Information on some recent applications of umbral extensions to discrete mathematics

Algebraic Quantum Codes: Linking Quantum Mechanics and Discrete Mathematics

We present a general framework of quantum error-correcting codes (QECCs) as a subspace of a complex Hilbert space and the corresponding error models. Then we illustrate how QECCs can be constructed using techniques from algebraic coding theory. Additionally, we discuss secondary constructions for QECCs, leading to propagation rules for the parameters of QECCs.

A survey of discrete methods in (algebraic) statistics for networks

Sampling algorithms, hypergraph degree sequences, and polytopes play a crucial role in statistical analysis of network data. This article offers a brief overview of open problems in this area of discrete mathematics from the point of view of a particular family of statistical models for networks called exponential random graph models. The problems and underlying constructions are also related to well-known concepts in commutative algebra and graph-theoretic concepts in computer science. We outline a few lines of recent work that highlight the natural connection between these fields and unify t…

离散数学

Discrete Mathematics

本研究提供一套采用Rota方法的初等计数代数组合学入门课程教学讲义，以组合构造和两种初等原理推导结果，内容自洽仅需高中数学基础，兼具组合与概率视角。

About a counterexample on contractible transformations of graphs

In [A. V. Ivashchenko, Contractible transformations do not change the homology groups of graphs, Discrete Mathematics 126 (1) (1994) p 159,170], Ivashchenko started with the study of contractible graphs, he began with this because they have application to molecular spaces. But in a second paper [A. V. Ivashchenko, Some properties of contractible transformations on graphs, Discrete Mathematics 133 (1) (1994) p 139,145], he made several mistakes. We show in this paper that the results in that second paper are false or the proofs are wrong.

Fundamentals of media theory

Media theory is a new branch of discrete applied mathematics originally developed in mid-nineties to deal with stochastic evolution of preference relations in political science and mathematical psychology. The theory focuses on a particular semigroup of `messages' acting as transformations of a set of `states', called a `medium', whose axioms are both strong and natural. The term `medium' stems from a particular application in which the transformations formalize the effects, on an individual, of `tokens' of information delivered by the environment--that is, the `medium'. However, many differen…

Tanglegrams: a reduction tool for mathematical phylogenetics

Many discrete mathematics problems in phylogenetics are defined in terms of the relative labeling of pairs of leaf-labeled trees. These relative labelings are naturally formalized as tanglegrams, which have previously been an object of study in coevolutionary analysis. Although there has been considerable work on planar drawings of tanglegrams, they have not been fully explored as combinatorial objects until recently. In this paper, we describe how many discrete mathematical questions on trees "factor" through a problem on tanglegrams, and how understanding that factoring can simplify analysis…

Marches aléatoires dans un cône et fonctions discrètes harmoniques

Random walks in cones have the double interest of being at the heart of many probabilistic problems and of being related to many mathematical fields, such as spectral theory, combinatorics, or discrete complex analysis. In this article, we present some key ideas associated with these processes: we will discuss their definition, the link with Brownian motion in cones, as well as some recent research topics such as the construction of discrete harmonic functions.