cyclic classes and an ergodic theorem in dynamic fuzzy systems this paper analyses cyclic behavior of dynamic fuzzy systems defined by fuzzy relations on a metric space for the system we give an ergodic theorem with a strata structure

a limit theorem in dynamic fuzzy systems with a monotone property abstract a dynamic fuzzy system defined on a compact state space is considered under a monotone property of the fuzzy relation we study the limit of a sequence of fuzzy states and obtain a convergence theorem the limit is characterized as the solution of a fuzzy relational equation a numerical example is given to comprehend our idea in this paper

duality in dynamic fuzzy systems this paper shows the resolvent equation and the co balayage theorem for a dynamic fuzzy system next we define a dual system for the dynamic fuzzy system and give a duality for snell s optimal stopping problem by the dual system

a multiobjective fuzzy stopping in a stochastic and fuzzy environment abstract in a stochastic and fuzzy environment a multiobjective fuzzy stopping problem is discussed the randomness and fuzziness are evaluated by probabilistic expectations and linear ranking functions respectively pareto optimal fuzzy stopping times are given under the assumption of regularity for stopping rules by using optimal stopping times

some new sequence spaces of fuzzy numbers defined by orlicz functions using a fuzzy metric using orlicz functions and a fuzzy metric some new sequence spaces are introduced in this paper using the concept that the distance between two points in a fuzzy metric space is a non negative upper semicontinuous normal and convex fuzzy number we study some basic properties of the new sequence spaces of fuzzy numbers also we investigate the relations between these spaces

λ difference sequence spaces of fuzzy numbers in this paper using the difference operator of order m and an orlicz function we introduce and examine some classes of sequences of fuzzy numbers we give the relations between the strongly cesaro type convergence and statistical convergence in these spaces furthermore we study some of their properties like completeness solidity symmetricity etc we also give some inclusion relations related to these classes

a general cone decomposition theory based on efficiency letk1 andk2 be two convex cones in some common vector space this paper is concerned with the question of finding a good decomposition with respect tok1 andk2 of a given element of the minkowski sumk1 k2 we propose the criterion of efficiency as a measure for the quality of a decomposition this criterion allows us to set up a framework from which a general cone decomposition theory is then derived

weakly convex sets and modulus of nonconvexity we consider a definition of a weakly convex set which is a generalization of the notion of a weakly convex set in the sense of vial and a proximally smooth set in the sense of clarke from the case of the hilbert space to a class of banach spaces with the modulus of convexity of the second order using the new definition of the weakly convex set with the given modulus of nonconvexity we prove a new retraction theorem and we obtain new results about continuity of the intersection of two continuous set valued mappings one of which has nonconvex images and new affirmative solutions of the splitting problem for selections we also investigate relationship between the new definition and the definition of a proximally smooth set and a smooth set

on the ideal convergence of sequences of fuzzy numbers in this paper we introduce and study the concepts of i convergence i convergence and i cauchy sequence for sequences of fuzzy numbers where i denotes the ideal of subsets of n the set of positive integers

a continuous time dynamic fuzzy system i a limit theorem abstract a continuous time dynamic fuzzy system with possibility transition is constructed and a differential of fuzzy states is given by paths a limit theorem for fuzzy states is discussed under a contractive condition on fuzzy relations

ranking of fuzzy sets based on the concept of existence abstract various approaches have been proposed for the comparison or ranking of fuzzy sets however due to the complexity of the problem a general method which can be used for any situation still does not exist this paper formalizes the concept of existence for the ranking of fuzzy sets many of the existing fuzzy ranking methods are shown to be some application of this concept an improved fuzzy ranking method is then introduced based on this concept this newly introduced method is expanded for treating both normal and nonnormal convex and nonconvex fuzzy sets emphasis is placed on the use of the subjectivity of the decision maker such as the optimistic or the pessimistic view points an improved procedure for obtaining linguistic conclusions is also developed finally some numerical examples are given to illustrate the approach

statistical cluster and extreme limit points of sequences of fuzzy numbers in this paper we investigate the relations between statistical limit superior or statistical limit inferior and statistical cluster points of a statistically bounded sequence of fuzzy numbers and show that the set of statistical cluster points of a sequence of this type might be empty

polyhedral approximations of strictly convex compacta we consider polyhedral approximations of strictly convex compacta in finite dimensional euclidean spaces such compacta are also uniformly convex we obtain the best possible estimates for errors of considered approximations in the hausdorff metric we also obtain new estimates of an approximate algorithm for finding the convex hulls

uniform convexity of ψ direct sums of banach spaces abstract let x and y be banach spaces and a continuous convex function on the unit interval 0 1 satisfying certain conditions let x y be the direct sum of x and y equipped with the associated norm with we show that x y is uniformly convex if and only if x y are uniformly convex and is strictly convex as a corollary we obtain that the l p q direct sum x p q y 1 q p not p q 1 nor is uniformly convex if and only if x y are where l p q is the lorentz sequence space these results extend the well known fact for the l p sum x p y 1 some other examples are also presented

the time average reward for some dynamic fuzzy systems abstract in this paper by using a fuzzy relation we define a dynamic fuzzy system with a bounded convex fuzzy reward on the positive orthant r n of an n dimensional euclidean space as a measure of the system s performance we introduce the time average fuzzy reward which is characterized by the limiting fuzzy state under the contractive properties of the fuzzy relation in the one dimensional case the average fuzzy reward is expressed explicitly by the functional equations concerning the extreme points of its cuts also a numerical example is given to illustrate the theoretical results

a continuous time dynamic fuzzy system ii fuzzy potentials abstract this paper is a sequel of yoshida 9 fuzzy potentials are introduced in a continuous time dynamic fuzzy system and a fuzzy relational differential equation is given to characterize the fuzzy potentials

on the splitting problem for selections abstract we investigate when does the repovs semenov splitting problem for selections have an affirmative solution for continuous set valued mappings in finite dimensional banach spaces we prove that this happens when images of set valued mappings or even their graphs are p sets in the sense of balashov or strictly convex sets we also consider an example which shows that there is no affirmative solution of this problem even in the simplest case in r 3 we also obtain affirmative solution of the approximate splitting problem for lipschitz continuous selections in the hilbert space

local characterization of strongly convex sets abstract strongly convex sets in hilbert spaces are characterized by local properties one quantity which is used for this purpose is a generalization of the modulus of convexity of a set we also show that lim e 0 e e 2 exists whenever is closed and convex

regularity and chaos in recurrent fuzzy systems abstract in this paper we shall present a mathematical definition of recurrent fuzzy systems and begin to systematically investigate the underlying theory involved unlike static fuzzy systems recurrent fuzzy systems are equipped with time delayed feedback of their output and allow representing knowledge based dynamic processes that may be stated in the form of if then rules we study their relationship to automata and show that they have an automaton like behavior when appropriately designed in other cases recurrent fuzzy system may exhibit chaotic behavior we present sufficient conditions for the occurrence of chaos in recurrent fuzzy systems that can easily be checked solely on the basis of the qualitative linguistically formulated models we also discuss the extent to which state graphs may be used for describing the behaviors of recurrent fuzzy systems

some properties of limit inferior and limit superior for sequences of fuzzy real numbers abstract the limit inferior and limit superior of a bounded sequence of fuzzy real numbers have been introduced by aytar et al 2008 1 in this paper we give a simplified expressions for limit inferior and limit superior the expressions is concise and convenient for use as a straightforward corollary of this expressions we can easily prove some properties of the limit inferior and limit superior

the dual theory of choice under risk in this essay a new theory of choice under risk is being proposed it is a theory which in a sense that will become clear is dual to expected utility theory hence the title dual theory risky prospects are evaluated in this theory by a cardinal numerical scale which resembles an expected utility except that the roles of payments and probabilities are reversed this theme the reversal of the roles of probabilities and payments will recur throughout the paper i should emphasize that playing games with probabilities masquerading as payments and payments masquerading as probabilities is not my object rather i hope to convince the reader that the dual theory has intrinsic economic significance and that in some areas its predictions are superior to those of expected utility theory while in other areas the reverse will be the case two reasons have prompted me to look for an alternative to expected utility theory the first reason is methodological in expected utility theory the agent s attitude towards risk and the agent s attitude towards wealth are forever bonded together at the level of fundamental principles risk aversion and diminishing marginal utility of wealth which are synonymous under expected utility theory are horses of different colors the former expresses an attitute towards risk increased uncertainty hurts while the latter expresses an attitude towards wealth the loss of a sheep hurts more when the agent is poor than when the agent is rich a question arises therefore as to whether these two notions can be kept separate from each other in a full fledged theory of cardinal utility the dual theory will have this property the second reason that leads me to look for an alternative to expected utility theory is empirical behavior patterns which are systematic yet inconsistent with expected utility theory have often been observed two prominent references among many others are allais 1953 and kahneman tversky 1979 so deeply

sequences in a fuzzy metric space in this paper we introduce the concept of bounded sequences in a fuzzy metric space x d min max and show that every convergent sequence in x is bounded we also discuss the space of convergent sequences in a complete fuzzy metric space x d min max

fuzzy sets probability and measurement abstract in recent years the problem of uncertainty modeling has received much attention from scholars in artificial intelligence and decision theory various formal settings including but not restricted to fuzzy sets and possibility measures have been proposed based on different intuitions and dealing with various kinds of uncertain data the two main research directions are upper and lower probabilities which convey the idea of imprecisely estimated probability measures and distorted probabilities for the descriptive assessment of partial belief possibility measures and thereby fuzzy sets stand at the crossroads of these new approaches traditional views interpretive settings and canonical experiments for the measurement of probability such as frequentist approaches betting theories comparative uncertainty relations are currently extended to the generalized uncertainty measures these works shed new light on various interpretations of fuzzy sets and clarify their links with probability theory conversely zadeh s logical point of view on fuzzy sets suggests a set theoretic perspective on uncertainty measures that brings together numerical quantification and logic

differentials of fuzzy functions abstract in this paper the radstrom embedding theorem proc amer math soc 3 1952 165 is generalized and is used to define the concept of the differential of a fuzzy function

a linear structure and convexity for relations in dynamic fuzzy systems abstract introducing a linear structure for a fuzzy relation we could prove the linearity of the fuzzy integral with respect to this relation it is known that the fuzzy integral does not have the linearity so this becomes an obstacle in developing the fuzzy dynamic system to avoid this we will impose some restrictive assumptions for a fuzzy relation defined on the positive orthant of an n dimensional euclidean space also fuzzy markov decision processes with deterministic stationary policies are formulated and its characterization is given by a recursive equation in the fuzzy state

remark on lacunary statistical convergence of fuzzy numbers fatih nuray fuzzy sets and systems 99 3 1998 269 proved the inclusion relations between the sets of statistically convergent and lacunary statistically convergent sequences of fuzzy numbers in this paper we show that nuray s condition is sufficient as well as necessary

sections of convex bodies and splitting problem for selections abstract let f 1 x y 1 and f 2 x y 2 be any convex valued lower semicontinuous mappings and let l y 1 y 2 y be any linear surjection the splitting problem is the problem of representation of any continuous selection f of the composite mapping l f 1 f 2 in the form f l f 1 f 2 where f 1 and f 2 are some continuous selections of f 1 and f 2 respectively we prove that the splitting problem always admits an approximate solution with f i being an e selection theorem 2 1 we also propose a special case of finding exact splittings whose occurrence is stable with respect to continuous variations of the data theorem 3 1 and we show that in general exact splittings do not exist even for the finite dimensional range

a limit theorem in some dynamic fuzzy systems abstract in this paper using a fuzzy relation we define a dynamic fuzzy system with a compact state space by introducing a contraction property we study the limit of a sequence of fuzzy states and obtain a theorem for the existence and uniqueness of the solution of a fuzzy relational equation as an application we deal with a dynamic fuzzy system with a terminal fuzzy gain and give some characterizations for the fuzzy expected gain a numerical example is given to comprehend our idea in this paper

a potential of fuzzy relations with a linear structure the unbounded case this paper is a sequel to yoshida et al 1993 in which the potential theory for linear fuzzy relations on the positive orthant r n is considered in the class of fuzzy sets with a compact support under the contractive assumption in this paper potential treatment for unbounded fuzzy sets is developed without the assumption of contraction and compactness the objective of this paper is to give the existence and the characterization of potentials for linear fuzzy relations under some reasonable conditions also introducing a partial order in fuzzy sets we prove riesz decomposition theorem in the fuzzy potential theory the proofs are shown by using only the linear structure and the monotonicity of fuzzy relations in the one dimensional case the potential and its cuts are explicitly calculated numerical examples are given to comprehend further discussions

the core of a sequence of fuzzy numbers in this paper based on level sets we define the limit inferior and limit superior of a bounded sequence of fuzzy numbers and prove some properties we extend the concept of the core of a sequence of complex numbers first introduced by knopp in 1930 to a bounded sequence of fuzzy numbers and prove that the core of a sequence of fuzzy numbers is the interval n m where n and m are extreme limit points of the sequence

an optimal stopping problem in dynamic fuzzy systems with fuzzy rewards abstract this paper deals with an optimal stopping problem in dynamic fuzzy systems with fuzzy rewards and shows that the optimal discounted fuzzy reward is characterized by a unique solution of a fuzzy relational equation we define a fuzzy expectation with a density given by fuzzy goals and we estimate discounted fuzzy rewards by the fuzzy expectation this paper characterizes the optimal fuzzy expected value and gives an optimal stopping time

on fuzzy number lattice r abstract we give a systematic development of fuzzy number lattice theory many of our results generalize to number lattice over the lattice algebra our first main result is that the real number chain r is a homomorphic image of the fuzzy number lattice r that is r r and the congruence class r is a component of r thus we can be specialized to describe the structure of the fuzzy number lattice r next we study the structure and representation of the congruence class

equilibria of recurrent fuzzy systems unlike static fuzzy systems recurrent fuzzy systems are equipped with feedback loops and thus exhibit dynamic behaviors the dynamics of a recurrent fuzzy system is largely determined by its rule base the dynamic behavior of a significant subclass of recurrent fuzzy systems may be immediately deduced from their rule base without need for analyzing their mathematical description their equilibrium points may be readily identified and their stability behaviors investigated based on their rule base the investigations involved lead to convergence theorems and other statements that preclude chaotic dynamics

on λ statistical convergence of difference sequences of fuzzy numbers in this paper the concept of strongly d l p 2 cesaro summability of a sequence of fuzzy numbers is introduced also some inclusion relations between the set of strongly d l p 2 cesaro convergent and d l 2 statistically convergent sequences of fuzzy numbers are given

uniformly convex subsets of the hilbert space with modulus of convexity of the second order we prove that in the hilbert space every uniformly convex set with modulus of convexity of the second order at zero is an intersection of closed balls of fixed radius we also obtain an estimate of this radius

a potential of fuzzy relations with a linear structure the contractive case abstract in this paper we develop a potential theory of fuzzy relations on the positive orthant in a euclidean space by introducing a linear structure for fuzzy relations the existence of a potential and its characterization by fuzzy relational equation are derived under the assumption of contraction and compactness in the one dimensional unimodal case a potential is given explicity also a numerical example is shown to illustrate our approaches

the extension of the knopp core theorem to the sequences of fuzzy numbers the concept of the core of a sequence of fuzzy numbers has been introduced by aytar et al in the core of a sequence of fuzzy numbers fuzzy sets and systems 159 24 2008 3369 3379 quite recently some matrix transformations on the classical sets of sequences of fuzzy numbers have been characterized by talo and basar in determination of the duals of classical sets of sequences of fuzzy numbers and related matrix transformations comput math appl 58 2009 717 733 in this paper we have proved a core theorem for sequences of fuzzy numbers which is analogous to the famous knoop core theorem for real sequences to achieve this goal we have studied some properties of the lim sup and lim inf of sequences of fuzzy numbers also we have characterized a class of positive regular matrices of fuzzy numbers such that these matrices leave the core of any bounded sequence of fuzzy numbers invariant

on statistically convergent double sequences of fuzzy numbers in this paper we continue the study the notion of statistically convergent and statistically cauchy for double sequences of fuzzy numbers

the recurrence of dynamic fuzzy systems abstract this paper analyses a recurrent behavior of dynamic fuzzy systems defined by fuzzy relations on a euclidean space by introducing a recurrence for crisp sets we prove probability theoretical properties for the fuzzy systems in the contractive case in kurano et al fuzzy sets and systems 51 1992 83 88 the existence of the maximum recurrent set is proved as another case we introduce a monotonicity for fuzzy relations which is extended from the linear structure in yoshida et al fuzzy sets and systems 66 1994 83 95 in the monotone case we prove the existence of the arcwise connected maximal recurrent sets

calibration results for non expected utility theories rabin 2000 proved that a low level of risk aversion with respect to small gambles leads to a high and absurd level of risk aversion with respect to large gambles rabin s arguments strongly depend on expected utility theory but we show that similar arguments apply to general non expected utility theories

mixture symmetry and quadratic utility the independence axiom of expected utility theory has recently been weakened to the betweenness axiom in this paper an even weaker axiom called mixture symmetry is presented the corresponding functional structure is such that utility is a betweenness functional on part of its domain and quadratic in probabilities elsewhere the experimental evidence against betweenness provides one motivation for the more general theory presented here another advantage of the mixture symmetric class of utility functions is that it is sufficiently flexible to permit the disentangling of attitudes towards risk and towards randomization

iteration algorithms for solving a system of fuzzy linear equations abstract in this paper we discuss the solution of a system of fuzzy linear equations x ax u and its iteration algorithms where a is a real n n matrix the unknown vector x and the constant u are all vectors consisting of n fuzzy numbers and the addition scale multiplication are defined by zadeh s extension principle after introducing a metric between two fuzzy vectors we prove that the system has unique solution if a we also give the convergence and the error estimation for using simple iteration to obtain the solution finally we give the convergence and the error estimation of successive iteration sequence for obtaining the solution

uniform convexity and the splitting problem for selections abstract we continue to investigate cases when the repovs semenov splitting problem for selections has an affirmative solution for continuous set valued mappings we consider the situation in infinite dimensional uniformly convex banach spaces we use the notion of polyak of uniform convexity and modulus of uniform convexity for arbitrary convex sets not necessary balls we study general geometric properties of uniformly convex sets we also obtain an affirmative solution of the splitting problem for selections of certain set valued mappings with uniformly convex images

the equivalence of convergences of sequences of fuzzy numbers and its applications to the characterization of compact sets in this paper with the help of the convergence of sequences of fuzzy numbers with respect to the lebesgue measure we study the relationship between convergences of sequences of fuzzy numbers with respect to the endograph metric the sendograph metric and the l p metric we prove that these convergences are equivalent under proper conditions in addition by applying our result we give a new characterization of compact sets in fuzzy number space with the sendograph metric

an approach to stopping problems of a dynamic fuzzy system we formulate a stopping problem for dynamic fuzzy systems concerning with fuzzy decision environment it could be regarded as a natural fuzzification of non fuzzy stopping problem with a deterministic dynamic system the validity of the approach by cuts of fuzzy sets will be discussed in constructing one step look ahead policy of an optimal fuzzy stopping time a numerical example is given to illustrate the theoretical results and to show a merit to use fuzzy stopping times in dynamic fuzzy systems

statistical limit inferior and limit superior for sequences of fuzzy numbers in this paper we extend the concepts of statistical limit superior and limit inferior as introduced by fridy and orhan statistical limit superior and limit inferior proc amer math soc 125 12 1997 3625 3631 12 to statistically bounded sequences of fuzzy numbers and give some fuzzy analogues of properties of statistical limit superior and limit inferior for sequences of real numbers

determination of the duals of classical sets of sequences of fuzzy numbers and related matrix transformations the convergence of a series of fuzzy sets was examined via zadeh s extension principle by m stojakovic and z stojakovic m stojakovic z stojakovic addition and series of fuzzy sets fuzzy sets and systems 83 1996 341 346 since the utilization of this approach is quite difficult in practice we prefer the idea of using the sum of the series of l level sets the main purpose of the present paper is to determine the a b and c duals of the classical sets of sequences of fuzzy numbers and is to give the necessary and sufficient conditions on an infinite matrix of fuzzy numbers transforming one of the classical sets to the another one

markov type fuzzy decision processes with a discounted reward on a closed interval abstract we formulate a new multi stage decision process with markov type fuzzy transition which is termed markov type fuzzy decision process in the general framework of the decision process both of state and action are assumed to be fuzzy itself the transition of states is defined using the fuzzy relation with markov property and the discounted total reward is described as a fuzzy number on a closed bounded interval to discuss the optimization problem a partial order of convex fuzzy numbers is introduced in this paper the discounted total reward associated with an admissible stationary policy is characterized by a unique fixed point of the contractive mapping moreover the optimality equation for the fuzzy decision model is derived under some continuity conditions also an illustrated example is given to explain the theoretical results and the computation in the paper

superharmonic fuzzy sets on recurrent sets in dynamic fuzzy systems abstract we analyse p superharmonic fuzzy sets on recurrent sets in dynamic fuzzy systems and we derive a simple estimation for the fuzzy sets this paper presents a method to calculate an optimal value for snell s optimal stopping problem