oscillation of neutral differential equation with positive and negative coefficients abstract in this paper we provide oscillation properties of every solution of the neutral differential equation with positive and negative coefficients x t r t x t r p t x t q t x t 0 where r t p t q t c t 0 r r 0 0 0

existence of nonoscillatory solutions of first order linear neutral delay differential equations in this paper the existence of nonoscillatory solutions of the first order neutral delay differential equations with variable coefficients and delays are studied some new sufficient conditions are given in particular conditions given in this paper are weaker than those known so the results in this paper have wider application than the existing ones

stability of runge kutta methods in the numerical solution of equation u t au t a 0 u t this paper is concerned with the stability analysis of the runge kutta methods for the equation u t au t a0u t the stability regions for the runge kutta methods are determined the conditions that the analytic stability region is contained in the numerical stability region are obtained and some numerical experiments are given

approximation of minimum weight spanners for sparse graphs a t spanner of a graph g is its spanning subgraph s such that the distance between every pair of vertices in s is at most t times their distance in g the sparsestt spanner problem asks to find for a given graph g and an integer t a t spanner of g with the minimum number of edges the problem is known to be np hard for all t 2 and even more it is np hard to approximate it with ratio o logn for every t 2 for t 5 the problem remains np hard for planar graphs and the approximability status of the problem on planar graphs was open we resolve this open issue by showing that the sparsestt spanner problem admits the efficient polynomial time approximation scheme eptas for every t 1 our result holds for a much wider class of graphs namely the class of apex minor free graphs which contains the classes of planar and bounded genus graphs moreover it is possible to extend our results to weighted apex minor free graphs when the maximum edge weight is bounded by some constant

positive solutions of p type retarded functional differential equations abstract for systems of retarded functional differential equations with unbounded delay and with finite memory sufficient and necessary conditions of existence of positive solutions on an interval of the form t 0 are derived a general criterion is given together with corresponding applications including a linear case too examples are inserted to illustrate the results

np hardness and fixed parameter tractability of the minimum spanner problem abstract for a positive integer t a t spanner of a graph g is a spanning subgraph in which the distance between every pair of vertices is at most t times of their distance in g in this paper we consider the problem of finding a t spanner with minimum number of edges in a given graph which we call minimum t spanner problem for t 2 minimum t spanner problem is known to be np hard in general graphs when the input graph is planar it is shown by brandes and handke in 1997 that this problem is np hard for t 5 since then the case of t 2 3 4 has been open for more than two decades the main contribution of this paper is to settle this open problem by showing the np hardness of minimum t spanner problem in planar graphs for t 2 3 4 as a byproduct we show the np hardness of the problem on degree bounded graphs which improves previously known degree bounds we also present a fixed parameter algorithm for this problem in which the number of removed edges is regarded as a parameter

the sparsest additive spanner via multiple weighted bfs trees abstract spanners are fundamental graph structures that sparsify graphs at the cost of small stretch in particular in recent years many sequential algorithms constructing additive all pairs spanners were designed providing very sparse small stretch subgraphs remarkably it was then shown that the known 6 spanner constructions are essentially the sparsest possible that is larger additive stretch cannot guarantee a sparser spanner which brought the stretch sparsity trade off to its limit distributed constructions of spanners are also abundant however for additive spanners while there were algorithms constructing 2 and 4 all pairs spanners the sparsest case of 6 spanners remained elusive we remedy this by designing a new sequential algorithm for constructing a 6 spanner with the essentially optimal sparsity of o n 4 3 edges we then show a distributed implementation of our algorithm answering an open problem in 12 a main ingredient in our distributed algorithm is an efficient construction of multiple weighted bfs trees a weighted bfs tree is a bfs tree in a weighted graph that consists of the lightest among all shortest paths from the root to each node we present a distributed algorithm in the congest model that constructs multiple weighted bfs trees in s d 1 rounds where s is the set of sources and d is the diameter of the network graph

controller and estimator for dynamic networks afek awerbuch plotkin and saks identified an important fundamental problem inherent to distributed networks which they called the resource controller problem consider first the problem in which one node called the root is required to estimate the number of events that occurred all over the network this counting problem can be viewed as a useful variant of the heavily studied and used task of topology update that deals with collecting all remote information the resource controller problem generalizes the counting problem such remote events are considered as requests and the counting node i e the root also issues permits for the requests that way the number of requests granted can be controlled bounded an efficient resource controller was constructed in the paper by afek et al and it can operate on a dynamic network assuming that the network is spanned by a tree that may only grow and only by allowing leaves to join the tree in contrast the resource controller presented here can operate under a more general dynamic model allowing the spanning tree of the network to undergo both insertions and deletions of both leaves and internal nodes despite the more dynamic network model we allow the message complexity of our controller is never more than the message complexity of the more restricted controller all the applications for the controller of afek et al can be used also with our controller moreover with the same message complexity our controller can handle these applications under the more general dynamic model mentioned above in particular the new controller can be transformed into an efficient size estimation protocol i e a protocol allowing all nodes to maintain a constant factor estimation of the number of nodes in the dynamically changing network informally the resulting new size estimation protocol uses o log 2n amortized message complexity per topological change assuming that the number of changes in the network size is not too small where n is the current number of nodes in the network an application of the size estimation of afek et al was to solve agreement in the case of initial faults fischer lynch and paterson and leader election under initial faults bar yehuda and kutten hence the controllers in this paper can be useful for these applications too

linearized oscillation of first order nonlinear neutral delay differential equations abstract in this paper we establish a equivalence of oscillation for the first order nonlinear neutral delay differential equation x t px t i 1 m q i f i x t i 0 and its corresponding linear equation x t px t i 1 m q i x t i 0 under a weaker condition on fi

generalized exponents of primitive two colored digraphs abstract a l colored digraph d l is primitive if there exists a nonnegative integer vector such that for each ordered pair of vertices x and y not necessarily distinct there exists an walk in d l from x to y the exponent of the primitive l colored digraph d l is defined to be the minimum value of the sum of all coordinates of taken over all such in this paper we generalize the concept of exponent of a primitive l colored digraph by introducing three types of generalized exponents further we study the generalized exponents of primitive two colored wielandt digraphs

stability and asymptotic properties of a system of functional differential equations with nonconstant delays the asymptotic properties of a real two dimensional differential system x t a t x t k 1 mb k t x q k t h t x t x q 1 t x q m t with unbounded nonconstant delays t q k t 0 satisfying lim t q k t are studied here a b k and h are supposed to be matrix functions and a vector function the conditions for the stability and asymptotic stability of solutions and the conditions under which all solutions tend to zero are given the methods are based on the transformation of the considered real system to one equation with complex valued coefficients asymptotic properties are studied by means of a lyapunov krasovskii functional the results generalize some previous ones where the asymptotic properties for two dimensional systems with one or more constant delays or one nonconstant delay were studied

oscillation in first order neutral differential equations with integrally small coefficients abstract consider the neutral differential equation with positive and negative coefficients d dt x t r t x t r p t x t q t x t 0 where p q r c t0 r r 0 and r we obtain some new sufficient conditions for the oscillation of solutions of the above equation without the restriction t 0 p s q s ds

a survey on interval routing we survey in this paper the classical results and also the most recent results in the field of interval routing a well known strategy to code in a compact way distributed routing algorithms these results are classified in several themes characterization compactness and shortest path dilation and stretch factor specific class of graphs interconnection networks bounded degree planar chordal rings random graphs etc and the other recent extensions proposed in the literature dead lock free congestion non deterministic and distributed problems related to interval routing for each of these themes we review the state of the art and propose several open problems

fault tolerant compact routing schemes for general graphs this paper considers compact fault tolerant routing schemes for weighted general graphs namely routing schemes that avoid a set of failed or forbidden edges we present a compact routing scheme capable of handling multiple edge failures assume a source node s contains a message m designated to a destination target t and assume a set f of edges crashes unknown to s our scheme routes the message to t provided that s and t are still connected in g f over a path whose length is proportional to the distance between s and t in g f to f 3 and to some poly log factor the routing table required at a node v is of size proportional to the degree of v in g and some poly log factor this improves on the previously known fault tolerant compact routing scheme for general graphs which was capable of overcoming at most 2 edge failures

delay differential equations with hill s type growth rate and linear harvesting for the equation email protected t r t n t 1 n t c b t n t a t n g t we obtain the following results boundedness of all positive solutions extinction and persistence conditions the proofs employ recent results in the theory of linear delay equations with positive and negative coefficients

convergence and asymptotic stability of galerkin methods for a partial differential equation with piecewise constant argument the paper deals with the convergence and asymptotic stability of galerkin methods for a partial differential equation with piecewise constant argument the optimal convergence orders are obtained for the semidiscrete and full discrete backward euler methods respectively both the discrete solutions are proved to be asymptotically stable under the condition that the analytical solution is asymptotically stable

distributed computing of efficient routing schemes in generalized chordal graphs efficient algorithms for computing routing tables should take advantage of particular properties arising in large scale networks two of them are of special interest low logarithmic diameter and high clustering coefficient high clustering coefficient implies the existence of few large induced cycles considering this fact we propose here a routing scheme that computes short routes in the class of k chordal graphs i e graphs with no induced cycles of length more than k in the class of k chordal graphs our routing scheme achieves an additive stretch of at most k 1 i e for all pairs of nodes the length of the route never exceeds their distance plus k 1 in order to compute the routing tables of any n node graph with diameter d we propose a distributed algorithm which uses o logn bit messages and takes o d time the corresponding routing scheme achieves the stretch of k 1 on k chordal graphs we then propose a routing scheme that achieves a better additive stretch of 1 in chordal graphs notice that chordal graphs are 3 chordal graphs in this case distributed computation of routing tables takes o min dd n time where d is the maximum degree of the graph our routing schemes use addresses of size logn bits and local memory of size 2 d 1 logn bits per node of degree d

multivalued fredholm type operators with abstract generalised inverses multivalued semi fredholm type operators with topologically complemented kernels and ranges are investigated in normed linear spaces it is shown that regularisers generalised inverses can be constructed for the classes given and that the operators considered can be characterised in terms of their regularisers continuity of the inverses is discussed and dual properties of adjoints are given

a new existence theory for positive periodic solutions to functional differential equations abstract this paper deals with a new existence theory for positive periodic solutions to a kind of nonautonomous functional differential equation by employing the fixed point theorem in cones applying the general theorems established to several biomathematical models the paper improves some previous results and obtains some new results

space efficient routing tables for almost all networks and the incompressibility method we use the incompressibility method based on kolmogorov complexity to determine the total number of bits of routing information for almost all network topologies in most models for routing for almost all labeled graphs theta n 2 bits are necessary and sufficient for shortest path routing by almost all graphs we mean the kolmogorov random graphs which constitute a fraction of 1 1 nc of all graphs on n nodes where c 0 is an arbitrary fixed constant there is a model for which the average case lower bound rises to omega n 2 log n and another model where the average case upper bound drops to o n log 2 n this clearly exposes the sensitivity of such bounds to the model under consideration if paths have to be short but need not be shortest if the stretch factor may be larger than 1 then much less space is needed on average even in the more demanding models full information routing requires theta n 3 bits on average for worst case static networks we prove an omega n 2 log n lower bound for shortest path routing and all stretch factors 2 in some networks where free relabeling is not allowed

spanners for bounded tree length graphs this paper concerns construction of additive stretched spanners with few edges for n vertex graphs having a tree decomposition into bags of diameter at most d i e the tree length d graphs for such graphs we construct additive 2 d spanners with o dn nlogn edges and additive 4 d spanners with o dn edges this provides new upper bounds for chordal graphs for which d 1 we also show a lower bound and prove that there are graphs of tree length d for which every multiplicative d spanner and thus every additive d 1 spanner requires w n 1 1 q d edges

stability analysis of runge kutta methods for unbounded retarded differential equations with piecewise continuous arguments abstract this paper deals with the asymptotical stability of the analytic solutions for the unbounded retarded differential equations with piecewise continuous arguments epca u t f t u t u t n t 0 u 0 u 0 where n z a sufficient condition that the differential equations are asymptotically stable is derived this paper is also concerned with the stability analysis of the runge kutta methods for equation u t au t bu t n t 0 u 0 u 0 where n z the conditions that the numerical solutions preserve the stability of the analytic solutions are obtained and some numerical experiments are given

a change of variables in the asymptotic theory of differential equations with unbounded delays in this paper the author investigates the asymptotic properties of solutions of the nonhomogeneous linear differential equation x t ax t bx t f t with nonzero real scalars a b and the unbounded lag using the change of the independent and dependent variable he relates the asymptotic behaviour of solutions of this equation to the asymptotic behaviour of solutions of auxiliary functional nondifferential equations

new oscillation criteria for linear delay differential equations abstract we obtain new sufficient conditions for the oscillation of all solutions of the first order linear delay differential equation x t p t x t 0 with p t 0 t t our results improve the known results in the literature which include those obtained recently in papers by elbert and stavroulakis li and yu et al

exponential solutions of equation ẏ t β t y t δ y t τ abstract asymptotic behaviour of solutions of first order differential equation with two deviating arguments of the form y t t y t y t is discussed for t a criterion for representing solutions in exponential form is proved as consequences inequalities for such solutions are given connections with known results are discussed and a sufficient condition for existence of unbounded solutions generalizing previous ones is derived an illustrative example is considered too

oscillation of first order delay differential equations abstract some new oscillation criteria for the first order delay differential equation x t p t x t 0 t t 0 t t are established in the case where t t p s ds 1 e and lim t t t p s ds 1 e an open problem by a elbert and i p stavroulakis 1995 proc amer math soc 123 1503 1510 is solved

positive and oscillating solutions of differential equations with delay in critical case abstract this article is devoted to the problem of existence of positive solution for common classes of nonlinear retarded functional differential equations some criteria of its existence are proved as well as some comparison results the obtained results are applied to the linear case moreover the existence of positive and oscillating solutions of a differential equation with delay x t a t x t in the critical case is considered some comparisons with known results are given

exponents of 2 coloring of symmetric digraphs a 2 coloring g1 g2 of a digraph is 2 primitive if there exist nonnegative integers h and k with h k 0 such that for each ordered pair u v of vertices there exists an h k walk in g1 g2 from u to v the exponent of g1 g2 is the minimum value of h k taken over all such h and k in this paper we consider 2 colorings of strongly connected symmetric digraphs with loops establish necessary and sufficient conditions for these to be 2 primitive and determine an upper bound on their exponents we also characterize the 2 colored digraphs that attain the upper bound and the exponent set for this family of digraphs on n vertices

oscillation criteria for first order neutral differential equations with positive and negative coefficients sufficient conditions for the oscillation of the equation ddt x t r t x t r p t x t q t x t 0 where p q r c t0 r r 0 r are obtained where r t tt q u du 1 is allowed to oscillate and the condition t0s p s q s s p u q u du ds is not necessary some examples are given which show that the results here are almost sharp

the characterization of operators preserving primitivity for matrix k tuples we obtain a complete characterization of surjective additive operators acting on the cartesian product of several matrix spaces over an antinegative semiring without zero divisors which map primitive matrix k tuples to primitive matrix k tuples

multidimensional interval routing schemes interval routing scheme k irs is a compact routing scheme on general networks it has been studied extensively and recently been implemented on the latest generation inmos transputer router chip in this paper we introduce an extension of the interval routing scheme k irs to the multi dimensional case k d mirs where k is the number of intervals and d is the number of dimensions whereas k irs only represents compactly a single shortest path between any two nodes with this new extension we are able to represent all shortest paths compactly this is useful for fault tolerance and traffic distribution in a network we study efficient representations of all shortest paths between any pair of nodes for general network topologies and for specific interconnection networks such as rings grids tori and hypercubes for these interconnection networks we show that for about the same space complexity as k irs we can represent all shortest paths in k d mirs as compared to only a single shortest path in k irs moreover tradeoffs are derived between the dimension d and the number of intervals k in multi dimensional interval routing schemes on hypercubes grids and tori

stability of runge kutta methods in the numerical solution of equation u t au t a0u t a1u t 1 this paper is concerned with the stability analysis of the runge kutta methods for the equation u t au t a 0u t a 1u t 1 the stability regions for the runge kutta methods are determined the conditions that the analytic stability region is contained in the numerical stability region are obtained and some numerical experiments are given

fault tolerant additive and µ α spanners graph spanners are sparse subgraphs that preserve the distances of the original graph up to some approximation ratio the spanner s stretch a number of algorithms are known for constructing sparse spanners with small multiplicative or additive stretch recently algorithms were introduced for constructing fault tolerant multiplicative spanners of general graphs this paper addresses the analogous problem of constructing fault tolerant additive and spanners for general graphs and presents a number of deterministic and randomized construction algorithms for such spanners

the improved linear multistep methods for differential equations with piecewise continuous arguments abstract this paper deals with the convergence of the linear multistep methods for the equation x t ax t a 0 x t numerical experiments demonstrate that the 2 step adams bashforth method is only of order p 0 when applied to the given equation an improved linear multistep methods is constructed it is proved that these methods preserve their original convergence order for ordinary differential equations odes and some numerical experiments are given

exponents of tuples of nonnegative matrices abstract the notions of irreducibility primitivity and exponent of a nonegative matrix are generalized to k tuples of non negative matrices of the same order it is shown that for each positive integer k the maximum exponent of a primitive k tuple of n by n nonnegative matrices is n k 1

new oscillation criteria for odd order neutral equations abstract in this paper new oscillation criteria are obtained for all solutions of the odd order neutral differential equation x t p t x t n q t x t 0 where p c t 0 r q c t 0 r 0 0 and n is odd our results do not need the usual hypothesis t 0 s n 1 q s ds some examples are given to demonstrate the advantage of our results than existing ones in the literature

the complexity of the characterization of networks supporting shortest path interval routing interval routing is a routing method that was proposed in order to reduce the size of the routing tables by using intervals and was extensively studied and implemented some variants of the original method were also defined and studied the question of characterizing networks which support optimal i e shortest path interval routing has been thoroughly investigated for each of the variants and under different models with only partial answers both positive and negative given so far in this paper we study the characterization problem under the most basic model the one unit cost and with the most restrictive memory requirements one interval per edge we prove that this problem is np hard even for the restricted class of graphs of diameter at most 3 our result holds for all variants of interval routing it significantly extends some related np hardness result and implies that unless p np partial characterization results of some classes of networks which support shortest path interval routing cannot be pushed further to lead to efficient characterizations for these classes

positive solutions of retarded functional differential equations abstract for systems of retarded functional differential equations with unbounded delay and with finite memory sufficient conditions of existence of positive solutions on an interval of the form t 0 are derived

existence of bounded nonoscillatory solutions of first order nonlinear neutral delay differential equations the aim of this paper is to study the following first order nonlinear neutral delay differential equation ddt x t c t x email protected h t f x email protected 1 x email protected 2 x email protected k g t t t 0 where c h email protected c t 0 r t 0 s email protected r for email protected j k 1 k and email protected c r k r by using the schauder and krasnoselskii fixed point theorems we establish the existence of uncountably many bounded nonoscillatory solutions for the above equation to dwell upon the importance and advantages of our results seven nontrivial examples are included

oscillations of first order neutral differential equations with variable coefficients abstract consider the first order neutral delay differential equation d dt x t p t x t q t x t t 0 t t 0 where 0 p q c t0 r and limt t t first we obtain a necessary and sufficient condition for the oscillation of all solutions of this equation and then some applications of this result are listed our results can improve many known results in the literature

oscillation criteria for first order impulsive differential equations with positive and negative coefficients some sufficient conditions are obtained for oscillation of all solutions of the first order impulsive differential equation with positive and negative coefficients x t r t x t r p t x t t q t x t s 0 t s 0 t t 0 x t k i k x t k k 1 2 our results improve the known results in the literature

explicit criteria for the existence of positive solutions for a scalar differential equation with variable delay in the critical case a scalar linear differential equation with time dependent delay x t a t x t t t is considered where t i t 0 t 0 r a i r 0 is a continuous function and t i r is a continuous function such that t t t t 0 t t 0 if t t 0 the goal of our investigation is to give sufficient conditions for the existence of positive solutions as t in the critical case in terms of inequalities on a and t a generalization of one known final in a certain sense result is given for the case of t being not a constant analysing this generalization we show e g that it differs from the original statement with a constant delay since it does not give the best possible result this is demonstrated on a suitable example

stability of differential equations with piecewise constant arguments of generalized type abstract in this paper we continue to consider differential equations with piecewise constant argument of generalized type epcag m u akhmet integral manifolds of differential equations with piecewise constant argument of generalized type nonlinear anal tma 66 2007 367 383 a deviating function of a new form is introduced the linear and quasilinear systems are under discussion the structure of the sets of solutions is specified necessary and sufficient conditions for stability of the zero solution are obtained our approach can be fruitfully applied to the investigation of stability oscillations controllability and many other problems of epcag some of the results were announced at the international conference on hybrid systems and applications university of louisiana lafayette 2006

comparison results for parabolic differential equations at resonance the objective of this paper is to develop systematically general comparison techniques for semilinear parabolic equations with periodic and homogeneous neumann boundary conditions these results are expected to play a significant role in the existence theory for parabolic boundary value problems at resonance

on nonoscillation of mixed advanced delay differential equations with positive and negative coefficients for a mixed advanced delay differential equation with variable delays and coefficients email protected t a t x g t b t x h t 0 t t 0 where a t 0 b t 0 g t t h t t explicit nonoscillation conditions are obtained

on oscillation properties of delay differential equations with positive and negative coefficients abstract for a scalar delay differential equation x t a t x h t b t x g t 0 a t 0 b t 0 h t t g t t a connection between the following properties is established nonoscillation of the differential equation and the corresponding differential inequalities positiveness of the fundamental function and existence of a nonnegative solution for a certain explicitly constructed nonlinear integral inequality a comparison theorem and explicit nonoscillation and oscillation results are presented

k primitivity of digraphs abstract let d be a directed graph digraph on n vertices the digraph d is said to be primitive if for some m between any ordered pair of vertices of d there is a directed walk of length m from the first vertex to the other here our focus is a generalization of primitivity called k primitivity where k arc colorings of digraphs are considered let k max n be the maximum k for which there exists a k coloring of some strong n tournament that is k primitive we show that n 1 2 k max n n 2 n 4

on the initial growth of interfaces in reaction diffusion equations with strong absorption we study the initial growth of the interfaces of non negative local solutions of the equation u t u m xx lambdau q when m greater than or equal to 1 and 0 c0 for some explicit c0 c0 lambda m q then the free boundary zeta t sup x u x t 0 is a heating front more precisely zeta t greater than or equal to at m q 2 1 q for any t small enough and for some a 0 if on the contrary u x 0 less than or equal to c x 2 m q with c 0 applications to solutions of the associated cauchy and dirichlet problems are also given

oscillation of impulsive neutral delay differential equations abstract in this paper effective sufficient conditions for the oscillation of all solutions of impulsive neutral delay differential equations of the form x t p t x t q t x t sgn x t 0 t t 0 x t k b k x t k k 1 2 are established our results reveal the fact that the oscillatory properties of all solutions of eqs and may be caused by the impulsive perturbations though the corresponding neutral delay differential equation without impulses i e eq admits a nonoscillatory solution it is also seen that the oscillatory behavior of all solutions of eq can be inherited by eqs and under certain impulsive perturbations some examples are also given to illustrate the applicability of the results obtained

a trade off between space and efficiency for routing tables two conflicting goals play a crucial role in the design of routing schemes for communication networks a routing scheme should use paths that are as short as possible for routing messages in the network while keeping the routing information stored in the processors local memory as succinct as possible the efficiency of a routing scheme is measured in terms of its stretch factor the maximum ratio between the length of a route computed by the scheme and that of a shortest path connecting the same pair of vertices most previous work has concentrated on finding good routing schemes with a small fixed stretch factor for special classes of network topologies in this paper the problem for general networks is studied and the entire range of possible stretch factors is examined the results exhibit a trade off between the efficiency of a routing scheme and its space requirements almost tight upper and lower bounds for this trade off are presented specifically it is proved that any routing scheme for general n vertex networks that achieves a stretch factor k 1 must use a total of o n 1 1 2 k 4 bits of routing information in the networks this lower bound is complemented by a family k k of hierarchical routing schemes for every k l for unit cost general networks which guarantee a stretch factor of o k require storing a total of o k 3 n 1 1 h log n bits of routing information in the network name the vertices with o log 2 n bit names and use o log n bit headers

interval routing in reliability networks in this paper we consider routing with compact tables in reliability networks more precisely we study interval routing on random graphs g b p obtained from a base graph b by independently removing each edge with a failure probability 1 p we focus on additive stretched routing for n node random graphs for which the base b is a square mesh and p 0 5 that is the percolation model at the critical phase we show a lower bound of log n 2 on the number of intervals required per edge for every additive stretch 0 on the other side our experimental results show that the size of the largest biconnected components is n0 827 and thus that there exists a trivial shortest path routing scheme using at most o n0 827 intervals per edge the results are extended to random meshes of higher dimension we show that asymptotically almost surely the number of intervals per edge for a random r dimensional mesh with n nodes is 16 r 2 1 r r 4 log n 1 1 r for every additive stretch 0 and for every integral dimension r 1 log2n

static and dynamic low congested interval routing schemes interval routing schemes irs have been extensively investigated in the past years with special emphasis on shortest paths besides their theoretical interest irs have practical applications as they have been implemented with wormhole routing in the last generation of inmos transputer router chips in this paper we consider irs that are optimal with respect to the congestion of the induced path system in fact wormhole routing is strongly influenced by the maximum number of paths that share a physical link and from low to moderate congestion it outperforms the packet switching technique we provide a general framework able to deal with the various congestion issues in irs in fact we will distinguish between static cases in which the source destination configurations are fixed and dynamic cases where they vary over time all these situations can be handled in a unified setting thanks to the notion of competitiveness introduced in this paper we first give some general results not related to specific traffic demands then in the one to all communication pattern we show that constructing competitive irs for a given network is an intractable problem both for the static and the dynamic case that is when the root vertex is fixed and when it can change along the time respectively finally both for one to all and all to all communication patterns we provide nicely competitive k irs for relevant topologies networks considered are chains trees rings chordal rings and multi dimensional grids and tori we consider both the directed congestion case in which there are pairwise opposite unidirectional links connecting two neighbor processors and the undirected congestion case in which two neighbors are connected by a single bi directional link

on k primitivity of two classes of digraphs in this paper we study two classes of primitive digraphs and show that there are k colorings that are k primitive

oscillation of forced neutral differential equations with positive and negative coefficients some sufficient conditions are obtained for the oscillation of forced neutral differential equations with positive and negative coefficients x t r t x t r p t x email protected q t x email protected f t t t 0 where r t p t q t c t 0 r r 0 t 0 s 0

the complexity of shortest path and dilation bounded interval routing abstract interval routing is a popular compact routing method for point to point networks which found industrial applications in novel transputer routing technology may and thompson transputers and routers components for concurrent machines inmos 1991 recently much effort is devoted to relate the efficiency measured by the dilation or the stretch factor to space requirements measured by the compactness or the total number of memory bits in a variety of compact routing methods eilam moran and zaks 10th international workshop on distributed algorithms wdag lecture notes in computer science vol 1151 springer berlin 1996 pp 191 205 fraigniaud and gavoille 8th annual acm symp on parallel algorithms and architectures spaa acm press new york 1996 gavoille and perennes proc sircco 96 carleton press 1996 pp 88 103 kranakis krizanc 13th annual symp on theoretical aspects of computer science stacs lecture notes in computer science vol 1046 springer berlin 1996 pp 529 540 meyer auf der heide and scheideler proc 37th symp on foundations of computer science focs november 1996 peleg and upfal j acm 36 1989 510 530 tse and lau proc sirocco 95 carleton press 1995 pp 123 134 we add new results in this direction for interval routing for the shortest path interval routing we apply a technique from flammini van leeuwen and marchetti spaccamela mfcs 95 lecture notes in computer science vol 969 springer berlin 1995 pp 37 49 to some interconnection networks shuffie exchange se cube connected cycles ccc butterfly bf and star s and get improved lower bounds on compactness in the form n 1 2 e any e 0 for se n log n for ccc and bf and n log log n log n 5 for s where n is the number of nodes in the corresponding network previous lower bounds for these networks were only constant fraigniaud and gavoille conpar 94 lecture notes in computer science vol 854 springer berlin 1994 pp 785 796 for the dilation bounded interval routing we give a routing algorithm with the dilation 1 5d and the compactness o n log n on n node networks with the diameter d it is the first nontrivial upper bound on the dilation bounded interval routing on general networks moreover we construct a network on which each interval routing with the dilation 1 5d 3 needs the compactness at least n it is an asymptotical improvement over the previous lower bounds in tse and lau proc sirocco 95 carleton press 1995 pp 123 134 and it is also better than independently obtained lower bounds in tse and lau proc computing the australasian theory symp cats 97 sydney australia february 1997

approximation algorithms for spanner problems and directed steiner forest we present an o nlogn approximation algorithm for the problem of finding the sparsest spanner of a given directed graph g on n vertices a spanner of a graph is a sparse subgraph that approximately preserves distances in the original graph more precisely given a graph g v e with nonnegative edge lengths d e r 0 and a stretchk 1 a subgraph h v e h is a k spanner of g if for every edge s t e the graph h contains a path from s to t of length at most email protected d s t the previous best approximation ratio was email protected n 2 3 due to dinitz and krauthgamer stoc 11 we also improve the approximation ratio for the important special case of directed 3 spanners with unit edge lengths from email protected n to o n 1 3logn the best previously known algorithms for this problem are due to berman raskhodnikova and ruan fsttcs 10 and dinitz and krauthgamer the approximation ratio of our algorithm almost matches dinitz and email protected s lower bound for the integrality gap of a natural linear programming relaxation our algorithm directly implies an o n 1 3logn approximation for the 3 spanner problem on undirected graphs with unit lengths an easy o n approximation algorithm for this problem has been the best known for decades finally we consider the directed steiner forest problem given a directed graph with edge costs and a collection of ordered vertex pairs find a minimum cost subgraph that contains a path between every prescribed pair we obtain an approximation ratio of o n 2 3 e for any constant e 0 which improves the o n email protected min n 4 5 m 2 3 ratio due to feldman kortsarz and nutov email protected 12

decentralized routing on spatial networks with stochastic edge weights we investigate algorithms to find short paths in spatial networks with stochastic edge weights our formulation of the problem of finding short paths differs from traditional formulations because we specifically do not make two of the usual simplifying assumptions 1 we allow edge weights to be stochastic rather than deterministic and 2 we do not assume that global knowledge of a network is available we develop a decentralized routing algorithm that provides en route guidance for travelers on a spatial network with stochastic edge weights without the need to rely on global knowledge about the network to guide a traveler our algorithm uses an estimation function that evaluates cumulative arrival probability distributions based on distances between pairs of nodes the estimation function carries a notion of proximity between nodes and thereby enables routing without global knowledge in testing our decentralized algorithm we define a criterion that makes it possible to discriminate among arrival probability distributions and we test our algorithm and this criterion using both synthetic and real networks

space lower bounds for low stretch greedy embeddings greedy geometric routing is an important paradigm for routing in communication networks it uses an embedding of the nodes of a network into points of a space e g r d equipped with a distance function e g the euclidean distance 2 and uses as address of each node the coordinates of the corresponding point the embedding has particular properties so that the routing decision for a packet is taken greedily based only on the addresses of the current node its neighbors and the destination node and the distances of the corresponding points in this way there is no need to keep extensive routing information e g routing tables at the nodes embeddings that allow for this functionality are called greedy embeddings even though greedy embeddings do exist for several spaces and distance functions they usually result in paths of high stretch i e significantly longer than the corresponding shortest paths in this paper we show that greedy embeddings in low dimensional euclidean spaces necessarily have high stretch in particular greedy embeddings of n node graphs with optimal stretch require at least n dimensions for distance 2 this result disproves a conjecture by maymounkov 2006 14 stating that greedy embeddings of optimal stretch are possible in euclidean spaces with polylogarithmic number of dimensions furthermore we present trade offs between the stretch and the number of dimensions of the host space our results imply that every greedy embedding into a space with polylogarithmic number of dimensions and euclidean distance has stretch log n log log n we extend this result for a distance function used by an o log n stretch greedy embedding presented by flury et al 2009 9 our lower bound implies that their embedding has almost best possible stretch

oscillation and nonoscillation in neutral equations with integrable coefficients we obtain respectively new sufficient conditions for the oscillation of all solutions and the existence of a positive solution of the neutral delay differential equation ddt t p t t q t t 0 t t0 where 0 p q c t0 r and t0 q s ds

existence of multiple positive periodic solutions of delayed predator prey models with functional responses in this paper by applying the continuation theorem of coincidence degree theory we establish some new criteria for the existence of multiple positive periodic solutions for the delayed predator prey model x t x t r t a t x t b t f x t y t y t y t c t f x email protected d t when functional response function f is monotonic or nonmonotonic

interval routing in some planar networks in this article we design optimal or near optimal interval routing schemes irs for short with small compactness for several classes of plane quadrangulations and triangulations by optimality or near optimality we mean that messages are routed via shortest or almost shortest paths we show that the subgraphs of the rectilinear grid bounded by simple circuits allow optimal irs with at most two circular intervals per edge 2 irs we extend this result to all plane quadrangulations in which all inner vertices have degrees 4 namely we establish that every such graph has an optimal irs with at most seven linear intervals per edge 7 lirs this leads to a 7 lirs with the stretch factor 2 for all plane triangulations in which all inner vertices have degrees 6 all routing schemes can be implemented in linear time

oscillation and existence of positive solutions for odd order neutral equations with integrally small coefficients abstract consider the odd order neutral differential equation formula we establish several new sufficient conditions for the oscillation of all solutions and the existence of a positive solution when p t 1 is allowed to oscillate and the usual divergent condition q s ds is not satisfied these conditions are almost sharp and improve some known results in the literature some examples are given to demonstrate the advantage of our results

a note on k primitive directed graphs abstract we consider the problem of which primitive directed graphs can be k colored to yield a k primitive directed graph if such a k coloring exists then certainly such a graph must have at least k cycles we prove that any primitive directed graph admits a 2 coloring that is 2 primitive by contrast for each k 4 we construct examples of primitive directed graphs having k cycles for which no k coloring is k primitive we also give some partial results for the case that k 3

stability and oscillations of numerical solutions for differential equations with piecewise continuous arguments of alternately advanced and retarded type this paper is concerned with the numerical properties of q methods for the solution of alternately advanced and retarded differential equations with piecewise continuous arguments using two q methods namely the one leg q method and the linear q method the necessary and sufficient conditions under which the analytic stability region is contained in the numerical stability region are obtained and the conditions of oscillations for the q methods are also obtained it is proved that oscillations of the analytic solution are preserved by the q methods furthermore the relationships between stability and oscillations are revealed some numerical experiments are presented to illustrate our results

positive solutions of the equation ẋ t c t x t τ in the critical case abstract the equation t c t x t is considered in the critical case for it the asymptotic behavior of dominant and subdominant solutions is studied a generalization is made and connections with known results are discussed

exponents of nonnegative matrix pairs the notions of primitivity and exponent of a square nonnegative matrix a are classical a is primitive provided there is a nonnegative integer k such that a k is entrywise positive and in the case a is primitive the exponent of a is the smallest such k fornasini and valcher have extended the notion of primitivity to pairs a b of square nonnegative matrices of the same order the pair a b is primitive provided there exist nonnegative integers h and k such that the sum of all products formed by words consisting of ha s and kb s is entrywise positive this paper defines the exponent of a nonnegative matrix pair to be the smallest value of h k over all such h and k it is then shown that the largest exponent of a primitive pair of n by n nonnegative matrices lies in the interval n 3 5n 2 2 3n 3 2n 2 2n 2 in addition the exponent of a pair of nonnegative matrices is related to properties of an associated twodimensional dynamical system 2001 elsevier science inc all rights reserved