Weight hierarchies of a family of linear codes associated with degenerate quadratic forms

We restrict a degenerate quadratic form $f$ over a finite field of odd characteristic to subspaces. Thus, a quotient space related to $f$ is introduced. Then we get a non-degenerate quadratic form induced by $f$ over the quotient space. Some related results on the subspaces and quotient space are obtained. Based on this, we solve the weight hierarchies of a family of linear codes related to $f.$Comment: 12 page

The Weight Hierarchies of Linear Codes from Simplicial Complexes

The study of the generalized Hamming weight of linear codes is a significant research topic in coding theory as it conveys the structural information of the codes and determines their performance in various applications. However, determining the generalized Hamming weights of linear codes, especially the weight hierarchy, is generally challenging. In this paper, we investigate the generalized Hamming weights of a class of linear code \C over \bF_q, which is constructed from defining sets. These defining sets are either special simplicial complexes or their complements in \bF_q^m. We determine the complete weight hierarchies of these codes by analyzing the maximum or minimum intersection of certain simplicial complexes and all $r$-dimensional subspaces of \bF_q^m, where 1\leq r\leq {\rm dim}_{\bF_q}(\C)

2D growth processes: SLE and Loewner chains

This review provides an introduction to two dimensional growth processes. Although it covers a variety processes such as diffusion limited aggregation, it is mostly devoted to a detailed presentation of stochastic Schramm-Loewner evolutions (SLE) which are Markov processes describing interfaces in 2D critical systems. It starts with an informal discussion, using numerical simulations, of various examples of 2D growth processes and their connections with statistical mechanics. SLE is then introduced and Schramm's argument mapping conformally invariant interfaces to SLE is explained. A substantial part of the review is devoted to reveal the deep connections between statistical mechanics and processes, and more specifically to the present context, between 2D critical systems and SLE. Some of the SLE remarkable properties are explained, as well as the tools for computing with SLE. This review has been written with the aim of filling the gap between the mathematical and the physical literatures on the subject.Comment: A review on Stochastic Loewner evolutions for Physics Reports, 172 pages, low quality figures, better quality figures upon request to the authors, comments welcom

Symmetry reduction in convex optimization with applications in combinatorics

This dissertation explores different approaches to and applications of symmetry reduction in convex optimization. Using tools from semidefinite programming, representation theory and algebraic combinatorics, hard combinatorial problems are solved or bounded. The first chapters consider the Jordan reduction method, extend the method to optimization over the doubly nonnegative cone, and apply it to quadratic assignment problems and energy minimization on a discrete torus. The following chapter uses symmetry reduction as a proving tool, to approach a problem from queuing theory with redundancy scheduling. The final chapters propose generalizations and reductions of flag algebras, a powerful tool for problems coming from extremal combinatorics

Geometric Analysis of Nonlinear Partial Differential Equations

This book contains a collection of twelve papers that reflect the state of the art of nonlinear differential equations in modern geometrical theory. It comprises miscellaneous topics of the local and nonlocal geometry of differential equations and the applications of the corresponding methods in hydrodynamics, symplectic geometry, optimal investment theory, etc. The contents will be useful for all the readers whose professional interests are related to nonlinear PDEs and differential geometry, both in theoretical and applied aspects