Related papers: On a generalization of Dehn's algorithm

We give a 1-sided randomised algorithm to detect when a permutation group of degree n, given by generators, contains the alternating group of degree n. This improves on standard methods, and on an algorithm of P. Cameron and J. Cannon.

Deterministic 2-head finite automata which are machines that process an input word from both ends are analyzed for their ability to perform reversible computations. This implies that the automata are backward deterministic, enabling unique…

Decompositions of the unitary group U(n) are useful tools in quantum information theory as they allow one to decompose unitary evolutions into local evolutions and evolutions causing entanglement. Several recursive decompositions have been…

In this document we achieve exact and asymptotic enumeration of words, compositions over a finite group, and/or integer compositions characterized by local restrictions and, separately, subsequence pattern avoidance. We also count…

We study groups of reversible cellular automata, or CA groups, on groups. More generally, we consider automorphism groups of subshifts of finite type on groups. It is known that word problems of CA groups on virtually nilpotent groups are…

We classify all closed, aspherical Riemannian manifolds M whose universal cover has indiscrete isometry group. One sample application is the theorem that any such M with word-hyperbolic fundamental group must be isometric to a negatively…

Due to the noncommutative nature of quaternions and octonions we introduce barred operators. This objects give the opportunity to manipulate appropriately the hypercomplex fields. The standard problems arising in the definitions of…

Universality is a key hypothesis in mechanistic interpretability -- that different models learn similar features and circuits when trained on similar tasks. In this work, we study the universality hypothesis by examining how small neural…

Descents in permutations or words are defined from the relative position of two consecutive letters. We investigate a statistic involving patterns of k consecutive letters, and show that it leads to Hopf algebras generalizing noncommutative…

The downward closure of a word language is the set of all (not necessarily contiguous) subwords of its members. It is well-known that the downward closure of any language is regular. While the downward closure appears to be a powerful…

A set X of partial words over a finite alphabet A is called unavoidable if every two-sided infinite word over A has a factor compatible with an element of X. Unlike the case of a set of words without holes, the problem of deciding whether…

The following discourse is inspired by the works on hyperbolic groups of Epstein, and Neumann/Reeves. Epstein showed that geometrically finite hyperbolic groups are biautomatic. Neumann/Reeves showed that virtually central extensions of…

Lie group theory states that knowledge of a $m$-parameters solvable group of symmetries of a system of ordinary differential equations allows to reduce by $m$ the number of equations. We apply this principle by finding some \emph{affine…

In the theory of algebraic groups, parabolic subgroups form a crucial building block in the structural studies. In the case of general linear groups over a finite field $F_q$, given a sequence of positive integers $n_1, ..., n_k$, where…

We shall characterize the structure of invertible substitutions on three-letter alphabet. We show that any invertible substitution, after some cyclic operation, can be written as a finite product of permutations and Fibonacci's…

We show, that groups, defined by wide class of automata, including all polynomial ones, act on the set of infinite words not paradoxical

We generalize the study of standard Lyndon loop words from [A.Negut, A.Tsymbaliuk, "Quantum loop groups and shuffle algebras via Lyndon words", Adv. Math. 439 (2024), Paper No. 109482] to a more general class of orders on the underlying…

Consider equipping an alphabet $\mathcal{A}$ with a group action that partitions the set of words into equivalence classes which we call patterns. We answer standard questions for the Penney's game on patterns and show non-transitivity for…

The normaliser problem takes as input subgroups $G$ and $H$ of the symmetric group $S_n$, and asks one to compute $N_G(H)$. The fastest known algorithm for this problem is simply exponential, whilst more efficient algorithms are known for…

We construct new examples of non-nil algebras with any number of generators, which are direct sums of two locally nilpotent subalgebras. As all previously known examples, our examples are contracted semigroup algebras and the underlying…