Related papers: Short presentations of finite simple groups

Two recent lower bounds on the compressibility of repetitive sequences, $\delta \le \gamma$, have received much attention. It has been shown that a length-$n$ string $S$ over an alphabet of size $\sigma$ can be represented within the…

We investigate the minimum number of cycles of specified lengths in planar $n$-vertex triangulations $G$. It is proven that this number is $\Omega(n)$ for any cycle length at most $3 + \max \{ {\rm rad}(G^*), \lceil…

Gruenberg and Linnell showed that the standard relation module of a free product of $n$ groups of the form $C_r \times \mathbb{Z}$ could be generated by just $n+1$ generators, raising the possibility of a relation gap. We explicitly give…

For a finite group $G$, the size of a minimum generating set of $G$ is denoted by $d(G)$. Given a finite group $G$ and an integer $k$, deciding if $d(G)\leq k$ is known as the minimum generating set (MIN-GEN) problem. A group $G$ of order…

An idea to present a classical Lie group of positive dimension by generators and relations sounds dubious, but happens to be fruitful. The isometry groups of classical geometries admit elegant and useful presentations by generators and…

In the first section of this senior thesis the author provides some new efficient algorithms for calculating with finite permutation groups. They cannot be found in the computer algebra system MAGMA, but they can be implemented there. For…

In the classic longest common substring (LCS) problem, we are given two strings $S$ and $T$, each of length at most $n$, over an alphabet of size $\sigma$, and we are asked to find a longest string occurring as a fragment of both $S$ and…

A $c$-short program for a string $x$ is a description of $x$ of length at most $C(x) + c$, where $C(x)$ is the Kolmogorov complexity of $x$. We show that there exists a randomized algorithm that constructs a list of $n$ elements that…

The question of characterizing the (finite) representable relation algebras in a ``nice" way is open. The class $\mathbf{RRA}$ is known to be not finitely axiomatizable in first-order logic. Nevertheless, it is conjectured that ``almost…

Let $G$ be an $n$-vertex graph, and $s,t$ vertices of $G$. We present an efficient algorithm which enumerates the set of minimal $st$-separators of $G$ in ascending order of cardinality, with a delay of $O(n^{3.5})$ per separator. In…

Given a sequence of integers, we want to find a longest increasing subsequence of the sequence. It is known that this problem can be solved in $O(n \log n)$ time and space. Our goal in this paper is to reduce the space consumption while…

Following (Kolpakov et al., 2013; Gawrychowski and Manea, 2015), we continue the study of {\em $\alpha$-gapped repeats} in strings, defined as factors $uvu$ with $|uv|\leq \alpha |u|$. Our main result is the $O(\alpha n)$ bound on the…

The Planar Graph Metric Compression Problem is to compactly encode the distances among $k$ nodes in a planar graph of size $n$. Two na\"ive solutions are to store the graph using $O(n)$ bits, or to explicitly store the distance matrix with…

Over a field of characteristic 0, the algebra of invariants of several $n\times n$ matrices under simultameous conjugation by $GL_n$ is generated by traces of products of generic matrices. Teranishi, 1986, found a minimal system of eleven…

In his initial paper on braids E.Artin gave a presentation with two generators for an arbitrary braid group. We give analogues of this Artin's presentation for various generalizations of braids.

A minimal presentation of the cohomology ring of the flag manifold $GL_n/B$ was given in [A. Borel, 1953]. This presentation was extended by [E. Akyildiz-A. Lascoux-P. Pragacz, 1992] to a non-minimal one for all Schubert varieties. Work of…

Let S=Sym(\Omega) be the group of all permutations of an infinite set \Omega. Extending an argument of Macpherson and Neumann, it is shown that if U is a generating set for S as a group, respectively as a monoid, then there exists a…

It is shown that if $G$ is a primitive permutation group on a set of size $n$, then any nilpotent quotient of $G$ has order at most $n^{\beta}$ and any solvable quotient of $G$ has order at most $n^{\alpha+1}$ where $\beta=\log 32/ \log 9$…

We find a family of groups generated by a pair of parabolic elements in which every relator must admit a long subword of a specific form. In particular, this collection contains groups in which the number of syllables of any relator is…

Let $G$ be the cyclic group of order $n$ and suppose ${\bf F}$ is a field containing a primitive $n^\text{th}$ root of unity. We consider the ring of invariants ${\bf F}[W]^G$ of a three dimensional representation $W$ of $G$ where $G…