on the effect of noise on the moore penrose generalized inverse associative memory monte carlo simulations of the continuous moore penrose generalized inverse associative memory kohonen l have shown that the noise to signal ratio is improved on recall in the autoassociative case as long as the number of vector pairs stored is less than the number of components per vector in the heteroassociative case however the noise to signal ratio may actually be greatly increased upon recall particularly as the number of vector pairs stored approaches the number of components per vector the increase in output noise to signal ratio in the heteroassociative case is found to be due to the fact that the inverse of the product of the key vector matrix with its transpose may increase without bound in spite of the fact that the key vectors are linearly independent

a quantitative comparison of the performance of three discrete distributed associative memory models monte carlo methods have been used to study the recall accuracy of three discrete distributed associative memory models a discrete correlation matrix model dcm the bayes discriminant rule model developed by murakami and aibara ma and a discrete version of the generalized inverse model of kohonen dgi the key input and data output vectors have nc components restricted to the values 1 and 1 the elements of the nc nc recall matrix may take on any floating point value auto associative key and data vectors identical and hetero associative operation have been examined with both noiseless perfect and noisy key vectors simulations have been performed over a range of nc from 20 to 100 and of nv the number of vector pairs stored from 1 to 50 the recall accuracy defined as the probability that a component of the output vector is correct depends strongly on the ratio nv nc the dgi model is the only one able to provide perfect hetero associative recall with perfect input for nv nc but is extremely sensitive to input noise if the probability that a component of the input vector is correct is 0 9 the three models will provide perfect output only if nv is less than about 0 15 nc for more noisy inputs the ma model provides the highest probability that a component of the output will be correct

the piecewise polynomial collocation method for nonlinear weakly singular volterra equations second kind volterra integral equations with weakly singular kernels typically have solutions which are nonsmooth near the initial point of the interval of integration using an adaptation of the analysis originally developed for nonlinear weakly singular fredholm integral equations we present a complete discussion of the optimal global and local order of convergence of piecewise polynomial collocation methods on graded grids for nonlinear volterra integral equations with algebraic or logarithmic singularities in their kernels

linear and nonlinear associative memories for parameter estimation this paper proposes the use of associative memories for obtaining preliminary parameter estimates for nonlinear systems for each parameter vector r in a selected training set the system equations are used to determine a vector s of system outputs an associative memory matrix m is then constructed which optimally in the least squares sense associates each system output vector s with its corresponding parameter vector i given any observed system output vector s an estimate i for the system parameters is obtained by setting i ms numerical experiments are reported which indicate the effectiveness of this approach especially for the nonlinear associative memory case in which the training vectors s include not only the system output levels but also products of these levels training with noisy output vectors is shown to improve the accuracy of the parameter estimates when the

on the oscillations appearing in numerical solutions of solvable and nonsolvable integral equations for thin wire antennas differences between certain solvable and nonsolvable ill posed integral equations with the same nonsingular kernel are discussed the main results come from constructing a solvable equation in the context of straight thin wire antennas the kernel of this equation is the usual approximate also called reduced kernel while its exact solution is the familiar sinusoidal current numerical solutions to this solvable equation are compared to corresponding numerical solutions of the usual halle n and pocklington equations with the approximate kernel it is known from previous publications that these last two equations are nonsolvable and that their numerical solutions present severe oscillations when the number of basis functions is sufficiently large it is found that the difficulties encountered in the former solvable equation are much less important compared to those of the nonsolvable ones the same conclusion is brought out from other integral equations arising in different contexts thin wire circular loop antenna method of auxiliary sources and straight wire antenna of infinite length we discuss the consistency of our results with picard s theorem the results in this paper supplement previous publications regarding the difficulties of numerically solving thin wire integral equations with the approximate kernel

exact associative neural memory dynamics utilizing boolean matrices the exact dynamics of shallow loaded associative neural memories are generated and characterized the boolean matrix analysis approach is employed for the efficient generation of all possible state transition trajectories for parallel updated binary state dynamic associative memories dams general expressions for the size of the basin of attraction of fundamental and oscillatory memories and the number of oscillatory and stable states are derived for discrete synchronous hopfield dams loaded with one two or three even dimensionality bipolar memory vectors having the same mutual hamming distances between them spurious memories are shown to occur only if the number of stored patterns exceeds two in an even dimensionality hopfield memory the effects of odd versus even dimensionality memory vectors on dam dynamics and the effects of memory pattern encoding on dam performance are tested an extension of the boolean matrix dynamics characterization technique to other more complex dams is presented

accurate and efficient algorithms for frequency domain scattering from a thin wire this paper contains a systematic study of numerical approximations for solving the exact kernel form of pocklington s integro differential equation for the current induced on a thin wire by an incident time harmonic electromagnetic field we consider various galerkin h p hp and adaptive h and collocation schemes and show that a sensible hp refinement strategy provides a very efficient way to solve the problem we also describe how the kernel itself a difficult singular integral can be evaluated reliably and efficiently

relatively weakly open sets in closed balls of banach spaces and real jb triples of finite rank we prove that given a real jb triple x there exists a nonempty relatively weakly open subset of the closed unit ball of x with diameter less than 2 if and only if the banach space of x is isomorphic to a hilbert space moreover we give the structure of real jb triples whose banach spaces are isomorphic to hilbert spaces such real jb triples are also characterized in two different purely algebraic ways

fixed points and best proximity points in contractive cyclic self maps satisfying constraints in closed integral form with some applications this paper investigates the existence of best proximity points and the potential fixed points of p cyclic self maps which are defined on the union of a finite set of subsets of a certain uniform metric space under several integral type contractive conditions the corresponding properties of the associated restricted composed maps from any of the subsets to itself are also dealt with in a formal context the above subsets of the uniform metric space are not assumed to necessarily intersect but the case when they intersect leads to particular results of the general case in the sense that best proximity points become fixed points common fixed points and best proximity points of pairs of self maps on the above subsets of the uniform metric space are also investigated

definitions properties and applications of finite part integrals by using standard calculus we discuss some definitions and properties of finite part integrals point out the essence of this concept and show how these integrals naturally arise in some integral equation applications a couple of new examples are also described

construction of a distributed associative memory on the basis of bayes discriminant rule the purpose of this correspondence is to propose a new construction method of distributed associative memory which operates with discrete valued signals in this method memorized pairs of vectors cue vectors and data vectors are recorded in the form of a matrix w and a vector t from an input vector x the data vector is recalled by an operation u xw t where x is a cue vector or a noisy cue vector and u is a quantizing function the methods of memorization and recall are similar to the associatron however the proposed model can recall the data vectors optimally in bayesian sense even when noisy cue vectors are given as the input vectors

numerical solution and its analysis for a system of hypersingular integral equations abstract a system of hypersingular integral equations occurs quite naturally in several branches of science and engineering during the formulation of many boundary value problems the analytical solution is known for the system of dominant equations however there are many real world problems such as crack problems occur in the field of fracture mechanics which may not be formulated as system of dominant equations therefore we propose a numerical method to find the approximate solution for such generalized form the convergence of the proposed method is proved in l n 2 space this convergence helps to derive theoretical error bound for the error between the exact and the approximate solution an application of the proposed method in finding numerical solution of hypersingular integral equation over the curves is shown finally the derived theoretical error bound is numerically calculated and validated with the help of numerical examples

the correlation memory matrix for parameter estimation dynamic systems described by nonlinear differential or difference equations are abound in the literature until recently estimation of system parameters has been mainly carried out using iterative search techniques such as conjugate gradient steepest descent and evolutionary algorithms these methods vary in terms of their noise immunity and robustness but they all tend to require a considerable number of iterations in order for the solution to converge especially in the presence of noise this is a disadvantage for systems requiring real time control in continuation of a series of papers utilizing linear associative memories lam for parameter estimation this paper proposes a non iterative single pass technique for estimating these parameters using the well known correlation memory matrix cmm previously used in pattern recognition the cmm method is used to estimate the parameters of the lotka volterra predator prey model and the van der pol oscillator it is shown that this class of linear associators is superior to other types of lam in that it provides equally good parameter estimates while requiring neither matrix inversion nor extensive training as nonlinear quadratic associations are incorporated into the cmm the method becomes more robust than the linear version

regularity theory and superalgebraic solvers for wire antenna problems we consider the problem of evaluating the current distribution j z that is induced on a straight wire antenna by a time harmonic incident electromagnetic field the scope of this paper is twofold one of its main contributions is a regularity proof for a straight wire occupying the interval 1 1 in particular for a smooth time harmonic incident field this theorem implies that j z i z sqrt 1 z 2 where i z is an infinitely differentiable function the previous state of the art in this regard placed i in the sobolev space w 1 p p 1 the second focus of this work is on numerics we present three superalgebraically convergent algorithms for the solution of wire problems two based on hallen s integral equation and one based on the pocklington integrodifferential equation both our proof and our algorithms are based on two main elements 1 a new decomposition of the kernel of the form g z f 1 z ln z f 2 z where f 1 z and f 2 z are analytic functions on the real line and 2 removal of the end point square root singularities by means of a coordinate transformation the hallen and pocklington based algorithms we propose converge superalgebraically faster than mathcal o n m and mathcal o m m for any positive integer m where n and m are the numbers of unknowns and the number of integration points required for construction of the discretized operator respectively in previous studies at most the leading order contribution to the logarithmic singular term was extracted from the kernel and treated analytically the higher order singular derivatives were left untreated and the resulting integration methods for the kernel exhibit mathcal o m 3 convergence at best a rather comprehensive set of tests we consider shows that in many cases to achieve a given accuracy the numbers n of unknowns required by our codes are up to a factor of five times smaller than those required by the best solvers previously available the required number m of integration points in turn can be several orders of magnitude smaller than those required in previous methods in particular four digit solutions were found in computational times of the order of four seconds and in most cases of the order of a fraction of a second on a contemporary personal computer much higher accuracies result in very small additional computing times

noetherian jordan banach algebras are finite dimensional w x sinclair and tullo proved in 10 that noetherian banach algebras that is complex associative banach algebras satisfying the ascending chain w x condition on left ideals are finite dimensional in 4 we extended this result to the alternative case for a jordan algebra the natural noetherian condition is the ascending chain condition on inner ideals here we show that noetherian jordan banach algebras are finite dimensional

on the use of overdetermined systems in the adaptive numerical solution of integral equations the residual error incurred when numerically solving integral equations for a number of electromagnetic radiation and scattering problems is calculated with the aid of an overdetermined system this error is systematically reduced by adaptively refining the model for the surface current error reduction is achieved by selectively shrinking cell dimensions h refinement increasing the order of the basis functions representing the current p refinement or a combination of both hp refinement the correlation between residual error and surface current error is investigated and found to be high the impact of edge singularities and curvature discontinuities is discussed

techniques for evaluating the uniform current vector potential at the isolated singularity of the cylindrical wire kernel the cylindrical wire kernel possesses a singularity which must be properly treated in order to evaluate the uniform current vector potential traditionally the singular part of the kernel is extracted resulting in a slowly varying function which is convenient for numerical integration this paper provides some new accurate and computationally efficient methods for evaluating the remaining singular integral it is shown that this double integral may be converted to a single integral which no longer possesses a singular integrand and consequently may be efficiently evaluated numerically this form of the integral is independent of the restrictions involving wire length and radius which are inherent in various approximations also presented is a highly convergent exact series representation of the integral which is valid except in the immediate vicinity of the singularity finally a new approximation is derived which is found to be an improvement over the classical thin wire approximation it is demonstrated that each of these methods provides extremely accurate as well as efficient results for a wide range of wire radii and field point locations

the h p and h p versions of the finite element method in 1 dimension this paper is the first one in the series of three which are addressing in detail the properties of the three basic versions of the finite element method in the one dimensional setting the main emphasis is placed on the analysis when the exact solution has singularity of x type the first part analyzes thep version the second theh version and generalh p version and the final third part addresses the problems of the adaptiveh p version

transient excitation of a straight thin wire segment a new look at an old problem the transient excitation of a straight thin wire segment is analyzed with the aid of a one dimensional integral equation for the current along the wire an almost exact derivation of that equation in which only the radial current on the end faces is approximated is given the integral equation obtained turns out to be identical to the reduced version of pocklington s equation on the basis of this derivation existing and new numerical solution techniques are critically reviewed pocklington s equation and hallen s equivalent form are solved directly by marching on in time as well as indirectly via a transformation to the frequency domain for pocklington s equation a conventional moment method discretization leads to a toeplitz matrix that is inverted with levinson s algorithm for hallen s equation the toeplitz structure is disturbed and the frequency domain constituents are determined with the aid of the conjugate gradient fft method illustrative numerical results are presented and discussed

the h p and h p versions of the finite element method in 1 dimension part ii the error analysis of the h and h p versions the paper is the second in the series of three devoted to the detailed analysis of the three basic versions of the finite element method in one dimensional setting the first part 1 analyzed thep version the present one concentrates on theh andh p versions

some new problems in numerical integration in this paper we examine the numerical computation of the multiple integrals generated by galerkin methods applied to two nonstandard hypersingular integral equations which are of interest by their own these equations are used to solve two classical electromagnetic problems that are briefly described

the time domain integral equation for a straight thin wire antenna with the reduced kernel is not well posed we show that the pocklington integral equation for time domain scattering from thin wire antennas is not mathematically well posed this has considerable implications for numerical solution schemes in particular our argument explains the observed occurrence of rapidly oscillating errors in numerical solutions as the numerical grid sizes are reduced

backscattering from a two dimensional rectangular crack using fie the numerical solution of a rectangular crack in a perfectly conducting surface is appropriate for non destructive testing ndt applications to model faults the paper presents a direct modeling technique for determining the h and e polarized backscattering signatures of a two dimensional crack in a metallic surface that is suitable for inverse scattering problem the governing field integral equations fie with logarithmic and hyper singular kernels is first discretized and solved by a collocation method based on chebyshev polynomials by using ad hoc quadrature rules the integral equation is then reduced to a linear system of algebraic equations this approach does not have the size or frequency limitations of the regular techniques such as modal expansion and quasi static manners the results are in good agreement with the entirely numerical and non reversible solution of a finite element method

a fast method for the solution of a hypersingular integral equation arising in a waveguide scattering problem in this paper we study the time harmonic electromagnetic scattering problem associated with a t junction between two rectangular waveguides this junction is composed by an infinite primary waveguide and by a semi infinite secondary waveguide with the same height and coupled through a common aperture a standard calculation reduces maxwell equations to a simpler scalar two dimensional non homogeneous helmholtz equation to which the domain decomposition technique is applied in each wave guide the scattered electric field satisfies a non homogeneous helmholtz equation with homogeneous boundary conditions an integral representation for this field is then obtained from which the corresponding expression for the scattered magnetic field follows by enforcing the continuity of the tangential component of the total electric and magnetic fields generated in the two separate waveguides at their interface we obtain a hypersingular integral equation defined on an interval from the solution of this equation the scattering matrix of the junction is easily computed the integral equation has two kernels given in terms of series expansions our analysis determines the relevant singular components and shows that besides the standard second order hypersingularity one of the kernels also has a fixed point second order hypersingularity at each endpoint of the interval of integration the equation is finally solved by means of a galerkin method whose implementation is performed quite efficiently so that the overall numerical method is very fast and accurate the complete numerical solution is presented in the form of parametric plots and convergence results are discussed copyright 2006 john wiley sons ltd

a new formulation of pocklington s equation for thin wires using the exact kernel pocklington s integro differential equation for thin wires with the exact kernel is reformulated using a second derivative formula for improper integrals this allows for analytical evaluation of the second derivatives resulting in a pure integral equation in a similar manner to what is done when using the approximate kernel however as opposed to using the approximate kernel the resulting integral equation developed here is numerically stable even with a simple pulse function point matching solution good convergence for the current is obtained using pulse functions and the severe unphysical oscillations of the current that are encountered when using pulse functions with the approximate kernel are avoided

linear algebra approach to neural associative memories and noise performance of neural classifiers the authors present an analytic evaluation of saturation and noise performance for a large class of associative memories based on matrix operations the importance of using standard linear algebra techniques for evaluating noise performance of associative memories is emphasized the authors present a detailed comparative analysis of the correlation matrix memory and the generalized inverse memory construction rules for auto associative memory and neural classifiers analytic results for the noise performance of neural classifiers that can store several prototypes in one class are presented the analysis indicates that for neural classifiers the simple correlation matrix memory provides better noise performance than the more complex generalized inverse memory