Order of current variance and diffusivity in the rate one totally asymmetric zero range process

We prove that the variance of the current across a characteristic is of order t^{2/3} in a stationary constant rate totally asymmetric zero range process, and that the diffusivity has order t^{1/3}. This is a step towards proving universality of this scaling behavior in the class of one-dimensional interacting systems with one conserved quantity and concave hydrodynamic flux. The proof proceeds via couplings to show the corresponding moment bounds for a second class particle. We build on the methods developed by Balazs-Seppalainen for asymmetric simple exclusion. However, some modifications were needed to handle the larger state space. Our results translate into t^{2/3}-order of variance of the tagged particle on the characteristics of totally asymmetric simple exclusion.Comment: 23 pages; some minor typos correcte

Lexicographic allocations and extreme core payoffs: the case of assignment games

We consider various lexicographic allocation procedures for coalitional games with transferable utility where the payoffs are computed in an externally given order of the players. The common feature of the methods is that if the allocation is in the core, it is an extreme point of the core. We first investigate the general relationship between these allocations and obtain two hierarchies on the class of balanced games. Secondly, we focus on assignment games and sharpen some of these general relationship. Our main result is the coincidence of the sets of lemarals (vectors of lexicographic maxima over the set of dual coalitionally rational payoff vectors), lemacols (vectors of lexicographic maxima over the core) and extreme core points. As byproducts, we show that, similarly to the core and the coalitionally rational payoff set, also the dual coalitionally rational payoff set of an assignment game is determined by the individual and mixed-pair coalitions, and present an efficient and elementary way to compute these basic dual coalitional values. This provides a way to compute the Alexia value (the average of all lemacols) with no need to obtain the whole coalitional function of the dual assignment game

Analytic combinatorics : functional equations, rational and algebraic functions

This report is part of a series whose aim is to present in a synthetic way the major methods and models in analytic combinatorics. Here, we detail the case of rational and algebraic functions and discuss systematically closure properties, the location of singularities, and consequences regarding combinatorial enumeration. The theory is applied to regular and context-free languages, finite state models, paths in graphs, locally constrained permutati- ons, lattice paths and walks, trees, and planar maps

Lexicographic allocations and extreme core payoffs: the case of assignment games

We consider various lexicographic allocation procedures for coalitional games with transferable utility where the payoffs are computed in an externally given order of the players. The common feature of the methods is that if the allocation is in the core, it is an extreme point of the core. We first investigate the general relationships between these allocations and obtain two hierarchies on the class of balanced games. Secondly, we focus on assignment games and sharpen some of these general relationships. Our main result shows that, similarly to the core and the coalitionally rational payoff set, also the dual coalitionally rational payoff set of an assignment game is determined by the individual and mixed-pair coalitions, and present an efficient and elementary way to compute these basic dual coalitional values. As a byproduct we obtain the coincidence of the sets of lemarals (vectors of lexicographic maxima over the set of dual coalitionally rational payoff vectors), lemacols (vectors of lexicographic maxima over the core) and extreme core points. This provides a way to compute the AL-value (the average of all lemacols) with no need to obtain the whole coalitional function of the dual assignment game

The approximate determinantal assignment problem

The Determinantal Assignment Problem (DAP) is one of the central problems of Algebraic Control Theory and refers to solving a system of non-linear algebraic equations to place the critical frequencies of the system to specied locations. This problem is decomposed into a linear and a multi-linear subproblem and the solvability of the problem is reduced to an intersection of a linear variety with the Grassmann variety. The linear subproblem can be solved with standard methods of linear algebra, whereas the intersection problem is a problem within the area of algebraic geometry. One of the methods to deal with this problem is to solve the linear problem and then and which element of this linear space is closer - in terms of a metric - to the Grassmann variety. If the distance is zero then a solution for the intersection problem is found, otherwise we get an approximate solution for the problem, which is referred to as the approximate DAP. In this thesis we examine the second case by introducing a number of new tools for the calculation of the minimum distance of a given parametrized multi-vector that describes the linear variety implied by the linear subproblem, from the Grassmann variety as well as the decomposable vector that realizes this least distance, using constrained optimization techniques and other alternative methods, such as the SVD properties of the so called Grassmann matrix, polar decompositions and mother tools. Furthermore, we give a number of new conditions for the appropriate nature of the approximate polynomials which are implied by the approximate solutions based on stability radius results. The approximate DAP problem is completely solved in the 2-dimensional case by examining uniqueness and non-uniqueness (degeneracy) issues of the decompositions, expansions to constrained minimization over more general varieties than the original ones (Generalized Grassmann varieties), derivation of new inequalities that provide closed-form non-algorithmic results and new stability radii criteria that test if the polynomial implied by the approximate solution lies within the stability domain of the initial polynomial. All results are compared with the ones that already exist in the respective literature, as well as with the results obtained by Algebraic Geometry Toolboxes, e.g., Macaulay 2. For numerical implementations, we examine under which conditions certain manifold constrained algorithms, such as Newton's method for optimization on manifolds, could be adopted to DAP and we present a new algorithm which is ideal for DAP approximations. For higher dimensions, the approximate solution is obtained via a new algorithm that decomposes the parametric tensor which is derived by the system of linear equations we mentioned before

Convex optimization methods for graphs and statistical modeling

Thesis (Ph. D.)--Massachusetts Institute of Technology, Dept. of Electrical Engineering and Computer Science, 2011.Cataloged from PDF version of thesis.Includes bibliographical references (p. 209-220).An outstanding challenge in many problems throughout science and engineering is to succinctly characterize the relationships among a large number of interacting entities. Models based on graphs form one major thrust in this thesis, as graphs often provide a concise representation of the interactions among a large set of variables. A second major emphasis of this thesis are classes of structured models that satisfy certain algebraic constraints. The common theme underlying these approaches is the development of computational methods based on convex optimization, which are in turn useful in a broad array of problems in signal processing and machine learning. The specific contributions are as follows: -- We propose a convex optimization method for decomposing the sum of a sparse matrix and a low-rank matrix into the individual components. Based on new rank-sparsity uncertainty principles, we give conditions under which the convex program exactly recovers the underlying components. -- Building on the previous point, we describe a convex optimization approach to latent variable Gaussian graphical model selection. We provide theoretical guarantees of the statistical consistency of this convex program in the high-dimensional scaling regime in which the number of latent/observed variables grows with the number of samples of the observed variables. The algebraic varieties of sparse and low-rank matrices play a prominent role in this analysis. -- We present a general convex optimization formulation for linear inverse problems, in which we have limited measurements in the form of linear functionals of a signal or model of interest. When these underlying models have algebraic structure, the resulting convex programs can be solved exactly or approximately via semidefinite programming. We provide sharp estimates (based on computing certain Gaussian statistics related to the underlying model geometry) of the number of generic linear measurements required for exact and robust recovery in a variety of settings. -- We present convex graph invariants, which are invariants of a graph that are convex functions of the underlying adjacency matrix. Graph invariants characterize structural properties of a graph that do not depend on the labeling of the nodes; convex graph invariants constitute an important subclass, and they provide a systematic and unified computational framework based on convex optimization for solving a number of interesting graph problems. We emphasize a unified view of the underlying convex geometry common to these different frameworks. We describe applications of both these methods to problems in financial modeling and network analysis, and conclude with a discussion of directions for future research.by Venkat Chandrasekaran.Ph.D

Convex Mathematical Programs for Relational Matching of Object Views

Automatic recognition of objects in images is a difficult and challenging task in computer vision which has been tackled in many different ways. Based on the powerful and widely used concept to represent objects and scenes as relational structures, the problem of graph matching, i.e. to find correspondences between two graphs is a part of the object recognition problem. Belonging to the field of combinatorial optimization graph matching is considered to be one of the most complex problems in computer vision: It is known to be NP-complete in the general case. In this thesis, two novel approaches to the graph matching problem are proposed and investigated. They are based on recent progress in the mathematical literature on convex programming. Starting out from describing the desired matchings by suitable objective functions in terms of binary variables, relaxations of combinatorial constraints and an adequate adaption of the objective function lead to continuous convex optimization problems which can be solved without parameter tuning and in polynomial time. A subsequent post-processing step results in feasible, sub-optimal combinatorial solutions to the original decision problem. In the first part of this thesis, the connection between specific graph-matching problems and the quadratic assignment problem is explored. In this case, the convex relaxation leads to a convex quadratic program , which is combined with a linear program for post-processing. Conditions under which the quadratic assignment representation is adequate from the computer vision point of view are investigated, along with attempts to relax these conditions by modifying the approach accordingly. The second part of this work focuses directly on the matching of subgraphs -- representing a model -- to a considerably larger scene graph. A bipartite matching is extended with a quadratic regularization term to take into account relations within each set of vertices. Based on this convex relaxation, post-processing and the application to computer vision are investigated and discussed. Numerical experiments reveal both the power and the limitations of the approach. For problems of sizes which occur in applications the approach is quite reasonable and often the combinatorial optimal solution is found. For larger instances the intrinsic combinatorial nature of the problem comes out and leads to sub-optimal solutions which, however, are still good