fixed time consensus of complex dynamical networks with nonlinear coupling and fuzzy state dependent uncertainties abstract in this paper the fixed time consensus problem for complex dynamical networks with nonlinear coupling and fuzzy state dependent uncertainties is discussed different from most existing publications the controller is designed as a fuzzy logic system which is applicable for complex dynamical networks with system uncertainties and unknown topology to ensure the fixed time consensus furthermore compared with the initial condition based finite time consensus the settling time is bounded by a constant within fixed time regardless of the initial condition based on the lyapunov stability theory and inequality techniques some sufficient criteria for fixed time consensus are obtained finally two numerical examples are granted to display the performance of the obtained results

determinants on von neumann algebras mahler measures and ljapunov exponents for an ergodic automorphism of a probability space a p consider the semidirect product von neumann algebra r l z let u be the unitary operator corresponding to the action of then the finite sums of the form i aiu i with ai in l form a strongly dense subalgebra of r in this paper we calculate the fuglede kadison determinant detr for a certain class of elements the answer is given in terms of the ljapunov exponents i of a matrix a over the ring l which is attached to by a simple receipe the formula has the form

optimal control for stochastic nonlinear singular system using neural networks in this paper optimal control for stochastic nonlinear singular system with quadratic performance is obtained using neural networks the goal is to provide optimal control with reduced calculus effort by comparing the solutions of the matrix riccati differential equation mrde obtained from the well known traditional runge kutta rk method and nontraditional neural network method to obtain the optimal control the solution of mrde is computed by feedforward neural network ffnn the accuracy of the solution of the neural network approach to the problem is qualitatively better the advantage of the proposed approach is that once the network is trained it allows instantaneous evaluation of solution at any desired number of points spending negligible computing time and memory the computation time of the proposed method is shorter than the traditional rk method an illustrative numerical example is presented for the proposed method

exponential synchronizationlike criterion for state dependent impulsive dynamical networks this paper focuses on the problem of the exponential synchronizationlike criteria for state dependent impulsive dynamical networks sidns two types of sufficient conditions which are applied to ensure every solution intersecting each impulsive surface exactly once are derived for each type of collision conditions combining with comparison principle and inequality techniques some sufficient conditions are obtained to ensure local exponential synchronizationlike for sidn moreover a quiet different impulsive strategy concerning the trigger rules of impulsive instants is proposed finally an example is given to demonstrate the effectiveness of our results

mixing automorphisms of compact groups and a theorem of schlickewei we prove that every mixing d by automorphisms of a compact connected abelian group is mixing of all orders

global asymptotic stability of recurrent neural networks with multiple time varying delays in this paper several sufficient conditions are established for the global asymptotic stability of recurrent neural networks with multiple time varying delays the lyapunov krasovskii stability theory for functional differential equations and the linear matrix inequality lmi approach are employed in our investigation the results are shown to be generalizations of some previously published results and are less conservative than existing results the present results are also applied to recurrent neural networks with constant time delays

synchronization points and associated dynamical invariants this paper introduces new invariants for multiparameter dynamical systems this is done by counting the number of points whose orbits intersect at time n under simultaneous iteration of finitely many endomorphisms we call these points synchronization points the resulting sequences of counts together with generating functions and growth rates are subsequently investigated for homeomorphisms of compact metric spaces toral automorphisms and compact abelian group epimorphisms synchronization points are also used to generate invariant measures and the distribution properties of these are analysed for the algebraic systems considered furthermore these systems reveal strong connections between the new invariants and problems of active interest in number theory relating to heights and greatest common divisors

a regularization parameter in discrete ill posed problems the tikhonov regularization method for discrete ill posed problems is considered for the practical choice of the regularization parameter alpha some authors use a plot of the norm of the regularized solution versus the norm of the residual vector for all alpha considered this paper contains an analysis of the shape of this plot and gives a theoretical justification for choosing the regularization parameter so it is related to the l corner of the plot considered in the logarithmic scale moreover a new criterion for choosing alpha is introduced independent of the shape of the plot which gives a new interpretation of the corner criterion mentioned above the existence of l corner is discussed

stability and synchronization of discrete time neural networks with switching parameters and time varying delays this paper is concerned with the problems of exponential stability analysis and synchronization of discrete time switched delayed neural networks using the average dwell time approach together with the piecewise lyapunov function technique sufficient conditions are proposed to guarantee the exponential stability for the switched neural networks with time delays benefitting from the delay partitioning method and the free weighting matrix technique the conservatism of the obtained results is reduced in addition the decay estimates are explicitly given and the synchronization problem is solved the results reported in this paper not only depend upon the delay but also depend upon the partitioning which aims at reducing the conservatism numerical examples are presented to demonstrate the usefulness of the derived theoretical results

stability analysis for neural networks with time varying delay based on quadratic convex combination in this paper a novel method is developed for the stability problem of a class of neural networks with time varying delay new delay dependent stability criteria in terms of linear matrix inequalities for recurrent neural networks with time varying delay are derived by the newly proposed augmented simple lyapunov krasovski functional different from previous results by using the first order convex combination property our derivation applies the idea of second order convex combination and the property of quadratic convex function which is given in the form of a lemma without resorting to jensen s inequality a numerical example is provided to verify the effectiveness and superiority of the presented results

abelian sandpiles and the harmonic model we present a construction of an entropy preserving equivariant surjective map from the d dimensional critical sandpile model to a certain closed shift invariant subgroup of mathbb t mathbb z d the harmonic model a similar map is constructed for the dissipative abelian sandpile model and is used to prove uniqueness and the bernoulli property of the measure of maximal entropy for that model

mahler measure and entropy for commuting automorphisms of compact groups we compute the joint entropy ofd commuting automorphisms of a compact metrizable group letr d u 1 1 d 1 1 be the ring of laurent polynomials ind commuting variables andm be anr d module then the dual groupx m ofm is compact and multiplication onm by each of thed variables corresponds to an action m of d by automorphisms ofx m every action of d by automorphisms of a compact abelian group arises this way iff r d our main formula shows that the topological entropy of alpha r d left langle f right rangle is given by h left alpha r d left langle f right rangle right log m left f right int limits 0 1 ldots int limits 0 1 log left f left e 2 pi it 1 ldots e 2 pi it d right right dt 1 ldots dt d where m f is the mahler measure off this reduces to the classical result for toral automorphisms via jensen s formula while the entropy of a single automorphism of a compact group is always the logarithm of an algebraic integer this no longer seems to hold for joint entropy of commuting automorphisms since values such as 7 3 4 2 occur if p is a non principal prime ideal we show h left alpha r d p right 0 using an analogue of the yuzvinskii thomas addition formula we computeh m for arbitraryr d modulesm and then the joint entropy for an action of d on a not necessarily abelian compact group using a result of boyd we characterize those m which have completely positive entropy in terms of the prime ideals associated tom and show this condition implies that m is mixing of all orders we also establish an analogue of berg s theorem proving that if m has finite entropy then haar measure is the unique measure of maximal entropy if and only if m has completely positive entropy finally we show that for expansive actions the growth rate of the number of periodic points equals the topological entropy

bowen topological entropy of subsets for amenable group actions abstract we extend the definition of bowen topological entropy of subsets to continuous action of amenable groups on a compact metrizable space we investigate how bowen topological entropy behaves for restricted actions of finite index subgroups and verify the validity of the subgroup formula we also show that the bowen topological entropy of subsets for amenable group actions can be determined via the local entropies of measures as applications we compute the bowen topological entropy of a subset for bernoulli shift over amenable groups

new criteria for global robust stability of delayed neural networks with norm bounded uncertainties in this paper we study the global asymptotic robust stability of delayed neural networks with norm bounded uncertainties by employing the lyapunov stability theory and homeomorphic mapping theorem we derive some new types of sufficient conditions ensuring the existence uniqueness and global asymptotic stability of the equilibrium point for the class of neural networks with discrete time delays under parameter uncertainties and with respect to continuous and slope bounded activation functions an important aspect of our results is their low computational complexity as the reported results can be verified by checking some properties of symmetric matrices associated with the uncertainty sets of the network parameters the obtained results are shown to be generalizations of some of the previously published corresponding results some comparative numerical examples are also constructed to compare our results with some closely related existing literature results

entropy and the variational principle for actions of sofic groups recently lewis bowen introduced a notion of entropy for measure preserving actions of a countable sofic group on a standard probability space admitting a generating partition with finite entropy by applying an operator algebra perspective we develop a more general approach to sofic entropy which produces both measure and topological dynamical invariants and we establish the variational principle in this context in the case of residually finite groups we use the variational principle to compute the topological entropy of principal algebraic actions whose defining group ring element is invertible in the full group c algebra

a variational principle for free semigroup actions abstract in this paper we introduce a notion of measure theoretical entropy for a finitely generated free semigroup action and establish a variational principle when the semigroup is generated by continuous self maps on a compact metric space and has finite topological entropy in the case of semigroups generated by ruelle expanding maps we prove the existence of equilibrium states and describe some of their properties of independent interest are the different ways we will present to compute the metric entropy and a characterization of the stationary measures

criteria for robust absolute stability of time varying nonlinear continuous time systems the present paper establishes results for the robust absolute stability of a class of nonlinear continuous time systems with time varying matrix uncertainties of polyhedral type and multiple time varying sector nonlinearities by using the variational method and the lyapunov second method criteria for robust absolute stability are obtained in different forms for the given class of systems specifically the parametric classes of lyapunov functions are determined which define the necessary and sufficient conditions of robust absolute stability the piecewise linear lyapunov functions of the infinity vector norm type are applied to derive an algebraic criterion for robust absolute stability in the form of solvability conditions of a set of matrix equations several simple sufficient conditions of robust absolute stability are given which become necessary and sufficient for special cases two examples are presented as applications of the present results to a particular second order system and to a specific class of systems with time varying interval matrices in the linear part

stability analysis of markovian jumping stochastic cohen grossberg neural networks with mixed time delays in this letter the global asymptotical stability analysis problem is considered for a class of markovian jumping stochastic cohen grossberg neural networks cgnns with mixed delays including discrete delays and distributed delays an alternative delay dependent stability analysis result is established based on the linear matrix inequality lmi technique which can easily be checked by utilizing the numerically efficient matlab lmi toolbox neither system transformation nor free weight matrix via newton leibniz formula is required two numerical examples are included to show the effectiveness of the result

fuglede kadison determinants and entropy for actions of discrete amenable groups consider a discrete group t and an element in the integral group ring zt then t acts from the left on the discrete additive group zt ztf by automorphisms of groups dualizing we obtain a left t action on the compact pontrjagin dual group xf zt ztf by continuous automorphisms of groups by definition xf is a closed subshift of s1 1 for measurable or topological actions of amenable groups a good entropy theory is available ow m thus from now on let t be a finitely generated discrete amenable group we are interested in determining the topological entropy hf of the t action on xf it agrees with the measure theoretic entropy of the t action on xf equipped with its haar probability measure in their paper fk from 1952 fuglede and kadison introduced a determinant for units in finite factor von neumann algebras more recently l ck has general ized their theory to nonunits and to group von neumann algebras which are not necessarily factors l2 he needed this to define l2 torsion for general covering spaces the von neumann algebra a t of the amenable group t is a certain completion of cr and hence we may consider the fuglede kadison l ck determinant detjvr of we refer to section 6 for the definition of a log strong f0lner sequence it is known that finitely generated virtually nilpotent groups have log strong f0lner sequences our main result is

almost block independence and bernoullicity of ℤ d actions by automorphisms of compact abelian groups we prove that a d action by automorphisms of a compact abelian group is bernoulli if and only if it has completely positive entropy the key ingredients of the proof are the extension of certain notions of asymptotic block independence from actions to d action and their equivalence with bernoullicity and a surprisingly close link between one of these asymptotic block independence properties for d actions by automorphisms of compact abelian groups and the product formula for valuations on global fields

stochastic relaxation gibbs distributions and the bayesian restoration of images we make an analogy between images and statistical mechanics systems pixel gray levels and the presence and orientation of edges are viewed as states of atoms or molecules in a lattice like physical system the assignment of an energy function in the physical system determines its gibbs distribution because of the gibbs distribution markov random field mrf equivalence this assignment also determines an mrf image model the energy function is a more convenient and natural mechanism for embodying picture attributes than are the local characteristics of the mrf for a range of degradation mechanisms including blurring nonlinear deformations and multiplicative or additive noise the posterior distribution is an mrf with a structure akin to the image model by the analogy the posterior distribution defines another imaginary physical system gradual temperature reduction in the physical system isolates low energy states annealing or what is the same thing the most probable states under the gibbs distribution the analogous operation under the posterior distribution yields the maximum a posteriori map estimate of the image given the degraded observations the result is a highly parallel relaxation algorithm for map estimation we establish convergence properties of the algorithm and we experiment with some simple pictures for which good restorations are obtained at low signal to noise ratios

robust exponential stability of uncertain delayed neural networks with stochastic perturbation and impulse effects this paper focuses on the hybrid effects of parameter uncertainty stochastic perturbation and impulses on global stability of delayed neural networks by using the ito formula lyapunov function and halanay inequality we established several mean square stability criteria from which we can estimate the feasible bounds of impulses provided that parameter uncertainty and stochastic perturbations are well constrained moreover the present method can also be applied to general differential systems with stochastic perturbation and impulses

deligne periods of mixed motives theory and the entropy of certain ℤⁿ actions for a coherent sheaf m on the split n torus gn a spec a 7n over a commutative ring a there is a natural en action on its group of global sections f m f g aa m if we give f m the discrete topology we obtain a en action on the compact abelian pontrjagin dual f m the dynamical properties of this action for a z have been investigated for some time especially by k schmidt his book sch gives a comprehensive account of the theory developed so far in particular one may ask for the entropy 0 h m oc of the en action on f m i which measures to what extent repeated application of the action scatters around points note that for any action of zn on a compact abelian group topological entropy and metric entropy with respect to haar measure coincide the results of lind schmidt and ward lsw see also sch chap v can be formulated as follows a z the entropy of m is finite if and only if m belongs to the category t of coherent torsion sheaves on gn by yuzvinskii s addition formula sch th 14 1 entropy defines a homomorphism from the grothendieck group of t to the reals h ko t ri so that it suffices to calculate h oz for irreducible closed subschemes z c n with z g n if z is not defined by a laurent polynomial then h oz 0 on the other hand for any

finite horizon h_ infty state estimation for periodic neural networks over fading channels the problem of finite horizon h infty state estimator design for periodic neural networks over multiple fading channels is studied in this paper to characterize the measurement signals transmitted through different channels experiencing channel fading a multiple fading channels model is considered for investigating the situation of correlated fading channels a set of correlated random variables is introduced specifically the channel coefficients are described by white noise processes and are assumed to be correlated two sufficient criteria are provided by utilizing a stochastic analysis approach to guarantee that the estimation error system is stochastically stable and achieves the prescribed h infty performance then the parameters of the estimator are derived by solving recursive linear matrix inequalities finally some simulation results are shown to illustrate the effectiveness of the proposed method

dimension and entropy in compact topological groups abstract we study the topological entropy h f of continuous endomorphisms f of compact like groups more specifically we consider the e spectrum e t o p k for a compact like group k namely the set of all values h f when f runs over the set e n d k of all continuous endomorphisms of k we pay particular attention to the class e of topological groups without continuous endomorphisms of infinite entropy i e e t o p k as well as the subclass e 0 of e consisting of those groups k with e t o p k 0 it turns out that the properties of the e spectrum and these two classes are very closely related to the topological dimension we show among others that a compact connected group k with finite dimensional commutator subgroup belongs to e if and only if dim k and we obtain a simple formula involving the entropy function for the dimension of an abelian topological group which is either locally compact or bounded in particular compact examples are provided to show the necessity of the compactness or commutativity conditions imposed for the validity of these results e g compact connected semi simple groups k with dim k and k e 0 or countably compact connected abelian groups with the same property since the class e is not stable under taking closed subgroups or quotients we study also the largest subclasses s e and q e respectively of e having these stability properties we provide a complete description of these two classes in the case of compact groups that are either abelian or connected the counterpart for s e 0 and q e 0 is done as well

stabilizing effects of impulses in discrete time delayed neural networks this brief studies the global exponential stability of the equilibrium point of discrete time delayed hopfield neural networks dhnns with impulse effects by using difference inequalities we shall consider the stabilizing effects of impulses when the corresponding impulse free dhnn is even not asymptotically stable the obtained results characterize the aggregated effects of impulses and deviation of the impulse free dhnn from its equilibrium point on the exponential stability of the whole system it is shown that because of effects of impulses the impulsive discrete time dhnn may be exponentially stable even if the evolution of impulse free component deviates from its equilibrium point exponentially

compact group automorphisms addition formulas and fuglede kadison determinants for a countable amenable group and an element f in the integral group ring z being invertible in the group von neumann algebra of we show that the entropy of the shift action of on the pontryagin dual of the quotient of z by its left ideal generated by f is the logarithm of the fuglede kadison determinant of f for the proof we establish an p version of rufus bowen s denition of topological entropy addition formulas for group extensions of countable amenable group actions and an approximation formula for the fuglede kadison determinant of f in terms of the determinants of perturbations of the compressions of f

stability analysis for stochastic cohen grossberg neural networks with mixed time delays in this letter the global asymptotic stability analysis problem is considered for a class of stochastic cohen grossberg neural networks with mixed time delays which consist of both the discrete and distributed time delays based on an lyapunov krasovskii functional and the stochastic stability analysis theory a linear matrix inequality lmi approach is developed to derive several sufficient conditions guaranteeing the global asymptotic convergence of the equilibrium point in the mean square it is shown that the addressed stochastic cohen grossberg neural networks with mixed delays are globally asymptotically stable in the mean square if two lmis are feasible where the feasibility of lmis can be readily checked by the matlab lmi toolbox it is also pointed out that the main results comprise some existing results as special cases a numerical example is given to demonstrate the usefulness of the proposed global stability criteria

mean square exponential stability of stochastic nonlinear delay systems abstractin this paper we are concerned with the stability of stochastic nonlinear delay systems different from the previous literature we aim to show that when the determinate nonlinear delay system is globally exponentially stable the corresponding stochastic nonlinear delay system can be mean square globally exponentially stable in particular we remove the linear growth condition and introduce a new polynomial growth condition for g x t x t t which overcomes the limitation of application scope and the boundedness of diffusion term form finally we provide an example to illustrate our results

robust exponential stability of markovian jump impulsive stochastic cohen grossberg neural networks with mixed time delays this paper is concerned with the problem of exponential stability for a class of markovian jump impulsive stochastic cohen grossberg neural networks with mixed time delays and known or unknown parameters the jumping parameters are determined by a continuous time discrete state markov chain and the mixed time delays under consideration comprise both time varying delays and continuously distributed delays to the best of the authors knowledge till now the exponential stability problem for this class of generalized neural networks has not yet been solved since continuously distributed delays are considered in this paper the main objective of this paper is to fill this gap by constructing a novel lyapunov krasovskii functional and using some new approaches and techniques several novel sufficient conditions are obtained to ensure the exponential stability of the trivial solution in the mean square the results presented in this paper generalize and improve many known results finally two numerical examples and their simulations are given to show the effectiveness of the theoretical results

local characteristics entropy and limit theorems for spanning trees and domino tilings via transfer impedances let g be a finite graph or an infinite graph on which mathbb z d acts with finite fundamental domain if g is finite let mathbf t be a random spanning tree chosen uniformly from all spanning trees of g if g is infinite methods from pemantle show that this still makes sense producing a random essential spanning forest of g a method for calculating local characteristics i e finite dimensional marginals of mathbf t from the transfer impedance matrix is presented this differs from the classical matrix tree theorem in that only small pieces of the matrix n dimensional minors are needed to compute small n dimensional marginals calculation of the matrix entries relies on the calculation of the green s function for g which is not a local calculation however it is shown how the calculation of the green s function may be reduced to a finite computation in the case when g is an infinite graph admitting a z d action with finite quotient the same computation also gives the entropy of the law of mathbf t these results are applied to the problem of tiling certain lattices by dominos the so called dimer problem another application of these results is to prove modified versions of conjectures of aldous on the limiting distribution of degrees of a vertex and on the local structure near a vertex of a uniform random spanning tree in a lattice whose dimension is going to infinity included is a generalization of moments to tree valued random variables and criteria for these generalized moments to determine a distribution

constrained restoration and the recovery of discontinuities the linear image restoration problem is to recover an original brightness distribution x sup 0 given the blurred and noisy observations y kx sup 0 b where k and b represent the point spread function and measurement error respectively this problem is typical of ill conditioned inverse problems that frequently arise in low level computer vision a conventional method to stabilize the problem is to introduce a priori constraints on x sup 0 and design a cost functional h x over images x which is a weighted average of the prior constraints regularization term and posterior constraints data term the reconstruction is then the image x which minimizes h a prominent weakness in this approach especially with quadratic type stabilizers is the difficulty in recovering discontinuities the authors therefore examine prior smoothness constraints of a different form which permit the recovery of discontinuities without introducing auxiliary variables for marking the location of jumps and suspending the constraints in their vicinity in this sense discontinuities are addressed implicitly rather than explicitly

mean dimension mean rank and von neumann luck rank we introduce an invariant called mean rank for any module m of the integral group ring of a discrete amenable group as an analogue of the rank of an abelian group it is shown that the mean dimension of the induced action on the pontryagin dual of m the mean rank of m and the von neumann luck rank of m all coincide as applications we establish an addition formula for mean dimension of algebraic actions prove the analogue of the pontryagin schnirelmnn theorem for algebraic actions and show that for elementary amenable groups with an upper bound on the orders of finite subgroups algebraic actions with zero mean dimension are inverse limits of finite entropy actions

exponential synchronization of linearly coupled neural networks with impulsive disturbances this brief investigates globally exponential synchronization for linearly coupled neural networks nns with time varying delay and impulsive disturbances since the impulsive effects discussed in this brief are regarded as disturbances the impulses should not happen too frequently the concept of average impulsive interval is used to formalize this phenomenon by referring to an impulsive delay differential inequality we investigate the globally exponential synchronization of linearly coupled nns with impulsive disturbances the derived sufficient condition is closely related with the time delay impulse strengths average impulsive interval and coupling structure of the systems the obtained criterion is given in terms of an algebraic inequality which is easy to be verified and hence our result is valid for large scale systems the results extend and improve upon earlier work as a numerical example a small world network composing of impulsive coupled chaotic delayed nn nodes is given to illustrate our theoretical result

algebraic entropy for amenable semigroup actions abstract we introduce two notions of algebraic entropy for actions of cancellative right amenable semigroups s on discrete abelian groups a by endomorphisms they extend the classical algebraic entropy for endomorphisms of abelian groups corresponding to the case s n we investigate the fundamental properties of the algebraic entropy and compute it in several examples paying special attention to the case when s is an amenable group for actions of cancellative right amenable monoids on torsion abelian groups we prove the so called addition theorem in the same setting we see that a bridge theorem connects the algebraic entropy with the topological entropy of the dual action by means of pontryagin duality so that we derive an addition theorem for the topological entropy of actions of cancellative left amenable monoids on totally disconnected compact abelian groups

inferring perceptual saliency fields from viewpoint dependent recognition data we present an algorithm for computing the relative perceptual saliencies of the features of a three dimensional object using either goodness of view scores measured at several viewpoints or perceptual similarities among several object views this technique addresses the inverse illposed version of the direct problem of predicting goodness of view scores or view point similarities when the object features are known on the basis of a linear model for the direct problem we solve the inverse problem using the method of regularization the critical assumption we make to regularize the solution is that perceptual salience varies slowly on the surface of the object the salient regions derived using this assumption empirically indicate what object structures are important in human three dimensional object perception a domain where theories typically have been based on somewhat ad hoc features

some remarks on the entropy for algebraic actions of amenable groups in this short note we study the entropy for algebraic actions of certain amenable groups the possible values for this entropy are studied various fundamental results about certain classes of amenable groups are reproved using elementary arguments and the entropy invariant we provide a natural decomposition of the entropy into summands contributed by individual primes and a summand corresponding to these results extend previous work by lind and ward on p adic entropy

an lp version of von neumann dimension for banach space representations of sofic groups abstract in 5 a gournay defined a notion of l p dimension for invariant subspaces of l q n with amenable the number dim l q l p v is dim v when p q and is preserved by a certain class of equivariant bounded linear isomorphisms we develop a notion of dim l p y where y is a banach space with a uniformly bounded action of a sofic group and is a sofic approximation in particular our definition makes sense for a large class of non amenable groups we also develop a notion of dim s p y with an r embeddable group and s p the space of finite dimensional schatten p class operators these numbers are invariant under bounded equivariant linear isomorphisms and under the natural translation action of dim l p l p v dim v and dim s p l p v dim v for 1 p 2 in particular this shows that l p v is not isomorphic to l p w as a representation of if dim v dim w and is r embeddable we discuss other concrete computations in a follow up paper including proving that our dimension agrees with von neumann dimension for representations contained in a multiple of the left regular representation

sofic mean dimension abstract we introduce mean dimensions for continuous actions of countable sofic groups on compact metrizable spaces these generalize the gromov lindenstrauss weiss mean dimensions for actions of countable amenable groups and are useful for distinguishing continuous actions of countable sofic groups with infinite entropy

representation of similarity in three dimensional object discrimination how does the brain represent visual objects in simple perceptual generalization tasks the human visual system performs as if it represents the stimuli in a low dimensional metric psychological space shepard 1987 in theories of three dimensional 3d shape recognition the role of feature space representations as opposed to structural biederman 1987 or pictorial ullman 1989 descriptions has long been a major point of contention if shapes are indeed represented as points in a feature space patterns of perceived similarity among different objects must reflect the structure of this space the feature space hypothesis can then be tested by presenting subjects with complex parameterized 3d shapes and by relating the similarities among subjective representations as revealed in the response data by multidimensional scaling shepard 1980 to the objective parameterization of the stimuli the results of four such tests accompanied by computational simulations support the notion that discrimination among 3d objects may rely on a low dimensional feature space representation and suggest that this space may be spanned by explicitly encoded class prototypes

homoclinic points of algebraic ℤ actions an algebraic zd action is an action of zd by continuous automorphisms of a compact abelian group the dynamics of a single group automorphism have been investigated in great detail over the past several decades more recently the study of algebraic zd actions for d 2 has revealed a striking interplay between these actions and commutative algebra in 2 we summarize those parts of this interaction needed here the purpose of this paper is to study the homoclinic points of algebraic zd actions let al be an algebraic zd action on the compact abelian group x and let ox denote the additive identity of x a point x e x is homoclinic for ce if c nx ox as llnll oo the set ai x of all homoclinic points for oc is clearly a subgroup of x which we call the homoclinic group of oz in 3 we discuss some elementary properties of the homoclinic group including counltability of za x whenever oz is expansive our two main results are contained in 4 these are that if oz is an expansive algebraic ed action then 1 a x is nontrivial if and only if oz has strictly positive entropy and 2 a x is nontrivial and dense in x if and only if oz has completely positive entropy the second result is proved by first establishing in lemma 4 5 the density of a x for certain principal expansive actions by use of fourier analysis for these actions a x is generated by a single fundamental homoclinic point which can be computed explicitly this lemma is then combined with some commutative algebra to prove 2 recent work of kaminker and putnam 3 11 has suggested a general duality in the k theory of c algebras for a principal expansive action al on x we show that a x is isomorphic to the dual group of x providing a class of examples to which their duality theory applies ruelle has investigated expansive topological zd actions which satisfy an orbit tracing property called specification see 12 and 13 showing that there is a thermodynamic formalism for such actions in 5 we show that expansive algebraic zd actions with completely positive entropy always satisfy very strong specification properties thereby providing an extensive class of examples to which the thermodynamic formalism applies

stability analysis of impulsive stochastic cohen grossberg neural networks with mixed time delays abstract in this paper the problem of stability analysis for a class of impulsive stochastic cohen grossberg neural networks with mixed delays is considered the mixed time delays comprise both the time varying and infinite distributed delays by employing a combination of the m matrix theory and stochastic analysis technique a sufficient condition is obtained to ensure the existence uniqueness and exponential p stability of the equilibrium point for the addressed impulsive stochastic cohen grossberg neural network with mixed delays the proposed method which does not make use of the lyapunov functional is shown to be simple yet effective for analyzing the stability of impulsive or stochastic neural networks with variable and or distributed delays we then extend our main results to the case where the parameters contain interval uncertainties moreover the exponential convergence rate index is estimated which depends on the system parameters an example is given to show the effectiveness of the obtained results

harmonic models and spanning forests of residually finite groups abstract we prove a number of identities relating the sofic entropy of a certain class of non expansive algebraic dynamical systems the sofic entropy of the wired spanning forest and the tree entropy of cayley graphs of residually finite groups we also show that homoclinic points and periodic points in harmonic models are dense under general conditions

exponential stability on stochastic neural networks with discrete interval and distributed delays this brief addresses the stability analysis problem for stochastic neural networks snns with discrete interval and distributed time varying delays the interval time varying delay is assumed to satisfy 0 d1 d t d2 and is described as d t d 1 h t with 0 h t d 2 d 1 based on the idea of partitioning the lower bound d 1 new delay dependent stability criteria are presented by constructing a novel lyapunov krasovskii functional which can guarantee the new stability conditions to be less conservative than those in the literature the obtained results are formulated in the form of linear matrix inequalities lmis numerical examples are provided to illustrate the effectiveness and less conservatism of the developed results

output controllability and optimal output control of state dependent switched boolean control networks in the present paper we investigate the output controllability and optimal output control problems of a state dependent switched boolean control network by using the semi tensor product the algebraic form of the system is obtained then output controllability problems of the system are discussed and some necessary and sufficient conditions are given next the mayer type optimal output control issue is considered and an algorithm is provided to find out the control sequence at last an example is given to show the effectiveness of the main results

global exponential stability of memristive neural networks with mixed time varying delays this article investigates the lagrange exponential stability and the lyapunov exponential stability of memristive neural networks with discrete and distributed time varying delays dmnns by means of inequality techniques theories of the m matrix and the comparison strategy the lagrange exponential stability of the underlying dmnns is considered in the sense of filippov and the globally exponentially attractive set is estimated through employing the m matrix and external input especially when the external input is not concerned the lyapunov exponential stability of the corresponding dmnns is developed immediately in the form of an m matrix which contains some published outcomes as special cases furthermore by constructing an m matrix based differential system the lyapunov exponential stability of the dmnns is studied which is less conservative than some existing ones finally three simulation examples are carried out to examine the validness of the theories

delay dependent stochastic stability of delayed hopfield neural networks with markovian jump parameters in this paper the problem of stochastic stability for a class of time delay hopfield neural networks with markovian jump parameters is investigated the jumping parameters are modeled as a continuous time discrete state markov process without assuming the boundedness monotonicity and differentiability of the activation functions some results for delay dependent stochastic stability criteria for the markovian jumping hopfield neural networks mjdhnns with time delay are developed we establish that the sufficient conditions can be essentially solved in terms of linear matrix inequalities

entropy determinants and l 2 torsion we show that for any amenable group and any z module m of type fl with vanishing euler characteristic the entropy of the natural action on the pontryagin dual of m is equal to the l 2 torsion of m as a particular case the entropy of the principal algebraic action associated with the module z z f is equal to the logarithm of the fuglede kadison determinant of f whenever f is a non zero divisor in z this conrms a conjecture of deninger as a key step in the proof we provide a general szeg o type approximation theorem for the fuglede kadison determinant on the group von neumann algebra of an amenable group as a consequence of the equality between l 2 torsion and entropy we show that the l 2 torsion of a non trivial amenable group with nite classifying space vanishes this was conjectured by luck finally we establish a milnor turaev formula for the l 2 torsion of a nite acyclic chain complex

robust stabilization of uncertain linear systems quadratic stabilizability and h sup infinity control theory the problem of robustly stabilizing a linear uncertain system is considered with emphasis on the interplay between the time domain results on the quadratic stabilization of uncertain systems and the frequency domain results on h sup infinity optimization a complete solution to a certain quadratic stabilization problem in which uncertainty enters both the state and the input matrices of the system is given relations between these robust stabilization problems and h sup infinity control theory are explored it is also shown that in a number of cases if a robust stabilization problem can be solved via lyapunov methods then it can be also be solved via h sup infinity control theory based methods

common asymptotic behavior of solutions and almost periodicity for discontinuous delayed and impulsive neural networks the paper considers a general neural network model with impulses at a given sequence of instants discontinuous neuron activations delays and time varying data and inputs it is shown that when the neuron interconnections satisfy an m matrix condition or a dominance condition then the state solutions and the output solutions display a common asymptotic behavior as time t it is also shown via a new technique based on prolonging the solutions of the delayed neural network to that it is possible to select a unique special solution that is globally exponentially stable and can be considered as the unique global attractor for the network finally this paper shows that for almost periodic data and inputs the selected solution is almost periodic moreover it is robust with respect to a large class of perturbations of the data analogous results also hold for periodic data and inputs a by product of the analysis is that a sequence of almost periodic impulses is able to induce in the generic case nonstationary almost periodic solutions in an otherwise globally convergent nonimpulsive neural network to the authors knowledge the results in this paper are the only available results on global exponential stability of the unique periodic or almost periodic solution for a general neural network model combining three main features i e impulses discontinuous neuron activations and delays the results in this paper are compared with several results in the literature dealing with periodicity or almost periodicity of some subclasses of the neural network model here considered and some hints for future work are given

homoclinic groups ie groups and expansive algebraic actions we give algebraic characterizations for expansiveness of algebraic actions of countable groups the notion of p expansiveness is introduced for algebraic actions and we show that for countable amenable groups a finitely presented algebraic action is 1 expansive exactly when it has finite entropy we also study the local entropy theory for actions of countable amenable groups on compact groups by automorphisms and show that the ie group determines the pinsker factor for such actions for an expansive algebraic action of a polycyclic by finite group on x it is shown that the entropy of the action is equal to the entropy of the induced action on the pontryagin dual of the homoclinic group the homoclinic group is a dense subgroup of the ie group the homoclinic group is nontrivial exactly when the action has positive entropy and the homoclinic group is dense in x exactly when the action has completely positive entropy

sofic mean length abstract given a length function l on the r modules of a unital ring r for each sofic group we define a mean length for every locally l finite r module relative to a bigger r module we establish an addition formula for the mean length we give two applications the first one shows that for any unital left noetherian ring r r is stably direct finite the second one shows that for any z module m the mean topological dimension of the induced action on the pontryagin dual of m coincides with the von neumann luck rank of m

finite time stability and settling time estimation of nonlinear impulsive systems abstract this paper studies the problem of finite time stability fts for nonlinear impulsive systems based on impulsive control theory several lyapunov based fts theorems involving stabilizing impulses and destabilizing impulses are established respectively our proposed results provide sufficient conditions for estimating the settling time with respect to suitable classes of impulse time sequences it is shown that the settling time of nonlinear impulsive systems depends not only on the initial state but also on the impulse effect as compared with the case without using stabilizing impulses a smaller bound of setting time can be derived when a fts system is subject to stabilizing impulses conversely a larger bound of settling time can be derived when the fts system is subject to destabilizing impulses as compared with the case without using destabilizing impulses as a special case we extend the ideas to nonlinear impulsive delay systems and derive some delay independent fts results examples and their simulations are given to demonstrate the applicability of the proposed results

stability analysis of quaternion valued neural networks with both discrete and distributed delays abstract the existence uniqueness and stability of the equilibrium of quaternion valued neural networks qvnns with both discrete and distributed delays are investigated in this paper the considered model is managed as a single entirety without decomposition based on homeomorphic mapping theorem and linear matrix inequality several sufficient criteria are derived to ascertain the aforementioned qvnns to be globally asymptotically stable and exponentially stable moreover provided criteria can be verified by the linear matrix inequality lmi toolbox in matlab finally one simulation example is demonstrated to verify the effectiveness of obtained results

weak expansiveness for actions of sofic groups in this paper we shall introduce h expansiveness and asymptotical h expansiveness for actions of sofic groups by definition each h expansive action of a sofic group is asymptotically h expansive we show that each expansive action of a sofic group is h expansive and for any given asymptotically h expansive action of a sofic group the entropy function with respect to measures is upper semi continuous and hence the system admits a measure with maximal entropy observe that asymptotically h expansive property was first introduced and studied by misiurewicz for z actions using the language of tail entropy and thus in the remaining part of the paper we shall compare our definitions of weak expansiveness for actions of sofic groups with the definitions given in the same spirit of misiurewicz s ideas when the group is amenable it turns out that these two definitions are equivalent in this setting

exponential stability in p th mean of solutions and of convergent euler type solutions of stochastic delay differential equations one concept of the stability of a solution of an evolutionary equation relates to the sensitivity of the solution to perturbations in the initial data there are other stability concepts notably those concerned with persistent perturbations results are presented on the stability in p th mean of solutions of stochastic delay differential equations with multiplicative noise and of stochastic delay difference equations the difference equations are of a type found in numerical analysis and we employ our results to obtain mean square stability criteria for the solution of the euler maruyama discretization of stochastic delay differential equations the analysis proceeds as follows we show that an inequality of halanay type derivable via comparison theory can be employed to derive conditions for p th mean stability of a solution we then produce a discrete analogue of the halanay type theory that permits us to develop a p th mean stability analysis of analogous stochastic difference equations the application of the theoretical results is illustrated by deriving mean square stability conditions for solutions and numerical solutions of a constant coefficient linear test equation

a remark on t valued cohomology groups of algebraic group actions we prove that for a weakly mixing algebraic action g x the nth cohomology group hn g x t after quotienting out the natural subgroup hn g t contains hn g x as a natural subgroup for n 1 if we further assume the diagonal actions 2 4 are t cocycle superrigid and h2 g x is torsion free as an abelian group then the above also holds true for n 2 applying it for principal algebraic actions when n 1 we show that h2 g zg is torsion free as an abelian group when g has property t as a direct corollary of sorin popa s cocycle superrigidity theorem we also use it when n 2 to answer negatively a question of sorin popa on the 2nd cohomology group of bernoulli shift actions of property t groups

fuglede kadison determinant theme and variations abstract we review the definition of determinants for finite von neumann algebras due to fuglede and kadison fuglede b kadison r 1952 ann math 55 520 530 and a generalization for appropriate groups of invertible elements in banach algebras from a paper by skandalis and the author 1984 after some discussion of k theory and whitehead torsion we indicate the relevance of these determinants to the study of torsion in topology contents are as follows 1 the classical setting 2 on von neumann algebras and traces 3 fuglede kadison determinant for finite von neumann algebras 4 motivating question 5 brief reminder of and bott periodicity 6 revisiting the fuglede kadison and other determinants 7 on whitehead torsion 8 a few lines on torsion