solving systems of linear equations over polynomials we consider a system of linear equations of the form a x x x b x where a x b x are given m n and m 1 matrices with entries from q x the ring of polynomials in the variable x over the rationals we provide a polynomial time algorithm to find the general solution of this system over q x this is accomplished by devising a polynomial time algorithm to find the triangular canonical form hermite form of the matrix a x using unimodular column operations as applications we are able to give polynomial time algorithms for finding the diagonal smith canonical form of a polynomial matrix testing whether two given matrices of rational entries are similar and for finding the invariant factors of a matrix of rationals

on a system of klein gordon type equations with acoustic boundary conditions we prove the existence uniqueness and uniform stabilization of global solutions for a generalized system of klein gordon type equations with acoustic boundary conditions on a portion of the boundary and the dirichlet boundary condition on the rest

uniform stabilization of wave equation with localized damping and acoustic boundary condition abstract in this paper we deal with stability of the nonlinear wave equation with acoustic boundary conditions to non locally reacting boundary

some nonlinear wave equations with acoustic boundary conditions abstract we prove the existence and uniqueness of global solutions to the mixed problem for the carrier equation u tt m u 2 dx u u t u t 0 with acoustic boundary conditions

density estimation and random variate generation using multilayer networks in this paper we consider two important topics density estimation and random variate generation we present a framework that is easily implemented using the familiar multilayer neural network first we develop two new methods for density estimation a stochastic method and a related deterministic method both methods are based on approximating the distribution function the density being obtained by differentiation in the second part of the paper we develop new random number generation methods our methods do not suffer from some of the restrictions of existing methods in that they can be used to generate numbers from any density provided that certain smoothness conditions are satisfied one of our methods is based on an observed inverse relationship between the density estimation process and random number generation we present two variants of this method a stochastic and a deterministic version we propose a second method that is based on a novel control formulation of the problem where a controller network is trained to shape a given density into the desired density we justify the use of all the methods that we propose by providing theoretical convergence results in particular we prove that the l sub spl infin convergence to the true density for both the density estimation and random variate generation techniques occurs at a rate o log log n n sup 1 spl epsi 2 where n is the number of data points and spl epsi can be made arbitrarily small for sufficiently smooth target densities this bound is very close to the optimally achievable convergence rate under similar smoothness conditions also for comparison the l sub 2 root mean square rms convergence rate of a positive kernel density estimator is o n sup 2 5 when the optimal kernel width is used we present numerical simulations to illustrate the performance of the proposed density estimation and random variate generation methods in addition we present an extended introduction and bibliography that serves as an overview and reference for the practitioner

how initial conditions affect generalization performance in large networks generalization is one of the most important problems in neural network research it is influenced by several factors in the network design such as network size weight decay factor and others we show here that the initial weight distribution for gradient decent training algorithms is one other factor that influences generalization the initial conditions guide the training algorithm to search particular places of the weight space for instance small initial weights tend to result in low complexity networks and therefore can effectively act as a regularization factor we propose a novel network complexity measure which is helpful in shedding insight into the phenomenon as well as in studying other aspects of generalization

the multidimensional wave equation with generalized acoustic boundary conditions i strong stability we study the wave equation on a domain of mathbb r d d geq 2 with dynamical boundary control applied on a part of the boundary and a dirichlet boundary condition on the remaining part we prove the asymptotic stability and nonuniform stability of the associated semigroup

the rank of a random matrix this work is concerned with the numerical rank of matrix in the matrix computations we conclude that a real random matrix has full rank with probability 1 and a rational random matrix has full rank with probability 1 too finally the applications of the numerical matrix are given

solving sparse linear equations over finite fields a coordinate recurrence method for solving sparse systems of linear equations over finite fields is described the algorithms discussed all require o n 1 omega n 1 log k n 1 field operations where n 1 is the maximum dimension of the coefficient matrix omega is approximately the number of field operations required to apply the matrix to a test vector and the value of k depends on the algorithm a probabilistic algorithm is shown to exist for finding the determinant of a square matrix also probabilistic algorithms are shown to exist for finding the minimum polynomial and rank with some arbitrarily small possibility of error

learning and convergence analysis of neural type structured networks a class of feedforward neural networks structured networks has recently been introduced as a method for solving matrix algebra problems in an inherently parallel formulation a convergence analysis for the training of structured networks is presented since the learning techniques used in structured networks are also employed in the training of neural networks the issue of convergence is discussed not only from a numerical algebra perspective but also as a means of deriving insight into connectionist learning bounds on the learning rate are developed under which exponential convergence of the weights to their correct values is proved for a class of matrix algebra problems that includes linear equation solving matrix inversion and lyapunov equation solving for a special class of problems the orthogonalized back propagation algorithm an optimal recursive update law for minimizing a least squares cost functional is introduced it guarantees exact convergence in one epoch several learning issues are investigated

existence and decay of solutions for a viscoelastic wave equation with acoustic boundary conditions abstract in this paper we consider a multi dimensional damped semilinear viscoelastic wave equation with variable coefficient and acoustic boundary conditions first we prove a local existence theorem by using the faedo galerkin approximations combined with a contraction mapping theorem second we show that under suitable conditions on the initial data and the relaxation function the solution exists globally in time and we prove the uniform decay rate of the energy

stochastic complexity and modeling on demontre un theoreme fondamental qui donne une borne inferieure pour la longueur de code et donc pour les erreurs de prediction on definit les notions d information a priori et d information utile dans les donnees

neural representation of space using sinusoidal arrays o keefe 1991 has proposed that spatial information in rats might be represented as phasors phase and amplitude of a sine wave encoding angle and distance to a landmark we describe computer simulations showing that operations on phasors can be efficiently realized by arrays of spiking neurons that re code the temporal dimension of the sine wave spatially some cells in motor and parietal cortex exhibit response properties compatible with this proposal

learning from examples in large neural networks a statistical mechanical theory of learning from examples in layered networks at finite temperature is studied when the training error is a smooth function of continuously varying weights the generalization error falls off asymptotically as the inverse number of examples by analytical and numerical studies of single layer perceptrons we show that when the weights are discrete the generalization error can exhibit a discontinuous transition to perfect generalization for intermediate sizes of the example set the state of perfect generalization coexists with a metastable spin glass state

decay rate estimates for wave equations of memory type with acoustic boundary conditions abstract a wave equation of memory type u u 0 t g t s u s d s 0 with acoustic boundary conditions is considered this work is devoted to investigating the influence of the kernel function g and proving general rates of decay of solutions when g does not necessarily decay exponentially

convergent algorithm for sensory receptive field development an unsupervised developmental algorithm for linear maps is derived which reduces the pixel entropy using the measure introduced in previous work at every update and thus removes pairwise correlations between pixels since the measure of pixel entropy has only a global minimum the algorithm is guaranteed to converge to the minimum entropy map such optimal maps have recently been shown to possess cognitively desirable properties and are likely to be used by the nervous system to organize sensory information the algorithm derived here turns out to be one proposed by goodall for pairwise decorrelation it is biologically plausible since in a neural network implementation it requires only data available locally to a neuron in training over ensembles of two dimensional input signals with the same spatial power spectrum as natural scenes networks develop output neurons with center surround receptive fields similar to those of ganglion cells in the retina some technical issues pertinent to developmental algorithms of this sort such as symmetry fixing are also discussed

determining the structure of the jordan normal form of a matrix by symbolic computation abstract an algorithm for determining the jordan normal form of a matrix is presented the algorithm implements the geometric approach given by gantmacher using this algorithm structures of the jordan blocks of the matrix can be completely determined by means of symbolic computation the approximation of eigenvalues of the matrix has no influence on this structure

redundancy reduction as a strategy for unsupervised learning a redundancy reduction strategy which can be applied in stages is proposed as a way to learn as efficiently as possible the statistical properties of an ensemble of sensory messages the method works best for inputs consisting of strongly correlated groups that is features with weaker statistical dependence between different features this is the case for localized objects in an image or for words in a text a local feature measure determining how much a single feature reduces the total redundancy is derived which turns out to depend only on the probability of the feature and of its components but not on the statistical properties of any other features the locality of this measure makes it ideal as the basis for a neural implementation of redundancy reduction and an example of a very simple non hebbian algorithm is given the effect of noise on learning redundancy is also discussed

polynomial stability for wave equations with acoustic boundary conditions and boundary memory damping we study wave equations with acoustic boundary conditions where only one memory damping acts on the acoustic boundary under some conditions on the memory kernel polynomial energy decay rates are established by using higher order energy estimates among some other techniques

parallel algorithms for algebraic problems fast parallel algorithms are presented for the following problems in symbolic manipulation of univariate polynomials computing all entries of the extended euclidean scheme of two polynomials over an arbitrary field gcd and 1cm of many polynomials factoring polynomials over finite fields and the squarefree decomposition of polynomials over fields of characteristic zero and over finite fields for the following estimates assume that the input polynomials have degree at most n and the finite field has p d elements the euclidean algorithm is deterministic and runs in parallel time o log 2 n all the other algorithms are probabilistic las vegas in the general case but when applicable to bf q or bf r they can be implemented deterministically over these fields the algorithms for gcd and lcm use parallel time o log 2 n the factoring algorithm runs in parallel time o log 2 n log 2 d 1 log p the algorithm for squarefree decomposition runs in parallel time o log 2 n

temporal evolution of generalization during learning in linear networks we study generalization in a simple framework of feedforward linear networks with n inputs and n outputs trained from examples by gradient descent on the usual quadratic error function we derive analytical results on the behavior of the validation function corresponding to the lms error function calculated on a set of validation patterns we show that the behavior of the validation function depends critically on the initial conditions and on the characteristics of the noise under certain simple assumptions if the initial weights are sufficiently small the validation function has a unique minimum corresponding to an optimal stopping time for training for which simple bounds can be calculated there exists also situations where the validation function can have more complicated and somewhat unexpected behavior such as multiple local minima at most n of variable depth and long but finite plateau effects additional results and possible extensions are briefly discussed

parallel algorithms for matrix normal forms abstract here we offer a new randomized parallel algorithm that determines the smith normal form of a matrix with entries being univariate polynomials with coefficients in an arbitrary field the algorithm has two important advantages over our previous one the multipliers relating the smith form to the input matrix are computed and the algorithm is probabilistic of las vegas type i e always finds the correct answer the smith form algorithm is also a good sequential algorithm our algorithm reduces the problem of smith form computation to two hermite form computations thus the smith form problem has complexity asymptotically that of the hermite form problem we also construct fast parallel algorithms for jordan normal form and testing similarity of matrices both the similarity and non similarity problems are in the complexity class rnc for the usual coefficient fields i e they can be probabilistically decided in polylogarithmic time using polynomially many processors

a fast las vegas algorithm for computing the smith normal form of a polynomial matrix abstract a las vegas probabilistic algorithm is presented that finds the smith normal form s q x n n of a nonsingular input matrix a z x n n the algorithm requires an expected number of o n 3 d d n 2 log a bit operations where a bounds the magnitude of all integer coefficients appearing in a and d bounds the degrees of entries of a in practice the main cost of the computation is obtaining a nonunimodular triangularization of a polynomial matrix of the same dimension and with similar size entries to those of the input matrix we show how to accomplish this in o n 5 d d log a log a bit operations using standard integer polynomial and matrix arithmetic these complexity results improve significantly on previous algorithms in both a theoretical and a practical sense

the numerical rank of a matrix and its applications in this paper it is proved that the probabilities of a real random matrix and a rational random matrix being non singular are 1 in the matrix computations respectively finally the applications for the least squares problem are given

constraints on synchronizing oscillator networks this paper investigates the constraints placed on some synchronized oscillator models by their underlying dynamics phase response graphs are used to determine the phase locking behaviors of three oscillator models these results are compared with idealized phase response graphs for single phase and multiple phase systems we find that all three oscillators studied are best suited to operate in a single phase system and that the requirements placed on oscillatory models for operation in a multiple phase system are not compatible with the underlying dynamics of oscillatory behavior for these types of oscillator models

a taxonomy of problems with fast parallel algorithms the class nc consists of problems solvable very fast in time polynomial in log n in parallel with a feasible polynomial number of processors many natural problems in nc are known in this paper an attempt is made to identify important subclasses of nc and give interesting examples in each subclass the notion of nc 1 reducibility is introduced and used throughout problem r is nc 1 reducible to problem s if r can be solved with uniform log depth circuits using oracles for s problems complete with respect to this reducibility are given for many of the subclasses of nc a general technique the parallel greedy algorithm is identified and used to show that finding a minimum spanning forest of a graph is reducible to the graph accessibility problem and hence is in nc 2 solvable by uniform boolean circuits of depth o log 2 n and polynomial size the class logcfl is given a new characterization in terms of circuit families the class det of problems reducible to integer determinants is defined and many examples given a new problem complete for deterministic polynomial time is given namely finding the lexicographically first maximal clique in a graph this paper is a revised version of s a cook 1983 in proceedings 1983 intl found comut sci conf lecture notes in computer science vol 158 pp 78 93 springer verlag berlin new york

coupled second order evolution equations with fading memory optimal energy decay rate abstract of concern is the cauchy problem for coupled second order semilinear evolution equations in a hilbert space with indirect memory damping we find an approach to obtain successfully an optimal rate of uniform decay for the system energy only under basic conditions on the memory kernels simultaneously the same rate is also obtained with less difficulty for the corresponding single memory dissipative second order evolution equations as can be seen our results essentially improve the previously related ones in the literature the abstract results are then applied to several concrete problems in the real world

worst case complexity bounds on algorithms for computing the canonical structure of finite abelian groups and the hermite and smith normal forms of an integer matrix an o s 5 m s 2 algorithm for computing the canonical structure of a finite abelian group represented by an integer matrix of size s this is the smith normal form of the matrix is presented moreover an o s 3 m s 2 algorithm for computing the hermite normal form of an integer matrix of size s is given the upper bounds derived on the computational complexity of the algorithms above improve the upper bounds given by kannan and bachem in siam j comput 8 1979 pp 499 507 and chou and collins in siam j comput 11 1982 pp 687 708

statistical theory of learning curves under entropic loss criterion the present paper elucidates a universal property of learning curves which shows how the generalization error training error and the complexity of the underlying stochastic machine are related and how the behavior of a stochastic machine is improved as the number of training examples increases the error is measured by the entropic loss it is proved that the generalization error converges to h0 the entropy of the conditional distribution of the true machine as h0 m 2t while the training error converges as h0 m 2t where t is the number of examples and m shows the complexity of the network when the model is faithful implying that the true machine is in the model m is reduced to m the number of modifiable parameters this is a universal law because it holds for any regular machine irrespective of its structure under the maximum likelihood estimator similar relations are obtained for the bayes and gibbs learning algorithms these learning curves show the relation among the accuracy of learning the complexity of a model and the number of training examples