matrix polynomials

detecting hyperbolic and definite matrix polynomials hyperbolic or more generally definite matrix polynomials are important classes of hermitian matrix polynomials they allow for a definite linearization and can therefore be solved by a standard algorithm for hermitian matrices they have only real eigenvalues which can be characterized as minmax and maxmin values of rayleigh functionals but there is no easy way to test if a given polynomial is hyperbolic or definite or not taking advantage of the safeguarded iteration which converges globally and monotonically to extreme eigenvalues we obtain an efficient algorithm that identifies hyperbolic or definite polynomials and enables the transformation to an equivalent definite linear pencil numerical examples demonstrate the efficiency of the approach

on the sign characteristic of hermitian linearizations in dl p abstract the computation of eigenvalues and eigenvectors of matrix polynomials is an important but difficult problem the standard approach to solve this problem is to use linearizations which are matrix polynomials of degree 1 that share the eigenvalues of p hermitian matrix polynomials and their real eigenvalues are of particular interest in applications attached to these eigenvalues is a set of signs called the sign characteristic from both a theoretical and a practical point of view it is important to be able to recover the sign characteristic of a hermitian linearization of p from the sign characteristic of p in this paper for a hermitian matrix polynomial p with nonsingular leading coefficient we describe in terms of the sign characteristic of p the sign characteristic of the hermitian linearizations in the vector space dl p mackey mackey mehl and mehrmann 2006 in particular we identify the hermitian linearizations in dl p that preserve the sign characteristic of p we also provide a description of the sign characteristic of the hermitian linearizations of p in the family of generalized fiedler pencils with repetition bueno dopico furtado and rychnovsky 2015

positive definite combination of symmetric matrices this correspondence presents a new algorithm to find a positive definite combination of some given real symmetric matrices when this problem has solutions we show that it can be written as an optimization problem an iterative algorithm using a relaxation type method is proposed to solve this optimization problem we also prove that this algorithm has a finite step global convergence property a comparison by computer simulations shows that our algorithm is much faster than another one generally proposed in literature to find such a combination

mobius transformations of matrix polynomials we discuss mobius transformations for general matrix polynomials over arbitrary fields analyzing their influence on regularity rank determinant constructs such as compound matrices and on structural features including sparsity and symmetry results on the preservation of spectral information contained in elementary divisors partial multiplicity sequences invariant pairs and minimal indices are presented the effect on canonical forms such as smith forms and local smith forms on relationships of strict equivalence and spectral equivalence and on the property of being a linearization or quadratification are investigated we show that many important transformations are special instances of mobius transformations and analyze a mobius connection between alternating and palindromic matrix polynomials finally the use of mobius transformations in solving polynomial inverse eigenproblems is illustrated

canonical forms for hermitian matrix pairs under strict equivalence and congruence after a brief historical review and an account of the canonical forms attributed to jordan and kronecker a systematic development is made of the simultaneous reduction of pairs of quadratic forms over the complex numbers and over the reals these reductions are by strict equivalence and by congruence and essentially complete proofs are presented some closely related results which can be derived from the canonical forms are also included they concern simultaneous diagonalization a new criterion for the existence of positive definite linear combinations of a pair of hermitian matrices and the canonical structures of matrices which are self adjoint in an indefinite inner product

finding a positive definite linear combination of two hermitian matrices abstract let a and b be n by n hermitian matrices over the complex field a result of au yeung 1 and stewart 8 states that if x a ib x 0 for all nonzero n vectors x then there is a linear combination of a and b which is positive definite in this article we present an algorithm which finds such a linear combination in a finite number of steps we also discuss the implementation of the algorithm in case a and b are real symmetric sparse matrices

optimal transition probability of reversible data hiding for general distortion metrics and its applications recently a recursive code construction rcc approaching the rate distortion bound of reversible data hiding rdh was proposed however to estimate the rate distortion bound or execute rcc one should first estimate the optimal transition probability matrix otpm by previous methods otpm can be effectively estimated only for some specific distortion metrics such as square error distortion or l 1 norm in this paper we proposed a unified framework of estimating the otpm for general distortion metrics with which we can calculate the rate distortion bound of rdh for general cases and extend rcc to improve state of the art rdh schemes based on any distortion metrics

expansion formulas for the inertias of hermitian matrix polynomials and matrix pencils of orthogonal projectors abstract this paper gives a group of expansion formulas for the inertias of hermitian matrix polynomials a a 2 i a 2 and a a 3 through some congruence transformations for block matrices where a is a hermitian matrix then the paper derives various expansion formulas for the ranks and inertias of some matrix pencils generated from two or three orthogonal projectors and hermitian unitary matrices as applications the paper establishes necessary and sufficient conditions for many matrix equalities to hold as well as many inequalities in the lowner partial ordering to hold

efficient reversible watermarking based on adaptive prediction error expansion and pixel selection prediction error expansion pee is an important technique of reversible watermarking which can embed large payloads into digital images with low distortion in this paper the pee technique is further investigated and an efficient reversible watermarking scheme is proposed by incorporating in pee two new strategies namely adaptive embedding and pixel selection unlike conventional pee which embeds data uniformly we propose to adaptively embed 1 or 2 bits into expandable pixel according to the local complexity this avoids expanding pixels with large prediction errors and thus it reduces embedding impact by decreasing the maximum modification to pixel values meanwhile adaptive pee allows very large payload in a single embedding pass and it improves the capacity limit of conventional pee we also propose to select pixels of smooth area for data embedding and leave rough pixels unchanged in this way compared with conventional pee a more sharply distributed prediction error histogram is obtained and a better visual quality of watermarked image is observed with these improvements our method outperforms conventional pee its superiority over other state of the art methods is also demonstrated experimentally

network approximation in the limit of small interparticle distance of the effective properties of a high contrast random dispersed composite we consider a high contrast two phase composite such as a ceramic polymer composite or a fiberglass composite our objective is to determine the dependence of the effective conductivity or the effective dielectric constant or the effective shear modulus of the composite on the random locations of the inclusions ceramic particles or fibers when the concentration of the inclusions is high we consider a two dimensional model and show that the continuum problem can be approximated by a discrete random network graph we use variational techniques to provide rigorous mathematical justification for this approximation in particular we have shown asymptotic equivalence of the effective constant for the discrete and continuum models in the limit when the relative interparticle distance goes to zero we introduce the geometrical interparticle distance parameter using voronoi tessellation and emphasize the relevance of this parameter due to the fact that for irregular non periodic geometries it is not uniquely determined by the volume fraction of the inclusions we use the discrete network to compute numerically for this purpose we employ a computer program which generates a random distribution of disks on the plane using this distribution we obtain the corresponding discrete network furthermore the computer program provides the distribution of fluxes in the network which is based on keller s formula for two closely spaced disks we compute the dependence of on the volume fraction of the inclusions v for monodispersed composites and obtaine results which are consistent with the percolation theory predictions for polydispersed composites random inclusions of two different sizes the dependence is not simple and is determined by the relative volume fraction v r of large and small particles we found some special values of v r for which the effective coefficient is significantly decreased the computer program which is based on our network model is very efficient and it allows us to collect the statistical data for a large number of random configurations

definite triples of hermitian matrices and matrix polynomials let a b and c be three n n nonzero hermitian matrices the triple a b c is called defnite if the convex hull of the joint numerical range f a b c x ax x bx x cx r3 x cn x x 1 does not contain 0 0 0 if the triple a b c is nondefinite then the numerical ranges of the matrix polynomials q a v3 b v2 c v1 v3 v2 v1 0 and l a 2 b ic 1 2 1 0 coincide with the whole complex plane providing no information as a consequence it is of particular interest to characterize a definite triple a b c and find the distance between 0 0 0 and the boundary f a b c the distance between a nondefinite triple a b c and the nearest definite triples with specified properties is also investigated moreover applications of definite triples on matrix polynomials of special interest are presented

the nearest definite pair for the hermitian generalized eigenvalue problem the generalized eigenvalue problem ax lambda bx has special properties when a b is a hermitian and definite pair given a general hermitian pair a b it is of interest to find the nearest definite pair having a specified crawford number delta 0 we solve the problem in terms of the inner numerical radius associated with the field of values of a ib we show that once the problem has been solved it is trivial to rotate the perturbed pair a da b db to a pair widetilde a widetilde b for which lambda min widetilde b achieves its maximum value delta which is a numerically desirable property when solving the eigenvalue problem by methods that convert to a standard eigenvalue problem by inverting b numerical examples are given to illustrate the analysis

geometry of the numerical range of matrices abstract some techniques for the study of the algebraic curve c a which generates the numerical range w a of an n n matrix a as its convex hull are developed these enable one to give an explicit point equation of c a and a formula for the curvature of c a at a boundary point of w a applied to the case of a nonnegative matrix a a simple relation is found between the curvature of the function a p 1 a a t p being the perron root at 1 2 and the curvature of w a at the perron root of 1 2 a a t a connection with 2 dimensional pencils of hermitian matrices is mentioned and a conjecture formulated

expansion embedding techniques for reversible watermarking reversible watermarking enables the embedding of useful information in a host signal without any loss of host information tian s difference expansion technique is a high capacity reversible method for data embedding however the method suffers from undesirable distortion at low embedding capacities and lack of capacity control due to the need for embedding a location map we propose a histogram shifting technique as an alternative to embedding the location map the proposed technique improves the distortion performance at low embedding capacities and mitigates the capacity control problem we also propose a reversible data embedding technique called prediction error expansion this new technique better exploits the correlation inherent in the neighborhood of a pixel than the difference expansion scheme prediction error expansion and histogram shifting combine to form an effective method for data embedding the experimental results for many standard test images show that prediction error expansion doubles the maximum embedding capacity when compared to difference expansion there is also a significant improvement in the quality of the watermarked image especially at moderate embedding capacities

transport properties of cylinder pairs and of the square array of cylinders we evaluate the multipole coefficients b 2n 1 in the potential expansion for a perfectly conducting cylinder pair separated by unit potential difference we obtain a simple expression for the b 2n 1 when the cylinders approach each other and show that this expression also holds for a square array of cylinders we derive a formula linking ratios of multipole coefficients from green9s theorem and use this to obtain an estimate for the effect of changing the cylinder conductivity from infinity to a large finite value on the array conductivity

hermitian quadratic matrix polynomials solvents and inverse problems a monic quadratic hermitian matrix polynomial l can be factorized into a product of two linear matrix polynomials say l i s i a for the inverse problem of finding a quadratic matrix polynomial with prescribed spectral data eigenvalues and eigenvectors it is natural to prescribe a right solvent a and then determine compatible left solvents s this problem is explored in the present paper the splitting of the spectrum between real eigenvalues and nonreal conjugate pairs plays an important role special attention is paid to the case of real symmetric quadratic polynomials and the allocation of the canonical sign characteristics as well as the eigenvalues themselves

eigenvalue based algorithm and analysis for nonconvex qcqp with one constraint a nonconvex quadratically constrained quadratic programming qcqp with one constraint is usually solved via a dual sdp problem or more s algorithm based on iteratively solving linear systems in this work we introduce an algorithm for qcqp that requires finding just one eigenpair of a generalized eigenvalue problem and involves no outer iterations other than the usually black box iterations for computing the eigenpair numerical experiments illustrate the efficiency and accuracy of our algorithm we also analyze the qcqp solution extensively including difficult cases and show that the canonical form of a matrix pair gives a complete classification of the qcqp in terms of boundedness and attainability and explain how to obtain a global solution whenever it exists

recursive histogram modification establishing equivalency between reversible data hiding and lossless data compression state of the art schemes for reversible data hiding rdh usually consist of two steps first construct a host sequence with a sharp histogram via prediction errors and then embed messages by modifying the histogram with methods such as difference expansion and histogram shift in this paper we focus on the second stage and propose a histogram modification method for rdh which embeds the message by recursively utilizing the decompression and compression processes of an entropy coder we prove that for independent identically distributed i i d gray scale host signals the proposed method asymptotically approaches the rate distortion bound of rdh as long as perfect compression can be realized i e the entropy coder can approach entropy therefore this method establishes the equivalency between reversible data hiding and lossless data compression experiments show that this coding method can be used to improve the performance of previous rdh schemes and the improvements are more significant for larger images