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# Axioms, Volume 12, Issue 5 (May 2023) – 98 articles

Cover Story (view full-size image): Studying mathematical inequalities is popular, and many mathematicians are working on improvements, extensions and generalizations of classical inequalities, such as Hardy’s, Steffensen’s, Opial’s, Boas’ ones, etc. We use Hardy’s inequality in a general setting over measured spaces with positive σ-finite measures and a non-negative kernel. We combined Hardy’s inequality with Montgomery’s identity and four new Green functions, which are continuous and convex functions with respect to the first variable. We present n-convex functions results, which are a more general class of functions that contains convex functions as a special case. We also derived related Grüss and Ostrowski-type inequalities. View this paper
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11 pages, 680 KiB
Article
On 3-Restricted Edge Connectivity of Replacement Product Graphs
by Yilan Cui, Jianping Ou and Saihua Liu
Axioms 2023, 12(5), 504; https://doi.org/10.3390/axioms12050504 - 22 May 2023
Cited by 1 | Viewed by 1059
Abstract
A 3-restricted edge cut is an edge cut of a connected graph that separates this graph into components, each having an order of at least 3. The minimum size of all 3-restricted edge cuts of a graph is called its 3-restricted edge connectivity. [...] Read more.
A 3-restricted edge cut is an edge cut of a connected graph that separates this graph into components, each having an order of at least 3. The minimum size of all 3-restricted edge cuts of a graph is called its 3-restricted edge connectivity. This work determines the upper and lower bounds on the 3-restricted edge connectivity of replacement product graphs and presents sufficient conditions for replacement product graphs to be maximally 3-restricted edge connected. As a result, the 3-restricted edge connectivity of replacement product graphs of some special graphs are determined. Full article
(This article belongs to the Special Issue Advances in Graph Theory and Combinatorial Optimization)
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12 pages, 1807 KiB
Article
Consistency of the Estimator for the Common Mean in Fixed-Effect Meta-Analyses
by Nanami Taketomi and Takeshi Emura
Axioms 2023, 12(5), 503; https://doi.org/10.3390/axioms12050503 - 21 May 2023
Cited by 1 | Viewed by 1416
Abstract
Fixed-effect meta-analyses aim to estimate the common mean parameter by the best linear unbiased estimator. Besides unbiasedness, consistency is one of the most fundamental requirements for the common mean estimator to be valid. However, conditions for the consistency of the common mean estimator [...] Read more.
Fixed-effect meta-analyses aim to estimate the common mean parameter by the best linear unbiased estimator. Besides unbiasedness, consistency is one of the most fundamental requirements for the common mean estimator to be valid. However, conditions for the consistency of the common mean estimator have not been discussed in the literature. This article fills this gap by clarifying conditions for making the common mean estimator consistent in fixed-effect meta-analyses. In this article, five theorems are devised, which state regularity conditions for the common mean estimator to be consistent. These theorems are novel applications of the classical large sample theory to meta-analyses. Numerical illustrations are also given to help understand the needs of the regularity conditions. Three real datasets illustrate the practical consequences of the devised theorems. This article concludes that the inconsistency of the common mean estimator occurs under some conditions in real meta-analyses. Full article
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26 pages, 504 KiB
Article
Inverse Unit Teissier Distribution: Theory and Practical Examples
by Najwan Alsadat, Mohammed Elgarhy, Kadir Karakaya, Ahmed M. Gemeay, Christophe Chesneau and M. M. Abd El-Raouf
Axioms 2023, 12(5), 502; https://doi.org/10.3390/axioms12050502 - 20 May 2023
Cited by 5 | Viewed by 1372
Abstract
In this paper, we emphasize a new one-parameter distribution with support as $\left[1,+\infty \right)$. It is constructed from the inverse method applied to an understudied one-parameter unit distribution, the unit Teissier distribution. Some properties are investigated, such as [...] Read more.
In this paper, we emphasize a new one-parameter distribution with support as $\left[1,+\infty \right)$. It is constructed from the inverse method applied to an understudied one-parameter unit distribution, the unit Teissier distribution. Some properties are investigated, such as the mode, quantiles, stochastic dominance, heavy-tailed nature, moments, etc. Among the strengths of the distribution are the following: (i) the closed-form expressions and flexibility of the main functions, and in particular, the probability density function is unimodal and the hazard rate function is increasing or unimodal; (ii) the manageability of the moments; and, more importantly, (iii) it provides a real alternative to the famous Pareto distribution, also with support as $\left[1,+\infty \right)$. Indeed, the proposed distribution has different functionalities but also benefits from the heavy-right-tailed nature, which is demanded in many applied fields (finance, the actuarial field, quality control, medicine, etc.). Furthermore, it can be used quite efficiently in a statistical setting. To support this claim, the maximum likelihood, Anderson–Darling, right-tailed Anderson–Darling, left-tailed Anderson–Darling, Cramér–Von Mises, least squares, weighted least-squares, maximum product of spacing, minimum spacing absolute distance, and minimum spacing absolute-log distance estimation methods are examined to estimate the unknown unique parameter. A Monte Carlo simulation is used to compare the performance of the obtained estimates. Additionally, the Bayesian estimation method using an informative gamma prior distribution under the squared error loss function is discussed. Data on the COVID mortality rate and the timing of pain relief after receiving an analgesic are considered to illustrate the applicability of the proposed distribution. Favorable results are highlighted, supporting the importance of the findings. Full article
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17 pages, 1423 KiB
Article
Dynamics in an n-Species Lotka–Volterra Cooperative System with Delays
by Zhao Jiang, Azhar Halik and Ahmadjan Muhammadhaji
Axioms 2023, 12(5), 501; https://doi.org/10.3390/axioms12050501 - 19 May 2023
Cited by 1 | Viewed by 1166
Abstract
We studied a class of generalized n-species non-autonomous cooperative Lotka–Volterra (L-V) systems with time delays. We obtained new criteria on the dynamic properties of the systems. First, we obtained the boundedness and permanence of the system using the inequality analysis technique and comparison [...] Read more.
We studied a class of generalized n-species non-autonomous cooperative Lotka–Volterra (L-V) systems with time delays. We obtained new criteria on the dynamic properties of the systems. First, we obtained the boundedness and permanence of the system using the inequality analysis technique and comparison method. Then, the existence of positive periodic solutions was investigated using the coincidence degree theory. The global attractivity of the system was obtained by constructing suitable Lyapunov functionals and utilizing Barbalat’s lemma. The existence and global attractivity of the periodic solutions were also obtained. Finally, we conducted two numerical simulations to validate the feasibility and practicability of our results. Full article
(This article belongs to the Special Issue Impulsive, Delay and Fractional Order Systems)
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22 pages, 2533 KiB
Article
An Analytic Solution for the Dynamic Behavior of a Cantilever Beam with a Time-Dependent Spring-like Actuator
by Jer-Rong Chang, Te-Wen Tu and Chun-Jung Huang
Axioms 2023, 12(5), 500; https://doi.org/10.3390/axioms12050500 - 19 May 2023
Viewed by 1367
Abstract
The purpose of this study is to derive an analytical solution for a cantilever beam with a novel spring-like actuator that behaves like a time-dependent spring and to study the dynamic behavior of the system. A time-dependent spring was set at the free [...] Read more.
The purpose of this study is to derive an analytical solution for a cantilever beam with a novel spring-like actuator that behaves like a time-dependent spring and to study the dynamic behavior of the system. A time-dependent spring was set at the free end of the cantilever beam to model the novel spring-like actuator. First, the boundary conditions were transformed from being nonhomogeneous to being homogeneous using the shifting function method. The solution of the analytic series was then obtained by using the expansion theorem method. The correctness of the proposed analytical solution was verified by comparing the results with those obtained via the separation of variables in the special extreme case of a constant spring coefficient. We took the free end of a cantilever beam with harmonic spring stiffness and an external periodic unit load as an example. The influence of the actuator parameters, such as the effect of the magnitude and the frequency of the time-dependent spring stiffness on the resonance frequency, was investigated. An important new result was found, i.e., that the resonance frequency is clearly dependent on the magnitude and the frequency of the spring-like actuator in the first two modes, but not in the third and fourth modes. In practical engineering applications, system resonance can be avoided by adjusting the magnitude and frequency of the actuator. Full article
(This article belongs to the Special Issue Computational and Mathematical Methods in Science and Engineering)
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16 pages, 294 KiB
Article
A New Hardy–Hilbert-Type Integral Inequality Involving One Multiple Upper Limit Function and One Derivative Function of Higher Order
by Bicheng Yang and Michael Th. Rassias
Axioms 2023, 12(5), 499; https://doi.org/10.3390/axioms12050499 - 19 May 2023
Viewed by 805
Abstract
Using weight functions and parameters, as well as applying real analytic techniques, we derive a new Hardy–Hilbert-type integral inequality with the homogeneous kernel $\frac{1}{{\left(x+y\right)}^{\lambda +n}}$ involving one multiple upper limit function and one derivative function of [...] Read more.
Using weight functions and parameters, as well as applying real analytic techniques, we derive a new Hardy–Hilbert-type integral inequality with the homogeneous kernel $\frac{1}{{\left(x+y\right)}^{\lambda +n}}$ involving one multiple upper limit function and one derivative function of higher order. Certain equivalent statements of the optimal constant factor related to some parameters are considered. A few particular inequalities and the case of reverses are also provided. Full article
(This article belongs to the Special Issue Fractional Calculus - Theory and Applications II)
17 pages, 461 KiB
Article
Mathematical Modeling and Exact Optimizing of University Course Scheduling Considering Preferences of Professors
by Mo Chen, Frank Werner and Mohammad Shokouhifar
Axioms 2023, 12(5), 498; https://doi.org/10.3390/axioms12050498 - 19 May 2023
Cited by 1 | Viewed by 2167
Abstract
University course scheduling (UCS) is one of the most important and time-consuming issues that all educational institutions face yearly. Most of the existing techniques to model and solve UCS problems have applied approximate methods, which differ in terms of efficiency, performance, and optimization [...] Read more.
University course scheduling (UCS) is one of the most important and time-consuming issues that all educational institutions face yearly. Most of the existing techniques to model and solve UCS problems have applied approximate methods, which differ in terms of efficiency, performance, and optimization speed. Accordingly, this research aims to apply an exact optimization method to provide an optimal solution to the course scheduling problem. In other words, in this research, an integer programming model is presented to solve the USC problem. In this model, the constraints include the facilities of classrooms, courses of different levels and compression of students’ curriculum, courses outside the faculty and planning for them, and the limited time allocated to the professors. The objective is to maximize the weighted sum of allocating available times to professors based on their preferences in all periods. To evaluate the presented model’s feasibility, it is implemented using the GAMS software. Finally, the presented model is solved in a larger dimension using a real data set from a college in China and compared with the current program in the same college. The obtained results show that considering the mathematical model’s constraints and objective function, the faculty courses’ timetable is reduced from 4 days a week to 3 working days. Moreover, master courses are planned in two days, and the courses in the educational groups do not interfere with each other. Furthermore, by implementing the proposed model for the real case study, the maximum teaching hours of the professors are significantly reduced. The results demonstrate the efficiency of the proposed model and solution method in terms of optimization speed and solution accuracy. Full article
(This article belongs to the Special Issue Optimization Algorithms and Applications)
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13 pages, 542 KiB
Article
An Analysis of the One-Phase Stefan Problem with Variable Thermal Coefficients of Order p
by Lazhar Bougoffa, Smail Bougouffa and Ammar Khanfer
Axioms 2023, 12(5), 497; https://doi.org/10.3390/axioms12050497 - 19 May 2023
Cited by 3 | Viewed by 1177
Abstract
Approximate solutions are obtained in implicit forms for the following general form of the nonlinear Stefan problem [...] Read more.
Approximate solutions are obtained in implicit forms for the following general form of the nonlinear Stefan problem $\frac{d}{dx}\left[\left(1+{\delta }_{1}{y}^{p}\right)\frac{dy}{dx}\right]+2x\left(1+{\delta }_{2}{y}^{p}\right)\frac{dy}{dx}=\frac{4}{Ste}\beta \left(x\right),\phantom{\rule{4pt}{0ex}}0 with $y\left(0\right)=1,\phantom{\rule{4pt}{0ex}}y\left(\lambda \right)=0,$ where $\lambda >0$ is a solution to the nonlinear equation ${y}^{\prime }\left(\lambda \right)=-\frac{2\lambda }{Ste}$, where ${\delta }_{i}>-1,\phantom{\rule{4pt}{0ex}}i=1,2,$$p>0,$ and $Ste$ is the Stefan number, which represents a phase-change problem with a nonlinear temperature-dependent thermal parameters (i.e., thermal conductivity and specific heat) on $\left(0,\lambda \right)$. Full article
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13 pages, 1129 KiB
Article
Conditional Tail Expectation and Premium Calculation under Asymmetric Loss
by Enrique Calderín-Ojeda, Emilio Gómez-Déniz and Francisco J. Vázquez-Polo
Axioms 2023, 12(5), 496; https://doi.org/10.3390/axioms12050496 - 19 May 2023
Cited by 2 | Viewed by 1396
Abstract
In this paper, we calculate premiums that are based on the Conditional Tail Expectation (CTE) and asymmetric loss functions to account for the risk of both underestimation and overestimation losses. After selecting an appropriate loss function, the premium is calculated as the quantity [...] Read more.
In this paper, we calculate premiums that are based on the Conditional Tail Expectation (CTE) and asymmetric loss functions to account for the risk of both underestimation and overestimation losses. After selecting an appropriate loss function, the premium is calculated as the quantity minimizing an objective function related to the conditional tail expectation of the loss. The premium satisfies desirable properties, i.e., it is a coherent risk measure, and it helps the practitioner to quantify the global risk of the insurer. Finally, this methodology is applied to quantify the risks associated to the total claims amount that are modelled via composite models and comparisons with the usual risk measures, i.e., Value-at-Risk (VaR) and Tail Value-at-Risk (TVaR) are carried out. Full article
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5 pages, 196 KiB
Editorial
Various Deep Learning Algorithms in Computational Intelligence
by Oscar Humberto Montiel Ross
Axioms 2023, 12(5), 495; https://doi.org/10.3390/axioms12050495 - 19 May 2023
Viewed by 1013
Abstract
Deep Learning (DL) is an essential topic of increasing interest in science, industry, and academia [...] Full article
(This article belongs to the Special Issue Various Deep Learning Algorithms in Computational Intelligence)
21 pages, 349 KiB
Article
New Applications of Fuzzy Set Concept in the Geometric Theory of Analytic Functions
by Alina Alb Lupaş
Axioms 2023, 12(5), 494; https://doi.org/10.3390/axioms12050494 - 19 May 2023
Cited by 1 | Viewed by 889
Abstract
Zadeh’s fuzzy set theory offers a logical, adaptable solution to the challenge of defining, assessing and contrasting various sustainability scenarios. The results presented in this paper use the fuzzy set concept embedded into the theories of differential subordination and superordination established and developed [...] Read more.
Zadeh’s fuzzy set theory offers a logical, adaptable solution to the challenge of defining, assessing and contrasting various sustainability scenarios. The results presented in this paper use the fuzzy set concept embedded into the theories of differential subordination and superordination established and developed in geometric function theory. As an extension of the classical concept of differential subordination, fuzzy differential subordination was first introduced in geometric function theory in 2011. In order to generalize the idea of fuzzy differential superordination, the dual notion of fuzzy differential superordination was developed later, in 2017. The two dual concepts are applied in this article making use of the previously introduced operator defined as the convolution product of the generalized Sălgean operator and the Ruscheweyh derivative. Using this operator, a new subclass of functions, normalized analytic in U, is defined and investigated. It is proved that this class is convex, and new fuzzy differential subordinations are established by applying known lemmas and using the functions from the new class and the aforementioned operator. When possible, the fuzzy best dominants are also indicated for the fuzzy differential subordinations. Furthermore, dual results involving the theory of fuzzy differential superordinations and the convolution operator are established for which the best subordinants are also given. Certain corollaries obtained by using particular convex functions as fuzzy best dominants or fuzzy best subordinants in the proved theorems and the numerous examples constructed both for the fuzzy differential subordinations and for the fuzzy differential superordinations prove the applicability of the new theoretical results presented in this study. Full article
12 pages, 505 KiB
Article
Quasi-Magical Fermion Numbers and Thermal Many-Body Dynamics
by Angelo Plastino, Diana Monteoliva and Angel Ricardo Plastino
Axioms 2023, 12(5), 493; https://doi.org/10.3390/axioms12050493 - 19 May 2023
Viewed by 1054
Abstract
This work scrutinizes, using statistical mechanics indicators, important traits displayed by quantum many-body systems. Our statistical mechanics quantifiers are employed, in the context of Gibbs’ canonical ensemble at temperature T. A new quantifier of this sort is also presented here. The present [...] Read more.
This work scrutinizes, using statistical mechanics indicators, important traits displayed by quantum many-body systems. Our statistical mechanics quantifiers are employed, in the context of Gibbs’ canonical ensemble at temperature T. A new quantifier of this sort is also presented here. The present discussion focuses attention on the role played by the fermion number N in many-fermion dynamics, that is, N is our protagonist. We have discovered discovers particular values of N for which the thermal indicators exhibit unexpected abrupt variations. Such a fact reflects an unanticipated characteristic of fermionic dynamics. Full article
(This article belongs to the Special Issue Advances in Quantum Theory and Quantum Computing)
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8 pages, 8852 KiB
Communication
Modulation Transfer between Microwave Beams: Angular Dependence of the Delay-Time
by Ilaria Cacciari and Anedio Ranfagni
Axioms 2023, 12(5), 492; https://doi.org/10.3390/axioms12050492 - 18 May 2023
Cited by 1 | Viewed by 765
Abstract
Further measurements of delay time, in the transfer of modulation between microwave beams, are reported even considering the angular dependence in the orientation of the ${F}_{1}$ c.w. launcher. The obtained results allow for a better interpretation of the observed phenomenology on the [...] Read more.
Further measurements of delay time, in the transfer of modulation between microwave beams, are reported even considering the angular dependence in the orientation of the ${F}_{1}$ c.w. launcher. The obtained results allow for a better interpretation of the observed phenomenology on the basis of a more sophisticated analysis, which takes into account the presence of pole singularities in field-amplitude evaluations according to the saddle point method. The model already presented in one of our previous publications, consisting of a photon–photon scattering mechanism as concomitant with a stochastic process, is then confirmed. Full article
(This article belongs to the Section Mathematical Physics)
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12 pages, 287 KiB
Article
Norms of a Product of Integral and Composition Operators between Some Bloch-Type Spaces
by Stevo Stević
Axioms 2023, 12(5), 491; https://doi.org/10.3390/axioms12050491 - 18 May 2023
Viewed by 853
Abstract
We present some formulas for the norm, as well as the essential norm, of a product of composition and an integral operator between some Bloch-type spaces of analytic functions on the unit ball, in terms of given symbols and weights. Full article
(This article belongs to the Special Issue Recent Advances in Functional Analysis and Operator Theory)
18 pages, 545 KiB
Article
Quaternionic Fuzzy Sets
by Songsong Dai
Axioms 2023, 12(5), 490; https://doi.org/10.3390/axioms12050490 - 18 May 2023
Viewed by 1128
Abstract
A novel concept of quaternionic fuzzy sets (QFSs) is presented in this paper. QFSs are a generalization of traditional fuzzy sets and complex fuzzy sets based on quaternions. The novelty of QFSs is that the range of the membership function is the set [...] Read more.
A novel concept of quaternionic fuzzy sets (QFSs) is presented in this paper. QFSs are a generalization of traditional fuzzy sets and complex fuzzy sets based on quaternions. The novelty of QFSs is that the range of the membership function is the set of quaternions with modulus less than or equal to one, of which the real and quaternionic imaginary parts can be used for four different features. A discussion is made on the intuitive interpretation of quaternion-valued membership grades and the possible applications of QFSs. Several operations, including quaternionic fuzzy complement, union, intersection, and aggregation of QFSs, are presented. Quaternionic fuzzy relations and their composition are also investigated. QFS is designed to maintain the advantages of traditional FS and CFS, while benefiting from the properties of quaternions. Cuts of QFSs and rotational invariance of quaternionic fuzzy operations demonstrate the particularity of quaternion-valued grades of membership. Full article
(This article belongs to the Special Issue Recent Advances in Fuzzy Sets and Related Topics)
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15 pages, 498 KiB
Article
An Energy-Efficient Optimal Operation Control Strategy for High-Speed Trains via a Symmetric Alternating Direction Method of Multipliers
by Shan Ma, Feng Ma and Chaoyu Tang
Axioms 2023, 12(5), 489; https://doi.org/10.3390/axioms12050489 - 18 May 2023
Cited by 1 | Viewed by 915
Abstract
Train operation control is of great importance in reducing train operation energy consumption and improving railway operation efficiency. This paper investigates the design of optimal control inputs for multiple trains on a single railway line with several stations. Firstly, a distributed optimal control [...] Read more.
Train operation control is of great importance in reducing train operation energy consumption and improving railway operation efficiency. This paper investigates the design of optimal control inputs for multiple trains on a single railway line with several stations. Firstly, a distributed optimal control problem for multiple train operation is formulated to reduce the energy consumption and improve the punctuality of trains. Then, we propose an efficient algorithm based on the framework of the symmetric alternating direction method of multipliers to solve this optimization problem. Finally, numerical simulations show that the method can obtain the optimal train control sequence in fewer iterative steps compared to the alternating direction multiplier method, thus illustrating the effectiveness of the algorithm. Full article
(This article belongs to the Special Issue Applied Mathematics in Energy and Mechanical Engineering)
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15 pages, 384 KiB
Article
New Algorithms for Dealing with Fractional Initial Value Problems
by Iqbal M. Batiha, Ahmad A. Abubaker, Iqbal H. Jebril, Suha B. Al-Shaikh and Khaled Matarneh
Axioms 2023, 12(5), 488; https://doi.org/10.3390/axioms12050488 - 18 May 2023
Cited by 6 | Viewed by 1106
Abstract
This work purposes to establish two small numerical modifications for the Fractional Euler method (FEM) and the Modified Fractional Euler Method (MFEM) to deal with fractional initial value problems. Two such modifications, which are named Improved Modified Fractional Euler Method 1 (IMFEM 1) [...] Read more.
This work purposes to establish two small numerical modifications for the Fractional Euler method (FEM) and the Modified Fractional Euler Method (MFEM) to deal with fractional initial value problems. Two such modifications, which are named Improved Modified Fractional Euler Method 1 (IMFEM 1) and Improved Modified Fractional Euler Method 2 (IMFEM 2), endeavor to further enhance FEM and MFEM in terms of attaining more accuracy. By utilizing certain theoretical results, the resultant error bounds of the proposed methods are analyzed and estimated. Several numerical comparisons are carried out to validate the efficiency of our proposed methods. Full article
(This article belongs to the Special Issue Numerical Analysis and Advanced Applications)
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22 pages, 453 KiB
Article
Monitoring the Weibull Scale Parameter Based on Type I Censored Data Using a Modified EWMA Control Chart
by Dan Yu, Li Jin, Jin Li, Xixi Qin, Zhichuan Zhu and Jiujun Zhang
Axioms 2023, 12(5), 487; https://doi.org/10.3390/axioms12050487 - 17 May 2023
Cited by 1 | Viewed by 1219
Abstract
In industrial production, the exponentially weighted moving average scheme is widely used to monitor shifts in product quality, especially small-to-moderate shifts. In this paper, we propose a modified one-sided EWMA scheme for Type I right-censored Weibull lifetime data for detecting shifts in the [...] Read more.
In industrial production, the exponentially weighted moving average scheme is widely used to monitor shifts in product quality, especially small-to-moderate shifts. In this paper, we propose a modified one-sided EWMA scheme for Type I right-censored Weibull lifetime data for detecting shifts in the scale parameter with the shape parameter fixed. A comparative analysis with existing cumulative sum and exponentially weighted moving average results from the literature is provided. The zero-state and steady-state behaviour of the new scheme are considered with regard to the average run length, the standard deviation of the run length, and other performance measures. Our simulation shows stronger power in detecting changes in the censored lifetime data using the modified scheme than that using the traditional exponentially weighted moving average scheme, and the new scheme is superior to the cumulative sum scheme in most situations. A real-data example further demonstrates the effectiveness of the proposed method. Full article
(This article belongs to the Special Issue Mathematical and Statistical Methods and Their Applications)
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16 pages, 403 KiB
Article
Quaternionic Shape Operator and Rotation Matrix on Ruled Surfaces
by Yanlin Li and Abdussamet Çalışkan
Axioms 2023, 12(5), 486; https://doi.org/10.3390/axioms12050486 - 17 May 2023
Cited by 20 | Viewed by 1339
Abstract
In this article, we examine the relationship between Darboux frames along parameter curves and the Darboux frame of the base curve of the ruled surface. We derive the equations of the quaternionic shape operators, which can rotate tangent vectors around the normal vector, [...] Read more.
In this article, we examine the relationship between Darboux frames along parameter curves and the Darboux frame of the base curve of the ruled surface. We derive the equations of the quaternionic shape operators, which can rotate tangent vectors around the normal vector, and find the corresponding rotation matrices. Using these operators, we examine the Gauss curvature and mean curvature of the ruled surface. We explore how these properties are found by the use of Frenet vectors instead of generator vectors. We provide illustrative examples to better demonstrate the concepts and results discussed. Full article
(This article belongs to the Special Issue Advances in Differential Geometry and Singularity Theory)
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12 pages, 804 KiB
Article
Tensor Eigenvalue and SVD from the Viewpoint of Linear Transformation
by Xinzhu Zhao, Bo Dong, Bo Yu and Yan Yu
Axioms 2023, 12(5), 485; https://doi.org/10.3390/axioms12050485 - 17 May 2023
Viewed by 1281
Abstract
A linear transformation from vector space to another vector space can be represented as a matrix. This close relationship between the matrix and the linear transformation is helpful for the study of matrices. In this paper, the tensor is regarded as a generalization [...] Read more.
A linear transformation from vector space to another vector space can be represented as a matrix. This close relationship between the matrix and the linear transformation is helpful for the study of matrices. In this paper, the tensor is regarded as a generalization of the matrix from the viewpoint of the linear transformation instead of the quadratic form in matrix theory; we discuss some operations and present some definitions and theorems related to tensors. For example, we provide the definitions of the triangular form and the eigenvalue of a tensor, and the theorems of the tensor QR decomposition and the tensor singular value decomposition. Furthermore, we explain the significance of our definitions and their differences from existing definitions. Full article
(This article belongs to the Special Issue Advances in Linear Algebra with Applications)
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28 pages, 378 KiB
Article
Dealing with Evaluative Expressions and Hate Speech Metaphors with Fuzzy Property Grammar Systems
by Adrià Torrens-Urrutia, Maria Dolores Jiménez-López and Susana Campillo-Muñoz
Axioms 2023, 12(5), 484; https://doi.org/10.3390/axioms12050484 - 16 May 2023
Cited by 2 | Viewed by 1296
Abstract
We introduce a Fuzzy Property Grammar System (FPGS), a formalism that integrates a Fuzzy Property Grammar into a linguistic grammar system to formally characterize metaphorical evaluative expressions. The main scope of this paper is to present the formalism of FPGS and to show [...] Read more.
We introduce a Fuzzy Property Grammar System (FPGS), a formalism that integrates a Fuzzy Property Grammar into a linguistic grammar system to formally characterize metaphorical evaluative expressions. The main scope of this paper is to present the formalism of FPGS and to show how it might provide a formal characterization of hate speech linguistic evaluative expressions with metaphors (as fuzzy concepts), together with evaluating their degree of linguistic violence. Linguistic metaphors are full of semantic coercions. It is necessary to formally characterize the context of the communication to acknowledge the extralinguistic constraints of the pragmatic domain, which establishes whether an utterance is violent. To show the applicability of our formalism, we present a proof of concept. By compiling and tagging a 3000-tweet corpus, we have extracted a lexicon of hate speech metaphors. Furthermore, we show how FPGS architecture can deal with different types of hate speech and can identify implicit violent figurative evaluative expressions by context and type. Although we are still in the experimental phase of our project and cannot present conclusive results at the computational level, the proof-of-concept results show that our formalism can achieve the desired outcome. Full article
(This article belongs to the Section Logic)
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19 pages, 2205 KiB
Review
Review of Quaternion Differential Equations: Historical Development, Applications, and Future Direction
by Alit Kartiwa, Asep K. Supriatna, Endang Rusyaman and Jumat Sulaiman
Axioms 2023, 12(5), 483; https://doi.org/10.3390/axioms12050483 - 16 May 2023
Viewed by 1942
Abstract
Quaternion is a four-dimensional and an extension of the complex number system. It is often viewed from various fields, such as analysis, algebra, and geometry. Several applications of quaternions are related to an object’s rotation and motion in three-dimensional space in the form [...] Read more.
Quaternion is a four-dimensional and an extension of the complex number system. It is often viewed from various fields, such as analysis, algebra, and geometry. Several applications of quaternions are related to an object’s rotation and motion in three-dimensional space in the form of a differential equation. In this paper, we do a systematic literature review on the development of quaternion differential equations. We utilize PRISMA (preferred reporting items for systematic review and meta-analyses) framework in the review process as well as content analysis. The expected result is a state-of-the-art and the gap of concepts or problems that still need to develop or answer. It was concluded that there are still some opportunities to develop a quaternion differential equation using a quaternion function domain. Full article
(This article belongs to the Special Issue Differential Equations and Related Topics)
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19 pages, 394 KiB
Article
A Characterization of Multipliers of the Herz Algebra
by Hans G. Feichtinger
Axioms 2023, 12(5), 482; https://doi.org/10.3390/axioms12050482 - 16 May 2023
Cited by 1 | Viewed by 900
Abstract
For the characterization of multipliers of ${\mathbit{L}}^{p}\left({\mathbb{R}}^{d}\right)$ or more generally, of ${\mathbit{L}}^{p}\left(G\right)$ for some locally compact Abelian group G, the so-called Figa-Talamanca–Herz algebra ${\mathbit{A}}_{p}\left(G\right)$ plays an important role. Following [...] Read more.
For the characterization of multipliers of ${\mathbit{L}}^{p}\left({\mathbb{R}}^{d}\right)$ or more generally, of ${\mathbit{L}}^{p}\left(G\right)$ for some locally compact Abelian group G, the so-called Figa-Talamanca–Herz algebra ${\mathbit{A}}_{p}\left(G\right)$ plays an important role. Following Larsen’s book, we describe multipliers as bounded linear operators that commute with translations. The main result of this paper is the characterization of the multipliers of ${\mathbit{A}}_{p}\left(G\right)$. In fact, we demonstrate that it coincides with the space of multipliers of $\left({\mathbit{L}}^{p}\left(G\right),\phantom{\rule{0.166667em}{0ex}}{\parallel ·\parallel }_{p}\right)$. Given a multiplier T of $\left({\mathbit{A}}_{p}\left(G\right),\parallel ·{\parallel }_{{\mathbit{A}}_{p}\left(G\right)}\right)$ and using the embedding $\left({\mathbit{A}}_{p}\left(G\right){,\parallel \phantom{\rule{0.166667em}{0ex}}·\phantom{\rule{0.166667em}{0ex}}\parallel }_{{\mathbit{A}}_{p}\left(G\right)}\right)↪\left({\mathbit{C}}_{\phantom{\rule{-1.66672pt}{0ex}}0}\left(G\right),\phantom{\rule{0.166667em}{0ex}}{\parallel \phantom{\rule{0.166667em}{0ex}}·\phantom{\rule{0.166667em}{0ex}}\parallel }_{\infty }\right)$, the linear functional $f↦\left[T\left(f\right)\left(0\right)\right]$ is bounded, and T can be written as a moving average for some element in the dual $\mathbit{P}\phantom{\rule{-1.66672pt}{0ex}}{\mathbit{M}}_{p}\left(G\right)$ of $\left({\mathbit{A}}_{p}\left(G\right),\parallel ·{\parallel }_{{\mathbit{A}}_{p}\left(G\right)}\right)$. A key step for this identification is another elementary fact: showing that the multipliers from $\left({\mathbit{L}}^{p}\left(G\right),\phantom{\rule{0.166667em}{0ex}}{\parallel \phantom{\rule{0.166667em}{0ex}}·\phantom{\rule{0.166667em}{0ex}}\parallel }_{p}\right)$ to $\left({\mathbit{C}}_{\phantom{\rule{-1.66672pt}{0ex}}0}\left(G\right),\phantom{\rule{0.166667em}{0ex}}{\parallel \phantom{\rule{0.166667em}{0ex}}·\phantom{\rule{0.166667em}{0ex}}\parallel }_{\infty }\right)$ are exactly the convolution operators with kernels in $\left({\mathbit{L}}^{q}\left(G\right),\phantom{\rule{0.166667em}{0ex}}{\parallel \phantom{\rule{0.166667em}{0ex}}·\phantom{\rule{0.166667em}{0ex}}\parallel }_{q}\right)$ for $1 and $1/p+1/q=1$. The proofs make use of the space of mild distributions, which is the dual of the Segal algebra $\left({\mathbit{S}}_{\phantom{\rule{-1.66672pt}{0ex}}0}\left(G\right),{\parallel \phantom{\rule{0.166667em}{0ex}}·\phantom{\rule{0.166667em}{0ex}}\parallel }_{{\mathbit{S}}_{\phantom{\rule{-1.66672pt}{0ex}}0}}\right)$, and the fact that multipliers T from ${\mathbit{S}}_{\phantom{\rule{-1.66672pt}{0ex}}0}\left(G\right)$ to ${\mathbit{S}}_{\phantom{\rule{-1.66672pt}{0ex}}0}^{\prime }\left(G\right)$ are convolution operators of the form $T:f↦\sigma \ast f$ for some uniquely determined $\sigma \in {\mathbit{S}}_{\phantom{\rule{-1.66672pt}{0ex}}0}^{\prime }$. This setting also allows us to switch from the description of these multipliers as convolution operators (by suitable pseudomeasures) to their description as Fourier multipliers, using the extended Fourier transform in the setting of $\left({\mathbit{S}}_{\phantom{\rule{-1.66672pt}{0ex}}0}^{\prime }\left(G\right),{\parallel \phantom{\rule{0.166667em}{0ex}}·\phantom{\rule{0.166667em}{0ex}}\parallel }_{{\mathbit{S}}_{\phantom{\rule{-1.66672pt}{0ex}}0}^{\prime }}\right)$. The approach presented here extends to other function spaces, but a more detailed discussion is left to future publications. Full article
(This article belongs to the Special Issue Time-Frequency Analysis, Distributions, and Operators)
13 pages, 10764 KiB
Article
Investigating the Number of Monte Carlo Simulations for Statistically Stationary Model Outputs
by Jiahang Zhang and Shengai Cui
Axioms 2023, 12(5), 481; https://doi.org/10.3390/axioms12050481 - 16 May 2023
Cited by 2 | Viewed by 2368
Abstract
The number of random fields required to capture the spatial variability of soil properties and their impact on the performance of geotechnical systems is often varied. However, the number of random fields required to obtain higher-order statistical moments of model outputs has not [...] Read more.
The number of random fields required to capture the spatial variability of soil properties and their impact on the performance of geotechnical systems is often varied. However, the number of random fields required to obtain higher-order statistical moments of model outputs has not yet been studied. This research aims to investigate the number of Monte Carlo simulations needed to achieve stationary higher-order statistics of a pore pressure head in an unsaturated soil slope under steady-state infiltration. The study recommends using at least 500 Monte Carlo samples for the probabilistic analysis of geotechnical engineering models. A more conservative choice for up to second-moment analysis is 1000 samples. The analysis reveals significant variations in skewness, which become stationary for all mesh grids when the number of samples exceeds 15,000. Kurtosis stabilizes only when the number of samples reaches 25,000. The pore pressure head in the unsaturated zone is less uncertain. Additionally, the probability density function of the pore pressure head follows a leptokurtic distribution. Full article
(This article belongs to the Special Issue Probabilistic and Statistical Analysis in Engineering Geology)
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25 pages, 1235 KiB
Article
Analysis of the Solution of a Model of SARS-CoV-2 Variants and Its Approximation Using Two-Step Lagrange Polynomial and Euler Techniques
by Muhammad Usman, Mujahid Abbas and Andrew Omame
Axioms 2023, 12(5), 480; https://doi.org/10.3390/axioms12050480 - 16 May 2023
Cited by 2 | Viewed by 1262
Abstract
In this paper, a vaccination model for SARS-CoV-2 variants is proposed and is studied using fractional differential operators involving a non-singular kernel. It is worth mentioning that variability in transmission rates occurs because of the particular population that is vaccinated, and hence, the [...] Read more.
In this paper, a vaccination model for SARS-CoV-2 variants is proposed and is studied using fractional differential operators involving a non-singular kernel. It is worth mentioning that variability in transmission rates occurs because of the particular population that is vaccinated, and hence, the asymptomatic infected classes are classified on the basis of their vaccination history. Using the Banach contraction principle and the Arzela–Ascoli theorem, existence and uniqueness results for the proposed model are presented. Two different numerical approaches, the fractional Euler and Lagrange polynomial methods, are employed to approximate the model’s solution. The model is then fitted to data associated with COVID-19 deaths in Pakistan between 1 January 2022 and 10 April 2022. It is concluded that our model is much aligned with the data when the order of the fractional derivative $\zeta =0.96$. The two different approaches are then compared with different step sizes. It is observed that they behave alike for small step sizes and exhibit different behaviour for larger step sizes. Based on the numerical assessment of the model presented herein, the impact of vaccination and the fractional order are highlighted. It is also noted that vaccination could remarkably decrease the spikes of different emerging variants of SARS-CoV-2 within the population. Full article
(This article belongs to the Special Issue Dynamical Systems: Theory and Applications in Mathematical Biology)
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14 pages, 475 KiB
Article
Matrix Representations for a Class of Eigenparameter Dependent Sturm–Liouville Problems with Discontinuity
by Shuang Li, Jinming Cai and Kun Li
Axioms 2023, 12(5), 479; https://doi.org/10.3390/axioms12050479 - 15 May 2023
Viewed by 899
Abstract
Matrix representations for a class of Sturm–Liouville problems with eigenparameters contained in the boundary and interface conditions were studied. Given any matrix eigenvalue problem of a certain type and an eigenparameter-dependent condition, a class of Sturm–Liouville problems with this specified condition was constructed. [...] Read more.
Matrix representations for a class of Sturm–Liouville problems with eigenparameters contained in the boundary and interface conditions were studied. Given any matrix eigenvalue problem of a certain type and an eigenparameter-dependent condition, a class of Sturm–Liouville problems with this specified condition was constructed. It has been proven that each Sturm–Liouville problem is equivalent to the given matrix eigenvalue problem. Full article
(This article belongs to the Special Issue Theory of Functions and Applications)
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13 pages, 306 KiB
Article
Relational Contractions Involving Shifting Distance Functions with Applications to Boundary Value Problems
by Ebrahem Ateatullah Algehyne, Mounirah Areshi and Faizan Ahmad Khan
Axioms 2023, 12(5), 478; https://doi.org/10.3390/axioms12050478 - 15 May 2023
Viewed by 1038
Abstract
This manuscript includes certain results on fixed points under a generalized contraction involving a pair of shifting distance functions in the framework of metric space endowed with a class of transitive relation. The results presented herein are illustrated by an example. Finally, we [...] Read more.
This manuscript includes certain results on fixed points under a generalized contraction involving a pair of shifting distance functions in the framework of metric space endowed with a class of transitive relation. The results presented herein are illustrated by an example. Finally, we apply our result to compute a unique solution of certain first order boundary value problems. Full article
(This article belongs to the Special Issue Differential Equations and Related Topics)
17 pages, 612 KiB
Article
A New Probability Distribution: Model, Theory and Analyzing the Recovery Time Data
by Huda M. Alshanbari, Omalsad Hamood Odhah, Zubair Ahmad, Faridoon Khan and Abd Al-Aziz Hosni El-Bagoury
Axioms 2023, 12(5), 477; https://doi.org/10.3390/axioms12050477 - 15 May 2023
Cited by 4 | Viewed by 1841
Abstract
Probability models are frequently used in numerous healthcare, sports, and policy studies. These probability models use datasets to identify patterns, analyze lifetime scenarios, predict outcomes of interest, etc. Therefore, numerous probability models have been studied, introduced, and implemented. In this paper, we also [...] Read more.
Probability models are frequently used in numerous healthcare, sports, and policy studies. These probability models use datasets to identify patterns, analyze lifetime scenarios, predict outcomes of interest, etc. Therefore, numerous probability models have been studied, introduced, and implemented. In this paper, we also propose a novel probability model for analyzing data in different sectors, particularly in biomedical and sports sciences. The probability model is called a new modified exponential-Weibull distribution. The heavy-tailed characteristics along with some other mathematical properties are derived. Furthermore, the estimators of the new modified exponential-Weibull are derived. A simulation study of the new modified exponential-Weibull model is also provided. To illustrate the new modified exponential-Weibull model, a practical dataset is analyzed. The dataset consists of seventy-eight observations and represents the recovery time after the injuries in different basketball matches. Full article
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16 pages, 698 KiB
Article
A Time-Varying Coefficient Double Threshold GARCH Model with Explanatory Variables
by Tongwei Zhang, Lianyan Fu, Dehui Wang and Zhuoxi Yu
Axioms 2023, 12(5), 476; https://doi.org/10.3390/axioms12050476 - 15 May 2023
Viewed by 1141
Abstract
In this article, we consider the nonparametric inference for the time-varying coefficient double-threshold generalized autoregressive conditional heteroscedastic models. The quasi-maximum exponential likelihood estimators (QMELEs) of the model’s parameters and the asymptotic properties of the estimators are obtained. The simulation study implies that the [...] Read more.
In this article, we consider the nonparametric inference for the time-varying coefficient double-threshold generalized autoregressive conditional heteroscedastic models. The quasi-maximum exponential likelihood estimators (QMELEs) of the model’s parameters and the asymptotic properties of the estimators are obtained. The simulation study implies that the distribution of the estimators is asymptotically normal. A real data application to stock returns is given. Both the simulations and real data example imply that the model and the QMELE are proper, compatible and accurately fit the financial time series data of the Nikkei 225. Full article
(This article belongs to the Special Issue Mathematical and Statistical Methods and Their Applications)
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13 pages, 1165 KiB
Article
An Interpretation of the Surface Temperature Time Series through Fuzzy Measures
by Rashmi Rekha Devi and Surajit Chattopadhyay
Axioms 2023, 12(5), 475; https://doi.org/10.3390/axioms12050475 - 15 May 2023
Cited by 3 | Viewed by 1289
Abstract
This paper reports a study to interpret the surface temperature based on time series and fuzzy measures. We demonstrated a method to identify the uncertainty around the surface temperature data concerning the summer monsoon in India. The random variables were standardized, and the [...] Read more.
This paper reports a study to interpret the surface temperature based on time series and fuzzy measures. We demonstrated a method to identify the uncertainty around the surface temperature data concerning the summer monsoon in India. The random variables were standardized, and the Dempster-Shafer Theory was used to generate common goals. Two criteria, represented as fuzzy numbers, were used for this purpose. We constructed three polynomials to illustrate a functional connection between time series and the measure of joint belief. The analysis of the obtained results showed that the certainty increased over time. It confirmed that the degree of the evidence is a more predictable parameter at a more extended period. Full article
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