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People solve different problems and know that some of them are simple, some are complex and some insoluble. The main goal of this work is to develop a mathematical theory of algorithmic complexity for problems. This theory is aimed at…
The calculus of classes and closure operations has proved to be a useful tool in group theory and has led to a deep theory in the study of finite soluble groups. More recently, parallel theories have started to be developed in various…
This article is a short introduction to the theory of the groups of points of elliptic curves over finite fields. It is concerned with the elementary theory and practice of elliptic curves cryptography, the new generation of public key…
For a finite group $G$, we introduce a generalization of norm relations in the group algebra $\mathbb Q[G]$. We give necessary and sufficient criteria for the existence of such relations and apply them to obtain relations between the…
We consider applications of Subset Product Problem (SPP) in number theory and cryptography. We obtain a probabilistic algorithm that attack SPP and we analyze it with respect time/space complexity and success probability. In fact we provide…
Using the theory of representations of the symmetric group, we propose an algorithm to compute the invariant ring of a permutation group. Our approach have the goal to reduce the amount of linear algebra computations and exploit a thinner…
Power circuits are data structures which support efficient algorithms for highly compressed integers. Using this new data structure it has been shown recently by Myasnikov, Ushakov and Won that the Word Problem of the one-relator Baumslag…
The selection of random subspaces plays a role in quantum information theory analogous to the role of random strings in classical information theory. Recent applications have included protocols achieving the quantum channel capacity and…
An important problem in the theory of cluster algebras is to compute the fundamental group of the exchange graph. A non-trivial closed loop in the exchange graph, for example, generates a non-trivial identity for the classical and quantum…
The theory of holographic algorithms, which are polynomial time algorithms for certain combinatorial counting problems, yields insight into the hierarchy of complexity classes. In particular, the theory produces algebraic tests for a…
This article will explore the K- and L-theory of group rings and their applications to algebra, geometry and topology. The Farrell-Jones Conjecture characterizes K- and L-theory groups. It has many implications, including the Borel and…
We present a quantum algorithm which identifies with certainty a hidden subgroup of an arbitrary finite group G in only a polynomial (in log |G|) number of calls to the oracle. This is exponentially better than the best classical algorithm.…
In this paper, we study groups of automorphisms of algebraic systems over a set of $p$-adic integers with different sets of arithmetic and coordinate-wise logical operations and congruence relations modulo $p^k,$ $k\ge 1.$ The main result…
The main purpose of this paper is to develop new algorithms for computing invariant rings in a general setting. This includes invariants of nonreductive groups but also of groups acting on algebras over certain rings. In particular, we…
This paper introduces a novel approach to understanding Galois theory, one of the foundational areas of algebra, through the lens of machine learning. By analyzing polynomial equations with machine learning techniques, we aim to streamline…
This note describes an application of the theory of generalised Burnside rings to algebraic representation theory. Tables of marks are given explicitly for the groups $S_4$ and $S_5$ which are of particular interest in the context of…
Entanglement entropy plays a variety of roles in quantum field theory, including the connections between quantum states and gravitation through the holographic principle. This article provides a review of entanglement entropy from a mixed…
We give an elementary theory of Henselian local rings and construct the Henselization of a local ring. All our theorems have an algorithmic content.
In this article we survey recent progress in the algorithmic theory of matrix semigroups. The main objective in this area of study is to construct algorithms that decide various properties of finitely generated subsemigroups of an infinite…
This is a survey of using Minsky machines to study algorithmic problems in semigroups, groups and other algebraic systems.