Axioms

13 pages, 293 KiB

Open AccessArticle

Theoretical Investigation of Fractional Estimations in Liouville–Caputo Operators of Mixed Order with Applications

by Pshtiwan Othman Mohammed, Alina Alb Lupas, Ravi P. Agarwal, Majeed A. Yousif, Eman Al-Sarairah and Mohamed Abdelwahed

Axioms 2024, 13(8), 570; https://doi.org/10.3390/axioms13080570 - 21 Aug 2024

Viewed by 495

Abstract

In this study, to approximate nabla sequential differential equations of fractional order, a class of discrete Liouville–Caputo fractional operators is discussed. First, some special functions are re-called that will be useful to make a connection with the proposed discrete nabla operators. These operators [...] Read more.

In this study, to approximate nabla sequential differential equations of fractional order, a class of discrete Liouville–Caputo fractional operators is discussed. First, some special functions are re-called that will be useful to make a connection with the proposed discrete nabla operators. These operators exhibit inherent symmetrical properties which play a crucial role in ensuring the consistency and stability of the method. Next, a formula is adopted for the solution of the discrete system via binomial coefficients and analyzing the Riemann–Liouville fractional sum operator. The symmetry in the binomial coefficients contributes to the precise approximation of the solutions. Based on this analysis, the solution of its corresponding continuous case is obtained when the step size

p_{0}

tends to 0. The transition from discrete to continuous domains highlights the symmetrical nature of the fractional operators. Finally, an example is shown to testify the correctness of the presented theoretical results. We discuss the comparison of the solutions of the operators along with the numerical example, emphasizing the role of symmetry in the accuracy and reliability of the numerical method. Full article

(This article belongs to the Special Issue Fractional Calculus - Theory and Applications II)

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13 pages, 316 KiB

Open AccessArticle

On the Čech-Completeness of the Space of τ-Smooth Idempotent Probability Measures

by Ljubiša D. R. Kočinac, Adilbek A. Zaitov and Muzaffar R. Eshimbetov

Axioms 2024, 13(8), 569; https://doi.org/10.3390/axioms13080569 - 21 Aug 2024

Viewed by 459

Abstract

For the set

I (X)

of probability measures on a compact Hausdorff space X, we propose a new way to introduce the topology by using the open subsets of the space X. Then, among other things, we give a [...] Read more.

For the set

I (X)

of probability measures on a compact Hausdorff space X, we propose a new way to introduce the topology by using the open subsets of the space X. Then, among other things, we give a new proof that for a compact Hausdorff space X, the space

I (X)

is also a compact Hausdorff space. For a Tychonoff space X, we consider the topological space

I_{τ} (X)

of

τ

-smooth idempotent probability measures on X and show that the space

I_{τ} (X)

is Čech-complete if and only if the given space X is Čech-complete. Full article

(This article belongs to the Special Issue Topics in General Topology and Applications)

15 pages, 337 KiB

Open AccessArticle

Extreme Behavior of Competing Risks with Random Sample Size

by Long Bai, Kaihao Hu, Conghua Wen, Zhongquan Tan and Chengxiu Ling

Axioms 2024, 13(8), 568; https://doi.org/10.3390/axioms13080568 - 21 Aug 2024

Viewed by 352

Abstract

The advances in science and technology have led to vast amounts of complex and heterogeneous data from multiple sources of random sample length. This paper aims to investigate the extreme behavior of competing risks with random sample sizes. Two accelerated mixed types of [...] Read more.

The advances in science and technology have led to vast amounts of complex and heterogeneous data from multiple sources of random sample length. This paper aims to investigate the extreme behavior of competing risks with random sample sizes. Two accelerated mixed types of stable distributions are obtained as the extreme limit laws of random sampling competing risks under linear and power normalizations, respectively. The theoretical findings are well illustrated by typical examples and numerical studies. The developed methodology and models provide new insights into modeling complex data across numerous fields. Full article

(This article belongs to the Special Issue Advances in Financial Mathematics)

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12 pages, 397 KiB

Open AccessArticle

Nonlocal Extensions of First Order Initial Value Problems

by Ravi Shankar

Axioms 2024, 13(8), 567; https://doi.org/10.3390/axioms13080567 - 21 Aug 2024

Viewed by 355

Abstract

We study certain Volterra integral equations that extend and recover first order ordinary differential equations (ODEs). We formulate the former equations from the latter by replacing classical derivatives with nonlocal integral operators with anti-symmetric kernels. Replacements of spatial derivatives have seen success in [...] Read more.

We study certain Volterra integral equations that extend and recover first order ordinary differential equations (ODEs). We formulate the former equations from the latter by replacing classical derivatives with nonlocal integral operators with anti-symmetric kernels. Replacements of spatial derivatives have seen success in fracture mechanics, diffusion, and image processing. In this paper, we consider nonlocal replacements of time derivatives which contain future data. To account for the nonlocal nature of the operators, we formulate initial “volume” problems (IVPs) for these integral equations; the initial data is prescribed on a time interval rather than at a single point. As a nonlocality parameter vanishes, we show that the solutions to these equations recover those of classical ODEs. We demonstrate this convergence with exact solutions of some simple IVPs. However, we find that the solutions of these nonlocal models exhibit several properties distinct from their classical counterparts. For example, the solutions exhibit discontinuities at periodic intervals. In addition, for some IVPs, a continuous initial profile develops a measure-valued singularity in finite time. At subsequent periodic intervals, these solutions develop increasingly higher order distributional singularities. Full article

(This article belongs to the Special Issue Difference, Functional, and Related Equations)

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17 pages, 383 KiB

Open AccessArticle

An Exploration of Ideals and Filters in Triangle Algebras

by Euclide Noumen, Fabrice Tchoua Yinga, Blaise Blériot Koguep Njionou and Chris Cornelis

Axioms 2024, 13(8), 566; https://doi.org/10.3390/axioms13080566 - 21 Aug 2024

Viewed by 329

Abstract

In the study of algebraic structures related to logical systems, ideals and filters have different meanings and are algebraic notions related to logical provable formulas. Unlike the classical Boolean lattice theory, ideals and filters are not dual notions in residuated lattices. An interesting [...] Read more.

In the study of algebraic structures related to logical systems, ideals and filters have different meanings and are algebraic notions related to logical provable formulas. Unlike the classical Boolean lattice theory, ideals and filters are not dual notions in residuated lattices. An interesting subclass of residuated lattices is the class of triangle algebras, which is an equational representation of interval-valued residuated lattices that provides an algebraic framework for using closed intervals as truth values in fuzzy logic. The main aim of this article is to introduce and study the concept of ideals in triangle algebras and investigate the connection between ideals and filters. We first point out that the construction procedure for the filter generated by a subset of a triangle algebra established by another study is incorrect, and we proceed to give an alternative characterization. Full article

(This article belongs to the Special Issue Advances in Applied Algebra and Related Topics)

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16 pages, 555 KiB

Open AccessArticle

Maximizing the Index of Signed Complete Graphs Containing a Spanning Tree with k Pendant Vertices

by Dan Li, Minghui Yan and Zhaolin Teng

Axioms 2024, 13(8), 565; https://doi.org/10.3390/axioms13080565 - 20 Aug 2024

Viewed by 367

Abstract

A signed graph

Σ = (G, σ)

consists of an underlying graph

G = (V, E)

with a sign function

σ : E \to {- 1, 1}

. Let

A (Σ)

be [...] Read more.

A signed graph

Σ = (G, σ)

consists of an underlying graph

G = (V, E)

with a sign function

σ : E \to {- 1, 1}

. Let

A (Σ)

be the adjacency matrix of

Σ

and

λ_{1} (Σ)

denote the largest eigenvalue (index) of

Σ

. Define

(K_{n}, H^{-})

as a signed complete graph whose negative edges induce a subgraph H. In this paper, we focus on the following question: which spanning tree T with a given number of pendant vertices makes the

λ_{1} (A (Σ))

of the unbalanced

(K_{n}, T^{-})

as large as possible? To answer the question, we characterize the extremal signed graph with maximum

λ_{1} (A (Σ))

among graphs of type

(K_{n}, T^{-})

. Full article

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16 pages, 337 KiB

Open AccessArticle

Probability Bracket Notation for Probability Modeling

by Xing M. Wang and Tony C. Scott

Axioms 2024, 13(8), 564; https://doi.org/10.3390/axioms13080564 - 20 Aug 2024

Viewed by 566

Abstract

Following Dirac’s notation in Quantum Mechanics (QM), we propose the Probability Bracket Notation (PBN), by defining a probability-bra (P-bra), P-ket, P-bracket, P-identity, etc. Using the PBN, many formulae, such as normalizations and expectations in systems of one or more random variables, can now [...] Read more.

Following Dirac’s notation in Quantum Mechanics (QM), we propose the Probability Bracket Notation (PBN), by defining a probability-bra (P-bra), P-ket, P-bracket, P-identity, etc. Using the PBN, many formulae, such as normalizations and expectations in systems of one or more random variables, can now be written in abstract basis-independent expressions, which are easy to expand by inserting a proper P-identity. The time evolution of homogeneous Markov processes can also be formatted in such a way. Our system P-kets are identified with probability vectors and our P-bra system is comparable with Doi’s state function or Peliti’s standard bra. In the Heisenberg picture of the PBN, a random variable becomes a stochastic process, and the Chapman–Kolmogorov equations are obtained by inserting a time-dependent P-identity. Also, some QM expressions in Dirac notation are naturally transformed to probability expressions in PBN by a special Wick rotation. Potential applications show the usefulness of the PBN beyond the constrained domain and range of Hermitian operators on Hilbert Spaces in QM all the way to IT. Full article

(This article belongs to the Special Issue Stochastic Processes in Quantum Mechanics and Classical Physics)

11 pages, 248 KiB

Open AccessArticle

An Integrated Integrable Hierarchy Arising from a Broadened Ablowitz–Kaup–Newell–Segur Scenario

by Wen-Xiu Ma

Axioms 2024, 13(8), 563; https://doi.org/10.3390/axioms13080563 - 19 Aug 2024

Viewed by 376

Abstract

This study introduces a

4 \times 4

matrix eigenvalue problem and develops an integrable hierarchy with a bi-Hamiltonian structure. Integrability is ensured by the zero-curvature condition, while the Hamiltonian structure is supported by the trace identity. Explicit derivations yield second-order and third-order integrable [...] Read more.

This study introduces a

4 \times 4

matrix eigenvalue problem and develops an integrable hierarchy with a bi-Hamiltonian structure. Integrability is ensured by the zero-curvature condition, while the Hamiltonian structure is supported by the trace identity. Explicit derivations yield second-order and third-order integrable equations, illustrating the integrable hierarchy. Full article

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

20 pages, 332 KiB

Open AccessArticle

Enriched

Z

-Contractions and Fixed-Point Results with Applications to IFS

by Ibrahim Alraddadi, Muhammad Din, Umar Ishtiaq, Mohammad Akram and Ioannis K. Argyros

Axioms 2024, 13(8), 562; https://doi.org/10.3390/axioms13080562 - 19 Aug 2024

Viewed by 387

Abstract

In this manuscript, we initiate a large class of enriched

(d, Z)

-

Z

-contractions defined on Banach spaces and prove the existence and uniqueness of the fixed point of these contractions. We also provide an example to support our [...] Read more.

In this manuscript, we initiate a large class of enriched

(d, Z)

-

Z

-contractions defined on Banach spaces and prove the existence and uniqueness of the fixed point of these contractions. We also provide an example to support our results and give an existence condition for the uniqueness of the solution to the integral equation. The results provided in the manuscript extend, generalize, and modify the existence results. Our research introduces novel fixed-point results under various contractive conditions. Furthermore, we discuss the iterated function system associated with enriched

(d, Z)

-

Z

-contractions in Banach spaces and define the enriched

Z

-Hutchinson operator. A result regarding the convergence of Krasnoselskii’s iteration method and the uniqueness of the attractor via enriched

(d, Z)

-

Z

-contractions is also established. Our discoveries not only confirm but also significantly build upon and broaden several established findings in the current body of literature. Full article

(This article belongs to the Special Issue Trends in Fixed Point Theory and Fractional Calculus)

4 pages, 175 KiB

Open AccessEditorial

Differential Geometry and Its Application, 2nd Edition

by Mića S. Stanković

Axioms 2024, 13(8), 561; https://doi.org/10.3390/axioms13080561 - 18 Aug 2024

Viewed by 491

Abstract

With this Editorial, we present a Special Issue of Axioms entitled ‘Differential Geometry and Its Application, 2nd edition’ [...] Full article

(This article belongs to the Special Issue Differential Geometry and Its Application, 2nd Edition)

21 pages, 4683 KiB

Open AccessArticle

A Novel Multi-Objective Dynamic Reliability Optimization Approach for a Planetary Gear Transmission Mechanism

by Shuiguang Tong, Xiaoyan Yan, Lechang Yang and Xianmiao Yang

Axioms 2024, 13(8), 560; https://doi.org/10.3390/axioms13080560 - 16 Aug 2024

Cited by 1 | Viewed by 494

Abstract

Planetary gear transmission mechanisms (PGTMs) are widely used in mechanical transmission systems due to their compact structure and high transmission efficiency. To implement the reliability design and optimization of a PGTM, a novel multi-objective dynamic reliability optimization approach is proposed. First, a multi-objective [...] Read more.

Planetary gear transmission mechanisms (PGTMs) are widely used in mechanical transmission systems due to their compact structure and high transmission efficiency. To implement the reliability design and optimization of a PGTM, a novel multi-objective dynamic reliability optimization approach is proposed. First, a multi-objective reliability optimization model is established. Furthermore, considering the strength degradation of gears during service, a dynamic reliability analysis is conducted based on the theory of nonlinear fatigue damage accumulation. In addition, to improve computing efficiency, a random forest surrogate model based on the particle swarm optimization algorithm is proposed. Finally, an adaptive multi-objective evolutionary algorithm based on decomposition (AMOEA/D) is designed to optimize the mechanism, along with an adaptive neighborhood updating strategy and a hybrid crossover operator. The feasibility and superiority of the proposed approach are verified through an NGW planetary gear reducer. The results show that the proposed surrogate model can reduce the calculation cost and has high accuracy. The AMOEA/D algorithm can improve transmission efficiency, reduce gear volume and ensure reliability at the same time. It can provide guidance for actual gear production. Full article

(This article belongs to the Special Issue Reliability and Risk of Complex Systems: Modelling, Analysis and Optimization)

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9 pages, 932 KiB

Open AccessArticle

Neutrosophic Analysis of Experimental Data Using Neutrosophic Graeco-Latin Square Design

by Pranesh Kumar, Mahdieh Moazzamigodarzi and Mohamadtaghi Rahimi

Axioms 2024, 13(8), 559; https://doi.org/10.3390/axioms13080559 - 16 Aug 2024

Viewed by 460

Abstract

Experimental designs are commonly used to produce valid, defensible, and supportable conclusions. Among commonly used block designs, the class of Latin square designs is used to study factors or treatment levels expressed as Latin letters and applying two blocking factors in rows and [...] Read more.

Experimental designs are commonly used to produce valid, defensible, and supportable conclusions. Among commonly used block designs, the class of Latin square designs is used to study factors or treatment levels expressed as Latin letters and applying two blocking factors in rows and columns to simultaneously control two sources of nuisance variability. Another block design in which the error can be controlled by blocking three nuisance factors is obtained by simply using two superimposed Latin square designs, with one using the Latin letters and the other using the Greek letters. Such a design is termed as a Graeco-Latin square (GLS) design. While observing or measuring data in field or lab experiments, it is often noted to have vague, incomplete, and imprecise data for whatsoever reasons. In this regard, researchers have proposed various emerging approaches, which are based on fuzzy, intuitionistic fuzzy, and neutrosophic logic, and provide deeper understanding, analysis, and interpretations of the data. In this paper, we provide a brief review of the history of GLS designs and propose a neutrosophic Graeco-Latin square design, its model, and the analysis. To illustrate this, we have considered an experimental study which analyzes the effects of different formulations of a rocket propellant, which are used in aircrew escape systems, on the observed burning rate. Full article

(This article belongs to the Special Issue The Application of Fuzzy Decision-Making Theory and Method)

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24 pages, 329 KiB

Open AccessArticle

Exploring Fixed-Point Theorems in k-Fuzzy Metric Spaces: A Comprehensive Study

by Muhammad Nazam, Seemab Attique, Aftab Hussain and Hamed H. Alsulami

Axioms 2024, 13(8), 558; https://doi.org/10.3390/axioms13080558 - 15 Aug 2024

Viewed by 861

Abstract

Recently, k -fuzzy metric spaces were introduced by connecting the degree of nearness of two points with k parameters

(t_{1}, t_{2}, t_{3}, \dots, t_{k})

and the authors presented an analogue of Grabiec’s fixed-point [...] Read more.

Recently, k -fuzzy metric spaces were introduced by connecting the degree of nearness of two points with k parameters

(t_{1}, t_{2}, t_{3}, \dots, t_{k})

and the authors presented an analogue of Grabiec’s fixed-point result in k-fuzzy metric spaces along with other necessary notions. The results presented only addressed continuous mappings. For discontinuous mappings, there is no result in k-fuzzy metric spaces. In this paper, we obtain some fixed-point results stating necessary conditions for the existence of fixed points of mappings eliminating the continuity requirement in k-fuzzy metric spaces. We illustrate the hypothesis of our findings with examples. We provide a common fixed-point theorem and fixed-point theorems for single-valued k-fuzzy Kannan type contractions. As an application, we use a fixed-point result to ensure the existence of solution of fractional differential equations. Full article

(This article belongs to the Special Issue Fixed Point Theory and Its Applications)

13 pages, 270 KiB

Open AccessArticle

Statistical Cesàro Summability in Intuitionistic Fuzzy n-Normed Linear Spaces Leading towards Tauberian Theorems

by Pradip Debnath

Axioms 2024, 13(8), 557; https://doi.org/10.3390/axioms13080557 - 15 Aug 2024

Viewed by 616

Abstract

The concept of summability is crucial in deriving formal solutions to partial differential equations. This paper explores the connection between the methods of statistical convergence of sequences and statistical Cesàro summability in intuitionistic fuzzy n-normed linear space (IFnNLS). While the existing literature [...] Read more.

The concept of summability is crucial in deriving formal solutions to partial differential equations. This paper explores the connection between the methods of statistical convergence of sequences and statistical Cesàro summability in intuitionistic fuzzy n-normed linear space (IFnNLS). While the existing literature covers Cesàro summability and its statistical variant in fuzzy, intuitionistic fuzzy, and classical normed spaces, this study stands out not only for its methodology but also for its comprehensive approach, encompassing a broader range of spaces and detailing the pathway from the statistical Cesàro summability method to statistical convergence. These results will lead us to Tauberian theorems in IFnNLS. Full article

27 pages, 381 KiB

Open AccessArticle

Different Types of Entropy Measures for Type-2 Fuzzy Sets

by Luis Magdalena, Carmen Torres-Blanc, Susana Cubillo and Jesus Martinez-Mateo

Axioms 2024, 13(8), 556; https://doi.org/10.3390/axioms13080556 - 14 Aug 2024

Viewed by 389

Abstract

In this work, we consider De Luca and Termini’s notion of non-probabilistic entropy, and we extend some entropy-like measures of the degree of fuzziness to type-2 fuzzy sets. With this aim, we first study different entropy measures proposed in the frameworks of fuzzy, [...] Read more.

In this work, we consider De Luca and Termini’s notion of non-probabilistic entropy, and we extend some entropy-like measures of the degree of fuzziness to type-2 fuzzy sets. With this aim, we first study different entropy measures proposed in the frameworks of fuzzy, intuitionistic, and interval-valued fuzzy sets. Then, we propose three possible novel axiomatizations for entropy in type-2 fuzzy sets. The proposed types of entropy measures evaluate how much a type-2 fuzzy set is non-crisp, non-fuzzy, and non-interval-valued fuzzy. This can also be interpreted as how far a type-2 fuzzy set is from a crisp, fuzzy, or interval-valued fuzzy set. The present contribution is also novel, since we considered the interpretation of type-2 fuzzy sets that is closest to Zadeh’s original conception. Full article

(This article belongs to the Special Issue Recent Advances in Fuzzy Sets and Related Topics)

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12 pages, 573 KiB

Open AccessArticle

Fuzzy Evaluation Model for Critical Components of Machine Tools

by Kuen-Suan Chen, Kai-Chao Yao, Chien-Hsin Cheng, Chun-Min Yu and Chen-Hsu Chang

Axioms 2024, 13(8), 555; https://doi.org/10.3390/axioms13080555 - 14 Aug 2024

Viewed by 418

Abstract

The rapid progression of emerging technologies like the Internet of Things (IoT) and Big Data analytics for manufacturing has driven innovation across various industries worldwide. Production data are utilized to construct a model for quality evaluation and analysis applicable to components processed by [...] Read more.

The rapid progression of emerging technologies like the Internet of Things (IoT) and Big Data analytics for manufacturing has driven innovation across various industries worldwide. Production data are utilized to construct a model for quality evaluation and analysis applicable to components processed by machine tools, ensuring process quality for critical components and final product quality for the machine tools. Machine tool parts often encompass several quality characteristics concurrently, categorized into three types: smaller-the-better, larger-the-better, and nominal-the-better. In this paper, an evaluation index for the nominal-the-better quality characteristic was segmented into two single-sided Six Sigma quality indexes. Furthermore, the process quality of the entire component product was assessed by n single-sided Six Sigma quality indexes. According to numerous studies, machine tool manufacturers conventionally base their decisions on small sample sizes (n), considering timeliness and costs. However, this often leads to inconsistent evaluation results due to significant sampling errors. Therefore, this paper established fuzzy testing rules using the confidence intervals of the q single-sided Six Sigma quality indices, serving as the fuzzy quality evaluation model for components of machine tools. Full article

(This article belongs to the Special Issue Fuzzy Sets, Simulation and Their Applications)

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19 pages, 511 KiB

Open AccessArticle

Modeling and Analysis of Monkeypox Outbreak Using a New Time Series Ensemble Technique

by Wilfredo Meza Cuba, Juan Carlos Huaman Alfaro, Hasnain Iftikhar and Javier Linkolk López-Gonzales

Axioms 2024, 13(8), 554; https://doi.org/10.3390/axioms13080554 - 14 Aug 2024

Cited by 1 | Viewed by 1070

Abstract

The coronavirus pandemic has raised concerns about the emergence of other viral infections, such as monkeypox, which has become a significant hazard to public health. Thus, this work proposes a novel time series ensemble technique for analyzing and forecasting the spread of monkeypox [...] Read more.

The coronavirus pandemic has raised concerns about the emergence of other viral infections, such as monkeypox, which has become a significant hazard to public health. Thus, this work proposes a novel time series ensemble technique for analyzing and forecasting the spread of monkeypox in the four highly infected countries with the monkeypox virus. This approach involved processing the first cumulative confirmed case time series to address variance stabilization, normalization, stationarity, and a nonlinear secular trend component. After that, five single time series models and three proposed ensemble models are used to estimate the filtered confirmed case time series. The accuracy of the models is evaluated using typical accuracy mean errors, graphical evaluation, and an equal forecasting accuracy statistical test. Based on the results, it is found that the proposed time series ensemble forecasting approach is an efficient and accurate way to forecast the cumulative confirmed cases for the top four countries in the world and the entire world. Using the best ensemble model, a forecast is made for the next 28 days (four weeks), which will help understand the spread of the disease and the associated risks. This information can prevent further spread and enable timely and effective treatment. Furthermore, the developed novel time series ensemble approach can be used to forecast other diseases in the future. Full article

(This article belongs to the Special Issue Modeling and Analysis of Complex Network)

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22 pages, 344 KiB

Open AccessArticle

New Improvements of the Jensen–Mercer Inequality for Strongly Convex Functions with Applications

by Muhammad Adil Khan, Slavica Ivelić Bradanović and Haitham Abbas Mahmoud

Axioms 2024, 13(8), 553; https://doi.org/10.3390/axioms13080553 - 14 Aug 2024

Viewed by 520

Abstract

In this paper, we use the generalized version of convex functions, known as strongly convex functions, to derive improvements to the Jensen–Mercer inequality. We achieve these improvements through the newly discovered characterizations of strongly convex functions, along with some previously known results about [...] Read more.

In this paper, we use the generalized version of convex functions, known as strongly convex functions, to derive improvements to the Jensen–Mercer inequality. We achieve these improvements through the newly discovered characterizations of strongly convex functions, along with some previously known results about strongly convex functions. We are also focused on important applications of the derived results in information theory, deducing estimates for

χ

-divergence, Kullback–Leibler divergence, Hellinger distance, Bhattacharya distance, Jeffreys distance, and Jensen–Shannon divergence. Additionally, we prove some applications to Mercer-type power means at the end. Full article

(This article belongs to the Special Issue Analysis of Mathematical Inequalities)

14 pages, 303 KiB

Open AccessArticle

MacWilliams Identities and Generator Matrices for Linear Codes over ℤ_p⁴[u]/(u² − p³β, pu)

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

Axioms 2024, 13(8), 552; https://doi.org/10.3390/axioms13080552 - 14 Aug 2024

Viewed by 514

Abstract

Suppose that

R = Z_{p^{4}} [u]

with

u^{2} = p^{3} β

and

p u = 0,

where p is a prime and

β

is a unit in

R .

Then, R is a local non-chain ring [...] Read more.

Suppose that

R = Z_{p^{4}} [u]

with

u^{2} = p^{3} β

and

p u = 0,

where p is a prime and

β

is a unit in

R .

Then, R is a local non-chain ring of order

p^{5}

with a unique maximal ideal

J = (p, u)

and a residue field of order

p .

A linear code C of length N over R is an R-submodule of

R^{N} .

The purpose of this article is to examine MacWilliams identities and generator matrices for linear codes of length N over

R .

We first prove that when

p \neq 2,

there are precisely two distinct rings with these properties up to isomorphism. However, for

p = 2

, only a single such ring is found. Furthermore, we fully describe the lattice of ideals of R and their orders. We then calculate the generator matrices and MacWilliams relations for the linear codes C over

R,

illustrated with numerical examples. It is important to address that there are challenges associated with working with linear codes over non-chain rings, as such rings are not principal ideal rings. Full article

(This article belongs to the Special Issue Computational Algebra, Coding Theory and Cryptography: Theory and Applications)

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13 pages, 2022 KiB

Open AccessArticle

Advanced Methods for Conformable Time-Fractional Differential Equations: Logarithmic Non-Polynomial Splines

by Majeed A. Yousif, Ravi P. Agarwal, Pshtiwan Othman Mohammed, Alina Alb Lupas, Rashid Jan and Nejmeddine Chorfi

Axioms 2024, 13(8), 551; https://doi.org/10.3390/axioms13080551 - 13 Aug 2024

Viewed by 577

Abstract

In this study, we present a numerical method named the logarithmic non-polynomial spline method. This method combines conformable derivative, finite difference, and non-polynomial spline techniques to solve the nonlinear inhomogeneous time-fractional Burgers–Huxley equation. The developed numerical scheme is characterized by a sixth-order convergence [...] Read more.

In this study, we present a numerical method named the logarithmic non-polynomial spline method. This method combines conformable derivative, finite difference, and non-polynomial spline techniques to solve the nonlinear inhomogeneous time-fractional Burgers–Huxley equation. The developed numerical scheme is characterized by a sixth-order convergence and conditional stability. The accuracy of the method is demonstrated with 3D mesh plots, while the effects of time and fractional order are shown in 2D plots. Comparative evaluations with the cubic B-spline collocation method are provided. To illustrate the suitability and effectiveness of the proposed method, two examples are tested, with the results are evaluated using

L_{2}

and

L_{\infty}

norms. Full article

(This article belongs to the Special Issue Recent Developments in Stability and Control of Dynamical Systems)

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19 pages, 293 KiB

Open AccessArticle

Generalization of Fixed-Point Results in Complex-Valued Bipolar Metric Space with Applications

by Amnah Essa Shammaky and Jamshaid Ahmad

Axioms 2024, 13(8), 550; https://doi.org/10.3390/axioms13080550 - 13 Aug 2024

Viewed by 471

Abstract

We undertake this study with the objective of introducing certain control functions in the contractive condition to prove fixed-point theorems in the framework of complex-valued bipolar metric spaces. The incorporation of control functions broadens the applicability of the contractive condition. This approach yields [...] Read more.

We undertake this study with the objective of introducing certain control functions in the contractive condition to prove fixed-point theorems in the framework of complex-valued bipolar metric spaces. The incorporation of control functions broadens the applicability of the contractive condition. This approach yields key results consistent with previous studies. In support of our results, we offer two insightful examples that demonstrate the concepts discussed. Additionally, we present the notion of interpolative contraction in this new and generalized metric space and prove fixed-point theorems for non-self mappings. To demonstrate the application of our approach, we reproduce key findings from several established studies in the field. Full article

(This article belongs to the Special Issue Fixed Point Theory and Its Related Topics IV)

14 pages, 293 KiB

Open AccessArticle

Extensions of Orders to a Power Set vs. Scores of Hesitant Fuzzy Elements: Points in Common of Two Parallel Theories

by Esteban Induráin, Ana Munárriz and M. Sergio Sara

Axioms 2024, 13(8), 549; https://doi.org/10.3390/axioms13080549 - 13 Aug 2024

Viewed by 491

Abstract

We deal with two apparently disparate theories. One of them studies extensions of orderings from a set to its power set. The other one defines suitable scores on hesitant fuzzy elements. We show that both theories have the same mathematical substrate. Thus, important [...] Read more.

We deal with two apparently disparate theories. One of them studies extensions of orderings from a set to its power set. The other one defines suitable scores on hesitant fuzzy elements. We show that both theories have the same mathematical substrate. Thus, important possibility/impossibility results concerning criteria for extensions can be transferred to new results on scores. And conversely, conditions imposed a priori on scores can give rise to new extension criteria. This enhances and enriches both theories. We show examples of translations of classical results on extensions in the context of scores. Also, we state new results concerning the impossibility of finding a utility function representing some kind of extension order if some restrictions are imposed on the utility function considered as a score. Full article

(This article belongs to the Special Issue Advances in Fuzzy MCDM, Hybrid Methods, Fuzzy Number Ranking and Their Applications)

23 pages, 364 KiB

Open AccessArticle

Existence of the Nontrivial Solution for a p-Kirchhoff Problem with Critical Growth and Logarithmic Nonlinearity

by Lixiang Cai and Qing Miao

Axioms 2024, 13(8), 548; https://doi.org/10.3390/axioms13080548 - 13 Aug 2024

Viewed by 513

Abstract

In this paper, we mainly study the p-Kirchhoff type equations with logarithmic nonlinear terms and critical growth: [...] Read more.

In this paper, we mainly study the p-Kirchhoff type equations with logarithmic nonlinear terms and critical growth:

\{\begin{matrix} - M (\int_{Ω} {|\nabla u|}^{p} d x) Δ_{p} u = {|u|}^{p^{*} - 2} u + λ {|u|}^{p - 2} u - {|u|}^{p - 2} u \ln {|u|}^{2} x \in Ω, \\ u = 0 x \in \partial Ω, \end{matrix}

where

Ω \subset ℝ^{N}

is a bounded domain with a smooth boundary,

2 < p < p^{*} < N

, and both

p

and

N

are positive integers. By using the Nehari manifold and the Mountain Pass Theorem without the Palais-Smale compactness condition, it was proved that the equation had at least one nontrivial solution under appropriate conditions. It addresses the challenges posed by the critical term, the Kirchhoff nonlocal term and the logarithmic nonlinear term. Additionally, it extends partial results of the Brézis–Nirenberg problem with logarithmic perturbation from p = 2 to more general p-Kirchhoff type problems. Full article

18 pages, 384 KiB

Open AccessArticle

Critical Permeability from Resummation

by Simon Gluzman

Axioms 2024, 13(8), 547; https://doi.org/10.3390/axioms13080547 - 11 Aug 2024

Viewed by 742

Abstract

Special calculation methods are presented for critical indices and amplitudes for the permeability of thin wavy channels dependent on the waviness. The effective permeability and wetted perimeter of the two-dimensional random percolating media are considered as well. A special mathematical framework is developed [...] Read more.

Special calculation methods are presented for critical indices and amplitudes for the permeability of thin wavy channels dependent on the waviness. The effective permeability and wetted perimeter of the two-dimensional random percolating media are considered as well. A special mathematical framework is developed to characterize the dependencies on porosities, critical points, and indices. Various approximation techniques are applied without involving popular lubrication approximation in any sense. In particular, the Borel summation technique is applied to the effective polynomial approximations with or without optimization. Minimal difference and minimal derivative optimal conditions are adapted to calculations of critical indices and amplitudes for the effective permeability of thin wavy channels. Critical indices, amplitudes, and thresholds are obtained for the effective permeability and wetted perimeter of the two-dimensional percolating random media. Closed-form expressions for all porosities, critical points, and indices are calculated from the polynomial approximations for the first time. Full article

(This article belongs to the Special Issue Computational and Mathematical Methods in Science and Engineering II)

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10 pages, 249 KiB

Open AccessArticle

Pathway to Fractional Integrals, Fractional Differential Equations, and Role of the H-Function

by Arak M. Mathai and Hans J. Haubold

Axioms 2024, 13(8), 546; https://doi.org/10.3390/axioms13080546 - 11 Aug 2024

Viewed by 530

Abstract

In this paper, the pathway model for the real scalar variable case is re-explored and its connections to fractional integrals, solutions of fractional differential equations, Tsallis statistics and superstatistics in statistical mechanics, the reaction-rate probability integral, Krätzel transform, pathway transform, etc., are explored. [...] Read more.

In this paper, the pathway model for the real scalar variable case is re-explored and its connections to fractional integrals, solutions of fractional differential equations, Tsallis statistics and superstatistics in statistical mechanics, the reaction-rate probability integral, Krätzel transform, pathway transform, etc., are explored. It is shown that the common thread in these connections is their H-function representations. The pathway parameter is shown to be connected to the fractional order in fractional integrals and fractional differential equations. Full article

(This article belongs to the Section Mathematical Physics)

55 pages, 1974 KiB

Open AccessReview

Exploring the Landscape of Fractional-Order Models in Epidemiology: A Comparative Simulation Study

by Ritu Agarwal, Pooja Airan and Ravi P. Agarwal

Axioms 2024, 13(8), 545; https://doi.org/10.3390/axioms13080545 - 11 Aug 2024

Viewed by 601

Abstract

Mathematical models play a crucial role in evaluating real-life processes qualitatively and quantitatively. They have been extensively employed to study the spread of diseases such as hepatitis B, COVID-19, influenza, and other epidemics. Many researchers have discussed various types of epidemiological models, including [...] Read more.

Mathematical models play a crucial role in evaluating real-life processes qualitatively and quantitatively. They have been extensively employed to study the spread of diseases such as hepatitis B, COVID-19, influenza, and other epidemics. Many researchers have discussed various types of epidemiological models, including deterministic, stochastic, and fractional order models, for this purpose. This article presents a comprehensive review and comparative study of the transmission dynamics of fractional order in epidemiological modeling. A significant portion of the paper is dedicated to the graphical simulation of these models, providing a visual representation of their behavior and characteristics. The article further embarks on a comparative analysis of fractional-order models with their integer-order counterparts. This comparison sheds light on the nuances and subtleties that differentiate these models, thereby offering valuable insights into their respective strengths and limitations. The paper also explores time delay models, non-linear incidence rate models, and stochastic models, explaining their use and significance in epidemiology. It includes studies and models that focus on the transmission dynamics of diseases using fractional order models, as well as comparisons with integer-order models. The findings from this study contribute to the broader understanding of epidemiological modeling, paving the way for more accurate and effective strategies in disease control and prevention. Full article

(This article belongs to the Special Issue Theory of Functions and Applications II)

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9 pages, 233 KiB

Open AccessArticle

One Turán Type Problem on Uniform Hypergraphs

by Linlin Wang and Sujuan Liu

Axioms 2024, 13(8), 544; https://doi.org/10.3390/axioms13080544 - 11 Aug 2024

Viewed by 379

Abstract

Let

n, m, p, r \in N

with

p \geq n \geq r

. For a hypergraph, if each edge has r vertices, then the hypergraph is called an r-graph. Define [...] Read more.

Let

n, m, p, r \in N

with

p \geq n \geq r

. For a hypergraph, if each edge has r vertices, then the hypergraph is called an r-graph. Define

e_{r} (n, m; p)

to be the maximum number of edges of an r-graph with p vertices in which every subgraph of n vertices has at most m edges. Researching this function constitutes a Turán type problem. In this paper, on the one hand, for fixed p, we present some results about the exact values of

e_{r} (n, m; p)

for small m compared to n; on the other hand, for sufficient large p, we use the combinatorial technique of double counting to give an upper bound of

e (n, m; p)

and obtain a lower bound of

e_{r} (n, m; p)

by applying the lower bound of the independent set of a hypergraph. Full article

19 pages, 386 KiB

Open AccessArticle

Optimal Investment Strategy for DC Pension Plan with Stochastic Salary and Value at Risk Constraint in Stochastic Volatility Model

by Zilan Liu, Huanying Zhang, Yijun Wang and Ya Huang

Axioms 2024, 13(8), 543; https://doi.org/10.3390/axioms13080543 - 10 Aug 2024

Viewed by 496

Abstract

This paper studies the optimal asset allocation problem of a defined contribution (DC) pension plan with a stochastic salary and value under a constraint within a stochastic volatility model. It is assumed that the financial market contains a risk-free asset and a risky [...] Read more.

This paper studies the optimal asset allocation problem of a defined contribution (DC) pension plan with a stochastic salary and value under a constraint within a stochastic volatility model. It is assumed that the financial market contains a risk-free asset and a risky asset whose price process satisfies the Stein–Stein stochastic volatility model. To comply with regulatory standards and offer a risk management tool, we integrate the dynamic versions of Value-at-Risk (VaR), Conditional Value-at-Risk (CVaR), and worst-case CVaR (wcCVaR) constraints into the DC pension fund management model. The salary is assumed to be stochastic and characterized by geometric Brownian motion. In the dynamic setting, a CVaR/wcCVaR constraint is equivalent to a VaR constraint under a higher confidence level. By using the Lagrange multiplier method and the dynamic programming method to maximize the constant absolute risk aversion (CARA) utility of terminal wealth, we obtain closed-form expressions of optimal investment strategies with and without a VaR constraint. Several numerical examples are provided to illustrate the impact of a dynamic VaR/CVaR/wcCVaR constraint and other parameters on the optimal strategy. Full article

(This article belongs to the Special Issue Stochastic Optimization and Metaheuristic Optimization: Theory and Applications)

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14 pages, 316 KiB

Open AccessArticle

Estimation of the Attraction Domain for the Quantum Systems Based on the Schrödinger Equation

by Hongli Yang, Guohui Yu and Ivan Ganchev Ivanov

Axioms 2024, 13(8), 542; https://doi.org/10.3390/axioms13080542 - 9 Aug 2024

Viewed by 476

Abstract

This paper investigates a quantum system described by the Schrödinger equation, utilizing the concept of the quantum Lyapunov function. The Lyapunov function is chosen based on the mean value of a virtual mechanical quantity, where different values of P, the mean value [...] Read more.

This paper investigates a quantum system described by the Schrödinger equation, utilizing the concept of the quantum Lyapunov function. The Lyapunov function is chosen based on the mean value of a virtual mechanical quantity, where different values of P, the mean value of the virtual mechanical quantity in the Lyapunov function, have an impact on the attractive domain of the quantum system. The selected primary optimization algorithms approximating matrix P are the particle swarm optimization (PSO) algorithm and the simulated annealing (SA) algorithm. This study examines the characteristics of the system’s attraction domain under these two distinct algorithms and establishes stability conditions for the nonlinear quantum system. We introduce a method to estimate the size of the attractive domain using the Lyapunov function approach, converting the attractive domain issue into an optimization challenge. Numerical simulations are conducted in various two-dimensional test systems and spin 1/2 particle systems. Full article

(This article belongs to the Special Issue The Advancement in Mathematical and Quantum Physics)

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10 pages, 270 KiB

Open AccessArticle

Exact Reliability and Signature Formulas for Linear m-Consecutive-k-out-of-n: F Systems

by Gökhan Gökdere and Ayse Bugatekin

Axioms 2024, 13(8), 541; https://doi.org/10.3390/axioms13080541 - 9 Aug 2024

Viewed by 432

Abstract

An m-consecutive-k-out-of-n: F (

m / C / k / n : F

) system consists of n linearly ordered components such that the system fails if and only if there are at least m nonoverlapping runs of k consecutive failed components. [...] Read more.

An m-consecutive-k-out-of-n: F (

m / C / k / n : F

) system consists of n linearly ordered components such that the system fails if and only if there are at least m nonoverlapping runs of k consecutive failed components. Our motivation in this work is to obtain efficient formulas for the signature and reliability of the

m / C / k / n : F

system with independent and identical (i.i.d) components that are easy to implement and have a low computational time. We demonstrate that the reliability formula derived for this system requires less computational time than the

m / C / k / n : F

system formula currently in use. For the minimal and maximal signatures of the

m / C / k / n : F

system, we provide precise equations. In addition, the average number of faulty components at the time of an

m / C / k / n : F

system failure and mean time to failure (MTTF) of an

m / C / k / n : F

system are analyzed through the system signature. Full article

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Axioms, Volume 13, Issue 8 (August 2024) – 77 articles

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