Axioms

Research

15 pages, 349 KiB

Open AccessArticle

New Results About Some Chain Conditions in Serre Conjecture Rings

by Bana Al Subaiei and Noômen Jarboui

Axioms 2024, 13(11), 778; https://doi.org/10.3390/axioms13110778 - 10 Nov 2024

Viewed by 806

Abstract

Let

φ : T \to T / I

be the natural homomorphism, where I is a nonzero ideal of an integral domain T. We define

R : = φ^{- 1} (D)

, where D is a subring of [...] Read more.

Let

φ : T \to T / I

be the natural homomorphism, where I is a nonzero ideal of an integral domain T. We define

R : = φ^{- 1} (D)

, where D is a subring of

T / I

. This paper aims to investigate the conditions under which the Serre conjecture ring

R ⟨ m ⟩

is a strong S-domain. Several examples are constructed to demonstrate both the scope and limitations of the results. Full article

(This article belongs to the Special Issue The Pythagorean Heritage: From Number Theory and Combinatorics to Artificial Intelligence)

8 pages, 270 KiB

Open AccessArticle

A Note on Extended Genus Fields of Kummer Extensions of Global Rational Function Fields

by Martha Rzedowski-Calderón and Gabriel Villa-Salvador

Axioms 2024, 13(11), 734; https://doi.org/10.3390/axioms13110734 - 24 Oct 2024

Viewed by 726

Abstract

We consider the generalization of the extended genus field of a prime degree cyclic Kummer extension of a global rational function field obtained by R. Clement in 1992 to general Kummer extensions. We observe that the same approach of Clement works in general. [...] Read more.

We consider the generalization of the extended genus field of a prime degree cyclic Kummer extension of a global rational function field obtained by R. Clement in 1992 to general Kummer extensions. We observe that the same approach of Clement works in general. Full article

(This article belongs to the Special Issue The Pythagorean Heritage: From Number Theory and Combinatorics to Artificial Intelligence)

15 pages, 286 KiB

Open AccessArticle

On a Matrix Formulation of the Sequence of Bi-Periodic Fibonacci Numbers

by Mustapha Rachidi, Elen V. P. Spreafico and Paula Catarino

Axioms 2024, 13(9), 590; https://doi.org/10.3390/axioms13090590 - 30 Aug 2024

Viewed by 812

Abstract

In this study, we investigate some new properties of the sequence of bi-periodic Fibonacci numbers with arbitrary initial conditions, through an approach that combines the matrix aspect and the fundamental Fibonacci system. Indeed, by considering the properties of the eigenvalues of their related [...] Read more.

In this study, we investigate some new properties of the sequence of bi-periodic Fibonacci numbers with arbitrary initial conditions, through an approach that combines the matrix aspect and the fundamental Fibonacci system. Indeed, by considering the properties of the eigenvalues of their related

2 \times 2

matrix, we provide a new approach to studying the analytic representations of these numbers. Moreover, the similarity of the associated

2 \times 2

matrix with a companion matrix, allows us to formulate the bi-periodic Fibonacci numbers in terms of a homogeneous linear recursive sequence of the Fibonacci type. Therefore, the combinatorial aspect and other analytic representations formulas of the Binet type for the bi-periodic Fibonacci numbers are achieved. The case of bi-periodic Lucas numbers is outlined, and special cases are exposed. Finally, some illustrative examples are given. Full article

(This article belongs to the Special Issue The Pythagorean Heritage: From Number Theory and Combinatorics to Artificial Intelligence)

13 pages, 283 KiB

Open AccessArticle

The Mean Square of the Hurwitz Zeta-Function in Short Intervals

by Antanas Laurinčikas and Darius Šiaučiūnas

Axioms 2024, 13(8), 510; https://doi.org/10.3390/axioms13080510 - 28 Jul 2024

Cited by 2 | Viewed by 809

Abstract

The Hurwitz zeta-function

ζ (s, α)

,

s = σ + i t

, with parameter

0 < α ⩽ 1

is a generalization of the Riemann zeta-function

ζ (s)

( [...] Read more.

The Hurwitz zeta-function

ζ (s, α)

,

s = σ + i t

, with parameter

0 < α ⩽ 1

is a generalization of the Riemann zeta-function

ζ (s)

(

ζ (s, 1) = ζ (s)

) and was introduced at the end of the 19th century. The function

ζ (s, α)

plays an important role in investigations of the distribution of prime numbers in arithmetic progression and has applications in special function theory, algebraic number theory, dynamical system theory, other fields of mathematics, and even physics. The function

ζ (s, α)

is the main example of zeta-functions without Euler’s product (except for the cases

α = 1

,

α = 1 / 2

), and its value distribution is governed by arithmetical properties of

α

. For the majority of zeta-functions,

ζ (s, α)

for some

α

is universal, i.e., its shifts

ζ (s + i τ, α)

,

τ \in R

, approximate every analytic function defined in the strip

{s : 1 / 2 < σ < 1}

. For needs of effectivization of the universality property for

ζ (s, α)

, the interval for

τ

must be as short as possible, and this can be achieved by using the mean square estimate for

ζ (σ + i t, α)

in short intervals. In this paper, we obtain the bound

O (H)

for that mean square over the interval

[T - H, T + H]

, with

T^{27 / 82} ⩽ H ⩽ T^{σ}

and

1 / 2 < σ ⩽ 7 / 12

. This is the first result on the mean square for

ζ (s, α)

in short intervals. In forthcoming papers, this estimate will be applied for proof of universality for

ζ (s, α)

and other zeta-functions in short intervals. Full article

(This article belongs to the Special Issue The Pythagorean Heritage: From Number Theory and Combinatorics to Artificial Intelligence)

13 pages, 320 KiB

Open AccessFeature PaperArticle

Ratio-Covarieties of Numerical Semigroups

by María Ángeles Moreno-Frías and José Carlos Rosales

Axioms 2024, 13(3), 193; https://doi.org/10.3390/axioms13030193 - 14 Mar 2024

Cited by 1 | Viewed by 1232

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 \in 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 \in R

and

S \neq \min (R)

, then

S \ {r (S)} \in 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 multiplicity m} .

Full article

(This article belongs to the Special Issue The Pythagorean Heritage: From Number Theory and Combinatorics to Artificial Intelligence)

15 pages, 344 KiB

Open AccessArticle

Quasi-Semilattices on Networks

by Yanhui Wang and Dazhi Meng

Axioms 2023, 12(10), 943; https://doi.org/10.3390/axioms12100943 - 30 Sep 2023

Cited by 4 | Viewed by 1734

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

Open AccessArticle

Some Variants of Integer Multiplication

by Francisco Javier de Vega

Axioms 2023, 12(10), 905; https://doi.org/10.3390/axioms12100905 - 23 Sep 2023

Viewed by 1326

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

Open AccessArticle

On Approximation by an Absolutely Convergent Integral Related to the Mellin Transform

by Antanas Laurinčikas

Axioms 2023, 12(8), 789; https://doi.org/10.3390/axioms12080789 - 14 Aug 2023

Cited by 2 | Viewed by 1209

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 \in 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

Open AccessArticle

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://doi.org/10.3390/axioms12080785 - 12 Aug 2023

Cited by 1 | Viewed by 1457

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

Open AccessArticle

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://doi.org/10.3390/axioms12050426 - 26 Apr 2023

Cited by 1 | Viewed by 1099

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 + i t_{k}^{α_{1}}), \dots, ζ (s + i t_{k}^{α_{r}}))

of the Riemann zeta function. Here,

{t_{k} : k \in N}

is the sequence of Gram numbers, and

α_{1}, \dots, α_{r}

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 \to \infty

, 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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