Axioms

21 pages, 321 KiB

Open AccessArticle

(Almost) Ricci Solitons in Lorentzian–Sasakian Hom-Lie Groups

by Esmaeil Peyghan, Leila Nourmohammadifar, Akram Ali and Ion Mihai

Axioms 2024, 13(10), 693; https://doi.org/10.3390/axioms13100693 - 4 Oct 2024

We study Lorentzian contact and Lorentzian–Sasakian structures in Hom-Lie algebras. We find that the three-dimensional

sl (2, R)

and Heisenberg Lie algebras provide examples of such structures, respectively. Curvature tensor properties in Lorentzian–Sasakian Hom-Lie algebras are investigated. If v is [...] Read more.

We study Lorentzian contact and Lorentzian–Sasakian structures in Hom-Lie algebras. We find that the three-dimensional

sl (2, R)

and Heisenberg Lie algebras provide examples of such structures, respectively. Curvature tensor properties in Lorentzian–Sasakian Hom-Lie algebras are investigated. If v is a contact 1-form, conditions under which the Ricci curvature tensor is v-parallel are given. Ricci solitons for Lorentzian–Sasakian Hom-Lie algebras are also studied. It is shown that a Ricci soliton vector field

ζ

is conformal whenever the Lorentzian–Sasakian Hom-Lie algebra is Ricci semisymmetric. To illustrate the use of the theory, a two-parameter family of three-dimensional Lorentzian–Sasakian Hom-Lie algebras which are not Lie algebras is given and their Ricci solitons are computed. Full article

(This article belongs to the Special Issue Differential Geometry and Its Application, 3rd Edition)

16 pages, 1116 KiB

Open AccessArticle

An Analysis of Type-I Generalized Progressive Hybrid Censoring for the One Parameter Logistic-Geometry Lifetime Distribution with Applications

by Magdy Nagy , Mohamed Ahmed Mosilhy, Ahmed Hamdi Mansi and Mahmoud Hamed Abu-Moussa

Axioms 2024, 13(10), 692; https://doi.org/10.3390/axioms13100692 - 4 Oct 2024

Abstract

Based on Type-I generalized progressive hybrid censored samples (GPHCSs), the parameter estimate for the unit-half logistic-geometry (UHLG) distribution is investigated in this work. Using maximum likelihood estimation (MLE) and Bayesian estimation, the parameters, reliability, and hazard functions of the UHLG distribution under GPHCSs [...] Read more.

Based on Type-I generalized progressive hybrid censored samples (GPHCSs), the parameter estimate for the unit-half logistic-geometry (UHLG) distribution is investigated in this work. Using maximum likelihood estimation (MLE) and Bayesian estimation, the parameters, reliability, and hazard functions of the UHLG distribution under GPHCSs have been assessed. Likewise, the computation is carried out for the asymptotic confidence intervals (ACIs). Furthermore, two bootstrap CIs, bootstrap-p and bootstrap-t, are mentioned. For symmetric loss functions, like squared error loss (SEL), and asymmetric loss functions, such as linear exponential loss (LL) and general entropy loss (GEL), there are specific Bayesian approximations. The Metropolis–Hastings samplers methodology were used to construct the credible intervals (CRIs). In conclusion, a genuine data set measuring the mortality statistics of a group of male mice with reticulum cell sarcoma is regarded as an application of the methods given. Full article

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

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16 pages, 813 KiB

Open AccessArticle

Starlikeness and Convexity of Generalized Bessel-Maitland Function

by Muhammad Umar Nawaz, Daniel Breaz, Mohsan Raza and Luminiţa-Ioana Cotîrlă

Axioms 2024, 13(10), 691; https://doi.org/10.3390/axioms13100691 - 4 Oct 2024

Abstract

The main objective of this research is to examine a specific sufficiency criteria for the starlikeness and convexity of order

δ

, k-uniform starlikeness, k-uniform convexity, lemniscate starlikeness and convexity, exponential starlikeness and convexity, uniform convexity of the Generalized Bessel-Maitland function. Applications of [...] Read more.

The main objective of this research is to examine a specific sufficiency criteria for the starlikeness and convexity of order

δ

, k-uniform starlikeness, k-uniform convexity, lemniscate starlikeness and convexity, exponential starlikeness and convexity, uniform convexity of the Generalized Bessel-Maitland function. Applications of these conclusions to the concept of corollaries are also provided. Additionally, an illustrated representation of these outcomes will be presented. So functional inequalities involving gamma function will be the main research tools of this exploration. The outcomes from this study generalize a number of conclusions from earlier studies. Full article

13 pages, 269 KiB

Open AccessArticle

Fixed Point Results of Fuzzy Multivalued Graphic Contractive Mappings in Generalized Parametric Metric Spaces

by Talat Nazir, Mujahid Abbas and Safeer Hussain Khan

Axioms 2024, 13(10), 690; https://doi.org/10.3390/axioms13100690 - 4 Oct 2024

Abstract

The aim of this paper is to introduce to a pair of fuzzy graphic rational F-contraction multivalued mappings and to study the necessary condition for the existence of common fixed points of fuzzy multivalued mappings in the setup of generalized parametric metric [...] Read more.

The aim of this paper is to introduce to a pair of fuzzy graphic rational F-contraction multivalued mappings and to study the necessary condition for the existence of common fixed points of fuzzy multivalued mappings in the setup of generalized parametric metric space endowed with a directed graph. A non-trivial example is presented to support the results presented herein. Our results improve and extend some recent results in the existing literature. Full article

(This article belongs to the Section Mathematical Analysis)

14 pages, 400 KiB

Open AccessArticle

A Modified Fractional Newton’s Solver

by Chih-Wen Chang, Sania Qureshi, Ioannis K. Argyros, Khair Muhammad Saraz and Evren Hincal

Axioms 2024, 13(10), 689; https://doi.org/10.3390/axioms13100689 - 4 Oct 2024

Abstract

Fractional calculus extends the conventional concepts of derivatives and integrals to non-integer orders, providing a robust mathematical framework for modeling complex systems characterized by memory and hereditary properties. This study enhances the convergence rate of the Caputo-based Newton’s solver for solving one-dimensional nonlinear [...] Read more.

Fractional calculus extends the conventional concepts of derivatives and integrals to non-integer orders, providing a robust mathematical framework for modeling complex systems characterized by memory and hereditary properties. This study enhances the convergence rate of the Caputo-based Newton’s solver for solving one-dimensional nonlinear equations. By modifying the order to

1 + η

, we provide a thorough analysis of the convergence order and present numerical simulations that demonstrate the improved efficiency of the proposed modified fractional Newton’s solver. The numerical simulations indicate significant advancements over traditional and existing fractional Newton-type approaches. Full article

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11 pages, 312 KiB

Open AccessArticle

Existence of Solutions for a Coupled Hadamard Fractional System of Integral Equations in Local Generalized Morrey Spaces

by Asra Hadadfard, Mohammad Bagher Ghaemi and António M. Lopes

Axioms 2024, 13(10), 688; https://doi.org/10.3390/axioms13100688 - 3 Oct 2024

Abstract

This paper introduces a new measure of non-compactness within a bounded domain of

R^{N}

in the generalized Morrey space. This measure is used to establish the existence of solutions for a coupled Hadamard fractional system of integral equations in generalized Morrey spaces. [...] Read more.

This paper introduces a new measure of non-compactness within a bounded domain of

R^{N}

in the generalized Morrey space. This measure is used to establish the existence of solutions for a coupled Hadamard fractional system of integral equations in generalized Morrey spaces. To illustrate the application of the main result, an example is presented. Full article

(This article belongs to the Topic Fractional Calculus: Theory and Applications, 2nd Edition)

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21 pages, 352 KiB

Open AccessArticle

Ternary Associativity and Ternary Lie Algebras at Cube Roots of Unity

by Viktor Abramov

Axioms 2024, 13(10), 687; https://doi.org/10.3390/axioms13100687 - 3 Oct 2024

Abstract

We propose a new approach to extend the notion of commutator and Lie algebra to algebras with ternary multiplication laws. Our approach is based on the ternary associativity of the first and second kinds. We propose a ternary commutator, which is a linear [...] Read more.

We propose a new approach to extend the notion of commutator and Lie algebra to algebras with ternary multiplication laws. Our approach is based on the ternary associativity of the first and second kinds. We propose a ternary commutator, which is a linear combination of six triple products (all permutations of three elements). The coefficients of this linear combination are the cube roots of unity. We find an identity for the ternary commutator that holds due to the ternary associativity of either the first or second kind. The form of this identity is determined by the permutations of the general affine group

G A (1, 5) \subset S_{5}

. We consider this identity as a ternary analog of the Jacobi identity. Based on the results obtained, we introduce the concept of a ternary Lie algebra at cube roots of unity and provide examples of such algebras constructed using ternary multiplications of rectangular and three-dimensional matrices. We also highlight the connection between the structure constants of a ternary Lie algebra with three generators and an irreducible representation of the rotation group. The classification of two-dimensional ternary Lie algebras at cube roots of unity is proposed. Full article

(This article belongs to the Special Issue Recent Advances in Representation Theory with Applications)

12 pages, 2012 KiB

Open AccessArticle

Kink Wave Phenomena in the Nonlinear Partial Differential Equation Representing the Transmission Line Model of Microtubules for Nanoionic Currents

by Safyan Mukhtar, Weaam Alhejaili, Mohammad Alqudah, Ali M. Mahnashi, Rasool Shah and Samir A. El-Tantawy

Axioms 2024, 13(10), 686; https://doi.org/10.3390/axioms13100686 - 2 Oct 2024

Abstract

This paper provides several new traveling wave solutions for a nonlinear partial differential equation (PDE) by applying symbolic computation and a new approach, the Riccati–Bernoulli sub-ODE method, in a computer algebra system. Herein, employing the Bäcklund transformation, we solve a nonlinear PDE associated [...] Read more.

This paper provides several new traveling wave solutions for a nonlinear partial differential equation (PDE) by applying symbolic computation and a new approach, the Riccati–Bernoulli sub-ODE method, in a computer algebra system. Herein, employing the Bäcklund transformation, we solve a nonlinear PDE associated with nanobiosciences and biophysics based on the transmission line model of microtubules for nanoionic currents. The equation introduced here in this form is suitable for critical nanoscience concerns like cell signaling and might continue to explain some of the basic cognitive functions in neurons. We employ advanced procedures to replicate the previously detected solitary waves. We offer our solutions in graphical forms, such as 3D and contour plots, using Mathematica. We can generalize the elementary method to other nonlinear equations in physics, requiring only a few steps. Full article

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

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17 pages, 291 KiB

Open AccessArticle

Crossed Modules and Non-Abelian Extensions of Differential Leibniz Conformal Algebras

by Hui Wu, Shuangjian Guo and Xiaohui Zhang

Axioms 2024, 13(10), 685; https://doi.org/10.3390/axioms13100685 - 2 Oct 2024

Abstract

In this paper, we introduce two-term differential

L e i b_{\infty}

-conformal algebras and give characterizations of some particular classes of such two-term differential

L e i b_{\infty}

-conformal algebras. Furthermore, we discuss the classification of the non-Abelian extensions in terms [...] Read more.

In this paper, we introduce two-term differential

L e i b_{\infty}

-conformal algebras and give characterizations of some particular classes of such two-term differential

L e i b_{\infty}

-conformal algebras. Furthermore, we discuss the classification of the non-Abelian extensions in terms of non-Abelian cohomology groups. Finally, we explore the inducibility of pairs of automorphisms and derive the analog Wells exact sequences under the circumstance of differential Leibniz conformal algebras. Full article

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

33 pages, 590 KiB

Open AccessArticle

Generalization of the Fuzzy Fejér–Hadamard Inequalities for Non-Convex Functions over a Rectangle Plane

by Hanan Alohali, Valer-Daniel Breaz, Omar Mutab Alsalami, Luminita-Ioana Cotirla and Ahmed Alamer

Axioms 2024, 13(10), 684; https://doi.org/10.3390/axioms13100684 - 2 Oct 2024

Abstract

Integral inequalities with generalized convexity play a vital role in both theoretical and applied mathematics. The theory of integral inequalities is one of the branches of mathematics that is now developing at the quickest rate due to its wide range of applications. We [...] Read more.

Integral inequalities with generalized convexity play a vital role in both theoretical and applied mathematics. The theory of integral inequalities is one of the branches of mathematics that is now developing at the quickest rate due to its wide range of applications. We define a new Hermite–Hadamard inequality for the novel class of coordinated ƛ-pre-invex fuzzy number-valued mappings (C-ƛ-pre-invex 𝘍 𝘕 𝘝 𝘔s) and examine the idea of C-ƛ-pre-invex 𝘍 𝘕 𝘝 𝘔s in this paper. Furthermore, using C-ƛ-pre-invex 𝘍 𝘕 𝘝 𝘔s, we construct several new integral inequalities for fuzzy double Riemann integrals. Several well-known results, as well as recently discovered results, are included in these findings as special circumstances. We think that the findings in this work are new and will help to stimulate more research in this area in the future. Additionally, unique choices lead to new outcomes. Full article

(This article belongs to the Special Issue Theory and Application of Integral Inequalities)

13 pages, 259 KiB

Open AccessArticle

Starlikeness, Convexity, Close-to-Convexity, and Quasi-Convexity for Functions with Fixed Initial Coefficients

by Mohanad Kadhim Ahmed Alkarafi, Ali Ebadian and Saeid Shams

Axioms 2024, 13(10), 683; https://doi.org/10.3390/axioms13100683 - 2 Oct 2024

Abstract

In this paper, we employ the theory of differential subordination to establish a theorem that delineates certain sufficient conditions for starlikeness, convexity, close-to-convexity, and quasi-convexity in relation to functions with fixed initial coefficients. Furthermore, we introduce some results derived from these conditions. Building [...] Read more.

In this paper, we employ the theory of differential subordination to establish a theorem that delineates certain sufficient conditions for starlikeness, convexity, close-to-convexity, and quasi-convexity in relation to functions with fixed initial coefficients. Furthermore, we introduce some results derived from these conditions. Building upon this framework, we derive an extension of Nunokawa’s lemma for analytic functions with fixed initial coefficients. Full article

(This article belongs to the Section Mathematical Analysis)

25 pages, 444 KiB

Open AccessArticle

Ulam–Hyers Stability and Simulation of a Delayed Fractional Differential Equation with Riemann–Stieltjes Integral Boundary Conditions and Fractional Impulses

by Xiaojun Lv, Kaihong Zhao and Haiping Xie

Axioms 2024, 13(10), 682; https://doi.org/10.3390/axioms13100682 - 1 Oct 2024

Abstract

In this article, we delve into delayed fractional differential equations with Riemann–Stieltjes integral boundary conditions and fractional impulses. By using differential inequality techniques and some fixed-point theorems, some novel sufficient assessments for convenient verification have been devised to ensure the existence and uniqueness [...] Read more.

In this article, we delve into delayed fractional differential equations with Riemann–Stieltjes integral boundary conditions and fractional impulses. By using differential inequality techniques and some fixed-point theorems, some novel sufficient assessments for convenient verification have been devised to ensure the existence and uniqueness of solutions. We further employ the nonlinear analysis to reveal that this problem is Ulam–Hyers (UH) stable. Finally, some examples and numerical simulations are presented to illustrate the reliability and validity of our main results. Full article

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

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16 pages, 315 KiB

Open AccessArticle

Second-Order Neutral Differential Equations with a Sublinear Neutral Term: Examining the Oscillatory Behavior

by Ahmed Alemam, Asma Al-Jaser, Osama Moaaz, Fahd Masood and Hamdy El-Metwally

Axioms 2024, 13(10), 681; https://doi.org/10.3390/axioms13100681 - 1 Oct 2024

Abstract

This article highlights the oscillatory properties of second-order Emden–Fowler delay differential equations featuring sublinear neutral terms and multiple delays, encompassing both canonical and noncanonical cases. Through the proofs of several theorems, we investigate criteria for the oscillation of all solutions to the equations [...] Read more.

This article highlights the oscillatory properties of second-order Emden–Fowler delay differential equations featuring sublinear neutral terms and multiple delays, encompassing both canonical and noncanonical cases. Through the proofs of several theorems, we investigate criteria for the oscillation of all solutions to the equations under study. By employing the Riccati technique in various ways, we derive results that expand the scope of previous research and enhance the cognitive understanding of this mathematical domain. Additionally, we provide three illustrative examples to demonstrate the validity and applicability of our findings. Full article

(This article belongs to the Special Issue Differential Equations and Related Topics, 2nd Edition)

16 pages, 331 KiB

Open AccessArticle

Existence Results for Differential Equations with Tempered Ψ–Caputo Fractional Derivatives

by Michal Pospíšil and Lucia Škripková Pospíšilová

Axioms 2024, 13(10), 680; https://doi.org/10.3390/axioms13100680 - 1 Oct 2024

Abstract

The method of the equivalent system of fractional integral equations is used to prove the existence results of a unique solution for initial value problems corresponding to various classes of nonlinear fractional differential equations involving the tempered

Ψ

–Caputo fractional derivative. These include [...] Read more.

The method of the equivalent system of fractional integral equations is used to prove the existence results of a unique solution for initial value problems corresponding to various classes of nonlinear fractional differential equations involving the tempered

Ψ

–Caputo fractional derivative. These include equations with their right side depending on ordinary as well as fractional-order derivatives, or fractional integrals of the solution. Full article

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

13 pages, 1975 KiB

Open AccessArticle

A Second-Order Numerical Method for a Class of Optimal Control Problems

by Kamil Aida-zade, Alexander Handzel and Efthimios Providas

Axioms 2024, 13(10), 679; https://doi.org/10.3390/axioms13100679 - 1 Oct 2024

Abstract

The numerical solution of optimal control problems through second-order methods is examined in this paper. Controlled processes are described by a system of nonlinear ordinary differential equations. There are two specific characteristics of the class of control actions used. The first one is [...] Read more.

The numerical solution of optimal control problems through second-order methods is examined in this paper. Controlled processes are described by a system of nonlinear ordinary differential equations. There are two specific characteristics of the class of control actions used. The first one is that controls are searched for in a given class of functions, which depend on unknown parameters to be found by minimizing an objective functional. The parameter values, in general, may be different at different time intervals. The second feature of the considered problem is that the boundaries of time intervals are also optimized with fixed values of the parameters of the control actions in each of the intervals. The special cases of the problem under study are relay control problems with optimized switching moments. In this work, formulas for the gradient and the Hessian matrix of the objective functional with respect to the optimized parameters are obtained. For this, the technique of fast differentiation is used. A comparison of numerical experiment results obtained with the use of first- and second-order optimization methods is presented. Full article

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

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21 pages, 393 KiB

Open AccessArticle

Spatio-Functional Nadaraya–Watson Estimator of the Expectile Shortfall Regression

by Mohammed B. Alamari, Fatimah A. Almulhim, Zoulikha Kaid and Ali Laksaci

Axioms 2024, 13(10), 678; https://doi.org/10.3390/axioms13100678 - 30 Sep 2024

Abstract

The main aim of this paper is to consider a new risk metric that permits taking into account the spatial interactions of data. The considered risk metric explores the spatial tail-expectation of the data. Indeed, it is obtained by combining the ideas of [...] Read more.

The main aim of this paper is to consider a new risk metric that permits taking into account the spatial interactions of data. The considered risk metric explores the spatial tail-expectation of the data. Indeed, it is obtained by combining the ideas of expected shortfall regression with an expectile risk model. A spatio-functional Nadaraya–Watson estimator of the studied metric risk is constructed. The main asymptotic results of this work are the establishment of almost complete convergence under a mixed spatial structure. The claimed asymptotic result is obtained under standard assumptions covering the double functionality of the model as well as the data. The impact of the spatial interaction of the data in the proposed risk metric is evaluated using simulated data. A real experiment was conducted to measure the feasibility of the Spatio-Functional Expectile Shortfall Regression (SFESR) in practice. Full article

(This article belongs to the Special Issue Advances in Functional and Topological Data Analysis)

17 pages, 294 KiB

Open AccessArticle

New Properties and Matrix Representations on Higher-Order Generalized Fibonacci Quaternions with q-Integer Components

by Can Kızılateş, Wei-Shih Du, Nazlıhan Terzioğlu and Ren-Chuen Chen

Axioms 2024, 13(10), 677; https://doi.org/10.3390/axioms13100677 - 30 Sep 2024

Abstract

In this paper, by using q-integers and higher-order generalized Fibonacci numbers, we define the higher-order generalized Fibonacci quaternions with q-integer components. We give some special cases of these newly established quaternions. This article examines q-calculus and quaternions together. We obtain [...] Read more.

In this paper, by using q-integers and higher-order generalized Fibonacci numbers, we define the higher-order generalized Fibonacci quaternions with q-integer components. We give some special cases of these newly established quaternions. This article examines q-calculus and quaternions together. We obtain a Binet-like formula, some new identities, a generating function, a recurrence relation, an exponential generating function, and sum properties of quaternions with quantum integer coefficients. In addition, we obtain some new identities for these types of quaternions by using three new special matrices. Full article

(This article belongs to the Special Issue Theory and Application of Integral Inequalities)

16 pages, 1632 KiB

Open AccessArticle

Algebraic and Geometric Methods for Construction of Topological Quantum Codes from Lattices

by Edson Donizete de Carvalho, Waldir Silva Soares, Jr., Douglas Fernando Copatti, Carlos Alexandre Ribeiro Martins and Eduardo Brandani da Silva

Axioms 2024, 13(10), 676; https://doi.org/10.3390/axioms13100676 - 30 Sep 2024

Abstract

Current work provides an algebraic and geometric technique for building topological quantum codes. From the lattice partition derived of quotient lattices

Λ^{'} / Λ

of index m combined with geometric technique of the projections of vector basis

Λ^{'}

over vector basis [...] Read more.

Current work provides an algebraic and geometric technique for building topological quantum codes. From the lattice partition derived of quotient lattices

Λ^{'} / Λ

of index m combined with geometric technique of the projections of vector basis

Λ^{'}

over vector basis

Λ

, we reproduce surface codes found in the literature with parameter

[[2 m^{2}, 2, | a | + | b |]]

for the case

Λ = Z^{2}

and

m = a^{2} + b^{2}

, where a and b are integers that are not null, simultaneously. We also obtain a new class of surface code with parameters

[[2 m, 2, | a | + | b |]]

from the

Λ = A_{2}

-lattice when m can be expressed as

m = a^{2} + a b + b^{2}

, where a and b are integer values. Finally, we will show how this technique can be extended to the construction of color codes with parameters

[[18 m, 4, 6 (| a | + | b |)]]

by considering honeycomb lattices partition

A_{2} / Λ^{'}

of index

m = 9 (a^{2} + a b + b^{2})

where a and b are not null integers. Full article

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

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22 pages, 9444 KiB

Open AccessArticle

A Robust and Optimal Iterative Algorithm Employing a Weight Function for Solving Nonlinear Equations with Dynamics and Applications

by Shahid Abdullah, Neha Choubey, Suresh Dara, Moin-ud-Din Junjua and Tawseef Abdullah

Axioms 2024, 13(10), 675; https://doi.org/10.3390/axioms13100675 - 30 Sep 2024

Abstract

This study introduces a novel, iterative algorithm that achieves fourth-order convergence for solving nonlinear equations. Satisfying the Kung–Traub conjecture, the proposed technique achieves an optimal order of four with an efficiency index

(I)

of

1.587

, requiring three function evaluations. An [...] Read more.

This study introduces a novel, iterative algorithm that achieves fourth-order convergence for solving nonlinear equations. Satisfying the Kung–Traub conjecture, the proposed technique achieves an optimal order of four with an efficiency index

(I)

of

1.587

, requiring three function evaluations. An analysis of convergence is presented to show the optimal fourth-order convergence. To verify the theoretical results, in-depth numerical comparisons are presented for both real and complex domains. The proposed algorithm is specifically examined on a variety of polynomial functions, and it is shown by the efficient and accurate results that it outperforms many existing algorithms in terms of speed and accuracy. The study not only explores the proposed method’s convergence properties, computational efficiency, and stability but also introduces a novel perspective by considering the count of black points as an indicator of a method’s divergence. By analyzing the mean number of iterations necessary for methods to converge within a cycle and measuring CPU time in seconds, this research provides a holistic assessment of both the efficiency and speed of iterative methods. Notably, the analysis of basins of attraction illustrates that our proposed method has larger sets of initial points that yield convergence. Full article

(This article belongs to the Special Issue The Numerical Analysis and Its Application)

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19 pages, 1007 KiB

Open AccessArticle

An RBF Method for Time Fractional Jump-Diffusion Option Pricing Model under Temporal Graded Meshes

by Wenxiu Gong, Zuoliang Xu and Yesen Sun

Axioms 2024, 13(10), 674; https://doi.org/10.3390/axioms13100674 - 29 Sep 2024

Abstract

This paper explores a numerical method for European and American option pricing under time fractional jump-diffusion model in Caputo scene. The pricing problem for European options is formulated using a time fractional partial integro-differential equation, whereas the pricing of American options is described [...] Read more.

This paper explores a numerical method for European and American option pricing under time fractional jump-diffusion model in Caputo scene. The pricing problem for European options is formulated using a time fractional partial integro-differential equation, whereas the pricing of American options is described by a linear complementarity problem. For European option, we present nonuniform discretization along time and the radial basis function (RBF) method for spatial discretization. The stability and convergence analysis of the discrete scheme are carried out in the case of European options. For American option, the operator splitting method is adopted which split linear complementary problem into two simple equations. The numerical results confirm the accuracy of the proposed method. Full article

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

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37 pages, 424 KiB

Open AccessArticle

The Robin Problems in the Coupled System of Wave Equations on a Half-Line

by Po-Chun Huang and Bo-Yu Pan

Axioms 2024, 13(10), 673; https://doi.org/10.3390/axioms13100673 - 29 Sep 2024

Abstract

This article investigates the local well-posedness of a coupled system of wave equations on a half-line, with a particular emphasis on Robin boundary conditions within Sobolev spaces. We provide estimates for the solutions to linear initial-boundary-value problems related to the coupled system of [...] Read more.

This article investigates the local well-posedness of a coupled system of wave equations on a half-line, with a particular emphasis on Robin boundary conditions within Sobolev spaces. We provide estimates for the solutions to linear initial-boundary-value problems related to the coupled system of wave equations, utilizing the Unified Transform Method in conjunction with the Hadamard norm while considering the influence of external forces. Furthermore, we demonstrate that replacing the external force with a nonlinear term alters the iteration map defined by the unified transform solutions, making it a contraction map in a suitable solution space. By employing the contraction mapping theorem, we establish the existence of a unique solution. Finally, we show that the data-to-solution map is locally Lipschitz continuous, thus confirming the local well-posedness of the coupled system of wave equations under consideration. Full article

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

17 pages, 2201 KiB

Open AccessArticle

Dementia Classification Approach Based on Non-Singleton General Type-2 Fuzzy Reasoning

by Claudia I. Gonzalez

Axioms 2024, 13(10), 672; https://doi.org/10.3390/axioms13100672 - 28 Sep 2024

Abstract

Dementia is the most critical neurodegenerative disease that gradually destroys memory and other cognitive functions. Therefore, early detection is essential, and to build an effective detection model, it is required to understand its type, symptoms, stages and causes, and diagnosis methodologies. This paper [...] Read more.

Dementia is the most critical neurodegenerative disease that gradually destroys memory and other cognitive functions. Therefore, early detection is essential, and to build an effective detection model, it is required to understand its type, symptoms, stages and causes, and diagnosis methodologies. This paper presents a novel approach to classify dementia based on a data set with some relevant patient features. The classification methodology employs non-singleton general type-2 fuzzy sets, non-singleton interval type-2 fuzzy sets, and non-singleton type 1 fuzzy sets. These advanced fuzzy sets are compared with traditional singleton fuzzy sets to evaluate their performance. The Takagi–Sugeno–Kang TSK inference method is used to handle fuzzy reasoning. In the process, the parameters of the membership functions (MFs) and rules are obtained using ANFIS, and non-singleton MFs are optimized with PSO. The results demonstrate that non-singleton general type-2 fuzzy sets improve classification accuracy compared to singleton fuzzy sets, demonstrating their ability to model the uncertainties inherent in the diagnosis of dementia. This improvement suggests that non-singleton fuzzy systems offer a more robust framework for developing effective diagnostic tools in the medical domain. Accurate classification of dementia is of utmost importance to improve patient care and advance medical research. Full article

(This article belongs to the Special Issue Advances in Mathematical Optimization Algorithms and Its Applications)

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25 pages, 773 KiB

Open AccessArticle

An Efficient and Stable Caputo-Type Inverse Fractional Parallel Scheme for Solving Nonlinear Equations

by Mudassir Shams and Bruno Carpentieri

Axioms 2024, 13(10), 671; https://doi.org/10.3390/axioms13100671 - 27 Sep 2024

Abstract

Nonlinear problems, which often arise in various scientific and engineering disciplines, typically involve nonlinear equations or functions with multiple solutions. Analytical solutions to these problems are often impossible to obtain, necessitating the use of numerical techniques. This research proposes an efficient and stable [...] Read more.

Nonlinear problems, which often arise in various scientific and engineering disciplines, typically involve nonlinear equations or functions with multiple solutions. Analytical solutions to these problems are often impossible to obtain, necessitating the use of numerical techniques. This research proposes an efficient and stable Caputo-type inverse numerical fractional scheme for simultaneously approximating all roots of nonlinear equations, with a convergence order of

2 ψ + 2

. The scheme is applied to various nonlinear problems, utilizing dynamical analysis to determine efficient initial values for a single root-finding Caputo-type fractional scheme, which is further employed in inverse fractional parallel schemes to accelerate convergence rates. Several sets of random initial vectors demonstrate the global convergence behavior of the proposed method. The newly developed scheme outperforms existing methods in terms of accuracy, consistency, validation, computational CPU time, residual error, and stability. Full article

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

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12 pages, 254 KiB

Open AccessArticle

A General Fixed Point Theorem for a Sequence of Multivalued Mappings in S-Metric Spaces

by Valeriu Popa and Alina-Mihaela Patriciu

Axioms 2024, 13(10), 670; https://doi.org/10.3390/axioms13100670 - 27 Sep 2024

Abstract

Over the years, the concept of metric space has been extended in several directions, and numerous common fixed point theorems for multivalued mappings in complete metric space have been demonstrated. In this paper, we prove a general fixed point theorem for a pair [...] Read more.

Over the years, the concept of metric space has been extended in several directions, and numerous common fixed point theorems for multivalued mappings in complete metric space have been demonstrated. In this paper, we prove a general fixed point theorem for a pair of multivalued mappings satisfying implicit relations, extending some results from the literature to S-metric spaces. As an application, we obtain new results for a sequence of multivalued mappings in S-metric spaces, generalizing some known results. Full article

14 pages, 244 KiB

Open AccessArticle

Some Identities Related to Semiprime Ideal of Rings with Multiplicative Generalized Derivations

by Ali Yahya Hummdi, Emine Koç Sögütcü, Öznur Gölbaşı and Nadeem ur Rehman

Axioms 2024, 13(10), 669; https://doi.org/10.3390/axioms13100669 - 27 Sep 2024

Abstract

This paper investigates the relationship between the commutativity of rings and the properties of their multiplicative generalized derivations. Let

F

be a ring with a semiprime ideal

Π

. A map

ϕ : F \to F

is classified as a multiplicative generalized derivation [...] Read more.

This paper investigates the relationship between the commutativity of rings and the properties of their multiplicative generalized derivations. Let

F

be a ring with a semiprime ideal

Π

. A map

ϕ : F \to F

is classified as a multiplicative generalized derivation if there exists a map

σ : F \to F

such that

ϕ (x y) = ϕ (x) y + x σ (y)

for all

x, y \in F

. This study focuses on semiprime ideals

Π

that admit multiplicative generalized derivations

ϕ

and G that satisfy certain differential identities within

F

. By examining these conditions, the paper aims to provide new insights into the structural aspects of rings, particularly their commutativity in relation to the behavior of such derivations. Full article

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

18 pages, 2801 KiB

Open AccessArticle

Maneuvering Object Tracking and Movement Parameters Identification by Indirect Observations with Random Delays

by Alexey Bosov

Axioms 2024, 13(10), 668; https://doi.org/10.3390/axioms13100668 - 26 Sep 2024

Abstract

The paper presents an approach to solving the problem of unknown motion parameters Bayesian identification for the stochastic dynamic system model with randomly delayed observations. The system identification and the object tracking tasks obtain solutions in the form of recurrent Bayesian relations for [...] Read more.

The paper presents an approach to solving the problem of unknown motion parameters Bayesian identification for the stochastic dynamic system model with randomly delayed observations. The system identification and the object tracking tasks obtain solutions in the form of recurrent Bayesian relations for a posteriori probability density. These relations are not practically applicable due to the computational challenges they present. For practical implementation, we propose a conditionally minimax nonlinear filter that implements the concept of conditionally optimal estimation. The random delays model source is the area of autonomous underwater vehicle control. The paper discusses in detail a computational experiment based on a model that is closely aligned with this practical need. The discussion includes both a description of the filter synthesis features based on the geometric interpretation of the simulated measurements and an impact analysis of the effectiveness of model special factors, such as time delays and model unknown parameters. Furthermore, the paper puts forth a novel approach to the identification problem statement, positing a random jumping change in the motion parameters values. Full article

(This article belongs to the Special Issue Stochastic Modeling and Analysis for Applications and Technologies)

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17 pages, 317 KiB

Open AccessArticle

The Bundle of Tensor Densities and Its Covariant Derivatives

by Joan Grandes Umbert and Tom Mestdag

Axioms 2024, 13(10), 667; https://doi.org/10.3390/axioms13100667 - 26 Sep 2024

Abstract

We construct the smooth vector bundle of tensor densities of arbitrary weight in a coordinate-independent way. We prove the general existence of a globally smooth tensor density field, as well as the existence of a globally smooth metric density for a pseudo-Riemannian manifold, [...] Read more.

We construct the smooth vector bundle of tensor densities of arbitrary weight in a coordinate-independent way. We prove the general existence of a globally smooth tensor density field, as well as the existence of a globally smooth metric density for a pseudo-Riemannian manifold, specifically. We study the coordinate description of a covariant derivative over densities, and define a natural extension of affine connections to densities. We provide an equivalent characterization, in the case of a pseudo-Riemannian manifold. Full article

(This article belongs to the Section Mathematical Analysis)

24 pages, 4236 KiB

Open AccessArticle

A New Contribution in Fractional Integral Calculus and Inequalities over the Coordinated Fuzzy Codomain

by Zizhao Zhou, Ahmad Aziz Al Ahmadi, Alina Alb Lupas and Khalil Hadi Hakami

Axioms 2024, 13(10), 666; https://doi.org/10.3390/axioms13100666 - 26 Sep 2024

Abstract

The correct derivation of integral inequalities on fuzzy-number-valued mappings depends on applying fractional calculus to fuzzy number analysis. The purpose of this article is to introduce a new class of convex mappings and generalize various previously published results on the fuzzy number and [...] Read more.

The correct derivation of integral inequalities on fuzzy-number-valued mappings depends on applying fractional calculus to fuzzy number analysis. The purpose of this article is to introduce a new class of convex mappings and generalize various previously published results on the fuzzy number and interval-valued mappings via fuzzy-order relations using fuzzy coordinated ỽ-convexity mappings so that the new version of the well-known Hermite–Hadamard (H-H) inequality can be presented in various variants via the fractional integral operators (Riemann–Liouville). Some new product forms of these inequalities for coordinated ỽ-convex fuzzy-number-valued mappings (coordinated ỽ-convex FNVMs) are also discussed. Additionally, we provide several fascinating non-trivial examples and exceptional cases to show that these results are accurate. Full article

(This article belongs to the Special Issue Theory and Application of Integral Inequalities)

11 pages, 268 KiB

Open AccessArticle

Hamiltonian Formulation for Continuous Systems with Second-Order Derivatives: A Study of Podolsky Generalized Electrodynamics

by Yazen M. Alawaideh, Alina Alb Lupas, Bashar M. Al-khamiseh, Majeed A. Yousif, Pshtiwan Othman Mohammed and Y. S. Hamed

Axioms 2024, 13(10), 665; https://doi.org/10.3390/axioms13100665 - 26 Sep 2024

Abstract

This paper presents an analysis of the Hamiltonian formulation for continuous systems with second-order derivatives derived from Dirac’s theory. This approach offers a unique perspective on the equations of motion compared to the traditional Euler–Lagrange formulation. Focusing on Podolsky’s generalized electrodynamics, the Hamiltonian [...] Read more.

This paper presents an analysis of the Hamiltonian formulation for continuous systems with second-order derivatives derived from Dirac’s theory. This approach offers a unique perspective on the equations of motion compared to the traditional Euler–Lagrange formulation. Focusing on Podolsky’s generalized electrodynamics, the Hamiltonian and corresponding equations of motion are derived. The findings demonstrate that both Hamiltonian and Euler–Lagrange formulations yield equivalent results. This study highlights the Hamiltonian approach as a valuable alternative for understanding the dynamics of second-order systems, validated through a specific application within generalized electrodynamics. The novelty of the research lies in developing advanced theoretical models through Hamiltonian formalism for continuous systems with second-order derivatives. The research employs an alternative method to the Euler–Lagrange formulas by applying Dirac’s theory to study the generalized Podolsky electrodynamics, contributing to a better understanding of complex continuous systems. Full article

(This article belongs to the Special Issue Mathematical Models and Simulations, 2nd Edition)

24 pages, 607 KiB

Open AccessArticle

Bivariate Length-Biased Exponential Distribution under Progressive Type-II Censoring: Incorporating Random Removal and Applications to Industrial and Computer Science Data

by Aisha Fayomi, Ehab M. Almetwally and Maha E. Qura

Axioms 2024, 13(10), 664; https://doi.org/10.3390/axioms13100664 - 26 Sep 2024

Abstract

In this paper, we address the analysis of bivariate lifetime data from a length-biased exponential distribution observed under Type II progressive censoring with random removals, where the number of units removed at each failure time follows a binomial distribution. We derive the likelihood [...] Read more.

In this paper, we address the analysis of bivariate lifetime data from a length-biased exponential distribution observed under Type II progressive censoring with random removals, where the number of units removed at each failure time follows a binomial distribution. We derive the likelihood function for the progressive Type II censoring scheme with random removals and apply it to the bivariate length-biased exponential distribution. The parameters of the proposed model are estimated using both likelihood and Bayesian methods for point and interval estimators, including asymptotic confidence intervals and bootstrap confidence intervals. We also employ different loss functions to construct Bayesian estimators. Additionally, a simulation study is conducted to compare the performance of censoring schemes. The effectiveness of the proposed methodology is demonstrated through the analysis of two real datasets from the industrial and computer science domains, providing valuable insights for illustrative purposes. Full article

(This article belongs to the Special Issue Applications of Bayesian Methods in Statistical Analysis)

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Axioms, Volume 13, Issue 10 (October 2024) – 39 articles

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