adaptive control design based on adaptive optimization principles recently we introduced an adaptive control design for linearly parameterized multi input nonlinear systems admitting a known control lyapunov function clf that depends on the unknown system parameters the main advantage of that design is that it overcomes the problem where the estimation model becomes uncontrollable at regions of the state space where the actual system is controllable however the resulted adaptive control design is quite complicated and moreover it exhibited poor transient behavior in various applications in this technical note we propose and analyze a new computationally efficient adaptive control design that overcomes the aforementioned shortcomings the proposed design is based on an adaptive optimization algorithm introduced recently by the author which makes sure that the parameters to be optimized which correspond to the controller parameters in this technical note are modified so as to both lead to a decrease of the function to be minimized and satisfy a persistence of excitation condition the main advantage of the proposed adaptive control design is that it can produce arbitrarily good transient performance outside a the regions of the state space where the actual system becomes uncontrollable and b a region of the parameter estimates space which shrinks exponentially fast it is also worth noting that the class of systems where the proposed algorithm is applicable is more general than that of our previous work however it has to be emphasized that due to the fact that the proposed algorithm involves the use of random sequences all the established stability and convergence results are guaranteed to hold with probability one

neural networks for functional approximation and system identification we construct generalized translation networks to uniformly approximate a class of nonlinear continuous functionals defined on l p 1 1 s for integer s 1 1 p 1 or c 1 1 s we obtain lower bounds on the possible order of approximation for such functionals in terms of any approximation process depending continuously upon a given number of parameters our networks almost achieve this order of approximation in terms of the number of parameters neurons involved in the network the training is simple and noniterative in particular we avoid any optimization such as that involved in the usual back propagation

longest fault free paths in star graphs with vertex faults the star graph sn has been recognized as an attractive alternative to the hypercube since s1 s2 and s3 have trivial structures we focus our attention on sn with n 4 in this paper let fv denote the set of faulty vertices in sn we show that when fv n 5 sn with n 6 contains a fault free path of length n 2 fv 2 n 2 fv 1 between arbitrary two vertices of even odd distance since sn is bipartite with two partite sets of equal size the path is longest for the worst case scenario the situation of n 4 and fv n 5 is also discussed

on the partitioning capabilities of feedforward neural networks with sigmoid nodes in this letter the capabilities of feedforward neural networks fnns on the realization and approximation of functions of the form g rl a which partition the rl space into polyhedral sets each one being assigned to one out of the c classes of a are investigated more specifically a constructive proof is given for the fact that fnns consisting of nodes having sigmoid output functions are capable of approximating any function g with arbitrary accuracy also the capabilities of fnns consisting of nodes having the hard limiter as output function are reviewed in both cases the two class as well as the multiclass cases are considered

hyper hamiltonian laceability on edge fault star graph the star graph possess many nice topological properties edge fault tolerance is an important issue for a network since the edges in the network may fail sometimes in this paper we show that the n dimensional star graph is n 3 edge fault tolerant hamiltonian laceable n 3 edge fault tolerant strongly hamiltonian laceable and n 4 edge fault tolerant hyper hamiltonian laceable all these results are optimal in a sense described in this paper

rate of convergence of basic multivariate neural network operators to the unit this chapter deals with the determination of the rate of convergence to the unit of some multivariate neural network operators namely the normalized bell and squashing type operators

universal approximation to nonlinear operators by neural networks with arbitrary activation functions and its application to dynamical systems the purpose of this paper is to investigate neural network capability systematically the main results are 1 every tauber wiener function is qualified as an activation function in the hidden layer of a three layered neural network 2 for a continuous function in s r sup 1 to be a tauber wiener function the necessary and sufficient condition is that it is not a polynomial 3 the capability of approximating nonlinear functionals defined on some compact set of a banach space and nonlinear operators has been shown and 4 the possibility by neural computation to approximate the output as a whole not at a fixed point of a dynamical system thus identifying the system

univariate fuzzy random neural network approximation operators in this article we study the rate of pointwise convergence in the q mean to thefuzzy random unit operator of very precise univariate fuzzy random neural network operators of cardaliaguet euvrard and squashing types these fuzzy random operators arise in a natural and common way among fuzzy random neural networks these rates are given through probabilistic jackson type inequalities involving the fuzzy random modulus of continuity of the engaged fuzzy random function or its fuzzy derivatives also several interesting new results in fuzzy random analysis are given of independent merit which are used then in the proofs of the main results of the paper

rate of convergence of basic neural network operators to the unit univariate case this chapter deals with the determination of the rate of convergence to the unit of some neural network operators namely the normalized bell and squashing type operators

multivariate adaptive regression splines a new method is presented for flexible regression modeling of high dimensional data the model takes the form of an expansion in product spline basis functions where the number of basis functions as well as the parameters associated with each one product degree and knot locations are automatically determined by the data this procedure is motivated by the recursive partitioning approach to regression and shares its attractive properties unlike recursive partitioning however this method produces continuous models with continuous derivatives it has more power and flexibility to model relationships that are nearly additive or involve interactions in at most a few variables in addition the model can be represented in a form that separately identifies the additive contributions and those associated with the different multivariable interactions

brief paper an adaptive optimization scheme with satisfactory transient performance adaptive optimization ao schemes based on stochastic approximation principles such as the random directions kiefer wolfowitz rdkw the simultaneous perturbation stochastic approximation spsa and the adaptive fine tuning aft algorithms possess the serious disadvantage of not guaranteeing satisfactory transient behavior due to their requirement for using random or random like perturbations of the parameter vector the use of random or random like perturbations may lead to particularly large values of the objective function which may result to severe poor performance or stability problems when these methods are applied to closed loop controller optimization applications in this paper we introduce and analyze a new algorithm for alleviating this problem mathematical analysis establishes satisfactory transient performance and convergence of the proposed scheme under a general set of assumptions application of the proposed scheme to the adaptive optimization of a large scale complex control system demonstrates the efficiency of the proposed scheme

quivers and matrix equations abstract we discuss methods for solving some familiar matrix equations the methods were derived from the theory of quivers a powerful tool in the representation theory of algebras it is not necessary that the reader understand quivers but we include a brief explanation for those who may be interested

degree of approximation results for feedforward networks approximating unknown mappings and their derivatives recently barron 1993 has given rates for hidden layer feedforward networks with sigmoid activation functions approximating a class of functions satisfying a certain smoothness condition these rates do not depend on the dimension of the input space we extend barron s results to feedforward networks with possibly nonsigmoid activation functions approximating mappings and their derivatives simultaneously our conditions are similar but not identical to barron s but we obtain the same rates of approximation showing that the approximation error decreases at rates as fast as n 1 2 where n is the number of hidden units the dimension of the input space appears only in the constants of our bounds

neural networks for optimal approximation of smooth and analytic functions we prove that neural networks with a single hidden layer are capable of providing an optimal order of approximation for functions assumed to possess a given number of derivatives if the activation function evaluated by each principal element satisfies certain technical conditions under these conditions it is also possible to construct networks that provide a geometric order of approximation for analytic target functions the permissible activation functions include the squashing function 1 e x 1 as well as a variety of radial basis functions our proofs are constructive the weights and thresholds of our networks are chosen independently of the target function we give explicit formulas for the coefficients as simple continuous linear functionals of the target function

universal approximation of multiple nonlinear operators by neural networks recently there has been interest in the observed capabilities of some classes of neural networks with fixed weights to model multiple nonlinear dynamical systems while this property has been observed in simulations open questions exist as to how this property can arise in this article we propose a theory that provides a possible mechanism by which this multiple modeling phenomenon can occur

hierarchical mixtures of experts and the em algorithm we present a tree structured architecture for supervised learning the statistical model underlying the architecture is a hierarchical mixture model in which both the mixture coefficients and the mixture components are generalized linear models glim s learning is treated as a maximum likelihood problem in particular we present an expectation maximization em algorithm for adjusting the parameters of the architecture we also develop an on line learning algorithm in which the parameters are updated incrementally comparative simulation results are presented in the robot dynamics domain

approximation by network operators with logistic activation functions this paper aims to study the construction and multivariate approximation of a class of network operators with logistic sigmoidal functions first a class of even and bell shaped function with support on r is constructed by using appropriate translation and combination of the logistic function then the constructed function is employed as activation function to construct a kind of so called cardaliaguet euvrard type network operators finally these network operators are used to approximate bivariate functions in c 1 1 2 and a jackson type theorem for the approximation errors is established

wiener s tauberian theorems for the fourier bessel transformation and uniform approximation by rbf networks of delsarte translates abstract we prove analogues of the wiener and wiener pitt tauberian theorems for the delsarte translation and the fourier bessel transformation through similar functional analytic techniques we give conditions on the kernel function or activation function for the family of radial basis function rbf neural networks obtained upon replacing the usual translation by the delsarte one with not necessarily the same smoothing factor in all kernel nodes to have the universal approximation property in a certain space of continuous functions endowed with the uniform norm

rbf based technique for statistical demodulation of pathological tremor this paper presents an innovative technique based on the joint approximation capabilities of radial basis function rbf networks and the estimation capability of the multivariate iterated hilbert transform iht for the statistical demodulation of pathological tremor from electromyography emg signals in patients with parkinson s disease we define a stochastic model of the multichannel high density surface emg by means of the rbf networks applied to the reconstruction of the stochastic process characterizing the disease modeled by the multivariate relationships generated by the karhunen loeve transform in hilbert spaces next we perform a demodulation of the entire random field by means of the estimation capability of the multivariate iht in a statistical setting the proposed method is applied to both simulated signals and data recorded from three parkinsonian patients and the results show that the amplitude modulation components of the tremor oscillation can be estimated with signal to noise ratio close to 30 db with root mean square error for the estimates of the tremor instantaneous frequency additionally the comparisons with a large number of techniques based on all the combinations of the rbf extreme learning machine backpropagation support vector machine used in the first step of the algorithm and iht empirical mode decomposition multiband energy separation algorithm periodic algebraic separation and energy demodulation used in the second step of the algorithm clearly show the effectiveness of our technique these results show that the proposed approach is a potential useful tool for advanced neurorehabilitation technologies that aim at tremor characterization and suppression

a constrained backpropagation approach for the adaptive solution of partial differential equations this paper presents a constrained backpropagation cprop methodology for solving nonlinear elliptic and parabolic partial differential equations pdes adaptively subject to changes in the pde parameters or external forcing unlike existing methods based on penalty functions or lagrange multipliers cprop solves the constrained optimization problem associated with training a neural network to approximate the pde solution by means of direct elimination as a result cprop reduces the dimensionality of the optimization problem while satisfying the equality constraints associated with the boundary and initial conditions exactly at every iteration of the algorithm the effectiveness of this method is demonstrated through several examples including nonlinear elliptic and parabolic pdes with changing parameters and nonhomogeneous terms

a simultaneous decomposition for seven matrices with applications abstract let h m n be the set of all m n matrices over the quaternion algebra h in this paper we construct a simultaneous decomposition for seven matrices with compatible sizes a h m n b h m p 1 c h m p 2 d h m p 3 e h q 1 n f h q 2 n and g h q 3 n as applications of the simultaneous matrix decomposition we give some solvability conditions general solutions as well as the range of ranks of the general solutions to the following two generalized sylvester matrix equations b x e c y f d z g a and b x w e c y f d z g a where a b c d e f and g are given quaternion matrices moreover we provide some numerical examples to illustrate the results of this paper

approximation of conditional densities by smooth mixtures of regressions this paper shows that large nonparametric classes of conditional multivariate densities can be approximated in the kullback leibler distance by different specifications of finite mixtures of normal regressions in which normal means and variances and mixing probabilities can depend on variables in the conditioning set covariates these models are a special case of models known as mixtures of experts in statistics and computer science literature flexible specifications include models in which only mixing probabilities modeled by multinomial logit depend on the covariates and in the univariate case models in which only means of the mixed normals depend flexibly on the covariates modeling the variance of the mixed normals by flexible functions of the covariates can weaken restrictions on the class of the approximable densities obtained results can be generalized to mixtures of general location scale densities rates of convergence and easy to interpret bounds are also obtained for different model specifications these approximation results can be useful for proving consistency of bayesian and maximum likelihood density estimators based on these models the results also have interesting implications for applied researchers

clf based control design for unknown multiinput nonlinear systems with good transient performance this technical note tackles the problem of constructing state feedback stabilizers that guarantee good transient closed loop performance when applied to general unknown nonlinear multiinput state feedback stabilizable systems an adaptive control lyapunov function based control scheme is proposed in order to address this problem mathematical analysis establishes that the proposed control scheme guarantees good closed loop transient performance outside the regions where the system is uncontrollable provided the controlled system admits a controllability like property

smooth function approximation using neural networks an algebraic approach for representing multidimensional nonlinear functions by feedforward neural networks is presented in this paper the approach is implemented for the approximation of smooth batch data containing the function s input output and possibly gradient information the training set is associated to the network adjustable parameters by nonlinear weight equations the cascade structure of these equations reveals that they can be treated as sets of linear systems hence the training process and the network approximation properties can be investigated via linear algebra four algorithms are developed to achieve exact or approximate matching of input output and or gradient based training sets their application to the design of forward and feedback neurocontrollers shows that algebraic training is characterized by faster execution speeds and better generalization properties than contemporary optimization techniques

approximation by neural networks with scattered data in this paper the approximation capability of feed forward neural networks fnns formed as k 1 nc k f x x k is considered where f is an even periodic continuous function and x k s are scattered data the best approximation error of trigonometric polynomials and the continuous modulus of the activation function f are introduced to describe the approximation capability of fnns

error bounds for functional approximation and estimation using mixtures of experts we examine some mathematical aspects of learning unknown mappings with the mixture of experts model mem specifically we observe that the mem is at least as powerful as a class of neural networks in a sense that will be made precise upper bounds on the approximation error are established for a wide class of target functions the general theorem states that spl par f f sub n spl par sub p spl les c n sup r d for f spl isin w sub p sup r l a sobolev class over 1 1 sup d and f sub n belongs to an n dimensional manifold of normalized ridge functions the same bound holds for the mem as a special case of the above the stochastic error in the context of learning from independent and identically distributed i i d examples is also examined an asymptotic analysis establishes the limiting behavior of this error in terms of certain pseudo information matrices these results substantiate the intuition behind the mem and motivate applications

longest fault free paths in star graphs with edge faults in this paper we aim to embed longest fault free paths in an n dimensional star graph with edge faults when n spl ges 6 and there are n 3 edge faults a longest fault free path can be embedded between two arbitrary distinct vertices exclusive of two exceptions in which at most two vertices are excluded since the star graph is regular of degree n 1 n 3 edge faults is maximal in the worst case when n spl ges 6 and there are n 4 edge faults a longest fault free path can be embedded between two arbitrary distinct vertices the situation of n 6 is also discussed

accuracy analysis for wavelet approximations constructive wavelet networks are investigated as a universal tool for function approximation the parameters of such networks are obtained via some direct monte carlo procedures approximation bounds are given typically it is shown that such networks with one layer of wavelons achieve an l sub 2 error of order o n sup spl rhod where n is the number of nodes d is the problem dimension and spl rho is the number of summable derivatives of the approximated function an algorithm is also proposed to estimate this approximation based on noisy input output data observed from the function under consideration unlike neural network training this estimation procedure does not rely on stochastic gradient type techniques such as the celebrated backpropagation and it completely avoids the problem of poor convergence or undesirable local minima

interpolation by neural network operators activated by ramp functions abstract in this paper a family of interpolation neural network operators are introduced here ramp functions as well as sigmoidal functions generated by central b splines are considered as activation functions the interpolation properties of these operators are proved together with a uniform approximation theorem with order for continuous functions defined on bounded intervals the relations with the theory of neural networks and with the theory of the generalized sampling operators are discussed

a system of quaternary coupled sylvester type real quaternion matrix equations abstract in this paper we establish some necessary and sufficient solvability conditions for a system of quaternary coupled sylvester type real quaternion matrix equations in terms of ranks and generalized inverses of matrices an expression of the general solution to this system is given when it is solvable also a numerical example is presented to illustrate the main result of this paper the findings of this paper widely generalize the known results in the literature the main results are also valid over the real number field and the complex number field

risk bounds for model selection via penalization performance bounds for criteria for model selection are devel oped using recent theory for sieves the model selection criteria are based on an empirical loss or contrast function with an added penalty term moti vated by empirical process theory and roughly proportional to the number of parameters needed to describe the model divided by the number of ob servations most of our examples involve density or regression estimation settings and we focus on the problem of estimating the unknown density or regression function we show that the quadratic risk of the minimum penal ized empirical contrast estimatoris bounded by an index of the accuracy of the sieve this accuracy index quanties the trade off among the candidate models between the approximation error and parameter dimension relative to sample size if we choose a list of models which exhibit good approximation prop erties with respect to different classes of smoothness the estimator can be simultaneously minimax rate optimal in each of those classes this is what is usually called adaptation the type of classes of smoothness in which one gets adaptation depends heavily on the list of models if too many models are involved in order to get accurate approximation of many wide classes of functions simultaneously it may happen that the estimator is only approx

complete memory structures for approximating nonlinear discrete time mappings this paper introduces a general structure that is capable of approximating input output maps of nonlinear discrete time systems the structure is comprised of two stages a dynamical stage followed by a memoryless nonlinear stage a theorem is presented which gives a simple necessary and sufficient condition for a large set of structures of this form to be capable of modeling a wide class of nonlinear discrete time systems in particular we introduce the concept of a complete memory a structure with a complete memory dynamical stage and a sufficiently powerful memoryless stage is shown to be capable of approximating arbitrarily wide class of continuous causal time invariant approximately finite memory mappings between discrete time signal spaces furthermore we show that any bounded input bounded output time invariant causal memory structure has such an approximation capability if and only if it is a complete memory several examples of linear and nonlinear complete memories are presented the proposed complete memory structure provides a template for designing a wide variety of artificial neural networks for nonlinear spatiotemporal processing

the errors of simultaneous approximation of multivariate functions by neural networks there have been many studies on the simultaneous approximation capability of feed forward neural networks fnns most of these however are only concerned with the density or feasibility of performing simultaneous approximations this paper considers the simultaneous approximation of algebraic polynomials employing taylor expansion and an algebraic constructive approach to construct a class of fnns which realize the simultaneous approximation of any smooth multivariate function and all of its derivatives we also present an upper bound on the approximation accuracy of the fnns expressed in terms of the modulus of continuity of the functions to be approximated

fault free longest paths in star networks with conditional link faults the star network which belongs to the class of cayley graphs is one of the most versatile interconnection networks for parallel and distributed computing in this paper adopting the conditional fault model in which each node is assumed to be incident with two or more fault free links we show that an n dimensional star network can tolerate up to 2n 7 link faults and be strongly fault free hamiltonian laceable where n 4 in other words we can embed a fault free linear array of length n 1 n 2 in an n dimensional star network with up to 2n 7 link faults if the two end nodes belong to different partite sets the same partite set the result is optimal with respect to the number of link faults tolerated it is already known that under the random fault model an n dimensional star network can tolerate up to n 3 faulty links and be strongly hamiltonian laceable for n 3

approximation bounds for smooth functions in c r sup d by neural and mixture networks we consider the approximation of smooth multivariate functions in c r sup d by feedforward neural networks with a single hidden layer of nonlinear ridge functions under certain assumptions on the smoothness of the functions being approximated and on the activation functions in the neural network we present upper bounds on the degree of approximation achieved over the domain r sup d thereby generalizing available results for compact domains we extend the approximation results to the so called mixture of expert architecture which has received considerable attention in recent years showing that the same type of approximation bound may be achieved

approximation of nonlinear systems with radial basis function neural networks a technique for approximating a continuous function of n variables with a radial basis function rbf neural network is presented the method uses an n dimensional raised cosine type of rbf that is smooth yet has compact support the rbf network coefficients are low order polynomial functions of the input a simple computational procedure is presented which significantly reduces the network training and evaluation time storage space is also reduced by allowing for a nonuniform grid of points about which the rbfs are centered the network output is shown to be continuous and have a continuous first derivative when the network is used to approximate a nonlinear dynamic system the resulting system is bounded input bounded output stable for the special case of a linear system the rbf network representation is exact on the domain over which it is defined and it is optimal in terms of the number of distinct storage parameters required several examples are presented which illustrate the effectiveness of this technique

generalized single hidden layer feedforward networks for regression problems in this paper traditional single hidden layer feedforward network slfn is extended to novel generalized slfn gslfn by employing polynomial functions of inputs as output weights connecting randomly generated hidden units with corresponding output nodes the significant contributions of this paper are as follows 1 a primal gslfn p gslfn is implemented using randomly generated hidden nodes and polynomial output weights whereby the regression matrix is augmented by full or partial input variables and only polynomial coefficients are to be estimated 2 a simplified gslfn s gslfn is realized by decomposing the polynomial output weights of the p gslfn into randomly generated polynomial nodes and tunable output weights 3 both p and s gslfn are able to achieve universal approximation if the output weights are tuned by ridge regression estimators and 4 by virtue of the developed batch and online sequential ridge elm br elm and osr elm learning algorithms high performance of the proposed gslfns in terms of generalization and learning speed is guaranteed comprehensive simulation studies and comparisons with standard slfns are carried out on real world regression benchmark data sets simulation results demonstrate that the innovative gslfns using br elm and osr elm are superior to standard slfns in terms of accuracy training speed and structure compactness

solving nonlinear equality constrained multiobjective optimization problems using neural networks this paper develops a neural network architecture and a new processing method for solving in real time the nonlinear equality constrained multiobjective optimization problem necmop where several nonlinear objective functions must be optimized in a conflicting situation in this processing method the necmop is converted to an equivalent scalar optimization problem sop the sop is then decomposed into several separable subproblems processable in parallel and in a reasonable time by multiplexing switched capacitor circuits the approach which we propose makes use of a decomposition coordination principle that allows nonlinearity to be treated at a local level and where coordination is achieved through the use of lagrange multipliers the modularity and the regularity of the neural networks architecture herein proposed make it suitable for very large scale integration implementation an application to the resolution of a physical problem is given to show that the approach used here possesses some advantages of the point of algorithmic view and provides processes of resolution often simpler than the usual techniques

near embeddings of hypercubes into cayley graphs on the symmetric group simulations of hypercube networks by certain cayley graphs on the symmetric group are investigated let q k be the familiar k dimensional hypercube and let s n be the star network of dimension n defined as follows the vertices of s n are the elements of the symmetric group of degree n two vertices x and y being adjacent if xo 1 i y for some i that is xy is an edge if x and y are related by a transposition involving some fixed symbol which we take to be spl i bold 1 this network has nice symmetry properties and its degree and diameter are sublogarithmic as functions of the number of vertices making it compare favorably with the hypercube network these advantages of s n motivate the study of how well it can simulate other parallel computation networks in particular the hypercube the first step in such a simulation is the construction of a one to one map f q k spl rarr s n of dilation d for d small that is one wants a map f such that images of adjacent points are at most distance d apart in s n an alternative approach best applicable when one to one maps are difficult or impossible to find is the construction of a one to many map g of dilation d defined as follows for each point x spl isin q k there is an associated subset g x spl sube v s n such that for each edge xy in q k every x spl isin g x is at most distance d in s n from some y spl isin g y such one to many maps allow one to achieve the low interprocessor communication time desired in the usual one to one embedding underlying a simulation this is done by capturing the local structure of q k inside of s n via the one to many embedding when the global structure cannot be so captured our results are the following 1 there exist the following one to many embeddings a f q k spl rarr s 3k 1 with dilation f 1 b f q 11k 2 spl rarr s 13k 2 with dilation f 2 2 there exists a one to one embedding f q n2 sup n spl minus 1 spl rarr s 2 sup n with dilation f 3

on the fault tolerant embedding of complete binary trees in the pancake graph interconnection network abstract this paper studies the problem of embedding complete binary trees cbt s into an n dimensional pancake graph p n with fault tolerant capability first a new embedding scheme is developed for mapping a source cbt with height m 2 n log m and dilation 2 onto the p n this scheme not only embeds a cbt whose height is very close to the largest possible one but also saves a lot of unused generators and generator products furthermore these unused generators and generator products are used to recover faulty nodes and embed multiple cbt s maximally near 2 3 nodes of the source cbt are allowed to be faulty at the same time and can be recovered by our scheme with dilation 4 alternatively a scheme which can embed a cbt with height m 2 n log m 1 is also given in this case if all nodes in the cbt are faulty they can be recovered in the smallest number of recovery steps and only with dilation 4

approximations of continuous functionals by neural networks with application to dynamic systems the paper gives several strong results on neural network representation in an explicit form under very mild conditions a functional defined on a compact set in c a b or l sup p a b spaces of infinite dimensions can be approximated arbitrarily well by a neural network with one hidden layer the results are a significant development beyond earlier work where theorems of approximating continuous functions defined on a finite dimensional real space by neural networks with one hidden layer were given all the results are shown to be applicable to the approximation of the output of dynamic systems at any particular time

representation of nonlinear random transformations by non gaussian stochastic neural networks the learning capability of neural networks is equivalent to modeling physical events that occur in the real environment several early works have demonstrated that neural networks belonging to some classes are universal approximators of input output deterministic functions recent works extend the ability of neural networks in approximating random functions using a class of networks named stochastic neural networks snn in the language of system theory the approximation of both deterministic and stochastic functions falls within the identification of nonlinear no memory systems however all the results presented so far are restricted to the case of gaussian stochastic processes sps only or to linear transformations that guarantee this property this paper aims at investigating the ability of stochastic neural networks to approximate nonlinear input output random transformations thus widening the range of applicability of these networks to nonlinear systems with memory in particular this study shows that networks belonging to a class named non gaussian stochastic approximate identity neural networks sainns are capable of approximating the solutions of large classes of nonlinear random ordinary differential transformations the effectiveness of this approach is demonstrated and discussed by some application examples

function approximation using generalized adalines this paper proposes neural organization of generalized adalines gadalines for data driven function approximation by generalizing the threshold function of adalines we achieve the k state transfer function of gadalines which responds a unitary vector of k binary values to the projection of a predictor on a receptive field a generative component that uses the k state activation of a gadaline to trigger k posterior independent normal variables is employed to emulate stochastic predictor oriented target generation the fitness of a generative component to a set of paired data mathematically translates to a mixed integer and linear programming since consisting of continuous and discrete variables the mathematical framework is resolved by a hybrid of the mean field annealing and gradient descent methods following the leave one out learning strategy the obtained learning method is extended for optimizing multiple generative components the learning result leads to parameters of a deterministic gadaline network for function approximation numerical simulations further test the proposed learning method with paired data oriented from a variety of target functions the result shows that the proposed learning method outperforms the mlp and rbf learning methods for data driven function approximation

boundary control of linear uncertain 1 d parabolic pde using approximate dynamic programming this paper develops a near optimal boundary control method for distributed parameter systems governed by uncertain linear 1 d parabolic partial differential equations pde by using approximate dynamic programming a quadratic surface integral is proposed to express the optimal cost functional for the infinite dimensional state space accordingly the hamilton jacobi bellman hjb equation is formulated in the infinite dimensional domain without using any model reduction subsequently a neural network identifier is developed to estimate the unknown spatially varying coefficient in pde dynamics novel tuning law is proposed to guarantee the boundedness of identifier approximation error in the pde domain a radial basis network rbn is subsequently proposed to generate an approximate solution for the optimal surface kernel function online the tuning law for near optimal rbn weights is created such that the hjb equation error is minimized while the dynamics are identified and closed loop system remains stable ultimate boundedness ub of the closed loop system is verified by using the lyapunov theory the performance of the proposed controller is successfully confirmed by simulation on an unstable diffusion reaction process

embedding an arbitrary binary tree into the star graph binary tree structures have been very useful in solving divide and conquer type of problems embedding binary trees into another network the host network helps in designing solutions for the host network using the known solutions on binary trees embedding arbitrary binary trees into networks in particular into the hypercube has been addressed in the literature the latter was achieved with load 1 and constant dilation the n star graph is a recently introduced interconnection network for massively parallel systems it enjoys symmetry and fault tolerance properties that make it a viable alternative to the hypercube in this paper we address the problem of embedding arbitrary binary trees into the n star graph this work is the first to present such an embedding the tree has e n 1 1 or fewer vertices the embedding leads to load 1 and constant dilation for all values of n it therefore enables the star graph to efficiently simulate an arbitrary binary tree with only a constant factor of communication delay

approximation of discontinuous signals by sampling kantorovich series abstract in this paper the behavior of the sampling kantorovich operators has been studied when discontinuous functions signals are considered in the above sampling series moreover the rate of approximation for the family of the above operators is estimated when uniformly continuous and bounded signals are considered finally several examples of duration limited kernels which satisfy the assumptions of the present theory have been provided and also the problem of the linear prediction by sampling values from the past is analyzed

embedding meshes and tori on double loop networks of the same size double loop networks are extensions of ring networks and are widely used in the design and implementation of local area networks and parallel processing architectures however embedding of other types of networks on double loop networks has not been well studied due to the topological complexity of double loop networks the traditional l shape designed to compute the diameter of double loop networks is not effective to solve the embedding problem we propose a novel tessellation approach to partition the geometric plane of double loop networks into a set of parallelogram tiles called p shape based on the characteristics of p shape we design a simple embedding scheme namely p shape embedding that embeds meshes and tori on double loop networks in a systematic way under p shape embedding we evaluate the embedding metrics of dilation average dilation and congestion which depend heavily on the parameters of p shape a main merit of p shape embedding is that a large fraction of embedded mesh torus edges have edge dilation 1 resulting in a low average dilation these are the first results to our knowledge for embedding meshes and tori on double loop networks which is of great significance due to the popularity of these architectures our p shape construction bridges between regular graphs and double loop networks and provides a powerful tool for studying double loop networks

cycle embedding in star graphs with more conditional faulty edges abstract the star graph is one of the most attractive interconnection networks the cycle embedding problem is widely discussed in many networks and edge fault tolerance is an important issue for networks since edge failures may occur when a network is put into use in this paper we investigate the cycle embedding problem in star graphs with conditional faulty edges we show that there exist fault free cycles of all even lengths from 6 to n in any n dimensional star graph s n n 4 with 3 n 10 faulty edges in which each node is incident with at least two fault free edges our result not only improves the previously best known result where the number of tolerable faulty edges is up to 2 n 7 but also extends the result that there exists a fault free hamiltonian cycle under the same condition

high order fuzzy approximation by fuzzywavelet type and neural network operators here are studied in terms of fuzzy high approximation to the unit several basic sequences of fuzzy wavelet type operators and fuzzy neural network operators these operators are fuzzy analogs of earlier studied real ones the produced results generalize earlier real ones into the fuzzy setting here the high order fuzzy pointwise convergence with rates to the fuzzy unit operator is established through fuzzy inequalities involving the fuzzy modulus of continuity of the nth order n 1 h fuzzy derivative of the engaged fuzzy number valued function at the end we present a related lp result for fuzzy neural network operators

embedding binary trees into crossed cubes the recently introduced interconnection network crossed cube has attracted much attention in the parallel processing area due to its many attractive features like the ordinary hypercube the n dimensional crossed cube is a regular graph with 2 sup n vertices and n2 sup n 1 edges the diameter of the crossed cube is approximately half that of the ordinary hypercube these advantages of the crossed cube motivated the study of how well it can simulate other networks such as the complete binary tree we show that the 2 sup n 1 node complete binary tree can be embedded into the n dimensional crossed cube with dilation 1

edge bipancyclicity of star graphs with faulty elements in this paper we investigate the fault tolerant edge bipancyclicity of an n dimensional star graph s n given a set f comprised of faulty vertices and or edges in s n with f n 3 and any fault free edge e in s n f we show that there exist cycles of every even length from 6 to n 2 f v in s n f containing e where n 3 our result is optimal because the star graph is both bipartite and regular with the common degree n 1 the length of the longest fault free cycle in the bipartite s n is n 2 f v in the worst case where all faulty vertices are in the same partite set we also provide some sufficient conditions from which longer cycles with length from n 2 f v 2 to n 2 f v can be embedded

the max product generalized sampling operators convergence and quantitative estimates abstract in this paper we study the max product version of the generalized sampling operators based upon a general kernel function in particular we prove pointwise and uniform convergence for the above operators together with a certain quantitative jackson type estimate based on the first order modulus of continuity of the function being approximated the proof of the proposed results are based on the definition of the so called generalized absolute moments by the proposed approach the achieved approximation results can be applied for several type of kernels not necessarily duration limited such as the sinc function the fejer kernel and many others examples of kernels with compact support for which the above theory holds can be given for example by the well known central b splines

output feedback based boundary control of uncertain coupled semilinear parabolic pde using neurodynamic programming in this paper neurodynamic programming based output feedback boundary control of distributed parameter systems governed by uncertain coupled semilinear parabolic partial differential equations pdes under neumann or dirichlet boundary control conditions is introduced first hamilton jacobi bellman hjb equation is formulated in the original pde domain and the optimal control policy is derived using the value functional as the solution of the hjb equation subsequently a novel observer is developed to estimate the system states given the uncertain nonlinearity in pde dynamics and measured outputs consequently the suboptimal boundary control policy is obtained by forward in time estimation of the value functional using a neural network nn based online approximator and estimated state vector obtained from the nn observer novel adaptive tuning laws in continuous time are proposed for learning the value functional online to satisfy the hjb equation along system trajectories while ensuring the closed loop stability local uniformly ultimate boundedness of the closed loop system is verified by using lyapunov theory the performance of the proposed controller is verified via simulation on an unstable coupled diffusion reaction process

the approximation operators with sigmoidal functions the aim of this paper is to investigate the error which results from the method of approximation operators with logarithmic sigmoidal function by means of the method of extending functions a class of feed forward neural network operators is introduced using these operators as approximation tools the upper bounds of errors in uniform norm approximating continuous functions are estimated also a class of quasi interpolation operators with logarithmic sigmoidal function is constructed for approximating continuous functions defined on the total real axis

multivariate hyperbolic tangent neural network approximation here we study the multivariate quantitative approximation of real and complex valued continuous multivariate functions on a box or r n n n by the multivariate quasi interpolation hyperbolic tangent neural network operators this approximation is derived by establishing multidimensional jackson type inequalities involving the multivariate modulus of continuity of the engaged function or its high order partial derivatives our multivariate operators are defined by using a multidimensional density function induced by the hyperbolic tangent function the approximations are pointwise and uniform the related feed forward neural network is with one hidden layer

embedding star networks into hypercubes the star interconnection network has recently been suggested as an alternative to the hypercube as hypercubes are often viewed as universal and capable of simulating other architectures efficiently we investigate embeddings of star network into hypercubes our embeddings exhibit a marked trade off between dilation and expansion for the n dimensional star network we exhibit 1 a dilation n 1 embedding of s sub n into h sub n where n log sub 2 n 2 a dilation 2 d 1 embedding of s sub n into h sub 2d n 1 where d log sub 2 n 2 3 a dilation 2d 2i embedding of s 2 sup i m into h 2 sup i d i2 sup i m 2i 1 where d log sub 2 m 4 a dilation l embedding of s sub n into h sub d where l 1 log sub 2 n and d n 1 l 5 a dilation k 1 k 2 2 embedding of s sub n into h n k 1 2 sup k 1 1 where k log sub 2 n 1 6 a dilation 3 embedding of s sub 2k 1 into h 2k sup 2 k and 7 a dilation 4 embedding of s sub 3k 2 into h 3k sup 2 3k 1 some of the embeddings are in fact optimum in both dilation and expansion for small values of n we also show that the embedding of s sub n into its optimum hypercube requires dilation spl omega log sub 2 n

embedding longest fault free paths onto star graphs with more vertex faults the n dimensional star graph sn belongs to a class of bipartite graphs and it is recognized as an attractive alternative to the hypercube since s1 s2 and s3 have trivial structures we focus our attention on sn with n 4 in this paper let f sn be the set of vertex faults previously it was shown that when f sn n 5 sn with n 6 can embed a longest fault free path of length at least n 2 f sn 1 respectively n 2 f sn 2 between two arbitrary vertices in different partite sets respectively the same partite set longest fault free paths in star graphs with vertex faults theoretical computer science 262 2001 215 227 in this paper we improve the above result by tolerating more faults f sn n 3 to embed a longest fault free path between two arbitrary vertices in sn n 4

adaptive classification of temporal signals in fixed weight recurrent neural networks an existence proof recurrent neural networks with fixed weights have been shown in practice to successfully classify adaptively signals that vary as a function of time in the presence of additive noise and parametric perturbations we address the question can this ability be explained theoretically we provide a mathematical proof that these networks have this ability even when parametric perturbations enter the signals nonlinearly the restrictions that we impose on the signals to be classified are that they satisfy an assumption of nondegeneracy and that noise amplitude is sufficiently small further we demonstrate that the recurrent neural networks may not only classify uncertain signals adaptively but also can recover the values of uncertain parameters of the signals up to their equivalence classes

fractional neural network approximation here we study the univariate fractional quantitative approximation of real valued functions on a compact interval by quasi interpolation sigmoidal and hyperbolic tangent neural network operators these approximations are derived by establishing jackson type inequalities involving the moduli of continuity of the right and left caputo fractional derivatives of the engaged function the approximations are pointwise and with respect to the uniform norm the related feed forward neural networks are with one hidden layer our fractional approximation results into higher order converges better than the ordinary ones

artificial neural networks for solving ordinary and partial differential equations we present a method to solve initial and boundary value problems using artificial neural networks a trial solution of the differential equation is written as a sum of two parts the first part satisfies the initial boundary conditions and contains no adjustable parameters the second part is constructed so as not to affect the initial boundary conditions this part involves a feedforward neural network containing adjustable parameters the weights hence by construction the initial boundary conditions are satisfied and the network is trained to satisfy the differential equation the applicability of this approach ranges from single ordinary differential equations ode to systems of coupled ode and also to partial differential equations pde in this article we illustrate the method by solving a variety of model problems and present comparisons with solutions obtained using the galerkin finite element method for several cases of partial differential equations with the advent of neuroprocessors and digital signal processors the method becomes particularly interesting due to the expected essential gains in the execution speed

multilayer potts perceptrons with levenberg marquardt learning this paper presents learning multilayer potts perceptrons mlpotts for data driven function approximation a potts perceptron is composed of a receptive field and a k state transfer function that is generalized from sigmoid like transfer functions of traditional perceptrons an mlpotts network is organized to perform translation from a high dimensional input to the sum of multiple postnonlinear projections each with its own postnonlinearity realized by a weighted k state transfer function mlpotts networks span a function space that theoretically covers network functions of multilayer perceptrons compared with traditional perceptrons weighted potts perceptrons realize more flexible postnonlinear functions for nonlinear mappings numerical simulations show mlpotts learning by the levenberg marquardt lm method significantly improves traditional supervised learning of multilayer perceptrons for data driven function approximation

multiobjective algebraic synthesis of neural control systems by implicit model following the advantages brought about by using classical linear control theory in conjunction with neural approximators have long been recognized in the literature in particular using linear controllers to obtain the starting neural control design has been shown to be a key step for the successful development and implementation of adaptive critic neural controllers despite their adaptive capabilities neural controllers are often criticized for not providing the same performance and stability guarantees as classical linear designs therefore this paper develops an algebraic synthesis procedure for designing dynamic output feedback neural controllers that are closed loop stable and meet the same performance objectives as any classical linear design the performance synthesis problem is addressed by deriving implicit model following algebraic relationships between model matrices obtained from the classical design and the neural control parameters additional linear matrix inequalities lmis conditions for closed loop exponential stability of the neural controller are derived using existing integral quadratic constraints iqcs for operators with repeated slope restricted nonlinearities the approach is demonstrated by designing a recurrent neural network controller for a highly maneuverable tailfin controlled missile that meets multiple design objectives including pole placement for transient tuning h infin and h 2 performance in the presence of parameter uncertainty and command input tracking

cycle embedding in star graphs with edge faults the star graph is one of the famous interconnection networks edge fault tolerance is an important issue for networks and the cycle embedding problem is widely discussed in many researches in this paper we show that the cycles of even lengths from 6 to n can be embedded into the n dimensional star graph when the number of edge fault in the graph does not exceed n 3 since the graph is bipartite and n 1 regular our result is optimal