Related papers: Algorithmic problems in Engel groups and cryptogra…

We call a group $G$ {\it algorithmically finite} if no algorithm can produce an infinite set of pairwise distinct elements of $G$. We construct examples of recursively presented infinite algorithmically finite groups and study their…

This survey focuses on the computational complexity of some of the fundamental decision problems in 3-manifold theory. The article discusses the wide variety of tools that are used to tackle these problems, including normal and almost…

We announce new methods for using prismatic cohomology to compute the K-groups of $\mathbb{Z}/p^n$ and related rings. We use computer algebra methods to compute these K-groups through a large range in specific cases and also obtain explicit…

We present a family of non-abelian groups for which the hidden subgroup problem can be solved efficiently on a quantum computer.

The root extraction problem in braid groups is the following: given a braid $\beta \in \mathcal{B}_n$ and a number $k\in \mathbb{N}$, find $\alpha\in \mathcal{B}_n$ such that $\alpha^k=\beta$. In the last decades, many cryptosystems such as…

Understanding the convergence process of neural networks is one of the most complex and crucial issues in the field of machine learning. Despite the close association of notable successes in this domain with the convergence of artificial…

It is well known that quantum computers can efficiently find a hidden subgroup $H$ of a finite Abelian group $G$. This implies that after only a polynomial (in $\log |G|$) number of calls to the oracle function, the states corresponding to…

While efficient algorithms are known for solving many important problems related to groups, no efficient algorithm is known for determining whether two arbitrary groups are isomorphic. The particular case of 2-nilpotent groups, a special…

This is an introduction to linear algebra and group theory. We first review the linear algebra basics, namely the determinant, the diagonalization procedure and more, and with the determinant being constructed as it should, as a signed…

Classifying states as entangled or separable is a highly challenging task, while it is also one of the foundations of quantum information processing theory. This task is higly nontrivial even for relatively simple cases, such as two-qutrit…

One of the most interesting questions about a group is if its word problem can be solved and how. The word problem in the braid group is of particular interest to topologists, algebraists and geometers, and is the target of intensive…

A recent trend in data mining has explored (hyper)graph clustering algorithms for data with categorical relationship types. Such algorithms have applications in the analysis of social, co-authorship, and protein interaction networks, to…

Extremal Graph Theory is a very deep and wide area of modern combinatorics. It is very fast developing, and in this long but relatively short survey we select some of those results which either we feel very important in this field or which…

Geometric methods proposed by Stallings for treating finitely generated subgroups of free groups were successfully used to solve a wide collection of decision problems for free groups and their subgroups. In the present paper we employ the…

In order to make the fundamental group, one of the most well known invariants in algebraic topology, more useful and powerful some researchers have introduced and studied various topologies on the fundamental group from the beginning of the…

We use the kernel category to give a finiteness condition for semigroups. As a consequence we provide yet another proof that finitely generated periodic semigroups of matrices are finite.

In this paper we study the relationships between the elementary abelian regular subgroups and the Sylow $2$-subgroups of their normalisers in the symmetric group $\mathrm{Sym}(\mathbb{F}_2^n)$, in view of the interest that they have…

We present a new group law defined on a subset of the projective plane $\mathbb{F}P^2$ over an arbitrary field $\mathbb{F}$, which lends itself to applications in Public Key Cryptography, in particular to a Diffie-Hellman-like key agreement…

There are three kinds of Lie superalgebras for each differentiable manifold. In this note, we shall show an application of the homology groups of those superalgebras in order to classify 4 dimensional Engel-like Lie algebras.

We give two applications of the 2-Engel relation, classically studied in finite and Lie groups, to the 4-dimensional topological surgery conjecture. The A-B slice problem, a reformulation of the surgery conjecture for free groups, is shown…