Intersection conductance and canonical alternating paths, methods for general finite Markov chains

A transfer matrix approach to the enumeration of colored links

We propose a transfer matrix algorithm for the enumeration of alternating link diagrams with external legs, giving a weight $n$ to each connected component. Considering more general tetravalent diagrams with self-intersections and tangencies allows us to treat topological (flype) equivalences. This is done by means of a finite renormalization scheme for an associated matrix model. We give results, expressed as polynomials in $n$, for the various generating functions up to order 19 (link diagrams), 15 (prime alternating tangles) and 11 (6-legged links) intersections. The limit $n\to\infty$ is s…

具有块交数 $0$ 和 $y\leq10$ 的旗传递点本原拟对称 $2$-设计

Flag-transitive point-primitive quasi-symmetric $2$-designs with block intersection numbers $0$ and $y\leq10$

研究旗传递、点本原且块交数为 $0$ 与 $2\leq y\leq10$ 的拟对称 $2$-设计，分类其自同构群类型，并排除交错群基柱、确定散在群情形。

On the number of alternating paths in random graphs

In the noisy channel model from coding theory, we wish to detect errors introduced during transmission by optimizing various parameters of the code. Bennett, Dudek, and LaForge framed a variation of this problem in the language of alternating paths in edge-colored complete bipartite graphs in 2016. Here, we extend this problem to the random graph $\mathbb{G}(n,p)$. We seek the alternating connectivity, $κ_{r,\ell}(G)$, which is the maximum $t$ such that there is an $r$-edge-coloring of $G$ such that any pair of vertices is connected by $t$ internally disjoint and alternating (i.e. no consecuti…

Finite Groups Generated in Low Real Codimension

We study the intersection lattice of the arrangement $\mathcal{A}^G$ of subspaces fixed by subgroups of a finite linear group $G$. When $G$ is a reflection group, this arrangement is precisely the hyperplane reflection arrangement of $G$. We generalize the notion of finite reflection groups. We say that a group $G$ is generated (resp. strictly generated) in codimension $k$ if it is generated by its elements that fix point-wise a subspace of codimension at most $k$ (resp. precisely $k$). If $G$ is generated in codimension two, we show that the intersection lattice of $\mathcal{A}^G$ is atomic. …

几乎对称的几乎完全交数值半群的结构

The structure of almost symmetric almost complete intersection numerical semigroups

本文证明几乎对称且几乎完全交的数值半群具有级联矩阵结构，并由此得到刚性定理、级联多项式及Herzog–Watanabe问题的肯定解答。

少顶点球面的非面刻画

A non-face characterization of spheres on few vertices

本研究给出顶点数为d+4的单纯d-球面的组合刻画，通过极小非面族的相交模式，证明其为球面当且仅当极小非面数为不小于3的奇数且满足特定相交与划分条件。