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A special issue of Mathematics (ISSN 2227-7390). This special issue belongs to the section "Algebra, Geometry and Topology".

Deadline for manuscript submissions: closed (30 June 2024) | Viewed by 6416

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Special Issue Editors

Prof. Dr. Alexei Semenov

E-Mail Website
Guest Editor

1. Faculty of Mechanics and Mathematics, Lomonosov Moscow State University, 119991 Moscow, Russia
2. Moscow Institute of Physics and Technology, 117303 Moscow, Russia
Interests: decidability of logical theories;definability in structures word combinatorics; symbolic dynamics;almost periodic sequences definability;reducts;svenonius theorem
Special Issues, Collections and Topics in MDPI journals

Prof. Dr. Alexei Kanel-Belov

E-Mail Website
Guest Editor

1. Department of Mathematics, Bar-Ilan University, Ramat Gan 5290002, Israel
2. Faculty of Mechanics and Mathematics, Lomonosov Moscow State University, 119991 Moscow, Russia
Interests: specht problem; dixmiere conjecture; pI-algebra; jacobian Conjecture interlocking structures; finitelly generated skew field; small cancellation.
Special Issues, Collections and Topics in MDPI journals

Special Issue Information

Dear Colleagues,

The combinatorics of words lies in the background of research in numerous fields of mathematics, computer science, and applications including:

Combinatorial theories of groups, monoids, and rings;
Noncommutative algebra;
Algebraic geometry;
Formal languages and automata theory;
Symbolic dynamics;
Mathematics and computer science education.

These are the major broad areas for our Special Issue. We will also include interesting topics such as definability theory, computability and algorithmic problems in algebra, the application of nonstandard analysis to quantization, and AI methods in mathematics.

Prof. Dr. Alexei Semenov
Prof. Dr. Alexei Kanel-Belov
Guest Editors

Manuscript Submission Information

Manuscripts should be submitted online at www.mdpi.com by registering and logging in to this website. Once you are registered, click here to go to the submission form. Manuscripts can be submitted until the deadline. All submissions that pass pre-check are peer-reviewed. Accepted papers will be published continuously in the journal (as soon as accepted) and will be listed together on the special issue website. Research articles, review articles as well as short communications are invited. For planned papers, a title and short abstract (about 100 words) can be sent to the Editorial Office for announcement on this website.

Submitted manuscripts should not have been published previously, nor be under consideration for publication elsewhere (except conference proceedings papers). All manuscripts are thoroughly refereed through a single-blind peer-review process. A guide for authors and other relevant information for submission of manuscripts is available on the Instructions for Authors page. Mathematics is an international peer-reviewed open access semimonthly journal published by MDPI.

Please visit the Instructions for Authors page before submitting a manuscript. The Article Processing Charge (APC) for publication in this open access journal is 2600 CHF (Swiss Francs). Submitted papers should be well formatted and use good English. Authors may use MDPI's English editing service prior to publication or during author revisions.

Keywords

combinatorics of words
combinatorial ring theory
combinatorial group theory
operator algebras
model theory in algebraic geometry
algorithmic problems in algebra
artificial intelligence
mathematical education

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Published Papers (5 papers)

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Research

Jump to: Review

26 pages, 413 KiB

Open AccessArticle

Combinatorial Estimations on Burnside Type Problems

by Anton Beletskiy and Ilya Ivanov-Pogodaev

Mathematics 2024, 12(5), 665; https://doi.org/10.3390/math12050665 - 24 Feb 2024

Viewed by 702

Abstract

The Burnside problem, formulated by W. Burnside in 1902, is one of the most well-known and important open questions in the field of Group Theory. Despite significant progress made in the past century towards solving this problem, its complete solution remains unknown. In this paper, we investigate one of the approaches to solving the Burnside problem based on the application of an iterative theory of small cancellations and canonical forms developed by E. Rips in recent years. We present a novel self-contained exposition of this theory and utilize it to obtain new estimates on the infiniteness of initial approximations of Burnside groups where only a finite number of periodic relations is used for relatively small odd exponents (

n > 120

). Full article

(This article belongs to the Special Issue Combinatorial Algebra, Computation, and Logic, 2nd Edition)

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11 pages, 274 KiB

Open AccessArticle

Kernel Words and Gap Sequences of the Tribonacci Word on an Infinite Alphabet

by Jiemeng Zhang

Mathematics 2023, 11(20), 4356; https://doi.org/10.3390/math11204356 - 20 Oct 2023

Viewed by 766

Abstract

In this paper, we propose a nuanced variation in the kernel words of the tribonacci sequence. Our primary objective is to investigate the intrinsic properties of the kernel words and associated gap sequences when the tribonacci sequence is expanded over an infinite alphabet. [...] Read more.

(This article belongs to the Special Issue Combinatorial Algebra, Computation, and Logic, 2nd Edition)

14 pages, 337 KiB

Open AccessArticle

Graded Rings Associated with Factorizable Finite Groups

by Mohammed M. Al-Shomrani and Najla Al-Subaie

Mathematics 2023, 11(18), 3864; https://doi.org/10.3390/math11183864 - 10 Sep 2023

Viewed by 938

Abstract

Let R be an associative ring with unity, X be a finite group, H be a subgroup of X, and

G

be a set of left coset representatives for the left action of H on X. In this article, we introduce [...] Read more.

Let R be an associative ring with unity, X be a finite group, H be a subgroup of X, and

G

be a set of left coset representatives for the left action of H on X. In this article, we introduce two different ways to put R into a non-trivial

G

-weak graded ring that is a ring graded by the set

G

which is defined with a binary operation ∗ and satisfying an algebraic structure with specific properties. The first one is by choosing a subset S of

G

such that S is a group under the ∗ operation and putting

R_{t} = 0

for all

t \in G

and

t \notin S

. The second way, which is the most important, is induced by combining the operation ∗ defined on

G

and the coaction

◁

of H on

G

. Many examples are provided. Full article

(This article belongs to the Special Issue Combinatorial Algebra, Computation, and Logic, 2nd Edition)

16 pages, 347 KiB

Open AccessArticle

Construction of Quantum Codes over the Class of Commutative Rings and Their Applications to DNA Codes

by Shakir Ali, Amal S. Alali, Elif Segah Oztas and Pushpendra Sharma

Mathematics 2023, 11(6), 1430; https://doi.org/10.3390/math11061430 - 15 Mar 2023

Cited by 3 | Viewed by 1802

Abstract

Let

k, m

be positive integers and

F_{2^{m}}

be a finite field of order

2^{m}

of characteristic 2. The primary goal of this paper is to study the structural properties of cyclic codes over the ring [...] Read more.

Let

k, m

be positive integers and

F_{2^{m}}

be a finite field of order

2^{m}

of characteristic 2. The primary goal of this paper is to study the structural properties of cyclic codes over the ring

S_{k} = \frac{F_{2^{m}} [v_{1}, v_{2}, \dots, v_{k}]}{⟨ v_{i}^{2} - α_{i} v_{i}, v_{i} v_{j} - v_{j} v_{i} ⟩}

, for

i, j = 1, 2, 3, \dots, k

, where

α_{i}

is the non-zero element of

F_{2^{m}}

. As an application, we obtain better quantum error correcting codes over the ring

S_{1}

(for

k = 1

). Moreover, we acquire optimal linear codes with the help of the Gray image of cyclic codes. Finally, we present methods for reversible DNA codes. Full article

(This article belongs to the Special Issue Combinatorial Algebra, Computation, and Logic, 2nd Edition)

Review

Jump to: Research

38 pages, 452 KiB

Open AccessReview

From Quantum Automorphism of (Directed) Graphs to the Associated Multiplier Hopf Algebras

by Farrokh Razavinia and Ghorbanali Haghighatdoost

Mathematics 2024, 12(1), 128; https://doi.org/10.3390/math12010128 - 30 Dec 2023

Viewed by 1168

Abstract

This is a noticeably short biography and introductory paper on multiplier Hopf algebras. It delves into questions regarding the significance of this abstract construction and the motivation behind its creation. It also concerns quantum linear groups, especially the coordinate ring of M_q [...] Read more.

K

[M_q(n)] is a quadratic algebra, and can be equipped with a multiplier Hopf ∗-algebra structure in the sense of quantum permutation groups developed byWang and an observation by Rollier–Vaes. In our next paper, we will propose the study of multiplier Hopf graph algebras. The current paper can be viewed as a precursor to this upcoming work, serving as a crucial intermediary bridging the gap between the abstract concept of multiplier Hopf algebras and the well-developed field of graph theory, thereby establishing connections between them! This survey review paper is dedicated to the 78th birthday anniversary of Professor Alfons Van Daele. Full article

(This article belongs to the Special Issue Combinatorial Algebra, Computation, and Logic, 2nd Edition)

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