optimal control problem for nonlinear stochastic difference second kind volterra equations abstract the problems of stability and optimal control for stochastic difference equations are receiving important attention now see for example 1 6 in this paper the necessary optimality condition for nonlinear stochastic difference second kind volterra equation are constructed for stochastic integral functional equations analogous results were obtained in 7

monotone methods for a finite difference system of reaction diffusion equation with time delay abstract the aim of this paper is to present some monotone iterative schemes for computing the solution of a system of nonlinear difference equations which arise from a class of nonlinear reaction diffusion equations with time delays the iterative schemes lead to computational algorithms as well as existence uniqueness and upper and lower bounds of the solution an application to a diffusive logistic equation with time delay is given

computational methods for discrete boundary value problems on considere des methodes constructives pour les problemes aux valeurs limites discrets y t 1 g t y t t i n f y 0 ou y et g t y sont des vecteurs a n dimensions i n est l ensemble des points discrets 0 1 n f y est un operatuer de b i n dans r n et b i n est l espace de toutes les fonctions n vectorielles reelles definies sur i n

initial value methods for discrete boundary value problems motivated by the work of angel and kalaba s and golberg 8 on twopoint boundary value problems for difference equations in this paper we shall study several initial value methods for multipoint boundary value problems for linear and nonlinear difference systems we generalize method of adjoints goodman and lance 9 method of complementary functions tifford 17 d an method of particular solutions miele 111 proposed for two point continuous problems to linear multipoint problems for difference equations next we show that method of adjoint and method of complementary functions can be used in an iterative way to solve nonlinear problems to justify the results obtained here several examples are also given the methods discussed here are similar to our earlier work l 3 on continuous problems we will take up the initial adjusting method of ojika kasue and welsh 12 15 18 for the discrete case later throughout the paper we shall assume the existence uniqueness and stability properties of solutions of the problems discussed and refer to 4 for some of these questions

monotone enclosure for a class of discrete boundary value problems without monotone nonlinearities abstract a new concept of a pair of upper and lower solutions is introduced for a class of discrete boundary value problems without monotone nonlinearities some comparison results are established an existence and enclosing theorem for the solutions is given in terms of upper and lower solutions a new monotone iterative scheme is proposed the numerical example is given

new sufficient conditions for the stability of slowly varying linear systems the frozen time approach is used to state some new sufficient conditions for the stability of linear time varying systems an upper bound on the norm of the time derivative of system matrix which under different assumptions on frozen time system eigenvalues guarantees asymptotic stability or exponential stability of the system is established

stability and asymptoticity of volterra difference equations a progress report we survey some of the fundamental results on the stability and asymptoticity of linear volterra difference equations the method of z transform is heavily utilized in equations of convolution type an example is given to show that uniform asymptotic stability does not necessarily imply exponential stabilty it is shown that the two notions are equivalent if the kernel decays exponentially for equations of nonconvolution type liapunov functions are used to find explicit criteria for stability moreover the resolvent matrix is defined to produce a variation of constants formula the study of asymptotic equivalence for difference equations with infinite delay is carried out in section 6 finally we state some problems

the capacity of the semideterministic relay channel corresp the capacity of the class of relay channels with sender x 1 a relay sender x 2 a relay receiver y 1 f x 1 x 2 and ultimate receiver y is proved to be c max min p x 1 x 2 i x 1 x 2 y h y 1 x 2 i x 1 y x 2 y 1

stability for a class of difference equations we consider the following non autonomous and nonlinear difference equations with unbounded delays x i 1 qx i j ia i jf i j x j i 0 1 2 x j f j

computational methods for discrete boundary value problems ii abstract for a given nonlinear discrete system together with nonlinear boundary conditions we use newton s iterative scheme and provide sufficient conditions for its convergence to the unique solution then the computational aspects of this iterative method on a floating point system are discussed this includes necessary and sufficient conditions for the convergence of the approximate newton s method an oscillatory state and the stopping criterion to make the analysis widely applicable all the results are proved component wise

high dimensional asymptotics of prediction ridge regression and classification we provide a unified analysis of the predictive risk of ridge regression and regularized discriminant analysis in a dense random effects model we work in a high dimensional asymptotic regime where p n to infty and p n to gamma 0 and allow for arbitrary covariance among the features for both methods we provide an explicit and efficiently computable expression for the limiting predictive risk which depends only on the spectrum of the feature covariance matrix the signal strength and the aspect ratio gamma especially in the case of regularized discriminant analysis we find that predictive accuracy has a nuanced dependence on the eigenvalue distribution of the covariance matrix suggesting that analyses based on the operator norm of the covariance matrix may not be sharp our results also uncover an exact inverse relation between the limiting predictive risk and the limiting estimation risk in high dimensional linear models the analysis builds on recent advances in random matrix theory

a converse result for the discrete memoryless relay channel with relay transmitter feedback a converse result is proved for the discrete memoryless relay channel in the presence of relay transmitter causal feedback the result subsumes the cover el gamal max flow min cut upper bound on the capacity of the one way relay channel and it is used to establish the partial feedback capacity of the semideterministic relay channel

asymptotic stability of q h fractional difference equations abstract asymptotic stability of linear nabla riemann liouville q h fractional difference equation is investigated in this paper a liapunov functional is constructed for the fractional difference equation the sufficient condition for the asymptotic stability of considered equations is proposed the results are illustrated with the corresponding numerical examples

a stable numerical method for volterra integral equations with discontinuous kernel numerical methods for volterra integral equations with discontinuous kernel need to be tuned to their peculiar form here we propose a version of the trapezoidal direct quadrature method adapted to such a type of equations in order to delineate its stability properties we first investigate about the behavior of the solution of a suitable basic test equation and then we find out under which hypotheses the trapezoidal direct quadrature method provides numerical solutions which inherit the properties of the continuous problem

coding for the feedback gel fand pinsker channel and the feedforward wyner ziv source we consider both channel coding and source coding with perfect past feedback feedforward in the presence of side information it is first observed that feedback does not increase the capacity of the gel fand pinsker channel nor does feedforward improve the achievable rate distortion performance in the wyner ziv problem we then focus on the gaussian case showing that as in the absence of side information feedback feedforward allows to efficiently attain the respective performance limits in particular we derive schemes via variations on that of schalkwijk and kailath these variants which are as simple as their origin and require no binning are shown to achieve respectively the capacity of costa s channel and the wyner ziv rate distortion function finally we consider the finite alphabet setting and derive schemes for both the channel and the source coding problems that attain the fundamental limits using variations on schemes of ahlswede and ooi and wornell and of martinian and wornell respectively

boundary value problems for general discrete systems on infinite intervals this paper continues the study of boundary value problems for a discrete system which in particular includes the prototype equation x k 1 f k x k equations with finite as well as infinite delays equations of neutral type and the discrete integral equations of volterra type while the finite discrete interval case regular as well as at resonance has been discussed in 1 2 in the present paper the infinite interval case will be addressed

asymptotic behavior of volterra difference equations abstract in this paper we investigate boundedness and stability properties of some classes of discrete volterra equations the main tool in this work is the use of a representation formula which allows us to express the solution of discrete volterra equations in terms of the resolvent matrix of the corresponding system of volterra difference equations

on the relay channel with receiver transmitter feedback an achievable rate for the discrete memoryless relay channel with receiver transmitter feedback is proposed based on block markov superposition encoding the achievable rate can also be extended to gaussian channels a second achievable rate for the gaussian relay channel based on a schalkwijk kailath type scheme is presented for some channels both achievable rates strictly improve upon all previously known achievable rates for the discrete memoryless relay channel also a converse result is provided

stability analysis of discrete recurrence equations of volterra type with degenerate kernels abstract stability criteria are derived for difference equations of volterra type with degenerate kernels the main tool in this analysis is the use of the new representation formula which allows us to express the solution of discrete volterra equation with degenerate kernel in terms of the fundamental matrix of the corresponding first order system of the difference equations

on the distribution of the largest eigenvalue in principal components analysis let x 1 denote the square of the largest singular value of an n p matrix x all of whose entries are independent standard gaussian variates equivalently x 1 is the largest principal component variance of the covariance matrix x x or the largest eigenvalue of a p variate wishart distribution on n degrees of freedom with identity covariance consider the limit of large p and n with n p gamma ge 1 when centered by mu p sqrt n 1 sqrt p 2 and scaled by sigma p sqrt n 1 sqrt p 1 sqrt n 1 1 sqrt p 1 3 the distribution of x 1 approaches the tracey widom law of order 1 which is defined in terms of the painleve ii differential equation and can be numerically evaluated and tabulated in software simulations show the approximation to be informative for n and p as small as 5 the limit is derived via a corresponding result for complex wishart matrices using methods from random matrix theory the result suggests that some aspects of large p multivariate distribution theory may be easier to apply in practice than their fixed p counterparts

boundedness of discrete volterra equations abstract investigation of the boundedness of the solutions of volterra equations with discrete time by means of the direct liapunov method and comparison theorems is performed boundedness conditions of solutions of some particular classes of equations are derived using such conditions stability of direct quadrature methods for second kind volterra equations is analyzed

some recent developments of numerov s method abstract this paper is a survey of some recent developments of numerov s method for solving nonlinear two point boundary value problems the survey consists of three different parts the existence uniqueness of a solution computational algorithm for computing a solution and some extensions of numerov s method the sufficient conditions for the existence and uniqueness of a solution are presented some of them are best possible various iterative methods are reviewed including picard s iterative method modified newton s iterative method monotone iterative method and accelerated monotone iterative method in particular two more direct monotone iterative methods are presented to save computational work each of these iterative methods not only gives a computational algorithm for computing a solution but also leads to an existence and uniqueness theorem the estimate on the rate of convergence of the iterative sequence is given the extensions of numerov s method to a coupled problem and a general problem are addressed the numerical results are presented to validate the theoretical analysis

monotone methods for a boundary value problem of second order discrete equation abstract the comparison and existence results of the upper and lower solutions are established for a boundary value problem of second order discrete equation a monotone iterative scheme is proposed the local as well as global existence and uniqueness of the solution are obtained the global convergence of the iteration is investigated the convergence rate of the iteration is analyzed and compared some examples are given

asymptotic behavior of solutions for a lotka volterra mutualism reaction diffusion system with time delays this paper is to investigate the asymptotic behavior of solutions for a time delayed lotka volterra n species mutualism reaction diffusion system with homogeneous neumann boundary condition it is shown under a simple condition on the reaction rates that the system has a unique bounded time dependent solution and a unique constant positive steady state solution and for any nontrivial nonnegative initial function the corresponding time dependent solution converges to the constant positive steady state solution as time tends to infinity this convergence result implies that the trivial steady state solution and all forms of semitrivial steady state solutions are unstable and moreover the system has no nonconstant positive steady state solution a condition ensuring the convergence of the time dependent solution to one of nonnegative semitrivial steady state solutions is also given

invariant and attracting sets of volterra difference equations with delays will be concerned with the invariant and asymptotic properties of volterra differ ence equations with delays sufficient conditions for determining the invariant and attracting sets of the equations are obtained examples are given to illustrate the obtained results 2003 elsevier science ltd all rights reserved

linearized stability analysis of discrete volterra equations abstract various different types of stability are defined in a unified framework for discrete volterra equations of the type x n f n n j 0 k n j x n n 0 under appropriate assumptions stability results are obtainable from those valid in the linear case k n j x n b n j x j and a linearized stability theory is studied here by using the fundamental and resolvent matrices several necessary and sufficient conditions for stability are obtained for solutions of the linear equation by considering the equations in various choices of banach space b the elements of which are sequences of vectors x n f n e d b n j e d e d n j 0 etc we show that the theory including a number of new results as well as results already known can be presented in a systematic framework in which results parallel corresponding results for classical volterra integral equations of the second kind

variations on a theme by schalkwijk and kailath schalkwijk and kailath 1966 developed a class of block codes for gaussian channels with ideal feedback for which the probability of decoding error decreases as a second order exponent in block length for rates below capacity this well known but surprising result is explained and simply derived here in terms of a result by elias 1956 concerning the minimum mean square distortion achievable in transmitting a single gaussian random variable over multiple uses of the same gaussian channel a simple modification of the schalkwijk kailath scheme is then shown to have an error probability that decreases with an exponential order which is linearly increasing with block length in the infinite bandwidth limit this scheme produces zero error probability using bounded expected energy at all rates below capacity a lower bound on error probability for the finite bandwidth case is then derived in which the error probability decreases with an exponential order which is linearly increasing in block length at the same rate as the upper bound

accelerated monotone iterative methods for a boundary value problem of second order discrete equations abstract an accelerated monotone iterative method for a boundary value problem of second order discrete equations is presented this method leads to an existence comparison theorem as well as a computational algorithm for the solutions the monotone property of the iterations gives improved upper and lower bounds of the solution in each iteration and the rate of convergence of the iterations is either quadratic or nearly quadratic depending on the property of the nonlinear function some numerical results are presented to illustrate the monotone convergence of the iterative sequences and the rate of convergence of the iterations

on the number of positive solutions of a nonlinear algebraic system abstract in this paper we study the nonlinear algebraic system of the form e x af x where 0 is a parameter x and f x denote the column vectors col x 1 x 2 x n and col f 1 x 1 f 2 x 2 f n x n respectively with fk r r k 1 2 n 1 n and n is a positive integer a aij n n is an n n matrix and all its entries are positive numbers many problems in various areas such as difference equations boundary value problems dynamical networks stochastic process numerical analysis etc can be converted to system e applying fixed point theorems we prove results on existence uniqueness multiplicity and nonexistence of positive solutions for e

bounds on capacity and minimum energy per bit for awgn relay channels upper and lower bounds on the capacity and minimum energy per bit for general additive white gaussian noise awgn and frequency division awgn fd awgn relay channel models are established first the max flow min cut bound and the generalized block markov coding scheme are used to derive upper and lower bounds on capacity these bounds are never tight for the general awgn model and are tight only under certain conditions for the fd awgn model two coding schemes that do not require the relay to decode any part of the message are then investigated first it is shown that the side information coding scheme can outperform the block markov coding scheme it is also shown that the achievable rate of the side information coding scheme can be improved via time sharing in the second scheme the relaying functions are restricted to be linear the problem is reduced to a single letter nonconvex optimization problem for the fd awgn model the paper also establishes a relationship between the minimum energy per bit and capacity of the awgn relay channel this relationship together with the lower and upper bounds on capacity are used to establish corresponding lower and upper bounds on the minimum energy per bit that do not differ by more than a factor of 1 45 for the fd awgn relay channel model and 1 7 for the general awgn model

bootstrapping spectral statistics in high dimensions author s lopes me blandino a aue a abstract 2019 biometrika trust statistics derived from the eigenvalues of sample covariance matrices are called spectral statistics and they play a central role in multivariate testing although bootstrap methods are an established approach to approximating the laws of spectral statistics in low dimensional problems such methods are relatively unexplored in the high dimensional setting the aim of this article is to focus on linear spectral statistics as a class of prototypes for developing a new bootstrap in high dimensions a method we refer to as the spectral bootstrap in essence the proposed method originates from the parametric bootstrap and is motivated by the fact that in high dimensions it is difficult to obtain a nonparametric approximation to the full data generating distribution from a practical standpoint the method is easy to use and allows the user to circumvent the difficulties of complex asymptotic formulas for linear spectral statistics in addition to proving the consistency of the proposed method we present encouraging empirical results in a variety of settings lastly and perhaps most interestingly we show through simulations that the method can be applied successfully to statistics outside the class of linear spectral statistics such as the largest sample eigenvalue and others

stability of backward differentiation formulas for volterra integro differential equations the numerical stability of the backward differentiation methods for linear systems of volterra integro differential equations with convolution kernel whose logarithmic norm is nonpositive is analyzed

a survey stability and boundedness of volterra difference equations abstract volterra difference equations arise in the modeling of many real phenomena in this survey stability and boundedness problems of some volterra nonlinear difference equations are investigated stability conditions and boundedness are formulated in terms of the characteristics of equations

stability of difference volterra equations direct liapunov method and numerical procedure abstract a procedure to construct liapunov functionals for discrete volterra equations is proposed using this procedure stability conditions are derived for general volterra difference equations some applications of the proposed procedure for obtaining stability conditions for linear multistep methods for volterra integro differential equations are presented

on discrete boundary value problems arising in transport phenomena for the discrete boundary value problems arising in transport processes we provide comparison results these results are used to develop monotone iterative methods for the construction of the maximal and minimal solutions in a sector the advantage of this technique is that the successive approximations are the solutions of the initial and terminal value problems numerical illustration showing the sharpness as well the importance of the obtained results is also included

stability of a class of delay difference equations on montre que les proprietes de stabilite a la fois locales et globales de la solution d equilibre positive du systeme a retard n i 1 s i n i t 1 s i g i n t h i peuvent se deduire de celles du systeme sans retard associe n i t 1 g i n t

a modified accelerated monotone iterative method for finite difference reaction diffusion convection equations this paper is concerned with monotone algorithms for the finite difference solutions of a class of nonlinear reaction diffusion convection equations with nonlinear boundary conditions a modified accelerated monotone iterative method is presented to solve the finite difference systems for both the time dependent problem and its corresponding steady state problem this method leads to a simple and yet efficient linear iterative algorithm it yields two sequences of iterations that converge monotonically from above and below respectively to a unique solution of the system the monotone property of the iterations gives concurrently improving upper and lower bounds for the solution it is shown that the rate of convergence for the sum of the two sequences is quadratic under an additional requirement quadratic convergence is attained for one of these two sequences in contrast with the existing accelerated monotone iterative methods our new method avoids computing local maxima in the construction of these sequences an application using a model problem gives numerical results that illustrate the effectiveness of the proposed method

finite difference solutions of reaction diffusion equations with continuous time delays abstract this paper is an extension of the monotone iterative methods for finite difference equations with discrete time delays to a class of nonlinear finite difference system with continuous time delays the system under consideration is a finite difference approximation of a class of reaction diffusion equations with continuous time delays in the nonlinear reaction under either dirichlet or neumann robin boundary conditions various monotone iterative schemes which depend on the property of the nonlinear reaction mechanism are developed for the computation of numerical solutions it is shown by the method of upper and lower solutions that the two sequences obtained from each iterative scheme converge monotonically from above and below respectively to a unique solution of the finite difference system applications are given to two model problems known as the diffusive logistic equation and the fisher s diffusion equation in population genetics

monotone iterative methods for a general class of discrete boundary value problems abstract in this paper we shall offer comparison results as well as monotone iterative schemes for the construction of solutions to a very general class of discrete boundary value problems the discrete system we consider includes in particular the n th order prototype systems finite as well as infinite discrete delay equations and discrete integral equations further the boundary conditions we consider include the initial terminal periodic and transport type problems numerical examples illustrating the usefulness of the proposed schemes to a variety of boundary value problems are also included

parallel monotone iterative relaxation methods for a class of two dimensional discrete boundary value problems abstract a parallel monotone iterative relaxation method for a class of two dimensional discrete boundary value problems is established and the sequence of iterations is shown to converge monotonically either from above or below to a solution of the problem this monotone convergence results yields a parallel computational algorithm as well as an existence comparison result for the solutions to compute the sequence of iterations the thomas algorithm can be used in the same fashion as for one dimensional problem the existence and comparison results of the upper and lower solutions are given the local as well as global existence uniqueness of the solution are obtained the global convergence of the iterations is investigated and the influence of the parameters on the rate of convergence of the iterations is analyzed numerical results are given to corroborate the analytical results

on nonlinear discrete boundary value problems mo x 1 x n 0 1 4 where n is a fixed positive integer 8 represent n by n constant matrices a t is an n by n matrix for each te 0 l h t belongs to ir for each nonnegative integer t u is in r 1 is a real parameter f and g are continuously differentiable f r i ir and g r i r for systems 1 1 1 2 we prove that if the linear system x t l a l x t h t 1 5 subject to

an algorithm for approximate solving of differential equations with maxima an algorithm for constructing two sequences of successive approximations of the solution of the initial value problem for nonlinear differential equations with maxima is given this algorithm is based on the monotone iterative technique it is proved that both sequences are monotonically convergent each term of the constructed sequences is a solution of an initial value problem for linear differential equations with maxima and it is a lower upper solution of the given problem both the scalar case and the multidimensional case are studied an example solved by computer realization of the suggested algorithm illustrates the practical application of the method

discrete systems on infinite intervals abstract we study superlinear boundary value problems for a discrete system which in particular includes the prototype equation x k 1 f k x k equations with finite as well as infinite delays equations of neutral type and the discrete integral equations of volterra type

stability of a general class of difference systems abstract in this paper we shall investigate several notions of stability of a general ordinary difference system using the prolific second method of lyapunov

a perspective on the numerical treatment of volterra equations abstract we discuss the properties and numerical treatment of various types of volterra and abel volterra integral and integro differential equations

periodicity and stability of linear volterra difference systems abstract we give sufficient conditions for a system of volterra difference equations to have a unique periodic solution then a criterion for uniform asymptotic stability is established

periodic solutions of a single species discrete population model with periodic harvest stock abstract we discuss a discrete population model describing single species growth with periodic harvest stock the theory of coincidence degree is applied to show that the model equation admits two periodic solutions under minor technical assumptions we show that one of these two periodic solutions is positive and attracts almost all positive solutions

general theorems for stability and boundedness for nonlinear functional discrete systems abstract we consider the nonlinear functional discrete system x n 1 g n x s 0 s n def g n x and obtain sufficient conditions for stability properties of the zero solution and for the boundedness of solutions

the ceo problem multiterminal source coding we consider a new problem in multiterminal source coding motivated by the following decentralized communication estimation task a firm s chief executive officer ceo is interested in the data sequence x t sub t 1 sup spl infin which cannot be observed directly perhaps because it represents tactical decisions by a competing firm the ceo deploys a team of l agents who observe independently corrupted versions of x t sub t 1 sup spl infin because x t is only one among many pressing matters to which the ceo must attend the combined data rate at which the agents may communicate information about their observations to the ceo is limited to say r bits per second if the agents were permitted to confer and pool their data then in the limit as l spl rarr spl infin they usually would be able to smooth out their independent observation noises entirely then they could use their r bits per second to provide the ceo with a representation of x t with fidelity d r where d spl middot is the distortion rate function of x t in particular with such data pooling d can be made arbitrarily small if r exceeds the entropy rate h of x t suppose however that the agents are not permitted to convene agent i having to send data based solely on his own noisy observations y sub i t we show that then there does not exist a finite value of r for which even infinitely many agents can make d arbitrarily small furthermore in this isolated agents case we determine the asymptotic behavior of the minimal error frequency in the limit as l and then r tend to infinity