fault tolerant cycles embedding in hypercubes with faulty edges let q n be an n dimensional hypercube with f e 5 in this paper we consider the faulty hypercube under the following two additional conditions 1 each vertex is incident to at least two fault free edges and 2 every 4 cycle does not have any pair of non adjacent vertices whose degrees are both two after removing the faulty edges we prove that there exists a fault free cycle of every even length from 4 to 2 n in q n our result improves the result by liu and wang 2014 in terms of the lengths of embedding cycles where under the same conditions a fault free hamiltonian cycle was constructed

embedding hamiltonian cycles in alternating group graphs under conditional fault model in this paper assuming that each node is incident with two or more fault free links we show that an n dimensional alternating group graph can tolerate up to 4n 13 link faults where n 4 while retaining a fault free hamiltonian cycle the proof is computer assisted the result is optimal with respect to the number of link faults tolerated previously without the assumption at most 2n 6 link faults can be tolerated for the same problem and the same graph

oscillation results for linear hamiltonian systems for linear hamiltonian systems this paper established some new oscillation criteria which improve previous results even for self adjoint second order differential systems the main tool used is a generalized riccati transformation

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

the t k diagnosability of the bc graphs processor fault diagnosis takes an important role in fault tolerant computing on multiprocessor systems there are two classical diagnosis strategies the precise strategy and the pessimistic strategy both of which are based on the well known pmc diagnostic model nevertheless the degree of diagnosability of the system is limited under these two strategies a better method called the t k diagnosis strategy is proposed by somani and peleg in which the identified fault set is allowed to contain at most k fault free processors using this diagnosis strategy the degree of diagnosability of the hypercube increases greatly as the number of the fault free processors in the fault set increases we study the t k diagnosability of so called bc graphs that include hypercubes crossed cubes mobius cubes and twisted cubes etc we show that any n dimensional bc graph is t n k k diagnosable when n spl ges 4 and 0 spl les k spl les n where t n k k 1 n 1 2 k 1 k 2 1 therefore the crossed cube the mobius cube and the twisted cube all have the same t k diagnosability as the hypercube as a result the algorithms developed for diagnosis on the hypercube may also be used to diagnose multiprocessor systems whose network topologies are based on bc graphs

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

analytic and geometric background of recurrence and non explosion of the brownian motion on riemannian manifolds we provide an overview of such properties of the brownian motion on complete non compact riemannian manifolds as recurrence and non explosion it is shown that both properties have various analytic characterizations in terms of the heat kernel green function liouville properties etc on the other hand we consider a number of geometric conditions such as the volume growth isoperimetric inequalities curvature bounds etc which are related to recurrence and non explosion

new kamenev type oscillation criteria for linear matrix hamiltonian systems some new kamenev type criteria have been obtained for the oscillation of the linear matrix hamiltonian system x a t x b t y y c t x a t y under the hypothesis a t b t b t 0 and c t c t are nxn real continuous matrix functions on the interval t 0 t 0 our results are different from most known ones in the sense that they are given in the form of lim t supg const rather than in the form of lim t sup l 1 where g is a positive linear functional on the linear space of nxn matrices with real entries consequently our results improve some previous results to some extent which can be seen by the examples given at the end of this paper

embedding two edge disjoint hamiltonian cycles into locally twisted cubes the n dimensional hypercube network q n is one of the most popular interconnection networks since it has simple structure and is easy to implement the n dimensional locally twisted cube ltq n an important variation of the hypercube has the same number of nodes and the same number of connections per node as q n one advantage of ltq n is that the diameter is only about half of the diameter of q n recently some interesting properties of ltq n have been investigated in the literature the presence of edge disjoint hamiltonian cycles provides an advantage when implementing algorithms that require a ring structure by allowing message traffic to be spread evenly across the interconnection network the existence of two edge disjoint hamiltonian cycles in locally twisted cubes has remained unknown in this paper we prove that the locally twisted cube ltq n with n 4 contains two edge disjoint hamiltonian cycles based on the proof of existence we further provide an o n2 n linear time algorithm to construct two edge disjoint hamiltonian cycles in an n dimensional locally twisted cube ltq n with n 4 where ltq n contains 2 n nodes and n2 n 1 edges

cover times for brownian motion and random walks in two dimensions lett x denote the rst hitting time of the disc of radius centered at x for brownian motion on the two dimensional torus t 2 we prove that sup x2t2t x j log j 2 2 as 0 the same applies to brownian motion on any smooth compact connected two dimensional riemannian manifold with unit area and no boundary as a consequence we prove a conjecture due to aldous 1989 that the number of steps it takes a simple random walk to cover all points of the lattice torus z 2 is asymptotic to 4n 2 logn 2 determining these asymptotics is an essential step toward analyzing the fractal structure of the set of uncovered sites before coverage is complete so far this structure was only studied non rigorously in the physics literature we also establish a conjecture due to kesten and r ev esz that describes the asymptotics for the number of steps needed by simple random walk in z 2 to cover the disc of radius n

oscillation of linear second order matrix differential equations in this paper sufficient conditions are obtained for oscillation of all nontrivial prepared symmetric solutions of a class of nonlinear second order matrix differential equations of the form p t y q t f y 0 t geqslant 0 and y q t f y 0 t geqslant 0

brownian motion on a pseudo sphere in minkowski space mathbb r l_v for a brownian motion moving on a pseudo sphere in minkowski space mathbb r l v of radius r starting from point x we obtain the distribution of hitting a fixed point on this pseudo sphere with l ge 3 by solving dirichlet problems the proof is based on the method of separation of variables and the orthogonality of trigonometric functions and gegenbauer polynomials

new oscillation criteria for linear matrix hamiltonian systems some new criteria have been obtained for the oscillation of the linear matrix hamiltonian system x a t x b t y y c t x a t y under the hypothesis a t b t b t 0 and c t c t are nxn real continuous matrix functions on the interval t 0 t 0 our results are independent of the main results obtained in our recent paper appl math comput 131 2002 357 in the sense that they are given in terms of the positive linear functional on the linear space of nxn matrices with real entries at the same time our results improve some previous results to a great extent which can be seen by the examples given at the end of this paper

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

fault free hamiltonian cycles in crossed cubes with conditional link faults the crossed cube which is a variation of the hypercube possesses some properties superior to the hypercube in this paper assuming that each node is incident with at least two fault free links we show that an n dimensional crossed cube contains a fault free hamiltonian cycle even if there are up to 2n 5 link faults the result is optimal with respect to the number of link faults tolerated we also verify that the assumption is practically meaningful by evaluating its occurrence probability which is very close to 1

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

weighted means and oscillation conditions for second order matrix differential equations on considere le systeme d equations differentielles d ordre 2 p t y t q t y t 0 pour t 0 ou p et q sont des matrices symetriques n n reelles sur 0 et y un vecteur colonne on suppose que p t est definie positive pour tout t tel que p t 1 on donne des conditions d oscillations

interval oscillation criteria for matrix differential systems with damping abstract this paper concerns the oscillation of solutions to leighton type for the matrix differential system with damping p t y r t y q t y 0 under the hypothesis p t p t 0 q t q t and r t r t are n n matrices of real valued continuous functions on the interval t 0

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

oscillation results for matrix differential systems with damping abstract in this paper we establish several new oscillation theorems of leighton type for the matrix differential system with damping p t y r t y q t y 0 under the hypothesis p t p t 0 q t q t and r t r t are n n matrices of real valued continuous functions on the interval t 0 our results are not contained in known results

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

oscillation results for selfadjoint differential systems abstract we consider the second order differential system 1 r t y q t y 0 where r q y are n n matrices with r t q t symmetric and r t positive definite for t a r t 0 t a we establish sufficient conditions in order that all prepared solutions y t of 1 are oscillatory that is det y t vanishes infinitely often on a the conditions involve the smallest and largest eigen values n r 1 t and 1 a t q s ds respectively the results obtained can be regarded as generalizing well known results of leighton and others in the scalar case

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

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

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

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

constructing edge disjoint spanning trees in twisted cubes the use of edge disjoint spanning trees or independent spanning trees in a network for data broadcasting has the benefits of increased of bandwidth and fault tolerance in this paper we propose an algorithm which constructs n edge disjoint spanning trees in the n dimensional twisted cube denoted by tq n because the n dimensional twisted cube is n regular the result is optimal with respect to the number of edge disjoint spanning trees constructed furthermore we also show that n2 of the n edge disjoint spanning trees constructed are independent spanning trees this algorithm runs in time o nlogn and can be parallelized to run in time o logn where n is the number of nodes in tq n

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

fault tolerance in k ary n cube networks the k ary n cube q n k is one of the most commonly used interconnection topologies for parallel and distributed computing systems let f n m be the minimum number of faulty nodes that make every n m dimensional subcube q n m k faulty in q n k under node failure models in this paper we prove that f n 0 1 f n 1 k for odd k 3 f n n 1 k n 1 for odd k 3 and k m f n m n 1m 1 k m n 2m 1 k m 1 for odd k 3

fault tolerance in bubble sort graph networks the bubble sort graph b n is one of the attractive underlying topologies for distributed systems f b n k resp f b n k is the minimum number of faulty nodes resp links that make every n k dimensional sub bubble sort graph faulty in b n under node resp link failure model in this paper we proved that f b n 0 f b n 0 1 f b n 1 f b n 1 n for n 4 f b n 2 f b n 2 n 2n 3 for n 6 f b n n 2 n 2 for n 3 f b n n 2 n 1 n 2 for n 3 f b n n 1 n for n 2 and f b n k f b n k k 1 2 k c k n for n 6 and 3 k n 3

safety levels an efficient mechanism for achieving reliable broadcasting in hypercubes we consider a distributed broadcasting algorithm for injured hypercubes using incomplete spanning binomial trees an injured hypercube is a connected hypercube with faulty nodes the incomplete spanning binomial tree proposed in this paper is a useful structure for implementing broadcasting in injured hypercubes it is defined as a sub tree of a regular spanning binomial tree that connects all the nonfaulty nodes we show that in an injured n dimensional hypercube with m faulty nodes there are at least 2 sup n 2 sup m source nodes called l nodes each of which can generate an incomplete spanning binomial tree a method is proposed to locate a large subset of the l node set using the concept of safety level the safety level of each node in an n dimensional hypercube can be easily calculated through n 1 rounds of information exchange among neighboring nodes an optimal broadcast initiated from a safe node is proposed when a nonfaulty source node is unsafe and there are at most n 1 faulty nodes in an injured n dimensional hypercube the proposed broadcasting scheme requires at most n 1 steps

riccati techniques and variational principles in oscillation theory for linear systems on etablit des conditions suffisantes pour que toutes les solutions soient oscillatoires on utilise les techniques de riccati et les principes variationnels

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

two node hamiltonicity of enhanced pyramid networks an enhanced pyramid network is an alternate hierarchical structure for a pyramid network this structure is created in a pyramid network by replacing each mesh with a torus at layers greater than one this work studies the fault tolerant hamiltonian problem on the enhanced pyramid network and demonstrates that an enhanced pyramid network with two faulty nodes is hamiltonian the result is optimal because edge connectivity and node connectivity of the enhanced pyramid network are both 4

oscillation of two dimensional linear second order differential systems abstract this article is concerned with the oscillatory behavior at infinity of the solution y a r 2 of a system of two second order differential equations y t q t y t 0 t a q is a continuous matrix valued function on a whose values are real symmetric matrices of order 2 it is shown that the solution is oscillatory at infinity if the largest eigenvalue of the matrix at q s ds tends to infinity as t this proves a conjecture of d hinton and r t lewis rocky mountain j math 10 1980 751 766 for the two dimensional case furthermore it is shown that considerably weaker forms of the condition still suffice for oscillatory behavior at infinity

hamiltonian connectivity of restricted hypercube like networks under the conditional fault model restricted hypercube like networks rhlns are an important class of interconnection networks for parallel computing systems which include most popular variants of the hypercubes such as crossed cubes mobius cubes twisted cubes and locally twisted cubes this paper deals with the fault tolerant hamiltonian connectivity of rhlns under the conditional fault model let g be an n dimensional rhln and email protected v g e g where n 7 and f 2n 10 we prove that for any two nodes u email protected v g f satisfying a simple necessary condition on neighbors of u and v there exists a hamiltonian or near hamiltonian path between u and v in g f the result extends further the fault tolerant graph embedding capability of rhlns

conditional edge fault hamiltonian connectivity of restricted hypercube like networks a graph g is considered conditional k edge fault hamiltonian connected if after k faulty edges are removed from g under the assumption that each node is incident to at least three fault free edges a hamiltonian path exists between any two distinct nodes in the resulting graph this paper focuses on the conditional edge fault hamiltonian connectivity of a wide class of interconnection networks called restricted hypercube like networks rhls an n dimensional rhl r h l n is proved to be conditional 2 n 7 edge fault hamiltonian connected for n 5 the technical theorem proposed in this paper is then applied to show that several multiprocessor systems including n dimensional crossed cubes n dimensional twisted cubes for odd n n dimensional locally twisted cubes n dimensional generalized twisted cubes n dimensional mobius cubes and recursive circulants g 2 n 4 for odd n are all conditional 2 n 7 edge fault hamiltonian connected

squeezed states generalized hermite polynomials and pseudo diffusion equation abstract we show that the wavefunctions pq n of the harmonic oscillator in the squeezed state representation have the generalized hermite polynomials as their natural orthogonal polynomials these wavefunctions lead to generalized poisson distribution pn pq which satisfy an interesting pseudo diffusion equation p n p q 1 4 2 p 2 1 2 2 q 2 p 2 p q in which the squeeze parameter plays the role of time th entropies sn have minima at the unsqueezed states 1 which means that squeezing or stretching decreases the correlation between momentum p and position q

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

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

linear transformation and oscillation criteria for hamiltonian systems abstract using a linear transformation similar to the kummer transformation some new oscillation criteria for linear hamiltonian systems are established these results generalize and improve the oscillation criteria due to i s kumari and s umanaheswaram i sowjaya kumari s umanaheswaram oscillation criteria for linear matrix hamiltonian systems j differential equations 165 2000 174 198 q yang et al q yang r mathsen s zhu oscillation theorems for self adjoint matrix hamiltonian systems j differential equations 190 2003 306 329 and s chen and z zheng shaozhu chen zhaowen zheng oscillation criteria of yan type for linear hamiltonian systems comput math appl 46 2003 855 862 these criteria also unify many of known criteria in literature and simplify the proofs

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

super connectivity and super edge connectivity for some interconnection networks let g v e be a k regular graph with connectivity k and edge connectivity l g is maximum connected if k k and g is maximum edge connected if l k moreover g is super connected if it is a complete graph or it is maximum connected and every minimum vertex cut is x v x e for some vertex email protected v and g is super edge connected if it is maximum edge connected and every minimum edge disconnecting set is v x v x e for some vertex email protected v in this paper we present three schemes for constructing graphs that are super connected and super edge connected applying these construction schemes we can easily discuss the super connected property and the super edge connected property of hypercubes twisted cubes crossed cubes mobius cubes split stars and recursive circulant graphs

oscillation criteria for second order linear matrix differential systems with damping using a generalization of sturm s comparison theorem some new oscillation criteria are established for the matrix differential system with damping p t y r t y q t y 0 under the hypothesis p t p t 0 q t q t y t are nxn matrices of real valued continuous functions on the interval t 0 and r t r t c 1 t 0 r n 2 our results are sharper than some previous results

oscillation theorems for linear matrix hamiltonian systems using a matrix linear transformation which preserves oscillatory property some new oscillation criteria for the following linear matrix hamiltonian system u a t t i u b t v v c t u t i a t v t t 0 are obtained which improve and generalize some recent results in literature

oscillation of linear hamiltonian systems abstract in this paper sufficient conditions have been obtained for the oscillation of a class of linear hamiltonian systems