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The Pythagorean Heritage: From Number Theory and Combinatorics to Artificial...
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The Pythagorean Heritage: From Number Theory and Combinatorics to Artificial Intelligence

A special issue of Axioms (ISSN 2075-1680). This special issue belongs to the section "Algebra and Number Theory".

Deadline for manuscript submissions: 29 July 2024 | Viewed by 5183

Special Issue Editors

Prof. Dr. Fabio Caldarola

E-Mail Website
Guest Editor
Department of Mathematics and Computer Science, Cubo 31/A, Università della Calabria, 87036 Rende, Italy
Interests: number theory; iwasawa theory; combinatorics; fibonacci numbers; mumber sequences; graph theory; unimaginable numbers; combinatorics on words; fractal geometry; polytopes; elliptic curves; cryptography; applied mathematics; cellular automata; mathematical models; chaos theory; nonlinear dynamics; shallow water
Special Issues, Collections and Topics in MDPI journals
Prof. Dr. Gianfranco d'Atri

E-Mail Website
Guest Editor
Department of Mathematics and Computer Science, University of Calabria, Arcavacata, Italy
Interests: probability theory; number theory; graph theory; applied mathematics; unimaginable numbers; blockchain technologies; artificial intelligence

Special Issue Information

Dear Colleagues,

The evolution of ideas born in the Pythagorean School in Magna Graecia (in today's Calabria, Italy), between the sixth and third centuries BC, can now be observed in the most varied fields of mathematical and scientific knowledge. The number, seen as arche or first principle of all things, occupied a truly privileged place: the Pythagoreans “assumed the elements of numbers to be the elements of everything, and the whole universe to be a proportion or number" (Aristotle). Examples of such influences are contained in the volume “From Pythagoras to Schützenberger: Unimaginable numbers” edited by us and others (see https://www.researchgate.net/publication/350631850_From_Pythagoras_to_Schutzenberger_Unimaginable_numbers). Themes with the same roots, but with even broader intentions than those of this SI, will be discussed in the “Special Pythagorean Stream” organized by us and others in the International Conference NUMTA 2023 (see https://www.numta.org/special-streams-and-sessions/ and the instructions contained therein for submissions).

Number theory, geometry, algebra, combinatorics, discrete mathematics, etc., but also computer science and information theory may be considered the evolution of the ideas developed by the Pythagoreans. This SI aims to collect broad and high-level contributions that show some kind of connection or root with Pythagorean, Magna Graecia or even Siceliot (e.g., Archimedean) mathematics. Works on Diophantine geometry, unimaginable numbers, theory of graphs, groups, rings, combinatorics of words, elliptic curves, cryptography, and mathematical aspects in blockchain and artificial intelligence are also welcome. 

Prof. Dr. Fabio Caldarola
Prof. Dr. Gianfranco d'Atri
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. Axioms is an international peer-reviewed open access monthly 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 2400 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

  • number theory
  • combinatorics
  • sequences of integers
  • Pythagorean triples
  • Pythagorean fields
  • Fibonacci numbers
  • combinatorics on words
  • unimaginable numbers
  • graph theory
  • group theory
  • commutative algebra
  • elliptic curves
  • cryptography
  • artificial intelligence

Published Papers (6 papers)

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Research

13 pages, 320 KiB  
Article
Ratio-Covarieties of Numerical Semigroups
by María Ángeles Moreno-Frías and José Carlos Rosales
Axioms 2024, 13(3), 193; https://0-doi-org.brum.beds.ac.uk/10.3390/axioms13030193 - 14 Mar 2024
Viewed by 643
Abstract
In this work, we will introduce the concept of ratio-covariety, as a family R of numerical semigroups that has a minimum, denoted by min(R), is closed under intersection, and if S∈R and [...] Read more.
In this work, we will introduce the concept of ratio-covariety, as a family R of numerical semigroups that has a minimum, denoted by min(R), is closed under intersection, and if S∈R and S≠min(R), then S\{r(S)}∈R, where r(S) denotes the ratio of S. The notion of ratio-covariety will allow us to: (1) describe an algorithmic procedure to compute R; (2) prove the existence of the smallest element of R that contains a set of positive integers; and (3) talk about the smallest ratio-covariety that contains a finite set of numerical semigroups. In addition, in this paper we will apply the previous results to the study of the ratio-covariety R(F,m)={S∣S is a numerical semigroup with Frobenius number F and multiplicitym}. Full article
(This article belongs to the Special Issue The Pythagorean Heritage: From Number Theory and Combinatorics to Artificial Intelligence)
15 pages, 344 KiB  
Article
Quasi-Semilattices on Networks
by Yanhui Wang and Dazhi Meng
Axioms 2023, 12(10), 943; https://0-doi-org.brum.beds.ac.uk/10.3390/axioms12100943 - 30 Sep 2023
Cited by 3 | Viewed by 1106
Abstract
This paper introduces a representation of subnetworks of a network Γ consisting of a set of vertices and a set of relations, where relations are the primitive structures of a network. It is proven that all connected subnetworks of a network Γ form [...] Read more.
This paper introduces a representation of subnetworks of a network Γ consisting of a set of vertices and a set of relations, where relations are the primitive structures of a network. It is proven that all connected subnetworks of a network Γ form a quasi-semilattice L(Γ), namely a network quasi-semilattice.Two equivalences σ and δ are defined on L(Γ). Each δ class forms a semilattice and also has an order structure with the maximum element and minimum elements. Here, the minimum elements correspond to spanning trees in graph theory. Finally, we show how graph inverse semigroups, Leavitt path algebras and Cuntz–Krieger graph C*-algebras are constructed in terms of relations. Full article
(This article belongs to the Special Issue The Pythagorean Heritage: From Number Theory and Combinatorics to Artificial Intelligence)
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15 pages, 337 KiB  
Article
Some Variants of Integer Multiplication
by Francisco Javier de Vega
Axioms 2023, 12(10), 905; https://0-doi-org.brum.beds.ac.uk/10.3390/axioms12100905 - 23 Sep 2023
Viewed by 790
Abstract
In this paper, we will explore alternative varieties of integer multiplication by modifying the product axiom of Dedekind–Peano arithmetic (PA). In addition to studying the elementary properties of the new models of arithmetic that arise, we will see that the truth or falseness [...] Read more.
In this paper, we will explore alternative varieties of integer multiplication by modifying the product axiom of Dedekind–Peano arithmetic (PA). In addition to studying the elementary properties of the new models of arithmetic that arise, we will see that the truth or falseness of some classical conjectures will be equivalently in the new ones, even though these models have non-commutative and non-associative product operations. To pursue this goal, we will generalize the divisor and prime number concepts in the new models. Additionally, we will explore various general number properties and project them onto each of these new structures. This fact will enable us to demonstrate that indistinguishable properties on PA project different properties within a particular model. Finally, we will generalize the main idea and explain how each integer sequence gives rise to a unique arithmetic structure within the integers. Full article
(This article belongs to the Special Issue The Pythagorean Heritage: From Number Theory and Combinatorics to Artificial Intelligence)
12 pages, 297 KiB  
Article
On Approximation by an Absolutely Convergent Integral Related to the Mellin Transform
by Antanas Laurinčikas
Axioms 2023, 12(8), 789; https://0-doi-org.brum.beds.ac.uk/10.3390/axioms12080789 - 14 Aug 2023
Viewed by 612
Abstract
In this paper, we consider the modified Mellin transform of the product of the square of the Riemann zeta function and the exponentially decreasing function, and we discuss its probabilistic and approximation properties. It turns out that this Mellin transform approximates the identical [...] Read more.
In this paper, we consider the modified Mellin transform of the product of the square of the Riemann zeta function and the exponentially decreasing function, and we discuss its probabilistic and approximation properties. It turns out that this Mellin transform approximates the identical zero in the strip {s∈C:1/2<σ<1}. Full article
(This article belongs to the Special Issue The Pythagorean Heritage: From Number Theory and Combinatorics to Artificial Intelligence)
11 pages, 281 KiB  
Article
New Equivalents of Kurepa’s Hypothesis for Left Factorial
by Aleksandar Petojević, Snežana Gordić, Milinko Mandić and Marijana Gorjanac Ranitović
Axioms 2023, 12(8), 785; https://0-doi-org.brum.beds.ac.uk/10.3390/axioms12080785 - 12 Aug 2023
Viewed by 681
Abstract
Kurepa’s hypothesis for the left factorial has been an unsolved problem for more than 50 years. In this paper, we have proposed new equivalents for Kurepa’s hypothesis for the left factorial. The connection between the left factorial and the continued fractions is given. [...] Read more.
Kurepa’s hypothesis for the left factorial has been an unsolved problem for more than 50 years. In this paper, we have proposed new equivalents for Kurepa’s hypothesis for the left factorial. The connection between the left factorial and the continued fractions is given. The new equivalent based on the properties of the integer part of real numbers is proven. Moreover, a new equivalent based on the properties of two well-known sequences is given. A new representation of the left factorial is listed. Since derangement numbers are closely related to Kurepa’s hypothesis, we made some notes about the derangement numbers and defined a new sequence of natural numbers based on the derangement numbers. In this paper, we indicate a possible direction for further research through solving quadratic equations. Full article
(This article belongs to the Special Issue The Pythagorean Heritage: From Number Theory and Combinatorics to Artificial Intelligence)
14 pages, 821 KiB  
Article
Joint Discrete Approximation of Analytic Functions by Shifts of the Riemann Zeta Function Twisted by Gram Points II
by Antanas Laurinčikas
Axioms 2023, 12(5), 426; https://0-doi-org.brum.beds.ac.uk/10.3390/axioms12050426 - 26 Apr 2023
Viewed by 631
Abstract
In this paper, a theorem is obtained on the approximation in short intervals of a collection of analytic functions by shifts [...] Read more.
In this paper, a theorem is obtained on the approximation in short intervals of a collection of analytic functions by shifts (ζ(s+itkα1),…,ζ(s+itkαr)) of the Riemann zeta function. Here, {tk:k∈N} is the sequence of Gram numbers, and α1,…,αr are different positive numbers not exceeding 1. It is proved that the above set of shifts in the interval [N,N+M], here M=o(N) as N→∞, has a positive lower density. For the proof, a joint limit theorem in short intervals for weakly convergent probability measures is applied. Full article
(This article belongs to the Special Issue The Pythagorean Heritage: From Number Theory and Combinatorics to Artificial Intelligence)
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