existence of infinitely many homoclinic orbits for fourth order difference systems containing both advance and retardation abstract by using the symmetric mountain pass theorem we establish some existence criteria to guarantee the fourth order difference system 4u n 2 q n u n f n u n 1 u n u n 1 have infinitely many homoclinic orbits where n z u r n q z r n n and f z r 3 n r are no periodic in n

subharmonic solutions and homoclinic orbits of second order discrete hamiltonian systems with potential changing sign in this paper a class of second order discrete hamiltonian systems with periodicity assumptions is considered based on the critical point theory some sufficient conditions for the existence of subharmonic solutions and homoclinic orbits are obtained the results obtained extend the results in 2007 by relaxing the assumptions on the sign of the potential

existence of infinitely many homoclinic orbits in discrete hamiltonian systems abstract by using the symmetric mountain pass theorem we establish some existence criteria to guarantee the second order self adjoint discrete hamiltonian system p n u n 1 l n u n w n u n 0 has infinitely many homoclinic orbits where n z u r n p l z r n n and w z r n r are no periodic in n our conditions on the potential w n x are rather relaxed

homoclinic orbits for second order self adjoint difference equations abstract in this paper we discuss how to use variational methods to study the existence of nontrivial homoclinic orbits of the following nonlinear difference equations p t u t 1 q t u t f t u t t z without any periodicity assumptions on p t q t and f providing that f t x grows superlinearly both at origin and at infinity or is an odd function with respect to x r and satisfies some additional assumptions

homoclinic orbits and subharmonics for nonlinear second order difference equations abstract we show the existence of a nontrivial homoclinic orbit and subharmonic solutions for a class of second order difference equations by applying the mountain pass theorem relying on ekeland s variational principle and the diagonal method and the homoclinic orbit as the limit of the subharmonics a completely new way is provided for dealing with the existence of solutions for difference equations

distributed verification and hardness of distributed approximation we study the verification problem in distributed networks stated as follows let h be a subgraph of a network g where each vertex of g knows which edges incident on it are in h we would l

polylog time and near linear work approximation scheme for undirected shortest paths shortest paths computations constitute one of the most fundamental network problems nonetheless known parallel shortest paths algorithms are generally inefficient they perform significantly more work product of time and processors than their sequential counterparts this gap known in the literature as the transitive closure bottleneck poses a long standing open problem our main result is an o mn e 0 s m n1 e 0 work polylog time randomized algorithm that computes paths within 1 o 1 polylog n of shortest from s source nodes to all other nodes in weighted undirected networks with n nodes and m edges for any fixed e 0 0 this work bound nearly matches the o d sm sequential time in contrast previous polylog time algorithms required nearly min o d n3 o d m2 work even when s 1 and previous near linear work algorithms required near o n time we also present faster sequential algorithms that provide good approximate distances only between distant vertices we obtain an o m sn n e 0 time algorithm that computes paths of weight 1 o 1 polylog n dist o w max polylog n where dist is the corresponding distance and w max is the maximum edge weight our chief instrument which is of independent interest are efficient constructions of sparse hop sets a d e hop set of a network g v e is a set e of new weighted edges such that mimimum weight d edge paths in v e e have weight within 1 e of the respective distances in g we construct hop sets of size o n1 e 0 where e o 1 polylog n and d o polylog n

homoclinic orbits for second order hamiltonian systems possessing superquadratic potentials received by the editors november 27 1990 1991 mathematics subject classification primary 34c99 58e99 58f99 the first author was supported by ministero p i gruppo 40 calcolo delle variazioni the second author s research was sponsored in part by the u s army research office under contract daal03 87 k 0043 the national science foundation under grant mcs 8110556 and the office of naval research under grant n00014 88 ko134 any reproduction for the purpose of the united states govemment is permitted

distributed exact shortest paths in sublinear time the distributed single source shortest paths problem is one of the most fundamental and central problems in the message passing distributed computing classical bellman ford algorithm solves it in o n time where n is the number of vertices in the input graph g peleg and rubinovich 49 showed a lower bound of d s n for this problem where d is the hop diameter of g whether or not this problem can be solved in o n time when d is relatively small is a major open question despite intensive research 10 17 33 41 45 that yielded near optimal algorithms for the approximate variant of this problem no progress was reported for the original problem in this article we answer this question in the affirmative we devise an algorithm that requires o n log n 5 6 time for d e o s n log n and o d1 3 n log n 2 3 time for larger d this running time is sublinear in n in almost the entire range of parameters specifically for d e o n log2 n we also generalize our result in two directions one is when edges have bandwidth b 1 and the other is the s sources shortest paths problem for both problems our algorithm provides bounds that improve upon the previous state of the art in almost the entire range of parameters in particular we provide an all pairs shortest paths algorithm that requires o n5 3 log 2 3 n time even for b e 1 for all values of d we also devise the first algorithm with non trivial complexity guarantees for computing exact shortest paths in the multipass semi streaming model of computation from the technical viewpoint our distributed algorithm computes a hopset g of a skeleton graph g of g without first computing g itself we then conduct a bellman ford exploration in g g while computing the required edges of g on the fly as a result our distributed algorithm computes exactly those edges of g that it really needs rather than computing approximately the entire g

existence of nontrivial homoclinic orbits for fourth order difference equations we obtain a sufficient condition for the existence of nontrivial homoclinic orbits for fourth order difference equations by using mountain pass theorem a weak convergence argument and a discrete version of lieb s lemma

monge ampere equations and moduli spaces of manifolds of circular type abstract a bounded manifold of circular type is a complex manifold m of dimension n admitting a bounded exhaustive real function u defined on m minus a point x o so that a it is a smooth solution on m x o to the monge ampere equation d d c u n 0 b x o is a singular point for u of logarithmic type and e u extends smoothly on the blow up of m at x o c d d c e u 0 at any point of m x o this class of manifolds naturally includes all smoothly bounded strictly linearly convex domains and all smoothly bounded strongly pseudoconvex circular domains of c n a set of modular parameters for bounded manifolds of circular type is considered in particular for each biholomorphic equivalence class of them it is proved the existence of an essentially unique manifold in normal form it is also shown that the class of normalizing maps for an n dimensional manifold m is a new holomorphic invariant with the following property it is parameterized by the points of a finite dimensional real manifold of dimension n 2 when m is a non convex circular domain while it is of dimension n 2 2 n when m is a strictly linearly convex domain new characterizations of the circular domains and of the unit ball are also obtained

fast homoclinic solutions for a class of damped vibration problems in this paper we deal with the existence and multiplicity of homoclinic solutions of the following damped vibration problemsu t q t u t l t u t w t u t 0 where l t and w t x are neither autonomous nor periodic in t our approach is variational and it is based on the critical point theory we prove existence and multiplicity results of fast homoclinic solutions under general growth conditions on the potential function moreover some open problems proposed by zhang and ruan are resolved our theorems appear to be the first such result

periodic and homoclinic solutions generated by impulses for asymptotically linear and sublinear hamiltonian system in this paper we get the existence of periodic and homoclinic solutions for a class of asymptotically linear or sublinear hamiltonian systems with impulsive conditions via variational methods however without impulses there is no homoclinic or periodic solution for the system considered in this paper moreover our results can be used to study the existence of periodic and homoclinic solutions of difference equations

homoclinic solutions for fractional discrete laplacian equations abstract in this paper we introduce a variational framework to a fractional difference equation on z driven by the fractional discrete laplacian and involving a coercive weight function and a positive parameter by means of the critical point theory we prove the existence of at least two nontrivial and nonnegative homoclinic solutions for big enough

a sublinear time distributed algorithm for minimum weight spanning trees this paper considers the question of identifying the parameters governing the behavior of fundamental global network problems many papers on distributed network algorithms consider the task of optimizing the running time successful when an o n bound is achieved on an n vertex network we propose that a more sensitive parameter is the network s diameter diam this is demonstrated in the paper by providing a distributed minimum weight spanning tree algorithm whose time complexity is sublinear in n but linear in diam specifically o diam n varepsilon cdot log n for varepsilon frac ln 3 ln 6 0 6131 our result is achieved through the application of graph decomposition and edge elimination by pipelining techniques that may be of independent interest

homoclinic orbits of a class of second order difference equations abstract in this paper we apply the variational method and the spectral theory of difference operators to investigate the existence of homoclinic orbits of the second order difference equation 2 x t 1 l t x t v x t x t 0 in the two cases that v t is superquadratic and subquadratic under the assumptions that l t is positive definite for sufficiently large t z we show that there exists at least one non trivial homoclinic orbit of the difference equation further if v t x is superquadratic and even with respect to x then it has infinitely many different non trivial homoclinic orbits at the end two illustrative examples are provided

first order elliptic systems and the existence of homoclinic orbits in hamiltonian systems and h r x r 2 r is smooth by a homoclinic or a doubly asymptotic orbit to y where y is a periodic solution of 1 we mean a solution x y of 1 such that i x 0 y t l 0 as i tl oo finding homoclinic orbits in systems like 1 can be quite difficult in the case when y is a hyperbolic orbit this is equivalent to showing that the stable and unstable manifold of y intersect in some situation this can be done by melnikov s method recently some progress has been made by applying variational methods coti zelati and ekeland showed in 3 using dual variational methods that 1 has at least one homoclinic orbit if h satisfies some convexity and growth assumptions in their proof the convexity assumption enters in a crucial way already in the set up of the variational problem variational methods have been used by benci and giannoni 2 who studied homoclinic orbits on compact riemannian manifolds and by p rabinowitz in 9 who proved existence of heteroclinic orbits on the n dimensional torus as well as their multiplicity in 2 and 9 only special hamiltonian of lagrangian type are studied the associated variational problem is then at least semi definite the

modular data and regularity of monge ampere exhaustions and of kobayashi distance regularity properties of intrinsic objects for a large class of stein manifolds namely of monge ampere exhaustions and kobayashi distance is interpreted in terms of modular data the results lead to a construction of an infinite dimensional family of convex domains with squared kobayashi distance of prescribed regularity properties a new sharp refinement of stoll s characterization of mathbb c n is also given

fast estimation of diameter and shortest paths without matrix multiplication in the recent past there has been considerable progress in devising algorithms for the all pairs shortest paths apsp problem running in time significantly smaller than the obvious time bound of o n3 unfortunately all the new algorithms are based on fast matrix multiplication algorithms that are notoriously impractical our work is motivated by the goal of devising purely combinatorial algorithms that match these improved running times our results come close to achieving this goal in that we present algorithms with a small additive error in the length of the paths obtained our algorithms are easy to implement have the desired property of being combinatorial in nature and the hidden constants in the running time bound are fairly small our main result is an algorithm which solves the apsp problem in unweighted undirected graphs with an additive error of 2 in time o n 2 5 sqrt log n this algorithm returns actual paths and not just the distances in addition we give more efficient algorithms with running time footnotesize o n 1 5 sqrt k log n n 2 log 2 n for the case where we are only required to determine shortest paths between k specified pairs of vertices rather than all pairs of vertices the starting point for all our results is an o m sqrt n log n algorithm for distinguishing between graphs of diameter 2 and 4 and this is later extended to obtaining a ratio 2 3 approximation to the diameter in time o m sqrt n log n n 2 log n unlike in the case of apsp our results for approximate diameter computation can be extended to the case of directed graphs with arbitrary positive real weights on the edges

high probability parallel transitive closure algorithms there is a straightforward algorithm for computing the transitive closure of an n node graph in o log 2 n time on an erew pram using n 3 log n processors or indeed with m n log n processors if serial matrix multiplication in m n time can be done this algorithm is within a log factor of optimal in work processor time product for solving the all pairs transitive closure problem for dense graphs however this algorithm is far from optimal when either a the graph is sparse or b we want to solve the single source transitive closure problem it would be ideal to have an mathcal nc algorithm for transitive closure that took about e processors for the single source problem on a graph with n nodes and e geqq n arcs or about en processors for the all pairs problem on the same graph while an algorithm that good cannot be offered algorithms with the following performance can be offered 1 for single source tilde o n varepsilon time with tilde o en 1 2 varepsil

homoclinic orbits in a first order superquadratic hamiltonian system convergence of subharmonic orbits abstract we consider the existence of homoclinic orbits for a first order hamiltonian system z jh z t z we assume h t z is of form h t z 1 2 az z w t z where a is a symmetric matrix with ja i r and w t z is 2 periodic in t and has superquadratic growth in z we prove the existence of a nontrivial homoclinic solution z t and subharmonic solutions zt t t n i e 2 t periodic solutions of hs such that zt t z t in cloc1 r r 2n as t

homoclinic orbits for first order hamiltonian systems abstract we establish the existence of homoclinic orbits for a class of first order time dependent hamiltonian systems z jh z t z without any periodicity assumption on h

existence and multiplicity of solutions for nonlinear second order difference boundary value problems in the present paper the following nonlinear second order difference boundary value problems d p n 1 du n 1 q n u n f n u n 0 n z 1 n u 0 u n p 0 du 0 p n du n are studied by using the critical theory and three new multiple results of solutions are obtained where z 1 n 1 2 n f z 1 n xr r is a continuous function in the second variable p q z 0 n r and p n 0

moduli for pointed convex domains a moduli space for the class of pointed strictly linearly convex domains in n is obtained it is shown that the space of pointed smoothly bounded strictly linearly convex domains with a fixed indicatrix is parameterized by a class of deformations of the cr structure of the boundary of the indicatrix these deformations are constructed by using the circular representation of a domain to pull back its complex structure tensor to the indicatrix a careful study of the pull back structure shows that the allowable deformations are parameterized by a class of complex hamiltonian vector fields the proof of this fact is based on the folland stein estimates for the bar partial b complex of the boundary of the indicatrix