Related papers: Combinatorics of Complex Maximal Determinant Matri…
Many enumeration problems in combinatorics, including such fundamental questions as the number of regular graphs, can be expressed as high-dimensional complex integrals. Motivated by the need for a systematic study of the asymptotic…
This PhD thesis has the following structure: Chapter 1 - General introduction; Chapter 2 - Preliminaries; Chapter 3 - The Replicated Transfer Matrix; Chapter 4 - Finite Size Corrections On Random Graphs; Chapter 5 - The Random Field Ising…
We provide an algorithm to construct unitary matrices over finite fields. We present various constructions of Hermitian self-dual code by means of unitary matrices, where some of them generalize the quadratic double circulant constructions.…
This article provides a comprehensive exploration of submodular maximization problems, focusing on those subject to uniform and partition matroids. Crucial for a wide array of applications in fields ranging from computer science to systems…
We present a novel algebraic combinatorial view on low-rank matrix completion based on studying relations between a few entries with tools from algebraic geometry and matroid theory. The intrinsic locality of the approach allows for the…
Effective therapy of complex diseases requires control of highly non-linear complex networks that remain incompletely characterized. In particular, drug intervention can be seen as control of signaling in cellular networks. Identification…
A long-term research proposal on the algebraic structure, the representations and the possible applications of paraparticle algebras is structured in three modules: The first part stems from an attempt to classify the inequivalent gradings…
This dissertation explores applications of discrete geometry in mathematical neuroscience. We begin with convex neural codes, which model the activity of hippocampal place cells and other neurons with convex receptive fields. In Chapter 4,…
Robust optimization is becoming increasingly important in machine learning applications. In this paper, we study a unified framework of robust submodular optimization. We study this problem both from a minimization and maximization…
The main theme of this thesis is the development of computational methods for classes of infinite-dimensional optimization problems arising in optimal control and information theory. The first part of the thesis is concerned with the…
We construct new pairs of orthogonal maximal abelian $*$-subalgebras of $M_6(\mathbb C)$, by classifying all self-adjoint complex Hadamard matrices of order 6. In particular, we exhibit a non-affine one-parameter family of non-equivalent…
This survey article is devoted to general results in combinatorial enumeration. The first part surveys results on growth of hereditary properties of combinatorial structures. These include permutations, ordered and unordered graphs and…
We give a combinatorial model for the bounded derived category of graded modules over the dual numbers in terms of arcs on the integer line with a point at infinity. Using this model we describe the lattice of thick subcategories of the…
We survey interactions between the topology and the combinatorics of complex hyperplane arrangements. Without claiming to be exhaustive, we examine in this setting combinatorial aspects of fundamental groups, associated graded Lie algebras,…
We address combinatorial problems that can be formulated as minimization of a partially separable function of discrete variables (energy minimization in graphical models, weighted constraint satisfaction, pseudo-Boolean optimization, 0-1…
A new class of structured matrices is presented and a closed form formula for their determinant is established. This formula has strong connections with the one for Vandermonde matrices.
We give lower bounds on the largest singular value of arbitrary matrices, some of which are asymptotically tight for almost all matrices. To study when these bounds are exact, we introduce several combinatorial concepts. In particular, we…
We show that the maximal determinant D(n) for $n \times n$ ${\pm 1}$-matrices satisfies $R(n) := D(n)/n^{n/2} \ge \kappa_d > 0$. Here $n^{n/2}$ is the Hadamard upper bound, and $\kappa_d$ depends only on $d := n-h$, where $h$ is the maximal…
This is a survey primarily about determining the border rank of tensors, especially those relevant for the study of the complexity of matrix multiplication. This is a subject that on the one hand is of great significance in theoretical…
The optimal branch number of MDS matrices has established their importance in designing diffusion layers for various block ciphers and hash functions. As a result, numerous matrix structures, including Hadamard and circulant matrices, have…