on the positive sum property and the computation of graver test sets in the first part of this paper we discuss the positive sum property a property inherent to lp ip and mip graver test sets and develop based on that property a common notational and algorithmic framework for graver test sets moreover we show how to find an initial feasible solution with the help of universal test sets and deal with termination of the augmentation algorithm that improves this solution to optimality

martingales on riemannian manifolds with prescribed limit abstract we are given a random variable on a riemannian manifold and we study the set of manifold valued martingales converging to this variable more precisely we are interested in uniqueness and existence theorems we use stochastic calculus tools for existence we restrict ourselves to wiener probability spaces and we apply malliavin calculus the relations with some nonlinear partial differential equations are discussed

the complexity of generic primal algorithms for solving general integer programs primal methods constitute a common approach to solving combinatorial optimization problems starting from a given feasible solution they successively produce new feasible solutions with increasingly better objective function value until an optimal solution is reached from an abstract point of view an augmentation problem is solved in each iteration that is given a feasible point these methods find an augmenting vector if one exists usually augmenting vectors with certain properties are sought to guarantee the polynomial running time of the overall algorithm in this paper we show that one can solve every integer programming problem in polynomial time provided one can efficiently solve the directed augmentation problem the directed augmentation problem arises from the ordinary augmentation problem by splitting each direction into its positive and its negative part and by considering linear objectives on these parts our main result states that in order to get a polynomial time algorithm for optimization it is sufficient to efficiently find for any linear objective function in the positive and negative part an arbitrary augmenting vector this result also provides a general framework for the design of polynomial time algorithms for specific combinatorial optimization problems we demonstrate its applicability by considering the min cost flow problem by giving a novel algorithm for linear programming over unimodular spaces and by providing a different proof that for 0 1 integer programming an efficient algorithm solving the ordinary augmentation problem suffices for efficient optimization our main result also implies that directed augmentation is at least as hard as optimization in other words for an np hard problem already the task of finding a directed augmenting vector in polynomial time is hopeless unless p np we illustrate this kind of consequence with the help of the knapsack problem

differentiable and analytic families of continuous martingales in manifolds with connection we prove that the derivative of a differentiable family x t a of continuous martingales in a manifold m is a martingale in the tangent space for the complete lift of the connection in m provided that the derivative is bicontinuous in t and a we consider a filtered probability space t 0 t 1 such that all the real martingales have a continuous version and a manifold m endowed with an analytic connection and such that the complexification of m has strong convex geometry we prove that given an analytic family a l a of random variable with values in m and such that l 0 x0 m there exists an analytic family a x a of continuous martingales such that x1 a l a for this we investigate the convexity of the tangent spaces t n m and we prove that any continuous martingale in any manifold can be uniformly approximated by a discrete martingale up to a stopping time t such that t 1 is arbitrarily small we use this construction of families of martingales in complex analytic manifolds to prove that every 1 measurable random variable with values in a compact convex set v with convex geometry in a manifold with a c1 connection is reachable by a v valued martingale

linear bounds for stochastic dispersion it has been suggested that stochastic flows might be used to model the spread of passive tracers in a turbulent fluid we define a stochastic flow by the equations begin eqnarray phi 0 x x d phi t x f dt phi t x end eqnarray where f t x is a field of semimartingales on x in mathbb r d for d geq 2 whose local characteristics are bounded and lipschitz the particles are points in a bounded set mathscr x and we ask how far the substance has spread in a time t that is we define phi t sup x in mathscr x sup 0 leq t leq t phi t x and seek to bound p phi t z without drift when f cdot x are required to be martingales although single points move on the order of sqrt t it is easy to construct examples in which the supremum phi t still grows linearly in time that is lim inf t to infty phi t t 0 almost surely we show that this is an upper bound for the growth that is we compute a finite constant k 0 depending on the bounds for the local characteristics such that limsup t to infty frac phi t t leq k 0 text almost surely a linear bound on growth holds even when the field itself includes a drift term

a strongly polynomial algorithm to solve combinatorial linear programs khachiyan and recently karmarkar gave polynomial algorithms to solve the linear programming problem these algorithms have a small theoretical drawback namely the number of arithmetic steps depends on the size of the input numbers we present a polynomial linear programming algorithm whose number of arithmetic steps depends only on the size of the numbers in the constraint matrix but is independent of the size of the numbers in the right hand side and objective vectors in particular it gives a polynomial algorithm for the minimum cost flow and multicommodity flow problems in which the number of arithmetic steps is independent of the size of the costs and capacities the algorithm makes use of an existing polynomial linear programming algorithm the problem of whether any algorithm has a running time that is independent even of the size of the numbers in the constraint matrix remains open

n fold integer programming in cubic time n fold integer programming is a fundamental problem with a variety of natural applications in operations research and statistics moreover it is universal and provides a new variable dimension parametrization of all of integer programming the fastest algorithm for n fold integer programming predating the present article runs in time o left n g a l right with l the binary length of the numerical part of the input and g a the so called graver complexity of the bimatrix a defining the system in this article we provide a drastic improvement and establish an algorithm which runs in time o n3l having cubic dependency on n regardless of the bimatrix a our algorithm works for separable convex piecewise affine objectives as well moreover it can be used to define a hierarchy of approximations for any integer programming problem

theory of a curved planar waveguide with robin boundary conditions a model of a thin straight strip with a uniformly curved section and with boundary requirements zeroing at the edges a linear superposition of the wave function and its normal derivative robin boundary condition is analyzed theoretically within the framework of the linear schrodinger equation and is applied to the study of the processes in the bent magnetic multilayers superconducting films and metallic ferrite filled waveguides in particular subband thresholds of the straight and curved parts of the film are calculated and analyzed as a function of the robin parameter 1 lambda with lambda being an extrapolation length entering robin boundary condition for the arbitrary robin coefficients which are equal on the opposite interfaces of the strip and for all bend parameters the lowest mode energy of the continuously curved duct is always smaller than its straight counterpart accordingly the bound state below the fundamental propagation threshold of the straight arms always exists as a result of the bend in terms of the superconductivity language it means an increased critical temperature of the curved film compared to its straight counterpart localized level dependence on the parameters of the curve is investigated with its energy decreasing with increasing bend angle and decreasing bend radius conditions of the bound state existence for the different robin parameters on the opposite edges are analyzed too in particular it is shown that the bound state below the first transverse threshold of the straight arm always exists if the inner extrapolation length is not larger than the outer one in the opposite case there is a range of the bend parameters where the curved film cannot trap the wave and form the localized mode for example for the fixed bend radius the bound state emerges from the continuum at some nonzero bend angle that depends on the difference of the two lengths lambda at the opposite interfaces various transport properties of the film such as interference blockade of the current flow at some special energies is also discussed with the special attention being paid to the transformation from the dirichlet to the neumann case as the extrapolation length lambda sweeps the positive axis

the simplex and policy iteration methods are strongly polynomial for the markov decision problem with a fixed discount rate we prove that the classic policy iteration method howard r a 1960 dynamic programming and markov processes mit cambridge and the original simplex method with the most negative reduced cost pivoting rule of dantzig are strongly polynomial time algorithms for solving the markov decision problem mdp with a fixed discount rate furthermore the computational complexity of the policy iteration and simplex methods is superior to that of the only known strongly polynomial time interior point algorithm ye y 2005 a new complexity result on solving the markov decision problem math oper res 30 3 733 749 for solving this problem the result is surprising because the simplex method with the same pivoting rule was shown to be exponential for solving a general linear programming problem klee v g j minty 1972 how good is the simplex method technical report o shisha ed inequalities iii academic press new york the simplex method with the smallest index pivoting rule was shown to be exponential for solving an mdp regardless of discount rates melekopoglou m a condon 1994 on the complexity of the policy improvement algorithm for markov decision processes informs j comput 6 2 188 192 and the policy iteration method was recently shown to be exponential for solving undiscounted mdps under the average cost criterion we also extend the result to solving mdps with transient substochastic transition matrices whose spectral radii are uniformly below one

test sets for integer programs in this paper i discuss various properties of the simplicial complex of maximal lattice free bodies associated with a matrix a if the matrix satisfies some mild conditions and is generic the edges of the complex form the minimal test set for the family of integer programs obtained by selecting a particular row of a as the objective function and using the remaining rows to impose constraints on the integer variables 1997 the mathematical programming society inc published by elsevier science b v

a bound for the number of different basic solutions generated by the simplex method in this short paper we give an upper bound for the number of different basic feasible solutions generated by the simplex method for linear programming problems lp having optimal solutions the bound is polynomial of the number of constraints the number of variables and the ratio between the minimum and the maximum values of all the positive elements of primal basic feasible solutions when the problem is primal nondegenerate it becomes a bound for the number of iterations the result includes strong polynomiality for markov decision problem by ye http www stanford edu yyye simplexmdp1 pdf 2010 and utilize its analysis we also apply our result to an lp whose constraint matrix is totally unimodular and a constant vector b of constraints is integral

sound propagation in rigid bends a multimodal approach the sound propagation in a waveguide with bend of finite constant curvature is analyzed using multimodal decomposition two infinite first order differential equations are constructed for the pressure and velocity in the bend projected on the local transverse modes a riccati equation for the impedance matrix is then derived which can be numerically integrated after truncation at a sufficient number of modes an example of validation is considered and results show the accuracy of the method and its suitability for the formulation of radiation conditions reflection and transmission coefficients are also computed showing the importance of higher order mode generation at the junction between the bend and the straight ducts the case of varying cross section curved ducts is also considered using multimodal decomposition

central limit theorem in negative curvature we prove a central limit theorem for the distance of the brownian point on the universal cover of a compact negatively curved riemannian manifold the technical point is a contraction property for the leafwise brownian motion along the stable foliation

boundary condition varying circle billiards and gratings the dirichlet singularity waves in a two dimensional domain with robin mixed boundary conditions that vary smoothly along the boundary exhibit unexpected phenomena if the variation includes a d point where the boundary condition is dirichlet vanishing wavefunction a variety of arguments indicate that the system is singular for a circle billiard the boundary condition fails to determine a discrete set of levels so the spectrum is continuous for a diffraction grating defined by periodically varying boundary conditions on the edge of a half plane the phase of a diffracted beam amplitude remains undetermined in both cases the wavefunction on the boundary has a singularity at a d point described by the polylogarithm function

robin boundary value problems on arbitrary domains we develop a theory of generalised solutions for elliptic boundary value problems subject to robin boundary conditions on arbitrary domains which resembles in many ways that of the dirichlet problem in particular we establish lp lq estimates which turn out to be the best possible in that framework we also discuss consequences to the spectrum of robin boundary value problems finally we apply the theory to parabolic equations

quadratic diameter bounds for dual network flow polyhedra both the combinatorial and the circuit diameter of polyhedra are of interest to the theory of linear programming for their intimate connection to a best case performance of linear programming algorithms we study the diameters of dual network flow polyhedra associated to b flows on directed graphs g v e g v e and prove quadratic upper bounds for both of them the minimum of v 1 cdot e v 1 e and frac 1 6 v 3 16 v 3 for the combinatorial diameter and frac v cdot v 1 2 v v 1 2 for the circuit diameter previously bounds on these diameters have only been known for bipartite graphs the situation is much more involved for general graphs in particular we construct a family of dual network flow polyhedra with members that violate the circuit diameter bound for bipartite graphs by an arbitrary additive constant

a scaling algorithm for optimizing arbitrary functions over vertices of polytopes in this paper we present a scaling algorithm for minimizing arbitrary functions over vertices of polytopes in an oracle model of computation which includes an augmentation oracle for the binary case when the vertices are 0 1 vectors we show that the oracle time is polynomial also this algorithm allows us to generalize some concepts of combinatorial optimization concerning performance bounds of greedy algorithms and leads to new bounds for the complexity of the simplex method

on the computational behavior of a polynomial time network flow algorithm a variation on the edmonds karp scaling approach to the minimum cost network flow problem is examined this algorithm which scales costs rather than right hand sides also runs in polynomial time large scale computational experiments indicate that the computational behavior of such scaling algorithms may be much better than had been presumed within several distributions of square dense capacitated transportation problems a cost scaling code scale exhibits linear growth in average execution time with the number of edges while two network simplex codes rnet and gnet exhibit greater than linear growth our experiments reveal that median and mean execution times are predictable with surprising accuracy for all of the three codes and all three distributions from which test problems were generated moreover for fixed problem size individual execution times appear to behave as though they are approximately lognormally distributed with constant variance the experiments also reveal sensitivity of the parameters in the models and in the models themselves to variations in the distribution of problems this argues for caution in the interpretation of such computational studies beyond the realm in which the computations were performed

a polynomial oracle time algorithm for convex integer minimization in this paper we consider the solution of certain convex integer minimization problems via greedy augmentation procedures we show that a greedy augmentation procedure that employs only directions from certain graver bases needs only polynomially many augmentation steps to solve the given problem we extend these results to convex n fold integer minimization problems and to convex 2 stage stochastic integer minimization problems finally we present some applications of convex n fold integer minimization problems for which our approach provides polynomial time solution algorithms

probability convexity and harmonic maps with small image i uniqueness and fine existence this paper uses probability theory to provide an approach to the dirichlet problem for harmonic maps the probabilistic tools used are those of manifold valued brownian motion and martingales probabilistic proofs are given of uniqueness continuous dependence on data and existence of generalized solutions finely harmonic maps for target domains which have convex geometry this class of domain includes all regular geodesic balls links are made with new results in other fields the martingale dirichlet problem riemannian centres of mass

barycenters of measures transported by stochastic flows we investigate the evolution of barycenters of masses transported by stochastic flows the state spaces under consideration are smooth affine manifolds with certain convexity structure under suitable conditions on the flow and on the initial measure the barycenter zt is shown to be a semimartingale and is described by a stochastic differential equation for the hyperbolic space the barycenter of two independent brownian particles is a martingale and its conditional law converges to that of a brownian motion on the limiting geodesic on the other hand for a large family of discrete measures on suitable cartan hadamard manifolds the barycenter of the measure carried by an unstable brownian flow converges to the busemann barycenter of the limiting measure

bound states in curved quantum waveguides a free quantum particle living on a curved planar strip of a fixed width d with dirichlet boundary conditions is studied it can serve as a model for electrons in thin films on a cylinder type substrate or in a curved quantum wire assuming that the boundary of is infinitely smooth and its curvature decays fast enough at infinity it is proved that a bound state with energy below the first transversal mode exists for all sufficiently small d a lower bound on the critical width is obtained using the birman schwinger technique

on augmentation algorithms for linear and integer linear programming from edmonds karp to bland and beyond motivated by bland s linear programming lp generalization of the renowned edmonds karp efficient refinement of the ford fulkerson maximum flow algorithm we analyze three closely related natural augmentation rules for lp and integer linear programming ilp augmentation algorithms for all three rules and in both contexts lp and ilp we bound the number of augmentations extending bland s discrete steepest descent augmentation rule i e choosing directions with the best ratio of cost improvement per unit 1 norm length and then making maximal augmentations in such directions from lp to ilp we i show that the number of discrete steepest descent augmentations is bounded by the number of elements in the graver basis of the problem matrix and ii give the first strongly polynomial time algorithm for n fold ilp for lp two of the rules can suffer from a zig zagging phenomenon and so in those cases we apply the rules more subtly to achieve good bounds our results improve on what is known fo

subspaces with well scaled frames abstract in this paper classes of subspaces are introduced that generalize the regular unimodular subspaces of rational coordinate space these spaces are shown to be of interest in the study of linear and integer programming fundamental properties of standard representative matrices for classes of subspaces are established certain subspaces are shown to be projectively equivalent to regular subspaces well known examples of totally unimodular matrices arising from graphs and intervals of the real line are generalized in this context a generalization of heller s theorem bounding the number of distinct columns of a rank r totally unimodular matrix is established in this setting