symmetric sums of squares over k subset hypercubes we consider the problem of finding sum of squares sos expressions to establish the non negativity of a symmetric polynomial over a discrete hypercube whose coordinates are indexed by the k element subsets of n for simplicity we focus on the case k 2 but our results extend naturally to all values of k ge 2 we develop a variant of the gatermann parrilo symmetry reduction method tailored to our setting that allows for several simplifications and a connection to flag algebras we show that every symmetric polynomial that has a sos expression of a fixed degree also has a succinct sos expression whose size depends only on the degree and not on the number of variables our method bypasses much of the technical difficulties needed to apply the gatermann parrilo method and offers flexibility in obtaining succinct sos expressions that are combinatorially meaningful as a byproduct of our results we arrive at a natural representation theoretic justification for the concept of flags as introduced by razborov in his flag algebra calculus furthermore this connection exposes a family of non negative polynomials that cannot be certified with any fixed set of flags answering a question of razborov in the context of our finite setting

new code upper bounds from the terwilliger algebra and semidefinite programming we give a new upper bound on the maximum size a n d of a binary code of word length n and minimum distance at least d it is based on block diagonalizing the terwilliger algebra of the hamming cube the bound strengthens the delsarte bound and can be calculated with semidefinite programming in time bounded by a polynomial in n we show that it improves a number of known upper bounds for concrete values of n and d from this we also derive a new upper bound on the maximum size a n d w of a binary code of word length n minimum distance at least d and constant weight w again strengthening the delsarte bound and yielding several improved upper bounds for concrete values of n d and w

asymptotics of particle trajectories in infinite one dimensional systems with collisions we investigate the asymptotic behavior of the trajectories of a tagged particle tp in an infinite one dimensional system of point particles the particles move independently when not in contact the only interactions are harris type generalized elastic collisions which prevent crossings this is achieved by relabeling the independent trajectories when they cross when these trajectories are differentiable as in particles with velocities undergoing ornstein uhlenbeck processes collisions correspond to exchange of velocities we prove very generally that the suitably scaled tp trajectory converges weakly to a simple gaussian process this extends the results of spitzer for new tonian particles to very general non crossing processes the proof is based on the consideration of the simpler process which counts the crossings of the origin by the independent trajectories

super brownian motion with reflecting historical paths we consider super brownian motion whose historical paths reflect from each other unlike those of the usual historical super brownian motion we prove tightness for the family of distributions corresponding to a sequence of discrete approximations but we leave the problem of uniqueness of the limit open we prove a few results about path behavior for processes under any limit distribution in particular we show that for any 0 a typical increment of a reflecting historical path over a small time interval t is not greater than t 3 4

a comparison of the sherali adams lov 341sz schrijver and lasserre relaxations for 0 1 programming sherali and adams sa90 lov asz and schrijver ls91 and recently lasserre las01b have proposed lift and project methods for constructing hierarchies of successive linear or semidefinite relaxations of a 0 1 polytope p subseteq or n converging to p in n steps lasserre s approach uses results about representations of positive polynomials as sums of squares and the dual theory of moments we present the three methods in a common elementary framework and show that the lasserre construction provides the tightest relaxations of p as an application this gives a direct simple proof for the convergence of the lasserre s hierarchy we describe applications to the stable set polytope and to the cut polytope

bernoulli property for a one dimensional system with localized interaction we consider a one dimensional system of particles on the half line 0 interacting through elastic collisions among themselves and with a wall at the origin on the first particle a constant forcee is acting no external forces act on the other particles all particles are identical except the first one which has a larger mass we prove that ife is such that the gibbs equilibrium state exists the corresponding equilibrium dynamical system is a bernoulli flow

a change of variable formula for the 2d fractional brownian motion of hurst index bigger or equal to 1 4 we prove a change of variable formula for the 2d fractional brownian motion of index h bigger of equal to 1 4 for h strictly bigger than 1 4 our formula coincides with that obtained by using the rough paths theory for h 1 4 the more interesting case there is an additional term that is a classical wiener integral against an independent standard brownian motion

on the motion of a charged particle interacting with an infinitely extended system we study the time evolution of a charged particle moving in a medium under the action of a constant electric field e in the framework of fully hamiltonian models we discuss conditions on the particle medium interaction which are necessary for the particle to reach a finite limit velocity we first consider the case when the charged particle is confined in an unbounded tube of 3 the electric field e is directed along the symmetry axis of the tube and the particle also interacts with an infinitely many particle system the background system initial conditions are chosen in a set which is typical for any reasonable thermodynamic equilibrium or non equilibrium state we prove that for large e and bounded interactions between the charged particle and the background the velocity v t of the charged particle does not reach a finite limit velocity but it increases to infinite as v t et c 0 1 t where c 0 is a constant independent of e as a corollary we obtain that if the initial conditions of the background system are distributed according to any gibbs state then the average velocity of the charged particle diverges as time goes to infinite this result is obtained for e large enough in comparison with the mean energy of the gibbs state we next study the one dimensional case in which the estimates can be improved we finally discuss at an heuristic level the existence of a finite limit velocity for unbounded interactions and give some suggestions about the case of small electric fields

reduction of symmetric semidefinite programs using the regular ast representation we consider semidefinite programming problems on which a permutation group is acting we describe a general technique to reduce the size of such problems exploiting the symmetry the technique is based on a low order matrix representation of the commutant centralizer ring of the matrix algebra generated by the permutation matrices we apply it to extending a method of de klerk et al that gives a semidefinite programming lower bound to the crossing number of complete bipartite graphs it implies that cr k 8 n 2 9299n 2 6n cr k 9 n 3 8676n 2 8n and for any m 9 lim n to infty frac rm cr k m n z m n geq 0 8594 frac m m 1 where z m n is the zarankiewicz number lfloor frac 1 4 m 1 2 rfloor lfloor frac 1 4 n 1 2 rfloor which is the conjectured value of cr k m n here the best factor previously known was 0 8303 instead of 0 8594

a sum of squares approximation of nonnegative polynomials we show that every real nonnegative polynomial f can be approximated as closely as desired in the l 1 norm of its coefficient vector by a sequence of polynomials f epsilon that are sums of squares the novelty is that each f epsilon has a simple and explicit form in terms of f and epsilon

cones of matrices and set functions and 0 1 optimization it has been recognized recently that to represent a polyhedron as the projection of a higher dimensional but simpler polyhedron is a powerful tool in polyhedral combinatorics a general method is developed to construct higher dimensional polyhedra or in some cases convex sets whose projection approximates the convex hull of 0 1 valued solutions of a system of linear inequalities an important feature of these approximations is that one can optimize any linear objective function over them in polynomial time in the special case of the vertex packing polytope a sequence of systems of inequalities is obtained such that the first system already includes clique odd hole odd antihole wheel and orthogonality constraints in particular for perfect and many other graphs this first system gives the vertex packing polytope for various classes of graphs including t perfect graphs it follows that the stable set polytope is the projection of a polytope with a polynomial number of facets an extension o

weak convergence of the scaled median of independent brownian motions we consider the median of n independent brownian motions denoted by m n t and show that sqrt n m n converges weakly to a centered gaussian process the chief difficulty is establishing tightness which is proved through direct estimates on the increments of the median process an explicit formula is given for the covariance function of the limit process the limit process is also shown to be holder continuous with exponent for all 1 4

numerical block diagonalization of matrix algebras with application to semidefinite programming semidefinite programming sdp is one of the most active areas in mathematical programming due to varied applications and the availability of interior point algorithms in this paper we propose a new pre processing technique for sdp instances that exhibit algebraic symmetry we present computational results to show that the solution times of certain sdp instances may be greatly reduced via the new approach

on brownian excursions in lipschitz domains part i local path properties a necessary and sufficient condition is given for a brownian excursion law in a lipschitz domain to share the local path properties with an excursion law in a halfspace this condition is satisfied for all boundary points of every c1 t domain ct 0 there exists a cl domain such that the condition is satisfied almost nowhere on the boundary a probabilistic interpretation and applications to minimal thinness and boundary behavior of green functions are given 0 introduction we will show that the local path properties of brownian excursions in cl domains a 0 are the same as the local path properties of brownian excursions in a half space theorem 3 1 this need not be the case if the domain is of class c proposition 3 1 the method of proof is based on the exit system theory of maisonneuve 12 we will study brownian excursion laws i e one of the ingredients of an exit system we will give a necessary and sufficient condition for an excursion law in a lipschitz domain to have the same local path properties as an excursion law in a half space theorems 2 1 and 2 2 this last result can be applied to obtain new criteria for minimal thinness theorem 4 1 and existence of a nondegenerate normal derivative for the green function theorem 4 2 local path properties of 1 dimensional brownian excursions have been known for some time they are the same as the local path properties of the 3 dimensional bessel process see 2 10 of ito and mckean 11 or ii 67 of williams 21 dvoretsky and erd6s 8 and shiga and watanabe 18 have given kolmogorov type tests for bessel processes local path properties of excursions of multidimensional brownian motion have been studied by shimura 19 excursions of 2 dimensional brownian motion from a line and burdzy 1 n dimensional case n 2 our theorem about minimal thinness generalizes results of essen and jackson 9 and burdzy 3 4 a new criterion for the existence of the nondegenerate normal derivative for the green function extends some results of widman 20 rippon 17 and burdzy 4 the present article mainly uses ideas from excursion theory see maisonneuve 12 williams 21 and burdzy 1 and potential theory see doob 7 received by the editors july 26 1985 1980 mathematics subject classification 1985 revision primary 60g17 60j50 60j65 secondary 31b25 60j45

strengthened semidefinite programming bounds for codes we give a hierarchy of semidefinite upper bounds for the maximum size a n d of a binary code of word length n and minimum distance at least d at any fixed stage in the hierarchy the bound can be computed to an arbitrary precision in time polynomial in n this is based on a result of de klerk et al math program 2006 about the regular representation for matrix algebras the delsarte bound for a n d is the first bound in the hierarchy and the new bound of schrijver ieee trans inform theory 51 2859 2866 2005 is located between the first and second bounds in the hierarchy while computing the second bound involves a semidefinite program with o n7 variables and thus seems out of reach for interesting values of n schrijver s bound can be computed via a semidefinite program of size o n3 a result which uses the explicit block diagonalization of the terwilliger algebra we propose two strengthenings of schrijver s bound with the same computational complexity

current fluctuations of a system of one dimensional random walks in random environment we study the current of particles that move independently in a common static random environment on the one dimensional integer lattice a two level fluctuation picture appears on the central limit scale the quenched mean of the current process converges to a brownian motion on a smaller scale the current process centered at its quenched mean converges to a mixture of gaussian processes these gaussian processes are similar to those arising from classical random walks but the environment makes itself felt through an additional brownian random shift in the spatial argument of the limiting current process

exploiting symmetries in sdp relaxations for polynomial optimization in this paper we study various approaches for exploiting symmetries in polynomial optimization problems within the framework of semidefinite programming relaxations our special focus is on constrained problems especially when the symmetric group is acting on the variables in particular we investigate the concept of block decomposition within the framework of constrained polynomial optimization problems show how the degree principle for the symmetric group can be computationally exploited and also propose some methods to efficiently compute the geometric quotient

brownian excursions and minimal thinness i on donne des criteres de type integral et un critere de type kolmogorov pour les excursions browniennes comme corollaire

symmetry in rlt type relaxations for the quadratic assignment and standard quadratic optimization problems the reformulation linearization technique rlt introduced in sherali h d adams w p 1990 a hierarchy of relaxations between the continuous and convex hull representations for zero one programming problems siam journal on discrete mathematics 3 3 411 430 provides a way to compute a hierarchy of linear programming bounds on the optimal values of np hard combinatorial optimization problems in this paper we show that in the presence of suitable algebraic symmetry in the original problem data it is sometimes possible to compute level two rlt bounds with additional linear matrix inequality constraints as an illustration of our methodology we compute the best known bounds for certain graph partitioning problems on strongly regular graphs

exploiting special structure in semidefinite programming a survey of theory and applications abstract semidefinite programming sdp may be seen as a generalization of linear programming lp in particular one may extend interior point algorithms for lp to sdp but it has proven much more difficult to exploit structure in the sdp data during computation we survey three types of special structures in sdp data 1 a common chordal sparsity pattern of all the data matrices this structure arises in applications in graph theory and may also be used to deal with more general sparsity patterns in a heuristic way 2 low rank of all the data matrices this structure is common in sdp relaxations of combinatorial optimization problems and sdp approximations of polynomial optimization problems 3 the situation where the data matrices are invariant under the action of a permutation group or more generally where the data matrices belong to a low dimensional matrix algebra such problems arise in truss topology optimization particle physics coding theory computational geometry and graph theory we will give an overview of existing techniques to exploit these structures in the data most of the paper will be devoted to the third situation since it has received the least attention in the literature so far

a representation of local time for lipschitz surfaces suppose thatd n n 2 is a lipschitz domain and letn t r be the number of excursions of brownian motion insided with diameter greater thanr which started before timet thenrn t r converges asr 0 to a constant multiple of local time on d a s and inl p for allp the limit need not exist or may be trivial 0 or in holder domains non tangentially accessible domains and domains whose boundaries have finite surface area

semidefinite code bounds based on quadruple distances let a n d be the maximum number of 0 1 words of length n any two having hamming distance at least d it is proved that a 20 8 256 which implies that the quadruply shortened golay code is optimal moreover it is shown that a 18 6 673 a 19 6 1237 a 20 6 2279 a 23 6 13674 a 19 8 135 a 25 8 5421 a 26 8 9275 a 27 8 17099 a 21 10 47 a 22 10 84 a 24 10 268 a 25 10 466 a 26 10 836 a 27 10 1585 a 28 10 2817 a 25 12 55 and a 26 12 96 the method is based on the positive semidefiniteness of matrices derived from quadruples of words this can be put as constraint in a semidefinite program whose optimum value is an upper bound for a n d the order of the matrices involved is huge however the semidefinite program is highly symmetric by which its feasible region can be restricted to the algebra of matrices invariant under this symmetry by block diagonalizing this algebra the order of the matrices will be reduced so as to make the program solvable with semidefinite programming software in the above range of values of n and d

global optimization with polynomials and the problem of moments we consider the problem of finding the unconstrained global minimum of a real valued polynomial p x mathbb r n to mathbb r as well as the global minimum of p x in a compact set k defined by polynomial inequalities it is shown that this problem reduces to solving an often finite sequence of convex linear matrix inequality lmi problems a notion of karush kuhn tucker polynomials is introduced in a global optimality condition some illustrative examples are provided

symmetry in semidefinite programs this paper is a tutorial in a general and explicit procedure to simplify semidefinite programs which are invariant under the action of a symmetry group the procedure is based on basic notions of representation theory of finite groups as an example we derive the block diagonalization of the terwilliger algebra of the binary hamming scheme in this framework here its connection to the orthogonal hahn and krawtchouk polynomials becomes visible

exploiting symmetry in copositive programs via semidefinite hierarchies copositive programming is a relative young field which has evolved into a highly active research area in mathematical optimization an important line of research is to use semidefinite programming to approximate conic programming over the copositive cone two major drawbacks of this approach are the rapid growth in size of the resulting semidefinite programs and the lack of information about the quality of the semidefinite programming approximations these drawbacks are an inevitable consequence of the intractability of the generic problems that such approaches attempt to solve to address such drawbacks we develop customized solution approaches for highly symmetric copositive programs which arise naturally in several contexts for instance symmetry properties of combinatorial problems are typically inherited when they are addressed via copositive programming as a result we are able to compute new bounds for crossing number instances in complete bipartite graphs

a semidefinite programming hierarchy for packing problems in discrete geometry packing problems in discrete geometry can be modeled as finding independent sets in infinite graphs where one is interested in independent sets which are as large as possible for finite graphs one popular way to compute upper bounds for the maximal size of an independent set is to use lasserre s semidefinite programming hierarchy we generalize this approach to infinite graphs for this we introduce topological packing graphs as an abstraction for infinite graphs coming from packing problems in discrete geometry we show that our hierarchy converges to the independence number

computation of the lasserre ranks of some polytopes over the years various lift and project methods have been proposed to construct hierarchies of successive linear or semidefinite relaxations of a 0 1 polytope p rn that converge to p in n steps many such methods have been shown to require n steps in the worst case in this paper we show that the method of lasserre also requires n steps in the worst case

convergence to a stationary state and diffusion for a charged particle in a standing medium we study a one dimensional semi infinite system of identical particles driven by a constant force acting only on the first particle particles interact through elastic collisions at time zero all particles are at rest and the interparticle distances are i i d r v s the support of the distribution being in d d 0 we show that if d is large enough the dynamics has a strong cluster property and prove for large times convergence to a limiting distribution for the system as seen from the first particle as well as existence of drift velocity and invariance principle for the motion of the first particle

drift and diffusion for a mechanical system we consider a mechanical system in the plane consisting of a vertical rod of length l with its center moving on the horizontal axis subject to elastic collisions with the particles of a free gas and to a constant forcef assuming a suitable initial measure we show that the evolution of the system as seen from the rod is described by an exponentially ergodic irreducible harris chain implying convergence to a stationary invariant measure ast we deduce that in the proper scaling the motion of the rod is described as a drift plus a diffusion we prove in conclusion that the diffusion is nondegenerate and that the drift is nonzero iff 0 and has the same sign off

meteor process on mathbb z d the meteor process is a model for mass redistribution on a graph the case of finite graphs was analyzed in billey et al on meteors earthworms and wimps ann appl probab 2014 this paper is devoted to the meteor process on mathbb z d the process is constructed and a stationary distribution is found convergence to this stationary distribution is proved for a large family of initial distributions the first two moments of the mass distribution at a vertex are computed for the stationary distribution for the one dimensional lattice mathbb z the net flow of mass between adjacent vertices is shown to have bounded variance as time goes to infinity an alternative representation of the process on mathbb z as a collection of non crossing paths is presented the distributions of a tracer particle in this system of non crossing paths are shown to be tight as time goes to infinity

central limit theorems for the one dimensional rayleigh gas with semipermeable barriers a version of the one dimensional rayleigh gas is considered a point particle of massm molecule confined to the unit interval 0 1 is surrounded by an infinite ideal gas of point particles of mass 1 atoms the molecule interacts with the atoms and with the walls via elastic collision central limit theorems are proved for a wide class of additive functionals of this system e g the number of collisions with the walls and the total length of the molecular path

geometric properties of 2 dimensional brownian paths let a be the set of all points of the plane visited by 2 dimensional brownian motion before time 1 with probability 1 all points of a are twist points except a set of harmonic measure zero twist points may be continuously approached in a only along a special spiral although negligible in the sense of harmonic measure various classes of cone points are dense in a with probability 1 cone points may be approached in a within suitable wedges

brownian excursions from hyperplanes and smooth surfaces on obtient une decomposition en produit gauche de la loi d excursion brownienne a n dimensions n 2 a partir d un hyperplan