Axioms

17 pages, 2621 KiB

Open AccessArticle

Inspection of a Time-Delayed Excited Dam** Duffing Oscillator

by Khalid Alluhydan, Galal M. Moatimid, T. S. Amer and A. A. Galal

Axioms 2024, 13(6), 416; https://doi.org/10.3390/axioms13060416 - 20 Jun 2024

Viewed by 252

This paper examines a time delay in position and velocity to minimize the nonlinear vibration of an excited Duffing oscillator (DO). This model is highly beneficial for capturing the nonlinear characteristics of many different applications in engineering. To achieve an estimated uniform solution [...] Read more.

This paper examines a time delay in position and velocity to minimize the nonlinear vibration of an excited Duffing oscillator (DO). This model is highly beneficial for capturing the nonlinear characteristics of many different applications in engineering. To achieve an estimated uniform solution to the problem under consideration, a modified homotopy perturbation method (HPM) is utilized. This adaptation produces a more accurate precise approximation with a numerical solution (NS). This is obtained by employing Mathematica software 12 (MS) in comparison with the analytical solution (AS). The comparison signifies a good match between the two methodologies. The comparison is made with the aid of the NS. Consequently, the work allows for a qualitative assessment of the results of a representative analytical approximation approach. A promising stability analysis for the unforced system is performed. The time history of the accomplished results is illustrated in light of a diverse range of physical frequency and time-delay aspects. The outcomes are theoretically discussed and numerically explained with a set of graphs. The nonlinear structured prototype is examined via the multiple-scale procedure. It investigates how various controlling limits affect the organization of vibration performances. As a key assumption, according to cubic nonlinearity, two significant examples of resonance, sub-harmonic and super-harmonic, are explored. The obtained modulation equations, in these situations, are quantitatively investigated with regard to the influence of the applied backgrounds. Full article

(This article belongs to the Section Mathematical Physics)

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21 pages, 320 KiB

Open AccessArticle

Fixed-Point Results of Generalized (ϕ,Ψ)-Contractive Map**s in Partially Ordered Controlled Metric Spaces with an Application to a System of Integral Equations

by Mohammad Akram, Salha Alshaikey, Umar Ishtiaq, Muhammad Farhan, Ioannis K. Argyros and Samundra Regmi

Axioms 2024, 13(6), 415; https://doi.org/10.3390/axioms13060415 - 20 Jun 2024

Viewed by 237

Abstract

In this manuscript, we prove numerous results concerning fixed points, common fixed points, coincidence points, coupled coincidence points, and coupled common fixed points for

(ϕ, Ψ)

-contractive map**s in the framework of partially ordered controlled metric spaces. Our findings introduce [...] Read more.

In this manuscript, we prove numerous results concerning fixed points, common fixed points, coincidence points, coupled coincidence points, and coupled common fixed points for

(ϕ, Ψ)

-contractive map**s in the framework of partially ordered controlled metric spaces. Our findings introduce a novel perspective on this mathematical context, and we illustrate the uniqueness of our findings through various explanatory examples. Also, we apply the main result to find the existence and uniqueness of the solution of the system of integral equations as an application. Full article

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

23 pages, 7396 KiB

Open AccessArticle

A Hybrid Method for Solving the One-Dimensional Wave Equation of Tapered Sucker-Rod Strings

by Jiaojian Yin and Hongzhang Ma

Axioms 2024, 13(6), 414; https://doi.org/10.3390/axioms13060414 - 20 Jun 2024

Viewed by 204

Abstract

Simulating surface conditions by solving the wave equation of a sucker-rod string is the theoretical basis of a sucker-rod pum** system. To overcome the shortcomings of the conventional finite difference method and analytical solution, this work describes a novel hybrid method that combines [...] Read more.

Simulating surface conditions by solving the wave equation of a sucker-rod string is the theoretical basis of a sucker-rod pum** system. To overcome the shortcomings of the conventional finite difference method and analytical solution, this work describes a novel hybrid method that combines the analytical solution with the finite difference method. In this method, an analytical solution of the tapered rod wave equation with a recursive matrix form based on the Fourier series is proposed, a unified pum** condition model is established, a modified finite difference method is given, a hybrid strategy is established, and a convergence calculation method is proposed. Based on two different types of oil wells, the analytical solutions are verified by comparing different methods. The hybrid method is verified by using the finite difference method simulated data and measured oil data. The pum** speed sensitivity and convergence of the hybrid method are studied. The results show that the proposed analytical solution has high accuracy, with a maximum relative error relative to that of the classical finite difference method of 0.062%. The proposed hybrid method has a high simulation accuracy, with a maximum relative area error relative to that of the finite difference method of 0.09% and a maximum relative area error relative to measured data of 1.89%. Even at higher pum** speeds, the hybrid method still has accuracy. The hybrid method in this paper is convergent. The introduction of the finite difference method allows the hybrid method to more easily converge. The novelty of this work is that it combines the advantages of the finite difference method and the analytical solution, and it provides a convergence calculation method to provide guidance for its application. The hybrid method presented in this paper provides an alternative scheme for predicting the behavior of sucker-rod pum** systems and a new approach for solving wave equations with complex boundary conditions. Full article

(This article belongs to the Topic Numerical Methods for Partial Differential Equations)

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21 pages, 487 KiB

Open AccessArticle

Hermite–Hadamard–Mercer-Type Inequalities for Three-Times Differentiable Functions

by Loredana Ciurdariu and Eugenia Grecu

Axioms 2024, 13(6), 413; https://doi.org/10.3390/axioms13060413 - 19 Jun 2024

Viewed by 283

Abstract

In this study, an integral identity is given in order to present some Hermite–Hadamard–Mercer-type inequalities for functions whose powers of the absolute values of the third derivatives are convex. Several consequences and three applications to special means are given, as well as four [...] Read more.

In this study, an integral identity is given in order to present some Hermite–Hadamard–Mercer-type inequalities for functions whose powers of the absolute values of the third derivatives are convex. Several consequences and three applications to special means are given, as well as four examples with graphics which illustrate the validity of the results. Moreover, several Hermite–Hadamard–Mercer-type inequalities for fractional integrals for functions whose powers of the absolute values of the third derivatives are convex are presented. Full article

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

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11 pages, 271 KiB

Open AccessArticle

Stability of the Stochastic Ginzburg–Landau–Newell Equations in Two Dimensions

by **g Wang and Yan Zheng

Axioms 2024, 13(6), 412; https://doi.org/10.3390/axioms13060412 - 19 Jun 2024

Viewed by 232

Abstract

This paper concerns the 2D stochastic Ginzburg–Landau–Newell equations with a degenerate random forcing. We study the relationship between stationary distributions which correspond to the original and perturbed systems and then prove the stability of the stationary distribution. This suggests that the complexity of [...] Read more.

This paper concerns the 2D stochastic Ginzburg–Landau–Newell equations with a degenerate random forcing. We study the relationship between stationary distributions which correspond to the original and perturbed systems and then prove the stability of the stationary distribution. This suggests that the complexity of stochastic systems is likely to be robust. The main difficulty of the proof lies in estimating the expectation of exponential moments and controlling nonlinear terms while working on the evolution triple

H^{2} \subset H^{1} \subset H^{0}

to obtain a bound on the difference between the original solution and the perturbed solution. Full article

(This article belongs to the Special Issue Stochastic and Statistical Analysis in Natural Sciences)

20 pages, 633 KiB

Open AccessArticle

Intuitionistic Fuzzy Granular Matrix: Novel Calculation Approaches for Intuitionistic Fuzzy Covering-Based Rough Sets

by **gqian Wang and **aohong Zhang

Axioms 2024, 13(6), 411; https://doi.org/10.3390/axioms13060411 - 18 Jun 2024

Viewed by 272

Abstract

Intuitionistic fuzzy (IF)

β

-minimal description operators can deal with noise data in the IF covering-based rough set theory. That is to say, they can be used to find data that we need in IF environments. For an IF

β

-covering approximation space [...] Read more.

Intuitionistic fuzzy (IF)

β

-minimal description operators can deal with noise data in the IF covering-based rough set theory. That is to say, they can be used to find data that we need in IF environments. For an IF

β

-covering approximation space (i.e., an IF environment) with a high cardinality, it would be tedious and complicated to use IF set representations to calculate them. Therefore, it is necessary to find a quick method to obtain them. In this paper, we present the notion of IF

β

-maximal description based on the definition of IF

β

-minimal description, along with the concepts of IF granular matrix and IF reduction. Moreover, we propose matrix calculation methods for IF covering-based rough sets, such as IF

β

-minimal descriptions, IF

β

-maximal descriptions, and IF reductions. Firstly, the notion of an IF granular matrix is presented, which is used to calculate IF

β

-minimal description. Secondly, inspired by IF

β

-minimal description, we give the notion of IF

β

-maximal description. Furthermore, the matrix representations of IF

β

-maximal descriptions are presented. Next, two types of reductions for IF

β

-covering approximation spaces via IF

β

-minimal and fuzzy

β

-minimal descriptions are presented, along with their matrix representations. Finally, the new calculation methods are compared with corresponding set representations by carrying out several experiments. Full article

(This article belongs to the Topic Fuzzy Number, Fuzzy Difference, Fuzzy Differential: Theory and Applications)

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11 pages, 268 KiB

Open AccessArticle

A Characterization of Normal Injective and Normal Projective Hypermodules

by Ergül Türkmen, Burcu Nİşancı Türkmen and Hashem Bordbar

Axioms 2024, 13(6), 410; https://doi.org/10.3390/axioms13060410 - 18 Jun 2024

Viewed by 258

Abstract

This study is motivated by the recently published papers on normal injective and normal projective hypermodules. We provide a new characterization of the normal injective and normal projective hypermodules by using the splitting of the short exact sequences of hypermodules. After presenting some [...] Read more.

This study is motivated by the recently published papers on normal injective and normal projective hypermodules. We provide a new characterization of the normal injective and normal projective hypermodules by using the splitting of the short exact sequences of hypermodules. After presenting some of their fundamental properties, we show that if a hypermodule is normal projective, then every exact sequence ending with it is splitting. Moreover, if a hypermodule is normal injective, then every exact sequence starting with it is splitting as well. Finally, we investigate the relationships between semisimple, simple, normal injective, and normal projective hypermodules. Full article

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

12 pages, 258 KiB

Open AccessArticle

Inverses and Determinants of Arrowhead and Diagonal-Plus-Rank-One Matrices over Associative Algebras

by Nevena Jakovčević Stor and Ivan Slapničar

Axioms 2024, 13(6), 409; https://doi.org/10.3390/axioms13060409 - 18 Jun 2024

Viewed by 254

Abstract

This article considers arrowhead and diagonal-plus-rank-one matrices in

F^{n \times n}

where

F \in {R, C, H}

and where

H

is a noncommutative algebra of quaternions. We provide unified formulas for fast determinants and inverses for considered matrices. [...] Read more.

This article considers arrowhead and diagonal-plus-rank-one matrices in

F^{n \times n}

where

F \in {R, C, H}

and where

H

is a noncommutative algebra of quaternions. We provide unified formulas for fast determinants and inverses for considered matrices. The formulas are unified in the sense that the same formula holds in both commutative and noncommutative associative fields or algebras, with noncommutative examples being matrices of quaternions and block matrices. Each formula requires

O (n)

arithmetic operations, as does multiplication of such matrices with a vector. The formulas are efficiently implemented using the polymorphism or multiple-dispatch feature of the Julia programming language. Full article

22 pages, 1201 KiB

Open AccessArticle

Generalization of Fermatean Fuzzy Set and Implementation of Fermatean Fuzzy PROMETHEE II Method for Decision Making via PROMETHEE GAIA

by Revathy Aruchsamy, Inthumathi Velusamy, Krishnaprakash Sanmugavel, Prasantha Bharathi Dhandapani and Kavikumar Ramasamy

Axioms 2024, 13(6), 408; https://doi.org/10.3390/axioms13060408 - 17 Jun 2024

Viewed by 322

Abstract

The Fermatean fuzzy set, in contrast to other generalizations of fuzzy sets like PFS and IFS, has a wide range of acceptance for both MF and NMF. In light of this, the Fermatean fuzzy set performs as an efficient, flexible, and comprehensive representation [...] Read more.

The Fermatean fuzzy set, in contrast to other generalizations of fuzzy sets like PFS and IFS, has a wide range of acceptance for both MF and NMF. In light of this, the Fermatean fuzzy set performs as an efficient, flexible, and comprehensive representation in situations that lack certainty. Here, the weaker forms of Fermatean fuzzy sets are introduced, and their traits are analyzed. Decomposition and continuity of the Fermatean fuzzy

α

-open set are also accustomed. With the goal of safeguarding our green environment, hiring the best supplier is of the utmost significance in the construction industry. Using outranking techniques, Visual PROMETHEE Academic Edition 1.4 is a live multi-criteria decision aid software program. It runs virtual analysis through GAIA and applies selected criteria to contrast parameters. It also saves them for possible export and editing. In this article, the PROMETHEE II method is applied for Fermatean fuzzy numbers with

F F (α, β)

-level for selecting the optimal green supplier for a construction company. Because of its ability to handle vagueness, the FF PROMETHEE II method emerges as a valuable tool in Multi-criteria decision making. Furthermore, this study assesses the efficacy of the proposed technique by comparing its results with those obtained through other established methods. Full article

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

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10 pages, 242 KiB

Open AccessReview

A Survey on the Oscillation of First-Order Retarded Differential Equations

by Ioannis P. Stavroulakis

Axioms 2024, 13(6), 407; https://doi.org/10.3390/axioms13060407 - 17 Jun 2024

Viewed by 252

Abstract

In this paper, a survey of the most interesting conditions for the oscillation of all solutions to first-order linear differential equations with a retarded argument is presented in chronological order, especially in the case when well-known oscillation conditions are not satisfied. The essential [...] Read more.

In this paper, a survey of the most interesting conditions for the oscillation of all solutions to first-order linear differential equations with a retarded argument is presented in chronological order, especially in the case when well-known oscillation conditions are not satisfied. The essential improvement and the importance of these oscillation conditions is also indicated. Full article

(This article belongs to the Special Issue Axioms and Methods for Handling Differential Equations and Inverse Problems)

16 pages, 312 KiB

Open AccessArticle

On Using Relative Information to Estimate Traits in a Darwinian Evolution Population Dynamics

by Eddy Kwessi

Axioms 2024, 13(6), 406; https://doi.org/10.3390/axioms13060406 - 16 Jun 2024

Viewed by 297

Abstract

Since its inception, evolution theory has garnered much attention from the scientific community for a good reason: it theorizes how various living organisms came to be and what changes are to be expected in a certain environment. While many models of evolution have [...] Read more.

Since its inception, evolution theory has garnered much attention from the scientific community for a good reason: it theorizes how various living organisms came to be and what changes are to be expected in a certain environment. While many models of evolution have been proposed to track changes in species’ traits, not much has been said about how to calculate or estimate these traits. In this paper, using information theory, we propose an estimation method for trait parameters in a Darwinian evolution model for species with one or multiple traits. We propose estimating parameters by minimizing the relative information in a Darwinian evolution population model using either a classical gradient ascent or a stochastic gradient ascent. The proposed procedure is shown to be possible in a supervised or unsupervised learning environment, similarly to what occurs with Boltzmann machines. Simulations are provided to illustrate the method. Full article

(This article belongs to the Special Issue Infinite Dynamical System and Differential Equations)

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14 pages, 293 KiB

Open AccessArticle

Some Results of Stochastic Differential Equations

by Shuai Guo, Wei Li and Guangying Lv

Axioms 2024, 13(6), 405; https://doi.org/10.3390/axioms13060405 - 16 Jun 2024

Viewed by 270

Abstract

In this paper, there are two aims: one is Schauder and Sobolev estimates for the one-dimensional heat equation; the other is the stabilization of differential equations by stochastic feedback control based on discrete-time state observations. The nonhomogeneous Poisson stochastic process is used to [...] Read more.

In this paper, there are two aims: one is Schauder and Sobolev estimates for the one-dimensional heat equation; the other is the stabilization of differential equations by stochastic feedback control based on discrete-time state observations. The nonhomogeneous Poisson stochastic process is used to show how knowing Schauder and Sobolev estimates for the one-dimensional heat equation allows one to derive their multidimensional analogs. The properties of a jump process is used. The stabilization of differential equations by stochastic feedback control is based on discrete-time state observations. Firstly, the stability results of the auxiliary system is established. Secondly, by comparing it with the auxiliary system and using the continuity method, the stabilization of the original system is obtained. Both parts focus on the impact of probability theory. Full article

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

30 pages, 399 KiB

Open AccessArticle

A New Class of Coordinated Non-Convex Fuzzy-Number-Valued Map**s with Related Inequalities and Their Applications

by Aleksandr Rakhmangulov, A. F. Aljohani, Ali Mubaraki and Saad Althobaiti

Axioms 2024, 13(6), 404; https://doi.org/10.3390/axioms13060404 - 16 Jun 2024

Viewed by 364

Abstract

Both theoretical and applied mathematics depend heavily on integral inequalities with generalized convexity. Because of its many applications, the theory of integral inequalities is currently one of the areas of mathematics that is evolving at the fastest pace. In this paper, based on [...] Read more.

Both theoretical and applied mathematics depend heavily on integral inequalities with generalized convexity. Because of its many applications, the theory of integral inequalities is currently one of the areas of mathematics that is evolving at the fastest pace. In this paper, based on fuzzy Aumann’s integral theory, the Hermite–Hadamard’s type inequalities are introduced for a newly defined class of nonconvex functions, which is known as

U

·

D

preinvex fuzzy number-valued map**s (

U

·

D

preinvex

F

·

N

·

V

·

M

s) on coordinates. Some Pachpatte-type inequalities are also established for the product of two

U

·

D

preinvex

F

·

N

·

V

·

M

s, and some Hermite–Hadamard–Fejér-type inequalities are also acquired via fuzzy Aumann’s integrals. Additionally, several new generalized inequalities are also obtained for the special situations of the parameters. Additionally, some of the interesting remarks are provided to acquire the classical and new exceptional cases that can be considered as applications of the main outcomes. Lastly, a few suggested uses for these inequalities in numerical integration are made. Full article

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

16 pages, 284 KiB

Open AccessArticle

On the Generalized Stabilities of Functional Equations via Isometries

by Muhammad Sarfraz, Jiang Zhou, Yong** Li and John Michael Rassias

Axioms 2024, 13(6), 403; https://doi.org/10.3390/axioms13060403 - 14 Jun 2024

Viewed by 259

Abstract

The main goal of this research article is to investigate the stability of generalized norm-additive functional equations. This study demonstrates that these equations are Hyers-Ulam stable for surjective functions from an arbitrary group G to a real Banach space B using the large [...] Read more.

The main goal of this research article is to investigate the stability of generalized norm-additive functional equations. This study demonstrates that these equations are Hyers-Ulam stable for surjective functions from an arbitrary group G to a real Banach space B using the large perturbation method. Furthermore, hyperstability results are investigated for a generalized Cauchy equation. Full article

(This article belongs to the Special Issue New Theory and Applications of Nonlinear Analysis, Fractional Calculus and Optimization, 2nd Edition)

18 pages, 485 KiB

Open AccessArticle

Weighted Convolution for Quaternion Linear Canonical Cosine Transform and Its Application

by Rongbo Wang and Qiang Feng

Axioms 2024, 13(6), 402; https://doi.org/10.3390/axioms13060402 - 14 Jun 2024

Viewed by 264

Abstract

Convolution plays a pivotal role in the domains of signal processing and optics. This paper primarily focuses on studying the weighted convolution for quaternion linear canonical cosine transform (QLCcT) and its application in multiplicative filter analysis. Firstly, we propose QLCcT by combining quaternion [...] Read more.

Convolution plays a pivotal role in the domains of signal processing and optics. This paper primarily focuses on studying the weighted convolution for quaternion linear canonical cosine transform (QLCcT) and its application in multiplicative filter analysis. Firstly, we propose QLCcT by combining quaternion algebra with linear canonical cosine transform (LCcT), which extends LCcT to Hamiltonian quaternion algebra. Secondly, we introduce weighted convolution and correlation operations for QLCcT, accompanied by their corresponding theorems. We also explore the properties of QLCcT. Thirdly, we utilize these proposed convolution structures to analyze multiplicative filter models that offer lower computational complexity compared to existing methods based on quaternion linear canonical transform (QLCT). Additionally, we discuss the rationale behind studying such transforms using quaternion functions as an illustrative example. Full article

(This article belongs to the Special Issue Fractional Calculus and Its Applications: Historical and Recent Developments)

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27 pages, 4691 KiB

Open AccessArticle

Enhanced Real-Life Data Modeling with the Modified Burr III Odds Ratio–G Distribution

by Haochong Yang, Mingfang Huang, **nyu Chen, Ziyan He and Shusen Pu

Axioms 2024, 13(6), 401; https://doi.org/10.3390/axioms13060401 - 14 Jun 2024

Viewed by 311

Abstract

In this study, we introduce the modified Burr III Odds Ratio–G distribution, a novel statistical model that integrates the odds ratio concept with the foundational Burr III distribution. The spotlight of our investigation is cast on a key subclass within this innovative framework, [...] Read more.

In this study, we introduce the modified Burr III Odds Ratio–G distribution, a novel statistical model that integrates the odds ratio concept with the foundational Burr III distribution. The spotlight of our investigation is cast on a key subclass within this innovative framework, designated as the Burr III Scaled Inverse Odds Ratio–G (B-SIOR-G) distribution. By effectively integrating the odds ratio with the Burr III distribution, this model enhances both flexibility and predictive accuracy. We delve into a thorough exploration of this distribution family’s mathematical and statistical properties, spanning hazard rate functions, quantile functions, moments, and additional features. Through rigorous simulation, we affirm the robustness of the B-SIOR-G model. The flexibility and practicality of the B-SIOR-G model are demonstrated through its application to four datasets, highlighting its enhanced efficacy over several well-established distributions. Full article

(This article belongs to the Special Issue Research on Stochastic Analysis and Applied Statistics)

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21 pages, 768 KiB

Open AccessArticle

A Conservative Difference Scheme for Solving the Coupled Fractional Schrödinger–Boussinesq System

by Yao Shi, Rian Yan and Tao Liu

Axioms 2024, 13(6), 400; https://doi.org/10.3390/axioms13060400 - 14 Jun 2024

Viewed by 236

Abstract

In this paper, a high-accuracy conservative implicit algorithm for computing the space fractional coupled Schrödinger–Boussinesq system is constructed. Meanwhile, the conservative nature, a priori boundedness, and solvability of the numerical solution are presented. Then, the proposed algorithm is proved to be second-order convergence [...] Read more.

In this paper, a high-accuracy conservative implicit algorithm for computing the space fractional coupled Schrödinger–Boussinesq system is constructed. Meanwhile, the conservative nature, a priori boundedness, and solvability of the numerical solution are presented. Then, the proposed algorithm is proved to be second-order convergence in temporal and fourth-order spatial convergence using the discrete energy method. Finally, some numerical experiments validate the effectiveness of the conservative algorithm and confirm the accuracy of the theoretical results for different choices of the fractional-order

α

. Full article

(This article belongs to the Section Mathematical Analysis)

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36 pages, 446 KiB

Open AccessFeature PaperArticle

Using Lie Sphere Geometry to Study Dupin Hypersurfaces in Rⁿ

by Thomas E. Cecil

Axioms 2024, 13(6), 399; https://doi.org/10.3390/axioms13060399 - 14 Jun 2024

Viewed by 433

Abstract

A hypersurface M in

R^{n}

or

S^{n}

is said to be Dupin if along each curvature surface, the corresponding principal curvature is constant. A Dupin hypersurface is said to be proper Dupin if each principal curvature has constant multiplicity on M [...] Read more.

A hypersurface M in

R^{n}

or

S^{n}

is said to be Dupin if along each curvature surface, the corresponding principal curvature is constant. A Dupin hypersurface is said to be proper Dupin if each principal curvature has constant multiplicity on M, i.e., the number of distinct principal curvatures is constant on M. The notions of Dupin and proper Dupin hypersurfaces in

R^{n}

or

S^{n}

can be generalized to the setting of Lie sphere geometry, and these properties are easily seen to be invariant under Lie sphere transformations. This makes Lie sphere geometry an effective setting for the study of Dupin hypersurfaces, and many classifications of proper Dupin hypersurfaces have been obtained up to Lie sphere transformations. In these notes, we give a detailed introduction to this method for studying Dupin hypersurfaces in

R^{n}

or

S^{n}

, including proofs of several fundamental results. We also give a survey of the results in the field that have been obtained using this approach. Full article

(This article belongs to the Special Issue Advances in Differential Geometry and Singularity Theory)

13 pages, 428 KiB

Open AccessArticle

Homotopy Analysis Transform Method for a Singular Nonlinear Second-Order Hyperbolic Pseudo-Differential Equation

by Said Mesloub and Hassan Eltayeb Gadain

Axioms 2024, 13(6), 398; https://doi.org/10.3390/axioms13060398 - 14 Jun 2024

Viewed by 280

Abstract

In this study, we employed the homotopy analysis transform method (HATM) to derive an iterative scheme to numerically solve a singular second-order hyperbolic pseudo-differential equation. We evaluated the effectiveness of the derived scheme in solving both linear and nonlinear equations of similar nature [...] Read more.

In this study, we employed the homotopy analysis transform method (HATM) to derive an iterative scheme to numerically solve a singular second-order hyperbolic pseudo-differential equation. We evaluated the effectiveness of the derived scheme in solving both linear and nonlinear equations of similar nature through a series of illustrative examples. The stability of this scheme in handling the approximate solutions of these examples was studied graphically and numerically. A comparative analysis with existing methodologies from the literature was conducted to assess the performance of the proposed approach. Our findings demonstrate that the HATM-based method offers notable efficiency, accuracy, and ease of implementation when compared to the alternative technique considered in this study. Full article

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

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21 pages, 2294 KiB

Open AccessArticle

Statistical Advancement of a Flexible Unitary Distribution and Its Applications

by Hugo S. Salinas, Hassan S. Bakouch, Fatimah E. Almuhayfith, Wilson E. Caimanque, Leonardo Barrios-Blanco and Olayan Albalawi

Axioms 2024, 13(6), 397; https://doi.org/10.3390/axioms13060397 - 14 Jun 2024

Viewed by 415

Abstract

A flexible distribution has been introduced to handle random variables in the unit interval. This distribution is based on an exponential transformation of the truncated positive normal distribution with two parameters and can effectively fit data with varying degrees of skewness and kurtosis. [...] Read more.

A flexible distribution has been introduced to handle random variables in the unit interval. This distribution is based on an exponential transformation of the truncated positive normal distribution with two parameters and can effectively fit data with varying degrees of skewness and kurtosis. Therefore, it presents an alternative for modeling this type of data. Several mathematical and statistical properties of this distribution have been derived, such as moments, hazard function, the Bonferroni curve, and entropy. Moreover, we investigate the characterizations of the proposed distribution based on its hazard function. Parameter estimation has been performed using both the maximum likelihood method and method of the moments. Because of this, we were able to determine the best critical region and the information matrix, facilitating the calculation of asymptotic confidence intervals. A simulation study is presented to analyze the behavior of the obtained estimators for different sample sizes. To demonstrate the suitability of the proposed distribution, applications and goodness-of-fit tests have been performed on two practical data sets. Full article

(This article belongs to the Special Issue Stochastic and Statistical Analysis in Natural Sciences)

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15 pages, 3160 KiB

Open AccessArticle

Packing Spheres into a Minimum-Height Parabolic Container

by Yuriy Stoyan, Georgiy Yaskov, Tetyana Romanova, Igor Litvinchev, José Manuel Velarde Cantú and Mauricio López Acosta

Axioms 2024, 13(6), 396; https://doi.org/10.3390/axioms13060396 - 13 Jun 2024

Viewed by 266

Abstract

Sphere packing consists of placing several spheres in a container without mutual overlap**. While packing into regular-shape containers is well explored, less attention is focused on containers with nonlinear boundaries, such as ellipsoids or paraboloids. Packing n-dimensional spheres into a minimum-height container [...] Read more.

Sphere packing consists of placing several spheres in a container without mutual overlap**. While packing into regular-shape containers is well explored, less attention is focused on containers with nonlinear boundaries, such as ellipsoids or paraboloids. Packing n-dimensional spheres into a minimum-height container bounded by a parabolic surface is formulated. The minimum allowable distances between spheres as well as between spheres and the container boundary are considered. A normalized Φ-function is used for analytical description of the containment constraints. A nonlinear programming model for the packing problem is provided. A solution algorithm based on the feasible directions approach and a decomposition technique is proposed. The computational results for problem instances with various space dimensions, different numbers of spheres and their radii, the minimal allowable distances and the parameters of the parabolic container are presented to demonstrate the efficiency of the proposed approach. Full article

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

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15 pages, 319 KiB

Open AccessArticle

Recent Advances in Proximity Point Theory Applied to Fractional Differential Equations

by Nabil Mlaiki, Dur-e-Shehwar Sagheer, Sana Noreen, Samina Batul and Ahmad Aloqaily

Axioms 2024, 13(6), 395; https://doi.org/10.3390/axioms13060395 - 13 Jun 2024

Viewed by 302

Abstract

This article introduces the concept of generalized

(ﬀ F, b, \overset{˘}{ϕ})

contraction in the context of

b

-metric spaces by utilizing the idea of

F

contraction introduced by Dariusz Wardowski. The main findings of the research focus on [...] Read more.

This article introduces the concept of generalized

(ﬀ F, b, \overset{˘}{ϕ})

contraction in the context of

b

-metric spaces by utilizing the idea of

F

contraction introduced by Dariusz Wardowski. The main findings of the research focus on the existence of best proximity points for multi-valued

(ﬀ F, b, \overset{˘}{ϕ})

contractions in partially ordered

b

-metric spaces. The article provides examples to illustrate the main results and demonstrates the existence of solutions to a second-order differential equation and a fractional differential equation using the established theorems. Additionally, several corollaries are presented to show that the results generalize many existing fixed-point and best proximity point theorems. Full article

(This article belongs to the Special Issue Research on Fixed Point Theory and Application)

18 pages, 406 KiB

Open AccessArticle

Inference of Constant-Stress Model of Fréchet Distribution under a Maximum Ranked Set Sampling with Unequal Samples

by Jia Liu, Liang Wang, Yogesh Mani Tripathi and Yuhlong Lio

Axioms 2024, 13(6), 394; https://doi.org/10.3390/axioms13060394 - 12 Jun 2024

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Abstract

This paper explores the inference for a constant-stress accelerated life test under a ranked set sampling scenario. When the lifetime of products follows the Fréchet distribution, and the failure times are collected under a maximum ranked set sampling with unequal samples, classical and [...] Read more.

This paper explores the inference for a constant-stress accelerated life test under a ranked set sampling scenario. When the lifetime of products follows the Fréchet distribution, and the failure times are collected under a maximum ranked set sampling with unequal samples, classical and Bayesian approaches are proposed, respectively. Maximum likelihood estimators along with the existence and uniqueness of model parameters are established, and the corresponding asymptotic confidence intervals are constructed based on asymptotic theory. Under squared error loss, Bayesian estimation and highest posterior density confidence intervals are provided, and an associated Monte-Carlo sampling algorithm is proposed for complex posterior computation. Finally, extensive simulation studies are conducted to demonstrate the performance of different methods, and a real-data example is also presented for applications. Full article

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17 pages, 383 KiB

Open AccessArticle

The Measurement Errors and Their Effects on the Cumulative Sum Schemes for Monitoring the Ratio of Two Correlated Normal Variables

by Wei Yang, Xueting Ji and Jiujun Zhang

Axioms 2024, 13(6), 393; https://doi.org/10.3390/axioms13060393 - 12 Jun 2024

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Abstract

Monitoring the ratio of two correlated normal random variables is often used in many industrial manufacturing processes. At the same time, measurement errors inevitably exist in most processes, which have different effects on the performance of various charting schemes. This paper comprehensively analyses [...] Read more.

Monitoring the ratio of two correlated normal random variables is often used in many industrial manufacturing processes. At the same time, measurement errors inevitably exist in most processes, which have different effects on the performance of various charting schemes. This paper comprehensively analyses the impacts of measurement errors on the detection ability of the cumulative sum (CUSUM) charting schemes for the ratio of two correlated normal variables. A thorough numerical assessment is performed using the Monte Carlo simulation, and the results indicate that the measurement errors negatively impact the performance of the CUSUM scheme for the ratio of two correlated normal variables. Increasing the number of measurements per set is not a lucrative approach for minimizing the negative impact of measurement errors on the performance of the CUSUM charting scheme when monitoring the ratio of two correlated normal variables. We consider a food formulation as an example that illustrates the quality control problems involving the ratio of two correlated normal variables in an industry with a measurement error. The results are presented, along with some suggestions for further study. Full article

(This article belongs to the Special Issue Stochastic and Statistical Analysis in Natural Sciences)

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16 pages, 292 KiB

Open AccessArticle

Weighted-Likelihood-Ratio-Based EWMA Schemes for Monitoring Geometric Distributions

by Yizhen Zhang, Hongxing Cai and Jiujun Zhang

Axioms 2024, 13(6), 392; https://doi.org/10.3390/axioms13060392 - 12 Jun 2024

Viewed by 265

Abstract

Monitoring the parameter of discrete distributions is common in industrial production. Also, it is often crucial to monitor the parameter of geometric distribution, which is often regarded as the nonconforming item rate. To enhance the detection of nonconforming item, we designed an exponentially [...] Read more.

Monitoring the parameter of discrete distributions is common in industrial production. Also, it is often crucial to monitor the parameter of geometric distribution, which is often regarded as the nonconforming item rate. To enhance the detection of nonconforming item, we designed an exponentially weighted moving average (EWMA) scheme based on the weighted likelihood ratio test (WLRT) method, and this scheme is denoted as the EWLRT scheme, specifically designed for monitoring the increase of the parameter in geometric distribution. Moreover, the optimal statistical design of the EWLRT scheme is presented when the shift is known. Results from numerical comparisons through Monte Carlo simulations indicates that the EWLRT scheme performs better than the competing schemes in some scenarios. Additionally, the designed scheme is characterized by its simplicity and ease of use, making it ideally suited for scenarios involving single observation. An example is illustrated to demonstrate the effectiveness of the EWLRT scheme. Full article

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22 pages, 533 KiB

Open AccessArticle

Fixed Time Synchronization of Stochastic Takagi–Sugeno Fuzzy Recurrent Neural Networks with Distributed Delay under Feedback and Adaptive Controls

by Yiran Niu, **aofeng Xu and Ming Liu

Axioms 2024, 13(6), 391; https://doi.org/10.3390/axioms13060391 - 11 Jun 2024

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Abstract

In this paper, the stochastic Takagi–Sugeno fuzzy recurrent neural networks (STSFRNNS) with distributed delay is established based on the Takagi–Sugeno (TS) model and the fixed time synchronization problem is investigated. In order to synchronize the networks, we design two kinds of controllers: a [...] Read more.

In this paper, the stochastic Takagi–Sugeno fuzzy recurrent neural networks (STSFRNNS) with distributed delay is established based on the Takagi–Sugeno (TS) model and the fixed time synchronization problem is investigated. In order to synchronize the networks, we design two kinds of controllers: a feedback controller and an adaptive controller. Then, we obtain the synchronization criteria in a fixed time by combining the Lyapunov method and the related inequality theory of the stochastic differential equation and calculate the stabilization time for the STSFRNNS. In addition, to verify the authenticity of the theoretical results, we use MATLABR2023A to carry out numerical simulation. Full article

(This article belongs to the Special Issue Recent Advances in Applied Mathematics and Artificial Intelligence)

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15 pages, 11097 KiB

Open AccessArticle

Bicubic Splines for Fast-Contracting Control Nets

by Kȩstutis Karčiauskas, Kyle Shih-Huang Lo, Erkan Gunpinar and Jörg Peters

Axioms 2024, 13(6), 390; https://doi.org/10.3390/axioms13060390 - 9 Jun 2024

Viewed by 351

Abstract

Merging parallel quad strips facilitates narrowing surface passages, and allows a design to transition to a simpler shape. While a number of spline surface constructions exist for the isotropic case where n pieces join, few existing spline constructions deliver a good shape for [...] Read more.

Merging parallel quad strips facilitates narrowing surface passages, and allows a design to transition to a simpler shape. While a number of spline surface constructions exist for the isotropic case where n pieces join, few existing spline constructions deliver a good shape for control nets that merge parameter lines. Additionally, untilrecently, none provided a good shape for fast-contracting polyhedral control nets. This work improves the state-of-the-art of piecewise polynomial spline surfaces accommodating fast-contracting control nets. The new fast-contracting (FC) surface algorithm yields the industry-preferred uniform degree bi-3 (bi-cubic). The surfaces are by default differentiable, have an improved shape, measured empirically as to highlight the line distribution, and require fewer pieces compared to existing methods. Full article

(This article belongs to the Special Issue Innovations in Geometric Modelling and CAD: Exploring Boundaries and Beyond)

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17 pages, 774 KiB

Open AccessArticle

Precise Obstacle Avoidance Movement for Three-Wheeled Mobile Robots: A Modified Curvature Tracking Method

by **angrong Wen and Yusheng Zhou

Axioms 2024, 13(6), 389; https://doi.org/10.3390/axioms13060389 - 8 Jun 2024

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Abstract

This paper proposes a precise motion control strategy for a three-wheeled mobile robot with two driven rear wheels and one steered front wheel so that an obstacle avoidance motion task is able to be well implemented. Initially, the motion laws under nonholonomic constraints [...] Read more.

This paper proposes a precise motion control strategy for a three-wheeled mobile robot with two driven rear wheels and one steered front wheel so that an obstacle avoidance motion task is able to be well implemented. Initially, the motion laws under nonholonomic constraints are expounded for the three-wheeled mobile robot in order to facilitate the derivation of its dynamic model. Subsequently, a prescribed target curve is converted into a speed target through the nonholonomic constraint of zero lateral speed. A modified dynamical tracking target that is aligned with the dynamic model is then developed based on the relative curvature of the prescribed curve. By applying this dynamical tracking target, path tracking precision is enhanced through appropriate selection of a yaw motion speed target, thus preventing speed errors from accumulating during relative curvature tracking. On this basis, integral sliding mode control and feedback linearization methods are adopted for designing robust controllers, enabling the accurate movement of the three-wheeled mobile robot along a given path. A theoretical analysis and simulation results corroborate the effectiveness of the proposed trajectory tracking control strategy in preventing off-target deviations, even with significant speed errors. Full article

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

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11 pages, 264 KiB

Open AccessArticle

On Modulus Statistical Convergence in Partial Metric Spaces

by Francisco Javier García-Pacheco and Ramazan Kama

Axioms 2024, 13(6), 388; https://doi.org/10.3390/axioms13060388 - 8 Jun 2024

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Abstract

Modulus statistical convergence has been studied in very different general settings such as topological spaces and uniform spaces. In this manuscript, modulus statistical convergence is defined and studied in partial metric spaces. Full article

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

18 pages, 543 KiB

Open AccessArticle

AHD-SLE: Anomalous Hyperedge Detection on Hypergraph Symmetric Line Expansion

by Yingle Li, Hongtao Yu, Haitao Li, Fei Pan and Shuxin Liu

Axioms 2024, 13(6), 387; https://doi.org/10.3390/axioms13060387 - 7 Jun 2024

Viewed by 359

Abstract

Graph anomaly detection aims to identify unusual patterns or structures in graph-structured data. Most existing research focuses on anomalous nodes in ordinary graphs with pairwise relationships. However, complex real-world systems often involve relationships that go beyond pairwise relationships, and insufficient attention is paid [...] Read more.

Graph anomaly detection aims to identify unusual patterns or structures in graph-structured data. Most existing research focuses on anomalous nodes in ordinary graphs with pairwise relationships. However, complex real-world systems often involve relationships that go beyond pairwise relationships, and insufficient attention is paid to hypergraph anomaly detection, especially anomalous hyperedge detection. Some existing methods for researching hypergraphs involve transforming hypergraphs into ordinary graphs for learning, which can result in poor detection performance due to the loss of high-order information. We propose a new method for Anomalous Hyperedge Detection on Symmetric Line Expansion (AHD-SLE). The SLE of a hypergraph is an ordinary graph with pairwise relationships and can be backmapped to the hypergraph, so the SLE is able to preserve the higher-order information of the hypergraph. The AHD-SLE first maps the hypergraph to the SLE; then, the information is aggregated by Graph Convolutional Networks (GCNs) in the SLE. After that, the hyperedge embedding representation is obtained through a backmap** operation. Finally, an anomaly function is designed to detect anomalous hyperedges using the hyperedge embedding representation. Experiments on five different types of real hypergraph datasets show that AHD-SLE outperforms the baseline algorithm in terms of Area Under the receiver operating characteristic Curve(AUC) and Recall metrics. Full article

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

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Axioms, Volume 13, Issue 6 (June 2024) – 78 articles

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