on the traveling salesman problem in binary hamming spaces given a subset x of vertices of the n cube i e the n dimensional hamming space we are interested in the solution of the traveling salesman problem namely the minimal length of a cycle passing through all vertices of x for a given number m we estimate the maximum of these lengths when x ranges over all possible choices of sets of m vertices asymptotically our estimates show that for a number m of vertices growing exponentially in n the maximum is attained for a code with maximal possible minimum distance

the binary weight distribution of the extended 2m 2m 4 code of the reed solomon code over gf 2m with generator polynomial x α x α2 x α3 abstract consider an n k linear code with symbols from gf 2 m if each code symbol is represented by a binary m tuple using a certain basis for gf 2 m we obtain a binary nm km linear code called a binary image of the original code in this paper we present a lower bound on the minimum weight of a binary image of a cyclic code over gf 2 m and the weight enumerator for a binary image of the extended 2 m 2 m 4 code of the reed solomon code over gf 2 m with generator polynomial x x 2 x 3 and its dual code where is a primitive element in gf 2 m

on flows associated to sobolev vector fields in wiener spaces an approach a la diperna lions in this paper we extend the diperna lions theory of flows associated to sobolev vector fields to the case of cameron martin valued vector fields in wiener spaces e having a sobolev regularity the proof is based on the analysis of the continuity equation in e and on uniform gaussian commutator estimates in finite dimensional spaces

ricci curvature of finite markov chains via convexity of the entropy we study a new notion of ricci curvature that applies to markov chains on discrete spaces this notion relies on geodesic convexity of the entropy and is analogous to the one introduced by lott sturm and villani for geodesic measure spaces in order to apply to the discrete setting the role of the wasserstein metric is taken over by a different metric having the property that continuous time markov chains are gradient flows of the entropy using this notion of ricci curvature we prove discrete analogues of fundamental results by bakry emery and otto villani further we show that ricci curvature bounds are preserved under tensorisation as a special case we obtain the sharp ricci curvature lower bound for the discrete hypercube

equations differentielles sur l espace de wiener et formules de cameron martin non lineaires proprietes algebriques des approximations sur r n une formule de perturbation des flots et convergence des approximations formule de changement de variable sur x et equation differentielle

a particle partition of unity method part iii a multilevel solver in this sequel to part i siam j sci comput 22 2000 pp 853 890 and part ii siam j sci comput 23 2002 pp 1655 1682 we focus on the efficient solution of the linear block systems arising from a galerkin discretization of an elliptic partial differential equation of second order with the partition of unity method pum we present a cheap multilevel solver for partition of unity pu discretizations of any order the shape functions of a pum are products of piecewise rational pu functions varphi i with supp varphi i omega i and higher order local approximation functions psi i n usually a local polynomial of degree leq p i furthermore they are noninterpolatory in a multilevel approach we have to cope with not only noninterpolatory basis functions but also with a sequence of nonnested spaces due to the meshfree construction hence injection or interpolatory interlevel transfer operators are not available for our multilevel pum therefore the remaining natural choice for the prolongation operators are l2 projections here we exploit the pum construction of the function spaces and a hierarchical construction of the pu itself to localize the corresponding projection problem this significantly reduces the computational costs associated with the setup and the application of the interlevel transfer operators the second main ingredient of our multilevel solver is the use of a block smoother to treat the local approximation functions psi i n for all n simultaneously the results of our numerical experiments in two and three dimensions show that the convergence rate of the proposed multilevel solver is independent of the number of patches card omega i the convergence rate is slightly dependent on the local approximation orders pi

studying the locator polynomials of minimum weight codewords of bch codes primitive binary cyclic codes of length n 2 sup m are considered a bch code with designed distance delta is denoted b n delta a bch code is always a narrow sense bch code a codeword is identified with its locator polynomial whose coefficients are the symmetric functions of the locators the definition of the code by its zeros set involves some properties for the power sums of the locators moreover the symmetric functions and the power sums of the locators are related to newton s identities an algebraic point of view is presented in order to prove or disprove the existence of words of a given weight in a code the principal result is the true minimum distance of some bch codes of length 255 and 511 which were not known the minimum weight codewords of the codes b n2 sup h 1 are studied it is proved that the set of the minimum weight codewords of the bch code b n 2 sup m 2 1 equals the set of the minimum weight codewords of the punctured reed muller code of length n and order 2 for any m

uniform energy distribution for an isoperimetric problem with long range interactions we study minimizers of a nonlocal variational problem the problem is a mathematical paradigm for the ubiquitous phenomenon of energy driven pattern formation induced by competing short and long range interactions the short range interaction is attractive and comes from an interfacial energy and the long range interaction is repulsive and comes from a nonlocal energy contribution in particular the problem is the sharp interface version of a problem used to model microphase separation of diblock copolymers a natural conjecture is that in all space dimensions minimizers are essentially periodic on an intrinsic scale however proving any periodicity result turns out to be a formidable task in dimensions larger than one in this paper we address a weaker statement concerning the distribution of energy for minimizers we prove in any space dimension that each component of the energy interfacial and nonlocal of any minimizer is uniformly distributed on cubes which are suciently large with respect to the intrinsic length scale moreover we also prove an l 1 bound on the optimal potential associated with the long range interactions this bound allows for an interesting interpretation note that the average volume fraction of the optimal pattern in a subsystem of size r fluctuates around the system average m the bound on the potential yields a rate of decay of these fluctuations as r tends to 1 this rate of decay is stronger than the one for a random checkerboard pattern in this sense the optimal pattern has less large scale variations of the average volume fraction than a pattern with a finite correlation

minimal change order and separability in linear codes a linear code is said to be in minimal change order if each codeword differs from its predecessor by a word of minimum weight a rule is presented for constructing such an order in the case in which the code has a basis of codewords with minimum weight some consequences concerning the ranking and separability are mentioned

existence and stability for fokker planck equations with log concave reference measure we study markov processes associated with stochastic differential equations whose non linearities are gradients of convex functionals we prove a general result of existence of such markov processes and a priori estimates on the transition probabilities the main result is the following stability property if the associated invariant measures converge weakly then the markov processes converge in law the proofs are based on the interpretation of a fokker planck equation as the steepest descent flow of the relative entropy functional in the space of probability measures endowed with the wasserstein distance

calculus and heat flow in metric measure spaces and applications to spaces with ricci bounds from below this paper is devoted to a deeper understanding of the heat flow and to the refinement of calculus tools on metric measure spaces x mathsf d mathfrak m our main results are a general study of the relations between the hopf lax semigroup and hamilton jacobi equation in metric spaces x d the equivalence of the heat flow in l 2 x mathfrak m generated by a suitable dirichlet energy and the wasserstein gradient flow of the relative entropy functional mathrm ent mathfrak m in the space of probability measures open image in new window the proof of density in energy of lipschitz functions in the sobolev space w 1 2 x mathsf d mathfrak m a fine and very general analysis of the differentiability properties of a large class of kantorovich potentials in connection with the optimal transport problem is the fourth achievement of the paper our results apply in particular to spaces satisfying ricci curvature bounds in the sense of lott and villani ann math 169 903 991 2009 and sturm acta math 196 65 131 2006 and acta math 196 133 177 2006 and require neither the doubling property nor the validity of the local poincare inequality

displacement convexity of generalized relative entropies abstract we investigate the m relative entropy which stems from the bregman divergence on weighted riemannian and finsler manifolds we prove that the displacement k convexity of the m relative entropy is equivalent to the combination of the nonnegativity of the weighted ricci curvature and the k convexity of the weight function we use this to show appropriate variants of the talagrand hwi and the logarithmic sobolev inequalities as well as the concentration of measures we also prove that the gradient flow of the m relative entropy produces a solution to the porous medium equation or the fast diffusion equation

on the probability of undetected error for the maximum distance separable codes in this paper we investigate the performance of maximum distance separable codes with symbols from gf q when they are used for pure error detection or for simultaneous error correction and detection over a q input and q output discret memoryless channel with symbol error probability e first we show that the probability of undetected error for an mds code used for pure error detection is upper bounded by q r and decreases monotonically as edecreases from q 1 q to 0 where r is the number of parity check symbols of the code then we show that the probability of undetected error for an mds code used for correcting t or fewer symbol errors is upper bounded by q r sum min i 0 max t min i max n q 1 i and decreases monotonically as e decreases from q 1 q to 0 these results show that the mds codes are effective for both pure error detection and simultaneous error correction and detection

monge kantorovitch measure transportation and monge ampere equation on wiener space let w h be an abstract wiener space assume two i i 1 2 probabilities on w w we give some conditions for the wasserstein distance between 1 and 2 with respect to the cameron martin space to be finite where the infimum is taken on the set of probability measures on w w whose first and second marginals are 1 and 2 in this case we prove the existence of a unique cyclically monotone map t iw with w h such that t maps 1 to 2 moreover if 2 then t is stochastically invertible i e there exists s w w such that s t iw 1 a s and t s iw 2 a s if in addition 1 then there exists a 1 convex function in the gaussian sobolev space such that these results imply that the quasi invariant transformations of the wiener space with finite wasserstein distance from can be written as the composition of a transport map t and a rotation i e a measure preserving map we give also 1 convex sub solutions and ito type solutions of the monge ampere equation on w

on generator matrices of codes the if class of the q ary linear codes of given length dimension and minimum weight is nonempty it is shown to contain a code whose generator matrix consists of words of minimum weight

integration by parts and quasi invariance for heat kernel measures on loop groups abstract integration by parts formulas are established both for wiener measure on the path space of a loop group and for the heat kernel measures on the loop group the wiener measure is defined to be the law of a certain loop group valued brownian motion and the heat kernel measures are time t t 0 distributions of this brownian motion a corollary of either of these integrations by parts formulas is the closability of the pre dirichlet form considered by b k driver and t lohrenz 1996 j functional anal 140 381 448 we also show that the heat kernel measures are quasi invariant under right under right and left translations by finite energy loops

gradient flows of the entropy for finite markov chains let k be an irreducible and reversible markov kernel on a finite set x we construct a metric w on the set of probability measures on x and show that with respect to this metric the law of the continuous time markov chain evolves as the gradient flow of the entropy this result is a discrete counterpart of the wasserstein gradient flow interpretation of the heat flow in rn by jordan kinderlehrer and otto 1998 the metric w is similar to but different from the l2 wasserstein metric and is defined via a discrete variant of the benamou brenier formula

wasserstein space over the wiener space the goal of this paper is to study optimal transportation problems and gradient flows of probability measures on the wiener space based on and extending fundamental results of feyel ustunel carrying out the program of ambrosio gigli savare we present a complete characterization of the derivative processes for certain class of absolutely continuous curves we prove existence of the gradient flow curves for the relative entropy w r t the wiener measure and identify these gradient flow curves with solutions of the ornstein uhlenbeck evolution equation

on the geometry of metric measure spaces ii we introduce a curvature dimension condition cd k n for metric measure spaces it is more restrictive than the curvature bound underline text curv left m text d m right k introduced in sturm k t 2006 on the geometry of metric measure spaces i acta math 196 65 131 which is recovered as the borderline case cd k the additional real parameter n plays the role of a generalized upper bound for the dimension for riemannian manifolds cd k n is equivalent to text ric m left xi xi right k left xi right 2 and dim m n the curvature dimension condition cd k n is stable under convergence for any triple of real numbers k n l the family of normalized metric measure spaces m d m with cd k n and diameter l is compact condition cd k n implies sharp version of the brunn minkowski inequality of the bishop gromov volume comparison theorem and of the bonnet myers theorem moreover it implies the doubling property and local scale invariant poincare inequalities on balls in particular it allows to construct canonical dirichlet forms with gaussian upper and lower bounds for the corresponding heat kernels

an adaptive hp version of the multilevel particle partition of unity method abstract this paper is concerned with the hp adaptive multilevel solution of second order elliptic partial differential equations using the meshfree particle partition of unity method the proposed refinement scheme automatically constructs new discretization points or particles the meshfree analogue of an adaptive h refinement and local approximation spaces with better local resolution a p refinement the refinement process is steered with the help of an a posteriori subdomain type error indicator we present results of numerical experiments in two and three space dimensions which demonstrated the overall efficiency of the proposed scheme

an algebraic treatment of essential boundary conditions in the particle partition of unity method this paper is concerned with the treatment of essential boundary conditions in meshfree methods in particular we focus on the particle partition of unity method ppum however the proposed technique is applicable to any partition of unity based approach we present an efficient scheme for the automatic construction of a direct splitting of a ppum function space into the degrees of freedom suitable for the approximation of the dirichlet data and the degrees of freedom that remain for the approximation of the pde by simple linear algebra notably our approach requires no restrictions on the distribution of the discretization points nor on the employed local approximation spaces we attain the splitting of the global function space from the respective direct splittings of the employed local approximation spaces hence the global splitting can be computed with sub linear complexity due to this direct splitting of the meshfree ppum function space we can implement a conforming local treatment of essential boundary data so that the realization of dirichlet boundary values in the meshfree ppum is straightforward the presented approach yields an optimally convergent scheme which is demonstrated by the presented numerical results

convergence of pointed non compact metric measure spaces and stability of ricci curvature bounds and heat flows aim of this paper is to discuss convergence of pointed metric measure spaces in absence of any compactness condition we propose various definitions show that all of them are equivalent and that for doubling spaces these are also equivalent to the well known measured gromov hausdorff convergence then we show that the curvature conditions cd k and rcd k are stable under this notion of convergence and that the heat flow passes to the limit as well both in the wasserstein and in the l framework we also prove the variational convergence of cheeger energies in the naturally adapted mosco sense and the convergence of the spectra of the laplacian in the case of spaces either uniformly bounded or satisfying the rcd k condition with k 0 when applied to riemannian manifolds our results allow for sequences with diverging dimensions

an analog of the 2 wasserstein metric in non commutative probability under which the fermionic fokker planck equation is gradient flow for the entropy let mathfrak c denote the clifford algebra over mathbb r n which is the von neumann algebra generated by n self adjoint operators qj j 1 n satisfying the canonical anticommutation relations qiqj qjqi 2 iji and let denote the normalized trace on mathfrak c this algebra arises in quantum mechanics as the algebra of observables generated by n fermionic degrees of freedom let mathfrak p denote the set of all positive operators rho in mathfrak c such that 1 these are the non commutative analogs of probability densities in the non commutative probability space mathfrak c tau the fermionic fokker planck equation is a quantum mechanical analog of the classical fokker planck equation with which it has much in common such as the same optimal hypercontractivity properties in this paper we construct a riemannian metric on mathfrak p that we show to be a natural analog of the classical 2 wasserstein metric and we show that in analogy with the classical case the fermionic fokker planck equation is gradient flow in this metric for the relative entropy with respect to the ground state we derive a number of consequences of this such as a sharp talagrand inequality for this metric and we prove a number of results pertaining to this metric several open problems are raised

shape optimization under uncertainty a stochastic programming perspective we present an algorithm for shape optimization under stochastic loading and representative numerical results our strategy builds upon a combination of techniques from two stage stochastic programming and level set based shape optimization in particular usage of linear elasticity and quadratic objective functions permits us to obtain a computational cost which scales linearly in the number of linearly independent applied forces which often is much smaller than the number of different realizations of the stochastic forces numerical computations are performed using a level set method with composite finite elements both in two and in three spatial dimensions

ricci curvature for metric measure spaces via optimal transport we dene a notion of a measured length space x having nonnegative n ricci curvature for n 2 1 1 or having1 ricci curvature bounded below byk fork2 r the denitions are in terms of the displacement convexity of certain functions on the associated wasserstein metric space p2 x of probability measures we show that these properties are preserved under measured gromov hausdor limits we give geometric and analytic consequences this paper has dual goals one goal is to extend results about optimal transport from the setting of smooth riemannian manifolds to the setting of length spaces a second goal is to use optimal transport to give a notion for a measured length space to have ricci curvature bounded below we refer to 11 and 44 for background material on length spaces and optimal transport respectively further bibliographic notes on optimal transport are in appendix f in the present introduction we motivate the questions that we address and we state the main results to start on the geometric side there are various reasons to try to extend notions of curvature from smooth riemannian manifolds to more general spaces a fairly general setting is that of length spaces meaning metric spaces x d in which the distance between two points equals the inmum of the lengths of curves joining the points in the rest of this introduction we assume that x is a compact length space alexandrov gave a good notion of a length space having curvature bounded below by k with k a real number in terms of the geodesic triangles in x in the case of a riemannian manifold m with the induced length structure one recovers the riemannian notion of having sectional curvature bounded below by k length spaces with alexandrov curvature bounded below by k behave nicely with respect to the gromovhausdor topology on compact metric spaces modulo isometries they form

localization and tensorization properties of the curvature dimension condition for metric measure spaces abstract this is an addendum to the paper k bacher k t sturm localization and tensorization properties of the curvature dimension condition for metric measure spaces j funct anal 259 2010 28 56 we prove the tensorization property for the curvature dimension condition add some detailed calculations including explicit dependence of constants and comment on assumptions and conjectures concerning the local to global statement in bacher and sturm 2010 1 and villani 2009 6 respectively

an upper diameter bound for compact ricci solitons with application to the hitchin thorpe inequality in this article stimulated by fernandez lopez and garcia rio we shall give an upper diameter bound for compact shrinking ricci solitons in terms of the range of the scalar curvature as well as in terms of the range of the potential function as an application we shall provide some new sufficient conditions for compact four dimensional shrinking ricci solitons to satisfy the hitchin thorpe inequality