stability of discrete breathers abstract we approach the problem of linear stability of discrete breathers in a rigorous way for networks of not necessarily identical oscillators with general type of coupling in the case of hamiltonian systems using symplectic signature theory we give general conditions on the spectrum of the monodromy map in the uncoupled limit such that the discrete breathers are l 2 linearly stable for weak enough coupling we consider also dissipative networks of oscillators in this case we prove that the discrete breathers are not only stable but also attract a neighbourhood of initial data for any choice of l p topology some examples are considered amongst which an instance of roto breather

independent spanning trees on even networks the use of multiple independent spanning trees ists for data broadcasting in networks provides a number of advantages including the increase of fault tolerance and bandwidth thus the designs of multiple ists on several classes of networks have been widely investigated in this paper we give an algorithm to construct ists on even networks and show that these ists are optimal for height and path lengths and each path in the ists has length at most the length of the shortest path 4 in the even network

dispersive estimate for quasi periodic schrodinger operators on 1 d lattices abstract consider the one dimensional discrete schrodinger operator h h q n q n 1 q n 1 v n q n n z with r d diophantine and v a real analytic function on t d r 2 z d for v sufficiently small we prove the dispersive estimate for every l 1 z 1 e i t h l k 0 ln e 0 a ln ln 2 t 2 d t 1 3 l 1 t 1 t 2 with a and k 0 two absolute constants and e 0 an analytic norm of v the estimate holds for every t d

an algorithm to construct independent spanning trees on parity cubes independent spanning trees have applications in networks such as reliable communication protocols one to all broadcasting reliable broadcasting and secure message distribution thus the designs of independent spanning trees in several classes of networks have been widely investigated however there is a conjecture on independent spanning trees any n connected graph has n independent spanning trees rooted at an arbitrary vertex this conjecture still remains open for n 5 in this paper by proposing an algorithm to construct n independent spanning trees rooted at any vertex we confirm the conjecture on n dimensional parity cube pq n a variant of n dimensional hypercube furthermore we prove that all independent spanning trees rooted at an arbitrary vertex constructed by our construction method are isomorphic and the height of each tree is n 1 for any integer n 2

on the independent spanning trees of recursive circulant graphs g cdm d with d 2 two spanning trees of a graph g are said to be independent if they are rooted at the same vertex r and for each vertex v r in g the two different paths from v to r one path in each tree are internally disjoint a set of spanning trees of g is independent if they are pairwise independent the construction of multiple independent spanning trees has many applications in network communication for instance it is useful for fault tolerant broadcasting and secure message distribution a recursive circulant graph g n d has n cd m vertices labeled from 0 to n 1 where d 2 m 1 and 1 2 where the number of independent spanning trees matches the connectivity of g cd m d

asymptotic stability of ground states in some hamiltonian pdes with symmetry we consider a ground state soliton of a hamiltonian pde we prove that if the soliton is orbitally stable then it is also asymptotically stable the main assumptions are transversal nondegeneracy of the manifold of the ground states linear dispersion in the form of strichartz estimates and nonlinear fermi golden rule we allow the linearization of the equation at the soliton to have an arbitrary number of eigenvalues the theory is tailor made for the application to the translational invariant nls in space dimension 3 the proof is based on the extension of some tools of the theory of hamiltonian systems reduction theory darboux theorem normal form to the case of systems invariant under a symmetry group with unbounded generators

a comparative study of penalization and phase field methods for the solution of the diffusion equation in complex geometries we present a comparative study of penalization and phase field methods for the solution of the diffusion equation in complex geometries embedded using simple cartesian meshes the two methods have been widely employed to solve partial differential equations in complex and moving geometries for applications ranging from solid and fluid mechanics to biology and geophysics their popularity is largely due to their discretization on cartesian meshes thus avoiding the need to create body fitted grids at the same time there are questions regarding their accuracy and it appears that the use of each one is confined by disciplinary boundaries here we compare penalization and phase field methods to handle problems with neumann and robin boundary conditions we discuss extensions for dirichlet boundary conditions and in turn compare with methods that have been explicitly designed to handle dirichlet boundary conditions the accuracy of all methods is analyzed using one and two dimensional benchmark problems such as the flow induced by an oscillating wall and by a cylinder performing rotary oscillations this comparative study provides information to decide which methods to consider for a given application and their incorporation in broader computational frameworks we demonstrate that phase field methods are more accurate than penalization methods on problems with neumann boundary conditions and we present an error analysis explaining this result

packing spanning trees we given an algorithm for packing spanning trees in a graph g v e with capacities on the edges the problem reduces to o v 2 maximum flow computations the algorithm is based on nash williams s proof of a min max relation for this problem

construction of optimal independent spanning trees on folded hypercubes the n dimensional folded hypercube fq n is an important variant of the n dimensional hypercube q n which is obtained from q n by adding an edge between any pair of vertices with complementary addresses the diameter of fq n is n 2 about half the diameter of q n a set of k 2 spanning trees rooted at the same vertex r in a graph g is said to be independent if for each vertex x other than r the k paths from r to x with one path in each spanning tree are internally disjoint by using independent spanning trees ists one can design fault tolerant broadcasting schemes and increase message security in a network recently yang et al proposed an algorithm which can be parallelized for constructing n 1 ists on fq n with the height of each spanning tree being n in this paper we propose an algorithm for constructing n 1 optimal ists on fq n in the sense that there is a shortest path between the only child of the root r and any other vertex in each spanning tree therefore the height of each spanning tree is n 2 1 moreover the algorithm runs in time o n 1 n and can be parallelized to run by using n 2 n processors on fq n in time o n

primal mixed formulations for reaction diffusion systems on deforming domains we propose a finite element formulation for a coupled elasticity reaction diffusion system written in a fully lagrangian form and governing the spatio temporal interaction of species inside an elastic or hyper elastic body a primal weak formulation is the baseline model for the reaction diffusion system written in the deformed domain and a finite element method with piecewise linear approximations is employed for its spatial discretization on the other hand the strain is introduced as mixed variable in the equations of elastodynamics which in turn acts as coupling field needed to update the diffusion tensor of the modified reaction diffusion system written in a deformed domain the discrete mechanical problem yields a mixed finite element scheme based on row wise raviart thomas elements for stresses brezzi douglas marini elements for displacements and piecewise constant pressure approximations the application of the present framework in the study of several coupled biological systems on deforming geometries in two and three spatial dimensions is discussed and some illustrative examples are provided and extensively analyzed

disjoint rooted spanning trees with small depths in debruijn and kautz graphs the problem of broadcasting long messages on store and forward communication networks where a processor node can send and receive messages simultaneously to and from all its neighbors was studied by bermond and fraigniaud in such networks the delays encountered by a message from a node v to all other nodes over a broadcast spanning tree is directly proportional to the length of the paths in the tree over which the message is sent furthermore the speed of the broadcast can be improved by the segmentation of the message at v into equal length segments and then the broadcast of these segments over arc disjoint broadcast spanning trees simultaneously these observations lead bermond and fraigniaud to look for the maximum number of arc disjoint spanning trees in a debruijn network rooted at an arbitrary node with small depths this paper improves and extends the results of the above authors

endpoint strichartz estimates we prove an abstract strichartz estimate which implies previously unknown endpoint strichartz estimates for the wave equation in dimension n 4 and the schrodinger equation in dimension n 3 three other applications are discussed local existence for a nonlinear wave equation and strichartz type estimates for more general dispersive equations and for the kinetic transport equation

long paths and cycles in hypercubes with faulty vertices a fault free path in the n dimensional hypercube q n with f faulty vertices is said to be long if it has length at least 2 n 2f 2 similarly a fault free cycle in q n is long if it has length at least 2 n 2f if all faulty vertices are from the same bipartite class of q n such length is the best possible we show that for every set of at most 2n 4 faulty vertices in q n and every two fault free vertices u and v satisfying a simple necessary condition on neighbors of u and v there exists a long fault free path between u and v this number of faulty vertices is tight and improves the previously known results furthermore we show for every set of at most n 2 10 n 2 1 faulty vertices in q n where n 15 that q n has a long fault free cycle this is a first quadratic bound which is known to be asymptotically optimal

internal modes of discrete solitons near the anti continuum limit of the dnls equation discrete solitons of the discrete nonlinear schrodinger dnls equation are compactly supported in the anti continuum limit of the zero coupling between lattice sites eigenvalues of the linearization of the dnls equation at the discrete soliton determine its spectral stability small eigenvalues bifurcating from the zero eigenvalue near the anti continuum limit were characterized earlier for this model here we analyze the resolvent operator and prove that it is bounded in the neighborhood of the continuous spectrum if the discrete soliton is simply connected in the anti continuum limit this result rules out the existence of internal modes neutrally stable eigenvalues of the discrete spectrum near the anti continuum limit

finding four independent trees motivated by a multitree approach to the design of reliable communication protocols itai and rodeh gave a linear time algorithm for finding two independent spanning trees in a 2 connected graph cheriyan and maheshwari gave an o v 2 algorithm for finding three independent spanning trees in a 3 connected graph in this paper we present an o v 3 algorithm for finding four independent spanning trees in a 4 connected graph we make use of chain decompositions of 4 connected graphs

breathers in nonlinear lattices existence linear stability and quantization abstract it has been proved by the principle of anti integrable limit renamed here anti continuous limit and the implicit function theorem that under rather weak hypotheses there exist spatially localized and time periodic solutions breathers in arrays of nonlinear coupled classical oscillators provided the coupling is not too strong the models can be translationally invariant or not and be of any dimension there are also time periodic solutions corresponding to arbitrary distributions on the lattice of such solutions multibreathers the whole set of these solutions can be labeled by a discrete coding sequence the condition that the frequency of the breather or multibreather solution has no harmonics in the phonon spectrum can be discarded for a large subset of spatially extended time periodic solutions called phonobreather which correspond to special codes which can be also spatially chaotic and where no oscillator is at rest these results extend almost without change to models with coupled rotors the existence of rotobreathers which are exact solutions consisting of a single rotor rotating while the other rotors oscillate is proved there are also multirotobreather solutions and phono rotobreathers etc it is also shown that the phase of the single breathers constituting a multibreather or multirotobreather solution can be submitted to a phase torsion these new dynamical structures carry a nonzero energy flow in models with two dimensions and more the phase torsion of multibreather solutions can generate new dynamical solutions with vortices in the energy flow the dynamics of a quantum electron coupled to a classical lattice is also studied within the same approach from the anti continuous limit the existence of two kinds of polarobreathers which are spatially exponentially localized dynamical solutions is proved for a small enough electronic transfer integral the polarobreathers of the first kind include the standard static polaron the lattice configuration is static time independent when the electron is in an excited state i e the electronic oscillator oscillates periodically they can be spatially chaotic a polarobreather of the second kind corresponds to a localized periodic lattice oscillation associated with a localized quasiperiodic electronic oscillation the most simple of these solutions are those for which both the lattice oscillation and the electron wave function are mostly concentrated close to the same single site the standard floquet analysis of the linear stability of the breather and multibreather solutions is shown to be related with the band spectrum property of the newton operator involved in the implicit function theorem the krein theory of bifurcations is reinterpreted from this point of view the linear stability of the single breather and rotobreather solutions is then proved at least at small enough coupling finally it is shown that in the quantum version of translationally invariant classical models where there cannot exist in principle any strictly localized excitation there are narrow bands of excitations consisting of bound states of many bosons which are the quantum counterpart of the classical breathers the bandwidth of these excitations goes exponentially to zero with the number of bound phonons this theory provides simultaneously existence theorems and a new powerful and accurate numerical method for calculating practically any of the solutions which has been formally predicted to exist these numerical applications are currently written elsewhere

coupling remeshed particle and phase field methods for the simulation of reaction diffusion on the surface and the interior of deforming geometries reaction diffusion systems on the surface and the interior of complex domains are potent models of growth in living organisms the simulation of these models requires numerical methods capable of handling large deformations and the accurate coupling of the evolution of substances in the lumen and on the surface of the deforming geometries here we develop a novel computational method to handle such problems by combining a remeshed particle method with a phase field method remeshed particle methods are well suited to discretizing deforming geometries while the phase field method is used to impose boundary conditions that effectuate the coupling of substances evolving in their lumen and on their surfaces we demonstrate that this hybrid method enables for the first time the accurate coupling of reaction diffusion on a deformable surface and its interior the method is validated on benchmark problems and the effect of lumen diffusion to a pattern forming reaction diffusion system on a deforming surface is

asymptotic stability of breathers in some hamiltonian networks of weakly coupled oscillators we consider a hamiltonian chain of weakly coupled anharmonic oscillators it is well known that if the coupling is weak enough then the system admits families of periodic solutions exponentially localized in space breathers in this paper we prove asymptotic stability in energy space of such solutions the proof is based on two steps first we use canonical perturbation theory to put the system in a suitable normal form in a neighborhood of the breather second we use dispersion in order to prove asymptotic stability the main limitation of the result rests in the fact that the nonlinear part of the on site potential is required to have a zero of order 8 at the origin from a technical point of view the theory differs from that developed for hamiltonian pdes due to the fact that the breather is not a relative equilibrium of the system

demonstration of the stability or instability of multibreathers at low coupling abstract whereas there exists a mathematical proof for one site breathers stability and an unpublished one for two site breathers the methods for determining the stability properties of multibreathers rely on numerical computation of the floquet multipliers or on the weak nonlinearity approximation leading to discrete nonlinear schrodinger equations here we present a set of multibreather stability theorems mst that provides a simple method to determine multibreathers stability in klein gordon systems these theorems are based in the application of degenerate perturbation theory to aubry s band theory we illustrate them with several examples

on dispersion of small energy solutions of the nonlinear klein gordon equation with a potential in this paper we study small amplitude solutions of nonlinear klein gordon equations with a potential under suitable smoothness and decay assumptions on the potential and a genericity assumption on the nonlinearity we prove that all small energy solutions are asymptotically free in cases where the linear system has at most one bound state the result was already proved by soffer and weinstein we obtain here a result valid in the case of an arbitrary number of possibly degenerate bound states the proof is based on a combination of birkhoff normal form techniques and dispersive estimates

dispersive estimate for two periodic discrete one dimensional schrodinger operator abstract for the two periodic discrete one dimensional schrodinger operator h n n 1 n 1 v n n n z with v n 2 v n for every n z and v 0 v 1 r we show the dispersive estimate e i t h l 181 1 v 0 v 1 2 1 4 1 t 1 3 l 1 l 1 z t r

independent spanning trees on multidimensional torus networks two spanning trees rooted at vertex r in a graph g are called independent spanning trees ists if for each vertex v in g vner the paths from vertex v to vertex r in these two trees are internally distinct if the connectivity of g is k the ist problem is to construct k ists rooted at each vertex the ist problem has found applications in fault tolerant broadcasting but it is still open for general graphs with connectivity greater than four in this paper we shall propose a very simple algorithm for solving the ist problem on multidimensional torus networks in our algorithm every vertex can determine its parent for a specific independent spanning tree only depending on its own label thus our algorithm can also be implemented in parallel systems or distributed systems very easily

independent spanning trees in crossed cubes multiple independent spanning trees ists can be used for data broadcasting in networks which can provide advantageous performances such as the enhancement of fault tolerance bandwidth and security however there is a conjecture on the existence of ists in graphs if a graph g is n connected n 1 then there are n ists rooted at an arbitrary vertex in g this conjecture has remained open for n 5 the n dimensional crossed cube cq n is a n connected graph with various desirable properties which is an important variant of the n dimensional hypercube in this paper we study the existence and construction of ists in crossed cubes we first give a proof of the existence of n ists rooted at an arbitrary vertex in cq n n 1 then we propose an o nlog 2n constructive algorithm where n 2 n is the number of vertices in cq n

edge independent spanning trees in augmented cubes edge independent spanning trees eists have important applications in networks such as reliable communication protocols one to all broadcasting and secure message distribution thus their designs in several classes of networks have been widely investigated the n dimensional augmented cube aqn is an important variant of the n dimensional hypercube it is 2n1 regular 2n1 connected n3 vertex symmetric and has diameter of n 2 in this paper by proposing an o nlogn algorithm that constructs 2n1 eists in aqn where n is the number of nodes in aqn we solve the eists problem for this class of graphs since aqn is 2n1 regular the result is optimal with respect to the number of eists constructed

smoothing properties and retarded estimates for some dispersive evolution equations smoothing properties in the form of space time integrability properties play an important role in the study of dispersive evolution equations a number of them follow from a combination of general arguments and specific estimates we present a general formulation which makes the separation between the two types of ingredients as clear as possible and we illustrate it with the examples of the schrodinger equation of the wave equation and of a class of 1 1 dimensional equations related to the benjamin ono equation of special interest for the cauchy problem are retarded estimates expressed in terms of those properties we derive a number of such estimates associated with the last example and we mention briefly an application of those estimates to the cauchy problem for the generalized benjamin ono equation

a kind of conditional vertex connectivity of cayley graphs generated by 2 trees let g be a graph then t v g is called an r k vertex cut if g t is disconnected and each vertex in v g t has at least k neighbors in g t the size of a smallest r k vertex cut is the r k vertex connectivity of g and is denoted by k k g in this paper we determine the numbers k 1 and k 2 for cayley graphs generated by 2 trees including the popular alternating group graphs

asymptotic stability of lattice solitons in the energy space orbital and asymptotic stability for 1 soliton solutions of the toda lattice equations as well as for small solitary waves of the fpu lattice equations are established in the energy space unlike analogous hamiltonian pdes the lattice equations do not conserve the adjoint momentum in fact the toda lattice equation is a bidirectional model that does not fit in with the existing theory for the hamiltonian systems by grillakis shatah and strauss to prove stability of 1 soliton solutions we split a solution around a 1 soliton into a small solution that moves more slowly than the main solitary wave and an exponentially localized part we apply a decay estimate for solutions to a linearized toda equation which has been recently proved by mizumachi and pego to estimate the localized part we improve the asymptotic stability results for fpu lattices in a weighted space obtained by friesecke and pego

many to many disjoint paths in faulty hypercubes this paper considers the problem of many to many disjoint paths in the hypercube q n with f v faulty vertices and f e faulty edges and obtains the following result for any integer k with 1 k n 1 any two sets s and t of k fault free vertices in different parts if f v f e n k 1 then there exist k disjoint fault free s t paths in q n which contains at least 2 n 2 f v vertices this result is optimal in the worst case

resonances radiation damping and instabilitym in hamiltonian nonlinear wave equations we consider a class of nonlinear klein gordon equations which are hamiltonian and are perturbations of linear dispersive equations the unperturbed dynamical system has a bound state a spatially localized and time periodic solution we show that for ge neric nonlinear hamiltonian perturbations all small amplitude so lutions decay to zero as time tends to infinity at an anomalously slow rate in particular spatially localized and time periodic solutions of the linear problem are destroyed by generic nonlinear hamiltonian perturbations via slow radiation of energy to infinity these solutions can therefore be thought of as metastable states the main mecha nism is a nonlinear resonant interaction of bound states eigenfunc tions and radiation continuous spectral modes leading to energy transfer from the discrete to continuum modes this is in contrast to the kam theory in which appropriate nonresonance conditions im ply the persistence of invariant tori a hypothesis ensuring that such a resonance takes place is a nonlinear analogue of the fermi golden rule arising in the theory of resonances in quantum mechanics the techniques used involve i a time dependent method developed by the authors for the treatment of the quantum resonance problem and perturbations of embedded eigenvalues ii a generalization of the hamiltonian normal form appropriate for infinite dimensional dis persive systems and iii ideas from scattering theory the arguments are quite general and we expect them to apply to a large class of

existence of solutions for schrodinger evolution equations we study the existence uniqueness and regularity of the solution of the initial value problem for the time dependent schrodinger equationi u t 1 2 u v t x u u 0 u0 we provide sufficient conditions onv t x such that the equation generates a unique unitary propagatoru t s and such thatu t s u0ec1 l2 c0 h2 n foru0eh2 n the conditions are general enough to accommodate moving singularities of type x 2 n 4 or x n 2 n 3

non linear wave and schrodinger equations i instability of periodic and quasiperiodic solutions we investigate stability of periodic and quasiperiodic solutions of linear wave and schrodinger equations under non linear perturbations we show in the case of the wave equations that such solutions are unstable for generic perturbations for the schrodinger equations periodic solutions are stable while the quasiperiodic ones are not we extend these results to periodic solutions of non linear equations

on the spectral theory and dispersive estimates for a discrete schrodinger equation in one dimension based on the recent work komech et al dispersive estimates for 1d discrete schrodinger and klein gordon equations appl anal 85 1487 2006 for compact potentials we develop the spectral theory for the one dimensional discrete schrodinger operator h v n 1 n 1 2 n vn n we show that under appropriate decay conditions on the general potential and a nonresonance condition at the spectral edges the spectrum of h consists of finitely many eigenvalues of finite multiplicities and the essential absolutely continuous spectrum while the resolvent satisfies the limiting absorption principle and the puiseux expansions near the edges these properties imply the dispersive estimates eithpa c h l 2 l 2 t 3 2 for any fixed 52 and any t 0 where pa c h denotes the spectral projection to the absolutely continuous spectrum of h in addition based on the scattering theory for the discrete jost solutions and the previous results by stefanov and kevrekidis asymptotic behaviour of small