Zachary Chase
We prove that for any distinct $x,y \in \{0,1\}^n$, there is a deterministic finite automaton with $\widetilde{O}(n^{1/3})$ states that accepts $x$ but not $y$. This improves Robson's 1989 upper bound of $\widetilde{O}(n^{2/5})$.
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For a given language $L$, we study the languages $X$ such that for all distinct words $u, v \in L$, there exists a word $x \in X$ that appears a different number of times as a factor in $u$ and in $v$. In particular, we are interested in the following question: For which languages $L$ does there exist a finite language $X$ satisfying the above condition? We answer this question for all regular languages and for all sets of factors of infinite words.
An Elementary Proof of the $\widetilde O(n^{1/3})$ Bound for Separating Words
针对分离单词问题，用有限差分和二阶实递推截断替代复分析估计，给出初等证明，得到 $O(n^{1/3}(\log n)^{7/3})$ 状态的显式上界。
Exact upper bounds for the minimum sizes of strong and weak separating path systems of cliques
研究确定了完全图强分离数和弱分离数的上界，改进了之前的结果。
An $N\times n$ matrix on $q$ symbols is called $\{w_1,\ldots,w_t\}$-separating if for arbitrary $t$ pairwise disjoint column sets $C_1,\ldots,C_t$ with $|C_i|=w_i$ for $1\le i\le t$, there exists a row $f$ such that $f(C_1),\ldots,f(C_t)$ are also pairwise disjoint, where $f(C_i)$ denotes the collection of components of $C_i$ restricted to row $f$. Given integers $N,q$ and $w_1,\ldots,w_t$, denote by $C(N,q,\{w_1,\ldots,w_t\})$ the maximal $n$ such that a corresponding matrix does exist. The determination of $C(N,q,\{w_1,\ldots,w_t\})$ has received remarkable attentions during the recent years…
A Tighter Upper Bound for the Number of Distinct Squares in Circular Words
研究圆周词中不同平方数的数量，改进上界至1.5n。
A power is a word of the form $\underbrace{uu...u}_{k \; \text{times}}$, where $u$ is a word and $k$ is a positive integer and a square is a word of the form $uu$. Fraenkel and Simpson conjectured in 1998 that the number of distinct squares in a word is bounded by the length of the word. This conjecture was proven recently by Brlek and Li. Besides, there exists a stronger upper bound for binary words conjectured by Jonoska, Manea and Seki stating that for a word of length $n$ over the alphabet $\left\{a, b\right\}$, if we let $k$ be the least of the number of a's and the number of b's and $k \…
Separating Notions of Graph Width: the Adaptive, Normal, Linear, Entropic, and Submodular Width
本文构造了一个32顶点的简单图，证明其五种图宽度两两不同，并精确计算除熵宽度外的所有宽度，利用不等式给出熵宽度的上下界及多拟阵构造。
A non-empty word $w$ is a border of the word $u$ if $\vert w\vert<\vert u\vert$ and $w$ is both a prefix and a suffix of $u$. A word $u$ with the border $w$ is closed if $u$ has exactly two occurrences of $w$. A word $u$ is privileged if $\vert u\vert\leq 1$ or if $u$ contains a privileged border $w$ that appears exactly twice in $u$. Peltomäki (2016) presented the following open problem: "Give a nontrivial upper bound for $B(n)$", where $B(n)$ denotes the number of privileged words of length $n$. Let $D(n)$ denote the number of closed words of length $n$. Let $q>1$ be the size of the alphabet…
A finite word $w$ of length $n$ contains at most $n+1$ distinct palindromic factors. If the bound $n+1$ is attained, the word $w$ is called rich. An infinite word $w$ is called rich if every finite factor of $w$ is rich. Let $w$ be a word (finite or infinite) over an alphabet with $q>1$ letters, let $F(w,n)$ be the set of factors of length $n$ of the word $w$, and let $F_p(w,n)\subseteq F(w,n)$ be the set of palindromic factors of length $n$ of the word $w$. We present several upper bounds for $| F(w,n)|$ and $| F_p(w,n)|$, where $w$ is a rich word. In particular we show that \[| F(w,n)| \leq …
We study two parameters obtained from the Euler characteristic by replacing the number of faces with that of induced and induced non-separating cycles. By establishing monotonicity of such parameters under certain homomorphism and edge contraction, we obtain new upper bounds on the chromatic number in terms of the number of induced cycles and the Hadwiger number in terms of the number of induced non-separating cycles. As an application, we show that a 3-connected graph with average degree $k\ge 2$ have at least $(k-1)|V|+Ck^{3}\log^{3/2}k$ induced non-separating cycles for some explicit consta…
The work in this article is concerned with two different types of families of finite sets: separating families and splitting families (they are also called "systems"). These families have applications in combinatorial search, coding theory, cryptography, and related fields. We define and study generalizations of these two notions, which we have named $n$-separating families and $n$-splitting families. For each of these new notions, we outline their basic properties and connections with the well-studied notions. We then spend the greatest effort obtaining lower and upper bounds on the minimal s…
It has been known since the 60's that any complete discrete $n$-state automaton admits a reset word of length not exceeding $αn^3+o(n^3)$ for some absolute constant $α$. J.-E. Pin and P. Frankl proved this statement with $α=1/6=0.1666...$ in 1982, and this bound remained best known until 2017, when M. Szykuła decreased its value to $α\approx0.1664$. In this note, we present a modification to the latest approach and develop a different counting argument which leads to a more substantial improvement of $α\leqslant 0.1654$.
A non-empty word $w$ is a \emph{border} of a word $u$ if $\vert w\vert<\vert u\vert$ and $w$ is both a prefix and a suffix of $u$. A word $u$ is \emph{privileged} if $\vert u\vert\leq 1$ or if $u$ has a privileged border $w$ that appears exactly twice in $u$. Peltomäki (2016) presented the following open problem: ``Give a nontrivial upper bound for $B(n)$'', where $B(n)$ denotes the number of privileged words of length $n$. Let $\ln^{[0]}{(n)}=n$ and let $\ln^{[j]}{(n)}=\ln{(\ln^{[j-1]}{(n)})}$, where $j,n$ are positive integers. We show that if $q>1$ is a size of the alphabet and $j\geq 3$ is…
The Tight Upper Bound on the Number of Distinct Squares in Circular Words
本文证明了长度为 $n$ 的循环单词中长度不超过 $n$ 的不同平方数至多为 $\lfloor 3n/2 \rfloor$，该上界紧，且与已知下界一致。
The words separation problem, originally formulated by Goralcik and Koubek (1986), is stated as follows. Let $Sep(n)$ be the minimum number such that for any two words of length $\le n$ there is a deterministic finite automaton with $Sep(n)$ states, accepting exactly one of them. The problem is to find the asymptotics of the function $Sep$. This problem is inverse to finding the asymptotics of the length of the shortest identity in full transformation semigroups $T_k$. The known lower bound on $Sep$ stems from the unary identity in $T_k$. We find the first series of identities in $T_k$ which a…
Density Characterization with The Upper Bound of Density of Fibonacci Word
本文研究斐波那契词的自然密度与结构关系，推导其密度上界、生成函数与组合公式，验证其密度趋近于1的结论并揭示相关增长与极限特性。
In this paper we consider the following problems: how many different subsets of Sigma^n can occur as set of all length-n factors of a finite word? If a subset is representable, how long a word do we need to represent it? How many such subsets are represented by words of length t? For the first problem, we give upper and lower bounds of the form alpha^(2^n) in the binary case. For the second problem, we give a weak upper bound and some experimental data. For the third problem, we give a closed-form formula in the case where n <= t < 2n. Algorithmic variants of these problems have previously bee…
A finite word $w$ is called \emph{rich} if it contains $\vert w\vert+1$ distinct palindromic factors including the empty word. Let $q\geq 2$ be the size of the alphabet. Let $R(n)$ be the number of rich words of length $n$. Let $d>1$ be a real constant and let $ϕ, ψ$ be real functions such that \begin{itemize}\item there is $n_0$ such that $2ψ(2^{-1}ϕ(n))\geq dψ(n)$ for all $n>n_0$, \item $\frac{n}{ϕ(n)}$ is an upper bound on the palindromic length of rich words of length $n$, and \item $\frac{x}{ψ(x)}+\frac{x\ln{(ϕ(x))}}{ϕ(x)}$ is a strictly increasing concave function. \end{itemize} We show …
We prove that there is a set of integers $A$ having positive upper Banach density whose difference set $A-A:=\{a-b:a,b\in A\}$ does not contain a Bohr neighborhood of any integer, answering a question asked by Bergelson, Hegyvári, Ruzsa, and the author, in various combinations. In the language of dynamical systems, this result shows that there is a set of integers $S$ which is dense in the Bohr topology of $\mathbb Z$ and which is not a set of measurable recurrence. Our proof yields the following stronger result: if $S\subseteq \mathbb Z$ is dense in the Bohr topology of $\mathbb Z$, then ther…
We consider the following problem: given that a finite automaton $M$ of $N$ states accepts at least one $k$-power-free (resp., overlap-free) word, what is the length of the shortest such word accepted? We give upper and lower bounds which, unfortunately, are widely separated.
Separating paths systems of almost linear size
本文证明每个 $n$ 顶点图存在大小为 $O(n \log^* n)$ 的分离路径系统，改进了先前 $O(n \log n)$ 的上界，并推进了关于 $O(n)$ 上界的猜想。
Let $Λ^{\ast}$ be the free monoid of (finite) words over a not necessarily finite alphabet $Λ$, which is equipped with some (partial) order. This ordering lifts to $Λ^{\ast}$, where it extends the divisibility ordering of words. The MacNeille completion of $Λ^{\ast}$ constitutes a complete lattice ordered monoid and is realized by the system of "closed" lower sets in $Λ^*$ (ordered by inclusion) or its isomorphic copy formed of the "closed" upper sets (ordered by reverse inclusion). Under some additional hypothesis on $Λ$, one can easily identify the closed lower sets as the finitely generated…
A \emph{power} is a word of the form $\underbrace{uu...u}_{k \; \text{times}}$, where $u$ is a word and $k$ is a positive integer; the power is also called a {\em $k$-power} and $k$ is its {\em exponent}. We prove that for any $k \ge 2$, the maximum number of different non-empty $k$-power factors in a word of length $n$ is between $\frac{n}{k-1}-Θ(\sqrt{n})$ and $\frac{n-1}{k-1}$. We also show that the maximum number of different non-empty power factors of exponent at least 2 in a length-$n$ word is at most $n-1$. Both upper bounds generalize the recent upper bound of $n-1$ on the maximum numb…
We use stack words to find a new, simple proof for the best known upper bound for the number of 3-stack sortable permutations of a given length. This is the first time that stack words are used to obtain such a result.
The number of frequencies of factors of length $n+1$ in a recurrent aperiodic infinite word does not exceed $3Δ\C(n)$, where $Δ\C (n)$ is the first difference of factor complexity, as shown by Boshernitzan. Pelantová together with the author derived a better upper bound for infinite words whose language is closed under reversal. In this paper, we further diminish the upper bound for uniformly recurrent infinite words whose language is invariant under all elements of a finite group of symmetries and we prove the optimality of the obtained upper bound.
Separating the edges of a graph by a linear number of paths
本研究针对图的边分离问题，将n阶图分离所有边所需的路径数上界从O(n log⋆n)改进为19n，解答了Katona的问题并证实了两项独立提出的猜想，证明过程初等自包含。
Improved upper and lower bounds on the number of square-free ternary words are obtained. The upper bound is based on the enumeration of square-free ternary words up to length 110. The lower bound is derived by constructing generalised Brinkhuis triples. The problem of finding such triples can essentially be reduced to a combinatorial problem, which can efficiently be treated by computer. In particular, it is shown that the number of square-free ternary words of length n grows at least as 65^(n/40), replacing the previous best lower bound of 2^(n/17).
Mapping words to powers by morphisms
本文刻画了能被单射态射映射至任意高次幂的字，证明了其余字的最高次幂线性上界，并证明判定字能否经非周期态射映射为 $n$ 次幂或非本原字是 NP-hard 且属于 PSPACE 的。
On the size of the neighborhoods of a word
该研究针对单词的莱文斯坦距离邻域相关问题，给出一元单词的压缩与超压缩邻域大小的精确公式，提出并证明了给定长度任意单词压缩邻域最大规模的新上界及猜想上界，可用于近似模式匹配算法的复杂度分析。
We develop a new tool, namely polynomial and linear algebraic methods, for studying systems of word equations. We illustrate its usefulness by giving essentially simpler proofs of several hard problems. At the same time we prove extensions of these results. Finally, we obtain the first nontrivial upper bounds for the fundamental problem of the maximal size of independent systems. These bounds depend quadratically on the size of the shortest equation. No methods of having such bounds have been known before.
The complexity of smooth words over binary alphabets
本文研究二元字母表上平滑词的因子复杂度，证明了有限平滑词是平滑词的因子，并证明了在偶数字母表上的复杂度阶，在任意二元字母表上的下界，改进了奇数字母表上的上界。
This note is an attempt to attack a conjecture of Fraenkel and Simpson stated in 1998 concerning the number of distinct squares in a finite word. By counting the number of (right-)special factors, we give an upper bound of the number of {\em $k$-powers} in a finite word for any integer $k\geq 3$. By {\em $k$-power}, we mean a word of the form $\underbrace{uu...u}_{k \; \text{times}}$.
The power graph of a group $G$ is the graph whose vertex set is $G$ and two distinct vertices are adjacent if one is a power of the other. This paper investigates the minimal separating sets of power graphs of finite groups. For power graphs of finite cyclic groups, certain minimal separating sets are obtained. Consequently, a sharp upper bound for their connectivity is supplied. Further, the components of proper power graphs of $p$-groups are studied. In particular, the number of components of that of abelian $p$-groups are determined.
Separating path systems for cubic graphs and for complete bipartite graphs
研究了立方图、平面图和平面二分图的强分离数，证明了2-退化图的强分离数不超过n，并给出了完全二分图的强分离数下界。
Subexponential upper bound on the number of rich words
针对有限字母表上长度为n的丰富字计数问题，本文构造了满足次指数增长的上界函数B(n)，首次给出了R(n)的次指数上界，填补了该领域的研究空白。
Asymptotic bounds for the number of closed and privileged words
本文研究闭词和特权词的计数，改进其渐近上下界，完全解决闭词渐近行为，并近乎解决特权词渐近行为。
We provide geometric methods and algorithms to verify, construct and enumerate pairs of words (of specified length over a fixed $m$-letter alphabet) that form identities in the semigroup $\ut{n}$ of $n\times n$ upper triangular tropical matrices. In the case $n=2$ these identities are precisely those satisfied by the bicyclic monoid, whilst in the case $n=3$ they form a subset of the identities which hold in the plactic monoid of rank $3$. To each word we associate a signature sequence of lattice polytopes, and show that two words form an identity for $\ut{n}$ if and only if their signatures a…
Maximum number of one-element commutation classes of a permutation
本文研究了排列的单元素交换类的最大数目，并通过上界证明了缩减词数目与交换类数目的关系猜想。
Finite alphabets of at least three letters permit the construction of square-free words of infinite length. We show that the entropy density is strictly positive and derive reasonable lower and upper bounds. Finally, we present an approximate formula which is asymptotically exact with rapid convergence in the number of letters.
It is a classical result that any permutation in the symmetric group can be generated by a sequence of adjacent transpositions. The sequences of minimal length are called reduced words, and in this paper we study the graphs of these reduced words, with edges determined by relations in the underlying Coxeter group. Recently, the diameter has been calculated for the longest permutation $n\ldots 21$ by Reiner and Roichman as well as Assaf. In this paper we find inductive formulas for the diameter of the graphs of 12-inflations and many 21-inflations. These results extend to the associated graphs …
In 1999 Lyngsø and Pedersen proposed a conjecture stating that every binary circular word of length $n$ with equal number of zeros and ones has an antipalindromic linear subsequence of length at least $\frac{2}{3}n$. No progress over a trivial $\frac{1}{2}n$ bound has been achieved since then. We suggest a palindromic counterpart to this conjecture and provide a non-trivial infinite series of circular words which prove the upper bound of $\frac{2}{3}n$ for both conjectures at the same time. The construction also works for words over an alphabet of size $k$ and gives rise to a generalization of…
An isoperimetric inequality for word overlap
该研究证明了在词集$A$和$B$中不存在词后缀与前缀重叠的情况下，两者的密度乘积的渐近上界为$(1+o(1))/(en)$。
Improved upper bound on the number of distinct k-decks for any k and alphabet size by counting the independent parameters
通过计数Lyndon词确定k-序列的仿射子空间维数，改进了不同k-序列数量的上界，并在前两个非平凡情形证明了匹配下界。
Separating hash families are useful combinatorial structures which are generalizations of many well-studied objects in combinatorics, cryptography and coding theory. In this paper, using tools from graph theory and additive number theory, we solve several open problems and conjectures concerning bounds and constructions for separating hash families. Firstly, we discover that the cardinality of a separating hash family satisfies a Johnson-type inequality. As a result, we obtain a new upper bound, which is superior to all previous ones. Secondly, we present a construction for an infinite class o…
On the Failure of the Upper Bound in the Refined BMV Conjecture and a Pinching Correction
本文分析了精化BMV猜想为何失效，提出了一种pinching修正方法，通过研究交换部分的性质，修正了原猜想中的上界问题，并给出了在两种字母情况下修正猜想的证明。
The sandglass conjecture beyond cancellative pairs
本文研究沙漏猜想中恢复对子族规模乘积的上界问题，通过将其与相消对的上界指数级分离，把已知最佳上界改进为 2.2543^n。
Short Synchronizing Words for Random Automata
本文证明了一致随机自动机以高概率存在长度为$O(n^{1/2}\log n)$的短同步字，通过引入$w$-树概念及概率方法证明随机自动机的强结构结果，从而确立了比以往更紧的上界。
We extend upper bounds on the quantum independence number and the quantum Shannon capacity of graphs to their counterparts in the commuting operator model. We introduce a von Neumann algebraic generalization of the fractional Haemers bound (over $\mathbb{C}$) and prove that the generalization upper bounds the commuting quantum independence number. We call our bound the tracial Haemers bound, and we prove that it is multiplicative with respect to the strong product. In particular, this makes it an upper bound on the Shannon capacity. The tracial Haemers bound is incomparable with the Lovász the…
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