dynamical stability conditions for recurrent neural networks with unsaturating piecewise linear transfer functions we establish two conditions that ensure the nondivergence of additive recurrent networks with unsaturating piecewise linear transfer functions also called linear threshold or semilinear transfer functions as hahnloser sarpeshkar mahowald douglas and seung 2000 showed networks of this type can be efficiently built in silicon and exhibit the coexistence of digital selection and analog amplification in a single circuit to obtain this behavior the network must be multistable and nondivergent and our conditions allow determining the regimes where this can be achieved with maximal recurrent amplification the first condition can be applied to nonsymmetric networks and has a simple interpretation of requiring that the strength of local inhibition match the sum over excitatory weights converging onto a neuron the second condition is restricted to symmetric networks but can also take into account the stabilizing effect of nonlocal inhibitory interactions we demonstrate the application of the conditions on a simple example and the orientation selectivity model of ben yishai lev bar or and sompolinsky 1995 we show that the conditions can be used to identify in their model regions of maximal orientation selective amplification and symmetry breaking

permitted and forbidden sets in discrete time linear threshold recurrent neural networks the concepts of permitted and forbidden sets enable a new perspective of the memory in neural networks such concepts exhibit interesting dynamics in recurrent neural networks this paper studies the basic theories of permitted and forbidden sets of the linear threshold discrete time recurrent neural networks the linear threshold transfer function has been regarded as an adequate transfer function for recurrent neural networks networks with this transfer function form a class of hybrid analog and digital networks which are especially useful for perceptual computations networks in discrete time can directly provide algorithms for efficient implementation in digital hardware the main contribution of this paper is to establish foundations of permitted and forbidden sets necessary and sufficient conditions for the linear threshold discrete time recurrent neural networks are obtained for complete convergence existence of permitted and forbidden sets as well as conditionally multiattractivity respectively simulation studies explore some possible interesting practical applications

exact penalization of mathematical programs with equilibrium constraints we study theoretical and computational aspects of an exact penalization approach to mathematical programs with equilibrium constraints mpecs in the first part we prove that a mangasarian fromovitz type condition ensures the existence of a stable local error bound at the root of a real valued nonnegative piecewise smooth function a specification to nonsmooth formulations of equilibrium constraints e g complementarity conditions or normal equations provides conditions which guarantee the existence of a nonsmooth exact penalty function for mpecs in the second part we study a trust region minimization method for a class of composite nonsmooth functions which comprises exact penalty functions arising from mpecs we prove a global convergence result for the general method and incorporate a penalty update rule a further specification results in an sqp trust region method for mpecs based on an ell 1 penalty function

on p matrices we present some necessary and sufficient conditions for a real matrix to be a p matrix they are based on the sign real spectral radius and regularity of a certain interval matrix we show that no minor can be left out when checking for p property furthermore a not necessarily exponential method for checking the p property is given

fixed point attractor analysis for a class of neurodynamics nearly all models in neural networks start from the assumption that the input output characteristic is a sigmoidal function on parameter space we present a systematic and feasible method for analyzing the whole spectrum of attractors all saturated all but one saturated all but twosaturated and so on of a neurodynamical system with a saturated sigmoidal function as its input output characteristic we present an argument that claims under a mild condition that only all saturated or all but one saturated attractors are observable for the neurodynamics for any given all saturated configuration all but one saturated configuration the article shows how to construct an exact parameter region r such that if and only if the parameters fall within r then is an attractor a fixed point of the dynamics the parameter region for an all saturated fixed point attractor is independent of the specific choice of a saturated sigmoidal function whereas for an all but one saturated fixed point it

generalization errors of the simple perceptron to find an exact form for the generalization error of a learning machine is an open problem even in the simplest case simple perceptron learning we introduce a new approach to tackle the problem the generalization error of the simple perceptron is expressed as a linear combination of extreme values of inputs with the help of extreme value theory in statistics we then obtain an exact form of the generalization error of the simple perceptron in the case of the worst learning generalization errors of the higher order perceptron taking the form of an inverse power law in the number of examples are also considered

convergence theorems for the kohonen feature mapping algorithms with vlrps abstract the convergence of the kohonen feature mapping algorithm with vanishing learning rate parameters vlrps is considered which includes the simple competitive learning algorithm as a special case a few examples show that the learning fails to converge to global minima in general then we present a novel approach which enables us to find out a new family of vlrps such that the corresponding learning algorithm converges to the set of global minima with probability one the new vlrps is a generalization of the well known rate parameters used in the simulated annealing a numerical example is also included to confirm our theoretical approach we believe that this discovery is of importance for a large class of learning algorithms in neural networks and statistics

robust stability analysis for interval cohen grossberg neural networks with unknown time varying delays in this paper robust stability problems for interval cohen grossberg neural networks with unknown time varying delays are investigated using linear matrix inequality m matrix theory and halanay inequality techniques new sufficient conditions independent of time varying delays are derived to guarantee the uniqueness and the global robust stability of the equilibrium point of interval cohen grossberg neural networks with time varying delays all these results have no restriction on the rate of change of the time varying delays compared to some existing results these new criteria are less conservative and are more convenient to check two numerical examples are used to show the effectiveness of the present results

nonsingularity and symmetry for linear normal maps normal maps are single valued generally nonsmooth functions expressing conditions for the solution of variational problems such as those of optimization or equilibrium normal maps arising from linear transformations are particularly important both in their own right and as predictors of the behavior of related nonlinear normal maps they are called locally or globally nonsingular if the functions appearing in them are local or global homeomorphisms satisfying a lipschitz condition we show here that when the linear transformation giving rise to such a normal map has a certain symmetry property the necessary and sufficient condition for nonsingularity takes a particularly simple and convenient form being simply a positive definiteness condition on a certain subspace

stability of a complex dynamical network model abstract in this paper firstly we propose a complex dynamical network model which is always uniformly asymptotically stable about its equilibrium then we give theoretical analysis of its uniform asymptotical stability by using lyapunov stability theory finally we show a numerical example to verify our theoretical result

spike output jitter mean firing time and coefficient of variation to understand how a single neurone processes information it is critical to examine the relationship between input and output marsalek koch and maunsell s study focused on output jitter standard deviation of output interpike interval found that for the integrate and fire if whereas an input distribution with a more slowly decreasing tail induces divergence of output jitter exact formulae for mean firing time are also obtained which enable us to estimate the coefficient of variation the i f model with leakage is also briefly considered

a note on smooth tests of goodness of fit for location scale families summary in this note we show that the test statistic given by rayner best 1986 for neyman s 1937 smooth test of goodness of fit for location scale families does not in general have the alleged chisquared distribution as a result gross error may result from using their test strategy the size of this error is obtained for the logistic distribution smooth test

estimate of exponential convergence rate and exponential stability for neural networks estimates of exponential convergence rate and exponential stability are studied for a class of neural networks which includes hopfield neural networks and cellular neural networks both local and global exponential convergence are discussed theorems for estimation of the exponential convergence rate are established and the bounds on the rate of convergence are given the domains of attraction in the case of local exponential convergence are obtained simple conditions are presented for checking exponential stability of the neural networks

activity invariant sets and exponentially stable attractors of linear threshold discrete time recurrent neural networks this technical note proposes to study the activity invariant sets and exponentially stable attractors of linear threshold discrete time recurrent neural networks the concept of activity invariant sets deeply describes the property of an invariant set by that the activity of some neurons keeps invariant all the time conditions are obtained for locating activity invariant sets under some conditions it shows that an activity invariant set can have one equilibrium point which attracts exponentially all trajectories starting in the set since the attractors are located in activity invariant sets each attractor has binary pattern and also carries analog information such results can provide new perspective to apply attractor networks for applications such as group winner take all associative memory etc

qualitative behaviour of some simple networks some classical neural network systems including the hartline ratliff system the linsker system and the general sigmoid dynamics are reconsidered within a more general class of dynamical systems for synchronous dynamics the existence uniqueness local and global stability of stationary points is investigated for asynchronous dynamics a convergence theorem is proved the application of the theory of quasimonotone flows leads to some insights so far not widespread in network theory

monostability and multistability for almost periodic solutions of fractional order neural networks with unsaturating piecewise linear activation functions since the unsaturating activation function is unbounded more complex dynamics may exist in neural networks with this kind of activation function in this article monostability and multistability results of almost periodic solutions are developed for fractional order neural networks with unsaturating piecewise linear activation functions some globally mittag leffler attractive sets are given and the existence of globally mittag leffler stable almost periodic solution is demonstrated by using ascoli arzela theorem in particular some sufficient conditions are provided to ascertain the multistability of almost periodic solutions based on locally positively invariant set it shows that there exists an almost periodic solution in each positively invariant set and all trajectories converge to this periodic trajectory in that rectangular area two illustrative examples are provided to demonstrate the effectiveness of the proposed sufficient criteria

homeomorphism conditions for coherently oriented piecewise affine mappings this article is mainly concerned with the homeomorphism problem for piecewise affine mappings pa maps i e mappings which coincide with an affine mapping on each polyhedron of some finite polyhedral subdivision of rn in the first part we prove that a pa map can be defined without referring to a subdivision of rn as a continuous mapping which coincides at every point x rn with at least one function from a finite collection of affine functions the second part studies the recession function of a pa map it is shown that the recession function is piecewise linear and that a coherently oriented pa map is a homeomorphism if and only if its recession function is a homeomorphism in the last part we prove that a coherently oriented pa map is a homeomorphism if it admits a corresponding polyhedral subdivision of rn such that for some number k 2 n every face of codimension k is contained in at most 2k polyhedra provided the subdivision contains at least one face of codimension k

capacity of the hopfield model for a given if neurons deviating from the memorized patterns are allowed we constructively show that if and only if all stored patterns are fixed points of the hopfield model if neurons are allowed with then where is the distribution function of the normal distribution the result obtained by amit and co workers only formally coincides with the latter case which indicates that the replica trick approach to the capacity of the hopfield model is only valid in the case

a new proof of robinson s homeomorphism theorem for pl normal maps abstract we study a certain class of piecewise linear functions from r n to r n namely robinson s normal maps induced by linear mappings and polyhedral convex sets called pl normal maps robinson s homeomorphism theorem characterizes the pl normal maps which are homeomorphisms as those which have nonzero determinants of the same sign on all pieces of linearity this paper presents a new shorter proof of the result pl normal systems include many optimization and equilibrium problems they arise from variational inequalities or equivalently generalized equations specified by linear maps and polyhedral convex sets unique continuous solvability of these systems which is important in theory and computation is captured by the homeomorphism result

an analysis on neural dynamics with saturated sigmoidal functions we propose a unified approach to study the relation between the set of saturated attractors and the set of system parameters of the hopfield model linsker s model and the dynamic link network dln which use saturated sigmoidal functions in its dynamics of the state or weight the key point for this approach is to rigorously derive a necessary and sufficient condition to test whether a given saturated state in the hopfield model or weight vector in linsker s model and the dln is stable or not for any given set of system parameters and used this to determine the complete regime in the parameter space over which the given state or weight is stable our approach allows us to give an exact characterization between the parameters and the capacity in the hopfield model to generalize our previous results on linsker s network and the dln to have a better understanding of the underlying mechanism among these models the method reported here could be adopted to analyze a variety of models in the field of the neural networks

lyapunov functions for neural nets with nondifferentiable input output characteristics i construct lyapunov functions for asynchronous dynamics and synch ronous dynamics of neural networks with nondifferentiable input output characteristics

multistability and new attraction basins of almost periodic solutions of delayed neural networks in this paper we investigate multistability of almost periodic solutions of recurrently connected neural networks with delays simply called delayed neural networks we will reveal that under some conditions the space rn can be divided into 2n subsets and in each subset the delayed n neuron neural network has a locally stable almost periodic solution furthermore we also investigate the attraction basins of these almost periodic solutions we reveal that the attraction basin of almost periodic trajectory is larger than the subset where the corresponding almost periodic trajectory is located in addition several numerical simulations are presented to corroborate the theoretical results

data driven smooth tests for composite hypotheses the classical problem of testing goodness of fit of a parametric family is reconsidered a new test for this problem is proposed and investigated the new test statistic is a combination of the smooth test statistic and schwarz s selection rule more precisely as the sample size increases an increasing family of exponential models describing departures from the null model is introduced and schwarz s selection rule is presented to select among them schwarz s rule provides the right dimension given by the data while the smooth test in the right dimension finishes the job theoretical properties of the selection rules are derived under null and alternative hypotheses they imply consistency of data driven smooth tests for composite hypotheses at essentially any alternative

boundedness and complete stability of complex valued neural networks with time delay in this paper the boundedness and complete stability of complex valued neural networks cvnns with time delay are studied some conditions to guarantee the boundedness of the cvnns are derived using local inhibition moreover under the boundedness conditions a compact set that globally attracts all the trajectories of the network is also given additionally several conditions in terms of real valued linear matrix inequalities lmis for complete stability of the cvnns are established via the energy minimization method and the approach that converts the complex valued lmis to real valued ones examples with simulation results are given to show the effectiveness of the theoretical analysis

closed form expressions for the moments of the binomial probability distribution this work develops closed form expressions for the raw and central moments of the binomial probability distribution for this i first derive a recursive formula for the raw moments from the moment generating function then it is shown that the recursion involved is essentially the same as for the stirling numbers of the second kind from this fact it is then possible to derive the closed formulae finally i discuss an application of these formulae to the analysis of neural associative memory

relationships of properties of piecewise affine maps over ordered fields abstract with an emphasis on general ordered fields we survey relationships of properties of piecewise linear maps on vector spaces the properties of interest are local and global variations of univalence openness onto condition nonexpansiveness and orientation and eigenvalue conditions on derivatives

multistability of discrete time recurrent neural networks with unsaturating piecewise linear activation functions this paper studies the multistability of a class of discrete time recurrent neural networks with unsaturating piecewise linear activation functions it addresses the nondivergence global attractivity and complete stability of the networks using the local inhibition conditions for nondivergence are derived which not only guarantee nondivergence but also allow for the existence of multiequilibrium points under these nondivergence conditions global attractive compact sets are obtained complete stability is studied via constructing novel energy functions and using the well known cauchy convergence principle examples and simulation results are used to illustrate the theory

linear threshold discrete time recurrent neural networks stability and globally attractive sets the stability of linear threshold dynamic neural networks is studied and a series of methods to obtain globally attractive sets is proposed a sufficient condition to judge whether an invariant set is a globally attractive set is also proposed this method requires only the solution to a class of linear matrix inequalities also two direct methods to obtain globally attractive sets are given the stability criteria presented are based on the proposed globally attractive sets some numerical examples are given to illustrate the effectiveness of the obtained results

piecewise smoothness local invertibility and parametric analysis of normal maps this paper is concerned with properties of the euclidean projection map onto a convex set defined by finitely many smooth convex inequalities and affine equalities under a constant rank constraint qualification we show that the projection map is piecewise smooth pc1 hence bouligand differentiable or directionally differentiable and a relatively simple formula is given for the b derivative these properties of the projection map are used to obtain inverse and implicit function theorems for associated normal maps using a new characterization of invertibility of a pc1 function in terms of its b derivative an extension of the implicit function theorem which does not require local uniqueness is also presented degree theory plays a major role in the analysis of both the locally unique case and its extension

selectable and unselectable sets of neurons in recurrent neural networks with saturated piecewise linear transfer function the concepts of selectable and unselectable sets are proposed to describe some interesting dynamical properties of a class of recurrent neural networks rnns with saturated piecewise linear transfer function a set of neurons is said to be selectable if it can be co unsaturated at a stable equilibrium point by some external input a set of neurons is said to be unselectable if it is not selectable i e such set of neurons can never be co unsaturated at any stable equilibrium point regardless of what the input is the importance of such concepts is that they enable a new perspective of the memory in rnns necessary and sufficient conditions for the existence of selectable and unselectable sets of neurons are obtained as an application the problem of group selection is discussed by using such concepts it shows that under some conditions each group is a selectable set and each selectable set is contained in some group thus groups are indicated by selectable sets of the rnns and can be selected by external inputs simulations are carried out to further illustrate the theory

on branching numbers of normal manifolds kuhn and lowen l and robinson 2 have provided deep and elegant results that characterize the homeomorphic members of certain classes of piecewise linear pl mappings the purpose of this paper is to show that the class of mappings examined by robinson is contained in the class of mappings examined by kuhn and lowen let 312 be a finite piecewise linear pl manifold of r and p r r be pl with respect to 3n i e p is affine on each set in 9x the branching number of 9x l is the maximum number of sets in 317 that can contain any common face of codimension 2 in l theorem 5 31 it is shown that if 312 has a branching number less than or equal to 4 then p is a homeomorphism if and only if it is coherently oriented i e the determinants of p on each set in 312 have the same nonzero sign let c be a nonempty polyhedral convex set in real n dimensional space ir and a belong to the space of real n x n matrices f robinson 2 defines a finite pl manifold 32 of k 2 proposition 2 41 called the normal manifold of c and the normal map a i t r induced by a c which is pl with respect to e 2 proposition 2 51 the main result 2 theorem 4 31 shows that ac is homeomorphic if and only if it is coherently oriented see ralph 3 for an alternative proof and robinson 4 for a much simpler proof in the special case that a is symmetric it is pointed out in 2 section l that if xc could be shown to have a branching number less than or equal to 4 then 2 theorem 4 31 would follow from l theorem 5 31 indeed showing that normal manifolds have this branching number property is precisely our goal throughout the paper c and a will be as above let q be the euclidean projection onto c that is rcc maps each point of ir to its nearest point in c with respect to the euclidean norm the identity in ir x is i

on neurodynamics with limiter function and linsker s developmental model the limiter function is used in many learning and retrieval models as the constraint controlling the magnitude of the weight or state vectors in this paper we developed a new method to relate the set of saturated fixed points to the set of system parameters of the models that use the limiter function and then as a case study applied this method to linsker s hebbian learning network we derived a necessary and sufficient condition to test whether a given saturated weight or state vector is stable or not for any given set of system parameters and used this condition to determine the whole regime in the parameter space over which the given state is stable this approach allows us to investigate the relative stability of the major receptive fields reported in linsker s simulations and to demonstrate the crucial role played by the synaptic density functions

piecewise affine bijections of rn and the equation sx tx y we prove two criteria for bijectivity of piecewise affine maps f of rn it is assumed that det f has the same nonzero sign everywhere the first criterion is based solely on the branching properties of the partition of rn into domains of linearity of f the second one concerns the map itself and depends on the images under f of certain two dimensional angles this generalizes and improves results of r schramm the methods are topological our results imply k g murty s solution of the linear complementarity problem and some other partly known facts which we show are equivalent to it for example the map x sx tx is bijective iff all principal minors of s 1t are strictly positive

analysis of cyclic dynamics for networks of linear threshold neurons the network of neurons with linear threshold lt transfer functions is a prominent model to emulate the behavior of cortical neurons the analysis of dynamic properties for lt networks has attracted growing interest such as multistability and boundedness however not much is known about how the connection strength and external inputs are related to oscillatory behaviors periodic oscillation is an important characteristic that relates to nondivergence which shows that the network is still bounded although unstable modes exist by concentrating on a general parameterized two cell network theoretical results for geometrical properties and existence of periodic orbits are presented although it is restricted to two dimensional systems the analysis can provide a useful contribution to analyze cyclic dynamics of some specific lt networks of high dimension as an application it is extended to an important class of biologically motivated networks of large scale the winner take all model using local excitation and global inhibition

normal maps inducted by linear transformations we study a certain piecewise linear manifold which we call the normal manifold associated with a polyhedral convex set and a family of continuous functions called normal maps that are induced on this manifold by continuous functions from rn to rn these normal maps occur frequently in optimization and equilibrium problems and the subclass of normal maps induced by linear transformations plays a key role our main result is that the normal map induced by a linear transformation is a lipschitzian homeomorphism if and only if the determinant of the map in each n cell of the normal manifold has the same nonzero sign

dynamic stability conditions for lotka volterra recurrent neural networks with delays the lotka volterra model of neural networks derived from the membrane dynamics of competing neurons have found successful applications in many winner take all types of problems this paper studies the dynamic stability properties of general lotka volterra recurrent neural networks with delays conditions for nondivergence of the neural networks are derived these conditions are based on local inhibition of networks thereby allowing these networks to possess a multistability property multistability is a necessary property of a network that will enable important neural computations such as those governing the decision making process under these nondivergence conditions a compact set that globally attracts all the trajectories of a network can be computed explicitly if the connection weight matrix of a network is symmetric in some sense and the delays of the network are in l2 space we can prove that the network will have the property of complete stability

convergence analysis of a discrete time recurrent neural network to perform quadratic real optimization with bound constraints presents a model of a discrete time recurrent neural network designed to perform quadratic real optimization with bound constraints the network iteratively improves the estimate of the solution always maintaining it inside of the feasible region several neuron updating rules which assure global convergence of the net to the desired minimum have been obtained some of them also assure exponential convergence and maximize a lower bound for the convergence degree simulation results are presented to show the net performance

about existence and uniqueness of the restricted total least squares estimation total least squares tls is an estimation method for the solutiona of the linear system tilde xa tilde y when both data sets tilde x and tilde y are subject to error the tls method minimizes the functional theta tls left x a right lambda left x tilde x right 2 left xa tilde y right 2 with weighting parameter in this paper the tls functional is analyzed by the technique of lagrangian multipliers the main part of the work deals with the case when the estimatea is restricted by an inequality of the formd a b 0 d a diagonal matrix it is shown that there exists a unique estimatea if the weighting parameter is chosen sufficiently large

sensitivity analysis of composite piecewise smooth equations this paper is a contribution to the sensitivity analysis of piecewise smooth equations a piecewise smooth function is a lipschitzian homeomorphism near a given point if and only if it is coherently oriented and has an invertible b derivative at this point we emphasise the role of functions of the typef g h whereg is piecewise smooth andh is smooth and present verifiable conditions which ensure that the functionf g tilde h is a lipschitzian homeomorphism near a given point for every tilde h sufficiently close toh with respect to thec1 topology

variational inequalities over perturbed polyhedral convex sets this paper provides conditions for existence of a locally unique lipschitzian solution of a linear variational inequality posed over a polyhedral convex set in rn under perturbation of either or both of the constant term in the variational inequality and the right hand side of the system of linear constraints defining its feasible set conditions for perturbation of just the constant term are well known here we show that a suitable extension of those conditions suffices for the more general case in which the right hand side of the constraints varies also as a consequence we obtain existence uniqueness and lipschitz continuity properties of solutions of nonlinear variational inequalities posed over perturbed polyhedral convex sets

convergence analysis of a deterministic discrete time system of oja s pca learning algorithm the convergence of oja s principal component analysis pca learning algorithms is a difficult topic for direct study and analysis traditionally the convergence of these algorithms is indirectly analyzed via certain deterministic continuous time dct systems such a method will require the learning rate to converge to zero which is not a reasonable requirement to impose in many practical applications recently deterministic discrete time ddt systems have been proposed instead to indirectly interpret the dynamics of the learning algorithms unlike dct systems ddt systems allow learning rates to be constant which can be a nonzero this paper will provide some important results relating to the convergence of a ddt system of oja s pca learning algorithm it has the following contributions 1 a number of invariant sets are obtained based on which we can show that any trajectory starting from a point in the invariant set will remain in the set forever thus the nondivergence of the trajectories is guaranteed 2 the convergence of the ddt system is analyzed rigorously it is proven in the paper that almost all trajectories of the system starting from points in an invariant set will converge exponentially to the unit eigenvector associated with the largest eigenvalue of the correlation matrix in addition exponential convergence rate are obtained providing useful guidelines for the selection of fast convergence learning rate 3 since the trajectories may diverge the careful choice of initial vectors is an important issue this paper suggests to use the domain of unit hyper sphere as initial vectors to guarantee convergence 4 simulation results will be furnished to illustrate the theoretical results achieved

globally generated vector bundles on p1 p1 p1 with low first chern classes abstract we classify globally generated vector bundles on p 1 p 1 p 1 with small first chern class i e c 1 a 1 a 2 a 3 a i 2 our main method is to investigate the associated smooth curves to globally generated vector bundles via the hartshorne serre correspondence

absolute stability conditions for discrete time recurrent neural networks an analysis of the absolute stability for a general class of discrete time recurrent neural networks rnn s is presented a discrete time model of rnn s is represented by a set of nonlinear difference equations some sufficient conditions for the absolute stability are derived using ostrowski s theorem and the similarity transformation approach for a given rnn model these conditions are determined by the synaptic weight matrix of the network the results reported in this paper need fewer constraints on the weight matrix and the model than in previously published studies

neyman type smooth tests for location scale families summary the smooth tests of neyman 1937 are modified to permit composite alternatives and a choice of orthonormal functions in their construction these orthonormal functions may be chosen to improve detection of particular alternatives the tests are easy to apply and have useful orthogonal components a test for exponentiality is examined and shown to perform well against its competitors

multiperiodicity and attractivity of delayed recurrent neural networks with unsaturating piecewise linear transfer functions this paper studies multiperiodicity and attractivity for a class of recurrent neural networks rnns with unsaturating piecewise linear transfer functions and variable delays using local inhibition conditions for boundedness and global attractivity are established these conditions allow coexistence of stable and unstable trajectories moreover multiperiodicity of the network is investigated by using local invariant sets it shows that under some interesting conditions there exists one periodic trajectory in each invariant set which exponentially attracts all trajectories in that region correspondingly simulations are carried out to illustrate the theories

on 2 4 complete intersection threefolds that contain an enriques surface we study nodal complete intersection threefolds of type 2 4 in mathbb p 5 which contain an enriques surface in its fano embedding we completely determine calabi yau birational models of a generic such threefold these models have hodge numbers h 11 2 h 12 32 we also describe calabi yau varieties with hodge numbers h 11 h 12 equal to 2 26 23 5 and 31 1 the last two pairs of hodge numbers are to the best of our knowledge new

the role of constraints in hebbian learning models of unsupervised correlation based hebbian synaptic plasticity are typically unstable either all synapses grow until each reaches the maximum allowed strength or all synapses decay to zero strength a common method of avoiding these outcomes is to use a constraint that conserves or limits the total synaptic strength over a cell we study the dynamic effects of such constraints two methods of enforcing a constraint are distinguished multiplicative and subtractive for otherwise linear learning rules multiplicative enforcement of a constraint results in dynamics that converge to the principal eigenvector of the operator determining unconstrained synaptic development subtractive enforcement in contrast typically leads to a final state in which almost all synaptic strengths reach either the maximum or minimum allowed value this final state is often dominated by weight configurations other than the principal eigenvector of the unconstrained operator multiplicative enforcement yields a graded receptive field in which most mutually correlated inputs are represented whereas subtractive enforcement yields a receptive field that is sharpened to a subset of maximally correlated inputs if two equivalent input populations e g two eyes innervate a common target multiplicative enforcement prevents their segregation ocular dominance segregation when the two populations are weakly correlated whereas subtractive enforcement allows segregation under these circumstances these results may be used to understand constraints both over output cells and over input cells a variety of rules that can implement constrained dynamics are discussed

hypersurfaces in projective schemes and a moving lemma let x s be a quasi projective morphism over an affine base we develop in this article a technique for proving the existence of closed subschemes h s of x s with various favorable properties we offer several applications of this technique including the existence of finite quasi sections in certain projective morphisms and the existence of hypersurfaces in x s containing a given closed subscheme c and intersecting properly a closed set f assume now that the base s is the spectrum of a ring r such that for any finite morphism z s pic z is a torsion group this condition is satisfied if r is the ring of integers of a number field or the ring of functions of a smooth affine curve over a finite field we prove in this context a moving lemma pertaining to horizontal 1 cycles on a regular scheme x quasi projective and flat over s we also show the existence of a finite surjective s morphism to psd for any scheme x projective over s when x s has all its fibers of a fixed dimension d

on the lipschitzian properties of polyhedral multifunctions in this paper we show that for a polyhedral multifunctionf rn rm with convex range the inverse functionf 1 is locally lower lipschitzian at every point of the range off equivalently lipschitzian on the range off if and only if the functionf is open as a consequence we show that for a piecewise affine functionf rn rn f is surjective andf 1 is lipschitzian if and only iff is coherently oriented an application via robinson s normal map formulation leads to the following result in the context of affine variational inequalities the solution mapping as a function of the data vector is nonempty valued and lipschitzian on the entire space if and only if the solution mapping is single valued this extends a recent result of murthy parthasarathy and sabatini proved in the setting of linear complementarity problems

on variational inequalities over polyhedral sets the results on regularity behavior of solutions to variational inequalities over polyhedral sets proved in a series of papers by robinson ralph and dontchev rockafellar in the 90s has long become classics of variational analysis but the available proofs are very complicated and practically do not use techniques of variational analysis the only exception is the proof by dontchev and rockafellar of their critical face regularity criterion in the paper we offer a different approach completely based on polyhedral geometry and a few basic principles of metric regularity theory it leads to new proofs that look simpler and shorter and in addition gives some clarifying geometrical information

selectively grouping neurons in recurrent networks of lateral inhibition winner take all networks have been proposed to underlie many of the brain s fundamental computational abilities however not much is known about how to extend the grouping of potential winners in these networks beyond single neuron or uniformly arranged groups of neurons we show that competition between arbitrary groups of neurons can be realized by organizing lateral inhibition in linear threshold networks given a collection of potentially overlapping groups with the exception of some degenerate cases the lateral inhibition results in network dynamics such that any permitted set of neurons that can be coactivated by some input at a stable steady state is contained in one of the groups the information about the input is preserved in this operation the activity level of a neuron in a permitted set corresponds to its stimulus strength amplified by some constant sets of neurons that are not part of a group cannot be coactivated by any input at a stable steady state we analyze the storage capacity of such a network for random groups the number of random groups the network can store as permitted sets without creating too many spurious ones in this framework we calculate the optimal sparsity of the groups maximizing group entropy we find that for dense inputs the optimal sparsity is unphysiologically small however when the inputs and the groups are equally sparse we derive a more plausible optimal sparsity we believe our results are the first steps toward attractor theories in hybrid analog digital networks

multistability analysis for recurrent neural networks with unsaturating piecewise linear transfer functions multistability is a property necessary in neural networks in order to enable certain applications e g decision making where monostable networks can be computationally restrictive this article focuses on the analysis of multistability for a class of recurrent neural networks with unsaturating piecewise linear transfer functions it deals fully with the three basic properties of a multistable network boundedness global attractivity and complete convergence this article makes the following contributions conditions based on local inhibition are derived that guarantee boundedness of some multistable networks conditions are established for global attractivity bounds on global attractive sets are obtained complete convergence conditions for the network are developed using novel energy like functions and simulation examples are employed to illustrate the theory thus developed

a proof of the branching number bound for normal manifolds abstract we prove that every face of dimension n 2 of the normal manifold induced by a polyhedron in r n is contained in precisely four distinct n dimensional polyhedra of the normal manifold this result implies that the branching number of the normal manifold is 4 and refines ralph s result which only gave 4 as an upper bound the proof is based on the well known fact that every ridge of a polyhedron is contained in precisely two facets

foundations of implementing the competitive layer model by lotka volterra recurrent neural networks the competitive layer model clm can be described by an optimization problem the problem can be further formulated by an energy function called the clm energy function in the subspace of nonnegative orthant the set of minimum points of the clm energy function forms the set of solutions of the clm problem solving the clm problem means to find out such solutions recurrent neural networks rnns can be used to implement the clm to solve the clm problem the key point is to make the set of minimum points of the clm energy function just correspond to the set of stable attractors of the recurrent neural networks this paper proposes to use lotka volterra rnns lv rnns to implement the clm the contribution of this paper is to establish foundations of implementing the clm by lv rnns the contribution mainly contains three parts the first part is on the clm energy function necessary and sufficient conditions for minimum points of the clm energy function are established by detailed study the second part is on the convergence of the proposed model of the lv rnns it is proven that interesting trajectories are convergent the third part is the most important it proves that the set of stable attractors of the proposed lv rnn just equals the set of minimum points of the clm energy function in the nonnegative orthant thus the lv rnns can be used to solve the problem of the clm it is believed that by establishing such basic rigorous theories more and interesting applications of the clm can be found

lagrange exponential stability for fuzzy cohen grossberg neural networks with time varying delays this paper focuses on the globally exponential stability in lagrange sense for takagi sugeno t s fuzzy cohen grossberg neural networks with time varying delays by employing lyapunov method and delay inequality technique we analyze two different types of activation functions which include both lipschitz function and general activation functions several easily verifiable sufficient criteria about linear matrix inequality form are obtained to guarantee the lagrange exponential stability of cohen grossberg neural networks with time varying delays which are represented by t s fuzzy models meanwhile the estimations of the globally exponentially attractive sets are given here the existence and uniqueness of the equilibrium points need not be considered finally two numerical examples with simulations are given to illustrate the effectiveness of the theoretical results

theorems of perron frobenius type for matrices without sign restrictions abstract the paper attempts to solve a problem which was called a challenge for the future in linear algebra appl we define and investigate a new quantity for real matrices the sign real spectral radius and derive various characterizations bounds and properties of it in certain aspects our quantity shows similar behavior to the perron root of a nonnegative matrix it is shown that our quantity also has intimate connections to the componentwise distance to the nearest singular matrix relations to the perron root of the entrywise absolute value of the matrix and to the number are given as well

nontrivial global attractors in 2 d multistable attractor neural networks attractor dynamics is a crucial problem for attractor neural networks as it is the underling computational mechanism for memory storage and retrieval in neural systems this brief studies a class of attractor network consisting of linearized threshold neurons and analyzes global attractors based on a parameterized 2 d model on the basis of previous results on nondegenerate and degenerate equilibria in mathematics we further elucidate all possible nontrivial global attractors our theoretical result provides precise descriptions on how the changes of network parameters affect the attractors distribution and landscape and it may give a feasible solution towards specifying attractors by specifying weights simulations are presented to illustrate the theoretical results