dynamical stability conditions for recurrent neural networks with unsaturating piecewise linear transfer functions we establish two conditions that ensure the nondivergence of additive recurrent networks with unsaturating piecewise linear transfer functions also called linear threshold or semilinear transfer functions as hahnloser sarpeshkar mahowald douglas and seung 2000 showed networks of this type can be efficiently built in silicon and exhibit the coexistence of digital selection and analog amplification in a single circuit to obtain this behavior the network must be multistable and nondivergent and our conditions allow determining the regimes where this can be achieved with maximal recurrent amplification the first condition can be applied to nonsymmetric networks and has a simple interpretation of requiring that the strength of local inhibition match the sum over excitatory weights converging onto a neuron the second condition is restricted to symmetric networks but can also take into account the stabilizing effect of nonlocal inhibitory interactions we demonstrate the application of the conditions on a simple example and the orientation selectivity model of ben yishai lev bar or and sompolinsky 1995 we show that the conditions can be used to identify in their model regions of maximal orientation selective amplification and symmetry breaking

risk sensitive and risk neutral multiarmed bandits for the multiarmed bandit the classic result is probabilistic each state of each bandit markov chain with rewards has an index that is determined by an optimal stopping time for that state s bandit and expected discounted income is maximized by playing at each epoch a bandit whose current state has the largest index our approach is analytic not probabilistic it uses pairwise comparison in place of stopping times a simple recursion assigns to each state of each bandit a utility and an amplification of future utility that depend solely on the data for that state s bandit these utilities and amplifications determine whether or not one state dominates another we show that it is optimal to play at each epoch any bandit whose current state is not dominated by the current states of the other bandits we obtain this result by a coherent analysis that encompasses three models one with risk averse exponential utility one with risk seeking exponential utility and one with linear utility and either stopping or discounting we also show that the risk seeking case and a model of nash nash p 1980 a generalized bandit problem j roy statist soc b42 165 169 are equivalent to each other

permitted and forbidden sets in discrete time linear threshold recurrent neural networks the concepts of permitted and forbidden sets enable a new perspective of the memory in neural networks such concepts exhibit interesting dynamics in recurrent neural networks this paper studies the basic theories of permitted and forbidden sets of the linear threshold discrete time recurrent neural networks the linear threshold transfer function has been regarded as an adequate transfer function for recurrent neural networks networks with this transfer function form a class of hybrid analog and digital networks which are especially useful for perceptual computations networks in discrete time can directly provide algorithms for efficient implementation in digital hardware the main contribution of this paper is to establish foundations of permitted and forbidden sets necessary and sufficient conditions for the linear threshold discrete time recurrent neural networks are obtained for complete convergence existence of permitted and forbidden sets as well as conditionally multiattractivity respectively simulation studies explore some possible interesting practical applications

a convex analytic approach to risk aware markov decision processes in classical markov decision process mdp theory we search for a policy that say minimizes the expected infinite horizon discounted cost expectation is of course a risk neutral measure which does not suffice in many applications particularly in finance we replace the expectation with a general risk functional and call such models risk aware mdp models we consider minimization of such risk functionals in two cases the expected utility framework and conditional value at risk a popular coherent risk measure later we consider risk aware mdps wherein the risk is expressed in the constraints this includes stochastic dominance constraints and the classical chance constrained optimization problems in each case we develop a convex analytic approach to solve such risk aware mdps in most cases we show that the problem can be formulated as an infinite dimensional linear program lp in occupation measures when we augment the state space we provide a discretization method and finite approximati

collocation methods for index 1 daes with a singularity of the first kind we study the convergence behavior of collocation schemes applied to approximate solutions of bvps in linear index 1 daes which exhibit a critical point at the left boundary such a critical point of the dae causes a singularity within the inherent ode system we focus our attention on the case when the inherent ode system is singular with a singularity of the first kind apply polynomial collocation to the original dae system and consider different choices of the collocation points such as equidistant gaussian or radau points we show that for a well posed boundary value problem for daes having a sufficiently smooth solution the global error of the collocation scheme converges with the order o h s where s is the number of collocation points superconvergence cannot be expected in general due to the singularity not even for the differential components of the solution the theoretical results are illustrated by numerical experiments

fixed point attractor analysis for a class of neurodynamics nearly all models in neural networks start from the assumption that the input output characteristic is a sigmoidal function on parameter space we present a systematic and feasible method for analyzing the whole spectrum of attractors all saturated all but one saturated all but twosaturated and so on of a neurodynamical system with a saturated sigmoidal function as its input output characteristic we present an argument that claims under a mild condition that only all saturated or all but one saturated attractors are observable for the neurodynamics for any given all saturated configuration all but one saturated configuration the article shows how to construct an exact parameter region r such that if and only if the parameters fall within r then is an attractor a fixed point of the dynamics the parameter region for an all saturated fixed point attractor is independent of the specific choice of a saturated sigmoidal function whereas for an all but one saturated fixed point it

generalization errors of the simple perceptron to find an exact form for the generalization error of a learning machine is an open problem even in the simplest case simple perceptron learning we introduce a new approach to tackle the problem the generalization error of the simple perceptron is expressed as a linear combination of extreme values of inputs with the help of extreme value theory in statistics we then obtain an exact form of the generalization error of the simple perceptron in the case of the worst learning generalization errors of the higher order perceptron taking the form of an inverse power law in the number of examples are also considered

markov decision processes with iterated coherent risk measures this paper considers a markov decision process in borel state and action spaces with the aggregated or say iterated coherent risk measure to be minimised for this problem we establish the bellman optimality equation as well as the value and policy iteration algorithms and show the existence of a deterministic stationary optimal policy the cost function while being allowed to be unbounded from below in the sense that its negative part needs be bounded by some nonnegative real valued possibly unbounded weight function can be arbitrarily unbounded from above and possibly infinitely valued

nonzero sum risk sensitive stochastic games on a countable state space the infinite horizon risk sensitive discounted cost and ergodic cost nonzero sum stochastic games for controlled markov chains with countably many states are analyzed for the discounted cost game we prove the existence of nash equilibrium strategies in the class of markov strategies under fairly general conditions under an additional weak geometric ergodicity condition and a small cost criterion the existence of nash equilibrium strategies in the class of stationary markov strategies is proved for the ergodic cost game the key nontrivial contributions in the ergodic part are to prove the existence of a particular form of a relative value function solution to a player s bellman equation and the continuity of this solution with respect to the opponent s strategies

multistability analysis of quaternion valued neural networks with time delays this paper addresses the multistability issue for quaternion valued neural networks qvnns with time delays by using the inequality technique sufficient conditions are proposed for the boundedness and the global attractivity of delayed qvnns based on the geometrical properties of the activation functions several criteria are obtained to ensure the existence of 81 n equilibrium points 16 n of which are locally stable two numerical examples are provided to illustrate the effectiveness of the obtained results

continuous attractors of a class of neural networks with a large number of neurons a class of simplified background neural networks model with a large number of neurons is proposed continuous attractors of the simplified model are studied in this paper it contains 1 when the background inputs are set to zero and the excitatory connections are in gaussian shape continuous attractors of the new network are obtained under some condition 2 when the background inputs are nonzero and the excitatory connections are still in gaussian shape continuous attractors are achieved under some appropriately selected condition 3 discussions and examples are used to illustrate the theories developed

the convergence of shooting methods for singular boundary value problems we investigate the convergence properties of single and multiple shooting when applied to singular boundary value problems particular attention is paid to the well posedness of the process it is shown that boundary value problems can be solved efficiently when a high order integrator for the associated singular initial value problems is available moreover convergence results for a perturbed newton iteration are discussed

multistability of two kinds of recurrent neural networks with activation functions symmetrical about the origin on the phase plane in this paper we investigate multistability of two kinds of recurrent neural networks with time varying delays and activation functions symmetrical about the origin on the phase plane one kind of activation function is with zero slope at the origin on the phase plane while the other is with nonzero slope at the origin on the phase plane we derive sufficient conditions under which these two kinds of n dimensional recurrent neural networks are guaranteed to have 2m 1 n equilibrium points with m 1 n of them being locally exponentially stable these new conditions improve and extend the existing multistability results for recurrent neural networks finally the validity and performance of the theoretical results are demonstrated through two numerical examples

a note on the sturmian theorem for singular boundary value problems in this paper we study properties of the eigenfunctions of singular boundary value problems the main results show that for a class of singular boundary value problems the sturmian oscillation theorem holds it is a generalization of the standard case to the singular case

numerical solution of singular boundary value problems with logarithmic singularities by pade approximation and collocation methods in this paper we present a method using quintic b spline and pade approximation for numerically solving singular boundary value problems for certain ordinary differential equation having singular coefficient we use the modified quintic b spline to approximate the first coefficients in the taylor expansion on which all the other coefficients depend moreover we introduce the superposed pade approximation the resulting perturbed initial value problem is then efficiently treated

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

activity invariant sets and exponentially stable attractors of linear threshold discrete time recurrent neural networks this technical note proposes to study the activity invariant sets and exponentially stable attractors of linear threshold discrete time recurrent neural networks the concept of activity invariant sets deeply describes the property of an invariant set by that the activity of some neurons keeps invariant all the time conditions are obtained for locating activity invariant sets under some conditions it shows that an activity invariant set can have one equilibrium point which attracts exponentially all trajectories starting in the set since the attractors are located in activity invariant sets each attractor has binary pattern and also carries analog information such results can provide new perspective to apply attractor networks for applications such as group winner take all associative memory etc

qualitative behaviour of some simple networks some classical neural network systems including the hartline ratliff system the linsker system and the general sigmoid dynamics are reconsidered within a more general class of dynamical systems for synchronous dynamics the existence uniqueness local and global stability of stationary points is investigated for asynchronous dynamics a convergence theorem is proved the application of the theory of quasimonotone flows leads to some insights so far not widespread in network theory

nontrivial global attractors in 2 d multistable attractor neural networks attractor dynamics is a crucial problem for attractor neural networks as it is the underling computational mechanism for memory storage and retrieval in neural systems this brief studies a class of attractor network consisting of linearized threshold neurons and analyzes global attractors based on a parameterized 2 d model on the basis of previous results on nondegenerate and degenerate equilibria in mathematics we further elucidate all possible nontrivial global attractors our theoretical result provides precise descriptions on how the changes of network parameters affect the attractors distribution and landscape and it may give a feasible solution towards specifying attractors by specifying weights simulations are presented to illustrate the theoretical results

monotone iterative techniques for time dependent problems with applications abstract in this paper we establish an abstract monotone iterative scheme for time dependent problems with or without impulses using the theory of semigroups of operators and the method of upper and lower solutions applications to population growth models and impulsive reaction diffusion systems are also discussed

capacity of the hopfield model for a given if neurons deviating from the memorized patterns are allowed we constructively show that if and only if all stored patterns are fixed points of the hopfield model if neurons are allowed with then where is the distribution function of the normal distribution the result obtained by amit and co workers only formally coincides with the latter case which indicates that the replica trick approach to the capacity of the hopfield model is only valid in the case

an analysis on neural dynamics with saturated sigmoidal functions we propose a unified approach to study the relation between the set of saturated attractors and the set of system parameters of the hopfield model linsker s model and the dynamic link network dln which use saturated sigmoidal functions in its dynamics of the state or weight the key point for this approach is to rigorously derive a necessary and sufficient condition to test whether a given saturated state in the hopfield model or weight vector in linsker s model and the dln is stable or not for any given set of system parameters and used this to determine the complete regime in the parameter space over which the given state or weight is stable our approach allows us to give an exact characterization between the parameters and the capacity in the hopfield model to generalize our previous results on linsker s network and the dln to have a better understanding of the underlying mechanism among these models the method reported here could be adopted to analyze a variety of models in the field of the neural networks

a numerical study for multiple solutions of a singular boundary value problem arising from laminar flow in a porous pipe with moving wall abstract this paper is concerned with multiple solutions of a singular nonlinear boundary value problem bvp on the interval 0 1 which arises in a study of the laminar flow in a porous pipe with an expanding or contracting wall for the singular nonlinear bvp the correct boundary conditions are derived to guarantee that its linearization has a unique smooth solution then a numerical technique is proposed to find all possible multiple solutions for the suction driven pipe flow with the expanding wall e g 2 we find a new solution numerically and classify it as a type vi solution the computed results agree well with what can be obtained by the bifurcation package auto in addition we also construct asymptotic solutions for a few cases of parameters which agree well with numerical solutions these serve as validations of our numerical results thus we believe that the numerical technique designed in the paper is reliable and may be further applied to solve a variety of nonlinear equations that arise from other flow problems

limit set dichotomy and multistability for a class of cooperative neural networks with delays recent papers have pointed out the interest to study convergence in the presence of multiple equilibrium points eps multistability for neural networks nns with nonsymmetric cooperative nonnegative interconnections and neuron activations modeled by piecewise linear pl functions one basic difficulty is that the semiflows generated by such nns are monotone but due to the horizontal segments in the pl functions are not eventually strongly monotone esm this notwithstanding it has been shown that there are subclasses of irreducible interconnection matrices for which the semiflows although they are not esm enjoy convergence properties similar to those of esm semiflows the results obtained so far concern the case of cooperative nns without delays the goal of this paper is to extend some of the existing results to the relevant case of nns with delays more specifically this paper considers a class of nns with pl neuron activations concentrated delays and a nonsymmetric cooperative interconnection matrix a and delay interconnection matrix a the main result is that when a a satisfies a full interconnection condition then the generated semiflows which are monotone but not esm satisfy a limit set dichotomy analogous to that valid for esm semiflows it follows that there is an open and dense set of initial conditions in the state space of continuous functions on a compact interval for which the solutions converge toward an ep the result holds in the general case where the nns possess multiple eps i e is a result on multistability and is valid for any constant value of the delays

a recurrent neural network for nonlinear optimization with a continuously differentiable objective function and bound constraints this paper presents a continuous time recurrent neural network model for nonlinear optimization with any continuously differentiable objective function and bound constraints quadratic optimization with bound constraints is a special problem which can be solved by the recurrent neural network the proposed recurrent neural network has the following characteristics 1 it is regular in the sense that any optimum of the objective function with bound constraints is also an equilibrium point of the neural network if the objective function to be minimized is convex then the recurrent neural network is complete in the sense that the set of optima of the function with bound constraints coincides with the set of equilibria of the neural network 2 the recurrent neural network is primal and quasiconvergent in the sense that its trajectory cannot escape from the feasible region and will converge to the set of equilibria of the neural network for any initial point in the feasible bound region 3 the recurrent neural network has an attractivity property in the sense that its trajectory will eventually converge to the feasible region for any initial states even at outside of the bounded feasible region 4 for minimizing any strictly convex quadratic objective function subject to bound constraints the recurrent neural network is globally exponentially stable for almost any positive network parameters simulation results are given to demonstrate the convergence and performance of the proposed recurrent neural network for nonlinear optimization with bound constraints

risk sensitive control with hara utility a control methodology based on the hyperbolic absolute risk averse hara utility function is presented as an alternative to the exponential of an integral approach to finding robust controllers the work is inspired by the intuition that hara controllers while being robust may give better performance than exponential controllers in normal situations the hara problem is shown to be equivalent to a certain differential game and the asymptotic properties of the hara problem and this differential game are studied as an example a linear quadratic hara problem is studied where the problem of finding a robust hara controller is proved to be equivalent to solving a standard linear quadratic problem for a system with a higher noise intensity this reveals an interesting relationship between robustness and uncertainty

boundedness and complete stability of complex valued neural networks with time delay in this paper the boundedness and complete stability of complex valued neural networks cvnns with time delay are studied some conditions to guarantee the boundedness of the cvnns are derived using local inhibition moreover under the boundedness conditions a compact set that globally attracts all the trajectories of the network is also given additionally several conditions in terms of real valued linear matrix inequalities lmis for complete stability of the cvnns are established via the energy minimization method and the approach that converts the complex valued lmis to real valued ones examples with simulation results are given to show the effectiveness of the theoretical analysis

closed form expressions for the moments of the binomial probability distribution this work develops closed form expressions for the raw and central moments of the binomial probability distribution for this i first derive a recursive formula for the raw moments from the moment generating function then it is shown that the recursion involved is essentially the same as for the stirling numbers of the second kind from this fact it is then possible to derive the closed formulae finally i discuss an application of these formulae to the analysis of neural associative memory

multistability of discrete time recurrent neural networks with unsaturating piecewise linear activation functions this paper studies the multistability of a class of discrete time recurrent neural networks with unsaturating piecewise linear activation functions it addresses the nondivergence global attractivity and complete stability of the networks using the local inhibition conditions for nondivergence are derived which not only guarantee nondivergence but also allow for the existence of multiequilibrium points under these nondivergence conditions global attractive compact sets are obtained complete stability is studied via constructing novel energy functions and using the well known cauchy convergence principle examples and simulation results are used to illustrate the theory

the application of runge kutta schemes to singular initial value problems a theory for explicit runge kutta schemes applied to the initial value problem for a first order system of differential equations with a singularity of the first kind is developed it is shown that in general the order of convergence is at most two but that the usual order up to a logarithmic term can be obtained for level three and four schemes applied to specific problems 1 introduction in this paper we examine the application of explicit runge kutta schemes to the numerical solution of the problem

selectable and unselectable sets of neurons in recurrent neural networks with saturated piecewise linear transfer function the concepts of selectable and unselectable sets are proposed to describe some interesting dynamical properties of a class of recurrent neural networks rnns with saturated piecewise linear transfer function a set of neurons is said to be selectable if it can be co unsaturated at a stable equilibrium point by some external input a set of neurons is said to be unselectable if it is not selectable i e such set of neurons can never be co unsaturated at any stable equilibrium point regardless of what the input is the importance of such concepts is that they enable a new perspective of the memory in rnns necessary and sufficient conditions for the existence of selectable and unselectable sets of neurons are obtained as an application the problem of group selection is discussed by using such concepts it shows that under some conditions each group is a selectable set and each selectable set is contained in some group thus groups are indicated by selectable sets of the rnns and can be selected by external inputs simulations are carried out to further illustrate the theory

global exponential convergence of recurrent neural networks with variable delays convergence analysis of recurrent neural networks is an important research direction in the field of neural networks novel methods to study the global exponential convergence of recurrent neural networks with variable delays are proposed a condition for global exponential stability which is independent of the delays is derived by the method of delayed inequalities analysis another condition for global exponential stability which depends on the delays is obtained via the method of constructing a suitable and interesting lyapunov functional

analysis of cyclic dynamics for networks of linear threshold neurons the network of neurons with linear threshold lt transfer functions is a prominent model to emulate the behavior of cortical neurons the analysis of dynamic properties for lt networks has attracted growing interest such as multistability and boundedness however not much is known about how the connection strength and external inputs are related to oscillatory behaviors periodic oscillation is an important characteristic that relates to nondivergence which shows that the network is still bounded although unstable modes exist by concentrating on a general parameterized two cell network theoretical results for geometrical properties and existence of periodic orbits are presented although it is restricted to two dimensional systems the analysis can provide a useful contribution to analyze cyclic dynamics of some specific lt networks of high dimension as an application it is extended to an important class of biologically motivated networks of large scale the winner take all model using local excitation and global inhibition

piecewise affine bijections of rn and the equation sx tx y we prove two criteria for bijectivity of piecewise affine maps f of rn it is assumed that det f has the same nonzero sign everywhere the first criterion is based solely on the branching properties of the partition of rn into domains of linearity of f the second one concerns the map itself and depends on the images under f of certain two dimensional angles this generalizes and improves results of r schramm the methods are topological our results imply k g murty s solution of the linear complementarity problem and some other partly known facts which we show are equivalent to it for example the map x sx tx is bijective iff all principal minors of s 1t are strictly positive

the vanishing discount approach in markov chains with risk sensitive criteria in this paper stochastic dynamic systems are studied modeled by a countable state space markov cost reward chain satisfying a lyapunov type stability condition for an infinite planning horizon risk sensitive exponential discounted and average cost criteria are considered the main contribution is the development of a vanishing discount approach to relate the discounted criterion problem with the average criterion one as the discount factor increases to one i e no discounting in comparison to the well established risk neutral case our results are novel and reveal several fundamental and surprising differences other contributions made include the use of convex analytic arguments to obtain appropriately convergent sequences and a verification theorem for the case of unbounded solutions to the average cost poisson equation arising in the risk sensitive case also of importance is the fact that our developments are very much self contained and employ only basic probabilistic and analysis principles

on neurodynamics with limiter function and linsker s developmental model the limiter function is used in many learning and retrieval models as the constraint controlling the magnitude of the weight or state vectors in this paper we developed a new method to relate the set of saturated fixed points to the set of system parameters of the models that use the limiter function and then as a case study applied this method to linsker s hebbian learning network we derived a necessary and sufficient condition to test whether a given saturated weight or state vector is stable or not for any given set of system parameters and used this condition to determine the whole regime in the parameter space over which the given state is stable this approach allows us to investigate the relative stability of the major receptive fields reported in linsker s simulations and to demonstrate the crucial role played by the synaptic density functions

discounted mdp s distribution functions and exponential utility maximization the present value of the rewards associated with a discrete time markov process has a probability distribution which depends on the initial state the first part of the paper applies fixed point theory to a system of equations for the distribution functions of the present value the second part of the paper expands the model to a markov decision process mdp and considers the maximization of the expected utility of the present value when the utility function is exponential

equilibria and their bifurcations in a recurrent neural network involving iterates of a transcendental function some practical models contain so complicated mathematical expressions that it is hard to determine the number and distribution of all equilibria not mentioning the qualitative properties and bifurcations of those equilibria the three node recurrent neural network system with two free weight parameters originally introduced by ruiz owens and townley in 1997 is such a system for which the equation of equilibria involves transcendental function and its iterates not computing coordinates of its equilibria in this paper we display an effective technique to determine the number and distribution of its equilibria without full information about equilibria our method enables to further study qualitative properties of those equilibria and discuss their saddle node pitchfork and hopf bifurcations by approximating center manifolds

stochastic optimization of forward recursive functions this note solves a finite horizon stochastic optimization problem with forward recursive criterion through dynamic programming the forward recursive criterion is wide it includes additive discounted multiplicative discounted risk sensitive minimum and terminal criteria the basic idea is to apply invariant imbedding method for the stochastic optimization the method incorporates recursive accumulation process into dynamics by expanding the original state space

attractivity analysis of memristor based cellular neural networks with time varying delays this paper presents new theoretical results on the invariance and attractivity of memristor based cellular neural networks mcnns with time varying delays first sufficient conditions to assure the boundedness and global attractivity of the networks are derived using state space decomposition and some analytic techniques it is shown that the number of equilibria located in the saturation regions of the piecewise linear activation functions of an n neuron mcnn with time varying delays increases significantly from 2n to 22n2 n 22n2 times compared with that without a memristor in addition sufficient conditions for the invariance and local or global attractivity of equilibria or attractive sets in any designated region are derived finally two illustrative examples are given to elaborate the characteristics of the results in detail

a characterization of the optimal certainty equivalent of the average cost via the arrow pratt sensitivity function this work is concerned with finite state irreducible markov decision chains satisfying continuity compactness requirements it is supposed that the system is driven by a decision maker with utility function u which aside mild conditions is arbitrary and the performance of a control policy is measured by the long run average cost criterion induced by u the main conclusions about this performance index are as follows i the optimal u average value function coincides with the optimal v average index for a certain exponential utility v and ii the average criteria associated with u and v have the same class of optimal stationary policies

iterated defect correction for the solution of singular initial value problems we investigate the convergence properties of the iterated defect correction idec method based on the implicit euler rule for the solution of singular initial value problems with a singularity of the first kind we show that the method retains its classical order of convergence which means that the sequence of approximations obtained during the iteration shows gradually growing order of convergence limited by the smoothness of the data and technical details of the procedure

average cost markov decision processes with weakly continuous transition probabilities this paper presents sufficient conditions for the existence of stationary optimal policies for average cost markov decision processes with borel state and action sets and weakly continuous transition probabilities the one step cost functions may be unbounded and the action sets may be noncompact the main contributions of this paper are i general sufficient conditions for the existence of stationary discount optimal and average cost optimal policies and descriptions of properties of value functions and sets of optimal actions ii a sufficient condition for the average cost optimality of a stationary policy in the form of optimality inequalities and iii approximations of average cost optimal actions by discount optimal actions

eigenspectrum bounds for semirandom matrices with modular and spatial structure for neural networks the eigenvalue spectrum of the matrix of directed weights defining a neural network model is informative of several stability and dynamical properties of network activity existing results for eigenspectra of sparse asymmetric random matrices neglect spatial or other constraints in determining entries in these matrices and so are of partial applicability to cortical like architectures here we examine a parameterized class of networks that are defined by sparse connectivity with connection weighting modulated by physical proximity i e asymmetric euclidean random matrices modular network partitioning and functional specificity within the excitatory population we present a set of analytical constraints that apply to the eigenvalue spectra of associated weight matrices highlighting the relationship between connectivity rules and classes of network dynamics

partially observable risk sensitive markov decision processes we consider the problem of minimizing a certainty equivalent of the total or discounted cost over a finite and an infinite time horizon that is generated by a partially observable markov decision process pomdp in contrast to a risk neutral decision maker this optimization criterion takes the variability of the cost into account it contains as a special case the classical risk sensitive optimization criterion with an exponential utility we show that this optimization problem can be solved by embedding the problem into a completely observable markov decision process with extended state space and give conditions under which an optimal policy exists the state space has to be extended by the joint conditional distribution of current unobserved state and accumulated cost in case of an exponential utility the problem simplifies considerably and we rediscover what in previous literature has been named information state however since we do not use any change of measure techniques here our approach is s

global exponential stability of neural networks with globally lipschitz continuous activations and its application to linear variational inequality problem this paper investigates the existence uniqueness and global exponential stability ges of the equilibrium point for a large class of neural networks with globally lipschitz continuous activations including the widely used sigmoidal activations and the piecewise linear activations the provided sufficient condition for ges is mild and some conditions easily examined in practice are also presented the ges of neural networks in the case of locally lipschitz continuous activations is also obtained under an appropriate condition the analysis results given in the paper extend substantially the existing relevant stability results in the literature and therefore expand significantly the application range of neural networks in solving optimization problems as a demonstration we apply the obtained analysis results to the design of a recurrent neural network rnn for solving the linear variational inequality problem vip defined on any nonempty and closed box set which includes the box constrained quadratic programming and the linear complementarity problem as the special cases it can be inferred that the linear vip has a unique solution for the class of lyapunov diagonally stable matrices and that the synthesized rnn is globally exponentially convergent to the unique solution some illustrative simulation examples are also given

discounted approximations for risk sensitive average criteria in markov decision chains with finite state space this work concerns markov decision processes with finite state space and compact action set the performance of a control policy is measured by a risk sensitive average cost criterion and under standard continuity compactness conditions it is shown that the discounted approximations converge to the optimal value function and that the superior and inferior limit average criteria have the same optimal value function these conclusions are obtained for every nonnull risk sensitivity coefficient and regardless of the communication structure induced by the transition law

more risk sensitive markov decision processes we investigate the problem of minimizing a certainty equivalent of the total or discounted cost over a finite and an infinite horizon that is generated by a markov decision process mdp in contrast to a risk neutral decision maker this optimization criterion takes the variability of the cost into account it contains as a special case the classical risk sensitive optimization criterion with an exponential utility we show that this optimization problem can be solved by an ordinary mdp with extended state space and give conditions under which an optimal policy exists in the case of an infinite time horizon we show that the minimal discounted cost can be obtained by value iteration and can be characterized as the unique solution of a fixed point equation using a sandwich argument interestingly it turns out that in the case of a power utility the problem simplifies and is of similar complexity than the exponential utility case however has not been treated in the literature so far we also establish the validity and convergence of the policy improvement method a simple numerical example namely the classical repeated casino game is considered to illustrate the influence of the certainty equivalent and its parameters finally the average cost problem is also investigated surprisingly it turns out that under suitable recurrence conditions on the mdp for convex power utility the minimal average cost does not depend on the parameter of the utility function and is equal to the risk neutral average cost this is in contrast to the classical risk sensitive criterion with exponential utility

process based risk measures and risk averse control of discrete time systems for controlled discrete time stochastic processes we introduce a new class of dynamic risk measures which we call process based their main feature is that they measure risk of processes that are functions of the history of a base process we introduce a new concept of conditional stochastic time consistency and we derive the structure of process based risk measures enjoying this property we show that they can be equivalently represented by a collection of static law invariant risk measures on the space of functions of the state of the base process we apply this result to controlled markov processes and we derive dynamic programming equations we also derive dynamic programming equations for multistage stochastic programming with decision dependent distributions

multiperiodicity and attractivity of delayed recurrent neural networks with unsaturating piecewise linear transfer functions this paper studies multiperiodicity and attractivity for a class of recurrent neural networks rnns with unsaturating piecewise linear transfer functions and variable delays using local inhibition conditions for boundedness and global attractivity are established these conditions allow coexistence of stable and unstable trajectories moreover multiperiodicity of the network is investigated by using local invariant sets it shows that under some interesting conditions there exists one periodic trajectory in each invariant set which exponentially attracts all trajectories in that region correspondingly simulations are carried out to illustrate the theories

finite state approximations to discounted and average cost constrained markov decision processes in this paper we consider the finite state approximation of a discrete time constrained markov decision process mdp under the discounted and average cost criteria using the linear programming formulation of the constrained discounted cost problem we prove the asymptotic convergence of the optimal value of the finite state model to the optimal value of the original model with further continuity condition on the transition probability we also establish a method to compute approximately optimal policies for the average cost instead of using the finite state linear programming approximation method we use the original problem definition to establish the finite state asymptotic approximation of the constrained problem and compute approximately optimal policies under lipschitz type regularity conditions on the components of the mdp we also obtain explicit rate of convergence bounds quantifying how the approximation improves as the size of the approximating finite state space increases

selectively grouping neurons in recurrent networks of lateral inhibition winner take all networks have been proposed to underlie many of the brain s fundamental computational abilities however not much is known about how to extend the grouping of potential winners in these networks beyond single neuron or uniformly arranged groups of neurons we show that competition between arbitrary groups of neurons can be realized by organizing lateral inhibition in linear threshold networks given a collection of potentially overlapping groups with the exception of some degenerate cases the lateral inhibition results in network dynamics such that any permitted set of neurons that can be coactivated by some input at a stable steady state is contained in one of the groups the information about the input is preserved in this operation the activity level of a neuron in a permitted set corresponds to its stimulus strength amplified by some constant sets of neurons that are not part of a group cannot be coactivated by any input at a stable steady state we analyze the storage capacity of such a network for random groups the number of random groups the network can store as permitted sets without creating too many spurious ones in this framework we calculate the optimal sparsity of the groups maximizing group entropy we find that for dense inputs the optimal sparsity is unphysiologically small however when the inputs and the groups are equally sparse we derive a more plausible optimal sparsity we believe our results are the first steps toward attractor theories in hybrid analog digital networks

risk sensitive control of discrete time markov processes with infinite horizon in this paper we study existence of solutions to the bellman equation corresponding to risk sensitive ergodic control of discrete time markov processes using three different approaches also for particular classes of systems asymptotics for vanishing risk factor is investigated showing that in the limit the optimal value for an average cost per unit time is obtained

discounted approximations to the risk sensitive average cost in finite markov chains abstract this work concerns with markov chains on a finite state space which is endowed with a cost function the evolution of the chain is observed by an agent with constant risk sensitivity and assuming that the state space is a communicating class the relation between the risk sensitive discounted and average performance criteria is studied it is proved that as the discount factor increases to 1 an appropriate normalization of the discounted value function converges to the average cost also it is shown that if the classical normalization used in the risk neutral case is applied in the risk sensitive context then the normalized discounted value function converges to an arithmetic mean of the average cost

risk averse dynamic programming for markov decision processes we introduce the concept of a markov risk measure and we use it to formulate risk averse control problems for two markov decision models a finite horizon model and a discounted infinite horizon model for both models we derive risk averse dynamic programming equations and a value iteration method for the infinite horizon problem we develop a risk averse policy iteration method and we prove its convergence we also propose a version of the newton method to solve a nonsmooth equation arising in the policy iteration method and we prove its global convergence finally we discuss relations to min max markov decision models

multistability analysis for recurrent neural networks with unsaturating piecewise linear transfer functions multistability is a property necessary in neural networks in order to enable certain applications e g decision making where monostable networks can be computationally restrictive this article focuses on the analysis of multistability for a class of recurrent neural networks with unsaturating piecewise linear transfer functions it deals fully with the three basic properties of a multistable network boundedness global attractivity and complete convergence this article makes the following contributions conditions based on local inhibition are derived that guarantee boundedness of some multistable networks conditions are established for global attractivity bounds on global attractive sets are obtained complete convergence conditions for the network are developed using novel energy like functions and simulation examples are employed to illustrate the theory thus developed

foundations of implementing the competitive layer model by lotka volterra recurrent neural networks the competitive layer model clm can be described by an optimization problem the problem can be further formulated by an energy function called the clm energy function in the subspace of nonnegative orthant the set of minimum points of the clm energy function forms the set of solutions of the clm problem solving the clm problem means to find out such solutions recurrent neural networks rnns can be used to implement the clm to solve the clm problem the key point is to make the set of minimum points of the clm energy function just correspond to the set of stable attractors of the recurrent neural networks this paper proposes to use lotka volterra rnns lv rnns to implement the clm the contribution of this paper is to establish foundations of implementing the clm by lv rnns the contribution mainly contains three parts the first part is on the clm energy function necessary and sufficient conditions for minimum points of the clm energy function are established by detailed study the second part is on the convergence of the proposed model of the lv rnns it is proven that interesting trajectories are convergent the third part is the most important it proves that the set of stable attractors of the proposed lv rnn just equals the set of minimum points of the clm energy function in the nonnegative orthant thus the lv rnns can be used to solve the problem of the clm it is believed that by establishing such basic rigorous theories more and interesting applications of the clm can be found

efficient mesh selection for collocation methods applied to singular bvps we describe a mesh selection strategy for the numerical solution of boundary value problems for singular ordinary differential equations this mesh adaptation procedure is implemented in our matlab code sbvp which is based on polynomial collocation we prove that under realistic assumptions our mesh selection strategy serves to approximately equidistribute the global error of the collocation solution thus enabling to reach prescribed tolerances efficiently moreover we demonstrate that this strategy yields a favorable performance of the code and compare its computational effort with other implementations of polynomial collocation

a novel approach to the numerical solution of boundary value problems on infinite intervals the classical numerical treatment of two point boundary value problems on infinite intervals is based on the introduction of a truncated boundary instead of infinity where appropriate boundary conditions are imposed then the truncated boundary allowing for a satisfactory accuracy is computed by trial motivated by several problems of interest in boundary layer theory here we consider boundary value problems on infinite intervals governed by a third order ordinary differential equation we highlight a novel approach to define the truncated boundary the main result is the convergence of the solution of our formulation to the solution of the original problem as a suitable parameter goes to zero in the proposed formulation the truncated boundary is an unknown free boundary and has to be determined as part of the solution for the numerical solution of the free boundary formulation a noniterative and an iterative transformation method are introduced furthermore we characterize the class of free boundary value problems that can be solved noniteratively a nonlinear flow problem involving two physical parameters and belonging to the characterized class of problems is then solved moreover the falkner skan equation with relevant boundary conditions is considered and representative results obtained by the iterative transformation method are listed for the homann flow all the obtained numerical results clearly indicate the effectiveness of our approach finally we discuss the possible extensions of the proposed approach and for the question of a priori error analysis

minimizing risk models in markov decision processes with policies depending on target values abstract this paper studies the minimizing risk problems in markov decision processes with countable state space and reward set the objective is to find a policy which minimizes the probability risk that the total discounted rewards do not exceed a specified value target in this sort of model the decision made by the decision maker depends not only on system s states but also on his target values by introducing the decision maker s state we formulate a framework for minimizing risk models the policies discussed depend on target values and the rewards may be arbitrary real numbers for the finite horizon model the main results obtained are i the optimal value functions are distribution functions of the target ii there exists an optimal deterministic markov policy and iii a policy is optimal if and only if at each realizable state it always takes optimal action in addition we obtain a sufficient condition and a necessary condition for the existence of finite horizon optimal policy independent of targets and we give an algorithm computing finite horizon optimal policies and optimal value functions for an infinite horizon model we establish the optimality equation and we obtain the structure property of optimal policy we prove that the optimal value function is a distribution function of target and we present a new approximation formula which is the generalization of the nonnegative rewards cases an example which illustrates the mistakes of previous literature shows that the existence of optimal policy has not been proved really in this paper we give an existence condition which is a sufficient and necessary condition for the existence of an infinite horizon optimal policy independent of targets and we point out that whether there exists an optimal policy remains an open problem in the general case