attractors and asymptotic stability for fuzzy dynamical systems abstract in this work we study the asymptotic properties of maps on fuzzy spaces which are extensions of maps on the main results are in section 4 see theorem 21 and we give an illustrative example in the last section

a strong consistency result for fuzzy relative frequencies interpreted as estimator for the fuzzy valued probability the unavoidable imprecision of measurements of continuous physical quantities can be modelled by using the concept of fuzzy numbers and fuzzy vectors concerning a quantitative usage of such data the classical concept of relative frequencies for real data has to be extended to so called fuzzy relative frequencies for fuzzy data whereby the fuzzy relative frequency of a set is a fuzzy number analogous to a dempster s interval valued probabilities induced by multi valued mappings fuzzy valued probabilities induced by fuzzy random vectors are considered and analyzed it is proved that fuzzy relative frequencies can be interpreted as strongly consistent estimator for the corresponding fuzzy valued probability

the s differentiability of a fuzzy valued mapping several approaches to define the differential of fuzzy valued mappings can be found in the literature in this paper we introduce a new concept of differential of a fuzzy valued mapping which is based on the notion of differential of the support function associated with such a mapping properties of the new concept as well as the relation between it and previous approaches are also analyzed in detail

cones and decomposition of sub and supermartingales abstract we study sub and supermartingales with values in the class f c of all convex fuzzy subsets of a banach space having compact support the terms sub and super are understood with respect to a partial order not necessarily inclusion in this general framework orders with respect to which a doob type decomposition can hold are naturally associated to convex subcones with a special property termed f c properness decomposition theorems into the sum or difference of a martingale and a compensator are proven in the sub case the martingale need exist only in a superspace of f c

radon nikodym theorem and vitali hahn saks theorem on fuzzy number measures in banach spacers abstract radon nikodym theorem is established on fuzzy number measures in banach spaces vitali hahn saks theorem is generalized to set valued measures and fuzzy number measures

estimation of a fuzzy regression model using fuzzy distances regression analysis is a powerful statistical tool that has many applications in different areas the problem of regression analysis under a fuzzy environment has been treated in the literature from different points of view and considering a variety of input output data crisp or fuzzy however we realize that in general most research papers have a conflict between the solution of the fuzzy regression problem using crisp distances minimizing a real error function and the interpretation of fuzzy data as possibility distributions the main aim of this paper is to develop a methodology to solve this problem introducing a fuzzy partial order and a family of fuzzy distance measures on the whole set of fuzzy numbers the new approach allows us to obtain linear and nonlinear models that reach the lowest fuzzy error the estimation process in general can be considered easier to apply in practice and it is not limited to triangular fuzzy numbers numerical examples are provided to illustrate the usefulness and applicability of these results and comparisons with existing methodologies show that the performance of the proposed solution is very satisfactory

weak laws of large numbers for fuzzy random variables abstract in this paper we establish weak laws of large numbers for sums of convex compactly uniformly integrable fuzzy random variables taking values in the space of fuzzy numbers in rp as a result weak laws of large numbers for sums of convexly tight fuzzy random variables are given

uncertain alternating renewal process and its application uncertain process is a sequence of uncertain variables indexed by time and space first this paper presents a kind of uncertain process known as the uncertain alternating renewal process whose alternating interarrival times are uncertain variables then it proves an uncertain alternating renewal theorem on the limit value of average working rate finally an application of the alternating renewal theorem is discussed

new families of orthogonal polynomials this paper provides with a generalization of the work by wimp and kiesel non linear recurrence relations and some derived orthogonal polynomials ann numer math 2 1995 169 180 who generated some new orthogonal polynomials from chebyshev polynomials of second kind we consider a class of polynomials p n x defined by p n x an x bn pn 1 x 1 an pn x n 0 1 2 a0 1 where the pk x are monic classical orthogonal polynomials satisfying the well known three term recurrence relation pn 1 x x n pn x n pn 1 x n 1 p1 x x 0 p0 x 1 we explicitly derive the sequences an and bn in general and illustrate by some concrete relevant examples

simulation of random fuzzy variables an empirical approach to statistical probabilistic studies with fuzzy experimental data in this paper we simulate different types of random fuzzy variables to get some conclusions concerning fuzzy valued random variables this simulation was carried out to illustrate certain limit results formalizing the convergence of the arithmetic mean of sampled fuzzy data to the population mean or an expected value of the random fuzzy variable like the well known strong law of large numbers and the law of iterated logarithm since the theoretical results have recently been proved this simulation analysis represents a complementary study in which we can determine sample sizes providing us with suitable approximations finally future directions regarding statistical inference with fuzzy data and other applications are commented

random fuzzy variables of second order and applications to statistical inference abstract this paper summarizes some results on random fuzzy variables with existing expectation and variance called random fuzzy variables of second order using the frechet principle and via support functions the embedding of convex fuzzy sets into a banach space of functions it especially presents a unified view on expectation and variance of random fuzzy variables these notions are applied in developing linear statistical inference with fuzzy data detailed investigations are presented concerning best linear unbiased estimation in linear regression models with fuzzy observations

fuzzy random renewal reward process and its applications this paper studies a renewal reward process with fuzzy random interarrival times and rewards under the independence associated with any continuous archimedean t norm the interarrival times and rewards of the renewal reward process are assumed to be positive fuzzy random variables whose fuzzy realizations are independent fuzzy variables under these conditions some limit theorems in mean chance measure are derived for fuzzy random renewal rewards in the sequel a fuzzy random renewal reward theorem is proved for the long run expected reward per unit time of the renewal reward process the renewal reward theorem obtained in this paper can degenerate to that of stochastic renewal theory finally some application examples are provided to illustrate the utility of the result

a parameterized l2 metric between fuzzy numbers and its parameter interpretation abstract when handling fuzzy number data it is a common practice to make use of a metric to quantify distances between fuzzy numbers several metrics have been suggested in the literature for this purpose when statistically analyzing fuzzy number valued data l 2 metrics become especially useful this paper introduces a new family of generalized l 2 metrics which take into account key features of the involved fuzzy numbers namely a measure of central location and two measures associated with the shape of the fuzzy numbers are used a crucial property related to these three measures is that necessary and sufficient conditions can be established for them to characterize fuzzy numbers furthermore the family of generalized l 2 metrics depends on one parameter a discussion is provided regarding the interpretation of this parameter which can guide selection of its value in practice

all automorphisms of the 3 nilpotent free group of countably infinite rank can be lifted abstract it is known from s andreadakis proc london math soc 3 15 1965 239 268 and s bachmuth trans amer math soc 122 1966 1 17 that for a free 3 nilpotent group f of rank n the map aut f aut f is not onto it is proved here that for f countably infinitely generated the map is onto

existence and uniqueness for solutions to fuzzy stochastic differential equations driven by local martingales under the non lipschitzian condition abstract this paper makes a research into a class of fuzzy stochastic differential equations fsdes driven by a continuous local martingale under the non lipschitzian condition such equations can be useful in modelling of hybrid systems where the phenomena are subjected to two kinds of uncertainties randomness and fuzziness simultaneously the solutions of fsdes are the fuzzy stochastic processes and their uniqueness is considered to be in a strong sense thus the existence and uniqueness of solutions to fsdes under the non lipschitzian condition is first proven and the continuity of solutions to fsdes with respect to the initial data or the coefficients of the equations is investigated

multiple geronimus transformations abstract we consider multiple geronimus transformations and show that they lead to discrete non diagonal sobolev type inner products moreover it is shown that every discrete sobolev inner product can be obtained as a multiple geronimus transformation a connection with geronimus spectral transformations for matrix orthogonal polynomials is also considered

stack filters the median and other rank order operators possess two properties called the threshold decomposition and the stacking properties the first is a limited superposition property which leads to a new architecture for these filters the second is an ordering property which allows an efficient vlsi implementation of the threshold decomposition architecture motivated by the success of rank order filters in a wide variety of applications and by the ease with which they can now be implemented we consider in this paper a new class of filters called stack filters they share the threshold decomposition and stacking properties of rank order filters but are otherwise unconstrained they are shown to form a very large class of easily implemented nonlinear filters which includes the rank order operators as well as all compositions of morphological operators the convergence properties of these filters are investigated using techniques similar to those used to determine root signal behavior of median filters the results obtained include necessary conditions for a stack filter to preserve monotone regions or edges in signals the output distribution for these filters is also found all the stack filters of window width 3 are determined along with their convergence properties among these filters are found two which we have named asymmetric median filters they share all the properties of median filters except that they remove impulses of one sign only that is one removes only positive going edges the other removes only negative going edges while the median filter removes impulses of both signs this investigation of the properties of stack filters thus produces several new useful and easily implemented filters

the nonfinite generation of aut g g free metabelian of rank 3 the group of automorphisms of the free metabelian group of rank 3 is not finitely generated let h be a free solvable group of rank n 2 aut h the automorphism group of h and inn h the group of inner automorphisms of h for n 2 it was shown by the authors and e formanek 4 theorem 1 that aut h inn h gl2 z z the ring of integers one consequence of this fact is that aut h is finitely generated f g the purpose of this paper is to prove the following contrasting theorem theorem if g is the free metabelian group of rank three then aut g is not finitely generated this result may also be compared with a theorem of l auslander 1 which states that the automorphism group of a polycyclic group is finitely presented if g denotes the commutator subgroup of g then the kernel of the natural homomorphism aut g 4 aut g g is called the group of ia automorphisms of g and will be denoted by ia g ia g which contains inn g was shown by the authors 3 to be non f g for g as in the theorem to prove that aut g is not f g we need to show that ia g is not f g as an aut g g gl3 z operator group the question arises concerning aut h where h is free metabelian of rank n 3 the immediate feeling that aut h is also non f g may be incorrect for definiteness let n 4 suppose g is free metabelian of rank 3 and consider aut g as embedded in aut h in an obvious manner the nonfinite generatiion of aut g comes from the existence of nontame automorphisms in aut g i e automorphisms of g f f which are not induced by automorphisms of the free group f of rank three and the necessity to include infinitely many nontame automorphisms in any generating set for aut g however the authors have discovered that many of the nontame automorphisms become tame when considered as elements in aut h whether this phenomenon is true for all nontame elements of aut g is unknown as yet but the question of the finite or nonfinite generation of aut h must be considered a difficult open problem with the interesting possibility of finite generation as the answer received by the editors december 31 1980 1980 mathematics subject classification primary 20f28

the cokernel of the johnson homomorphisms of the automorphism group of a free metabelian group in this paper we determine the cokernel of the k th johnson homomorphisms of the automorphism group of a free metabelian group for k 2 and n 4 as a corollary we obtain a lower bound on the rank of the graded quotient of the johnson filtration of the automorphism group of a free group furthermore by using the second johnson homomorphism we determine the image of the cup product map in the rational second cohomology group of the ia automorphism group of a free metabelian group and show that it is isomorphic to that of the ia automorphism group of a free group which is already determined by pettet finally by considering the kernel of the magnus representations of the automorphism group of a free group and a free metabelian group we show that there are non trivial rational second cohomology classes of the ia automorphism group of a free metabelian group which are not in the image of the cup product map

the l average value and the fuzzy expectation of a fuzzy random variable abstract the concepts of fuzzy random variable and the associated fuzzy expected value have been introduced by puri and ralescu as an extension of measurable set valued functions random sets and of the aumann integral of these functions respectively on the other hand the average function has been suggested by campos and gonzalez as an appropriate function to rank fuzzy numbers in this paper we are going to analyze some useful properties concerning the average value of the expectation of a fuzzy random variable and some practical implications of these properties are also commented on

set valued choquet integrals revisited this paper performs a further investigation on the set valued choquet integral given by jang and kwon fuzzy sets and systems 112 2000 233 some incorrect properties are revised it is pointed out that the definition for convergence of sequence of sets or set valued functions is unreasonable by using the usual kuratowski s convergence some new convergence theorems for the set valued choquet integral are shown meanwhile some other properties are obtained

a unified approach to asymptotic behaviors for the autoregressive model with fuzzy data we propose a unified estimator for the autoregressive model with fuzzy input output variables based on the least squares method the least squares estimation is investigated in presence of a unified r distance defined on the space of fuzzy numbers we investigate asymptotic properties of the unified estimator under some simple conditions as well as a generalization which reduces to the asymptotic properties under those distances when the distances are the special cases of r distance some simulation studies are included to compare the asymptotic properties of estimators formed under several distances being the special cases of r distance

almost periodic solutions of fuzzy systems we prove the existence of almost periodic solutions and asymptotically almost periodic solutions for the fuzzy functional differential equations moreover we consider uniform stable and uniformly asymptotically stable of almost periodic solutions for the fuzzy system

on strong laws of large numbers for random upper semicontinuous functions a short proof of the strong law of large numbers for random upper semicontinuous functions fuzzy random variables with compact support is given this approach allows us to deduce many results on convergence of fuzzy random variables from the relevant results on random sets that appear as their level sets

overview on the development of fuzzy random variables this paper presents a backward analysis on the interpretation modelling and impact of the concept of fuzzy random variable after some preliminaries the situations modelled by means of fuzzy random variables as well as the main approaches to model them are explained we also summarize briefly some of the probabilistic studies concerning this concept as well as some statistical applications

strong convergence for weighted sums of fuzzy random sets in this paper we establish some results on strong convergence for sums and weighted sums of uniformly integrable fuzzy random sets taking values in the space of upper semicontinuous fuzzy sets in r p

on borel measurability and large deviations for fuzzy random variables fuzzy random variables are a special case of measurable mappings taking on values in a non separable metric space due to that non separability the two main definitions in the literature turn out not to be equivalent we give an easy necessary and sufficient condition for their equivalence an intermediate result of independent value is a characterization of separable subsets of that metric space several probabilistic applications are proposed notably a generalization to this setting of the large deviations principle of bolthausen

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

continuity of functions defined on the space of fuzzy sets in this paper we investigate the continuity of many useful functions defined on the space f rp of normal and upper semicontinuous fuzzy sets in rp with compact support

strong law of large numbers for t normed arithmetics we prove a strong law of large numbers for sequences of pairwise i i d fuzzy random variables addition is given by the extension principle associated with a general continuous triangular norm our result includes the slln for levelwise sums obtained when the chosen triangular norm is the minimum

laguerre freud equations for the recurrence coefficients of d semi classical orthogonal polynomials of class one abstract the laguerre freud equations giving the recurrence coefficients n n of orthogonal polynomials with respect to a d semi classical linear form are derived d is the difference operator the limit when 0 are also investigated recovering known results applications to generalized meixner polynomials of class one are also treated

the intersection operation in light of joint observables and bell inequalities in operational probability theory operational probability theory appears as a smooth extension of classical probability theory that fulfills quantum physics needs in this paper we first explain the need to extend the concept of the intersection operation in the light of the concept of joint observables then we tried to determine the basic features that characterize any possible definition of the intersection operation furthermore we tried to know if there is a relation between the definitions of the intersection operation of both fuzzy and operational probability theories these relations are used first to find a suitable suggestion to define the intersection operation second they are used besides the studies made on bell inequalities in fuzzy probability theory to find the conditions that should be satisfied by the intersection operation to ensure the possibility of the violation of bell inequalities in operational probability theory

b fuzzy probabilities abstract we treat b fuzzy probabilities in nonstandard frameworks by the methods based on boolean valued models and especially boolean powers these formal methods apart from the difficulties associated with the formal model itself give a simple and powerful theory for fuzzy probability spaces and fuzzy random variables essentially everything is reduced to a transfer principle however one can also work using classical methods of proofs we first give the necessary background on nonstandard frameworks within these frameworks we develop boolean fuzzy probabilities and fuzzy random variables

fuzziness and randomness in an optimization framework abstract a semi infinite approach for linear programming in the presence of fuzzy random variable coefficients is described as a byproduct a way for dealing with optimization problems including both fuzzy and random data is obtained numerical examples are provided for the sake of illustration

strong laws of large numbers for independent fuzzy set valued random variables abstract in this paper we shall present strong laws of large numbers slln s for independent not necessary identically distributed fuzzy set valued random variables whose base space is a separable banach space or an euclidean space in the sense of the extended hausdorff metric d h we apply the method to the sequence of independent identically distributed fuzzy set valued random variables to give a simple proof of slln s

limit theorems for sums of random fuzzy sets we consider limit theorems for counting processes generated by minkowski sums of random fuzzy sets using support functions we prove almost sure convergence for a renewal process indexed by fuzzy sets in an inner product vector space we also get convergence for the associated containment renewal function

the median of a random fuzzy number the 1 norm distance approach in quantifying the central tendency of the distribution of a random fuzzy number or fuzzy random variable in puri and ralescu s sense the most usual measure is the aumann type mean which extends the mean of a real valued random variable and preserves its main properties and behavior although such a behavior has very valuable and convenient implications extreme values or changes of data entail too much influence on the aumann type mean of a random fuzzy number this strong influence motivates the search for a more robust central tendency measure in this respect this paper aims to explore the extension of the median to random fuzzy numbers this extension is based on the 1 norm distance and its adequacy will be shown by analyzing its properties and comparing its robustness with that of the mean both theoretically and empirically

linear programming with fuzzy random variable coefficients abstract linear programming with fuzzy random variable coefficients is introduced by discussing a practical engineering problem in order to study the solution of the linear programming with fuzzy random variable coefficients we discuss the simplex algorithm for linear programming with random variable coefficients furthermore the solution and distribution problem of this new fuzzy random programming are studied

fuzzy random variables expectation and martingales abstract banach space valued fuzzy random variables on a complete finite measure space f are studied their fuzzy valued integral is defined and its properties are examined some connections between fuzzy integral and conditional expectation are given the definition and some properties of fuzzy martingales are investigated

jensen and chebyshev inequalities for pseudo integrals of set valued functions set valued functions are an important mathematical notion and play a crucial role in several practical areas at the same time pseudo analysis as a background allows extension of some classical mathematical notions to the forms that are highly applicable in some nonstandard situations this paper focuses on pseudo integration of set valued functions which is generalization of aumann s research and corresponding extensions of the jensen and chebyshev integral inequalities to the set valued case since the integral inequalities in question are widely used in various aspects of mathematics the main motivation for the presented research lies in the possibility of expanding the applicability of these inequalities by combining the properties of set valued functions with pseudo analysis

orthogonal polynomials associated with an inverse quadratic spectral transform let p n n 0 be a sequence of monic orthogonal polynomials with respect to a quasi definite linear functional u and q n n 0 a sequence of polynomials defined by q n x p n x s np n 1 x t np n 2 x n 1 with t n 0 for n 2 we obtain a new characterization of the orthogonality of the sequence q n n 0 with respect to a linear functional v in terms of the coefficients of a quadratic polynomial h such that h x v u we also study some cases in which the parameters s n and t n can be computed more easily and give several examples finally the interpretation of such a perturbation in terms of the jacobi matrices associated with p n n 0 and q n n 0 is presented

compatibility between fuzzy set operations and level set operations applications to fuzzy difference abstract three types of binary operation between any two fuzzy sets are proposed in this paper the concept of 0 level set of fuzzy set plays an important role for defining the binary operations between two fuzzy sets according to the decomposition theorem of fuzzy sets the consistency between the membership functions and level sets should be rigorously verified when the binary operation between two fuzzy sets is defined by using their level sets the consistency is investigated thoroughly in this paper for the purpose of guaranteeing the existence of such binary operations moreover we also propose the so called hausdorff difference between any two fuzzy sets to define the difference of any two fuzzy numbers

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

convergence of set valued sub and supermartingales in the kuratowski mosco sense the purpose of this paper is to prove some convergence theorems of closed and convex set valued sub and supermartingales in the kuratowski mosco sense to get submartingale convergence theorems we give sufficient conditions for the kudo aumann integral and hiai umegaki conditional expectation to be closed both for compact convex set valued random variables and for closed convex set valued random variables we also give an example of a bounded closed convex set valued random variable whose kudo aumann integral is not closed

approximating integrably bounded fuzzy random variables in terms of the generalized hausdorff metric abstract consider the class of the mappings from a euclidean space to the unit interval 0 1 that is the class of the fuzzy sets of this space which are upper semicontinuous the closure of their supports is compact and the inverse images of the singleton are nonempty sets when the metric d which generalizes the hausdorff metric is defined on the preceding class the resulting metric space is complete but nonseparable consequently for this nonseparability given a probability space and an integrably bounded fuzzy random variable associated with it one cannot ensure the existence of a sequence of simple fuzzy random variables d converging almost surely to the former one however we are going to show in this paper that for any given integrably bounded fuzzy random variable we can guarantee the existence of a sequence of integrably fuzzy random variables whose level functions are simple random compact sets and which d converges at every point to the former variable on the basis of this result we will obtain in this paper a constructive definition of integrably bounded fuzzy random variables and an operational way to compute their fuzzy expected values

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

algebraic metric and probabilistic properties of convex combinations based on the t normed extension principle the strong law of large numbers consider the strong law of large numbers for t normed averages of fuzzy random variables in the uniform metric d that probabilistic property is known to hold when the t norm is the minimum and to fail when the t norm is the product we prove that it is characterized by an algebraic property of the t norm that of being eventually idempotent and by a metric property of the space of fuzzy sets that it becomes a convex combination space we show that the equivalence holds not only for euclidean or banach spaces but in the more general setting of convex combination spaces

fuzzy random variables abstract in this paper we define the concepts of fuzzy random variable and the expectation of a fuzzy random variable the new definition of expectation generalizes the integral of a set valued function we derive some properties of these new concepts by considering a suitable generalization of the hausdorff metric we derive the lebesgue dominated convergence type theorem

hukuhara derivative of the fuzzy expected value in this paper we establish sufficient conditions to guarantee the exchange of the fuzzy expected value for a fuzzy random variable depending on certain parameter and its hukuhara derivative with respect to the parameter main results are obtained for two different sets of general conditions one of them requiring measurability conditions with respect to the product space the fuzzy random variable is defined from and the second one just requiring conditions over the projections of the fuzzy random variables

convexification in limit laws of random sets in banach spaces a strong law of large numbers for random compact sets in a banach space with the minkowski addition is provided by puri and ralescu 6 and among other results by gine hahn and zinn 2 and hess 3 this follows the finite dimensional version introduced in 1 the latter however allows nonconvex sets while the generalizations to the banach space setting were proved for convex sets a convexification argument stated in the lemma below shows that the convexity assumption can be dropped the proof reveals information about the nature of the convergence since writing this note we learned that f hiai 4 has independently established this result using a different method some notations let b be a banach space with norm 1111 the minkowski addition of two compact subsets k and l is k l x y x e k y e li and ak is the set tax x e k the closed convex hull of k is denoted cok it is a linear operation the hausdorff distance h k l is defined as max maxxekminyel 11 x y 11 maxyelminxek 11 x y y convergence of sets in this paper is meant in the hausdorff metric

limit theorems for fuzzy random variables a strong law of large numbers and a central limit theorem are proved for independent and identically distributed fuzzy random variables whose values are fuzzy sets with compact levels the proofs are based on embedding theorems as well as on probability techniques in banach space