decompositions and range for additive fuzzy measures the aim of the present paper is to prove the existence of hahn and lebesgue decompositions for additive fuzzy measures in the sense of 3 it is shown that the radon nikodym theorem can be carried over from classical measures to additive fuzzy measures with their corresponding integrals these results are applied to obtain a generalization of liapounoff s convexity theorem 12 whose utility in the theory of fuzzy games is argued in the last section

panconnectivity and pancyclicity of hypercube like interconnection networks with faulty elements in this paper we deal with the graph g email protected g 1 obtained from merging two graphs g 0 and g 1 with n vertices each by n pairwise nonadjacent edges joining vertices in g 0 and vertices in g 1 the main problems studied are how fault panconnectivity and fault pancyclicity of g 0 and g 1 are translated into fault panconnectivity and fault pancyclicity of g email protected g 1 respectively many interconnection networks such as hypercube like interconnection networks can be represented in the form of g email protected g 1 connecting two lower dimensional networks g 0 and g 1 applying our results to a class of hypercube like interconnection networks called restricted hl graphs we show that in a restricted hl graph g of degree m 3 each pair of vertices are joined by a path in email protected f of every length from 2m 3 to v email protected f 1 for any set f of faulty elements vertices and or edges with f m 3 and there exists a cycle of every length from 4 to v email protected f for any fault set f with f m 2

path embedding in star graphs abstract the star graph interconnection network has been introduced as an attractive alternative to the hypercube network in this paper we consider the path embedding problem in star graphs assume that n 4 we prove that paths of all even lengths from d x y to n 2 can be embedded between two arbitrary vertices x and y from the same partite set in the n dimensional star graph in addition paths of all odd lengths from d x y to n 1 can be embedded between two arbitrary vertices x and y from different partite sets in the n dimensional star graph except that if x and y are adjacent there is no path of length 3 between them the result is optimal in the sense that paths of all possible lengths are found in star graphs

entropy of a fuzzy process abstract using an additive fuzzy measure 1 in 5 a general concept of entropy of a fuzzy partition has been considered this concept is used to define the entropy of a fuzzy process in this paper some properties of this entropy are summarized these results could be used to build a fuzzy ergodic theory in a further paper the entropy of a measure preserving transformation of a fuzzy measure space will be studied

edge bipancyclicity of star graphs under edge fault tolerant abstract the star graph s n is one of the most famous interconnection networks it has been shown by li t k li cycle embedding in star graphs with edge faults appl math comput 167 2005 891 900 that s n contains a cycle of length from 6 to n when the number of fault edges in the graph does not exceed n 3 in this paper we improve this result by showing that for any edge subset f of s n with f n 3 every edge of s n f lies on a cycle of every even length from 6 to n provided n 3

bipanconnectivity of balanced hypercubes the balanced hypercube proposed by wu and huang is a variant of the hypercube network in this paper paths of various lengths are embedded into balanced hypercubes a bipartite graph g is bipanconnected if for two arbitrary nodes x and y of g with distance d x y there exists a path of length l between x and y for every integer l with d x y l v g 1 and l d x y 0 mod 2 we prove that the n dimensional balanced hypercube bh n is bipanconnected for all n 1 this result is stronger than that obtained by xu et al which shows that the balanced hypercube is edge bipancyclic and hamiltonian laceable

edge pancyclicity and hamiltonian laceability of the balanced hypercubes abstract the balanced hypercube bh n is a variant of the hypercube q n huang and wu proved that bh n has better properties than q n with the same number of links and processors in particularly they showed that there exists a cycle of length 2 l in bh n for all l 2 l 2 n in this paper we improve this result by showing that bh n is edge pancyclic which means that for arbitrary edge e there exists a cycle of even length from 4 to 2 2 n containing e in bh n we also show that the balanced hypercubes are hamiltonian laceable

super connectivity of balanced hypercubes abstract the reliability of an interconnection network is an important issue for multiprocessor systems in this paper we study a reliability measure called super connectivity in the balanced hypercube bh n which is a variant of the hypercube we show that the super connectivity of bh n is 4 n 4 and the super edge connectivity of bh n is 4 n 2 for n 2 that is to become a disconnected graph containing no isolated vertex we need to remove at least 4 n 4 vertices resp 4 n 2 edges from bh n

the balanced hypercube a cube based system for fault tolerant applications in this paper we present an interconnection structure called the balanced hypercube which is a variation of the standard hypercube with desirable properties of strong connectivity regularity and symmetry the proposed structure is a special type of load balanced graph designed to tolerate processor failure in balanced hypercubes each processor has a backup matching processor that shares the same set of neighboring nodes therefore tasks that run on a faulty processor can be reactivated in the backup processor to provide efficient system reconfiguration other properties of balanced hypercubes are examined it is also shown that odd dimensional balanced hypercubes have smaller diameters than that of standard hypercubes as an application of balanced hypercubes we show a fault tolerant embedding of rings in balanced hypercubes

on some properties of solutions of the biharmonic equation in the present paper the properties of the linear complex operator l f zfz z fz which is defined on the class of complex valued c1 functions in the plane are investigated it is shown that harmonicity and biharmonicity are invariant under the linear operatorl results concerning starlikeness and convexity of biharmonic functions versus the corresponding harmonic functions are considered the operator l can be manipulated to express the conditions in the definitions of starlikeness and convexity in a convenient way

the structural properties and weak convergence of fuzzy orthogonal measures the aim of this paper is to study the fuzzy measures proposed by butnariu 1 3 first the concept of fuzzy orthogonal measure on a regular fuzzy measurable space is presented then the structural characteristics of fuzzy orthogonal measure which shows the relation between the classical finite measure and the fuzzy orthogonal measure is demonstrated finally the weak convergence of fuzzy orthogonal measures is investigated

fault hamiltonicity and fault hamiltonian connectivity of the arrangement graphs the arrangement graph a sub n k is a generalization of the star graph there are some results concerning fault hamiltonicity and fault hamiltonian connectivity of the arrangement graph however these results are restricted in some particular cases and thus are less completed we improve these results and obtain a stronger and simpler statement let n k spl ges 2 and f spl sube v a sub n k spl cup e a sub n k we prove that a sub n k f is hamiltonian if f spl les k n k 2 and a sub n k f is hamiltonian connected if f spl les k n k 3 these results are optimal

symmetric property and reliability of balanced hypercube huang and wu in ieee transactions on computers 46 1997 484 490 introduced the balanced hypercube bh n as an interconnection network topology for computing systems and they proved that bh n is vertex transitive however some other symmetric properties say edge transitivity and arc transitivity of bh n remained unknown in this paper we solve this problem and prove that bh n is an arc transitive cayley graph using this we also investigate some reliability measures including super connectivity cyclic connectivity etc in bh n first we prove that every minimum edge cut of bh n n ge 2 isolates a vertex and every minimum vertex cut of bh n n ge 3 isolates a vertex this is stronger than that obtained by wu and huang which shows the connectivity and edge connectivity of bh n are 2n second yang applied mathematics and computation 219 2012 970 975 proved that for n ge 2 the super connectivity of bh n is 4n 4 and the super edge connectivity of bh n is 4n 2 in this paper we proved that bh n n ge 2 is super lambda prime but not super kappa prime that is every minimum super edge cut of bh n n ge 2 isolates an edge but the minimum super vertex cut of bh n n ge 2 does not isolate an edge third we also obtain that for n ge 2 the cyclic connectivity of bh n is 4n 4 and the cyclic edge connectivity of bh n is 4 2n 2 that is to become a disconnected graph which has at least two components containing cycles we need to remove at least 4n 4 vertices resp 4 4n 2 edges from bh n n ge 2

edge fault tolerant bipanconnectivity of hypercubes this paper shows that for any two distinct vertices u and v with distance d in the hypercube q n n 3 with at most 2n 5 faulty edges and each vertex incident with least two fault free edges there exist fault free uv paths of length in q n for every with d 4 2 n 1 and d 0 mod2 this result improves some known results on edge fault bipanconnectivity of hypercubes the proof is based on the recursive structure of q n

additive fuzzy measures and integrals iii abstract a point of view concerning fuzzy measures is explained to this end a new concept of disjointness for fuzzy is introduced and studied also a concept of an additive class of fuzzy sets is defined to be a class of fuzzy sets closed under some additive operations the fuzzy measures are defined to be sum preserving real functions over such additive classes some basic properties of the fuzzy measures are derived in contrast with other homonymous concepts studied in literature our fuzzy measures lead to an additive fuzzy integral see the part ii of the paper

pancyclicity and bipancyclicity of conditional faulty folded hypercubes a graph is said to be pancyclic if it contains cycles of every length from its girth to its order inclusive and a bipartite graph is said to be bipancyclic if it contains cycles of every even length from its girth to its order the pancyclicity or the bipancyclicity of a given network is an important factor in determining whether the network s topology can simulate rings of various lengths an n dimensional folded hypercube fq n is an attractive variant of an n dimensional hypercube q n that is obtained by establishing some extra edges between the vertices of q n fq n for any odd n is known to be bipartite in this paper we explore the pancyclicity and bipancyclicity of fq n for any fq n n 2 with at most 2n 3 faulty edges where each vertex is incident to at least two fault free edges we prove that there exists a fault free cycle of every even length from 4 to 2 n and when n 2 is even we prove there also exists a fault free cycle of every odd length from n 1 to 2 n 1 the result is optimal with respect to the number of faulty edges tolerated

hamiltonian paths passing through prescribed edges in balanced hypercubes abstract the balanced hypercube was proposed as a novel interconnection network for large scale parallel systems it is known that the balanced hypercube is edge bipancyclic given a set of edges p with p n 1 and two vertices x and y in different bipartite sets we show that there exists a hamiltonian path from x to y passing through p in b h n if and only if p is a linear forest and neither x nor y is an internal vertex of p and x and y are not end vertices of a component of p simultaneously consequently the balanced hypercube contains a hamiltonian cycle passing through a set p of at most n prescribed edges if and only if p is a linear forest moreover our result improves some known results

the bipanconnectivity and m panconnectivity of the folded hypercube the interconnection network considered in this paper is the folded hypercube that is an attractive variance of the well known hypercube the folded hypercube is superior to the hypercube in many criteria such as diameter connectivity and fault diameter in this paper we study the path embedding aspects bipanconnectivity and m panconnectivity of the n dimensional folded hypercube a bipartite graph is bipanconnected if each pair of vertices x and y are joined by the bipanconnected paths that include a path of each length s satisfying and is even where n is the number of vertices and denotes the shortest distance between x and y a graph is m panconnected if each pair of vertices x and y are joined by the paths that include a path of each length ranging from m to n 1 in this paper we introduce a new graph called the path of ladders by presenting algorithms to embed the path of ladders into the folded hypercube we show that the n dimensional folded hypercube is bipanconnected for n is an odd number we also show that the n dimensional folded hypercube is strictly n 1 panconnected for n is an even number that is each pair of vertices are joined by the paths that include a path of each length ranging from n 1 to n 1 and the value n 1 reaches the lower bound of the problem

matching preclusion for balanced hypercubes matching preclusion is a measure of robustness in the event of edge failure in interconnection networks the matching preclusion number of a graph g is the minimum number of edges whose deletion leaves the resulting graph without a perfect matching or an almost perfect matching and the conditional matching preclusion number of g is the minimum number of edges whose deletion leaves the resulting graph with no isolated vertices and without a perfect matching or an almost perfect matching in this paper we consider balanced hypercubes we obtain that an n dimension balanced hypercube bh n has the matching preclusion number 2n and mainly prove that for the balanced hypercube bh n each matching preclusion set of cardinality 2n is trivial and the conditional matching preclusion number of balanced hypercube is 4n 2 whenever n 2

edge disjoint hamiltonian cycles in k ary n cubes and hypercubes solutions for decomposing a higher dimensional torus to edge disjoint lower dimensional tori in particular edge disjoint hamiltonian cycles are obtained based on the coding theory approach first lee distance gray codes in z sub k sup n are presented and then it is shown how these codes can directly be used to generate edge disjoint hamiltonian cycles in k ary n cubes further some new classes of binary gray codes are designed from these lee distance gray codes and using these new classes of binary gray codes edge disjoint hamiltonian cycles in hypercubes are generated

fault tolerant ring embedding in de bruijn networks a method of embedding a ring in a d ary de bruijn multiprocessor network in the event of multiple node processor failures is presented in particular the algorithm guarantees that a 2 sup n n 1 node ring will be found in a binary de bruijn network with a single faulty node where 2 sup n is the total number of nodes in the network it is also shown that a d sup n 1 node ring can always be found in the presence of d 1 link failures when d is a prime power and the network contains d sup n nodes the latter is accomplished by constructing d 1 edge disjoint cycles each of length d sup n 1 a modification of the graph that allows it to admit a hamiltonian cycle in the event of d 1 edge failures is also discussed

schwarzian derivative integral means and the affine and linear invariant families of biharmonic mappings abstract in this paper we discuss the properties of the schwarzian derivative integral means and the affine and linear invariant families of biharmonic mappings first we introduce the schwarzian derivative s f for biharmonic mappings f z 2g h and obtain several necessary and sufficient conditions for s f to be analytic second we introduce the subordination of biharmonic mappings and obtain inequalities for integral means of subordinate biharmonic mappings finally we introduce the affine and linear invariant families of biharmonic mappings and prove several estimates related to the jacobian of functions in these invariant families

fuzzy measurability and integrability developpement d une nouvelle theorie relative au probleme de la mesure des objets flous continuite des fonctions sigma additives sur une classe sigma additive d ensembles flous extension de la notion d integrale floue methode de calcul des integrales floues problemes ouverts

conditional edge fault hamiltonicity of augmented cubes the augmented cube is a variation of hypercubes it possesses many superior properties in this paper we show that for any n dimensional augmented cube n 3 with faulty edges up to 4n 8 in which each vertex is incident to at least two fault free edges there exists a fault free hamiltonian cycle our result is optimal with respect to the number of faulty edges tolerated

biorthogonal series solution of stokes flow problems in sectorial regions in this paper we develop an eigenfunction expansion method for solving stokes flow problems in sectorial cavities that arise in fluid dynamics the method leads to the development of a set of eigenfunctions adjoint eigenfunctions biorthogonality conditions and an algorithm for the computation of the coefficients of the eigenfunction expansion the resulting infinite system of linear equations is then solved by truncation these biorthogonality conditions are properties satisfied by the eigenfunctions and adjoint eigenfunctions which are used to compute the coefficients of the eigenfunction expansion solution the biorthogonality conditions are derived for a class of fourth order boundary value problems with variable coefficients that arise from separating variables of the governing stokes equation the method is applied to the slow steady flow in a two dimensional sectorial cavity the fluid is set into motion by the uniform translation of a covering plate or belt