optimal multiplicity in local bifurcation theory i generalized generic eigenvalues abstract in this paper we introduce a class of eigenvalues for a family of operators depending on a real parameter for this class of eigenvalues we define a multiplicity concept which is the best in local bifurcation theory in part ii we shall reduce the general situation to the one considered here

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

positive solutions of the dirichlet problem for the prescribed mean curvature equation abstract we discuss existence and multiplicity of positive solutions of the prescribed mean curvature problem div u 1 u 2 f x u in u 0 on in a general bounded domain r n depending on the behavior at zero or at infinity of f x s or of its potential f x s 0 s f x t d t our main effort here is to describe in a way as exhaustive as possible all configurations of the limits of f x s s 2 at zero and of f x s s at infinity which yield the existence of one two three or infinitely many positive solutions either strong or weak or bounded variation solutions are considered our approach is variational and combines critical point theory the lower and upper solutions method and elliptic regularization

double grow up and global convergence of solutions for a parabolic prescribed mean curvature problem abstract the paper is concerned with a parabolic mean curvature type problem with a varying parameter we study large time behavior of global solutions for different ranges of and initial data and present some new asymptotic results about global convergence and infinite time blow up in particular it is shown that for suitable ranges of parameter and initial data there exists a double grow up phenomenon the solution itself blows up at every interior point and its gradient blows up at the boundary of the domain as t we also establish an interesting connection between global convergence and the non classical solution of the associated stationary problem if initial data are smaller than the non classical solution then the solutions must decay to zero in c 1 norm as t

constructing the nearly shortest path in crossed cubes a graph g is panconnected if each pair of distinct vertices u v v g are joined by a path of length l for all d g u v

positive solutions of a one dimensional indefinite capillarity type problem a variational approach abstract we prove the existence and the multiplicity of positive solutions of the one dimensional capillarity type problem u 1 u 2 a x f u u 0 0 u 1 0 where a l 1 0 1 changes sign and f 0 0 is continuous and has a power like behavior at the origin and at infinity our approach is variational and relies on a regularization procedure that yields bounded variation solutions which are of class w loc 2 1 and hence satisfy the equation pointwise almost everywhere on each open interval where the weight function a has a constant sign

embedding hamiltonian paths in augmented cubes with a required vertex in a fixed position it is proved that there exists a path p l x y of length l if d a q n x y l 2 n 1 between any two distinct vertices x and y of aq n obviously we expect that such a path p l x y can be further extended by including the vertices not in p l x y into a hamiltonian path from x to a fixed vertex z or a hamiltonian cycle in this paper we prove that there exists a hamiltonian path r x y z l from x to z such that d r x y z l x y l for any three distinct vertices x y and z of aq n with n 2 and for any d a q n x y l 2 n 1 d a q n y z furthermore there exists a hamiltonian cycle s x y l such that d s x y l x y l for any two distinct vertices x and y and for any d a q n x y l 2 n 1

subsystems time scales and multimodeling through a couple of naive examples the control theorists are invited to re examine the role of modeling in the study of large scale dynamic systems instead of assuming the existence of n diagonally dominant blocks it is possible to justify one strongly coupled slow core and n weakly coupled fast subsystems this structure is exhibited with a physically meaningful choice of state variables the controls are introduced following the recent concept of multimodeling

classical and non classical solutions of a prescribed curvature equation we discuss existence and multiplicity of positive solutions of the one dimensional prescribed curvature problem u root 1 u 2 lambda f t u u 0 0 u 1 0 depending on the behaviour at the origin and at infinity of the potential integral u 0 f t s ds besides solutions in w 2 w 1 0 1 we also consider solutions in w loc 2 1 0 1 which are possibly discontinuous at the endpoints of 0 1 our approach is essentially variational and is based on a regularization of the action functional associated with the curvature problem c 2007 elsevier inc all rights reserved

the panpositionable panconnectedness of augmented cubes a graph g is panconnected if for any two distinct vertices x and y of g it contains an x y path of length l for each integer l satisfying d g x y

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

infinite time gradient blow up for parabolic prescribed mean curvature equations abstract we prove some results on the existence of infinite time gradient blow up phenomena for parabolic prescribed mean curvature equations over bounded mean convex domains in r n

a quasilinear parabolic perturbation of the linear heat equation abstract this paper studies a quasilinear perturbation through the mean curvature flow operator of the classical linear heat equation the mean curvature has the effect of maintaining bounded all classical positive steady states of the model

global structure of radial positive solutions for a prescribed mean curvature problem in a ball abstract in this paper we are concerned with the global structure of radial positive solutions of boundary value problem div n v f x v 0 in b r v 0 on b r where n y y 1 y 2 y r n is a positive parameter b r x r n x r and denotes the euclidean norm in r n all results depending on the behavior of nonlinear term f near 0 are obtained by using global bifurcation techniques

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

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 bipancycle connectivity of the hypercube paths and cycles are popular interconnection networks due to their simplicity and low degrees and therefore an efficient way of embedding them into an interconnection network is of particular importance many path embedding aspects and cycle embedding aspects such as hamiltonicity hamiltonian connectivity panconnectivity bipanconnectivity pancyclicity vertex pancyclicity and edge pancyclicity have been proposed and extensively studied in fact these discussions are meaningful for interconnection networks of resource allocated systems or heterogeneous computing systems in this paper we propose a new cycle embedding aspect called bipancycle connectivity that combines the concepts of vertex pancyclicity and bipanconnectivity a bipartite graph is bipancycle connected if an arbitrary pair of vertices x y is contained by common cycles called bipanconnected cycles that include a cycle of every even length ranging from minc x y to n where minc x y denotes the length of the minimum cycle that contains x and y and n is the number of vertices in this paper we introduce a new graph called the cycle of ladders we show that the hypercube is bipancycle connected by presenting algorithms to embed the cycle of ladders into the hypercube

bifurcation of positive solutions for a one dimensional indefinite quasilinear neumann problem abstract we study the structure of the set of the positive regular solutions of the one dimensional quasilinear neumann problem involving the curvature operator u 1 u 2 a x f u u 0 0 u 1 0 here r is a parameter a l 1 0 1 changes sign and f c r we focus on the case where the slope of f at 0 f 0 is finite and non zero and the potential of f is superlinear at infinity but also the two limiting cases where f 0 0 or f 0 are discussed we investigate in some special configurations the possible development of singularities and the corresponding appearance in this problem of bounded variation solutions

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

asymptotic stability of nonlinear multiparameter singularly perturbed systems sufficient conditions are derived to guarantee the asymptotic stability of a class of nonlinear singularly perturbed systems with several perturbation parameters of the same order estimates of the region of attraction and bounds on the small parameters are obtained

existence and regularity of extremal solutions for a mean curvature equation we study a class of mean curvature equations mu h up where m denotes the mean curvature operator and for p 1 we show that there exists an extremal parameter such that this equation admits a minimal weak solutions for all 0 while no weak solutions exists for weak solutions will be defined as critical points of a suitable functional in the radially symmetric case we then show that minimal weak solutions are classical solutions for all 0 and that another branch of classical solutions exists in a neighborhood of

complete path embeddings in crossed cubes crossed cubes are popular variants of hypercubes in this paper we study path embeddings between any two distinct nodes in crossed cubes we prove two important results in the n dimensional crossed cube a for any two nodes all paths whose lengths are greater than or equal to the distance between the two nodes plus 2 can be embedded between the two nodes with dilation 1 b for any two integers n 2 and l with 1

singular perturbations with origin eigenvalues abstract the aim of this study is to apply singular perturbation theory to systems with origin eigenvalues if the reduced system has origin eigenvalues with jordan block sizes equal to one then this guarantees the stability of the original system however if these sizes are greater than one then the reduced system does not give any information about the stability of the original system so another reduced system is obtained using the second derivative of the fast part and it is shown that under some specified conditions stability or instability of this system implies the stability or instability of the original system

the maximum principle and the existence of principal eigenvalues for some linear weighted boundary value problems abstract in this work we deal with the problem of the existence and uniqueness of principal eigenvalues for some linear weighted boundary value problems associated to a general second order uniformly elliptic operator for a large class of sign definited weights we characterize whether the boundary value problem admits a principal eigenvalue or not

embedding binary trees into crossed cubes the recently introduced interconnection network crossed cube has attracted much attention in the parallel processing area due to its many attractive features like the ordinary hypercube the n dimensional crossed cube is a regular graph with 2 sup n vertices and n2 sup n 1 edges the diameter of the crossed cube is approximately half that of the ordinary hypercube these advantages of the crossed cube motivated the study of how well it can simulate other networks such as the complete binary tree we show that the 2 sup n 1 node complete binary tree can be embedded into the n dimensional crossed cube with dilation 1

embedding of cycles in arrangement graphs arrangement graphs have been proposed as an attractive interconnection topology for large multiprocessor systems the authors study these graphs by proving the existence of hamiltonian cycles in any arrangement graph they also prove that an arrangement graph contains cycles of all lengths ranging between 3 and the size of the graph they show that an arrangement graph can be decomposed into node disjoint cycles in many different ways

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

bounded components of positive solutions of abstract fixed point equations mushrooms loops and isolas abstract in this work a general class of nonlinear abstract equations satisfying a generalized strong maximum principle is considered in order to study the behavior of the bounded components of positive solutions bifurcating from the curve of trivial states u 0 at a nonlinear eigenvalue 0 with geometric multiplicity one since the unilateral theorems of rabinowitz j funct anal 7 1971 487 theorems 1 27 and 1 40 are not true as originally stated cf the very recent counterexample of dancer bull london math soc 34 2002 533 in order to get our main results the unilateral theorem of lopez gomez spectral theory and nonlinear functional analysis research notes in mathematics vol 426 crc press boca raton fl 2001 theorem 6 4 3 is required our analysis fills some serious gaps existing is some published papers that were provoked by a direct use of rabinowitz s unilateral theory actually the abstract theory developed in this paper cannot be covered with the pioneering results of rabinowitz 1971 since in rabinowitz s context any component of positive solutions must be unbounded by a celebrated result attributable to dancer arch rational mech anal 52 1973 181

two spanning disjoint paths with required length in generalized hypercubes given two pairs and of vertices of a graph g v e and two integers l 1 and l 2 with l 1 l 2 v g 2 g is said to be satisfying the 2rp property if there exist two disjoint paths p 1 and p 2 such that 1 p 1 is a path joining u to v with l p 1 l 1 2 p 2 is a path joining x to y with l p 2 l 2 and 3 p email protected p 2 spans g where l p denotes the length of path p in this paper we show that an r dimensional generalized hypercube denoted by g m r m r 1 m 1 satisfies the 2rp property except some special conditions where m i 4 for all 1

a weighted quasilinear equation related to the mean curvature operator abstract this paper studies a weighted quasilinear perturbation through the mean curvature flow operator of the classical linear heat equation the mean curvature has the effect of maintaining bounded all classical positive steady states of the model though their derivatives must be somewhere unbounded the dynamics of the positive solutions of the model is fully described

properties of the principal eigenvalues of a general class of non classical mixed boundary value problems in this paper we characterize the existence of principal eigenvalues for a general class of linear weighted second order elliptic boundary value problems subject to a very general class of mixed boundary conditions our theory is a substantial extension of the classical theory by p hess and t kato 1980 comm partial differential equations5 999 1030 in obtaining our main results we must give a number of new results on the continuous dependence of the principal eigenvalue of a second order linear elliptic boundary value problem with respect to the underlying domain and the boundary condition itself these auxiliary results complement and in some sense complete the theory of d daners and e n dancer 1997 j differential equations138 86 132 the main technical tool used throughout this paper is a very recent characterization of the strong maximum principle in terms of the existence of a positive strict supersolution due to h amann and j lopez gomez 1998 j differential equations146 336 374

singular perturbations and time scale methods in control theory survey 1976 1983 recent progress in the use of singular perturbation and two time scale methods of modeling and design for control systems is reviewed over 350 references are organized into major problem areas representative issues and results are discussed with a view to outlining research directions and indicating potential areas of application the survey is aimed at engineers and applied mathematicians interested in model order reduction separation of time scales and allied simplified methods of control system analysis and design the exposition does not assume prior knowledge of singular perturbation methods