formulating the mixed integer fractional posynomial programming the mixed integer fractional posynomial programming mifpp problem arises from the summation minimization of several quotient terms appearing in the objective function subject to given constraints all decision variables of the problem could be binary integer and or continuous with without absolute value functions this paper addresses the problem of providing an optimization approach which could certainly obtain a solution as close as possible to a global optimum in order to solve the complex mifpp problem several strategies are used including linear programming relaxation modified goal programming logarithmic piecewise function lpf and so on in addition the proposed model solves the undefined problem of the logarithm of zero or negative in the lpf this significantly improved the utility of lpf in real applications finally an illustrative example is included to demonstrate the solution procedure of the proposed model

mutually independent bipanconnected property of hypercube a graph is denoted by g with the vertex set v g and the edge set e g a path p hv0 v1 vmi is a sequence of adjacent vertices two paths with equal length p1 h u1 u2 umi and p2 h v1 v2 vmi from a to b are independent if u1 v1 a um vm b and ui vi for 2 6 i 6 m 1 paths with equal length fpig n i 1 from a to b are mutually independent if they are pairwisely independent let u and v be two distinct vertices of a bipar

fuzzy mathematical programming for multi objective linear fractional programming problem the present paper deals with a solution procedure for multi objective linear fractional programming problems an equivalent multi objective linear programming form of the problem has been formulated in the proposed methodology using fuzzy set theoretic approach a procedure has been explored the proposed solution procedure has also been used to solve numerical examples

the einstein equation for signature type changing spacetimes we give a local existence and uniqueness theorem for solutions of einstein s equations with dust energy momentum tensor in the class of m dimensional analytic transverse signature type changing spacetimes where the initial condi tions are given on the hypersurface of signature type change we also prove a similar theorem in the case that the energy momentum tensor represents a scalar field

panpositionable hamiltonicity and panconnectivity of the arrangement graphs abstract the arrangement graph an k is a generalization of the star graph it is more flexible in its size than the star graph there are some results concerning hamiltonicity and pancyclicity of the arrangement graphs in this paper we propose a new concept called panpositionable hamiltonicity a hamiltonian graph g is panpositionable if for any two different vertices x and y of g and for any integer l satisfying d x y l v g d x y there exists a hamiltonian cycle c of g such that the relative distance between x and y on c is l a graph g is panconnected if there exists a path of length l joining any two different vertices x and y with d x y l v g 1 we show that an k is panpositionable hamiltonian and panconnected if k 1 and n k 2

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

two disjoint cycle cover bipancyclicity of balanced hypercubes abstract a bipartite graph g is two disjoint cycle cover r1 r2 bipancyclic if for any even integer l satisfying r1 l r2 there exist two vertex disjoint cycles c1 and c2 in g such that v c 1 l and v c 2 v g l where v g denotes the number of vertices in g in this paper we study the two disjoint cycle cover bipancyclicity of the n dimensional balanced hypercube bhn which is a hypercube variant network and is superior to hypercube due to having a smaller diameter as a consequence we show that bhn is two disjoint cycle cover 4 2 2 n 1 bipancyclic for n 2

two node disjoint paths in balanced hypercubes the balanced hypercube bhn proposed by wu and huang is a variation of the hypercube it has been proved that the balanced hypercube is a node transitive and bipartite graph assume that the nodes are divided into two bipartite node sets x and y u and x are two different nodes in x and v and y are two different nodes in y in this paper we prove that there exist two node disjoint paths p x y and r u v in bhn and v p x y v r u v v bhn where n 1 the hamiltonian laceability of bhn which was proved by xu et al is also obtained from the corollary of our result

transverse type changing pseudo riemannian metrics and the extendability of geodesics we study the geodesics and pre geodesics of a smooth manifold with smooth pseudo riemannian metric which changes bilinear type i e the signature changes on a hypersurface we classify all geodesics and pre geodesics that cross the hypersurface of type change transversely we then apply these results to the eikonal partial differential equation to find geometric conditions for the local existence or non existence of smooth transverse solutions

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

edge pancyclicity and path embeddability of bijective connection graphs an n dimensional bijective connection graph in brief bc graph is a regular graph with 2 n nodes and n2 n 1 edges the n dimensional hypercube crossed cube mobius cube etc are some examples of the n dimensional bc graphs in this paper we propose a general method to study the edge pancyclicity and path embeddability of the bc graphs first we prove that a path of length l with dist x n x y 2 y in x n where x n n 4 is a n dimensional bc graph satisfying the three specific conditions and v x n is the node set of x n furthermore by this result we can claim that x n is edge pancyclic lastly we show that these results can be applied to not only crossed cubes and mobius cubes but also other bc graphs except crossed cubes and mobius cubes so far the research on edge pancyclicity and path embeddability has been limited in some specific interconnection architectures such as crossed cubes mobius cubes

fuzzy approaches for multiple criteria linear fractional optimization a comment abstract in this note a fallacy of the second solution procedure for fuzzy multiple criteria linear fractional optimization described by luhandjula 1984 is noted

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

multiple objective linear fractional programming this paper presents a simplex based solution procedure for the multiple objective linear fractional programming problem by 1 departing slightly from the traditional notion of efficiency and 2 augmenting the feasible region as in goal programming the solution procedure solves for all weakly efficient vertices of the augmented feasible region the article discusses the difficulties that must be addressed in multiple objective linear fractional programming and motivates the solution algorithm that is developed

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

cycle embedding of augmented cubes a graph g v e is vertex pancyclic if for every vertex u and any integer l ranging from a positive constant l to v g contains a cycle c of length l such that u is in c in this paper we show that an n dimensional augmented cube where n 2 is vertex pancyclic with l 3

a restricted class of multiobjective linear fractional programming problems abstract in this paper a restricted class of multiobjective linear fractional programming problems in the sense that the denominators are identical is investigated a numerical example is presented to illustrate the solution procedure

taylor series approach to fuzzy multiobjective linear fractional programming this paper presents the use of a taylor series for fuzzy multiobjective linear fractional programming problems fmolfp the taylor series is a series expansion that a representation of a function in the proposed approach membership functions associated with each objective of fuzzy multiobjective linear fractional programming problem transformed by using a taylor series are unified thus the problem is reduced to a single objective practical applications and numerical examples are used in order to show the efficiency and superiority of the proposed approach

conditional edge fault tolerant hamiltonicity of dual cubes the dual cube is an interconnection network for linking a large number of nodes with a low node degree it uses low dimensional hypercubes as building blocks and keeps the main desired properties of the hypercube a dual cube dc n has n 1 links per node where n is the degree of a cluster n cube and one more link is used for connecting to a node in another cluster in this paper assuming each node is incident with at least two fault free links we show a dual cube dc n contains a fault free hamiltonian cycle even if it has up to 2n 3 link faults the result is optimal with respect to the number of tolerant edge faults

hamiltonian cycle embedding for fault tolerance in balanced hypercubes abstract the balanced hypercube bh n defined by wu and huang is a variant of the hypercube network yang proposed that fault tolerance of balanced hypercube bh n is an important issue in parallel computing which needs further study yang 2010 24 in this paper we prove that there exists a fault free hamiltonian path between any two adjacent vertices in bh n with 2 n 2 faulty edges as a corollary we derive that for any fault free edge e there exists a fault free hamiltonian cycle containing e in bh n with 2 n 2 faulty edges which is optimal in the sense of the number of faulty edges

fault tolerant strong menger edge connectivity and 3 extra edge connectivity of balanced hypercubes abstract a connected graph g is called strongly menger edge connected if for any two distinct vertices x y of g there are min deg g x deg g y edge disjoint paths between x and y the g g 0 extra edge connectivity of the connected graph g denoted by g g is the minimal cardinality of a set of edges of g if exists whose deletion disconnects g and each remaining component contains more than g vertices in this paper we show that the n dimensional balanced hypercube b h n which is a variant of hypercube q n is still strongly menger edge connected even when there are 2 n 4 faulty vertices resp 2 n 2 faulty edges for n 2 moreover under the restricted condition that each vertex has at least two fault free edges we show that b h n is still strongly menger edge connected even when there are 6 n 8 faulty edges for n 2 these results are all optimal with respect to the maximum number of tolerated edge faults furthermore we showed that the 3 extra edge connectivity of b h n is 8 n 8 for n 2

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

on some super fault tolerant hamiltonian graphs a k regular hamiltonian and hamiltonian connected graph g is super fault tolerant hamiltonian if g remains hamiltonian after removing at most k 2 nodes and or edges and remains hamiltonian connected after removing at most k 3 nodes and or edges a super fault tolerant hamiltonian graph has a certain optimal flavor with respect to the fault tolerant hamiltonicity and hamiltonian connectivity in this paper we investigate a construction scheme to construct super fault tolerant hamiltonian graphs in particularly twisted cubes crossed cubes and mobius cubes are all special cases of this construction scheme therefore they are all super fault tolerant hamiltonian graphs

goal programming with linear fractional criteria abstract in this paper we present an extension of goal programming to include linear fractional criteria the extension forms a natural link between goal programming gp and multiple objective linear fractional programming molfp

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

embedding a hamiltonian cycle in the crossed cube with two required vertices in the fixed positions abstract a hamiltonian graph g is said to be panpositionably hamiltonian if for any two distinct vertices x and y of g there is a hamiltonian cycle c of g having dc x y l for any integer l satisfying d g x y l v g 2 where dg x y respectively dc x y denotes the distance between vertices x and y in g respectively c and v g denotes the total number of vertices of g as the importance of hamiltonian properties for data communication among units in an interconnected system the panpositionable hamiltonicity involves more flexible message transmission in this paper we study this property with respect to the class of crossed cubes which is a popular variant of the hypercube network

conditional edge fault pancyclicity of augmented cubes the augmented cube aq n proposed by choudum and sunitha 7 is a variation of the hypercube q n and possesses many superior properties that the hypercube does not contain in this paper we show that any n dimensional augmented cube with at most 4n 12 faulty edges contains cycles of lengths from 3 to 2 n under the condition that every node is incident with at least two fault free edges where n 3 ma et al 21 obtained the same result but with the number of faulty edges up to 2n 3 our result improves ma et email protected s result in terms of the number of fault tolerant edges

fuzzy approaches for multiple objective linear fractional optimization abstract the problem of finding a solution to a multiple objective linear fractional program arises in several real world situations in this paper we advocate that fuzzy sets theory provides a basis for solving this problem with sufficient consistency and rigorousness after representing imprecise aspirations of the decision maker by structured linguistic variables or converting the original problem via approximations or change of variables into a multiple objective linear program techniques of fuzzy linear programming may be used to reach a satisfactory solution it is shown that under reasonable restrictions this solution is efficient pareto optimal for the original problem numerical examples are also included for illustration

fault tolerant cycle embedding in the faulty hypercubes the hypercube is one of the best known interconnection networks embedding cycles of all possible lengths in faulty hypercubes has received much attention let f v respectively f e denote the set of faulty vertices respectively faulty edges and f v respectively f e denote the number of faulty vertices respectively faulty edge in an n dimensional hypercube q n let f e denote the number of faulty nodes and or faulty edges incident with the end vertices of an edge email protected e q n in this paper we assume that each node is incident with at least three fault free neighbors and at least three fault free edges under this assumption we show that every fault free edge lies on a fault free cycle of every even length from 4 to 2 n 2 f v if f v f e 2n 7 and f e 5 under our condition our result not only improves the previously best known result of hsieh et al s y hsieh t h shen edge bipancyclicity of a hypercube with faulty vertices and edges discrete applied mathematics 156 10 2008 1802 1808 where f v f e

the construction of spinor fields on manifolds with smooth degenerate metrics we examine some of the subtleties inherent in formulating a theory of spinors on a manifold with a smooth degenerate metric we concentrate on the case where the metric is singular on a hypersurface that partitions the manifold into lorentzian and euclidean domains we introduce the notion of a complex spinor fibration to make precise the meaning of continuity of a spinor field and give an express ion for the components of a local spinor connection that is valid in the absence of a frame of local orthonormal vectors these considerations enable one to construct a dirac equation for the discussion of the behavior of spinors in the vicinity of the metric degeneracy we conclude that the theory contains more freedom than the spacetime dirac theory and we discuss some of the implications of this for the continuity of conserved currents

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

multiple objective linear fractional programming a fuzzy set theoretic approach abstract a method for optimizing a multiobjective linear fractional programming problem is developed which yields always an efficient solution

mixed type surfaces with bounded gaussian curvature in three dimensional lorentzian manifolds abstract a mixed type surface is a connected regular surface in a lorentzian 3 manifold with non empty spacelike and timelike point sets the induced metric of a mixed type surface is a signature changing metric and their lightlike points may be regarded as singular points of such metrics in this paper we investigate the behavior of gaussian curvature at a non degenerate lightlike point of a mixed type surface to characterize the boundedness of gaussian curvature at a non degenerate lightlike points we introduce several fundamental invariants along non degenerate lightlike points such as the lightlike singular curvature and the lightlike normal curvature moreover using the results by pelletier and steller we obtain the gauss bonnet type formula for mixed type surfaces with bounded gaussian curvature

cycle embedding in star graphs with more conditional faulty edges abstract the star graph is one of the most attractive interconnection networks the cycle embedding problem is widely discussed in many networks and edge fault tolerance is an important issue for networks since edge failures may occur when a network is put into use in this paper we investigate the cycle embedding problem in star graphs with conditional faulty edges we show that there exist fault free cycles of all even lengths from 6 to n in any n dimensional star graph s n n 4 with 3 n 10 faulty edges in which each node is incident with at least two fault free edges our result not only improves the previously best known result where the number of tolerable faulty edges is up to 2 n 7 but also extends the result that there exists a fault free hamiltonian cycle under the same condition

various cycles embedding in faulty balanced hypercubes consider the balanced hypercube bh n with f e 2 n 3 faulty edges prove that every edge of bh n f e lies on fault free cycles of even lengths from 6 to 22n prove that the lower limit of the length 6 is sharp quite a lot of interconnection networks are served as the underlying topologies of large scale multiprocessor systems the hypercube is one of the most popular interconnection networks in this paper we consider the balanced hypercube which is a variant of the hypercube huang and wu showed that the balanced hypercube has better properties than hypercube with the same number of links and processors let f e be the set of faulty edges in an n dimensional balanced hypercube bh n where n 2 in this paper we consider bh n with f e 2 n 3 faulty edges and prove that every fault free edge lies on a fault free cycle of every even length from 6 to 2 2 n in bh n f e furthermore we prove that the lower limit of the length 6 is sharp by giving a counter example

linear array and ring embeddings in conditional faulty hypercubes the n dimensional hypercube qn is a graph having 2n vertices labeled from 0 to 2n 1 two vertices are connected by an edge if their binary labels differ in exactly one bit position in this paper we consider the faulty hypercube qn with n 3 that each vertex of qn is incident to at least two nonfaulty edges based on this requirement we prove that qn contains a hamiltonian path joining any two different colored vertices even if it has up to 2n 5 edge faults moreover we show that there exists a path of length 2n 2 between any two the same colored vertices in this faulty qn furthermore we also prove that the faulty qn still contains a cycle of every even length from 4 to 2n inclusive

a compromise procedure for the multiple objective linear fractional programming problem abstract a generalization of a well known multiple objective linear fractional programming molfp problem the multiple objective fractional programming mofp problem is formulated a concept of multiple objective programming mop problem corresponding to mofp is introduced and some relations between those problems are examined based on these results a compromise procedure for molfp problem is proposed a numerical example is given to show how the procedure works

weighting method for bi level linear fractional programming problems in this paper we consider the solution of a bi level linear fractional programming problem bllfpp by weighting method a non dominated solution set is obtained by this method in this article decision makers dms provide their preference bounds to the decision variables that is the upper and lower bounds to the decision variables they control we convert the hierarchical system into scalar optimization problem sop by finding proper weights using the analytic hierarchy process ahp so that objective functions of both levels can be combined into one objective function here the relative weights represent the relative importance of the objective functions

fault tolerant path embedding in folded hypercubes with both node and edge faults the folded hypercube fq n is a well known variation of the hypercube structure fq n is superior to q n in many measurements such as diameter fault diameter connectivity and so on let email protected fq n resp email protected fq n denote the set of faulty nodes resp faulty edges in fq n in the case that all nodes in fq n are fault free it has been shown that fq n contains a fault free path of length 2 n 1 resp 2 n 2 between any two nodes of odd resp even distance if email protected fq n n 1 where n 1 is odd and fq n contains a fault free path of length 2 n 1 between any two nodes if email protected fq n n 2 where n 2 is even in this paper we extend the above result to obtain two further properties which consider both node and edge faults as follows 1 fq n contains a fault free path of length at least 2 n 2 email protected fq n 1 resp 2 n 2 email protected fq n 2 between any two fault free nodes of odd resp even distance if email protected fq n email protected fq n n 1 where n 1 is odd 2 fq n contains a fault free path of length at least 2 n 2 email protected fq n 1 between any two fault free nodes if email protected fq n email protected fq n n 2 where n 2 is even