strong laws of large numbers for adapted arrays of set valued and fuzzy valued random variables in banach space in this paper we establish some strong laws of large numbers for double adapted arrays of set valued and fuzzy valued random variables in separable banach space our results are obtained with and without conditionally compactly uniformly integrable hypothesis and they are more general than some related results

a strong law of large numbers for fuzzy random sets abstract an slln for fuzzy random variables in rp including other convergence theorems has been proved by klement puri and ralescu the present paper is motivated by the definitions and properties developed by puri and ralescu and concerned with an slln for banach spaced valued fuzzy random sets as an easy application of the previous results

embedding problem of fuzzy number space part ii abstract using a concrete structure into which we embed the fuzzy number space e1 several necessary and sufficient conditions of fuzzy set valued functions are given by means of abstract function theory

noncompact qualitative games with application to equilibria abstract in 2003 s y chang maximal elements in noncompact spaces with application to equilibria proc amer math soc 132 2 2003 535 541 obtained a maximal theorem for l s majorized correspondences in noncompact spaces in this paper we extend this result to obtain new existence theorems for qualitative games applying it we have more refined existence theorems for both abstract economies and generalized n person games than before

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

on the formalization of fuzzy random variables abstract in this paper we develop a discussion on the mathematical formalization of the concept of fuzzy random variable this discussion is mainly focused on finding an adequate notion of measurability to be coherent with the notions on the space these random elements take values

hamiltonian path embedding and pancyclicity on the mobius cube with faulty nodes and faulty edges a graph g v e is said to be pancyclic if it contains fault free cycles of all lengths from 4 to v in g let f sub v and f sub e be the sets of faulty nodes and faulty edges of an n dimensional mobius cube mq sub n respectively and let f f sub v spl cup f sub e a faulty graph is pancyclic if it contains fault free cycles of all lengths from 4 to v f sub v in this paper we show that mq sub n f contains a fault free hamiltonian path when f spl les n 1 and n spl ges 1 we also show that mq sub n f is pancyclic when f spl les n 2 and n spl ges 2 since mq sub n is regular of degree n both results are optimal in the worst case

the 4 set tree connectivity of n k star networks abstract the tree connectivity as a generalization of the traditional connectivity can serve to measure the capability of connection for vertices in a network the n k star graph s n k can be used to model the topological structure of a large scale parallel processing system we show in this article that the 4 set tree connectivity of s n k is n 2 that is there exist n 2 internally disjoint trees connecting x y z and w in s n k for four arbitrary vertices x y z and w of s n k two known results about the 3 set tree connectivity of star graphs and n k star graphs are immediate consequences of our result

characterization of compact subsets of fuzzy sets a characterization of compact subsets is presented for the metric space of normal fuzzy convex fuzzy sets on the base space rn the metric for which is the supremum over the hausdorff distances between corresponding level sets it is shown that a closed subset is compact if and only if it is uniformly support bounded and the corresponding set of support functions is equileftcontinuous in the membership grade variable

topological properties on the space of fuzzy sets abstract in this paper we introduce a new metric on the space f rp of fuzzy sets and prove that f rp is separable and complete also a characterization of compact subsets of f rp is presented and some examples are given

intersection theorems and their applications in general almost convex spaces abstract in this paper we extend the concept of the almost convex condition and establish an approximation property for convex continuous correspondences we use the approximation property to generalize von neumann s intersection theorems and related topics we obtain as applications some new equilibrium theorems for the existence of a maximal element generalized games and qualitative games of u majorized correspondences

measurablility concepts for fuzzy mappings abstract in a series of recent works alternative measurability concepts for fuzzy mappings were introduced and studied in this paper we show that some of these concepts are equivalent or strongly inter related

a characterization of compact subsets of fuzzy number space abstract in this paper a characterization of compact subsets on the space of fuzzy numbers endowed with the skorokhod topology is presented this characterization is an analogue of the arzela ascoli theorem which characterizes compact subsets of the space c 0 1

equilibria of noncompact abstract economies in general almost convex strategy spaces abstract we first extend the concept of almost convex condition and establish a fixed point theorem for correspondences with convex values only on an almost convex subset for their ranges this generalizes both results of himmelberg c j himmelberg fixed points of compact multifunctions j math anal appl 38 1972 205 207 and the result of jafari and sehgal f jafari v m sehgal an extension to a theorem of himmelberg j math anal appl 327 2007 298 301 furthermore applying it we have existence theorems for equilibria of noncompact abstract economies in general almost convex strategy spaces our theorems generalize the corresponding results of zhou jianxin zhou on the existence of equilibrium for abstract economies j math anal appl 193 1995 839 858 tan and wu k k tan z wu a note on abstract economies with upper semicontinuous correspondence appl math lett 11 5 1998 21 22 in several ways in particular we answer the question raised by zhou in the reference cited above in the affirmative with weaker hypotheses

on embedding cycles into faulty twisted cubes the twisted cube tq n is an alternative to the popular hypercube network recently some interesting properties of tq n were investigated in this paper we study the pancycle problem on faulty twisted cubes let f e and f v be the numbers of faulty edges and faulty vertices in tq n respectively we show that with f e f v 3

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

inequalities and nash equilibria abstract this paper concentrates on the problem of the existence of equilibrium points for non cooperative generalized n person games n person games of normal form and their related inequalities we utilize the k k m lemma to obtain a theorem and then use it to obtain a new fan type inequality and minimax theorems various new equilibrium point theorems are derived with the necessary and sufficient conditions and with strategy spaces with no fixed point property examples are given to demonstrate that these existence theorems cover areas where other existence theorems break down

n 3 edge fault tolerant weak pancyclicity of n k star graphs the n k star graphs are a generalized version of n star graphs which belong to the class of cayley graphs and have been recognized as an attractive alternative to hypercubes for building massively parallel computers recently chen et al showed that n k star graphs are 6 weak vertex pancyclic for k 4 and email protected k

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

conditional matching preclusion for the arrangement graphs the matching preclusion number of a graph is the minimum number of edges whose deletion results in a graph that has neither perfect matchings nor almost perfect matchings for many interconnection networks the optimal sets are precisely those induced by a single vertex recently the conditional matching preclusion number of a graph was introduced to look for obstruction sets beyond those induced by a single vertex it is defined to be the minimum number of edges whose deletion results in a graph with no isolated vertices that has neither perfect matchings nor almost perfect matchings in this paper we find this number and classify all optimal sets for the arrangement graphs one of the most popular interconnection networks

a strong law of large numbers for fuzzy random variables abstract a limit of a sequence of fuzzy numbers is defined and its some properties are shown based on these concept and properties an independent sequence of fuzzy random variables is considered and a strong law of large numbers for fuzzy random variables is shown

integrals of fuzzy number valued functions abstract using the representation theorem of fuzzy numbers we give an alternative approach for the integrals of fuzzy number valued functions and for the radon nikodym theorem of the fuzzy number valued measures

the skorokhod topology on space of fuzzy numbers in this paper we introduce the skorokhod metric on the space f r of fuzzy numbers and prove that f r is separable and complete

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

compactness and convexity on the space of fuzzy sets ii abstract in this paper we establish some characterizations of convex and relatively compact subsets of the space f r p of normal upper semicontinuous and compactly supported fuzzy sets in rp equipped with the skorokhod metric

on the convergence of fuzzy sets three kind of convergences are defined on a space of fuzzy sets relationships between these convergences are studied some comments on the fixed point property are represented

fault tolerance of n k star networks this paper considers a refined measure s h for the fault tolerance of a network and for the generalized star network s n k determines s h s n k n h k 2 1 for 2 k n 1 and 0 h n k which implies that at least n h k 2 1 vertices of s n k have to be removed to get a disconnected graph without vertices of degree less than h this work generalizes some known results when the n k star graph is used to model the topological structure of a large scale parallel processing system this result can provide a more accurate measure for the fault tolerance of the system

a law of large numbers for random sets abstract we show a law of large numbers for random sets taking values in a class of subsets of a banach space which is at least larger than the class of compact subsets of the space if the banach space is of infinite dimension to see this we first define the hausdorff distance on the family of all the non empty subsets of a banach space and give some properties of it motivated by the fact that a study of fuzzy sets is to study of set valued analysis

some new topological properties of the triangular pyramid networks abstract the triangular pyramid proposed by razavi and sarbazi azad the triangular pyramid routing and topological properties information sciences 180 2010 2328 2339 is a hierarchical structure based on triangular meshes in this paper we show that the triangular pyramid shares some nice symmetry properties of the pyramid we also determine the connectivity of the triangular pyramid and prove that it is 1 fault tolerant vertex pancyclic

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

law of large numbers for uncertain random variables the law of large numbers in probability theory states that the average of random variables converges to its expected value in some sense under some conditions sometimes random factors and human uncertainty exist simultaneously in complex systems and a concept of uncertain random variable has been proposed to study this type of complex systems this paper aims to provide a law of large numbers for uncertain random variables which states that the average of uncertain random variables converges in distribution to an uncertain variable as a byproduct the convergence of a sequence of uncertain variables is also studied

convergence in distribution for level continuous fuzzy random sets in this paper we give the relationships between tightness and convergence in distribution for level continuous fuzzy random sets to this end we first obtain some characterizations of compact subsets of the space of level continuous fuzzy sets in r p and establish the tightness for level continuous fuzzy random sets

fault tolerant embedding of cycles of various lengths in k ary n cubes the k ary n cube has been one of the most popular interconnection networks for distributed memory parallel systems in this paper we study the problem of embedding cycles of various lengths into faulty k ary n cubes it is proved that a faulty k ary n cube with f v faulty vertices and f e faulty edges admits a fault free cycle of every even length from 4 to k n 2f v if k 4 is even and f v f e 4 is even and f v f e 2n 3 the results are optimal with respect to the number of vertex and edge faults tolerated

a generalized strong law of large numbers strong laws of large numbers have been stated in the literature for measurable functions taking on values on different spaces in this paper a strong law of large numbers which generalizes some previous ones like those for real valued random variables and compact random sets is established this law is an example of a strong law of large numbers for borel measurable nonseparably valued elements of a metric space

weak vertex pancyclicity of n k star graphs the n k star graph s n k for short is an attractive alternative to the hypercube and also a generalized version of the n star it is isomorphic to the n star n complete graph if k n 1 k 1 jwo et al have already demonstrated in 1991 that an n star contains a cycle of every even length from 6 to n this work shows that every vertex in an s n k lies on a cycle of length l for every email protected email protected n n k when email protected email protected n 4 and n 6 additionally for email protected email protected n 2 each vertex in an s n k is contained in a cycle of length ranged from 6 to n n k moreover each constructed cycle of an available length in an s n k can contain a desired 1 edge

a general approach to deriving the g good neighbor conditional diagnosability of interconnection networks abstract we discuss a general approach to deriving the g good neighbor conditional diagnosability of interconnection networks as demonstrative examples we derive the 1 and 2 good neighbor conditional diagnosabilities of the arrangement graphs under both the commonly adopted pmc and mm model we also derive the general g good neighbor conditional diagnosability of the n k star graphs under the pmc model for g 1 n k and under the mm model for g 2 n k as well as that of the related graphs such as the star graph the alternating group graph and the alternating group network

constructing vertex disjoint paths in n k star graphs this work describes a novel routing algorithm for constructing a container of width n 1 between a pair of vertices in an n k star graph with connectivity n 1 since lin et al t c lin d r duh h c cheng wide diameter of n k star networks in proceedings of the international conference on computing communications and control technologies vol 5 2004 pp 160 165 already calculated the wide diameters in n n 1 star and n 1 star graphs this study only considers an n k star with 2

the conditional fault diagnosability of n k star graphs abstract the growing size of the multiprocessor system increases its vulnerability to component failures it is crucial to locate and replace the faulty processors to maintain a system s high reliability the fault diagnosis is the process of identifying faulty processors in a system through testing this paper shows that the largest connected component of the survival graph contains almost all the remaining vertices when a lot of faulty vertices occur in the n k star graph s n k n 5 k 3 n k 2 based on this fault resiliency it establishes the conditional fault diagnosability of s n k under the comparison model

vertex bipancyclicity of the generalized honeycomb tori assume that m n and s are integers with m 2 n 4 email protected s 6 and is even besides ght m n s is vertex bipancyclic in some special cases the result is optimal in the sense that the absence of cycles of certain lengths on some ght m n s s is inevitable due to their hexagonal structure

kolmogorov s strong law of large numbers for fuzzy random variables in this paper kolmogorov s strong law of large numbers for sums of independent and level wise identically distributed fuzzy random variables is obtained

some properties of a new metric on the space of fuzzy numbers abstract this paper deals with some properties of a new metric here called the hausdorff skorokhod metric previously introduced by joo and kim on the space of fuzzy numbers in the euclidean space r k we first prove that convergence in the hausdorff skorokhod metric implies convergence in the sendograph metric also we show that the hausdorff skorokhod metric is not absolutely homogeneous and not translation invariant and investigate the continuity of operations and convexity on the space of fuzzy numbers equipped with the hausdorff skorokhod metric

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

measurability for fuzzy valued functions in this paper we characterize the borel field of the space of fuzzy sets endowed with the skorokhod metric as a result the measurability for a fuzzy valued function is discussed

supremum metric on the space of fuzzy sets and common fixed point theorems for fuzzy mappings this paper generalizes a classical result about the space of bounded closed sets with the hausdorff metric and establishes the completeness of cb x with respect to the completeness of the metric space x where cb x is the class of fuzzy sets with nonempty bounded closed a cut sets equipped with the supremum metric d which takes the supremum on the hausdorff distances between the corresponding a cut sets in addition some common fixed point theorems for fuzzy mappings are proved and two examples are given to illustrate the validity of the main results in fixed point theory

on the existence of equilibrium for abstract economies abstract in this paper several existence theorems for abstract economies have been established they are natural modifications strict generalizations when strategy spaces are metrizable of the main results of yannelis and prabhakar 23 tulcea 22 and ding kim and tan 10 for infinite dimensional casas for the finite dimensional case we show that the topological conditions on correspondences in shafer and sonnenschein s result 20 can be much weaker along the proofs several propositions are established which are of independent interest we also show that many existence results in fixed point theory maximization theory and generalized quasi variational inequalities can be encompassed

the strong law of large numbers for fuzzy random variables abstract sequences of independent and identically distributed fuzzy random variables are considered it is shown that the strong law of large numbers holds also for fuzzy random variables this result is used to give a consistent estimator for the expectation of a fuzzy random variable

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

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