Axioms

9 pages, 224 KiB

Open AccessArticle

On Some Unification Theorems: Yang–Baxter Systems; Johnson–Tzitzeica Theorem

by Florin Felix Nichita

Axioms 2025, 14(3), 156; https://doi.org/10.3390/axioms14030156 - 21 Feb 2025

This paper investigates the properties of the Yang–Baxter equation, which was initially formulated in the field of theoretical physics and statistical mechanics. The equation’s framework is extended through Yang–Baxter systems, aiming to unify algebraic and coalgebraic structures. The unification of the algebra structures [...] Read more.

This paper investigates the properties of the Yang–Baxter equation, which was initially formulated in the field of theoretical physics and statistical mechanics. The equation’s framework is extended through Yang–Baxter systems, aiming to unify algebraic and coalgebraic structures. The unification of the algebra structures and the coalgebra structures leads to an extension for the duality between finite dimensional algebras and finite dimensional coalgebras to the category of finite dimensional Yang–Baxter structures. In the same manner, we attempt to unify the Tzitzeica–Johnson theorem and its dual version, obtaining a new theorem about circle configurations. Full article

(This article belongs to the Special Issue New Perspectives in Lie Algebras)

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14 pages, 280 KiB

Open AccessArticle

Spacelike Hypersurfaces in de Sitter Space

by Yanlin Li, Mona Bin-Asfour, Kholoud Saad Albalawi and Mohammed Guediri

Axioms 2025, 14(3), 155; https://doi.org/10.3390/axioms14030155 - 21 Feb 2025

Abstract

A closed conformal vector field in de Sitter space

S_{1}^{n + 1} (\bar{c})

induces a vector field on a spacelike hypersurface M of

S_{1}^{n + 1} (\bar{c})

, referred to as the induced vector field on M [...] Read more.

A closed conformal vector field in de Sitter space

S_{1}^{n + 1} (\bar{c})

induces a vector field on a spacelike hypersurface M of

S_{1}^{n + 1} (\bar{c})

, referred to as the induced vector field on M. This article investigates the characterization of compact spacelike hypersurfaces in de Sitter space without assuming the constancy of the mean curvature. Specifically, we establish that under certain conditions, a compact spacelike hypersurface in

S_{1}^{n + 1} (\bar{c})

is a sphere, that is, a totally umbilical hypersurface with constant mean curvature. We also present a different characterization of compact spacelike hypersurfaces in de Sitter space as spheres by using a lower bound on the integral of the Ricci curvature of the compact hypersurface in the direction of the induced vector field. We also consider de Sitter space as a Robertson–Walker space and provide several characterizations of spheres within its spacelike hypersurfaces. Full article

(This article belongs to the Special Issue Advances in Differential Geometry and Mathematical Physics)

12 pages, 269 KiB

Open AccessArticle

An Analysis of the Continuum Hypothesis

by Andrew Powell

Axioms 2025, 14(3), 154; https://doi.org/10.3390/axioms14030154 - 20 Feb 2025

Abstract

This paper analyzes the Continuum Hypothesis, that the cardinality of a set of real numbers is either finite, countably infinite, or the same as the cardinality of the set of all real numbers. It argues (i) that the real numbers are as similar [...] Read more.

This paper analyzes the Continuum Hypothesis, that the cardinality of a set of real numbers is either finite, countably infinite, or the same as the cardinality of the set of all real numbers. It argues (i) that the real numbers are as similar to the natural numbers as possible in the sense that the relationship between any general method of deciding membership of a set of real numbers and the cardinality of the set should be a natural generalization of the case of the same relationship in the case of a set of natural numbers; and (ii) that CH is a very strong choice principle that is maximally efficient as a principle for deciding whether a real number is in a set of real numbers in the sense that it is uniform in deciding membership for every real number in a countable number of steps. The approach taken is to formulate principles equivalent to or weaker than the Continuum Hypothesis and to use techniques from computer science (infinite binary search), information theory, and set theory to prove theorems that support theses (i) and (ii). Full article

21 pages, 309 KiB

Open AccessArticle

Research on Group Scheduling with General Logarithmic Deterioration Subject to Maximal Completion Time Cost

by Jin-Da Miao, Dan-Yang Lv, Cai-Min Wei and Ji-Bo Wang

Axioms 2025, 14(3), 153; https://doi.org/10.3390/axioms14030153 - 20 Feb 2025

Abstract

Single-machine group scheduling with general logarithmic deterioration is investigated, where the actual job processing (resp. group setup) time is a non-decreasing function of the sum of the logarithmic job processing (resp. group setup) times of the jobs (resp. groups) already processed. Under some [...] Read more.

Single-machine group scheduling with general logarithmic deterioration is investigated, where the actual job processing (resp. group setup) time is a non-decreasing function of the sum of the logarithmic job processing (resp. group setup) times of the jobs (resp. groups) already processed. Under some optimal properties, it is shown that the maximal completion time (i.e., makespan) cost is solved in polynomial time and the optimal algorithm is presented. In addition, an extension of the general weighted deterioration model is given. Full article

(This article belongs to the Special Issue Mathematical Optimizations and Operations Research)

24 pages, 398 KiB

Open AccessArticle

Objective Posterior Analysis of kth Record Statistics in Gompertz Model

by Zoran Vidović and Liang Wang

Axioms 2025, 14(3), 152; https://doi.org/10.3390/axioms14030152 - 20 Feb 2025

Abstract

The Gompertz distribution has proven highly valuable in modeling human mortality rates and assessing the impacts of catastrophic events, such as plagues, financial crashes, and famines. Record data, which capture extreme values and critical trends, are particularly relevant for analyzing such phenomena. In [...] Read more.

The Gompertz distribution has proven highly valuable in modeling human mortality rates and assessing the impacts of catastrophic events, such as plagues, financial crashes, and famines. Record data, which capture extreme values and critical trends, are particularly relevant for analyzing such phenomena. In this study, we propose an objective Bayesian framework for estimating the parameters of the Gompertz distribution using record data. We analyze the performance of several objective priors, including the reference prior, Jeffreys’ prior, the maximal data information (MDI) prior, and probability matching priors. The suitability and properties of the resulting posterior distributions are systematically examined for each prior. A detailed simulation study is performed to assess the effectiveness of various estimators based on the performance criteria. To demonstrate the practical application of the methodology, it is applied to a real-world dataset. This study contributes to the field by providing a thorough comparative evaluation of objective priors and showcasing their impact and applicability in parameter estimation for Gompertz distribution based on record values. Full article

(This article belongs to the Special Issue Stochastic Modeling and Its Analysis)

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15 pages, 560 KiB

Open AccessArticle

Characterization Results of Extremization Models with Interval Values

by Savin Treanţă and Omar Mutab Alsalami

Axioms 2025, 14(3), 151; https://doi.org/10.3390/axioms14030151 - 20 Feb 2025

Abstract

The present paper investigates new connections and characterization results on interval-valued minimization models. Specifically, we describe the solution set of the considered control problem with mixed constraints by employing the solution set associated with a class of controlled split variational inequalities. These equivalence [...] Read more.

The present paper investigates new connections and characterization results on interval-valued minimization models. Specifically, we describe the solution set of the considered control problem with mixed constraints by employing the solution set associated with a class of controlled split variational inequalities. These equivalence results are also accompanied by suitable numerical experiments. Full article

(This article belongs to the Special Issue New Perspectives in Operator Theory and Functional Analysis)

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14 pages, 280 KiB

Open AccessArticle

On the Complete Indeterminacy and the Chaoticity of the Generalized Heun Operator in Bargmann Space

by Abdelkader Intissar

Axioms 2025, 14(3), 150; https://doi.org/10.3390/axioms14030150 - 20 Feb 2025

Abstract

In 1998, we gave a complete scattering analysis of the cubic Heun operator

H = a^{*} (a + a^{*}) a

acting on Bargmann space, where a and

a^{*}

are the standard Bose annihilation and creation operators satisfying the [...] Read more.

In 1998, we gave a complete scattering analysis of the cubic Heun operator

H = a^{*} (a + a^{*}) a

acting on Bargmann space, where a and

a^{*}

are the standard Bose annihilation and creation operators satisfying the commutation relation

[a, a^{*}] = I

. We used the boundary conditions at infinity to give a description of all maximal dissipative extensions in Bargmann space of the minimal Heun’s operator H. The characteristic functions of the dissipative extensions were computed, and some completeness theorems were obtained for the system of generalized eigenvectors of this operator. In this paper, we study the deficiency numbers of the generalized Heun’s operator

H^{p, m} = a^{*^{p}} (a^{m} + a^{*^{m}}) a^{p}; (p, m = 1, 2, \dots)

acting on Bargmann space. In particular, here we find some conditions on the parameters p and m such that

H^{p, m}

is completely indeterminate. It follows from these conditions that

H^{p, m}

is entirely of minimal type. Then, we show that

H^{p, m}

and

H^{p, m} + H^{*^{p, m}}

(where

H^{*^{p, m}}

is the adjoint of

H^{p, m}

) are connected to the chaotic operators. Full article

(This article belongs to the Section Mathematical Physics)

15 pages, 349 KiB

Open AccessArticle

Convergence Analysis for Cayley Variational Inclusion Problem Involving XOR and XNOR Operations

by Arifuzzaman, Syed Shakaib Irfan and Iqbal Ahmad

Axioms 2025, 14(3), 149; https://doi.org/10.3390/axioms14030149 - 20 Feb 2025

Abstract

In this article, we introduce and study a generalized Cayley variational inclusion problem incorporating XOR and XNOR operations. We establish an equivalent fixed-point formulation and demonstrate the Lipschitz continuity of the generalized Cayley approximation operator. Furthermore, we analyze the existence and convergence of [...] Read more.

In this article, we introduce and study a generalized Cayley variational inclusion problem incorporating XOR and XNOR operations. We establish an equivalent fixed-point formulation and demonstrate the Lipschitz continuity of the generalized Cayley approximation operator. Furthermore, we analyze the existence and convergence of the proposed problem using an implicit iterative algorithm. The iterative algorithm and numerical results presented in this study significantly enhance previously known findings in this domain. Finally, a numerical result is provided to support our main result and validate the proposed algorithm using MATLAB programming. Full article

(This article belongs to the Special Issue Numerical Analysis and Optimization)

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33 pages, 4434 KiB

Open AccessArticle

Enumerating the Number of Spanning Trees of Pyramid Graphs Based on Some Nonahedral Graphs

by Ahmad Asiri and Salama Nagy Daoud

Axioms 2025, 14(3), 148; https://doi.org/10.3390/axioms14030148 - 20 Feb 2025

Abstract

The enumeration of spanning trees in various graph forms has been made easier by the study of electrically equivalent transformations, which was motivated by Kirchhoff’s work on electrical networks. In this work, using knowledge of difference equations, the electrically equivalent transformations and rules [...] Read more.

The enumeration of spanning trees in various graph forms has been made easier by the study of electrically equivalent transformations, which was motivated by Kirchhoff’s work on electrical networks. In this work, using knowledge of difference equations, the electrically equivalent transformations and rules of weighted generating function are used to calculate the explicit formulas of the number of spanning trees of novel pyramid graph types based on some nonahedral graphs. Lastly, we compare our graphs’ entropy with that of other average-degree graphs that have been researched. Full article

(This article belongs to the Section Algebra and Number Theory)

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Axioms, Volume 14, Issue 3 (March 2025) – 9 articles

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