pullback and pushout constructions in c algebra theory abstract a systematic study of pullback and pushout diagrams is conducted in order to understand restricted direct sums and amalgamated free products of c algebras particular emphasis is given to the relations with tensor products both with the minimal and the maximal c tensor norm thus it is shown that pullback and pushout diagrams are stable under tensoring with a fixed algebra and stable under crossed products with a fixed group general tensor products between diagrams are also investigated the relations between the theory of extensions and pullback and pushout diagrams are explored in some detail the crowning result is that if three short exact sequences of c algebras are given with appropriate morphisms between the sequences allowing for pullback or pushout constructions at the levels of ideals algebras and quotients then the three new c algebras will again form a short exact sequence under some mild extra conditions as a generalization of a theorem of t a loring it is shown that each morphism between a pair of c algebras combined with its extension to the stabilized algebras gives rise to a pushout diagram this result has applications to corona extendibility and conditional projectivity finally the pullback and pushout constructions are applied to the class of noncommutative cw complexes defined by s eilers t a loring and g k pedersen j reine angew math 1998 499 101 143 to show that this category is stable under tensor products and under restricted direct sums

stability of asynchronous systems with poisson transitions we consider the stability of an asynchronous system modeled as a collection of coupled processes with independent poisson state transitions using lyapunov functions sufficient conditions for both mean square and almost sure convergence independent of state transition rates are developed for the linear case and weaker transition rate dependent conditions are derived for the linear and nonlinear case an illustrative example is discussed and simulated

strongly self absorbing c algebras say that a separable unital c algebra v c is strongly self absorbing if there exists an isomorphism v such that lx ai e approximately unitarily equivalent homomorphisms we study this class of algebras which includes the cuntz algebras 2 ooo the uhf algebras of infinite type the jiang su algebra z and tensor products of 00 with uhf algebras of infinite type given a strongly self absorbing c algebra v we characterise when a separable c algebra absorbs v tensorially i e is p stable and prove closure properties for the class of separable testable c algebras finally we compute the possible if groups and prove a number of classification results which suggest that the examples listed above are the only strongly self absorbing c algebras

compact lie group action with the continuous rokhlin property abstract in this paper we study the continuous rokhlin property of c dynamical systems using techniques of equivariant kk theory and quantum group theory in particular we determine the kk equivalence class and give a classification of kirchberg g algebras when the g is a compact lie group with the hodgkin condition

on discrete time diagonal and d stability abstract this paper introduces four discrete time analogs of different types of matrix stability diagonal simultaneous vertex and d stability the last three being defined in terms of a certain associated polytope of matrices the diagonal stability of any vertex of this polytope is shown to imply its simultaneous stability and hence d stability of the vertices it is shown by a counterexample that d stability as in the continuous time case is not equivalent to diagonal stability several important classes for which this equivalence is true are identified it is shown that simultaneous stability of the vertices is equivalent to the simultaneous stability of the whole polytope and it is conjectured that this equivalence holds without the requirement of simultaneity some other conjectures relating the four types of stability are made and it is shown that in the 2 x 2 case all four are equivalent

on solving constrained optimization problems with neural networks a penalty method approach deals with the use of neural networks to solve linear and nonlinear programming problems the dynamics of these networks are analyzed in particular the dynamics of the canonical nonlinear programming circuit are analyzed the circuit is shown to be a gradient system that seeks to minimize an unconstrained energy function that can be viewed as a penalty method approximation of the original problem next the implementations that correspond to the dynamical canonical nonlinear programming circuit are examined it is shown that the energy function that the system seeks to minimize is different than that of the canonical circuit due to the saturation limits of op amps in the circuit it is also noted that this difference can cause the circuit to converge to a different state than the dynamical canonical circuit to remedy this problem a new circuit implementation is proposed

the spatial rokhlin property for actions of compact quantum groups abstract we introduce the spatial rokhlin property for actions of coexact compact quantum groups on c algebras generalizing the rokhlin property for both actions of classical compact groups and finite quantum groups two key ingredients in our approach are the concept of sequentially split homomorphisms and the use of braided tensor products instead of ordinary tensor products we show that various structure results carry over from the classical theory to this more general setting in particular we show that a number of c algebraic properties relevant to the classification program pass from the underlying c algebra of a rokhlin action to both the crossed product and the fixed point algebra towards establishing a classification theory we show that rokhlin actions exhibit a rigidity property with respect to approximate unitary equivalence regarding duality theory we introduce the notion of spatial approximate representability for actions of discrete quantum groups the spatial rokhlin property for actions of a coexact compact quantum group is shown to be dual to spatial approximate representability for actions of its dual discrete quantum group and vice versa

on the convergence of parallel asynchronous block iterative computations abstract this paper considers the convergence problem of parallel asynchronous block iterative computation schemes a new mathematical state space model for a class of nonlinear time varying parallel iterative schemes is proposed using this model which generalizes several models of the chazan miranker type together with large scale systems and liapunov techniques it is shown that the well known quasidominance condition on a certain aggregated matrix guarantees exponential convergence of this class of methods

gelfand numbers of operators with values in a hilbert space in the paper we prove two inequalities involving gelfand numbers of operators with values in a hilbert space the first inequality is a rademacher version of the main result in pa to 1 which relates the gelfand numbers of an operator from a banach spacex intol 2 n with a certain rademacher average for the dual operator the second inequality states that the gelfand numbers of an operatoru froml 1 n into a hilbert space satisfy the inequality k 1 2 c k u leqq c parallel u parallel log 1 n k 1 2 wherec is a universal constant several applications of these inequalities in the geometry of banach spaces are given

donsker classes of sets we study the central limit theorem clt and the law of large numbers lln for empirical processes indexed by a countable class of sets c the main result of purely measure theoretical nature relates different ways to measure the size of c it relies on a new rearrangement inequality that has been inspired by techniques used in the local theory of banach spaces as an application we give sharp necessary conditions for the clt that are in some sense the best possible we also obtain a way to compute the rate of convergence in the lln

exponential stability of impulsive high order hopfield type neural networks with time varying delays this paper considers the problems of global exponential stability and exponential convergence rate for impulsive high order hopfield type neural networks with time varying delays by using the method of lyapunov functions some sufficient conditions for ensuring global exponential stability of these networks are derived and the estimated exponential convergence rate is also obtained as an illustration an numerical example is worked out using the results obtained

symmetries of the car algebra a automorphism of period two of the uhf c algebra of type 2 is constructed for which the fixed point algebra is not approximately finite dimensional the construction also yields a diagonal maximal commutative c subalgebra whose spectrum is not totally disconnected variations and consequences of the construction are discussed

stability analysis of neural net controllers using fuzzy neural networks neural networks can only be trained with a crisp and finite data set therefore the approximation quality of a trained network is hard to verify so a common way in proving stability of a trained neural net controller is to demonstrate the existence of a lyapunov function in this article we propose a new method how stability of a neural net controller used as a closed loop feedback controller can be proven instead of finding a lyapunov function conditions for a fuzzy training set are developed if a fuzzy neural net is trained using this training set special properties of fuzzy neural nets can be used for estimating the generalization error after defuzzification of the fuzzy net finite stability of the process can be concluded

uniform limit theorems for harris recurrent markov chains we study uniform limit theorems for regenerative processes and get strong law of large numbers and central limit theorem of this type then we apply those results to harris recurrent markov chains based on some ideas of k athreya p ney and e nummelin

cellular neural field and its convergence analysis a new continuum model complementary to traditional cellular neural networks is introduced in this note we consider a cellular neural field formed by infinitely many cellular neurons and modelled by an integrodifferential equation then using lasalle s invariance principle on banach space we show that the field quantity will asymptotically converge to an equilibrium state under the condition that all equilibria of the system are isolated from the practical sense the convergence indicates the essential capability of retrieving message from original raw data

multiplicative perturbations of stable and convergent operators abstract some familiar classes of stable hilbert space operators are studied to determine how they overlap and where the unitary similarity classes of their members lie analogous but less familiar classes of convergent operators are examined with the same aim the classes considered are often sets of products m a where m is a given set of diagonal or hermitian matrices and a is a single matrix the a s for which m a is a set of stable or convergent operators are sometimes characterized

a new approach to least squares estimation with applications on montre que les conditions d entropie sur la classe 7b g des fonctions de regression impliquent une consistance l 2 forte de l estimateur des moindres carres

the central limit theorem for empirical processes under local conditions the case of radon infinitely divisible limits without gaussian component on obtient des resultats de convergence faible pour des processus empiriques indices par des classes f de fonctions dans le cas de limites de radon purement de poisson infiniment divisibles

quantitative characterizations of exponential convergence property of nonlinear discrete dynamical systems in this technical note we consider the exponential convergence property of the trajectories of nonlinear discrete dynamical systems with exponential stability based on an invariant quantity derived from lyapunov functions of the systems the essential and quantitative relationship between exponential convergence property of trajectories and the lyapunov functions is revealed

partially asynchronous parallel algorithms for network flow and other problems the problem of computing a fixed point of a nonexpansive function f is considered sufficient conditions are provided under which a parallel partially asynchronous implementation of the iteration x f x converges these results are then applied to i quadratic programming subject to box constraints ii strictly convex cost network flow optimization iii an agreement and a markov chain problem iv neural network optimization and v finding the least element of a polyhedral set determined by a weakly diagonally dominant leontief system finally simulation results illustrating the attainable speedup and the effects of asynchronism are presented

c algebras of real rank zero abstract the concept of real rank of a c algebra is introduced as a non commutative analogue of dimension it is shown that real rank zero is equivalent to the previously defined conditions fs and hp and that it is invariant under strong morita equivalence in particular under stable isomorphism real rank zero is also invariant under inductive limits and split extensions and the class may well be regarded as the conceptual completion of the af algebras in some cases notably when the algebra is matroid it is shown that the multiplier algebra also has real rank zero although that is not true in general by a result of g j murphy this implies a weyl von neumann type result for self adjoint multiplier elements in these cases

rokhlin dimension and c dynamics we develop the concept of rokhlin dimension for integer and for finite group actions on c algebras our notion generalizes the so called rokhlin property which can be thought of as rokhlin dimension 0 we show that finite rokhlin dimension is prevalent and appears in cases in which the rokhlin property cannot be expected the property of having finite rokhlin dimension is generic for automorphisms of mathcal z stable c algebras where mathcal z denotes the jiang su algebra moreover crossed products by automorphisms with finite rokhlin dimension preserve the property of having finite nuclear dimension and under a mild additional hypothesis also preserve mathcal z stability in topological dynamics our notion may be interpreted as a topological version of the classical rokhlin lemma automorphisms arising from minimal homeomorphisms of finite dimensional compact metrizable spaces always have finite rokhlin dimension the latter result has by now been generalized by szabo to the case of free and aperiodic mathbb z d actions on compact metrizable and finite dimensional spaces

bootstrapping general empirical measures it is proved that the bootstrapped central limit theorem for empirical processes indexed by a class of functions mathscr f and based on a probability measure p holds a s if and only if mathscr f in mathrm clt p and int f 2 dp infty where f sup f in mathscr f f and it holds in probability if and only if mathscr f in mathrm clt p thus for a large class of statistics no local uniformity of the clt about p is needed for the bootstrap to work consistency of the bootstrap the bootstrapped law of large numbers is also characterized these results are proved under certain weak measurability assumptions on mathscr f

a simple separable exact c algebra not anti isomorphic to itself we give an example of an exact stably finite simple separable c algebra d which is not isomorphic to its opposite algebra moreover d has the following additional properties it is stably finite approximately divisible has real rank zero and stable rank one has a unique tracial state and the order on projections over d is determined by traces it also absorbs the jiang su algebra z and in fact absorbs the 3 uhf algebra we can also explicitly compute the k theory of d namely k 0 d cong mathbb z tfrac 1 3 with the standard order and k1 d 0 as well as the cuntz semigroup of d namely w d cong mathbb z tfrac 1 3 sqcup 0 infty

localization of vc classes beyond local rademacher complexities abstract in statistical learning the excess risk of empirical risk minimization erm is controlled by comp n f n where n is a size of a learning sample comp n f is a complexity term associated with a given class f and 1 2 1 interpolates between slow and fast learning rates in this paper we introduce an alternative localization approach for binary classification that leads to a novel complexity measure fixed points of the local empirical entropy we show that this complexity measure gives a tight control over comp n f in the upper bounds under bounded noise our results are accompanied by a minimax lower bound that involves the same quantity in particular we practically answer the question of optimality of erm under bounded noise for general vc classes

dimension and stable rank in the k theory of c algebras in topological k theory which can be viewed as the algebraic side of the theory of vector bundles some of the interesting properties which one investigates are for example the conditions under which bundles must possess trivial direct summands or the extent to which the cancellation property for direct whitney sums of bundles holds such properties turn out to be controlled in part by the dimension of the base space and results describing the nature of this control are among what are frequently called stability results 17 during the past few years there has been a sudden flowering of a theory for c algebras see surveys 35 36 37 since c algebras are profitably thought of as non commutative locally compact spaces with the finitely generated projective modules being the appropriate generalization of vector bundles according to swan s theorem 40 it would be natural to look for stability results for c algebras but until now the theory has been focussed on the c groups of c algebras and there has been little discussion of stability properties presumably in part for lack of an appropriate concept of dimension for c algebras one of the main objectives of this paper is to introduce a concept of dimension for c algebras which directly generalizes the classical concept of dimension for compact spaces 16 20 22 and to develop some techniques for calculating it notably for c algebras which are obtained as crossed product algebras for an action of the group of integers in algebraic k theory which is the generalization of topological c theory to rings a substantial number of stability properties have been discovered 2 39 42 48 within that theory the concept which has played a role most analogous to that of dimension is the bass stable rank but despite its many successes the bass stable rank has often been difficult to calculate even for relatively uncomplicated rings another main objective of the present paper is to make accessible for application to c algebras some of the stability results from algebraic c theory this comes about from the fact that the concept of dimension which we introduce dominates the bass stable rank i was led to seek stability results for c algebras by my desire to determine whether the cancellation property for projective modules holds for the irrational rotation c algebras discussed in 30 which as elliott 11 has suggested can appropriately be called 2 dimensional non commutative tori in more concrete terms the question is whether two projections in such an algebra which have the same trace must be unitarily equivalent the present paper provides the general theory which is needed for the affirmative answer which i obtained in a subsequent paper i will derive the

mathcal c _ 0 x algebras stability and strongly self absorbing mathcal c algebras we study permanence properties of the classes of stable and so called mathcal d stable mathcal c algebras respectively more precisely we show that a mathcal c 0 x algebra a is stable if all its fibres are provided that the underlying compact metrizable space x has finite covering dimension or that the cuntz semigroup of a is almost unperforated a condition which is automatically satisfied for mathcal c algebras absorbing the jiang su algebra mathcal z tensorially furthermore we prove that if mathcal d is a k 1 injective strongly self absorbing mathcal c algebra then a absorbs mathcal d tensorially if and only if all its fibres do again provided that x is finite dimensional this latter statement generalizes results of blanchard and kirchberg we also show that the condition on the dimension of x cannot be dropped along the way we obtain a useful characterization of when a mathcal c algebra with weakly unperforated cuntz semigroup is stable which allows us to show that stability passes to extensions of mathcal z absorbing mathcal c algebras

a clt for empirical processes involving time dependent data for stochastic processes xt t e we establish sufficient conditions for the empirical process based on ixt y pr xt y t e y r to satisfy the clt uniformly in t e y r corollaries of our main result include examples of classical processes where the clt holds and we also show that it fails for brownian motion tied down at zero and e 0 1

finite group actions on c algebras with the rohlin property i basic properties of finite group actions with the rohlin property on unital c algebras are investigated a characterization of finite group actions with the rohlin property on the cuntz algebra 02 is given in terms of central sequences which may be considered as an equivariant version of e kirchberg and n c phillips s characterization of 02 a large class of symmetries on 02 are classified in terms of the fixed point algebras for conjugacy and the crossed products for cocycle conjugacy model actions of symmetries of 02 are constructed for given k theoretical invariants

entropy and the combinatorial dimension we solve talagrand s entropy problem the l2 covering numbers of every uniformly bounded class of functions are exponential in its shattering dimension this extends dudley s theorem on classes of 0 1 valued functions for which the shattering dimension is the vapnik chervonenkis dimension in convex geometry the solution means that the entropy of a convex body k is controlled by the maximal dimension of a cube of a fixed side contained in the coordinate projections of k this has a number of consequences including the optimal elton s theorem and estimates on the uniform central limit theorem in the real valued case

approximate homogeneity is not a local property it is shown that the ah algebras satisfy a certain splitting property at the level of k theory with torsion coefficients the splitting property is used to prove the following i there are locally homogeneous c algebras which are not ah algebras ii the class of ah algebras is not closed under countable inductive limits iii there are real rank zero split quasidiagonal extensions of ah algebras which are not ah algebras

a non parametric entropy based approach to detect changes in climate extremes summary the paper focuses primarily on temperature extremes measured at 24 european stations with at least 90 years of data here the term extremes refers to rare excesses of daily maxima and minima as mean temperatures in this region have been warming over the last century it is automatic that this positive shift can be detected also in extremes after removing this warming trend we focus on the question of determining whether other changes are still detectable in such extreme events as we do not want to hypothesize any parametric form of such possible changes we propose a new non parametric estimator based on the kullback leibler divergence tailored for extreme events the properties of our estimator are studied theoretically and tested with a simulation study our approach is also applied to seasonal extremes of daily maxima and minima for our 24 selected stations

guaranteed convergence in a class of hopfield networks a class of symmetric hopfield networks with nonpositive synapses and zero threshold is analyzed in detail it is shown that all stationary points have a one to one correspondence with the minimal vertex covers of certain undirected graphs that the sequential hopfield algorithm as applied to this class of networks converges in at most 2n steps n being the number of neurons and that the parallel hopfield algorithm either converges in one step or enters a two cycle in one step the necessary and sufficient condition on the initial iterate for the parallel algorithm to converge in one step are given a modified parallel algorithm which is guaranteed to converge in 3n 2 steps x being the integer part of x for an n neuron network of this particular class is also given by way of application it is shown that this class naturally solves the vertex cover problem simulations confirm that the solution provided by this method is better than those provided by other known methods

a generalized convergence theorem for neural networks a neural network model is presented in which each neuron performs a threshold logic function the model always converges to a stable state when operating in a serial mode and to a cycle of length at most 2 when operating in a fully parallel mode this property is the basis for the potential applications of the model such as associative memory devices and combinatorial optimization the two convergence theorems for serial and fully parallel modes of operation are reviewed and a general convergence theorem is presented that unifies the two known cases new relations between the neural network model and the problem of finding a minimum cut in a graph are obtained

central limit theorems for stochastic processes under random entropy conditions necessary and sufficient conditions are found for the weak convergence of the row sums of an infinitesimal row independent triangular array nj of stochastic processes indexed by a set s to a sample continuous gaussian process when the array satisfies a random entropy condition analogous to one used by gine and zinn 1984 for empirical processes this entropy condition is satisfied when s is a class of sets or functions with the vapnik ervonenkis property and each nj f fd nj is of the form njc for some reasonable random finite signed measure vnj as a result we obtain necessary and sufficient conditions for the weak convergence of possibly non i i d partial sum processes and new sufficient conditions for empirical processes indexed by vapnik ervonenkis classes special cases include prokhorov s 1956 central limit theorem for empirical processes and shorack s 1979 theorems on weighted empirical processes

lower bounds for the empirical minimization algorithm in this correspondence we present a simple argument that proves that under mild geometric assumptions on the class f and the set of target functions t the empirical minimization algorithm cannot yield a uniform error rate that is faster than 1 radic k in the function learning setup this result holds for various loss functionals and the target functions from t that cause the slow uniform error rate are clearly exhibited

consistency for the least squares estimator in nonparametric regression we shall study the general regression model y g o x e where x and e are independent the available information about g 0 can be expressed by g 0 g for some class g as an estimator of g 0 we choose the least squares estimator we shall give necessary and sufficient conditions for consistency of this estimator in terms of basically geometric properties of g our main tool will be the theory of empirical processes

model selection in nonparametric regression model selection using a penalized data splitting device is studied in the context of nonparametric regression finite sample bounds under mild conditions are obtained the resulting estimates are adaptive for large classes of functions

donsker classes and random geometry on donne des conditions geometriques aleatoires pour que le processus empirique indice par la classe des fonctions de carre integrable satisfasse le theoreme central limite

stability analysis of hopfield type neural networks the paper applies several concepts in robust control research such as linear matrix inequalities edge theorem parameter dependent lyapunov function and popov criteria to investigate the stability property of hopfield type neural networks the existence and uniqueness of an equilibrium is formulated as a matrix determinant problem an induction scheme is used to find the equilibrium to verify whether the determinant is nonzero for a class of matrix a numerical range test is proposed several robust control techniques in particular linear matrix inequalities are used to characterize the local stability of the neural networks around the equilibrium the global stability of the hopfield neural networks is then addressed using a parameter dependent lyapunov function technique all these results are shown to generalize existing results in verifying the existence uniqueness of the equilibrium and local global stability of hopfield type neural networks

coupled samples in simulation assume that we wish to generate two samples of n independent identically distributed random variables x1 xn and y1 yn where x1 and y1 have densities f and g respectively if these samples are used in a simulation and f is close to g it is sometimes desirable to have close simulation results this can be achieved by insisting that both samples agree in most of their components that is xi yi for as many i as possible under the given distributional constraints samples with this property are said to be optimally coupled in this paper we propose and study various methods of coupling two samples a sequence of samples and an infinite family of samples

crossed products by finite group actions with the rokhlin property we prove that a number of classes of separable unital c algebras are closed under crossed products by finite group actions with the rokhlin property including a ai algebras at algebras and related classes characterized by direct limit decompositions using semiprojective building blocks b simple unital ah algebras with slow dimension growth and real rank zero c c algebras with real rank zero or stable rank one d simple c algebras for which the order on projections is determined by traces e c algebras whose quotients all satisfy the universal coefficient theorem f c algebras with a unique tracial state along the way we give a systematic treatment of the derivation of direct limit decompositions from local approximation conditions by homomorphic images which are not necessarily injective

on characterizations of exponential stability of nonlinear discrete dynamical systems on bounded regions in this paper we discuss the quantitative characterization problems of the exponential stability and trajectory convergence property of nonlinear discrete dynamical systems on a bounded set of the state space through introducing two new concepts the lip constant of a nonlinear operator and strongly equivalent metrics of a norm we show that the nonlinear discrete dynamical systems are exponentially stable on the bounded set if and only if the corresponding nonlinear operator is contractive under some strongly equivalent metrics or if and only if the lip constant of the nonlinear operator is less than one in the latter case we further show that the infimum of the exponential bounds of trajectories of the system equals exactly to the lip constant based on the obtained results we clearly explain how trajectory convergence properties of the systems are determined quantitatively by the lip constant and strongly equivalent metrics the obtained results not only are of importance in understanding the essence of exponential stability and trajectory convergence properties of nonlinear discrete dynamical systems on bounded sets of the state space but also provide some new useful criteria of testing exponential stability and estimating convergence speed of trajectories of the systems

the tracial rokhlin property for actions of finite groups on c algebras we define tracial analogs of the rokhlin property for actions of finite groups approximate representability of actions of finite abelian groups and approximate innerness we prove the following four analogs of related nontracial results bullet the crossed product of an infinite dimensional simple separable unital c algebra with tracial rank zero by an action of a finite group with the tracial rokhlin property again has tracial rank zero bullet an outer action of a finite abelian group on an infinite dimensional simple separable unital c algebra has the tracial rokhlin property if and only if its dual is tracially approximately representable and is tracially approximately representable if and only if its dual has the tracial rokhlin property bullet if a strongly tracially approximately inner action of a finite cyclic group on an infinite dimensional simple separable unital c algebra has the tracial rokhlin property then it is tracially approximately representable bullet an automorphism of an infinite dimensional simple separable unital c algebra a with tracial rank zero is tracially approximately inner if and only if it is the identity on k 0 a mod infinitesimals

tracially z absorbing c algebras abstract we study a tracial notion of z absorption for simple unital c algebras we show that if a is a c algebra for which this property holds then a has almost unperforated cuntz semigroup and if in addition a is nuclear and separable we show this property is equivalent to having a a z we furthermore show that this property is preserved under forming certain crossed products by actions satisfying a tracial rokhlin type property

the law of the iterated logarithm in uniformly convex banach spaces we give necessary and sufficient conditions for a random vari able x with values in a uniformly convex banach space b to satisfy the law of the iterated logarithm precisely we show that a b valued random vari able x satisfies the compact law of the iterated logarithm if and only if a 2 i2 x oo the family i x 2 x 6 b i 1 is uni formly integrable and sn a n 0 in probability 1 introduction let b be a banach space with topological dual b and norm by random variable with values in b we mean a measurable map x from some probability space 0 7 p into b such that the image of p by x defines a radon probability measure on the borel cr field of b we may assume equivalently that x takes almost surely its values in a separable subspace of b given a b valued random variable x we denote by xn new a sequence of inde pendent copies of x and let as usual sn xx h h xn we write lt to denote the function max l logi and use l2t to denote l lt t g r we set further or 2nl2n 1 2 the random variable x is said to satisfy the bounded law of the iterated logarithm lil if the sequence sn an n s is bounded with probability one and the compact lil if this sequence is almost surely relatively compact in b it is well known that if x satisfies the compact lil there is a compact convex and symmetric set k in b such that

asynchronous block iterative methods for almost linear equations abstract this paper gives convergence conditions for asynchronous block iterative methods for the solution of the almost linear equation ax f x where a is a linear operator f a block diagonal lipschitz continuous operator and x a vector in terms of a splitting of a and the lipschitz constant of f the methods used are a combination of the contraction mapping approach using a vectorial norm and a large scale systems approach using vector difference inequalities the load flow equations for a power system are almost linear in the above sense and considerable speedup can be obtained on a four transputer machine

existence and stability of a unique equilibrium in continuous valued discrete time asynchronous hopfield neural networks it is shown that the assumption of d stability of the interconnection matrix together with the standard assumptions on the activation functions guarantee the existence of a unique equilibrium under a synchronous mode of operation as well as a class of asynchronous modes for the synchronous mode these assumptions are also shown to imply local asymptotic stability of the equilibrium for the asynchronous mode of operation two results are derived first it is shown that symmetry and stability of the interconnection matrix guarantee local asymptotic stability of the equilibrium under a class of asynchronous modes this is referred to as local absolute asymptotic stability second it is shown that under the standard assumptions if the nonnegative matrix whose elements are the absolute values of the corresponding elements of the interconnection matrix is stable then the equilibrium is globally absolutely asymptotically stable under a class of asynchronous modes the results obtained are discussed from the points of view of their applications robustness and their relationship to earlier results