Axioms

36 pages, 458 KiB

Open AccessFeature PaperArticle

Bilinear Optimal Control for a Nonlinear Parabolic Equation Involving Nonlocal-in-Time Term

by Gisèle Mophou, Arnaud Fournier and Célia Jean-Alexis

Axioms 2025, 14(1), 38; https://doi.org/10.3390/axioms14010038 (registering DOI) - 4 Jan 2025

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We study a bilinear optimal control problem for an evolution equation with a nonlinear term that depends on both the state and its time integral. First, we establish existence and uniqueness results for this evolution equation. Then, we derive weak maximum principle results [...] Read more.

We study a bilinear optimal control problem for an evolution equation with a nonlinear term that depends on both the state and its time integral. First, we establish existence and uniqueness results for this evolution equation. Then, we derive weak maximum principle results to improve the regularity of the state equation. We proceed by formulating an optimal control problem aimed at steering the system’s state to a desired final state. Finally, we demonstrate that this optimal control problem admits a solution and derive the first- and second-order optimality conditions. Full article

(This article belongs to the Special Issue Advances in Mathematical Optimal Control and Applications)

14 pages, 290 KiB

Open AccessArticle

On the Normalized Laplacian Spectrum of the Zero-Divisor Graph of the Commutative Ring \({\mathbb{Z}_{p_1^{T_1} p_2^{T_2}}}\)

by Ali Al Khabyah, Nazim and Nadeem Ur Rehman

Axioms 2025, 14(1), 37; https://doi.org/10.3390/axioms14010037 (registering DOI) - 4 Jan 2025

Viewed by 76

Abstract

The zero-divisor graph of a commutative ring

R

with a nonzero identity, denoted by

Γ (R)

, is an undirected graph where the vertex set

Z {(R)}^{*}

consists of all nonzero zero-divisors of

R

. Two distinct vertices [...] Read more.

The zero-divisor graph of a commutative ring

R

with a nonzero identity, denoted by

Γ (R)

, is an undirected graph where the vertex set

Z {(R)}^{*}

consists of all nonzero zero-divisors of

R

. Two distinct vertices a and b in

Γ (R)

are adjacent if and only if

a b = 0

. The normalized Laplacian spectrum of zero-divisor graphs has been studied extensively due to its algebraic and combinatorial significance. Notably, Pirzada and his co-authors computed the normalized Laplacian spectrum of

Γ (Z_{n})

for specific values of

n

in the set

{p q, p^{2} q, p^{3}, p^{4}}

, where p and q are distinct primes satisfying

p < q

. Motivated by their work, this article investigates the normalized Laplacian spectrum of

Γ (Z_{n})

for a more general class of

n

, where

n

is represented as

p_{1}^{T_{1}} p_{2}^{T_{2}}

, with

p_{1}

and

p_{2}

being distinct primes

(p_{1} < p_{2})

, and

T_{1}, T_{2}

are positive integers. Full article

20 pages, 742 KiB

Open AccessArticle

Parameters Determination via Fuzzy Inference Systems for the Logistic Populations Growth Model

by Yuney Gorrin-Ortega, Selene Lilette Cardenas-Maciel, Jorge Antonio Lopez-Renteria and Nohe Ramon Cazarez-Castro

Axioms 2025, 14(1), 36; https://doi.org/10.3390/axioms14010036 - 3 Jan 2025

Viewed by 244

Abstract

This study addresses the fuzzy parameters (coefficient) determination for the logistic population growth model, proposing a novel methodology based on fuzzy logic concepts. Population dynamics are often modeled using differential equations whose parameters represent critical ecological information, where the parameters determination is a [...] Read more.

This study addresses the fuzzy parameters (coefficient) determination for the logistic population growth model, proposing a novel methodology based on fuzzy logic concepts. Population dynamics are often modeled using differential equations whose parameters represent critical ecological information, where the parameters determination is a problem itself. Unlike those approaches, the proposed methodology leverages ecosystem variables as inputs to a fuzzy inference system, which then generates fuzzy coefficients that better capture the inherent uncertainties in population dynamics. The approach was tested on a case study involving marine fish populations, where the fuzzy coefficients for growth rate and carrying capacity were calculated and integrated into the logistic model. The results illustrate that the fuzzy model with the proposed coefficients provide a robust framework for modeling population growth, preserving the increasing trajectory of the population under different scenarios. This method allows for the incorporation of expert knowledge and linguistic variables into the model, offering a more flexible and accurate representation of real-world ecosystems. The study concludes that this methodology significantly enhances the model’s applicability and predictive power, particularly in situations where precise data are not available. Full article

(This article belongs to the Special Issue Advances in Dynamical Systems and Control)

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20 pages, 269 KiB

Open AccessArticle

Nonrelativistic Approximation in the Theory of a Spin-2 Particle with Anomalous Magnetic Moment

by Alina Ivashkevich, Viktor Red’kov and Artur Ishkhanyan

Axioms 2025, 14(1), 35; https://doi.org/10.3390/axioms14010035 - 3 Jan 2025

Viewed by 320

Abstract

We start with the 50-component relativistic matrix equation for a hypothetical spin-2 particle in the presence of external electromagnetic fields. This equation is hypothesized to describe a particle with an anomalous magnetic moment. The complete wave function consists of a two-rank symmetric tensor [...] Read more.

We start with the 50-component relativistic matrix equation for a hypothetical spin-2 particle in the presence of external electromagnetic fields. This equation is hypothesized to describe a particle with an anomalous magnetic moment. The complete wave function consists of a two-rank symmetric tensor and a three-rank tensor that is symmetric in two indices. We apply the general method for performing the nonrelativistic approximation, which is based on the structure of the

50 \times 50

matrix

Γ^{0}

of the main equation. Using the 7th-order minimal equation for the matrix

Γ^{0}

, we introduce three projective operators. These operators permit us to decompose the complete wave function into the sum of three parts: one large part and two smaller parts in the nonrelativistic approximation. We have found five independent large variables and 45 small ones. To simplify the task, by eliminating the variables related to the 3-rank tensor, we have derived a relativistic system of second-order equations for the 10 components related to the symmetric tensor. We then take into account the decomposition of these 10 variables into linear combinations of large and small ones. In accordance with the general method, we separate the rest energy in the wave function and specify the orders of smallness for different terms in the arising equations. Further, after performing the necessary calculations, we derive a system of five linked equations for the five large variables. This system is presented in matrix form, which has a nonrelativistic structure, where the term representing additional interaction with the external magnetic field through three spin projections is included. The multiplier before this interaction contains the basic magnetic moment and an additional term due to the anomalous magnetic moment. The latter characteristic is treated as a free parameter within the hypothesis. Full article

(This article belongs to the Special Issue Mathematical Aspects of Quantum Field Theory and Quantization)

21 pages, 353 KiB

Open AccessArticle

On Value Distribution for the Mellin Transform of the Fourth Power of the Riemann Zeta Function

by Virginija Garbaliauskienė, Audronė Rimkevičienė, Mindaugas Stoncelis and Darius Šiaučiūnas

Axioms 2025, 14(1), 34; https://doi.org/10.3390/axioms14010034 - 3 Jan 2025

Viewed by 214

Abstract

In this paper, the asymptotic behavior of the modified Mellin transform

Z_{2} (s)

,

s = σ + i t

, of the fourth power of the Riemann zeta function is characterized by weak convergence of probability measures in the [...] Read more.

In this paper, the asymptotic behavior of the modified Mellin transform

Z_{2} (s)

,

s = σ + i t

, of the fourth power of the Riemann zeta function is characterized by weak convergence of probability measures in the space of analytic functions. The main results are devoted to probability measures defined by generalized shifts

Z_{2} (s + i φ (τ))

with a real increasing to

+ \infty

differentiable functions connected to the growth of the second moment of

Z_{2} (s)

. It is proven that the mass of the limit measure is concentrated at the point expressed as

h (s) \equiv 0

. This is used for approximation of

h (s)

by

Z_{2} (s + i φ (τ))

. Full article

12 pages, 1032 KiB

Open AccessArticle

Fractal Continuum Maxwell Creep Model

by Andriy Kryvko, Claudia del C. Gutiérrez-Torres, José Alfredo Jiménez-Bernal, Orlando Susarrey-Huerta, Eduardo Reyes de Luna and Didier Samayoa

Axioms 2025, 14(1), 33; https://doi.org/10.3390/axioms14010033 - 2 Jan 2025

Viewed by 207

Abstract

In this work, the fractal continuum Maxwell law for the creep phenomenon is introduced. By mapping standard integer space-time into fractal continuum space-time using the well-known Balankin’s approach to variable-order fractal calculus, the fractal version of Maxwell model is developed. This methodology employs [...] Read more.

In this work, the fractal continuum Maxwell law for the creep phenomenon is introduced. By mapping standard integer space-time into fractal continuum space-time using the well-known Balankin’s approach to variable-order fractal calculus, the fractal version of Maxwell model is developed. This methodology employs local fractional differential operators on discontinuous properties of fractal sets embedded in the integer space-time so that they behave as analytic envelopes of non-analytic functions in the fractal continuum space-time. Then, creep strain

ε (t)

, creep modulus

J (t)

, and relaxation compliance

G (t)

in materials with fractal linear viscoelasticity can be described by their generalized forms,

ε^{β} (t), J^{β} (t) and G^{β} (t)

, where

β = d i m S / d i m H

represents the time fractal dimension, and it implies the variable-order of fractality of the self-similar domain under study, which are

d i m S

and

d i m H

for their spectral and Hausdorff dimensions, respectively. The creep behavior depends on beta, which is characterized by its geometry and fractal topology: as beta approaches one, the fractal creep behavior approaches its standard behavior. To illustrate some physical implications of the suggested fractal Maxwell creep model, graphs that showcase the specific details and outcomes of our results are included in this study. Full article

(This article belongs to the Special Issue Fractal Analysis and Mathematical Integration)

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23 pages, 1299 KiB

Open AccessArticle

Competing Risks in Accelerated Life Testing: A Study on Step-Stress Models with Tampered Random Variables

by Hanan Haj Ahmad, Ehab M. Almetwally and Dina A. Ramadan

Axioms 2025, 14(1), 32; https://doi.org/10.3390/axioms14010032 - 2 Jan 2025

Viewed by 254

Abstract

This study introduces a novel approach to accelerated life test experiments by examining competing risk factors using the Tampered Random Variable (TRV) model. This approach remains extensively unexplored in current research. The methodology is implemented for a simple step-stress life test (SSLT) model [...] Read more.

This study introduces a novel approach to accelerated life test experiments by examining competing risk factors using the Tampered Random Variable (TRV) model. This approach remains extensively unexplored in current research. The methodology is implemented for a simple step-stress life test (SSLT) model and accounts for various causes of failure. The Power Chris–Jerry (PCJ) distribution is utilized to model the lifetimes of units under different stress levels, incorporating unique shape parameters while maintaining a fixed-scale parameter. This study employs the TRV model to integrate constant tampering coefficients for each failure cause within step-stress data analysis. Maximum-likelihood estimates for model parameters and tampering coefficients are derived from SSLT data, and some confidence intervals are presented based on the Type-II censoring scheme. Furthermore, Bayesian estimation is applied to the parameters, supported by appropriate prior distributions. The robustness of the proposed method is validated through comprehensive simulations and real-world applications in different scientific domains. Full article

(This article belongs to the Special Issue Methods and Applications of Advanced Statistical Analysis, 2nd Edition)

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18 pages, 326 KiB

Open AccessFeature PaperArticle

Group-Valued Multisets

by Andrei Alexandru and Gabriel Ciobanu

Axioms 2025, 14(1), 31; https://doi.org/10.3390/axioms14010031 - 1 Jan 2025

Viewed by 179

Abstract

Hybrid sets are defined as multisets having also negative multiplicities, i.e. as functions from a crisp set to the group of all integers. In this article, we introduce a significant advancement in hybrid sets through the concept of group-valued multisets. These multisets [...] Read more.

Hybrid sets are defined as multisets having also negative multiplicities, i.e. as functions from a crisp set to the group of all integers. In this article, we introduce a significant advancement in hybrid sets through the concept of group-valued multisets. These multisets map elements of a set X to an arbitrary group, ensuring that each multiplicity has an inverse. This framework allows us to explore deeper relationships and correlations among the multiplicities of the elements within X. By involving the finitely supported sets, we study the new defined group-valued multisets over infinite universes of discourse in a finitary manner. After presenting the algebraic groups in the framework of finitely supported sets, we study the finitely supported group-valued multisets. We provide a finitary characterization of group-valued multisets over infinite universes of discourse, and obtain new results that generalize the properties of hybrid sets obtained in the Zermelo–Fraenkel framework. Full article

15 pages, 673 KiB

Open AccessArticle

Analytical Relations and Statistical Estimations for Sums of Powered Integers

by Stan Lipovetsky

Axioms 2025, 14(1), 30; https://doi.org/10.3390/axioms14010030 - 1 Jan 2025

Viewed by 230

Abstract

Finding analytical closed-form solutions for the sums of powers of the first n positive integers is a classical problem of number theory. Analytical methods of constructing such sums produce complicated formulae of polynomials of a higher order, but they can be presented via [...] Read more.

Finding analytical closed-form solutions for the sums of powers of the first n positive integers is a classical problem of number theory. Analytical methods of constructing such sums produce complicated formulae of polynomials of a higher order, but they can be presented via the first two power sums. The current paper describes new presentations of the power sums and their extensions from polynomial to algebraic functions. Particularly, it shows that power sums of any higher order can be expressed just by a value of the arithmetic progression of the first power sum, or by the second power sum, or approximately by any another power sum. Regression modeling for the estimation of the powered sums is also considered, which is helpful for finding approximate values of long sums for big powers. Several problems based on the relations between sums of different powers in explicit forms are suggested for educational purposes. Full article

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Figure 1

Figure 1
Dependence of <math display="inline"><semantics> <mrow> <mi>y</mi> <mo>=</mo> <mstyle scriptlevel="0" displaystyle="true"> <mfrac> <mrow> <msub> <mrow> <mi>S</mi> </mrow> <mrow> <mi>p</mi> <mo>,</mo> <mi>n</mi> </mrow> </msub> </mrow> <mrow> <msup> <mrow> <mo>(</mo> <mi>n</mi> <mo>+</mo> <mn>0.5</mn> <mo>)</mo> </mrow> <mrow> <mi>p</mi> </mrow> </msup> </mrow> </mfrac> </mstyle> </mrow> </semantics></math> by n for a dozen p values. Full article ">Figure 2
Dependence of <math display="inline"><semantics> <mrow> <mi>y</mi> <mo>=</mo> <mstyle scriptlevel="0" displaystyle="true"> <mfrac> <mrow> <msub> <mrow> <mi>S</mi> </mrow> <mrow> <mi>p</mi> <mo>,</mo> <mi>n</mi> </mrow> </msub> </mrow> <mrow> <msup> <mrow> <mo>(</mo> <mi>n</mi> <mo>+</mo> <mn>0.5</mn> <mo>)</mo> </mrow> <mrow> <mi>p</mi> </mrow> </msup> </mrow> </mfrac> </mstyle> </mrow> </semantics></math> by p for a set of even n values. Full article ">

14 pages, 490 KiB

Open AccessArticle

About Stabilization of the Controlled Inverted Pendulum Under Stochastic Perturbations of the Type of Poisson’s Jumps

by Leonid Shaikhet

Axioms 2025, 14(1), 29; https://doi.org/10.3390/axioms14010029 - 31 Dec 2024

Viewed by 206

Abstract

The classical problem of stabilization of the controlled inverted pendulum is considered in the case of stochastic perturbations of the type of Poisson’s jumps. It is supposed that stabilized control depends on the entire trajectory of the pendulum. Linear and nonlinear models of [...] Read more.

The classical problem of stabilization of the controlled inverted pendulum is considered in the case of stochastic perturbations of the type of Poisson’s jumps. It is supposed that stabilized control depends on the entire trajectory of the pendulum. Linear and nonlinear models of the controlled inverted pendulum are considered, and the stability of the zero and nonzero equilibria is studied. The obtained results are illustrated by examples with numerical simulation of solutions of the equations under consideration. Full article

(This article belongs to the Special Issue Advances in Mathematical Optimal Control and Applications)

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23 pages, 362 KiB

Open AccessArticle

Mappings of a Bounded Dirichlet Integral: The Modulus Method

by Mihai Cristea

Axioms 2025, 14(1), 28; https://doi.org/10.3390/axioms14010028 - 31 Dec 2024

Viewed by 177

Abstract

We study the geometric properties of some classes of mappings for which an inverse Poletsky modular inequality holds. In these classes of mappings, we give some extensions of the theorems of Lindelőf and Fatou from the classical complex analysis. We also find some [...] Read more.

We study the geometric properties of some classes of mappings for which an inverse Poletsky modular inequality holds. In these classes of mappings, we give some extensions of the theorems of Lindelőf and Fatou from the classical complex analysis. We also find some conditions for the existence of injective minimizers for mappings of biconformal energy. Full article

(This article belongs to the Section Mathematical Analysis)

20 pages, 499 KiB

Open AccessArticle

Definition of Triangular Norms and Triangular Conorms on Subfamilies of Type-2 Fuzzy Sets

by Pablo Hernández-Varela, Francisco Javier Talavera, Susana Cubillo, Carmen Torres-Blanc and Jorge Elorza

Axioms 2025, 14(1), 27; https://doi.org/10.3390/axioms14010027 - 31 Dec 2024

Viewed by 276

Abstract

In certain stages of the application of a type-2 fuzzy logic system, it is necessary to perform operations between input or output fuzzy variables in order to compute the union, intersection, aggregation, complement, and so forth. In this context, operators that satisfy the [...] Read more.

In certain stages of the application of a type-2 fuzzy logic system, it is necessary to perform operations between input or output fuzzy variables in order to compute the union, intersection, aggregation, complement, and so forth. In this context, operators that satisfy the axioms of t-norms and t-conorms are of particular significance, as they are applied to model intersection and union, respectively. Furthermore, the existence of a range of these operators allows for the selection of the t-norm or t-conorm that offers the optimal performance, in accordance with the specific context of the system. In this paper, we obtain new t-norms and t-conorms on some important subfamilies of the set of functions from

[0, 1]

to

[0, 1]

. The structure of these families provides a more solid algebraic foundation for the applications. In particular, we define these new operators on the subsets of the functions that are convex, normal, and normal and convex, as well as the functions taking only the values 0 or 1 and the subset of functions whose support is a finite union of closed intervals. These t-norms and t-conorms are generalized to the type-2 fuzzy set framework. Full article

(This article belongs to the Special Issue Advances in Fuzzy Logic with Applications)

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13 pages, 258 KiB

Open AccessArticle

Analyzing Uniqueness of Solutions in Nonlinear Fractional Differential Equations with Discontinuities Using Lebesgue Spaces

by Farva Hafeez, Mdi Begum Jeelani and Nouf Abdulrahman Alqahtani

Axioms 2025, 14(1), 26; https://doi.org/10.3390/axioms14010026 - 31 Dec 2024

Viewed by 261

Abstract

We explore the existence and uniqueness of solutions to nonlinear fractional differential equations (FDEs), defined in the sense of RL-fractional derivatives of order

η \in (1, 2)

. The nonlinear term is assumed to have a discontinuity at zero. By [...] Read more.

We explore the existence and uniqueness of solutions to nonlinear fractional differential equations (FDEs), defined in the sense of RL-fractional derivatives of order

η \in (1, 2)

. The nonlinear term is assumed to have a discontinuity at zero. By employing techniques from Lebesgue spaces, including Holder’s inequality, we establish uniqueness theorems for this problem, analogous to Nagumo, Krasnoselskii–Krein, and Osgood-type results. These findings provide a fundamental framework for understanding the properties of solutions to nonlinear FDEs with discontinuous nonlinearities. Full article

(This article belongs to the Special Issue Fractional Calculus—Theory and Applications, 3rd Edition)

35 pages, 452 KiB

Open AccessArticle

The Theory and Applications of Hölder Widths

by Man Lu and Peixin Ye

Axioms 2025, 14(1), 25; https://doi.org/10.3390/axioms14010025 - 31 Dec 2024

Viewed by 231

Abstract

We introduce the Hölder width, which measures the best error performance of some recent nonlinear approximation methods, such as deep neural network approximation. Then, we investigate the relationship between Hölder widths and other widths, showing that some Hölder widths are essentially smaller than [...] Read more.

We introduce the Hölder width, which measures the best error performance of some recent nonlinear approximation methods, such as deep neural network approximation. Then, we investigate the relationship between Hölder widths and other widths, showing that some Hölder widths are essentially smaller than

n

-Kolmogorov widths and linear widths. We also prove that, as the Hölder constants grow with n, the Hölder widths are much smaller than the entropy numbers. The fact that Hölder widths are smaller than the known widths implies that the nonlinear approximation represented by deep neural networks can provide a better approximation order than other existing approximation methods, such as adaptive finite elements and

n

-term wavelet approximation. In particular, we show that Hölder widths for Sobolev and Besov classes, induced by deep neural networks, are

O (n^{- 2 s / d})

and are much smaller than other known widths and entropy numbers, which are

O (n^{- s / d})

. Full article

(This article belongs to the Special Issue Numerical Computation, Approximation of Functions and Applied Mathematics, 2nd Edition)

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18 pages, 2240 KiB

Open AccessArticle

The Characteristic Relation in Two-Dimensional Type I Intermittency

by Juan Colman and Sergio Elaskar

Axioms 2025, 14(1), 24; https://doi.org/10.3390/axioms14010024 - 31 Dec 2024

Viewed by 239

Abstract

To explore intermittency in discrete systems with two or more degrees of freedom, we analyze the general characteristics of type I intermittency within a two-dimensional map. This investigation is carried out numerically, concentrating on the system’s attractors, bifurcation diagrams, and the characteristic relation [...] Read more.

To explore intermittency in discrete systems with two or more degrees of freedom, we analyze the general characteristics of type I intermittency within a two-dimensional map. This investigation is carried out numerically, concentrating on the system’s attractors, bifurcation diagrams, and the characteristic relation associated with type I intermittency. We present two methods for determining the laminar interval and the channel structure. Our computations yield numerical results for the average laminar length as a function of the control parameter, which we then compare with findings from intermittency in one-dimensional maps. We observe a strong agreement between the numerical data and the theoretical predictions. Full article

(This article belongs to the Special Issue Advancements in Applied Mathematics and Computational Physics)

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13 pages, 480 KiB

Open AccessArticle

On Trees with a Given Number of Vertices of Fixed Degree and Their Two Bond Incident Degree Indices

by Abeer M. Albalahi, Muhammad Rizwan, Akhlaq A. Bhatti, Ivan Gutman, Akbar Ali, Tariq Alraqad and Hicham Saber

Axioms 2025, 14(1), 23; https://doi.org/10.3390/axioms14010023 - 30 Dec 2024

Viewed by 255

Abstract

This paper is mainly concerned with the study of two bond incident degree (BID) indices, namely the variable sum exdeg index

S E I_{a}

and the general zeroth-order Randić index

{}^{0}R_{α}

. The minimum values of

S E I_{a}

[...] Read more.

This paper is mainly concerned with the study of two bond incident degree (BID) indices, namely the variable sum exdeg index

S E I_{a}

and the general zeroth-order Randić index

{}^{0}R_{α}

. The minimum values of

S E I_{a}

and

{}^{0}R_{α}

in the class of all trees of fixed order containing no vertex of even degree are obtained for

a > 1

and

α \in [0, 1]

; also, the maximum value of

{}^{0}R_{α}

in the mentioned class is determined for

0 < α < 1

. Moreover, in the family of all trees of fixed order and with a given number of vertices of even degrees, the extremum values of

S E I_{a}

and

{}^{0}R_{α}

are found for every real number

α \notin {0, 1}

and

a > 1

. Furthermore, in the class of all trees of fixed order and with a given number of vertices of maximum degree, the minimum values of

S E I_{a}

and

{}^{0}R_{α}

are determined when

a > 1

and

α

does not belong to the closed interval

[0, 1]

; in the same class, the maximum values of

{}^{0}R_{α}

are also found for

0 < α < 1

. The graphs that achieve the obtained extremal values are also determined. Full article

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Figure 1
The trees T and <math display="inline"><semantics> <mrow> <msup> <mi>T</mi> <mo>′</mo> </msup> <mo>=</mo> <mrow> <mo>(</mo> <mi>T</mi> <mo>−</mo> <mrow> <mo>{</mo> <mi>x</mi> <msub> <mi>b</mi> <mn>1</mn> </msub> <mo>,</mo> <mi>x</mi> <msub> <mi>b</mi> <mn>2</mn> </msub> <mo>}</mo> </mrow> <mo>)</mo> </mrow> <mo>+</mo> <mrow> <mo>{</mo> <msub> <mi>b</mi> <mn>1</mn> </msub> <mi>w</mi> <mo>,</mo> <msub> <mi>b</mi> <mn>2</mn> </msub> <mi>w</mi> <mo>}</mo> </mrow> </mrow> </semantics></math> used in Lemma 1. Full article ">Figure 2
The tree <math display="inline"><semantics> <msup> <mi>T</mi> <mo>′</mo> </msup> </semantics></math> considered in Example 1. Full article ">Figure 3
The trees T and <math display="inline"><semantics> <mrow> <msup> <mi>T</mi> <mo>′</mo> </msup> <mo>=</mo> <mrow> <mo>(</mo> <mi>T</mi> <mo>−</mo> <mrow> <mo>{</mo> <mi>x</mi> <msub> <mi>b</mi> <mn>1</mn> </msub> <mo>}</mo> </mrow> <mo>)</mo> </mrow> <mo>+</mo> <mrow> <mo>{</mo> <msub> <mi>b</mi> <mn>1</mn> </msub> <mi>w</mi> <mo>}</mo> </mrow> </mrow> </semantics></math> considered in Lemma 2. Full article ">Figure 4
The trees T and <math display="inline"><semantics> <mrow> <msup> <mi>T</mi> <mo>′</mo> </msup> <mo>=</mo> <mrow> <mo>(</mo> <mi>T</mi> <mo>−</mo> <mrow> <mo>{</mo> <msub> <mi>w</mi> <mn>1</mn> </msub> <mi>y</mi> <mo>,</mo> <mo>…</mo> <mo>,</mo> <msub> <mi>w</mi> <mrow> <msub> <mi>d</mi> <mi>y</mi> </msub> <mo>−</mo> <mn>2</mn> </mrow> </msub> <mi>y</mi> <mo>}</mo> </mrow> <mo>)</mo> </mrow> <mo>+</mo> <mrow> <mo>{</mo> <msub> <mi>w</mi> <mn>1</mn> </msub> <mi>x</mi> <mo>,</mo> <mo>…</mo> <mo>,</mo> <msub> <mi>w</mi> <mrow> <msub> <mi>d</mi> <mi>y</mi> </msub> <mo>−</mo> <mn>2</mn> </mrow> </msub> <mi>x</mi> <mo>}</mo> </mrow> </mrow> </semantics></math> considered in Case 1 of Lemma 3. Full article ">Figure 5
The trees T and <math display="inline"><semantics> <mrow> <msup> <mi>T</mi> <mrow> <mo>″</mo> </mrow> </msup> <mo>=</mo> <mrow> <mo>(</mo> <mi>T</mi> <mo>−</mo> <mrow> <mo>{</mo> <msub> <mi>w</mi> <mn>1</mn> </msub> <mi>y</mi> <mo>,</mo> <mo>…</mo> <mo>,</mo> <msub> <mi>w</mi> <mrow> <msub> <mi>d</mi> <mi>y</mi> </msub> <mo>−</mo> <mn>1</mn> </mrow> </msub> <mi>y</mi> <mo>}</mo> </mrow> <mo>)</mo> </mrow> <mo>+</mo> <mrow> <mo>{</mo> <msub> <mi>w</mi> <mn>1</mn> </msub> <mi>x</mi> <mo>,</mo> <mo>…</mo> <mo>,</mo> <msub> <mi>w</mi> <mrow> <msub> <mi>d</mi> <mi>y</mi> </msub> <mo>−</mo> <mn>1</mn> </mrow> </msub> <mi>x</mi> <mo>}</mo> </mrow> </mrow> </semantics></math> considered in Case 2 of Lemma 3. Full article ">Figure 6
The trees T and <math display="inline"><semantics> <mrow> <msup> <mi>T</mi> <mrow> <mo>(</mo> <mn>1</mn> <mo>)</mo> </mrow> </msup> <mo>=</mo> <mrow> <mo>(</mo> <mi>T</mi> <mo>−</mo> <mrow> <mo>{</mo> <mi>x</mi> <mi>y</mi> <mo>}</mo> </mrow> <mo>)</mo> </mrow> <mo>+</mo> <mrow> <mo>{</mo> <mi>y</mi> <mi>w</mi> <mo>}</mo> </mrow> </mrow> </semantics></math> considered in the proof of Lemma 4. Full article ">

15 pages, 475 KiB

Open AccessArticle

On the Construction of a Two-Step Sixth-Order Scheme to Find the Drazin Generalized Inverse

by Keyang Zhang, Fazlollah Soleymani and Stanford Shateyi

Axioms 2025, 14(1), 22; https://doi.org/10.3390/axioms14010022 - 30 Dec 2024

Viewed by 241

Abstract

This study introduces a numerically efficient iterative solver for computing the Drazin generalized inverse, addressing a critical need for high-performance methods in matrix computations. The proposed two-step scheme achieves sixth-order convergence, distinguishing it as a higher-order method that outperforms several existing approaches. A [...] Read more.

This study introduces a numerically efficient iterative solver for computing the Drazin generalized inverse, addressing a critical need for high-performance methods in matrix computations. The proposed two-step scheme achieves sixth-order convergence, distinguishing it as a higher-order method that outperforms several existing approaches. A rigorous convergence analysis is provided, highlighting the importance of selecting an appropriate initial value to ensure robustness. Extensive numerical experiments validate the analytical findings, showcasing the method’s superior speed and efficiency, making it an advancement in iterative solvers for generalized inverses. Full article

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

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24 pages, 332 KiB

Open AccessArticle

Intuitionistic Hesitant Fuzzy Rough Aggregation Operator-Based EDAS Method and Its Application to Multi-Criteria Decision-Making Problems

by Muhammad Kamraz Khan, Muhammad Sajjad Ali Khan, Kamran and Ioan-Lucian Popa

Axioms 2025, 14(1), 21; https://doi.org/10.3390/axioms14010021 - 30 Dec 2024

Viewed by 252

Abstract

The fundamental notions of the intuitionistic hesitant fuzzy set (IHFS) and rough set (RS) are general mathematical tools that may easily manage imprecise and uncertain information. The EDAS (Evaluation based on Distance from Average Solution) approach has an important role in decision-making (DM) [...] Read more.

The fundamental notions of the intuitionistic hesitant fuzzy set (IHFS) and rough set (RS) are general mathematical tools that may easily manage imprecise and uncertain information. The EDAS (Evaluation based on Distance from Average Solution) approach has an important role in decision-making (DM) problems, particularly in multi-attribute group decision-making (MAGDM) scenarios, where there are many conflicting criteria. This paper aims to introduce the IHFR-EDAS approach, which utilizes the IHF rough averaging aggregation operator. The aggregation operator is crucial for aggregating intuitionistic hesitant fuzzy numbers into a cohesive component. Additionally, we introduce the concepts of the IHF rough weighted averaging (IHFRWA) operator. For the proposed operator, a new accuracy function (AF) and score function (SF) are established. Subsequently, the suggested approach is used to show the IHFR-EDAS model for MAGDM and its stepwise procedure. In conclusion, a numerical example of the constructed model is demonstrated, and a general comparison between the investigated models and the current methods demonstrates that the investigated models are more feasible and efficient than the present methods. Full article

(This article belongs to the Special Issue Advances in Fuzzy Logic with Applications)

15 pages, 2338 KiB

Open AccessArticle

A Comparative Study and Numerical Solutions for the Fractional Modified Lorenz–Stenflo System Using Two Methods

by Mohamed Elbadri, Mohamed A. Abdoon, Abdulrahman B. M. Alzahrani, Rania Saadeh and Mohammed Berir

Axioms 2025, 14(1), 20; https://doi.org/10.3390/axioms14010020 - 30 Dec 2024

Viewed by 216

Abstract

This paper provides a solution to the new fractional-order Lorenz–Stenflo model using the adaptive predictor–corrector approach and the

ρ

-Laplace New Iterative Method (

L_{ρ} N I M

), representing an extensive comparison between both techniques with RK4 related to accuracy and [...] Read more.

This paper provides a solution to the new fractional-order Lorenz–Stenflo model using the adaptive predictor–corrector approach and the

ρ

-Laplace New Iterative Method (

L_{ρ} N I M

), representing an extensive comparison between both techniques with RK4 related to accuracy and error analysis. The results show that the suggested approaches allow one to be more accurate in analyzing the dynamics of the system. These techniques also produce results that are comparable to the results of other approximate techniques. The techniques can, thus, be used on a wider class of systems in order to provide more accurate results. These techniques also appropriately identify chaotic attractors in the system. These techniques can be applied to solve various numerical problems arising in science and engineering in the future. Full article

(This article belongs to the Special Issue Fractional Differential Equation and Its Applications)

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28 pages, 1678 KiB

Open AccessArticle

Handling Multicollinearity and Outliers in Logistic Regression Using the Robust Kibria–Lukman Estimator

by Adewale F. Lukman, Suleiman Mohammed, Olalekan Olaluwoye and Rasha A. Farghali

Axioms 2025, 14(1), 19; https://doi.org/10.3390/axioms14010019 - 30 Dec 2024

Viewed by 236

Abstract

Logistic regression models encounter challenges with correlated predictors and influential outliers. This study integrates robust estimators, including the Bianco–Yohai estimator (BY) and conditionally unbiased bounded influence estimator (CE), with the logistic Liu (LL), logistic ridge (LR), and logistic KL (KL) estimators. The resulting [...] Read more.

Logistic regression models encounter challenges with correlated predictors and influential outliers. This study integrates robust estimators, including the Bianco–Yohai estimator (BY) and conditionally unbiased bounded influence estimator (CE), with the logistic Liu (LL), logistic ridge (LR), and logistic KL (KL) estimators. The resulting estimators (LL-BY, LL-CE, LR-BY, LR-CE, KL-BY, and KL-CE) are evaluated through simulations and real-life examples. KL-BY emerges as the preferred choice, displaying superior performance by reducing mean squared error (MSE) values and exhibiting robustness against multicollinearity and outliers. Adopting KL-BY can lead to stable and accurate predictions in logistic regression analysis. Full article

(This article belongs to the Special Issue Mathematical Tools and Techniques Applicable to Probability Theory and Statistics II)

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17 pages, 4615 KiB

Open AccessArticle

Analysis of Bulk Queueing Model with Load Balancing and Vacation

by Subramani Palani Niranjan, Suthanthiraraj Devi Latha, Sorin Vlase and Maria Luminita Scutaru

Axioms 2025, 14(1), 18; https://doi.org/10.3390/axioms14010018 - 30 Dec 2024

Viewed by 245

Abstract

Data center architecture plays an important role in effective server management network systems. Load balancing is one such data architecture used to efficiently distribute network traffic to the server. In this paper, we incorporated the load-balancing technique used in cloud computing with power [...] Read more.

Data center architecture plays an important role in effective server management network systems. Load balancing is one such data architecture used to efficiently distribute network traffic to the server. In this paper, we incorporated the load-balancing technique used in cloud computing with power business intelligence (BI) and cloud load based on the queueing theoretic approach. This model examines a bulk arrival and batch service queueing system, incorporating server overloading and underloading based on the queue length. In a batch service system, customers are served in groups following a general bulk service rule with the server operating between the minimum value

‘ a ’

and the maximum value

‘ b ’

. But in certain situations, maintaining the same extreme values of the server is difficult, and it needs to be changed according to the service request. In this paper, server load balancing is introduced for a batch service queueing model, which is the capacity of the server that can be adjusted, either increased or decreased, based upon the service request by the customer. On service completion, if the service request is not enough to start any of the services, the server will be assigned to perform a secondary job (vacation). After vacation completion based upon the service request, the server will start regular service, overload or underload. Cloud computing using power BI can be analyzed based on server load balancing. The function that determines the probability of the queue size at any given time is derived for the specified queueing model using the supplementary variable technique with the remaining time as the supplementary variable. Additionally, various system characteristics are calculated and illustrated with suitable numerical examples. Full article

(This article belongs to the Special Issue Mathematical and Statistical Methods and Their Applications, 2nd Edition)

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14 pages, 307 KiB

Open AccessArticle

The Singularity of the K₄ Homeomorphic Graph

by Haicheng Ma

Axioms 2025, 14(1), 17; https://doi.org/10.3390/axioms14010017 - 30 Dec 2024

Viewed by 209

Abstract

Let G be a finite simple graph and let

A (G)

be its adjacency matrix. Then, G is singular if

A (G)

is singular. The singularity of graphs is of certain interest in graph theory and algebraic combinatorics. For [...] Read more.

Let G be a finite simple graph and let

A (G)

be its adjacency matrix. Then, G is singular if

A (G)

is singular. The singularity of graphs is of certain interest in graph theory and algebraic combinatorics. For positive integers

a_{i} \geq 2

,

i = 1, 2, \dots, 6

. Insert

a_{1} - 2

,

a_{2} - 2

,

a_{3} - 2

,

a_{4} - 2

,

a_{5} - 2

and

a_{6} - 2

vertices in the six edges of the complete graph

K_{4}

, respectively, then the resulting graph is called the

K_{4}

homeomorphic graph, denoted by

K (a_{1}, a_{2}, a_{3}, a_{4}, a_{5}, a_{6})

. In this paper, we give the necessary and sufficient condition for the singularity of

K (a_{1}, a_{2}, a_{3}, a_{4}, a_{5}, a_{6})

, and we also show that the probability of a

K_{4}

homeomorphic graph

K (a_{1}, a_{2}, a_{3}, a_{4}, a_{5}, a_{6})

being a singular graph is equal to

\frac{193}{512}

. Full article

(This article belongs to the Special Issue Recent Developments in Graph Theory)

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Figure 1
A weighting on the grid graph <math display="inline"><semantics> <mrow> <msub> <mi>P</mi> <mn>7</mn> </msub> <mo>×</mo> <msub> <mi>P</mi> <mn>7</mn> </msub> </mrow> </semantics></math> and the wheel graph <math display="inline"><semantics> <msub> <mi>W</mi> <mn>8</mn> </msub> </semantics></math> that satisfies Equation <math display="inline"><semantics> <mrow> <mo>(</mo> <mn>1</mn> <mo>)</mo> </mrow> </semantics></math>. Full article ">Figure 2
The <math display="inline"><semantics> <msub> <mi>K</mi> <mn>4</mn> </msub> </semantics></math> homeomorphic graph <math display="inline"><semantics> <mrow> <mi>K</mi> <mo>(</mo> <msub> <mi>a</mi> <mn>1</mn> </msub> <mo>,</mo> <msub> <mi>a</mi> <mn>2</mn> </msub> <mo>,</mo> <msub> <mi>a</mi> <mn>3</mn> </msub> <mo>,</mo> <msub> <mi>a</mi> <mn>4</mn> </msub> <mo>,</mo> <msub> <mi>a</mi> <mn>5</mn> </msub> <mo>,</mo> <msub> <mi>a</mi> <mn>6</mn> </msub> <mo>)</mo> </mrow> </semantics></math>. Full article ">Figure 3
The graph X and Y, and the spanning Sachs subgraphs of these two graphs. Full article ">Figure 4
The graph <math display="inline"><semantics> <msub> <mi>G</mi> <mn>1</mn> </msub> </semantics></math>. Full article ">Figure 5
The graph <math display="inline"><semantics> <msub> <mi>G</mi> <mn>2</mn> </msub> </semantics></math>. Full article ">Figure 6
Two perfect matchings in Figure <math display="inline"><semantics> <mrow> <mi>K</mi> <mo>(</mo> <mn>3</mn> <mo>,</mo> <mn>3</mn> <mo>,</mo> <mn>3</mn> <mo>,</mo> <mn>3</mn> <mo>,</mo> <mn>4</mn> <mo>,</mo> <mn>4</mn> <mo>)</mo> </mrow> </semantics></math> (1,2). Two perfect matchings in Figure <math display="inline"><semantics> <mrow> <mi>K</mi> <mo>(</mo> <mn>3</mn> <mo>,</mo> <mn>3</mn> <mo>,</mo> <mn>4</mn> <mo>,</mo> <mn>3</mn> <mo>,</mo> <mn>3</mn> <mo>,</mo> <mn>4</mn> <mo>)</mo> </mrow> </semantics></math> (3,4). Full article ">

27 pages, 28870 KiB

Open AccessArticle

A Novel Procedure in Scrutinizing a Cantilever Beam with Tip Mass: Analytic and Bifurcation

by Asma Alanazy, Galal M. Moatimid, T. S. Amer, Mona A. A. Mohamed and M. K. Abohamer

Axioms 2025, 14(1), 16; https://doi.org/10.3390/axioms14010016 - 30 Dec 2024

Viewed by 236

Abstract

An examination was previously derived to conclude the understanding of the response of a cantilever beam with a tip mass (CBTM) that is stimulated by a parameter to undergo small changes in flexibility (stiffness) and tip mass. The study of this problem is [...] Read more.

An examination was previously derived to conclude the understanding of the response of a cantilever beam with a tip mass (CBTM) that is stimulated by a parameter to undergo small changes in flexibility (stiffness) and tip mass. The study of this problem is essential in structural and mechanical engineering, particularly for evaluating dynamic performance and maintaining stability in engineering systems. The existing work aims to study the same problem but in different situations. He’s frequency formula (HFF) is utilized with the non-perturbative approach (NPA) to transform the nonlinear governing ordinary differential equation (ODE) into a linear form. Mathematica Software 12.0.0.0 (MS) is employed to confirm the high accuracy between the nonlinear and the linear ODE. Actually, the NPA is completely distinct from any traditional perturbation technique. It simply inspects the stability criteria in both the theoretical and numerical calculations. Temporal histories of the obtained results, in addition to the corresponding phase plane curves, are graphed to explore the influence of various parameters on the examined system’s behavior. It is found that the NPA is simple, attractive, promising, and powerful; it can be adopted for the highly nonlinear ODEs in different classes in dynamical systems in addition to fluid mechanics. Bifurcation diagrams, phase portraits, and Poincaré maps are used to study the chaotic behavior of the model, revealing various types of motion, including periodic and chaotic behavior. Full article

(This article belongs to the Special Issue Applied Nonlinear Dynamical Systems in Mathematical Physics)

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12 pages, 257 KiB

Open AccessArticle

Factor Rings with Algebraic Identities via Generalized Derivations

by Ali Yahya Hummdi, Zakia Z. Al-Amery and Radwan M. Al-omary

Axioms 2025, 14(1), 15; https://doi.org/10.3390/axioms14010015 - 30 Dec 2024

Viewed by 192

Abstract

The current article focuses on studying the behavior of a ring

ℜ / Π

when ℜ admits generalized derivations

Ψ

and

Ω

with associated derivations

ϕ

and

δ

, respectively. These derivations satisfy specific differential identities involving

Π

, where

Π

is a [...] Read more.

The current article focuses on studying the behavior of a ring

ℜ / Π

when ℜ admits generalized derivations

Ψ

and

Ω

with associated derivations

ϕ

and

δ

, respectively. These derivations satisfy specific differential identities involving

Π

, where

Π

is a prime ideal of an arbitrary ring ℜ, not necessarily prime or semiprime. Furthermore, we explore some consequences of our findings. To emphasize the necessity of the primeness of

Π

in the hypotheses of our various theorems, we provide a list of examples. Full article

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

16 pages, 280 KiB

Open AccessArticle

Properties of Generalized Bronze Fibonacci Sequences and Their Hyperbolic Quaternions

by Engin Özkan, Hakan Akkuş and Alkan Özkan

Axioms 2025, 14(1), 14; https://doi.org/10.3390/axioms14010014 - 29 Dec 2024

Viewed by 446

Abstract

In this study, we establish some properties of Bronze Fibonacci and Bronze Lucas sequences. Then we find the relationships between the roots of the characteristic equation of these sequences with these sequences. What is interesting here is that even though the roots change, [...] Read more.

In this study, we establish some properties of Bronze Fibonacci and Bronze Lucas sequences. Then we find the relationships between the roots of the characteristic equation of these sequences with these sequences. What is interesting here is that even though the roots change, equality is still maintained. Also, we derive the special relations between the terms of these sequences. We give the important relations among these sequences, positive and negative index terms, with the sum of the squares of two consecutive terms being related to these sequences. In addition, we present the application of generalized Bronze Fibonacci sequences to hyperbolic quaternions. For these hyperbolic quaternions, we give the summation formulas, generating functions, etc. Moreover, we obtain the Binet formulas in two different ways. The first is in the known classical way and the second is with the help of the sequence’s generating functions. In addition, we calculate the special identities of these hyperbolic quaternions. Furthermore, we examine the relationships between the hyperbolic Bronze Fibonacci and Bronze Lucas quaternions. Finally, the terms of the generalized Bronze Fibonacci sequences are associated with their hyperbolic quaternion values. Full article

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

15 pages, 429 KiB

Open AccessArticle

On the Uniform Projection Problem in Descriptive Set Theory

by Vladimir Kanovei and Vassily Lyubetsky

Axioms 2025, 14(1), 13; https://doi.org/10.3390/axioms14010013 - 29 Dec 2024

Viewed by 333

Abstract

For every \({\nn\ge1})\ ≥ 1, generic models of ZFC will be presented for either of the following two sentences: 1. There exists a linear \({\is1{\nn+2}})\ set not equal to the projection of any uniform planar \({\fp1{\nn+2}})\ set. 2. There exists a linear \({\id1{\nn+2}})\ [...] Read more.

For every \({\nn\ge1})\ ≥ 1, generic models of ZFC will be presented for either of the following two sentences: 1. There exists a linear \({\is1{\nn+2}})\ set not equal to the projection of any uniform planar \({\fp1{\nn+2}})\ set. 2. There exists a linear \({\id1{\nn+2}})\ set not equal to the projection of any uniform planar \({\fp1{\nn+1}})\ set. Ensuing consistency and independence corollaries are discussed. Full article

15 pages, 1807 KiB

Open AccessArticle

A First Application of the Backward Technique in Social Sciences: Exploring Demographic Noise in a Model with Three Personality Types

by Roberto Macrelli, Margherita Carletti and Vincenzo Fano

Axioms 2025, 14(1), 9; https://doi.org/10.3390/axioms14010009 - 27 Dec 2024

Viewed by 308

Abstract

In the realm of dynamical systems described by deterministic differential equations used in biomathematical modeling, two types of random events influence the populations involved in the model: the first one is called environmental noise, due to factors external to the system; the second [...] Read more.

In the realm of dynamical systems described by deterministic differential equations used in biomathematical modeling, two types of random events influence the populations involved in the model: the first one is called environmental noise, due to factors external to the system; the second one is called demographic noise, deriving from the inherent randomness of the modeled phenomenon. When the populations are small, only space-discrete stochastic models are capable of describing demographic noise; when the populations are large, these discrete models converge to continuous models described by stochastic ordinary differential systems, maintaining the essence of intrinsic noise. Moving forward again from a continuous stochastic framework, we get to the continuous deterministic setting described by ordinary differential equations if we assume that noise can be neglected. The inverse process has recently been explored in the literature by means of the so-called “backward technique” in a biological context, starting from a system of continuous ordinary differential equations and going “backward” to the reconstruction and numerical simulation of the underlying discrete stochastic process, that models the demographic noise intrinsic to the biological phenomenon. In this study, starting from a predictable, deterministic system, we move beyond biology and explore the effects of demographic noise in a novel model arising from the social sciences. Our field will be psychosocial, that is, the connections and processes that support social relationships between individuals. We consider a group of individuals having three personality types: altruistic, selfish, and susceptible (neutral). Applying the backward technique to this model built on ordinary differential equations, we demonstrate how demographic noise can act as a switching factor, i.e., moving backward from the deterministic continuous model to the discrete stochastic process using the same parameter values, a given equilibrium switches to a different one. This highlights the importance of addressing demographic noise when studying complex social interactions. To our knowledge, this is also the first time that the backward technique has been applied in social contexts. Full article

(This article belongs to the Special Issue Advances in Mathematical Modeling and Related Topics)

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13 pages, 517 KiB

Open AccessFeature PaperArticle

First and Second Integrals of Hopf–Langford-Type Systems

by Vassil M. Vassilev and Svetoslav G. Nikolov

Axioms 2025, 14(1), 8; https://doi.org/10.3390/axioms14010008 - 27 Dec 2024

Viewed by 214

Abstract

The work examines a seven-parameter, three-dimensional, autonomous, cubic nonlinear differential system. This system extends and generalizes the previously studied quadratic nonlinear Hopf–Langford-type systems. First, by introducing polar coordinates in its phase space, we show that the regarded system can be reduced to a [...] Read more.

The work examines a seven-parameter, three-dimensional, autonomous, cubic nonlinear differential system. This system extends and generalizes the previously studied quadratic nonlinear Hopf–Langford-type systems. First, by introducing polar coordinates in its phase space, we show that the regarded system can be reduced to a two-dimensional Liénard system, which corresponds to a second-order Liénard equation. Then, we present (in explicit form) polynomial first and second integrals of Liénard systems of the considered type identifying those values of their parameters for which these integrals exist. It is also proved that a generic Liénard equation is factorizable if and only if the corresponding Liénard system admits a second integral of a special form. It is established that each Liénard system corresponding to a Hopf–Langford system of the considered type admits such a second integral, and hence, the respective Liénard equation is factorizable. Full article

(This article belongs to the Special Issue Complex Networks and Dynamical Systems)

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15 pages, 298 KiB

Open AccessArticle

Finite Local Rings of Length 4

by Sami Alabiad, Alhanouf Ali Alhomaidhi and Nawal A. Alsarori

Axioms 2025, 14(1), 12; https://doi.org/10.3390/axioms14010012 - 27 Dec 2024

Viewed by 224

Abstract

This paper presents a comprehensive characterization of finite local rings of length 4 and with residue field

F_{p^{m}},

where p is a prime number. Such rings have an order of

p^{4 m}

elements. The current paper provides the structure [...] Read more.

This paper presents a comprehensive characterization of finite local rings of length 4 and with residue field

F_{p^{m}},

where p is a prime number. Such rings have an order of

p^{4 m}

elements. The current paper provides the structure and classification, up to isomorphism, of local rings consisting of

p^{4 m}

elements. We also give the exact number of non-isomorphic classes of these rings with fixed invariants

p, n, m, k .

In particular, we have listed all finite local rings of 4-length and of order

p^{8}

and

256 .

Full article

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

12 pages, 248 KiB

Open AccessArticle

Solutions of Cauchy Problems for the Caudrey–Dodd–Gibbon–Kotera–Sawada Equation in Three Spatial and Two Temporal Dimensions

by Yufeng Zhang and Linlin Gui

Axioms 2025, 14(1), 11; https://doi.org/10.3390/axioms14010011 - 27 Dec 2024

Viewed by 268

Abstract

A.S. Fokas has obtained integrable nonlinear partial differential equations (PDEs) in 4 + 2 dimensions by complexifying the independent variables. In this work, the complexification of the independent variables of the 2 + 1-dimensional Caudrey–Dodd–Gibbon–Kotera–Sawada (CDGKS) equation yields the 4 + 2 integrable [...] Read more.

A.S. Fokas has obtained integrable nonlinear partial differential equations (PDEs) in 4 + 2 dimensions by complexifying the independent variables. In this work, the complexification of the independent variables of the 2 + 1-dimensional Caudrey–Dodd–Gibbon–Kotera–Sawada (CDGKS) equation yields the 4 + 2 integrable extension of the CDGKS equation. Then, by transforming two temporal variables, the CDGKS equation in three dimensions is reduced, and the Lax pairs of the corresponding equations are given. Finally, the solutions of Cauchy problems for the CDGKS equation in three spatial and two temporal dimensions are constructed by introducing a novel nonlocal d-bar formalism, in which several new long derivative operators,

D_{x}

,

D_{y}

, and

D_{t}

, are constructed for the study of the initial value problem for the CDGKS equation. Some significant propositions and results are presented in this paper. Full article

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