strong allee effect in a diffusive predator prey system with a protection zone article history a reaction diffusion predator prey system with strong allee effect and a protection zone for the prey is considered dynamics and steady state solutions of the system are analyzed in particular it is shown that the overexploitation phenomenon can be avoided if the allee effect threshold is low and the protection zone is large

reliability evaluation of bc networks reliability evaluation of interconnection network is important to the design and maintenance of multiprocessor systems extra connectivity determination and faulty networks structure analysis are two important aspects for the reliability evaluation of interconnection networks an n dimensional bijective connection network in brief bc network denoted by xn is an n regular graph with 2n vertices and n2n 1 edges the hypercubes mobius cubes crossed cubes and twisted cubes are some examples of the bc networks by exploring the boundary problem of the bc networks we prove that when n 4 and 0 h n 4 the h extra connectivity of an n dimensional bc network xn is kh xn n h 1 1 2h h 3 furthermore there exists a large connected component and the remaining small components have at most h vertices in total if the total number of faulty vertices is strictly less its h extra connectivity as an application the results on the h extra connectivity and structure of faulty networks on hypercubes mobius cubes crossed cubes and twisted cubes are obtained

many to many two disjoint path covers in restricted hypercube like graphs a disjoint path cover dpc for short of a graph is a set of pairwise internally disjoint paths that altogether cover every vertex of the graph given a set s of k sources and a set t of k sinks a many to many k dpc between s and t is a disjoint path cover each of whose paths joins a pair of source and sink it is classied as paired if each source of s must be joined to a designated sink oft or unpaired if there is no such constraint in this paper we show that every m dimensional restricted hypercube like graph with at most m 3 faulty vertices and or edges being removed has a paired and unpaired 2 dpc joining arbitrary two sources and two sinks where m 5 the bound m 3 on the number of faults is optimal for both paired and unpaired types

effect of cross diffusion in the diffusion prey predator model with a protection zone ii abstract in the current paper we continue the mathematical analysis studied in li and wu 2017 15 and oeda 2011 22 and further study the effect of cross diffusion for the predator on the stationary problem the existence of positive solutions is first established by the bifurcation theory we next discuss the limiting behavior of positive solutions when the intrinsic growth rate of the predator species tends to infinity moreover as the prevention coefficient tends to infinity we obtain two shadow systems and give the complete limiting characterization of positive solutions

hamiltonian connectivity of the wk recursive network with faulty nodes a graph g is said to be hamiltonian connected if there is a hamiltonian path between every two distinct nodes of g let f denote the set of faulty nodes of g then g is f node hamiltonian connected if g f is hamiltonian connected we use k d t to denote a wk recursive network of level t each of whose basic modules is a d node complete graph compared with other networks it is rather difficult to construct a hamiltonian path between two arbitrary nodes in a faulty wk recursive network in this paper we show that k d t is d 4 node hamiltonian connected since the connectivity of k d t is d 1 the result is optimal in the worst case

some uniqueness and exact multiplicity results for a predator prey model in this paper we consider positive solutions of a predator prey model with diffusion and under homogeneous dirichlet boundary conditions it turns out that a certain parameter m in this model plays a very important role a good understanding of the existence stability and number of positive solutions is gained when m is large in particular we obtain various results on the exact number of positive solutions our results for large m reveal interesting contrast with that for the well studied casem 0 i e the classical lotka volterra predator prey model

effect of a protection zone in the diffusive leslie predator prey model abstract in this paper we consider the diffusive leslie predator prey model with large intrinsic predator growth rate and investigate the change of behavior of the model when a simple protection zone 0 for the prey is introduced as in earlier work y du j shi a diffusive predator prey model with a protection zone j differential equations 229 2006 63 91 y du x liang a diffusive competition model with a protection zone j differential equations 244 2008 61 86 we show the existence of a critical patch size of the protection zone determined by the first dirichlet eigenvalue of the laplacian over 0 and the intrinsic growth rate of the prey so that there is fundamental change of the dynamical behavior of the model only when 0 is above the critical patch size however our research here reveals significant difference of the model s behavior from the predator prey model studied in y du j shi a diffusive predator prey model with a protection zone j differential equations 229 2006 63 91 with the same kind of protection zone we show that the asymptotic profile of the population distribution of the leslie model is governed by a standard boundary blow up problem and classical or degenerate logistic equations

on fundamental solutions of generalized schrodinger operators abstract we consider the generalized schrodinger operator where is a nonnegative radon measure in r n n 3 assuming that satisfies certain scale invariant kato conditions and doubling conditions we establish the following bounds for the fundamental solution of in r n ce e 2 d x y x y n 2 x y ce e 1 d x y x y n 2 where d x y is the distance function for the modified agmon metric m x dx2 associated with we also study the boundedness of the corresponding riesz transforms 1 2 on lp r n dx

structure connectivity and substructure connectivity of k ary n cube networks abstract the k ary n cube is one of the most attractive interconnection networks for parallel and distributed computing system in this paper we investigate the fault tolerant capabilities of k ary n cubes with respect to the structure connectivity and substructure connectivity let h is a connected graph the structure connectivity of a graph g denoted by g h is the minimum cardinality of a set of connected subgraphs in g whose deletion disconnects the graph g and every element in the set is isomorphic to h the substructure connectivity of a graph g denoted by s g h is the minimum cardinality of a set of connected subgraphs in g whose deletion disconnects the graph g and every element in the set is isomorphic to a connected subgraph of h we show q n k h and s q n k h for each h k1 k1 1 k1 2 k1 3

a survey of the theory of hypercube graphs abstract we present a comprehensive survey of the theory of hypercube graphs basic properties related to distance coloring domination and genus are reviewed the properties of the n cube defined by its subgraphs are considered next including thickness coarseness hamiltonian cycles and induced paths and cycles finally various embedding and packing problems are discussed including the determination of the cubical dimension of a given cubical graph

lee distance and topological properties of k ary n cubes in this paper we consider various topological properties of a k ary n cube q sub n sup k using lee distance we feel that lee distance is a natural metric for defining and studying a q sub n sup k after defining a q sub n sup k graph using lee distance we show how to find all disjoint paths between any two nodes given a sequence of radix k numbers a function mapping the sequence to a gray code sequence is presented and this function is used to generate a hamiltonian cycle embedding the graph of a mesh and the graph of a binary hypercube into the graph of a q sub n sup k is considered using a k ary gray code we show the embedding of a k n sub 1 spl times k n sub 2 spl times spl times k n sub m dimensional mesh into a q sub n sup k where n spl sigma sub i l sup m n sub i then using a single digit 4 ary reflective gray code we demonstrate embedding a q sub n into q sub n 2 sup 4 we look at how lee distance may be applied to the problem of resource placement in a q sub n sup k by using a lee distance error correcting code although the results in this paper are only preliminary lee distance error correcting codes have not been applied previously to this problem finally we consider how lee distance can be applied to message routing and single node broadcasting in a q sub n sup k in this section we present two single node broadcasting algorithms that are optimal when single port and multi port i o is used

embedding of tori and grids into twisted cubes the hypercube is one of the most popular interconnection networks since it has a simple structure and is easy to implement an n dimensional twisted cube tq n is an important variation of hypercube q n and preserves many of its desirable properties the problem of how to embed a family of disjoint meshes or tori into a host graph has attracted great attention in recent years however there is no systematic method proposed to generate the desired meshes and tori in tq n in this paper we develop two systematic linear time algorithms for embedding disjoint multi dimensional tori into tq n n 7 as follows 1 for a positive integer m with email protected email protected email protected n 4 a family of 2 m disjoint k dimensional tori of size 2 s 1x2 s 2x x2 s k each can be embedded with unit dilation where k 2 and i 1 ks email protected n m and 2 for a positive integer m with email protected email protected n 5 a family of 2 m disjoint k dimensional tori of size 2 s 1x2 s 2x x2 s k each can be embedded with unit dilation where k 2 s i 2 i 1 ks email protected n m and max 1 i k s i n 2m moreover we also provide similar embedding results for meshes and hypercubes our results mean that a family of torus structured mesh structured or hypercube structured parallel algorithms can be executed on the same twisted cube efficiently and in parallel

on the spanning connectivity and spanning laceability of hypercube like networks let u and v be any two distinct nodes of an undirected graph g which is k connected for email protected email protected k a w containerc u v of a k connected graph g is a set of w disjoint paths joining u and v a w container c u v of g is a w container if it contains all the nodes of g a graph g is w connected if there exists a w container between any two distinct nodes a bipartite graph g is w laceable if there exists a w container between any two nodes from different parts of g let g 0 v 0 e 0 and g 1 v 1 e 1 be two disjoint graphs with v 0 v 1 let e v f v email protected v 0 f v v 1 and f v 0 v 1 is a bijection let g g email protected g 1 v email protected v 1 e email protected e email protected e the set of n dimensional hypercube like graph h n is defined recursively as a h 1 k 2 k 2 complete graph with two nodes and b if g 0 and g 1 are in h n then g g email protected g 1 is in h n 1 let b n email protected h n and g is bipartite and n n h n b n in this paper we show that every graph in b n is w laceable for every w email protected email protected n it is shown that a constructed n n graph h can not be 4 connected in addition we show that every graph in n n is w connected for every w email protected email protected 3

fault tolerant embedding of paths in crossed cubes the crossed cube cq n is an important variant of the hypercube q n and possesses many desirable properties for interconnection networks this paper shows that in cq n with f v faulty vertices and f e faulty edges there exists a fault free path of length between any two distinct fault free vertices for each satisfying 2 n email protected email protected email protected 2 n f v 1 provided that f v f email protected n 3 where the lower bound of and the upper bound of f v f e are tight for some n moreover this result improves the known result that cq n is n 3 hamiltonian connected

fault tolerance of hypercube like networks spanning laceability under edge faults abstract given two vertices u and v in a connected undirected graph g a w container c u v is a set of w internally vertex disjoint paths between u and v spanning all the vertices in g a bipartite graph g is w laceable if there exists a w container between any two vertices belonging to different partitions of g in 8 33 a class b n of bipartite graphs called hypercube like bipartite networks was defined in 22 lin et al showed that every graph in b n is w laceable for every 1 w n we define a graph is f edge fault tolerant w laceable if g f is w laceable for any arbitrary subset f of edges of g with f f in this paper we show that every graph in b n is f edge fault tolerant w laceable for every 0 f n 2 and 1 w n f which generalize lin s result we also give generalization of two other results in 22 27

on positive solutions of some pairs of differential equations this paper is a sequel to the author s paper 6 which studied positive solutions of a pair of differential equations introduced by conway gardner and smoller 4 their equations were equations for population problems in biology their significance is discussed further in 141 the purpose of the present paper is to obtain further results for the conwat gardner smoller equations and to show how the methods in 6 can be modified to decide exactly when the classical predator prey system as in blat and brown 2 has a strictly positive solution we also show how the methods can be applied to a number of other problems in particular our results answer some questions in conway s secondly we show how the asymptotic methods developed in 6 can be used to study the existence and uniqueness of strictly positive solutions of competing species problems our results suggest that this is a much more complicated problem than the predator prey problem we do obtain a condition which is except for a few special cases necessary and sufficient for the existence of a strictly positive solution however the condition is complicated and rather implicit thirdly we prove the uniqueness of the strictly positive solution in the conway smoller system if d is small and n 1 that is the case of ordinary differential equations this partially answers a question in 4 our proof is a local result and can be used in other situations for example it can be used to show that in certain cases of asocial nonlinearities where our notation follows 4 sect 4 there are exactly two strictly positive solutions for small d we also show how our method applies to more general predator prey problems in section 1 we find necessary and sufficient conditions for the existence of strictly positive solutions of predator prey systems in section 2 we study competing species models by iteration and asymptotic methods finally in section 3 we prove our uniqueness results in an appendix we briefly discuss how the methods of section 3 apply to more general models

bmo spaces related to schrodinger operators with potentials satisfying a reverse holder inequality we identify the dual space of the hardy type space related to the time independent schrodinger operator v with v a potential satisfying a reverse holder inequality as a bmo type space we prove the boundedness in this space of the versions of some classical operators associated to hardy littlewood semigroup and poisson maximal functions square function fractional integral operator we also get a characterization of in terms of carlesson measures

parabolic equations with vmo coefficients in sobolev spaces with mixed norms an lq lp theory of divergence and non divergence form parabolic equations is presented the main coefficients are supposed to belong to the class vmox which in particular contains all measurable functions depending only on t the method of proving simplifies the methods previously used in the case p q

many to many disjoint path covers in the presence of faulty elements a many to many k disjoint path cover k dpc of a graph g is a set of k disjoint paths joining k sources and k sinks in which each vertex of g is covered by a path it is called a paired many to many disjoint path cover when each source should be joined to a specific sink and it is called an unpaired many to many disjoint path cover when each source can be joined to an arbitrary sink in this paper we discuss about paired and unpaired many to many disjoint path covers including their relationships application to strong hamiltonicity and necessary conditions and then we give a construction scheme for paired many to many disjoint path covers in the graph h0 oplus h1 obtained from connecting two graphs h0 and h1 with v h0 v h1 by v h1 pairwise nonadjacent edges joining vertices in h0 and vertices in h1 where h0 g0 oplus g1 and h1 g2 oplus g3 for some graphs gj using the construction we show that every m dimensional restricted hl graph and recursive circulant g 2m 4 with f or less faulty elements have a paired k dpc for any f and k ges 2 with f 2k les m

fault tolerant cycle embedding in alternating group graphs abstract cycle embedding is an important issue in evaluating the efficiency of interconnection networks and is also an extension of the theoretical research on hamiltonicity in this paper we study the fault tolerant hamiltonicity of alternating group graphs which were recently proposed as interconnection topologies for parallel and distributed systems a cycle of length l is said to be an l cycle let f be a set of faulty vertices in an n dimensional alternating group graph ag n and let m be the number of vertices in ag n f a previous result in j m chang j s yang y l wang y cheng panconnectivity fault tolerant hamiltonicity and hamiltonian connectivity in alternating group graphs networks 44 2004 302 310 showed that for n 4 ag n f is hamiltonian i e ag n f possesses a hamiltonian cycle if f n 2 and ag n f is hamiltonian connected i e every two vertices of ag n f are connected by a hamiltonian path if f n 3 in this paper we show the following results i ag n f is pancyclic i e ag n f contains an l cycle for each l with 3 l m if f n 2 ii ag n f is vertex pancyclic i e every vertex of ag n f is contained in an l cycle for each l with 3 l m if f n 3 and iii ag n f is edge 4 pancyclic i e every edge of ag n f is contained in an l cycle for each l with 4 l m if f n 4 the first result is an improvement over the previous one and the last two results are new characterization of fault tolerant pancyclicity on ag n all the results we achieved are the best possible in the sense that the number of faulty vertices cannot be increased

global morrey regularity of strong solutions to the dirichlet problem for elliptic equations with discontinuous coefficients abstract well posedness is proved in the space w2 p w1 p0 for the dirichlet problem equation omitted if the principal coefficients aij x of the uniformly elliptic operator belong to vmo l

one to one disjoint path covers on k ary n cubes the k ary n cube q n k is one of the most popular interconnection networks let n 2 and k 3 it is known that q n k is a nonbipartite resp bipartite graph when k is odd resp even in this paper we prove that there exist r vertex disjoint paths p i email protected email protected r 1 between any two distinct vertices u and v of q n k when k is odd and there exist r vertex disjoint paths r i email protected email protected r 1 between any pair of vertices w and b from different partite sets of q n k when k is even such that i 0 r 1p i or i 0 r 1r i covers all vertices of q n k for email protected email protected 2n in other words we construct the one to one r disjoint path cover of q n k for any r with email protected email protected 2n the result is optimal since any vertex in q n k has exactly 2n neighbors

one to one disjoint path covers on alternating group graphs the alternating group graph denoted by ag n is one of the popular interconnection networks which has many attractive properties in this paper we prove that for any two distinct nodes and there exist m node disjoint paths for any integer n 3 with 1 m 2 n 4 whose union covers all the nodes of ag n for any node of ag n has exactly 2 n 4 neighbors 2 n 4 is the maximum number of node disjoint paths can be constructed in ag n

super spanning connectivity on wk recursive networks abstract the wk recursive network has received much attention due to its many favorable properties such as small diameter large connectivity and high degree of scalability and expandability in this paper we consider the super spanning connectivity properties of the wk recursive network we use k d t to denote the wk recursive network of level t each of which basic modules is a d vertex complete graph where d 1 and t 1 we prove that for any two distinct vertices and there exist m node disjoint paths whose union covers all the vertices of k d t for d 4 t 1 and 1 m d 1 since the connectivity of k d t is d 1 the result is optimal in the worst case

fault diagnosability of arrangement graphs the growing size of the multiprocessor system increases its vulnerability to component failures it is crucial to locate and to 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 in this paper we show that the largest connected component of the survival graph contains almost all of the remaining vertices in the n k arrangement graph a n k when the number of moved faulty vertices is up to twice or three times the traditional connectivity using this fault resiliency we establish the conditional diagnosability of a n k under the comparison model and prove that the conditional diagnosability of a n k is 3k 2 n k 3 for k 4 n k 2 and the conditional diagnosability of a n n 1 is 3n 7 for n 5

paths and cycles identifying vertices in twisted cubes the hypercube is one of the most popular interconnection networks since it has simple structure and is easy to implement the twisted cube is an important variation of the hypercube and preserves many of its desirable properties karpovsky et al introduced the concept of identifying codes to model fault detection in multiprocessor systems and honkala et al developed an identifying code by using cycles to identify the faulty processors in the hypercube in this paper we study the vertex identification problem on the twisted cube we first propose an interesting construction scheme to build paths and cycles and furthermore apply a minimum number of paths and cycles to identify the faulty processors of the twisted cube

a diffusive competition model with a protection zone this paper is concerned with a two species diffusive competition model with a protection zone for the weak competitor our mathematical results imply that when the protection zone is above a certain critical patch size determined by the birth rate of the weak competitor the weak species almost always survives but it cannot survive when the protection zone is below the critical size and its competitor is strong enough while this is the main feature of the model the actual dynamical behavior of the reaction diffusion system is more complicated the key to reveal the main feature of the system lies in a detailed analysis of the attracting regions of its steady state solutions our mathematical analysis shows that compared with the predator prey model discussed in yihong du junping shi a diffusive predator prey model with a protect zone j differential equations 226 2006 63 91 the protection zone has some essentially different effects on the fine dynamics of the competition model

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

an efficient algorithm to construct disjoint path covers of dcell networks data center networks have been becoming more and more important with the development of cloud computing for any two integers k 0 and n 2 the k dimensional dcell with n port switches d k n has been proposed for one of the most important data center networks as a server centric data center network structure d k n can support millions of servers with outstanding network capacity and provide good fault tolerance by only using commodity switches a disjoint path cover has significant applications in data center networks in this paper we prove that d k n is one to one r disjoint path coverable for any integer 1 r n k 1 except for d 1 2 moreover we propose an o t k algorithm for finding a one to one r disjoint path cover in d k n for any integer 1 r n k 1 where t k is the number of servers in d k n

a type of perfect matchings extend to hamiltonian cycles in k ary n cubes abstract kreweras conjectured that every perfect matching in a hypercube q n for n 2 extends to a hamiltonian cycle of q n fink confirmed the conjecture to be true the k ary n cube q n k is a generalization of the hypercube however the analogous result does not necessarily hold for q n k we can find a perfect matching in q 2 6 which is not contained in any hamiltonian cycle of q 2 6 in this paper we investigate the existence of a hamiltonian cycle passing through a perfect matching in q n k for an integer n 2 and an even integer k 6 we prove that every perfect matching in q n k consisting of edges in the same dimension can be extended to a hamiltonian cycle of q n k

on positive solutions of some pairs of differential equations ii on etudie l existence de solutions strictement positives de systemes predateur proie on demontre l unicite de la solution strictement positive dans le systeme de conway smoller

on the indices of fixed points of mappings in cones and applications abstract assume that k is a cone in a banach space and a k k is completely continuous we obtain a formula for the index in k of a fixed point of a under the assumption that a linearization exists and satisfies an invertibility condition we then use this formula to generalize some results of amann on the number of fixed points of a to the case where k has empty interior

a lower bound on the number of hamiltonian cycles through a prescribed edge in a crossed cube the crossed cube proposed by efe is one of the most notable variations of the hypercube but some properties of the former are superior to those of the latter for example the diameter of the crossed cube is almost the half that of the hypercube in this paper we consider the problem of embedding a hamiltonian cycle passing through a prescribed edge in crossed cubes a concept termed the cycle pattern is used to construct a linear algorithm for embedding a structural hamiltonian cycle so the cycle passes through an arbitrary given edge in the crossed cube further we give the necessary and sufficient conditions for determining what kind of permutation generates a hamiltonian cycle pattern of the crossed cube as a result we obtain a lower bound for the number of hamiltonian cycles through a given edge in an n dimensional crossed cube our work extends some recently obtained results

solvability for schrodinger equations with discontinuous coefficients abstract we consider the l p solvability for divergence and non divergence form schrodinger equations with discontinuous coefficients as an application we give the global morrey regularity for divergence and non divergence form schrodinger operators with vmo coefficients in a bounded domain

fault tolerant edge and vertex pancyclicity in alternating group graphs abstract in j m chang j s yang fault tolerant cycle embedding in alternating group graphs appl math comput 197 2008 760 767 the authors claim that every alternating group graph agn is n 4 fault tolerant edge 4 pancyclic which means that if the number of faults f n 4 then every edge in agn f is contained in a cycle of length l for every 4 l n 2 f they also claim that agn is n 3 fault tolerant vertex pancyclic which means that if f n 3 then every vertex in agn f is contained in a cycle of length l for every 3 l n 2 f their proofs are not complete they left a few important things unexplained in this paper we fulfill these gaps and present another proofs that agn is n 4 fault tolerant edge 4 pancyclic and n 3 fault tolerant vertex pancyclic

effect of cross diffusion on the stationary problem of a prey predator model with a protection zone abstract this paper is concerned with the stationary problem of a prey predator cross diffusion system with a protection zone for the prey we discuss the existence and non existence of coexistence states of the two species by using the bifurcation theory as a result it is shown that the cross diffusion for the prey has beneficial effects on the survival of the prey when the intrinsic growth rate of the predator is positive we also study the asymptotic behavior of positive stationary solutions as the cross diffusion coefficient of the prey tends to infinity

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

embedding graphs onto the supercube in this paper we consider the supercube a new interconnection network derived from the hypercube the supercube introduced by a sen 1989 has the same diameter and connectivity as a hypercube but can be realized for any number of nodes not only powers of 2 we study the supercube s ability to execute parallel programs using graph embedding techniques we show that complete binary trees and bidimensional meshes with a side length power of 2 are spanning subgraphs of the supercube we then prove that the supercube is hamiltonian and when the number of nodes is not a power of 2 it contains all cycles of length greater than 3 as subgraphs

the g good neighbor conditional diagnosability of hypercube under pmc model abstract processor fault diagnosis plays an important role in multiprocessor systems for reliable computing and the diagnosability of many well known networks has been explored for example hypercubes crossed cubes mobius cubes and twisted cubes of dimension n all have diagnosability n the conditional diagnosability of n dimensional hypercube q n is proved to be 4 n 2 1 under the pmc model in this paper we study the g good neighbor conditional diagnosability of q n under the pmc model and show that it is 2 g n g 2 g 1 for 0 g n 3 the g good neighbor conditional diagnosability of q n is several times larger than the classical diagnosability

on w1 p estimates for elliptic equations in divergence form first we will apply the method to study w 1 p regularity for a nonlinear elliptic operator in divergence form we would like to point out that in the particular case of a linear elliptic equation this method gives an alternative proof to the classical one which uses the general theory of singular integrals the utility of the method that we describe below is that hypotheses a and b are obtained directly by studying the deviation of the coefficients of a from the coefficients of a0 and this is usually not a difficult task a more interesting application of this kind of approximation method is to elliptic homogenization problems for which we obtain results that give w 1 p estimates with a weak hypothesis of regularity in the coefficients for instance we are able to study the following cases

on vertex pancyclicity and edge pancyclicity of the wk recursive network abstract in this paper we study the pancyclic properties of the wk recursive networks we show that a wk recursive network with amplitude w and level l is vertex pancyclic for w 6 that is each vertex on them resides in cycles of all lengths ranging from 3 to n where n is the size of the interconnection network on the other hand we also investigate the m edge pancyclicity of the wk recursive network we show that the wk recursive network is strictly 3 2 l 1 edge pancyclic for w 7 and l 1 that is each edge on them resides in cycles of all lengths ranging from 3 2 l 1 to n and the value 3 2 l 1 reaches the lower bound of the problem

hamiltonian connected graphs for a simple graph g let ncd g min n u n v d w u v w v g uv e g wvorwu e g in this paper we prove that if ncd g v g then either g is hamiltonian connected or g belongs to a well characterized class of graphs the former results by dirac ore and faudree et al are extended

a diffusive predator prey model with a protection zone in this paper we study the effects of a protection zone 0 for the prey on a diffusive predator prey model with holling type ii response and no flux boundary condition we show the existence of a critical patch size described by the principal eigenvalue d 0 of the laplacian operator over 0 with homogeneous dirichlet boundary conditions if the protection zone is over the critical patch size i e if d 0 is less than the prey growth rate then the dynamics of the model is fundamentally changed from the usual predator prey dynamics in such a case the prey population persists regardless of the growth rate of its predator and if the predator is strong then the two populations stabilize at a unique coexistence state if the protection zone is below the critical patch size then the dynamics of the model is qualitatively similar to the case without protection zone but the chances of survival of the prey species increase with the size of the protection zone as generally expected our mathematical approach is based on bifurcation theory topological degree theory the comparison principles for elliptic and parabolic equations and various elliptic estimates

hamiltonian properties of twisted hypercube like networks with more faulty elements twisted hypercube like networks thlns are a large class of network topologies which subsume some well known hypercube variants this paper is concerned with the longest cycle in an n dimensional n d thln with up to 2n 9 faulty elements let g be an n d thln n 7 let f be a subset of v g e g f 2n 9 we prove that g f contains a hamiltonian cycle if d g f 2 and g f contains a near hamiltonian cycle if d g f 1 our work extends some previously known results

global orlicz estimates for non divergence elliptic operators with potentials satisfying a reverse holder condition abstract in this paper we establish the global orlicz regularity for non divergence form schrodinger operators with new bmo coefficients in r n

uniqueness and multiplicity of positive solutions for a diffusive predator prey model in the heterogeneous environment this is the second part of our study on the spatially heterogeneous predator prey model where the interaction is governed by a crowley martin type functional response in part i we have proved that when the predator competition is strong i e k is large the model has at most one positive steady state solution for any this part is denoted to study the effect of saturation our result shows that the large saturation coefficient i e large m can not only lead to the uniqueness of positive solutions but also lead to the multiplicity of positive solutions moreover the stability of the corresponding positive solutions is also completely obtained this work can be regarded as a supplement of ref 10

efficient unicast in bijective connection networks with the restricted faulty node set the connectivity is an important criteria to measure the fault tolerant performance of a graph however the connectivity based on the condition of the set of arbitrary faulty nodes is generally lower in this paper in order to heighten this measure we introduce the restricted connectivity into bijective connection networks first we prove that the probability that all the neighbors of an arbitrary node becomes faulty in any n dimensional bijective connection network x n is very low when n becomes sufficient large then we give a constructive proof that under the condition that each node of an n dimensional bijective connection network x n has at least one fault free neighbor its restricted connectivity is 2n 2 about half of the connectivity of x n finally by our constructive proof we give an o n algorithm to get a reliable path of length at most n 3 log 2 f 1 between any two fault free nodes in an n dimensional bijective connection network in particular since the family of bc networks contains hypercubes crossed cubes mobius cubes etc our algorithm is appropriate for these cubes

sufficient conditions for graphs to be spanning connected abstract a graph g is t connected if there exist t internally disjoint u v paths between any two vertices u and v whose union spans g in this sense t connectedness is a natural extension of hamiltonicity in this paper we provide a sufficient condition for graphs to be t connected by generalizing a classic result given by chavatal 7 furthermore as byproducts we extend some known results concerning fault tolerant hamiltonicity and minimum cardinality of edges we also establish analogous results for balanced bipartite graphs

elliptic equations with bmo coefficients in lipschitz domains in this paper we study inhomogeneous dirichlet problems for elliptic equations in divergence form optimal regularity requirements on the coefficients and domains for the w 1 p 1 p estimates are obtained the principal coefficients are supposed to be in the john nirenberg space with small bmo semi norms the domain is supposed to have lipschitz boundary with small lipschitz constant these conditions for the w 1 p theory do not just weaken the requirements on the coefficients they also lead to a more general geometric condition on the domain

on global bifurcation for quasilinear elliptic systems on bounded domains general second order quasilinear elliptic systems with nonlinear boundary conditions on bounded domains are formulated into nonlinear mappings between sobolev spaces it is shown that the linearized mapping is a fredholm operator of index zero this and the abstract global bifurcation theorem of jacobo pejsachowicz patrick j rabier degree theory for c1 fredholm mappings of index 0 j anal math 76 1998 289 319 allow us to carry out bifurcation analysis directly on these elliptic systems at the abstract level we establish a unilateral global bifurcation result that is needed when studying positive solutions finally we supply two examples of cross diffusion population model and chemotaxis model to demonstrate how the theory can be applied

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

subsolution estimates and harnack s inequality for schrodinger operators let be an open set in r n and q 0 a function which is subjected to a certain integrability condition given p 0 we wish to show that there exists a number c 0 depending only on n p and a local norm of q such that non negative solutions of q v 0 satisfy v x c br x v p 1 p for all x and all r which are sufficiently small the oblique stroke indicates that the mean value is taken over the open ball b r x we give a new non probabilistic proof for q element of the kato class k loc n it is convenient to derive this estimate with p 2 for continuous positive functions v h loc 1

interior estimates in morrey spaces for strong solutions to nondivergence form equations with discontinuous coefficients abstract let us consider the nondivergence form elliptic equation a ij u x i x j in this paper we show that if the known term belongs to the morrey space l p then the second derivatives of the w 2 p solution u belong to the same space next we derive a c 1 loc regularity result that is related to a recent work by caffarelli

unpaired many to many disjoint path covers in restricted hypercube like graphs for two disjoint vertex sets s s 1 s k and t t 1 t k of a graph an unpaired many to many k disjoint path cover joining s and t is a set of pairwise vertex disjoint paths p 1 p k that altogether cover every vertex of the graph in which p i is a path from s i to some t j for 1 i j k a family of hypercube like interconnection networks called restricted hypercube like graphs includes most nonbipartite hypercube like networks found in the literature such as twisted cubes crossed cubes mobius cubes recursive circulant g 2 m 4 of odd m etc in this paper we show that every m dimensional restricted hypercube like graph m 5 with at most f faulty vertices and or edges being removed has an unpaired many to many k disjoint path cover joining arbitrary disjoint sets s and t of size k each subject to k 2 and f k m 1 the bound m 1 on f k is the maximum possible

interior and boundary w1 q estimates for quasi linear elliptic equations of schrodinger type abstract we consider nonlinear elliptic equations that are naturally obtained from the elliptic schrodinger equation u v u 0 in the setting of the calculus of variations and obtain l q estimates for the gradient of weak solutions in particular we generalize a result of shen 1995 39 in the nonlinear setting by using a different approach this allows us to consider discontinuous coefficients with a small bmo semi norm and non smooth boundaries which might not be lipschitz continuous

topological properties of twisted cube twisted cube tq n is derived by changing some connections of hypercube q n according to specific rules recently many topological properties of this variation cube are studied in this paper we prove that its connectivity is n its wide diameter and fault diameter are n 2 2 furthermore we show that tq n is a pancyclic network that is cycles of an arbitrary length at least four

a novel algorithm to embed a multi dimensional torus into a locally twisted cube this paper addresses how to embed a multi dimensional torus of maximal size of into an n dimensional locally twisted cube the major contribution of this paper is that for n 4 every k dimensional torus of size 2 s 1x2 s 2x x2 s k satisfying i 1 ks i n can be embedded into an n dimensional locally twisted cube with dilation 2 and unit expansion further an embedding algorithm can be constructed based on our embedding method and the time complexity of this algorithm is linear with respect to the size of the locally twisted cube the embedding is optimal in the sense that it has unit expansion

an efficient fault tolerant routing algorithm in bijective connection networks with restricted faulty edges in this paper we study fault tolerant routing in bijective connection networks with restricted faulty edges first we prove that the probability that all the incident edges of an arbitrary node become faulty in an n dimensional bijective connection network denoted by x n is extremely low when n becomes sufficient large then we give an o n algorithm to find a fault free path of length at most email protected log 2 f 1 between any two different nodes in x n if each node of x n has at least one fault free incident edge and the number of faulty edges is not more than 2n 3 in fact we also for the first time provide an upper bound of the fault diameter of all the bijective connection networks with the restricted faulty edges since the family of bc networks contains hypercubes crossed cubes mobius cubes etc all the results are appropriate for these cubes

fault free hamiltonian cycles in twisted cubes with conditional link faults the n dimensional twisted cube denoted by tq n a variation of the hypercube possesses some properties superior to the hypercube in this paper assuming that each vertex is incident with at least two fault free links we show that tq n can tolerate up to 2n 5 edge faults while retaining a fault free hamiltonian cycle the result is optimal with respect to the number of edge faults tolerated

a dynamic programming algorithm for simulation of a multi dimensional torus in a crossed cube the torus is a popular interconnection topology and several commercial multicomputers use a torus as the basis of their communication network moreover there are many parallel algorithms with torus structured and mesh structured task graphs have been developed if one network can embed a mesh or torus network the algorithms with mesh structured or torus structured can also be used in this network thus the problem of embedding meshes or tori into networks is meaningful for parallel computing in this paper we prove that for n 6 and 1 m n 2 1 a family of 2m disjoint k dimensional tori of size 2 s 1 2 s 2 2 s k each can be embedded in an n dimensional crossed cube with unit dilation where each si 2 i 1 k s i n m and max1 i k si 3 if n is odd and m n 3 2 otherwise max1 i k si n 2m 1 a new concept cycle skeleton is proposed to construct a dynamic programming algorithm for embedding a desired torus into the crossed cube furthermore the time complexity of the algorithm is linear with respect to the size of desired torus as a consequence a family of disjoint tori can be simulated on the same crossed cube efficiently and in parallel

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

2 solvability of the dirichlet problem for nondivergence elliptic equations with vmo coefficients we prove a well posedness result in the class w 2 p w 0 1 p for the dirichlet problem lu f a e in u 0 on we assume the coefficients of the elliptic nondivergence form equation that we study are in vmo l

many to many disjoint path covers in k ary n cubes the class of k ary n cubes represents the most commonly used interconnection topology for parallel and distributed computing systems embedding of disjoint paths has attracted much attention in the parallel processing in disjoint path problems the many to many disjoint path problem is the most generalized one this paper considers the problem of many to many disjoint path covers in the k ary n cube qnk with even k 4 and obtains the following result let m be an integer with 1 m 2n 1 for any two sets s and t of m vertices in different partite sets qnk has m disjoint s t paths containing all vertices of qnk and our result is optimal

the spanning connectivity of folded hypercubes a k container of a graph g is a set of k internally disjoint paths between u and v a k container of g is a k container if it contains all vertices of g a graph g is k connected if there exists a k container between any two distinct vertices and a bipartite graph g is k laceable if there exists a k container between any two vertices u and v from different partite sets of g for a given k a k connected graph respectively bipartite graph g is f edge fault tolerant spanning connected respectively laceable if g f is w connected for any w with 1 3 is odd and f 2 is even and f

fault free hamiltonian cycles passing through a linear forest in ternary n cubes with faulty edges a linear forest in a graph is a subgraph each component of which is a path in this paper we investigate the existence of a hamiltonian cycle passing through a linear forest in a ternary n cube qn3 n 2 with faulty edges let f be a faulty edge set of qn3 and l be a prescribed linear forest in qn3 f if e l 2n 1 and f n e l 2 1 then there is a hamiltonian cycle passing through l in qn3 f

fault tolerant hamiltonian connectivity of the wk recursive networks abstract many research on the wk recursive network has been published during the past several years due to its favorite properties in this paper we consider the fault tolerant hamiltonian connectivity of the wk recursive network we use k d t to denote the wk recursive network of level t each of which basic modules is a d vertex complete graph where d 1 and t 1 the fault tolerant hamiltonian connectivity h f g is defined to be the maximum integer k such that g is k fault tolerant hamiltonian connected if g is hamiltonian connected and is undefined otherwise in this paper we prove that h f k d t d 4 if d 4

constructing edge disjoint spanning trees in locally twisted cubes the use of edge disjoint spanning trees for data broadcasting and scattering problem in networks provides a number of advantages including the increase of bandwidth and fault tolerance in this paper we present an algorithm for constructing n edge disjoint spanning trees in an n dimensional locally twisted cube since the n dimensional locally twisted cube is regular with the common degree n the number of constructed trees is optimal

one to one disjoint path covers in digraphs abstract a one to one k disjoint directed path cover k ddpc for short of a digraph d is a set of k disjoint directed paths joining source with sink that cover all the vertices of the digraph let 0 d m i n d d be the minimum semi degree of d we show that every digraph d of sufficiently large order n with 0 d n k 1 2 contains a one to one k ddpc for any given one distinct source sink the undirected version and a ore type degree conditions of this result was proved earlier 1