the effect of dissipation on solutions of the generalized korteweg de vries equation recent numerical simulations of the generalized korteweg de vries equation u l u p u x u xxx 0 indicate that for p 4 smooth solutions of the initial value problem may form singularities in finite time it is the purpose of this paper to ascertain what effect dissipation has on the instability of solitary waves and the associated blow up phenomena that are related to this singularity formation two different dissipative mechanisms are appended to in our study a burgers type term u xx and a simple zeroth order term u for both of these types of dissipation it is found that for small values of the positive parameters and solutions continue to form singularities in finite time however for given initial data u o it appears there are critical values c and c such that if c or c the solution associated with u o is globally defined and decays as t in the case wherein the singularity formation is averted by larger values of or a simple analysis shows the solution to approach its mean value exponentially fast theoretical analysis in the case when u o is a perturbed solitary wave leads to a conjecture about how c and c depend on the amplitude and spread of u o the numerical simulations indicate the analysis to be surprisingly sharp in predicting the qualitative dependence of c and c on u o

backward perturbation analysis of certain characteristic subspaces this paper gives optimal backward perturbation bounds and the accuracy of approximate solutions for subspaces associated with certain eigenvalue problems such as the eigenvalue problemax x the generalized eigenvalue problem ax bx and the singular value decomposition of a matrixa this paper also gives residual bounds for certain eigenvalues generalized eigenvalues and singular values

minimum probability of error for asynchronous gaussian multiple access channels consider a gaussian multiple access channel shared by k users who transmit asynchronously independent data streams by modulating a set of assigned signal waveforms the uncoded probability of error achievable by optimum multiuser detectors is investigated it is shown that the k user maximum likelihood sequence detector consists of a bank of single user matched filters followed by a viterbi algorithm whose complexity per binary decision is o 2 k the upper bound analysis of this detector follows an approach based on the decomposition of error sequences the issues of convergence and tightness of the bounds are examined and it is shown that the minimum multiuser error probability is equivalent in the iow noise region to that of a single user system with reduced power these results show that the proposed multiuser detectors afford important performance gains over conventional single user systems in which the signal constellation carries the entire burden of complexity required to achieve a given performance level

on condition numbers of the spectral projections associated with periodic eigenproblems abstract in this paper we study and analyze absolute and relative condition numbers of the spectral projections for regular periodic eigenproblems the main contribution is to derive explicit expressions of the condition numbers of the j th left and right spectral projections numerical examples are given to illustrate the proposed condition numbers for the spectral projections associated with periodic eigenproblems

a note on the backward errors for hermite eigenvalue problems we discuss the backward error analysis of the hermite eigenproblems three kinds of problems are included generalized eigenproblem matrix polynomial eigenproblem and multiparameter eigenproblem in the case of the approximate eigenvalue being real these problems have already been solved by higham and higham siam j matrix anal 20 1998 493 tisseur lin alg appl 309 2002 339 and hachstenbach and plestenjak lin alg appl 375 2003 63 this note as a complement of their results considers the complex case

inverse eigenproblem for centrosymmetric and centroskew matrices and their approximation in this paper we first give the solvability condition for the following inverse eigenproblem iep given a set of vectors xi i 1m in cn and a set of complex numbers i i 1m find a centrosymmetric or centroskew matrix c in rn n such that xi i 1m and i i 1m are the eigenvectors and eigenvalues of c respectively we then consider the best approximation problem for the ieps that are solvable more precisely given an arbitrary matrix b in rn n we find the matrix c which is the solution to the iep and is closest to b in the frobenius norm we show that the best approximation is unique and derive an expression for it

on the generalized bi skew symmetric solutions of a linear matrix equation and its procrust problems in this paper the solvability conditions and the explicit expressions of the generalized bisymmetric and bi skew symmetric solutions of the matrix equation ax b are respectively established by applying two methods then the maximal and minimal ranks of the solutions are derived if the solvability conditions are not satisfied the generalized bisymmetric and bi skew symmetric least squares solutions of the matrix equation are considered and the generalized bisymmetric and bi skew symmetric least squares solutions with the minimum norm are also obtained in addition two algorithms are provided to compute the generalized bi skew symmetric least squares solution and some examples are given to illustrate that the algorithms are feasible

limited linear cancellation of multiuser interference in ds cdma asynchronous systems this paper presents a linear cancellation detector for the gaussian channel operating over limited time intervals of the received signal in a similar way to a multiinput multioutput finite impulse response fir filter the parameters defining the detector become time invariant and the conditions to be met by the signatures are stated a bound for the multiuser interference due to the limited time correlation and upper and lower bounds for the error probability are obtained the theoretical bounds and numerical results show that the detector is adequate for systems intended for many users whose amplitudes can be restricted to a given range

bounds for the spectrum of a two parameter matrix eigenvalue problem abstract we consider the two parameter eigenvalue problem t j v j 1 a j 1 v j 2 a j 2 v j 0 where j c t j a j k j k 1 2 are matrices bounds for the spectral radius of that problem are suggested our main tool is a norm estimate for the operator inverse to the operator a 11 a 22 a 12 a 21 where means the tensor product in addition by virtue of that norm estimate and the ostrowsky schneider theorem we establish a condition that provides the conservation of the number of the eigenvalues of the considered problem in a half plane

the inverse eigenproblem with a submatrix constraint and the associated approximation problem for r s symmetric matrices abstract let r r n n and s r n n be nontrivial involutions i e r r 1 i and s s 1 i a matrix a r n n is called r s symmetric if r a s a this paper presents a r s symmetric matrix solution to the inverse eigenproblem with a leading principal submatrix constraint the solvability condition of the constrained inverse eigenproblem is also derived the existence the uniqueness and the expression of the r s symmetric matrix solution to the best approximation problem of the constrained inverse eigenproblem are achieved respectively an algorithm is presented to compute the r s symmetric matrix solution to the best approximation problem two numerical examples are given to illustrate the effectiveness of our results

perturbation of eigenvalues for periodic matrix pairs via the bauer fike theorems abstract in earlier papers the bauer fike technique was applied to the ordinary eigenvalue problem a x x the generalized eigenvalue problem a x b x and the matrix polynomial eigenvalue problem k 0 m k a k x 0 general multiple eigenvalues were dealt with and condition numbers were obtained for individual as well as clusters of eigenvalues in this paper we shall generalize the technique to the eigenvalue problem for regular periodic matrix pairs perturbation to eigenvalues simple or multiple finite or infinite will be considered for perturbations of any size for small perturbations clusters of eigenvalues will be considered

minimal norm static feedbacks using dissipative hamiltonian matrices abstract in this paper we characterize the set of static state feedbacks that stabilize a given continuous linear time invariant system pair using dissipative hamiltonian matrices this characterization results in a parametrization of feedbacks in terms of skew symmetric and symmetric positive semidefinite matrices and leads to a semidefinite program that computes a static state stabilizing feedback this characterization also allows us to propose an algorithm that computes minimal norm static feedbacks the theoretical results extend to the static output feedback sof problem and we also propose an algorithm to compute the minimal norm sof we illustrate the effectiveness of our algorithm compared to state of the art methods for the sof problem on numerous numerical examples from the compleib library

positon solutions of the sine gordon equation the concept of positons is introduced for the sine gordon equation exploiting its covariance under darboux transformations positon solutions represent potentials in the associated linear system corresponding to a vanishing reflection coefficient and a transmission coefficient identically one the mutual interaction of positons and the interaction of positons with solitons is considered

on a class of solutions of kdv abstract practically every book on the inverse scattering transform method for solving the cauchy problem for kdv and other integrable systems refers to this method as nonlinear fourier transform if this is indeed so the method should lead to a nonlinear analogue of the fourier expansion formula u t x u k e i kx k t dk in this paper a special class of solutions of kdv whose role is similar to that of e i kx k t is discussed the theory of these solutions referred to here as harmonic breathers is developed and it is shown that these solutions may be used to construct more general solutions of kdv similarly to how the functions e i kx t are used to perform the same task in the theory of fourier transform a nonlinear superposition formula for general solutions of kdv similar to the fourier expansion formula is conjectured

on the kahan parlett jiang theorem a globally optimal backward perturbation error for two sided invariant subspaces abstract a crucial question in large non hermitian eigenproblems is whether or not any residual bounds can be very useful in practical situations for this problem the famous kahan parlett jiang theorem kahan et al 1982 15 seems to be the best compromise between generality and sharpness in essence it presents an a posteriori error bound on approximate solutions to eigenproblems which provides a powerful tool to evaluate quality of the computed two sided invariant subspaces of large scale non hermitian matrices however the perturbation error determined by this theorem is only locally optimal rather than globally optimal in this work we revisit this problem and derive a globally optimal backward perturbation error for given two sided approximate invariant subspaces our theorem enhances the kahan parlett jiang theorem and numerical experiments demonstrate the effectiveness of our theoretical results

semidefinite inverse eigenvalue problems with prescribed entries and partial eigendata in this paper we study the semidefinite inverse eigenvalue problem of reconstructing a real n by n matrix c such that it is nearest to the original pre estimated real n by n matrix c o in the frobenius norm and satisfies the measured partial eigendata where the required matrix c should preserve the symmetry positive semidefiniteness and the prescribed entries of the pre estimated matrix c o we propose the alternating direction method of multipliers for solving the semidefinite inverse eigenvalue problem where three related iterative algorithms are presented we also extend our method to the case of lower bounds numerical experiments are reported to illustrate the efficiency of the proposed method for solving semidefinite inverse eigenvalue problems

symmetric matrix pencils abstract a significant number of matrix eigenvalue problems quadratic or linear are best reformulated as pencils a m in which both a and m are real and symmetric some examples are given and then the canonical forms are re examined to explain the role of the sign characteristic attached to real eigenvalues in addition we examine the limitations on the use of the rayleigh quotient functional x ax x mx in describing the eigenvalues this sheds new light on the class of definite pencils and the stability of their eigenvalues under perturbations the reduction of indefinite pencils to useful sparse forms is mentioned

supertransparent potentials for the dirac equation an extension of the classical darboux transformations is applied to the one dimensional dirac equation in order to construct von neumann wigner potentials allowing embedded eigenvalues these potentials lead to a novel type of scattering problem with a trivial s matrix composed of vanishing reflection coefficients and a trivial transmission coefficient related topics like the underlying symmetry of the dirac equation and the connection with positon solutions of nonlinear evolution equations are discussed

cauchy type determinants and integrable systems it is well known that the sylvester matrix equation ax xb c has a unique solution x if and only if 0 spec a spec b the main result of the present article are explicit formulas for the determinant of x in the case that c is one dimensional for diagonal matrices a b we reobtain a classical result by cauchy as a special case the formulas we obtain are a cornerstone in the asymptotic classification of multiple pole solutions to integrable systems like the sine gordon equation and the toda lattice we will provide a concise introduction to the background from soliton theory an operator theoretic approach originating from work of marchenko and carl and discuss examples for the application of the main results

structured mapping problems for linearly structured matrices abstract given an appropriate class of structured matrices s we characterize matrices x and b for which there exists a matrix a s such that a x b and determine all matrices in s mapping x to b we also determine all matrices in s mapping x to b and having the smallest norm we use these results to investigate structured backward errors of approximate eigenpairs and approximate invariant subspaces and structured pseudospectra of structured matrices

kronecker bases for linear matrix equations with application to two parameter eigenvalue problems abstract the general solutions of the homogeneous matrix equation axct bxdt 0 and the system of the matrix equations ax by 0 xct ydt 0 are described in terms of kronecker canonical forms i e in terms of kronecker invariants and kronecker bases for pairs of matrices a b and c d a canonical form for a pair of commuting matrices e f such that e2 f2 ef 0 is discussed these results are applied to construct a canonical basis for the second root subspace of a two parameter eigenvalue problem the corresponding relations for canonical invariants are given

the interacton equation given the lie algebra lfp of vector fields of a free particle then in a joint field of interacting particles an interacton is defined as a direct summand isomorphic to lfp for several soliton equations it is shown how interactons can be obtained from various methods of symmetry analysis and how their dynamics can be expressed in terms of self interaction alone this procedure leads to new integrable systems furthermore in case of nonlinear schrodinger equations it is shown how the dynamics of interactons can be formulated in terms of self consistent potentials with antlinear parts

negaton and positon solutions of the kdv and mkdv hierarchy we give a systematic classification and a detailed discussion of the structure motion and scattering of the recently discovered negaton and positon solutions of the korteweg de vries hierarchy there are two distinct types of negaton solutions which we label and where n 1 is the order of the wronskian used in the derivation for negatons the number of singularities and zeros is finite and they show very interesting time dependence the general motion is in the positive x direction except for certain negatons which exhibit one oscillation around the origin in contrast there is just one type of positon solution which we label for positons one gets a finite number of singularities for n odd but an infinite number for even values of n the general motion of positons is in the negative x direction with periodic oscillations negatons and positons retain their identities in a scattering process and their phase shifts are discussed we obtain a simple explanation of all phase shifts by generalizing the notions of mass and centre of mass to singular solutions finally it is shown that negaton and positon solutions of the kdv hierarchy can be used to obtain corresponding new solutions of the modified kdv hierarchy

multisoliton multipositon multinegaton and multiperiodic solutions of a coupled volterra lattice system and their continuous limits this paper aims to find new explicit solutions including multisoliton multipositon multinegaton and multiperiodic for a coupled volterra lattice system this coupled lattice system is an integrable discrete version of the coupled korteweg devries kdv equation which has many physical applications the dynamical properties of these new solutions are discussed in detail we also prove that the theory of the coupled volterra lattice system including the lax pair the darboux transformation and explicit solutions yield the corresponding theory of the coupled kdv equation in the continuous limit

a note on stewart s theorem for definite matrix pairs abstract let a and b be n n hermitian matrices the matrix pair a b is called definite pair and the corresponding eigenvalue problem ax bx is definite if c a b inf x 1 h a ib x 0 in this note we develop a uniform upper bound for differences of corresponding eigenvalues of two definite pairs and so improve a result which is obtained by g w stewart 2 moreover we prove that this upper bound is a projective metric in the set of n n definite pairs

nonlocal symmetries and the kaup kupershmidt equation computational and geometric aspects of nonlocal infinitesimal symmetries of nonlinear partial differential equations are considered in particular the relation of nonlocal symmetries with classical generalized and internal symmetries is briefly discussed a nonlocal symmetry for the kaup kupershmidt equation is introduced and studied in some detail some explicit particular solutions are found with its help and a darboux like transformation is also obtained

root vectors for geometrically simple two parameter eigenvalues a class of multiparameter eigenvalue problems involving generally non self adjoint and unbounded operators is studied bases for lower order root subspaces at geometrically simple eigenvalues of fredholm type of arbitrary finite index are computed in terms of the underlying multiparameter system

a note on the approximation of two sided rayleigh quotient to eigenvalues abstract we consider the two sided rayleigh quotient introduced by ostrowski as convergence acceleration tool for eigenvalue solvers an interesting property of the two sided rayleigh quotient is established its approximation to eigenvalues depends on the eigenvalue condition number this fact is valid not only for matrix but also for matrix polynomial multiparameter system and periodic matrix pairs

eigenvalues of rayleigh quotient matrices support thata is ann n hermitian matrix andx 1 y 1 aren 1 matrices satisfyingx 1 h x 1 y 1 h y 1 i 1 l n such that the column space x 1 is an eigenspace ofa and y 1 is an approximation of x 1 let a 1 x 1 h ax 1 h 1 y 1 h ay 1 r ay 1 y 1 h 1 and theta arccos x 1 h y 1 y 1 h x 1 1 mathord left vphantom 1 2 right kern nulldelimiterspace 2 geqslant 0 the matricesa 1 andh 1 are called the rayleigh quotient matrices ofa with respect tox 1 andy 1 respectively this paper derives new upper bounds of the difference between h 1 and a 1 the set of the eigenvaluesh 1 anda 1 in terms of the quantities sin and or r in a simple way here denotes a unitarily invariant norm

backward errors for eigenvalue and singular value decompositions we present bounds on the backward errors for the symmetric eigenvalue decomposition and the singular value decomposition in the two norm and in the frobenius norm through different orthogonal decompositions of the computed eigenvectors we can define different symmetric backward errors for the eigenvalue decomposition when the computed eigenvectors have a small residual and are close to orthonormal then all backward errors tend to be small consequently it does not matter how exactly a backward error is defined and how exactly residual and deviation from orthogonality are measured analogous results hold for the singular vectors we indicate the effect of our error bounds on implementations for eigenvector and singular vector computation in a more general context we prove that the distance of an appropriately scaled matrix to its orthogonal qr factor is not much larger than its distance to the closest orthogonal matrix

completion problems and scattering problems for dirac type differential equations with singularities abstract a completion problem to recover a rational matrix function which is j unitary on the line is treated a dirac type system with singularities on the semi axis is recovered explicitly from its left reflection coefficient the close relation between these two problems is discussed

on the alternating direction method of multipliers for nonnegative inverse eigenvalue problems with partial eigendata we consider the nonnegative inverse eigenvalue problem with partial eigendata which aims to find a nonnegative matrix such that it is nearest to a pre estimated nonnegative matrix and satisfies the prescribed eigendata in this paper we propose several iterative schemes based on the alternating direction method of multipliers for solving the nonnegative inverse problem we also extend our schemes to the symmetric case and the cases of prescribed lower bounds and of prescribed entries numerical tests including a practical engineering application in vibrations show the efficiency of the proposed iterative schemes

a numerical algorithm for computing a basis for the root subspace at a nonderogatory eigenvalue of a multiparameter system abstract a numerical algorithm for the calculation of a basis for the root subspace at a nonderogatory eigenvalue of a finite dimensional multiparameter system is given it is based on the algorithm proposed by kosir 1 the perturbation theory for the algorithm is studied and a new stopping criterion is proposed at the end two numerical examples are given

on the structure of commutative matrices ii a finite set a of n n nilpotent commutative matrices that have one dimensional joint kernel is considered the theorem due to suprunenko and tyshkevich that the algebra a generated by a and the identity matrix has dimension equal to n is proved a canonical basis for a is given and related structure constants are discussed

the hermitian p k 1 anti reflexive solutions of a linear matrix equation in this paper we first establish the necessary and sufficient conditions for the existence and the explicit expressions of the hermitian p k 1 anti reflexive solutions of the matrix equation a x b and meanwhile the best approximation solution is considered then if the solvability conditions are not satisfied the least squares hermitian p k 1 anti reflexive solutions and the least squares hermitian p k 1 anti reflexive solutions with the minimum norm of the above matrix equation are respectively derived in addition two algorithms are shown to compute the least squares hermitian p k 1 anti reflexive solutions and the corresponding numerical examples are also given to illustrate the feasibility of the algorithms

residual bounds of approximate solutions of the algebraic riccati equation let tilde x geq 0 approximate the unique hermitian positive semi definite solution x to the algebraic riccati equation are where f g geq 0 a f is stabilizable and a g is detectable let be the residual of the are with respect to tilde x and define the linear operator vec l by by applying a new forward perturbation bound to the optimal backward perturbation corresponding to the approximate solution tilde x we obtained the following result if a f tilde x is stable and if 4 vec l 1 vec l 1 hat r f 1 for any unitarily invariant norm then

on stability of invariant subspaces of commuting matrices we study the stability of joint invariant subspaces of a finite set of commuting matrices we generalize some of the results of gohberg lancaster and rodman for the single matrix case for sets of two or more commuting matrices we exhibit some phenomena different from the single matrix case we show that each root subspace is a stable invariant subspace that each invariant subspace of a root subspace of a nonderogatory eigenvalue is stable and that even in the derogatory case the eigenspace is stable if it is one dimensional we prove that a pair of commuting matrices has only finitely many stable invariant subspaces at the end we discuss the stability of invariant subspaces of an algebraic multiparameter eigenvalue problem

continuity properties of the 1 inverse and perturbation bounds for the drazin inverse there are known results about the continuity of the important pseudoinverses we extend some well known results to the 1 inverse as for the drazin inverse which is not a 1 inverse we present new perturbation bounds

on the variation of the spectrum of a normal matrix abstract let a and a be two n n normal matrices with spectra and j then by the hoffman wielandt theorem there is a permutation of 1 2 n such that n j 1 j j 2 a a f where f denotes the frobenius norm however if a is normal but a nonnormal it may be asked how to relate the eigenvalues of a to those of a an answer is given in this paper there is a permutation of 1 2 n such that n j 1 j j 2 n a a f and the factor n is best possible as a corollary we have j j n a a 2 for the spectral norm 2 thus the known upper bound 2n 1 a a 2 is reduced by a factor of about two

inexact and incremental bilinear lanczos components algorithms for high dimensionality reduction and image reconstruction recently matrix based methods have gained wide attentions in pattern recognition and machine learning communities the generalized low rank approximations of matrices glram and the bilinear lanczos components blc algorithm are two popular algorithms that treat data as the native two dimensional matrix patterns however these two algorithms often require heavy computation time and memory space in practice especially for large scale problems in this paper we propose inexact and incremental bilinear lanczos components algorithms for high dimensionality reduction and image reconstruction we first introduce the thick restarting strategy to the blc algorithm and present a thick restarted lanczos components trblc algorithm in this algorithm we use the ritz vectors as approximations to dominant eigenvectors instead of the lanczos vectors in our implementation the iterative matrices are neither formed nor stored explicitly thanks to the characteristic of the lanczos procedure then we explore the relationship between the reconstruction error and the accuracy of the ritz vectors so that the computational complexities of eigenpairs can be reduced significantly as a result we propose an inexact thick restarted lanczos components inex trblc algorithm moreover we investigate the problem of incremental generalized low rank approximations of matrices and propose an incremental and inexact trblc incr trblc algorithm numerical experiments illustrate the superiority of the new algorithms over the glram algorithm and its variations as well as the blc algorithm for some real world image reconstruction and face recognition problems highlightswe proposed an inexact algorithm for image reconstructionthis algorithm is efficient both in storage and running time we analyze its convergence and backward errors an incremental version of the algorithm is given to handle dynamic increased data set

eigenvector free solutions to ax b with px xp and xh sx constraints abstract the matrix equation ax b with px xp and xh sx constraints is considered where p is a given hermitian involutory matrix and s 1 by an eigenvalue decomposition of p we equivalently transform the constrained problem to two well known constrained problems and represent the solutions in terms of the eigenvectors of p using moore penrose generalized inverses of the products generated by matrices a b and p the involved eigenvectors can be released and eigenvector free formulas of the general solutions are presented similar strategy is applied to the equations ax b xc d with the same constraints

the geometric approach for linear periodic discrete time systems abstract recently the disturbance localization problem and the problems of designing disturbance decoupled observers and of giving a geometric characterization of invariant zeros were solved for linear periodic discrete time systems through an extension of the geometric approach based on the notions of controlled invariant and conditionally invariant subspaces to periodic ones here the existing periodic geometric theory is supplemented with the notions of outer reachable subspace and controllability subspace and with some further results on the previously introduced notions of inner reachable controllable subspace and outer controllable subspace in addition it is shown that any such periodic geometric theory can be restated in an equivalent time invariant form which is just the same theory written for a suitable time invariant system having an extended state space

nonnegative inverse eigenvalue problems with partial eigendata in this paper we consider the inverse problem of constructing an n n real nonnegative matrix a from the prescribed partial eigendata we first give the solvability conditions for the inverse problem without the nonnegative constraint and then discuss the associated best approximation problem to find a nonnegative solution we reformulate the inverse problem as a monotone complementarity problem and propose a nonsmooth newton type method for solving its equivalent nonsmooth equation under some mild assumptions the global and quadratic convergence of our method is established we also apply our method to the symmetric nonnegative inverse problem and to the cases of prescribed lower bounds and of prescribed entries numerical tests demonstrate the efficiency of the proposed method and support our theoretical findings

relative eigenvalues of a definite matrix pair abstract let a and b be hermitian matrices and let c a b min x 2 1 xh a ib x the matrix pair a b is called a definite pair and the corresponding eigenvalue problem ax bx is definite if c a b 0 the relationship between the eigenvalues of a b and those of a definite pair of the form xh1ax1 xh1bx1 is studied in this paper the eigenvalues of xh1ax1 xh1bx1 are said to be the relative eigenvalues of a b with respect to the subspace r x1 from the main result of this paper one can deduce a corresponding conclusion on hermitian matrices

complexiton solutions to integrable equations abstract complexiton solutions or complexitons for short are exact solutions newly introduced to integrable equations are newly introduced starting with the solution classification for a linear differential equation the korteweg de vries equation and the toda lattice equation are considered as examples to exhibit complexiton structures of nonlinear integrable equations the crucial step in the solution process is to apply the wronskian and casoratian techniques for hirota s bilinear equations correspondence between complexitons of the korteweg de vries equation and complexitons of the toda lattice equation is provided

near far resistance of multiuser detectors in asynchronous channels we consider an asynchronous code division multiple access environment in which the receiver has knowledge of the signature waveforms of all the users under the assumption of white gaussian background noise we compare detectors by their worst case bit error rate in a low background noise near far environment where the received energies of the users are unknown to the receiver and are not necessarily similar

the rayleigh ritz method refinement and arnoldi process for periodic matrix pairs we extend the rayleigh ritz method to the eigen problem of periodic matrix pairs assuming that the deviations of the desired periodic eigenvectors from the corresponding periodic subspaces tend to zero we show that there exist periodic ritz values that converge to the desired periodic eigenvalues unconditionally yet the periodic ritz vectors may fail to converge to overcome this potential problem we minimize residuals formed with periodic ritz values to produce the refined periodic ritz vectors which converge under the same assumption these results generalize the corresponding well known ones for rayleigh ritz approximations and their refinement for non periodic eigen problems in addition we consider a periodic arnoldi process which is particularly efficient when coupled with the rayleigh ritz method with refinement the numerical results illustrate that the refinement procedure produces excellent approximations to the original periodic eigenvectors

negatons positons rational like solutions and conservation laws of the korteweg de vries equation with loss and non uniformity terms solitons negatons positons rational like solutions and mixed solutions of a non isospectral equation the korteweg de vries equation with loss and non uniformity terms are obtained through the wronskian technique the non isospectral characteristics of the motion behaviours of some solutions are described with some figures made by using mathematica we also derive an infinite number of conservation laws of the equation

asymptotics of the multipositon soliton τ function of the korteweg de vries equation and the supertransparency the asymptotics of the function generating the n positon m soliton solution of the korteweg de vries equation is calculated this allows to prove that solitons do not experience any phase shift in a collision with positons the positons themselves survive mutual collisions unchanged this phenomenon is called the supertransparency or super reflectionless property of the multipositon solutions the linear aspects of this phenomenon are also discussed it is demonstrated that positons acquire two additional but always finite phase shifts in collision with solitons this result admits a natural extension to any number of solitons and positons in interaction

modulating properties of harmonic breather solutions of kdv we show that a class of solutions of kdv can be used to modulate miscellaneous wavepackets similarly to how the trigonometric functions are used in linear theory to perform the same task

structured backward errors for kkt systems karush kuhn tucker kkt systems are linear systems with coefficient matrices of the form graphics where h is symmetric a normwise structured backward error for kkt systems is defined and a com

perturbation theorems for the generalized eigenvalue problem abstract given a matrix pair z a b the perturbation of its eigenvalues is studied considering two pairs z w as points of the grassman manifold g n 2 n and its eigenvalues as points in g 1 2 the projective complex plane the distance of the spectra measured in the chordal metric in g 1 2 is bounded by some distance of the matrix pairs in g n 2 n analogs of the bauer fike theorem henrici s theorem and the hoffman weilandt theorem are obtained from which the classical results can be derived

the difference periodic ricati equation for the periodic prediction problem gives a comprehensive treatment of several important aspects of the discrete time periodic riccati equation dpre arising from the prediction problem for linear discrete time periodic systems the authors analyze the symmetric periodic positive semidefinite spps solution of the dpre under appropriate assumptions of stabilizability and detectability of the periodic system among the results obtained are necessary and sufficient conditions for the existence and uniqueness of the spps solution and the stability of the resulting closed loop system some of these results can be seen as extensions of the corresponding results for the time invariant case however a number of them contain contributions to the time invariant case as well the paper also gives a numerical algorithm based on an iterative linearization procedure for computing the spps solution the algorithm is a periodic version of kleinman s algorithm for the time invariant case

on estimating the separation of two regular matrix pairs let a b and c d be two n times n n n and m times m m m regular matrix pairs respectively the main contribution of this paper is to derive lower and upper bounds of the separation between a b and c d in terms of their generalized schur decompositions and weierstrass canonical forms the results are illustrated by numerical examples

perturbation analysis for the eigenproblem of periodic matrix pairs this paper is devoted to perturbation analysis for the eigenproblem of periodic matrix pairs a j e j j 1 k we first study perturbation expansions of periodic deflating subspaces and eigenv

periodic descriptor systems solvability and conditionability the authors consider discrete time linear periodic descriptor systems and study the concepts of solvability and conditionability introduced by luenberger they prove that solvability is equivalent to conditionability just as in the time invariant case we give a characterization of solvability conditionability in terms of a cyclic matrix pencil and furthermore propose a simple test via the periodic schur decomposition to check for either property this could lead to further systematic study of these systems

on a class of alternating coefficient matrices quadratic eigenvalue problem in this paper we consider a class of alternating coefficient matrices quadratic eigenvalue problem aqep appearing in a wide range of applications at first the aqep is reformulated as a generalized eigenvalue problem gep by applying tisseur s linearization technique then we point out a property about skew symmetric tridiagonal matrix eigenvalue problem consequently we give that the corresponding gep can be solved by applying the cholesky ql algorithm or lanczos algorithm without complex arithmetic the structured backward error for qep would be discussed because of the special structure of the polynomial eigenvalue problems we develop structured condition number the upper bound and the lower bound are given some computational experiments and some concluding remarks are provided

an algebraic approach to multiparameter spectral theory root subspaces for multiprameter eigenvalue problems are described using coalgebraic techniques an algorithm is given to construct bases for the root subspaces

structured backward errors for generalized saddle point systems abstract in this paper we investigate structured backward errors for three kinds of generalized saddle point systems where the matrix is not symmetric and its 2 2 block is not zero and has perturbations computable formula of backward errors are derived the expressions are useful for testing the stability of practical algorithms

backward error condition numbers and pseudospectra for the multiparameter eigenvalue problem we define and evaluate the normwise backward error and condition numbers for the multiparameter eigenvalue problem mep the pseudospectrum for the mep is defined and characterized we show that the distance from a right definite mep to the closest non right definite mep is related to the smallest unbounded pseudospectrum some numerical results are given

finite dimensional multiparameter spectral theory the nonderogatory case abstract a completeness theorem for nonderogatory eigenvalues of multiparameter systems is proved in a finite dimensional setting also a basis for the second root subspace of a simple eigenvalue is given weakly elliptic multiparameter systems are introduced it is shown that simple real eigenvalues of such systems are nonderogatory

on a class of solutions of the sine gordon equation in this paper we consider solutions of the sine gordon equation that are nonlinear analogs of solutions of the klein gordon equation just like the latter may be combined together by means of the fourier integral to form more complicated solutions of the klein gordon equation the former may also interact to produce more complicated solutions of the sine gordon equation by means of a nonlinear analog of the fourier integral also derived in the paper

minimization problems for r s symmetric and r s skew symmetric matrices let r cm m and s cn n be nontrivial involutions thus r r 1 i and s s 1 i we say that a cm n is r s symmetric r s skew symmetric if ras a ras a let s be the class of m n r s symmetric matrices or the class of m n r s skew symmetric matrices let z cn q and w cm q we study the following problems i give necessary and sufficient conditions for the existence of an a s such that az w and find all such matrices if the conditions are met ii find z w mina s az w and characterize the class s z w a s az w z w iii if b cm n is arbitrary find z w b mina s z w a b and find a s z w such that a b z w b we obtain explicit formulas for z w b z w and all the matrices in question

two parameter eigenvalue problems for matrices abstract the linked pair of two parameter eigenvalue problems t m 1 v m 1 2 v m 2 x m 0 x m 0 x m h m is considered for hermitian matrices t m v m1 v m2 acting on on the finite dimensional hilbert spaces h m m 1 2 we dispense with customary definiteness conditions although for m 1 2 we assume the existence of a linear combination of v m1 v m2 which is definite an embedding technique is used to study eigenvalues r 2 of specified variational index the transition between real and nonreal eigenvalues and algebraic multiplicity