determining the inertia of a matrix pencil as a function of the parameter abstract we determine the inertia of a linear real symmetric matrix pencil a t a tb of order n as a function of the parameter t finding the critical values where the inertia of a t changes is reduced to determining the real eigenvalues of a not necessarily symmetric matrix the order of this matrix is at most r s r 1 where r s and r 1 are the ranks of a b and a b respectively we illustrate the method by means of a numerical example then we reduce determining the inertia of a quadratic real symmetric matrix pencil a t a tb 1 t 2 b 2 to the linear case our results are extensions of those by caron and gould

maximization and minimization of the rank and inertia of the hermitian matrix expression a bx bx with applications abstract we give in this paper a group of closed form formulas for the maximal and minimal ranks and inertias of the linear hermitian matrix function a bx bx with respect to a variable matrix x as applications we derive the extremal ranks and inertias of the matrices x x where x is a solution to the matrix equation axb c and then give necessary and sufficient conditions for the matrix equation axb c to have hermitian definite and re definite solutions in addition we give closed form formulas for the extremal ranks and inertias of the difference x 1 x 2 where x 1 and x 2 are hermitian solutions of two matrix equations a 1 x 1 a 1 c 1 and a 2 x 2 a 2 c 2 and then use the formulas to characterize relations between hermitian solutions of the two equations

on inertia and schur complement in optimization partitioned symmetric matrices in particular the hessian of the lagrangian play a fundamental role in nonlinear optimization for this type of matrices s p han and o fujiwara recently presented an inertia theorem under a certain regularity assumption we prove that this theorem is true without any regularity assumption then we consider matrix extensions preserving the sign of the determinant such extensions are shown to be related with the positive definiteness of some schur complement under a regularity assumption this shows from the viewpoint of linear algebra the equivalence of strong stability in the sense of m kojima and strong regularity in the sense of s m robinson finally we discuss the inertia of a typical one parameter family of symmetric matrices occurring in various places in optimization augmented lagrangians focal point theory etc

stochastic hopfield neural networks hopfield 1984 proc natl acad sci usa 81 3088 92 showed that the time evolution of a symmetric neural network is a motion in state space that seeks out minima in the system energy i e the limit set of the system in practice an eural network is often subject to environmental noise it is therefore useful and interesting to find out whether the system still approaches some limit set under stochastic perturbation in this paper we will give a number of useful bounds for the noise intensity under which the stochastic neural network will approach its limit set

computing the inertias in symmetric matrix pencils abstract we define and study generalized pencil eigenvalues for a pencil p s t of real symmetric matrices s and t this allows us to find all occurring inertias among pencil members as bt theoretically and numerically computed examples based on a matlab code are included

robust reduced order h output feedback control of retarded stochastic linear systems a method for designing reduced order output feedback controllers for linear time invariant retarded systems with state multiplicative wiener type noise is introduced which achieves a minimum bound on the h performance level a cost function is defined that is the expected value of the standard h performance cost with respect to the stochastic parameters a solution is thus obtained for the reduced order h output feedback control problem for the stationary case via the input output approach where the system is replaced by a non retarded one that contains deterministic norm bounded uncertainties the results achieved for the nominal case are extended to the uncertain case where the system matrices reside in a given polytope a numerical example taken from the field of aircraft control is given that demonstrates the applicability of the theory developed

applications of cs decomposition in linear combinations of two orthogonal projectors abstract we study the spectrum and the rank of a linear combination of two orthogonal projectors we characterize when this linear combination is ep diagonalizable idempotent tripotent involutive nilpotent generalized projector and hypergeneralized projector also we derive the moore penrose inverse of a linear combination of two orthogonal projectors in a particular case the main tool used here is the cs decomposition

on the product of orthogonal projectors abstract in this paper we introduce a canonical form to represent the product of two orthogonal projectors this form is useful for deriving many known results which are scattered in the literature we summarize many of those results and present new ones

root spreads for polynomials and hermitian matrix pencils abstract beginning with a polynomial with real roots a family of polynomials with degree reduced by one is defined using a nonnegative matrix then the root spreads of the original polynomial and polynomials in the family are compared there are applications to the spread of the roots of a hermitian matrix pencil and the root spreads of its principal subpencils having one fewer row and column

computational study of the percolation phenomena in inhomogeneous two and three dimensional systems in this paper the results of computer investigation of the percolation processes in inhomogeneous lattices are discussed the inhomogeneity is simulated by a random distribution of obstacles differing in size and number the influence of obstacles on the parameters critical concentration average number of sites in finite clusters percolation probability critical exponents and fractal and spectral dimensions of a percolation cluster characterizing the percolation in the system is analysed it is demonstrated that all these parameters essentially depend on the linear size and relative area of the obstacles

global asymptotic stability of recurrent neural networks with multiple time varying delays in this paper several sufficient conditions are established for the global asymptotic stability of recurrent neural networks with multiple time varying delays the lyapunov krasovskii stability theory for functional differential equations and the linear matrix inequality lmi approach are employed in our investigation the results are shown to be generalizations of some previously published results and are less conservative than existing results the present results are also applied to recurrent neural networks with constant time delays

idempotency of linear combinations of three idempotent matrices two of which are commuting abstract the considerations of the present paper were inspired by baksalary o m baksalary idempotency of linear combinations of three idempotent matrices two of which are disjoint linear algebra appl 388 2004 67 78 who characterized all situations in which a linear combination p c 1 p 1 c 2 p 2 c 3 p 3 with c i i 1 2 3 being nonzero complex scalars and p i i 1 2 3 being nonzero complex idempotent matrices such that two of them p 1 and p 2 say are disjoint i e satisfy condition p 1 p 2 0 p 2 p 1 is an idempotent matrix in the present paper by utilizing different formalism than the one used by baksalary the results given in the above mentioned paper are generalized by weakening the assumption expressing the disjointness of p 1 and p 2 to the commutativity condition p 1 p 2 p 2 p 1

the role of coefficients of a general spde on the stability and convergence of a finite difference method in this paper for the approximate solution of stochastic partial differential equations spdes of ito type the stability and application of a class of finite difference method with regard to the coefficients in the equations is analyzed the finite difference methods discussed here will be either explicit or implicit and a comparison between them will be reported we prove the consistency and stability of these methods and investigate the influence of the multiplier particularly multiplier of the random noise in mean square stability from stochastic version of lax richtmyer the convergence of these methods under some conditions are established numerical experiments are included to show the efficiency of the methods

stability analysis for neural networks with time varying delay based on quadratic convex combination in this paper a novel method is developed for the stability problem of a class of neural networks with time varying delay new delay dependent stability criteria in terms of linear matrix inequalities for recurrent neural networks with time varying delay are derived by the newly proposed augmented simple lyapunov krasovski functional different from previous results by using the first order convex combination property our derivation applies the idea of second order convex combination and the property of quadratic convex function which is given in the form of a lemma without resorting to jensen s inequality a numerical example is provided to verify the effectiveness and superiority of the presented results

observer based control design of semi markovian jump systems with uncertain probability intensities and mode transition dependent sojourn time distribution abstract this paper deals with the problem of h observer based control for a class of continuous time semi markovian jump systems s mjss with more detailed observational information first to explore the impact of uncertain probability intensities a mathematical analysis is accomplished from which some useful inequality conditions on the sum of transition rates trs are obtained further to come up with more accurate bounds of trs the mode transition dependent probability distribution of sojourn time is imposed on the mechanism of forming trs lastly by devising a compatible relaxation process that can embrace all the conditions found in our derivation the resultant observer based stabilization conditions are formulated in terms of linear matrix inequalities

any hermitian matrix is a linear combination of four projections abstract any hermitian matrix a is a linear combination of four projections the number of projections can be reduced to three or two according as rank a 7 or 3

stochastic functional partial differential equations existence uniqueness and asymptotic decay property existence and uniqueness of strong solutions for a class of stochastic functional differential equations in hilbert spaces are established sufficient conditions which guarantee the transference of mean square and pathwise exponential stability from stochastic partial differential equations to stochastic functional partial differential equations are studied the stability results derived are also applied to stochastic ordinary differential equations with hereditary characteristics in particular as a direct consequence our main results improve some of those by mao shah in which it was proved that under certain conditions pathwise exponential stability is transferred from non delay equations to delay equations if the constant time lag appearing in the problem is sufficiently small while in our treatment the transference actually holds for arbitrary bounded delay variables not only in finite but in infinite dimensions

stochastic functional evolution equations with monotone nonlinearity existence and stability of the mild solutions abstract in this paper we study a class of semilinear functional evolution equations in which the nonlinearity is demicontinuous and satisfies a semimonotone condition we prove the existence uniqueness and exponentially asymptotic stability of the mild solutions our approach is to apply a convenient version of burkholder inequality for convolution integrals and an iteration method based on the existence and measurability results for the functional integral equations in hilbert spaces an ito type inequality is the main tool to study the uniqueness p th moment and almost sure sample path asymptotic stability of the mild solutions we also give some examples to illustrate the applications of the theorems and meanwhile we compare the results obtained in this paper with some others appeared in the literature

a note on linear combinations of commuting tripotent matrices abstract the purpose of this note is to characterize all situations in which a linear combination of two commuting tripotent matrices is also a tripotent matrix in the case of real scalars and real symmetric matrices this problem admits an interesting statistical interpretation namely it is equivalent to the question of when a linear combination of two quadratic forms in normal variables each distributed as a difference of two independent 2 variables is also distributed as such a difference

once more on algebras generated by two projections we consider von neumann algebras generated by two arbitrary orthoprojections on a hilbert space a canonical decomposition is obtained for elements a of these algebras in terms of the operator angle between the ranges of the above mentioned projections this decomposition leads to explicit descriptions and formulas for kernels ranges spectra and essential spectra generalized inverses and other objects related to a

dynamical behavior of delayed reaction diffusion hopfield neural networks driven by infinite dimensional wiener processes in this paper we focus on the long time behavior of the mild solution to delayed reaction diffusion hopfield neural networks drdhnns driven by infinite dimensional wiener processes we analyze the existence uniqueness and stability of this system under the local lipschitz function by constructing an appropriate lyapunov krasovskii function and utilizing the semigroup theory some easy to test criteria affecting the well posedness and stability of the networks such as infinite dimensional noise and diffusion effect are obtained the criteria can be used as theoretic guidance to stabilize drdhnns in practical applications when infinite dimensional noise is taken into consideration meanwhile considering the fact that the standard brownian motion is a special case of infinite dimensional wiener process we undertake an analysis of the local lipschitz condition which has a wider range than the global lipschitz condition two samples are given to examine the availability of the results in this paper simulations are also given using the matlab

the inverse inertia problem for graphs cut vertices trees and a counterexample abstract let g be an undirected graph on n vertices and let s g be the set of all real symmetric n n matrices whose nonzero off diagonal entries occur in exactly the positions corresponding to the edges of g the inverse inertia problem for g asks which inertias can be attained by a matrix in s g we give a complete answer to this question for trees in terms of a new family of graph parameters the maximal disconnection numbers of a graph we also give a formula for the inertia set of a graph with a cut vertex in terms of inertia sets of proper subgraphs finally we give an example of a graph that is not inertia balanced which settles an open problem from the october 2006 aim workshop on spectra of families of matrices described by graphs digraphs and sign patterns we also determine some restrictions on the inertia set of any graph

stochastic stability properties of jump linear systems jump linear systems are defined as a family of linear systems with randomly jumping parameters usually governed by a markov jump process and are used to model systems subject to failures or changes in structure the authors study stochastic stability properties in jump linear systems and the relationship among various moment and sample path stability properties it is shown that all second moment stability properties are equivalent and are sufficient for almost sure sample path stability and a testable necessary and sufficient condition for second moment stability is derived the lyapunov exponent method for the study of almost sure sample stability is discussed and a theorem which characterizes the lyapunov exponents of jump linear systems is presented finally for one dimensional jump linear system it is proved that the region for delta moment stability is monotonically converging to the region for almost sure stability at delta down arrow 0 sup

an optimization problem for nonlinear steklov eigenvalues with a boundary potential abstract in this paper we analyze an optimization problem for the first nonlinear steklov eigenvalue plus a boundary potential with respect to the potential function which is assumed to be uniformly bounded and with fixed l 1 norm

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

regularity of the free boundary in an optimization problem related to the best sobolev trace constant in this paper we study the regularity properties of a free boundary problem arising in the optimization of the best sobolev trace constant in the immersion h 1 omega hookrightarrow l q partial omega for functions that vanish in a subset of omega this problem is also related to a minimization problem for steklov eigenvalues

a constrained shape optimization problem in orlicz sobolev spaces abstract in this manuscript we study the following optimization problem given a bounded and regular domain r n we look for an optimal shape for the w vanishing window on the boundary with prescribed measure over all admissible profiles in the framework of the orlicz sobolev spaces associated to constant for the sobolev trace embedding in this direction we establish existence of minimizer profiles and optimal sets as well as we obtain further properties for such extremals finally we also place special emphasis on analyzing the corresponding optimization problem involving an a vanishing hole inside the domain with volume constraint

stability of nonlinear stochastic evolution equations abstract criteria for boundedness asymptotic stability of sample paths given by solutions to nonlinear stochastic evolution equations are presented the analysis is based on a functional ito formula liapunov and related functionals and generalization of methods developed in finite dimensions applications to parabolic ito equations are given

global asymptotic stability of stochastic bam neural networks with distributed delays and reaction diffusion terms this paper is concerned with global asymptotic stability of a class of reaction diffusion stochastic bi directional associative memory bam neural networks with discrete and distributed delays based on suitable assumptions we apply the linear matrix inequality lmi method to propose some new sufficient stability conditions for reaction diffusion stochastic bam neural networks with discrete and distributed delays the obtained results are easy to check and improve upon the existing stability results an example is also given to demonstrate the effectiveness of the obtained results

optimal boundary holes for the sobolev trace constant in this paper we study the problem of minimizing the sobolev trace rayleigh quotient u w1 p p u lq p among functions that vanish in a set contained on the boundary of given boundary measure we prove existence of extremals for this problem and analyze some particular cases where information about the location of the optimal boundary set can be given moreover we further study the shape derivative of the sobolev trace constant under regular perturbations of the boundary set

synchronization of coupled reaction diffusion neural networks with time varying delays via pinning impulsive controller in this paper global exponential synchronization stability in an array of linearly diffusively coupled reaction diffusion neural networks with time varying delays is investigated by adding impulsive controller to a small fraction of nodes pinning impulsive controller in order to overcome the difficulty resulting from the fact that the impulsive controller affects only the dynamical behaviors of the controlled nodes a new analysis method is developed by using the developed method two known lemmas on stability of delayed functional differential equation with and without impulses and lyapunov stability theory several novel and easily verified synchronization criteria guaranteeing the whole network will be pinned to a homogenous solution are derived moreover the effects of the pinning impulsive controller and the dynamics of the uncontrolled nodes and the diffusive couplings on the synchronization process are explicitly expressed in the obtained criteria our results also show that we can always de

global robust asymptotic stability analysis of bam neural networks with time delay and impulse an lmi approach the global robust asymptotic stability of bi directional associative memory bam neural networks with constant or time varying delays and impulse is studied an approach combining the lyapunov krasovskii functional with the linear matrix inequality lmi is taken to study the problem some a criteria for the global robust asymptotic stability which gives information on the delay dependent property are derived some illustrative examples are given to demonstrate the effectiveness of the obtained results

exponential stability for stochastic reaction diffusion bam neural networks with time varying and distributed delays abstract in this paper we study the stability for a class of stochastic bidirectional associative memory bam neural networks with reaction diffusion and mixed delays the mixed delays considered in this paper are time varying and distributed delays based on a new lyapunov krasovskii functional and the poincare inequality as well as stochastic analysis theory a set of novel sufficient conditions are obtained to guarantee the stochastically exponential stability of the trivial solution or zero solution the obtained results show that the reaction diffusion term does contribute to the exponentially stabilization of the considered system moreover two numerical examples are given to show the effectiveness of the theoretical results

rank equalities for idempotent matrices with applications we show by elementary block matrix operations a variety of rank equalities related to sums of two idempotent matrices and give various consequences and applications

brief paper markovian jump delayed hopfield networks with multiplicative noise a non symmetric version of hopfield networks subject to state multiplicative noise pure time delay and markov jumps is considered such networks arise in the context of visuo motor control loops and may therefore be used to mimic their complex behavior in this paper we adopt the lur e postnikov systems approach to analyze the stochastic stability and the l 2 gain of generalized hopfield networks including these effects

rank equalities for idempotent and involutary matrices abstract we establish several rank equalities for idempotent and involutary matrices in particular we obtain new formulas for the rank of the difference the sum the product and the commutator of idempotent or involutary matrices extensions to scalar potent matrices are also included our matrices are complex and are not necessarily hermitian

existence results for the p laplacian with nonlinear boundary conditions in this paper we study the existence of nontrivial solutions for the problem pu u p 2u in a bounded smooth domain rn with a nonlinear boundary condition given by u p 2 u f u on the boundary of the domain the proofs are based on variational and topological arguments

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

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

on idempotency of linear combinations of idempotent matrices p 1 p 2 and p 3 being any three different nonzero mutually commutative nxn idempotent matrices and c 1 c 2 and c 3 being nonzero scalars the problem of characterizing some situations where a linear combination of the form p c 1p 1 c 2p 2 or p c 1p 1 c 2p 2 c 3p 3 is also an idempotent matrix has been considered moreover two interesting results about the idempotency of linear combinations of 2x2 idempotent matrices and 3x3 idempotent matrices have been given a statistical interpretation of the idempotency problem considered in this study has also been pointed out

stability and l_ 2 performance analysis of stochastic delayed neural networks this brief focuses on the robust mean square exponential stability and l2 performance analysis for a class of uncertain time delay neural networks perturbed by both additive and multiplicative stochastic noises new mean square exponential stability and l2 performance criteria are developed based on the delay partition lyapunov krasovskii functional method and generalized finsler lemma which is applicable to stochastic systems the analytical results are established without involving any model transformation estimation for cross terms additional free weighting matrices or tuning parameters numerical examples are presented to verify that the proposed approach is both less conservative and less computationally complex than the existing ones

the inertia of a hermitian pencil abstract let a and b be hermitian matrices and p a b where r 2 using parametric dependence of the eigenvalues we study the inertia of p under variation of and under small hermitian perturbations in particular we give semicontinuous dependence results for the set of where inertia of p is discontinuous

on the algebra generated by two projections the von neumann algebra generated by a pair of projections on a separable hilbert space occurs naturally in the study of quantum hall systems by a systematic use of kato s dual pair of self adjoint operators corresponding to the two projections the argument referring to the structure of the algebra can be simplified and the existence of an action of the group su 2 on the algebra via inner automorphisms can be shown also the algebra can be given an r structure with respect to which it is r commutative and an extension of the classical commutative differential calculus becomes possible

equalities and inequalities for inertias of hermitian matrices with applications abstract the inertia of a hermitian matrix is defined to be a triplet composed of the numbers of the positive negative and zero eigenvalues of the matrix counted with multiplicities respectively in this paper we show some basic formulas for inertias of 2 2 block hermitian matrices from these formulas we derive various equalities and inequalities for inertias of sums parallel sums products of hermitian matrices submatrices in block hermitian matrices differences of outer inverses of hermitian matrices as applications we derive the extremal inertias of the linear matrix expression a bxb with respect to a variable hermitian matrix x in addition we give some results on the extremal inertias of hermitian solutions to the matrix equation ax b as well as the extremal inertias of a partial block hermitian matrix

finite time synchronization of sampled data markovian jump complex dynamical networks with additive time varying delays based on dissipative theory abstract this paper is concerned with the problem of finite time synchronization issue of sampled data markovian jump complex dynamical networks mjcdns with additive time varying delays based on dissipative theory sufficient conditions to guarantee the finite time stability of mjcdns with additive time varying delays are presented sampled data control with stochastically varying sampling periods is considered the closed loop system is not just finite time bounded but also satisfies the dissipativity conditions further we handled a nonuniform sampled data controller by utilizing the properties of kronecker product combined with the appropriate lyapunov functionals technique a novel delay dependent finite time stability of dissipativity rule is derived in terms of linear matrix inequalities lmis to ensure that the delayed complex dynamical systems to be dissipative finally numerical examples are given to represent the applicability of the proposed approach

exponential and almost sure exponential stability of stochastic fuzzy delayed cohen grossberg neural networks in this paper we study a class of stochastic fuzzy delayed cohen grossberg neural networks two kinds of stability are discussed in our investigation one is exponential stability in the mean square and the other is almost sure exponential stability first some sufficient conditions are derived to guarantee the exponential stability in the mean square for the considered system based on the lyapunov krasovskii functional stochastic analysis theory and the ito s formula as well as the dynkin formula then we further investigate the almost sure exponential stability by employing the nonnegative semi martingale convergence theorem moreover we prove that the addressed system is both almost sure exponentially stable and exponentially stable in the mean square under suitable conditions finally three numerical examples are also given to show the effectiveness of the theoretical results in particular the simulation figures establish that fuzzy systems do have more advantages than non fuzzy systems

idempotency of linear combinations of two idempotent matrices abstract a complete solution is established to the problem of characterizing all situations where a linear combination of two different idempotent matrices p 1 and p 2 is also an idempotent matrix including naturally three such situations known in the literature viz if the combination is either the sum p 1 p 2 or one of the differences p 1 p 2 p 2 p 1 and appropriate additional conditions are fulfilled the solution asserts that in the particular case where p 1 and p 2 are complex matrices such that p 1 p 2 is hermitian these three situations exhaust the list of all possibilities and that this list extends when the above assumption on p 1 and p 2 is violated a statistical interpretation of the idempotency problem considered in this note is also pointed out

a property of orthogonal projectors abstract it is shown that if p 1 and p 2 are orthogonal projectors then a product having p 1 and p 2 as its factors is equal to another such product if and only if p 1 and p 2 commute in which case all products involving p 1 and p 2 reduce to the orthogonal projector p 1 p 2 this is a generalization of a result by baksalary and baksalary linear algebra appl 341 2002 129 with the proof based on a simple property of powers of hermitian nonnegative definite matrices

lyapunov stability analysis for nonlinear delay systems abstract sufficient conditions ensuring that a nonlinear system with disturbances having a delay is globally asymptotically stable independent of delay are given the proof carried out relies extensively on a characterization of the stability property in terms of lyapunov function the result is applied to some biological systems and neural networks a stabilizing memoryless controller for a second order system with state delay is also proposed

the behavior of the best sobolev trace constant and extremals in thin domains in this paper we study the asymptotic behavior of the best sobolev trace constant and extremals for the immersion w 1 p o l q o in a bounded smooth domain when it is contracted in one direction we find that the limit problem when rescaled in a suitable way is a sobolev type immersion in weighted spaces over a projection of o w 1 p p o a l q p o b for the special case p q this problem leads to an eigenvalue problem with a nonlinear boundary condition we also study the convergence of the eigenvalues and eigenvectors in this case