measuring the strangeness of strange attractors we study the correlation exponent v introduced recently as a characteristic measure of strange attractors which allows one to distinguish between deterministic chaos and random noise the exponent v is closely related to the fractal dimension and the information dimension but its computation is considerably easier its usefulness in characterizing experimental data which stem from very high dimensional systems is stressed algorithms for extracting v from the time series of a single variable are proposed the relations between the various measures of strange attractors and between them and the lyapunov exponents are discussed it is shown that the conjecture of kaplan and yorke for the dimension gives an upper bound for v various examples of finite and infinite dimensional systems are treated both numerically and analytically

approximation schemes for infinite linear programs this paper presents approximation schemes for an infinite linear program in particular it is shown that under suitable assumptions the program s optimum value can be approximated by the values of finite dimensional linear programs and that in addition every accumulation point of a sequence of optimal solutions for the approximating programs is an optimal solution for the original problem

generalization in a multi state neural network the generalization ability of an extremely dilute feedback neural network with multi state neurons is studied by means of a deterministic noiseless parallel dynamics the overlap with any one of a macroscopic number of binary full activity concepts is determined when the network is trained with examples of variable activity according to a hebbian learning algorithm that favours stable symmetric mixture states explicit results about the phase diagram and the generalization error are obtained for a network with three state neurons which remain inactive below a threshold it is shown that the generalization ability can be considerably enhanced either by training the network with low activity examples or by means of a moderate increase in

robustness against random dilution in attractor neural networks the authors study the robustness of attractor neural networks against random disruption of a fraction of the synaptic couplings for the maximally stable network msn the authors determine the effect of different degrees of dilution on the overall storage capacity the size of the basins of attraction and the attractor overlap comparison with corresponding results for the hopfield model indicates that although msn is more robust at low degrees of dilution the hopfield network becomes more robust at high dilution

dynamics of ising random bond models neural network and random anisotropy axis model a discrete time retrieval dynamics for a class of random bond models with infinite range interactions is studied in a unified picture the hopfield model of neural networks with the hebb learning rule is considered together with the ising random anisotropy axis model in the strong anisotropy limit the main overlap magnetization the residual overlap and its dispersion as well as the correlation between the gaussian component of the residual overlap and the initial value of the latter obtained in the first step of a recursion relation are used to infer the structure of the recursion for large times a crucial assumption is the strong stationarity of the gaussian component the dynamics are discussed for finite alpha p n the storage ratio in the neural network problem or the ratio of random axis components per site in the limit where both p and n go to infinity the long time behaviour of the theory is shown to yield the equilibrium solution of an earlier work in mean field theory for a tri modal distribution of random axis components explicit results for the basins of attraction of either a ferromagnetic or a spin glass phase are obtained as well as the relaxation time with a square root power law decay near saturation

thermodynamic properties of fully connected q ising neural networks the thermodynamic and retrieval properties of fully connected q ising networks are studied in the replica symmetric mean field approximation in particular capacity gain parameter and capacity temperature phase diagrams are derived for q 3 4 and q infinity and different distributions of the stored patterns furthermore the optimal gain function is determined in order to obtain the best performance where appropriate the results are compared with the diluted and layered versions of these models

onset of super retrieval phase and enhancement of the storage capacity in neural networks of nonmonotonic neurons analogue neural networks of associative memory with continuous time dynamics are studied for nonmonotonic transfer functions using the method of self consistent signal to noise analysis the hebb learning rule with unbiased random patterns is assumed for the synaptic couplings a novel phenomenon is found to occur as a result of a phase transition concerning the property of the local field distribution in retrieval states of the newly found phase which the authors refer to as the super retrieval phase noise in the local field vanishes and the memory retrieval without errors ensures even for an extensive number of memory patterns stored under the local learning rule the storage capacity is obtained as a function of the parameter representing the degree of nonmonotonicity of the transfer functions with the result that an enhancement of the storage capacity can also occur

the geometry of optimal transportation introduction 113 1 summary of main results 120 2 background on optimal measures 126 part i strictly convex costs 133 3 existence and uniqueness of optimal maps 133 4 characterization of the optimal map 137 part ii costs which are strictly concave as a function of d i s t a n c e 141 5 the role of optimal maps 141 6 uniqueness of optimal solutions 144 part iii appendices 148 a legendre transforms and conjugate costs 148 b examples of c concave potentials 152 c regularity of c concave potentials 154 references 159

categorization in a layered neural network abstract the layered feedforward network of domany meir and kinzel is extended to investigate the ability to recognize any one of a macroscopic number of concepts ancestors when the network is trained through a finite number of examples descendents of each concept by means of a generalized hebbian learning rule between cells on two consecutive layers learning curves describing the generalization error are obtained as well as phase diagrams and basins of attraction that exhibit the possible coexistence of a generalization phase a retrieval phase of examples and a spin glass phase

categorization ability in a biologically motivated neural network abstract in this work we study the categorization properties of an extreme and asymmetrically diluted version of the hopfield model for associative memory when the effect of the refractory period of the neurons is taken into account in the dynamics of the system the simplest way of modeling these refractory periods is by means of a time dependent threshold that acts only on those neurons that emit a signal and favors them to be at rest during a given time interval the dynamic equations are derived in the limit of extreme dilution using an approach that explicitly preserves the dependence of the system on its whole history the categorization error is analyzed for different values of the parameters in particular we confront our analytical results with numerical simulations for the noiseless case t 0 when the number of examples or their correlations increases the system always categorizes independently of the amplitude of the potential that mimics the effect of the refractory period

on the parallel dynamics for the little hopfield model we propose a method algorithm for calculation of the explicit formulas for evolution of the main and the residual overlaps it allows us to confirm the gardner derrida mottishaw second step formula for the main overlap and to go beyond to the next steps we discuss the dynamical status of the amit gutfreund sompolinsky formula for the main overlap and some computersimulation results

in search of an optimal dilution algorithm for feedforward networks in a recent paper the authors presented a dilution algorithm that yields a storage capacity per synapse alpha eff larger than 2 in this letter two new algorithms with even higher alpha eff values are introduced they study a one step dilution algorithm where the perceptron is used to select a fraction of the couplings to be removed for the remaining bonds the perceptron of optimal stability is relearned they further compare simulation data from an iterative version of the one step dilution algorithm with phase space volume calculations

categorization in a hopfield network trained with weighted examples i finite number of concepts abstract we consider the categorization problem in a hopfield network with a finite number of concepts and trained with s examples of weight 1 s we find that the retrieval capacity of an example with weight 1 and the corresponding categorization error depends also on the arithmetic mean m 1 s 1 2 s of the other weights for 1 m 1 we find that the line of first order transitions between the retrieval and categorization phases ends at a critical point in the s t plane when two solutions are present the global minimum of the free energy corresponds to the solution with the highest weight

generalization in a hopfield network with noise the generalization ability of the hopfield model of neural networks trained with examples is studied in mean field theory in the presence of synaptic noise although the latter improves the generalization ability for a finite number of concepts it does not for a macroscopic number of them nevertheless the network performance is still robust against synaptic noise numerical simulations are performed to verify the mean field theory predictions

retrieval and chaos in layered q ising neural networks using a probabilistic approach we study the parallel dynamics of theq ising layered networks for arbitraryq by introducing auxiliary thermal fields we express the stochastic dynamics within the gain function formulation of the deterministic dynamics evolution equations are derived for arbitraryq at both zero and finite temperatures an explicit analysis of the fixed point equations is carried out for bothq 3 andq the retrieval properties are discussed in terms of the gain parameter the storage capacity and the temperature using the time evolution of the distance between two network configurations we investigate the possibility of microscopic chaos chaotic behavior is always present for arbitrary finiteq however in the limitq the existence of chaos depends on the parameters of the system

complex behaviour in a production distribution model abstract the purpose of this paper is to show how the cascaded structure of a production distribution chain can produce a wide variety of dynamic behaviours despite the apparent simplicity of the chain the bounded rationality of human decision making interferes with the information feedbacks nonlinearities and time delays of the system to produce exceedingly complex behaviour when simulated with realistic parameter values the model can exhibit stationary periodic quasiperiodic as well as chaotic and hyperchaotic motions lyapunov exponents are used to characterize the different types of behaviour repeated simulation reveals how the profile of operating cost is related to the modes of behaviour and to the conditions that generate chaos detailed maps of parameter space graphically show the influence of inventory control policies upon dynamics and costs the analyses illustrate why it is so difficult for decision makers to navigate in low cost policy parameter regions suggestions are made for decision heuristics that avoid high cost unstable solutions

the roles of stability and symmetry in the dynamics of neural networks the authors study the retrieval phase of spin glass like neural networks considering that the dynamics should depend only on gauge invariant quantities they propose that two such parameters characterising the symmetry of the neural net s connections and the stabilities of the patterns are responsible for most of the dynamical effects this is supported by a numerical study of the shape of the basins of attraction for a one pattern neural network model the effects of stability and symmetry on the short time dynamics of this model are studied analytically and the full dynamics for vanishing symmetry is shown to be exactly solvable

estimating correlation dimension from a chaotic time series when does plateau onset occur abstract suppose that a dynamical system has a chaotic attractor a with a correlation dimension d 2 a common technique to probe the system is by measuring a single scalar function of the system state and reconstructing the dynamics in an m dimensional space using the delay coordinate technique the estimated correlation dimension of the reconstructed attractor typically increases with m and reaches a plateau on which the dimension estimate is relatively constant for a range of large enough m values the plateaued dimension value is then assumed to be an estimate of d 2 for the attractor in the original full phase space in this paper we first present rigorous results which state that for a long enough data string with low enough noise the plateau onset occurs at m ceil d 2 where ceil d 2 standing for ceiling of d 2 is the smallest integer greater than or equal to d 2 we then numerical examples illustrating the theoretical prediction in addition we discuss new findings showing how practical factors such as a lack of data and observational noise can produce results that may seem to be inconsistent with the theoretically predicted plateau onset at m ceil d 2

parallel dynamics of the neural network with the pseudoinverse coupling matrix the authors investigate the parallel dynamics of the neural network with the pseudoinverse coupling matrix based on an exact dynamic theory they develop an approximate treatment for long timescales the temporal development of the overlap of correlation functions and of the remanent magnetization are investigated they present results for deterministic and stochastic dynamics large scale numerical simulations supplement their analytical findings

the three state layered neural network with finite dilution abstract the dynamics and the stationary states of an exactly solvable three state layered feed forward neural network model with asymmetric synaptic connections finite dilution and low pattern activity are studied in extension of a recent work on a recurrent network detailed phase diagrams are obtained for the stationary states and for the time evolution of the retrieval overlap with a single pattern it is shown that in spite of instabilities for low thresholds there is a gradual improvement in network performance with increasing threshold up to an optimal stage the robustness to synaptic noise is checked and the effects of dilution and of variable threshold on the information content of the network are also established

feed forward chains of recurrent attractor neural networks with finite dilution near saturation abstract a stationary state replica analysis for a dual neural network model that interpolates between a fully recurrent symmetric attractor network and a strictly feed forward layered network studied by coolen and viana is extended in this work to account for finite dilution of the recurrent hebbian interactions between binary ising units within each layer gradual dilution is found to suppress part of the phase transitions that arise from the competition between recurrent and feed forward operation modes of the network despite that a long chain of layers still exhibits a relatively good performance under finite dilution for a balanced ratio between inter layer and intra layer interactions

practical method for determining the minimum embedding dimension of a scalar time series abstract a practical method is proposed to determine the minimum embedding dimension from a scalar time series it has the following advantages 1 does not contain any subjective parameters except for the time delay for the embedding 2 does not strongly depend on how many data points are available 3 can clearly distinguish deterministic signals from stochastic signals 4 works well for time series from high dimensional attractors 5 is computationally efficient several time series are tested to show the above advantages of the method

simulating highly diluted neural networks the asymmetric diluted hopfield model is studied through numerical simulations and reliable results are obtained even when the network sizes are not exponentially large the mean final overlap the mean convergence time and a quantity that gives information on the relative importance of spurious states are measured as a function of the load capacity alpha

categorization in the pseudo inverse neural network we investigate analytically the emergence of the categorization ability in the pseudo inverse attractor neural network more pointedly we consider the problem of learning an extensive number of concepts by storing a finite number of examples s of each concept we find that there is a critical value beyond which the categorization error as measured by the average fraction of unstable sites in the concepts decreases monotonically with s

storing an extensive number of grey toned patterns in a neural network using multistate neurons grey toned patterns are pictures composed of pixels of several shades of grey the ability of neural networks using multistate neurons to store such patterns is systematically investigated if conventional generalizations of hopfield networks using analogue or soft neurons are considered it is impossible to stabilize these grey tones nevertheless it is shown that it can be done with networks that use neurons which have only a discrete set of possible activities this is demonstrated for the pseudo inverse rule for the synaptic couplings where only the stability of the patterns shrinks with increasing number q of grey tones one wants to store if the patterns are uncorrelated one can use the hebb rule and in this case the mean field theory is presented applying this rule the storage capacity decreases as q 2 with the number of grey tones

on the parallel dynamics of the q state potts and q ising neural networks using a probabilistic approach the parallel dynamics of theq state potts andq ising neural networks are studied at zero and at nonzero temperatures evolution equations are derived for the first time step and arbitraryq these formulas constitute recursion relations for the exact parallel dynamics of the extremely diluted asymmetric versions of these networks an explicit analysis including dynamical capacity temperature diagrams and the temperature dependence of the overlap is carried out forq 3 both types of models are compared

optimal solution of a monge kantorovitch transportation problem abstract suppose that p and q are probabilities on a separable banach space e it is known that if p q satisfies certain regularity conditions and a random variable x has law p then there exists a function f e e such that the function f x has the law q and the random pair x f x is an optimal coupling for the monge kantorovitch problem in this paper we provide an approximation of the function f when the law q is discrete thenwe extend this main result to any law q the proofs are based on a relationship between optimal couplings and nonlinear equations

information processing in three state neural networks we introduce networks of three state neurons where the additional state embodies the absence of information their dynamical behavior is studied from the standpoint of information processing these networks display strong pattern completion capabilities moreover inference naturally occurs between coherent patterns

neural networks a biased overview an overview of recent activity in the field of neural networks is presented the long range aim of this research is to understand how the brain works first some of the problems are stated and terminology defined then an attempt is made to explain why physicists are drawn to the field and their main potential contribution in particular in recent years some interesting models have been introduced by physicists a small subset of these models is described with particular emphasis on those that are analytically soluble finally a brief review of the history and recent developments of single and multilayer perceptrons is given bringing the situation up to date regarding the central immediate problem of the field search for a learning algorithm that has an associated convergence theorem

parallel dynamics of fully connected q ising neural networks using a probabilistic approach the parallel dynamics of fully connected q ising neural networks is studied for arbitrary q a novel recursive scheme is set up to determine the time evolution of the order parameters through the evolution of the distribution of the local field taking into account all feedback correlations in contrast to extremely diluted and layered network architectures the local field is no longer normally distributed but contains a discrete part as an illustrative example an explicit analysis is carried out for the first four time steps for the case of the q 2 and q 3 model the results are compared with extensive numerical simulations and excellent agreement is found finally equilibrium fixed point equations are derived and compared with the thermodynamic approach based upon the replica symmetric mean field approximation

predicting the future advantages of semilocal units in investigating gaussian radial basis function rbf networks for their ability to model nonlinear time series we have found that while rbf networks are much faster than standard sigmoid unit backpropagation for low dimensional problems their advantages diminish in high dimensional input spaces this is particularly troublesome if the input space contains irrelevant variables we suggest that this limitation is due to the localized nature of rbfs to gain the advantages of the highly nonlocal sigmoids and the speed advantages of rbfs we propose a particular class of semilocal activation functions that is a natural interpolation between these two families we present evidence that networks using these gaussian bar units avoid the slow learning problem of sigmoid unit networks and very importantly are more accurate than rbf networks in the presence of irrelevant inputs on the mackey glass and coupled lattice map problems the speedup over sigmoid networks is so dramatic that the difference in training time between rbf and gaussian bar networks is minor gaussian bar architectures that superpose composed gaussians gaussians of gaussians to approximate the unknown function have the best performance we postulate that an interesing behavior displayed by gaussian bar functions under gradient descent dynamics which we call automatic connection pruning is an important factor in the success of this representation

exact solutions to the transportation problem on the line given distributions w of production and v of consumption on the line the monge kantorovich problem is to decide which producer should supply each consumer in order to minimize the total transportation costs here cost will be assumed to be a strictly concave function of the distance which translates into an economy of scale for longer trips and may encourage cross hauling the resulting solutions display a hierarchical structure that reflects a striking separation into local and global scales also found in the real world moreover this structure can be exploited to reduce the infinite dimensional linear problem to a convex minimization in m variables where 2 m 2 counts the number of times that w v changes sign a combinatorial algorithm is then derived which yields exact solutions by optimizing a certain finite sequence of convex separable network flows

retrieval and chaos in extremely diluted q ising neural networks using a probabilistic approach the deterministic and the stochastic parallel dynamics of aq ising neural network are studied at finiteq and in the limitq exact evolution equations are presented for the first time step these formulas constitute recursion relations for the parallel dynamics of the extremely diluted asymmetric versions of these networks an explicit analysis of the retrieval properties is carried out in terms of the gain parameter the loading capacity and the temperature the results for theq network are compared with those for theq 3 andq 4 models possible chaotic microscopic behavior is studied using the time evolution of the distance between two network configurations for arbitrary finiteq the retrieval regime is always chaotic in the limitq the network exhibits a dynamical transition toward chaos

extracting qualitative dynamics from experimental data abstract we consider the notion of qualitative information and the practicalities of extracting it from experimental data our approach based on a theorem of takens draws on ideas from the generalized theory of information known as singular system analysis due to bertero pike and co workers we illustrate our technique with numerical data from the chaotic regime of the lorenz model

layered neural networks with non monotonic transfer functions abstract we investigate storage capacity and generalization ability for two types of fully connected layered neural networks with non monotonic transfer functions random patterns are embedded into the networks by a hebbian learning rule one of them is a layered network in which a non monotonic transfer function of even layers is different from that of odd layers the other is a layered network with intra layer connections in which the non monotonic transfer function of inter layer is different from that of intra layer and inter layered neurons and intra layered neurons are updated alternately we derive recursion relations for order parameters for those layered networks by the signal to noise ratio method we clarify that the storage capacity and the generalization ability for those layered networks are enhanced in comparison with those with a conventional monotonic transfer function when non monotonicity of the transfer functions is selected optimally we also point out that some chaotic behavior appears in the order parameters for the layered networks when non monotonicity of the transfer functions increases

interpreting supply chain dynamics a quasi chaos perspective abstract chaotic phenomena chaos amplification and other interesting nonlinear behaviors have been observed in supply chain systems chaos can be defined theoretically if the dynamics under study are produced only by deterministic factors however deterministic settings rarely present themselves in reality in fact real data are typically unknown how can the chaos theory and its related methodology be applied in the real world when the demand is stochastic the interpretation and distribution of the lyapunov exponents derived from the effective inventory at different supply chain levels are not similar to those under deterministic demand settings are the observed dynamics of the effective inventory random chaotic or simply quasi chaos in this study we investigate a situation whereby the chaos analysis is applied to a time series as if its underlying structure deterministic or stochastic is unknown the result shows clear distinction in chaos characterization between the two categories of demand process deterministic vs stochastic it also highlights the complexity of the interplay between stochastic demand processes and nonlinear dynamics therefore caution should be exercised in interpreting system dynamics when applying chaos analysis to a system of unknown underlying structure by understanding this delicate interplay decision makers have the better chance to tackle the problem correctly or more effectively at the demand end or the supply end

a probabilistic approach to parallel dynamics for the little hopfield model presents new results on a probabilistic approach to parallel dynamics of the little hopfield model the authors propose a truncated auxiliary dynamics method to control a feedback noise in this symmetrical neural network with full connection it allows them to propose an ansatz for derivation of the explicit recurrence relations for the main and residual noisy overlap evolution for arbitrary discrete moment t

a general duality theorem for marginal problems given probability spaces xi ai pi i 1 2 letm p1 p2 denote the set of all probabilities on the product space with marginalsp1 andp2 and leth be a measurable function on x1 x2 a1 a2 in order to determine supfh dp where the supremum is taken overp inm p1 p2 a general duality theorem is proved only the perfectness of one of the coordinate spaces is imposed without any further topological or tightness assumptions an example without any further topological or tightness assumptions an example is given to show that the assumption of perfectness is essential applications to probabilities with given marginals and given supports stochastic order and probability metrics are included

neural networks that use three state neurons the dynamics of a neural network which uses three state neurons 1 0 and 1 is solved exactly in the limit of non symmetric and highly dilute synapses recursion relations for the activity the fraction of non zero neurons and overlap of the network with a given pattern are derived which have three generic kinds of fixed points a retrieval fixed point a chaotic fixed point which corresponds to non zero activity but no overlap and a zero fixed point where all the neurons go to the 0 state as the non retrieval fixed points both have activities different from the retrieval fixed point one can easily tell whether a pattern has been recovered an analysis of which fixed points occur as a function of the thresholds and the storage ratio of the system yields remarkably rich phase diagrams optimising the threshold level can be very important especially when low activity patterns are stored a similar analysis can be applied to biased networks using two state 1 0 neurons finally one finds that mixture states which have an overlap with two patterns can be stabilised by a threshold in the networks using three state neurons this property allows larger higher activity memories to be naturally constructed out of smaller ones

stability analysis of constrained inventory systems with transportation delay stability is a fundamental design property of inventory systems however the often exploited linearity assumptions in the current literature create a major gap between theory and practice in this paper the stability of a constrained production and inventory system with a forbidden returns constraint that is a non negative order rate is studied via a piecewise linear model an eigenvalue analysis and a simulation investigation the apviobpcs automatic pipeline variable inventory and order based production control system and epviobpcs estimated pipeline variable inventory and order based production control system replenishment policies are adopted surprisingly all kinds of non linear dynamical behaviours of systems can be observed in these simple models exact expressions of the asymptotic stability boundaries and lyapunovian stability boundaries are derived when actual and perceived transportation lead time is 1 and 2 periods long respectively asymptotically stable regions in the non linear forbidden return systems are identical to the stable regions in its unconstrained counterpart however regions of bounded fluctuations that continue forever including both periodicity and chaos exist in the parametrical plane outside the asymptotically stable region simulation shows a complex and delicate structure in these regions the results suggest that accurate lead time information is essential to eliminate inventory drift and instability and that ordering policies have to be designed properly in accordance with the actual lead time to avoid these fluctuations and divergence

order parameter evolution in a feedforward neural network we consider layered neural networks in which the weights are trained with the pseudoinverse rule to store a set of random patterns using many body diagrammatic techniques the evolution in the network can be described by the overlap order parameter m and the noise parameter delta looping effects are shown to be significant in contrast to a previous conjecture order parameter pairs corresponding to various input conditions are found to collapse on a universal curve

categorization in the symmetrically dilute hopfield network a symmetrically dilute hopfield model with a hebbian learning rule is used to study the effects of gradual dilution and of synaptic noise on the categorization ability of an attractor neural network with hierarchically correlated patterns in a two level structure of ancestors and descendants categorization consists in recognizing the ancestors when the network has been trained exclusively with the descendants we consider a macroscopic number of ancestors each with a finite number of descendants and take into account the stochastic noise produced by the former in an equilibrium study of the network by means of replica symmetric mean field theory phase diagrams are obtained that exhibit a categorization a spin glass and a paramagnetic phase as well as the dependence of the order parameters on the relevant quantities the de almeida thouless lines that limit the validity of the replica symmetric results are also obtained it is shown that gradual dilution increases considerably the region where a stable categorization phase may be found

a practical method for calculating largest lyapunov exponents from small data sets detecting the presence of chaos in a dynamical system is an important problem that is solved by measuring the largest lyapunov exponent lyapunov exponents quantify the exponential divergence of initially close state space trajectories and estimate the amount of chaos in a system we present a new method for calculating the largest lyapunov exponent from an experimental time series the method follows directly from the definition of the largest lyapunov exponent and is accurate because it takes advantage of all the available data we show that the algorithm is fast easy to implement and robust to changes in the following quantities embedding dimension size of data set reconstruction delay and noise level furthermore one may use the algorithm to calculate simultaneously the correlation dimension thus one sequence of computations will yield an estimate of both the level of chaos and the system complexity

dilution and sparse coding in threshold linear nets the storage capacity of an autoassociative memory with extremely diluted connectivity and with threshold linear elementary units is studied in its dependence on the graded structure and on the sparseness of the coding scheme and on the form of the learning rule used as the coding becomes sparse more patterns can be stored and the difference in capacity measured for a given number of modifiable synapses per unit between fully connected and highly diluted systems vanishes graded non binary codings especially when used with learning rules nonlinear in their post synaptic factor further increase the number of patterns that can be stored by making their retrieved representation even sparser

on solutions to the mass transfer problem this paper studies the monge kantorovich mass transfer mt problem on metric spaces with possibly unbounded cost function conditions are given under which the mt problem is solvable and furthermore an optimal solution can be obtained as the weak limit of a sequence of optimal solutions to suitably approximating mt problems

parallel dynamics of extremely diluted symmetric q ising neural networks the parallel dynamics of extremely diluted symmetric q ising neural networks is studied for arbitrary q using a probabilistic approach in spite of the extremely diluted architecture the feedback correlations arising from the symmetry prevent a closed form solution in contrast with the extremely diluted asymmetric model a recursive scheme is found determining the complete time evolution of the order parameters taking into account all feedback it is based upon the evolution of the distribution of the local field as in the fully connected model as an illustrative example an explicit analysis is carried out for the q 2 and q 3 model these results agree with and extend the partial results existing for q 2 for q 2 the analysis is entirely new finally equilibrium fixed point equations are derived and a capacity gain function diagram is obtained

mean field dynamics of sequence processing neural networks with finite connectivity abstract a recent dynamic mean field theory for sequence processing in fully connected neural networks of hopfield type is extended and analyzed here for a symmetrically diluted network with finite connectivity near saturation equations for the dynamics and the stationary states are obtained for the macroscopic observables and the precise equivalence is established with the single pattern retrieval problem in a layered feed forward network with finite connectivity

reconstruction expansion as a geometry based framework for choosing proper delay times abstract the quality of attractor reconstruction using the method of delays is known to be sensitive to the delay parameter here we develop a new computationally efficient approach to choosing that quantifies reconstruction expansion from the identity line of the embedding space we show that reconstruction expansion is related to the concept of reconstruction signal strength and that increased expansion corresponds to diminished effects of measurement error thus reconstruction expansion represents a simple geometrical framework for choosing furthermore we describe the role of dynamical error in attractor expansion and argue that algorithms for determining should be considered as attempts at estimating an upper bound to the optimal delay

on the dynamics of analogue neurons with nonsigmoidal gain functions abstract the characteristic properties of the macroscopic retrieval dynamics of analogue neurons with hebbian coupling strengths are studied usign the shape of the gain function as a modeling parameter already at low loading a rich diversity in dynamical behaviour is observed covering the full range from point attractors to chaotic dynamics the attractors which are not of the fixed point type are interpreted as an intermediate phase between the well known retrieval states and the zero state enabling a waiting mode in the system using a probabilistic approach it is shown that these features persist in extremely and asymmetrically diluted systems at sufficiently low loading a number of generic examples are worked out in detail illustrating some properties of networks governed by nonmonotonic piecewise linear gain functions the critical storage level above which the chaotic behaviour is absent is numerically determined

layered neural networks exact solutions for the dynamics of layered feedforward neural networks are presented these networks are expected to respond to an input by going through a sequence of preassigned states on the various layers the family of networks considered has a variety of interlayer couplings linear and non linear hebbian hebbian with gaussian synaptic noise and with various kinds of dilution in addition the authors also solve the problem of layered networks with the pseudoinverse projector matrix of couplings in all cases the solutions take the form of layer to layer recursions for the mean overlap with a random key pattern and for the width of the embedding field distribution the dynamics is governed by the fixed points of these recursions for all cases non trivial domains of attraction of the memory states are found and graphically displayed

a dilution algorithm for neural networks a dilution algorithm to enlarge the storage capacity per synapse alpha eff of neural networks is proposed the algorithm is a hybrid method where hebb s rule is used to select a fraction of couplings to be removed afterwards the perceptron of optimal stability for the remaining couplings is learned the authors present an analytical calculation and the results of the numerical simulations in comparison with the fully connected or the randomly diluted perceptron the effective storage capacity alpha eff is remarkably enlarged

generalization and chaos in a layered neural network the generalization performance of a multi state and a graded response layered attractor neural network trained with examples of low activity is established exactly for monotonic and non monotonic input output functions complex behaviour is found which goes from fixed point attractors to chaos through a cascade of bifurcations depending on an appropriate threshold or cut off parameter the effect of the irregular behaviour on the generalization curves is explicitly demonstrated and phase diagrams for the recognition ratio of concepts in terms of the threshold cut off exhibit ordered generalization disordered paramagnetic or self sustained activity and chaotic phases

strong duality of the monge kantorovich mass transfer problem in metric spaces this paper studies the monge kantorovich mass transfer mt problem on metric spaces and with an unbounded cost function conditions are given under which the strong duality condition holds that is mt and its dual mt are both solvable and their optimal values coincide

characterization of strange attractors a new measure of strange attractors is introduced which offers a practical algorithm to determine their character from the time series of a single observable the relation of this new measure to fractal dimension and information theoretic entropy is discussed

determining lyapunov exponents from a time series we present the first algorithms that allow the estimation of non negative lyapunov exponents from an experimental time series lyapunov exponents which provide a qualitative and quantitative characterization of dynamical behavior are related to the exponentially fast divergence or convergence of nearby orbits in phase space a system with one or more positive lyapunov exponents is defined to be chaotic our method is rooted conceptually in a previously developed technique that could only be applied to analytically defined model systems we monitor the long term growth rate of small volume elements in an attractor the method is tested on model systems with known lyapunov spectra and applied to data for the belousov zhabotinskii reaction and couette taylor flow

1 dimensional attractor neural networks we solve a class of attractor neural network models with a mixture of 1d nearest neighbour interactions and infinite range interactions which are both of a hebbian type form our solution is based on a combination of mean field methods transfer matrices and 1d random field techniques and is obtained both for boltzmann type equilibrium following sequential glauber dynamics and peretto type equilibrium following parallel dynamics competition between the alignment forces mediated via short range interactions and those mediated via infinite range ones is found to generate novel phenomena such as multiple locally stable pure states first order transitions between recall states 2 cycles and non recall states and domain formation leading to extremely long relaxation times we test our results against numerical simulations and simple benchmark cases and find excellent agreement

quantifying the bullwhip effect in a supply chain with stochastic lead time abstract in a recent paper dejonckheere disney lambrecht and towill european journal of operational research 147 2003 567 used control systems engineering transfer functions frequency response spectral analysis to quantify the bullwhip effect in the present paper we like chen ryan drezner and simchi levi management science 46 2000 436 use the statistical method but our method extends dejonckheere et al and chen et al in that we include stochastic lead time and provide expressions for quantifying the bullwhip effect both with information sharing and without information sharing we use iid demands in a k stage supply chain for both by contrast chen et al provide lower bounds using autoregressive demand for information sharing and for information not sharing with zero safety factor for stocks dejonckheere et al validate chen et al s results for a 2 stage supply chain without information sharing using both autoregressive and iid normally distributed demands we estimate the mean and variance of lead time demand ltd from historical ltd data rather than from the component period demands and lead time nevertheless we also calculate the variance amplification like chen et al but with gamma lead times with constant lead times which chen et al used our method yields lower variance amplification as for the effect of information we find that the variance increases nearly linearly in echelon stage with information sharing but exponentially in echelon stage without information sharing