subgradient based neural networks for nonsmooth nonconvex optimization problems this paper presents a subgradient based neural network to solve a nonsmooth nonconvex optimization problem with a nonsmooth nonconvex objective function a class of affine equality constraints and a class of nonsmooth convex inequality constraints the proposed neural network is modeled with a differential inclusion under a suitable assumption on the constraint set and a proper assumption on the objective function it is proved that for a sufficiently large penalty parameter there exists a unique global solution to the neural network and the trajectory of the network can reach the feasible region in finite time and stay there thereafter it is proved that the trajectory of the neural network converges to the set which consists of the equilibrium points of the neural network and coincides with the set which consists of the critical points of the objective function in the feasible region a condition is given to ensure the convergence to the equilibrium point set in finite time moreover under suitable assumptions the coincidence between the solution to the differential inclusion and the ldquoslow solutionrdquo of it is also proved furthermore three typical examples are given to present the effectiveness of the theoretic results obtained in this paper and the good performance of the proposed neural network

on the global convergence of a class of functional differential equations with applications in neural network theory abstract we study a system of retarded functional differential equations which generalise both the hopfield neural network model as well as hybrid network models of the cellular neural network type our main results give sufficient conditions for the global asymptotic stability of such systems and are milder than previously known conditions for the hybrid models when specialised to neural networks our models allow us to consider several different types of activation functions including piecewise linear sigmoids and unbounded activations as well as the usualc smooth sigmoids these issues are vital in the applications we also study neural network models with nonconstant delaysr t

convergence of neural networks for programming problems via a nonsmooth lojasiewicz inequality this paper considers a class of neural networks nns for solving linear programming lp problems convex quadratic programming qp problems and nonconvex qp problems where an indefinite quadratic objective function is subject to a set of affine constraints the nns are characterized by constraint neurons modeled by ideal diodes with vertical segments in their characteristic which enable to implement an exact penalty method a new method is exploited to address convergence of trajectories which is based on a nonsmooth lstrokojasiewicz inequality for the generalized gradient vector field describing the nn dynamics the method permits to prove that each forward trajectory of the nn has finite length and as a consequence it converges toward a singleton furthermore by means of a quantitative evaluation of the lstrokojasiewicz exponent at the equilibrium points the following results on convergence rate of trajectories are established 1 for nonconvex qp problems each trajectory is either exponentially convergent or convergent in finite time toward a singleton belonging to the set of constrained critical points 2 for convex qp problems the same result as in 1 holds moreover the singleton belongs to the set of global minimizers and 3 for lp problems each trajectory converges in finite time to a singleton belonging to the set of global minimizers these results which improve previous results obtained via the lyapunov approach are true independently of the nature of the set of equilibrium points and in particular they hold even when the nn possesses infinitely many nonisolated equilibrium points

compatibility conditions for time dependent partial differential equations and the rate of convergence of chebyshev and fourier spectral methods abstract compatibility conditions for partial differential equations pdes are an infinite set of relations between the initial conditions the pde and the boundary conditions which are necessary and sufficient for the solution to be c that is infinitely differentiable everywhere on the computational domain including the boundaries since the performance of chebyshev spectral and spectral element methods is dramatically reduced when the solution is not c one would expect that the compatibility conditions would be a major theme in the spectral literature instead it has been completely ignored therefore we pursue three goals here first we present a proof of the compatibility conditions in a simplified form that does not require functional analysis second we analyze the connection between the compatibility conditions and the rate of convergence of chebyshev methods lastly we describe strategies for slightly adjusting initial conditions so that the compatibility conditions are satisfied

sum accelerated pseudospectral methods finite differences and sech weighted differences this work continues our previous studies of algorithms for accelerating the convergence of pseudospectral derivative series in order to obtain new differentiation schemes which have the sparsity and low cost of finite differences but the accuracy of spectral methods we develop a general theoretical framework for difference schemes finite differences are close to optimum but can be bettered by a new scheme we have dubbed sech weighted differences through numerical examples we show that sech weighted differences are effective in contrast non linear accelerations like pade approximants and the levin u transform so popular in other applications are inferior for approximating derivatives to linear accelerations like finite differences sech weighted differences and the euler method

exponential stability of impulsive high order hopfield type neural networks with time varying delays this paper considers the problems of global exponential stability and exponential convergence rate for impulsive high order hopfield type neural networks with time varying delays by using the method of lyapunov functions some sufficient conditions for ensuring global exponential stability of these networks are derived and the estimated exponential convergence rate is also obtained as an illustration an numerical example is worked out using the results obtained

hadamard s theorem for locally lipschitzian maps if the derivativef p of a continuously differentiable mapffrom r into itself is an invertible linear transformation for each p then by the classical inverse function theorem f is a local homeomorphism every p has a neighborhood u mapped homeomorphically onto its image f u a natural and intriguing question under what further conditions will f be a homeomorphism of r onto itself was answered in one way by hadamard 8 in 1906 using a condition on the norms b max bh h i of the inverses b f p

neural network for nonsmooth nonconvex constrained minimization via smooth approximation a neural network based on smoothing approximation is presented for a class of nonsmooth nonconvex constrained optimization problems where the objective function is nonsmooth and nonconvex the equality constraint functions are linear and the inequality constraint functions are nonsmooth convex this approach can find a clarke stationary point of the optimization problem by following a continuous path defined by a solution of an ordinary differential equation the global convergence is guaranteed if either the feasible set is bounded or the objective function is level bounded specially the proposed network does not require 1 the initial point to be feasible 2 a prior penalty parameter to be chosen exactly 3 a differential inclusion to be solved numerical experiments and comparisons with some existing algorithms are presented to illustrate the theoretical results and show the efficiency of the proposed network

a lag averaged generalization of euler s method for accelerating series abstract euler s method is an efficient accelerator for an alternating series that is one whose terms a j are such that sign a j sign a j 1 however its effectiveness deteriorates rapidly as the period p of the oscillations in degree j increases to reduce this nonuniformity we introduce an algorithm which begins with the recursive computation of a triangular array of numbers the diagonal elements define a generalization of euler s method each horizontal row defines a delayed euler method which is even more effective the key element is that the array is generated by repeatedly averaging partial sums of the original series with a lag that is a distance in degree between the partial sums that are combined which optimally is half the period p of the oscillations in the series we illustrate the algorithm with the fourier series for a function with a shock wave jump discontinuity the terms a j 1 j sin jx vary from strictly alternating p 2 at x 0 to monotonic p as x the lag averaged euler scheme with and without delay is much more efficient than its classical counterpart especially near the discontinuities we present an analytical convergence theory for both the standard and generalized euler methods

stability analysis for neural dynamics with time varying delays by using the usual additive neural network model a delay independent stability criterion for neural dynamics with perturbations of time varying delays is derived we extend previously known results obtained by gopalsamy and he 1994 to the time varying delay case and present decay estimates of solutions of neural networks the asymptotic stability is global in the state space of neuronal activations from the techniques used in this paper it is shown that our criterion ensures stability of neural dynamics even when the delay functions vary violently with time our approach provides an effective method for the stability analysis of neural dynamics with delays

morse theory and finiteness properties of groups we examine the finiteness properties of certain subgroups of right angled artin groups in particular we find an example of a group that is of type fp z but is not finitely presented

almost sure exponential stability of recurrent neural networks with markovian switching this paper presents new stability results for recurrent neural networks with markovian switching first algebraic criteria for the almost sure exponential stability of recurrent neural networks with markovian switching and without time delays are derived the results show that the almost sure exponential stability of such a neural network does not require the stability of the neural network at every individual parametric configuration next both delay dependent and delay independent criteria for the almost sure exponential stability of recurrent neural networks with time varying delays and markovian switching parameters are derived by means of a generalized stochastic halanay inequality the results herein include existing ones for recurrent neural networks without markovian switching as special cases finally simulation results in three numerical examples are discussed to illustrate the theoretical results

improved global robust stability of interval delayed neural networks via split interval generalizations abstract the problem of global robust stability of hopfield type delayed neural networks with the intervalized network parameters is revisited recently a computationally tractable i e linear matrix inequality lmi based global robust stability criterion derived from an earlier criterion based on dividing the given interval into more that two intervals has been presented in the present paper generalizations i e division of the given interval into m intervals where m is an integer greater than or equal to 2 is considered and some new lmi based global robust stability criteria are derived it is shown that in some cases m 2 may not suffice i e m 2 may be needed to realize the improvement an example showing the effectiveness of the proposed generalization is given the paper also provides a complete and systematic explanation of the split interval idea

synchronization of competitive neural networks with different time scales abstract in this paper the exponential synchronization problem for a class of competitive neural networks is investigated moreover without assuming the active functions to be differentiable and bounded some exponential synchronization criteria are devised by lyapunov functionals linear matrix inequality lmi approach the leibniz newton formula the conditions are less conservative than existing ones reported in the literature for delayed neural networks the applicability of the conditions are validated by an illustrative example

existence and global exponential stability of almost periodic solution for cellular neural networks with variable coefficients and time varying delays in this paper we study cellular neural networks with almost periodic variable coefficients and time varying delays by using the existence theorem of almost periodic solution for general functional differential equations introducing many real parameters and applying the lyapunov functional method and the technique of young inequality we obtain some sufficient conditions to ensure the existence uniqueness and global exponential stability of almost periodic solution the results obtained in this paper are new useful and extend and improve the existing ones in previous literature

acceleration of algebraically converging fourier series when the coefficients have series in powers of 1 n if the coefficients in a fourier cosine series f x f email protected n 0 a ncos nx decrease as a small negative power of n then one may need millions of terms to sum the series to high accuracy we show that if the a n are known analytically and have a power series in 1 n then it is straightforward to approximate f x as a series of what we shall the lanczos krylov lk functions we describe the similar methodology for sine series general fourier series are merely the sum of a cosine series with a sine series and thus are implicitly handled too for cosine coefficients that involve only even powers of n and sine coefficients that are functions of odd powers of n the lk functions may be expressed in terms of bernoulli polynomials the lk functions for cosine coefficients involving odd powers of n and for sine coefficients in even powers of n are not known explicitly these are also known as clausen functions we provide rapidly convergent series to compute these clausen functions to high accuracy our method includes the endpoint subtraction ideas of lanczos and krylov but is more general the sum n 1 1 n 1 1 email protected cos nx where l 0 is a constant arises in phase transitions in absorbed monolayers on metal surfaces it is easily summed by our method which correctly incorporates the logarithmic singularities at x email protected

deriving sufficient conditions for global asymptotic stability of delayed neural networks via nonsmooth analysis ii in this paper we obtain new sufficient conditions ensuring existence uniqueness and global asymptotic stability gas of the equilibrium point for a general class of delayed neural networks dnns via nonsmooth analysis which makes full use of the lipschitz property of functions defining dnns based on this new tool of nonsmooth analysis we first obtain a couple of general results concerning the existence and uniqueness of the equilibrium point then those results are applied to show that existence assumptions on the equilibrium point in some existing sufficient conditions ensuring gas are actually unnecessary and some strong assumptions such as the boundedness of activation functions in some other existing sufficient conditions can be actually dropped finally we derive some new sufficient conditions which are easy to check comparison with some related existing results is conducted and advantages are illustrated with examples throughout our paper spectral properties of the matrix a a sup spl tau play an important role which is a distinguished feature from previous studies here a and a sup spl tau are respectively the feedback and the delayed feedback matrix defining the neural network under consideration

a new criterion for global robust stability of interval delayed neural networks a novel criterion for the global robust stability of hopfield type interval neural networks with delay is presented an example showing the effectiveness of the present criterion is given

numerical comparisons of nonlinear convergence accelerators as part of a continuing program of numerical tests of convergence accelerators we have compared the iterated aitken s a2 method wynn s algorithm brezinski s 0 algorithm and levin s u transform on a broad range of test problems linearly convergence alternating monotone and irregular sign series logarithmically convergent series power method and bernoulli method sequences alternating and monotone asymptotic series and some perturba tion series arising in applications in each category either the algorithm or the u transform gives the best results of the four methods tested in some cases differences among methods are slight and in others they are quite striking 1 introduction we reported in 19 on a numerical test program that compared eleven different scalar convergence accelerators on a broad range of infinite series problems in an attempt to find an across the board accelerator that would handle both linear and logarithmic convergence and both alternating and monotone series or sequences the best candidates to emerge from that study were levin s u transform 14 and brezinski s 0 algorithm 6 we also provided some theoretical support for that conclusion in terms of the wide range of series summed exactly by these methods and by proving that 02 and the levin transforms all accelerate linear convergence the other levin transforms t and v produced essentially equivalent numerical results to u except that t fails on logarithmic convergence we have continued the test program started with 19 to include monotone linearly convergent series series with irregular sign patterns specifically fourier series and sequences whose terms do not depend explicitly on the previous term and or the index e g bernoulli s method since convergence accelerators are often known to transform divergent series to convergent ones we have also included in our test program divergent power series asymptotic series and a few perturbation series the methods considered in this paper are the u and 0 algorithms noted above and

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

exponential stability of discrete time neural networks with delay and impulses abstract in this paper we investigate the exponential stability of discrete time neural networks with impulses and time varying delay the discrete time neural networks are derived by discretizing the corresponding continuous time counterparts with different discretization methods the impulses are classified into three classes input disturbances stabilizing and neutral type the impulses are neither helpful for stabilizing nor destabilizing the neural networks and then by using the excellent ideology introduced recently by chen and zheng w h chen w x zheng global exponential stability of impulsive neural networks with variable delay an lmi approach ieee trans circuits syst i 56 6 2009 1248 1259 the connections between the impulses and the utilized lyapunov function are fully explored with respect to each type of impulse novel techniques that used to realize the ideology in discrete time situation are proposed and it is shown that they are essentially different from the continuous time case several criteria for global exponential stability of the discrete time neural networks are established in terms of matrix inequalities and based on these theoretical results numerical simulations are given to compare the capability of different discretization methods

euler classes and bredon cohomology for groups with restricted families of finite subgroups we explore some of the special features with respect to bredon cohomology for groups whose finite subgroups are all either nilpotent or p groups or cyclic p groups we get some results on dimensions and also a formula for the equivariant euler class for certain groups we consider the generalization for bredon cohomology of the properties of being duality or poincare duality and study their behavior under p power index extensions with coefficients in a field of characteristic p

global exponential stability of multitime scale competitive neural networks with nonsmooth functions in this paper we study the global exponential stability of a multitime scale competitive neural network model with nonsmooth functions which models a literally inhibited neural network with unsupervised hebbian learning the network has two types of state variables one corresponds to the fast neural activity and another to the slow unsupervised modification of connection weights based on the nonsmooth analysis techniques we prove the existence and uniqueness of equilibrium for the system and establish some new theoretical conditions ensuring global exponential stability of the unique equilibrium of the neural network numerical simulations are conducted to illustrate the effectiveness of the derived conditions in characterizing stability regions of the neural network

cohomological finiteness conditions for elementary amenable groups it is proved that every elementary amenable group of type fp admits a cocompact classifying space for proper actions

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

global stability for cellular neural networks with time delay a sufficient condition related to the existence of a unique equilibrium point and its global asymptotic stability for cellular network networks with delay dcnns is derived it is shown that the condition relies on the feedback matrices and is independent of the delay parameter furthermore this condition is less restrictive than that given in the literature

trigonometric approximation of periodic functions belonging to lip ω t p class analysis of signals or time functions are of great importance because it convey information or attributes of some phenomenon in this paper we consider the functions signals defined in terms of their integral modulus of continuity and approximate them by matrix means of their trigonometric fourier series the error of approximation obtained in this paper depends on the integral modulus of continuity of the signal we also discuss an example to show the application of the result in the situation when the fourier series of the signal has gibbs phenomenon we also discuss some corollaries of our theorems

robust synchronization of an array of coupled stochastic discrete time delayed neural networks this paper is concerned with the robust synchronization problem for an array of coupled stochastic discrete time neural networks with time varying delay the individual neural network is subject to parameter uncertainty stochastic disturbance and time varying delay where the norm bounded parameter uncertainties exist in both the state and weight matrices the stochastic disturbance is in the form of a scalar wiener process and the time delay enters into the activation function for the array of coupled neural networks the constant coupling and delayed coupling are simultaneously considered we aim to establish easy to verify conditions under which the addressed neural networks are synchronized by using the kronecker product as an effective tool a linear matrix inequality lmi approach is developed to derive several sufficient criteria ensuring the coupled delayed neural networks to be globally robustly exponentially synchronized in the mean square the lmi based conditions obtained are dependent not only on the lower bound but also on the upper bound of the time varying delay and can be solved efficiently via the matlab lmi toolbox two numerical examples are given to demonstrate the usefulness of the proposed synchronization scheme

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

a generalized lmi based approach to the global asymptotic stability of delayed cellular neural networks a novel linear matrix inequality lmi based criterion for the global asymptotic stability and uniqueness of the equilibrium point of a class of delayed cellular neural networks cnns is presented the criterion turns out to be a generalization and improvement over some previous criteria

common asymptotic behavior of solutions and almost periodicity for discontinuous delayed and impulsive neural networks the paper considers a general neural network model with impulses at a given sequence of instants discontinuous neuron activations delays and time varying data and inputs it is shown that when the neuron interconnections satisfy an m matrix condition or a dominance condition then the state solutions and the output solutions display a common asymptotic behavior as time t it is also shown via a new technique based on prolonging the solutions of the delayed neural network to that it is possible to select a unique special solution that is globally exponentially stable and can be considered as the unique global attractor for the network finally this paper shows that for almost periodic data and inputs the selected solution is almost periodic moreover it is robust with respect to a large class of perturbations of the data analogous results also hold for periodic data and inputs a by product of the analysis is that a sequence of almost periodic impulses is able to induce in the generic case nonstationary almost periodic solutions in an otherwise globally convergent nonimpulsive neural network to the authors knowledge the results in this paper are the only available results on global exponential stability of the unique periodic or almost periodic solution for a general neural network model combining three main features i e impulses discontinuous neuron activations and delays the results in this paper are compared with several results in the literature dealing with periodicity or almost periodicity of some subclasses of the neural network model here considered and some hints for future work are given

a proof that the discrete singular convolution dsc lagrange distributed approximating function ldaf method is inferior to high order finite differences finite differences approximate the mth derivative of a function u x by a series j n nd j m u x j where the x j are the grid points the closely related discrete singular convolution dsc and lagrange distributed approximating function ldaf methods treated here as a single algorithm approximate derivatives in the same way as finite differences but with different numerical weights that depend upon a free parameter a by means of fourier analysis and error theorems we show that the dsc is worse than the standard finite differences in differentiating exp ikx for all k when a a f d where a f d 1 n 1 with n as the stencil width is the value of the dsc parameter that makes its weights most closely resemble those of finite differences for a

a nonsmooth version of newton s method newton s method for solving a nonlinear equation of several variables is extended to a nonsmooth case by using the generalized jacobian instead of the derivative this extension includes the b derivative version of newton s method as a special case convergence theorems are proved under the condition of semismoothness it is shown that the gradient function of the augmented lagrangian forc2 nonlinear programming is semismooth thus the extended newton s method can be used in the augmented lagrangian method for solving nonlinear programs

convergence of newton s method for convex best interpolation in this paper we consider the problem of finding a convex function which interpolates given points and has a minimal l2 norm of the second derivative this problem reduces to a system of equations involving semismooth functions we study a newton type method utilizing clarke s generalized jacobian and prove that its local convergence is superlinear for a special choice of a matrix in the generalized jacobian we obtain the newton method proposed by irvine et al and settle the question of its convergence by using a line search strategy we present a global extension of the newton method considered the efficiency of the proposed global strategy is confirmed with numerical experiments

some groups of type vf a group is of type vf if it contains a finite index subgroup which has a finite classifying space we construct groups of type vf in which the centralizers of some elements of finite order are not of type vf and groups of type vf containing infinitely many conjugacy classes of finite subgroups from these examples it follows that a group g of type vf need not admit a finite type or finite classifying space for proper actions sometimes also called the universal proper g space we construct groups g for which the minimal dimension of a universal proper g space is strictly greater than the virtual cohomological dimension of g each of our groups embeds in some general linear group over the rational integers applications to algebraic k theory of group algebras and topological k theory of group c algebras are also considered the groups are constructed as finite extensions of bestvina brady groups

an analysis of global asymptotic stability of delayed cellular neural networks in this paper a new sufficient condition is given for the uniqueness and global asymptotic stability of the equilibrium point for delayed cellular neural networks dcnns this condition imposes constraints on the feedback and delayed feedback matrices of a dcnn independently of the delay parameter this result is also compared with the previous results derived in the literature

stability in asymmetric hopfield nets with transmission delays sufficient conditions are derived for the delay independent stability of the equilibria in hopfield s graded response networks of the type dxi t dt bixi t nj 1aijfj jxj t ij fi t i 1 2 n when the external inputs fi are held temporally uniform a generalization to continuously distributed delays is briefly indicated several illustrative examples are numerically simulated and the results of simulations are graphically displayed

optimal filter and mollifier for piecewise smooth spectral data we discuss the reconstruction of piecewise smooth data from its pseudo spectral information spectral projections enjoy superior resolution provided the function is globally smooth while the presence of jump discontinuities is responsible for spurious o 1 gibbs oscillations in the neighborhood of edges and an overall deterioration of the convergence rate to the unacceptable first order classical filters and mollifiers are constructed to have compact support in the fourier frequency and physical time spaces respectively and are dilated by the projection order or the width of the smooth region to maintain this compact support in the appropriate region here we construct a noncompactly supported filter and mollifier with optimal joint time frequency localization for a given number of vanishing moments resulting in a new fundamental dilation relationship that adaptively links the time and frequency domains not giving preference to either space allows for a more balanced error decomposition which when minimized yields an optimal filter and mollifier that retain the robustness of classical filters yet obtain true exponential accuracy