the local well posedness and global solution for a modified periodic two component camassa holm system abstract the local well posedness and global solution for a modified periodic two component camassa holm system on the circle s are established in the sobolev space h s h s 2 with s 7 2 which are different from that of the two component camassa holm system

approximating the domatic number a set of vertices in a graph is a dominating set if every vertex outside the set has a neighbor in the set the domatic number problem is that of partitioning the vertices of a graph into the maximum number of disjoint dominating sets let n denote the number of vertices delta the minimum degree and delta the maximum degree we show that every graph has a domatic partition with 1 o 1 delta 1 ln n dominating sets and moreover that such a domatic partition can be found in polynomial time this implies a 1 o 1 ln n approximation algorithm for domatic number since the domatic number is always at most delta 1 we also show this to be essentially best possible namely extending the approximation hardness of set cover by combining multiprover protocols with zero knowledge techniques we show that for every epsilon 0 a 1 epsilon ln n approximation implies that np subseteq dtime n o log log n this makes domatic number the first natural maximization problem known to the authors that is provably approximable to within polylogarithmic factors but no better we also show that every graph has a domatic partition with 1 o 1 delta 1 ln delta dominating sets where the o 1 term goes to zero as delta increases this can be turned into an efficient algorithm that produces a domatic partition of omega delta ln delta sets

blow up and global solutions to a new integrable model with two components abstract we will discuss a new integrable model which describes the motion of fluid the present work is mainly concerned with global existence and blow up phenomena which are largely due to the application of conservation laws for this integrable equations moreover a new blow up criterion for nonperiodic case is also established via the associated potential some interesting examples are also given to illustrate the application of our results the precise blow up rate is also investigated finally we will emphasize the relations of classical camassa holm equation and our model by analyzing the existence of global solutions

persistence properties and unique continuation of solutions of the camassa holm equation it is shown that a strong solution of the camassa holm equation initially decaying exponentially together with its spacial derivative must be identically equal to zero if it also decays exponentially at a later time in particular a strong solution of the cauchy problem with compact initial profile can not be compactly supported at any later time unless it is the zero solution

computational simulation of flux trapping and vortex pinning in type ii superconductors effects of pointwise thickness variations and material inhomogeneities in the dynamics of vorticies are investigated by means of appropriately formulated time dependent ginzburg landau tdgl based models for inhomogeneous thin films having constant thickness and homogeneous thin films having variable thickness the resulting models are solved numerically using a nonstandard finite difference method for spatial discretization and a modified forward euler method for temporal discretization in particular vortex pinning and flux trapping phenomena in both types of films are investigated our numerical simulations indicate that the dynamics of vortices are mostly affected by inhomogeneities introduced in the chemical composition of a material as compared to variations introduced in thickness of the material

distributed δ 1 coloring in sublogarithmic rounds we give a new randomized distributed algorithm for 1 coloring in the local model running in o log 2o log log n rounds in a graph of maximum degree this implies that the 1 coloring problem is easier than the maximal independent set problem and the maximal matching problem due to their lower bounds of min log n log log n log log log by kuhn moscibroda and wattenhofer podc 04 our algorithm also extends to list coloring where the palette of each node contains 1 colors we extend the set of distributed symmetry breaking techniques by performing a decomposition of graphs into dense and sparse parts

the cauchy problem for a two component generalized θ equations abstract in this paper we investigate a two component system which arises in shallow water theory the present work is mainly concerned with blow up phenomenon persistence properties and infinite propagation speed for this new kind of system firstly we establish sufficient conditions on the initial data to guarantee wave breaking for this system moreover the persistence properties of the strong solutions are also analyzed finally we intend to investigate infinite propagation speed in the sense that the corresponding solution u x t does not have compact spatial support for t 0 though u 0 c 0 r while the other one x t is always compactly supported under the assumption of its initial compact support condition

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

locality in distributed graph algorithms this paper concerns a number of algorithmic problems on graphs and how they may be solved in a distributed fashion the computational model is such that each node of the graph is occupied by a processor which has its own id processors are restricted to collecting data from others which are at a distance at most t away from them in t time units but are otherwise computationally unbounded this model focuses on the issue of locality in distributed processing namely to what extent a global solution to a computational problem can be obtained from locally available data three results are proved within this model a 3 coloring of an n cycle requires time omega log n this bound is tight by previous work of cole and vishkin any algorithm for coloring the d regular tree of radius r which runs for time at most 2r 3 requires at least omega sqrt d colors in an n vertex graph of largest degree delta an o delta 2 coloring may be found in time o log n

analysis and approximation of the ginzburg landau model of superconductivity the authors consider the ginzburg landau model for superconductivity first some well known features of superconducting materials are reviewed and then various results concerning the model the resultant differential equations and their solution on bounded domains are derived then finite element approximations of the solutions of the ginzburg landau equations are considered and error estimates of optimal order are derived

what can be computed locally the purpose of this paper is a study of computation that can be done locally in a distributed network where locally means within time or distance independent of the size of the network locally checkable labeling lcl problems are considered where the legality of a labeling can be checked locally e g coloring the results include the following there are nontrivial lcl problems that have local algorithms there is a variant of the dining philosophers problem that can be solved locally randomization cannot make an lcl problem local i e if a problem has a local randomized algorithm then it has a local deterministic algorithm it is undecidable in general whether a given lcl has a local algorithm however it is decidable whether a given lcl has an algorithm that operates in a given time t any lcl problem that has a local algorithm has one that is order invariant the algorithm depends only on the order of the processor ids

tri hamiltonian duality between solitons and solitary wave solutions having compact support a simple scaling argument shows that most integrable evolutionary systems which are known to admit a bi hamiltonian structure are in fact governed by a compatible trio of hamiltonian structures we demonstrate how their recombination leads to integrable hierarchies endowed with nonlinear dispersion that supports compactons solitary wave solutions having compact support or cusped and or peaked solitons a general algorithm for effecting this duality between classical solitons and their nonsmooth counterparts is illustrated by the construction of dual versions of the modified korteweg de vries equation the nonlinear schr odinger equation the integrable boussinesq system used to model the two way propagation of shallow water waves and the ito system of coupled nonlinear wave equations these hierarchies include a remarkable variety of interesting integrable nonlinear differential equations textcopyright 1996 the american physical society

fast approximation algorithms for fractional packing and covering problems this paper presents fast algorithms that find approximate solutions for a general class of problems which we call fractional packing and covering problems the only previously known algorithms for solving these problems are based on general linear programming techniques the techniques developed in this paper greatly outperform the general methods in many applications and are extensions of a method previously applied to find approximate solutions to multicommodity flow problems our algorithm is a lagrangian relaxation technique an important aspect of our results is that we obtain a theoretical analysis of the running time of a lagrangian relaxation based algorithm we give several applications of our algorithms the new approach yields several orders of magnitude of improvement over the best previously known running times for algorithms for the scheduling of unrelated parallel machines in both the preemptive and the nonpreemptive models for the job shop problem for the held and karp bound for the tr

sweeping algorithms for inverting the discrete ginzburg landau operator the ginzburg landau equations we study arise in the modeling of superconductivity one widely used method of discretizing the equations together with the associated periodic boundary conditions in the case of a rectangle of dimension 2 leads to a five point stencil solving the system means inverting a sparse matrix of dimension n 2 where n is the number of grid points on each side of the rectangle we propose a method that is similar to the shooting technique in the numerical solution of ordinary differential equations for small n the method requires inverting a full matrix of dimension 2n when n is large an iterative procedure combining partial sweeping and the technique of divide and conquer domain decomposition is appropriate

distributed coloring algorithms for triangle free graphs vertex coloring is a central concept in graph theory and an important symmetry breaking primitive in distributed computing whereas degree graphs may require palettes of 1 colors in the worst case it is well known that the chromatic number of many natural graph classes can be much smaller in this paper we give new distributed algorithms to find k coloring in graphs of girth 4 triangle free graphs girth 5 and trees the parameter k can be at most 1 4 o 1 ln in triangle free graphs and at most 1 o 1 ln in girth 5 graphs and trees where o 1 is a function of specifically for sufficiently large we can find such a coloring in o k log n time moreover for any we can compute such colorings in roughly logarithmic time for triangle free and girth 5 graphs and in o log log log n time on trees as a byproduct our algorithm shows that the chromatic number of triangle free graphs is at most 4 o 1 ln which improves on jamall s recent bound of 67 o 1 ln finally we show that 1 coloring for triangle free graphs can be obtained in sublogarithmic time for any

a two component generalization of the degasperis procesi equation we present two different hamiltonian extensions of the degasperis procesi equation to two component equations the construction is based on the observation that the second hamiltonian operator of the degasperis procesi equation could be considered as the dirac reduced poisson tensor of the second hamiltonian operator of the boussinesq equation the first extension is generated by the hamiltonian operator which is a dirac reduced operator of the generalized but degenerated second hamiltonian operator of the boussinesq equation the second one is obtained by the n 2 supersymmetric extension of the aforementioned method as the byproduct of this procedure we obtain the hamiltonian system of interacting equations which contains the camassa holm and degasperis procesi equations

a distributed 2 e approximation for vertex cover in o log δ e log log δ rounds we present a simple deterministic distributed 2 e approximation algorithm for minimum weight vertex cover which completes in o log elog log rounds where is the maximum degree in the graph for any e 0 that is at most o 1 for a constant e this implies a constant approximation in o log log log rounds which contradicts the lower bound of kmw10

approximating the minimum vertex cover in sublinear time and a connection to distributed algorithms for a given graph g over n vertices let opt g denote the size of an optimal solution in g of a particular minimization problem e g the size of a minimum vertex cover a randomized algorithm will be called an a approximation algorithm with an additive error for this minimization problem if for any given additive error parameter e 0 it computes a value opt such that with probability at least 2 3 it holds that opt g opt a opt g en assume that the maximum degree or average degree of g is bounded in this case we show a reduction from local distributed approximation algorithms for the vertex cover problem to sublinear approximation algorithms for this problem this reduction can be modified easily and applied to other optimization problems that have local distributed approximation algorithms such as the dominating set problem we also show that for the minimum vertex cover problem the query complexity of such approximation algorithms must grow at least linearly with the average degree d of the graph this lower bound holds for every multiplicative factor a and small constant e as long as d o n a in particular this means that for dense graphs it is not possible to design an algorithm whose complexity is o n

primal dual rnc approximation algorithms for set cover and covering integer programs we build on the classical greedy sequential set cover algorithm in the spirit of the primal dual schema to obtain simple parallel approximation algorithms for the set cover problem and its generalizations our algorithms use randomization and our randomized voting lemmas may be of independent interest fast parallel approximation algorithms were known before for set cover though not for the generalizations considered in this paper

asymptotic profiles of solutions to the two component camassa holm system abstract we discuss the two component camassa holm system which arises in shallow water theory this paper is concerned with the asymptotic description of solutions in the following sense the corresponding solution to initial data with algebraic decay at infinity will retain this property at infinity in its lifespan

discrete gauge invariant approximations of a time dependent ginzburg landau model of superconductivity we present here a mathematical analysis of a nonstandard difference method for the numerical solution of the time dependent ginzburg landau models of superconductivity this type of method has been widely used in numerical simulations of the behavior of superconducting materials we also illustrate some of their nice properties such as the gauge invariance being retained in discrete approximations and the discrete order parameter having physically consistent pointwise bound

deterministic local algorithms unique identifiers and fractional graph colouring in the fractional graph colouring problem the task is to schedule the activities of the nodes so that each node is active for 1 time unit in total and at each point of time the set of active nodes forms an independent set we show that for any 1 there exists a deterministic distributed algorithm that finds a fractional graph colouring of length at most 1 in any graph in one synchronous communication round here is the maximum degree of the graph the result is near tight as there are graphs in which the optimal solution has length 1 the result is of course too good to be true the usual definitions of scheduling problems fractional graph colouring fractional domatic partition etc in a distributed setting leave a loophole that can be exploited in the design of distributed algorithms the size of the local output is not bounded our algorithm produces an output that seems to be perfectly good by the usual standards but it is impractical as the schedule of each node consists of a very large number of short periods of activity more generally the algorithm demonstrates that when we study distributed algorithms for scheduling problems we can choose virtually any trade off between the following three parameters t the running time of the algorithm the length of the schedule and the maximum number of periods of activity for any single node here is the objective function of the optimisation problem while captures the subjective quality of the solution if we study for example bounded degree graphs we can trivially keep t and constant at the cost of a large or we can keep and constant at the cost of a large t our algorithm shows that yet another trade off is possible we can keep t and constant at the cost of a large

a simple parallel algorithm for the maximal independent set problem two basic design strategies are used to develop a very simple and fast parallel algorithms for the maximal independent set mis problem the first strategy consists of assigning identical copies o

local computation lower and upper bounds the question of what can be computed and how efficiently is at the core of computer science not surprisingly in distributed systems and networking research an equally fundamental question is what can be computed in a distributed fashion more precisely if nodes of a network must base their decision on information in their local neighborhood only how well can they compute or approximate a global optimization problem in this paper we give the first polylogarithmic lower bound on such local computation for optimization problems including minimum vertex cover minimum connected dominating set maximum matching maximal independent set and maximal matching in addition we present a new distributed algorithm for solving general covering and packing linear programs for some problems this algorithm is tight with the lower bounds whereas for others it is a distributed approximation scheme together our lower and upper bounds establish the local computability and approximability of a large class of problems characterizing how much local information is required to solve these tasks

lower bounds for local approximation in the study of deterministic distributed algorithms it is commonly assumed that each node has a unique o log n bit identifier we prove that for a general class of graph problems local algorithms constant time distributed algorithms do not need such identifiers a port numbering and orientation is sufficient our result holds for so called simple po checkable graph optimisation problems this includes many classical packing and covering problems such as vertex covers edge covers matchings independent sets dominating sets and edge dominating sets we focus on the case of bounded degree graphs and show that if a local algorithm finds a constant factor approximation of a simple po checkable graph problem with the help of unique identifiers then the same approximation ratio can be achieved on anonymous networks as a corollary of our result we derive a tight lower bound on the local approximability of the minimum edge dominating set problem by prior work there is a deterministic local algorithm that achieves the approximation factor of 4 1 2 in graphs of maximum degree this approximation ratio is known to be optimal in the port numbering model our main theorem implies that it is optimal also in the standard model in which each node has a unique identifier our main technical tool is an algebraic construction of homogeneously ordered graphs we say that a graph is r homogeneous if its nodes are linearly ordered so that an fraction of nodes have pairwise isomorphic radius r neighbourhoods we show that there exists a finite r homogeneous 2k regular graph of girth at least g for any

analysis and convergence of a covolume approximation of the ginzburg landau model of superconductivity in this paper we present the mathematical analysis of a covolume method for the approximations of the ginzburg landau gl model for superconductivity a nice feature of this approach is that the gauge invariance properties are retained in discrete approximations based on triangular grids we also use properties of discrete vector fields to study issues such as the gauge choices and their enforcement

a new series of dense graphs of high girth let k 1 be an odd integer t b k 2 4 c and q be a prime power we construct a bipartite q regular edge transitive graph cd k q of order v 2qk t 1 and girth g k 5 if e is the the number of edges of cd k q then e v 1 k t 1 these graphs provide the best known asymptotic lower bound for the greatest number of edges in graphs of order v and girth at least g g 5 g 6 11 12 for g 24 this represents a slight improvement on bounds established by margulis and lubotzky phillips sarnak for 5 g 23 g 6 11 12 it improves on or ties existing bounds