on common diagonal lyapunov solutions abstract several recent results regarding common diagonal lyapunov solutions are further explored here the first one attributed to redheffer and revisited by shorten and narendra reduces the diagonal stability of a matrix to common diagonal lyapunov solutions on two matrices of order one less we present a shorter purely matrix theoretic proof of this result along with its extensions the second one concerns two different necessary and sufficient conditions due to oleng narendra and shorten for a pair of 2 2 matrices to share a common diagonal lyapunov solution we show that these two conditions are connected directly to each other

on discrete time diagonal and d stability abstract this paper introduces four discrete time analogs of different types of matrix stability diagonal simultaneous vertex and d stability the last three being defined in terms of a certain associated polytope of matrices the diagonal stability of any vertex of this polytope is shown to imply its simultaneous stability and hence d stability of the vertices it is shown by a counterexample that d stability as in the continuous time case is not equivalent to diagonal stability several important classes for which this equivalence is true are identified it is shown that simultaneous stability of the vertices is equivalent to the simultaneous stability of the whole polytope and it is conjectured that this equivalence holds without the requirement of simultaneity some other conjectures relating the four types of stability are made and it is shown that in the 2 x 2 case all four are equivalent

random antagonistic matrices the ensemble of antagonistic matrices is introduced and studied in antagonistic matrices the entries and are real and have opposite signs or are both zero and the diagonal is zero this generalization of antisymmetric matrices is suggested by the linearized dynamics of competitive species in ecology

stability analysis of diagonally equipotent matrices diagonally equipotent matrices are diagonally dominant matrices for which dominance is never strict in any coordinate they appear e g as laplacian matrices of signed graphs we show in this paper that for this class of matrices it is possible to provide a complete characterization of the stability properties based only on the signs of the entries of the matrices

stability of acyclic matrices abstract necessary and sufficient conditions for stability of acyclic matrices whose principal minors are nonnegative are given also a characterization is given of additive d stable acyclic matrices

new conditions on global stability of cohen grossberg neural networks in this letter we discuss the dynamics of the cohen grossberg neural networks we provide a new and relaxed set of sufficient conditions for the cohen grossberg networks to be absolutely stable and exponentially stable globally we also provide an estimate of the rate of convergence

lmi characterization of structural and robust stability abstract this paper introduces several stability conditions for a given class of matrices expressed in terms of linear matrix inequalities lmi being thus simply and efficiently computable diagonal and simultaneous stability both characterized by polytopes of matrices are addressed using this approach a method particularly attractive to test a given matrix for d stability is proposed lyapunov parameter dependent functions are built in order to reduce conservativeness of the stability conditions the key idea is to relate hurwitz stability with a positive realness condition

p matrix properties injectivity and stability in chemical reaction systems in this paper we examine matrices which arise naturally as jacobians in chemical dynamics we are particularly interested in when these jacobians are p matrices up to a sign change ensuring certain bounds on their eigenvalues precluding certain behavior such as multiple equilibria and sometimes implying stability we first explore reaction systems and derive results which provide a deep connection between system structure and the p matrix property we then examine a class of systems consisting of reactions coupled to an external rate dependent negative feedback process and characterize conditions which ensure that the p matrix property survives the negative feedback the techniques presented are applied to examples published in the mathematical and biological literature

global stability of cellular neural networks with constant and variable delays this paper gives new conditions ensuring global asymptotic stability and global exponential stability for cellular neural networks with constant delay and variable delay respectively these conditions are derived by using the essence of piecewise linearity of the output function of cellular neural networks and by constructing lyapunov functions and functionals furthermore these conditions are significantly weaker than those given in existing literature

flexible scaling model for use in random field simulation of hydraulic conductivity a new fractal scaling model is proposed for use in aquifer heterogeneity modeling and simulation the new model which is obtained by subordination of fractional brownian motion provides a unifying framework for scaling models of heterogeneity in that it includes previous scaling models as special cases and can also be tuned continuously between these models choosing the subordinator to be a lognormal distribution results in a non gaussian scaling model with a moderate degree of variability that matches data better than previous fractal scaling models thus providing more accurate heterogeneity representations for use in stochastic flow and transport predictions two new stochastic simulation algorithms are constructed from the new models one based on sequential simulation algorithms and the other based on probability field simulation both algorithms can be made conditional on available measurements

stronger than lyapunov notions of matrix stability or how flowers help solve problems in mathematical ecology abstract persistent patterns of interactions in a multi component system e g intra and inter species relations in a community of n interacting species may imply a number of formalizations as special stronger than lyapunov notions of matrix stability like d stability qualitative stability volterra lyapunov stability and others a variety of these notions each having a certain motivation with regard to uncertainties inherent in model applications constitute a hierarchical topology sometimes very intricate and not yet well understood in a formal space of real n n matrices as visible forms of this hierarchy matrix flowers are suggested where petals correspond to subsets of particular stability kinds whose visible inclusion intersection represent logical implication junction the flowers are constructed under a few simple conventions and in the absence of ready characterizations to draw a petal often poses a challenging mathematical problem whose solution may reveal a new biological knowledge of a general nature particular petals concern the topics of strong and weak interactions in a community key species in its structure diffusion instability in its spatial dynamics where the flower plays a heuristic role in formulating a new problem or and stimulating a new application

multiplicative perturbations of stable and convergent operators abstract some familiar classes of stable hilbert space operators are studied to determine how they overlap and where the unitary similarity classes of their members lie analogous but less familiar classes of convergent operators are examined with the same aim the classes considered are often sets of products m a where m is a given set of diagonal or hermitian matrices and a is a single matrix the a s for which m a is a set of stable or convergent operators are sometimes characterized

absolute stability analysis in cellular neural networks with variable delays and unbounded delay abstract in this paper the existence and uniqueness of the equilibrium point and absolute stability of a class of cellular neural networks with globally lipschitz continue activation functions are investigated the neural networks contain both variable and unbounded delays the necessary and sufficient condition for the existence and uniqueness of the equilibrium point of the neural networks is obtained by constructing proper vector liapunov functions nonlinear integrodifferential inequalities involving both variable delays and unbounded delay which are corresponding to the neural networks are established by using m matrix theory and qualitative property of the integrodifferential inequalities the sufficient conditions for absolute stability global asymptotic stability are obtained

global exponential stability of high order hopfield type neural networks this paper studies the problem of global exponential stability for a class of high order hopfield type neural networks by utilizing lyapunov functions several sufficient conditions for the global exponential stability of equilibrium point are obtained the exponential convergence rate of the equilibrium point is also estimated as a special case several new criteria on global exponential stability for the corresponding first order hopfield neural networks are obtained those criteria generalize some of the recent results reported in the literature

notes on ω and τ matrices abstract results on and matrices are surveyed the question whether an matrix with positive leading principal minors is a matrix is answered negatively other open questions are discussed new conjectures and questions are introduced

not all gkk τ matrices are stable abstract hermitian positive definite totally positive and nonsingular m matrices enjoy many common properties in particular a positivity of all principal minors b weak sign symmetry c eigenvalue monolonicity d positive stability the class of gkk matrices is defined by properties a and b whereas the class of nonsingular matrices by a and c it was conjectured that a b d d carlson j res nat bur standards sect b 78 1974 1 2 a c d g m engel and h schneider linear and multilinear algebra 4 1976 155 176 a b a property stronger than d r varga numerical methods in linear algebra 1978 pp 5 15 a b c d d hershkowitz linear algebra appl 171 1992 161 186 we describe a class of unstable gkk matrices thus disproving all four conjectures

positivity of principal minors sign symmetry and stability the relation between positivity of principal minors sign symmetry and stability of matrices is studied it is shown that for sign symmetric matrices having positive principal minors is equivalent to stability to d stability and to having a positive scaling into a stable matrix the relation between spectra of matrices some of whose powers have positive principal minors and matrices whose corresponding powers have positive sums of principal minors of each order is studied as well it is shown that for matrices of order less than 4 these two classes share the same spectra the relation of these classes and stability is studied in particular for sign symmetric matrices and for anti sign symmetric matrices

on schur d stable matrices abstract it is shown that vertex stability implies schur d stability for real 2 2 matrices and real n n tridiagonal matrices additional results describing the class of n n complex schur d stable matrices are given

recent directions in matrix stability abstract matrix stability has been intensively investigated in the past two centuries we review work that has been done in this topic focusing on the great progress that has been achieved in the last decade or two we start with classical stability criteria of lyapunov routh and hurwitz and lienard and chipart we then study recently proven sufficient conditions for stability with particular emphasis on p matrices we investigate conditions for the existence of a stable scaling for a given matrix we review results on other types of matrix stability such as d stability additive d stability and lyapunov diagonal stability we discuss the weak principal submatrix rank property shared by lyapunov diagonally semistable matrices we also discuss the uniqueness of lyapunov scaling factors maximal lyapunov scaling factors cones of real positive semidefinite matrices and their applications to matrix stability and inertia preserving matrices

characterization of acyclic d stable matrices abstract necessary and sufficient conditions for d stability of acyclic matrices are given special cases of this characterization are recent results on tridiagonal matrices

localization of the spectra of p and p0 matrices abstract it is well known that if s 1 n is a spectrum of a p matrix then arg i n we show that if the number of elements of s in the right half plane or in the left half plane is given then the above bound can be improved namely there exists such that arg i n when s has exactly one element in the right half plane it is shown that barg i 2 3 independently of n

on two classes of matrices with positive diagonal solutions to the lyapunov equation abstract we characterize real indecomposable quasi jacobi matrices of class d i e those which satisfy the lyapunov equation pa a p q with p diagonal and both p and q positive definite the subclass d 2 of class d when also q is diagonal is also characterized in the case of general indecomposable real matrices

global exponential stability and global convergence in finite time of delayed neural networks with infinite gain this paper introduces a general class of neural networks with arbitrary constant delays in the neuron interconnections and neuron activations belonging to the set of discontinuous monotone increasing and possibly unbounded functions the discontinuities in the activations are an ideal model of the situation where the gain of the neuron amplifiers is very high and tends to infinity while the delay accounts for the finite switching speed of the neuron amplifiers or the finite signal propagation speed it is known that the delay in combination with high gain nonlinearities is a particularly harmful source of potential instability the goal of this paper is to single out a subclass of the considered discontinuous neural networks for which stability is instead insensitive to the presence of a delay more precisely conditions are given under which there is a unique equilibrium point of the neural network which is globally exponentially stable for the states with a known convergence rate the conditions are easily testable and independent of the delay moreover global convergence in finite time of the state and output is investigated in doing so new interesting dynamical phenomena are highlighted with respect to the case without delay which make the study of convergence in finite time significantly more difficult the obtained results extend previous work on global stability of delayed neural networks with lipschitz continuous neuron activations and neural networks with discontinuous neuron activations but without delays

some remarks on matrix stability with application to turing instability abstract an example of a 4 4 matrix is given that provides a counterexample to a result on turing diffusion driven instability and also answers negatively a conjecture on strong stability such instability is shown to arise from nonreal eigenvalues the relevance and the connection of our example to classes of matrix stability known in the literature are discussed

on the uniqueness of the lyapunov scaling factors sufficient conditions are given for a lyapunov diagonally semistable matrix a to have a unique up to scalar multiplication lyapunov scaling factor suppose d is a lyapunov scaling factor of a a necessary and sufficient condition is given for the uniqueness of d the results extend work of hershkowitz and schneider

on global asymptotic stability of recurrent neural networks with time varying delays in this paper by constructing a new lyapunov functional and using m matrix and topological degree tool problem of the global asymptotic stability gas is discussed for a class of recurrent neural networks with time varying delays some simple and new sufficient conditions are obtained ensuring existence uniqueness of the equilibrium point and its gas of the neural networks some previous works are improved in addition this condition does not require the activation functions to be differentiable bounded and monotone nondecreasing and the weight connected matrices to be symmetric the neural network model considered in this paper include the delayed hopfield neural networks bidirectional associative memory networks and delayed cellular neural networks as its special cases

scaling a canonical design problem for networks motivated by decentralized control and numerical methods applications we explore the scaling problem that is we consider the design of a diagonal matrix k so as to i place the eigenvalues of the product kg in the olhp and ii optimize a scalar performance measure c kg of interest for a square topology matrix g we first illustrate our methodology for scaling design which combines optimization machinery with graph algebra notions for a class of topology matrices and a specific performance measure in particular for symmetric positive definite topologies and for a dominant eigenvalue ratio performance measure we find that the optimally scaled system kg has a special structure the right eigenvectors associated with the largest and smallest eigenvalues have corresponding entries of identical magnitude and hence differ only in sign pattern using this structure we develop both explicit and algorithmic methods for computing the optimal scaling and cost after illustrating our methodology with this

generalized lyapunov approach for convergence of neural networks with discontinuous or non lipschitz activations the paper considers a class of additive neural networks where the neuron activations are modeled by discontinuous functions or by continuous non lipschitz functions some tools are developed which enable us to apply a lyapunov like approach to differential equations with discontinuous right hand side modeling the neural network dynamics the tools include a chain rule for computing the time derivative along the neural network solutions of a nondifferentiable lyapunov function and a comparison principle for this time derivative which yields conditions for exponential convergence or convergence in finite time by means of the lyapunov like approach a general result is proved on global exponential convergence toward a unique equilibrium point of the neural network solutions moreover new results on global convergence in finite time are established which are applicable to neuron activations with jump discontinuities or neuron activations modeled by means of continuous non lipschitz holder functions

a new characterization of simultaneous lyapunov diagonal stability via hadamard products abstract a well known characterization by kraaijevanger 14 for lyapunov diagonal stability states that a real square matrix a is lyapunov diagonally stable if and only if a s is a p matrix for any positive semidefinite s with nonzero diagonal entries this result is extended here to a new characterization involving similar hadamard multiplications for simultaneous lyapunov diagonal stability on a set of matrices among the main ingredients for this extension are a new concept called p sets and a recent result regarding simultaneous lyapunov diagonal stability by berman goldberg and shorten 2

a comprehensive review of stability analysis of continuous time recurrent neural networks stability problems of continuous time recurrent neural networks have been extensively studied and many papers have been published in the literature the purpose of this paper is to provide a comprehensive review of the research on stability of continuous time recurrent neural networks including hopfield neural networks cohen grossberg neural networks and related models since time delay is inevitable in practice stability results of recurrent neural networks with different classes of time delays are reviewed in detail for the case of delay dependent stability the results on how to deal with the constant variable delay in recurrent neural networks are summarized the relationship among stability results in different forms such as algebraic inequality forms m matrix forms linear matrix inequality forms and lyapunov diagonal stability forms is discussed and compared some necessary and sufficient stability conditions for recurrent neural networks without time delays are also discussed concluding remarks and future directions of stability analysis of recurrent neural networks are given

maximal lyapunov scaling factors and their applications in the study of lyapunov diagonal semistability of block triangular matrices abstract for a lyapunov diagonally semistable matrix a we introduce the concept of maximal lyapunov scaling factor that is a lyapunov scaling factor d of a such that the range of ad da t is of maximal dimension we investigate properties of maximal lyapunov scaling factors and apply them in proving a characterization for block triangular lyapunov diagonally semistable matrices in terms of lyapunov diagonal semistability of the diagonal blocks the latter result is used to prove that a block triangular matrix is lyapunov diagonally semistable if and only if every 2 2 block principal submatrix of it is lyapunov diagonally semistable

asymptotic stability of systems with saturation constraints in the present paper the authors establish new sufficient conditions for the global asymptotic stability of linear systems operating on the unit hypercube in r sup n and of linear systems with partial state saturation constraints they also determine a set of necessary and sufficient conditions for the global asymptotic stability of second order linear systems with saturation constraints on both states using the above results systems of the type considered herein are widely used in several areas of applications including control systems signal processing and artificial neural networks

analysis of complete stability for discrete time cellular neural networks with piecewise linear output functions this letter discusses the complete stability of discrete time cellular neural networks with piecewise linear output functions under the assumption of certain symmetry on the feedback matrix a sufficient condition of complete stability is derived by finite trajectory length because the symmetric conditions are not robust the complete stability of networks may be lost under sufficiently small perturbations the robust conditions of complete stability are also given for discrete time cellular neural networks with multiple equilibrium points and a unique equilibrium point these complete stability results are robust and available

convex invertible sets and matrix sign function it is proved that in two dimensional unital real algebras a step by step construction of minimal convex invertible sets terminates in two steps an analogous construction of minimal convex conditionally invertible sets of the complex field also terminates in two steps and the complex field is the only finite dimensional complex algebra with this property connections are made with the matrix sign function the matrix sign function of a matrix belongs to the convex invertible set generated by the matrix an example is given to show that the matrix sign function cannot always be obtained in two steps by following the step by step construction

lyapunov diagonal semistability of real h matrices we characterize lyapunov diagonally stable real h matrices and those real h matrices which are lyapunov diagonally semistable but not lyapunov diagonally stable called lyapunov diagonally near stable the latter characterization is given in terms of the principal submatrix rank property defined here we apply our results to the numerical abscissas of real matrices one of our main tools is a slight strengthening of classical results of ostrowski which we derive from a fundamental theorem of wielandt

an adaptive ghost fluid finite volume method for compressible gas water simulations an adaptive ghost fluid finite volume method is developed for one and two dimensional compressible multi medium flows in this work it couples the real ghost fluid method gfm c w wang t g liu b c khoo a real ghost fluid method for the simulation of multi medium compressible flow siam j sci comput 28 2006 278 302 and the adaptive moving mesh method h z tang t tang moving mesh methods for one and two dimensional hyperbolic conservation laws siam j numer anal 41 2003 487 515 h z tang t tang p w zhang an adaptive mesh redistribution method for non linear hamilton jacobi equations in two and three dimensions j comput phys 188 2003 543 572 and thus combines their advantages this work shows that the local mesh clustering in the vicinity of the material interface can effectively reduce both numerical and conservative errors caused by the gfm around the material interface and other discontinuities besides the improvement of flow field resolution the adaptive gfm also largely increases the computational efficiency several numerical experiments are conducted to demonstrate robustness and efficiency of the current method they include several 1d and 2d gas water flow problems involving a large density gradient at the material interface and strong shock interface interactions the results show that our algorithm can capture the shock waves and the material interface accurately and is stable and robust even for solutions with large density and pressure gradients