Nhan Bao Ho
We study the periodicity of nim-sequences for subtraction games having subtraction sets with three elements. In particular, we give solutions in several cases, and we describe how these subtraction sets can be augmented by additional numbers without changing the nim-sequences. The paper concludes with a conjecture on ultimately bipartite subtraction games.
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Subtraction games with FES sets of size 3
本文推广了AngelaSiegel关于减法对策的工作，其中减法集是N \ X，对于某个有限集X. Siegel证明了对于任何有限集X，G序列最终是算术周期的，并且如果|X| = 1或2，则它是纯算术周期的。本文证明了当|X| = 3时，G序列是纯算术周期的.已知对于|X| \geq 4，序列不总是纯算术周期的。
Infinite Subtraction Games with Periodic Outcomes and Aperiodic SG Values
研究无限减法游戏，发现部分特例的结果序列周期、SG值序列有界非周期，丰富了组合博弈论相关内容。
On sequences arising from randomizing subtraction games
本文研究随机化单堆减法博弈中获胜概率序列的行为，利用线性递推关系刻画这些序列并分析其极限性质，完全解决了元素少于3个的减法集合情形并部分解决一般情形。
Comply subtraction games avoiding arithmetic progressions
公平减法游戏的非负整数已经研究了许多和详细讨论，例如显着的工作赢得方式康威，Berlekamp和盖伊。我们描述如何遵守这些游戏的变化，类似于那些介绍了Holshouser，Reiter，史密斯，Stennician，可以被定义为有其套的获胜位置相同的知名集避免等差数列，如$x+z=2y$，研究了Szerkeres，Erd \H OS和图兰，以及许多其他人，从而探索一个新的领域相结合的想法从组合游戏和组合数论。我们所考虑的集合是贪婪的，也就是说，在我们的例子中：递归地将一个新的非负整数包含到集合中，当且仅当它不与较小的项形成一个三项等差数列。众所周知，这样得到的集合等价于下面的对数线性时间闭表达式：每个获胜位置只包含基数为3的扩展中的数字0和1。事实上，这一套是不可能的一套获胜的立场，一个经典的减法游戏，在某种意义上介绍了最近由杜尚和里戈。因此，我们的减法游戏的服从规则的推广可以被看作是解决新的集合类作为堆游戏的获胜位置。在这方面的$\star$ -运营商不变减法游戏介绍了拉尔森，Hegarty和Fraenkel。我们为我们的游戏定义了一个类似的操作符。我们的服从博弈可以推广到几个维度。在二维空间中，获胜的位置可以用某些贪婪的排列来表示，这些排列避免了等差数列，其中之一是最近由Hegarty引入的;而其他的排列则推广了经典的组合游戏，如Nim和Wythoff Nim。
Tie-breaking in self interest cumulative subtraction games
本文研究了自我利益累积减法游戏中，通过引入tie-breaking规则单调性，证明了在两动作减法集情况下，友好玩家优于对抗性玩家。
Additive sink subtraction
本文研究加法沉降减法游戏，证明其nim序列具有显式线性或二次周期，提出与经典墙减法的对偶性猜想。
We introduce a class of normal play partizan games, called Complementary Subtraction. Let $A$ denote your favorite set of positive integers. This is Left's subtraction set, whereas Right subtracts numbers not in $A$. The Golden Nugget Subtraction Game has the $A$ and $B$ sequences, from Wythoff's game, as the two complementary subtraction sets. As a function of the heap size, the maximum size of the canonical forms grows quickly. However, the value of the heap is either a number or, in reduced canonical form, a switch. We find the switches by using properties of the Fibonacci word and standard…
On the compatible sets expansion of the Tutte polynomial
本文回答了Kochol关于拟阵Tutte多项式相容集展开与内部-外部活动公式关系的疑问，将其作为对Las Vergnas拟阵透视三元Tutte多项式展开公式推广的特例，并给出了双射的直接证明。
Partizan subtraction games are combinatorial games where two players, say Left and Right, alternately remove a number n of tokens from a heap of tokens, with $n \in S_L$ (resp. $n \in S_R$) when it is Left's (resp. Right's) turn. The first player unable to move loses. These games were introduced by Fraenkel and Kotzig in 1987, where they introduced the notion of dominance, i.e. an asymptotic behavior of the outcome sequence where Left always wins if the heap is sufficiently large. In the current paper, we investigate the other kinds of behaviors for the outcome sequence. In addition to dominan…
Subtraction games in more than one dimension
本文研究多维公平向量减法博弈，通过推广P到P原理和着色自动机求解二维三移动规则集实例，证明行列最终周期性，并提出正则性相关猜想。
Counting independent sets in expanding bipartite regular graphs
本文针对满足顶点扩张性质的一般正则二部图，利用聚合物模型和簇展开给出独立集数量的渐近展开，并证明其涵盖超立方体、中间层图及有界二部正则基图的笛卡尔积，同时提供顶点扩张的强界。
Given a positive integer $M$ and a real number $q \in (1,M+1]$, an expansion of a real number $x \in \left[0,M/(q-1)\right]$ over the alphabet $A=\{0,1,\ldots,M\}$ is a sequence $(c_i) \in A^{\mathbb N}$ such that $x=\sum_{i=1}^{\infty}c_iq^{-i}$. Generalizing many earlier results, we investigate in this paper the topological properties of the set $U_q$ consisting of numbers $x$ having a unique expansion of this form, and the combinatorial properties of the set $U_q'$ consisting of their corresponding expansions. We also provide shorter proofs of the main results of Baker in [B] by adapting th…
In recent years, high dimensional expanders have been found to have a variety of applications in theoretical computer science, such as efficient CSPs approximations, improved sampling and list-decoding algorithms, and more. Within that, an important high dimensional expansion notion is \emph{cosystolic expansion}, which has found applications in the construction of efficiently decodable quantum codes and in proving lower bounds for CSPs. Cosystolic expansion is considered with systems of equations over a group where the variables and equations correspond to faces of the complex. Previous works…
Simplices in large sets and directional expansion in ergodic actions
本文研究遍历 $\mathbb{Z}^r$-作用沿循环子群的方向扩张，并在谱条件下证明正测度可测集可被显著扩张，进而推出正上密度集合中 $r$-单纯形体积含无限等差数列。
We study random compact subsets of R^3 which can be described as "random Menger sponges". We use those random sets to construct a pair of compact sets A and B in R^3 which are of the same positive measure, such that A can be covered by finitely many translates of B, B can be covered by finitely many translates of A, and yet A and B are not equidecomposable. Furthermore, we construct the first example of a compact subset of R^3 of positive measure which is not a domain of expansion. This answers a question of Adrian Ioana.
Unique-neighbor Expanders with Better Expansion for Polynomial-sized Sets
本文通过三部乘积、广义存在性论证及新归约技术，构造了三类强显式唯一邻居扩张图，在保留线性大小集合扩张性的同时，显著提升了对多项式大小集合的唯一邻居扩张性能。
Definable sets containing productsets in expansions of groups
本文研究群一阶扩张中可定义集的积集性质，分别在稳定公式、稳定群及 distal 扩张情形下，利用 Banach 密度、coheir 独立性与正则引理给出刻画和可定义版本。
Recent works have shown that expansion of pseudorandom sets is of great importance. However, all current works on pseudorandom sets are limited only to product (or approximate product) spaces, where Fourier Analysis methods could be applied. In this work we ask the natural question whether pseudorandom sets are relevant in domains where Fourier Analysis methods cannot be applied, e.g., one-sided local spectral expanders. We take the first step in the path of answering this question. We put forward a new definition for pseudorandom sets, which we call ``double balanced sets''. We demonstrate th…
Kahale proved that linear sized sets in $d$-regular Ramanujan graphs have vertex expansion $\sim\frac{d}{2}$ and complemented this with construction of near-Ramanujan graphs with vertex expansion no better than $\frac{d}{2}$. However, the construction of Kahale encounters highly local obstructions to better vertex expansion. In particular, the poorly expanding sets are associated with short cycles in the graph. Thus, it is natural to ask whether high-girth Ramanujan graphs have improved vertex expansion. Our results are two-fold: 1. For every $d = p+1$ for prime $p$ and infinitely many $n$, we…
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