existence of nonexpansive retractions for amenable semigroups of nonexpansive mappings and nonlinear ergodic theorems in banach spaces in this paper we study nonlinear ergodic properties for an amenable semigroup of nonexpansive mappings in a banach space we prove that ifsis an amenable semigroup and s tt t s is a nonexpansive semigroup on a closed convex subsetcin a uniformly convex banach spaceesuch that the setf s of common fixed points of s is nonempty then there exists a nonexpansive retractionpfromcontof s such thatptt ttp pfor eacht sandpx co ttx t s for eachx c in this case there exists a net a of finite averages of s such that for eacht sand for each bounded subsetbofc lim a ttx a x 0 and lim tta x a x 0 uniformly forx b also if the norm ofeis frechet differentiable then for eachx c pxis the unique common fixed point in s s co ttsx t s furthermore if is an asymptotically invariant net of means then for eachx c t x converges weakly topx

nonlinear ergodic theorems of nonexpansive type mappings abstract let s be a semitopological semigroup let c be a closed convex subset of a uniformly convex banach space e whose norm is frechet differentiable and t t t s be a continuous representation of s as almost asymptotically nonexpansive type mapping of c into c such that the common fixed point set f of in c is nonempty in this paper we prove that if s is right reversible then for each x c the closed convex set s s c o t t x t s f consists of at most one point we also prove that if s is reversible then the intersection s s c o t t x t s f is nonempty for each x c if and only if there exists a nonexpansive retraction p of c onto f such that p t t t t p p for all t s and px is in the closed convex hull of t t x t s for each x c

on the dynamics of navier stokes and euler equations this is a detailed study on certain dynamics of navier stokes and euler equations via a combination of analysis and numerics we focus upon two main aspects a zero viscosity limit of the spectra of linear navier stokes operator b heteroclinics conjecture for euler equation its numerical verification melnikov integral and simulation and control of chaos due to the difficulty of the problem for the full navier stokes and euler equations we also propose and study two simpler models of them

an upper bound on the minimum number of monomials required to separate dichotomies of 1 1 n it is known that any dichotomy of 1 1 n can be learned separated with a higher order neuron polynomial function with 2n inputs monomials in general less than 2n monomials are sufficient to solve a given dichotomy in spite of the efforts to develop algorithms for finding solutions with fewer monomials there have been relatively fewer studies investigating maximum density n the minimum number of monomials that would suffice to separate an arbitrary dichotomy of 1 1 n this article derives a theoretical upper bound for this quantity superseding previously known bounds the main theorem here states that for any binary classification problem in 1 1 n n 1 one can always find a polynomial function solution with 2n 2n 4 or fewer monomials in particular any dichotomy of 1 1 n can be learned by a higher order neuron with a fan in of 2n 2n 4 or less with this result for the first time a deterministic ratio bound independent of n is established as n 2n 0 75 the main theorem is constructive so it provides a deterministic algorithm for achieving the theoretical result the study presented provides the basic mathematical tools and forms the basis for further analyses that may have implications for neural computation mechanisms employed in the cerebral cortex

minimal sign representation of boolean functions algorithms and exact results for low dimensions boolean functions bfs are central in many fields of engineering and mathematics such as cryptography circuit design and combinatorics moreover they provide a simple framework for studying neural computation mechanisms of the brain many representation schemes for bfs exist to satisfy the needs of the domain they are used in in neural computation it is of interest to know how many input lines a neuron would need to represent a given bf a common bf representation to study this is the so called polynomial sign representation where and 1 are associated with true and false respectively the polynomial is treated as a real valued function and evaluated at its parameters and the sign of the polynomial is then taken as the function value the number of input lines for the modeled neuron is exactly the number of terms in the polynomial this letter investigates the minimum number of terms that is the minimum threshold density that is sufficient to represent a given bf and more generally aims to find the maximum over this quantity for all bfs in a given dimension with this work for the first time exact results for four and five variable bfs are obtained and strong bounds for six variable bfs are derived in addition some connections between the sign representation framework and bent functions are derived which are generally studied for their desirable cryptographic properties

an ergodic theorem for semigroups of nonexpansive mappings in a hilbert space abstract j b baillon c r acad sci paris ser a 280 1975 1511 1514 proved an ergodic theorem for a single nonexpansive mapping in a hilbert space which is a nonlinear version of von neumann s mean ergodic theorem in this paper we study the ergodic behavior of a semigroup of nonexpansive mappings we try to find a sequence of means on the semigroup generalizing the cesaro means on n such that the corresponding sequence of nonexpansive mappings converges to a projection onto the set of common fixed points our method of proof is an appropriate modification of a pazy s proof israel j math 26 1977 197 204 of baillon s theorem

decomposition and construction of fuzzy sets and their applications to the arithmetic operations on fuzzy quantities abstract the membership function of a fuzzy set can be constructed from a family of subsets of a universal set via the form of resolution identity assuming that the family is nested in this paper we are going to consider the weak case of assuming that only a countable subfamily is nested using this generalized consideration we can overcome the ambiguity of arithmetic operations on fuzzy quantities two well known additions of fuzzy quantities in vector space have been adopted in the literature one is based on the extension principle by directly considering the membership functions of course this addition always exists another one uses the level sets without considering the membership functions however the existence of this addition cannot be guaranteed in this paper we are going to propose two other additions of fuzzy quantities in vector space one generalizes the extension principle and the other one also uses the level sets without considering the membership functions such that this generalized addition always exists we can show that these four additions are equivalent

extension principles for fuzzy set theory in several cases we show that it is possible to extend a notion in classical mathematics by identifying each fuzzy subset with the continuous chain of its closed cuts and by applying this notion to these cuts in particular this idea is applied to extend functions from subsets into subsets for instance closure operators and functions from sets into real numbers for instance measures

a natural interpretation of fuzzy mappings in this paper we present a new and natural interpretation of fuzzy mappings which is a direct extension of the natural interpretation of fuzzy sets and fuzzy relations shown in our previous paper a natural interpretation of fuzzy sets and fuzzy relations fuzzy sets and systems 128 2002 135 we interpret fuzzy mappings as mappings in a heyting valued model for intuitionistic set theory and present a characterization of fuzzy mappings with membership functions which seems to be different from all kinds of known definitions this interpretation is quite natural for it deduces naturally various properties of fuzzy mappings and it makes clear the meaning of the famous extension principle by zadeh which is said to be very important in fuzzy theory

extension principle in fuzzy optimization the paper deals with the generalization of the well known extension principle to set to set mappings multifunctions in the first part of the paper an image of a fuzzy set is defined and several properties for special classes of multifunctions are derived the second part of the paper is devoted to fuzzy optimization using the first part a new concept of solution of the fuzzy optimization problem is introduced and then compared with the usual one

possibilistic linear programming a brief review of fuzzy mathematical programming and a comparison with stochastic programming in portfolio selection problem in this paper we review some fuzzy linear programming methods and techniques from a practical point of view in the rst part the general history and the approach of fuzzy mathematical programming are introduced using a numerical example some models of fuzzy linear programming are described in the second part of the paper fuzzy mathematical programming approaches are compared to stochastic programming ones the advantages and disadvantages of fuzzy mathematical programming approaches are exemplied in the setting of an optimal portfolio selection problem finally some newly developed ideas and techniques in fuzzy mathematical programming are briey reviewed c 2000 elsevier science b v all rights reserved

nonlinear ergodic theorems for general curves defined on general semigroups let g be a algebra semigroup and let h be a real hilbert space let f g be a set of all mappings from g into h in this paper we first try to provide a nonlinear ergodic retraction theorem for the mappings from g to h moreover we introduce the notion of asymptotically nonexpansive type curves and give the ergodic convergence theorems for this kind of curves using the main results we can also prove the ergodic convergence theorem for the almost orbits of semitopological semigroups of non lipschitzian mappings without convexity

on the level convergence of a sequence of fuzzy numbers abstract in this paper we first give a simplified proof of the existence theorem of supremum and infimum in fuzzy number space e 1 established by wu and wu j math anal appl 210 1997 499 and improve the expressions of the supremum and infimum as a straightforward corollary of this result we obtain a necessary and sufficient condition under which for a bounded sequence of fuzzy numbers u n the pair of functions sup n u n and sup n u n can determine a fuzzy number secondly we give a necessary and sufficient condition for a sequence of fuzzy numbers u n to be levelwise convergent in e 1 and generalize some important theorems in real number spaces to fuzzy number spaces finally we prove the existence of supremum and infimum for the level continuous fuzzy valued function on a closed interval and give a necessary and sufficient condition under which its supremum and infimum can be attained

on xor lemmas for the weight of polynomial threshold functions abstract a multilinear polynomial p is said to sign represent a boolean function f 1 1 n 1 1 if f x s g n p x for all x 1 1 n in this paper we consider the length and weight of polynomials sign representing boolean functions of the form k f the xor of k copies of f on disjoint sets of variables firstly we show that for an infinite family of functions f a naive construction does not yield a shortest polynomial sign representing k f more precisely we give a construction of polynomials sign representing k and n whose length is strictly smaller than the k th power of the minimum length of a polynomial sign representing and n for every k 2 and n 2 except for k n 2 previously such polynomials were known only for n 2 sezener and oztop 2015 a similar result for the weight is also provided secondly we introduce a parameter v f of a boolean function f and show that the k th root of the minimum weight of a polynomial sign representing k f converges between v f and v f 2 as k goes to infinity

on the existence of fixed points and ergodic retractions for lipschitzian semigroups in hilbert spaces and especially nonexpansive in the case of k 1 let s be a semitopological semigroup and let s c s e s be a continuous representation of s as lipschitzian mappings of c into itself which is said to be a lipschitzian semigroup on c a lipschitzian semigroup s t s e s with lipschitz constants k s e s is called a uniformly k lipschitzian semigroup and a nonexpansive semigroup when k k and k 1 for all s e s respectively goebel and kirk 7 first called a mapping t of c into itself a uniformly k lipschitzian mapping on c in the case when s t n e n is a lipschitzian semigroup with k k for all n e n lifschitz 12 obtained a fixed point theorem for uniformly k lipschitzian mappings on c with k 5 and this result was extended to uniformly k lipschitzian semigroups which are left reversible by downing and ray 6 lau l l showed that a nonexpansive semigroup has a common fixed point if there exists a left invariant mean on the space rug s of bounded right uniformly continuous functions on the semigroup furthermore ishihara and takahashi 9 extended lau s result to uniformly k lipschitzian semigroups and also gave a simpler proof of downing and ray s theorem on the other hand the first nonlinear ergodic theorem for nonexpansive mappings was established by baillon 11 let t be a nonexpansive mapping of c into itself and suppose that the set f t of fixed points of t is nonempty the the cesaro means

semigroup of nonexpansive mappings on a hilbert space soit s un semi groupe semi topologique soit h un espace de hilbert et s t s s s une representation continue de s comme applications non dilatation sur un sous ensemble convexe clos c de h dans c soit x c on etudie la convergence de t s x s s vers un point fixe commun z de s dans c

weak convergence and non linear ergodic theorems for reversible semigroups of non lipschitzian mappings abstract letgbe a semitopological semigroup letcbe a closed convex subset of a uniformly convex banach spaceewith afrechetdifferentiable norm and t tt t g be a continuous representation ofgas asymptotically nonexpansive type mappings ofcinto itself such that the common fixed point setf t of t incis nonempty we prove in this paper that ifgis right reversible then for every almost orbitu of t formula consists of at most one point further formula is nonempty for eachx cif and only if there exists a nonexpansive retractionpofcontof t such thatpts tsp pfor alls gandp x is in the closed convex hull of tsx s g x c this result is applied to study the problem of weak convergence of the net u t t g to a common fixed point of t

asymptotic behavior of uniformly asymptotically almost nonexpansive curves in a hilbert space abstract we extend the notion of uniformly asymptotically almost nonexpansive curves which was first introduced in rouhani and kim j nonlinear convex anal 4 2003 175 for a commutative semigroup to a general semitopological semigroup study the asymptotic behavior of such curves in a hilbert space and apply our results to study the asymptotic behavior for almost orbits of semigroups of asymptotically nonexpansive type mappings over a general semitopological semigroup in a hilbert space since in this case the semigroup is not directed the proofs in rouhani and kim j nonlinear convex anal 4 2003 175 do not extend to this case and new methods have to be used for the proofs in addition to rouhani j math anal appl 147 1990 465 j math anal appl 151 1990 226 nonlinear anal 44 2001 627 nonlinear anal 51 2002 735 and rouhani and kim j nonlinear convex anal 4 2003 175 our results extend and unify many previously known results j math anal appl 127 1987 206 proc amer math soc 104 1988 431 nonlinear anal 29 1997 539 topol methods nonlinear anal 5 1995 305 j math anal appl 105 1985 514 nonlinear anal 26 1996 1411 pacific j math 126 1987 277 j math anal appl 206 1997 451 chinese ann math 6 1996 729 nonlinear anal 14 1990 69 j math anal appl 85 1982 172 proc amer math soc 81 1981 253 proc amer math soc 97 1986 55 canad j math 44 1992 880 j math anal appl 142 1989 242 proc amer math soc 117 1993 385

inequalities in banach spaces with applications well known that many problems in a hilbert space h are solved by applying the following identity ilax 1 a yi12 a412 1 llyl12 a 1 mx yl12 1 1 for all x y in h and 0 5 i 5 1 therefore one natural method to solve problems in a banach space xis to establish equalities and usually inequalities in xanalogous to 1 1 this is in fact shown by some authors recent work see for example reich l poffald and reich 2 lim 3 lim et af 4 prus and smarzewski 5 smarzewski 6 and reich 21 the purpose of this paper is to continue carrying out investigations in this line more precisely in section 3 we establish some inequalities analogous to 1 l in either uniformly convex or uniformly smooth banach spaces as corollaries we obtain new characteristics of p uniform convexity and q uniform smoothness of a banach space x via the functionals i lip and ii llq respectively we give in section 4 applications of the inequalities obtained in section 3 to fixed points of uniformly lipschitzian mappings and to best approximation theory

asymptotic behavior of almost orbits of reversible semigroups of lipschitzian mappings abstract let c be a nonempty closed convex subset of a hilbert space h g a right reversible semitopological semigroup and s s t t g a continuous representation of g as lipschitzian mappings of c into itself then we deal with the asymptotic behavior of an almost orbit u t t g of s s t t g we first prove that if limsupt kt 1 then the closed convex subset s e g co u t t s f s consists of at most one point where kt is the lipschitzian constant of s t t g then we apply this result to study the problem of weak convergence of the net u t t g