Axioms

17 pages, 302 KiB

Open AccessArticle

Existence and Uniqueness Results of Fractional Differential Inclusions and Equations in Sobolev Fractional Spaces

by Safia Meftah, Elhabib Hadjadj, Mohamad Biomy and Fares Yazid

Axioms 2023, 12(11), 1063; https://doi.org/10.3390/axioms12111063 - 20 Nov 2023

Viewed by 830

In this work, by using the iterative method, we discuss the existence and uniqueness of solutions for multiterm fractional boundary value problems. Next, we examine some existence and uniqueness returns for semilinear fractional differential inclusions and equations for multiterm problems by using some [...] Read more.

In this work, by using the iterative method, we discuss the existence and uniqueness of solutions for multiterm fractional boundary value problems. Next, we examine some existence and uniqueness returns for semilinear fractional differential inclusions and equations for multiterm problems by using some notions and properties on set-valued maps and give some examples to explain our main results. We explore and use in this paper the fundamental properties of set-valued maps, which are needed for the study of differential inclusions. It began only in the mid-1900s, when mathematicians realized that their uses go far beyond a mere generalization of single-valued maps. Full article

(This article belongs to the Special Issue Differential and Dynamic Equations on Time Scales and Their Applications)

28 pages, 549 KiB

Open AccessArticle

The Weighted, Relaxed Gradient-Based Iterative Algorithm for the Generalized Coupled Conjugate and Transpose Sylvester Matrix Equations

by Xiaowen Wu, Zhengge Huang, Jingjing Cui and Yanping Long

Axioms 2023, 12(11), 1062; https://doi.org/10.3390/axioms12111062 - 20 Nov 2023

Viewed by 737

Abstract

By applying the weighted relaxation technique to the gradient-based iterative (GI) algorithm and taking proper weighted combinations of the solutions, this paper proposes the weighted, relaxed gradient-based iterative (WRGI) algorithm to solve the generalized coupled conjugate and transpose Sylvester matrix equations. With the [...] Read more.

By applying the weighted relaxation technique to the gradient-based iterative (GI) algorithm and taking proper weighted combinations of the solutions, this paper proposes the weighted, relaxed gradient-based iterative (WRGI) algorithm to solve the generalized coupled conjugate and transpose Sylvester matrix equations. With the real representation of a complex matrix as a tool, the necessary and sufficient conditions for the convergence of the WRGI algorithm are determined. Also, some sufficient convergence conditions of the WRGI algorithm are presented. Moreover, the optimal step size and the corresponding optimal convergence factor of the WRGI algorithm are given. Lastly, some numerical examples are provided to demonstrate the effectiveness, feasibility and superiority of the proposed algorithm. Full article

(This article belongs to the Section Mathematical Analysis)

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14 pages, 398 KiB

Open AccessArticle

Algebrodynamics: Shear-Free Null Congruences and New Types of Electromagnetic Fields

by Vladimir V. Kassandrov, Joseph A. Rizcallah and Ivan A. Matveev

Axioms 2023, 12(11), 1061; https://doi.org/10.3390/axioms12111061 - 20 Nov 2023

Viewed by 784

Abstract

We briefly present our version of noncommutative analysis over matrix algebras, the algebra of biquaternions (

B

) in particular. We demonstrate that any

B

-differentiable function gives rise to a null shear-free congruence (NSFC) on the

B

-vector space

C M

and [...] Read more.

We briefly present our version of noncommutative analysis over matrix algebras, the algebra of biquaternions (

B

) in particular. We demonstrate that any

B

-differentiable function gives rise to a null shear-free congruence (NSFC) on the

B

-vector space

C M

and on its Minkowski subspace

M

. Making use of the Kerr–Penrose correspondence between NSFC and twistor functions, we obtain the general solution to the equations of

B

-differentiability and demonstrate that the source of an NSFC is, generically, a world sheet of a string in

C M

. Any singular point, caustic of an NSFC, is located on the complex null cone of a point on the generating string. Further we describe symmetries and associated gauge and spinor fields, with two electromagnetic types among them. A number of familiar and novel examples of NSFC and their singular loci are described. Finally, we describe a conservative algebraic dynamics of a set of identical particles on the “Unique Worldline” and discuss the connections of the theory with the Feynman–Wheeler concept of “One-Electron Universe”. Full article

(This article belongs to the Special Issue Computational Mathematics and Mathematical Physics)

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31 pages, 434 KiB

Open AccessArticle

Optimized Self-Similar Borel Summation

by Simon Gluzman and Vyacheslav I. Yukalov

Axioms 2023, 12(11), 1060; https://doi.org/10.3390/axioms12111060 - 20 Nov 2023

Viewed by 1163

Abstract

The method of Fractional Borel Summation is suggested in conjunction with self-similar factor approximants. The method used for extrapolating asymptotic expansions at small variables to large variables, including the variables tending to infinity, is described. The method is based on the combination of [...] Read more.

The method of Fractional Borel Summation is suggested in conjunction with self-similar factor approximants. The method used for extrapolating asymptotic expansions at small variables to large variables, including the variables tending to infinity, is described. The method is based on the combination of optimized perturbation theory, self-similar approximation theory, and Borel-type transformations. General Borel Fractional transformation of the original series is employed. The transformed series is resummed in order to adhere to the asymptotic power laws. The starting point is the formulation of dynamics in the approximations space by employing the notion of self-similarity. The flow in the approximation space is controlled, and “deep” control is incorporated into the definitions of the self-similar approximants. The class of self-similar approximations, satisfying, by design, the power law behavior, such as the use of self-similar factor approximants, is chosen for the reasons of transparency, explicitness, and convenience. A detailed comparison of different methods is performed on a rather large set of examples, employing self-similar factor approximants, self-similar iterated root approximants, as well as the approximation technique of self-similarly modified Padé–Borel approximations. Full article

(This article belongs to the Special Issue Mathematical Analysis and Applications IV)

15 pages, 330 KiB

Open AccessFeature PaperArticle

A Strong Maximum Principle for Nonlinear Nonlocal Diffusion Equations

by Tucker Hartland and Ravi Shankar

Axioms 2023, 12(11), 1059; https://doi.org/10.3390/axioms12111059 - 18 Nov 2023

Viewed by 838

Abstract

We consider a class of nonlinear integro-differential equations that model degenerate nonlocal diffusion. We investigate whether the strong maximum principle is valid for this nonlocal equation. For degenerate parabolic PDEs, the strong maximum principle is not valid. In contrast, for nonlocal diffusion, we [...] Read more.

We consider a class of nonlinear integro-differential equations that model degenerate nonlocal diffusion. We investigate whether the strong maximum principle is valid for this nonlocal equation. For degenerate parabolic PDEs, the strong maximum principle is not valid. In contrast, for nonlocal diffusion, we can formulate a strong maximum principle for nonlinearities satisfying a geometric condition related to the flux operator of the equation. In our formulation of the strong maximum principle, we find a physical re-interpretation and generalization of the standard PDE conclusion of the principle: we replace constant solutions with solutions of zero flux. We also consider nonlinearities outside the scope of our principle. For highly degenerate conductivities, we demonstrate the invalidity of the strong maximum principle. We also consider intermediate, inconclusive examples, and provide numerical evidence that the strong maximum principle is valid. This suggests that our geometric condition is sharp. Full article

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

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13 pages, 326 KiB

Open AccessArticle

Quantum Chromodynamics and the Hyperbolic Unitary Group SUh(3)

by Nikolay Popov and Ivan Matveev

Axioms 2023, 12(11), 1058; https://doi.org/10.3390/axioms12111058 - 18 Nov 2023

Viewed by 737

Abstract

The paper shows that it is possible to construct quantum chromodynamics as a rigorous theory on the basis of employment of hyperbolic unitary group

S U h (3)

, which is a symmetry group for the three-dimensional complex space of the [...] Read more.

The paper shows that it is possible to construct quantum chromodynamics as a rigorous theory on the basis of employment of hyperbolic unitary group

S U h (3)

, which is a symmetry group for the three-dimensional complex space of the hyperbolic type. Such an approach allows researchers to discover a profound connection between conserved color charges of the quarks and the symmetries of the hyperbolic three-dimensional complex space. Further, it allows a correct introduction of the Hermitian operators to describe the eight gluons, which are carriers of strong interactions. Full article

(This article belongs to the Special Issue Computational Mathematics and Mathematical Physics)

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19 pages, 1334 KiB

Open AccessArticle

New Two-Level Time-Mesh Difference Scheme for the Symmetric Regularized Long Wave Equation

by Jingying Gao, Qingmei Bai, Siriguleng He and Eerdun Buhe

Axioms 2023, 12(11), 1057; https://doi.org/10.3390/axioms12111057 - 17 Nov 2023

Viewed by 789

Abstract

The paper introduces a new two-level time-mesh difference scheme for solving the symmetric regularized long wave equation. The scheme consists of three steps. A coarse time-mesh and a fine time-mesh are defined, and the equation is solved using an existing nonlinear scheme on [...] Read more.

The paper introduces a new two-level time-mesh difference scheme for solving the symmetric regularized long wave equation. The scheme consists of three steps. A coarse time-mesh and a fine time-mesh are defined, and the equation is solved using an existing nonlinear scheme on the coarse time-mesh. Lagrange’s linear interpolation formula is employed to obtain all preliminary solutions on the fine time-mesh. Based on the preliminary solutions, Taylor’s formula is utilized to construct a linear system for the equation on the fine time-mesh. The convergence and stability of the scheme is analyzed, providing the convergence rates of

O (τ_{F}^{2} + τ_{C}^{4} + h^{4})

in the discrete

L_{\infty}

-norm for

u (x, t)

and in the discrete

L_{2}

-norm for

ρ (x, t)

. Numerical simulation results show that the proposed scheme achieves equivalent error levels and convergence rates to the nonlinear scheme, while also reducing CPU time by over half, which indicates that the new method is more efficient. Furthermore, compared to the earlier time two-mesh method developed by the authors, the proposed scheme significantly reduces the error between the numerical and exact solutions, which means that the proposed scheme is more accurate. Additionally, the effectiveness of the new scheme is discussed in terms of the corresponding conservation laws and long-time simulations. Full article

(This article belongs to the Special Issue Differential Equations and Asymptotic Analysis: Recent Advances and Applications)

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20 pages, 1190 KiB

Open AccessArticle

High-Effectiveness and -Accuracy Difference Scheme Based on Nonuniform Grids for Solving Convection–Diffusion Equations with Boundary Layers

by Fang Tian, Mingjing Wang and Yongbin Ge

Axioms 2023, 12(11), 1056; https://doi.org/10.3390/axioms12111056 - 16 Nov 2023

Viewed by 782

Abstract

In this paper, some rational high-accuracy compact finite difference schemes on nonuniform grids (NRHOC) are introduced for solving convection–diffusion equations. The derived NRHOC schemes not only can suppress the oscillatory property of numerical solutions but can also obtain a high-accuracy approximate solution, and [...] Read more.

In this paper, some rational high-accuracy compact finite difference schemes on nonuniform grids (NRHOC) are introduced for solving convection–diffusion equations. The derived NRHOC schemes not only can suppress the oscillatory property of numerical solutions but can also obtain a high-accuracy approximate solution, and they can effectively solve the convection–diffusion problem with boundary layers by flexibly adjusting the discrete grid, which can be obtained with the singularity in the computational region. Three numerical experiments with boundary layers are conducted to verify the accuracy of the proposed NRHOC schemes. We compare the computed results with the analytical solutions, the results of the rational high-accuracy compact finite difference schemes on uniform grids (RHOC), and the other schemes in the literature. For all test problems, good computed results are obtained with the presented NRHOC schemes. It is shown that the presented NRHOC schemes have a better resolution for the solution of convection-dominated problems. Full article

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18 pages, 1324 KiB

Open AccessArticle

Assessment of the Water Distribution Networks in the Kingdom of Saudi Arabia: A Mathematical Model

by Aiman Albarakati, Asifa Tassaddiq and Rekha Srivastava

Axioms 2023, 12(11), 1055; https://doi.org/10.3390/axioms12111055 - 16 Nov 2023

Viewed by 958

Abstract

Graph theory is a branch of mathematics that is crucial to modelling applicable systems and networks using matrix representations. In this article, a novel graph-theoretic model was used to assess an urban water distribution system (WDS) in Saudi Arabia. This graph model is [...] Read more.

Graph theory is a branch of mathematics that is crucial to modelling applicable systems and networks using matrix representations. In this article, a novel graph-theoretic model was used to assess an urban water distribution system (WDS) in Saudi Arabia. This graph model is based on representing its elements through nodes and links using a weighted adjacency matrix. The nodes represent the points where there can be a water input or output (sources, treatment plants, tanks, reservoirs, consumers, connections), and links represent the edges of the graph that carry water from one node to another (pipes, pumps, valves). Four WDS benchmarks, pumps, tanks, reservoirs, and external sources were used to validate the framework at first. This validation showed that the worst-case scenarios for vulnerability were provided by the fault sequence iterating the calculation of the centrality measurements. The vulnerability framework’s application to the Saudi Arabian WDS enabled the identification of the system’s most vulnerable junctions and zones. As anticipated, the regions with the fewest reservoirs were most at risk from unmet demand, indicating that this system is vulnerable to the removal of junctions and pipes that are intricately associated with their neighbours. Different centrality metrics were computed, from which the betweenness centrality offered the worst vulnerability prediction measures. The aspects and zones of the WDS that can more significantly impact the water supply in the event of a failure were identified by the vulnerability framework utilising attack tactics. Full article

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

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23 pages, 832 KiB

Open AccessArticle

Stress–Strength Reliability Analysis for Different Distributions Using Progressive Type-II Censoring with Binomial Removal

by Ibrahim Elbatal, Amal S. Hassan, L. S. Diab, Anis Ben Ghorbal, Mohammed Elgarhy and Ahmed R. El-Saeed

Axioms 2023, 12(11), 1054; https://doi.org/10.3390/axioms12111054 - 15 Nov 2023

Viewed by 1008

Abstract

In the statistical literature, one of the most important subjects that is commonly used is stress–strength reliability, which is defined as

δ = P [W < V]

, where V and W are the strength and stress random variables, respectively, and

δ

is [...] Read more.

In the statistical literature, one of the most important subjects that is commonly used is stress–strength reliability, which is defined as

δ = P [W < V]

, where V and W are the strength and stress random variables, respectively, and

δ

is reliability parameter. Type-II progressive censoring with binomial removal is used in this study to examine the inference of

δ = P [W < V]

for a component with strength V and being subjected to stress W. We suppose that V and W are independent random variables taken from the Burr XII distribution and the Burr III distribution, respectively, with a common shape parameter. The maximum likelihood estimator of

δ

is derived. The Bayes estimator of

δ

under the assumption of independent gamma priors is derived. To determine the Bayes estimates for squared error and linear exponential loss functions in the lack of explicit forms, the Metropolis–Hastings method was provided. Utilizing comprehensive simulations and two metrics (average of estimates and root mean squared errors), we compare these estimators. Further, an analysis is performed on two actual data sets based on breakdown times for insulating fluid between electrodes recorded under varying voltages. Full article

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

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18 pages, 713 KiB

Open AccessArticle

The Multivariable Zhang–Zhang Polynomial of Phenylenes

by Niko Tratnik

Axioms 2023, 12(11), 1053; https://doi.org/10.3390/axioms12111053 - 15 Nov 2023

Cited by 2 | Viewed by 873

Abstract

The Zhang–Zhang polynomial of a benzenoid system is a well-known counting polynomial that was introduced in 1996. It was designed to enumerate Clar covers, which are spanning subgraphs with only hexagons and edges as connected components. In 2018, the generalized Zhang–Zhang polynomial of [...] Read more.

The Zhang–Zhang polynomial of a benzenoid system is a well-known counting polynomial that was introduced in 1996. It was designed to enumerate Clar covers, which are spanning subgraphs with only hexagons and edges as connected components. In 2018, the generalized Zhang–Zhang polynomial of two variables was defined such that it also takes into account 10-cycles of a benzenoid system. The aim of this paper is to introduce and study a new variation of the Zhang–Zhang polynomial for phenylenes, which are important molecular graphs composed of 6-membered and 4-membered rings. In our case, Clar covers can contain 4-cycles, 6-cycles, 8-cycles, and edges. Since this new polynomial has three variables, we call it the multivariable Zhang–Zhang (MZZ) polynomial. In the main part of the paper, some recursive formulas for calculating the MZZ polynomial from subgraphs of a given phenylene are developed and an algorithm for phenylene chains is deduced. Interestingly, computing the MZZ polynomial of a phenylene chain requires some techniques that are different to those used to calculate the (generalized) Zhang–Zhang polynomial of benzenoid chains. Finally, we prove a result that enables us to find the MZZ polynomial of a phenylene with branched hexagons. Full article

(This article belongs to the Special Issue Spectral Graph Theory, Molecular Graph Theory and Their Applications)

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14 pages, 969 KiB

Open AccessArticle

Exotic Particle Dynamics Using Novel Hermitian Spin Matrices

by Timothy Ganesan

Axioms 2023, 12(11), 1052; https://doi.org/10.3390/axioms12111052 - 15 Nov 2023

Viewed by 873

Abstract

In this work, an analogue to the Pauli spin matrices is presented and investigated. The proposed Hermitian spin matrices exhibit four symmetries for spin-1/n particles. The spin projection operators are derived, and the electrodynamics for hypothetical spin-1/2 fermions are explored using the [...] Read more.

In this work, an analogue to the Pauli spin matrices is presented and investigated. The proposed Hermitian spin matrices exhibit four symmetries for spin-1/n particles. The spin projection operators are derived, and the electrodynamics for hypothetical spin-1/2 fermions are explored using the proposed spin matrices. The fermionic quantum Heisenberg model is constructed using the proposed spin matrices, and comparative studies against simulation results using the Pauli spin matrices are conducted. Further analysis of the key findings as well as discussions on extending the proposed spin matrix framework to describe hypothetical bosonic systems (spin-1 particles) are provided. Full article

(This article belongs to the Special Issue Differential Equations and Dynamical Systems—Theory and Applications)

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15 pages, 817 KiB

Open AccessArticle

A Hyperparameter Self-Evolving SHADE-Based Dendritic Neuron Model for Classification

by Haichuan Yang, Yuxin Zhang, Chaofeng Zhang, Wei Xia, Yifei Yang and Zhenwei Zhang

Axioms 2023, 12(11), 1051; https://doi.org/10.3390/axioms12111051 - 15 Nov 2023

Viewed by 972

Abstract

In recent years, artificial neural networks (ANNs), which are based on the foundational model established by McCulloch and Pitts in 1943, have been at the forefront of computational research. Despite their prominence, ANNs have encountered a number of challenges, including hyperparameter tuning and [...] Read more.

In recent years, artificial neural networks (ANNs), which are based on the foundational model established by McCulloch and Pitts in 1943, have been at the forefront of computational research. Despite their prominence, ANNs have encountered a number of challenges, including hyperparameter tuning and the need for vast datasets. It is because many strategies have predominantly focused on enhancing the depth and intricacy of these networks that the essence of the processing capabilities of individual neurons is occasionally overlooked. Consequently, a model emphasizing a biologically accurate dendritic neuron model (DNM) that mirrors the spatio-temporal features of real neurons was introduced. However, while the DNM shows outstanding performance in classification tasks, it struggles with complexities in parameter adjustments. In this study, we introduced the hyperparameters of the DNM into an evolutionary algorithm, thereby transforming the method of setting DNM’s hyperparameters from the previous manual adjustments to adaptive adjustments as the algorithm iterates. The newly proposed framework, represents a neuron that evolves alongside the iterations, thus simplifying the parameter-tuning process. Comparative evaluation on benchmark classification datasets from the UCI Machine Learning Repository indicates that our minor enhancements lead to significant improvements in the performance of DNM, surpassing other leading-edge algorithms in terms of both accuracy and efficiency. In addition, we also analyzed the iterative process using complex networks, and the results indicated that the information interaction during the iteration and evolution of the DNM follows a power-law distribution. With this finding, some insights could be provided for the study of neuron model training. Full article

(This article belongs to the Special Issue Mathematical Modelling of Complex Systems)

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13 pages, 327 KiB

Open AccessArticle

Fixed Point Theorems in Rectangular b-Metric Space Endowed with a Partial Order

by Andrei Horvat-Marc, Mariana Cufoian, Adriana Mitre and Ioana Taşcu

Axioms 2023, 12(11), 1050; https://doi.org/10.3390/axioms12111050 - 15 Nov 2023

Viewed by 855

Abstract

The purpose of this paper is to present some fixed-point results for self-generalized contractions in ordered rectangular b-metric spaces. We also provide some examples that illustrate the non-triviality and richness of this area of research. Full article

(This article belongs to the Special Issue Differential Equations and Related Topics)

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20 pages, 5301 KiB

Open AccessArticle

Novel Method for Ranking Generalized Fuzzy Numbers Based on Normalized Height Coefficient and Benefit and Cost Areas

by Thi Hong Phuong Le and Ta-Chung Chu

Axioms 2023, 12(11), 1049; https://doi.org/10.3390/axioms12111049 - 13 Nov 2023

Viewed by 761

Abstract

This paper proposes a method for ranking generalized fuzzy numbers, which guarantees that both horizontal and vertical values are important parameters affecting the final ranking score. In this method, the normalized height coefficient is introduced to evaluate the influence of the height of [...] Read more.

This paper proposes a method for ranking generalized fuzzy numbers, which guarantees that both horizontal and vertical values are important parameters affecting the final ranking score. In this method, the normalized height coefficient is introduced to evaluate the influence of the height of fuzzy numbers on the final ranking score. The higher the normalized height coefficient of a generalized fuzzy number is, the higher its ranking. The left and right areas are presented to calculate the impact of the vertical value on the final ranking score. The left area is considered the benefit area. The right area is considered the cost area. A generalized fuzzy number is preferred if the benefit area is larger and the cost area is smaller. The proposed method can be employed to rank both normal and non-normal fuzzy numbers without normalization or height minimization. Numerical examples and comparisons with other methods highlight the feasibility and robustness of the proposed method, which can overcome the shortcomings of some existing methods and can support decision-makers in selecting the best alternative. Full article

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

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12 pages, 270 KiB

Open AccessArticle

Multi-Objective Non-Linear Programming Problems in Linear Diophantine Fuzzy Environment

by Salma Iqbal, Naveed Yaqoob and Muhammad Gulistan

Axioms 2023, 12(11), 1048; https://doi.org/10.3390/axioms12111048 - 13 Nov 2023

Viewed by 877

Abstract

Due to various unpredictable factors, a decision maker frequently experiences uncertainty and hesitation when dealing with real-world practical optimization problems. At times, it’s necessary to simultaneously optimize a number of non-linear and competing objectives. Linear Diophantine fuzzy numbers are used to address the [...] Read more.

Due to various unpredictable factors, a decision maker frequently experiences uncertainty and hesitation when dealing with real-world practical optimization problems. At times, it’s necessary to simultaneously optimize a number of non-linear and competing objectives. Linear Diophantine fuzzy numbers are used to address the uncertain parameters that arise in these circumstances. The objective of this manuscript is to present a method for solving a linear Diophantine fuzzy multi-objective nonlinear programming problem (LDFMONLPP). All the coefficients of the nonlinear multi-objective functions and the constraints are linear Diophantine fuzzy numbers (LDFNs). Here we find the solution of the nonlinear programming problem by using Karush-Kuhn-Tucker condition. A numerical example is presented. Full article

(This article belongs to the Special Issue Recent Developments in Fuzzy Control Systems and Their Applications)

15 pages, 288 KiB

Open AccessArticle

On Hybrid Hyper k-Pell, k-Pell–Lucas, and Modified k-Pell Numbers

by Elen Viviani Pereira Spreafico, Paula Catarino and Paulo Vasco

Axioms 2023, 12(11), 1047; https://doi.org/10.3390/axioms12111047 - 11 Nov 2023

Viewed by 899

Abstract

Many different number systems have been the topic of research. One of the recently studied number systems is that of hybrid numbers, which are generalizations of other number systems. In this work, we introduce and study the hybrid hyper k-Pell, hybrid hyper [...] Read more.

Many different number systems have been the topic of research. One of the recently studied number systems is that of hybrid numbers, which are generalizations of other number systems. In this work, we introduce and study the hybrid hyper k-Pell, hybrid hyper k-Pell–Lucas, and hybrid hyper Modified k-Pell numbers. In order to study these new sequences, we established new properties, generating functions, and the Binet formula of the hyper k-Pell, hyper k-Pell–Lucas, and hyper Modified k-Pell sequences. Thus, we present some algebraic properties, recurrence relations, generating functions, the Binet formulas, and some identities for the hybrid hyper k-Pell, hybrid hyper k-Pell–Lucas, and hybrid hyper Modified k-Pell numbers. Full article

(This article belongs to the Special Issue Advances in Generalized Hypergeometric Functions, Integral Transforms and Number Theory)

15 pages, 390 KiB

Open AccessArticle

Complex Generalized Representation of Gamma Function Leading to the Distributional Solution of a Singular Fractional Integral Equation

by Asifa Tassaddiq, Rekha Srivastava, Ruhaila Md Kasmani and Rabab Alharbi

Axioms 2023, 12(11), 1046; https://doi.org/10.3390/axioms12111046 - 10 Nov 2023

Cited by 1 | Viewed by 896

Abstract

Firstly, a basic question to find the Laplace transform using the classical representation of gamma function makes no sense because the singularity at the origin nurtures so rapidly that

Γ (z) e^{- s z}

cannot be integrated over positive real numbers. Secondly, [...] Read more.

Firstly, a basic question to find the Laplace transform using the classical representation of gamma function makes no sense because the singularity at the origin nurtures so rapidly that

Γ (z) e^{- s z}

cannot be integrated over positive real numbers. Secondly, Dirac delta function is a linear functional under which every function

f

is mapped to

f (0)

. This article combines both functions to solve the problems that have remained unsolved for many years. For instance, it has been demonstrated that the power law feature is ubiquitous in theory but challenging to observe in practice. Since the fractional derivatives of the delta function are proportional to the power law, we express the gamma function as a complex series of fractional derivatives of the delta function. Therefore, a unified approach is used to obtain a large class of ordinary, fractional derivatives and integral transforms. All kinds of q-derivatives of these transforms are also computed. The most general form of the fractional kinetic integrodifferential equation available in the literature is solved using this particular representation. We extend the models that were valid only for a class of locally integrable functions to a class of singular (generalized) functions. Furthermore, we solve a singular fractional integral equation whose coefficients have infinite number of singularities, being the poles of gamma function. It is interesting to note that new solutions were obtained using generalized functions with complex coefficients. Full article

(This article belongs to the Special Issue Advances in Hypergeometric Series, Orthogonal Polynomials and Their Natural Extensions)

16 pages, 306 KiB

Open AccessArticle

Homotopes of Quasi-Jordan Algebras

by Reem K. Alhefthi, Akhlaq A. Siddiqui and Fatmah B. Jamjoom

Axioms 2023, 12(11), 1045; https://doi.org/10.3390/axioms12111045 - 10 Nov 2023

Viewed by 775

Abstract

The notion of quasi-Jordan algebras was originally proposed by R. Velasquez and R. Fellipe. Later, M. R. Bremner provided a modification called K-B quasi-Jordan algebras; these include all Jordan algebras and all dialgebras, and hence all associative algebras. Any quasi-Jordan algebra is special [...] Read more.

The notion of quasi-Jordan algebras was originally proposed by R. Velasquez and R. Fellipe. Later, M. R. Bremner provided a modification called K-B quasi-Jordan algebras; these include all Jordan algebras and all dialgebras, and hence all associative algebras. Any quasi-Jordan algebra is special if it is isomorphic to a quasi-Jordan subalgebra of some dialgebras. Keeping in view the pivotal role of homotopes in the theory of Jordan algebras, we begin a study of the homotopes of quasi-Jordan algebras; among other related results, we show that the homotopes of any special quasi-Jordan algebra are special quasi-Jordan algebras and that the homotopes of a K-B quasi-Jordan algebra is a quasi-Jordan algebra. In the sequel, we also give some open problems. Full article

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

18 pages, 3442 KiB

Open AccessArticle

Research on Repeated Quantum Games with Public Goods under Strong Reciprocity

by Simo Sun, Yadong Shu, Jinxiu Pi and Die Zhou

Axioms 2023, 12(11), 1044; https://doi.org/10.3390/axioms12111044 - 10 Nov 2023

Viewed by 804

Abstract

We developed a repeated quantum game of public goods by using quantum entanglement and strong reciprocity mechanisms. Utilizing the framework of quantum game analysis, a comparative investigation incorporating both entangled and non-entangled states reveals that the player will choose a fully cooperative strategy [...] Read more.

We developed a repeated quantum game of public goods by using quantum entanglement and strong reciprocity mechanisms. Utilizing the framework of quantum game analysis, a comparative investigation incorporating both entangled and non-entangled states reveals that the player will choose a fully cooperative strategy when the expected cooperation strategy of the competitor exceeds a certain threshold. When the entanglement of states is not considered, the prisoner’s dilemma still exists, and the cooperating party must bear the cost of defactoring the quantum strategy themselves; when considering the entanglement of states, the benefits of both parties in the game are closely related, forming a community of benefits. By signing a strong reciprocity contract, the degree of cooperation between the game parties can be considered using the strong reciprocity entanglement contract mechanism. The party striving to cooperate does not have to bear the risk of the other party’s defector, and to some extent, it can solve the prisoner’s dilemma problem. Finally, taking the public goods green planting industry project as an example, by jointly entrusting a third party to determine and sign a strong reciprocity entanglement contract, both parties can ensure a complete quantum strategy to maximize cooperation and achieve Pareto optimality, ultimately enabling the long-term and stable development of the public goods industry project. Full article

(This article belongs to the Special Issue Advances in Quantum Theory and Quantum Computing)

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20 pages, 436 KiB

Open AccessArticle

Conversion of Unweighted Graphs to Weighted Graphs Satisfying Properties

R

and

- SR

by Xiaolong Shi, Saira Hameed, Sadia Akhter, Aysha Khan and Maryam Akhoundi

Axioms 2023, 12(11), 1043; https://doi.org/10.3390/axioms12111043 - 09 Nov 2023

Viewed by 935

Abstract

Spectral graph theory is like a special tool for understanding graphs. It helps us find patterns and connections in complex networks, using the magic of eigenvalues. Let

G

be the graph and

A (G)

be its adjacency matrix, then

G

is [...] Read more.

Spectral graph theory is like a special tool for understanding graphs. It helps us find patterns and connections in complex networks, using the magic of eigenvalues. Let

G

be the graph and

A (G)

be its adjacency matrix, then

G

is singular if the determinant of the adjacency matrix

A (G)

is 0, otherwise it is nonsingular. Within the realm of nonsingular graphs, there is the concept of property

R

, where each eigenvalue’s reciprocal is also an eigenvalue of

G

. By introducing multiplicity constraints on both eigenvalues and their reciprocals, it becomes property

SR

. Similarly, the world of nonsingular graphs reveals property

- R

, where the negative reciprocal of each eigenvalue also finds a place within the spectrum of

G

. Moreover, when the multiplicity of each eigenvalue and its negative reciprocal is equal, this results in a graph with a property of

- SR

. Some classes of unweighted nonbipartite graphs are already constructed in the literature with the help of the complete graph

K_{n}

and a copy of the path graph

P_{4}

satisfying property

R

but not

SR

. This article takes this a step further. The main aim is to construct several weighted classes of graphs which satisfy property

R

but not

SR

. For this purpose, the weight functions are determined that enable these nonbipartite graph classes to satisfy the

- SR

and

R

properties, even if the unweighted graph does not satisfy these properties. Some examples are presented to support the investigated results. These examples explain how certain weight functions make these special types of graphs meet the properties

R

or

- SR

, even when the original graphs without weights do not meet these properties. Full article

(This article belongs to the Special Issue Spectral Graph Theory, Molecular Graph Theory and Their Applications)

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30 pages, 1906 KiB

Open AccessArticle

Schröder-Based Inverse Function Approximation

by Roy M. Howard

Axioms 2023, 12(11), 1042; https://doi.org/10.3390/axioms12111042 - 08 Nov 2023

Viewed by 820

Abstract

Schröder approximations of the first kind, modified for the inverse function approximation case, are utilized to establish general analytical approximation forms for an inverse function. Such general forms are used to establish arbitrarily accurate analytical approximations, with a set relative error bound, for [...] Read more.

Schröder approximations of the first kind, modified for the inverse function approximation case, are utilized to establish general analytical approximation forms for an inverse function. Such general forms are used to establish arbitrarily accurate analytical approximations, with a set relative error bound, for an inverse function when an initial approximation, typically with low accuracy, is known. Approximations for arcsine, the inverse of x − sin(x), the inverse Langevin function and the Lambert W function are used to illustrate this approach. Several applications are detailed. For the root approximation of a function, Schröder approximations of the first kind, based on the inverse of a function, have an advantage over the corresponding generalization of the standard Newton–Raphson method, as explicit analytical expressions for all orders of approximation can be obtained. Full article

(This article belongs to the Special Issue Advanced Approximation Techniques and Their Applications)

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13 pages, 304 KiB

Open AccessArticle

Investigation of the F* Algorithm on Strong Pseudocontractive Mappings and Its Application

by Felix D. Ajibade, Francis Monday Nkwuda, Hussaini Joshua, Taiwo P. Fajusigbe and Kayode Oshinubi

Axioms 2023, 12(11), 1041; https://doi.org/10.3390/axioms12111041 - 08 Nov 2023

Viewed by 829

Abstract

In the context of uniformly convex Banach space, this paper focuses on examining the strong convergence of the

F^{*}

iterative algorithm to the fixed point of a strongly pseudocontractive mapping. Furthermore, we demonstrate through numerical methods that the

F^{*}

iterative algorithm [...] Read more.

In the context of uniformly convex Banach space, this paper focuses on examining the strong convergence of the

F^{*}

iterative algorithm to the fixed point of a strongly pseudocontractive mapping. Furthermore, we demonstrate through numerical methods that the

F^{*}

iterative algorithm converges strongly and faster than other current iterative schemes in the literature and extends to the fixed point of a strong pseudocontractive mapping. Finally, under a nonlinear quadratic Volterra integral equation, the application of our findings is shown. Full article

(This article belongs to the Special Issue Advanced Approximation Techniques and Their Applications)

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46 pages, 2854 KiB

Open AccessArticle

The Branch-and-Bound Algorithm in Optimizing Mathematical Programming Models to Achieve Power Grid Observability

by Nikolaos P. Theodorakatos, Rohit Babu and Angelos P. Moschoudis

Axioms 2023, 12(11), 1040; https://doi.org/10.3390/axioms12111040 - 08 Nov 2023

Cited by 2 | Viewed by 2095

Abstract

Phasor Measurement Units (PMUs) are the backbone of smart grids that are able to measure power system observability in real-time. The deployment of synchronized sensors in power networks opens up the advantage of real-time monitoring of the network state. An optimal number of [...] Read more.

Phasor Measurement Units (PMUs) are the backbone of smart grids that are able to measure power system observability in real-time. The deployment of synchronized sensors in power networks opens up the advantage of real-time monitoring of the network state. An optimal number of PMUs must be installed to ensure system observability. For that reason, an objective function is minimized, reflecting the cost of PMU installation around the power grid. As a result, a minimization model is declared where the objective function is defined over an adequate number of constraints on a binary decision variable domain. To achieve maximum network observability, there is a need to find the best number of PMUs and put them in appropriate locations around the power grid. Hence, maximization models are declared in a decision-making way to obtain optimality satisfying a guaranteed stopping and optimality criteria. The best performance metrics are achieved using binary integer, semi-definite, and binary polynomial models to encounter the optimal number of PMUs with suitable PMU positioning sites. All optimization models are implemented with powerful optimization solvers in MATLAB to obtain the global solution point. Full article

(This article belongs to the Special Issue Applied Optimization for Solving Real-World Problems)

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14 pages, 1046 KiB

Open AccessArticle

Solution of Two-Dimensional Solute Transport Model for Heterogeneous Porous Medium Using Fractional Reduced Differential Transform Method

by Manan A. Maisuria, Priti V. Tandel and Trushitkumar Patel

Axioms 2023, 12(11), 1039; https://doi.org/10.3390/axioms12111039 - 08 Nov 2023

Viewed by 773

Abstract

This study contains a two-dimensional mathematical model of solute transport in a river with temporally and spatially dependent flow, explicitly focusing on pulse-type input point sources with a fractional approach. This model is analyzed by assuming an initial concentration function as a declining [...] Read more.

This study contains a two-dimensional mathematical model of solute transport in a river with temporally and spatially dependent flow, explicitly focusing on pulse-type input point sources with a fractional approach. This model is analyzed by assuming an initial concentration function as a declining exponential function in both the longitudinal and transverse directions. The governing equation is a time-fractional two-dimensional advection–dispersion equation with a variable form of dispersion coefficients, velocities, decay constant of the first order, production rate coefficient for the solute at the zero-order level, and retardation factor. The solution of the present problem is obtained by the fractional reduced differential transform method (FRDTM). The analysis of the initial retardation factor has been carried out via plots. Also, the influence of initial longitudinal and transverse dispersion coefficients and velocities has been examined by graphical analysis. The impact of fractional parameters on pollution levels is also analyzed numerically and graphically. The study of convergence for the FRDTM technique has been conducted to assess its efficacy and accuracy. Full article

(This article belongs to the Special Issue Fractional Calculus and the Applied Analysis)

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25 pages, 3202 KiB

Open AccessArticle

Fuzzy Algorithm Applied to Factors Influencing Competitiveness: A Case Study of Brazil and Peru through Affinities Theory

by Luciano Barcellos-Paula, Aline Castro-Rezende and Daniela Fantoni Alvares

Axioms 2023, 12(11), 1038; https://doi.org/10.3390/axioms12111038 - 08 Nov 2023

Viewed by 1403

Abstract

Innovation plays a crucial role in the economy of nations worldwide. In Latin America, countries foster competitiveness through public and private incentives to support innovation. Moreover, entrepreneurship incentives seek to improve countries’ performance, although factors such as low business growth rates and informality [...] Read more.

Innovation plays a crucial role in the economy of nations worldwide. In Latin America, countries foster competitiveness through public and private incentives to support innovation. Moreover, entrepreneurship incentives seek to improve countries’ performance, although factors such as low business growth rates and informality can compromise it. Despite the efforts, there are several difficulties in achieving competitiveness, and few studies in developing countries. Therefore, the article explores the relationship between the factors that influence competitiveness, especially the role of innovation and entrepreneurship in Brazil and Peru. The research uses quantitative-qualitative methodology through modeling and simulation and a case study. The authors use the Affinities Theory to verify the relationship between the indicators that make up the competitiveness landscape and its most significant and attractive factors, adapting the methodology established by the International Institute for Management Development (IMD) World Competitiveness ranking. As a result, this algorithm allows us to know the relationships between five factors of economic attractiveness and four competitiveness indicators. As its main contributions, the study advances the frontier of knowledge about innovation and entrepreneurship, as few studies explore competitiveness in developing countries. Also, it offers a detailed explanation of the application of this algorithm, allowing researchers to reproduce this methodology in other scenarios. Practically, it might support policymakers in formulating development strategies and stimuli for business competitiveness. In addition, academic and business leaders can strengthen university-business collaboration with applied research in innovation and entrepreneurship. One limitation would be the number of countries participating in the research. The authors suggest future lines of research. Full article

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

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18 pages, 1552 KiB

Open AccessArticle

A Regularized Tseng Method for Solving Various Variational Inclusion Problems and Its Application to a Statistical Learning Model

by Adeolu Taiwo and Simeon Reich

Axioms 2023, 12(11), 1037; https://doi.org/10.3390/axioms12111037 - 06 Nov 2023

Viewed by 897

Abstract

We study three classes of variational inclusion problems in the framework of a real Hilbert space and propose a simple modification of Tseng’s forward-backward-forward splitting method for solving such problems. Our algorithm is obtained via a certain regularization procedure and uses self-adaptive step [...] Read more.

We study three classes of variational inclusion problems in the framework of a real Hilbert space and propose a simple modification of Tseng’s forward-backward-forward splitting method for solving such problems. Our algorithm is obtained via a certain regularization procedure and uses self-adaptive step sizes. We show that the approximating sequences generated by our algorithm converge strongly to a solution of the problems under suitable assumptions on the regularization parameters. Furthermore, we apply our results to an elastic net penalty problem in statistical learning theory and to split feasibility problems. Moreover, we illustrate the usefulness and effectiveness of our algorithm by using numerical examples in comparison with some existing relevant algorithms that can be found in the literature. Full article

(This article belongs to the Section Hilbert’s Sixth Problem)

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11 pages, 289 KiB

Open AccessArticle

Vector-Valued Analytic Functions Having Vector-Valued Tempered Distributions as Boundary Values

by Richard D. Carmichael

Axioms 2023, 12(11), 1036; https://doi.org/10.3390/axioms12111036 - 06 Nov 2023

Viewed by 793

Abstract

Vector-valued analytic functions in

C^{n}

, which are known to have vector-valued tempered distributional boundary values, are shown to be in the Hardy space

H^{p}, 1 \leq p < 2

, if the boundary value is in the vector-valued [...] Read more.

Vector-valued analytic functions in

C^{n}

, which are known to have vector-valued tempered distributional boundary values, are shown to be in the Hardy space

H^{p}, 1 \leq p < 2

, if the boundary value is in the vector-valued

L^{p}, 1 \leq p < 2

, functions. The analysis of this paper extends the analysis of a previous paper that considered the cases for

2 \leq p \leq \infty

. Thus, with the addition of the results of this paper, the considered problems are proved for all

p, 1 \leq p \leq \infty

. Full article

(This article belongs to the Special Issue Recent Advances in Complex Analysis and Applications)

16 pages, 328 KiB

Open AccessArticle

On Unique Determination of Polyhedral Sets

by Luca Rondi

Axioms 2023, 12(11), 1035; https://doi.org/10.3390/axioms12111035 - 05 Nov 2023

Viewed by 864

Abstract

In this paper, we develop in detail the geometric constructions that lead to many uniqueness results for the determination of polyhedral sets, typically scatterers, by a finite minimal number of measurements. We highlight how unique continuation and a suitable reflection principle are enough [...] Read more.

In this paper, we develop in detail the geometric constructions that lead to many uniqueness results for the determination of polyhedral sets, typically scatterers, by a finite minimal number of measurements. We highlight how unique continuation and a suitable reflection principle are enough to proceed with the constructions without any other assumption on the underlying partial differential equation or the boundary condition. We also aim to keep the geometric constructions and their proofs as simple as possible. To illustrate the applicability of this theory, we show how several uniqueness results present in the literature immediately follow from our arguments. Indeed, we believe that this theory may serve as a roadmap for establishing similar uniqueness results for other partial differential equations or boundary conditions. Full article

(This article belongs to the Special Issue Mathematical Methods in Waves-Based Inverse Problems at Different Scales)

16 pages, 369 KiB

Open AccessArticle

Lax Extensions of Conical

I

-Semifilter Monads

by Gao Zhang and Shao-Qun Zhang

Axioms 2023, 12(11), 1034; https://doi.org/10.3390/axioms12111034 - 05 Nov 2023

Viewed by 751

Abstract

For a quantale

I,

the unit interval endowed with a continuous triangular norm, we introduce the canonical, op-canonical and Kleisli extensions of the conical

I

-semifilter monad to

I - Rel .

It is proved that the op-canonical extension coincides with the [...] Read more.

For a quantale

I,

the unit interval endowed with a continuous triangular norm, we introduce the canonical, op-canonical and Kleisli extensions of the conical

I

-semifilter monad to

I - Rel .

It is proved that the op-canonical extension coincides with the Kleisli extension. Full article

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Axioms, Volume 12, Issue 11 (November 2023) – 61 articles

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