conditional diagnosability of optical multi mesh hypercube networks under the comparison diagnosis model abstract due to integrated positive features of both hypercubes and tori optical multi mesh hypercube ommh networks are regarded as a class of promising optical interconnection topologies the notion of conditional diagnosability helps enhance the self diagnosing capability of multicomputers this paper determines the conditional diagnosabilities of ommh networks under the maeng malek comparison model

existence of solitary waves and periodic waves for a perturbed generalized bbm equation abstract the existence of solitary waves and periodic waves for a perturbed generalized bbm equation is established by using geometric singular perturbation theory attention goes to perturbations of the hamiltonian vector field with an elliptic hamiltonian of degree four exhibiting a cuspidal loop it is proven that the wave speed c 0 h is decreasing on h 0 1 12 by analyzing the ratio of abelian integrals the upper and lower bounds of the limit wave speed are given moreover the relation between the wave speed and the wavelength of traveling waves is obtained

on the number of limit cycles in small perturbations of a class of hyper elliptic hamiltonian systems abstract this paper deals with small perturbations of a class of hyper elliptic hamiltonian systems of degree 6 which is a lienard system of the form x y y q 1 x e y q 2 x with q 1 and q 2 polynomials of degree 5 and 4 respectively it is shown that this system can undergo degenerate hopf bifurcation and poincare bifurcation from which at most three limit cycles can emerge in the plane for e sufficiently small and the limit cycles can only surround an equilibrium inside i e the system can have the configuration 3 0 0 of limit cycles for some values of the parameters where 3 0 0 stands for three limit cycles surrounding an equilibrium and no limit cycles surrounding any of the two or three equilibria

the g good neighbor conditional diagnosability of star graphs under the pmc and mm model diagnosability plays an important role in measuring the fault tolerance and reliability of interconnection networks the g good neighbor conditional faulty set is a special faulty set that every fault free vertex should have at least g fault free neighbors the g good neighbor conditional diagnosability is a metric that can give the maximum cardinality of g good neighbor conditional faulty set that the system is guaranteed to identify this paper shows that the g good neighbor conditional diagnosabilities of star graph sn under the pmc and mm model are ng g 1 1 for n4 and 0gn2 our study shows the g good neighbor conditional diagnosability of star graph is several times larger than the classical diagnosability of star graph

the smallest upper bound on the number of zeros of abelian integrals abstract in this paper we develop a new method to estimate the smallest upper bound on the number of isolated zeros of abelian integrals and give an algebraic criterion for the case of abelian integrals along energy level ovals of potential systems as applications of our main result we first obtain a criterion guaranteeing that two abelian integrals have the chebyshev property with accuracy one and give an example of hyperelliptic abelian integrals then we get a criterion that any nontrivial linear combination of three abelian integrals has at most two isolated zeros using this criterion we prove that the smallest upper bound is two for the number of isolated zeros of a non algebraic abelian integral along the oval y 2 1 2 1 x 2 ln x 1 2 h with 0 h which corresponds to the perturbation of a quadratic reversible system with two centers moreover we derive all possible configurations of limit cycles from the poincare bifurcation of this perturbation to our knowledge these problems can not be solved by other known approaches

an 0 n 2 5 fault identification algorithm for diagnosable systems consider a system composed of n independent processors each of which tests a subset of the others it is assumed that at most tp of these processors are permanently faulty and that the outcome of a test is reliable if and only if the processor which performed the test is fault free such a system is said to be tp diagnosable if given any complete collection of test results the set of faulty processors can be uniquely identified in this paper it is shown that tp diagnosable systems due to their robust interconnection structure possess heretofore unknown graph theoretic properties relative to vertex cover sets and maximum matchings an 0 n2 5 algorithm is given which exploits these properties to identify the set of faulty processors in a tp diagnosable system the algorithm is shown to be correct complete not based on any conjecture and superior to any other known fault identification algorithm for the general class of tp diagnosable systems

the inverse integrating factor and the poincare map this work is concerned with planar real analytic differential systems with an analytic inverse integrating factor defined in a neighborhood of a regular orbit we show that the inverse integrating factor defines an ordinary differential equation for the transition map along the orbit when the regular orbit is a limit cycle we can determine its associated poincare return map in terms of the inverse integrating factor in particular we show that the multiplicity of a limit cycle coincides with the vanishing multiplicity of an inverse integrating factor over it we also apply this result to study the homoclinic loop bifurcation we only consider homoclinic loops whose critical point is a hyperbolic saddle and whose poincare return map is not the identity a local analysis of the inverse integrating factor in a neighborhood of the saddle allows us to determine the cyclicity of this polycycle in terms of the vanishing multiplicity of an inverse integrating factor over it our result also applies in the particular case in which the saddle of the homoclinic loop is linearizable that is the case in which a bound for the cyclicity of this graphic cannot be determined through an algebraic method

on the number of limit cycles near a homoclinic loop with a nilpotent singular point abstract in this article we study the expansion of the first melnikov function appearing by perturbing an integrable and reversible system with a homoclinic loop passing through a nilpotent singular point and obtain formulas for computing the first coefficients of the expansion based on these coefficients we obtain a lower bound for the maximal number of limit cycles near the homoclinic loop moreover as an application of our main results we consider a type of integrable and reversible polynomial systems obtaining at least 3 4 or 5 limit cycles respectively

finite cyclicity of graphics with a nilpotent singularity of saddle or elliptic type abstract in this paper we prove finite cyclicity of several of the most generic graphics through a nilpotent point of saddle or elliptic type of codimension 3 inside c families of planar vector fields in some cases our results are independent of the exact codimension of the point and depend only on the fact that the nilpotent point has multiplicity 3 by blowing up the family of vector fields we obtain all the limit periodic sets we calculate two different types of dulac maps in the blown up family and develop a general method to prove that some regular transition maps have a nonzero higher derivative at a point the finite cyclicity theorems are derived by a generalized derivation division method introduced by roussarie

perturbations from an elliptic hamiltonian of degree four abstract the paper deals with lienard equations of the form x y y p x yq x with p and q polynomials of degree respectively 3 and 2 attention goes to perturbations of the hamiltonian vector field with an elliptic hamiltonian of degree 4 exhibiting a cuspidal loop it is proven that the least upper bound for the number of zeros of the related elliptic integral is four and this upper bound is a sharp one this permits to prove the existence of lienard equations of type 3 2 with at least four limit cycles the paper also contains a complete result on the respective number of small and large limit cycles

limit cycle bifurcations by perturbing a cuspidal loop in a hamiltonian system abstract in this paper we first study the analytical property of the first melnikov function for general hamiltonian systems exhibiting a cuspidal loop and obtain its expansion at the hamiltonian value corresponding to the loop then by using the first coefficients of the expansion we give some conditions for the perturbed system to have 4 5 or 6 limit cycles in a neighborhood of loop as an application of our main results we consider some polynomial lienard systems and find 4 5 or 6 limit cycles

generalized hopf bifurcation and its dual generalized homoclinic bifurcation we show the duality between the generalized hopf bifurcation ghb and the generalized homoclinic bifurcation ghb this duality is twofold 1 the poincare normal forms at a weak focus and at a weak saddle 2 the bifurcation diagrams of the ghb and the ghb since the ghb is well known the ghb may be considered as the heart of the article

strong rabin numbers of folded hypercubes the strong rabin number of a network w of connectivity k is the minimum l so that for any k 1 nodes s d1 d2 dk of w there exist k node disjoint paths from s to d1 d2 dk respectively whose maximal length is not greater than l where s d1 d2 dk and d1 d2 dk are not necessarily distinct in this paper we show that the strong rabin number of a k dimensional folded hypercube is k 2 1 where k 2 is the diameter of the k dimensional folded hypercube each node disjoint path we obtain has length not greater than the distance between the two end nodes plus two this paper solves an open problem raised by liaw and chang

rg conditional diagnosability a novel generalized measure of system level diagnosis abstract system level diagnosis has become an important diagnosis method for multiprocessor systems among all system level diagnosis measures diagnosability is relatively small the conditional diagnosability constraint that each vertex has at least one good neighbor is relatively conservative when the dimension is far greater than 1 and g good neighbor conditional diagnosability does not consider this restriction on faulty vertices therefore a thorough study of diagnosability under the condition that each vertex has at least g good neighbors is an appealing subject motivated by r g vertex connectivity in this paper we introduce a novel generalized system level diagnosis measure named r g conditional diagnosability which assumes that every processor has at least g good neighbors the popular conditional diagnosability is a special case of r g conditional diagnosability when g 1 then we determine that the r g conditional diagnosability of n dimensional hypercube q n under the preparata metze chien pmc model is 2 2 g n 2 g 2 2 g 1 1

quadratic lienard equations with quadratic damping beijing univ dept math beijing 100871 peoples r china beijing univ math inst beijing 100871 peoples r china dumortier f limburgs univ ctr univ campus b 3590 diepenbeek belgium

cyclicity of periodic annulus and hopf cyclicity in perturbing a hyper elliptic hamiltonian system with a degenerate heteroclinic loop abstract in this paper we study the cyclicity of periodic annulus and hopf cyclicity in perturbing a quintic hamiltonian system the undamped system is hyper elliptic non symmetric with a degenerate heteroclinic loop which connects a hyperbolic saddle to a nilpotent saddle we rigorously prove that the cyclicity is 3 for periodic annulus when the weak damping term has the same degree as that of the associated hamiltonian system this result provides a positive answer to the open question whether the annulus cyclicity is 3 or 4 when the smooth polynomial damping term has degree n first a transformation based on the involution of the hamiltonian is introduced and then we analyze the coefficients involved in the bifurcation function to show that the hopf cyclicity is 2 n 1 3 further for piecewise smooth polynomial damping with a switching manifold at the y axis we consider the damping terms to have degrees l and n respectively and prove that the hopf cyclicity of the origin is 3 l 2 n 4 3 3 n 2 l 4 3 when l n n l

the g good neighbor conditional diagnosability of twisted hypercubes under the pmc and mm model abstract connectivity and diagnosability are important parameters in measuring the fault tolerance and reliability of interconnection networks the g good neighbor conditional faulty set is a special faulty set that every fault free vertex should have at least g fault free neighbors the rg vertex connectivity of a connected graph g is the minimum cardinality of a g good neighbor conditional faulty set x v g such that g x is disconnected the g good neighbor conditional diagnosability is a metric that can give the maximum cardinality of g good neighbor conditional faulty set that the system is guaranteed to identify the twisted hypercube is a new variant of hypercubes with asymptotically optimal diameter introduced by x d zhu in this paper we first determine the rg vertex connectivity of twisted hypercubes then establish the g good neighbor conditional diagnosability of twisted hypercubes under the pmc model and mm model respectively

the generalized homoclinic bifurcation abstract in this article we study a family x of vector fields having at 0 a homoclinic loop with multiplicity n we give conditions for the displacement function to have the same zeros as all those of p x 0 1 x n 1 xn 1 xn in a positive neighborhood of x 0 where is continuous these conditions determine the versality of the family x in a neighborhood of the loop to obtain the result we use the concept of chebychev systems

limit cycle bifurcations near a 2 polycycle or double 2 polycycle of planar systems abstract in this paper we study the bifurcation problem of limit cycles for a near hamiltonian system near a 2 polycycle or double 2 polycycle we obtain new results on the number of limit cycles by using the method of stability changing of a homoclinic loop this method also provides a new way to find alien limit cycles especially we found some alien limit cycles of new type

the triangular pyramid routing and topological properties in this paper a new topology for multicomputer interconnection networks based on triangular mesh is proposed the new network referred to as the triangular pyramid or tripy for short has l levels of triangular mesh we study some basic important properties of the proposed network as well as introduce a routing algorithm for the tripy network based on the routing of triangular meshes we prove that this form of pyramidal network is hamiltonian hamiltonian connected and pancyclic we also prove that the proposed network is 6 colorable and conduct a brief comparison of the tripy and its traditional pyramid counterpart our results show that the proposed network has higher scalability connectivity and total network bandwidth while preserving the important properties of the traditional pyramid network

diagnosability of hypercubes and enhanced hypercubes under the comparison diagnosis model a sengupta and a dahbura 1992 discussed how to characterize a diagnosable system under the comparison diagnosis model proposed by j maeng and m malek 1981 and a polynomial algorithm was given to identify the faulty processors provided that the system s diagnosability is known however for a general system the determination of its diagnosability is not algorithmically easy this paper proves that for the important hypercube structured multiprocessor systems n cubes the diagnosability under the comparison model is n when n spl ges 5 the paper also studies the diagnosability of enhanced hypercube which is obtained by adding 2 sup n 1 more links to a regular hypercube of 2 sup n processors it is shown that the augmented communication ability among processors also increases the system s diagnosability under the comparison model we prove that the diagnosability is n 1 for an enhanced hypercube when n spl ges 6

conditional diagnosability of alternating group graphs let an be the alternating group of degree n with n 3 set s 1 2 i 1 i 2 3 i n the alternating group graph denoted by agn is defined as the cayley graph on an with respect to s jwo et al networks 23 1993 315 326 introduced alternating group graph agn as an interconnection network topology for computing systems conditional diagnosability a new measure of diagnosability introduced by lai et al ieee transactions on computers 54 2 2005 165 175 can better measure the diagnosability of regular interconnection networks this paper determines that under pmc model the conditional diagnosability of agn is 4 for n 4 and 6n 18 for each n 5

the loop quantities and bifurcations of homoclinic loops abstract the stability and bifurcations of a homoclinic loop for planar vector fields are closely related to the limit cycles for a homoclinic loop of a given planar vector field a sequence of quantities the homoclinic loop quantities were defined to study the stability and bifurcations of the loop among the sequence of the loop quantities the first nonzero one determines the stability of the homoclinic loop there are formulas for the first three and the fifth loop quantities in this paper we will establish the formula for the fourth loop quantity for both the single and double homoclinic loops as applications we present examples of planar polynomial vector fields which can have five or twelve limit cycles respectively in the case of a single or double homoclinic loop by using the method of stability switching

on conditional diagnosability of the folded hypercubes the n dimensional folded hypercube fq n a variation of the hypercube proposed by ahmed et al a el amawy s latifi properties and performance of folded hypercubes ieee transactions on parallel and distributed systems 2 3 1991 31 42 is an n 1 regular n 1 connected graph conditional diagnosability a new measure of diagnosability introduced by lai et al pao lien lai jimmy j m tan chien ping chuang lih hsing hsu conditional diagnosability measures for large multiprocessor systems ieee transactions on computers 54 2 2005 165 175 can better measure the diagnosability of regular interconnection networks this paper determines that under pmc model the conditional diagnosability of fq n t c fq n is 4n 3 when n 5 or n 8 t c fq 3 3 t c fq 4 7

conditional diagnosability of arrangement graphs under the pmc model abstract processor fault diagnosis has played an important role in measuring the reliability of a multiprocessor system and the diagnosabilities of many well known multiprocessor systems have been investigated the conditional diagnosability has been widely accepted as a measure of diagnosability by assuming an additional condition that any fault set can t contain all the neighbors of any node in a multiprocessor system this paper considers the conditional diagnosability of an n k arrangement graph a n k a flexible interconnection network model for multiprocessor systems under the classical pmc diagnostic model and determines that the conditional diagnosability of a n k k 2 n k 2 is 4 k 4 n k 3 which is about four times its traditional diagnosability

on fault identification in diagnosable systems this paper begins by giving a characterization of t1 t1 diagnosable systems then a class of t0 diagnosable systems denoted by d n t0 x is considered it is shown for any member of this class that 1 necessary and sufficient conditions for t1 t1 diagnosability are greatly simplified 2 optimal diagnosis algorithms of time complexity 0 nt0 exist and most importantly 3 given the test results any set f of faults with f t1 can be identified to within a set f with f f and f min t1 f 1

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

the g good neighbor diagnosability of n k star graphs many large scale multiprocessor or multi computer systems take interconnection networks as underlying topologies fault diagnosis is especially important to identify fault tolerability of such systems the g good neighbor conditional diagnosability such that every fault free node has at least g fault free neighbors is a novel measure of diagnosability in this paper we show that the g good neighbor diagnosability of the n k star graph s n k under the pmc model 2 k n 1 and 1 g n k and the comparison model 2 k n 1 and 2 g n k is n g k 1 1 respectively in addition we derive that 1 good neighbor diagnosability of s n k under the comparison model is n k 2 for 3 k n 1 and n 4 as a supplement we also derive that the g good neighbor diagnosability of the n 1 star graph s n 1 1 g n 2 1 and n 4 under the pmc model and the comparison model is n 2 1 respectively we explore fault diagnosability of multiprocessor systems based on combinatorial network theory we establish the g good neighbor diagnosability of the n k star graph s n k under the pmc model we establish the g good neighbor diagnosability of the n k star graph s n k under the comparison model

on g good neighbor conditional diagnosability of n k star networks abstract the g good neighbor conditional diagnosability is a new measure for fault diagnosis of systems xu et al 2017 27 determined the g good neighbor conditional diagnosability of n k star networks s n k i e t g s n k with 1 k n 1 for 1 g n k under the pmc model and the mm model in this paper we determine t g s n k for all the remaining cases with 1 k n 1 for 1 g n 1 under the two models from which we can obtain the g good neighbor conditional diagnosability of the star graph obtained by li et al 2017 16 for 1 g n 2

limit cycle bifurcations near a double homoclinic loop with a nilpotent saddle of order m abstract in this paper we study the explicit expansion of the first order melnikov function near a double homoclinic loop passing through a nilpotent saddle of order m in a near hamiltonian system for any positive integer m m 1 we derive the formulas of the coefficients in the expansion which can be used to study the limit cycle bifurcations for near hamiltonian systems in particular for m 2 we use the coefficients to consider the limit cycle bifurcations of general near hamiltonian systems and give the existence conditions for 10 11 13 15 and 16 11 13 and 16 respectively limit cycles in the case that the homoclinic loop is of cuspidal type smooth type respectively and their distributions as an application we consider a near hamiltonian system with a nilpotent saddle of order 2 and obtain the lower bounds of the maximal number of limit cycles

on reliability of the folded hypercubes in this paper we explore the 2 extra connectivity and 2 extra edge connectivity of the folded hypercube fq n we show that k 2 fq n 3n 2 for n 8 and l 2 fq n 3n 1 for n 5 that is for n 8 resp n 5 at least 3n 2 vertices resp 3n 1 edges of fq n are removed to get a disconnected graph that contains no isolated vertices resp edges when the folded hypercube is used to model the topological structure of a large scale parallel processing system these results can provide more accurate measurements for reliability and fault tolerance of the system

pp graphics with a nilpotent elliptic singularity in quadratic systems and hilbert s 16th problem abstract this paper is part of the program launched in j differential equations 110 1 1994 86 to prove the finiteness part of hilbert s 16th problem for quadratic system which consists in proving that 121 graphics have finite cyclicity among quadratic systems we show that any pp graphic through a multiplicity 3 nilpotent singularity of elliptic type which does not surround a center has finite cyclicity such graphics may have additional saddles and or saddle nodes altogether we show the finite cyclicity of 15 graphics of j differential equations 110 1 1994 86 in particular we prove the finite cyclicity of a pp graphic with an elliptic nilpotent singular point together with a hyperbolic saddle with hyperbolicity 1 which appears in generic 3 parameter families of vector fields and hence belongs to the zoo of kotova and stanzo concerning the hilbert 16th problem american mathematical society translation series 2 vol 165 american mathematical society providence ri 1995 pp 155 201

the 1 2 good neighbor conditional diagnosabilities of regular graphs abstract fault diagnosis of systems is an important area of study in the design and maintenance of multiprocessor systems in 2012 peng et al proposed a new measure for the fault diagnosis of systems namely g good neighbor conditional diagnosability which requires that any fault free vertex has at least g fault free neighbors in the system the g good neighbor conditional diagnosabilities of a graph g under the pmc model and the mm model are denoted by t g pmc g and t g mm g respectively in this paper we first determine that t g pmc g t g mm g if g 2 second we establish a general result on the 1 2 good neighbor conditional diagnosabilities of some regular graphs as applications the 1 2 good neighbor conditional diagnosabilities of bc graphs folded hypercubes and four classes of cayley graphs namely unicyclic transposition graphs wheel transposition graphs complete transposition graphs and tree transposition graphs are determined under the pmc model and the mm model in addition we determine the r2 connectivities of bc graphs and folded hypercubes

obstructions and hypersurface sections minimally elliptic singularities we study the obstruction space t 2 for minimally elliptic surface singularities we apply the main lemma of our previous paper 3 which relates t 2 to deformations of hypersurface sections to use this we classify general hypersurface sections of minimally elliptic singularities as in the rational singularity case there is a simple formula for the minimal number of generators for t 2 as a module over the local ring this number is in many cases e g for cusps of hilbert modular surfaces equal to the vector space dimension of t 2

perturbations of a class of hyper elliptic hamiltonian systems of degree seven with nilpotent singular points in this paper we study the number of isolated zeros of abelian integrals associated to system x y y x 3 x 2 1 2 under the perturbations of 0 1 x 2 2 x 4 3 x 6 y y where 0 1 and i r i 0 1 2 3 we find a special transformation such that four generating elements are reduced to special three ones and then the algebraic method can be used the bounds and sharp bounds are obtained

perturbation from an elliptic hamiltonian of degree four iv figure eight loop the paper deals with lienard equations of the form x y y p x yq x with p and q polynomials of degree respectively 3 and 2 attention goes to perturbations of the hamiltonian vector fields with an elliptic hamiltonian of degree four exhibiting a figure eight loop it is proved that the least upper bound of the number of zeros of the related elliptic integral is five and this is a sharp bound multiplicity taken into account moreover if restricting to the level curves inside a saddle loop or outside the figure eight loop the sharp upper bound is respectively two or four also the multiplicity of the zeros is at most four this is the last one in a series of papers on this subject the results of this paper together with j differential equations 176 2001 114 j differential equations 175 2001 209 j differential equations to be published largely finish the study of the cubic perturbations of the elliptic hamiltonians of degree four and presumably provide a complete description of the number and the possible configurations of limit cycles for cubic lienard equations with small quadratic damping as a special case we obtain a configuration of four limit cycles surrounding three singularities together with a small limit cycle which surrounds one of the singularities

perturbations from an elliptic hamiltonian of degree four i saddle loop and two saddle cycle abstract the paper deals with lienard equations of the form x y y p x yq x with p and q polynomials of degree respectively 3 and 2 attention goes to perturbations of the hamiltonian vector fields with an elliptic hamiltonian of degree 4 and especially to the study of the related elliptic integrals besides some general results the paper contains a complete treatment of the saddle loop case and the two saddle cycle case it is proven that the related elliptic integrals have at most two zeros respectively one zero the multiplicity taken into account the bifurcation diagram of the zeros is also obtained

on reliability of the folded hypercubes in terms of the extra edge connectivity for a graph g and a non negative integer g the g extra edge connectivity of g is the minimum cardinality of a set of edges in g if it exists whose deletion disconnects g and each remaining component will have at least g vertices the extra edge connectivity is an important parameters for the reliability evaluation of interconnection networks in this paper we explore g extra edge connectivity g fq n of the folded hypercube fq n for g n denote g by i 0 s 2 t i where t 0 log 2 g and t i log 2 g r 0 i 1 2 t r we show that g fq n g n 1 i 0 s t i 2 t i i 0 s 2 i 2 t i for n 6 this result generalizes the previous results by zhu et al 2007 for 3 fq n and by hsieh and tsai in press for 4 fq n and so on

complete bifurcation diagram and global phase portraits of lienard differential equations of degree four abstract li and llibre in j differential equations 252 2012 3142 3162 proved that a lienard system of degree four d x d t y a x b x 2 c x 3 x 4 d y d t x has at most one limit cycle moreover the limit cycle is stable and hyperbolic if it exists based on their works the aim of this paper is to give the complete bifurcation diagram and global phase portraits in the poincare disc of this system further first we analyze the equilibria at both finity and infinity then a necessary and sufficient condition for existence of separatrix loop is founded by the rotation property moreover a necessary and sufficient condition of the existence of limit cycles is also obtained finally we show that the complete bifurcation diagram includes one hopf bifurcation surface and one bifurcation surface of separatrix loop

relating diagnosability strong diagnosability and conditional diagnosability of strong networks an interconnection network s diagnosability is an important measure of its self diagnostic capability based on the classical notion of diagnosability strong diagnosability and conditional diagnosability were proposed later to better reflect the networks self diagnostic capability under more realistic assumptions in this paper we study a class of interconnection networks called strong networks which are n regular n 1 connected and with cn number no more than n 3 we build a relationship among the three diagnosability measures for strong networks under both pmc and rm mm ast models given a strong network g with diagnosability t we prove that g is strongly t diagnosable if and only if g s conditional diagnosability is greater than t a simple check can show that almost all well known regular interconnection networks are strong networks the significance of this paper s result is that it reveals an important relationship between strong and conditional diagnosabilities and the proof of strong diagnosability for many interconnection networks under rm mm ast or pmc model is not necessary if their conditional diagnosability can be shown to be strictly larger than their diagnosability

an approach to conditional diagnosability analysis under the pmc model and its application to torus networks a general technique is proposed for determining the conditional diagnosability of interconnection networks under the pmc model several graph invariants are involved in the approach such as the length of the shortest cycle the minimum number of neighbors p resp p over all p vertex subsets resp cycles and a variant of connectivity called the r super connectivity an n dimensional torus network is defined as a cartesian product of n cycles ck1 ckn where ckj is a cycle of length kj for 1 j n the proposed technique is applied to the two or higher dimensional torus networks and their conditional diagnosabilities are established completely the conditional diagnosability of every torus network g is equal to 4 g 1 excluding the three small ones c3 c3 c3 c4 and c4 c4 in addition p g as well as 4 g is derived for 2 p 4 and the r super connectivity is also derived for 1 r 3

a system with three limit cycles appearing in a hopf bifurcation and dying in a homoclinic bifurcation the cusp of order 4 abstract in this paper we give an example of a family of polynomial vector fields with three limit cycles appearing simultaneously on a hopf bifurcation h of order 3 and vanishing simultaneously in a homoclinic loop bifurcation hl of order 3 the region with three limit cycles is a topological 3 simplex the system is a generalization of bogdanov s system at the same time we give the bifurcation diagram for the universal unfolding of the cusp of order 4 this bifurcation diagram is a cone it contains a cone on the bifurcation diagram with the three limit cycles inside a 3 simplex region plus saddle node and cusp bifurcations of lower order

bifurcations of limit cycles for a quintic hamiltonian system with a double cuspidal loop in this work we consider the number of limit cycles that can bifurcate from periodic orbits located inside a double cuspidal loop of the quintic hamiltonian vector field x h y x x 3 x 2 1 y under small perturbations of the form e a bx 2 cx 4 y y where 0 e 1 and a b c are real constants using picard fuchs equations for related abelian integrals asymptotic expansion of these integrals about critical level curves of h and some geometric properties of the curves defined by ratios of two especial integrals we show that the least upper bound for the number of limit cycles appeared in this bifurcation is two

bifurcations of limit cycles from quintic hamiltonian systems with an eye figure loop abstract in this paper we consider lienard equations of the form x y y x 2 x 3 x 5 e x 2 x 4 y where 0 e 1 r 3 and is bounded we prove that the least upper bound for the number of zeros of the related abelian integrals i h h x 2 x 4 y d x is 2 taking into account their multiplicities for h 0 1 6 and this upper bound is a sharp one this implies that the number of limit cycles bifurcated from periodic orbits in the vicinity of the center of the unperturbed system for e 0 inside an eye figure loop is less than or equal to 2

a proof of wang kooij s conjectures for a cubic lienard system with a cusp in this paper the global dynamics of a cubic lienard system with a cusp is studied to follow wang and kooij 1992 13 who proved that the maximum number of limit cycles is 2 and stated two conjectures about the curves of the cuspidal loop bifurcation and the double limit cycle bifurcation we give positive answers to those two conjectures and further properties of those bifurcation curves such as monotonicity and smoothness finally associated with previous results we obtain the complete bifurcation diagram and all phase portraits and demonstrate some numerical examples

diagnosable evaluation of dcc linear congruential graphs under the pmc diagnostic model a dcc disjoint consecutive cycles linear congruential graph g f n consists of n nodes and is generated by a set of linear functions f with special properties it was proved that g f n is a 2t regular graph and has connectivity 2t where t f and 1 3 and 2

exact bound on the number of zeros of abelian integrals for two hyper elliptic hamiltonian systems of degree 4 abstract this paper concerns the exact bound on the number of zeros of abelian integrals associated with two hyper elliptic hamiltonian systems of degree 4 the upper and lower bounds for the two systems have been obtained in several previous works however the sharp bounds are still unknown in this paper we provide a proof to show that the exact bound is 3 for both systems the basic idea of our method is to bound the parameter space in r 3 to obtain two parameter sets which might yield maximal 4 zeros of abelian integrals corresponding to the two sets further we show that the existence of 4 zeros on the two sets is not possible and thus the sharp bound is 3

on the number of critical points of the period in the study of subharmonic bifurcations from a planar hamiltonian system it is often assumed that the period of a one parameter family of periodic orbits of the unperturbed system is a monotone function of the parameter see e g hale and taboas s chow and hale es however this hypothesis is in many instances difficult to verify for example it is well known that every orbit in the first quadrant of the vollterra lotka equation is periodic it is only in recent years that the monotonicity of the periods of these periodic orbits has been shown hsu 9 and waldvogel 151 such problems also occur in the study of bifurcation of steady state solutions of a reaction diffusion equation in one space variable the steady state solutions satisfy the equation u f u o uer 1 1 on some interval in 14 smoller and wasserman showed that if f u u u u u cc

edge fault tolerance analysis of a class of interconnection networks fault tolerant measures have played an important role in the reliability of an interconnection network edge connectivity restricted edge connectivity extra edge connectivity and super edge connectivity of many well known interconnection networks have been explored in this paper we study the 2 extra edge connectivity of a special class of graphs g g0 g1 m proposed by chen et al appl math comput 140 2003 245 254 then by showing that several well known interconnection networks such as hyper

conditional diagnosability and strong diagnosability of split star networks under the pmc model an interconnection network s diagnosability is an important measure of its self diagnostic capability the classical problems of fault diagnosis are explored widely the conditional diagnosability is proposed by lai et al as a new measure of diagnosability which can better measure the diagnosability of regular interconnection networks the conditional diagnosability is an important indicator of the robustness of a multiprocessor system in presence of failed processors furthermore a multiprocessor system is strongly t diagnos able if it is t diagnosable and can achieve diagnosability t 1 except for the case where a node s neighbors are all faulty the conditional diagnosability and strong diagnosability were proposed later to better reflect the networks self diagnostic capability under more realistic assumptions in this paper we determine the conditional diagnosability of an n dimensional split star network denoted as s n 2 a well known interconnection network model for multiprocessor systems under the pmc preparata metze and chien model we show that the conditional diagnosability of s n 2 n 4 is 8 n 23 which is about four times of its traditional diagnosability as a byproduct the strong diagnosability of s n 2 is also obtained we investigate the combinatorial properties and fault tolerant properties of the split star network we show that as long as a fault set f has 6 n 17 or fewer nodes s n 2 f excluding a subset of at most two nodes contains one large connected component we show that as long as a fault set f has 8 n 25 or fewer nodes s n 2 f excluding a subset of at most three nodes contains one large connected component we show that the classic diagnosability of s n 2 n 2 under the pmc model is 2 n 3 we also establish that the conditional diagnosability and the strong diagnosability of s n 2 are 8 n 23 and 2 n 3 under the pmc model

on conditional fault tolerance and diagnosability of hierarchical cubic networks fault tolerance is especially important for interconnection networks since the growing size of networks increases their vulnerability to component failures a classical measure for the fault tolerance of a network in the case of vertex failures is its connectivity given a network based on a graph g and a positive integer h the r h connectivity of g is the minimum cardinality of a set of vertices in g if any whose deletion disconnects g and the minimum degree of every connected component is at least h this paper investigates the r h connectivity h 1 2 of the hierarchical cubic network hcn n n 2 and shows that 1 hcn n 2 n 2 hcn n 4 n 4 respectively furthermore the paper establishes the conditional diagnosability of hcn n under the pmc diagnostic model

on the stability of double homoclinic and heteroclinic cycles in this paper we give a criterion for the stability of planar double homoclinic and heteroclinic cycles with one or two saddles in some degenerate case

adaptive system level diagnosis for hypercube multiprocessors using a comparison model fault detection and localization are important in multiprocessor computing systems several methods of system self diagnosis have been developed including the model proposed by preparata metze and chien known as the pmc model and a comparison model proposed by maeng and malek the mm model adaptive diagnosis is a practical system level diagnostic scheme whose main design objectives are to reduce the number of test rounds and the total number of tests a number of interesting research results have been presented for adaptive diagnosis using the classical pmc model however adaptive diagnosis using the mm model has been discussed only in relation to completely connected systems and torus systems in this paper we consider the problem of adaptive fault diagnosis in an n dimensional hypercube using the mm model the hypercube is an important interconnection network and an n dimensional hypercube has been shown to be n diagnosable using the mm model the goal is to correctly identify the status of all processors assuming that the number of faulty vertices does not exceed the hypercube dimension for a cycle of n vertices we show that at least four test rounds are necessary for complete diagnosis if n is not a multiple of three we propose a scheme that completely diagnoses a 4 dimensional hypercube in at most seven test rounds and an n dimensional hypercube for n 5 in at most six test rounds and at most n 2n 3 8n 2 tests where n 2 n

a general technique to establish the asymptotic conditional diagnosability of interconnection networks we develop a general and demonstrably widely applicable technique for determining the asymptotic conditional diagnosability of interconnection networks prevalent within parallel computing under the comparison diagnosis model we apply our technique to replicate yet extend existing results for hypercubes and k ary n cubes before going on to obtain new results as regards folded hypercubes pancake graphs and augmented cubes in particular we show that the asymptotic conditional diagnosability of folded hypercubes fq n is 3n 2 pancake graphs p n is 3n 7 and augmented cubes aq n is 6n 17 we demonstrate how our technique is independent of structural properties of the interconnection network g in question and essentially only dependent upon the minimal size of the neighbourhood of a path of length 2 in g the number of neighbours any two distinct vertices of g have in common and the minimal degree of any vertex in g

saddle quantities and applications abstract in this paper we make the connection between the theoretical study of the generalized homoclinic loop bifurcation ghb and the practical computational aspects for this purpose we first compare the dulac normal form with the joyal normal form these forms were both used to prove the ghb theorem but the second one is far more practical from the algorithmic point of view we then show that the information carried by these normal forms can be computed in a much simpler way using what we shall call dual lyapunov constants the coefficients of a normal form or the dual lyapunov quantities are particular cases of what we shall call saddle quantities we calculate the saddle quantities for quadratic systems and we show that no more than three limit cycles can appear in a homoclinic loop bifurcation we also study the homoclinic loop bifurcation of order 5 appearing in a 6 parameter family close to a hamiltonian system to our knowledge this is the first time that one can find a complete description of a ghb of such high order finally we calculate the saddle quantities for a symmetric cubic vector field and we deduce a bound for the number of limit cycles that appear in a ghb

a criterion for determining the monotonicity of the ratio of two abelian integrals abstract a new method to determine the monotonicity of the ratio of two abelian integrals is given we find two criterion functions defined directly by the functions which appear in the abelian integrals the monotonicities of the criterion functions imply the monotonicity of the ratio of the abelian integrals by using this criterion several known results can be deduced easier and some new results are obtained

conditional diagnosability of matching composition networks under the mm model diagnosability is a critical metric for determining the reliability of a multiprocessor system in 2005 lai et al proposed a new measure for the fault diagnosis of a system i e conditional diagnosability in which it is assumed that at least one of the neighbors of an arbitrary node in the system is not faulty in this paper we obtain a sufficient condition for a class of networks called matching composition networks mcns which are conditionally t diagnosable under the mm model then we apply the sufficient condition to show the conditional diagnosability of bijective connection bc networks finally we show that the sufficient condition can be applied to networks other than bc networks

hilbert s 16th problem for quadratic vector fields abstract we discuss here a systematic approach towards a positive answer to hilbert s 16th problem for quadratic systems namely the existence of a uniform bound for the number of limit cycles of a quadratic system the method is the following describe the limit periodic sets surrounding the origin in a family of quadratic vector fields and prove that they have finite cyclicity in this paper we give the list of all graphics and degenerate graphics that should be considered and describe their general features we also indicate how to find or where to find concrete examples of these limit periodic sets

constructing one to many disjoint paths in folded hypercubes routing functions have been shown effective in deriving disjoint paths in the hypercube in this paper using a minimal routing function k 1 disjoint paths from one node to another k 1 distinct nodes are constructed in a k dimensional folded hypercube whose maximal length is not greater than the diameter plus one which is minimum in the worst case for the general case the maximal length is nearly optimal spl les the maximal distance between the two end nodes of these k 1 paths plus two as a by product the rabin number of the folded hypercube is obtained which is an open problem raised by s c liaw and g j chang 1999

an explicit expression of the first liapunov and period constants with applications in this paper we study systems in the plane having a critical point with pure imaginary eigenvalues and we search for effective conditions to discern whether this critical point is a focus or a center in the case of it being a center we look for additional conditions in order to be isochronous we stress that the essential differences between the techniques used in this work and the more usual ones are basically two the elimination of the integration constants when we consider primitives of functions see also remark 3 2 and the fact that we maintain the complex notation in the whole study thanks to these aspects we reach with relative ease an expression of the first three liapunov constants v3 v5 andv7 and of the first two period ones p2andp4 for a general system as far as we know this is the first time that a general and compact expression ofv7has been given moreover the use of a computer algebra system is only needed in the computation ofv7andp4 these results are applied to give a classification of centers and isochronous centers for certain families of differential equations

the cyclicity of quadratic reversible systems with a center of genus one and non morsean point in this paper we investigate a quadratic reversible system with non morsean point it is proved that the cyclicity of the period annulus under quadratic perturbations is less or equal to four

conditional matching preclusion sets abstract the matching preclusion number of a graph is the minimum number of edges whose deletion results in a graph that has neither perfect matchings nor almost perfect matchings for many interconnection networks the optimal sets are precisely those induced by a single vertex in this paper we look for obstruction sets beyond these sets we introduce the conditional matching preclusion number of a graph it is the minimum number of edges whose deletion results in a graph with no isolated vertices that has neither perfect matchings nor almost perfect matchings we find this number and classify all optimal sets for several basic classes of graphs