construction of invariant whiskered tori by a parameterization method part ii quasi periodic and almost periodic breathers in coupled map lattices we construct quasi periodic and almost periodic solutions for coupled hamiltonian systems on an infinite lattice which is translation invariant the couplings can be long range provided that they decay moderately fast with respect to the distance for the solutions we construct most of the sites are moving in a neighborhood of a hyperbolic fixed point but there are oscillating sites clustered around a sequence of nodes the amplitude of these oscillations does not need to tend to zero in particular the almost periodic solutions do not decay at infinity we formulate an invariance equation solutions of this equation are embeddings of an invariant torus on which the motion is conjugate to a rotation we show that if there is an approximate solution of the invariance equation that satisfies some non degeneracy conditions there is a true solution close by the proof of this emph a posteriori theorem is based on a nash moser iteration which does not use transformation theory simpler versions of the scheme were developed in e fontich r de la llave y sire emph j differential equations bf 246 3136 2009 one technical tool important for our purposes is the use of weighted spaces that capture the idea that the maps under consideration are local interactions using these weighted spaces the estimates of iterative steps are similar to those in finite dimensional spaces in particular the estimates are independent of the number of nodes that get excited using these techniques given two breathers we can place them apart and obtain an approximate solution which leads to a true solution nearby by repeating the process infinitely often we can get solutions with infinitely many frequencies which do not tend to zero at infinity

the fuzzy riemann integral and its numerical integration the fuzzy riemann integral and its numerical integration are proposed in this paper the level set of this fuzzy riemann integral is a closed interval whose end points are the classical riemann integrals thus we provide a numerical method to approximate the fuzzy riemann integral by invoking the simpson s rule we fit the end points closed interval of the level set of the fuzzy riemann integral as polynomials with variable in a least squares sense finally the membership function of the fuzzy riemann integral can be transformed into nonlinear programming problem and can be solved by any current optimizer

gaussian quadratures for approximate of fuzzy integrals abstract in this paper the integration formulas of gaussian methods with positive coefficient for fuzzy integrations are discussed and then are followed by convergence theorem the proposed algorithms are illustrated and compared with newton cot s methods t allahviranloo newton cot s methods for integration of fuzzy functions appl math comp in press by solving some numerical examples

gaussian quadratures for approximate of fuzzy multiple integrals in this paper the multiple integration formulas of gaussian methods with positive coefficient for fuzzy integrations are discussed and then are followed by convergence theorem the proposed algorithms are illustrated by solving some numerical examples

open fuzzy cubature rule with application to nonlinear fuzzy volterra integral equations in two dimensions abstract in this paper we propose an open fuzzy cubature formula and as application we construct an iterative numerical method for solving nonlinear fuzzy volterra integral equations in two dimensions the convergence of the method is proved by providing the error estimate and it is tested on two numerical experiments

on the fuzzy initial value problem abstract the initial value problem x t f t x t x 0 x 0 with fuzzy initial value and with deterministic or fuzzy function f is considered two different approaches viz the extension principle and the use of extremal solutions of deterministic initial value problems are applied generalizations to fuzzy integral equations and fuzzy functional differential equations are indicated

uniqueness of the srb measure for piecewise expanding weakly coupled map lattices in any dimension we prove the existence of a unique srb measure for a wide range of multidimensional weakly coupled map lattices these include piecewise expanding maps with diffusive coupling

numerical solution of two dimensional nonlinear hammerstein fuzzy integral equations based on optimal fuzzy quadrature formula in this paper our aim is to provide an efficient iterative method of successive approximations to approximate solution of linear and nonlinear two dimensional hammerstein fuzzy integral equations by defining and developing an optimal quadrature formula for classes of two dimensional fuzzy number valued functions of lipschitz type after the introduction of the optimal formula we prove the convergence of the method of successive approximations used to approximate the solution of two dimensional hammerstein fuzzy integral equations and investigate the numerical stability of the presented method with respect to the choice of the first iteration finally some illustrative numerical experiments confirm the theoretical results and demonstrate the accuracy of the method

short communication optimal fuzzy quadrature formula for classes of fuzzy number valued functions of lipschitz type in this paper we found out the optimal fuzzy quadrature formulas with given knots and the optimal fuzzy quadrature formulas among all fuzzy quadrature formulas for classes of fuzzy number valued functions of lipschitz type we also obtain the exact values of the errors of these optimal fuzzy quadrature formulas

dynamical systems on lattices with decaying interaction ii hyperbolic sets and their invariant manifolds abstract this is the second part of the work devoted to the study of maps with decay in lattices here we apply the general theory developed in fontich et al 2011 3 to the study of hyperbolic sets in particular we establish that any close enough perturbation with decay of an uncoupled lattice map with a hyperbolic set has also a hyperbolic set with dynamics on the hyperbolic set conjugated to the corresponding of the uncoupled map we also describe how the decay properties of the maps are inherited by the corresponding invariant manifolds

the supremum and infimum of the set of fuzzy numbers and its application abstract in this paper we prove that the bounded set of fuzzy numbers must exist supremum and infimum and give the concrete representation of supremum and infimum as an application we obtain that the continuous fuzzy valued function on a closed interval exists supremum and infimum and give the precise representation we also show that the bounded fuzzy valued function on a closed interval can define the lower and upper sums and the lower and upper integrals of riemann and riemann stieltjes by the usual way

fuzzy integral equations abstract we introduce a definition of the integral of a fuzzy valued function that is only slightly different from the usual one yet that is more intuitive and that can be applied to a larger class of functions we show that our definition is equivalent to the extension principle for functions of this class we then apply integration theory of fuzzy valued functions to integral equations we define a measure of the fuzziness of a fuzzy valued function and show that using this measure we can view the integration of fuzzy functions as a fuzzy linear transformation from a certain subspace of fuzzy valued functions to the space of fuzzy numbers we then show how this can be used to study fredholm integral equations of the second kind

elementary fuzzy calculus abstract in 2 dubois and prade introduced the notion of fuzzy numbers and defined the basic operations of addition subtraction multiplication and division a slightly modified definition of fuzzy numbers was presented in 4 and in that paper a metric was defined for this family of fuzzy sets another less restrictive definition of fuzzy numbers can be found in 1 in the present paper we consider fuzzy numbers from a somewhat different perspective basically we shall view fuzzy numbers in a topological vector space setting using the customary vector space operations together with the metric given in 4 we will define differentiation and integration of fuzzy valued functions in ways that parallel closely the corresponding definitions for real differentiation and integration

on numerical solution of two dimensional nonlinear urysohn fuzzy integral equations based on fuzzy haar wavelets in the present paper using fuzzy haar wavelets we introduce an iterative method for numerical solving two dimensional nonlinear urysohn fuzzy integral equations 2dnufie we define a useful metric to prove the convergence and error estimate of the method moreover we investigate the numerical stability of the presented numerical successive approximations finally numerical experiments show the accuracy of the method

application of simpson quadrature rule and iterative method for solving nonlinear fuzzy delay integral equations abstract in this study firstly we present an iterative method by using simpson quadrature formula for solving nonlinear fuzzy delay integral equations in addition the convergence analysis of the proposed method is investigated also we prove numerical stability analysis of the method by means of choosing the perturbation in the first iteration finally numerical examples to illustrate applicability of this method are presented

iterative method for numerical solution of two dimensional nonlinear fuzzy integral equations in this paper we prove the convergence of the method of successive approximations used to approximate the solution of two dimensional nonlinear fuzzy fredholm integral equations of the second kind also we present an iterative procedure based on quadrature formula to solve such equations the error estimation of the proposed method is given in terms of uniform and partial modulus of continuity finally some numerical experiments confirm the theoretical results and illustrate the accuracy of the method

numerical solutions of the nonlinear fuzzy hammerstein volterra delay integral equations in this paper an iterative numerical method that solves nonlinear fuzzy hammerstein volterra integral equations with constant delay is developed by using the error estimates a practical stopping criterion of the algorithm is obtained the convergence and the numerical stability of the method is proved and its accuracy is illustrated on three numerical examples the study of this integral equation is important because it has as a particular case the fuzzy variant of a mathematical model from epidemiology

dynamical systems on lattices with decaying interaction i a functional analysis framework abstract we consider weakly coupled map lattices with a decaying interaction that is we consider systems which consist of a phase space at every site such that the dynamics at a site is little affected by the dynamics at far away sites we develop a functional analysis framework which formulates quantitatively the decay of the interaction and is able to deal with lattices such that the sites are manifolds this framework is very well suited to study systematically invariant objects one obtains that the invariant objects are essentially local we use this framework to prove a stable manifold theorem and show that the manifolds are as smooth as the maps and have decay properties i e the derivatives of one of the coordinates of the manifold with respect to the coordinates at far away sites are small other applications of the framework are the study of the structural stability of maps with decay close to uncoupled possessing hyperbolic sets and the decay properties of the invariant manifolds of their hyperbolic sets in the companion paper by fontich et al 2011 10

fuzzy differential equations abstract this paper deals with fuzzy set valued mappings of a real variable whose values are normal convex upper semicontinuous and compactly supported fuzzy sets in rn we study differentiability and integrability properties of such functions and give an existence and uniqueness theorem for a solution to a fuzzy differential equation

theory and applications of fuzzy volterra integral equations formulations of fuzzy integral equations in terms of the aumann integral do not reflect the behavior of corresponding crisp models consequently they are ill adapted to describe physical phenomena even when vagueness and uncertainty are present a similar situation for fuzzy odes has been obviated by interpretation in terms of families of differential inclusions the paper extends this formalism to fuzzy integral equations and shows that the resulting solution sets and attainability sets are fuzzy and far better descriptions of uncertain models involving integral equations the investigation is restricted to volterra type equations with mildly restrictive conditions but the methods are capable of extensive generalization to other types and more general assumptions the results are illustrated by integral equations relating to control models with fuzzy uncertainties

about a numerical method of successive interpolations for two point boundary value problems with deviating argument a new numerical method for two point boundary value problems with deviating argument is obtained the method uses the fixed point technique the trapezoidal quadrature rule and a birkhoff interpolation procedure the convergence of the method is proved without smoothness conditions the kernel function being only lipschitzian in each argument the interpolation procedure is used only on the points where the argument is modified a stopping criterion of the algorithm is obtained and the accuracy of the method is illustrated on four numerical examples of pantograph type

existence and comparison theorems to volterra fuzzy integral equation in e n d abstract kaleva 1987 studied the measurability and integrability for the fuzzy set valued mappings of a real variable whose values are in the fuzzy number space e n d in this paper by using the embedding theorem of kaleva 1990 on e n d the darbo fixed points theorem in the convex cone c i e n where i denotes a bounded closed interval is given further the criteria for the existence and comparison of solutions to the volterra fuzzy integral equation in e n d are established under the compactness type conditions

equilibrium measures for coupled map lattices existence uniqueness and finite dimensional approximations we consider coupled map lattices of hyperbolic type i e chains of weakly interacting hyperbolic sets attractors over multi dimensional lattices we describe the thermodynamic formalism of the underlying spin lattice system and then prove existence uniqueness mixing properties and exponential decay of correlations of equilibrium measures for a class of holder continuous potential functions with a sufficiently small holder constant we also study finite dimensional approximations of equilibrium measures in terms of lattice systems approximations and lattice spin systems d approximations we apply our results to establish existence uniqueness and mixing property of srb measures as well as obtain the entropy formula

construction of fixed length insertion deletion correcting runlength limited codes an algorithm is presented for the construction of fixed length insertion deletion correcting runlength limited rll codes in this construction binary d k constrained codewords are generated by codewords of a q ary lee metric based code it is shown that this new construction always yields better codes than known constructions the use of a q ary lee 1987 metric code q odd is based on the assumption that an error insertion deletion or peak shift has maximal size q 1 2 it is shown that a decoding algorithm for the lee metric code can be extended so that it can also be applied to insertion deletion correcting rll codes furthermore such an extended algorithm can also correct some error patterns containing errors of size more than q 1 2 as a consequence if s denotes the maximal size of an error then in some cases the alphabet size of the generating code can be s 1 instead of 2 spl middot s 1

numerical solutions of fuzzy differential and integral equations using the embedding method numerical procedures for solving fuzzy differential equations fdes and fuzzy integral equations fies with arbitrary kernels have been investigated sufficient conditions for convergence of the proposed algorithms are given and their applicability is illustrated with examples this work and its conclusions may narrow the gap between the theoretical research on fdes and fies and the practical applications already existing in the design of various fuzzy dynamical systems

quadrature rules for integrals of fuzzy number valued functions in this paper we introduce some quadrature rules for the henstock integral of fuzzy number valued mappings by giving error bounds for mappings of bounded variation and of lipschitz type we also consider generalizations of classical quadrature rules such as midpoint type trapezoidal and three point type quadrature finally we study fine quadrature rules and we present some numerical applications

on the existence and uniqueness of solutions of fuzzy volterra fredholm integral equations abstract in this paper we study the problem of existence and uniqueness of solutions of volterra fredholm integral equation for the fuzzy set valued mappings

sinai ruelle bowen measures for lattice dynamical systems for weakly coupled expanding maps on the unit circle bricmont and kupiainen showed that the sinai ruelle bowen srb measure exists as a gibbs state via thermodynamic formalism we prove that this srb measure is indeed the unique equilibrium state for a holder continuous potential function on the infinite dimensional phase space for a more general class of lattice systems that are small perturbations of the uncoupled map lattice we present the variational principle the entropy formula and the formula for the potential function for the srb measures for coupled map lattices with nearest neighbor interactions we give an explicit formula of the potential function for the srb measure and consequently obtain the entropy in terms of coupling parameters

on henstock integral of fuzzy number valued functions i abstract in this paper we define and discuss the fh integral for fuzzy number valued functions using a concrete structure into which we embed the fuzzy number space e 1 several necessary and sufficient conditions of integrability for fuzzy number valued functions are given by means of abstract function theory

the entropy formula for srb measures of lattice dynamical systems we give a detailed proof of the entropy formula for srb measures of coupled hyperbolic attractors over integer lattices we show that the topological pressure for the potential function of the srb measure is zero

smooth dependence of thermodynamic limits of srb measures the thermodynamic limits of srb measures for uniformly hyperbolic sets are smoothly dependent on the map in an appropriate functional space

the improper fuzzy riemann integral and its numerical integration abstract the improper fuzzy riemann integral and its numerical integration are proposed in this paper under the setting of the improper fuzzy riemann integral we can find the expectation of fuzzy random variable the level set of the improper fuzzy riemann integral is a closed interval whose end points are the improper riemann integrals thus we provide a numerical method to approximate the improper fuzzy riemann integral by invoking the simpson s rule we also fit the end points closed interval of the level set of the improper fuzzy riemann integral as polynomials with variable in a least squares sense finally the membership function of the improper fuzzy riemann integral can be transformed into nonlinear programming problem and be solved by any presented optimizer