finite time extinction for the strongly damped nonlinear schrodinger equation in bounded domains abstract we prove the finite time extinction property u t 0 on for any t t for some t 0 for solutions of the nonlinear schrodinger problem i u t u a u 1 m u f t x on a bounded domain of r n n 3 a c with im a 0 the damping case and under the crucial assumptions 0 m 1 and the dominating condition 2 m im a 1 m re a we use an energy method as well as several a priori estimates to prove the main conclusion the presence of the non lipschitz nonlinear term in the equation introduces a lack of regularity of the solution requiring a study of the existence and uniqueness of solutions satisfying the equation in some different senses according to the regularity assumed on the data

building blocks for designing arbitrarily smooth subdivision schemes with conic precision since subdivision schemes featured by high smoothness and conic precision are strongly required in many application contexts in this work we define the building blocks to obtain new families of non stationary subdivision schemes enjoying such properties to this purpose we firstly derive a non stationary extension of the lane riesenfeld algorithm and we exploit the resulting class of schemes to design a non stationary family of alternating primal dual subdivision schemes all featured by reproduction of 1 x e t x e t x t 0 i r then we focus our attention on interpolatory subdivision schemes with conic precision that can be obtained as a byproduct of the above classes in particular we present a novel construction of a family of non stationary interpolatory 2 n point schemes which generalizes the well known dubuc deslauriers family in such a way the n th n 2 family member reproduces 2 n 3 e t x e t x t 0 i r and keeps the original smoothness of its stationary counterpart unchanged non stationary extension of lane riesenfeld algorithm new family of alternating primal dual subdivision schemes reproducing conics new family of non stationary interpolatory 2 n point schemes reproducing conics explicit formulation and recurrence relations analysis of the main properties of the above families

a family of non stationary subdivision schemes reproducing exponential polynomials we present a family of non stationary subdivision schemes taking the exponential quasi spline as the basic limit function reaching up to interpolatory refinable functions the exponential quasi spline generalizes the exponential b spline the associated non stationary subdivision scheme can reproduce the desired number of exponential polynomials due to the built in parameter permitting a wide range of tension control our scheme can avoid the undesirable artifact generated by usual non stationary interpolatory schemes on the irregularly spaced region of control points under a natural condition on the rate of convergence of the shape parameters all the subdivision schemes in the family have the same smoothness and approximation order as those of the asymptotically equivalent stationary schemes we present numerical examples to verify that our subdivision schemes have high flexibility and versatility in design

pairs of companion matrices and their simultaneous reduction to complementary triangular forms abstract a polynomial approach is described to deal with problems of the following type given two companion matrices a and z when does there exist a nonsingular matrix s such that s 1 as is upper triangular and s 1 zs is lower triangular and the eigenvalues of a and z appear in the main diagonals of s 1 as and s 1 zs in a prescribed ordering such problems are related to complete minimal factorizations in systems theory

common invariant subspace and commuting matrices abstract let k be a perfect field l be an extension field of k and a b m n k if a has n distinct eigenvalues in l that are explicitly known then we can check if a b are simultaneously triangularizable over l now we assume that a b have a common invariant proper vector subspace of dimension k over an extension field of k and that a the characteristic polynomial of a is irreducible over k let g be the galois group of a we show the following results i if k 1 n 1 then a b commute ii if 1 k n 1 and g s n or g a n then ab ba iii if 1 k n 1 and n is a prime number then ab ba yet when n 4 k 2 we show that a b do not necessarily commute if g is not s 4 or a 4 finally we apply the previous results to solving a matrix equation

a criterion for the existence of common invariant subspaces of matrices it is shown that square matrices a and b have a common invariant subspace w of dimension k 1 if and only if for some scalar s a c si and b c si are invertible and their kth compounds have a common eigenvector which is a grassmann representative for w t he applicability of this criterion and its ability to yield a basis for the common invariant subspace are investigated 2001 elsevier science inc all rights reserved

simultaneous reduction to triangular and companion forms of pairs of matrices the case rank i az 1 abstract this paper is concerned with simultaneous reduction to triangular and companion forms of pairs of matrices a z with rank i az 1 special attention is paid to the case where a is a first and z is a third companion matrix two types of simultaneous triangularization problems are considered 1 the matrix a is to be transformed to upper triangular and z to lower triangular form 2 both a and z are to be transformed to the same upper triangular form the results on companions are made coordinate free by characterizing the pairs a z for which there exists an invertible matrix s such that s 1 as is of first and s 1 zs is of third companion type one of the main theorems reads as follows if rank i az 1 and 1 for every eigenvalue of a and every eigenvalue of z then a and z admit simultaneous reduction to complementary triangular forms

reproduction of exponential polynomials by multivariate non stationary subdivision schemes with a general dilation matrix we study scalar multivariate non stationary subdivision schemes with a general integer dilation matrix we characterize the capability of such schemes to reproduce exponential polynomials in terms of simple algebraic conditions on their symbols these algebraic conditions provide a useful theoretical tool for checking the reproduction properties of existing schemes and for constructing new schemes with desired reproduction capabilities and other enhanced properties we illustrate our results with several examples

monotonicity method applied to the complex ginzburg landau and related equations abstract global existence of unique strong solutions is established for the complex ginzburg landau equation tu i u i u p 1u u 0 where 0 0 r p 1 and 1 2 p p 1 the key is a new inequality in monotonicity methods it is based on the sectorial estimates of in lp 1 and the nonlinear operator u u p 1u appearing in the equation the key inequality also yields the global existence of unique strong solutions of the nonlinear schrodinger type equation with monotone nonlinearity tu i u u p 1u 0 for all p 1

simultaneous reduction to triangular forms after extension with zeroes abstract this paper is concerned with pairs of m m matrices a z for which there exists an invertible m m matrix s such that s 1 as is upper triangular and s1zs is lower triangular such pairs are said to admit simultaneous reduction to complementary triangular forms in particular if such a similariy s does not exist for the pair a z tollowing question is taken into consideration does there exist a positive integer r such that the pair of m r m r matrices a o o o r z o o o r admits simultaneous reduction to complementary triangular forms so that the pair a z obtains the property after extension with zeroes it is shown that the answer to this question is mixed an example of a pair is given which obtains the property after extension with one zero this example is put into contrast with a large number of results that answers the question negatively i e that describes situations where the pair of extended matrices admits simultaneous reduction to complementary triangular forms if and only if the original pair does

minimal representations of semiseparable kernels and systems with separable boundary conditions abstract the simplest representations of triangular parts of finite rank kernels are analysed criteria for uniqueness up to similarity are given the results are applied to the problem of minimal realization of systems with separable boundary conditions

simultaneous reduction of sets of matrices under similarity abstract let s be a set of n n matrices over a field f and a the algebra generated by s over f the problem of deciding whether the elements of s can be simultaneously reduced to block triangular form with the diagonal blocks of some specified size is considered and an account is given of various methods used to attack the problem most of the techniques use representation theory to obtain information on a the problems of simultaneous triangularization existence of common eigenvectors etc are also considered the aim of the paper is to survey the methods used to attack these problems and to give some typical results the paper does not contain many new results

a characterization of m accretivity and an application to nonlinear schrodinger type equations on caracterise la m accretivite et la monotonicite cyclique et on demontre le theoreme d existence des equations de type schrodinger non lineaires par la methode des semigroupes

a family of non uniform subdivision schemes with variable parameters for curve design in this paper we present non uniform subdivision schemes with variable parameter sequences a locally different tension parameter is set at each edge of the initial control polygon to control locally the shape of the resulting curve such that the scheme becomes non uniform due to the variable parameters the scheme can reproduce locally different analytic curves such as conics lissajous trigonometric and catenary curves hence blending curves including such analytic components can be successfully generated we discuss the convergence and smoothness of the proposed non uniform schemes and present some numerical results to demonstrate their advantages in geometric modeling furthermore as an application we propose a chamfering algorithm which can be used in designing automobile and mechanical products

algebraic conditions on non stationary subdivision symbols for exponential polynomial reproduction we present an accurate investigation of the algebraic conditions that the symbols of a non singular univariate binary non stationary subdivision scheme should fulfill in order to reproduce spaces of exponential polynomials a subdivision scheme is said to possess the property of reproducing exponential polynomials if for any initial data uniformly sampled from some exponential polynomial function the scheme yields the same function in the limit the importance of this property is due to the fact that several curves obtained by combinations of exponential polynomials such as conic sections spirals or special trigonometric and hyperbolic functions are considered of interest in geometric modeling since the space of exponential polynomials trivially includes standard polynomials this work extends the theory on polynomial reproduction to the non stationary context a significant application of the derived algebraic conditions on the subdivision symbols is the construction of new non stationary subdivision schemes with specific reproduction properties

existence and decay estimates of solutions to complex ginzburg landau type equations abstract this paper deals with the initial boundary value problem denoted by cgl for the complex ginzburg landau type equation u t i u i u q 1 u u 0 with initial data u 0 l p in the case 1 q 1 2 p n where is bounded or unbounded in r n 0 r there are a lot of studies on local and global existence of solutions to cgl including the physically relevant case q 3 and 0 this paper gives existence results with precise properties of solutions and rigorous proof from a mathematical point of view the physically relevant case can be considered as a special case of the results moreover in the case 0 local and global existence of solutions with the decay estimate u t l p c 1 e c 2 t c 1 c 2 are positive constants is obtained under some conditions the key to the local existence is to construct a semigroup e t i and its l p l q estimate on the other hand the key to the global existence is to derive estimates for solutions by using a kind of interpolation inequality with re v p 2 v i v

similarity and reduction for time varying linear systems with well posed boundary conditions the classical theory of similarity and reduction of causal and anti causal systems is generalized and extended to time varying linear systems with well posed boundary conditions two main problems are solved the first is to determine to what extent a system with boundary conditions can be simplified by similarity and reduction the second problem is to find the invariants of such a simplification special attention is given to the time invariant case

shape controlled interpolatory ternary subdivision ternary subdivision schemes compare favorably with their binary analogues because they are able to generate limit functions with the same or higher smoothness but smaller support in this work we consider the two issues of local tension control and conics reproduction in univariate interpolating ternary refinements we show that both these features can be included in a unique interpolating 4 point subdivision method by means of non stationary insertion rules that do not affect the improved smoothness and locality of ternary schemes this is realized by exploiting local shape parameters associated with the initial polyline edges

common eigenvectors of two matrices abstract a computable criterion is given for two square matrices to possess a common eigenvector as well as a criterion for one matrix to have an eigenvector lying in a given subspace some applications are discussed

the inviscid limit for the complex ginzburg landau equation abstract we study the inviscid limit of the complex ginzburg landau equation we observe that the solutions for the complex ginzburg landau equation converge to the corresponding solutions for the nonlinear schrodinger equation we give its convergence rate we estimate the integral forms of solutions for two equations

all optimal hankel norm approximations of linear multivariable systems and their l error bounds the problem of approximating a multivariable transfer function g s of mcmillan degree n by s of mcmillan degree k is considered a complete characterization of all approximations that minimize the hankel norm is derived the solution involves a characterization of all rational functions s f s that minimize where s has mcmillan degree k and f s is anticavisal the solution to the latter problem is via results on balanced realizations all pass functions and the inertia of matrices all in terms of the solutions to lyapunov equations it is then shown that where k 1 g s is the k l st hankel singular value of g s and for one class of optimal hankel norm approximations the method is not computationally demanding and is applied to a 12 state model

well posedness and energy decay estimates in the cauchy problem for the damped defocusing schrodinger equation abstract in this paper we study the existence at the h 1 level as well as the stability for the damped defocusing schrodinger equation in r d the considered damping coefficient is time dependent and may vanish at infinity to prove the existence we employ the method devised by ozsari kalantarov and lasiecka 27 which is based on monotone operators theory in particular when d 1 or d 2 we obtain the uniqueness decay estimates for the l 2 level and h 1 l p 2 level energies are established with the help of direct multipliers method coupled with refined energy estimates and a lower semi continuity argument

from approximating subdivision schemes for exponential splines to high performance interpolating algorithms in this work we construct three novel families of approximating subdivision schemes that generate piecewise exponential polynomials and we show how to convert these into interpolating schemes of great interest in curve design for their ability to reproduce important analytical shapes and to provide highly smooth limit curves with a controllable tension in particular throughout this paper we will focus on the derivation of 6 point interpolating schemes that turn out to be unique in combining vital ingredients like c 2 continuity simplicity of definition ease of implementation user independency tension control and ability to reproduce salient trigonometric and transcendental curves

global attractors for the complex ginzburg landau equation abstract this paper concerns the long time behavior of the following complex ginzburg landau equations u t i u i u p 2 u u 0 without any restriction on p 2 under the assumptions 1 4 we first prove the well posedness of strong solutions for the complex ginzburg landau equations and then the existence of absorbing sets in l 2 h 0 1 l p and h 2 l 2 p 1 respectively for the semigroup s t t 0 generated by 1 1 1 3 is established finally we prove the existence of global attractors in l 2 and h 0 1 for the semigroup s t t 0 generated by 1 1 1 3 by the sobolev compactness embedding theorem and prove the existence of global attractor in l p for the semigroup s t t 0 generated by 1 1 1 3 using interpolation inequality

complementary triangular forms of pairs of matrices realizations with prescribed main matrices and complete factorization of rational matrix functions abstract the problem considered is the following given two square matrices a and z when does there exist an invertible matrix s such that s 1 as is upper triangular and s 1 zs is lower triangular several sufficient conditions for such an s to exist are presented in some cases the conditions are also necessary attention is paid to the order in which the eigenvalues of a and z may appear on the diagonals of s 1 as and s 1 zs respectively there is an intimate connection with the notions of realization and minimal factorization from systems theory

factorization and job scheduling a connection via companion based matrix functions a connection is made between two sets of problems the first set involves factorization problems of specific rational matrix functions the companion based matrix functions the second set is concerned with variants of the two machine flow shop problem 2mfsp from job scheduling theory in particular it is shown that with each companion based matrix function one can associate an instance of 2mfsp and vice versa the latter can be done in such a way that the factorization properties of the companion based matrix function correspond to the combinatorial properties of the instance of 2mfsp

on condensed forms for partially commuting matrices abstract two complex n n matrices a and b are said to be partially commuting if a and b have a common eigenvector we propose a condensed form for such matrices that can be obtained from a and b by a finite rational computation the condensed form is a pair of block triangular matrices with the sizes of the blocks being uniquely defined by the original matrices we then show how to obtain additional zeros inside the diagonal blocks of a condensed form by using the generalized lanczos procedure as given by elsner and ikramov this procedure can also be considered as a finite rational process we point out several applications of the constructions above it turns out that for laffey pairs of matrices i e for matrices a b such that rank a b 1 the condensed form is a pair of 2 2 block triangular matrices using this fact we show an economical way to find a spanning set for the matrix algebra generated by laffey matrices a and b another application concerns so called k self adjoint matrices we examine such matrices in the unitary space as well as in a krein space of defect 1 as an appendix we give a new description of the shemesh subspace of matrices a and b this is the maximal common invariant subspace of a and b on which these matrices commute

an algorithmic approach to simultaneous triangularization abstract we present and prove the validity of an algorithm constructing a simultaneous triangularization of a set on n matrices in c n n to do so we first prove that if a set of matrices has a common block decomposition then the set of matrices has a simultaneous triangularization if and only if the blocks on the diagonal have a simultaneous triangularization

deciding polyhedrality of spectrahedra spectrahedra are linear sections of the cone of positive semidefinite matrices which as convex bodies generalize the class of polyhedra in this paper we investigate the problem of recognizing when a spectrahedron is polyhedral we generalize and strengthen results of m v ramana polyhedra spectrahedra and semidefinite programming in topics in semidefinite and interior point methods fields inst commun 18 ams providence ri 1998 pp 27 38 regarding the structure of spectrahedra and we devise a normal form of representations of spectrahedra this normal form is effectively computable and leads to an algorithm for deciding polyhedrality

upper triangularization of matrices by permutations and lower triangular similarity transformations abstract let a be an arbitary square matrix as is well known there exists an invertible matrix s such that s 1as is upper triangular the present paper is concerned with the observation that s can always be chosen in the form s l where is a permutation matrix and l is lower triangular assuming that the eigenvalues of a are given the matrices l and u l 1 1a l are constructed in an explicit way the construction gives insight into the freedom one has in choosing the permutation matrix two cases where can be chosen to be the identity matrix are discussed a diagonable a a low order toeplitz matrix there is a connection with systems theory

similarity invariants for pairs of upper triangular toeplitz matrices abstract the problem considered is that of simultaneous reduction to simple forms of pairs of upper triangular toeplitz matrices in this context simple matrices are those that are equal to or differ only slightly from a power of the upper triangular nilpotent jordan block the problem is related to that of complementary triangularization of pairs of matrices and therefore has a background in systems theory the main results are concerned with existence and uniqueness of the reduced form also a similarity invariant for pairs of upper triangular toeplitz matrices is obtained the invariant consists of a pair involving a complex number and an integer the results can be generalized to a class of matrices strictly larger than that formed by the upper triangular toeplitz matrices

weak and strong solutions of the complex ginzburg landau equation abstract the generalized complex ginzburg landau equation t a ra 1 iv a 1 i a 2 a has been proposed and studied as a model for turbulent dynamics in nonlinear partial differential equations it is a particularly interesting model in this respect because it is a dissipative version of the hamiltonian nonlinear schrodinger equation possessing solutions that form localized singularities in finite time in this paper we investigate existence and regularity of solutions to this equation subject to periodic boundary conditions in various spatial dimensions appropriately defined weak solutions are established globally in time and unique strong solutions are found locally a new collection of a priori estimates are presented and we discuss the relationship of our results for the complex ginzburg landau equation to analogous issues for fluid turbulence described by the incompressible navier stokes equations

half range solutions of indefinite sturm liouville problems abstract for diffusion equations of indefinite sturm liouville type we develop two equivalent methods of constructing explicit representations for the solutions the first method is based on an eigenfunction expansion whereas the second uses a wiener hopf type integral equation and factorization some illustrative examples are worked out

random repeated quantum interactions and random invariant states we consider a generalized model of repeated quantum interactions where a system mathcal h is interacting in a random way with a sequence of independent quantum systems mathcal k n n geq 1 two types of randomness are studied in detail one is provided by considering haar distributed unitaries to describe each interaction between mathcal h and mathcal k n the other involves random quantum states describing each copy mathcal k n in the limit of a large number of interactions we present convergence results for the asymptotic state of mathcal h this is achieved by studying spectral properties of random quantum channels which guarantee the existence of unique invariant states finally this allows to introduce a new physically motivated ensemble of random density matrices called the asymptotic induced ensemble

weakly damped focusing nonlinear schrodinger equations with dirichlet control abstract in this article we consider the weakly damped focusing nonlinear schrodinger equations on bounded domains at the natural h 1 energy level with dirichlet control acting on a portion of the boundary we introduce the dynamic extension method for homogenizing the inhomogeneous boundary input then we construct approximate solutions using monotone operator theory a hidden trace regularity is proved to control the norm of the solutions in a global sense this allows the use of compactness techniques by which we prove the existence of weak solutions finally using multiplier techniques we prove the exponential decay of solutions under the assumption that the boundary control also decays in a similar fashion

linear spanning sets for matrix spaces abstract necessary and sufficient conditions are given on matrices a b and s having entries in some field f and suitable dimensions such that the linear span of the terms a i s b j over f is equal to the whole matrix space this result is then used to determine the cardinality of subsets of f a s f b when f is a finite field

on minimality and stable minimality of time varying linear systems with well posed boundary conditions for time varying linear systems with well posed boundary conditions the concept of stable minimality is introduced a complete characterization is given of stably minimal systems with analytic coefficients

exponential pseudo splines looking beyond exponential b splines abstract pseudo splines are a rich family of functions that allows the user to meet various demands for balancing polynomial reproduction i e approximation power regularity and support size such a family includes as special members b spline functions universally known for their usefulness in different fields of application when replacing polynomial reproduction by exponential polynomial reproduction a new family of functions is obtained this new family is here constructed and called the family of exponential pseudo splines it is the nonstationary counterpart of polynomial pseudo splines and includes exponential b splines as a special subclass in this work we provide a computational strategy for deriving the explicit expression of the laurent polynomial sequence that identifies the family of exponential pseudo spline nonstationary subdivision schemes for this family we study its symmetry properties and perform its convergence and regularity analysis finally we also show that the family of primal exponential pseudo splines fills in the gap between exponential b splines and interpolatory cardinal functions this extends the analogous property of primal pseudo spline stationary subdivision schemes

common invariant subspaces of two matrices abstract a finite rational procedure of the shemesh type is proposed to check whether given complex n n matrices a and b have a common invariant subspace of dimension k 2 k n the only restriction is that at least one of a and b must have distinct eigenvalues this fact can also be verified by a finite rational computation

stability of simultaneously block triangularisable switched systems with partial state reset abstractwe study the stability of a certain class of switched systems where discontinuous jumps resets on some of the state components are allowed at the switching instants it is known that if all components of the state are available for reset the system can be stabilisable by an adequate choice of resets however this question may have negative answer if there are forbidden state components for reset we give a sufficient condition for the stabilisability of a switched system under arbitrary switching by partial state reset in terms of a block simultaneous triangularisability condition based on this sufficient condition we show that the particular class of systems with partially commuting stable system matrices is stabilisable by partial state reset we also provide an algorithm that allows testing whether a switched system belongs to this particular class of systems

common eigenvector approach to exact order reduction for roesser state space models of multidimensional systems abstract the well known popov belevitch hautus pbh tests play an important role in the kalman decomposition of 1 d systems and reveal the relationship among the eigenvalues the eigenvectors and the reducibility of a given 1 d state space model this paper is to try to generalize the pbh tests to the n d case for the exact reducibility of n d roesser models by exploiting the so called common eigenvectors specifically the notion of constrained common eigenvectors is introduced for the first time which provides insight into the relationship between reducibility and multiple eigenvalues based on this result new reducibility conditions and the corresponding reduction procedure are developed for n d roesser models it will be shown that this common eigenvector approach is applicable to a larger class of roesser models for which the existing approaches may not be applied to do further order reduction a grobner basis approach is proposed to compute such a constrained common eigenvector which also leads to an equivalent reducibility condition moreover a generalization to the state delay case is also given so that the eigenvalues of both the system matrix and the state delay system matrix can be treated simultaneously

on the kneser property for the complex ginzburg landau equation and the lotka volterra system with diffusion as we have seen in the previous chapters when we consider the cauchy problem of a differential equation and uniqueness fails to hold or it is not known to hold then we have to consider a set of solutions corresponding to a given initial data

analysis of asymptotically equivalent binary subdivision schemes non stationary uniform subdivision schemes are presented including some important examples as the up function and exponential splines the multivariate analysis of subdivision schemes is extended to a class of non stationary schemes which are asymptotically equivalent to converging stationary or non stationary schemes

a causal realization theory part i linear deterministic systems we study acausal linear systems their controllability and observability properties and the weighting patterns that they realize a complete classification is given of all minimal real analytic realizations of a given weighting pattern and of all minimal autonomous realizations of a stationary weighting pattern

application of shemesh theorem to quantum channels completely positive maps are useful in modeling the discrete evolution of quantum systems spectral properties of operators associated with such maps are relevant for determining the asymptotic dynamics of quantum systems subjected to multiple interactions described by the same quantum channel we discuss a connection between the properties of the peripheral spectrum of completely positive and trace preserving map and the algebra generated by its kraus operators a a1 ak by applying the shemesh and amitsur levitzki theorems to analyse the structure of the algebra a a1 ak one can predict the asymptotic dynamics for a class of operations completely positive maps are useful in modeling the discrete evolution of quantum systems spectral properties of operators associated with such maps are relevant for determining the asymptotic dynamics of quantum systems subjected to multiple interactions described by the same quantum channel we discuss a connection between the properties of the peripheral spectrum of completely positive and trace preserving map and the algebra generated by its kraus operators a a1 ak by applying the shemesh and amitsur levitzki theorems to analyse the structure of the algebra a a1 ak one can predict the asymptotic dynamics for a class of operations

on matrices with common invariant cones with applications in neural and gene networks motivated by a differential continuous time switching model for gene and neural networks we investigate matrix theoretic problems regarding the relative location and topology of the dominant eigenvectors of words constructed multiplicatively from two matrices a and b these problems are naturally associated with the existence of common invariant subspaces and common invariant proper cones of a and b the commuting case and the two dimensional case are rich and considered analytically we also analyze and recast the problem of the existence of a common invariant polyhedral cone in a multilinear framework as well as present necessary conditions for the existence of low dimensional common invariant cones

solving the two dimensional cis problem by a rational algorithm abstract the cis problem is formulated as follows let p be a fixed integer 1 p for given n n compex matrices a and b can one verify whether a and b have a common invariant subspace of dimension p by a procedure employing a finite number of arithmetical operations we describe an algorithm solving the cis problem for p 2 unlike the algorithm proposed earlier by the second and third authors the new algorithm does not impose any restrictions on a and b moreover when a and b generate a semisimple algebra the algorithm is able to solve the cis problem for any p 1

quasicomplete factorization and the two machine flow shop problem a connection is made between the two machine flow shop problem 2mfsp from job scheduling theory and the issue of quasicomplete factorization of rational matrix functions a quasicomplete factorization is a factorization into elementary i e degree one factors such that the number of factors is minimal for a companion based matrix function w the number of factors in a quasicomplete factorization of w is related in a simple way to the minimum makespan of an instance j of 2mfsp which can be associated to w as a consequence of this result variants of the 2mfsp and other types of factorization can be related too

upper triangularization of matrices by lower triangular similarities abstract this paper is concerned with the following questions given a square matrix a when does there exist an invertible lower triangular matrix l such that l 1 al is upper triangular and if so what can be said about the order in which the eigenvalues of a may appear on the diagonal of l 1 al the motivation for considering these questions comes from systems theory in fact they arise in the study of complete factorizations of rational matrix functions there is also an intimate connection with the problem of complementary triangularization of pairs of matrices discussed elsewhere by the first author

a note on the matrix equation xa ax xp abstract another more elementary proof is given of proposition 2 3 in a recent paper by dietrich burde this proposition says that n n matrices a and x over an algebraically closed field k of characteristic zero satisfying the equality in the title of this paper can be simultaneously triangularized the proposed proof is based on the shemesh criterion for two matrices to have a common eigenvector

companion based matrix functions description and minimal factorization companion based matrix functions are rational matrix functions admitting a minimal realization involving state space matrices that are first companions necessary and sufficient conditions are given for a rational matrix function to be companion based minimal factorization of such functions is discussed in detail it is shown that the property of being companion based is hereditary with respect to minimal factorization also the issue of minimal factorization is reduced to a division problem for pairs of monic polynomials of the same degree in this context a connection with the euclidean algorithm is made the results apply to canonical wiener hopf factorization as well as to complete factorization the analysis of the latter leads to a combinatorial problem involving the eigenvalues of the state space matrices the algorithmic aspects of this problem are intimately related to the two machine flow shop problem and johnson s rule from job scheduling theory

antisymmetric hamiltonians variational resolutions for navier stokes and other nonlinear evolutions the theory of anti self dual asd lagrangians introduced in 6 is developed further to allow for a variational resolution of nonlinear pdes of the form u au u f 0 where is a convex lower semicontinuous function on a reflexive banach space x f x a d a x x is a positive linear operator and where d x x is a nonlinear operator that satisfies suitable continuity and antisymmetry properties asd lagrangians on path spaces also yield variational resolutions for nonlinear evolution equations of the form u t u t au t u t f 0 starting at u 0 u0 in both stationary and dynamic cases the equations associated to the proposed variational principles are not derived from the fact that they are critical points of the action functional but because they are also zeroes of the lagrangian itself for that we establish a general nonlinear variational principle that has many applications in particular to navier stokes type equations to generalized choquard pekar schrodinger equations with nonlocal terms as well as to complex ginsburg landau type initial value problems the case of navier stokes evolutions is more involved and will be dealt with in 9 the general theory of antisymmetric hamiltonians and its applications is developed in detail in an upcoming monograph 7 2007 wiley periodicals inc

simultaneous reduction of matrices in the presence of a nonderogatory matrix abstract some known results on the question when are two matrices simultaneously similar to matrices in upper triangular form are simplified in the case when one of the matrices is nonderogatory

upper triangular similarity of upper triangular matrices abstract we consider the following equivalence relation in the set of all complex upper triangular n n matrices a and b are called u similar if there exists an invertible upper triangular matrix s such that a s 1 bs if a b are u similar then they must have the same diagonal and the same jordan form it is known that for n 6 there are infinitely many mutually non u similar nilpotent upper triangular matrices with the same jordan form we introduce an appropriate generalization of the jordan block called an irreducible matrix and we prove that each upper triangular matrix is u similar to a generalized direct sum of irreducible blocks where the location and the order of the blocks is fixed and each block is determined uniquely up to u similarity

monotonicity method applied to complex ginzburg landau type equations abstract global existence and smoothing effect are established for the complex ginzburg landau type equation in which the real and imaginary parts of the complex coefficient are multiplied by nonlinear terms with different powers the proof is based on monotonicity methods described by subdifferential operators the key lies in two modified inequalities

on the existence of a common eigenvector for all matrices in the commutant of a single matrix the main purpose of this paper is to study common invariant subspaces of any matrix in the centralizer of a given matrix a mn f where f denotes an algebraically closed field in particular we obtain a necessary and sufficient condition for the existence of a common eigenvector for all the matrices in this set

standing waves of the complex ginzburg landau equation abstract we prove the existence of nontrivial standing wave solutions of the complex ginzburg landau equation t e i e i k on a bounded domain with dirichlet boundary conditions our result requires that 0 be sufficiently small

wiener hopf factorization inverse fourier transforms and exponentially dichotomous operators abstract for a certain class of matrix functions that are analytic on the real line but not at infinity an explicit method of factorization and of constructing inverse fourier transforms is developed this method is applied to invert wiener hopf integral equations on the half line and the full line the results obtained extend analogous results of the authors for rational matrix functions and for functions that are analytic on the real line and at infinity the analysis is based on an infinite dimensional realization theorem which involves operators that are a direct sum of two infinitesimal generators of c 0 semigroups of negative exponential type one of which has support on the negative half line and the other on the positive half line the latter operators are called exponentially dichotomous and the study of their properties forms an essential part of the paper

schur complements and state space realizations motivated by state space realizations of transfer functions from system theory a number of operations on schur complements are introduced and studied these operations are equivalence extension multiplication inversion and factorization together they form an algebraic framework which is of independent interest and also useful in solving problems in analysis

existence of standing waves for the complex ginzburg landau equation abstract we study the existence of standing wave solutions of the complex ginzburg landau equation gl t e i i e i 0 in r n where 0 n 2 4 0 and r we show that for any 2 2 there exists e 0 such that gl has a non trivial standing wave solution if e analogous result is obtained in a ball r n for 1 where 1 is the first eigenvalue of the laplace operator with dirichlet boundary conditions

the common invariant subspace problem an approach via grobner bases let a be an n n matrix it is a relatively simple process to construct a homogeneous ideal generated by quadrics whose associated projective variety parametrizes the one dimensional invariant subspaces of a given a finite collection of n n matrices one can similarly construct a homogeneous ideal again generated by quadrics whose associated projective variety parametrizes the one dimensional subspaces which are invariant subspaces for every member of the collection grobner basis techniques then provide a finite rational algorithm to determine how many points are on this variety in other words a finite rational algorithm is given to determine both the existence and quantity of common one dimensional invariant subspaces to a set of matrices this is then extended for each d to an algorithm to determine both the existence and quantity of common d dimensional invariant subspaces to a set of matrices 2004 published by elsevier inc

the cauchy problem in local spaces for the complex ginzburg landau equation i compactness methods we continue the study of the initial value problem for the complex ginzburg landau equation with a 0 b 0 g 0 in initiated in a previous paper i we treat the case where the initial data and the solutions belong to local uniform spaces more precisely to spaces of functions satisfying local regularity conditions and uniform bounds in local norms but no decay conditions or arbitrarily weak decay conditions at infinity in in i we used compactness methods and an extended version of recent local estimates 3 and proved in particular the existence of solutions globally defined in time with local regularity of the initial data corresponding to the spaces l r for r 2 or h 1 here we treat the same problem by contraction methods this allows us in particular to prove that the solutions obtained in i are unique under suitable subcriticality conditions and to obtain for them additional regularity properties and uniform bounds the method extends some of those previously applied to the nonlinear heat equation in global spaces to the framework of local uniform spaces

uniform decay rates for the energy of weakly damped defocusing semilinear schrodinger equations with inhomogeneous dirichlet boundary control abstract in this paper we study the open loop stabilization as well as the existence and regularity of solutions of the weakly damped defocusing semilinear schrodinger equation with an inhomogeneous dirichlet boundary control first of all we prove the global existence of weak solutions at the h 1 energy level together with the stabilization in the same sense it is then deduced that the decay rate of the boundary data controls the decay rate of the solutions up to an exponential rate secondly we prove some regularity and stabilization results for the strong solutions in h 2 sense the proof uses the direct multiplier method combined with monotonicity and compactness techniques the result for weak solutions is strong in the sense that it is independent of the dimension of the domain the power of the nonlinearity and the smallness of the initial data however the regularity and stabilization of strong solutions are obtained only in low dimensions with small initial and boundary data

a simple approach to the cauchy problem for complex ginzburg landau equations by compactness methods abstract this paper is concerned with the cauchy problem cgl in l 2 r n for complex ginzburg landau equations with laplacian and nonlinear term u q 2 u multiplied by the complex coefficients i and i respectively q 2 0 0 r the global existence of strong solutions to cgl is established without any upper restriction on q 2 but with some restriction on and the result corresponds to ginibre and velo 1996 3 proposition 5 1 which is technically proved by combining convolution regularizing methods with compactness localizing methods while our proof here is fairly simplified the key to our proof is the cauchy problem cgl r which is cgl with replaced with v r where v r x x r 2 x r v r x 0 x r the solvability of cgl r is a direct consequence of okazawa and yokota 2002 16 theorem 4 1 taking the limit of global strong solutions to cgl r as r yields a global strong solution to cgl the result gives also an unbounded version of okazawa and yokota 2002 16 theorem 1 1 with p 2 for the initial boundary value problem on bounded domains

optimal hankel norm model reductions and weiner hopf factorization i the canonical case we consider the problem of approximating a given stable rational matrix function g s of mcmillan degree n by a function s f s where has mcmillan degree l n and f is antistable it is known that the minimum possible l norm of the error g f l or equivalently the minimum possible hankel norm of the stable part of the error g h is equal to the l 1 st hankel singular value l 1 g s of g we give an explicit linear fractional map parametrization for the class of all functions s f s as above which satisfy g f l l 1 g s the coefficients of the linear fractional map are completely determined by the matrices a b and c in a realization g s c si a 1b for g s and the unique solutions of a pair of lyapunov equations involving these matrices note that without loss of generality g 0 the basic idea is to use the approach of ball and helton to reduce the problem to a symmetric wiener hopf factorization problem which in turn can be solved by applying a result of kaashoek and ran the results obtained here are equivalent to the results of glover but our analysis gives an alternative more geometric approach

transfer functions and operator theory abstract the notion of a transfer function from systems theory serves as an important tool in dealing with certain problems in operator and matrix theory this is illustrated with a variety of material taken from existing literature a striking feature of the results is their high degree of explicitness

the cauchy problem in local spaces for the complex ginzburg landau equation ii contraction methods we continue the study of the initial value problem for the complex ginzburg landau equation with a 0 b 0 g 0 in initiated in a previous paper i we treat the case where the initial data and the solutions belong to local uniform spaces more precisely to spaces of functions satisfying local regularity conditions and uniform bounds in local norms but no decay conditions or arbitrarily weak decay conditions at infinity in in i we used compactness methods and an extended version of recent local estimates 3 and proved in particular the existence of solutions globally defined in time with local regularity of the initial data corresponding to the spaces l r for r 2 or h 1 here we treat the same problem by contraction methods this allows us in particular to prove that the solutions obtained in i are unique under suitable subcriticality conditions and to obtain for them additional regularity properties and uniform bounds the method extends some of those previously applied to the nonlinear heat equation in global spaces to the framework of local uniform spaces

unobservable subspaces and common roots of several polynomials abstract for an m n matrix polynomial p c is called an eigenvalue if there exists 0 x c n such that p x 0 in this case x is an eigenvector of p corresponding to the eigenvalue the pair x is called a root of p given a set of matrix polynomials p1 p2 p1 of sizes m1 n m2 n m1 n and a monic n n polynomial q a criterion is derived for the existence of a common root of these polynomials this is done by constructing a specific unobservable subspace having its dimension equal to the number of common eigenvalues of these polynomials counted with their common multiplicities these notions are related to the lengths of the common maximal jordan chains in the particular case where q t ti a the above subspace is spanned by the jordan chains which are common to a p1 p2 p1