short note conservative method for the reduction of the number of particles in the monte carlo simulation method for kinetic equations a method for the reduction of the number of particles in the weighted monte carlo method for kinetic equations is described the method randomly redistributes statistical weights over the particle ensemble in such a way that the weight of some particles becomes 0 i e the particles are cancelled while all physically relevant macroscopic moments e g mass energy momentum of the ensemble do not change the method has been applied to the spatially uniform relaxation of a gas of hard spheres

fuzzy piecewise multilinear and piecewise linear systems as universal approximators in sobolev norms a piecewise multilinear and a piecewise linear model for fuzzy systems are introduced and their approximation capabilities investigated classical results on the approximation of regular functions are improved showing that fuzzy systems may keep their semantic structure while approximating to any degree of accuracy not only sufficiently regular functions but also their derivatives

lower bounds for the condition number of vandermonde matrices we derive lower bounds for the condition number of then n vandermonde matrixvn x in the cases where the node vectorxt x1 x2 xn has positive elements or real elements located symmetrically with respect to the origin the bounds obtained grow exponentially inn witho 2n ando 2n 2 respectively we also compute the optimal spectral condition numbers ofvn x for the two node configurations including the optimal nodes and compare them with the bounds obtained

duals of compact lie groups realized in the cuntz algebras and their actions on c algebras abstract as a first step towards a new duality theorem for compact groups we consider a representation category t g of a compact lie group g u d c whose objects are the tensor powers of the defining representation and whose arrows are the intertwiners we associate to t g in a natural way a c algebra og which can be identified with the fixed points of the cuntz algebra od under the natural action of g og carries a canonical endomorphism g and t g can be reconstructed from og g if g su d then og is simple we give conditions for an endomorphism of a unital c algebra a to determine an action of t su d on a such actions correspond to monomorphisms of osu d into a with natural intertwining properties

a clustering technique for the identification of piecewise affine systems we propose a new technique for the identification of discrete time hybrid systems in the piecewise affine pwa form this problem can be formulated as the reconstruction of a possibly discontinuous pwa map with a multi dimensional domain in order to achieve our goal we provide an algorithm that exploits the combined use of clustering linear identification and pattern recognition techniques this allows to identify both the affine submodels and the polyhedral partition of the domain on which each submodel is valid avoiding gridding procedures moreover the clustering step used for classifying the datapoints is performed in a suitably defined feature space which allows also to reconstruct different submodels that share the same coefficients but are defined on different regions measures of confidence on the samples are introduced and exploited in order to improve the performance of both the clustering and the final linear regression procedure

punctured haag duality in locally covariant quantum field theories we investigate a new property of nets of local algebras over 4 dimensional globally hyperbolic spacetimes called punctured haag duality this property consists in the usual haag duality for the restriction of the net to the causal complement of a point p of the spacetime punctured haag duality implies haag duality and local definiteness our main result is that if we deal with a locally covariant quantum field theory in the sense of brunetti fredenhagen and verch then also the converse holds the free klein gordon field provides an example in which this property is verified

on multivariate lagrange interpolation lagrange interpolation by polynomials in several variables is studied through a finite difference approach we establish an interpolation formula analogous to that of newton and a remainder formula both of them in terms of finite differences we prove that the finite difference admits an integral representation involving simplex spline functions in particular this provides a remainder formula for lagrange interpolation of degree n of a function f which is a sum of integrals of certain n 1 st directional derivatives of f multiplied by simplex spline functions we also provide two algorithms for the computation of lagrange interpolants which use only addition scalar multiplication and point evaluation of polynomials

constructive approximation to multivariate function by decay rbf neural network it is well known that single hidden layer feedforward networks with radial basis function rbf kernels are universal approximators when all the parameters of the networks are obtained through all kinds of algorithms however as observed in most neural network implementations tuning all the parameters of the network may cause learning complicated poor generalization overtraining and unstable unlike conventional neural network theories this brief gives a constructive proof for the fact that a decay rbf neural network with n 1 hidden neurons can interpolate n 1 multivariate samples with zero error then we prove that the given decay rbfs can uniformly approximate any continuous multivariate functions with arbitrary precision without training the faster convergence and better generalization performance than conventional rbf algorithm bp algorithm extreme learning machine and support vector machines are shown by means of two numerical experiments

constructive approximate interpolation by neural networks we present a type of single hidden layer feedforward neural networks with sigmoidal nondecreasing activation function we call them ai nets they can approximately interpolate with arbitrary precision any set of distinct data in one or several dimensions they can uniformly approximate any continuous function of one variable and can be used for constructing uniform approximants of continuous functions of several variables all these capabilities are based on a closed expression of the networks

reduction of the number of particles in the stochastic weighted particle method for the boltzmann equation different ideas for reducing the number of particles in the stochastic weighted particle method for the boltzmann equation are described and discussed the corresponding error bounds are obtained and numerical tests for the spatially homogeneous boltzmann equation presented it is shown that if an appropriate reduction procedure is used then any effect on the accuracy of the numerical scheme is negligible

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

a geometric approach to maximum speed n dimensional continuous linear interpolation in rectangular grids an algorithm for the linear interpolation of multi input functions sampled on rectangular grids is presented a geometric approach is adopted and the mathematics is thoroughly developed we show that the algorithm is optimum in fact when the number n of inputs grows to infinity its computational requirement is o n log n which is the same as the lower bound on the cost of continuous linear interpolation procedures

on the local minima free condition of backpropagation learning it is shown that if there are p noncoincident input patterns to learn and a two layered feedforward neural network having p 1 sigmoidal hidden neuron and one dummy hidden neuron is used for the learning then any suboptimal equilibrium point of the corresponding error surface is unstable in the sense of lyapunov this result leads to a sufficient local minima free condition for the backpropagation learning

representations of nets of c algebras over s 1 in recent times a new kind of representations has been used to describe superselection sectors of the observable net over a curved spacetime taking into account the effects of the fundamental group of the spacetime using this notion of representation we prove that any net of c algebras over s 1 admits faithful representations and when the net is covariant under diff s 1 it admits representations covariant under any amenable subgroup of diff s 1

feedforward artificial neural networks for solving discrete multiple criteria decision making problems decision making involves choosing some course of action among various alternatives in almost all decision making problems there are several criteria for judging possible alternatives the main concern of the decision maker dm is to fulfill his conflicting goals while satisfying the constraints of the system in this paper we present an adaptive feedforward artificial neural network af ann approach to solve discrete multiple criteria decision making mcdm problems the af ann is used to capture and represent the dm s preferences and then to select the most desirable alternative the af ann can adjust and improve its representation as more information from the dm becomes available we begin with the assumption that an af ann topology is given i e specific numbers of nodes and links are predetermined to adjust the parameters of the af ann we present an iterative learning algorithm consisting of two steps a generating a direction and b a one dimensional search along that direction we then present a methodology to obtain the most appropriate af ann topology and set its parameters the procedure starts with a small number of nodes and links and then adaptively increases the number of nodes and links until the proper topology is obtained furthermore when the set of training patterns alternatives with their associated evaluations by the dm changes the af ann model can adapt itself by re training or expanding the existing model some illustrative examples are presented to solve discrete mcdm problems by an af ann we show how to incorporate basic properties of efficiency concavity and convexity into the af ann we formulate the mcdm problems and use the af ann to rank the set of discrete alternatives where each alternative is associated with a set of conflicting and noncommensurate criteria we present a method for solving discrete mcdm problems through af anns which consists of a formulating and assessing the utility function by eliciting information from the dm and then training the af ann and b ranking a rating alternatives by using the trained af ann model some computational experiments are presented to show the effectiveness of the method

generalized polynomial decomposable multiple attribute utility functions for ranking and rating multiple criteria discrete alternatives1all rights reserved this study or parts thereof may not be reproduced in any form without the written permission of the authors 1 this paper describes a class of nonlinear polynomial functions which are far more complex than linear and additive functions and other types of well known utility functions although the assessment of such nonlinear polynomial functions may require more time and possibly more cognitive burden the advantage of using more complex and nonlinear functions is that more complex preferential behaviors can be represented and hence a larger set of decision problems can be solved in this paper we develop a generalized decomposable multiattribute utility function gdmauf for representing the decision maker s dm s preferential behavior we show that any polynomial function can be represented by the gdmauf and that such a function can be assessed to represent the dm s behavioral preferences or so called the utility function our approach is based on the use of holistic responses numerical rankings and rating of alternatives from which we assess the gradients of the dm s unknown utility function from assessed gradients we then construct a generalized decomposable multiattribute utility function that mimics and represent the dm s preferential behavior this assessed generalized decomposable multiattribute utility function can be used directly to rank and rate all alternatives without any further interaction from the dm we also demonstrate that the generalized utility function gdmauf is more general and flexible than some other well known utility structures such as additive multiplicative multilinear quasi concave and quasi convex the underlying assumptions for gdmauf are that it is differentiable and that it can be expressed in a polynomial form some classes extensions and properties of the generalized utility function are explored to simplify its assessment several examples and experimental results are provided

backpropagation with homotopy when training a feedforward neural network with backpropagation rumelhart et al 1986 local minima are always a problem because of the nonlinearity of the system there have been several ways to attack this problem for example to restart the training by selecting a new initial point to perform the preprocessing of the input data or the neural network here we propose a method which is efficient in computation to avoid some local minima

on lagrange and hermite interpolation in rk a method for the construction of a set of data of interpolation in several variables is given the resulting data which are either function values or directional derivatives values give rise to a space of polynomials in such a way that unisolvence is guaranteed the interpolating polynomial is calculated using a procedure which generalizes the newton divided differences formula for a single variable

linear independence of internal representations in multilayer perceptrons identifies the linear independence of the internal representation of the multilayer perceptron as an essential property for exact learning the sigmoidal hidden unit activation function has the ability to produce linearly independent outputs as a result the minimum number of hidden units for a set of specified input is the number of patterns less the rank of the input patterns in addition the basis of many training algorithms is shown to inherently increase the number of linearly independent vectors in the internal representations thereby increasing the likelihood of exact learning

a monte carlo methods for identification and sensitivity analysis of coagulation processes a stochastic simulation algorithm is presented to calculate parametric derivatives of solutions of a population balance equation the dispersed system is approximated by an n particle stochastic weighted ensemble the derivatives are accounted for through infinitesimal deviation of the statistical weights that are recalculated at each coagulation thus all the parametric derivatives can be calculated along one trajectory of the process given n sufficiently large we use an operator splitting technique to account for surface growth of the particles the obtained solution is in good agreement with the available analytical solutions as soon as the parametric derivatives are known the gradient based methods can be applied to the control and identification of the coagulation process the extension of the proposed technique to a multi dimensional case is straightforward

the generally covariant locality principle a new paradigm for local quantum field theory a new approach to the model independent description of quantum field theories will be introduced in the present work the main feature of this new approach is to incorporate in a local sense the principle of general covariance of general relativity thus giving rise to the concept of a locally covariant quantum field theory such locally covariant quantum field theories will be described mathematically in terms of covariant functors between the categories on one side of globally hyperbolic spacetimes with isometric embeddings as morphisms and on the other side of algebras with unital injective monomorphisms as morphisms moreover locally covariant quantum fields can be described in this framework as natural transformations between certain functors the usual haag kastler framework of nets of operator algebras over a fixed spacetime background manifold together with covariant automorphic actions of the isometry group of the background spacetime can be re gained from this new approach as a special case examples of this new approach are also outlined in case that a locally covariant quantum field theory obeys the time slice axiom one can naturally associate to it certain automorphic actions called relative cauchy evolutions which describe the dynamical reaction of the quantum field theory to a local change of spacetime background metrics the functional derivative of a relative cauchy evolution with respect to the spacetime metric is found to be a divergence free quantity which has as will be demonstrated in an example the significance of an energy momentum tensor up to addition of scalar functions for the locally covariant quantum field theory furthermore we discuss the functorial properties of state spaces of locally covariant quantum field theories that entail the validity of the principle of local definiteness

training pi sigma network by online gradient algorithm with penalty for small weight update a pi sigma network is a class of feedforward neural networks with product units in the output layer an online gradient algorithm is the simplest and most often used training method for feedforward neural networks but there arises a problem when the online gradient algorithm is used for pi sigma networks in that the update increment of the weights may become very small especially early in training resulting in a very slow convergence to overcome this difficulty we introduce an adaptive penalty term into the error function so as to increase the magnitude of the update increment of the weights when it is too small this strategy brings about faster convergence as shown by the numerical experiments carried out in this letter

a random discrete velocity model and approximation of the boltzmann equation an approximation procedure for the boltzmann equation based on random choices of collision pairs from a fixed velocity set and on discrete velocity models is designed in a suitable limit the procedure is shown to converge to the time discretized and spatially homogeneous boltzmann equation

k theoretic duality for shifts of finite type we will study the stable and unstable ruelle algebras associated to a hyperbolic homeomorphism of a compact space to do this we will describe a notion of k theoretic duality for algebras which generalizes spanier whitehead duality in topology a criterion for checking that it holds is presented as an application it is shown that the ruelle algebras which are associated to subshifts of finite type and agree with cuntz krieger algebras in this case satisfy this criterion and hence are dual

classification methodologies of multilayer perceptrons with sigmoid activation functions this paper studies the classification mechanisms of multilayer perceptrons mlps with sigmoid activation functions safs the viewpoint is presented that in the input space the hyperplanes determined by the hidden basis functions with values 0 s do not play the role of decision boundaries and such hyperplanes do not certainly go through the marginal regions between different classes for solving an n class problem a single hidden layer perceptron with at least log2 n 1 2 hidden nodes is needed the final number of hidden neurons is still related to the sample distribution shapes and regions but not to the number of samples and input dimensions as a result an empirical formula for optimally selecting the initial number of hidden nodes is proposed the ranks of response matrixes of hidden layers should be taken as a main basis for pruning or growing the existing hidden neurons a structure fixed perceptron ought to learn more than one round from different starting weight points for one classification task and only the group of weights and biases that has the best generalization performance should be reserved finally three examples are given to verify the above viewpoints

comments on can backpropagation error surface not have local minima in the above paper yu ieee trans neural networks vol 3 no 6 p 1019 21 1992 claims to prove that local minima do not exist in the error surface of backpropagation networks being trained on data with t distinct input patterns when the network is capable of exactly representing arbitrary mappings on t input patterns the commenter points out that the proof presented is flawed so that the resulting claims remain unproved in reply yu points out that the undesired phenomenon that was sited can be avoided by simply imposing the arbitrary mapping capacity of the network on lemma 1 in the article

constructive neural networks with piecewise interpolation capabilities for function approximations this paper proposes a constructive neural network with a piecewise linear or nonlinear local interpolation capability to approximate arbitrary continuous functions this neural network is devised by introducing a space tessellation which is a covering of the euclidean space by nonoverlapping hyperpolyhedral convex cells in the proposed neural network a number of neural network granules nng s are processed in parallel and repeated regularly with the same structures each nng does a local mapping with an interpolation capability for a corresponding hyperpolyhedral convex cell in a tessellation the plastic weights of the nng can be calculated to implement the mapping for training data consequently this reduces training time and alleviates the difficulties of local minima in training in addition the interpolation capability of the nng improves the generalization for the new data within the convex cell the proposed network requires additional neurons for tessellation over the standard multilayer neural networks this increases the network size but does not slow the retrieval response when implemented by parallel architecture

upper bounds on the number of hidden neurons in feedforward networks with arbitrary bounded nonlinear activation functions it is well known that standard single hidden layer feedforward networks slfns with at most n hidden neurons including biases can learn n distinct samples x sub i t sub i with zero error and the weights connecting the input neurons and the hidden neurons can be chosen almost arbitrarily however these results have been obtained for the case when the activation function for the hidden neurons is the signum function this paper rigorously proves that standard single hidden layer feedforward networks slfns with at most n hidden neurons and with any bounded nonlinear activation function which has a limit at one infinity can learn n distinct samples x sub i t sub i with zero error the previous method of arbitrarily choosing weights is not feasible for any slfn the proof of our result is constructive and thus gives a method to directly find the weights of the standard slfns with any such bounded nonlinear activation function as opposed to iterative training algorithms in the literature

convergence of the stochastic weighted particle method for the boltzmann equation this paper studies convergence of the stochastic weighted particle method for the boltzmann equation first the method is extended by introducing new stochastic reduction procedures in order to control the number of simulation particles then under rather general conditions convergence to the solution of the boltzmann equation is proved finally numerical experiments are performed illustrating both convergence and considerable variance reduction for the specific problem of calculating tails of the velocity distribution

the c0 x algebra of a net and index theory abstract given a connected and locally compact hausdorff space x with a good base k we assign in a functorial way a c 0 x algebra to any precosheaf of c algebras a defined over k afterwards we consider the representation theory and the kasparov k homology of a and interpret them in terms respectively of the representation theory and the k homology of the associated c 0 x algebra when a is an observable net over the spacetime x in the sense of algebraic quantum field theory this yields a geometric description of the recently discovered representations affected by the topology of x

a simple method to derive bounds on the size and to train multilayer neural networks a new derivation is presented for the bounds on the size of a multilayer neural network to exactly implement an arbitrary training set namely the training set can be implemented with zero error with two layers and with the number of the hidden layer neurons equal to hash 1 or p 1 the derivation does not require the separation of the input space by particular hyperplanes as in previous derivations the weights for the hidden layer can be chosen almost arbitrarily and the weights for the output layer can be found by solving hash 1 1 linear equations the method presented exactly solves m the multilayer neural network training problem for any arbitrary training set

on simultaneous approximations by radial basis function neural networks it is shown in this paper by a constructive method that any multivariate function and all its existing derivatives can be simultaneously approximated by a radial basis function rbf neural network where the assumptions on the rbfs are relatively mild by considering the translations of rbfs it is proved that an o m 1 2 order of simultaneous approximations to a certain class of functions can be achieved by using a translation network with just m neurons

deterministic convergence of an online gradient method for neural networks the online gradient method has been widely used as a learning algorithm for neural networks we establish a deterministic convergence of online gradient methods for the training of a class of nonlinear feedforward neural networks when the training examples are linearly independent we choose the learning rate to be a constant during the training procedure the monotonicity of the error function in the iteration is proved a criterion for choosing the learning rate is also provided to guarantee the convergence under certain conditions similar to those for the classical gradient methods an optimal convergence rate for our online gradient methods is proved

presheaves of superselection structures in curved spacetimes we show that superselection structures on curved spacetimes that are expected to describe quantum charges affected by the underlying geometry are categories of sections of presheaves of symmetric tensor categories when an embedding functor is given the superselection structure is a tannaka type dual of a locally constant group bundle which hence becomes a natural candidate for the role of the gauge group indeed we show that any locally constant group bundle with suitable structure group acts on a net of c algebras fulfilling normal commutation relations on an arbitrary spacetime we also give examples of gerbes of c algebras defined by wightman fields and constructed using projective representations of the fundamental group of the spacetime which we propose as solutions for the problem that existence and uniqueness of the embedding functor are not guaranteed

approximating polynomial functions by feedforward artificial neural networks capacity analysis and design polynomial functions are used in many applications in this paper we address the capacity of feedforward artificial neural networks fanns in approximating polynomial functions instead of studying the capacity of a fann with infinitely available hidden nodes which was proved to be a universal approximator we provide the capacity results of a fann with finite hidden nodes in approximating polynomial functions first we show that there is a relationship between the capacity of a fann in approximating polynomial functions and the number of hidden nodes used in the fann then we describe a procedure to realize a fann in approximating polynomial functions two examples are given to show the procedure several experiments are reported which verifies that a fann with a certain number of hidden nodes has the capability in learning given polynomial functions an extension of the approach for solving multiple criteria decision making problems is discussed the experiments also show that the proposed algorithm for training a fann performs accurately

the mwf method a convergence theorem for homogeneous one dimensional case the mwf numerical method for kinetic equations was presented by s motta and j wick in 1992 and recently extended by the authors to systems of kinetic equations the basic idea of the method consists in rewriting the kinetic equation in a conservation law in divergence form redefining the collisions as a flux and formally to transform the problem into a collisionless one in all tested cases the numerical results are in agreement with the exact solutions but a convergence proof of the method to the best of our knowledge is missing in this paper we present our investigation on the sufficient conditions that the collision operator may satisfy to guarantee a convergence proof of the method in the homogeneous one dimensional case this investigation is of both theoretical and applied interest

hermite interpolation by neural networks in this paper we study hermite interpolation by single layer feedforward networks we provide upper bound estimates for the minimum number of hidden units that can solve the hermite interpolation problem exactly we consider both the univariate and the multivariate cases

dimension reduction by local principal component analysis reducing or eliminating statistical redundancy between the components of high dimensional vector data enables a lower dimensional representation without significant loss of information recognizing the limitations of principal component analysis pca researchers in the statistics and neural network communities have developed nonlinear extensions of pca this article develops a local linear approach to dimension reduction that provides accurate representations and is fast to compute we exercise the algorithms on speech and image data and compare performance with pca and with neural network implementations of nonlinear pca we find that both nonlinear techniques can provide more accurate representations than pca and show that the local linear techniques outperform neural network implementations

interpolation by ridge polynomials and its application in neural networks we consider multivariate interpolation at arbitrary points and provide effective interpolation formulas by using simple ridge polynomials which essentially possess the nature of univariate polynomials in computation and show that ridge polynomials with least degree can be found for interpolation purpose which a computational algorithm is provided correspondingly we then apply the interpolation results to design a computational algorithm for training feedforward neural networks

a new light on nets of c algebras and their representations the present paper deals with the question of representability of nets of c algebras whose underlying poset indexing the net is not upward directed a particular class of nets called c net bundles is classified in terms of c dynamical systems having as group the fundamental group of the poset any net of c algebras has a canonical morphism into a c net bundle the enveloping net bundle which generalizes the notion of universal c algebra given by fredenhagen to nonsimply connected posets this allows a classification of nets in particular we call injective those nets such that the canonical morphism is faithful injectivity turns out to be equivalent to the existence of faithful representations we further relate injectivity to a generalized cech cocycle of the net and this allows us to give examples of nets exhausting the above classification

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

a generalized collision mechanism for stochastic particle schemes approximating boltzmann type equations abstract a generalized collision mechanism for boltzmann type stochastic particle schemes is developed this mechanism is based on the idea of random weight transfer originating from random discrete velocity models the problem of applying the new degrees of freedom for the purpose of variance reduction is studied some results of numerical experiments are given

dirac quantum fields on a manifold on globally hyperbolic lorentzian manifolds we construct field operators which satisfy the dirac equation and have a causal anticommutator ambiguities in the construction are removed by formulating the theory in terms of c algebras of local observables a generalized form of the haag kastler axioms is verified introduction a lorentzian manifold is a four dimensional manifold m together with a pseudo riemannian metric g of signature such manifolds are widely used as models for space time in particular in the absence of boundaries and gravitational fields one uses m r4 and g q minkowski metric which has components ab diag l 1 1 1 we are interested in formulating quantum field theories on general lorentzian manifolds this means giving up many ideas familiar from the usual minkowski space treatments thus there are no poincare transformations on m no vacuum no particle states and so forth all one is left with are the field equations these are to be solved taking as data a representation of the ccr or car over a space like hypersurface ccr canonical commutation relations car canonical anticommutation relations for linear equations this quantum problem can be reduced to a corresponding classical problem for which solutions can be constructed in favorable cases in the nonlinear case very singular formal solutions can be exhibited which one might hope to make rigorous the procedure sketched above is fraught with ambiguities what hypersurface should one take what representation of the ccr car and so forth in general there is no natural choice and one should show that all choices lead to the same theory to do this in a fundamental way it seems necessary to abandon the field operators themselves and instead consider the c algebras that they generate this algebraic approach was originally formulated for minkowski space and seems all the more appropriate here in the haag kastler approach 10 one focuses on the c algebras associated with local regions of space time the algebras of local observables the original formulation on minkowski space consisted of a list of structural properties or axioms which such algebras ought to obey in a recent paper 6 we have proposed a generalization of these axioms to lorentzian manifolds the scalar field was given as an example received by the editors july 23 1980 and in revised form december 11 1980 1980 mathematics subject classification primary 81a17 supported by national science foundation phy77 21740 1982 american mathematical society 0002 9947 82 0000 0 3 12 04 75

universal approximation using incremental constructive feedforward networks with random hidden nodes according to conventional neural network theories single hidden layer feedforward networks slfns with additive or radial basis function rbf hidden nodes are universal approximators when all the parameters of the networks are allowed adjustable however as observed in most neural network implementations tuning all the parameters of the networks may cause learning complicated and inefficient and it may be difficult to train networks with nondifferential activation functions such as threshold networks unlike conventional neural network theories this paper proves in an incremental constructive method that in order to let slfns work as universal approximators one may simply randomly choose hidden nodes and then only need to adjust the output weights linking the hidden layer and the output layer in such slfns implementations the activation functions for additive nodes can be any bounded nonconstant piecewise continuous functions g r r and the activation functions for rbf nodes can be any integrable piecewise continuous functions g r r and rg x dx 0 the proposed incremental method is efficient not only for sflns with continuous including nondifferentiable activation functions but also for slfns with piecewise continuous such as threshold activation functions compared to other popular methods such a new network is fully automatic and users need not intervene the learning process by manually tuning control parameters

osculatory interpolation in mathbb r n in this work we study the hermite osculatory interpolation problems in several variables we give general lagrange formulae for the solution of these problems and exhibit examples where these formulae apply in the context of p m correct hermite problems we finally give an inductive characterization of a family of p m correct hermite problems which we call totally reducible

identification of piecewise affine systems via mixed integer programming this paper addresses the problem of identification of hybrid dynamical systems by focusing the attention on hinging hyperplanes and wiener piecewise affine autoregressive exogenous models in which the regressor space is partitioned into polyhedra with affine submodels for each polyhedron in particular we provide algorithms based on mixed integer linear or quadratic programming which are guaranteed to converge to a global optimum for the special case where the estimation data only seldom switches between the different submodels we also suggest a way of trading off between optimality and complexity by using a change detection approach

global optimization of statistical functions with simulated annealing abstract many statistical methods rely on numerical optimization to estimate a model s parameters unfortunately conventional algorithms sometimes fail even when they do converge there is no assurance that they have found the global rather than a local optimum we test a new optimization algorithm simulated annealing on four econometric problems and compare it to three common conventional algorithms not only can simulated annealing find the global optimum it is also less likely to fail on difficult functions because it is a very robust algorithm the promise of simulated annealing is demonstrated on the four econometric problems

robust single hidden layer feedforward network based pattern classifier in this paper a new robust single hidden layer feedforward network slfn based pattern classifier is developed it is shown that the frequency spectrums of the desired feature vectors can be specified in terms of the discrete fourier transform dft technique the input weights of the slfn are then optimized with the regularization theory such that the error between the frequency components of the desired feature vectors and the ones of the feature vectors extracted from the outputs of the hidden layer is minimized for the linearly separable input patterns the hidden layer of the slfn plays the role of removing the effects of the disturbance from the noisy input data and providing the linearly separable feature vectors for the accurate classification however for the nonlinearly separable input patterns the hidden layer is capable of assigning the dfts of all feature vectors to the desired positions in the frequency domain such that the separability of all nonlinearly separable patterns are maximized in addition the output weights of the slfn are also optimally designed so that both the empirical and the structural risks are well balanced and minimized in a noisy environment two simulation examples are presented to show the excellent performance and effectiveness of the proposed classification scheme

on a natural homotopy between linear and nonlinear single layer networks in this paper we formulate a homotopy approach for solving for the weights of a network by smoothly transforming a linear single layer network into a nonlinear perceptron network while other researchers have reported potentially useful numerical results based on heuristics related to this approach the work presented here provides the first rigorous exposition of the deformation process results include a complete description of how the weights relate to the data space a proof of the global convergence and validity of the method and a rigorous formulation of the generalized orthogonality theorem to provide a geometric perspective of the solution process this geometric interpretation clarifies conditions resulting in the appearance of local minima and infinite weights in network optimization procedures and the similarities of and differences between optimizing the weights in a nonlinear network and optimizing the weights in a linear network the results provide a strong theoretical foundation for quantifying performance bounds on finite neural networks and for constructing globally convergent optimization approaches on finite data sets

on the uniqueness of weights in single layer perceptrons in this paper the geometric formulation of the single layer perceptron weight optimization problem previously described by coetzee et al 1993 1996 is combined with results from other researchers on nonconvex set projections to describe sufficient conditions for uniqueness of weight solutions it is shown that the perceptron data surface is pseudoconvex and has infinite folding allowing for the specification of a region of desired vectors having unique projections purely in terms of the local curvature of the data surface no information is therefore required regarding the global curvature or size of the data surface these results in principle allow for a posteriori evaluation of whether a weight solution is unique or globally optimal and for a priori scaling of desired vector values to ensure uniqueness through analysis of the input data the practical applicability of these results from a numerical perspective is evaluated on some carefully chosen examples

an analysis of the elastic net approach to the traveling salesman problem this paper analyzes the elastic net approach durbin and willshaw 1987 to the traveling salesman problem of finding the shortest path through a set of cities the elastic net approach jointly minimizes the length of an arbitrary path in the plane and the distance between the path points and the cities the tradeoff between these two requirements is controlled by a scale parameter k a global minimum is found for large k and is then tracked to a small value in this paper we show that 1 in the small k limit the elastic path passes arbitrarily close to all the cities but that only one path point is attracted to each city 2 in the large k limit the net lies at the center of the set of cities and 3 at a critical value of k the energy function bifurcates we also show that this method can be interpreted in terms of extremizing a probability distribution controlled by k the minimum at a given k corresponds to the maximum a posteriori map bayesian estimate of the tour under a natural statistical interpretation the analysis presented in this paper gives us a better understanding of the behavior of the elastic net allows us to better choose the parameters for the optimization and suggests how to extend the underlying ideas to other domains

neural networks for localized approximation we prove that feedforward artificial neural networks with a single hidden layer and an ideal sigmoidal response function cannot provide localized approximation in a euclidean space of dimension higher than one we also show that networks with two hidden layers can be designed to provide localized approximation since wavelet bases are most effective for local approximation we give a discussion of the implementation of spline wavelets using multilayered networks where the response function is a sigmoidal function of order at least two

on the geometry of feedforward neural network error surfaces many feedforward neural network architectures have the property that their overall input output function is unchanged by certain weight permutations and sign flips in this paper the geometric structure of these equioutput weight space transformations is explored for the case of multilayer perceptron networks with tanh activation functions similar results hold for many other types of neural networks it is shown that these transformations form an algebraic group isomorphic to a direct product of weyl groups results concerning the root spaces of the lie algebras associated with these weyl groups are then used to derive sets of simple equations for minimal sufficient search sets in weight space these sets which take the geometric forms of a wedge and a cone occupy only a minute fraction of the volume of weight space a separate analysis shows that large numbers of copies of a network performance function optimum weight vector are created by the action of the equioutput transformation group and that these copies all lie on the same sphere some implications of these results for learning are discussed

a theory of bundles over posets in algebraic quantum field theory the spacetime manifold is replaced by a suitable base for its topology ordered under inclusion we explain how certain topological invariants of the manifold can be computed in terms of the base poset we develop a theory of connections and curvature for bundles over posets in search of a formulation of gauge theories in algebraic quantum field theory

the mwf method for kinetic equations system many physical or biological phenomena deal with the dynamics of interacting entities these class of phenomena are well described in physics using a kinetic approach based on boltzmann equation a generalized kinetic theory has been proposed to extend this approach to biological scenarios an analytical solution of boltzmann equation can be found only in very simple cases so numerical methods become extremely relevant the particle method is a class of numerical methods used to find a numerical solution of boltzmann equations the mwf method for kinetic equations was firstly proposed by s motta and j wick in 1992 here we show that the mwf method can be extended to system of boltzamm equations

topology and geometry of single hidden layer network least squares weight solutions in this paper the topological and geometric properties of the weight solutions for multilayer perceptron mlp networks under the mse error criterion are characterized the characterization is obtained by analyzing a homotopy from linear to nonlinear networks in which the hidden node function is slowly transformed from a linear to the final sigmoidal nonlinearity two different geometric perspectives for this optimization process are developed the generic topology of the nonlinear mlp weight solutions is described and related to the geometric interpretations error surfaces and homotopy paths both analytically and using carefully constructed examples these results illustrate that although the natural homotopy provides a practically valuable heuristic for training it suffers from a number of theoretical and practical difficulties the linear system is a bifurcation point of the homotopy equations and solution paths are therefore generically discontinuous bifurcations and infinite solutions further occur for data sets that are not of measure zero these results weaken the guarantees on global convergence and exhaustive behavior normally associated with homotopy methods however the analyses presented provide a clear understanding of the relationship between linear and nonlinear perceptron networks and thus a firm foundation for development of more powerful training methods the geometric perspectives and generic topological results describing the nature of the solutions are further generally applicable to network analysis and algorithm evaluation

on the history of multivariate polynomial interpolation multivariate polynomial interpolation is a basic and fundamental subject in approximation theory and numerical analysis which has received and continues receiving not deep but constant attention in this short survey we review its development in the first 75 years of this century including a pioneering paper by kronecker in the 19th century

multivariate hermite interpolation by algebraic polynomials a survey abstract this is a survey of that theory of multivariate lagrange and hermite interpolation by algebraic polynomials which has been developed in the past 20 years its purpose is not to be encyclopedic but to present the basic concepts and techniques which have been developed in that period of time and to illustrate them with examples it takes classical hermite interpolation as a starting point but then successively broadens the assumptions so that finally interpolation of arbitrary functionals and the theory of singularities from algebraic geometry is discussed

a new duality theory for compact groups the tannaka krein duality theory characterizes the category g of finite dimensional continuous unitary representations of a compact group as a subcategory of the category of hilbert spaces we prove a more powerful result characterizing g as an abstract category every strict symmetric monoidalc category with conjugates which has subobjects and direct sums and for which thec algebra of endomorphisms of the monoidal unit reduces to the complex numbers is isomorphic to a category g for a compact groupg unique up to isomorphism

the essential order of simultaneous approximation for neural networks by establishing both upper and lower bound estimations on simultaneous approximation order the essential simultaneous approximation order of the network which can simultaneously approximate function and its derivatives is estimated and the theorem of saturation the largest capacity of simultaneous approximation is proved these results can precisely characterize the simultaneous approximation ability of the network and the relationship among the rate of simultaneous approximation the topological structure of hidden layer and the properties of approximated function

bundles of c categories ii c dynamical systems and dixmier douady invariants we introduce a cohomological invariant arising from a class in nonabelian cohomology this invariant generalises the dixmier douady class and encodes the obstruction to a c algebra bundle being the fixed point algebra of a gauge action as an application the duality breaking for group bundles vs tensor c categories with nonsimple unit is discussed in the setting of nistor troitsky gauge equivariant k theory there is a map assigning a nonabelian gerbe to a tensor category and triviality of the gerbe is equivalent to the existence of a dual group bundle at the c algebraic level this corresponds to studying c algebra bundles with fibre a fixed point algebra of the cuntz algebra and in this case our invariant describes the obstruction to finding an embedding into the cuntz pimsner algebra of a vector bundle

numerical study of a stochastic weighted particle method for a model kinetic equation a stochastic weighted particle method is applied to a model nonlinear kinetic equation a detailed study of various numerical approximations is presented the main effect achieved by the new method is an artificial increase of the relative number of simulation particles with prescribed velocities

nonlinear adaptive control using networks of piecewise linear approximators presents a stable nonparametric adaptive control approach using a piecewise local linear approximator the continuous piecewise linear approximator is developed and its universal approximation capability is proved the controller architecture is based on adaptive feedback linearization plus sliding mode control a time varying activation region is introduced for efficient self organization of the approximator during operation we modify the adaptive control approach for piecewise linear approximation and self organizing structures in addition we provide analyses of asymptotic stability of the tracking error and parameter convergence for the proposed adaptive control scheme with the online self organizing structure the method with a deadzone is also discussed to prevent a high frequency input which might excite the unmodeled dynamics in practical applications the application of the piecewise linear adaptive control method is demonstrated by a computational simulation

quantum charges and spacetime topology the emergence of new superselection sectors a new form of superselection sectors of topological origin is developed by that it is meant a new investigation that includes several extensions of the traditional framework of doplicher haag and roberts in local quantum theories at first we generalize the notion of representations of nets of c algebras then we provide a brand new view on selection criteria by adopting one with a strong topological flavour we prove that it is coherent with the older point of view hence a clue to a genuine extension in this light we extend roberts cohomological analysis to the case where 1 cocycles bear non trivial unitary representations of the fundamental group of the spacetime equivalently of its cauchy surface in the case of global hyperbolicity a crucial tool is a notion of group von neumann algebras generated by the 1 cocycles evaluated on loops over fixed regions one proves that these group von neumann algebras are localized at the bounded region where loops start and end and to be factorial of finite type i all that amounts to a new invariant in a topological sense which can be defined as the dimension of the factor we prove that any 1 cocycle can be factorized into a part that contains only the charge content and another where only the topological information is stored this second part much resembles what in literature is known as geometric phases indeed by the very geometrical origin of the 1 cocycles that we discuss in the paper they are essential tools in the theory of net bundles and the topological part is related to their holonomy content at the end we prove the existence of net representations