positivity of solutions of elliptic equations with nonlocal terms in this paper we study a nonlocal problem for a second order partial differential equation which depends on a parameter n we prove the existence of critical values 0 and 0 such that for all and for all non negative right hand sides our problem has nonnegative solutions we obtain a formula for 0 the maximal possible value of and find the exact value of for spherical we also study the corresponding eigenvalue problem at the end of the paper we consider the application of our results to the nonlinear system describing the distribution of temperature and potential in a microsensor

the evolution of a random vortex filament we study an evolution problem in the space of continuous loops in a three dimensional euclidean space modeled upon the dynamics of vortex lines in 3d incompressible and inviscid fluids we establish existence of a local solution starting from holder regular loops with index greater than 1 3 when the holder regularity of the initial condition x is smaller or equal to 1 2 we require x to be a rough path in the sense of lyons rev mat iberoamericana 14 1998 215 310 system control and rough paths 2002 oxford univ press the solution will then live in an appropriate space of rough paths in particular we can construct local solution starting from almost every brownian loop

sharp estimates and saturation phenomena for a nonlocal eigenvalue problem abstract we determine the shape which minimizes among domains with given measure the first eigenvalue of a nonlocal operator consisting of a perturbation of the standard dirichlet laplacian by an integral of the unknown function we show that this problem displays a saturation behaviour in that the corresponding value of the minimal eigenvalue increases with the weight affecting the average up to a finite critical value of this weight and then remains constant this critical point corresponds to a transition between optimal shapes from one ball as in the faber krahn inequality to two equal balls

efficient elementary and restricted non elementary route pricing abstract column generation is involved in the current most efficient approaches to routing problems set partitioning formulations model routing problems by considering all possible routes and selecting a subset that visits all customers these formulations often produce tight lower bounds and require column generation for their pricing step the bounds in the resulting branch and price are tighter when elementary routes are considered but this approach leads to a more difficult pricing problem balancing the pricing with route relaxations has become crucial for the efficiency of the branch and price for routing problems recently the ng routes relaxation was proposed as a compromise between elementary and non elementary routes the ng routes are non elementary routes with the restriction that when following a customer the route is not allowed to visit another customer that was visited before if they belong to a dynamically computed set the larger the size of these sets the closer the ng route is to an elementary route this work presents an efficient pricing algorithm for ng routes and extends this algorithm for elementary routes therefore we address the shortest path problem with resource constraint spprc and the elementary shortest path problem with resource constraint espprc the proposed algorithm combines the decremental state space relaxation technique dssr with completion bounds we apply this algorithm for the generalized vehicle routing problem gvrp and for the capacitated vehicle routing problem cvrp demonstrating that it is able to price elementary routes for instances up to 200 customers a result that doubles the size of the espprc instances solved to date

integration with respect to fractal functions and stochastic calculus i the classical lebesgue stieltjes integral b a fdg of real or complex valued functions on a finite interval a b is extended to a large class of integrands f and integrators g of unbounded variation the key is to use composition formulas and integration by part rules for fractional integrals and weyl derivatives in the special case of holder continuous functions f and g of summed order greater than 1 convergence of the corresponding riemann stieltjes sums is proved the results are applied to stochastic integrals where g is replaced by the wiener process and f by adapted as well as anticipating random functions in the anticipating case we work within slobodeckij spaces and introduce a stochastic integral for which the classical ito formula remains valid moreover this approach enables us to derive calculation rules for pathwise defined stochastic integrals with respect to fractional brownian motion

flots et series de taylor stochastiques we study the expansion of the solution of a stochastic differential equation as an infinite sum of iterated stochastic stratonovitch integrals this enables us to give a universal and explicit formula for any invariant diffusion on a lie group in terms of lie brackets as well as a universal and explicit formula for the brownian motion on a riemannian manifold in terms of derivatives of the curvature tensor the first of these formulae contains and extends to the non nilpotent case the results of doss 6 sussmann 17 yamato 18 fliess and normand cyrot 7 krener and lobry 19 and kunita 11 on the representation of solutions of stochastic differential equations

a rough path perspective on renormalization abstract we develop the algebraic theory of rough path translation particular attention is given to the case of branched rough paths whose underlying algebraic structure connes kreimer grossman larson makes it a useful model case of a regularity structure in the sense of hairer pre lie structures are seen to play a fundamental rule which allow a direct understanding of the translated i e renormalized equation under consideration this construction is also novel with regard to the algebraic renormalization theory for regularity structures due to bruned hairer zambotti 2016 the links with which are discussed in detail

total blow up versus single point blow up traditional thermal explosion theory is used to describe reaction initiation in condensed explosives and is limited formally to nondeformable materials kassoy and poland 4 significantly extended this theory to develop an ignition model for a reactive gas in a bounded container in order to describe the induction period during this induction period there is a spatially homogeneous pressure rise in the system which causes a com pressive heating effect in the constant volume container mathematically this compressibility of the gas is expressed by means of an integral term in the induction model for the temperature perturbation 0 x t this model is given by e ak 1 j e x t dx y x 2 0 x 0 d x 2 0

complicated dynamics of scalar reaction diffusion equations with a nonlocal term we consider the dynamics of scalar equations ut uxx f x u c x u 0 x l where denotes some weighted spatial average and dinchlet boundary conditions are assumed prescribing f c appropriately it is shown that complicated dynamics can occur specifically linearisations at equilibria can have any number of purely imaginary eigenvalues moreover the higher order terms of the reduced vector field in an associated centre manifold can be prescribed arbitrarily up to any finite order these results are in marked contrast with the case 0 where bounded solutions are known to converge to equilibrium

diffusion models with blow up a number of physical situations including chemical reactions electrical heating and fluid flow give rise to nonlinear diffusion problems in this paper models are derived and results relating to the blow up of the solutions are given some of the proofs are outlined and the physical significance of the results is noted

spectral analysis of a class of nonlocal elliptic operators related to brownian motion with random jumps let d c r d be a bounded domain and let p d denote the space of probability measures on d consider a brownian motion in d which is killed at the boundary and which while alive jumps instantaneously at an exponentially distributed random time with intensity 0 to a new point according to a distribution p d from this new point it repeats the above behavior independently of what has transpired previously the generator of this process is an extension of the operator l defined by l u with the dirichlet boundary condition where v is a nonlocal centering potential defined by v u u d u d the operator l is symmetric only in the case that is normalized lebesgue measure thus only in that case can it be realized as a selfadjoint operator the corresponding semigroup is compact and thus the spectrum of l consists exclusively of eigenvalues as is well known the principal eigenvalue gives the exponential rate of decay in t of the probability of not exiting the domain by time t we study the behavior of the eigenvalues our main focus being on the behavior of the principal eigenvalue for the regimes 1 and 1 we also consider conditions on that guarantee that the principal eigenvalue is monotone increasing or decreasing in

levy area of wiener processes in banach spaces the goal of this paper is to construct canonical levy area processes for banach space valued brownian motions via dyadic approximations the significance of the existence of canonical levy area processes is that a stochastic integration theory can be established for such brownian motions in banach spaces existence of flows for stochastic differential equations with infinite dimensional noise then follows via the results of lyons and lyons and qian see e g system control and rough paths 2000 oxford univ press this investigation involves a careful analysis on the choice of tensor norms motivated by the applications to infinite dimensional stochastic differential equations

a construction of the rough path above fractional brownian motion using volterra s representation this note is devoted to construct a rough path above a multidimensional fractional brownian motion b with any hurst parameter h lesser than 1 2 by means of its representation as a volterra gaussian process this approach yields some algebraic and computational simplifications with respect to a recent preprint by unterberger where the construction of a rough path over b was first introduced

flow of diffeomorphisms induced by a geometric multiplicative functional in this paper it is shown that the unique multiplicative functional solution to a differential equation driven by a geometric multiplicative functional consitutes a flow of local diffeomorphisms in the case where the driving geometric multiplicative functional is generated by a brownian motion the result in particular presents an answer to an open problem proposed in ikeda and watanabe 4

holder continuous rough paths by fourier normal ordering we construct in this article an explicit geometric rough path over arbitrary d dimensional paths with finite 1 variation for any alpha in 0 1 the method may be coined as fourier normal ordering since it consists in a regularization obtained after permuting the order of integration in iterated integrals so that innermost integrals have highest fourier frequencies in doing so there appear non trivial tree combinatorics which are best understood by using the structure of the hopf algebra of decorated rooted trees in connection with the chen or multiplicative property and of the hopf shuffle algebra in connection with the shuffle or geometric property holder continuity is proved by using besov norms the method is well suited in particular in view of applications to probability theory see the companion article 34 for the construction of a rough path over multidimensional fractional brownian motion with hurst index 1 4 or 35 for a short survey in that case

an improved hybrid ant local search algorithm for the partition graph coloring problem in this paper we propose a hybrid ant colony optimization aco algorithm for the partition graph coloring problem pgcp given an undirected graph g v e whose nodes are partitioned into a given number of the sets the goal of the pgcp is to find a subset v v that contains exactly one node from each cluster and a coloring for v so that in the graph induced by v two adjacent nodes have different colors and the total number of used colors is minimal our hybrid algorithm is obtained by executing a local search procedure after every aco iteration the performance of our algorithm is evaluated on a set of instances commonly used as benchmark and the computational results show that compared to state of the art algorithms our improved hybrid aco algorithm achieves solid results in very short run times

blowup in a partial differential equation with conserved first integral a reaction diffusion equation with a nonlocal term is studied the nonlocal term acts to conserve the spatial integral of the unknown function as time evolves such equations give insight into biological and chemical problems where conservation properties predominate the aim of the paper is to understand how the conservation property affects the nature of blowup the equation studied has a trivial steady solution that is proved to be stable existence of nontrivial steady solutions is proved and their instability established numerically blowup is proved for sufficiently large initial data by using a comparison principle in fourier space the nature of the blowup is investigated by a combination of asymptotic and numerical calculations

rough evolution equations we generalize lyons rough paths theory in order to give a pathwise meaning to some nonlinear infinite dimensional evolution equation associated to an analytic semigroup and driven by an irregular noise as an illustration we discuss a class of linear and nonlinear 1d spdes driven by a space time gaussian noise with singular space covariance and brownian time dependence

stochastic analysis rough path analysis and fractional brownian motions in this paper we show by using dyadic approximations the existence of a geometric rough path associated with a fractional brownian motion with hurst parameter greater than 1 4 using the integral representation of fractional brownian motions we furthermore obtain a skohorod integral representation of the geometric rough path we constructed by the results in ly1 a stochastic integration theory may be established for fractional brownian motions and strong solutions and a wong zakai type limit theorem for stochastic differential equations driven by fractional brownian motions can be deduced accordingly the method can actually be applied to a larger class of gaussian processes with covariance functions satisfying a simple decay condition

stochastic calculus with respect to gaussian processes in this paper we develop a stochastic calculus with respect to a gaussian process of the form b t int t 0 k t s dw s where w is a wiener process and k t s is a square integrable kernel using the techniques of the stochastic calculus of variations we deduce change of variable formulas for the indefinite integrals and we study the approximation by riemann sums the particular case of the fractional brownian motion is discussed

controlling rough paths abstract we formulate indefinite integration with respect to an irregular function as an algebraic problem which has a unique solution under some analytic constraints this allows us to define a good notion of integral with respect to irregular paths with holder exponent greater than 1 3 e g samples of brownian motion and study the problem of the existence uniqueness and continuity of solution of differential equations driven by such paths we recover young s theory of integration and the main results of lyons theory of rough paths in holder topology

approximation algorithms for solving the constrained arc routing problem in mixed graphs abstract given a mixed graph g with vertex set v let e and a denote the sets of edges and arcs respectively we use q and z to denote the sets of positive rational numbers and positive integers respectively for any connected mixed graph g v e a w l u with a length function w e a q and two integer functions l u e a z satisfying l e u e for each e e a we are asked to determine a minimum length tour t traversing each e e a at least l e and at most u e times this new constrained arc routing problem generalizes the mixed chinese postman problem let n v and m e a denote the number of vertices and edges including arcs respectively using network flow techniques we design a 1 1 l 0 approximation algorithm in time o n 2 m 3 log n to solve this constrained arc routing problem such that l e u e holds for each edge e e l e u e holds for each arc e a and l 0 min l e e e in addition we present two optimal combinatorial algorithms in times o n 3 and o nm 2 log n to solve this problem for the cases a and e respectively

a nonlocal sturm liouville eigenvalue problem a nonlocal eigenvalue problem of the form u a x u bu u with homogeneous dirichlet boundary conditions is considered where b is a rank one bounded linear operator and x belongs to some bounded interval on the real line the behaviour of the eigenvalues is studied using methods of linear perturbation theory in particular some results are given which ensure that the spectrum remains real a sturm type comparison result is obtained finally these results are applied to the study of some nonlocal reaction diffusion equations

dynamics of a reaction diffusion equation with nonlocal inhibition abstract in this paper we investigate the dynamics of a nonlocal scalar reaction diffusion equation which corresponds to the limiting equation for a singularly perturbed activator inhibitor system the equation is a nonlocal perturbation of the chafee infante problem it is shown that the nonlocal equation generates a gradient flow whose dynamics can differ from the local problem in several interesting ways stable heterogeneous equilibria having any number of zeros on the spatial domain can be obtained by an appropriate choice of parameters the steady state equation exhibits an interesting family of secondary bifurcations near double critical points by analyzing the center manifolds dynamics near the double critical points it is found that a sequence of secondary bifurcations can result in a transfer of stability between two equilibria by reversing a heteroclinic connection between them as a consequence the number of zeros of a solution the lap number can increase along trajectories

on stratonovich and skorohod stochastic calculus for gaussian processes in this article we derive a stratonovich and skorohod type change of variables formula for a multidimensional gaussian process with low holder regularity typically lower than 1 4 to this aim we combine tools from rough paths theory and stochastic analysis

stochastic calculus for fractional brownian motion with hurst exponent h a rough path method by analytic extension the d dimensional fractional brownian motion fbm for short b t b 1 t b d t t r with hurst exponent 0 1 is a d dimensional centered self similar gaussian process with covariance e b i s b i t 1 2 i j lsl 2 ltl 2 t s 2 the long standing problem of defining a stochastic integration with respect to fbm and the related problem of solving stochastic differential equations driven by fbm has been addressed successfully by several different methods although in each case with a restriction on the range of either d or the case corresponds to the usual stochastic integration with respect to brownian motion while most computations become singular when gets under various threshhold values due to the growing irregularity of the trajectories as 0 we provide here a new method valid for any d and for 1 4 by constructing an approximation e t e 0 of fbm which allows to define iterated integrals and then applying the geometric rough path theory the approximation relies on the definition of an analytic process z on the cut plane z c r of which fbm appears to be a boundary value and allows to understand very precisely the well known see 5 but as yet a little mysterious divergence of levy s area for 1 4

ramification of rough paths the stack of iterated integrals of a path is embedded in a larger algebraic structure where iterated integrals are indexed by decorated rooted trees and where an extended chen s multiplicative property involves the durr connes kreimer coproduct on rooted trees this turns out to be the natural setting for a non geometric theory of rough paths

improved lower bounds and exact algorithm for the capacitated arc routing problem in the capacitated arc routing problem carp a subset of the edges of an undirected graph has to be serviced at least cost by a fleet of identical vehicles in such a way that the total demand of the edges serviced by each vehicle does not exceed its capacity this paper describes a new lower bounding method for the carp based on a set partitioning like formulation of the problem with additional cuts this method uses cut and column generation to solve different relaxations of the problem and a new dynamic programming method for generating routes an exact algorithm based on the new lower bounds was also implemented to assess their effectiveness computational results over a large set of classical benchmark instances show that the proposed method improves most of the best known lower bounds for the open instances and can solve several of these for the first time

global asymptotic behaviour in some functional parabolic equations the global asymptotic behavior in some functional parabolic equations was presented the sufficient condition for the global convergence to a spatially uniform equilibrium in the system of functional partial differential equations was provided results were observed to be valid for all non negative spatio temporal kernels which also satisfied a uniform normailzation condition

a lagrangian based approach for the asymmetric generalized traveling salesman problem this paper presents an optimal approach for the asymmetric generalized traveling salesman problem gtsp the gtsp is defined on a directed graph in which the nodes are grouped into m predefined mutually exclusive and exhaustive sets with the arc set containing no intraset arcs the problem is to find a minimum cost m arc directed cycle which includes exactly one node from each set our approach employs a lagrangian relaxation to compute a lower bound on the total cost of an optimal solution the lower bound and a heuristically determined upper bound are used to identify and remove arcs and nodes which are guaranteed not to be in an optimal solution finally we use an efficient branch and bound procedure which exploits the multiple choice structure of the node sets we present computational results for the optimal approach tested on a series of randomly generated problems the results show success on a range of problems with up to 104 nodes

stochastic calculus for fractional brownian motion i theory in this paper a stochastic calculus is given for the fractional brownian motions that have the hurst parameter in 1 2 1 a stochastic integral of ito type is defined for a family of integrands so that the integral has zero mean and an explicit expression for the second moment this integral uses the wick product and a derivative in the path space some ito formulae or change of variables formulae are given for smooth functions of a fractional brownian motion or some processes related to a fractional brownian motion a stochastic integral of stratonovich type is defined and the two types of stochastic integrals are explicitly related a square integrable functional of a fractional brownian motion is expressed as an infinite series of orthogonal multiple integrals

the geometry of the space of branched rough paths we construct an explicit transitive free action of a banach space of holder functions on the space of branched rough paths which yields in particular a bijection between theses two spaces this endows the space of branched rough paths the structure of a principal homogeneous space over a banach space and allows to characterize its automorphisms the construction is based on the baker campbell hausdorff formula and on the hairer kelly map which allows to describe branched rough paths in terms of anisotropic geometric rough paths

etude de l algebre de lie double des arbres enracines decores the double lie algebra ld of rooted trees decorated by a set d is introduced generalising the construction of connes and kreimer it is shown that it is a simple lie algebra its derivations and its automorphisms are described as well as some central extensions finally the category of lowest weight modules is introduced and studied

hopf algebras renormalization and noncommutative geometry we explore the relation between the hopf algebra associated to the renormalization of qft and the hopf algebra associated to the ncg computations of tranverse index theory for foliations

on logistic diffusion equations with nonlocal interaction terms abstract this paper is concerned with logistic diffusion equations with nonlocal interaction terms appearing in population biology we intend to study effects of nonlocal terms and discuss the similarity and difference between local problems and nonlocal problems mainly the stationary problem is investigated for a certain class of nonlocal terms a constructive approach is proposed to look for positive stationary solutions and the unique existence of such a positive solution is established the analysis of stationary solutions depends on the spectrum for the linearized operator around the stationary solution however the linearized operator contains a nonlocal term which makes the spectral analysis delicate and difficult putting some additional assumptions we will derive the asymptotic stability of the unique positive solution and furthermore its global attractivity finally it will be seen that some arguments are valid to show the unique existence of a positive stationary solution for a considerably general class of nonlocal terms

stability and hopf bifurcation in a diffusive logistic population model with nonlocal delay effect abstract a reaction diffusion model with logistic type growth nonlocal delay effect and dirichlet boundary condition is considered and combined effect of the time delay and nonlocal spatial dispersal provides a more realistic way of modeling the complex spatiotemporal behavior the stability of the positive spatially nonhomogeneous positive equilibrium and associated hopf bifurcation are investigated for the case of near equilibrium bifurcation point and the case of spatially homogeneous dispersal kernel

further spectral properties of uniformly elliptic operators that include a non local term in this paper we consider the eigenvalues of a class of non local differential operators the real eigenvalues of certain examples of these operators have previously been plotted as functions of a fundamental parameter in the operator using appropriate computer software here we show how the complex eigenvalues of such operators may also be plotted we use the plots of both the real and the complex eigenvalues to motivate a number of new theoretical results regarding the structure of the spectrum which we subsequently prove under suitable hypotheses

a branch and cut algorithm for the symmetric generalized traveling salesman problem we consider a variant of the classical symmetric traveling salesman problem in which the nodes are partitioned into clusters and the salesman has to visit at least one node for each cluster this np hard problem is known in the literature as the symmetric generalized traveling salesman problem gtsp and finds practical applications in routing scheduling and location routing in a companion paper fischetti et al fischetti m j j salazar p toth 1995 the symmetric generalized traveling salesman polytope networks 26 113 123 we modeled gtsp as an integer linear program and studied the facial structure of two polytopes associated with the problem here we propose exact and heuristic separation procedures for some classes of facet defining inequalities which are used within a branch and cut algorithm for the exact solution of gtsp heuristic procedures are also described extensive computational results for instances taken from the literature and involving up to 442 nodes are reported

an efficient transformation of the generalized vehicle routing problem abstract the generalized vehicle routing problem gvrp is the problem of designing optimal delivery or collection routes subject to capacity restrictions from a given depot to a number of predefined mutually exclusive and exhaustive clusters in this paper we describe an efficient transformation of the gvrp into a capacitated arc routing problem carp for which an exact algorithm and several approximate procedures are reported in literature it constitutes the only known approach for solving the gvrp