VC bounds on the cardinality of nearly orthogonal function classes

一种用于最大基数匹配的近线性时间分布式算法

A Nearly Linear-Time Distributed Algorithm for Maximum Cardinality Matching

本文提出 CONGEST 模型下最大基数匹配的随机 $\tilde{O}(μ(G))$ 轮算法，并通过完全去中心化的 alternating base trees 方法在 $\tilde{O}(\ell)$ 轮内高概率找到长度为 $\ell$ 的增广路径。

图的1-近乎边独立数

The $1$-nearly edge independence number of a graph

该研究聚焦图的1-近乎边独立数$α_1'(G)$，推导了给定顶点数下图的该参数紧上下界，刻画了对应最小参数值的一般图与连通图，并提出后续研究的开放问题。

On the number of SQSs, latin hypercubes and MDS codes

It is established that the logarithm of the number of latin $d$-cubes of order $n$ is $Θ(n^{d}\ln n)$ and the logarithm of the number of pairs of orthogonal latin squares of order $n$ is $Θ(n^2\ln n)$. Similar estimations are obtained for systems of mutually strong orthogonal latin $d$-cubes. As a consequence, it is constructed a set of Steiner quadruple systems of order $n$ such that the logarithm of its cardinality is $Θ(n^3\ln n)$ as $n\rightarrow\infty$ and $n\ {\rm mod}\ 6= 2\ {\rm or}\ 4$.

Induced bisecting families for hypergraphs

Two $n$-dimensional vectors $A$ and $B$, $A,B \in \mathbb{R}^n$, are said to be \emph{trivially orthogonal} if in every coordinate $i \in [n]$, at least one of $A(i)$ or $B(i)$ is zero. Given the $n$-dimensional Hamming cube $\{0,1\}^n$, we study the minimum cardinality of a set $\mathcal{V}$ of $n$-dimensional $\{-1,0,1\}$ vectors, each containing exactly $d$ non-zero entries, such that every `possible' point $A \in \{0,1\}^n$ in the Hamming cube has some $V \in \mathcal{V}$ which is orthogonal, but not trivially orthogonal, to $A$. We give asymptotically tight lower and (constructive) upper …

图的1-近乎顶点独立数

The $1$-nearly vertex independence number of a graph

该研究定义了图的1-近乎顶点独立集与对应独立数α₁，给出其显式公式的若干情形，证明了n阶图α₁的紧上下界，并完整刻画了达到等号的极值图。

Optimizers of three-point energies and nearly orthogonal sets

This paper is devoted to spherical measures and point configurations optimizing three-point energies. Our main goal is to extend the classic optimization problems based on pairs of distances between points to the context of three-point potentials. In particular, we study three-point analogues of the sphere packing problem and the optimization problem for $p$-frame energies based on three points. It turns out that both problems are inherently connected to the problem of nearly orthogonal sets by Erdős. As the outcome, we provide a new solution of the Erdős problem from the three-point packing p…

有限域上更大的近似正交集

Larger Nearly Orthogonal Sets over Finite Fields

该研究针对有限域上的k-近似正交集构造问题，证明了特征为p的有限域上存在规模达d的δk/logk次方的最优近似正交集，并给出两个推广结果，采用概率、谱论证与超图容器方法完成证明。

Identities for classical group characters of nearly rectangular shape

We derive several identities that feature irreducible characters of the general linear, the symplectic, the orthogonal, and the special orthogonal groups. All the identities feature characters that are indexed by shapes that are "nearly" rectangular, by which we mean that the shapes are rectangles except for one row or column that might be shorter than the others. As applications we prove new results in plane partitions and tableaux enumeration, including new refinements of the Bender-Knuth and MacMahon (ex-)conjectures.

有限域上的近正交集

Nearly Orthogonal Sets over Finite Fields

本文研究有限域上的 $k$-近正交集，结合概率与谱方法证明其大小可达 $d^{\delta k^{1/(p-1)}/\log k}$，并在二元域上得到接近紧的下界。

Nearly orthogonal vectors and small antipodal spherical codes

How can $d+k$ vectors in $\mathbb{R}^d$ be arranged so that they are as close to orthogonal as possible? In particular, define $θ(d,k):=\min_X\max_{x\neq y\in X}|\langle x,y\rangle|$ where the minimum is taken over all collections of $d+k$ unit vectors $X\subseteq\mathbb{R}^d$. In this paper, we focus on the case where $k$ is fixed and $d\to\infty$. In establishing bounds on $θ(d,k)$, we find an intimate connection to the existence of systems of ${k+1\choose 2}$ equiangular lines in $\mathbb{R}^k$. Using this connection, we are able to pin down $θ(d,k)$ whenever $k\in\{1,2,3,7,23\}$ and establ…

delta定理：忠实正交图表示的维度界

The Delta Theorem: a dimension bound for faithful orthogonal graph representations

本文证明了Maehara的猜想及相关猜想，通过概率构造和运算符推理，展示了图的忠实正交表示的维度界及SAP属性。