discrete time orthogonal spline collocation methods for schrodinger equations in two space variables two discrete time orthogonal spline collocation schemes are formulated and analyzed for solving the linear time dependent schrodinger equation in two space variables these are crank nicolson and alternating direction implicit adi schemes employing c1 piecewise polynomial spaces of arbitrary degree ge 3 in each space variable the stability of the schemes is examined and optimal order a priori l2 and h1 error estimates at each time step are derived parallel implementation of the adi scheme is discussed

split step orthogonal spline collocation methods for nonlinear schrodinger equations in one two and three dimensions abstract split step orthogonal spline collocation osc methods are proposed for one two and three dimensional nonlinear schrodinger nls equations with time dependent potentials firstly the nls equation is split into two nonlinear equations and one or more one dimensional linear equations commonly the nonlinear subproblems could be integrated directly and accurately but it fails when the time dependent potential cannot be integrated exactly in this case we propose three approximations by using quadrature formulae but the split order is not reduced discrete time osc schemes are applied for the linear subproblems in numerical experiments many tests are carried out to prove the reliability and efficiency of the split step osc ssosc methods solitons in one two and three dimensions are well simulated and conservative properties and convergence rates are demonstrated we also apply the ways of solving the nonlinear subproblems to the split step finite difference ssfd methods and the time splitting spectral tssp methods and the approximate ways still work well finally we apply the ssosc methods to solve some problems of bose einstein condensates

estimation of the domain of attraction for discrete time linear impulsive control systems with input saturation abstract this paper discusses the domain of attraction and stability analysis for discrete time linear impulsive systems with input saturation the first goal is to give a stable condition for the system the second is to present a strategy to find an inner estimate of the domain of attraction via establishing lyapunov functions and using other mathematical methods finally we present several examples to illustrate the validity of the conditions we have derived

pinning impulsive synchronization of reaction diffusion neural networks with time varying delays this paper investigates the exponential synchronization of reaction diffusion neural networks with time varying delays subject to dirichlet boundary conditions a novel type of pinning impulsive controllers is proposed to synchronize the reaction diffusion neural networks with time varying delays by applying the lyapunov functional method sufficient verifiable conditions are constructed for the exponential synchronization of delayed reaction diffusion neural networks with large and small delay sizes it is shown that synchronization can be realized by pinning impulsive control of a small portion of neurons of the network the technique used in this paper is also applicable to reaction diffusion networks with neumann boundary conditions numerical examples are presented to demonstrate the effectiveness of the theoretical results

impulsive stabilization and impulsive synchronization of discrete time delayed neural networks this paper investigates the problems of impulsive stabilization and impulsive synchronization of discrete time delayed neural networks ddnns two types of ddnns with stabilizing impulses are studied by introducing the time varying lyapunov functional to capture the dynamical characteristics of discrete time impulsive delayed neural networks didnns and by using a convex combination technique new exponential stability criteria are derived in terms of linear matrix inequalities the stability criteria for didnns are independent of the size of time delay but rely on the lengths of impulsive intervals with the newly obtained stability results sufficient conditions on the existence of linear state feedback impulsive controllers are derived moreover a novel impulsive synchronization scheme for two identical ddnns is proposed the novel impulsive synchronization scheme allows synchronizing two identical ddnns with unknown delays simulation results are given to validate the effectiveness of the proposed criteria of impulsive stabilization and impulsive synchronization of ddnns finally an application of the obtained impulsive synchronization result for two identical chaotic ddnns to a secure communication scheme is presented

globally asymptotical stability of discrete time analog neural networks some globally asymptotical stability criteria for the equilibrium states of a general class of discrete time dynamic neural networks with continuous states are presented using a diagonal lyapunov function approach the neural networks are assumed to have the asymmetrical weight matrices throughout the paper the resulting criteria are described by the diagonal stability of some matrices associated with the network parameters some novel stability conditions represented by either the existence of the positive diagonal solutions of the lyapunov equations or some inequalities are given using the equivalence between the diagonal stability and the schur stability for a nonnegative matrix some simplified global stability conditions are also presented finally some examples are provided for demonstrating the effectiveness of the global stability conditions presented

approximation on attraction domain of cohen grossberg neural networks abstract in this paper approximations of attraction domains of the asymptotically stable equilibrium points of some typical cohen grossberg neural networks are achieved most cohen grossberg neural networks are highly nonlinear systems which makes it difficult to approximate their attraction domain under some weak assumptions we are allowed to employ the optimal lyapunov method to obtain a lyapunov function for asymptotically stable equilibrium points of a given cohen grossberg neural network with the help of this lyapunov function we approximate the corresponding attraction domain by the iterative expansion approach numerical simulations also illustrate that the approximation obtained is really part of the attraction domain

multiple periodic solutions in impulsive hybrid neural networks with delays abstract the existence of multiple periodic solutions and their exponential stability are investigated for impulsive hybrid hopfield type neural networks with both time dependent and distributed delays using the leray schauder fixed point theorem and lyapunov functionals the criteria given are easily verifiable possess many adjustable parameters and depend on impulses providing flexibility for the analysis and design of delayed neural networks with impulse effects examples are given

decremental single source shortest paths on undirected graphs in near linear total update time in the decremental single source shortest paths sssp problem we want to maintain the distances between a given source node s and every other node in an n node m edge graph g undergoing edge deletions while its static counterpart can be solved in near linear time this decremental problem is much more challenging even in the undirected unweighted case in this case the classic o mn total update time of even and shiloach 16 has been the fastest known algorithm for three decades at the cost of a 1 e approximation factor the running time was recently improved to n2 o 1 by bernstein and roditty 9 in this article we bring the running time down to near linear we give a 1 e approximation algorithm with m1 o 1 expected total update time thus obtaining near linear time moreover we obtain m1 o 1 log w time for the weighted case where the edge weights are integers from 1 to w the only prior work on weighted graphs in o mn time is the mn0 9 o 1 time algorithm by henzinger et al 18 19 which works for directed graphs with quasi polynomial edge weights the expected running time bound of our algorithm holds against an oblivious adversary in contrast to the previous results which rely on maintaining a sparse emulator our algorithm relies on maintaining a so called sparse h e hop set introduced by cohen 12 in the pram literature an h e hop set of a graph ge v e is a set f of weighted edges such that the distance between any pair of nodes in g can be 1 e approximated by their h hop distance given by a path containing at most h edges on g e v e f our algorithm can maintain an no 1 e hop set of near linear size in near linear time under edge deletions it is the first of its kind to the best of our knowledge to maintain approximate distances using this hop set we extend the monotone even shiloach tree of henzinger et al 20 and combine it with the bounded hop sssp technique of bernstein 4 5 and m dry 27 these two new tools might be of independent interest

stability analysis of a delayed hopfield neural network in this paper we study a class of neural networks which includes bidirectional associative memory networks and cellular neural networks as its special cases by brouwer s fixed point theorem a continuation theorem based on gains and mawhin s coincidence degree matrix theory and inequality analysis we not only obtain some different sufficient conditions ensuring the existence uniqueness and global exponential stability of the equilibrium but also estimate the exponentially convergent rate our results are less restrictive than previously known criteria and can be applied to neural networks with a broad range of activation functions assuming neither differentiability nor strict monotonicity

on equilibria stability and instability of hopfield neural networks existence and uniqueness of equilibrium as well as its stability and instability of a continuous time hopfield neural network are studied a set of new and simple sufficient conditions are derived

orthogonal spline collocation methods for partial differential equations abstract this paper provides an overview of the formulation analysis and implementation of orthogonal spline collocation osc also known as spline collocation at gauss points for the numerical solution of partial differential equations in two space variables advances in the osc theory for elliptic boundary value problems are discussed and direct and iterative methods for the solution of the osc equations examined the use of osc methods in the solution of initial boundary value problems for parabolic hyperbolic and schrodinger type systems is described with emphasis on alternating direction implicit methods the osc solution of parabolic and hyperbolic partial integro differential equations is also mentioned finally recent applications of a second spline collocation method modified spline collocation are outlined

global exponential stability of impulsive discrete time neural networks with time varying delays this paper studies the problem of global exponential stability and exponential convergence rate for a class of impulsive discrete time neural networks with time varying delays firstly by means of the lyapunov stability theory some inequality analysis techniques and a discrete time halanay type inequality technique sufficient conditions for ensuring global exponential stability of discrete time neural networks are derived and the estimated exponential convergence rate is provided as well the obtained results are then applied to derive global exponential stability criteria and exponential convergence rate of impulsive discrete time neural networks with time varying delays finally numerical examples are provided to illustrate the effectiveness and usefulness of the obtained criteria

a class of conservative orthogonal spline collocation schemes for solving coupled klein gordon schrodinger equations abstract a class of discrete time orthogonal spline collocation schemes for solving coupled klein gordon schrodinger equations with initial and boundary conditions are considered these schemes are constructed by using piecewise cubic hermite interpolations in space combined with finite difference methods in time it is proved that the schemes have the conservation laws of discrete energy and possess second order accuracy in maximum norm and fourth order in l 2 norm for time and space respectively the conservation laws and rate of convergence are verified in numerical experiments moreover the propagations of solitary waves and collisions of two head on solitary waves are also well simulated

stability and bifurcation in a neural network model with two delays abstract a simple neural network model with two delays is considered linear stability of the model is investigated by analyzing the associated characteristic transcendental equation for the case without self connection it is found that the hopf bifurcation occurs when the sum of the two delays varies and passes a sequence of critical values the stability and direction of the hopf bifurcation are determined by applying the normal form theory and the center manifold theorem an example is given and numerical simulations are performed to illustrate the obtained results

estimate of exponential convergence rate and exponential stability for neural networks estimates of exponential convergence rate and exponential stability are studied for a class of neural networks which includes hopfield neural networks and cellular neural networks both local and global exponential convergence are discussed theorems for estimation of the exponential convergence rate are established and the bounds on the rate of convergence are given the domains of attraction in the case of local exponential convergence are obtained simple conditions are presented for checking exponential stability of the neural networks

fully dynamic maximal matching in o log n update time we present an algorithm for maintaining maximal matching in a graph under addition and deletion of edges our algorithm is randomized and it takes expected amortized o log n time for each edge update where n is the number of vertices in the graph while there exists a trivial o n time algorithm for each edge update the previous best known result for this problem is due to ivkovic and lloyd lecture notes in comput sci 790 springer verlag london 1994 pp 99 111 for a graph with n vertices and m edges they gave an o n m 0 7072 update time algorithm which is sublinear only for a sparse graph for the related problem of maximum matching onak and rubinfeld proceedings of stoc 10 cambridge ma 2010 pp 457 464 designed a randomized algorithm that achieves expected amortized o log 2 n time for each update for maintaining a c approximate maximum matching for some unspecified large constant c in contrast we can maintain a factor 2 approximate maximum matching in expected amortized o log n time per update as a direct corollary of the maximal matching scheme this in turn also implies a 2 approximate vertex cover maintenance scheme that takes expected amortized o log n time per update

fast direct solvers for piecewise hermite bicubic orthogonal spline collocation equations fast direct methods are proposed for the solution of linear systems arising when orthogonal spline collocation with piecewise hermite bicubics is employed for the approximate solution of poisson s

discrete time orthogonal spline collocation methods for the nonlinear schrodinger equation with wave operator in this paper discrete time orthogonal spline collocation schemes are proposed for the nonlinear schrodinger equation with wave operator these schemes are constructed by using orthogonal spline collocation approaches combined with finite difference methods the conservative property the convergence and the stability of these methods are theoretically analyzed and also verified by extensive numerical experiments in addition some interesting phenomena which require further theoretical analysis are discussed numerically

singular perturbation margin and generalised gain margin for linear time invariant systems in this paper we propose a singular perturbation margin spm and a generalised gain margin ggm as stability metrics for single input single output siso linear time invariant lti systems from the view of singular perturbations and regular perturbations which have bijective correspondences with the classical phase margin pm and the gain margin gm respectively both of the numerical and analytical time domain spm and ggm assessment methods are provided and relationships between the singular perturbation parameter pm of the perturbed system pm and spm of the nominal system and the monotonically increasing phase of the fast system are also revealed these results make it possible to assess the pm of the nominal system in the time domain for siso lti systems using the spm with a standardised testing system called pm gauge as demonstrated by examples the concepts of spm and ggm can be used as metrics of stability margins for linear time varying systems and nonlinear systems

solving convex programming problems with equality constraints by neural networks abstract this paper presents a neural network approach for solving convex programming problems with equality constraints after defining the energy function and neural dynamics of the proposed neural network we show the existence of an equilibrium point at which the neural dynamics becomes asymptotically stable it is shown that under proper conditions an optimal solution of the underlying convex programming problems is an equilibrium point of the neural dynamics and vise versa the configuration of the proposed neural network with an exact layout is provided for solving linear programming problems the operational characteristics of the neural network are demonstrated by numerical examples

paper maximal lyapunov functions and domains of attraction for autonomous nonlinear systems in this paper we present various theoretical and computational methods for estimating the domain of attraction of an autonomous nonlinear system these methods are based on the concept of a maximal lyapunov function which is introduced in this paper a partial differential equation characterizing a maximal lyapunov function is derived and the relationships of this equation as compared to zubov s partial differential equation are discussed an iterative procedure is given for solving the new partial differential equation this procedure yields lyapunov function candidates that are rational functions rather than polynomials the method is applied to four two dimensional examples and one three dimensional example and it is shown that the estimates obtained using this method are in many cases substantially better than those obtained using known methods

configurations of steady states for hopfield type neural networks the dependence of the steady states on the external input vector i for the continuous time and discrete time hopfield type neural networks of n neurons is discussed conditions for the existence of one or several paths of steady states are derived it is shown that in some conditions for an external input i there may exist at least 2 n exponentially stable steady states called configuration of steady states and their regions of attraction are estimated this means that there exist 2 n paths of exponentially stable steady states defined on a certain set of input values conditions assuring the transfer of a configuration of exponentially stable steady states to another configuration of exponentially stable steady states by successive changes of the external input are obtained these results may be important for the design and maneuvering of hopfield type neural networks used to analyze associative memories

periodic solutions for a two neuron network with delays in this paper we use the coincidence degree theory to establish new results on the existence of w periodic solutions for a two neuron network model

control procedures using domains of attraction abstract a technique of gradual approximation of the domain of attraction in the case of nonlinear continuous and discrete analytical dynamical systems is presented the technique is based on the gradual extension of the embryo of the optimal lyapunov function and gives at each step a new open and non empty piece of the domain of attraction for a nonlinear dynamical system with control defined by an analytical autonomous system of differential equations or difference equations using domains of attraction it is shown that two steady states which belong to an analytic path of asymptotically stable steady states can be transferred gradually one in the other by successive changes of the values of the control parameters numerical examples are given for the alflex reentry vehicle and for a discrete dynamical system considered recently in the literature

methods for determination and approximation of the domain of attraction abstract in this paper an r analytical function and the sequence of its taylor polynomials which are lyapunov functions different from those of vanelli and vidyasagar automatica 21 1 1985 69 80 is presented in order to determine and approximate the domain of attraction of the exponentially asymptotically stable zero steady state of an autonomous r analytical system of differential equations the analytical function and the sequence of its taylor polynomials are constructed by recurrence formulae using the coefficients of the power series expansion of f at 0

stability and bifurcation analysis on a discrete time neural network using techniques developed by kuznetsov to discrete time systems we study the stability of the equilibrium 0 0 and neimark sacker bifurcation also called hopf bifurcation for map of a discrete time neural network system the obtained results are less restrictive and improve upon the existing ones on neimark sacker bifurcation of discrete time neural network with special classes of transfer functions the theoretical analyses are verified by numerical simulations our results have potential applications in neural networks

oscillation susceptibility analysis along the path of longitudinal flight equilibriums abstract in this paper the oscillation susceptibility of an aircraft in a longitudinal flight with constant forward velocity is analyzed in different flight models conditions which ensure such a flight and equations governing the flight are presented the stability of the equilibriums appearing is analyzed and the existence of hopf bifurcations and saddle node bifurcations is researched for two aircrafts in a simplified model it is shown that saddle node bifurcations are present and there are no hopf bifurcations it is shown that for the elevator deflection there are two turning points e e having the property that if e e e then the angle of attack and the pitch rate q oscillate with the same period while the pitch angle increases decreases tending to the behavior of the aircraft is simulated in the simplified model when the elevator deflection e varies in the range e e and when e leaves this range for one of the aircrafts the analysis is performed also in the not simplified model showing the differences between the results obtained in different models

on a planar system modelling a neuron network with memory abstract we consider a delay differential equation modelling a network of two neurons with memory the situations without self connections and two delays and with self connections and two equal delays are addressed in terms of local stability and bifurcation analysis in the first case the dynamical behavior is studied by taking one of the delays as the bifurcating parameter namely the hopf bifurcation whose existence was proven in former works occurs as the delay crosses some critical values and is completely described conditions ensuring the stability of the periodic cycles are given when self connections are present hopf and pitchfork bifurcations co exist and the equation describing the flow on the center manifold is derived

exponential stability of discrete time hopfield neural networks in this paper some sufficient conditions for the local and global exponential stability of the discrete time hopfield neural networks with general activation functions are derived which gen eralize those existing results by means of m matrix theory and some inequality analysis techniques the exponential convergence rate of the neural networks to the equilibrium is estimated and for the local exponential stability the basin of attraction of the stable equilibrium is also characterized 2004 elsevier ltd all rights reserved

a new approach for estimating the attraction domain for hopfield type neural networks in this letter using methods proposed by e kaslik st balint and their colleagues we develop a new method expansion approach for estimating the attraction domain of asymptotically stable equilibrium points of hopfield type neural networks we prove theoretically and demonstrate numerically that the proposed approach is feasible and efficient the numerical results that obtained in the application examples including the network system considered by e kaslik l br escu and st balint indicate that the proposed approach is able to achieve better attraction domain estimation

efficient orthogonal spline collocation methods for solving linear second order hyperbolic problems on rectangles piecewise hermite bicubic orthogonal spline collocation laplace modified and alternating direction schemes for the approximate solution of linear second order hyperbolic problems on rectangles are analyzed the schemes are shown to be unconditionally stable and of optimal order accuracy in the h 1 and discrete maximum norms for space and time respectively implementations of the schemes are discussed and numerical results presented which demonstrate the accuracy and rate of convergence using various norms

h 1 norm error bounds for piecewise hermite bicubic orthogonal spline collocation schemes for elliptic boundary value problems two piecewise hermite bicubic orthogonal spline collocation schemes are considered for the approximate solution of elliptic nonhomogeneous dirichlet boundary value problems on rectangles in the first scheme the nonhomogeneous dirichlet boundary condition is approximated by means of the piecewise hermite cubic interpolant while the piecewise cubic interpolant at the boundary gauss points is used in the second scheme the piecewise hermite bicubic interpolant of the exact solution of the boundary value problem is employed as a comparison function to show that the h 1 norm of the error for each scheme is o h 3

improved dynamic reachability algorithms for directed graphs we obtain several new dynamic algorithms for maintaining the transitive closure of a directed graph and several other algorithms for answering reachability queries without explicitly maintaining a transitive closure matrix among our algorithms are i a decremental algorithm for maintaining the transitive closure of a directed graph through an arbitrary sequence of edge deletions in o mn total expected time essentially the time needed for computing the transitive closure of the initial graph such a result was previously known only for acyclic graphs ii two fully dynamic algorithms for answering reachability queries the first is deterministic and has an amortized insert delete time of o m sqrt n and worst case query time of o sqrt n the second is randomized and has an amortized insert delete time of o m 0 58 n and worst case query time of o m 0 43 this significantly improves the query times of algorithms with similar update times iii a fully dynamic algorithm for maintaining the transitive closure of an acyclic graph the algorithm is deterministic and has a worst case insert time of o m constant amortized delete time of o 1 and a worst case query time of o n log n our algorithms are obtained by combining several new ideas one of which is a simple sampling idea used for detecting decompositions of strongly connected components with techniques of even and shiloach j acm 28 1981 pp 1 4 italiano inform process lett 28 1988 pp 5 11 henzinger and king proceedings of the 36 th annual symposium on foundations of computer science milwaukee wi 1995 pp 664 672 and frigioni et al acm j exp algorithmics 6 2001 electronic

global attractivity of a synchronized periodic orbit in a delayed network abstract we consider an artificial neural network where the signal transmission is of a digital mcculloch pitts nature and is delayed due to the finite switching speed of neurons amplifiers for a particular connection topology we show that all solutions starting from nonoscillatory initial states will be eventually synchronized and stabilized at a unique limit cycle and hence such a network can be used as a synchronized oscillator

a note on stability of analog neural networks with time delays this note presents a generalized sufficient condition which guarantees stability of analog neural networks with time delays the condition is derived using a lyapunov functional and the stability criterion is stated as the equilibrium of analog neural networks with delays is globally asymptotically stable if the product of the norm of connection matrix and the maximum neuronal gain is less than one

existence and stability of equilibria of the continuous time hopfield neural network the qualitative properties of the continuous time hopfield neural network are investigated by using the cauchy existence theorem of ordinary differential equations a set of new and simple sufficient conditions guaranteeing the existence and uniqueness of equilibrium is derived furthermore global and asymptotical stability conditions are established by means of lyapunov method

global exponential stability of impulsive delay difference equation abstract in this paper an impulsive delay difference equation is considered by establishing an impulsive delay difference inequality and using the properties of cone and eigenspace of the spectral radius of nonnegative matrices some new sufficient conditions for global exponential stability of the impulsive delay difference equation are obtained the results extend and improve the earlier publications an example is given to demonstrate the effectiveness of the theory

hopf bifurcating periodic orbits in a ring of neurons with delays in this paper we consider a ring of neurons with self feedback and delays the linear stability of the model is investigated by analyzing the associated characteristic transcendental equation based on the normal form approach and the center manifold theory we derive the formula for determining the properties of hopf bifurcating slowly oscillating periodic orbits for a ring of neurons with delays including the direction of hopf bifurcation stability of the hopf bifurcating slowly oscillating periodic orbits and so on moreover by means of the symmetric bifurcation theory of delay differential equations coupled with representation theory of standard dihedral groups we not only investigate the effect of synaptic delay of signal transmission on the pattern formation but also obtain some important results about the spontaneous bifurcation of multiple branches of periodic solutions and their spatio temporal patterns

improved global exponential stability for delay difference equations with impulses abstract in this paper by means of constructing an impulsive delay difference inequality we study the global exponential stability for a class of delay difference equations with impulses a new criterion ensuring the global exponential stability of the equilibrium point is obtained which is less restrictive and conservative than that given in the earlier reference an illustrative example is given to demonstrate the effectiveness and advantage of the obtained result